The Oxford handbook of neo-Riemannian music theories 9780195321333, 0195321332

Table of contents :
Acknowledgments
Preface
Part I. Intellectual Contexts
Chapter 1.: The Reception of Hugo Riemann's Music Theory
Ludwig Holtmeier
Chapter 2.: The Nature of Harmony: A Translation and Commentary
Benjamin Steege
Chapter 3.: What is a Function?
Brian Hyer
Chapter 4.: Riemann and Melodic Analysis: Studies in Folk-Musical Tonality
Matthew Gelbart and Alexander Rehding
Part II. Dualism
Chapter 5.: The Problem of Harmonic Dualism: A Translation and Commentary
Ian Bent
Chapter 6.: Harmonic Dualism as Historical and Structural Imperative
Henry Klumpenhouwer
Chapter 7.: Dualistic Forms
Alexander Rehding
Chapter 8.: Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music
Dmitri Tymoczko
Part III. Tone Space
Chapter 9.: From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combinatorial Conception of Interval
Edward Gollin
Chapter 10.: On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564)
Suzannah Clark
Chapter 11.: Tonal Pitch Space and the (neo-) Riemannian Tonnetz
Richard Cohn
Part IV. Harmonic Space
Chapter 12.: Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's "Systematik der Harmonieschritte"
Nora Engebretsen
Chapter 13.: On a Transformational Curiosity in Riemann's Schematisirung der Dissonanzen
Edward Gollin
Chapter 14.: Chromaticism and the Question of Tonality
David Kopp
Part V. Temporal Space
Chapter 15.: Perspectives on Riemann's Mature Theory of Meter
William E. Caplin
Chapter 16.: Reading Between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas
Scott Burnham
Chapter 17.: Metric Freedoms in Brahms's Songs: A Translation and Commentary
Paul Berry
Part VI. Transformation, Analysis, Criticism
Chapter 18.: Riemannian Analytical Values, Paleo- and Neo-
Steven Rings
Chapter 19.: Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit
Robert C. Cook
Chapter 20.: Three Short Essays on Neo-Riemannian Theory
Daniel Harrison
Glossary
Selected Bibliography
Index

Citation preview

The Oxford Handbook of Neo-Riemannian Music Theories

The Oxford Handbook of Neo-Riemannian Music Theo­ ries   The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music Online Publication Date: Sep 2012

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Oxford University Press, Inc., publishes works that further Oxford University's objective of excellence in research, scholarship, and education. Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Copyright © 2011 by Oxford University Press, Inc. Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York 10016 www.oup.com Oxford is a registered trademark of Oxford University Press. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data The Oxford handbook of neo-Riemannian music theories / edited by Edward Gollin and Alexander Rehding. p. cm. Includes bibliographical references and index. ISBN 978-0-19-532133-3 1. Music theory. 2. Riemann, Hugo, 1849–1919— Criticism and interpretation. I. Gollin, Edward. II. Rehding, Alexander. Page 1 of 2

The Oxford Handbook of Neo-Riemannian Music Theories MT6.O89 2011 781—dc22 2010017175 1 3 5 7 9 8 6 4 2 Printed in the United States of America on acid-free paper

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Dedication

Dedication   The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music Online Publication Date: Sep 2012

Dedication (p. v)

To M. J. M. and Z. G. for their patience and understanding.

—E. G. To B. R. C. with gratitude. —A.R.

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Acknowledgments

Acknowledgments   The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music Online Publication Date: Sep 2012

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Acknowledgments

In addition to the named contributors whose words and ideas fill these pages, the often less visible efforts of many others have made this volume possible. We wish to thank Suzanne Ryan at Oxford University Press for her support and encouragement of this handbook, as well as the music editorial staff at Oxford for their careful work. Frank Lehman, Thomas Lin, and Rowland Moseley provided valuable assistance with the music examples. The index was compiled by Frank Lehman. Research support was provided by Harvard University, Williams College and the Oakley Center for the Humanities and So­ cial Sciences, the Newhouse Center at Wellesley College, the National Endowment for the Humanities, the American Council of Learned Societies, and the Guggenheim Foundation. Wayne Alpern and the Mannes Institute—in particular the 2001 Institute on Historical Music Theory—were responsible for bringing together a number of the scholars whose work is here represented, and for sowing the seeds that eventually developed into the present book. We wish to thank the staff at the Eda Kuhn Loeb Music Library at Harvard University, the Sawyer Library at Williams College, and the Staatsbibliothek in Berlin for their assis­ tance. We are grateful also to Google, and in particular to Google Books, which has pro­ vided an invaluable service, making accessible (and searchable) many of the often ob­ scure nineteenth- and early-twentieth-century journals and treatises discussed herein.

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Preface

Preface   The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music Online Publication Date: Sep 2012

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Preface

The theoretical work of Hugo Riemann (1849–1919) has attracted much attention in re­ cent years. The past decade has seen a new comprehensive biography of Riemann by Michael Arntz; Tatjana Böhme-Mehner and Klaus Mehner have edited and contributed to a collection of studies by German scholars; and Alexander Rehding has produced a mono­ graph placing Riemann's work in its cultural and intellectual context.1 Yet in North Ameri­ ca, the study of Riemann's work itself has been dwarfed in recent years by a new theoreti­ cal and analytical subdiscipline named for Riemann and inspired by his ideas. In numer­ ous academic conference panels, specialized meetings, journal and book publications, and doctoral dissertations, the renewal of Riemann's ideas has reinvigorated the discipline of music theory, offering the prospect of establishing a new music-theoretical paradigm, to complement and stand alongside the two well-established systems of Heinrich Schenker and Allen Forte. The approaches that are now grouped together under the name neo-Riemannian theory first emerged over twenty-five years ago. The theory first grew out of the work of David Lewin and Brian Hyer, who treated certain functional triadic relationships in Riemann's harmonic theory as mathematical transformations acting on triads, using those transfor­ mations (and their attendant group structures) to model structural relations in late-nine­ teenth-century music.2 Subsequent work by John Clough and Richard Cohn, bringing as­ pects of set theory to bear on the materials and relations of tonal music, led to a number of striking insights about the structural properties of diatonic scales and triads—in partic­ ular, the realization that the familiar triads of Western music, long valued as ideal acousti­ cal objects, are also in many ways ideal mathematical objects from the perspective of voice leading.3 Clough, together with Lewin, Cohn, Jack Douthett, and others, convened a series of conferences at the State University of New York at Buffalo, beginning in the 1990s, to share their work and ideas. The field of neo-Riemannian theory was truly born out of the activities of this “Buffalo working group.”4 Neo-Riemannian theory has since overflowed the vessel of the Buffalo conferences, further developed by some of its initial practitioners, but also by generations of graduate students inspired by their research. It seems, therefore, that the time is ripe, given the maturation of the neo-Riemannian Page 1 of 3

Preface project and of the field of historical music theory, for a reassessment of Riemann's theo­ ries—old and new—in light of these developments. The present volume has a twofold intent: to provide contemporary perspectives on Riemann's scholarship and to illustrate the way the Riemannian perspective shapes and informs contemporary analytical and theoretical scholarship. The essays collected within were chosen to refocus attention somewhat toward the theories of (p. x) the original Rie­ mann, and to bring a historical dimension to the neo-Riemannian project. In the spirit of broadening its outlook, then, it seemed advisable not to maintain a strict distinction in this book between the essays focusing on the “historical Riemann” and those that fall squarely into the field of contemporary neo-Riemannian theory. In so far as all these es­ says are fed by the recent renewed interest in Riemann's ideas, regardless of their out­ look, they all constitute aspects of a broadly conceived field of neo-Riemannian studies. The chapters are divided into six parts, which address particular aspects of Riemann's work or the analytical traditions that have arisen therefrom. Part 1 explores Riemann's legacy and the intellectual, cultural, and philosophical traditions within which his work arose and became transmitted. Parts 2–5 address particular components of Riemann's theoretical project: dualism, tone relations and spaces, harmonic relations and spaces, and rhythmic-metric theories. The final part critically explores the analytical practices of Riemannian and neo-Riemannian theory, and their ability to interact and communicate with other analytical approaches. To some degree these sectional divisions overlap. Riemann's spatial conception of tone relations, manifest in the now-canonical Tonnetz, are clearly related to aspects of harmonic relations and voice-leading, yet the structures of the two kinds of musical objects, the structures of their relational systems, and the psy­ chological/cognitive distinctions that attend their perception (i.e., the perception of dis­ tance or relatedness in the two systems) justify their separate treatment. Similarly, issues of form impinge both on temporal and harmonic aspects of Riemann's work, and conse­ quently aspects of form are discussed in multiple sections. Theoretical and analytical essays in the volume are interspersed with annotated transla­ tions of a number of works and essays by Riemann that had not previously been available in English. The translation of these key documents—many of which known to the largely Anglophone neo-Riemannian community only through secondary sources—we believe, will provide a fuller picture of Riemann and his ideas, and may well provide further impe­ tus to future developments in neo-Riemannian theory. For all their quirkiness, Riemann's multifaceted theoretical writings, we believe, have much else to offer that may be of in­ terest to contemporary analytical discourse. The essays assembled in this volume are de­ signed both to provide an overview and to guide future research in this direction. Throughout the volume, certain music-theoretical terminology has been left untranslated: Klang, Harmonieschritt, Tonvorstellungen, and the like. For readers new to Riemann's ideas and to nineteenth-century German dualism, a glossary has been included to define key terms and provide a way into the individual essays, which explore the terms and con­ cepts in greater detail. Page 2 of 3

Preface The essays in this volume look both backward and forward: forward, in summarizing and exploring trends that have emerged over the last twenty-five years with the view to pro­ viding impetus for further projects; and backward, in examining the source concepts from which these ideas have emerged, not only to provide them with a historical background, but also to make familiar other aspects of Riemann's work that have not yet received the critical attention they deserve and (p. xi) that may well lead to further areas of investiga­ tion. The emphasis on reconnecting neo-Riemannian ideas with their source concepts is designed, on the one hand, to familiarize readers who know Riemann's theories only through neo-Riemannian accounts with the original ideas, and on the other, to expand the realms of inquiry of neo-Riemannian theory through cross-pollination with ideas that are as yet underexplored. At this stage in the development of neo-Riemannian theory, given that many of its particu­ lar analytical technologies have been fairly thoroughly explored, it seems that there is a possibility for new issues to take center stage: How can the question of tonality best be answered? To what extent is the neo-Riemannian approach engaged in canonizing a new repertoire of chromatic music? How does such a repertoire interact with the tonal/atonal divisions that the Schenkerian and pitch-class set paradigms had promoted? And more broadly, what is the nature of musical experience in a neo-Riemannian framework? The essays in this volume are designed to foster engagement with such wider-reaching ques­ tions and to lead to ever new ones, further expanding the resources that Riemann's ideas have given to music-theoretical discourse.

Notes: (1.) Michael Arntz, Hugo Riemann (1849–1919): Leben, Werk und Wirkung (Cologne: Alle­ gro, 1999); Tatjana Böhme-Mehner and Klaus Mehner eds., Hugo Riemann (1849–1919): Musikwissenschaftler mit Universalanspruch (Cologne: Böhlau Verlag, 2001); Alexander Rehding, Hugo Riemann and the Birth of Modern Musical Thought (Cambridge: Cam­ bridge University Press, 2003). (2.) See, for example, Lewin's “A Formal Theory of Generalized Tonal Functions,” Journal of Music Theory 26.1 (1982), 23–60, often considered the article that initiated the neoRiemannian enterprise; also Brian Hyer's dissertation, “Tonal Intuitions in Tristan und Isolde,” (Ph.D. diss., Yale University, 1989). (3.) A seminal article on the topic is Richard Cohn's “Neo-Riemannian Operations, Parsi­ monious Trichords, and Their Tonnetz Representations,” Journal of Music Theory 41.1 (1997), 1–66. (4.) For a more extended history of neo-Riemannian theory, see Richard Cohn, “An Intro­ duction to Neo-Riemannian Theory: A Survey and Historical Perspective,” Journal of Mu­ sic Theory 42.2 (1998), 167–180.

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Notes

Notes   The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music Online Publication Date: Sep 2012

Notes (1.) Michael Arntz, Hugo Riemann (1849–1919): Leben, Werk und Wirkung (Cologne: Alle­ gro, 1999); Tatjana Böhme-Mehner and Klaus Mehner eds., Hugo Riemann (1849–1919): Musikwissenschaftler mit Universalanspruch (Cologne: Böhlau Verlag, 2001); Alexander Rehding, Hugo Riemann and the Birth of Modern Musical Thought (Cambridge: Cam­ bridge University Press, 2003). (2.) See, for example, Lewin's “A Formal Theory of Generalized Tonal Functions,” Journal of Music Theory 26.1 (1982), 23–60, often considered the article that initiated the neoRiemannian enterprise; also Brian Hyer's dissertation, “Tonal Intuitions in Tristan und Isolde,” (Ph.D. diss., Yale University, 1989). (3.) A seminal article on the topic is Richard Cohn's “Neo-Riemannian Operations, Parsi­ monious Trichords, and Their Tonnetz Representations,” Journal of Music Theory 41.1 (1997), 1–66. (4.) For a more extended history of neo-Riemannian theory, see Richard Cohn, “An Intro­ duction to Neo-Riemannian Theory: A Survey and Historical Perspective,” Journal of Mu­ sic Theory 42.2 (1998), 167–180.

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Contributors

Contributors   The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music Online Publication Date: Sep 2012

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Contributors

IAN BENT is an emeritus professor of music of Columbia University and Honorary Professor in the History of Music Theory of Cambridge University, U.K. His publica­ tions include Analysis and Music Analysis in the Nineteenth Century; he has served as editor of Music Theory in the Age of Romanticism, and coeditor of the translations of Schenker's Meisterwerk and Tonwille. He is currently coordinator of the online edi­ tion of all Schenker's correspondence, diaries, and lesson books: Schenker Docu­ ments Online.

PAUL BERRY is an assistant professor (adjunct) of music history at the Yale School of Music. His current work centers on historical, critical, and analytic approaches to nineteenth-century chamber music and song, particularly that of Brahms, Schubert, and Schumann. Related focuses include rhetorical studies, connections between biog­ raphy and historiography, and theorizing and contextualizing the kinesthetics of per­ formance.

SCOTT BURNHAM is Scheide Professor of Music History at Princeton University. He is the author of Beethoven Hero, translator of A. B. Marx, Musical Form in the Age of Beethoven, and coeditor of Beethoven and His World. Forthcoming writings include “Late Styles,” for Rethinking Schumann, and “Intimacy and Impersonality in Late Beethoven,” for New Paths.

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Contributors WILLIAM E. CAPLIN is James McGill Professor of Music Theory in the Schulich School of Music, McGill University. His 1998 book Classical Form: A Theory of Formal Functions for the Instrumental Music of Haydn, Mozart, and Beethoven won the 1999 Wallace Berry Book Award from the Society for Music Theory. In addition to his work on musical form, he has published on the history of harmonic and rhythmic theory in the eighteen th and nineteenth centuries, including a chapter in the Cambridge Histo­ ry of Western Music Theory.

SUZANNAH CLARK is Gardner Cowles Associate Professor at Harvard University. She previously taught at Oxford University. Her research interests include the history of tonal theory, the analysis and criticism of Schubert's music, as well as the history and analysis of trouvères chansons and thirteenth-century French motets.

RICHARD COHN is Battell Professor of Music Theory at Yale University and editor of the Oxford Studies in Music Theory series. His book on triadic progressions in nine­ teenth-century music is forthcoming from Oxford University Press, and a book on geometric modeling of metric states is in preparation.

ROBERT C. COOK teaches music theory at the University of Iowa. His inter­ ests include chromaticism, contextual music, and languages and practices of analysis. He was educated at the University of Chicago and Northwestern University. (p. xviii)

NORA ENGEBRETSEN, an associate professor of music theory at Bowling Green State University, holds a Ph.D. in music theory from the State University of New York (SUNY) at Buffalo. Her research interests include chromatic harmony, transforma­ tional theory, and the history of theory. Her work has appeared in Music Theory Spec­ trum, Theoria, the Journal of Music Theory Pedagogy, and collections published by the University of Rochester Press and Stockholm University Press.

MATTHEW GELBART is an assistant professor of music in the department of art his­ tory and music at Fordham University. His research interests include eighteenth- and nineteenth- century music, how we label and sort the music we listen to, and rock

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Contributors music. He is the author of The Invention of “Folk Music” and “Art Music”: Emerging Categories from Ossian to Wagner.

EDWARD GOLLIN is an associate professor of music theory at Williams College. His research and publications have addressed topics in the history of music theory, trans­ formational and neo-Riemannian theory, and the analysis of twentieth-century music with particular focus on the music of Béla Bartók.

DANIEL HARRISON is the Allen Forte Professor of Music Theory at Yale University, where he is also chair of the department of music. He is the author of Harmonic Function in Chromatic Music, and has published on tonal-music topics in Journal of Music Theory, Music Theory Spectrum, Musical Quarterly, Theory and Practice, and Music Analysis, among other venues.

LUDWIG HOLTMEIER is a professor of music theory at the “Hochschule für Musik” in Freiburg. He is one of the editors of the journal Musik & Ästhetik and president of the Gesellschaft für Musik und Ästhetik. His recent publications include Richard Wagner und seine Zeit, Reconstructing Mozart, Musiktheorie zwischen Historie und System­ atik, “From ‘Musiktheorie’ to ‘Tonsatz’: National Socialism and German Music Theory after 1945,” and “Heinichen, Rameau and the Italian Thoroughbass Tradition: Con­ cepts of Tonality and Chord in the Rule of the Octave.”

BRIAN HYER is a professor of music at the University of Wisconsin, Madison. He has written widely on the anthropology of European music and its theories from the eigh­ teenth through the twentieth centuries.

HENRY KLUMPENHOUWER is a professor of music at the University of Alberta and former editor of Music Theory Spectrum. His published work involves the analysis of atonal music, the history of music theory, and analytical methodology.

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Contributors DAVID KOPP is an associate professor in the department of composition and theory at the Boston University School of Music. He is the author of Chromatic Transforma­ tions in Nineteenth-Century Music and articles in the Journal of Music (p. xix) Theory and Music Theory Online, among other publications. As a pianist, he has recorded for the New World Records, CRI, ARTBSN, and Arsis labels.

ALEXANDER REHDING is Fanny Peabody Professor of Music at Harvard University and editor of Acta musicologica. His interest focuses on music theory and nineteenthcentury history. Publications include the books Hugo Riemann and the Birth of Mod­ ern Musical Thought and Music and Monumentality. Recipient of a Guggenheim Fel­ lowship, Rehding is currently working on a study of the engagements of nineteenthcentury science with musical aesthetics.

STEVEN RINGS is an assistant professor of music and the humanities at the Universi­ ty of Chicago. His research focuses on transformational theory, phenomenology, and questions of musical interpretation and meaning. Before turning his attention to mu­ sic theory, he was active as a concert classical guitarist in the United States and Por­ tugal. His book Tonality and Transformation is forthcoming from Oxford University Press.

BENJAMIN STEEGE is an assistant professor of the history and theory of music at Stony Brook University, with interests in the histories of music theory and science, and in early modernism. He is currently writing a book exploring the relationship of Hermann von Helmholtz to music theory and discourses of aurality.

DMITRI TYMOCZKO is a composer and music theorist who teaches at Princeton Uni­ versity. His music has been performed by ensembles throughout the country, and he has been the recipient of a Rhodes scholarship, a Guggenheim fellowship, and numer­ ous other awards. His book, A Geometry of Music, has just been published by Oxford University Press; it will be followed shortly by an album of pieces combining classical and jazz ideas.

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The Reception of Hugo Riemann's Music Theory

The Reception of Hugo Riemann's Music Theory   Ludwig Holtmeier The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0001

Abstract and Keywords This article traces the reception of Hugo Riemann's ideas and examines their gradual transformations in the hands of his contemporaries and successors. By placing Riemann's legacy in a wider context of music-theoretical traditions, the article demonstrates that, di­ vergent to the common belief, there existed a middle ground between Viennese scale-de­ gree theory and Riemannian function theory, voice-leading approaches and Klang-based approaches. In this article, theorists such as Georg Capellen, Rudolf Louis, and Johannes Schreyer are discussed and seen as the missing links between the two great musical-theo­ retical centres of Vienna and Leipzig. Keywords: Hugo Riemann, Riemann's legacy, music-theoretical traditions, scale-degree theory, Riemannian func­ tion theory, voice-leading approaches, Klang-based approaches

One day a young, particularly enthusiastic student, to whom I gradually began to explain such problems, asked me: Have we erected a monument to the ingenious architect of this glorious logical harmonic system? That will happen some day, I replied, but as always, only after he will have passed on to the ancestors.1

If we are to believe Michael Arntz's biography, we would have to imagine Hugo Riemann as a kindly, avuncular figure, who worked tirelessly to put food on the family table, an un­ worldly scientist who lived only for his research2—a lovably quirky figure, something from a novel by Jean Paul. Not everything in this image, however, corresponds to reality.3 For example, his unparalleled productivity, which caused astonishment among his contempo­ raries and which makes it difficult even today to gain a unified sense of his theory, sug­ gests that Riemann cannot have been purely concerned with his ideas. Rather, it implies that he had a considerable interest in power and influence, in implementing his ideas so­ cietally and, above all, institutionally. Bernhard Ziehn's criticism is not merely “exception­ ally harsh” but also to the point:4 “no sooner would anyone have the audacity to wish that the slightest detail of [Riemann's] ideas were a little different, or—banish the thought!— point out to the most famous music teacher of all times some of his intellectual somer­

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The Reception of Hugo Riemann's Music Theory saults, than Herr Doctor would assault that unfortunate person with his quick quill as though he had committed patricide…. He demanded absolute submission.”5 Riemann was extremely sensitive to criticism, and he bore a grudge. This feature may have been more damaging to him personally and to the dissemination of his (p. 4) theory than many others. Not only his enemies, who were at the receiving end, but also his sup­ porters, propagandists, and even his closest friends suffered under this trait. Emil Ergo, an enthusiastic Belgian acolyte of Riemann, loyal to the master to the point of self-denial,6 worked hard to further develop the ideas of his “friend Riemann.”7 But Ergo pointed out, not without justification, that if Riemann had been true to his theory, he should have de­ scribed the Tristan chord as a secondary dominant, and not a functional subdominant. (“Riemann definitely saw some things too quickly and too indistinctly.”8) Ergo also criti­ cized Riemann's theory of phrasing in a public forum and made suggestions for improve­ ment.9 In response, Riemann retreated into resentful silence.10 Another faithful disciple, Johannes Schreyer, who had further developed features of Riemann's functional harmonic theory, suffered a similar fate.11 Like most of Riemann's supporters, he had distanced himself from dualism: “Even though we owe much enlightenment and stimulus to Riemann's writings, I cannot convince myself of the necessity to notate the minor harmo­ ny as under-Klang, as he requires.”12 Schreyer's “monistic” revision of Klangschlüssel notation found no favor with Riemann.13 As he communicated in a letter of 1903 to Schreyer, he had “no esteem for attempts at mediation such as yours.”14 As a conse­ quence, the relationship between the two cooled down considerably. What shines through underneath Riemann's thin skin is considerable ambition: his ex­ treme sensitivity betrays a striving for power, influence, and recognition. It is hard to overlook how much time and energy Riemann spent on propaganda for his ideas. His sub­ sequent concentration on music-historical research should not obscure the fact that it was above all on the practical disciplines of harmony and phrasing that he intended to leave his mark. He was not primarily interested in playing a role in the small, closed acad­ emic world of science and research, but he was eager to exert a lasting influence on mu­ sic history writ large—on practicing musicians and how they thought in and about music.15 He propagated the ever-same ideas in forever new guises: tutors, simplified tu­ tors, catechisms, introductions, compendia, handbooks, practical editions are tirelessly tossed out on the market as though new ideas would succeed simply by virtue of their vol­ ume.16 The prefaces of his pedagogical works leave no doubt about his ultimate mission: that his theory be granted admission to the “higher pedagogical institutions,”17 that his Handbuch der Harmonielehre succeed in replacing the harmony textbook that had been his reference point right from the beginning—Ernst Friedrich Richter's Harmonielehre, “a book spread throughout the whole civilized world.”18 Riemann's constant complaints about lacking reception, slow sales, about resistance, “insurmountable obstacles,” “silent disregard,”19 which can be found in all the prefaces to his pedagogical works, offer a glimpse into his frustration over never having achieved a genuinely popular harmony tu­ tor. His vehement and unjust response to the theory of harmony by Louis and Thuille can,

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The Reception of Hugo Riemann's Music Theory not least, be explained as a premonition on Riemann's part that it was their work that was destined to assume the mantle of Richter's Harmonielehre.20 The essential tool of Riemann's “propaganda” was his Musiklexikon, as Arntz has pointed out.21 Not only did it serve Riemann to promulgate his ideas but also to reward or punish his henchmen. Anyone who was with Riemann was met with a (p. 5) benevolent reception. Traces of Riemann's worldwide network are inscribed in the Lexikon. His acolytes dissem­ inated, as translators and publishers, Riemann's ideas throughout the world:22 Julius En­ gel, Peter Ivanovich Jürgenson, and Nikolai Kashkin in Russia, Emil Ergo in Belgium and the Netherlands, Michel Calvocoressi and Georges Humbert in France and Switzerland, Giacomo Settaccioli in Italy, Heinrich Bewerunge and John Shedlock in England and Ire­ land, John Comfort Fillmore in America, Henryk Bissing Schytte in Denmark, and Jan Ur­ bánek in the Czech lands.23 After Riemann's death, in the late twenties, central pedagogi­ cal works by Riemann would also appear in Spanish, the last important language of the “civilized world” still missing, and would in this way enter into all of Latin America.24 Hugo Riemann was a music-theoretical entrepreneur, as there had been few before or af­ ter him. It is useful to bear this in mind, especially as Riemann not only did not have a monument erected in his memory—despite the fervent hope expressed by Emil Ergo ini­ tially—but rather his entire pedagogical opus fell into oblivion, while paradoxically the theory of function rose to fame as the paradigm of harmony teaching at many conserva­ toires, universities, and music schools in central Europe. The fact that he became a his­ torical figure after 1945, without much relevance for contemporary practice or research, is a separate story with its own reasons, as we shall discuss later. Many have warned against overestimating Riemann's influence.25 It would be equally misguided, however, to underestimate him.

Riemann's Legacy The extent of Riemann's influence on applied (or “practical”) music theory is deceptive.26 This is primarily because of the dearth of any direct, “pure” continuation of Riemann's theories. In contrast to, say, Simon Sechter or Heinrich Schenker, Riemann did not suc­ ceed in forming a school. One crucial reason for this was Riemann's universalist ap­ proach, his attempt to develop a theory, “which would turn the long-desired union of mu­ sicology and practical music pedagogy into reality.”27 Two areas of the reception of Riemann's music theory relevant for practical music theory can be discerned: (1) the the­ ory of phrasing, and (2) the theory of harmony, which can in turn be subdivided into theo­ ries of (a) practical harmony and (b) dualistic derivation. The theory of phrasing, an essential part of Riemann's theoretical system, cannot be dealt with here. During Riemann's lifetime it occupied a central position in music-theoretical discourse, which, however, it gradually lost beginning as early as the 1910s. Only a few of Riemann's successors took it up, while it has disappeared completely from “modern” the­ ories of function and the general music-theoretical discussion.28 It is on the field of practi­ cal harmony that Riemann's theory had its most lasting impact. Page 3 of 53

The Reception of Hugo Riemann's Music Theory In the above division into ideal types, the “theory of dualistic derivation” is synonymous with the notion of “science” (or rather, its more inclusive German correlate, Wissenschaft). In this form of derivation, Riemann takes up the Leipzig (p. 6) tradition of dualism.29 The most far-reaching part is not only its integration into his theory of the imaginations of tone (Lehre von den Tonvorstellungen),30 but also his attempt to transfer “scientific” dual­ ism to his practical teaching, in line with his holistic method: the old theoretical notions of the upper and lower Klänge are, for the first time, consistently integrated into a practi­ cal theory of composition.31 The main stream of Riemann reception—and by this I mean his practical harmony tutors, which were widely disseminated—did not follow Riemann in his dualistic ideas. Almost all of Riemann's successors settled on a “monistic” variant of the theory of functions.32 The term “monism” appears to have been coined by the music theorist Georg Capellen, who spearheaded the criticisms of dualism.33 In his 1901 article, “Die Unmöglichkeit und Überflüssigkeit der dualistischen Molltheorie Riemanns” (“The impossibility and redun­ dancy of Riemann's dualistic theory of minor”) Capellen, like many others, attacked Riemann's system where it seemed least protected—namely, in the problem of the “root” of the minor chord. Capellen reproaches Riemann for theoretical inconsistencies: “Just like the other dualists, [Riemann had] not had the guts to think through the identity of the first scale degree and the root in minor consistently and to take it to the next level.”34 Riemann maintained a distinction, Capellen argued, between the generator of the minor chord, its “principal tone” (Hauptton), that is, the tone from which the lower sonority is formed, and the “root” (Grundton), corresponding to classical fundamental bass theory. It is the old prob­ lem of dualism: while the generator of the major triad is also perceived as its root, the generator of the minor triad is heard, due to “the normal perception of chordal weight,”35 as the fifth of the triad, and in Riemann's theory of composition it is consequently treated as such.36 Riemann could not convincingly rebut Capellen's reproach with his dualistic, “dialectical” explanation, arguing that the criticism was wrongly directed at the level of concrete experience, when it actually belonged in the realm of the abstract idea—in short, that Capellen confused spiritual essence with sensual appearance. In Riemann's re­ form project, however, Capellen's reproach becomes a real problem, as the dualistic con­ cepts, at least partially, become manifest compositional concepts. Riemann cannot simply withdraw into the safe haven of ideal construction, in which Ernst Kurth later considered dualism—after its demise, so to speak: the idea of dualism, Kurth argues, was in essence a “theory of projection” and as such grows out of a higher theoretical concept—that of analogy. From the perspective of “an approach that is more independent of the physical foundations of music theory,” dualism, “as a two-sided projection, can gain a foundation that falls into the realm of psychology.” This dual symmetry, he argued, offers “a unified theory of harmony so remarkable and valuable that the basis of tonality by means of chordal projection could still carry justificatory power, even when the foundation was partly converted from the physical (the real existence of undertones) to the abstract realm.”37 Page 4 of 53

The Reception of Hugo Riemann's Music Theory When Kurth distinguishes between a “sensually perceptible and a formal part of the sonic structure (Klanggerüst),”38 and restricts the relevancy of Riemann's dualism to the latter, however, he returns the theory to the splendid isolation of the abstract idea, from which Riemann's attempt at synthesis was precisely trying to (p. 7) remove it. The basic synthet­ ic character of Riemann's reform of harmonic theory can also explain why the disciples of the theory of function were not satisfied by Riemann's later “psychologizing” retreat, where he argued that What distinguishes major from minor comes down to the essence of major conso­ nance being the simplest ratios in the increase in speed of vibration, that of minor consonance, by contrast, being the simplest ratios in the enlargement of the vi­ brating mass…. By this means, the principle of major can succinctly be seen to lie in growing intensity, and the principle of minor in accumulating mass.39 With the admission that the undertone series did not exist, the minor chord can no longer be derived as a physical and physiological empirical fact from the theoretical foundations: the traditional level of mediation would yet again have to step between theory and prac­ tice, which was precisely what Riemann was trying to overcome. The synthesis between dualistic theory and compositional practice had been fragile from the beginning, and it was Riemann himself who created the conditions for its later rup­ ture. The gash opened up when he introduced what was to become the hallmark of his theory of functions: the symbolic shorthand labels with which the theory of function oper­ ated.40 When Ary Belinfante criticizes Riemann in 1904 for effectively renouncing the po­ larity of major and minor,41 Riemann's defense—“that these names [harmonic functions], far from being coined by me, have been in general use ever since Rameau; that I have re­ tained them with the same justification as [I have] the symbols [for] major, minor, paral­ lel, root”—does not sound very convincing.42 Why then was the complex, strictly dualistic system of sonority and harmonic root progressions (“schlichter Quintklang/schritt,” “Gegenquintklang/schritt,” “Seitenwechselklang/schritt,” “schlichter Terzschritt,” “Gegenterzschritt,” “Terzwechsel,” etc.) introduced at all, which marked precisely the idea of a polar cadential progression, and which was no less than the transference of du­ alistic theory to dualistic chordal relations?43 With the new taxonomy of harmonic func­ tions, Riemann returns to a traditional practice, which stands unmediated beside his radi­ cal dualistic theory. Without quite realizing it, Riemann himself observes that the intro­ duction of the famous function symbols actually marks a surrender and concession to an intransigent and overpowering tradition: “Although one of my personal students once as­ sured me that he no longer troubles himself with the terminology of harmonic root pro­ gressions since I introduced function symbols, I know for sure that this student is merely no longer concerned with names and labels, but he is far from considering the dominants in major and minor as equivalent.”44 With the introduction of function symbols, the entire system of dualistic progressions and, alongside it, the dualistic understanding of caden­ tial progressions became defunct, even during Riemann's lifetime.45 The monistic reac­

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The Reception of Hugo Riemann's Music Theory tion, which further questioned the notion of a dualistic minor model, was nothing but a logical consequence. Although the preeminent and successful strain of Riemann reception largely developed free from dualism, dualistic theories continued to exist.46 The Leipzig music theorist and composer Stephan Krehl could be considered the most successful popularizer of orthodox dualism in Riemann's sense.47 Not least the authority of his (p. 8) position as professor of theory at the Leipzig conservatory might have contributed to the wide dissemination of some of his theoretical works. Krehl's Formenlehre of 1902–1903 and above all his Allge­ meine Musiklehre of 1904 are worth mentioning, both of which appeared in the popular and affordable Leipzig series of Göschen pocketbooks and were reissued numerous times. Despite Krehl's insistence that his Allgemeine Musiklehre gave only “a rough treatment of the material,” it quickly moves into deeper waters, addressing the complex questions of dualistic theory and its notation.48 The unfettered propaganda of dualistic theory (“In the theory of harmony we must first discuss the only logical dualistic interpretation of chords and the terminology established for this purpose,”)49 is probably the reason that Krehl's three-volume Harmonielehre of 1921 and his Tonalitätslehre of 1922, which appears to anticipate the idea of “polarism,” had a very limited readership. In fact, Krehl's oeuvre displays some traces of a dualistic parochialism, of a kind we encounter as well in Sigrid Karg-Elert and his students Fritz Reuter and above all Paul Schenk.50 By 1933, Krehl's ap­ proach was already so outmoded that the reissue of his Allgemeine Musiklehre retained his name but actually contained a completely new book by Richard Hernried, in which du­ alism was merely presented as a historical movement, and in which he claims—paradoxi­ cally, but quite correctly, “the most important tool that the teachers of dualism, above all Hugo Riemann, have left us is the taxonomy of harmonic function.”51 Henried's edition, albeit theoretically quite sophisticated, was replaced as early as 1940—in the prevailing spirit of the times—by another sturdy Allgemeine Musiklehre by H. J. Moser, which was didactically and ideologically marked by the youth music movement. This version carries Krehl's only in its subtitle and is devoid of the last vestiges of harmonic dualism.52 This tendency is representative of the gradual decline of harmonic dualism. It appears that Riemannian ideas lived on predominantly in such esoteric approaches as Hans Kayser's harmonically driven neo-Pythagoreanism.53 The dualism of overtone and under­ tone series is here revived from a number-based perspective. It can effortlessly be inte­ grated into Kayser's “matrix of upper partials” (Teiltonkoordinatensystem), which he de­ rives from Thimus's “Lambdoma.” This kind of numerological, sometimes mystical, after­ life of Riemann's dualism can be observed variously (for instance, in the work of Victor Goldschmidt, Joseph Mathias Hauer, and Othmar Steinbauer). In this way, dualism experi­ enced a radical decline in social prestige and descended from the belle étage of German music theory to the gutters of esoteric and sectarian circles.

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The Reception of Hugo Riemann's Music Theory

Monistic Modes of Reception If Riemann's dualistic reform of music theory can be said to have failed, upon what is its influence on practical harmony founded? Three aspects must be considered above all: (p. 9)

1. The concept of tone representation (Klangvertretung) 2. The concept of the applied or secondary dominant (Zwischendominanten) 3. The taxonomy of harmonic function This sequence should be understood as a hierarchy. The idea of representation has greater promulgation than the idea of secondary dominants, which in turn enjoys wider dissemination than that of function symbols. It is not surprising that the idea of tone rep­ resentation found broad acceptance, as the basic notion of three cadential principal har­ monies and other subordinate harmonies were already common currency in theories of harmony and figured bass of the eighteenth century.54 And Riemann's term “representa­ tive,” likewise, has a prehistory. Sechter used it, among other things, to declare the triad and the seventh chord, in Rameau's sense, as a form of the fifth scale degree (without root), and sometimes simply to describe a chord above an “intermediate foundation” (Zwischenfundament).55 One can, however, hardly speak of a systematic theo­ ry of tonal representation, neither in Sechter nor in other of Riemann's predecessors.56 In fact, there is hardly any theory of harmony after Riemann that had not taken up this fun­ damental idea, which is inextricably related to Riemann's absolute view of cadential pro­ gressions.57 And yet, the theory of representation has also faced criticisms and modifica­ tions, not least as the system of Parallelklänge is the last bastion into which dualistic thought has withdrawn.

Applied Dominants So self-evident is the principle of applied dominants as a staple of theories of practical harmony that it might seem surprising to see it included in this list. Needless to say, the idea of applied dominants also has its prehistory, but in fact neither the term nor any ana­ lytical symbols for the phenomenon existed before Riemann.58 The notion of the sec­ ondary dominant is a central component of the theory of functions and is closely related to Riemann's modern understanding of modulation. To get a sense of its attraction, it is important first to consider the preeminent notion of modulation, from which the theory of function set itself off in the second half of the nineteenth century. On a trip to Germany, having just become familiar with Riemann's reform efforts in the fields of phrasing and harmony, I stayed with him for a few days in Ham­ burg (1887). We were talking about Richter's theory of harmony (the vehicle of my initial education in this discipline), and when he said: “there are abominable exer­ cises in this book,” I did not quite understand what he meant.59

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The Reception of Hugo Riemann's Music Theory

Ex. 1.1. Emil Ergo cites an example from E. F. Richter's influential Lehrbuch der Harmonie (p. 90, no. 192), about which Ergo writes, “Let us now com­ pare this with the descriptions of the modulations (!) Richter supplies for this example: ‘Measure 3 presents a modulation to D minor, since C♯-E-G-B♭ no longer belongs to C major but undeniably to D minor. In measure 4 it is doubtful whether the C-major tri­ ad, which is foreign to the prevailing key (D minor), belongs to C major or to the following G major, while the modulation to A minor in measure 5 is unmistak­ able.’ ”

Emil Ergo soon began to understand why Riemann objected to Richter's exercises. It was particularly his notion of modulation, which was antiquated in Riemann's view: “A modu­ lation happens when a harmony foreign to the previous key occurs.”60 (p. 10) Ergo, a keen pupil of Riemann, interprets an example of modulation by Richter in the sense of function theory and mocks the interpretation given by the old theory of “figured bass,” as shown in example 1.1. Richter represents a traditional notion of modulation, which had been commonly accept­ ed since the late eighteenth century. Ergo's polemic does not do justice to this old concep­ tion of modulation, which is less based on the notion of a sonority than one of the diatonic scale, but Richter himself shows hardly any awareness of the traditions in which he moves in his entirely ahistorical work.61 The basic condition of the classic concept of mod­ ulation is the notion of “relation” (Verwandtschaft). “Relation” denotes a demarcated area of diatonic scales that are related to a central (fundamental) scale: in C major, for in­ stance, related keys (ignoring, for now, the hierarchical relationships between each oth­ er) are D minor, E minor, F major, G major, A minor (plus, with restrictions, the flatted seventh scale degree: B♭ major).62 The classic concept of modulation unfolds against the background of this diatonic “matrix.” Within the reach of this matrix, in which one “di­ verts” into closely related diatonic scales via intersections, as it were, neither the notion of a secondary dominant, nor that of a modulatory pivot chord, are necessary.63 Consequently, for Richter as well as for Sechter, enharmonicism plays a subsidiary role. Even though representations of harmonic space had fundamentally changed in the sec­ ond half of the eighteenth century—witness Albrechtsberger's Inganni, Vogler's Summe der Harmonik, or indeed Gottfried Weber's Tonnetz64—and a “limitless” harmonic space was available in principle, theorists nonetheless retained an approach to modulation that was modeled on diatonic relations: in leaping from one scale degree to another, as it were, it was merely the diatonic framework of reference that was being altered. It goes without saying that this classic concept of modulation, which furthermore implies the tra­ ditional, clear-cut distinction of harmonic progressions in both major and minor modes, Page 8 of 53

The Reception of Hugo Riemann's Music Theory was hardly useful in interpreting Liszt's or Wagner's “Romantic (p. 11) harmony.” Prema­ ture criticisms of Riemann's system tend to overlook that Riemann started his career as a Wagnerian and aimed, right from the beginning, to construct a harmonic system that would not shy away from Wagner's “Romantic harmony.”65 Contemporary attempts to em­ ploy classic scale-degree and fundamental-bass theories66 in the analysis of “chromatic harmony” mostly led to a diagnosis of derisory amounts of “modulatory” processes and a tremendous welter of figures and symbols below the chords.67 In the eyes of its support­ ers, function theory was predestined “to demonstrate how superficial are judgments that assert: ‘Wagner is always modulating!’ ”68 Applied dominants and Hauptmann's concept of the “major-minor key”69 allow the theory of functions to interpret harmonically rich progressions within one key without having to invoke modulations. The idea of the ap­ plied dominant was the necessary harmonic linking module, so to speak, within a concep­ tion of tonality that had distanced itself from a narrow diatonic notion of relations.70 Nevertheless, the idea of applied dominants spread slowly. It was only thanks to Ernst Kurth that the concept became common knowledge and was, just like the idea of tone representation, gradually accepted by almost all German and many non-German post-Rie­ mannian harmonic theories.71 Numerous theories of harmony, which in their author's eyes are based entirely on the theory of scale degrees—among them popular German the­ ory books such as those by Lehmacher/Schröder or Dachs/Söhner—are in essence more practical theories of function operating with Roman numerals than they are genuine theo­ ries of scale degrees or fundamental bass. Applied dominants (also known as “parentheti­ cal dominants” [Klammerdominante], “intermediate fifths” [Zwischenfünf  ], or indicated by symbols such as [V], V/V, V/II, etc.) have been adopted by many practical theories of scale degrees. Kurth himself avails himself in this way of a function-based theory of scale degrees, which is perhaps best called a theory of functions in the guise of scale degrees. This mixture is typical of Riemann reception in the first half of the twentieth century: the diverse elements of once distinctly divided schools—Viennese fundamental bass/Weber's theory of scale degrees on the one hand, Leipzig dualistic functions on the other—begin to merge. It is therefore best not to speak of the theory of scale degrees or the theory of harmonic function in the context of Riemann reception.

The Taxonomy of Functions When we speak of the theory of functions, we usually mean its symbols. They have be­ come the hallmark of the theory of functions. It is, however, conspicuous that the most in­ teresting of Riemann's adherents and successors—Halm, Louis, Kurth, Eugen Schmitz, Fritz Rögely, and Heinrich van Eycken72—did not adopt the taxonomy of functions. The idea of “apparent consonances” (Scheinkonsonanzen) was, to be sure, quite attractive for most of them and was developed further in productive ways by Louis above all. A feeling of discontent, however, prevailed with many (p. 12) theorists vis-à-vis Riemann's deriva­ tions of relatives and leading-tone changes (Parallel- and Leittonwechselklänge), the lat­ ter of which Grabner would subsequently rename “opposites” (Gegenklänge). In this it is not the much-discussed question of hierarchical subordination that is the decisive ques­ Page 9 of 53

The Reception of Hugo Riemann's Music Theory tion but rather the fact that Riemann's theory of function refuses to conceptualize rela­ tives and leading-tone changes as diatonic representatives. It is in the concept of relatives and leading-tone changes that the modern theory of func­ tions, as it is practiced today, carries with it the legacy of dualism, albeit not always con­ sciously so. Dualism had always involved more than the derivation of the minor triad for Riemann. Harmonic dualism, he writes in his Musik-Lexikon, is “the pursuit of the twofold (dual) relation of tones, in the major and the minor senses.”73 This definition must be un­ derstood in a much broader sense than that which Riemann is prepared to underwrite. Philosophically, Riemann's dualism is actually a monistic principle: everything is derived from this one primordial principle, from this “Ur-eine,” the primordial entity, in which the major-minor relation is the governing principle. To this day, theorists of function deter­ mine the so-called Stellvertreterklänge (representative sonorities) strictly in a dualistic sense: in minor tonalities Grabner's Gegenklänge lie below the main functions to which they relate, in major above, while in major relatives are below, in major above their main function. Even function theories that otherwise assume strictly anti-Riemannian positions, such as Wilhelm Maler's, which (particularly in the völkisch 1941 version of his Beitrag zur Harmonielehre, adapted to the prevailing National-Socialist ideology) emphasizes that his theory has nothing to do with “unmusical mental gymnastics” and “Hugo Riemann's unworldly construals,” rehearse this polar concept without any objections. This is the main practical difference between the theories of functions and all so-called theories of scale degrees. Stellvertreter are not actually diatonic sounds, even if they ap­ pear as such at first sight. The derivation of these Stellvertreterklänge from the principal harmonies by replacing the fifth with the sixth (and the root with the seventh, respective­ ly) serve to justify the concept of “apparent consonances” (Scheinkonsonanzen) as disso­ nances: the argument behind this justification is strictly dualistic—in order to derive the relative in major, for instance, the fifth is replaced with the upper sixth, in minor the un­ der-fifth is replaced with the under-sixth. The exchanged intervals are always absolute: major sixths in parallels, minor in leading-tone changes. This is why the leading-tone change of a subdominant D minor is always D–F–B♭ and never the diatonic D–F–B. Rie­ mann offers a very simple solution: “Relatives are all those pairs of Klänge that are in the relationship of tonics to their relative keys, which we…derive from the self-same third by adding the upper or the lower fifth.”74 He calls the under-E Klang (A minor) the Terzwech­ selklang or Parallelklang of c+. What is meant here is not the relation between scale de­ grees but rather the major-minor relations between autonomous sonorities. The represen­ tative may have its “origin” in the scale, but its dualistic determination is not contingent on it. For practical composition, this concept has important consequences: not only does the theory convey the impression in its concepts that minor was the “opposite” of major, but in the major-minor polarity the (p. 13) second scale degree in minor becomes the black hole of the theory of function. In the system of major-minor relations, this “scale de­ gree” does not occur.75 It is primarily for these reasons that the theory of functions gained acceptance only slowly and encountered much resistance. It was only its specific development in National Socialism that led to its monopoly, which allowed Maler's func­

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The Reception of Hugo Riemann's Music Theory tion symbols to reach virtually all institutions of higher education after the Second World War.76

Pedagogical Reform and the Theory of Func­ tions: Vers une analyse fonctionelle Analysis is the best part of the study of composition.77 The partial eclecticism of post-Riemann theories of function is closely related to the cul­ tural and intellectual movements at the turn of the century. Robert Wason rightly talks about a fin-de-siècle “New Empiricism” in his groundbreaking study Viennese Harmonic Theory.78 With the clear task of cultural criticism in mind, the new practical theories of harmony turn against what they perceive as the leaden deserts of nineteenth-century the­ ory. The theories of harmony of that time reveal a much greater resemblance to their eighteenth-century counterparts: they are oriented by the work of art and operate with numerous examples from the repertoire, which had completely disappeared from the the­ ory books of Hauptmann, Richter, Sechter, and Riemann. Schreyer's battle cry that analy­ sis should be in the center of any theory of harmony became the much-commended and oft-quoted catchphrase of music-theoretical discourse.79 Even Dahlhaus refers to Schreyer's Harmonielehre—without discussing it in any detail—as the “first analysis cur­ riculum in the history of music.”80 This “New Empiricism” is closely related to culture-critical movements, which were di­ verse and in themselves divided, but which can all be subsumed under the notion of “life reform” and which concerned all areas of cultural and social life without exception: Lebensphilosophie (philosophy of life), Jugendbewegung (youth movement), Reformpäda­ gogik (reform pedagogy), Lebensreform-Bewegung (life-reform movement), Kun­ sterziehungsbewegung (art-education movement), Nietzscheanismus (Nietzscheanism) are only the best known of these partial areas. While it is difficult to find a common de­ nominator for all without generalizing unduly, one finds in these movements a common critical position, if not indeed a hostility, toward rationalism, an emphasis on experience and spiritual understanding as opposed to theory, on concrete examples as opposed to ab­ stract knowledge.81 Almost all of the influential function-based theories of harmony of the twentieth century are connected with the life-reform movement, whether it is the great “reform-pedagogi­ cal” theories of harmony of the 1910s of August Halm and Johannes Schreyer, or the “Schopenhauerian” theory of Rudolf Louis, or the “youth-musical” or “völkisch” ones of Hermann Grabner and Wilhelm Maler of the 1920s, (p. 14) 1930s and 1940s.82 A consider­ able part of the views taken in Ernst Kurth's music-theoretical oeuvre, in which various music-theoretical and life-reforming tendencies of his age converge, can still be consid­ ered from the angle of Riemann's legacy. The impact of Kurth's theory, however, exceeds that of its immediate predecessors so much so that it becomes almost nonsensical to

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The Reception of Hugo Riemann's Music Theory speak of succession in this context: Kurth himself becomes the reference point of almost all textbooks of function theory of the 1920s and 1930s. Felix Diergarten has shown how Riemann's theory of functions was transformed in Jo­ hannes Schreyer's holistic theory of harmony, marked by the “art-education movement.” Schreyer's theory is the only one among the important function-theoretical efforts of the 1910s that takes over Riemann's taxonomy of functions. The reasons he cites are almost identical to those used by Grabner almost twenty years later: “It was particularly impor­ tant to the present author to explain the formula T–S6–D7–T as early as possible and to demonstrate that 1. all progressions used in music are but derivations from these funda­ mental sonorities and 2. it is possible to analyze with this plain formula the most complex modern compositions.”83 Function symbols were used to serve for radical simplification. Grabner later calls the “functional recognition” of a chord the “reduction of a complicat­ ed sonic structure to its simplest form.” With this, the reform-pedagogical theory of func­ tions takes up an aspect of Riemann's oeuvre, which was situated, as it were, beyond the abysses of the dualistic discussion in theory—musical analysis. Riemann wrote numerous harmonic analyses, worked empirically in the sense of “reform pedagogy,” while his har­ mony tutors disregard analysis almost completely. To link the teaching of harmony with Riemann-style analysis was the openly stated goal: But while Riemann declares as the goal of his harmony teaching (cf. his Handbuch der Harmonielehre, 3rd edition, vii) getting his pupil to “write a four-part composi­ tion in the four vocal clefs as well as for transposing instruments in a few minutes, or to realize a chorale with figured bass at the piano in four parts in transposition without reflection,” we consider our supreme task the introduction to an under­ standing of the masterworks.84 The observation that Roman numerals make it “not quite impossible, but rather cumber­ some to analyze whole compositions harmonically,” because this required “also acciden­ tals for the scale-degree figure,”85 is not merely a commonplace of the theory of func­ tions: the economy of function symbols, particularly in the analysis of harmonically com­ plex music, is surely one of its strongest qualities. On the basis of the economy of its ba­ sic elements, both Schreyer and Grabner were hoping to be able to begin their teaching of composition immediately with analytical exercises. However, it is important to bear in mind—and this will be discussed later—that in Schreyer, Ergo, and Rögely, function sym­ bols are listed alongside Riemann's Klangschlüssel, his shorthand taxonomy for chords (albeit in a monistically modified form), which as the actual “reductive notation” plays an even greater part than the function symbols themselves. In Schreyer's case, the interplay between (p. 15) Klangschlüssel and reductional sketch fulfills the analytical function that later on function symbols alone will take on. The idea of harmonic reduction is central to Schreyer's theories. The production of a har­ monic reduction is central to both the analysis and the teaching of practical composition. A typical compositional exercise in Schreyer, which also aimed to understand a particular compositional style, looked as follows: the vantage point was a concrete work, of which a Page 12 of 53

The Reception of Hugo Riemann's Music Theory harmonic reduction had to be sketched. In general, this implied writing a two-part har­ monic skeleton. This skeleton then had to be “composed out” by the pupil in three parts, as shown in example 1.2. Not only is the pronounced connection between contrapuntal and harmonic thinking rem­ iniscent of Schenker, but Schreyer also worked with analytical “levels.” Schreyer's analy­ sis of Mozart's much-analyzed “Dissonance” quartet (K. 465) may serve as an example. Schreyer first reproduces the score, followed by two analytical levels; a third level is dis­ cussed only in the very concise explanatory text. Schreyer adopts Riemann's maxim “that [in this work] only the correct understanding of the suspensions reveals the harmonic progressions.”86 Reduction B, shown in example 1.3, presents the first layer, in which Schreyer rhythmically dissolves, so to speak, the “stratified” dissonances and puts them in their actual metric position. The explanatory text adds yet another level of reduction and relates the whole composi­ tion to a deeper (chromatic) fauxbourdon texture, shown in example 1.4, which Schreyer —who obviously could not know this rather modern analytical term—describes as a “suc­ cession of sixth chords (in the sense of figured-bass terminology).”87 Example 1.5 shows the “final” reductional sketch, which interprets this opening in harmonic terms. The principle of Schreyer's reductive notation can clearly be recognized in the example: the small notes signify that these are (hierarchically subordinate) dissonances that do not belong to the actual harmony. He simplifies the complex structure into a clear (in essence three-part) skeletal structure of diatonic “progressions” (Züge). He is careful to bring out the relevant relation of downward progression, which lies at a deeper level, from G to B in measures 6–13 and of the same ‘canonic’ and overlaid progression in measures 12–16. A structural element like this is completely disregarded in Riemann's motivic-thematic an­ alytical approach. Schreyer's final reduction (in C), working out the harmonic “back­ ground,” by no means makes the previous reductions redundant: analysis is the way from one analytical level to the next, and the interplay between them. This analysis, an exemplar of Schreyer's approach, is explicitly directed against Riemann's model, particularly as promoted in Große Kompositionslehre. Schreyer juxta­ poses his complex theory of harmonic levels with Riemann's spelling out of vertical sonic elements. For Riemann, the opening is “in C minor, the second [phrase] in B♭ minor; both merely circumscribe the simple formula °T–D.”88

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Ex. 1.2. Schreyer's analytical method. The fugal theme from Bach's two-part fugue in C minor (a) is turned into a skeletal harmonic sketch (b), which is then in turn composed out in three parts (c).

Ex. 1.3. “Reduction B” from Johannes Schreyer's Harmonielehre dissolves the stratified dissonances from the opening of Mozart's “Dissonance” Quartet.

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Ex. 1.4. The next reductive level in Schreyer's Har­ monielehre leaves a fauxbourdon texture.

Ex. 1.5. The final level, “Reduction C” of Schreyer's Harmonielehre adds an interpretive layer.

Ex. 1.6. Riemann's harmonic interpretation of the opening of Mozart's “Dissonance” quartet.

Ex. 1.7. Schreyer's alternative interpretation of the same passage.

Riemann's harmonic analysis of the first for measures of the quartet is as shown in exam­ ple 1.6. Example 1.7 presents the example in staff notation using Grabner/Maler's more contemporary function symbols. Whereas for Schreyer, many of the chords are merely “apparent harmonies” (Scheinharmonien), Klänge that emerge on (p. 16) (p. 17) (p. 18) the basis of melodic processes, or on the basis of a (temporal) disjuncture between melod­ ic and harmonic identity, Riemann's analysis can be guided only by the structures as they really sound, which leads to the harmonic interpretation of a temporary added sixth chord in minor, in which the apparent dominant G major appears as though it were a G minor altered into major. It should be added, in Riemann's defense, that this insistence on real Klänge or on the potential of “local harmonies” also constitutes an essential and fu­ ture-oriented element of music theory.89 At any rate, it would be unthinkable for Riemann Page 15 of 53

The Reception of Hugo Riemann's Music Theory to take apart the harmonies contrapuntally, as Schreyer does as early as his first level of reduction. With the final reduction, however, Schreyer's interpretation is still unfinished: having ar­ rived at the deepest structural level, he argues that Riemann's vantage point itself, the assumption that the movement began in C minor, as well as his idea of a “descent of keys,” was wrong. The “motley succession of harmonies,” which Riemann's analysis pro­ duces, would be out of style: it is “entirely un-Mozartian,”90 as Schreyer argues. The whole introduction, he argues instead, is in C major and is nothing but a nonsounding composed-out pedal point (which Rudolf Louis would call “idealized pedal”) on the domi­ nant: “[The introduction] should be understood in the sense of the dominant of C major, as a bold pedal on G, which however follows a strictly logical development.”91 It is easily to overlook, on the basis of Schreyer's critique, that the principle of reduction was once inscribed into the very idea of the theory of functions. It is no exaggeration to consider the practice of relating complex harmonic and metric structures back to a basic skeleton as one of its most central original ideas. In some ways, one could conclude that Schreyer argues against the late Riemann by using arguments of the early Riemann, for all Schrey­ er does is to productively continue Riemann's early, reductive efforts from his Hamburg years. Example 1.8 shows Riemann's reduction of J. S. Bach's Fugue in A minor from the WellTempered Clavier.92 Riemann chose it “because it almost never presents the (p. 19) chords directly but always masked by scalar movement.”93 Riemann indicates in detail the func­ tion of the separate melodic tones: neighbor notes, passing notes, consonant skips, in­ complete neighbors (verlassene Wechselnote), échappée (springender Durchgang), ap­ parent passing note (fingierter Durchgang), syncopation, prepared dissonance, anticipa­ tion, and so forth. Schreyer's thinking was particularly affected by Riemann's usage of the Klangschlüssel (which, for Riemann, was of course a strictly dualistic concept), indicated below the bass note. In this example, we can see that in early Riemann an important as­ pect ovf the Klangschlüssel was its capacity to interpret even longer passages with inde­ pendent voice-leading as the unfolding of a single underlying harmony. And within this “unfolded” Klang, “local” passing harmonies can occur. In the “harmonic skeleton” found at the end of Riemann's analysis in example 1.9, these “local” Klänge have completely disappeared. As contrapuntal voice-leading phenomena, they are merely secondary. Compared with the later, mature Riemannian theory, the con­ trast is stark: for the thirty-three-year-old theorist “there was no difference between har­ mony and counterpoint.”94 Chordal relations, he goes on to argue, are the essential core of counterpoint. Like the young Riemann, Schreyer too turns against “this fragmented approach to art.” It is possible, he argued, to “prove historically that the separation of harmony and counter­ point, in strict and free forms,”95 was the chief culprit in the promulgation of a method that had not advanced in 150 years and had caused a perplexing rift between “theory and practice.”96 Page 16 of 53

The Reception of Hugo Riemann's Music Theory Schreyer radicalized Riemann's idea of Klangschlüssel notation. Example 1.10a shows a neighbor-note figure, a soprano clausula, which Schreyer places at the beginning of his Harmonielehre.97 It is in this linear movement that Schreyer identifies the germ cell of all harmonic progressions. Example 1.10b shows further how Schreyer integrates the idea of the neighbor note into the concept of Klangschlüssel.98 All these examples move within the boundaries outlined by Riemann. In Example 1.10c, however, the passage shown un­ der (f) is barely a harmonic progression that Riemann would have recognized as the un­ folding of a Klang.99 Schreyer's commentary accompanying these examples indicates to what extent the origi­ nal significance of Riemann's concepts has been transformed in Schreyer's hands. The no­ tion of the “representative” (Stellvertreter), as well as that of the apparent consonance (Scheinkonsonanz), is associated with a different meaning. He introduces a concept for such Klänge that later on denotes even further reaching harmonic processes: Scheinhar­ monie, or “apparent harmony.” Coined in analogy to Riemann's concept of Scheinkonso­ nanz, it indicates Klänge that assume, within a specific structural context, a different sig­ nificance from the expected one. The harmonic reduction of Felix Mendelssohn's Song without Words op. 52, no. 2, shown in example 1.11a, is a good example of how Schreyer translates Riemann's Klangschlüssel into graphic analytical representation. All the pitches that are not a component of the Klang indicated by the Klangschlüssel are conceived as melodic representatives, so-called Einstellungen (modifications). They are indicated by means of grace notes, which Schrey­ er calls “vicariate” (Vikariat).

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Ex. 1.8. Riemann's melodic analysis of the opening of Bach's Fugue in A minor. Riemann explains that W indicates neighbor note, D passing note, A consonant skip, W˘ incomplete neighbor, fD apparent passing note, S syncopation (prepared dissonance), ↘ antici­ pation. He points out, under NB, that “the D is most curious here: if it is not actually a D♯, with D follow­ ing only in the next measure (in which case we would have 1 1/2 measures of b+ [in German: h+] and °b [°h] would become a suspension b64), then the D is a kind of anticipation from the following chord (e7), which should therefore be understood in the sense of b+ as 3〉 (flatted third).”

With detailed graphic elaborations that build on a Riemannesque harmonic skele­ ton, Schreyer achieves a fine-tuned functional hierarchy. The reduction in example 1.11b interprets measures 22–28 as a dominant pedal on F. But this interpretation, Schreyer ar­ gues, caused “more problems to the ear” than the version shown in example 1.11b.100 It is worth underlining that Schreyer is interested not in promoting the one or the other in­ terpretation as the only correct one, but (p. 21) rather in showing that there are two inter­ locking harmonic levels: the dominant pedal is modified by a superimposed subdominant pedal. In particular, the concept of the “parenthesis,” which occupies a central position in Schreyer's analytical method, shows most clearly how much further Schreyer developed Riemann's “harmonic skeleton.” The chordal progression in square brackets (measures 16–17) does not, for Schreyer, constitute a progression of independent Klänge, on ac­ (p. 20)

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The Reception of Hugo Riemann's Music Theory count of its “episodic character,”101 but rather is a “parenthetical” composing-out of a tonic Klang.

Ex. 1.9. The harmonic skeleton of Bach's A minor fugue, in Riemann's analysis.

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Ex. 1.10. Schreyer speculates that the melody origi­ nated in (a) the combination of a tone with its diaton­ ic neighbor. This, Schreyer contends, leads us straight into the relationship between consonance and dissonance. In (b) and (c) this principle is worked into a variety of increasingly complex fourpart harmonic textures.

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The Reception of Hugo Riemann's Music Theory

Ex. 1.11. (a) Schreyer produces a harmonic reduc­ tion of Mendelssohn's Song without Words no. 20 and (b) an alternative interpretation of measures 22–28.

Schreyer elaborates this concept using an exam­ ple from the transition to the finale of Beethoven's Fifth Symphony, which he considers a (p. 22)

(p. 23)

(p. 24)

(p. 25)

(p. 26)

(p. 27)

“parenthesis writ large.”102 Schreyer works down from the surface to the deepest level in four sketches, shown in example 1.12, outlining the process of reduction. The first sketch (numbered 183) adheres fairly closely to Beethoven's original musical text, except that each quarter note corresponds to one whole measure of the original—that is, one mea­ sure of the sketch corresponds to four measures in Beethoven. Schreyer's first stage of reduction almost always follow an idealized three-part texture, with the top part indicat­ ing Beethoven's melodic lines. In the subsequent sketches, the “harmonic quintessence” (reached at 186 in example 1.12d) is worked out ever more clearly, “by jet­ tisoning all redundant incidental elements (especially the many tendrilous suspensions.”103 The second sketch (184) is already a condensed version of the harmonic activities, and bears a certain resemblance to an eighteenth-century thorough-bass “skeleton” (Albrechtsberger): both harmonic and melodic repetitions have been eliminat­ ed in this sketch. The semitonal encircling figure A♭–G–F♯–G–A♭ comes to the fore, which is accompanied by a voice exchange of the middle parts (F♯–G–A–G–F♯). The most impor­ tant aspect of this sketch is the reduction of the syncopated rhythms of the first sketch to a simple rhythmic form. By this means, he underlines the function of G as a pure passing tone—or rather, a pure passing harmony—and, consequently, the passage up to the domi­ nant pedal G can be explained as unfoldings of a single harmony. This sketch, moreover, emphasizes the scalar structure of the upper voice. The third sketch (185), meanwhile, reduces the entire transition further to a basic two-part texture. The small note head C in Page 22 of 53

The Reception of Hugo Riemann's Music Theory the putative third inner part merely serves to illustrate that the whole melodic progres­ sion of these fifty measures are essentially nothing but the melodic unfolding of a soprano clausula. The final sketch presents the final resolution: the dominant pedal is reduced to the embellished quarter note of the fourth beat. This embellishment, however, is nothing but a “vicariate” in Schreyer's theory, a contrapuntal modification of the underlying Klangschlüssel-based harmony. It is in this way that the entire transition can ultimately be reduced to a simple cadential combination of a bass and soprano clausula. Just how far-reaching the consequences of Schreyer's concept of parentheses are can be seen in his analyses of Liszt, Wagner, and Chopin. For the “parenthetical” composing-out of a Klang also allows its enharmonic reinterpretation. The reduction of a passage from Liszt's Valse impromptu, shown in example 1.13, is for Schreyer a pedal point that has been extended by means of “parenthesis.”104 In this parenthetical composed-out form, it has been reinterpreted enharmonically into the third of the D7 chord. Even though this harmonic turn “reaches the limits of tonal understanding” “the pedal sharply displays the tonal harmonies.” Both dominants, including the six-five chord on G♯, should be under­ stood “as suspensions to T.”105

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Ex. 1.12a–d. Schreyer analyzes the last movement of Beethoven's Fifth symphony in four stages of reduc­ tion.

Schreyer's Lehrbuch der Harmonie, especially in the second edition of 1905, is doubtless one of the most important and most independent documents of German music theory of the early twentieth century. It is astonishing and regrettable that in (p. 28) (p. 29) the sub­ Page 24 of 53

The Reception of Hugo Riemann's Music Theory sequent evolution of the theory of function these reductive aspects are completely lost: the later tendency to add heaps of figured-bass annotations above the function symbols are actually a contradiction in terms. Riemann's own theories, as became clear apropos of his Mozart analysis, tend toward total verticalization. Later function theorists such as Grabner, Maler, and Distler would finally succumb to this tendency. By contrast, elements in which analytical thinking that would aim to a melodic interpretation of harmonic events are rarely found in post-Riemannian theories.106

Ex. 1.13. Schreyer's harmonic reduction of a pas­ sage from Liszt's Valse impromptu.

Ex. 1.14. Riemann's version of a voice exchange, from his Handbuch der Harmonielehre.

Example 1.14, from the third edition of Riemann's Handbuch, is reminiscent of what Sechter called “voice exchange” (Stimmtausch): “For the duration of one and the same fundamental chord, the voices can swap their parts.”107 The sonorities of (p. 30) the sec­ ond and third measures are not interpreted as full-fledged vertical events, in the sense of autonomous functions, but rather as linear-melodic ones. It is a hallmark of Sechter's the­ ories, and those of his successors, that the concept of the passing (or neighbor) note cap­ tures not merely a melodic dimension but also has a harmonic dimension, which becomes a central element of his theory of harmony: counterpoint and harmony converge in the concept of the passing note. One could even claim that the whole of Schenker's theory is the result of setting this idea of the passing note as an absolute. Robert Wason has im­ pressively described this moment in Viennese music theory. Just how important it also is in Schreyer's theory has been emphasized by Diergarten in the harmonic reduction of Liszt's first Consolation.108 Schreyer, however, is not the only theorist of functions who has spent a lot of time think­ ing about the contrapuntal interpretation of harmonic processes. Emil Ergo's considera­ tions concerning his analysis of the Tannhäuser overture even led further than what Carl Mayrberger, who expanded Sechter's concept of the passing note considerably, hoped to capture in his concept of the “passing chord”:109 in analyzing harmonic progressions, Er­ go argues, one has to sharply “distinguish between appearance and essence.”110

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Ex. 1.15. Emil Ergo analyzes Wagner's Tannhäuser overture.

Ex. 1.16. Ergo juxtaposes two alternative functional interpretations of the Tannhäuser overture.

Ex. 1.17. Ergo analyzes the theme from the last movement of Beethoven's Fifth Symphony.

Examples 1.15 and 1.16 show two of Ergo's function-based interpretations of the opening of the overture. The version of example 1.16, marked (a), in which every chord of the triplet figure in the second measure receives its own harmonic label, corresponds, ac­ cording to Ergo, to traditional Riemannian interpretation. If we took, however, Riemann's definition from his 1894 Vereinfachte Harmonielehre at face value, “that the actual carri­ ers of harmonic effect are the downbeats,”111 and consistently applied this idea to har­ monic theory, we would have to introduce a notion of “ornamental chords” or “passing chords.”112 Ergo demonstrates this by means of an analysis of the theme from the finale of Beethoven's Fifth Symphony, shown in example 1.17, in which he argues that the domi­ nants on weak beats are nothing but ornamental chords (b). The first four measures in their entirety are an (p. 31) unfolding of the tonic (a) “although the whole harmony of D7 is always clearly represented by the whole orchestra.”113 But Ergo goes further than that. Not only should the second chord of the triplet group from Tannhäuser, the “swiftly passing, ‘unaccented’ S-chord,” be interpreted in the sense of an embellishing chord, but also the tonic six-four starting the group was no indepen­ dent element but only an apparent harmony. The whole measure could also be represent­ ed as shown in example 1.18.114 Page 26 of 53

The Reception of Hugo Riemann's Music Theory

Ex. 1.18. Ergo's alternative interpretations of the triplet motive from the Tannhäuser Overture.

Ex. 1.19. Ergo's synoptic view of the opening of the Tannhäuser Overture.

If one continues this principle, then measures 2 and 3 show nothing but the har­ monic embellishment of the relative minor of the tonic (Tonikaparallele). Ergo works out (p. 32)

this radical function-theoretical reduction so that finally the harmonic analysis of the opening of the overture looks as shown in example 1.19. Given our current knowledge of the sources, it is impossible to say whether Schreyer's and Ergo's reduction techniques go back directly to ideas taken from Viennese fundamen­ tal-bass theories.115 Nor can the question be answered whether Schreyer's theories of harmony could in any way have influenced Schenker in turn. Ergo was a profound con­ noisseur of the German and French traditions. However, Sechter's work is never men­ tioned in his writings on Wagner analysis. It is possible, though, that Viennese harmonic thought could have been mediated, albeit unconsciously, by the popular Harmonielehre of Louis/Thuille. Nor did Schreyer ever mention Sechter. It is more likely that Schreyer and Ergo would have continued some of Riemann's ideas on passing notes from his early Neue Schule der Melodik (1883) and his sporadic comments on the passing note in Handbuch der Harmonielehre. What can be said with certainty, however, is that the most successful and most important theory of harmony of the first half of the twentieth century brought together the traditions of Vienna and Leipzig—the Harmonielehre of Rudolf Louis and Ludwig Thuille.

Synthesis: Rudolf Louis Just before Louis/Thuille's theory of harmony was published, a long announcement ap­ peared in the monthly Süddeutsche Monatshefte. This text is a significantly expanded ver­ sion of the preface. In it, Rudolf Louis explains his aim and the special (p. 33) approach of his study. He explains that he takes a “strictly empirical standpoint,” his theory is orien­ tated by analysis and the experience of the work of art:116 “For harmony, as we under­ stand it, the starting point is analysis, as faithful and exhaustive as possible, and uninflu­ enced by any theoretical prejudice, of that which the musician of our time and our culture actually hears in musical sounds and the connections between them.”117 Louis is trying to set himself apart from what he considers “dilettantish” analytical attempts of reform ped­ agogy of his time. The aspirations for “systematic” penetration—or, simply put, for theory Page 27 of 53

The Reception of Hugo Riemann's Music Theory —remain valid, he claimed, even if “any theorizing is a problematic undertaking,” since there is no “theory that does full justice to reality.” This is “precluded due to the nature of the relationship between subject and object: for the peculiar power of our spirit resides in precisely the fact that it is able to think the particular in the general, and diversity within unity, while everything—even the least significant particular—has its essence in being unique and incomparable, something that, strictly speaking, immediately ceases to be that which it is once we subsume it under a general term.”118 Thus his theory of harmony was meant to be the opposite of those music-theoretical works that dominated the nine­ teenth century: A few more or less correct observations give rise to a thought, and from this thought a theory is then spun out “deductively” without paying much attention to the nature of the object itself. The thought is spoken “in Hegelian,” and is left to its own devices and its own motion…. In this way speculative theory comes about…. Thanks to its regulatory architectonics, its clean symmetry and the smooth parallelism of its parts, it becomes the more compelling the more it resists the temptation to do justice to the facts, and the more it satisfies itself with the erection of a fantasy building.119 The exemplar of this mode of thought for him is Moritz Hauptmann: “the way in which he constructs harmony and meter along the lines of the triple-jump scheme of Hegel's dialec­ tical method will forever remain a deterrent example, showing into what wilderness even a theorist who is intimately familiar with his object can get entangled if he is captured by the suggestive force of a pre-formed opinion.”120 Louis harbors “great admiration and sincere gratitude” for Riemann. He is the “most bril­ liant representative” of his subject, “a German Fétis.” But then he adds criticism. [Riemann] would doubtless have been the most suitable authority to make the cer­ tain results of a purely theoretical harmony available for musical teaching in a fruitful way. Indeed, all his later publications on harmony have been dedicated to this very purpose. If these publications…now meet with relatively little success with real musicians, we have to assume that the cause for this failure must be sought exclusively in the highly speculative tendencies of Riemann's thinking.121 Louis notes an “unfortunate passion of mental construction, a predilection for premature generalization and analogizing, a pre-eminence of subjective factors in his theorizing, which in the final analysis has its cause in a—please excuse the harsh expression—lack of respect for the facts.”122 The conception of minor in dualism (p. 34) has “something se­ ductive for speculative minds,” but “for the unassuming musician it is unacceptable, in­ deed, it is basically intolerable…. The conception of minor sonorities is a phantasmago­ ria.”123 Louis does no less than to formulate a manifesto of music-theoretical “New Empiricism”: “The theory of harmony,” he already emphasized in his doctoral dissertation (in philoso­ phy), “is not a science…. Its principles are, to speak with Kant, not constitutive, but only Page 28 of 53

The Reception of Hugo Riemann's Music Theory regulative.”124 As a Realdialektiker,125 he is convinced that—simplifying somewhat—ratio­ nal thinking is, so to speak, an incidental product of a more fundamental “drive-guided” dynamic of the will. Any thinking, and particularly thinking about music, he argues, is subjected to a “psychic” dynamic. No discussion of music may lose contact with this psy­ chic basis: speaking about music is for Louis always speaking about humankind. Its psy­ chological tendency reveals Louis's theory of harmony as the direct predecessor of Kurth. Without exception, all Kurthian themes can be found at least touched on in Louis. Louis's theory, however, is much less radical in its phenomenological tendency; its ambition is more theoretical. For the “dynamic of the will” is merely the Schopenhauerian side of “re­ al-dialectical” thinking. For Realdialektik believes—hence the name—that contradictions and oppositions of being and thought cannot be led toward any Hegelian synthesis. The complexity of Louis's argumentation is based on this particular dialectical model: as a subject-centered theory of hearing and experiencing, it never does violence to the ob­ jects, the phenomena, but always considers them in their own rational logic and dynamic. The contradiction of subject and object is itself forever endured again and again: its only solution is “balance.”126 Louis's criticism had found Riemann's weak spot. And what happened, happened as it had to. Riemann immediately recognized that his claim to universal hegemony was seriously challenged.127 His review of the book, which appeared almost concurrently with Louis's publication in Süddeutsche Monatshefte, was “dualistic,” as Louis observed sarcastically. Riemann noted, “swiftly and succinctly,” that the book was “one of the most interesting publications in the field of music theory.” It had “cleared out” old mistakes but had avoid­ ed “pouring out the baby with the bathwater.” Riemann praised the “foundation of care­ fully chosen examples from the latest compositions by Richard Wagner, Franz Liszt, Anton Bruckner, Richard Strauss, Max Schillings, Ernst Böhe, Emanuel Chabrier, Ludwig Thuille, etc.” Further, he extolled the “reduction of the entire essence of harmony to the principle of tonality and the three tonal functions,” and the “theory of tonal representa­ tion.”128 But all of a sudden, the tone changed to irony. It would “doubtless be considered a particular advantage of the book that it preserves the good old figured bass and, only where this fails, it draws on Gottfried Weber's scale degree labels for chords, which have now been tried and tested for almost a hundred years.”129 Finally, the review changes in­ to a vehement attack, which even moves the prior praise into a very different light: Louis/ Thuille's theory of harmony is so excellent, Riemann argues, because it is entirely written “on the basis of my views, and—except for a few irrelevant details—replicates what I have posited.” The “core of the theory” is the “theory of tonal functions of harmony, which Louis/Thuille's book (p. 35) repeats in such breadth that it is not quite understandable why they would not also make use of the convenient shorthand symbols T for tonic, D for dominant and S for subdominant.”130 Even the rules of voice-leading, deduced from the theory of tonal functions, Louis allegedly took from Riemann. In short, Riemann accuses Louis of plagiarism and of intellectual theft: “Given the great dependence of the book on my works, I would have expected to be referred to in the preface.”131 Louis's attitude is not “fair,” he argues, and closes with the statement: “I shall leave it to other expert critics

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The Reception of Hugo Riemann's Music Theory to determine whether I have gone too far in emphasizing the dependence of this book on my ideas or not.”132 It is obvious that Riemann's review was the result of “a wholly abnormal state of anger and embitterment” and can be “fully explained psychologically,” as Louis puts it in his re­ sponse.133 But is the reproach justified in any way? It is not a coincidence, nor is it a con­ cession to the market, that Louis does not adopt Riemann's function symbols. Although he recognized (and utilized) the potential that the musical space of the Tonnetz opened up, in his view the principle of dualism did, as we saw above, too much violence to the diatonic foundations of harmony. Louis tries to refine the idea of the autonomy of Klänge—as stacked-up thirds on the diatonic scale degrees, as derived in Sechter's fundamental-bass theory—in light of the theory of tonal representation. For Louis, representatives are not generally “apparent consonances,” as they are for Riemann. For Riemann, any chord root­ ed on the second scale degree in major is a dissonance, which appears only as a “physical consonance.”134 For Louis, by contrast, representatives may have the character of appar­ ent consonances (that is, actual dissonances); in many situations, however, they are au­ tonomous sonorities, which bear only an “idealized” relation to the principal function. To hone Riemann's theory of representation, he links it ingeniously with the “linear” theory of changing notes, suspensions and passing notes taken from the Viennese fundamentalbass tradition with which Louis, as a second-generation student of Bruckner's,135 was inti­ mately familiar.136 For this purpose, he replaces the concept of the “apparent conso­ nance” with that of the “conceptual dissonance” (Auffassungsdissonanz). Conceptual dis­ sonances are “chords that are always consonant outside of the context in which they ap­ pear, but that can occasionally be used in such a way that they are dissonant with respect to the understanding of the broader harmonic context.”137 In a manner of speaking, Louis turns Riemann's concept phenomenologically upside down: “appearance” is the effect for Riemann, which obscures the true (theoretical) essence of the Klang, while for Louis it is the (context-free) structure, which blocks the effect of the Klang: “apparent consonances” sound consonant but are dissonant; “conceptual dissonances” sound dissonant but look consonant. For Louis, the conflict is no longer between structural essence and sonic ap­ pearance but occurs only on the level of perception: the effect of sounds is defined as a conflict of (context-free) sonic autonomy and each harmonic contextualization.138 The fun­ damental ambition of Louis's theory of harmony is to mediate between both. “Sonic au­ tonomy” (Klangautonomie) represents (in the sense of an ideal type) the “vertical” Rie­ mannian heritage, while contextualization represents the “linear” Sechterian legacy. In this light, the notion of conceptual dissonance is virtually identical to Sechter's concept (p. 36) of the “artistified composition” (gekünstelter Satz).139 Louis returns to the same examples as Sechter (bordering on citation) to introduce his concept, using the “gekün­ stelter Quartsextakkord” and of the “gekünstelter Sextakkord”140 (meaning cadential double-suspension six-four, and suspended sixth chords, respectively), that is to say: chords that are actually the product of dissonant linear voice-leading procedures and only “look” consonant. As chords that are emancipated from real voice-leading contexts, they turn into Louis's conceptual dissonances: “The clearest manifestation of the subdominant significance of the triad of the second scale degree is in its second inversion, as a sixth Page 30 of 53

The Reception of Hugo Riemann's Music Theory chord…. The chord gains a certain resemblance to a conceptually dissonant sixth chord in which the sixth precedes the fifth of the subdominant as a suspension, or follows it as a passing tone.”141 The contrapuntal interpretation of harmonic procedures is Louis's most central concern: It should not be forgotten that even in musical creation the most diverse require­ ments can arise and can come into conflict with one another. That which, from a purely harmonic viewpoint, would be pure nonsense can be possible if it appears somewhat melodically-contrapuntally motivated, and vice versa: voice leading that is melodically requisite (for instance a resolved leading tone) can be evaded with­ out harm if a harmonic advantage can be gained by this irregular progression (such as the completion of the chord, which might only be attainable in this way) to compensate for the melodic awkwardness.142 Here we encounter an aspect for which there is only little room in Riemann's theory of functions. Louis develops the Viennese theory of passing tones to a degree of differentia­ tion comparable to Schreyer's analytical reductive technique that is today surpassed only by Schenker's theories. When Riemann casually praises the “clearing of the view for the distinction between principal forms and accidental subsidiary forms, as they arise from figurative changing notes,”143 this goes to show only how little meaning this fundamental aspect of Louis's theory holds for him. In the second part of the theory of harmony, Chromatik und Enharmonik, Louis transfers the concept of the passing tone of Viennese fundamental-bass theory to the modern (chro­ matic) harmony of his age.144 Using an example of Max Schillings, shown in example 1.20, Louis explains a phenomenon that he calls “free suspension” (freie Vorhaltsbildung).145 Louis argues that the example shows “nothing but four triads, on E, C, F, and D,” in which the entry of the C-major and F-major chords is delayed by a “chro­ matic suspension.” We see “two six-four-two chords, derived through free suspensions, which would be completely wrongly understood if they were explained as inversions of ac­ tual seventh chords (E♭–G–B–D♭ and A♭–C–E–G♭).”146 As passing phenomena, the chromatic “chords” are subordinate to the central Klänge and belong to a different level of the structure. Louis coined the term “intermediate harmo­ ny” (Zwischenharmonie) to describe this situation. In example 1.20, we encounter “pass­ ing chords” characterized by stepwise (or semitonal) motion.147

Ex. 1.20. Max Schillings’ Meergruß as an example of what Rudolf Louis calls “free suspension.”

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The Reception of Hugo Riemann's Music Theory Louis further refines the concept of “intermediate harmony,” as shown in example 1.21. In that example, he argues, we do not find a “raised subdominant,” but only a “pass­ ing motion on the same continued foundation E” within a harmony. What is unusual here is the “leap of the top voice from F♯ to C♯,” for the concept of passing motion implies “that all parts move stepwise.”148 Louis then goes on to discuss the example in detail and introduces the concept of the “interpolated chord” (eingeschobener Accord)149 as a spe­ cial case of the intermediate harmony, “as it were, an intermediate harmony in parenthe­ ses.”150 Louis's definition of this term suggests that he was familiar with Johannes Schreyer's writings and his concept of the “parenthesis”: the concept signifies a chord, Louis explained, that arises “when, due to passing motion, harmonic structures appear that would also be intelligible as independent chords. In such cases the passing tones concerned may as well be continued as though they were constituents (chordal tones) of those harmonies imagined to be independent.”151 (p. 37)

Ex. 1.21. Louis uses Max Schillings’ Ingwelde to demonstrate his concept of intermediate harmony. The first half of the example analyzes the original; the second half shows the underlying schema.

The final interpretation of the effect of this phenomenon clearly shows the traces of the transference of Riemann's notion of the apparent consonance to the concept of passing motion:

(p. 38)

In the above example F♯ is the dissonant passing note with reference to the foun­ dation E, but it is a consonant constituent (third) of the passing chord D♯–F♯–A♯– C♯. The dissonance of the F♯…only exists conceptually. This F♯ therefore has the same kind of freedom that we have allowed any merely conceptual dissonance…. We could make this allowance because in any such apparent constructs this latter conceptualization (in the sense of the “accidental” chord), even though the har­ monic context suggests that it is not essential, always plays into our perception in the second place, and resonates more or less strongly.152 In this way, in the connection between the concept of apparent consonance, derived from the theory of functions, and that of passing motion from Viennese fundamental-bass theo­ ry, Louis develops his idea of “intermediate harmonies.” In this, the “conceptual disso­ nance” is a phenomenal dimension, while “passing motion” is essentially a structural term; in Louis's explanation above, the mention of “harmonic context” and “perception” respectively indicates this difference. “Accidental chord,” “changing-note chord,” “ideal­ ized pedal point,” “continued voice,” “passing chords,” “interpolated chords,” “intermedi­ Page 32 of 53

The Reception of Hugo Riemann's Music Theory ate chords,” “free suspensions,” and so forth—any of these terms represents a contrapun­ tal or voice-leading interpretation of harmonic procedures. As a logical consequence of this kind of thinking, Louis (like his contemporaries Schreyer, Halm, Schmitz, Rögely, and Eycken) displays considerable reticence vis-à-vis Riemann's concept of the applied dominant. In his conception of tonality, Louis remains loyal to Sechter's diatonic approach. The idea that any chromatic structure, no matter how com­ plex, can be related to a diatonic scaffold, is persistently discernible in his harmonic in­ terpretations. Louis is not amenable to the notion that the inner tension of Klänge (based on dominant character or leading tones) would triumph one-sidedly over identification by means of “root position.” When Louis considers bass motion a more essential criterion for function than the morphology of chords (or chordal tension), he reveals himself as follow­ ing the tradition of fundamental-bass teaching—for him, bass motion constitutes the to­ ken of a hierarchical understanding of chords: a raised fourth scale degree is a derivative form, and therefore structurally subordinate to its diatonic alternate, even if the Klang built on it constitutes the focus of our musical perception. But even in Louis's understanding of chromaticism, the idea of scale degree is mixed in with elements of dualistic and function-theoretical ideas of tonality. The reason that Louis does not fail with regard to chromatic harmony, unlike his predecessors of the fundamen­ tal-bass theory, is related to the fact that he gives up the strict separation of major and minor modes that is prevalent in the fundamental-bass tradition. Louis continues, it is true, to maintain the diatonic foundation of the scale, but the derivation of the scale is “dualistic”: “First of all, for us, the more recent generation of composers, who do not de­ rive the key from the scale, but from its constituent principal triadic harmonies, the con­ cept of diatonicism itself is vastly expanded in every possible sense.”153 In this way, Louis distinguishes between five tonal “genders” in which major and minor modes are mixed: the two major forms, pure major and minor-major (including the minor subdominant), as well as three (p. 39) minor forms, pure minor (including the minor dominant), major-mi­ nor (“normal minor,” with the major dominant), and Dorian minor (with the major sub­ dominant). Diatonicism itself has become chromatic. No further explanation is necessary to understand that Riemann's reproaches are ground­ less. Louis's indication concerning the significance of the second part of his Har­ monielehre has been affirmed by subsequent developments, which has confirmed it as perhaps the most important contribution to the discussion of chromaticism and enhar­ monicism in the first half of the twentieth century.154 In Louis/Thuille's theory of harmony, the two predominant traditions of theoretical harmony converge in an exemplary fashion —and result in fully independent views. Rudolf Louis's “Munich” theory of harmony was in its time the culmination of the “Viennese” fundamental bass tradition as well as the “Leipzig” theory of function.

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The Reception of Hugo Riemann's Music Theory

Epilogue In the first half of the twentieth century, few music theorists were as popular, beyond the narrow disciplinary confines, as Ernst Kurth. And it was thanks to Kurth that many of Riemann's central theoretical ideas gained considerable popularity: almost all important later developments of Riemannian concepts did not refer back to Riemann's writings di­ rectly but took a detour via Kurth. But for its popularization, Riemann's theory of function paid a price: as Hermann Erpf pointed out in his Studien zur Harmonie- und Klangtechnik der neueren Musik—one of the last music-theoretical works of the twentieth century that makes a serious effort to productively continue the theory of functions—Kurth “is reluc­ tant to use precise, well-defined terms because he is anxious to avoid a detrimental par­ ticularization of the phenomenon,”155 which highly specific terminology can cause. In his efforts to compensate for this, Erpf went on to argue, Kurth was constantly in danger “of moving away, particularly in his most subtle descriptions of musical connections, from the individual concrete situation, with the view to making them fit into other contexts as well.” This would lead to a situation in which he returned to talking about “ ‘music in gen­ eral,’ but not the individual musical passage.”156 Kurth's work was characterized, para­ doxically, on the one hand by an experiential analytical prose and on the other by a “natu­ ralistic” and “psychologistic” music-theoretical system with strong metaphysical leanings. The strange coexistence of these two irreconcilable aspects side by side is the heavy bur­ den that Riemann's legacy had to bear among the following generations. Even Erpf, who so sharply analyzed the weaknesses of Kurth's system, gets caught in these contradictions. The only type of music theory that still has any justification for its existence in the 1930s, he argued, is a “historical-descriptive approach”: “As far as con­ crete music theory is concerned, ‘music in general’ is not up for debate at present.” Rather, “the only task of the discipline of music theory is to” turn to “a (p. 40) certain, giv­ en music.”157 Its purpose, he continued, is to “identify the characteristics of musical structure of a given historical style, purely from the given situation, without any desire for a speculative reason.” What should take the place of a unified, overriding theory of tonality—to which Riemann had adhered—is a historical, detailed, “comparative theory of the structural features of stylistic attitudes.”158 Consequently, Erpf does not write a Harmonielehre but a monograph on contemporary harmony, which—as Erpf argues convincingly—“has no autonomous rules but is, for bet­ ter or worse, contingent on, built on, or opposed to the functional hearing of classical mu­ sic.”159 The methods and analytical processes, however, of Erpf's “comparative theory of structural features” remain opaque: the nebulous idea of the “pure given situations,” of “immediate cognition,” of “that which can be experienced in the real sounding world” are considered, with next to no explanation, as equivalent to Riemannian categories. In a “theory of hearing, it does not matter whether these givens are ‘founded in nature,’ ” Erpf argues, and yet in his writings the triad, the major-minor polarity, and the relation be­ tween the tonic and its dominants turn into anthropological facts. In large parts, Erpf's book reads like a large-scale rebuttal of his own perceptive introduction. Its more than 200 pages constitute a conceptual battlefield, which is virtually unparalleled even in the Page 34 of 53

The Reception of Hugo Riemann's Music Theory overconceptualized world of function theory. Erpf is not being unduly modest when he claims that he was the first “to consistently work through” Riemann's “demand to consid­ er all tones of functional progressions as root, third or fifth of a major or minor triad.”160 With this work, any kind of autonomous, interval-based concept of dissonance disappears from functional thinking and is replaced by a welter of independent, partly highly com­ plex, categories of Klänge. Where Erpf fails is with his actual stated ambition: the concep­ tual clarity that he found wanting in Kurth's music theory does not shine forth in his oeu­ vre. Erpf's attempts to develop the theory of function further were not continued.161 The lega­ cy of the theory of functions was taken up predominantly by Hermann Grabner and his pupil Wilhelm Maler. Grabner simplified symbols and terminology of the theory of func­ tions and adopted from Kurth, whose faithful supporter he was until the rise of National Socialism, a conception of applied dominants. Grabner developed the basic foundations of the modern theory of functions, which determines the practical theory of harmony and harmonic analysis at many higher institutions, conservatories, and musicology depart­ ments in Germany up to the present day.162 Both Maler and Grabner were formed, both aesthetically and politically, by the youth mu­ sic movement. Grabner is a typical representative of the “older generation,” which be­ came radically politicized by the experience of the First World War. Maler was typical of the younger generation. Both stood on the safe grounds of the German music-theoretical tradition but pursued primarily pedagogical aims. They were particularly concerned with a progressive musical pedagogy, which—as August Halm promoted—focused on the analy­ sis of musical works right from the start. Function symbols were supposed to serve for a radical simplification; they are the basic building blocks of analysis and strive for unam­ biguous simplicity. The theoretical writings that Grabner and Maler published before the Second World War are open-minded, following the pulse of their age. In the first edition of Grabner's influential Allgemeine Musiklehre (1924), which continues to be reissued up to the present day, there are numerous musical examples by Schoenberg, Mahler, Braunfels, Schreker, and so forth. It welcomes the “dissolution of tonality,” which it considers a his­ torically necessary step, and argues its case on the basis of Kurth's concept of “linear counterpoint.” Maler's influential Beitrag zur Harmonielehre appeared in 1931 as a size­ (p. 41)

able three-volume work. It aspired to be a historically restricted theory of “cadential har­ mony of major-minor tonality.” Maler makes reference to the writings of Kurth, Schoen­ berg, and Erpf.163 Maler's prose is wholly in the service of the explanation of examples: the book is written, as it were, around the music examples, among them works by Mahler, Strauss, Schoenberg, Stravinsky, Berg, Debussy, and Satie. Subsequent editions of both works have little in common with their original versions. Grabner and Maler radically adapted their teachings to the prevailing political condi­ tions: after 1933, not only do all traces of Jewish theorists (among them Kurth and Schenker) and of all Jewish and “culture-Bolshevist” composers disappear, but also—and Page 35 of 53

The Reception of Hugo Riemann's Music Theory this is decisive for the postwar history of German music theory—the original concept of an analysis-centered music theory was given up. In its stead, a new antitheoretical “prac­ tice” appeared, whose guiding image was folk song. Maler goes so far as not to reissue the crucial volume of music examples in the second edition (1941) but only the earlier “exercise book,” accompanied by a volume with numerous songs of the National Socialist movement. Grabner-Maler's monistic function symbols, originally conceived as a medium of music analysis, turn into modern “figured-bass” annotations of an elementary, practical theory of folk song. All that remained of the attempts at simplification in the service of analysis was simplification. After the war, little changed in the basic character of these textbooks. Both authors remained faithful to the National Socialist versions of their theo­ ry books. It was only through them that the function symbols become prevalent in Ger­ man institutions. With the view to the first half of the twentieth century, it is difficult to capture this devel­ opment in terms other than a decline in music-theoretical reflection. Indeed, one might think that music-theoretical discourse after 1945 has virtually ground to a halt—and with it, needless to say, the further development of Riemannian thought. The theory of func­ tions, it seems, found in Rudolf Louis a culmination point it never managed to reach again.

Notes: (1.) Emil Ergo, Ueber Richard Wagners Harmonik und Melodik: Ein Beitrag zur Wagner­ ischen Harmonik (Leipzig: Breitkopf und Härtel, 1914), 71. (2.) Michael Arntz, Hugo Riemann (1849–1919): Leben, Werk und Wirkung (Cologne: Alle­ gro, 1999). (3.) See also the reviews by Edward Gollin in Journal of the American Musical Society 56.1 (2003), 192–198, and by Alexander Rehding in Music Theory Spectrum 24.2 (2002), 283–293. Arntz's omission of Riemann's theories leads to consequences for his biography, as he overlooks some important sources, such as the substantial correspondence with Jo­ hannes Schreyer. (4.) In many essential points, Ziehn assumed a contrary position that amounts to an im­ portant critique of Riemann. See also my “Bernhard Ziehn,” in Die Musik in Geschichte und Gegenwart (Personenteil), vol. 17, cols. 1467–1469, ed. Ludwig Finscher (Kassel: Bärenreiter, 1994–2007). (5.) “[S]obald es Jemand unternimmt, auch nur das winzigste Zipfelchen seiner Geis­ testhaten ein ganz klein wenig anders zu wünschen, oder gar (Entsetzlich!) dem berühmtesten Musikgelehrten aller Zeiten seine Purzelbäume unter die Nase zu reiben, so fällt der Herr Doctor mit seiner fixen Feder den Unglücklichen an, als ob dieser min­ destens einen Vatermord begangen hätte…. Er verlangt unbedingte Unterwerfung.” Bern­ hard Ziehn, “Der Weise aus Großmehlra,” in Allgemeine Musik-Zeitung 17 (1890), 355– 561. Cited in Arntz, Hugo Riemann, 260. Riemann himself claimed he was “ill-famed” be­ Page 36 of 53

The Reception of Hugo Riemann's Music Theory cause of his “anti-critiques” (ibid.). In reality, however, none of the critics lacked respect­ fulness—Ziehn was a genuine exception. (6.) In the Netherlands and the Flemish parts of Belgium, Riemann had a large communi­ ty of supporters, spearheaded by Emil Ergo. On the occasion of Riemann's birthdays a number of “Riemann festivals” were held, on July 18, 1898 (“commemorating the 25 years that Riemann had given to the cause”) and on his sixtieth birthday, July 18, 1909. See Ergo, Wagner, 89. (7.) Ibid., 86. (8.) “Riemann hat entschieden etwas zu schnell und undeutlich gesehen.” Ibid., 133. The functional interpretation of the Tristan chord as a secondary dominant does not originate with Ernst Kurth, as is often claimed. Kurth knew—and, for obvious reasons, dismissed— Ergo's writings on Wagner. It would be overstating the case to claim that Kurth had adopted Ergo's interpretation. “Applied dominant” (Zwischendominantik) is a central term of Kurth's theory. (9.) Emil Ergo, “Die Taktlehre der Tonkunst,” Zeitschrift der IMG (1911), 180–188. (10.) For Ergo, “who had fought tirelessly for [Riemann's] good ideas since 1886,” this si­ lence is “mysterious” and he asks himself with noticeable disappointment whether his “propaganda had not been in the interest of the cause.” See Ergo, Wagner, 141. (11.) See Felix Diergarten, “Riemann-Rezeption und Reformpädagogik: Der Musiktheo­ retiker Johannes Schreyer,” Zeitschrift der Gesellschaft für Musiktheorie, Bd. 2 (2005, 763–770) and his “Johannes Schreyer,” in Die Musik in Geschichte und Gegenwart: Allge­ meine Enzyklopädie der Musik (henceforth MGG 2), (Personenteil), vol. 15, cols. 44–45. (12.) “So viel Belehrung und Anregung wir den Schriften Riemanns verdanken, so wenig können wir uns jedoch von der Notwendigkeit überzeugen, die Moll-Harmonie als Unterk­ lang zu notieren, wie er es fordert.” Schreyer, Von Bach bis Wagner: Ein Beitrag zur Psy­ chologie des Musikhörens (Dresden: Holze und Pahl, 1903), 13. (13.) In analogy with Riemann's c+, Schreyer designates the minor triad as c−, erroneous­ ly attributing it to Franz Marschner's Die Klangschrift: Ein Beitrag zur einheitlichen Gestaltung der Harmonielehre (Vienna: im Selbstverlag/Kreisel und Gröger, 1894). The minus sign in combination with a letter designating a minor chord is first found in Otto Kraushaar, Der accordliche Gegensatz und die Begründung der Skala (Kassel: C. Luck­ hardt, 1852), a dualist music theorist from Leipzig. There the sign is written before the letter and designates an upper or lower Klang. Stephan Krehl adopts this designation. See Diergarten, “Schreyer.” (14.) “[Ich habe] für Vermittlungsversuche wie den Ihren keinerlei Wertschätzung.” Let­ ter to Schreyer of March 1903, in Nachlass Schreyer, Sächsische Landesbibliothek Dres­

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The Reception of Hugo Riemann's Music Theory den (Mscr. Dresd. App. 721, V239). The letters to Johannes Schreyer constitute what is probably Riemann's most extensive surviving correspondence. (15.) In his own entry in various editions of the Musik-Lexikon, Riemann lists as his princi­ pal achievement his “reformatory efforts…in the field of music theory.” It has repeatedly been pointed out that Riemann's historical research often aimed to provide the “empirical proof” for his theories. See Scott Burnham, “Method and Motivation in Hugo Riemann's History of Harmonic Theory,” in Music Theory Spectrum 14.1 (1992), 1–14, and Alexander Rehding, Hugo Riemann and the Birth of Modern Musical Thought (Cambridge: Cam­ bridge University Press, 2003). (16.) Obviously, this also constitutes one of the reasons for the successes of his theory. (17.) Hugo Riemann, Handbuch der Harmonielehre, 3rd ed. (Leipzig: Breitkopf und Här­ tel, 1898), vi. (18.) Ergo, Wagner, 51. Richter's Harmonielehre became the bête noire of all progressive theorists (later joined by Jadassohn's works). Thus Rudolf Louis writes, in “Unsere Har­ monielehre,” in Süddeutsche Monatshefte 3.2 (1906), 431: “One is puzzled by the ques­ tion of how it is possible that a textbook can gain a kind of monopoly for more than one whole lifespan…without betraying even an inkling of the most important task of harmony textbooks—that of guiding in the student in understanding the meaning of harmonic rela­ tions. The prestige that the authority of the Leipzig Conservatoire lent this textbook has had the most harmful effect on the theoretical education of a whole generation of musi­ cians, had not the influence of the conservatoires of Vienna (Sechter, Bruckner) and Mu­ nich (Rheinberger) made itself felt as a happy counterbalance of ever-increasing impor­ tance.” (19.) Riemann, Handbuch, vi. (20.) Louis and Thuille will be discussed below. (21.) Arntz, Riemann, 217–218. (22.) The relationships with these people were in most cases more personal than mere business relations. (23.) Even though the theory of functions exerted its influence far beyond the national and linguistic boundaries, its influence remained by and large confined to Central Eu­ rope. It never became an “international” theory, such as Sorge/Vogler/Weber's Stufenthe­ orie. Nonetheless, Russian music theory, above all, is strongly influenced by the theory of functions up to the present day. This reception history remains to be written. (24.) To give a sense of the extent of these translation activities: during Riemann's life­ time, the Elementos de estética musical (1914) appeared with Daniel Jorro in Madrid. Furthermore, Labor (Barcelona/Buenos Aires) published Manual del pianista (1928), Dic­ tado musical (Educación sistemática del oido) (1928), Fraseo musical (1928), Teoría gen­ Page 38 of 53

The Reception of Hugo Riemann's Music Theory eral de la música (Buenos Aires 1928), Composición musical (1929), Reducción al piano de la partitura de orquesta (Buenos Aires 1929), Manual del organista (1929), Bajo cifrado (1929, 2nd ed. 1943), Armonía y modulación (1930), Historia de la música (1930, 2nd ed. 1959), Compendio de instrumentación (1930). (25.) See Robert Wason, Viennese Harmonic Theory from Albrechtsberger to Schenker and Schoenberg (Ann Arbor, MI: UMI Press, 1985), 115. “The influence of Hugo Riemann has been addressed, if not somewhat overemphasized.” (26.) For reasons of space, I will chiefly discuss the German reception of Riemann. But this is not to say that it had no influence beyond linguistic and geographical borders. See also n. 24 above. (27.) Riemann, Musikalische Syntaxis: Grundriß einer harmonischen Satzbildungslehre (Leipzig, 1877), vi. (28.) Among the former group are Johannes Schreyer and Emil Ergo. On the theory of phrasing, see Hans-Joachim Hinrichsen, Musikalische Interpretation: Hans von Bülow (Stuttgart: Steiner, 1999), 235–236; Hartmut Krones, “Hugo Riemanns Überlegungen zu Phrasierung und Artikulation,” in Tatjana Böhme-Mehner and Klaus Mehner, eds., Hugo Riemann (1849–1919): Musikwissenschaftler mit Universalanspruch (Cologne: Böhlau, 2001), 93–115. (29.) This includes, above all, Moritz Hauptmann, Arthur von Oettingen, Otto Tiersch, Ot­ to Kraushaar, Adolf Thürlings, and Oscar Paul. See Henry Klumpenhouwer, “Dualist Tonal Space and Transformation in Nineteenth-Century Musical Thought,” in Thomas Chris­ tensen, ed., The Cambridge History of Western Music Theory (Cambridge: Cambridge University Press, 2002), 456–476. (30.) As formulated in Musikalische Syntaxis. (31.) In this, Riemann here is clearly indebted to the tradition of German Harmonielehre: German theorists going back at least to Gotfried Weber had concentrated on the connec­ tion between “science” and “practice” as the most important element. (32.) This is not the place to discuss the protracted history of monism. (33.) Ernst Kurth found that Capellen's works were “among all the recent theoretical ap­ proaches…the most remarkable and in its practical results the most valuable ones.” See Ernst Kurth, Romantische Harmonik und ihre Krise in Wagners “Tristan” (Berlin: Max Hesse, 1920), 18. He was particularly influenced by Capellen's “Klanglehre,” which was in turn based on Stumpf's concept of “tone fusion,” or Klangverschmelzung, see Georg Capellen, Fortschrittliche Harmonie- und Melodielehre (Leipzig: Breitkopf und Härtel, 1908). See also David Walter Bernstein, The Harmonic Theory of Georg Capellen (Ann Ar­ bor, MI: UMI Press, 1986). The most recent MGG no longer contains an entry on Capellen.

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The Reception of Hugo Riemann's Music Theory (34.) Capellen, “Die Unmöglichkeit und Überflüssigkeit der dualistischen Molltheorie Rie­ manns,” Neue Zeitschrift für Musik 68 (1901), 541. “[Riemann hat] ebensowenig wie die übrigen Dualisten die Schlussfolgerung der Identität von Prim und Grundton in Moll zu ziehen gewagt.” (35.) Ernst Kurth, Die Voraussetzungen der theoretischen Harmonik und der tonalen Darstellungssysteme (Bern: M. Drechsel, 1913), 20. (36.) See Riemann, Grundriß der Musikwissenschaft, 4th ed. (Leipzig: Breitkopf und Här­ tel, 1918), 86. (37.) Ernst Kurth, Voraussetzungen, 19–20. “[Nimmt man eine] von der Forderung der physikalisch gegebenen Grundlagen unabängigere Betrachtungsweise der Musiktheorie [ein, so vermag der Dualismus] als zweiseitige ‘Projektion’ eine in das Psychologische fall­ ende Unterlage zu gewinnen…. [Die duale Symmetrie ist] eine so bemerkenswerte und wertvolle Vereinheitlichung der Harmonik, dass die Fundierung der Tonalität durch Akko­ rdprojektion an sich immer noch eine Rechtfertigung in sich tragen könnte, wenn auch die Grundlage aus dem Physikalischen (der reellen Existenz von Untertönen) teilweise in das Abstrakte übertragen würde.” On the actual “energetic” reformulation of dualism in Kurth, see the chapter “Zur Theorie des Dualismus” in Romantische Harmonik, the excel­ lent overview in Roland Ploeger, “Zum Problem Monismus-Dualismus,” in Studien zur sys­ tematischen Musiktheorie, 2nd ed. (Eutin: Petersen-Mickelsen, 2002), 92–93, as well as Helga de la Motte, “Musikalische Logik: Über das System von Hugo Riemann” in Musik­ theorie (=Handbuch der systematischen Musikwissenschaft vol. 2) (Laaber: Laaber, 2004), 203–223. Even August Halm, whose (monistic) Harmonielehre had a great influ­ ence on Kurth, used a similar argument: “The contradiction that resides in the ‘upper’ root may be the reason for the softer, less energetic character of the minor mode as com­ pared with major. However, this effect is the only thing that reaches our consciousness in this upside-down structure…. The minor Klang is just as much a tendency as the major Klang, the major third at the bottom is the downward leading tone. Let us therefore say: major is in its essence upper dominant, minor is subdominant.” Halm, Harmonielehre (Leipzig: Breitkopf und Härtel, 1900), 75–76. (38.) Kurth, Voraussetzungen, 21. (39.) Riemann, “Das Problem des harmonischen Dualismus,” 43. “dass der unterschei­ dende Charakter von Dur und Moll geradezu darauf zurückzuführen ist, dass die Durkon­ sonanz in den einfachsten Verhältnissen der Steigerung der Schwingungsgeschwindigkeit ihr Wesen hat, die Mollkonsonanz dagegen auf den einfachsten Verhältnissen der Ver­ grösserung der schwingenden Masse…. beruht, so dass man kurzweg das Durprinzip in der wachsenden Intensität und das Mollprinzip in dem zunehmenden Volumen sehen kann.” (40.) They are first used in his Vereinfachte Harmonielehre oder die Lehre von den tonalen Funktionen der Harmonie (London: Augener, 1893), that is twenty years after Riemann's first publications on the theory of harmony. In Skizze einer neuen Methode der Page 40 of 53

The Reception of Hugo Riemann's Music Theory Harmonielehre they appear in the “fully revised” third edition, which from then on bears the title Handbuch der Harmonielehre (Leipzig: Breitkopf und Härtel, 1898). (41.) Ary Belinfante, “De leer der tonalen functien in conflict met die der polaire tegen­ stelling,” Orgaan van de Vereeniging van Muziek Onderwijzers (1904). Sigfrid Karg-Elert argues correspondingly, and removes Riemann's inconsistency: in his “polaristic” theory of functions, the cadence itself is also dualistic. (42.) Riemann, “Das Problem des harmonischen Dualismus,” Neue Zeitschrift für Musik 72 (1905), 70, cited from the reprint (Leipzig: Breitkopf und Härtel, 1905), 35. “dass diese Namen gar nicht von mir gewählte, sondern seit Rameau gebräuchliche sind, und ich sie mit demselben Recht beibehalten habe wie die Bezeichnung Dur und Moll, Paral­ lele, Grundton.” (43.) Riemann's theory of root progressions was effectively a dualistic adaptation of fun­ damental bass. See also Thomas Christensen, “The Schichtenlehre of Hugo Riemann,” In Theory Only 6 (1982), 39–40. (44.) Riemann, “Das Problem,” 70. “Obgleich mir einmal einer meiner persönlichen Schüler versichert hat, dass er sich mit der Terminologie der Schritte nicht mehr den Kopf beschwere seit ich die Funktionszeichen eingeführt habe, so weiss ich doch ganz genau, dass der Betreffende sich nur nicht mehr um die Namen sorgte, aber weit entfernt war, nunmehr die Dominanten in Dur und in Moll für gleichbedeutend anzusehen.” (45.) Not until the advent of neo-Riemannian theory does any successor of Riemann re­ turn to the system of root progressions. (46.) The arguments of supporters and detractors were virtually identical. See Heinrich Schenker, Harmonielehre (Stuttgart: Cotta, 1906), vii–viii. (47.) Holtmeier, “Krehl,” in MGG (Personenteil), vol. 10, cols. 654–656. (48.) Stephan Krehl, Allgemeine Musiklehre (Leipzig: Göschen, 1904), 6. (49.) Krehl, Harmonielehre [Tonalitätslehre] (Leipzig: Vereinigung wissenschaftlicher Ver­ leger, 1922), 11. (50.) See Holtmeier, “From ‘Musiktheorie’ to ‘Tonsatz’: National Socialism and German Music Theory after 1945,” Music Analysis 2/3 (2004), 245–266. I am aware that I do not do justice by giving Karg-Elert's highly complex theory short shrift. Beyond the dualistic discussion of “intonation relations,” it is hard to come to an adequate appraisal. KargElert's theories remained without succession or impact, no matter how much his pupils Reuter and Schenk might have tried to promote the pure theory. See Günther Hartmann, Karg-Elerts Harmonologik: Vorstufen und Stellungnahmen (Bonn: Orpheus, 1999), and Thomas Schinköth, ed., Sigfrid Karg-Elert und seine Leipziger Schüler (Hamburg: von Bockel, 1999).

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The Reception of Hugo Riemann's Music Theory (51.) Robert Hernried (Stephan Krehl), Allgemeine Musiklehre (Leipzig: W. de Gruyter, 1933), 135. “Das Wichtigste, das uns die Lehrer des Dualismus, vor allem Hugo Riemann, gegeben haben, sind die Funktionsbezeichnungen.” (52.) A particular and rarely examined direction of dualism is pursued in Robert Mayrhofer's Naturklangtheorie (theory of natural sonority, which focuses on the “primor­ dial cell” of the major third, onto which harmonic sonorities are built up dualistically in both directions (based on Riemann's derivation of Parallelklänge). Hermann Wetzel, a Berlin-based Riemann supporter, returned to Mayrhofer's idea and developed it further, see Wetzel, Elementartheorie der Musik: Einführung in die Theorie der Melodik, Har­ monik, Rhythmik und der musikalischen Formen- und Vortragslehre (Leipzig: Breitkopf und Härtel, 1911). (53.) Holtmeier, “Kayser,” in MGG (Personenteil), vol. 9, cols. 1565–1568. (54.) Louis writes, with the view to Riemann, that it is “an old insight that the only prima­ ry harmonic elementary relations are the mutual relations between tonic, dominant and subdominant. For the traditional distinction between principal and subsidiary harmonies is based on precisely this insight.” “Even the understanding of these subsidiary har­ monies as representatives of the principal harmonies pre-dates Riemann.” Louis, “Zu Hugo Riemanns Besprechung der Louis-Thuilleschen Harmonielehre,” Süddeutsche Monatshefte 4.1 (1907), 614–615. On fundamental bass theories, see Holtmeier, “Har­ monik/Harmonielehre” in Helga de la Motte and Christian Utz, eds., Lexikon der systema­ tischen Musiktheorie (Laaber: Laaber, forthcoming). (55.) “Any seventh chord can be the representative of a ninth chord.” Simon Sechter, Die Grundsätze der musikalischen Komposition: Erste Abtheilung: Die richtige Folge der Grundharmonien oder vom Fundamentalbass und dessen Umkehrungen und Stel­ lvertretern (Leipzig: Breitkopf und Härtel, 1853), 83. (56.) See Wason, Viennese Harmonic Theory, 40. (57.) The fact that Heinrich Schenker would choose Louis/Thuille's Harmonielehre to demonstrate the nonsensicality of the function-theoretical idea of representation leads one to suspect that he never seriously studied the work, which shows more than merely incidental parallels to his thought. (58.) This discussion could be traced back to the controversy on the question of whether the popular rule of the octave with a “leading-tone” sixth chord on the sixth scale degree merely marks a “digression” or a “special case” of the scale (key). See David Kellner, Treulicher Unterricht im Generalbaß (Hamburg: C. Herold, 1743), 33. In fact, the as­ sumption that the rule of the octave consists of a combination of two keys is one of the basic assumptions of Rameau's music theory See Ludwig Holtmeier, “Heinichen, Rameau and the Italian Thoroughbass Tradition: Concepts of Tonality and Chord in the Rule of the Octave,” in Journal of Music Theory 51/1 (2007): 5–49. In Georg Andreus Sorge's, France­ santonio Valotti's and above all Georg Joseph Vogler's music theories, the raised fourth Page 42 of 53

The Reception of Hugo Riemann's Music Theory scale degree was counted as an integral component of the key. In 1789, Johann Gottlieb Portmann speaks of “Wechseldominante.” See his Leichtes Lehrbuch der Harmonie, Com­ position und des Generalbasses (Darmstadt: printed by J. J. Will, 1789). (59.) Ergo, Wagner, 48. “Als ich damals, kaum mit den Bestrebungen Riemanns (Phrasierung und Harmonielehre) bekannt geworden, auf einer gelegentlichen Reise in Deutschland einige Tage als Gast bei ihm—damals (1887) in Hamburg—verweilte, und wir uns über Richters Harmonielehre (das Vehikel meiner ersten Erziehung in dieser Wis­ senschaft) sprachen, und er sagte: ‘es stehen abscheuliche Uebungen darin,’ so begriff ich das nicht recht.” Ernst Friedrich Richter's Lehrbuch der Harmonie first appeared in 1853, the same year as two other central theoretical works of the middle of the nine­ teenth century: Simon Sechter's Die Grundsätze der musikalischen Komposition and Moritz Hauptmann's System der Harmonik und Metrik. It is not implausible to assert that Richter's theory of harmony was the most successful practical harmony tutor of all times. Until 1953, it went through thirty-five editions and was translated into almost all Euro­ pean languages. In France and England it was groundbreaking. It is mainly thanks to Richter's theory of harmony that (Weber's) Roman numerals were accepted throughout the world. (60.) Ernst Friedrich Richter, Lehrbuch der Harmonie (Leipzig: Breitkopf und Härtel, 1853), 90. (61.) “Ahistorical” is meant in a dual sense here: neither does his tutor have a single ex­ ample from the musical repertoire, nor does Richter himself say a word on the origin of his harmonic thinking. (62.) See also Wason, Viennese Harmonic Theory, and Wolfgang Budday, Harmonielehre Wiener Klassik: Theorie-Satztechnik-Werkanalyse (Stuttgart: Berthold & Schwerdtner, 2002). (63.) The harmonic “theory” of this system is the rule of the octave. (64.) Johann Georg Albrechtsberger, Inganni/Trugschlüsse (Vienna: C. F. Peters, n.d. [1814]); Georg Joseph Vogler, “Summe der Harmonik” in Betrachtungen der Mannheimer Tonschule 3/1 (Mannheim: Bossler, 1790), 1–117. (65.) See Riemann's “Hie Wagner! Hie Schumann!” [1880], in Präludien und Studien (reprint Hildesheim: Georg Olms, 1967), 3: 204–214. It has become a commonplace that Riemann's theory is particularly useful for the interpretation of classical harmony. I, by contrast, tend to think that approaches with a more diatonic understanding of tonality would be more suitable for this. (66.) Unlike scale-degree theory (Gottfried Weber), fundamental-bass theory (Simon Sechter) not only comprises reference to thinking in underlying scale-degrees and chordal inversions, but also chordal progressions that nearly exclusively move along fifths and thirds. Page 43 of 53

The Reception of Hugo Riemann's Music Theory (67.) See Carl Mayrberger, “Die Harmonik Richard Wagner's an den Leitmotiven des Vor­ spiels zu Tristan und Isolde erläutert,” Bayreuther Blätter 4 (1881), 169–180; Johann Em­ merich Hasel, Die Grundsätze des Harmoniesystems (Vienna: Kratochwill, 1892). (68.) Ergo, Wagner, 39. (69.) Moritz Hauptmann, Die Natur der Harmonik und Metrik, 2nd ed. (Leipzig: Breitkopf und Härtel, 1873), 36–37. (70.) Hermann Grabner writes, in Sechter's tradition: “The emergence of intermediary dominants can be related back to the tendency of fifth- and third-related Klänge, to as­ sume tonic significance. (Schenker speaks of ‘tonicization.’)” Grabner, Die Funktionstheo­ rie Hugo Riemanns und ihre Bedeutung für die praktische Analyse (Munich: O. Halbreit­ er, 1923), 31. (71.) See below. (72.) Eugen Schmitz, Harmonielehre als Theorie, Ästhetik und Geschichte der musikalis­ chen Harmonik (Kempten/Munich: J. Kösel, 1911). “Riemann's theory of functions repre­ sents for Schmitz ‘the safest Ariadne thread of orientation in the convoluted maze of mod­ ern harmony’: Riemann's ‘insight that all harmonic motion is nothing but a greatly ex­ panded cadence’ makes it possible ‘to uncover the underlying fundamental laws in as pre­ cise and simple a form as possible.’ However, in Schmitz's theory of harmony there are no function symbols: in the labeling of his music examples, Schmitz does not follow Schreyer's example (who used Riemann's symbols) but the example of R. Louis (to whose Munich circle he belonged) and employed Roman numerals.” Cited from Felix Diergarten, “Eugen Schmitz,” in MGG (Personenteil), vol. 14, cols. 1473–1480. Fritz Rögely (Har­ monielehre, Berlin: Habel, 1910) did not reject function symbols, but preferred a (monis­ tic) Klangschlüssel, which he, wrongly, attributes to Stephan Krehl (ibid., 27). Heinrich van Eycken's Harmonielehre (Leipzig: Hofmeister, 1911) was published posthumously by Hugo Leichtentritt and Oskar Wappenschmitt. Eycken claims to have invented the theory of representation independently of Riemann. His theory of harmony operates with scale degrees but occasionally uses function symbols. (73.) Riemann “Dualismus, harmonischer,” in Musik-Lexikon, 8th ed. (Berlin: Max Hesse, 1916), 264. “die Durchführung einer zweifachen (dualen) Verwandtschaft der Töne, der im Dursinne und der im Mollsinne.” (74.) Riemann, Musik-Lexikon, 10th ed., ed. Alfred Einstein (Berlin: M. Hesse, 1922), 945: “Parallelklänge sind alle Klangpaare, welche im Verhältnis der Toniken von Parallel­ tonarten stehen, welche wir…aus einer und derselben Terz durch Hinzufügen der Oberbzw. Unterquint entwickelt haben.” (75.) The seventh chord of the second scale degree is then an added sixth chord with the sixth in the bass. See Holtmeier, “Ist die Funktionslehre am Ende?,” Tijdschrift voor Muziektheorie 5 (1999): 72–77. Page 44 of 53

The Reception of Hugo Riemann's Music Theory (76.) See my “From ‘Musiktheorie’ to ‘Tonsatz.’ ” (77.) “Die Analyse ist des Kompositionsstudiums bester Teil.” Riemann, cited from Jo­ hannes Schreyer, Harmonielehre: Völlig umgearbeitete Ausgabe der Schrift ‘Von Bach bis Wagner’ (Dresden: Holze & Pahl, 1905), 189. (78.) Wason, Viennese Harmonic Theory, 101. I owe a number of important impulses to Wason's work. (79.) Schreyer, Harmonielehre, 3. (80.) Carl Dahlhaus, Die Geschichte der Musiktheorie im 18. und 19. Jahrhundert, Zweit­ er Teil (Darmstadt: Wissenschaftliche Buchgesellschaft, 1989), 101. In this context, Joseph Leibrock's Akkordlehre (Leipzig: J. Klinkhardt, 1875) and Bernhard Ziehn's Har­ monie- und Modulationslehre (Berlin: R. Sulzer, 1887) should be mentioned, which both led this trend. But also Ebenezer Prout's works can be considered as pathbreaking (see Holtmeier, MGG 2, vol. 13, cols. 1001). Edgardo Codazzi and Guglielmo Andreoli's Man­ uale di Armonia (Milan: L. F. Cogliati, 1903) contains over 900 musical examples. (81.) The two-volume exhibition catalogue, Die Lebensreform: Entwürfe zur Neugestal­ tung von Leben und Kunst um 1900, ed. Kai Buchholz et al. (Darmstadt: Institut Mathildenhöhe/Häusser, 2001), offers a good introduction. See also Hilmar Höckner, Die Musik in der deutschen Jugendbewegung (Wolfenbüttel: Georg Kallmeyer, 1927), Jo­ hannes Hodek, Musikalisch-pädagogische Bewegung zwischen Demokratie und Faschis­ mus: Zur Konkretisierung der Faschismus-Kritik Th. W. Adornos (Weinheim: Beltz, 1977); Ulrich Günther, “Jugendmusikbewegung und reformpädagogische Bewegung,” in Die Ju­ gendmusikbewegung, ed. Wilhelm Scholz and Waltraut Jonas-Corrieri (Wolfenbüttel: Möseler, 1980), 160–184; Anselm Ernst and Wolfgang Rüdiger, “Reformtendenzen in der Musikpädagogik (1900–1933): Resümee und Ausblick auf die Gegenwart,” in Visionen und Aufbrüche: Zur Krise der modernen Musik 1908–1933, ed. Günther Metz (Kassel: G. Bosse, 1994), 375–380; Dorothea Kolland, Die Jugendmusikbewegung: “Gemein­ schaftsmusik,” Theorie und Praxis (Stuttgart: Metzler, 1979). (82.) See Holtmeier, “From ‘Musiktheorie’ to ‘Tonsatz.’ ” (83.) Grabner, Lehrbuch der musikalischen Analyse (Leipzig: C. F. Kahnt, 1926), 5. “Es kam dem Verfasser besonders darauf an, dem Leser so bald als möglich die Formel T S6 D7 T zu erklären und den Nachweis zu führen: 1. dass alle in der Musik gebrauchten Zusammenklänge nur Absenker dieser Stammklänge sind und 2. dass es möglich ist, mit dieser schlichten Formel die kompliziertesten modernen Kompositionen zu analysieren.” (84.) Schreyer, Harmonielehre, iv. “Während aber Riemann als Ziel der Harmonielehre bezeichnet (vergl. sein Handbuch der Harmonielehre, 3. Auflage, Seite VII), den Schüler dahin zu bringen, ‘einen korrekten vierstimmigen Satz in den vier Singschlüsseln wie auch für transponierende Orchesterinstrumente in wenigen Minuten auszuarbeiten oder einen bezifferten Choral ohne Besinnen transponiert am Klavier vierstimmig zu spielen,’ Page 45 of 53

The Reception of Hugo Riemann's Music Theory betrachten wir als ihre wichtigste Aufgabe die Einführung in das Verständnis der Meister­ werke.” (emphasis in original). (85.) “Klangstufen,” Riemann-Lexikon 10th ed., ed. Alfred Einstein, 639. “[Mit römischen Stufenzahlen ist] es zwar nicht unmöglich, aber sehr umständlich, ganze Tonstücke har­ monisch zu analysieren, [da man] es der Aufnahme auch von Versetzungszeichen für die Stufenzahl bedarf.” (86.) Riemann, Große Kompositionslehre, Vol. 1: Der homophone Satz, 1: 493. (87.) Schreyer, Harmonielehre, 195. (88.) Riemann, Große Kompositionslehre, I:493. “[Der Beginn steht] in C-moll, der zweite in B-moll, beide umschreiben nur die einfache Formel °T–D.” (89.) Holtmeier, “Der Tristanakkord und die Neue Funktionstheorie,” Musiktheorie 17 (2002), 361–365. (90.) Schreyer, Harmonielehre, 196. “Sie ist zu verstehen im Sinne der Dominante von CDur als ein kühner, aber streng logisch sich entwickelnder Orgelpunkt auf G.” (91.) Ibid. (92.) Hugo Riemann, Neue Schule der Melodik (Hamburg: K. Grädener, 1883), 148–149. (93.) Ibid., 155. “weil sie die Akkorde fast nie direct, sondern überall durch Skalenbewe­ gung maskiert gibt.” (94.) Ibid., VI, “zwischen Harmonielehre und Kontrapunkt [besteht] ein Unterschied nicht.” (95.) Schreyer, Harmonielehre, 11. (96.) Ibid. (97.) Ibid., 12. (98.) Ibid., 45. (99.) Ibid., 46. (100.) Ibid., 59. (101.) Ibid., 60. (102.) Ibid., 141–142. (103.) Ibid., 141, “unter Hinweglassung alles überflüssigen Beiwerks (besonders der vie­ len rankenartigen Vorhalte).” (104.) Ibid., 215. Page 46 of 53

The Reception of Hugo Riemann's Music Theory (105.) Ibid., 216. “[Obwohl diese harmonische Wendung] bis an die Grenze der tonalen Auffassung [geht, prägt] der Orgelpunkt die tonalen Harmonien scharf aus. [Beide Domi­ nanten müssten als] Vorhalte zur T verstanden werden.” (106.) The inchoate ideas toward an advanced notion of the passing note in Neue Schule der Melodik were not developed any further by Riemann. (107.) Sechter, Die Grundsätze, 35. “Während der Dauer eines und desselben Fundamen­ talaccordes können die Stimmen ihre Antheile vertauschen.” (108.) Diergarten, “Riemann-Rezeption und Reformpädagogik.” (109.) Carl Mayrberger, Lehrbuch der musikalischen Harmonik (Pressburg/Leipzig: Gus­ tav Heckenast, 1878), 130–131. (110.) Ergo, Wagner, 28. (111.) Ibid., 29. Ergo pursued exactly the thought that Christensen presents when he con­ siders the possibilities of large-scale functional analysis. With regard to Riemann's theory of phrasing, he writes, “It seems to me that this offers a potentially fruitful approach for segmenting large-span harmonic successions which could be attributed to individual functions.” (Christensen, “The Schichtenlehre,” 44.) (112.) Ergo, Wagner, 28. (113.) Ibid., 29. (114.) Ibid. (115.) The topic was “in the air.” Capellen dedicated a treatise to it early on (Die Ab­ hängigkeitsverhältnisse in der Musik [Leipzig: C. F. Kahnt, 1904]), and even an orthodox dualist such as Stephan Krehl deals with the topic of “passing sonorities” (See his Har­ monielehre [Tonalitätslehre]). (116.) It is quite certain that by far the most substantial part of the Harmonielehre was authored by Rudolf Louis. Louis mentions repeatedly that Ludwig Thuille merely con­ tributed the examples, and there is no reason to doubt his statements (see also Max Schillings, “Besprechung der Harmonielehre von Rudolf Louis und Ludwig Thuille,” Die Musik 23 [1906/1907], 365–369). Thuille, who suddenly died in the year that Har­ monielehre (1907) appeared, was very popular and well-regarded in Munich, while Louis, critic for the Münchner Neueste Nachrichten had made himself numerous enemies as a Wagnerian, New-German and Reger-hater. Again and again, attempts were made to belit­ tle Louis's contribution. Thus the only Thuille biography states, “About the respective con­ tributions of both authors various suppositions have been voiced, which are not quite ac­ curate, suggesting that the theoretical part was more or less exclusively Louis's work, while Thuille only contributed exercises and examples (which in their rich selection, rang­ ing from Monteverdi to Rich. Strauss and Thuille himself constitute one of the most valu­ able aspects of the book). By contrast, it should be emphasized that the theoretical views Page 47 of 53

The Reception of Hugo Riemann's Music Theory represented in the book, too, are just as much the intellectual property of Thuille as they are Louis's, and have long been used in this form and arrangement in his teaching. The book is the result of many lengthy discussions, whose final formulation—but no more than that—were taken on by Louis, who might also well have contributed the music-historical and philosophical-aesthetic excursions, as these areas were alien to Thuille's interests.” Friedrich Munter, Ludwig Thuille (Munich: Drei Masken Verlag, 1923), 109–110. Besides the fact that it is precisely the “music-historical and philosophical-aesthetic excursions,” that is to say Louis's fundamentally theoretical approach that places the Harmonielehre high above all other theories of harmony of the time, the contract with the publisher also confirms that Louis had the lion's share of the work on this book. A letter by Louis to Thuille's widow indicates that two-thirds of the royalties went to Louis. (See letter of May 23, 1913, Bayerische Staatsbibliothek, Sign. Ana 493.I.3). See also Holtmeier, “Louis,” MGG 2 (Personenteil), vol. 11, cols. 513–515. A short discussion of the controversy is also found in Daniel Harrison, Harmonic Function in Chromatic Music: A Renewed Dualist Theory and An Account of Its Precedents (Chicago: University of Chicago Press, 1994), 298–299. (117.) Louis/Thuille, Harmonielehre, IV. “Für die Harmonik, wie wir sie fassen, ist der Ausgangspunct die möglichst treue und erschöpfende, durch keinerlei theoretisches Vorurteil beeinflußte Analyse dessen, was der Musiker unserer Zeit und unserer Cultur bei den musikalischen Zusammenklängen und ihren Verbindungen tatsächlich hört.” (118.) Louis, “Unsere Harmonielehre,” 430–431. “[auch wenn] alles Theoretisieren ein problematisches Unternehmen sei, [da es keine] Theorie gebe, die dem Realen völlig gerecht werde. [Das ist] durch die Natur des Verhältnisses von Subject und Object von vornherein ausgeschlossen: denn die eigentümliche Macht unseres Geistes liegt eben darin, daß das Einzelne im Allgemeinen, das Viele im Einen zu denken vermag, während jedes, auch das geringste Einzelne sein eigentliches Wesen daran hat, ein Einziges und Unvergleichliches zu sein, etwas, was strenggenommen sofort aufgehört das zu sein, was es ist, wenn wir es unter einen Allgemeinbegriff subsumieren.” (119.) Ibid., 433. “Einige mehr oder minder richtige Beobachtungen geben Veranlassung zur Entstehung eines Gedankens und aus diesem Gedanken wird dann die Theorie ‘de­ duktiv’ herausgesponnen, ohne dass auf die Natur des Gegenstandes selbst weiter viel Rücksicht genommen würde. Der Gedanke wird, Hegelisch gesprochen, seiner Selbstbe­ wegung überlassen…. So kommt die spekulative Theorie zustande, die…durch ihre reg­ uläre Architektonik, die saubre Symmetrie und den glatten Parallelismus ihrer Teilglieder um so mehr für sich einnehmen wird, je mehr sie darauf verzichtet, den Tatsachen gerecht zu werden, und mit der Aufrichtung eines Fantasiegebäudes sich begnügt.” (120.) Ibid. “die Art und Weise, wie er Harmonik und Metrik nach dem Dreiklapp-Schema von Hegels dialektischer Methode konstruiert, wird immer ein abschreckendes Beispiel dafür bieten konnen, auf welche Abwege selbst ein mit seinem Stoffe innigst vertrauter Theoretiker geraten kann, wenn er unter dem suggestiven Zwang einer vorgefaßten Mei­ nung steht.” Page 48 of 53

The Reception of Hugo Riemann's Music Theory (121.) Ibid., 434. “[Riemann] wäre zweifellos die geeignete Persönlichkeit gewesen, die gesicherten Ergebnisse der rein theoretischen Harmonik für die musikalische Unterricht­ spraxis in fruchtbringender Weise nutzbar zu machen; und in der Tat sind ja alle seine späteren harmonietheoretischen Veröffentlichungen diesem Zwecke gewidmet gewesen. Wenn diesen Veröffentlichungen nun trotzdem—und namentlich bei wirklichen Musikern —ein verhältnismäßig so geringer Erfolg beschieden war, so dürfte die Ursache dieses Mißerfolgs wohl ausschließlich in der extrem spekulativen Veranlagung des Rie­ mannschen Denkens zu suchen sein.” (122.) Ibid., “unselige Leidenschaft des gedanklichen Konstruierens, jene Vorliebe für vor­ eiliges Verallgemeinern und Analogisieren, jenes Ueberwuchern des subjektiven Faktors beim Theoretisieren, das letzten Endes seinen Grund hat in einem—man verzeihe die Härte des Ausdrucks—Mangels an Ehrfurcht vor den Tatsachen.” (123.) Ibid., 434–435. “[Die Mollauffassung des Dualismus hat] für spekulative Köpfe et­ was verlockendes, [sie ist] für den unvoreingenommenen Musiker…unannehmbar, ja im Grunde genommen, eigentlich indiskutabel…Die Mollauffassung von Zusammenklängen ist ein Phantasiegebilde.” (124.) Louis, Der Widerspruch in der Musik: Bausteine zu einer Ästhetik der Tonkunst auf realdialektischer Grundlage (Leipzig: Breitkopf und Härtel, 1893), 3. “Harmonielehre ist keine Wissenschaft…Ihre Principien sind, Kantisch gesprochen, nicht konstitutiv sondern nur regulativ.” (125.) Louis was a “pupil” of Julius Bahnsen, whose biography he also published posthu­ mously. (126.) Realdialektik is “pessimistic” insofar as it does not offer any development toward a better state. (127.) Louis, “Zu Hugo Riemanns Besprechung,” 614. (128.) Riemann, “Eine neue Harmonielehre: Harmonielehre von Rudolf Louis u. Ludwig Thuille,” Süddeutsche Monatshefte 4.1 (1907): 500. (129.) Ibid., 501. “zweifellos [wird] als ein besonderer Vorzug des Buches empfunden wer­ den, daß dasselbe die gute alte Generalbaßbezifferung konserviert und nur wo diese ver­ sagt, Gottfried Webers nun auch seit beinahe hundert Jahren akkreditierte Bezeichnung der Akkorde nach Stufen der Skala heranzieht.” (130.) Ibid., 502. “[Der ‘Kern seiner Lehre’ ist die] Lehre von den tonalen Funktionen der Harmonie, welche das Buch von Louis und Thuille dermaßen in extenso reproduziert, daß man eigentlich nicht recht begreift, weshalb nicht auch die so bequem abkürzenden Ze­ ichen T für Tonika, D für Dominante u. S für Subdominante zur Anwendung gebracht sind.”

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The Reception of Hugo Riemann's Music Theory (131.) Ibid. “Bei der großen Abhängigkeit des Buches von meinen Arbeiten durfte ich wohl erwarten, daß die Vorrede auf mich Bezug genommen hätte.” (132.) Ibid., 504. “Ich überlasse es anderen Fachkritiern, festzustellen, ob ich in der Beto­ nung der Abhängigkeit des Werks von mir zu weit gegangen bin oder nicht.” The question of how much this review did harm to the appreciation of Louis and his Harmonielehre is interesting in light of its strange reception history (omnipresent in practice, almost com­ pletely absent in music-theoretical literature). Riemann, at any rate, rarely missed an op­ portunity to point out the “theft.” He later calls the Harmonielehre “die Thuille/ Louis'sche.” (133.) Louis, “Zu Hugo Riemanns Besprechung,” 620. (134.) Riemann, Handbuch der Harmonielehre, 88. (135.) Louis was taught composition by the Bruckner student Friedrich Klose, who was al­ so his close friend. (Ernst Kurth even counts Louis among Bruckner's direct students, see his Bruckner 2 vols. [Berlin: M. Hesse, 1925], 1: 142.) Louis's Anton Bruckner (Berlin: G. Müller, 1904), is a rare example of a genuinely critical biography in the early twentieth century. The passages especially in which Louis sharply critiques Sechter's theory are in the tradition of his teacher (see Friedrich Klose, Meine Lehrjahre bei Bruckner: Erin­ nerungen und Betrachtungen [Regensburg: Bosse, 1925]) and breathe Louis's critical spirit. Ernst Kurth reacted to this book with unbridled disgust—and indeed, it is difficult to imagine a greater contrast between the “emotional” panegyric of the mature Kurth and the “intellectual” tendencies of Louis's dialectical analysis (see Kurth, Bruckner, 1: 196). (136.) Holtmeier, “Sechter, Simon” MGG 2 (Personenteil) 15: 497–500. (137.) Louis/Thuille, Harmonielehre 4th ed. (Stuttgart: C. Grüninger, 1913), 46. “Accorde, die außerhalb des musikalischen Zusammenhangs jederzeit consonieren, gelegentlich so gebraucht werden können, daß sie fur die harmonische Auffassung dissonieren.” (138.) See also Holtmeier, “Tristan,” 364. (139.) Sechter, Grundharmonien, 14. (140.) Ibid., 17. (141.) Louis/Thuille, Harmonielehre, 93. “Am klarsten offenbart sich die Unterdomi­ nantbedeutung des Dreiklangs der II. Stufe, wenn er in der ersten Umkehrung als Sextac­ cord auftritt…. Der Accord bekommt dann eine gewisse Ähnlichkeit mit einem auffas­ sungsdissonanten Sextaccord, bei dem die Sext der Unterdominantquint als Vorhalt vor­ angeht, oder ihr als Durchgang nachfolgt.” Carl Dahlhaus uses the terms “Scheinkonso­ nanz” and “Auffassungsdissonanz” as synonymous and ascribes both to Riemann. See Dahlhaus, “Über einige theoretische Voraussetzungen der musikalischen Analyse,” Bericht über den 1. Internationalen Kongreß für Musiktheorie, ed. Peter Rummenhöller, Friedrich Christoph Reininghaus, and Habakuk Traber (Stuttgart: Ichthys, 1972), 162. Page 50 of 53

The Reception of Hugo Riemann's Music Theory (142.) Louis, Unsere Harmonielehre, 436. “Man darf eben nun nicht vergessen, daß auch beim musikalischen Schaffen die allerverschiedensten Anforderungen zutage und miteinander in Konflikt treten können. Was rein harmonisch betrachtet ein Unding ist, kann möglich werden, wenn es etwas melodisch-kontrapunktisch motiviert erscheint, und umgekehrt: eine melodische geforderte Stimmführung (z. B. eine Leittonauflösung) kann ohne Schaden unterbleiben, wenn ein durch die unregelmäßige Fortschreitung zu erre­ ichender harmonischer Vorteil (etwa die nur so zu ermöglichende Vollständigkeit des Akkords) den melodischen Nachteil ausgleicht.” (143.) Riemann, “Eine neue Harmonielehre,” 500. “[Die] Freimachung des Blicks für die Unterscheidung von Hauptformen und mehr zufälligen Nebengebilden, wie sie besonders durch figurative Wechselnoten entstehen.” (144.) Hermann Grabner writes in Die Funktionstheorie Hugo Riemanns, 6: “The reader will not have missed that the tendency of this book is an attempt at mediation between Sechter and Riemann.” (145.) Louis/Thuille, Harmonielehre, 289. (146.) Ibid. “[Man hat es hier] mit nichts weiter zu tun als mit vier Dreiklängen über e, c, f und d, [der Eintritt des C-Dur und F-Dur Dreiklangs ist aber duch] chromatische Vorhalts­ bildung verzögert. [Man sieht] durch freie Vorhaltbildung zwei Sekundakkorde entstehen, die man ganz falsch verstehen würde, wenn man sie etwa als Umkehrungen von eigentlichen Septakkorden (es-g-h-des bezw. as-c-e-ges) auffassen wollte.” (147.) Louis/Thuille, Harmonielehre, 290. (148.) Ibid. (149.) Ibid. (150.) Ibid. (151.) Ibid., 293. “wenn durch Durchgangsbewegung solche Harmoniebildungen entste­ hen, die auch als selbständige Akkorde denkbar sind. In derartigen Fällen können die be­ treffenden Durchgangstöne sehr wohl auch in der Weise weitergeführt werden, als ob sie Bestandteile (Accordtöne) jener als selbständig gedachten Harmonien wären.” (152.) Ibid., 290. “Im obigen Beispiel ist das fis dissonierende Durchgangsnote in Bezug auf das Fundament E, aber es ist consonanter Bestandteil (Terz) des Durchgangsaccords dis-fas-a-cis. Die Dissonanz des fis…existiert nur in der Auffassung. Es kommt also diesem fis dieselbe Art der Freiheit zu, die wir jeder bloßen Auffassungsdissonanz zugebilligt haben…und zubilligen konnten, weil bei allen derartigen Scheinbildungen jene zweite Auffassung (nämlich im Sinn des ‘zufällig’ entstandenen Accords), wenn sie auch für den harmonischen Zusammenhang als wesentlich nicht in Betracht kommt, in secundärer Weise für unser Empfinden eben doch mit hereinspielt und mehr oder minder leise anklingt.” Page 51 of 53

The Reception of Hugo Riemann's Music Theory (153.) Louis/Thuille, Harmonielehre, 215. “Zunächst einmal hat sich für uns Neuere, die wir die Tonart nicht aus der Tonleiter, sondern aus den constituierenden Hauptdreiklän­ gen ableiten, der Begriff der Diatonik überhaupt und in jedem Sinne ganz wesentlich er­ weitert.” (154.) Daniel Harrison, Harmonic Function in Chromatic Music; also idem, “Supplement to the Theory of Augmented Sixth Chords,” Music Theory Spectrum 17.2 (1995): 189–195. (155.) Hermann Erpf, Studien zur Harmonie- und Klangtechnik der neueren Musik (Leipzig: Breitkopf und Härtel, 1927), 10. “[Kurth scheut] geradzu zurück vor exakter, isolierender Begriffsbildung, weil er deren störende Vereinzelung des Phänomens vermei­ den will.” (156.) Ibid., “dass gerade seine subtilsten Zusammenhangsbeschreibungen sich häufig so sehr von der individuellen konkreten Gegebenheit des beschriebenen Falles entfernen, dass sie auch auf andere Zusammenhänge passen. Dann war also wieder von ‘Musik überhaupt’ aber nicht von der einzelnen vorliegenden Musik die Rede.” (157.) Erpf, Studien zur Harmonie- und Klangtechnik, 10. “Die ‘Musik überhaupt’ [ist] von Seiten der konkreten Musiktheorie vorläufig nicht zu denken. [Vielmehr ist] allein Auf­ gabe der Disziplin ‘Musiktheorie’, [sich] mit bestimmter, gegebener Musik [zu beschäfti­ gen].” (158.) Ibid., 6. “[Die Theorie] habe für einen bestimmten historischen Stil die Merkmale seines Satzes rein aus den Gegebenheiten, ohne jedes Bedürfnis einer spekulativen Be­ gründung festzustellen, [um eine] vergleichende Theorie der Satztechnik der Stilhaltun­ gen [zu entwickeln].” (159.) Ibid., 29. (160.) Ibid., 42. (161.) In this context, one might mention Sigfrid Karg-Elert's Polaristische Funktionstheo­ rie, which cannot be discussed in this space. See Harrison, Harmonic Function in Chro­ matic Music, 315–316. (162.) I have dealt with the history of the theory of functions during the Third Reich in some depth in my article “From ‘Musiktheorie’ to ‘Tonsatz.’ ” (163.) Wilhelm Maler, Beitrag zur durmolltonalen Harmonielehre (Leipzig: F. E. C. Leuk­ ert, 1931), iv.

Ludwig Holtmeier

Ludwig Holtmeier is a professor of music theory at the “Hochschule für Musik” in Freiburg. He is one of the editors of the journal Musik & Ästhetik and president of the Gesellschaft für Musik und Ästhetik. His recent publications include Richard Wagner und seine Zeit, Reconstructing Mozart, Musiktheorie zwischen Historie und Page 52 of 53

The Reception of Hugo Riemann's Music Theory Systematik, “From ‘Musiktheorie’ to ‘Tonsatz’: National Socialism and German Music Theory after 1945,” and “Heinichen, Rameau and the Italian Thoroughbass Tradition: Concepts of Tonality and Chord in the Rule of the Octave.”

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“The Nature of Harmony”: A Translation and Commentary

“The Nature of Harmony”: A Translation and Commen­ tary   Benjamin Steege The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0002

Abstract and Keywords This article provides and introduction and translation of Riemann's “The Nature of Har­ mony”. The translation in this article provides an easy access to an important Riemann's own theoretical evolution, which was written at the moment when a budding psychologi­ cal perspective was beginning to supersede Riemann's earlier acoustical and physiologi­ cal perspective. Just as Riemann attempts to place his theoretical program within a his­ torical trajectory, the article locates his work within the wider and broader historical and intellectual discourse of nineteenth-century physics, physiology, and psychology, high­ lighting the implied and overt polemics with Helmholtz and others that course through its pages. Keywords: Nature of Harmony, theoretical evolution, Riemann, theoretical program, Helmholtz

AT the end of “The Nature of Harmony,” Riemann encourages readers “to attempt something new, to venture, rather than, as hitherto, to seek something new.”1 On a rhetor­ ical level, the elements of enterprise, risk, and novelty in this phrase could hardly be more modern, even modernist. To be sure, there is a certain modesty in Riemann's eleva­ tion of “attempting” or “essaying” (versuchen) above mere “seeking” (suchen). The quaintness of the wordplay here somewhat mutes the musicologist's reformist self-por­ trayal: it finesses as a lexical distinction what might have been more dramatically ex­ pressed as a fundamental contrast between the production of something from scratch and the discovery of something, already made, out there in the world. Still, a palpable sense of excitement at the very idea of the New remains. And the lingering uncertainty, inher­ ent in the notion of a “venture” or “wager” (related to Riemann's German wagen), calls to mind other modernizing projects of the nineteenth and twentieth centuries, with the enor­ mous gambles and trade-offs they necessarily entailed. But all of this prompts the ques­ tion: what exactly did Riemann, no enthusiast of aesthetic novelty or sociopolitical mod­ ernization for their own sake, mean here by invoking the New?

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“The Nature of Harmony”: A Translation and Commentary Alexander Rehding has discussed at length how Riemann's imagination of music's devel­ opment heralded a particular experience or situation of the modern. Riemann's moderni­ ty, Rehding suggests, must be observed not only in his ubiquitous historicist (p. 56) devel­ opmental narratives, which tend to create a unique halo around the present as a moment of heightened reflexivity—where to be “reflexive” involves a tendency to become peculiar­ ly conscious of the historical trajectory that has brought about the musical objects before one, but also, ironically, to be less than fully observant of the radical historicity of one's present perspectives on those objects.2 Rather, in addition to the keen historicist con­ sciousness Riemann exemplified, his theoretical project was always situated in a temporal orientation that projects into the future a feeling of present responsibility to shape that future. Thus, even when “progress” or “progressiveness” are not invoked as such, Rie­ mann can best be read as future-oriented, in spite of the more readily apparent back­ ward-looking characteristics of much of his writing and research.3 But the “future,” for all the responsibility Riemann felt toward it, is different from the idea of “newness” he is invoking in the case of the following essay. For one thing, the piece seems to truck less in regulatory gestures with regard to compositional or analyti­ cal practice than many other Riemann writings from this and later periods. The question in the present case, then, may be less one of how the essay relates to a broad view of the future of German musical culture than one of the specific character of the novelty that Riemann felt his project introduced into discussions of music theory's role in that future. Aside from the sheer originality of his arguments in, for example, the recent Musikalische Syntaxis (1877)—which gives the technical details of a theory for which “The Nature of Harmony” provides historical and metatheoretical background—Riemann is claiming that his approach to harmony somehow engages with a newness unique to his intellectual and cultural moment. By way of introducing Riemann's essay, then, it seems fair to suggest that the New here is conditioned by a structure of thinking and feeling that might be thought of as peculiarly modern in ways not captured by notions of historicism or reflexiv­ ity alone. Complementing the ethical concern for past and future is a sense for what Fredric Jame­ son has recently called the “ontology of the present.” That is, investing as much rhetori­ cal interest as he does in the novelty of his cultural moment, Riemann participates in the special “libidinal charge” of modernity, “a unique kind of intellectual excitement not nor­ mally associated with other forms of conceptuality.” To the extent that Riemann devotes attention explicitly to the past, and implicitly to the future, as a way of paradoxically ex­ panding the significance of the present, he seems to “concentrate a promise within a present of time and to offer a way of possessing the future more immediately within that present itself.”4 In “The Nature of Harmony,” it is not only the narrative structure of the essay itself, with its express historicist arrival at the modern conception of the triad, that betokens such a charged “promise,” but also, as we will see, the very conceptual units Riemann develops near the essay's conclusion to bear the burden of the overloaded ex­ pectations for present experience.

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“The Nature of Harmony”: A Translation and Commentary Toward the beginning of Riemann's essay, we find him proposing to explore how what he calls “exact science” might provide useful knowledge, which music theory (p. 57) could develop “in entirely new directions.” The notion that abstract science is best appreciated for what “utility” it can offer to everyday life and culture was and is commonplace. But the offhand comment becomes more interesting in conjunction with his later distinction between the values of “attempting” (versuchen) and “discovering” (suchen). The latter, for Riemann, is apparently an activity proper to science, whereas the former implies a genuine act of creativity. It does not take long to realize that such distinctions are code for his effort to differentiate himself from Hermann von Helmholtz, the older physicistphysiologist who haunts so many texts by Riemann. Here, Helmholtz plays the role of “seeker” against Riemann's “venturer,” or, more pointedly, “discoverer” against his “cre­ ator.” From a narrative standpoint, this division of labor works out conveniently since Rie­ mann ultimately wants to convey a particular view of his own position in the history of music theory, a view that depends on such a rigorous demarcation of roles. Much like Helmholtz himself twenty years earlier, Riemann parses the disciplinary terrain of music theory into three areas: physics, physiology, and psychology.5 The division is not merely a quasi-spatial one. Rather, a temporal series is involved here, as Riemann's wording sug­ gests: The natural science of music extends immediately to the investigation of the na­ ture of sounding bodies and is then part of physics, and specifically acoustics; if it pursues tone farther on its path into the human ear and examines the tone sensa­ tions excited by it, then it is part of physiology; if it concerns itself finally with the nature of tone representations and their combination, then it enters the area of psychology. The journey from sounding bodies (tönende Körper) to tone sensations (Tonempfindungen) to the mental representations, or “imaginations,” of tone (Tonvorstellungen) is a familiar one on many counts. Not only did the Empfindung-Vorstellung distinction already have a long and lofty history in the German philosophical tradition (a history we need not re­ hearse here); but simultaneously, the path Riemann maps out from a bird's-eye view ex­ actly coincides with the imagined journey of the tone itself: from the exterior world (physics), to the human body in its full corporeality (physiology), and “beyond” to the hu­ man imagination itself (psychology). The extreme directionality of this movement “inward” ought to be regarded with some suspicion. How is it that Riemann can allow his image of the perceptual process to struc­ ture not only his reading of the disciplinary configuration of contemporary science but al­ so, more adventurously, his distribution of music-historical events in a series that views the emergence of an authentic role for “imagination” as the great achievement of the modern moment?6 For this is indeed what the overall form of “The Nature of Harmony” would suggest. Departing from the Pythagorean mythology of the hammers, moving through Zarlino's monochord divisions and Rameau's fascination with the phenomenon of harmonic overtones—all elements of exteriority and extensivity—we eventually reach Helmholtz, whose unprecedented concern for the physiological mechanics of sensation as Page 3 of 37

“The Nature of Harmony”: A Translation and Commentary such marked (p. 58) a slim threshold onto what is taken to be an intellectually richer area of psychological inquiry, terrain Riemann claims as his own. Needless to say, this narra­ tive of a transhistorical coming-to-consciousness is hardly unique to Riemann, though the multifaceted ways in which it structures his narrative—as disciplinary terrain, as synopsis of tone perception, and as history of music theory—are certainly remarkable. Given Riemann's view of the relationships among disciplines here, it seems worthwhile to consider some further features of the physics-physiology-psychology series, which might otherwise be read as innocuous or self-evident. First, there is a certain historical truth to this particular ordering, insofar as the successive emergence of the three disciplines were in fact widely understood in the nineteenth century to have followed the very chronology Riemann posits. The emergence of physiology in the 1850s and 1860s as a ful­ ly institutionalized experimental discipline—as opposed to a speculative, or a merely em­ pirical one—was a major historical event with profound, and often still unrecognized, im­ plications for European and, later, North American culture. Fundamental mid-century in­ novations by researchers such as Johannes Müller, Claude Bernard, and Helmholtz him­ self, all under the rubric of “physiology” or “organic physics,” brought about a way of ap­ proaching organisms—and with it, a view of the “human” itself—that would have been un­ recognizable to scientists just a generation or so earlier.7 The new cultural environment has often been characterized as one of increasing mechanization, both in its apparent re­ turn to pre-Romantic views of the person as a “human machine” and also in the relatively sudden appearance of a battery of experimental apparatus previously deemed typical of the more venerable and continuous discipline of physics than of research on living matter.8 Yet whatever retrospective affinities the machinic and materialist rhetoric of some nineteenth-century scientists and philosophers may have had with the Enlighten­ ment physics of a Julien Offray de La Mettrie (author of L’Homme machine, 1750) or a Wolfgang von Kempelen (inventor of a “speaking machine” and “chess-playing Turk,” among other famous devices), it is important to emphasize those features of mid-nine­ teenth-century physiology that were particular to its historical moment. Prominent among these would be the initially controversial view that organic substances (vegetal and ani­ mal) could not be considered metaphysically different from inorganic substances; that is, they were not animated by any “special forces” that could not be observed under ordi­ nary physical conditions in various experimental apparatus.9 This view, much exaggerat­ ed, parodied, and misunderstood by both supporters and detractors, enabled the more immediately relevant corollary that studying any given element of human life—including not just activities traditionally understood as “mechanical” like self-propulsion and diges­ tion, but also experiences like fatigue, illusion, and, indeed, “normal” perception (seen as continuous with all these other elements)—would require rigorous methods of isolating the specific physical and physiological processes under consideration. In rejecting both the dream of a self-propelled mechanical device cut off from the circulation of forces in the world and also metaphysical speculation about “life forces” and the like, the new “or­ ganic physics,” unlike either Enlightenment materialism or even some strains of idealist Naturphilosophie, was (p. 59) keenly attuned to the finitude of the human.10 Since the “subject” in itself had been deemed unknowable in the Kantian tradition, it fell to the pur­ Page 4 of 37

“The Nature of Harmony”: A Translation and Commentary veyors of modern knowledge, in Helmholtz's view, to focus their energy on the dense and messy peripheries or boundary zones of subjective experience, including especially every­ thing that went into producing the subject's everyday experience of the world.11 Sensory physiology in the wake of “organic physics” might be said to have shifted attention from the core question of what it meant to be a subject, to how one might study the particular qualities of the subject's experience. Experimental physiology, then, as developed by Helmholtz and a relatively small group of fellow travelers in the late 1850s, conspicuously and self-consciously took on a very re­ stricted and controlled range of objects. Given the rigor with which Helmholtz's cohort adhered to the Kantian ethical imperative of disciplinary critique and self-limitation—lest they repeat the old metaphysicians’ error of overstepping the boundaries of what can tru­ ly be known—it is perhaps not surprising if their scientific project struck some observers as fatally narrow and ironically even dehumanizing, in spite of its conscientious efforts not to eliminate, but to circumscribe and even preserve the distinctively human spirit from the scope of the experimental gaze. In short, the particular “modernity” of physiolo­ gy after around 1850 inhered not merely in the superficial trappings of empiricism (with its associated rejection of speculative metaphysics and dogma), experimentalism, machin­ ism, and so forth, for these had already been associated with various scientific projects and personae for many generations. Rather, it was the disciplinary bracketing of an in­ creasingly narrow spectrum of knowable experience—the side effect of a more vigilant policing, since Kant, of the boundaries between subject and object, noumena and phe­ nomena—that produced a radically different way of viewing the natural and human worlds than what had gone before. The suddenly expanded role of controlled experiment on human perception was central to this shift, but should not be identified with the event itself. As Riemann's assigning the task of studying “sensation” to physiology correctly implies, Empfindung was indeed the elemental unit of the new discipline (at least when it occu­ pied itself with sensory processes as opposed to motor ones); all its energy was devoted to isolating these units, breaking down the complex events of perception into their least relations. In 1878, Helmholtz proudly wrote, “I believe that we must regard the most es­ sential progress of recent times to be the breakup of the concept of intuition (Anschauung) into the elementary processes of thought, which is still lacking in Kant.”12 In the specific case of studying hearing, this largely meant finding ways of making otherwise obscure musical phenomena empirically available. It meant rigging up apparatus to produce over­ tone-free fundamentals, or to highlight individual upper partials, or to amplify combina­ tion tones, or to draw listeners’ attention to acoustic beats (or the lack thereof), and so on. In short, Helmholtzian acoustics was largely a matter of shifting the modality of “nor­ mal” listening so as to bring into focus a range of phenomena habitually screened out of it. The first two parts of the three-part Lehre von den Tonempfindungen are arguably de­ voted almost entirely to bringing about just such a shift of focus.13

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“The Nature of Harmony”: A Translation and Commentary In the apparent obsession with minutiae such a perspective entails, it is easy to lose sight of the significance of the larger gesture involved. For Helmholtz, the prerequi­ site for a genuinely modern and well-grounded theory of music was, surprisingly, a radi­ cal defamiliarization of its elements. Empfindung, for all its association in Romantic aes­ thetics with something like culturally mediated “affect,” was to be made into a pointillis­ tic event stripped of context and connotation. But what would be the function of such a defamiliarizing gesture? If the sensation of an individual tone could be made a kind of ex­ periential ground zero, with no history and no immediately self-revealing future, one might hope to (re)construct a rational—that is, nominally unprejudiced—system of rela­ tions among such tones, including scalar and chordal contexts that “made sense” in terms of the tones’ specific material. As the physicist and music theorist Arthur von Oettingen put it, “The conscious perception of musical tone can only be dissolved into a multiplicity of singular sensations through an intentional, strenuous attention. This analysis of the Klang is the foundation of the theory of music.”14 (p. 60)

Oettingen's concise (though not entirely clarifying) synopsis of the relation of music theo­ ry to a mode of perception inaugurated in modern experimental physiology does not make explicit all one would need to know in order to proceed from “the analysis of the Klang” to “theory.” But it does exemplify how the professional scientists in Riemann's story ap­ proached the “conscious perception” of sensory aggregates with a peculiar skepticism perhaps only proper to those schooled daily in a critical empiricism. Oettingen implies that “perception” (Wahrnehmung) is not entirely to be trusted, in spite of the German word's constituent root wahr with its suggestion of reference to the “truth.” (Wahrnehmen might be rendered crudely as “to take for true.”) Rather, what is normally taken as “tone” without further ado must be dissolved into its particular individ­ ualities—overtones, combination tones, and so forth—in order to reach an unimpeachable contact with the authentic immediacy sensation seemed to promise. Helmholtz likewise expressed distrust not only for Wahrnehmung, but also for Vorstel­ lung. Both terms, whatever their privileged positions in traditional German philosophical discourse, stood for what Helmholtz considered a utilitarian form of perception, where groups of consistently associated sensations (the facial features of a loved one, the har­ monic spectrum of a particular wind instrument or of a friend's voice) would allow, uncon­ sciously, for recognition of given objects and thus enable the normal moment-to-moment functioning one otherwise takes for granted. For Helmholtz, Vorstellung, or the mental representation formed by the agglomeration of constituent sensations, was treated al­ most as a necessary evil: “I am of the opinion that it cannot possibly make sense to speak about any truth of our mental representations (Vorstellungen) other than practical truth.”15 And when it came to defining Vorstellung for musical contexts, Helmholtz was essentially forced to split the “facts” of perception into two incommensurable moments: Now what does the ear do? Does it analyze [a complex sound wave into simple tones], or does it grasp it as a whole?—The answer to this can vary according to (p. 61) the sense of the question, for we must differentiate here between two things: namely, in the first place, the sensation (Empfindung) in the auditory Page 6 of 37

“The Nature of Harmony”: A Translation and Commentary nerves as they occur without the intervention of intellectual activity, and the rep­ resentation (Vorstellung) we form as a result of this sensation. We must, that is, differentiate between the material ear of the body and the mental ear of the imagi­ nation (Vorstellungsvermögen).16 Helmholtz's writing here indicates that Riemann was not entirely fair in implying that Vorstellung was a novel concern for the post-Helmholtz turn to more a psychologically ori­ ented view of music theory. Helmholtz was always concerned to point out how listening was constituted of these two moments together, though each of the two incommensurable “ears” remained formatively deaf to the other's simultaneous experience. Yet it remains safe to say that the valence of “representation” shifted subtly but funda­ mentally in the twenty-some years between Helmholtz's first musical publications and Riemann's. The skepticism toward representation—a learned product of habit—and the great stock one finds Helmholtz placing in “sensation” as the site of perceptual truth are notably absent from Riemann's thinking. Indeed, while Helmholtz pointedly identifies rep­ resentation as the moment at which one is not hearing clearly, it is just the opposite for Riemann. Vorstellung, an imaginative act of “placing something before” the mind's eye, becomes the emblem of a more reliable engagement with the logical relations among things. Riemann was emphatic that Empfindung is always a matter of a crude passivity—it is something that happens to someone, as when one “suffers” the actions of another— while Vorstellung involved a certain taking control: If listening to music is a selecting-out from chordal material presented to the ear according to simple principles…., then it is no longer a physical suffering (ein fy­ sisches Erleiden), but rather a logical activity. It is precisely a matter of represent­ ing, a uniting, separating, comparing, relating-to-one-another of representations, which only share their name with the “representations of forms” elicited by visual impressions, but otherwise appear in totally different quality—tone representa­ tions (Tonvorstellungen).17 The critique of sensation here is cast in the form of a kind of antiparticularism. Where Helmholtz had been concerned to reveal the inner workings of tone in infinitesimal detail —almost for the sake of detail itself—Riemann aims to construct a kind of parallel uni­ verse to the raw sensory interventions of Helmholtzian listening, a universe he famously designated, from the beginning of his career, the “logical” (but which he instead refers to in “The Nature of Harmony” as the “psychological”). The highly mediated and formalist nature of this mode of musical thought is aptly characterized as one of “consolidation” by the mathematician and philosopher Moritz Wilhelm Drobisch in an influential logic text (from which Riemann quotes heavily at the beginning of Musikalische Syntaxis): All thinking is in general a consolidation of the many and manifold into a unity. What is consolidated are not real objects but rather representations, and not even (p. 62)

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that, insofar as they are (subjectively) our representations, products of our

“The Nature of Harmony”: A Translation and Commentary intellectual activity, but rather (objectively) with regard to what is represented in them, the thought.18 Riemann can be seen acting on the impulse to “consolidate” throughout “The Nature of Harmony.” Indeed, this thematic of bringing the multiple under the influence of some uni­ ty practically constitutes a kind of metanarrative for the essay, alongside that of the “com­ ing to consciousness” associated with his historicist vision of the advent of a psychologi­ cal music theory. Riemann celebrates, successively, the apparent acknowledgment of oc­ tave equivalence in classical Greek music notation, the “double form” of the “consonant chord” as theorized in Zarlino's “arithmetic” and “harmonic” proportions (“major” and “minor” purportedly understood as in some sense permutations of the “same” thing), the implicit recognition of the invertibility of intervals in thoroughbass notation (though only among upper voices), Rameau's landmark appreciation of the significance of full chordal inversion, and Hauptmann's “epoch-making” discovery (or “rediscovery”) of the principle of harmonic dualism—a progression that views the drawn-out, intergenerational process of ever greater theoretical generality as the privileged history of musical thought. Riemann's quest to construct major and minor triads as springing at once from the same theoretical source in equal and opposite orientations is written here as an epic effort to peel away from any hint of the particularism that Helmholtz enjoyed in the exhilarating positivity of sensation. In this stream of events, representation, in Drobisch's sense of consolidation, is both a motivating force and an ultimate goal. Only the failure of Baroque figured-bass notation to synthesize the three inversions of a triad (or the four inversions of a seventh chord) into a higher conceptual unity seems to irk Riemann more than Helmholtz's persistence in differentiating among sonorities based on the otherwise trivial nuances of quality induced by the physiological effects of beats. But why the impatience with particularity? On the level of theoretical argument, one might point to the widespread negative reaction to Helmholtz's physiological theory of consonance, represented perhaps most clearly by Rudolf Hermann Lotze (the Göttingen philosopher who approved Riemann's dissertation there in 1873). When Helmholtz sup­ posed that the age-old theoretical notions of dissonance and consonance were really just relative quanta in a gradation of nervous stimulation without any logical principle for dis­ tinguishing one categorically from the other, Lotze felt that he was being misled by a physicist's tendency to think in extensive (as opposed to intensive) terms. What one ob­ serves in the physical measurement of a thing is only a difference in degree: frequency (or other sorts of quantity) imagined in a one-dimensional spectrum. But, as Lotze point­ ed out in his Geschichte der Aesthetik in Deutschland (cited by Riemann in the present es­ say), when it comes to the effect on the “soul” (Seele), the quality of experience is of a dif­ ferent order, which does not seem to engage properties of a “more-or-less” type.19 This was, according to Riemann, the “Achilles’ heel” of Helmholtz's music-theoretical work. To the extent that it mistook a difference in kind for a mere difference in degree, (p. 63) Helmholtz's sensory particularism needed to be replaced with what Riemann here calls a “principled difference” between consonance and dissonance, as well as between major and minor. Page 8 of 37

“The Nature of Harmony”: A Translation and Commentary But, again, it is worth looking beyond the surface of theoretical argument. To some de­ gree, Lotze's, Drobisch's, and Riemann's turn toward a purportedly “higher” level of per­ ception is spurred by a broader cultural ambivalence about the decontextualization wrought by sensory physiology. Lotze, for one, was more than willing to pathologize per­ ceptual modalities that appeared to resist any form of logical and psychological holism. On one occasion, he described how an overeager ear for “sensory impression” (essentially synonymous with “sensation”) might lead someone into a morally dangerous forgetful­ ness of their own personality: The comparison of two sensory impressions, for example the pitch of two different tones…demands the greatest possible holding-off of all other representational processes that might dim the purity of sensation…. Someone tuning the strings of a piano with the most strained attention to his task has a minimum of self-con­ sciousness…. But someone attentively considering a decision to be taken must, at the same time, bring at least a certain memory of his personality to bear on this reflection. The unselfconscious absorption in a single thought as well as the dis­ connected flux of many thoughts are conditions that can be united with the healthy condition of intellectual life only when they cease for a moment. “Prolonged distraction, no less than prolonged narrowing of the thought process,” Lotze concluded, are “the first stages of a disturbance of the soul.”20 Of course, it is not that sensation was wholly absent from Riemann's universe. For a period of time, he was only too eager to pursue his own “experiments,” and “The Nature of Harmony” gives some sense, in passing, of his effort to devise apparatus as ingeniously modest as any of Helmholtz's: a vibrating tuning fork held lightly against a resonant surface, for example, miraculously produces the octave, twelfth, double octave, and so forth below its funda­ mental. Riemann would have immediately recognized these as the elusive undertones he struggled to make empirically available to others.21 But whatever the status of his earnest empiricism, any broader impulse to encourage attunement to sensation in itself was ulti­ mately deemed, at best, a Siegfried-like obliviousness to the broader moral and spiritual context of one's subjectivity, and, at worst, a capitulation to the “physical suffering” of corporeal experience. It may well be that the positive evaluation of Empfindung Helmholtz demonstrated would have appeared a historical anomaly from Riemann's perspective. One does not find any other major writer on music investing so much theoretical capital in the notion, until Schoenberg's striking celebration of a modernist Empfindungswelt in the Harmonielehre of 1911 (as if reading a kind of message-in-a-bottle from Helmholtz, perhaps transmitted by the Viennese physicist and science popularizer Ernst Mach).22 In any case, whatever critical historical perspectives one brings to it, the fact is that Riemann's critique of Empfindung and elevation of Vorstellung remain in a certain sense contemporary. For all the patent modernity of Helmholtz's antidogmatic empiricism, liberal progressivism, and direct participation in Germany's (p. 64) belated industrial revolution, it can be difficult to resist Riemann's largely subcutaneous conviction that there is something strangely premodern about any theory of music, logic, aesthetics, or what have you, that does not Page 9 of 37

“The Nature of Harmony”: A Translation and Commentary make representation the final arbiter of intellectual and cultural value. The directionality of the sensation-representation pair contains a certain persuasion in itself. This may have to do with the way in which sensation does not always seem to require the “human” in a conventional sense. In humanist terms (at least partly recognizable as Riemann's own), sensation is very often understood as “given” rather than produced, whether au­ tonomously or dialectically. Representation, on the other hand, has long been taken not only as the “process of bringing a thing before one's self, and thereby imagining it (the German word is the same), perceiving it, thinking or intuiting it,” but also, more boldly, as “taking possession of it,” as Jameson writes (in reference to Heidegger, who equated Vorstellung with the more assertive etwas in Besitz nehmen).23 This taking possession of musical objects, then, would be the willful act of making them contemporary with oneself that gives Riemann's sense of the New its characteristic modernity. Or to configure the relationship somewhat differently, we might recall Foucault's still provocative proposal in The Order of Things that the historical moment of “representa­ tion” had in fact ended around 1800, to be replaced by the historical moment of the “hu­ man,” along with (or indeed through) what would become known as the human sciences. Making sense of Riemann in terms of this exasperatingly abstract periodization seems disorienting at first, until one realizes that it is not so much that representation altogeth­ er ceases to operate with the emergence of the special form of being (that is, the human) Foucault so dramatically sketches; rather, representation becomes secondary to that be­ ing, enabled by it rather than enabling it as a kind of epiphenomenon. The gesture of tak­ ing possession, which Riemann's notion of a logical theory of Tonvorstellung seems to en­ tail, thus becomes a reenactment of an older gesture of inverting the priority of represen­ tation and being. Helmholtz is to be seen here as the proponent of an experience of modernity where representation (which he understands largely, but not exclusively, in terms of sensation) does not depend on its being “possessed” by a subject—nor even for a subject to be necessarily in full possession of itself. Riemann, then, wants us to read his own work as having (at last) installed the truly modern subject in the empty space sup­ posedly left by previously theorists, while Helmholtz becomes a kind of remnant of the proto- or even premodern that ironically continues to haunt the modern. If one takes “modernity” in this sense to be a narrative structure rather than a onetime break, event, or unitary period (again Jameson's suggestion), then it is productive to read Riemann's “retelling” the story of the emerging human as something more than a belated reiteration of what Foucault had dated to a full century before Riemann's earliest work. Instead, modernity is characterized precisely by that very retelling (among others, such as the perpetually reiterated break from the organic to the mechanical that one encoun­ ters from at least the eighteenth century all the way up to the present day). The New, fi­ nally, is not just what is patently novel in the (p. 65) here-and-now, but rather an element of a perpetually recurrent temporal structure deployed in this case to vivify the way in which Riemann might share a historical moment with his immediate predecessors even as he claims to supersede them.

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“The Nature of Harmony”: A Translation and Commentary

The Nature of Harmony, by Dr. Hugo Riemann in Hamburg24 The nearly incalculable number of harmony manuals (or harmony treatises) falls into two groups: theoretical harmony treatises or systems of harmony, and practical harmony trea­ tises or thoroughbass schools. Both may boast a very comprehensive literature; yet the number of guides intended directly for practice and leading to multi-part composition by way of figured bass is considerably greater than that of purely theoretical systematic works on the significance of harmonies and their relations to one another. True, it has be­ come common in recent times to launch practical methods with a theoretical introduction or to assimilate theoretical observations into single chapters. But one must keep the two modes of treating harmonic theory separate. The practical harmony treatise, exercise in the proper connection of chords in multi-part composition, is part of the actual teaching of art, of instruction in the technique of composition; the theory of harmony on the other hand is a part of musicology, specifically of the natural science of music, which is related to the teaching of art only insofar as it can make use of its positive results. The natural science of music extends immediately to the investigation of the nature of sounding bodies and is then part of physics, and specifically acoustics; if it pursues tone farther on its path into the human ear and examines the tone sensations excited by it, then it is part of physiology; if it concerns itself finally with the nature of tone representa­ tions and their combination, then it enters the area of psychology. From the results of in­ vestigation in all three research fields, in that of physics, physiology, and psychology, arise the elements of an exact theory of the nature of harmony, whose specific task is to provide, to the practical teaching of musical composition, the ways and means for sub­ suming the particular in the general, for the identification of advanced perspectives and rules, and above all for a systematic procedure free of arbitrary elements. Contributions to such a theory, supported by natural scientific research, at first flowed very sparingly, and only the recent general ascendancy of the natural sciences has brought us a good bit forward. The interest of practicing musicians in the young science is still fairly spotty and not very intensive; yet this is hardly surprising when one consid­ ers that not even twenty years have elapsed since the theory had been fleshed out enough that one could contemplate its use in practice as a system. The more the positive utility of the shift in perspective on harmonic relations emerges, the keener will be the general in­ terest in the scientific grounding of the (p. 66) fundamentals of music. In the following, I will attempt to indicate briefly what use the practical teaching of music has until now al­ ready made of exact science and to what extent it can still draw utility from it in entirely new directions. The oldest principles of exact harmonic theory are the mathematical definitions of the consonant intervals, adduced through the investigation of the nature of sounding bodies, which can be derived from the teaching of Pythagoras but are doubtless much older than him. The familiar tale of the different weights of the blacksmith's hammers, which were Page 11 of 37

“The Nature of Harmony”: A Translation and Commentary supposed to have revealed to Pythagoras the numerical ratios of the intervals, is physical­ ly false at its core and thus badly enough invented. Pythagoras may well have adopted the elements of the mathematical theory of intervals, known from string measurements, from Egyptian priests along with the rest of his number philosophy. The practical musician can justifiably ask: “Of what use is it to art and artists to know that the string lengths of two octave-related tones, assuming equal mass and tension, stand in the ratio 1:2, or those of the fifth 2:3, and those of the fourth 3:4?” Certainly no direct utility grows from the prac­ tice and knowledge of these numbers in themselves; only the instrument-builder, in the correct measurement of instruments, and the player of an instrument, in the discovery of correct application, can profit from it. But much greater is the indirect utility of the knowledge, resulting from these mathematical definitions, that the tones forming the mu­ sical interval of an octave stand in the mathematically simplest of all ratios and that this simple ratio must assert itself for our sensation (in a manner to be explained shortly), since we actually set octave-related tones in closer connection to one another than tones of any other combination. We cannot forget that the theory of music would be completely impracticable if it could not consider different tones under a common aspect; one such aspect is the principle of octave equivalence, which one would have achieved only by great effort without the aid of mathematics. Greek notation already acknowledges—al­ though, to be sure, only in its latest constituents: the five highest steps of the system—the designation of octave-related tones by the same sign (through differentiation by an octave mark, just as we now distinguish c´ from c); the Western tone system has indicated oc­ tave-related tones with the same names since at least the ninth or tenth century. Fifths and fourths, the simplest ratios aside from the octave, play a correspondingly important role in musical practice. Fundamental tone, fourth, fifth, and octave are the pillars of an­ cient as well as of modern scales; they were the only fixed tones in the ancient system, while seconds, thirds, sixths, and sevenths assumed different values in the chromatic and enharmonic genera. I do not need to emphasize the significance of fifths in modern chord and scale theory; particularly in scale theory, everything revolves around them right up to M. Hauptmann. The ancients did not yet recognize the consonance of the third 4:5; they theoretically defined the tone in their scale corresponding to our third scale-degree as an octave displacement of the fourth fifth (C–G–D–A–E) and considered it to be a dissonance by virtue of its complex mathematical ratio (64:81). The Arabs have the merit of having enriched exact theory with the concept of the consonant third (cf. my Studien zur Geschichte der Notenschrift, pp. 77–85); the so-called “messel-theory” of the Arabic-Per­ sian (p. 67) theorists, which demonstrates the interval theory on a string divided into twelve equal parts, reckons not only the major third 4:5 and the minor 5:6, but also the major sixth 3:5 and the minor sixth 5:8 among the consonances, which almost allows one to conclude that they no longer made only unison music like the Greeks, but also recog­ nized the significance of the consonant chords.25 The oldest versions of the “messel-theo­ ry” known to us (though they probably date back to considerably older times) belong to the turn of the thirteenth and fourteenth centuries, that is, to a time in which the West in­ deed already had a fairly developed musical practice of multi-part composition (Discant­ us, Fauxbourdon) but had not yet arrived theoretically at knowledge of the consonance of the third. The man who first declared the consonance of the major third in the West was Page 12 of 37

“The Nature of Harmony”: A Translation and Commentary Ludovico Fogliani (Musica theorica, 1529). Yet he believed he was establishing something new just as little as did Gioseffo Zarlino, who did the same thing in 1558 in his Istitutioni harmoniche. Both referred to the Greek theorists Didymus and Ptolemy, who, among vari­ ous possible divisions of the fourth (tetrachord divisions), also gave the division with the third 4:5 (consisting of the major whole tone 8:9 and the minor 9:10); but it still remained far from the Greeks to consider the consonance of the third, and it was not necessary for either Zarlino or Fogliano to renounce the originality of their idea for the Greeks’ sake. But Zarlino went farther. He is not called the “Father of Harmonic Theory” without rea­ son; for it is he who gave the world the concept of the consonant chord and indeed even in its double form as major chord and minor chord. While musical practice had long since discovered by empirical means the fundamentals of multi-part composition (consonant harmonies), theory and the practically oriented teach­ ing of art completely lacked concepts for the definition of these formations. Zarlino (Isti­ tutione harmoniche, Books I.30 and III.31) compared two methods of string division, the harmonic (divisione harmonica) and the arithmetic (divisione aritmetica); by “harmonic division” of the string, he meant the derivation of the pitches of one half, one third, one quarter, one fifth, and one sixth of the string; by “arithmetic division,” in contrast, the de­ rivation of the pitches of multiples of a smallest part = 1 : 2 : 3 : 4 : 5 : 6. The series of string lengths 1 : 1/2 : 1/3 : 1/4 : 1/5 : 1/6 corresponds, when we take C3 as 1, to the tones C3, C4, G4, C5, E5, G5, that is, tones that collectively belong to the C-major chord; the se­ ries 1 : 2 : 3 : 4 : 5 : 6 in contrast results, if we take G5 as a starting point, in the tones G5, G4, C4, G3, E♭3, C3, that is, tones that collectively belong to the minor chord under G5 (=

C minor):

In other words: the minor chord according to Zarlino is mathematically the polar opposite of the major chord (Zarlino also uses the expressions divisione harmonica and di­ visione aritmetica as shorthand for the major chord and minor chord themselves). (p. 68)

Whether Zarlino himself made the ingenious discovery cannot be proven; but I am not aware of any older theorists who mention it. Unfortunately, Zarlino's great thought remained a theorem; whether it was not sufficient­ ly noticed or whether it was not understood, in any case, it disappeared for two full cen­ turies, in order to resurface only in 1754 with Tartini. Thoroughbass figuration, which appeared in the literature only a few decades later but was already developing in practice at the time, was probably the reason that the seeds of a rational harmonic theory indicated here did not develop further, but rather fell into Page 13 of 37

“The Nature of Harmony”: A Translation and Commentary complete oblivion. A comprehensive naming of even the simplest and most common chords did not exist at that time; if such a naming had not been established on Zarlino's dualistic explanation of the major consonance and minor consonance—that is, an explana­ tion based on two opposed principles—then the theory would have been steered along the same paths on which the latest efforts also aspire to steer, namely that of the thoroughgo­ ing dualism of the major relations and the minor relations, the former considered from below and the latter from above.26 Instead, thoroughbass linked the interpretation of all simultaneities to the lowest voice and constructed chords from the bottom up. The Italian organists, who had to accompany choral singing in rehearsal or in perfor­ mance, probably made use of this shorthand figured notation as early as the middle of the sixteenth century. The limited notation, at the time, of mensural music without bar lines, and with varying note values in the individual voices to boot, made playing from a score in the modern sense impossible, and scores were therefore not written or printed at all, and directors and organists had to try to create an overview of the work by other means. The Italians accomplished this by placing the notes on top of one another somewhat in the manner of a score and then indicated with numbers above the lowest voice which in­ tervals the higher voices should form with it; the Germans had long had another means for better clarity in the so-called organ tablature. To grasp the significance of thorough­ bass for the practice of compositional teaching at this time, one must keep in mind that the previous era had only seen in the chords an accidental convergence of several voices in consonant intervals, and that even Glarean (1547) was of the view that polyphonic composition was a coupling of several voices moving in different modes and that in the same piece, for example, a plagal mode is found in the discantus and an authentic mode in the bass. The idea of considering and naming the simultaneously sounding tones from a consolidating perspective was foreign to an era that knew neither the dominating melody nor the supporting bass and viewed and treated the four or five voices of contra­ puntal composition as completely independent individuals. The first half of the sixteenth century is indeed the era of the highest blossoming of imitative contrapuntal style; only the second half brought the clarified composition of a Palestrina and an Orlando di Lasso, and its end brought the new musical style of homophonic composition. It is hard to resist the idea that the discovery, at precisely this time, of (p. 69) the chord concept and the thereby altered conception of music in many parts won decisive influence over musical production. As mentioned, thoroughbass was the very first thoroughly developed chordal nomencla­ ture and signified tremendous progress for theoretical knowledge as well as for practice itself. The essential character of thoroughbass is well known; it designates every tone with a number, which corresponds to its degree reckoned in diatonic order, starting from the bass tone, but it generally sets the intervals greater than an octave equal to those an octave or more closer to the bass—in other words, it recognizes identity between octaverelated tones, on the condition that the interval in relation to the bass tone remain the same from above and below; for since it always takes some bass tone as its point of de­ parture, it has no way of expressing the fact that the fifth C–G and the fourth G–C are just Page 14 of 37

“The Nature of Harmony”: A Translation and Commentary as much the same interval27 as the following chords have the same significance and same

sign:

Hence, while the inversions of the intervals of the upper parts do not change significance for thoroughbass, such interval equivalence is impossible when one voice creates this in­ terval in relation to the bass. It can hardly be denied that figured bass is an incomplete means for the theoretical representation of harmonies; but it was the first that people came to know and was therefore extremely beneficial. It was not long until additional ab­ breviations arose in the practical treatment of bass figures, which greatly simplified har­ monic thinking. That chords formed from the third and fifth were especially common was immediately noticed and they were given a special significance by virtue of the fact that they were assumed when no sign at all was written above the bass. Only if the third or fifth, as given in the notation, were to be altered were accidentals and numbers neces­ sary. But the chord of the third and fifth, or, as we now call it, the triad, required by the absence of signs could just as easily be a major or minor chord as a diminished triad: Thus figured bass occasioned the development of chord theory in a direction completely different from where it had been leading the most erudite and famous theorist of his cen­ tury, Zarlino, through the establishment of the polar opposition of the major and minor chords. That C–E–G and E–G–C and G–C–E as well as other (more extended and multipart) simultaneities of the three tones C, E and G have the same harmonic significance certainly arose from Zarlino's conception as clearly as could be desired, and (p. 70) also for the minor chord the various rearrangements appear as identical formations. In con­ trast, it was impossible from the standpoint of thoroughbass to arrive at this knowledge; rather, Zarlino's thought, even when it had already achieved wide dissemination, was pushed into the background, since for thoroughbass it seemed to have more to do with a melodic conception of simultaneities than with a truly harmonic one. According to figured thoroughbass, the minor chord is something that is not differentiated from the major chord; and yet according to figured thoroughbass, the major sixth chord is something dif­ ferent from the chord of the third and fifth, the triad, of the same chord. The budding knowledge in Zarlino of the diverse significances of the harmonies and the equivalence of their inversions was thus stifled, and the single positive gain was the possibility of an ab­ breviated notation for all simultaneities in connection with the figures, such as: sixth chord (6), six-four chord

, seventh chord (7), six-five chord

, etc. That the sixth chord

is an inversion of the triad, that the six-five chord is an inversion of the seventh, was only noticed nearly 150 years later, after Zarlino's pertinent idea had been totally forgotten. Thoroughbass spread throughout Europe with lightning speed once it first emerged around 1600 in the printed works of Italian composers, and it captured much terrain even from German tablature, since the latter did not contain the elements of chord nomencla­ ture. The rapid blossoming of opera, the oratorio, and instrumental music, moreover, Page 15 of 37

“The Nature of Harmony”: A Translation and Commentary pushed the thought of reforming and improving theory into the background, and for over a century, one was content with the practically very useful bass figures, which, as is well known, came to play an outstanding role to the extent that the organ or the harpsichord became integral components of the accompaniment in church as well as in the theater; the part of the organist or harpsichordist, and indeed of the theorbist or gambist, was nothing more than a figured bass from which the accompanist had to develop a correct multi-part composition. Thoroughbass was therefore an important art until past the mid­ dle of the previous century. Theory received a powerful new impetus toward rational development in 1722 from JeanPhilippe Rameau,28 a man of great significance in the history of French opera as well. Rameau can be considered the discoverer of the overtones. He noticed that a sounding string allows one to hear not only its proper tone (the tone which is demanded by a nota­ tion and which is normally considered to sound uniquely), but also simultaneously its twelfth (i.e., the fifth above the octave) and seventeenth (the major third above the dou­ ble octave); in other words, that what we normally hold to be a simple tone is rather a complex of several tones and is indeed a major chord; for from, say, C2, the twelfth above is G3 and the seventeenth is E4; that is, we have the complete C-major chord:

For a musician of Rameau's talent this discovery was more than a curiosity; it was a reve­ lation. Indeed, the phenomenon of the overtones was not entirely unknown (p. 71) before Rameau; Mersenne (1636) had already pointed to it, and Sauveur (1701) had explained it scientifically, and had even emphasized its significance for the knowledge of the princi­ ples of harmony; yet it first came to be known in broader circles and obtained a practical significance for the theory of art through Rameau's theory of fundamental bass, which was based upon it. A clever musician like Rameau sensed clearly that the grounding of the major consonance in an acoustical phenomenon was not fully sufficient for the construction of a scientific system of harmony; but his attempt to verify a corresponding phenomenon also for the minor consonance failed. Whether Rameau took Zarlino's mathematical explanation of both Klang-principles29 as his point of departure in such a quest is not known; in any case, he attempted to ground the minor chord in the phenomenon of undertones, in con­ trast to that of the overtones. Specifically, he discovered that those strings of which a sounding tone is an overtone (in other words, as Rameau says: those of the undertwelfth and underseventeenth) vibrate forcefully as long as the pertinent tone (Rameau's “générateur”) vibrates, while strings tuned differently remain completely at rest. Al­ though he was not able to hear out this lower string's tone from the sounding mass, he still assumed that it must be contained in it, and believed he had found the principle of the minor consonance in the so-called phenomenon of sympathetic vibration; for the un­ dertwelfth and underseventeenth produce the minor chord under the generating principal

Page 16 of 37

“The Nature of Harmony”: A Translation and Commentary tone in just the same way as the overtwelfth and overseventeenth represent the major chord over the principal tone:

Unfortunately, Rameau learned from the physicist d’Alembert that this sympathetic vibra­ tion of lower strings did not produce their proper tone (the tone of the whole string), but rather (in the manner of the harmonics of string instruments) the nodal points make it vi­ brate in so many aliquot parts that they intensify only the generating tone. A few years ago, I demonstrated that Rameau's observation was still not without significance for the explanation of the minor consonance (Musikalische Syntaxis, 1877); for the sympatheti­ cally vibrating strings do produce their proper tone in addition to the generating princi­ pal tone, though to be sure considerably more weakly. Rameau had to abandon the scientific grounding of the minor consonance, then, and found it necessary to base his system one-sidedly on the major principle. Thus the minor chord was a modification of the major chord for him—that is, a simultaneity not given by nature, a less perfect consonance. His physical explanation of consonance thus in the end remained behind Zarlino's merely mathematical explanation. Only in one point did his system represent great progress in theoretical knowledge; Rameau enunciated for the first time in unambiguous terms that all (p. 72) possible permutations of the chord—by oc­ tave displacement of the individual tones, by inversion of the interval from above and be­ low even with respect to the bass, by octave doubling, etc.—do not alter its harmonic sig­ nificance; that is, he did what was indeed near to hand but unattainable from the stand­ point of the thoroughbass method: he rendered chords that are composed of like-named tones identical, regardless of which of them is the bass tone; he created the theory of the inversion of chords. But this was an extraordinary stroke of genius; for with that, the ap­ paratus of the theory of harmony was fundamentally simplified in a single stroke. The tri­ ad, sixth chord, and six-four chord now appeared as different forms of the same harmony, just as did the seventh chord, six-five chord, four-three chord, and second chord. This may have been intuited for a long time, but nobody had as yet articulated it. It is much to be regretted that Rameau was hindered by d’Alembert from a dualistic con­ struction of the theory of harmony; for one can deduce how fine his harmonic instinct was from two further peculiarities of his system, namely from the conception of the dimin­ ished triad as a dominant seventh chord with omitted root, for example: B–D–F as a G-ma­ jor chord with the minor seventh (F) but an omitted root (F):

and, further, from the construction of the added-sixth chord (accord de la sixte ajoutée); specifically, he explains the six-five chord, F–A–C–D in C major, as a subdominant chord (F major) with an added major sixth (D):

Page 17 of 37

“The Nature of Harmony”: A Translation and Commentary and not as an inversion of the seventh chord D–F–A–C, as one would expect and is gener­ ally done today. No musician can deny that the effect of the diminished triad as well as of the chord F–A–C–D (both in C major) fully corresponds to this explanation; the sense of the harmony will be fundamentally altered neither by the addition of the omitted G nor by the omission of the added D (though the diminished triad can also be conceived in anoth­ er context as a half-diminished-[under-]seventh chord; for example, B–D–F in A minor as B–D–F–[A]). This conception makes explicit the intuition of an idea to which I will return, namely that dissonant chords are to be interpreted as alterations of consonant chords but not as fundamental formations in themselves. For the outward presentation of his system, Rameau needed a representational means other than thoroughbass, for the tone in whose sense the harmony is to be grasped, is a different one from the root in all inverted chords. Yet, as we have already seen, he stuck with the construction of chords from the bottom up (for minor (p. 73) chords too), and therefore expressed all chords, as in thoroughbass, as if they were resting on a bass tone; he called this bass tone the “son fondamentale” (fundamental tone) and the whole succes­ sion of fundamental tones “basse fondamentale” (fundamental bass). The following con­ spectus will serve to illustrate the difference between figured bass and the Ramellian fun­

damental bass:

The fundamental bass furnished a means to consider the relation of successive harmonies from a synoptic perspective and to find the fundamental laws of harmonic phrase compo­ sition; along this line, Rameau established the rule that the fundamental bass may only progress in perfect fifths and fourths, or (major and minor) thirds. If this rule does not ap­ pear entirely adequate even today, it still effectively contains an indication of the most im­ portant perspective for the judgment of Klang successions, namely the recognition of a third relationship between Klänge in addition to the generally acknowledged fifth rela­ tionship. As rich as Rameau's system is in attempts at a rational theory of harmony, it cannot be designated as such in its totality. The natural grounding of the major consonance as well as the ingenious derivation of the diminished triad and the major chord with major sixth remain too isolated within an otherwise fully arbitrary schematic construction. The ideas mentioned could easily have been missing in the same system without its failing; only the simplification of the chord theory inaugurated by thoroughbass would remain, the theory of chord inversion. It was this, then, that obtained direct and enduring influence on the entire further development of the theory of harmony; the theory of inversion resurfaces in the systems of Calegari (Tratto del sistema armonico di F. A. Calegari [† 1740], first pub­ Page 18 of 37

“The Nature of Harmony”: A Translation and Commentary lished in 1829 by Balbi), Vallotti (Della scienza teorica e pratica della moderna musica, 1779), Kirnberger (Die Kunst des reinen Satzes, 1774–79), Abbé Vogler (Handbuch der Harmonielehre, 1802), and everything that followed. The weak point of Rameau's system, the abortive development of connections to the physicalist theory of Klänge, was readily noticed, and Vallotti rejected the one-sided grounding of the major consonance in an acoustic phenomenon and derived the diatonic scale from the higher overtones, among which he found the minor chord as well as the major chord. It was made clear by d’Alembert (Eléments de musique théorique et pratique, suivant les principes de M. Rameau, 1752; translated into German by Marpurg in 1757) that, among other things, the overtones observed by Rameau (twelfth and seventeenth) (p. 74) do not stand alone but are rather only the elements most immediately apparent to the ear out of a series of tones —the higher the weaker—which, with respect to Zarlino's harmonic division, correspond to string lengths, but, with respect to vibrational frequencies, correspond to the natural

integer series 1, 2, 3, 4, 5, 6, etc.

Vallotti found the scale between the eighth and sixteenth overtones, the major chord be­ tween 4:5:6 and the minor chord between 10:12:15. Of course, an actual grounding of consonance did not result from this, since for Rameau, this grounding consists in the in­ terpretation of the tones of a Klang in the sense of the fundamental tone; G3 and E4 are consonant with C2 because they merge into it; one cannot interpret E5–G5–B5 as giving a sense of C2, however, without destroying the consonance of the chord. Kirnberger took the overtones as an explanation of the major consonance, allowed the in­ consistency of its continuation to remain, gave up the maverick explanation of the dimin­ ished triad and of the added-sixth chord, and retained the system of inversions in such a form that thoroughbass method did not need to be altered in any way; that is, he lined up the major chord, the minor chord, and the diminished triad (as figured bass had previous­ ly), and drew from them four types of seventh chord as source chords: the major chord with major and minor seventh, and the minor chord and diminished triad with minor sev­ enth. Kirnberger's system remained with incidental modifications in practical handbooks until the present day. Since Rameau, the punctum saliens for the differentiation of the source chords, inversions, and suspension chords, etc., has been construction in thirds; that is, chords that can been constructed as a series of (major and minor) thirds over their bass tone were seen as source chords; those that can be arranged in thirds through inversion (that is, choosing a different tone than the bass) appeared as inverted source chords; and finally those that cannot be represented as a series of thirds in any way were to be seen as incidental constructions, as suspension chords. With this construction from thirds, one went beyond the seventh to ninth, eleventh, and thirteenth chords, which one could naturally demonstrate only in elliptical form. From such monster chords—especially as rendered well-nigh horrifying in J. H. Knecht's system (see Allgemeine Musikalische Page 19 of 37

“The Nature of Harmony”: A Translation and Commentary Zeitung 1, 1798–1799)—there has been a retreat recently, and we are generally content to concede the ninth chords a somewhat conditional entitlement as source chords. I have already mentioned that, having been forgotten for 200 years, the dualistic concep­ tion of harmony, first posited by Zarlino, was taken up again by Tartini, the famous violin virtuoso.30 It is not improbable that Tartini thoroughly studied and understood Zarlino; not only does he derive the major consonance from the harmonic division of the string and the minor consonance from the arithmetic division, (p. 75) but he also sees in the mi­ nor not a different species of third (that is, not the minor one, as it is viewed by thorough­ bass) but rather only a different positioning of the only third to be considered (the major one), which is positioned against the lower tone of the fifth in the major chord and against the higher tone in the minor chord:

But Tartini was a contemporary of Rameau and did not simply adhere to Zarlino's view­ point. The question, first raised by Rameau, of the grounding of consonance in acoustical phenomena actively preoccupied him and he was able to see a new aspect in it. To be sure, in grounding the minor consonance, he remained satisfied with the suggestion of a polar opposition to the major consonance in Zarlino's conception (opposition between the harmonic and arithmetic divisions), but he considerably deepened the explanation of the major consonance by not ignoring, as did other theorists, the fact that other, higher over­ tones existed than the sixth, but he attempted to go at least as far as the seventh. As is well known, the seventh overtone is a minor seventh, which is a little flat in comparison with the minor seventh of the tempered twelve-note system. Tartini momentously assert­ ed the consonance of the major chord with a natural seventh, a view one also finds even in the latest promoter of the exact theory of harmony, Helmholtz; but that even the sev­ enth chord tuned with mathematical purity as 4:5:6:7 is a musical consonance, can never be accepted from science by art, although it cannot be denied, on the other hand, that it does exceed even the major chord in equal temperament in physical euphony—that is, as far as the undisturbed fusion of the vibration patterns. Kirnberger and Fasch in Berlin at­ tempted, a few decades after Tartini, to assert the use of the natural seventh for our prac­ tice of music, but with little success; for naturally it cannot make sense to introduce an untempered seventh next to a tempered third and tempered fifth, while it would remain possible for anyone to regard the natural seventh as having been introduced into our tone system with the meaning of a (dissonant) basic interval, tempered just like any other in­ terval.31 As is well known, Tartini is also the discoverer of the combination tones or “Tartini tones,” named after him; in fact, his Trattato first appeared in 1754, while Sorge had al­ ready pointed to the existence of combination tones in 1740 in his Vorgemach musikalis­ cher Komposition; but Tartini discovered the combination tones in 1714 and introduced them to his violin school, opened in Padua in 1728, as the touchstone of pure intonation in chords (cf. my Studien zur Geschichte der Notenschrift, 1878, p. 101). As Tartini correctly observes, the phenomenon of combination tones coincides with that of overtones insofar as the lower tones, which become audible when two tones sound together, are none other Page 20 of 37

“The Nature of Harmony”: A Translation and Commentary than the tones of an overtone series in which the pertinent interval, right down to the fun­ damental, can be correlated with the smallest ordinal numbers. Just like Rameau with the overtone series, Tartini only observed the combination tones incompletely and heard only (p. 76) the lowest combination tone, which always corresponded to the fundamental tone of the series and was initially (in the Trattato) identified by him in error as an octave too high but was correctly identified in the text De principii. We now know that the entire overtone series of this fundamental is audible, not only the tones that are lower than the given interval, but also those falling within the interval and higher, so that the relation­ ship of the two phenomena is in any case evident. The fifth 2:3 (C3–G3) produces only a lower combination tone, namely one corresponding to the integer 1, the underoctave (C2) of the interval's lower tone; the fourth 3:4 (G3–C4) audibly produces 1 and 2 (C2–C3); the third 4:5 (C4–E4) the tones 1, 2, and 3 (C2–C3–G3), etc. These lower combination tones have special significance for the interpretation of the major chord; for the first time, they provide the theory of chord inversion its true scientific foundation, since the triad C4–E4– G4 as well as the sixth chord E4–G4–C5 and the six-four chord G3–C4–E4 find their point of

unity in the combination tone C2:

Not the close position

but rather the open position

proves to be the typical form of the major chord. Another type of combination tone has first been graced with the appropriate recognition only recently, namely the coincident overtones (A. v. Oettingen, Harmoniesystem in dualer Entwickelung, 1866); among the higher overtones of an interval or chord (that is, the overtones of the individual chord tones and the combination tones of the overtones), the first common overtone of the chord tones sounds especially loud to the ear.32 Since its ordinal number is found by mul­ tiplying the ordinal numbers of the interval tones in the overtone series, one may call them multiplication tones. Thus the major third 4:5 (C4–E4) has the multiplication tone 4 × 5 = 20 (E6), the major sixth 3:5 (G3–E4) has the multiplication tone 3 × 5 = 15 (B5), the minor third 5:6 (E4–G4) has the multiplication (p. 77) tone 5 × 6 = 30 (B6). Just as the mi­ nor third E4–G4 and the major sixth G3–E4 are completed by the combination tone C2 to form a major chord, the same intervals are completed by the multiplication tones B6 and B5, respectively, to form a minor chord. A. v. Oettingen (Professor of Physics at the Uni­ versity of Dorpat) sees in the multiplication tones, or the “phonic overtones” as he calls

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“The Nature of Harmony”: A Translation and Commentary them, the natural cohesiveness of the minor chord; B6 is the common overtone of the fol­

lowing series of tones:

Thus we have here the complete undertone series, the counterpart of the overtone series, of equally foundational significance for the minor chord as the overtone series is for the major chord. The tones of this series fuse into a unity in relation to this highest tone in the same thorough way that the tones of the overtone series do in relation to the funda­ mental. I have already explained above how the musical conception can be brought into accord with the tones of this series that do not belong to the E-minor chord (7, 9, 11, 13, 14, etc) in connection with the corresponding overtones. Yet just as we cannot explain the consonance of the major chord through combination tones alone, but rather regard the phenomenon of the overtones as its actual foundation, we also need an opposing phenomenon of undertones for a fully satisfactory explanation of the minor consonance. If such a phenomenon has not yet been able to be established with sufficient certainty, we do not lack signs that minor relationships are reckoned by the perceiving mind along the same lines from the top as major ones are from the bottom. I have already pointed out that the phenomenon of sympathetic vibration furnishes the undertone series; the phenomenon of ringing tones (Klirrtöne) also belongs here. If you hold the base of a vibrating tuning fork only very lightly without affixing it firmly, or if you set a loosely fastened metal plate into vigorous vibration, you hear the underoctave or un­ dertwelfth, even perhaps the underfifteenth, underseventeenth or other lower undertones of the fork or plate instead of its proper tone. But it is even probable that each tone, with an intensity decreasing in proportion to the lowness of the pitch, always produces a series of undertones corresponding to the series of overtones, though the former are even more difficult to perceive—to single out, that is, from the mental representation of the Klang (Klangvorstellung)—than are the overtones. To that end, I have adduced all manner of material that has been observed on this point without having been refuted. (Musikalische Logik, 1873, p. 12; “Die objective Existenz der Untertöne in der Schallwelle,” 1875, spe­ cial publication of the Allgemeine Deutsche Musikzeitung; Foreword and Addendum to Musikalische Syntaxis, 1877; cf. also the article, “Untertöne,” from my Musik-Lexikon). Whatever one thinks about this or that of my proofs, the fact remains that the ma­ jor and minor consonances are, according to their mathematical-physical relations, strict (p. 78)

opposites of one another. The question then is whether the physiology of hearing and the psychology of tone representations can explain a similar reciprocity of major and minor and whether they are able to recognize the principle of this mathematical-physical theory of tone as their own.

Page 22 of 37

“The Nature of Harmony”: A Translation and Commentary The physiology of tone sensations has only recently undergone a more thoroughgoing re­ vision, particularly by the physicist and physiologist of outstanding merit, Heinrich Helmholtz [sic].33 The book however by no means contains only physiological investiga­ tions, as one might assume from the title, but rather spans the entire area of the scientif­ ic study of tone, from the generation and propagation of sound to the concatenation of chord ideas (Klangvorstellungen); in other words, it occupies itself not in the least only with mathematical-physical investigations and it extends all the way into psychology and aesthetics. Thus the theory of overtones and combination tones is treated thoroughly and the differences of tone color are explained by the differences in composition of Klänge out of overtones; these investigations are especially valuable for the theory of instrument construction and also explain, among other things, the mixture stops of the organ (Quint, Terz, Mixtur, Cornett, etc), which have been in practical use since long before knowledge of the composition of complex tones, and which collectively had no other purpose than to reinforce individual overtones and thus the fundamental sound of the organ's basic ranks. The specifically physiological chapters in Helmholtz's book include one on the analysis of complex tones by the ear, or the singling out of individual overtones from the wave mo­ tion that still strikes the ear as a single vibrational form; further, one on the perception of the different tone colors (which rest on the same principles); and one on the euphony of the different chord types. As fine and intelligent as the investigations and observations on both of the first named problems are, they must still be designated in their entirety as hy­ potheses and are so designated by Helmholtz. The hypothesis suggests that in the inner ear, a complicated apparatus exists with various-sized and more or less tautly stretched fibers, which are set in motion according to the law of sympathetic vibration and excite the nerve endings leading away from them. The entire apparatus is of microscopic pro­ portions. For music theory, there can be no discussion of a positive result of this hypothe­ sis, regardless of whether it pertains to the Membrana basilaris or the fibers of Corti; for the time being, it is not even beyond doubt that it constitutes a positive result for natural science itself. The most vulnerable chapter of Helmholtz's music theoretical work is the one on conso­ nance and dissonance, which concepts Helmholtz tries to explain from a physiological standpoint as difference in euphony. He locates the essence of dissonance in the presence of beats; that is, regular, rapidly recurring intensifications of a Klang, which are felt as unpleasantly disturbing. Consonance, according to Helmholtz, is the complete absence of beats or at least their limitation to a very small number. The major chord appears most free of beats, but the minor consonance is an obscuring of physiological consonance; in general, a complete scale of (p. 79) decreasing euphony can be constructed according to the scale of beats, beginning from the undisturbed fusion of a chord manifesting the rela­ tions of the first overtones:

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“The Nature of Harmony”: A Translation and Commentary up to the harshest dissonances and musically most impossible discordances, such that for neither the major chord and minor chord nor for consonance and dissonance is anything produced other than a difference in degree of euphony. This most unsatisfactory result immediately met the most vigorous opposition; right away, the already-named A. v. Oettingen34 entered the field and demanded a principled differ­ ence for major and minor; he found it in the completely opposed mode of construction of both species of consonance; that is, he constructed the minor consonance as the antipode of the major consonance in the manner already shown and also followed through with this opposition between major and minor in scale theory and chord theory. He rightly pointed out that the physical euphony of the major consonance is inferior when one compares the arrangement of the undertones with that of the overtones. The minor chord:

is just as free of beats as the major chord in the form given above and fuses in the most complete manner into the unity of the highest tone. The disaccord of the common combi­ nation tone F0 against the A-minor chord in the arrangement given here correlates with the discord of the corresponding overtone B5 against the C-major chord in the arrange­ ment given above. Oettingen also indicates the satisfactory results to be attained for the differentiation between consonance and dissonance. But this would lead us beyond the area of the physiology into psychology. With equally keen intellect, the ingenious Göttingen philosopher Hermann Lotze (Geschichte der Ästhetik in Deutschland, 1868) found the Achilles heel of the Helmholtz­ ian system; like Oettingen, he demanded a principled differentiation between major and minor and between consonance and dissonance; he demanded something other than a gradated difference in euphony for the many different types of dissonance. In the mean­ time, a number of more recent music-theoretical writings have followed, which have in­ terpreted Oettingen's and Lotze's reproofs against Helmholtz's theory differently and sought a satisfactory solution to the problem,35 whereby Oettingen's almost completely fleshed out system formed the basis of further investigations. If I have not dedicated a single word so far to Moritz Hauptmann, whose epochmaking work Die Natur der Harmonik und Metrik appeared in 1853, that was so as to do his work and its merits fuller justice without interruption. Hauptmann was at once an em­ (p. 80)

inently musical figure and a deep philosophical thinker. Harmonic dualism, the polar op­ position between major and minor, which two of the most ingenious older theorists found through mathematical construction, he generated anew through philosophical specula­ tion, long after the intellectual labor of his two predecessors had passed into oblivion in libraries. If, in studying the history of harmonic theory today, we find that Hauptmann's idea of considering the minor chord as a major chord developed negatively and set on its head—an idea that caused such a great sensation—had already been posited 300 years earlier by Zarlino, we must not deny Hauptmann the original discovery of this thought. We would commit a great injustice were we to contest Hauptmann's independent discov­ Page 24 of 37

“The Nature of Harmony”: A Translation and Commentary ery of this dual harmonic principle. As far as the further development of the theory, Hauptmann is the originator of this idea. It is just as unlikely that a theorist of the first half of our century would have generated his wisdom from Zarlino or Tartini, as it is that it would occur to a theorist of our time to draw on those “old masters.” All we loyal stu­ dents of Hauptmann, who adhere to the letter of his teaching (Köhler, Paul, Rischbieter), O. Tiersch who seeks a compromise between Hauptmann and Helmholtz (System und Methode der Harmonielehre, 1868), as well as the strict dualists (Oettingen, Thürlings, myself and, with reservations, Hostinsky), who have become more Hauptmannian than Hauptmann himself—all of us adopted harmonic dualism as a new concept from Haupt­ mann. That there were any early defenders of this idea at all, I first brought to light again in 1875 with regard to Tartini (“Die objective Existenz….”) and in 1881 with regard to Zarlino (“Zarlino als harmonischer Dualist,” Monatshefte für Musikgeschichte). Though Helmholtz did not accept harmonic dualism and still positions himself at least in resistance to it, Hauptmann's system exerted the greatest influence even on him, as be­ comes clear from the evidence of the third and musically most valuable section—entitled “The Affinity of Tones”—of his treatise on tone sensations. One senses everywhere here the essential lapidary thought of Hauptmann's work: “There are three directly intelligible intervals: I. the octave, II. the fifth, III. the (major) third” (Natur der Harmonik, p. 21). This thought is truly great and epoch-making and contains within it everything that exact theory has since been able to develop. The minor third, the fourth, the sixth, and all other intervals do not exist for Hauptmann; for him, they are not entities subsisting in them­ selves and in themselves significant, but rather merely products, combinations of the es­ sential concepts: octave, fifth, and third. I do not wish to suppress the fact that this recog­ nition was not absolutely new; the mathematicians had already known for several cen­ turies that all musical intervals can be expressed as products and powers of the numbers 2, 3, and 5. Ancient theory only recognized two essential intervals: octave and fifth, and derived all other intervals from them, the second (C–D) as an octave displacement of the second fifth (C–G–D), the third (C–E) as an octave displacement of the fourth fifth (C–G– D–A–E), etc. Since Fogliani's and Zarlino's construction of (p. 81) the consonance of the third, so definitive for the West, the third has also been taken into account and, for exam­ ple, the major seventh (C–B) has been defined as the third of the fifth (C–G–B), the aug­ mented fourth (C–F♯) as the third of the second fifth (C–G–D–F♯), the augmented fifth (C– G♯) as the third of the third, and so forth. But this progress in knowledge first entered the harmony manuals through Hauptmann. It was Hauptmann who introduced a differentia­ tion between fifth-related and third-related tones, not in notation but in a nomenclature for tones in letters. The significance of Hauptmann's familiar upper- and lower-case toneletters is that two similarly named tones, one of which is indicated with a large and one with a small letter, differ from each other in the mathematical determination of their pitches by the so-called syntonic comma or comma of Didymus. As already mentioned above, Didymus presented a tetrachordal division:

Page 25 of 37

“The Nature of Harmony”: A Translation and Commentary That is, he introduced two different whole tones, 10:9 and 9:8; the difference between the two (10/9 : 9/8) is the comma of Didymus 80:81. In our major scale C:D is the whole tone 8:9 (D is the second fifth of C, thus [3/2]2 = 9/4, or, in close position, 9/8) and D:E is the whole tone 9:10 (E is the third of C, thus 5/4; d:e = 9/8 : 5/4 = 40/36 = 10/9); the third E stands in relation to the fourth fifth (C–G–D–A–E) as 80:81 (since [3/2]4 is 81/16, or, in close position, 81/64; 5/4 : 81/64 = 80:81). Hauptmann indicated C with an upper-case character, the third e with a lower-case one and the fourth fifth E again with upper-case; in general, all tones indicated in upper-case represent a chain of fifths, while the intervals indicated through an alternation of upper- and lower-case letters imply a third-progres­ sion; the tones indicated with lower-case characters stand in relation to each other again in a fifth-chain:

Hauptmann represents the key as constituted from the tones of the three chords on the tonic, dominant, and subdominant: In C major, then, we have the two fifth-series F–C–G–D and a–e–b, the former comprising tones that are related to C by fifths, the latter from tones that are related to it by thirds. The eminently important result of this arrangement, though, is the recognition of the third-relationship of chords and keys. Even [A. B.] Marx wondered that the keys of E ma­ jor and A major were immediately intelligible following C major, while D major and B♭ ma­ jor sound foreign and incoherent in relation to (p. 82) C major; indeed, since Marx still had no knowledge of the third relation, he had to wonder at the fact that the key of the fourth fifth appeared better connected than that of the second fifth. Yet E major is not at all the key of the fourth fifth, but rather the key of the third. Although Beethoven had al­ ready introduced the second theme of the first movement of the C-major Sonata, op. 53, in E major, it was only Hauptmann who enunciated the third-relationship of keys and thereby disposed of the problem once and for all. In more recent music, the juxtaposition of third-related keys along with fifth-related keys has since been vernacularized, even if some theorists’ dogged clinging to old traditions and blindness against Hauptmann's in­ genious progress even today still regard it as somewhat abnormal, or at most as only ex­ ceptionally admissible. Hauptmann's letter nomenclature for tones, with its differentia­ tion between fifth- and third-related tones, has been further perfected by Helmholtz and Oettingen, such that one now distinguishes between thirds below and thirds above, and

Page 26 of 37

“The Nature of Harmony”: A Translation and Commentary third relationships of the first and second, etc., degrees (in the manner devised by Oettin­

gen and altered by Helmholtz):

Instead of the large and small letters, we thus now use the unambiguous comma-lines (E is one comma lower than E, C♯, etc).

is a comma higher than A♭, C♯ is two commas lower than

Yet Helmholtz did not only comprehend Hauptmann's theory in its full breadth and, apart from the insufficient explanation of the minor consonance and of the difference between consonance and dissonance, give it a scientific basis, but he also developed the theory es­ sentially further with the establishment of a concept that has opened entirely new per­ spectives: the concept of Klang representation (Klangvertretung). If all theorists had intu­ ited it, still none had spoken aloud that we conceive tones as representatives of Klänge. Indeed, according to Helmholtz, Klänge are just overtone sonorities—major chords—and in fact, for the construction of the minor chord C–E♭–G, he introduces the Klänge C (for C– G) and E♭ (for E♭–G); that is, the minor chord, according to Helmholtz, combines parts of two different Klänge. (Hostinsky follows him in this and goes even further.) A. v. Oettin­ gen, though, gave Helmholtz's concept of Klang representation an unparalleled breadth by positioning the minor chord along with the major as an actual Klang, as whose repre­ sentative a tone can appear. The principle of Klang representation no longer belongs with­ in physics, nor within physiology, but rather in psychology. If experience shows that we are able to understand a tone as the (p. 83) representative of a minor chord just as easily as we can understand it as the representative of a major chord (without one Klang or the other actually sounding), then this is a scientific fact on which we can build further just as well as we can build upon acoustical phenomena. Once we penetrate to this knowl­ edge, a physical grounding for the minor chord no longer matters much to us. The psy­ chological fact of the understanding of tones in the sense of Klänge remains fixed, and in­ deed each tone can be understood as the representative of three major chords and three minor chords; it can be the root, fifth, or third in either the major or the minor direction. It is by no means more difficult for us to comprehend a single E as the root of its under-Klang (A–C–E), or the fifth of a B under-Klang (E–G–B), or the third of a G♯ under-Klang (C♯ –E–G♯), as it is to comprehend it as the root of an E over-Klang (E–G♯ –B), or the fifth of an A over-Klang (C♯ –E), or finally as the third of a C over-Klang (C–E–G). There are no other Klänge as whose representative E can be understood; it can occur in a great number of other Klänge only as a foreign, consonance-disturbing tone, as the sev­ enth, for example, of an F♯-major chord, as the added sixth of a G-major chord, etc.

Page 27 of 37

“The Nature of Harmony”: A Translation and Commentary Through this most recent progress of scientific knowledge, the theory of harmony has de­ veloped from a theory of the mathematical ratios of musical intervals into a theory of tone representations and their concatenation, while acoustics and the physiology of hearing have regained the status of supplementary sciences, which they deserve and which they have certainly, from the perspective of the musician, always had.36 The musician greets this turn of events with joy, for the representations of tones, of chords are familiar to him, and he understands a theory pertaining to them immediately, so long as the chosen termi­ nology is understandable, while he feels a deep rift between the calculations of the physi­ cist and the nervous stimuli of the physiologist on the one hand, and his conceptions of music on the other. The same rift subsists between the second and third sections of Helmholtz's Lehre von den Tonempfindungen; in the third section, all is bright light, bril­ liant knowledge, true musical understanding, while in the second, a connection with liv­ ing music is striven for in vain and only the already-discussed thoroughly unsuccessful ex­ planation of consonance and dissonance presents itself. The mistake Helmholtz made is now easy to recognize; he sought to explain from the na­ ture of sounding bodies concepts that can only be explained from the nature of the per­ ceiving mind. Consonance and dissonance are musical concepts, but not definite forms of sound waves. But we cannot forget that this knowledge could only be won after physical and physiological investigation; the impossibility of physics and physiology being used to achieve a grounding of musical concepts could only be seen once psychology could enter in its own right. We know today that there is no absolute consonance, that even a chord which is, according to physical and physiological formulations, the most undisturbed and euphonious can be a dissonance musically (for example, the six-four chord). In order not to extend the reach of this sketch unduly, I must be brief and can only indi­ cate in broad strokes the form that the theory of harmony has achieved through the intro­ duction of the concept of Klang representation. As long as a tone is not determined according to the significance of its Klang, we have the simplest musical representation of tone; the conception of tone is empty and un­ satisfying, it comprises only the simple tone along with its octaves above and below; in­ deed, only seldom do we encounter with complete purity the mental representation of a tone not further determined, but tend much more to understand the first tone of a piece as the root either in the major or the minor direction. The representation of interval is (p. 84)

considerably richer in content; while tone can be conceived in six different ways, only two different conceptions exist for the (consonant) interval: C–G is either the representative of a C over-Klang or of a G under-Klang; for we could only be compelled to understand a tone in an interval as a dissonance by further added tones, which would give the second tone a definite significance as the representative of a Klang to which the former did not belong. But even the full concept of musical Klang is capable of yet further determina­ tions and is not in itself conclusive. If one thinks of the C-major chord in F major or in G major or in C major, the mental representation is different each time. Helmholtz rightly remarks (Lehre von den Tonempfindungen, 4th ed., p. 471) that a consonant chord as such is not yet entirely able to conclude a piece of music; this consonant chord must Page 28 of 37

“The Nature of Harmony”: A Translation and Commentary rather be the tonic, if it is really to have a conclusive effect. When Helmholtz states fur­ ther that earlier theorists had been entirely clear on this point, I must make an exception for Tartini, who emphasized (Trattato, p. 112) that all tones in a key are dissonant with the exception of those belonging to the tonic Klang. In other words: the tonic chord alone is uniquely a musical consonance in the strictest sense of the word—a chord that is capa­ ble of concluding and conditions no further progression, in C major the chord C–E–G, in G major G–B–D, in A minor A–C–E, etc. The G-major chord is not a perfect consonance in C major, from which it immediately follows that the seventh can be added to it without changing its significance and indeed without essentially changing its physical sonority (Klang); the F-major chord is also not a perfect consonance in C major and can appear with its major sixth without changing its effect. The effect of these chords is, then, a dis­ sonant one, or better put, the mental representation of these chords contains something that disturbs their consonance, and this something is nothing other than their relatedness to the C-major chord. For to comprehend a Klang within the context of a specific key means to understand it as a related Klang, as the subsidiary Klang of another, in exactly the same way that to comprehend a tone in the context of a particular Klang means to comprehend it not in isolation but rather in its relationship to a root, or as a root itself in relation to other tones. If I think to myself of a C-major chord in the context of the key of C major, it is itself the tonic, center, conclusive chord, its mental representation thus con­ tains nothing that opposes its consonance, it appears calm, pure, simple; but if I think to myself of a G-major chord in the context of the key of C major, then I think of it as the Klang of the over-fifth of the C-major Klang; that is, the C-major chord itself enters the mental representation as the Klang by which the significance of the G-major chord is de­ termined as something departing from it—the center of the mental representation, then, lies outside it, so to speak; that is, an element of unrest occurs in it, the desire for a pro­ gression to the C-major (p. 85) chord, dissonance. It is just the same with the F-major chord, and indeed generally with every Klang in the key. Yet this modern notion of key or, as we say in distinction from the old notion of mode, of tonality, is not bound to the scale; Klänge using tones foreign to the scale could also be comprehended in the sense of a ton­ ic and thereby receive their own characteristic meaning, such as above all the third-Klänge (the E-major chord and A♭-major chord in C major) and minor-third-Klänge (the E♭-major chord and A-major chord in C major). Indeed, the matter of tone relationships allows one further elaboration, namely that of the relation of keys to one another. Just as the root of a Klang relates to its subsidiary tones (the fifth and third and the more distant relatives), just as the principal Klang relates to its subsidiary Klänge (the Klang of the fifth scale degree, the Klang of the third, etc), the principal key relates to the subsidiary keys (the key of the fifth, the key of the third, etc). In a piece in C major, the G-major key plays the same role or a similar role to that played in a brief C-major cadence by the G-major chord or the tone G in a C-major arpeggio, or the tones D and B in a C-major scale; that is, it operates to dissonant effect and cannot justify its own existence, but rather its justification is conditional and is permanently without entitlement. The primal laws of chord succession, as of key succession (modula­ tion) thus present themselves in a direct manner from the expansion of the simplest musi­ Page 29 of 37

“The Nature of Harmony”: A Translation and Commentary cal concepts, consonance and dissonance. Psychology teaches us that several mental rep­ resentations cannot co-exist in the comprehending mind, but rather one dominates and the other appears in opposition to it, disturbing it. This rule proves itself in the most thor­ ough way in musical imagination; it provides the key for the true definition of the notions of consonance and dissonance, which physicists and physiologists have sought in vain. By it, we achieve not only a principled differentiation for consonance and dissonance, but al­ so simultaneously the qualitative differentiations, demanded by Lotze, among the differ­ ent species of dissonance. Consonance is the unified comprehension of tones represent­ ing one and the same Klang in the context of this Klang; in contrast, dissonance is the op­ position to the Klang forming the principal content of mental representation, the distur­ bance of the unification of the same by one or several tones, which represent other Klänge. The Klänge that are simultaneously represented in the dissonant chord thus do not appear as coordinated but rather one appears as the principal content of the mental representation and the other as a mere modification of it. This modification naturally dif­ fers according to the relatedness of the Klang represented as a dissonance. It is a fact, established through centuries of experience, but also easily confirmed through psychological experimentation—hence, a law—that only one major or minor chord can be the principal Klang (the tonic) of a key, though not a diminished triad or a seventh chord or some other kind of chord formation; one must therefore wonder that the theorists did not long ago arrive at the insight that all species of dissonant chords are not comprehen­ sible in themselves, but rather become so in the context of a consonance, except where one or two other tones are added to the tones of a Klang (seventh chord, added-sixth chord, ninth chord), or where for one chord tone, another neighboring one enters leading to (p. 86) it (suspension chords), or where one tone of a Klang is itself chromatically al­ tered so that it leads to a tone of another Klang (altered chords). Up until the present day, one instead construed dissonant chords as essential formations, as root chords in just the same way as the major and minor chord; the blame for this is due to Kirnberger, who did not understand and did not further develop Rameau's first steps toward a derivation of dissonant chords from consonant ones, but rather held onto chord classifications of thor­ oughbass that could not be accommodated to the progress of harmonic knowledge. If I succeed in carrying the harmonic theory I have sketched here forward into a complete system, the theory of harmony will become a true exercise in musical thought, for it moves from the simplest itself to the more complicated and induces one to attempt something new, to venture, rather than, as hitherto, to seek something new. In my Musikalische Syntaxis and Skizze einer neuen Methode der Harmonielehre, I have made an attempt to carry out the system, and I tried especially in the first book to achieve gen­ eral perspectives for the formation of harmonic phrases closed in themselves. Yet I em­ phasize once again in conclusion that some elements in the external apparatus of my pre­ sentation, in the formation of rules, may be new, but that on the other hand the principal perspectives, the fundamental concepts, do not originate with me, but rather, to the ex­ tent that they cannot be derived from older theorists (especially Rameau), originate with

Page 30 of 37

“The Nature of Harmony”: A Translation and Commentary the three greatest instigators of knowledge of the essence of harmony: Moritz Haupt­ mann, Heinrich [sic] Helmholtz, and Arthur von Oettingen.

Notes: (1.) Riemann, “Die Natur der Harmonik,” Sammlung musikalischer Vorträge (Leipzig: Bre­ itkopf und Härtel, 1882), 190. “…Neues zu versuchen, zu wagen, statt wie bisher, Neues zu suchen.” (2.) This formulation might seem to confuse the “historicist” and “presentist” categories that are traditionally taken as dialectically opposed. It has recently become easier to rec­ ognize—particularly in the wake of postcolonial and subaltern studies—how readily his­ toricist narratives tend to suppose a privileged point of historical arrival (often, though not always, their own present) by hewing to the structural “not yet” that is required to maintain the intellectual distance such narratives claim as a motivating asset. Historicism (even when it does not explicitly rely on the developmentalism that structures the vast majority of nineteenth-century historical writings) cannot be seen as entirely free from the more obvious presentist distortions Thomas Christensen diagnoses in “Music Theory and Its Histories,” in David W. Bernstein and Christopher Hatch, eds., Music Theory and the Exploration of the Past (Chicago: University of Chicago Press, 1993), 9–39. See Dipesh Chakrabarty, Provincializing Europe: Postcolonial Thought and Historical Difference (Princeton: Princeton University Press, 2000), esp. 6–16. (3.) See the discussion of the “responsibilities of music theory” in Rehding, Hugo Rie­ mann and the Birth of Modern Musical Thought (Cambridge: Cambridge University Press, 2003), 36–66. (4.) Frederic Jameson, A Singular Modernity: Essay on the Ontology of the Present (Verso: London, 2002), 34–35. (5.) In 1863 (about a decade before the advent of what is now recognizable as modern “psychology,” and thus without explicitly referring to that discipline as such), the first sentence of Helmholtz's monumental treatise on physiological acoustics and music theory had read, “The present book attempts to unite the boundaries of sciences, which, though oriented toward one another through many natural points of contact, have hitherto stood quite apart from one another—namely the boundaries between, on the one hand, physical and physiological acoustics, and on the other hand musicology and aesthetics.” “Das vor­ liegende Buch sucht die Grenzgebiete von Wissenschaften zu vereinigen, welche, obgle­ ich durch viele natürliche Beziehungen auf einander hingewiesen, bisher doch ziemlich getrennt neben einander gestanden haben, die Grenzgebiete nämlich einerseits der physikalischen und physiologischen Akustik, andererseits der Musikwissenschaft und Aes­ thetik.” Helmholtz, Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik (Braunschweig: Vieweg und Sohn, 1863), 1.

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“The Nature of Harmony”: A Translation and Commentary (6.) Admittedly, this question emphasizes only one possible translation of the problematic term Vorstellung. Yet, as Brian Hyer notes, Riemann would later put even greater empha­ sis on Phantasie, which arguably gives a more pointed sense of “imagination” as a form of higher thinking. See Hyer, “Reimag(in)ing Riemann,” Journal of Music Theory 39.1 (Spring 1995), 101–138. Hyer is referring here to “Ideen zu einer ‘Lehre von den Ton­ vorstellungen,’” Jahrbuch der Musikbibliothek Peters 21/22 (1914/1915), 1–26. (7.) A useful introduction to the cultural-historical shifts involved here is the essay collec­ tion by Timothy Lenoir, Instituting Science: The Cultural Production of Scientific Disci­ plines (Stanford: Stanford University Press, 1997). (8.) This familiar view is exemplified by classics such as Dolf Sternberger, Panorama, oder Ansichten vom 19. Jahrhundert (Hamburg: H. Govert, 1938); Sigfried Giedion, Mechaniza­ tion Takes Command: A Contribution to Anonymous History (New York: Oxford University Press, 1948), and Anson Rabinbach, The Human Motor: Energy, Fatigue, and the Origins of Modernity (New York: Basic Books, 1990). (9.) See Timothy Lenoir, The Strategy of Life: Teleology and Mechanics in NineteenthCentury German Biology (Chicago: University of Chicago Press, 1982). (10.) The notion of “finitude” as an unprecedented characteristic of nineteenth-century modernity is famously developed in Michel Foucault, The Order of Things (New York: Vin­ tage Books, 1970). (11.) A good sense of Helmholtz's methodological positions can be gained from his re­ spectively early and late essays, “On the Interaction of the Natural Forces” (1854) and “The Facts in Perception” (1878), in David Cahan, ed., Science and Culture: Popular and Philosophical Essays (Chicago: University of Chicago Press, 1995), 96–126 and 342–380. (12.) Helmholtz, “The Facts in Perception,” in Cahan, ed., Science and Culture, 364. “Als wesentlichsten Fortschritt der neueren Zeit glaube ich die Auflösung des Begriffs der An­ schauung in die elementaren Vorgänge des Denkens betrachten zu müssen, die bei Kant noch fehlt.” Helmholtz, “Die Thatsachen in der Wahrnehmung,” in Vorträge und Reden, vol. 2 (Braunschweig: Vieweg und Sohn, 1884), 248. Helmholtz may be referring here to work by Wilhelm Wundt, widely considered the founder of modern psychology, particular­ ly as exemplified in his unprecedented experimental laboratory in Leipzig, opened in 1879, and in his monumental Grundzüge der physiologischen Psychologie of 1874. But as the title of Wundt's three-volume book implies, psychology at this point was still very much modeled on physiology (so that the physics-physiology-psychology series in fact amounts to a cogent summary of the gradual transference of experimental technique from one set of “objects,” to the next, to the last). (13.) The cultural politics and epistemological pitfalls of this project are discusssed at greater length in Benjamin Steege, “Material Ears: Hermann von Helmholtz, Attention, and Modern Aurality,” (Ph.D. diss., Harvard University, 2007).

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“The Nature of Harmony”: A Translation and Commentary (14.) Arthur von Oettingen, Harmoniesystem in dualer Entwickelung: Studien zur Theorie der Musik (Dorpat and Leipzig: W. Gläser, 1866), 24. “Die bewusste Wahrnehmung eines Klanges wird nur durch eine beabsichtigte angestrengte Aufmerksamkeit in eine Summe von Einzelempfindungen aufgelöst. Diese Analyse des Klanges, ist das Fundament der Theorie der Musik.” (15.) Helmholtz, Handbuch der physiologischen Optik, vol. 3 (Leipzig: Leopold Voss, 1867), 443. “Ich meine…daß es gar keinen möglichen Sinn haben kann, von einer an­ deren Wahrheit unserer Vorstellungen zu sprechen, als von einer praktischen.” (16.) Helmholtz, “On the Physiological Causes of Harmony in Music” (1857), in David Ca­ han, ed., Science and Culture, 63–64. “Was thut nun das Ohr, löst es sie auf, oder fasst es sie als Ganzes?—Die Antwort darauf kann nach dem Sinne der Frage verschieden ausfall­ en, denn wir müssen hier Zweierlei unterscheiden, nämlich erstens die Empfindung im Hörnerven, wie sie sich ohne Einmischung geistiger Thätigkeit entwickelt, und die Vorstellung, welche wir in Folge dieser Empfindung uns bilden. Wir müssen also gleich­ sam unterscheiden das leibliche Ohr des Körpers, und das geistige Ohr des Vorstel­ lungsvermögens.” Helmholtz, “Ueber die physiologischen Ursachen der musikalischen Harmonie,” in Vorträge und Reden, 1: 103. (17.) “Ist also das Musikhören ein Auswählen aus dem zu Gehör gebrachten Klangmateri­ al nach einfachen…Gesichtspunkten, so ist es kein fysisches Erleiden mehr, sondern eine logische Aktivität. Es ist eben ein Vorstellen, ein vereinen, trennen, vergleichen, aufeinan­ der-beziehen von Vorstellungen, die freilich mit den durch Gesichtseindrücke her­ vorgerufenen Gestaltvorstellungen nur den Namen gemein haben, übrigens aber von to­ tal verschiedener Qualität erscheinen—Tonvorstellungen.” Riemann, Musikalische Syn­ taxis. Grundriß einer harmonischen Satzbildungslehre (Leipzig: Breitkopf und Härtel, 1877), viii. (18.) Moritz Wilhelm Drobisch, Neue Darstellung der Logik nach ihren einfachsten Ver­ hältnissen. Mit Rücksicht auf Mathematik und Naturwissenschaft, 3rd ed. (Leipzig: L. Voss, 1863), 5. “Jedes Denken ist im Allgemeinen ein Zusammenfassen eines Vielen und Mannigfaltigen in eine Einheit. Das was zusammengefasst wird, sind aber nicht wirkliche Gegenstände, sondern Vorstellungen und auch diese nicht, sofern sie (subjektiv) unsere Vorstellungen, Produkte unserer Geistesthätigkeit sind, sondern (objektiv) hinsichtlich Dessen, was in ihnen vorgestellt wird, das Gedachte.” Quoted in Riemann, Musikalische Syntaxis, 1–2. Drobisch would have been familiar to contemporary music theorists for his treatise on temperaments, Über musikalische Tonbestimmung und Temperatur (Leipzig: Weidmann, 1852). (19.) Lotze, Geschichte der Aesthetik in Deutschland (Munich: J. G. Cotta, 1868), 279–82. Lotze's other, less compelling, critique is that Helmholtz offered no way of expressing how consonance might be thought of in positive terms, since he described it merely as an effect of the (relative or absolute) absence of beats. But this dogmatically asserts a prede­

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“The Nature of Harmony”: A Translation and Commentary termined desideratum—that consonance be valued “positive”—without accepting the pos­ sibility that empiricist perspectives, by their very nature, tend to flip such valences. (20.) Rudolf Hermann Lotze, Medicinische Psychologie, oder Physiologie der Seele (Leipzig: Weidmann, 1852), 506–507. “[D]ie Vergleichung zweier sinnlicher Eindrücke dagegen, der Höhe verschiedener Töne etwa,…erfordert vielmehr die grösste mögliche Abhaltung alles andern Vorstellungsverlaufs, der die Reinheit der Empfindung trüben könnte…. Wer die Saiten eines Clavieres stimmt, hat bei der angestrengtesten Aufmerk­ samkeit auf seinen Gegenstand ein Minimum des Selbstbewusstseins;…wer mit Aufmerk­ samkeit dagegen einen zu wählenden Entschluss überlegt, soll wenigstens zugleich eine bestimmte Erinnerung seiner Persönlichkeit zu dieser Reflexion hinzubringen. Sowohl das selbstbewusstlose Versenken in eine einzige Vorstellung, als die unverbundene Flucht vieler sind Zustände, die nur, wo sie momentan sich einstellen, mit der gesunden Bestim­ mung des geistigen Lebens vereinbar sind; eine dauernde Zerstreuung sowohl, als eine Verengung des Gedankenlaufs werden wir dagegen später als Anfangspunkte der Seelen­ störungen kennen lernen.” (21.) Riemann's reference to Klirrtöne (which might be best translated as “rasping tones”) may be based on fairly dated material he could have unearthed in professional journals from the 1830s and 1840s. See, for example, August Seebeck, “Ueber Klirrtöne,” in An­ nalen der Physik und Chemie, n.s., 10 (1837), 539–547. On Riemann's “moonshine experi­ ments,” see Rehding, Hugo Riemann and the Birth of Modern Musical Thought, 15–35. (22.) Arnold Schoenberg, Harmonielehre, 3rd ed. (Vienna: Universal Edition, 1922), 15ff. To date, the connection remains underexplored. But see Steven Cahn, “Variations in Man­ ifold Time: Historical Consciousness in the Music and Writings of Arnold Schoenberg,” Ph.D. dissertation (Stony Brook University, 1996), esp. 433–462; and Albert Cramer, “Mu­ sic for the Future: Sounds of Early-Twentieth-Century Psychology and Language in Works of Schoenberg, Webern, and Berg, 1908 to the First World War,” Ph.D. dissertation (Uni­ versity of Pennsylvania, 1997). (23.) Jameson, A Singular Modernity, 46, Heidegger, Nietzsche, vol. 2 (Pfullingen: Neske, 1961), 151 (quoted in Jameson). (24.) Lecture held at Hamburg Conservatory, February 4, 1882. [Published in Waldersee's Sammlung musikalischer Vorträge 4 (Leipzig: Breitkopf und Härtel, 1882), 159–90.] (25.) [Contrary to some accounts, Messel does not derive from the German “cognate” messen (to measure), but rather appears to be a rough transliteration of the Arabic math­ ar, which connotes “comparison” or, in a mathematical context, “proportion.” Riemann's principal source here is Raphael Georg Kiesewetter's classic study, Die Musik der Araber, nach Originalquellen dargestellt (Leipzig: Breitkopf & Härtel, 1842). In his Studien zur Geschichte der Notenschrift, Riemann explained further: “By ‘messel,’ the [Arabic and Persian] authors mean the unit according to which the string lengths of the lower tone of an interval are measured; this unit is the string length of the interval's upper tone.” Geschichte der Notenschrift (Leipzig: Breitkopf & Härtel, 1878), 78. (“Unter dem Messel Page 34 of 37

“The Nature of Harmony”: A Translation and Commentary verstehen nun die Autoren die Masseinheit, nach welcher die Saitenlänge des tieferen Tones eines Intervalls gemessen wird; diese Masseinheit ist die Saitenlänge des höheren Intervalltones.”) In other words, if a string is divided in twelfths, the “messel” refers to the basic unit of one twelfth of a string, which when plucked would produce a pitch one fifth and three octaves above the pitch of the whole string. The remaining available string lengths created by a twelve-fold division would produce the following intervals above the pitch of the whole string: a fifth and two octaves (2/12 = 1/6), two octaves (3/12 = 1/4), a twelfth (4/12 = 1/3), a minor tenth (5/12), an octave (6/12 = 1/2), a (just) major sixth (7/12), a fifth (8/12 = 2/3), a fourth (9/12 = 3/4), a minor third (10/12 = 5/6), and a (small) minor second (11/12). Riemann was delighted that this procedure produced the same in­ tervals familiar from the undertone series. It is clear, however, that the respective rela­ tionships of the generative Klang (see below) and the “messel” unit of 1/12 with their in­ terval complexes are quite different: whereas the undertones have a downward direction­ ality, the intervals of the “messel-theory” are still conceived in reference to an underlying lowest tone, corresponding to the whole string length. But see n. 26.] (26.) That the conception of tone relationships from above downward was the only cur­ rent one in antiquity and also among the Arabs, and that it gradually gave way to the in­ verted conception which is now predominant, I have proven at length in my Studien zur Geschichte der Notenschrift (chapter 3: “Die Umbildung der Auffassung im Mollsinne in die Auffassung im Dursinne,” 72–95). (27.) [Riemann's phrase is “ebensogut das gleiche Verhältnis sind.” In other words, there is no way of clarifying explicitly that an interval in the upper parts is interpreted as equiv­ alent under octave inversion, despite the fact that it is not so interpreted if one of its notes appears in the bass. In J. C. Fillmore's 1886 translation of this essay, no longer widely available, this phrase is confusingly translated in precisely the opposite sense: “… there is no possible way of expressing the fact that the fifth C–G and the fourth G–C are not exactly the same interval.” Fillmore, trans., The Nature of Harmony (Philadelphia: Presser, 1886), 9. But Riemann is not criticizing the system's inability to say that intervals of fourth and fifth in the upper voices are different; rather, he is interested precisely in how it renders moot the need to say that fourth and fifth are the same. As his subsequent discussion shows, he values theoretical perspectives that favor greater generality, provid­ ed they do not privilege some types of equivalence (between different qualities of triad) at the expense of others (between different inversions of a single triad).] (28.) Traité d’harmonie reduite à ses principes naturels; a series of writings elaborating on this followed until 1760. (29.) [Following Alexander Rehding and Ian Bent among others, I have chosen not to translate the idiosyncratic German Klang since none of the standard English options quite do justice to its particular meaning in Riemann's writing. “Sonority” implies a far too lim­ ited notion of immediate sensation; “chord,” on the other hand, points too much toward a purely theoretical abstraction, and would obscure Riemann's separate use of the more

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“The Nature of Harmony”: A Translation and Commentary straightforward Akkord. In contrast, Klang sits somewhere right between the sensory and the abstract, capturing an idea for which there is no satisfactory English equivalent.] (30.) Trattato di musica secondo la vera scienza dell’ armonia (1754), and De’ principj dell’ armonia musicale contenuta nel diatonico genere (1767). (31.) [Fillmore creatively misconstrues Riemann's meaning here, rendering “…während es jedermann unbenommen ist, die natürliche Septime als in unser Tonsystem mit der Be­ deutung eines (dissonanten) Grundintervalls eingeführt anzusehen, nur sogut temperirt wie alle anderen Intervalle,” as: “…and nobody has proposed to use the natural seventh in our system as a fundamental (dissonant) interval, without being tempered like the rest.” Fillmore, trans., The Nature of Harmony, 18.] (32.) [Oettingen makes no assertion about the loudness of the first common (i.e., phonic) overtone.] (33.) Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik (1863, 4th ed. 1877). (34.) Harmoniesystem in dualer Entwickelung (1866). (35.) Dr. Adolf Thürlings, Die beiden Tongeschlechter und die neuere musikalische Theo­ rie (1877); Dr. Ottokar Hostinsky, Die Lehre von den musikalischen Klängen (1879); and my already mentioned Musikalische Logik and Musikalische Syntaxis, as well as the Sk­ izze einer neuen Methode der Harmonielehre (1880). (36.) [Perhaps drawing on Helmholtz's earlier use of the term Vorstellung to draw a dis­ tinction with Empfindung, Riemann here introduces the concept of Tonvorstellung that will be developed further in later essays, in which he elaborates a “Lehre von den Ton­ vorstellungen.” Following Robert Wason and Elizabeth West Marvin, the term “imagina­ tion of tone” has become the standard translation, though it is usually pointed out that the German Vorstellung can be rendered as either imagination or representation. In the present context, early on in Riemann's thinking, the concept of “representation” seems to capture better Riemann's ideas in their immediate intellectual context here. The use of “representation” as a rough equivalent for Vorstellung is conventional in translating rele­ vant (particularly neo-Kantian) philosophical discourses, in relation to which both Helmholtz and Riemann formulated their core aesthetic and epistemological beliefs. Ex­ tending this usage to the present text makes Riemann's early relationship to this intellec­ tual tradition more explicit for English readers. For a fuller consideration of the problems of translating this key term, see Wason and Marvin, “Riemann's ‘Ideen zu einer “Lehre von den Tonvorstellungen”’: An Annotated Translation,” Journal of Music Theory 36 (1992), 69–117, esp. 72–74.]

Benjamin Steege

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“The Nature of Harmony”: A Translation and Commentary Benjamin Steege is an assistant professor of the history and theory of music at Stony Brook University, with interests in the histories of music theory and science, and in early modernism. He is currently writing a book exploring the relationship of Her­ mann von Helmholtz to music theory and discourses of aurality.

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What is a Function?

What is a Function?   Brian Hyer The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0003

Abstract and Keywords This article discusses Hugo Riemann's notion of a tonal or harmonic function, which he first introduced into musical-theoretical discourse, Vereinfachte Harmonielehre in 1893. Riemann's notion of a tonal function refers to either the chords or properties of chords, classified as: tonic, dominant, and subdominant. In this article, the focus is on the equa­ tion of “function” with “meaning”, because it is in this connection that the term “func­ tion” occurs for the first time, and because the equation forms the core of the later refer­ ences to the idea. What follows is a critical appraisal rather than history of the concept. The aim in this article is to consider within broad but specific historical boundaries, the discursive potential of the term in Riemann's theoretical writings. Keywords: tonal function, harmonic function, Vereinfachte Harmonielehre, chords, tonic, dominant, subdominant, equation of function

Funktion/Bedeutung AS is well known, we owe the notion of a tonal or harmonic function to Hugo Riemann (1849–1919), who first introduced it into music-theoretical discourse in the subtitle of Vereinfachte Harmonielehre (1893), which reads (in the authorized translation) or the Theory of the Tonal Functions of Chords.1 Despite the attention the subtitle draws to a term Riemann knew would be unfamiliar to his readers, he almost never uses it in the treatise: it occurs just three times in the main text and never once in conjunction with a formal definition. The closest we come is at the very end of the preface, where Riemann introduces two principles intended to further “explain and develop the title of the book.”2 I. There are only two kinds of klangs: overklangs and underklangs; all dissonant chords are to be conceived, explained, and designated as modifications of overk­ langs and underklangs.

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What is a Function? II. There are only three kinds of tonal functions [Funktionen] in harmony (mean­ ings [Bedeutungen] within the key), namely that of the tonic, dominant, and sub­ dominant. In the change of these functions lies the essence of modulation.3 “Es giebt nur . . .”—in each case, the principle effects a reduction of harmonic phenomena to a mere few “kinds,” of chords in general to the “dual” major and minor triad (or Klang) in the first, and of chords belonging to a key to three determinate “tonal functions” in the second. It is this double reduction to which the main title of the book refers: “by keeping both these principles well in mind, we succeed in giving the theory of harmony a form thoroughly simple and easily comprehensible”—harmony simplified. If we run the subtitle together with the second principle, which is what Riemann himself does, it becomes clear that the notion of a tonal function refers either to chords (p. 93) or properties of chords (the wording is ambiguous), and that there are three of them, coinciding with the tonic, dominant, and subdominant. Riemann's parenthetical insertion in the second principle moreover indicates that whatever else a tonal function may be, it is synonymous with a “meaning within the key.” I want to focus on this equation of “function” with “meaning,” because it is in this connec­ tion that the term “function” occurs for the first time, and because the equation forms the core of all later references to the idea. If only by virtue of the numerous editions in which Riemann's writings appeared, the equation of function with meaning occurs repeatedly in his later theoretical writings, with one or the other term—“function” or “meaning”—oc­ curring between parentheses each time. Here, for instance, is the idea as it appears in the context of an entry on “Funktionsbezeichnung” (the arcane alphanumeric notation Riemann uses to designate the functional meanings of harmonies) in the seventh edition of the Musik-Lexikon (1909): The functional notation [Funktionsbezeichnung] of harmonies concerns the desig­ nation [Andeutung] of the differing meanings [Bedeutung] (functions [Funktion]) chords acquire in compositional logic according to their positions [relative to] the prevailing tonic.4 While Riemann uses the occasion to add to and subtract information from earlier formula­ tions of the idea (such as the one in Vereinfachte Harmonielehre), most of the language is familiar from before. Once again, tonal functions are something that chords possess, though he no longer singles out the dominant and subdominant, choosing instead to stress that chords in general assume an indefinite number of different (verschiedene) meanings in relation to the tonic. And though Riemann now identifies the notion of a tonal function as an element in some larger, more encompassing musical “logic,” he con­ tinues to equate it with “meaning,” this time reversing the terms so that the stress (and priority) falls on “meaning,” which “function” is then brought in to narrow and delimit: he seems to be describing the same basic idea as before in very nearly the same terms. Judging from its enormous historical success, readers appear to have had little trouble with the neologism; it must have seemed to them that “function” merely named a concept the contents of which were familiar musical entities.5 His use of “function” to pin down Page 2 of 51

What is a Function? and ground the word “meaning” would also seem to suggest that Riemann no longer re­ gards “function” as a problematic term; if anything, “meaning” appears to be the term that requires clarification. A tonal function, in contrast, seems to have been understood, since its inception, as a neutral descriptor for whatever it is that the tonic, dominant, and subdominant all are. At the same time, one senses a nagging discursive suspicion that the notion of a tonal function is nevertheless unclear and ambiguous; the very brevity of the definitions Riemann gives for a concept he regards as one of the two main principles of his harmonic theories suggests that the definitions themselves are in some sense inade­ quate or incomplete. “What is a tonal (or harmonic) function?” asks more than one recent commentator,6 and even though the question is meant to be rhetorical, for which (p. 94) answers are understood to be forthcoming, the mere fact that it is posed at all belies a concern that the concept is far from self-evident. Nor have such doubts always been mut­ ed. In an often read if not influential critique of tonal function—it would appear to have dissuaded no one from using the term—Carl Dahlhaus argues that because Riemann was unable to distinguish tonal functions from chords his theories could therefore reformulat­ ed without any reference to their cardinal concept.7 While the notion of tonal function moved to the center of his harmonic theories in Vereinfachte Harmonielehre, Riemann never gave an integral theoretical description of the idea. It has to be reconstituted, rather, from the traces it leaves in his theoretical writings, from comments scattered here and there, requiring us to piece the evidence together and fill in the gaps. In once again posing the question of what a tonal function is, I will thus follow Dahlhaus in being more kritisch than systematisch or begriffsgeschichtlich: what follows is a critical appraisal rather than a history of the concept. I will be reconsidering a number of by now wellknown texts, but the discussion will be schematic, intended to supplement rather than displace other accounts of the idea. The aim is less to recover the concept, to register its emergence and trace its mutations over historical time (though I will be doing some of both), as to consider, within broad but specific historical boundaries, the discursive poten­ tial of the term in Riemann's theoretical writings.

Vorstellung We can, as a point of departure, ground the notion of a tonal function in our understand­ ing of the tonic, dominant, and subdominant, which, considering Riemann's disinclination to elaborate the concept, is what his own readers must have done. In the absence of a clear definition, Riemann relies on instantiation: the tonic, dominant, and subdominant, that is, are all instances of what tonal functions are understood to be. In one of the later editions of the Handbuch der Harmonielehre (1887), he writes: Our theory of the tonal functions of harmony is nothing other than an extension of the Fétisian concept of tonality. The precisely determinable relations of all har­ monies to a tonic has found its most concise and intelligible expression in the des­ ignation of all chords as more or less clearly modified appearance-forms [Erschei­

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What is a Function? nungsform] of the three main pillars [Hauptsäulen] of the harmonic-logical con­ struction: the tonic itself and its two dominants.8

Ex. 3.1a. Imperfect Cadence, after Rameau, Généra­ tion harmonique (1737).

Ex. 3.1b. Perfect Cadence, after Rameau, Génération harmonique (1737).

It is unclear to Dahlhaus whether the metaphor of a “main pillar” refers to a tonal func­ tion or to a chord, though the question would seem to assume that there's a difference. I know of no occasion, however, on which Riemann explicitly applies the term “function” to anything other than a chord or its properties. As Alexander Rehding notes, in an entry on the “Dominante” in the eighth edition of the Musik-Lexikon (1916) Riemann identifies “the essential pillars of tonal harmony” specifically (p. 95) with “the tonic, dominant, and subdominant chords.”9 Assuming that “die drei Hauptsäulen” and “die eigentlichen Pfeil­ er der tonalen Harmonik” are identical, the three tonal functions would seem to coincide with the tonic, dominant, and subdominant as harmonies: the subtitle of Harmony Simpli­ fied, once again, reads “The Theory of the Tonal Functions of Chords.”

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What is a Function? Although Riemann regarded his harmonic theories as an elaboration of Fétis's concept of “tonalité,” the tonic, dominant, and subdominant themselves came down to him from Rameau. In Génération harmonique (1737), Rameau conceived relations between har­ monies in terms of harmonic successions, which he called “cadences.” In the imperfect cadence, the fundamental bass ascends a perfect fifth, as it does from the subdominant to the tonic in example 3.1a, which uses Riemann's notation to designate the subdominant and tonic. In the perfect cadence, the fundamental bass descends a perfect fifth, as it does from the dominant to the tonic in example 3.1b. In aggregate, these three harmonies —the tonic, dominant, and subdominant—comprise the “mode,” a system of relational harmonies with the tonic at its center. Within the mode, then, the subdominant is a domi­ nant below the tonic, an underdominant: “There are [in the mode] but three fundamen­ tals,” writes Rameau, “the tonic, its dominant, which is the perfect fifth above, and the subdominant, which is its perfect fifth below.”10 A factor crucial to the operation of this “sistême musical” was the addition of dissonances to the dominant and subdominant: Rameau added a major sixth (D in example 3.1a) to the subdominant, a minor seventh (F in example 3.1b) to the dominant, both of which resolve to the same octave-equivalent major or minor third above the tonic; the actual tone of resolution determines whether the mode is major (as in example 3.1) or minor. These added dissonances grace the tonic, dominant, and subdominant with distinct chordal identities, differentiating them from one another and endowing the subdominant and dominant with characteristic musical behav­ iors: (p. 96) the added dissonances increase the pressure on the dominant and subdomi­ nant to move to the tonic. Rameau often describes these harmonic relations in quasi-New­ tonian language: the tonic, that is, exerts a gravitational pull on the dominant and sub­ dominant, an invisible force that binds these three harmonies together.11 He was con­ cerned, then, both with the identities of harmonies (as tonics, dominants, or subdomi­ nants) and their succession, coordinating harmonic succession with conditions of conso­ nance and dissonance, tension and repose.

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What is a Function?

Ex. 3.2a. The three tonal functions in major, from Riemann, Vereinfachte Harmonielehre (1893), p. 8.

Ex. 3.2b. The three tonal functions in minor, from Riemann, Vereinfachte Harmonielehre (1893), p. 8.

As harmonies, the tonic, dominant, and subdominant each assert a fundamental that uni­ fies the intervals above and lends its pitch class to the entire perceptual structure. We do not hear C–E–G, that is, as a collection of three individual pitches, but as a single, homo­ geneous object, wholly imbued with C-ness. In this regard, the triad mirrors the acousti­ cal properties of its fundamental, which (as Rameau so often noted) was not simple, but Page 6 of 51

What is a Function? “harmonious,” composite, consisting of a fundamental together with its overtones. In the harmonic generation of the triad, Riemann recognized what he rechristened as the Klang, an abstraction that serves as the conceptual substrate for the idea of a tonal function. The Klang differs from the trias harmonica, however, in that it is neither major nor minor, but both at the same time; I will in fact translate it as “the major/minor triad.” It also differs from the trias harmonica in its “dual” generation of the major triad above the “prime” and minor triad below. In order to derive the dominant and subdominant from the tonic or principal Klang, Riemann extends perfect fifths in both directions from the tonic prime and places triads either above them (in major) or below them (in minor). His rationale for this is once again the dual harmonic series: the dominant is, for Riemann, the “nearest re­ lated” triad on the “overtone side,” while the subdominant is the “nearest related” triad on the “undertone side.”12 Example 3.2, from Vereinfachte Harmonielehre, compares the results in major and minor.13 Example 3.2a is for the most part straightforward: it flanks the tonic in C major with its subdominant below and the dominant above; the spatial or­ ganization of the diagram reflects the registral intuitions of below and above on which the entire arrangement relies. Riemann describes the same arrangement in example 3.2b, however, as the normal one for A minor, even though the prime of the minor tonic, ac­ cording to the tenets of dualism, is or should be E. In both cases, then—in minor as well as major—the subdominant resides below the tonic, the dominant above; the disposition of the tonic and its two dominants remains the same in minor. If the tonic, dominant, and subdominant are understood to be tonal functions, then the notion (p. 97) of function can­ not be dual: Riemann does not extend the dominant below the tonic in minor, or the sub­ dominant above, as the stringent requirements of dualism would demand. However en­ tangled the two become in the discursive complications of Riemann's theoretical writings, and despite the strain with which he attempts to bring them together, function and dual­ ism remain separate theories.14 What differentiates the dominant from the subdominant, and both of them from the tonic, are the “characteristic dissonances” added to the dominant and subdominant. The tonic, in contrast, occurs without a dissonance: Since the dominants are never perfectly consonant, in so far as they are always conceived [vorgestellt] and judged from the tonic (thus, so to speak always togeth­ er with the latter), it cannot be wondered at that they, far oftener than the tonic, appear with additional [tones] which make their meaning [Bedeutung] still clearer, and remove all danger of misunderstanding…. These characteristic dissonances are notes in each case borrowed from the other dominant.15 As derivations from the tonic, the dominant and subdominant represent departures from its consonance; the addition of actual dissonances to the dominant and subdominant merely underlines their conceptual dissonance. As in Rameau, the minor seventh added to the dominant represents the subdominant fundamental; the major sixth added to the subdominant likewise represents the dominant perfect fifth—the subdominant is, for that reason, conceived as being more remote that the dominant from the tonic. In each case, Riemann understands the characteristic dissonance as having been “borrowed” from the Page 7 of 51

What is a Function? other dominant; characteristic dissonances are thus extensions of what Riemann calls Klangvertretung, or “triad representation,” the idea that tones or intervals are to be un­ derstood in terms of one or more tonal functions. In the case of the dominant, the added seventh derives from the subdominant and thus disturbs or modifies the consonance of the dominant to which it has been added. Owing to the addition of these characteristic dissonances, the tonic, dominant, and subdominant assume qualitatively distinct chordal identities, allowing us, for instance, to recognize a particular chord as a dominant (as op­ posed to a subdominant or tonic) even in the absence of the tonic, but also allowing the dominant and subdominant to articulate, if only partially, the web of relations that (p. 98) bind them together. Although this is not an aspect of the dominant and subdominant that captures Riemann's imagination, it is crucial to our own understanding of what the domi­ nant and subdominant—not to mention tonal function in general—are: the tonic, domi­ nant, and subdominant, that is, are representational constructs. Both the dominant and subdominant represent the tonic, though not, of course, in the same way: the dominant represents the tonic lying a perfect fifth below, while the subdominant represents the ton­ ic lying a perfect fifth above.16 In both cases, the tonic is not to be understood as an inert triad, but rather as a referent within a larger complex of relational harmonies that also in­ cludes the dominant and subdominant, what Riemann (following common usage) calls the “Tonart.” In this context, the tonic triad, too, serves in a representational capacity, repre­ senting not itself, but rather its position as a locus or referent within the aggregate as a whole, something far more abstract and extensive than a given major or minor triad. It is understood in quasi-registral terms as occurring between the subdominant below and the dominant above, thus forming a center within the aggregate: “the combination of ele­ ments of two Klänge which stand to reach other in the relation of two dominants”—the in­ corporation within the dominant of a tone from the subdominant and vice versa—“points to a Klang, the tonic, lying between and mediating the two, thus making their relation in­ telligible.”17 The dominant and subdominant, either alone or together, “point” to a tonic that lies be­ tween them; the dominant and subdominant, that is, represent the tonic, standing for it in its absence and bringing it to mind—the tonic itself need not be physically present. The tonic, rather, exists as an ideal construct, an object in consciousness rather than a con­ crete major or minor triad. As representations, the dominant, and subdominant combine a material form, a chord, and a more extensive, immaterial content, an idea, for which it stands. A dominant is a mental image or schema—a concept—that involves, besides the actual chord, the relations it articulates with other harmonies in the Tonart. Riemann's most extended reflection on the conceptual nature of the dominant occurs amid a discus­ sion of “Dissonanz” near the end of “Die Natur der Harmonik” (1882): When I think of the C major triad in the sense [im Sinne] of C major, [I think of] the tonic itself, the center, a conclusive chord, the idea [Vorstellung] of which thus contains nothing contradicting its consonance, [and which] appears calm, pure, and simple; when in contrast I then think of the G major triad in the sense [im Sinne] of C major, I think of it as the chord a perfect fifth above the C major triad; that is, the C major triad belongs in the idea [Vorstellung] as the triad that deter­ Page 8 of 51

What is a Function? mines the meaning [Bedeutung] of the G major triad and from which the G major triad forms a departure—the center of the idea [Vorstellung], as it were, lies out­ side the G major triad.18 On this occasion, Riemann explicitly describes the dominant as a mental “idea”—a Vorstellung—that includes, in addition to the dominant triad, the tonic triad to which the dominant triad refers: the tonic triad belongs together with the dominant triad in the same idea, even though it is the dominant triad, a material form, about which Riemann “thinks”—the actual “center of the idea,” as Riemann puts it, lies in (p. 99) the mind, “out­ side” its material form. As his use of words such as “idea” and “meaning” indicate, the dominant, for Riemann, is a psychological construct, a specifically mental concept. He even writes about it in the first person, framing the entire discussion in terms of a cogni­ tive subject, a formal “I”—“denke ich….” Although these lines from “Die Natur der Harmonik” predate the first use of “function” in Vereinfachte Harmonielehre by over a decade, it seems clear, given his eventual equation of “function” with “meaning,” that Riemann has precisely the same idea in mind. He de­ scribes the C major triad as that which determines the “meaning” of the G major triad and renders it intelligible, using the very same word—Bedeutung—he would later use as a synonym for “function”: for Riemann, a chord's function and its meaning are one and the same. We can recognize most of the elements of his later definitions of tonal function in his earlier theoretical writings, in “Die Natur der Harmonik,” and even before. In an arti­ cle on “Funktion” in the fifth edition of the Musik-Lexikon (1900), Riemann himself points out that he had attempted to articulate the idea that all harmonies could be understood as modifications of tonic, dominant, and subdominant functions as long ago as “Musikalis­ che Logik” (1872), a goal he felt he only attained two decades later in Vereinfachte Har­ monielehre (1893).19 In addition to the term “tonic” and cognates for “dominant” (Über­ dominante) and “subdominant” (Unterdominante) one finds, in “Musikalische Logik,” a clear articulation of the idea that the dominant and subdominant assume their different meanings in relation to the tonic, and of the notion that these meanings are in some sense logical in nature. In describing “the mutual coherence of chords within a key,” he writes on the very first page of “the differing meanings [Bedeutungen] of these chords in rela­ tion to one another, their logical meaning [Bedeutung] in musical structure.”20 Here Rie­ mann enters into circulation language that turns up three decades later in his MusikLexikon entry on “Funktionsbezeichnung” (and on numerous other occasions besides), which he once again defines as “the designation of the differing meanings (functions) chords acquire in compositional logic according to their positions [relative to] the prevail­ ing tonic”21—little more than a paraphrase. His insistence on the notion that harmonies have “logical meanings” never wavered in the course of his career. It is his use of “mean­ ing,” as much or more than “function,” that we need to track: where “meaning” goes, “function” follows.

Page 9 of 51

What is a Function?

Ex. 3.3a. “Die große Cadenz,” after Riemann, “Musikalische Logik” (1872).

Ex. 3.3b. “Die große Cadenz,” after Riemann, “Musikalische Logik” (1872), abbreviated, four-chord version with medial tonic elided.

In “Musikalische Logik,” the logic Riemann has in mind is essentially dialectical. He bases his consideration of “harmonic logic” on the three dialectical moments of what he terms the große Cadenz, which he represents in the form of C–F–C–G–C, a conflation of the pla­ gal cadence C–F–C and the authentic cadence C–G–C in which the last term of the former coincides with the first term of the latter.22 Example 3.3a is an approximate transcription the große Cadenz as Riemann describes it. Riemann uses letter names and Roman numer­ als to designate harmonies as scale degrees. He does not use musical notation and gives us no information at all concerning the upper voices; I am giving him the benefit of the doubt on the soprano. Example 3.3a also presents the cadence in common time. For Rie­ mann, the second C in both the plagal and authentic cadence “delimits” the progression and thus determines the meter. In the combination of C–F | C and C–G | C, the second “duration” follows and (p. 100) completes the first, determining the meter of the entire progression to be compound duple.23 In the dialectical movement of the grosse Cadenz, the initial tonic forms a thetic moment that undergoes modification or alienation in an an­ tithetic moment before finding affirmation in a concluding, synthetic moment. Example 3.3 brackets these three moments and labels them “T” for These, “A” for Antithese, or “S” for Synthese, terms Riemann borrowed from Fichte.24 As befitting the antithetic moment, the middle term in this dialectic—the second C—appears in mediated form, in the six-four position, with G in the bass. The antithetic motion from F to C and synthetic motion from G to C form what Riemann calls “transitional moments” in the cadential dialectic: in the große Cadenz, “thesis is the initial tonic, antithesis the lower dominant with the tonic sixfour, synthesis the upper dominant with the concluding ground-position tonic.”25 In this scheme, the dominant and subdominant are once again relational in nature, though here relations are modeled in terms of harmonic successions: Riemann describes the antithetic moment, that is, as the lower dominant “with”—in relation to—the medial tonic, and the Page 10 of 51

What is a Function? synthetic moment as the upper dominant together “with” the terminal tonic. As an upper dominant in relation to F, however, the medial C negates the tonic meaning of the initial C; it therefore adds “nothing new” to the cadence and can be eliminated, leaving the fourchord progression C–F–G–C, here transcribed as example 3.3b. “On the basis of this new four-chord model,” notes Rehding, Riemann then “located the dialectical moments in the chords themselves.”26 Riemann writes (in boldface) that “thetic is the tonic, antithetic the underdominant, synthetic the overdominant.”27 In both the antithetic and synthetic mo­ ments, in other words, Riemann elides the referential tonic and puts the upper or lower dominant in its place: in the progression C–F–G–C, F thus (p. 101) forms the “real antithe­ sis” to C, “and in what follows we will always understand as antithesis the self-sufficient appearance of the underdominant.”28 (G, then, not C, forms the “synthesis” of the caden­ tial dialectic. Although there was precedent for understanding the dominant in these terms, this is the price Riemann pays for identifying each moment in the dialectic with one of the three Hauptklänge; the tonic no longer completes the cadence but represents an unmediated return, rather, to its point of departure, a new [or another] “thesis.”) As a consequence, form diverges from content: the subdominant's “appearance” diverges from its “meaning,” the relations that hold it in place and refer it to the tonic. In Riemann, “dominant” thus names two distinct musical/conceptual phenomena—an appearance and a meaning, a chord and a concept—with two different ontologies. It is this ontological dif­ ference that underlies Dahlhaus's complaint that there's no clear distinction between functions and chords in Riemann. He's right, there's not, but that doesn’t mean that a function and a chord are the same: unlike a chord, a function has no material, extracon­ ceptual existence. Because of this, appearance and meaning assume separate ontologies in which meaning transcends appearance: a dominant's meaning comes to seem more extensive than its ap­ pearance, its material form. In pulling apart from its appearance, however, the dominant's meaning undergoes “hypostasis,” the conversion of an abstraction into a seemingly real thing. A relation between harmonies thus turns into an ostensibly concrete tonal function, with attributes all its own. As Charles S. Peirce—one of Riemann's contem­ poraries—explains it, a logical proposition undergoes a formal operation in which the predicate becomes a substantive, the subject of another proposition: “thus, we transform the proposition, ‘honey is sweet,’ into ‘honey possesses sweetness.’ ”29 Hence the proposi­ tion “the G major triad is the dominant in C major” becomes “the G major triad possesses dominant-ness,” a “quality existing absolutely,”30 allowing us to think of dominant-ness separately, apart from the G major triad. In the historical reception of Riemann's harmon­ ic theories, this transmutation of a meaning into an immaterial substance has often been invested with enormous metaphysical significance. A tonal function, that is, is understood as a quasi-platonic essence that underlies a chord and ensures its intelligibility. A chord, in contrast—a material form—becomes the mere expression (a medium or conveyer) of a tonal function, which glows from within: the dominant function, as it were, imbues a chord with dominant-ness. In an important meditation on tonal function, Daniel Harrison pushes the metaphysical import of the idea to an extreme, likening tonal function in gen­ eral to the discovery “that one can play the car radio without having the engine Page 11 of 51

What is a Function? running.”31 It would appear, in other words, to be an invisible source of power, without which the music, one assumes, would cease to sound, or to be intelligible. In the case of the dominant and subdominant, a function derives its power, by proxy, from the tonic, which once again lies at the center of the idea and yields some of its presence to its upper and lower dominants. The notion of a tonal function thus assumes a position well within a metaphysics of presence that casts a long shadow over the main discursive traditions of Western musical modernism. As Peter Rummenhöller has noted, for all the pendulum swings (p. 102) of his theoretical writings, Riemann's theories were stamped in “a her­ itage of German idealism Riemann could never have denied.”32 The reification of tonal function gives rise, in Riemann's theories, to enormous abstrac­ tion. As an aggregate, the tonic, dominant, and subdominant—whether we consider them to be material appearance-forms or meanings—are self-contained. The dominant and sub­ dominant refer to the tonic, but also (via their characteristic dissonances) to one another, closing the circuit: “This method of connecting one dominant with a note of the dominant lying on the other side of the tonic circumscribes the key in the same manner as does the succession of the two dominants.”33 The “Tonart” is, as Riemann puts it, “geschlossen,”34 and it is within this conceptual enclosure that the global reduction of harmonic phenome­ na to three tonal functions occurs. Between them, the tonic, dominant, and subdominant functions partition this enclosure without remainder: without exception, all harmonies must have either tonic, dominant, or subdominant meanings. In Vereinfachte Har­ monielehre, Riemann believed that he had in fact “succeeded in characterizing every chord…as a more or less equivalent representative [Vertreter] of some tonic, dominant, or subdominant”—and here he should have added—“function.”35 The tonic, dominant, and subdominant functions thus enter into a continuum of meaning. In combination with the principle of “apparent consonance” (Scheinkonsonanz), Riemann uses “triad representation” (Klangvertretung) as a means of extending tonic, dominant, or subdominant meanings to other harmonies within the Tonart. Riemann thus defines three Tonverwandtschaften or “tone relations” between triads: the Parallele, the Leittonwechsel, and the Variante.36 The parallel and leittonwechsel occur together for the first time in Vereinfachte Harmonielehre: although the variant is implicit in the functional notation of Vereinfachte Harmonielehre, Riemann did not give it its own name until 1914.37 These three tone-relations are operations that transform one triad into another on the basis of common intervals between them. According to Riemann, triads are “related”—verwandt— when they have a tone or interval in common: Triads, whether major or minor, are related through their constitutive tones, which are neither identical nor derivable from perfect-fifth and major-third [bass] progressions; C major, for instance, is related with A minor [°e] through a com­ mon C and E.38 Although here Riemann writes about related triads in terms of tones (pitch classes) rather than intervals, he identifies those tones in terms of the intervals each one forms with the

Page 12 of 51

What is a Function? prime in the major/minor triad: E is no mere tone, or frequency, but a major third (in rela­ tion to C); G is likewise a perfect fifth. The perfect fifth and major third are, together with the prime, the constitutive ele­ ments of the triad, whether major or minor. The differing effects of the prime, per­ fect fifth, and major third in the major/minor triad are peculiar to each one and not further definable.39 Assuming we can in fact regard the “constitutive elements” of the major/minor triad as in­ tervals as well as tones, Riemann deploys the prime, major third, and perfect (p. 103) fifth (though notice that Riemann gives them in the order in which the prime generates them) as combinatorial intervals, rearranging them within the perfect fifth to create the major or minor triad of the opposite modal gender; in this scheme, the minor third arises as the complement of the major third within the perfect fifth.

Ex. 3.4a. Parallele (Tonverwandtschaft).

Ex. 3.4b. Leittonwechsel (Tonverwandtschaft).

Page 13 of 51

What is a Function?

Ex. 3.4c. Variante (Tonverwandtschaft).

The parallel thus relates a C major triad with an A minor triad, as in Example 3.4a, and vice versa (Riemann often refers to “pairs” of parallel triads); it preserves the common major third, in this case C–E, stemmed together on the upper staff, complementing it with a minor third—shown here in the form of the bass motion from C to A on the lower staff— in order to fill out the perfect fifth. The leittonwechsel relates a C major triad to an E mi­ nor triad and vice versa, as in example 3.4b; it preserves a minor third, here E–G, comple­ menting it with the major third, shown here as the bass motion from C to E. The variant, finally, relates a C major to a C minor triad, as in example 3.4c, and vice versa; it re­ arranges or inverts the major and minor third within a common perfect fifth, here C–G. All three tone-relations are their own inverses: each one self-undoes when applied a sec­ ond time. The variant of C major, for instance, is thus C minor, whose own variant, howev­ er, is once again C major—this accounts for the “vice versa” effect. Riemann associates the parallel and leittonwechsel in particular with characteris­ tic melodic progressions. The parallel thus replaces the perfect fifth above the major prime with an octave of the minor prime, as shown in example 3.4a; Riemann often uses these horizontal “guides” to draw attention to certain features of the voice-leading. As shown in example 3.4b, the leittonwechsel likewise substitutes an octave of the major prime with its own “leading tone,” the perfect fifth above the minor prime. For Riemann, these melodic progressions are more important in determining the tone-relation than bass motion. Indeed, because Riemann generates the minor triad from the prime down, the actual harmonic progression in example 3.4b, for instance, is from a major triad above C to a minor triad below B, where the tonic prime is exchanged for its leading tone; this is the exchange from which the leittonwechsel gets its name. The bass motion from C to E, in contrast, has no role in determining the harmonic progression. It is for this rea­ son that the common intervals preserved between major and minor triads supersede bass motion in determining tone-relations. As it often does, the variant stands somewhat apart from the parallel and leittonwechsel in this respect: although we often associate the vari­ ant with the melodic semitone (the one between E and E♭ in example 3.4c) resulting from (p. 104)

its inversion of the major and minor third within the perfect fifth, in practice (meaning the historical repertoire Riemann had in mind) variant-related triads rarely progress di­ Page 14 of 51

What is a Function? rectly to one another; more often, the major will substitute for—appear in place of—the minor and vice versa, rendering considerations of voice-leading between them superflu­ ous. In each case, these tone-relations conform to the principle of triad representation, in which “a tone or interval acquires its particular meaning when grasped in terms of this or that major or minor triad.”40 In tandem with triad representation, the principle of appar­ ent consonance determines the interactions between these tone-relations and the tonic, dominant, and subdominant triads. When applied to the three Hauptklänge, these tone-re­ lations transform them into Nebenklänge, understood as “simultaneous representatives of two of the three primary harmonies, where one triad is intelligible as the principal con­ tent (consonance) and the other as outside interference (dissonance).”41 As the tonic par­ allel in C major, an A minor triad can be understood at once as representing both the ton­ ic C major triad and the subdominant F major triad. It derives its meaning as a tonic through the C–E major third it has in common with the C major tonic, while the minor third A–C represents the F major subdominant triad. In this case, the C major tonic, through triad representation, forms the “principal content” of the A minor tonic parallel, while the F major subdominant forms “outside interference,” a dissonant, “foreign addi­ tion” that disturbs the otherwise “apparent consonance” of the A minor tonic parallel. A triad that “appears” to be consonant, then, is in fact a “dissonant” representation of more than one primary triad. Tones—and the intervals between them—“are only genuinely con­ sonant when understood in some actual context as components of one and the same ma­ jor or minor triad.”42 For Riemann, consonance and dissonance are conceptual rather than acoustical or perceptual phenomena; dissonance results from the conflict between different functional (p. 105) meanings, while consonance results from the absence of dis­ sonance—consonance in Riemann is negatively determined. It is through the interaction of triad representation and apparent consonance that the tonic, dominant, and subdominant, as functions, enter into a continuum. As crucial as I believe this continuum is to our understanding of tonal function, it is not, however, an idea that Riemann himself articulates. Although it is implicit in the reduction of all har­ monies to three functional meanings, the closest Riemann comes to actually making the idea explicit is a brief but remarkable passage from the article on “Dissonance” in the Musik-Lexikon. After discussing harmonic (characteristic) dissonances, he turns, some­ what surprisingly, to melodic dissonances, which, according to Riemann, often result in apparent consonances; ultimately, even melodic dissonances—non harmonic tones—are to be given harmonic interpretations. Passing tones, in particular, often result in Paralleland Leittonwechselklänge, which Riemann considers in turn: (a) The substitution of the major sixth for the perfect fifth above in the major triad and below in the minor triad results in the parallel of a given triad. In C major thence arises an apparent A minor triad (Tp, the parallel triad of the tonic, or tonic parallel), D minor triad (Sp), and E minor triad (Dp). (b) The substitution of the leading tone for the prime (from below [〈] in major, from above [〉] in minor) likewise results…in the leading-tone change (in C ma­ Page 15 of 51

What is a Function? jor: = E minor, = A minor, major, = B major [!]).43

=B minor [!]; in A minor:

= F major,

=C

Under (b), Riemann systematically applies the leittonwechsel to the three primary triads in C major and A minor. Under (a), he does the same with the parallel, but in major only. (The way in which both the form and the terms of the demonstration change between [a] and [b] is fairly typical of Riemann's discursive style.) Example 3.5 completes the demon­ stration, collates all the data, and transcribes it into musical notation; in this case, ties are used to indicate common intervals. The result, in both cases, is a continuum of func­ tional meaning, in which the three primary meanings overlap. In example 3.5a, the tonic parallel A minor coincides with the subdominant leittonwechsel on one side of the tonic, while the tonic leittonwechsel E minor coincides with the dominant parallel on the other. In example 3.5b, the tonic leittonwechsel F major coincides with the subdominant parallel on one side of the tonic, while on the other the tonic parallel E♭ major coincides with the dominant leittonwechsel. In both major and minor, the subdominant, tonic, and dominant occur in the same order and together exhaustively partition the continuum: where the tonic leaves off, the dominant (or subdominant) begins, leaving no gaps between them.

Ex. 3.5a. Tonal continuum in major, after Riemann, “Dissonanz” (1916), p. 250.

Ex. 3.5b. Tonal continuum in major, after Riemann, “Dissonanz” (1916), p. 250.

The functional continuum, with its three primary meanings, could be likened in this re­ spect to another familiar continuum, the color spectrum, with its three primary colors. The analogy, however, is imperfect. In the color spectrum, there is no central term, no ze­ ro-th color, and each color shades into the next: the color spectrum is in fact often imag­ ined as a circle or wheel in which red transitions to yellow, yellow transitions to blue, and blue transitions back to red, all without a (p. 106) break. In the functional continuum, in contrast, the location of the tonic at the center of the continuum is crucial to its opera­ tion: our intuition of a dominant above and a subdominant below relies on a locus tonic. Nor does the subdominant transition to the dominant, or vice versa: whereas the color spectrum is circular, the functional continuum is linear, and open at both ends—the two extremes do not join. No matter how the parallel, leittonwechsel, and variant are applied to dominant or subdominant-related harmonies, one cannot, without recourse to enhar­ monic equivalence (a crucial qualification), transform a subdominant into a dominant and Page 16 of 51

What is a Function? come out on the other side, the musical equivalent of turning base metal into gold.44 Within the functional continuum, however, transitions between functional meanings are smooth and continuous, precisely because those functional meanings overlap: the fact that the tonic parallel in major, for instance, coincides with the subdominant leittonwech­ sel ensures the smooth transition from the tonic to the subdominant. In effect, the Neben­ klänge intervene and mediate between the Hauptklänge, allowing for a seamless transi­ tion in functional meaning from the tonic to the dominant on either side. To be sure, the functional continuum does not appear to be continuous: the various harmonies in example 3.5, that is, would appear to be discrete, each one lying a certain distance from the next. Example 3.5, however, is a representation of the functional continuum and not the contin­ uum an sich; the functional continuum is, like the notion of function itself, an ideal con­ struct, a continuum of abstract functional meanings rather than concrete, material har­ monies. A comparison of example 3.5a with example 3.5b indicates that the tonic parallel occurs below the tonic in major but above the tonic in minor; the tonic leittonwechsel likewise occurs above the tonic in major but below the tonic in minor. Given our definitions of the parallel and leittonwechsel, the fact that parallel and leittonwechsel-related triads occur on opposite sides of the subdominant, tonic, and dominant in major and minor is a direct consequence of the inversion of the major and minor third within the tonic perfect fifth. Although major and minor (p. 107) appear in this regard to be mirror inversions of one an­ other, the arrangement is reciprocal rather than dual. The inversion of functional mean­ ings around the tonic, dominant, and subdominant does not alter the fact that the sub­ dominant occurs below the tonic and the dominant above in minor as well as major: there is nothing dual about the three functions themselves. Again, tonal function and harmonic dualism are separate theories: although Riemann attempted to “dualize” his harmonic theories, he was never able to complete the program, due to the numerous conceptual difficulties to which dualism gives rise, but also due to a pedagogical imperative to re­ main in some contact with received discursive conventions. While the exclamation points that punctuate the dominant leittonwechsel in example 3.5a and the subdominant leitton­ wechsel in example 3.5b express Riemann's surprise that the leittonwechsel, in both cas­ es, produces nondiatonic major/minor triads (a matter that will not further detain us), the real scandal here is the fact that he designates minor triads, on each and every occasion, from the prime up. He designates the tonic leittonwechsel in C major, that is, as “E moll” rather than “°b,” the subdominant leittonwechsel as “A moll” rather than “°e,” the domi­ nant leittonwechsel as “B moll” rather than “°f ♯.” Given the importance he ascribes to du­ alism (in the preface to Vereinfachte Harmonielehre) as one of the two principles underly­ ing his harmonic theories, this is more than a mere concession to standard musical nomenclature. It amounts, rather, to parapraxis, giving the lie to a dualism that renders his functional notation nearly illegible and his harmonic theories incoherent.

Page 17 of 51

What is a Function?

Ex. 3.6. Two Terzwechsel.

A distinction must also be made between tonal function and harmonic succession. The distinction is not a sharp one: in his earlier theories Riemann develops the two concerns —for harmonic meaning and harmonic succession—together. As we have seen, in “Musikalische Logik” Riemann worked out both ideas in terms of a dialectic of cadential moments in which an initial tonic gives rise in succession to the subdominant and domi­ nant before returning to itself at the end; the tonic, subdominant, and dominant derive their meanings in part from the order in which each one occurs in the grosse Cadenz. Over the next two decades, however, the two concerns diverge from one another and un­ dergo separate development, reemerging, after Vereinfachte Harmonielehre, as two sepa­ rate theories: that of the three tonale Funktionen on the one hand and what Riemann called the Harmonieschritte on the other. First outlined in the Skizze einer neuen Meth­ ode der Harmonielehre (1880), the Harmonieschritte appear in mature form in the Hand­ buch der Harmonielehre (1887) (a revision of the Skizze einer neuen Methode der Har­ monielehre) and Vereinfachte Harmonielehre (1893).45 In Vereinfachte Harmonielehre, the Harmonieschritte comprise eleven Schritte (steps) and their dual Wechsel (changes). Riemann intended the Harmonieschritte as an alternative to the fundamental-bass pro­ gressions of traditional scale-degree theories, using them to measure intervals not be­ tween fundamentals, but between dual primes. Hence the progression from a tonic to its leittonwechsel in major was not one of a major third (as it is when measured in scale de­ grees), but rather that of minor second between the prime of c+ (to (p. 108) use Riemann's Klangschlüssel) and the prime of °b. As such, the Harmonieschritte thus com­ pete with the Tonverwandtschaften, which constitute not a mere alternative conception or notation of fundamental-bass progressions, but a radically different means of chord con­ nection, one based on invariant or preserved dyadic intervals rather than directed inter­ vals between groundtones. For our purposes, however, the crucial point is that even though the Harmonieschritte operate on functionally determined harmonies, the Har­ monieschritte are not themselves functional. The Harmonieschritte are neutral with re­ spect to harmonic function: a Terzwechsel (a Terzschritt combined with a Seitenwechsel, the latter of which inverts a triad around its prime) is a Terzwechsel whether it connects a C major tonic with an A minor tonic parallel (c+ and °e in Klangschlüssel), as in example 3.6a, or an F major subdominant (f+) with a D minor subdominant parallel (°a), as in ex­ ample 3.6b; a Terzschritt can be realized by any number of different functional harmonies. In neither Riemann's Funktionstheorie nor his Harmonieschritte, however, is there an ex­ press concern for what we often think of as the dynamic properties of harmonic succes­ sion. I raise the issue because it answers to one of the two main colloquial uses for the Page 18 of 51

What is a Function? term “function” in current music-theoretical discourse, the notion that function has to do with harmonic behavior, the way in which chords “move.” In this sense, the locution “functions as” is understood as the equivalent of “acts like.” A chord, that is, functions as a dominant when or because it resolves (for instance) to the tonic. While it is sometimes assumed that this use of the term derives from Riemann, it is not, again, an aspect of har­ monic succession to which he devotes much attention: he does discuss chord resolution, but does so more in terms of correct or proper voice-leading than of the properties of this or that tonal function. In his Funktionstheorie, rather, the three tonal functions are dispo­ sitional in nature. As tonal functions, the tonic, dominant, and subdominant are inert, if not immobile: neither the dominant nor the subdominant conveys a strong urge to resolve or push to the tonic. Dissonances are added to the dominant and subdominant less to compel their succession to the tonic than to give them distinct chordal identities and thus clarify their functional meanings. A chord's function, rather, is a function (in the more general sense) of its pitch-class content and its relation to the tonic. The subdominant may in fact succeed to the dominant, and the dominant may in fact succeed to (p. 109) the tonic, but neither is required to do so, nor does succession factor into the determination of their functional meanings. Riemann conceives of a Schritt or Wechsel, in contrast, as an operation one performs on a triad, like a move in a chess game, rather than a move the triad itself requires. Although Riemann intended the Harmonieschritte as a means of con­ straining harmonic succession, the Harmonieschritte are more taxonomic in nature: be­ cause the Schritte, in combination with the Wechsel, allow a given prime to move to any other pitch class within the octave, there is virtually no succession between a pair of ma­ jor or minor triads the Harmonieschritte rule out. The real motivation behind the Har­ monieschritte would appear to have been one of classification, of the need to bring each and every harmonic succession under a determinant concept, under one of the defined Schritte or Wechsel. Even though individual Schritte and Wechsel are perforce ordered in time, the Harmonieschritte in general are all but ateleological: in C major, an F major tri­ ad could just as easily proceed via a Leittonschritt to an E major or via a Gegenterzwech­ sel to a D♭ major triad as it could via a Ganztonschritt to a G major triad—there is no mechanism within the system that would lead us to prefer one of these successions (or destinations) over the others. Indeed, the fact that the Wechsel are all their own inverses suggests that within a given Wechsel the balance might almost tip at any moment in ei­ ther direction: that a Gegenterzwechsel from an F major triad to a D♭ major triad would be more inclined to return to the F major triad and thus neutralize all sense of forward progress than to move on to some other major or minor triad. What counts, rather, is mere consecution: harmonies do not so much go to one another as exist side by side. As far as tonal function goes, then, it is not what a chord does that matters, but what it is: a functional designation names a chord's being. For Riemann, there can be no chord with­ out being, and being derives from a chord's participation in the Tonalität, from the bor­ rowed glory of the tonic—this is what the reduction of all harmonies to three tonal func­ tions sums to. This answers to the other main colloquial use of the term, where “functional” (now as an adjective) is understood to be synonymous with “tonal.” In the strong form of this usage, harmonies are functional whenever we can classify them Page 19 of 51

What is a Function? specifically as tonics, dominants, and subdominants or their transformational modifica­ tions; although Riemann never uses the term in its adjectival form, this is the usage that comes closest, I believe, to his own way of thinking. In the weak form, in contrast, har­ monies are functional whenever we can assign them to a particular key, whether using Riemann's discursive labels or, more commonly, the Roman numerals of scale-degree the­ ories. At least in Great Britain and North America, this is the usage most prevalent at the present moment, and has been for some time.

Ex. 3.7. “Skizze der Analyse” of the third movement from the Piano Sonata in A Major op.  101 (1816), from Riemann, L. van Beethoven's sämtliche KlavierSolosonaten: Aesthetische und formal-technische Analyse mit historischen Notizen (1919), pp. 281– 282.

For Riemann, functional notation determines the truth of a chord, and the truth of a chord is its function. One gets the definite impression from reading Riemann that he re­ gards the invention of functional notation as his most important contribution to the theo­ ry of harmony, an innovation outranking (p. 110) (p. 111) the concept of function itself, for which he was more than willing to give credit to others (as he does, for instance, in n. 9 above): in later editions of the Musik-Lexikon, Riemann includes detailed entries for Funk­ tionsbezeichnung but none for Funktion.46 As far as Riemann is concerned, functional no­ tation is a metalanguage more precise, consistent, and wissenschaftlich than actual musi­ cal notation, for which it substitutes. As evidence, example 3.7 reproduces Riemann's an­ notated Skizze of the slow prolegomenon to the fugal finale of the Piano Sonata in A Ma­ jor, op. 101 (1816), chosen almost at random from the third volume of his “aesthetic and formal-technical” commentaries on Beethovens sämtliche Klavier-Solosonaten (1919).47 I am less interested in the present context in the analytical content of the diagram or its adequacy to the music (though it sounds odd to be saying so) than I am in its form. In this diagram, Riemann includes functional notation below a thematic outline of the move­ ment, given for the most part in the treble. He uses equals signs to indicate modulations. At the beginning of the Nachsatz, “= Dp” on the downbeat of measure 5 thus indicates that we are now to hear the half-cadential major-turned-minor dominant in A minor at the end of measure 4 as the parallel minor (in Riemann's parlance) of the dominant in C ma­ Page 20 of 51

What is a Function? jor, on which the music will full cadence in measure 8; he does not otherwise indicate the new tonic, which, for Riemann, is implicit in the functional designation of the chord as “Dp”—there is only one Tonart in which an E major triad can function as the variant of the dominant parallel. He uses other annotations (dotted bar lines, numbers in parentheses, slurs) to indicate the movement's metrical organization and delineate its “Phrasierung.” What I most want to draw attention to is the way in which the functional notation re­ places the bass and inner voices; the functional notation itself accompanies the soprano, supplying the music's underlying sense far more clearly (for Riemann) than the actual musical notation does. Riemann, that is, turns his functional notation against the music, using it to abstract its inner harmonic essence from the outer figuration of the musical surface. His harmonic theories thus operate as an extension of a post-Romantic aesthetic that isolates harmony as the origin and essence of the purely musical in music, that with­ out which music ceases to be musical. Hence the fixation in Riemann with the naming and labeling of harmonies, one that continues down to our own time: institutionalized in academic music curricula, the mania for naming and labeling chords, often pursued as an end in itself, rages on the undergraduate classroom (where it shows no signs of abating) and in a good deal of professional writing on Western music. In Riemann, Funktionsbeze­ ichnung becomes an alternate form of musical notation, preferable to the actual score in its logic and rigor; the same could be said for our use of Roman numerals and scale-de­ gree theories as a putatively neutral descriptive language, a musicological lingua franca. For Riemann, functional notation indicates the true content of the music, the succession of meanings that underlie and accompany the music and guarantee its intelligibility.

(p. 112)

Begriff/Objekt

In asking what a tonal or harmonic function is, a number of Riemann's recent commenta­ tors echo, without realizing it, a question posed in his own time. “Was ist eine Funktion?” asked Gottlob Frege (1848–1925) in the title of a little-noticed essay published in 1904.48 Frege, of course, asked the question not about the tonal function, but about the function in mathematics. “Although it has been in continual use for a long time,” notes Frege, “it is even now not beyond all doubt what the word ‘function’ stands for.”49 As far as Frege was concerned, the mathematical function was generally misunderstood, a situation that threatened to hinder the further development of mathematical logic as a discipline. An answer to his question was needed in order to provide arithmetic, in particular, with a se­ cure, scientific foundation, but also to move the discipline as a whole onto new philosoph­ ical grounds, as mapped out in his own writings. It could well be from mathematics (if probably not from Frege) that Riemann appropriat­ ed the term “function” as a metaphor for the differing meanings of the tonic, dominant, and subdominant.50 The historical evidence, however, is largely circumstantial. Riemann never wrote about the term's derivation or how he came to use it in this connection. It simply appeared, unannounced, in the subtitle of Vereinfachte Harmonielehre (1893), and though he would later recognize Vereinfachte Harmonielehre as the summation of his har­ Page 21 of 51

What is a Function? monic theories, he never reflected on the term or singled it out for further consideration. In its cumulative mass, however, the evidence is compelling, if not decisive. The crucial evidence is of course discursive, the mere fact that Riemann uses the term at all: as far as I can determine, “function” does not occur as a technical term in the harmonic theories of any of Riemann's immediate predecessors. His use of the term is consistent, moreover, with the general mathematicization of his theoretical writing, his predilection for express­ ing pitch relations, for instance, in terms of arithmetical fractions (of perfect fifths and major thirds),51 and his use of alphanumeric Funktionsbezeichnungen to designate the functional meanings of harmonies. Riemann's invention of functional notation parallels Frege's earlier invention of a Begriffschrift, a notation or “concept writing” meant to iso­ late the formal constituents and operations of logic from the ambiguities of the discursive language in which one would otherwise be forced to present them. For confirmation, we can look to the language with which Riemann surrounds and contextualizes the term. As we have seen, function occurs in close conjunction with other logical terms, with meaning in particular, as it does, famously, in Frege; both function and meaning serve as formal categories in Riemann's harmonic theories. “The functional notation of harmonies,” once more, “concerns the indication of the differing meanings (functions) chords acquire in compositional logic according to their positions [relative to] the prevailing tonic.”52 Above all, it is Riemann's insistence that his harmonic theories constitute a specif­ ically musical logic that allies the notion of a tonal function with the function in mathe­ matics. It is no exaggeration to say that the notion of a musical logic, which Riemann in­ herited from romantic musical aesthetics, was the central concern of all his musicological activities, where it operates as a regulative ideal; though the force and meaning Riemann attributed to musical logic changed over time, it remained a constant focus of attention from “Musikalische Logik” (1872) until very near the end of his career, when it gave ground to Tonvorstellung, the notion of a “tonal idea” or “image.”53 In the context of his harmonic theories, the concept of musical logic underwent a radical change in orientation from a quasi-Hegelian logic of cadential progressions in the 1870s to a pseudo-mathemat­ ical logic of tone-relations after about 1891. While I would be wary of ascribing this reori­ entation to Frege's influence, it is significant that Riemann abandons a dialectical logic not for the neo-Aristotelian, subject/predicate logic of mainstream academic logic, but for a reasonably close approximation of the far more radical function/argument logic we now attribute specifically to Frege.54 In fact, “function” first appears (in 1893) as a technical term in Riemann's music-theoretical writings almost immediately after the publication of (p. 113)

three articles that would later secure Frege's posthumous reputation as a philosopher and in which he addressed precisely those categories—function and meaning—that Rie­ mann might well have appropriated from mathematical logic: “Funktion und Be­ griff” (1891), “Über Begriff und Objekt” (1892), and “Über Sinn und Bedeutung” (1892).55 The historical and biographical parallels are altogether remarkable: Frege and Riemann were more than mere contemporaries. After two years at Jena, Frege left for Göttingen in 1871, where he completed a dissertation “Über eine geometrische Darstellung der imag­ inären Gebilde in der Ebene” under the mathematician Ernst Schering; he defended the Page 22 of 51

What is a Function? dissertation in August of 1873.56 Although he matriculated at Leipzig in 1871, Riemann also moved on to Göttingen after his own dissertation, “Über das musikalische Hören,” was rejected in the spring of 1873. In Göttingen, Hermann Lotze, a celebrated philoso­ pher and one of the leading lights in the “return to Kant,” agreed to sponsor the disserta­ tion, which Riemann defended in October. While there is no actual evidence that Frege and Riemann ever met, it is hard to believe that their paths never crossed. The question of geographical proximity aside (Göttingen was a small town), Frege had studied the phi­ losophy of religion with Lotze and was a personal acquaintance.57 Just how much contact Riemann would have had with Lotze, on the other hand, is harder to ascertain: he never attended any of Lotze's lectures, nor was he overly familiar with his writings; Michael Arntz tells of Riemann holed up in a local Gasthaus, cramming Lotze's Geschichte der Äs­ thetik in Deutschland (1868) prior to his defense.58 While a chance encounter with Frege could well have awakened or deepened an interest on Riemann's part in mathematical logic—having published an (p. 114) essay on musical logic the year before, Riemann would have had much to discuss with Frege—it is unlikely that Riemann would have been current with Frege's work in 1893. (Even if they had met in 1872, there is no record of correspondence between them, from either end, and it is hard to believe that Riemann would have been familiar with Frege's writings absent some personal contact between them.) Although Frege would exert an enormous influence on Edmund Husserl, Bertrand Russell, Ludwig Wittgenstein, and Rudolf Carnap (all of whom he either knew or corresponded with), he was otherwise largely ignored, remaining an obscure figure in his own lifetime.59 More to the point, there are important differences between Frege's understanding of the mathematical function and what we can recon­ struct of Riemann's, and those differences, on balance, outweigh the similarities. This does not entirely rule out the possibility of Riemann's having borrowed the mathematical function from Frege: he could have misunderstood Frege, understood him only partially, or even have disagreed with him, after all, and the close (and remarkable) concurrence in their writings of function with meaning—a conjunction we associate specifically with Frege—should give us pause. It seems to me more cautious, however, and in the end more interesting, to use Frege's ideas as a foil for Riemann's. Frege elaborated his ideas far more completely than Riemann ever did, providing us with a rich and powerful con­ ceptual vocabulary with which to interrogate notions that would remain undertheorized in Riemann.60 In the following I will thus be interleaving their ideas, and though in so do­ ing I will be developing a particular interpretation of Riemann, I will for the most part at­ tempt no more than a mere reading—a lecture, as the French would say—of Frege. I do this because Frege is a remarkably clear and persuasive advocate of his own ideas, but al­ so to be circumspect: I will no more than touch on the wider philosophical issues to which his ideas have given rise. Riemann, not Frege, will be the focus of attention here. As Dahlhaus notes, Ernst Kirsch was the first observer to connect the tonal function in Riemann with the mathematical function.61 In Wesen und Aufbau der Lehre von den har­ monischen Funktionen (1928), Kirsch considers the connection as though it were com­ mon knowledge. As he explains it, the function in mathematics concerns the “depen­ dence” of one “variable magnitude” on another, “independent” one.62 He expresses this in Page 23 of 51

What is a Function? the formula y = f(x), where y designates the dependent and x the independent variable. It is just this understanding of the mathematical function, however, that Frege wished to set aside. Without going into detail, Frege contends that magnitudes are not mathematical objects and for that reason are not to be admitted into considerations of pure logic.63 As Frege sees it, “magnitude” operates (in cases like this one) as a mere euphemism for “number,” noting, however, that the alternative expression—“indefinite number”—makes no more sense, since there are and can be no indefinite (or variable) numbers: if we re­ place x or y with the proper names of numbers, what we in fact get are definite numbers. We write the letters x and y, rather, “in order to achieve generality.”64 Frege believes that it is altogether reasonable to speak of (p. 115) “indefiniteness” in this connection, but that the word should be used not as an adjective, to describe “number,” but as an adverb, to qualify “indicate”—“we cannot say that ‘n’ designates an indefinite number, but we can say that it indicates numbers indefinitely.” While this may strike some readers as an over­ ly fine distinction, the precision and rigor with which he uses language allows him to make a number of absolutely crucial distinctions. For Frege, the real problem comes when y = f(x) is then read as “y is a function of x.” There are two mistakes here: first, rendering the equals-sign as a copula; [sec­ ond], confusing the function with its value for an argument. From these mistakes has arisen the opinion that the function is a number, although a variable or indefi­ nite one. We have seen, on the contrary, that there are no such [variable or indefi­ nite] numbers at all, and that functions are fundamentally different from numbers.65 “Confusing the function with its value for an argument”—as we will see, this is the crux of the problem concerning the notion of a tonal function. And though it is unclear whether Kirsch read the functional equation as “y is a function of x,” doing so would have been consistent with his explanation: he conceptualizes the mathematical function, that is, al­ most entirely in terms of its two “variables,” expressing no discernable interest, in con­ trast, in the letter f. What the functional equation does, rather, is determine that for every x there is a single y, thus correlating an “x-range” with a “y-range.” As Frege notes, “the heart of the matter” lies in the word “correlation.”66 The difference between functions and numbers, that is, lies in the distinction “between form and content, sign and the thing signified.”67 He re­ gards the content of an expression as its “meaning,” the object to which the expression refers. What is expressed in the equation “2·23 + 2 = 18” is that the right-hand complex of signs has the same meaning [Bedeutung] as the left-hand one. I must here com­ bat the view that, e.g, 2 + 5 and 3 + 4 are equal but not the same. This view is grounded in the same confusion of form and content, sign and thing signified. It is as though one wanted to regard the sweet-smelling violet as differing from Viola odorata because the names sound different.68

Page 24 of 51

What is a Function? A rose by any other name would smell as sweet. The meaning of a mathematical expres­ sion, however, is not the same as a function, for then a function would just be a number. A function lies, rather, in the form of an expression. We can thus recognize the same func­ tion—the same form—in the following three expressions, all of them Frege's, where the dot is to be read in each case as the sign for multiplication:

Each of these expressions “means,” “stands for,” or “refers to”—all perfectly viable trans­ lations of bedeuten—a different number, but all of them have the same general form, which can be written as (p. 116)

“2·x3 + x” also refers to a number, but once again does so “indefinitely,” referring to no one number in particular. Here the letter “x” designates the argument, which we can re­ place with definite numbers to generate the three expressions above, all of which express the same function with different arguments. For Frege, the function of “2·x3 + x” is what's present in the expression “over and above” the letter “x.” To get at this, he rewrites the expression as

where the empty parentheses designate places into which an argument must be inserted in order to complete or “saturate” the function. I am concerned to show that the argument does not belong with a function, but goes together with the function to make up a complete whole; for a function by it­ self must be called incomplete, in need of supplementation, or “unsaturated” [ungesättigt]. And in this respect functions differ fundamentally from numbers. Since such is the essence of functions, we can explain why, on the one hand, we recognize the same function in “2·13 + 1” and “2·23 + 2,” even though these expressions stand for [bedeuten] different numbers, whereas, on the other hand, we do not find one and the same function in “2·13 + 1” and “4 – 1” in spite of their equal numerical values…. We now see how people are easily led to regard the form of an expression as what is essential to a function.69 A number, in other words, is complete unto itself, whereas a function is not. A function, rather, is hollow at the core, and requires filling in to gain meaning. Once completed with an argument, the function assumes a value, which corresponds to the number—the mean­ ing—for which the completed expression stands. Hence, for an argument of “1,” the ex­ pression “2·x3 + x” equals “3,” which Frege names “the value of the function for a given argument.”70 In this case, “3” is the value of the function “2·x3 + x” for the argument “1.” Page 25 of 51

What is a Function? We can thus rewrite the above expressions using different arguments, where the expres­ sion on either side of the equals sign designates the same object, or number: “2·13 + 1 = 3,” “2·23 + 2 = 18,” “2·43 + 4 = 132.” Frege greatly extended the domain of “what can occur as an argument” so that “not merely numbers, but objects in general, are now admissible; and here persons must as­ suredly be counted as objects.”71 This allowed him to recognize linguistic statements as mathematical functions in which substantives (proper names) serve as arguments and values. “The linguistic form of an equation,” writes Frege, “is a statement,” which we can imagine as being “split up” into two parts, one of them—the argument—“complete in it­ self,” the other—the function—“in need of supplementation.”72 Hence the statement “the capital of the German Empire” separates into “the capital of” and “the German Empire,” where the genitive form—corresponding (in translation) to the “of”—goes with “the capi­ tal.” In this case, “the capital of” constitutes the function, which is “unsaturated” and can be rewritten (though Frege himself does not do so) as “the capital of ( ),” where the va­ cant space between parentheses indicates the place into which an argument must be in­ serted. As he (p. 117) adds in “On Concept and Object,” the need for supplementation is essential to the function: “for not all parts of a thought can be complete; at least one must be unsaturated or predicative; otherwise they would not hold together.”73 It is only when the parentheses are then filled in with a proper name, as with “the German Empire,” that the expression becomes complete: “if we take the German Empire as the argument, we get Berlin as the value of the function.”74 “When we have thus admitted objects without restriction as arguments and values of functions,” however, “the question arises what it is that we are here calling an object.” I regard a regular definition as impossible, since we have here something too sim­ ple to admit of logical analysis. It is only possible to indicate what is meant. Here I can only say briefly: an object is anything that is not a function, so that an expres­ sion for it does not contain any empty place.75 Unlike a function, an object requires no filling in. An object is whatever a function is not, too primitive to be given a positive definition. Frege, however, would also later say the very same thing about a function. In “What is a Function?,” he writes that the unsaturatedness of functional signs answers to something “in the functions themselves.” They too may be called “unsaturated,” and in this way we mark them out as funda­ mentally different from numbers. Of course this is no definition; but likewise none is here possible. I must confine myself to hinting at what I have in mind by means of a metaphorical expression, and here I rely on my reader's agreeing to meet me half-way.76 An agreement, I might add, that Riemann seems to have assumed. It underlies his re­ liance on instantiation in defining a tonal function;77 for Riemann, a tonal function is a Page 26 of 51

What is a Function? logical primitive, too simple to be given a genuine definition. And as we have seen, he feels the same way about the component intervals of the major/minor triad, the effects of which “are peculiar to each one and not further definable.”78 Although Riemann feels a strong compulsion to ground his theories in the basic facts of the musical material (in his Harmonik) or of musical cognition (in the Metrik), the means he uses of doing so is causal association, not formal logic: his music-theoretical constructions do not rely on formal de­ finition or proof. Indeed, there is no sense in Riemann of what a formal definition would even look like: there is a less than wissenschaftlich disinclination to break down complex notions like that of a tonal function, notions that (in this case) cannot be derived from the acoustical properties of pitch, into their constituent elements. When Riemann does give definitions, one senses that he does so mostly for propaedeutic reasons, in the spirit of a lexicon entry: to ease the reader along, or to make an abstraction more concrete. What Frege is getting at in this last citation, however, is the idea that even though a func­ tion requires completion with an argument, the argument itself “does not form part of the designation of the function.”79 It is for this reason impossible to isolate or separate a function out from the expression in which it (p. 118) occurs: “the function-sign cannot oc­ cur on one side of an equation by itself”80—one cannot write out an expression determin­ ing what a given function is on its own, on one side of an equals-sign. Hence the impor­ tance Frege places (in “What is a Function?”) on “correlation.” A function, that is, is noth­ ing but the correlation of an x-range (a listing of all the possible arguments for a given function) with a y-range (a listing of all the corresponding values). For the function “the capital of,” we would then get, among others, the correlations “Germany” and “Berlin,” “England” and “London,”81 “France” and “Paris,” where the names of countries serve as arguments. Giving the values first— Berlin is the capital of Germany. London is the capital of England. Paris is the capital of France.

—and so on. But if a country or a person can serve as an argument in a mathematical function, so, then, can a major or minor triad. We can thus use the mathematical function in Frege to reconstruct and complete the concept of a tonal function in Riemann, conceiving the dom­ inant, for instance, as a functional expression on the model of “the capital of x.” Hence “the dominant of C major” splits up into the function “the dominant of” and the argument “C major,” where C major refers to the tonic, a proper name. As before, “the dominant of” is unsaturated and requires filling in, as in “the dominant of ( ),” where the empty space between parentheses designates the place where an argument is to go. When we plug “C major” into the blank, we get a complete thought, “the dominant of (C major).” G major is the value of the function for the argument C major, an expression that can be cast in the form of an identity statement, “the dominant of C major is G major,” where both sides of the equation refer to (bedeuten) the same object, a G major triad. We can perform the same demonstration, moreover, for the tonic and subdominant. Page 27 of 51

What is a Function? Riemann's fundamental inspiration, then, was to recognize that the tonic, dominant, and subdominant could all assume the form of the mathematical function. But because Rie­ mann never mentions mathematics in connection with the tonal function, it is difficult to determine what his actual understanding of the mathematical function would have been. In the context of Frege's comments on the mathematical function, though, it is crucial to realize that Riemann's use of the term is not even the least bit metaphorical: as tonal functions, the tonic, dominant, and subdominant correlate arguments with values, which is precisely what the mathematical function does—the tonic, dominant, and subdominant are, in this very real sense, mathematical functions. As I have worded it, though, the equation “the dominant of C major is G major” would ap­ pear to describe an object as a dominant and not the function itself: leaving the argument out for the nonce, we can reword the sentence to read “the dominant is G major,” which, as Dahlhaus reminds us, is how musicians—Riemann among them—are used to thinking of the dominant. The issue is crucial, because it underlies (p. 119) the confusion as to whether the dominant refers to a chord, an object, or to a tonal function, a concept. For Frege, “objects and concepts are fundamentally different and cannot stand in for one an­ other.”82 An object is something that “falls under” a concept, a relation Frege regards as “irreversible.” A concept can subsume an object, that is, but the reverse is never true: an object can never subsume a concept. And if neither an object or a function can be de­ fined, nor can a concept: One cannot require that everything be defined, any more than one can require that a chemist decompose every substance. What is simple cannot be decom­ posed, and what is logically simple cannot have a proper definition…. On the intro­ duction of a name for something logically simple, a definition is not possible. There is nothing for it but to lead the reader or hearer, by means of hints, to un­ derstand the words as is intended.83 In the absence of a proper definition, the best Frege can do is to characterize a concept as having a “predicative nature,” arising from “the [same] need of supplementation, the [same] unsaturatedness” he regards as “the essential feature of a function.”84 In order to distinguish between a concept and a function, he proposes an informal test, a rule of thumb in which “the singular definite article always indicates an object, whereas the in­ definite article accompanies a concept word.”85 On this criterion, however, “the domi­ nant” once again fails as a concept, since the definite article would appear to designate an object rather than a concept—it refers to a G major triad, for instance, not as a domi­ nant in general, but as a particular dominant, the one in C major. “In logical discussions,” however, “one quite often needs to say something about a concept, and to express this in the form usual for such predications—namely, to make what is said about the concept into the content of the grammatical predicate.” Consequently, one would expect that the meaning [Bedeutung] of the grammatical subject would be the concept; but the concept as such cannot play this part, in

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What is a Function? view of its predicative nature; it must first be converted into an object, or, more precisely, an object must go proxy for it.86 This, then, is what happens when we speak of G major as “the dominant” in C major: the value of the function for a given argument is made to “go proxy” for a correlation we can demonstrate but not (according to Frege) define—we convert a concept into an object and in the process once again “confuse the function with its value for an argument,” us­ ing “the dominant” as a proper name for both. For this reason, among others, Frege re­ gards concepts as being “unmanageable.” Yet he persists in believing that in distinguish­ ing between concept and object he has “got hold of a distinction of highest importance.” I admit that there is a quite peculiar obstacle in the way of an understanding with my reader. By a kind of necessity of language, my expressions, taken literally, sometimes miss my thought; I mention an object, when what I intend is a concept. I fully realize that in such cases I was relying upon a reader who would be ready to meet me halfway—who does not begrudge a pinch of salt.87 (p. 120)

However peculiar language's drift toward concretion, the distinction between con­

cept and object is nevertheless absolute. Frege warns against the hypostasis of concepts, the process (which we observed earlier [on pp. 101–102] in connection with the notion of a tonal function as a platonic essence) whereby abstractions are transformed into objects: in Die Grundlagen der Arithmetik (1884), he writes that “it is a mere illusion to suppose that a concept can be made an object without altering it,” without the concept undergo­ ing a radical transformation in being.88 In regarding unsaturatedness as the essential feature of both the function and the con­ cept, Frege partially collapses the distinction between them. For Frege, a concept just is a one-argument function, in which the meaning (Bedeutung) of the expression as a whole (in contrast to those of its separate parts) is not a number, as it is in the function proper, but a truth-value.89 When completed with an argument and a value, the meaning (Bedeu­ tung) of the expression “London is the capital of England” as a whole is what Frege calls “the True,” as opposed to “Paris is the capital of Belgium” (the example is mine), the meaning for which would be “the False.” It becomes clear in this context just how radical Frege's notion of meaning or reference—Bedeutung—is: for Frege, all true sentences have the same meaning or referent, which is “the True.”90 As he understands it, the no­ tion of meaning or reference is not particular to individual words, concepts, or sentences, but a general attribute all true (or all false) sentences share and share alike. This will soon loom as an important issue for us; I am flagging it now for future reference. Frege regards a concept under which an object falls as a “first-level concept” (ein Begriff erster Stufe) and in distinguishing a first-level from a “second-level concept” he gives us the means with which to grasp and articulate the ontological distinction between the dominant qua chord and the dominant qua concept, a tonal function. “Second-level con­ cepts,” writes Frege, “which concepts fall under, are essentially different from first-level concepts, which objects fall under.”91 Second-level concepts, in other words, are concepts under which other concepts fall. “The dominant of C major” is thus a first-level concept, Page 29 of 51

What is a Function? because an object—a G major triad—falls under it. When we imagine a dominant in the abstract, in contrast, away from any given major or minor triad, we imagine it as a second-lev­ el concept, a concept that subsumes other first-level concepts, all other pairings of argu­ ments and values according to the dominant function. It is the dominant as a second-level concept we have in mind when we airily reflect on its most general properties (such as the presence of dissonance or its putative need to resolve) or on the reduction of all har­ monies (in Riemann) to three tonal functions. In this sense, a tonal function is nothing other than a second-level concept, a concept under which other concepts fall. And though I have insisted on the abstraction of the tonal function as a concept, the distinction be­ tween first and second-level concepts is nonetheless a real one. It arises not from the need for methodological rigor, as a means of categorizing knowledge, but rather con­ forms to reality, to the way the world really is: “for it is not made arbitrarily, but founded deep in the nature of things.”92

(p. 121)

Sinn/Bedeutung

So far, we have drawn on distinctions Frege makes between “function,” “concept,” and “object” as a means of giving content to Riemann's identification of the tonic, dominant, and subdominant as tonal functions. As is clear from his titles, these distinctions are the overlapping concerns of Frege's essays on “Funktion und Begriff” (1891) and “Über Be­ griff und Objekt” (1892). I now need to introduce two further sets of distinctions: that be­ tween “sense” and “reference,” the main concern of his majesterial essay “Über Sinn und Bedeutung” (1892), and that between “thought” and “idea,” the focus of a late essay, “Der Gedanke” (1918). In order to manage these distinctions, I would like to introduce the fol­ lowing table, which lists their terms along with their usual translations: Funktion

function

Begriff

concept

Objekt

object

Sinn

sense

Bedeutung

meaning or reference

Gedanke

thought

Vorstellung

idea

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What is a Function? Beyond grouping these distinctions into sets, the table is unorganized. The terms them­ selves are heterogeneous (both within sets and between them) and nonhierarchical: no term or distinction is preliminary to or subsumes any of the others, nor do any of the terms align or form a larger pattern. If nothing else, these three sets of distinctions will serve as a conceptual outline of the remaining discussion. It is clear from his writings that what Riemann understood as a tonal function far exceed­ ed the narrower confines of the mathematical function in Frege. As we have seen, Rie­ mann confuses a function with its value for an argument, leading him to regard an object —rather than a relation—as a dominant or subdominant. But he also attributes properties to the dominant the mathematical function is unable to capture: the mathematical func­ tion does not answer to a number of attributes we tend to regard as essential to the tonal function—the notion that the dominant and subdominant, for instance, have distinct chordal identities and therefore distinctive sounds, or that the dominant, in particular, re­ quires resolution to the tonic. In running all these attributes together, Riemann folds what Frege calls the “sense” (Sinn) of an expression into its “referent” or “meaning” (Be­ deutung), the object to which the expression refers. In a celebrated example, Frege notes that the morning star and the evening star have different senses—the one bright at dawn, the other bright at dusk—but the same meaning or referent, the planet Venus.93 This dis­ tinction, drawn in precisely the same terms, underlies the dreadful ambiguity when, in the third act of Wagner's Tannhäuser (p. 122) (1845), Wolfram sings to the evening star. As he begins strumming, Wolfram apostrophizes Elisabeth, “ein sel’ger Engel,” drawing on Marian imagery more appropriate, however, to the morning star—ave Maris stella. But be­ cause the morning star, as Frege reminds us, is the evening star, Wolfram also sings, un­ knowingly, to Venus. It is Tannhäuser who picks up on the difference between the sense and referent of Wolfram's song. He staggers into the clearing to the last strains of the harp, gasping “Ich hörte Harfenschlag, wie klang er traurig! Der kam wohl nicht von ihr!” The brief wisp of Venusberg music that swirls through the orchestra makes it clear that Tannhäuser, in a shattering irony, ascribes a very different sense than Wolfram does to the same heavenly body. “From identity of reference,” in other words, “does not follow identity in the thought expressed.”94 In this case, “the different expressions correspond to different conceptions and aspects, but nevertheless always to the same thing.”95 Unfortunately, the distinction between sense and reference raises intractable problems of translation. In connection with Riemann, I have translated Bedeutung as “meaning,” and will continue to do so. Although it sounds odd to talk about the “meaning” of a chord—we ourselves are far more inclined to talk about what a chord is than about what it means —“meaning” is the synonym Riemann gives for “function,” and “meaning” is what he means; the official translation of Vereinfachte Harmonielehre renders Bedeutung as both “meaning” and “significance,” which I take to be the same. In connection with the distinc­ tion between sense and reference in Frege, however, I am for the moment translating it as “reference.” Both “meaning” and “reference” are viable translations for Bedeutung, though in the abstract “meaning” is the more natural one and in recent decades has be­ come the standard translation for Bedeutung in Frege.96 “Über Sinn und Bedeutung,” however, was influentially translated as “On Sense and Reference,” and it is in those Page 31 of 51

What is a Function? terms that the distinction is best known: over a century later, “On Sense and Reference” remains the most widely read single essay in the philosophy of language. Frege uses “meaning,” moreover, in a technical and highly restricted sense: it sounds odd, in German as well as English, to speak of the sentences “no men are mortal” and “2 + 2 = 5” as hav­ ing the same “meaning,” which is “the False,” but such is the consequence of equating the meaning of a sentence with its truth-value, in contrast to the meaning of a functionexpression, which is an object—a number. A further complication is the fact that what modern readers normally understand as the “meaning” of a function or concept is what, in the context of the sense/reference distinction, Frege regarded as its “sense.” In his own consideration of the distinction, Bertrand Russell indeed translated Sinn as “mean­ ing” and Bedeutung as “denotation.”97 For Frege, the sense of an expression is what we associate with it over and above its actual meaning, the object to which it refers. Whether we regard it as a first-level function or a second-level concept, the dominant or subdomi­ nant—in Riemann's time as well as our own—involves more than an inert correlation of arguments and values. To it we attach all our intuitions concerning dissonance, the pres­ ence of the leading tone, chromatic alterations and melodic displacements, resolution, its various uses, and so on, as its sense. The failure to distinguish between sense and reference underlies another of Dahlhaus's objections to the notion of a tonal function in Riemann. Using Kirsch's (p. 123) formula for the mathematical function, Dahlhaus defines the subdominant as “F = S(C),” or “F major is the subdominant of C major.” He then notes that the same formula can also be used to define scale degree IV, “F = IV(C).” “It is [thus] possible to understand the term ‘subdomi­ nant’…as a designation for a scale degree—not a function—and so to formulate function theory without the concept of a function.” On this basis, he concludes “that function theo­ ry is a more rigorous scale-degree theory in which the number of degrees is reduced to three (I, IV, and V),” and that the function concept is for that reason “superfluous.”98 Now, if function theories indeed give a “more rigorous” account of the same musical realities scale-degree theories do, one would think that Dahlhaus would have been persuaded to consider the scale degree, rather than the tonal function, “superfluous.” But that's beside the point. As far as his demonstration goes, Dahlhaus is in fact right: expressed as mathe­ matical functions, the subdominant generates the same value scale-degree IV does for the same argument; both “F = S(C)” and “F = IV(C)” stand for (bedeutet) the same major tri­ ad. Frege would have been the first to point out, however, that an identity in the thought expressed does not follow from an identity in reference: the subdominant and scale-de­ gree IV each differ in “mode of presentation,” a difference not expressed in the functional correlation of arguments with values. It is expressed, rather, in the different senses we at­ tach to the two thoughts: the sense of a dominant extended below the tonic in the sub­ dominant as against that of the fourth scale degree above I in IV, which is very different.99 In a groundbreaking essay on Parsifal, David Lewin argues that “the nature and logic of Riemannian tonal space are not isomorphic with the nature and logic of scale-degree space. The musical objects and relations that Riemann isolates and discusses are not sim­ ply the old objects and relations dressed up in new packages with new labels; they are es­ Page 32 of 51

What is a Function? sentially different objects and relations, embedded in an essentially different geometry.”100 Lewin goes on to contrast the ordinal arithmetic of scale degrees in the Gothic music of the Grail brotherhood with the non-euclidean geometries of tonal func­ tions in the Arabic music of Klingsor's magic garden. For Lewin, Wagner's music in fact relies “on our willingness to suppose an isomorphism of Riemann space with [scale] de­ gree space,”101 an isomorphism the music sometimes pries apart, to memorable effect. As a brief demonstration of the general nonisomorphism between scale degrees and tonal functions, we can consider the subdominant parallel. In major, the parallel of the subdom­ inant corresponds to scale-degree II: in C major, the parallel of the subdominant F major is D minor, or II. In minor, however, the parallel of the subdominant corresponds to scaledegree VI: in C minor, the parallel of the subdominant F minor is A♭ major, or VI. The same functional meaning, the subdominant parallel, corresponds to two different scale degrees, II in major and VI in minor. The sense we associate with individual scale degrees and tonal functions includes an awareness of the different musical geometries within which each one participates. We can now see why a proper definition, for Frege, involves the association of a concept expression with a sense as well as a referent (or meaning); a referent alone is inadequate to the complexity of a concept expression. In “On Sense and Reference,” Frege introduces a further distinction between thought and idea that displaces sense when his attention moves from concepts and ob­ jects to sentences. For Frege, “a sentence contains a thought,” which consists, as we have seen, of a completed concept. A thought, that is, is a concept that has been saturated or filled in, supplemented with an argument in order to return a value. He once again uses the term in a restricted sense: “by thought [Gedanke] I understand not the subjective per­ formance of thinking [Denken] but its objective content, which is capable of being the common property of several thinkers.”102 Frege pares thought of its normal cognitive as­ sociations and reduces it to the assertion of truth: (p. 124)

In order to avoid any misunderstanding and prevent the blurring of the boundary between psychology and logic, I assign to logic the task of discovering the laws of truth, not the laws of taking things to be true or of thinking. The meaning [Bedeu­ tung] of the word “true” is spelled out in the laws of truth.103 All attempts to explain truth in terms of correspondence break down, because correspon­ dence implies, for Frege, an essential noncoincidence between an idea and the reality to which it corresponds. Yet when truth is defined as “the correspondence of an idea with something real…it is essential precisely that the reality shall be distinct from the idea,” in which case, however, “there can be no complete correspondence, no complete truth.”104 Truth is absolute; there are no partial truths. Indeed, any attempt to define truth will de­ scend into infinite regress: For in a definition certain characteristics would have to be specified. And in appli­ cation to any particular case the question would always arise whether it were true

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What is a Function? that the characteristics were present. So we should be going round in a circle. So it seems likely that the content of the word “true” is sui generis and indefinable.105 Truth, rather, resides in the logical coherence of our assertions, their adherence to “the laws of truth,” which the goal of logic is to work out in the form of concepts, definitions, axioms, assertions, and inferences. He thus moves the question of truth from an episte­ mological to a logical foundation. A thought is objective in that it poses the question of truth, a question that arises a priori, in terms of pure reason, prior to sensible intuition: “The thought, in itself imperceptible by the senses, gets clothed in the perceptible garb of a sentence, and thereby we are en­ abled to grasp it. We say a sentence expresses a thought.”106 In a sentence we can more­ over discern both a “content,” some proposition concerning concepts and objects, and an “assertion,” the thought that is said to be true. Whether or not the assertion is true is im­ material. For Frege, a thought is a formal construct: a false thought is no less a thought than a true one. A thought is also objective in that other persons can grasp it.107 As Frege notes above, thoughts are the “common property” of those who think them and are in that sense au­ tonomous, belonging to no one person in particular. In this sense, a thought contrasts with what Frege calls an “image” or “idea,” a Vorstellung: (p. 125) If the meaning [Bedeutung] of [an expression] is an object perceivable by the sens­ es, my idea [Vorstellung] of it is an internal image, arising from memories of sense impressions which I have had, and acts, both internal and external, which I have performed. Such an idea is often imbued with feeling; the clarity of its separate parts varies and oscillates…. The idea is subjective: one man's idea is not that of another.108 He returns to the idea over two decades later, in “Der Gedanke,” elaborating at much greater length: Even an unphilosophical man soon finds it necessary to recognize an inner world distinct from the outer world, a world of sense impressions, of creations of his imagination, of sensations, of feelings and moods, a world of inclinations, wishes, and decisions. For brevity's sake, I want to use the word “idea” to cover all these occurrences, except [for] decisions.109 Although ideas often arise from sense perception, ideas—like thoughts—are themselves imperceptible. Unlike thoughts, however, ideas are subjective and belong to the contents of a personal consciousness: “no two men,” writes Frege, “have the same idea.”110 As he notes in “On Sense and Reference,” “an exact comparison” between the ideas of two peo­ ple “is not possible, because we cannot have both ideas together in the same conscious­ ness.”111 He then pauses to summarize:

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What is a Function? The following analogy will perhaps clarify these relationships. Somebody observes the moon through a telescope. I compare the moon itself to the [referent] [Bedeu­ tung]; it is the object of the observation, mediated by the real image projected by the object glass in the interior of the telescope, and by the retinal image of the ob­ server. The former I compare to the sense, the latter is like the idea or intuition [Anschauung]. The optical image in the telescope is indeed one-sided and depends upon the standpoint of observation; but it is still objective, inasmuch as it can be used by several observers. At any rate it could be arranged for several [observers] to use it simultaneously. But each one would have his own retinal image. On ac­ count of the diverse shapes of the observers’ eyes, even a geometrical congruence could hardly be achieved, and an actual coincidence would be out of the question.112 The image in the mirror is like a sense, or a thought; it can be seen by several different observers at once and in that sense belongs to all of them at the same time. An idea, on the other hand, is like the retinal image each observer forms of the image in the mirror and is unique to the personal circumstances of each individual: “we are not owners of our thoughts as we are owners of our ideas.”113 Thoughts occupy an intermediate rung on the ontological ladder, belonging neither to the outer world (because thoughts are impercep­ tible) nor to the world within (because thoughts are nevertheless objective). As Frege notes, “the content of a sentence often goes beyond the thought expressed by it.” A sentence, that is, “often contains, over and above a thought and assertion, a third component not covered by the assertion. This is often meant to act on the feelings and moods of the hearer, or to arouse his imagination.”114 Frege is (p. 126) not the least bit dismissive of those “feelings and moods,” recognizing that “what is essential depends on one's purpose: to a mind concerned with the beauties of language, what is trivial to the logician may be just what is important.” Nevertheless, “what is called mood, atmosphere, illumination in a poem, what is portrayed by intonation and rhythm,” does not concern what is true or false and therefore “does not belong to the thought.” Much of “Der Gedanke” is indeed given over to a devastating attack on the idealist reduction of thoughts to (psychologistic) ideas. For Frege, the proper domain of logic are thoughts, ob­ jective assertions of truth. Ideas, in contrast, fail the requirement of “the sharp delimita­ tion of concepts”115 and thus are to be excluded from the science of logic: an idea, once again, “varies and oscillates” in clarity and as such is of no use in defining concepts, de­ termining meanings, forming propositions, and drawing inferences. In his own notes to “On Sense and Reference,” he explains that “the laws of logic are first and foremost laws in the realm of meanings [Bedeutungen] and only relate indirectly to sense.” If it is a question of the truth of something—and truth is the goal of logic—we also have to inquire after meanings [Bedeutungen]; we have to throw aside proper names that do not designate or name an object, though they may have a sense; we have to throw aside concept words that do not have a meaning [Bedeutung]. These are not such as, say, contain a contradiction—for there is nothing at all wrong in a concept's being empty—but such as have vague boundaries. It must be determi­ Page 35 of 51

What is a Function? nate for every object whether it falls under a concept or not; a concept word which does not meet this requirement…is meaningless [bedeutungslos].116 “Meaningless,” that is, for the purposes of logic: here Frege folds the normal, colloquial sense of “meaning” over its narrower, technical one. Above all, Frege is anxious to make a clear and decisive distinction between logic and psychology. In the introduction to The Foundations of Arithmetic, he vows as one of his three guiding principles “always to separate sharply the psychological from the logical, the subjective from the objective.”117 His most extended consideration of the issue occurs at the beginning of “Der Gedanke”: From the laws of truth there follow prescriptions about asserting, thinking, judg­ ing, inferring, And we may very well speak of laws of thought in this way too. But there is at once a danger here of confusing different things. People may very well interpret the expression “law of thought” by analogy with “law of nature” and then have in mind general features of thinking as a mental occurrence. A law of thought in this sense would be a psychological law. And so they might come to be­ lieve that logic deals with the mental process of thinking…. That would be misun­ derstanding the task of logic, for truth has not been given its proper place.118 And with regard to truth, “it is not a matter of what happens but what is.” Logic, for Frege, has to do with the objective content of thought and not thinking as a mental process. This is an important point, because it is in this regard that Riemann differs most from Frege: whereas Frege purges logic of the psychological, on occasion the tonal func­ tion in Riemann verges on full-blown psychologism.119 I want to be cautious about this as­ sertion, because not all the evidence points in the same direction. I do not feel the claim can be easily made, for instance, of Vereinfachte Har­ monielehre, which I have otherwise treated as the crucial document in the discursive ar­ ticulation of the tonal function. In “On Concept and Object,” Frege complains that “the word ‘concept’ is used in various ways; its sense is sometimes logical, sometimes psycho­ logical, and sometimes perhaps a confused mixture of both.”120 In Vereinfachte Har­ monielehre, the concept of a tonal function occurs in a logical sense, though even this claim is difficult to evaluate, since Riemann never identifies the tonal function as a con­ cept: indeed, the very notion of a concept passes without formal theoretical recognition in Vereinfachte Harmonielehre. In Fregian terms, it is implicit in the reduction of all har­ (p. 127)

monies to three tonal functions: the tonal function, that is, is a second-level concept in that the tonic, dominant, and subdominant all fall under it. Riemann, however, does not recognize it as such: he does not specifically describe the tonal function—or for that mat­ ter the tonic, dominant, or subdominant—as a concept, nor does he attribute to it proper­ ties, psychological or otherwise. It is altogether possible, in other words, that the tonal function, as Riemann conceives it in Vereinfachte Harmonielehre, would in fact meet Frege's narrow criteria as to what counts as a concept: in this case, the tonic, dominant, and subdominant can be understood as functional expressions pairing arguments with values, what Frege understands as first-level concepts. Riemann, that is, does not appear Page 36 of 51

What is a Function? to regard the tonic, dominant, and subdominant as psychological constructs, as objects in consciousness. His use of the tonal function is, if anything, anti-psychologistic in nature, in a manner that might well have met with Frege's approval. As first-level concepts, the tonic, dominant, and subdominant give rise to more or less set-theoretical elaborations of transformationally related harmonies as chordal complexes in which each tonal function determines a set class. Riemann writes in this vein that “our principal aim was directed toward strengthening the feeling of tonality and extending to the utmost the circle of har­ monic formations intelligible in their relation to one and the same tonic.”121 It is in this sense that Rehding describes the tonal function, insightfully, as “not itself a chord, but an interpretation of a chord.”122 A logical understanding of the tonal function as a mere map­ ping of arguments onto values is consistent with latter-day appropriations of Riemann's harmonic theories, which have taken far greater interest in the mathematical properties of the Tonverwandtschaften and the collections of harmonies arising from them than the notion of a tonal function per se.123 More on which anon. Earlier, in “Die Natur der Harmonik” (1882), in contrast, Riemann endows the concept of a dominant with explicitly psychological properties. Here again are the relevant lines: When…I then think of the G major triad in the sense of C major, I think of it asthe chord a perfect fifth above the C major triad; that is, the C major triad belongs in the idea [Vorstellung] as the triad that determines the meaning [Bedeutung] of the G major triad and from which the G major triad forms a departure—the center of the idea [Vorstellung]…lies outside the G major triad.124 Here the notion of an “idea,” a Vorstellung, stands in for the that of the concept: an “idea” is a mental image under which the dominant (in this case) falls. It is a conceptu­ al enclosure within which the dominant as well as the tonic occur as objects in conscious­ ness. And though Riemann describes the dominant as an “idea,” the “idea” he has in mind is nevertheless objective in the crucial sense that more than one person can grasp it. As an “idea,” the dominant is subjective in that it arises as an “internal image” from “sense impressions,”125 but it is also objective in that it can (and often does) become the “com­ mon property” of several people. It thus comes closer to what Frege recognized as a “thought” than to what he understood as an “idea,” even though Riemann uses the same word to describe the dominant—Vorstellung—that Frege does to designate an “idea.” Al­ though it can be no coincidence that both Riemann and Frege would use the same word (p. 128)

to describe a psychological object, Riemann and Frege assign it different meanings, dif­ ferent Bedeutungen. Riemann's more immediate concern, however, is with “the subjective performance of thinking,” with cognition as a mental occurrence: “denke ich….” His deliberations on the dominant as an “idea” take place, moreover, within a broader consideration of the emerg­ ing discipline of musicology.126 For Riemann, the theory of harmony is part of a larger sci­ ence of music, which rests on an “investigation into musical nature” extending from acoustics through perception to cognition, from “Physis,” that is, through “Physiologie” to “Psychologie.”127 Psychology, in particular, “concerns itself with the nature of tonal ideas Page 37 of 51

What is a Function? [Tonvorstellungen] and their combinations,” Riemann using the same word in “tonal idea”—Tonvorstellung—he does in describing the dominant as an “idea.”128 If, as it ap­ pears, Riemann largely cleanses Vereinfachte Harmonielehre of the psychological (the historical coincidence of Vereinfachte Harmonielehre with the sustained attack on psy­ chologism in Frege's writings can only give rise to further, if idle, speculation), he circles back to it in the “psychological turn” of his later writings.129 In the “Ideen zu einer ‘Lehre von den Tonvorstellungen’ ” (1914), Riemann picks up where he had left off in “Die Natur der Harmonik,” with the notion of a “tonal idea” or “representation.”130 Once again, the notion of an “idea,” in this case a “tonal” one, stands in for the concept. Now, however, it moves to the center of attention, where it functions as a methodological category, a means of organizing the discursive field. As a formal concept, however, the tonal idea nevertheless remains undefined: we are given no clear sense of what a tonal idea is, or what, precisely, falls under it—it never becomes an object of theoretical reflection. It is clear that the tonal idea has to do with “the logical activity of musical hearing,”131 and that this hearing is inner, prior to music as phenomenal sound, but its “boundaries” (as Frege would put it) remain “vague.” In the absence of a definition, Riemann again relies on instantiation. He locates the tonal idea, that is, in the elements of musical logic, “in the harmonic concepts [Begriffe] tonic, dominant, and subdominant and the rhythmic con­ cepts [Begriffe] of heavy and light durations.” In looking back on his “Lehre von den tonalen Funktionen der Harmonie,” he thus explicitly identifies the tonic, dominant, and subdominant functions as concepts, using the same word for concepts—Begriffe—that Frege does. I nevertheless attribute less significance to his casual designation of the ton­ ic, (p. 129) dominant, and subdominant as “concepts,” however—something he was reluc­ tant to do in Vereinfachte Harmonielehre—than to the actual musical phenomena he iden­ tifies as tonal ideas. For Riemann, “the key to the innermost essence of music” lies not in “an examination [Untersuchung] of the elements of music as sound,” but rather in “the determination [Feststellung] of the elements of music as conceptualized [der vorgestell­ ten Musik].”132 Whether music is heard or only merely imagined, its inner essence lies in its “elements,” in the musical material. As before, a “tonal idea,” as Riemann seems to be conceiving it, is an amalgam of a Fregian “thought” and an “idea.” It is both objective and subjective at the same time: objective in that another person can hear and grasp it, but subjective in that it is nevertheless a mental occurrence, an object in consciousness. As a concept, the “tonal idea” would thus appear to be what Frege would have dismissed as a “confused mixture” of the logical and the psychological. For Riemann, then, musical logic is the domain of the musically thinkable. And within the bounds of the harmonic, what can be musically thought are the three tonal functions and their transformational modifications. Like the axioms of geometry, the truths of this har­ monic logic arise from pure intuition, which underlies our perception of objects in space: our intuitions of the dominant and subdominant are intuitions of a dominant above and a subdominant below the tonic. As abstract as those intuitions turn out to be, without them, as Frege have would put it, “to think at all seems no longer possible.”133 The tonic, domi­ nant, and subdominant in this sense are, to paraphrase Joan Weiner,134 “written into” mu­ sical perception: without them we could not perceive music at all. Our intuitions of them Page 38 of 51

What is a Function? are a priori and synthetic, and receive their justification through appeals to the empirical facts of the matter. Writes Frege: Empirical propositions hold good of what is physically or psychologically actual, the truths of geometry govern all that is spatially intuitable, whether actual or product of our fancy. The wildest visions of delirium, the boldest inventions of leg­ end and poetry, where animals speak and stars stand still, where men are turned to stone and trees turn into men, where the drowning haul themselves up out of swamps by their own topknots—all these remain, so long as they remain intu­ itable, still subject to the axioms of geometry. Conceptual thought alone can after a fashion shake off this yoke, when it assumes, say, a space of four dimensions or positive curvature. To study such conceptions is not useless by any means; but it is to leave the ground of intuition entirely behind. If we do make use of intuition even here, as an aid, it is still the same old intuition of Euclidean space, the only one whose structures we can intuit.135 Frege's comments about our doubtful ability to intuit four dimensions and positive curva­ ture are both suggestive and ironic with regard to Riemann's adumbration of a non-Eu­ clidean geometry in the Verwandtschaftstabelle toward the end of the “Ideen zu einer ‘Lehre von den Tonvorstellungen,’ ”136 motion across or around which induces just those mind-bending intuitions. As tonal ideas, the primitive “elements” of a harmonic logic, the tonic, dominant, and sub­ dominant are the harmonic “noumena behind the sounding (p. 130) phenomena,”137 the veritable subject matter of musical cognition, the forms within which musical thinking oc­ curs: The alpha and omega of music as art lies not in music as it sounds but rather in the mental representation [Vorstellung] of tone relations, which lives in the musi­ cal imagination [Tonphantasie] of the creative artist prior to its inscription in mu­ sical notation and subsequent re-emergence in the imagination [Tonphantasie] of the hearer. Both the registration [Festlegung] of artistic creation in musical nota­ tion and the actual realization of the work in performance are but means of trans­ planting a musical experience from the imagination [Phantasie] of the composer into that of the musical listener.138 In the communication of tonal ideas from the composer's imagination into the mind of the listener, musical notation serves as a medium, a form of mediation, through which the noumena immanent to the music are expressed. It is the essence of these tonal ideas that Riemann distills in his functional notation, which we can look on as traces of the cognitive activities of a listener engaged in continual acts of judgment concerning the harmonic truth of the music. His function ciphers admit to a secret sense that accompanies the mu­ sic and guarantees its intelligibility, the very image of our own musical hearing, and of the musical hearing of others. We can thus regard each functional designation—each hi­ eroglyph—in example 3.7 (p. 110) as a proposition, an assertion of functional equivalence in the form of x = y. As what Frege would call an “assertonic” sentence, however, the Page 39 of 51

What is a Function? function expression goes beyond a mere statement or declaration to an assertion of truth: the sentence “the chord on the downbeat of measure 1 is the (major) dominant in A mi­ nor” is equivalent to—has the same “meaning” (Bedeutung) as—the sentence “it is true that the chord on the downbeat of measure 1 is the (major) dominant in A minor.” Frege: With every property of a thing there is tied up a property of a thought, namely truth…. The sentence “I smell the scent of violets” has just the same content as the sentence “It is true that I smell the scent of violets.” So it seems, then, that nothing is added to the thought by my ascribing to it the property of truth.”139 An assertion of truth is built into the sentence. Indeed, with the aid of Frege's own con­ ceptual notation, we could express the logical content of the sentence “The chord on the downbeat of measure 1 is the (major) dominant in A minor” as where the horizontal or “content” stroke indicates that what follows is a proposition, the vertical or “judgment” stroke indicates that the proposition is judged to be true, and the “hollow” in the content stroke carves out the empty place into which the argument is to be inserted, the blank space requiring saturation. Using Klangschlüssel for both argu­ ment and value and Riemann's uppercase “D” for the dominant function (Frege would have used a combination of Greek and Fraktur), the expression can be read “e+ has the property D for the argument °e.” Each (p. 131) functional designation in example 3.7— each capital letter—would for Frege involve (a) “the grasp of a thought,” a consequence of all the factors leading to the determination of a particular function, (b) the “act of judg­ ment” proper, “the acknowledgement of the truth of [the] thought,” and (c) “the manifes­ tation of this judgment” in the form of an “assertion.”140 For Riemann, musical hearing boils down to a wearying succession of true/false asser­ tions about the meaning of each and every chord in relation to the prevailing tonic in which each proposition, each assertion about what a chord is, is understood as an affir­ mation of truth, a judgment to be performed again with every change of harmony. Hence we are to read the first “D” in example 3.7 as “the chord on the downbeat of measure 1 is the (major) dominant in under e” (= A minor), the following “°T ” as “the chord on the up­ beat of measure 1 is the tonic in A minor” (or more provocatively “the [minor] tonic in A major”), and so on, in saecula saeculorum, amen. As logic goes, the one we encounter in functional notation is exceedingly rudimentary, go­ ing no further than simple, perseverative assertions of truth—a weak form of logic. In the harmonic domain, functional notation reduces musical logic to the mere subsumption of objects under concepts, of which there are but three. With the possible exception of mod­ ulation, musical logic would appear to be a logic without subordination: without negation or the conditional, without universal and existential qualification, without conjunction and disjunction, without inference. In comparison with—why not?—German, what passes for musical logic in Riemann's harmonic theories is not so much impoverished as emaciated, so thin as to be negligible, a logic hardly worthy of the name, much less the effort. I do not mean this as a categorical assertion about musical logic in general, or to suggest that Page 40 of 51

What is a Function? nothing can ever be musically logical. Nor is this the place or now the time to examine claims others have made concerning the elaboration of a musical logic. I am merely as­ serting, rather, that however useful mathematical logic is in conceptualizing tonal func­ tion in Riemann, the reduction of all harmonies to tonic, dominant, and subdominant functions neither embodies nor gives rise to a musical logic in the stronger sense of the term. To the extent that it serves as the vehicle or medium of this musical logic, tonal function is a problematic concept, as it was in the context of Riemann's own harmonic theories: Riemann was never able to integrate the notion of tonal function with other aspects of his harmonic theories, their adherence to just intonation and harmonic dualism to name but two, factors that combined to prevent him from realizing the full algebraic potential of his tone-relations, the functional transformations so dear to a newer generation of his read­ ers one fin-de-siècle later. Indeed, what has allowed these readers to isolate and develop this aspect of his harmonic theories has been an opportune willingness to for the most part jettison tonal function as Riemann imagined it and to reconceive harmonic relations in fully transformational terms, in terms more consistent, that is, with the mathematical function in Frege. In this sense what we have come to call neo-Riemannianism is, both ironically and poetically, much less Riemannian than Fregean.

Notes: (1.) Riemann, Vereinfachte Harmonielehre, oder Die Lehre von den tonalen Funktionen der Akkorde (London: Augener, 1893), trans. Henry Bewerunge as Harmony Simplified, or The Theory of Tonal Functions of Chords (London: Augener, 1895). (2.) Riemann, Harmony Simplified, 8. (3.) Riemann, Vereinfachte Harmonielehre, 9. “I. Es giebt nur zwei Arten von Klängen: Oberklänge und Unterklänge; alle dissonanten Akkorde sind aufzufassen, zu erklären und zu bezeichnen als Modifikationen von Ober- und Unterklängen. II. Es giebt nur dreierlei tonale Funktionen der Harmonie (Bedeutungen innerhalb der Tonart), nämlich die der Tonika, Dominante und Subdominante. In der Veränderung dieser Funktionene beruht das Wesen der Modulation” (translation modified). I have altered the translation of Be­ deutung from “significance” to “meaning” for reasons that will become clear later. I am not trying to make a distinction between “significance” and “meaning,” both of which are reasonable translations for Bedeutung, and which I regard as synonymous. (4.) Riemann, Musik-Lexikon, 7th ed. (Leipzig: Max Hesse, 1909), 441. “Funktionsbezeich­ nung der Harmonien ist die Andeutung der verschiedenartigen Bedeutung (Funktion), welche die Akkorde nach ihrer Stellung zur jeweiligen Tonika für die Logik des Tonsatzes haben.” Cited in Carl Dahlhaus, “Über den Begriff der tonalen Funktion,” in Beiträge zur Musiktheorie des 19. Jahrhunderts, ed. Martin Vogel (Regensburg: Gustav Bosse, 1966), 93. I have converted the singulars Bedeutung and Funktion into the plurals meanings and functions. Page 41 of 51

What is a Function? (5.) Although he does not trace the historical fortunes of the tonal function per se, Ludwig Holtmeier considers the historical reception of Riemann's harmonic theories in general in “Grundzüge der Riemann-Rezeption,” in Musiktheorie, ed. Helga de la Motte-Haber and Oliver Schwab-Felisch (Laaber: Laaber Verlag, 2005), 230–262. (6.) “What is a harmonic function?”: Daniel Harrison, Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of Its Predecessors (Chicago: Universi­ ty of Chicago Press, 1994), 36; Kevin Mooney, “The ‘Table of Relations’ and Music Psy­ chology in Hugo Riemann's Harmonic Theory” (Ph.D. diss., Columbia University, 1996), 102. Although “harmonic function” has become the standard locution, Riemann consis­ tently describes functions as being “tonal” rather than “harmonic,” and it is his usage that I will be following. (7.) Dahlhaus, “Über den Begriff der tonalen Funktion,” 94. (8.) Riemann, Handbuch der Harmonielehre (1887), 6th ed. (Leipzig: Breitkopf & Härtel, 1917), 214. “Unsere Lehre von den tonalen Funktionen der Harmonie ist nichts anderes als der Ausbau des Fétis'schen Begriffes der Tonalität. Die festgehaltene Beziehung aller Harmonien auf eine Tonika hat ihren denkbar prägnantesten Ausdruck gefunden in der Bezeichnung aller Akkorde als mehr oder minder stark modifizierte Erscheinungsform der drei Hauptsäulen des harmonisch-logischen Aufbaues: der Tonika selbst und ihrer beiden Dominanten.” Cited in Dahlhaus, “Über den Begriff der tonalen Funktion,” 93. (9.) Riemann, “Dominante,” Musik-Lexikon, 8th ed. (1916), 254. “Doch griffen einige The­ oretiker (Daube) früh auf Rameau zurück und stellten die drei Akkorde Tonika, Subdomi­ nante und Dominante als die eigentlichen Pfeiler der tonalen Harmonik fest.” Cited in Alexander Rehding, Hugo Riemann and the Birth of Modern Musical Thought (Cambridge: Cambridge University Press, 2003), 58 (emphasis his). (10.) Jean-Philippe Rameau, Génération harmonique (Paris: Chez Prault fils, 1737), 171. (11.) See Thomas Christensen, Rameau and Musical Thought in the Enlightenment (Cambridge: Cambridge University Press, 1995), 129–132. (12.) Riemann, Harmony Simplified, 7. (13.) Also transcribed in Rehding, Hugo Riemann and the Birth of Modern Musical Thought, 52. (14.) As Rehding notes in Hugo Riemann and the Birth of Modern Musical Thought, 52, this criticism was made even at the time, most notably by Ary Belinfante; see also Henry Klumpenhouwer's and Ian Bent's contributions to this volume. For Riemann's response, see Hugo Riemann and the Birth of Modern Musical Thought, 8. (15.) Riemann, Harmony Simplified, 55; Vereinfachte Harmonielehre, 60–61. “Da nämlich die Dominanten insofern nie vollkommen konsonant sind, als sie stets von der Tonika aus vorgestellt und beurteilt werden (also sozusagen stets mit dieser zusammen), so ist es Page 42 of 51

What is a Function? nicht verwunderlich, dass sie ungleich häufiger als die Tonika mit Zusatztönen er­ scheinen, die ihre Bedeutung noch unzweifelhafter hinstellen und z. B. für die Folge T+– S+ oder °T–°D die Gefahr des Missverstehens (nämlich der Auffassung im Sinne eines zurückgeschehenden [retrograden] schlichten Quintschrittes, also eines | Schlusses, bei dem dann der gemeinte Gegenquintklang Tonika wäre) beseitigen. Diese charakteristis­ chen Dissonanzen sind Töne, die jedesmal der anderen Dominante entnommen sind.” (16.) Riemann describes the dominant in precisely these terms in the “Clarification of Ter­ minology and Notation,” appended to the German edition of Vereinfachte Harmonielehre, v. “Dominante (abgekürzt D) heisst der eine Quint höher als die Tonika liegende Klang” (emphasis added). He gives no definition for the subdominant. (17.) Riemann, Harmony Simplified, 56; Vereinfachte Harmonielehre, 61. “Diese Art der Verbindung einer Dominante mit einem Tone der auf der andern Seite der Tonika gelege­ nen Dominante umschreibt in ähnlicher Weise die Tonart, wie es die Folgen der beiden Dominanten [tun]. Der Zusammenklang von Elementen zweier Klänge, die im Verhältnis zweier Dominanten…stehen, weist ebenso auf den zwischen ihnen liegenden, ihr Ver­ ständnis vermittelnden Klang (also die Tonika) hin, wie die Folgen dieser beiden Klänge.” (18.) Riemann, “Die Natur der Harmonik,” in Sammlung musikalischer Vorträge, vol. 4, ed. Paul Waldersee (Leipzig: Breitkopf & Härtel, 1882), 188. “Denke ich mir den c-Du­ rakkord im Sinne der c-Durtonart, so ist er selbst Tonika, Centrum, schlußfähiger Akkord, seine Vorstellung enthält also nichts seiner Konsonanz Widersprechendes, erscheint ruhig, rein, einfach; denke ich mir dagegen den g-Durakkord im Sinne der c-Durtonart, so denke ich ihn mir als Klang der Oberquinte des c-Durakkordes, d.h. der c-Durakkord selb­ st geht mit in die Vorstellung ein als derjenige Klang, an welchem sich die Bedeutung des g-Durakkordes bestimmt als etwas von ihm Abweichendes—das Centrum der Vorstellung liegt also sozusagen außer ihr.” Cited in Rehding, Hugo Riemann and the Birth of Modern Musical Thought, 71–72. See also Benjamin Steege's contribution to this volume. (19.) Riemann, Musik-Lexikon, 5th ed. (Leipzig: Max Hesse, 1900), as translated in Rehd­ ing, Hugo Riemann and the Birth of Modern Musical Thought, 188–189. (20.) Riemann, “Musikalische Logik” (1872), in Präludien und Studien: Gesammelte Auf­ sätze zur Aesthetik, Theorie, und Geschichte der Musik, vol. 3 (Leipzig: Seeman, 1901), 1. Hauptmann “hat wohl die Zusammengehörigkeit der Akkorde in einer Tonart begriffen, aber durchaus nicht die verschiedene Bedeutung dieser Akkorde gegeneinander, ihre lo­ gische Bedeutung in musikalischem Satzgefüge.” I have translated this and a number of following passages from “Musikalische Logik” with reference to Kevin Mooney's “Musical Logic: A Contribution to the Theory of Music,” Journal of Music Theory 44.1 (2000), 100– 126. (21.) See n. 4 above. (22.) On the dialectics of the große Cadenz, see Harrison, Harmonic Function in Chromat­ ic Music, 266–270; Mooney, “The ‘Table of Relations’ and Music Psychology in Hugo Page 43 of 51

What is a Function? Riemann's Harmonic Theory,” 112–117; Adolf Nowak, “Wandlungen des Begriffs ‘musikalische Logik’ bei Hugo Riemann,” in Hugo Riemann (1849–1919): Musikwis­ senschaftler mit Universalansprach, ed. Tatjana Böhme-Mehner and Klaus Mehner (Cologne: Böhlau Verlag, 2001), 38–43; and Rehding, Hugo Riemann and the Birth of Modern Musical Thought, 68–71. (23.) Riemann, “Musikalische Logik,” 12. (24.) See Kevin Mooney, “Riemann's Debut as a Music Theorist,” Journal of Music Theory 44.1 (2000), 97.; and Rehding, Hugo Riemann and the Birth of Modern Musical Thought, 70. (25.) Riemann, “Musikalische Logik,” 3. (26.) Rehding, Hugo Riemann and the Birth of Modern Musical Thought, 71. (27.) Riemann, “Musikalische Logik,” 3. (28.) Ibid, 4. (29.) See Charles S. Peirce, “The Simplest Mathematics” (1902), in Collected Papers of Charles Sanders Peirce, ed. Charles Hartshorne and Paul Weiss (Cambridge: Harvard University Press, 1933), 4: 195–196. (30.) A remark the OED attributes to John Clark Murray's Outline of Sir William Henry Hamilton's Philosophy (1870). (31.) Harrison, Harmonic Function in Chromatic Music, 11. Harrison returns to the image elsewhere in the book. (32.) Peter Rummenhöller, “Die fluktuierende Theoriebegriff Riemanns,” in Hugo Rie­ mann (1849–1919), 34. (33.) Riemann, Harmony Simplified, 55–56; see n. 17. above. (34.) Riemann, “Musikalische Logik,” 4 and 8. (35.) Riemann, Harmony Simplified, 141; Vereinfachte Harmonielehre, 155. “Unser Haup­ taugenmerk war aber auf die Stärkung des Tonalitätsgefühls gerichtet, auf die grösst­ möglichste Erweiterung des Kreises der noch von einer Tonika aus verständlichen Har­ moniebildungen. Unsere Formeln für die tonalen Funktionen wurden allmählich ziemlich kompliziert, vereinfachten sich jedoch wieder, als wir zur Annahme von Zwischenkaden­ zen übergingen. Es gelang uns in der That, jeden Akkord nicht nur als Umbildung eines Dur- oder Moallakkordes, sondern auch als mehr oder minder gleichwertigen Vertreter einer Tonika, Dominante oder Subdominante zu charakterisieren.” (36.) I will henceforth naturalize these three terms as normal English words. This presents no problems for the Parallele and Variante, both of which have cognates. The Leittonwechsel, however, translates as “leading-tone change,” which has never gained Page 44 of 51

What is a Function? currency. I will use the term in unitalicized lowercase to underline its like status as a tone-relation to the parallel and variant. (37.) See Rehding, Hugo Riemann and the Birth of Modern Musical Thought, 55. (38.) Riemann, “Verwandt,” Musik-Lexikon, 8th ed. (1916), 1176. “Klänge (Dur- oder Mol­ lakkord) sind v. durch ihre konstitutiven Töne, die entweder identisch oder aber auf Quint- und Terzschritte zurückführbar sind; z.B. ist c+ verwandt mit °e durch die Gemein­ samkeit von c und e.” (39.) Ibid. “Quinte und (große) Terz sind neben der Prim die konstituierenden Elemente des Klanges (Dur- oder Mollakkord…). Die Verschiedenheit der Wirkung der Prim, Quinte oder Terz eines Klanges ist eine durchaus spezifische, nicht weiter definierbare.” (40.) Riemann, “Klangvertretung,” Musik-Lexikon, 8th ed. (1916), 553. “Klangvertretung ist die besondere Bedeutung, die ein Ton oder Intervall gewinnt, je nachdem es im Sinne dieses oder jenes Klanges gefaßt wird.” (41.) “Musikalische Logik,” in Geschichte der Musiktheorie im IX–XIX. Jahrhundert (Leipzig: Max Hesse, 1898), 504. “So sind z.B. die “Nebendreiklänge” der Tonart solche gleichzeitige Vertretungen je zweier der drei Hauptharmonien, von denen stets die eine als Hauptinhalt (Konsonanz), die andere als fremder Zusatz (Dissonanz) verständlich ist.” (42.) Riemann, “Konsonanz,” Musik-Lexikon, 8th ed. (1916), 576. “Doch im konkreten Falle [sind] Töne nur dann konsonant, wenn sie wirklich nach dem tonalen Zusammen­ hang als Bestandteile eines und desselben Klanges verstanden werden.” (43.) Riemann, “Dissonanz,” Musik-Lexikon, 8th ed. (1916), 250. “Diese sogenannten Durchgangstöne oder Wechselnoten sind leichter aufzufassen, wenn sie auf rhythmisch leichte, als wenn sie auf schwere Zeiten eintreten, am schwersten, wenn sie nach rück­ wärts nicht vollen melodischen Anschluß haben, sondern springend eintreten. Einige der­ selben führen unter Umständen zur Entstehung von scheinkonsonanten Harmonien, deren momentane Auffassung als wirkliche Klänge besonders reizvolle Nebenformen der Harmonien ergibt, nämlich—a) die Sexte des Durakkordes und die Untersexte des Mol­ lakkordes bei fehlender Quinte (für diese eintretend), ergibt den für den betr. Klang in­ nerhalb der Tonart stellvertretenden Parallelklang. In C dur entstehen so scheinbar der A moll-Akkord (Tp, d.h. Parallelklang der Tonika, Tonikaparallele), D moll-Akkord (Sp) und E moll-Akkord (Dp).—b) Der Leitton (für Dur der von unten [〈] für Moll der von oben [〉]) statt der Prim ergibt den ebenfalls zur Stellvertretung befähigten und in erhöhtem Maße der Harmoniebewegung Reiz gebende Leittonwechselklang (in C dur: = E moll, = A moll, = B moll [!], in A moll: = F dur, = C dur, = B dur [!]).” The bracketed inser­ tions are Riemann's. Rehding translates the entire entry in an appendix to Hugo Riemann and the Birth of Modern Musical Thought, 186–188. (44.) We can of course imagine music that does just that, but in all cases the effect would be an illusion, one that relies on the magical (because arbitrary) mediation of enharmonic equivalence. Page 45 of 51

What is a Function? (45.) On the Harmonieschritte, see Mooney, “The ‘Table of Relations’ and Music Psycholo­ gy in Hugo Riemann's Harmonic Theory,” 237–247; and Henry Klumpenhouwer, “Dualist Tonal Space and Transformation in Nineteenth-Century Musical Thought,” in The Cam­ bridge History of Western Music Theory, ed. Thomas Christensen (Cambridge: Cam­ bridge University Press, 2002), 465–471. (46.) Riemann did include an entry for “Funktion” in the fourth edition of the MusikLexikon (1894), following directly after the publication of Vereinfachte Harmonielehre (1893), but he never repeats it, replacing it in the fifth (1900) and all subsequent editions with an article on “Funktionsbezeichnung.” (47.) Riemann, L. van Beethoven's sämtliche Klavier-Solosonaten: Aesthetische und for­ mal-technische Analyse mit historischen Notizen (1919), 2nd ed. (Berlin: Max Hesse, 1920), 281–282. (48.) As it happens, “What is a function?” is a question asked as often of Frege as it is of Riemann. In addition Frege's asking it himself, Joan Weiner poses the question in Frege (Oxford: Oxford University Press, 1999), 37, as does Danielle MacBeth in Frege's Logic (Cambridge: Harvard University Press, 2005), 79, and Richard L. Mendelsohn in The Phi­ losophy of Gottlob Frege (Cambridge: Cambridge University Press, 2005), 8. (49.) Gottlob Frege, “What is a Function?” (1904), in Peter Geach and Max Black, Transla­ tions from the Philosophical Writings of Gottlob Frege (Oxford: Blackwell, 1952), 107. (50.) In an article that appeared after preparation of the present volume, Trevor Pearce explores Riemann's concept of function in the broader epistemological context of neoKantianism. See his “Tonal Functions and Active Synthesis: Hugo Riemann, German Psy­ chology, and Kantian Epistemology,” Intégral 22 (2008), 81–116. (51.) See, for instance, the article on “Tonbestimmung” in the Musik-Lexikon, 8th ed. (1916), 1133–1138, and the discussion of the Verwandtschaftstabelle in the “Ideen zu ein­ er ‘Lehre von den Tonvorstellungen’ ” (1914), Jahrbuch der Musikbibliothek Peters 21–22. (1914–15), 21–22. On the origins of this notation, see the contribution by Edward Gollin in chapter 9. of the present volume. (52.) See n. 4 above, emphasis added. (53.) See Helga de la Motte-Haber, “Musikalische Logik: Über das System Hugo Rie­ manns,” in Musiktheorie, ed. Helga de la Motte-Haber and Oliver Schwab-Felisch (Laaber: Laaber Verlag, 2005), 203–223. (54.) On Riemann's brief attempt to develop a subject/predicate logic in Musikalische Syn­ taxis (1877), see Rehding, Hugo Riemann and the Birth of Modern Musical Thought, 113– 114. (55.) Of the many introductions to Frege, those I have found most helpful, in order of in­ creasing length and detail, are Michael Dummett's “Gottlob Frege (1848–1925),” in A Page 46 of 51

What is a Function? Companion to Analytic Philosophy, ed. A. P. Martinich and David Sosa (Oxford: Blackwell, 2001), 6–20; Joan Weiner, Frege (op cit); and Michael Beaney, Frege Making Sense (London: Duckworth, 1996). (56.) Historical assessments of Frege's career are given in Terrell Ward Bynum, “On the Life and Work of Gottlob Frege,” in Gottlob Frege: Conceptual Notation and Related Arti­ cles, ed. Terrell Ward Bynum (Oxford: Oxford University Press, 1972), 1–54; and Hans D. Sluga, Gottlob Frege (London: Routledge & Kegan Paul, 1980). (57.) Gottfried Gabriel considers Frege in relation to neo-Kantianism in “Frege, Lotze, and the Continental Roots of Early Analytic Philosophy,” in From Frege to Wittgenstein: Per­ spectives on Early Analytic Philosophy, ed. Erich H. Reck (Oxford: Oxford University Press, 2002), 39–51. (58.) On Riemann's dissertation and the Göttingen affair, see Michael Arntz, Hugo Rie­ mann: (1849–1919): Leben, Werk und Wirkung (Cologne: Concerto, 1999), 67–69. On Lotze's ideas as background, see Rehding, Hugo Riemann and the Birth of Modern Musi­ cal Thought, 118. (59.) In view of the number of times Frege is cited in Russell and Whitehead's Principia Mathematica (1913) and Wittgenstein's Logisch-Philosophische Abhandlung (1921), what have become almost obligatory assertions concerning Frege's obscurity seem overstated; if indeed few people read Frege, the same cannot be said of Russell or Wittgenstein. (60.) Rehding considers Frege in relation to Riemann in Hugo Riemann and the Birth of Modern Musical Thought, 82–85. (61.) “Über den Begriff der tonalen Funktion,” 94. (62.) Ernst Kirsch, Wesen und Aufbau der Lehre von den harmonischen Funktionen: Ein Beitrag zur Theorie der Relationen der musikalische Harmonie (Leipzig: Breitkopf und Härtel, 1928), 13. (63.) Frege, “What is a Function?” (1904), in Translations from the Philosophical Writings, 107. (64.) Ibid, 110. (65.) Ibid, 115. (66.) Ibid, 115. (67.) Frege, “Function and Concept” (1891), trans. Peter Geach (1952), repr. in The Frege Reader, ed. Michael Beaney (Oxford: Blackwell, 1997), 131. (68.) Ibid, 131–132. (modified). (69.) Ibid, 133–134. Page 47 of 51

What is a Function? (70.) Ibid, 134. (71.) Ibid, 140. (72.) Ibid, 139. (73.) Frege, “On Concept and Object” (1892), trans. Peter Geach (1952), reprinted in The Frege Reader, 193. (74.) Frege, “Function and Concept,” 140. (75.) Ibid. (76.) Frege, “What is a Function?,” 115. (77.) See n. 8 above. (78.) See n. 39 above. (79.) Frege, “What is a Function?,” 113. (80.) Ibid, 114. (81.) See Frege, “Function and Concept,” 139. (82.) In Frege, “Comments on Sinn and Bedeutung” (1892), trans. Peter Long and Roger White, reprinted in The Frege Reader, 174–175. (83.) FREGE, “On Concept and Object,” 182. (84.) Ibid, 186. (85.) Ibid, 184. (86.) Ibid, 185. (87.) Ibid, 192. (88.) Frege, The Foundations of Arithmetic (1884), trans. J. L. Austin (Oxford: Blackwell, 1953), x. (89.) See Frege, “Function and Concept,” 139. And succinctly in “Comments on Sinn and Bedeutung,” 173. “On the view expressed [in ‘Function and Concept’] a concept is a func­ tion of one argument, whose value is always a truth-value.” (90.) Frege, “On Sense and Reference” (1892), trans. Max Black (1952), repr. in The Frege Reader under the title “On Sinn and Bedeutung,” 159. (91.) Frege, “On Concept and Object,” 189. (92.) Frege, “Function and Concept,” 148. Page 48 of 51

What is a Function? (93.) Frege, “On Sense and Reference,” 152. (94.) FREGE, “Function and Concept,” 138. (95.) Ibid, 132. (96.) I draw here on Michael Beaney's discussion of “The Translation of Bedeutung” in his introduction to The Frege Reader, 36–46. Issues of translation in Frege have garnered a secondary literature of their own. (97.) Bertrand Russell, “On Denoting” (1905), in Logic and Knowledge: Essays 1901– 1950, ed. Robert Charles Marsh (London: Allen & Unwin, 1956), 39–56. (98.) Dahlhaus, “Über den Begriff der tonalen Funktion,” 94. (99.) See David Lewin, “Music Theory, Phenomenology, and Modes of Perception,” Music Perception 3.4 (1986); reprinted in Lewin, Studies in Music with Text (New York: Oxford University Press, 2006), 66–67. (100.) David Lewin, “Amfortas's Prayer to Titurel and the Role of D in Parsifal: The Tonal Spaces of the Drama and the Enharmonic C♭/B,” 19th-Century Music 7.3 (1984), 345; reprinted in Lewin, Studies in Music with Text, 194. (101.) Ibid, 196. (102.) Frege, “On Sense and Reference,” 156. (103.) “Thought” (1918), trans. Peter Geach and R. T. Stoothoff (1984), reprinted in The Frege Reader, 326. (104.) Ibid, 327. (105.) Ibid. (106.) Ibid, 328. (107.) Ibid, 341. (108.) Frege, “On Sense and Reference,” 154 (modified). (109.) Frege, “Thought,” 334. (110.) Ibid, 335. (111.) Frege, “On Sense and Reference,” 155. (112.) Ibid. (113.) Frege, “Thought,” 341. (114.) Ibid, 330–331. Page 49 of 51

What is a Function? (115.) Frege, “Function and Concept,” 141. (116.) Frege, “Comments on Sinn and Bedeutung,” 178. (117.) Frege, The Foundations of Arithmetic, x. (118.) Frege, “Thought,” 325. (119.) For an over-the-top exposition, see Charles Travis, “Psychologism,” in The Oxford Handbook of Philosophy of Language, ed. Ernest Lepore and Barry C. Smith (Oxford: Ox­ ford University Press, 2006), 103–126. (120.) Frege, “On Concept and Object,” 181. (121.) Riemann, Harmony Simplified, 141. (122.) Rehding, Hugo Riemann and the Birth of Modern Musical Thought, 57. (123.) I am thinking here of neo-Riemannianism, which has its origins in David Lewin's “A Formal Theory of Generalized Tonal Functions,” Journal of Music Theory 26.1 (1982), 23– 60, and “Amfortas's Prayer to Titurel and the Role of D in Parsifal” (op. cit.). Neo-Rieman­ nianism received further impetus in some of Lewin's other writings as well as in numer­ ous publications of Richard Cohn, who have the movement its name. A notable and impor­ tant exception to the rule is Daniel Harrison's “genetic” interpretation of tonal function in Harmonic Function in Chromatic Music. (124.) Riemann, “Die Natur der Harmonik,” 188. See n. 18 above. (125.) See nn. 106 and 107 above. (126.) On the broader disciplinary ambitions of “Die Natur der Harmonik,” see Rehding, Hugo Riemann and the Birth of Modern Musical Thought, 89–91. (127.) Riemann, “Natur der Harmonik,” 159–160. (128.) Vorstellung, in German, ranges wider in meaning than any one of the translations we might reasonably assign it: “idea,” “conception,” “presentation,” “representation,” “imagination,” “image,” all of which I will be drawing on; it is not possible to give it a con­ sistent translation without misconstruing Riemann. For a nuanced discussion, see Mooney, “The ‘Table of Relations’ and Music Psychology in Hugo Riemann's Harmonic Theory,” 130–135. On the notion of Tonvorstellung more generally, see Klaus Mehner, “Hugo Riemanns ‘Ideen zu einer Lehrer von den Tonvorstellungen,’ ” in Hugo Riemann (1849–1919), 49–57. (129.) Rehding, Hugo Riemann and the Birth of Modern Musical Thought, 160. (130.) “Ideen zu einer ‘Lehre von den Tonvorstellungen’ ” (1914), Jahrbuch der Musikbib­ liothek Peters 21–22 (1914–1915), 1–26, trans. Robert Wason and Elizabeth West Marvin

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What is a Function? as “Ideas for a Study ‘On the Imagination of Tone,’ ” Journal of Music Theory 36.1 (1992), 81–117. (131.) Ibid, 1. (132.) Ibid, 2. See n. 136 below. (133.) Frege, The Foundations of Arithmetic, 21. (134.) See Weiner, Frege, 13. Weiner uses the idea to introduce the following citation (on p. 14), which is given here in an expanded form. (135.) Frege, Foundations of Arithmetic, 20. (136.) Riemann, “Ideen zu einer ‘Lehre von den Tonvorstellungen,’ ” 20. (137.) Rehding, Hugo Riemann and the Birth of Modern Musical Thought, pp. 166–167, modified to change the singular to the plural. (138.) Riemann, “Ideen zu einer ‘Lehre von den Tonvorstellungen,’ ”2. “Nicht die wirklich erklingende Musik sondern vielmehr die in der Tonphantasie des schaffenden Künstlers vor der Aufzeichnung in Noten lebende und wieder in der Tonphantasie des Hörers neu erstehende Vorstellung der Tonverhältnisse [ist] das Alpha und das Omega der Tonkunst. Sowohl die Festlegung der tonkünstlerischen Schöpfungen in Notenzeichen als die klin­ gende Ausführung der Werke sind nur Mittel, die musikalischen Erlebnisse aus der Phan­ tasie des Komponisten in die des musikalischen Hörers zu verpflanzen. Hat man diese grundlegenden Gedanken begriffen, so leuchtet ein, daß die induktive Methode der Ton­ physiologie und Tonpsychologie von Anfang an auf einem verkehrten Wege geht, wenn sie ihren Augsang nimmt von der Untersuchung der Elemente der klingenden Musik, anstatt von der Feststellung der Elemente der vorgestellten Musik.” (139.) Frege, “Thought,” 328. (140.) Ibid, 329.

Brian Hyer

Brian Hyer is a professor of music at the University of Wisconsin, Madison. He has written widely on the anthropology of European music and its theories from the eigh­ teenth through the twentieth centuries.

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Riemann and Melodic Analysis: Studies in Folk-Musical Tonality

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality   Matthew Gelbart and Alexander Rehding The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0004

Abstract and Keywords This article discusses Riemann's Folkloristische Tonalitätsstudien or “Studies in Folk-Mu­ sical Tonality”. While his study at a glance seemed to contradict his earlier beliefs on tonality as a natural and universal system, his study however, sought to reinforce the nat­ ural basis of the tonal system by looking and examining non-European repertoires. Riemann's Folkloristische Tonalitätsstudien is unique among his studies in that it is his only treatise that does not start out from the assumption of tonal triadic harmonies; in­ stead, it approaches musical structures from a strictly melodic angle. In this article, the focus is on Riemann's melodic analysis and evolutionary history of scalar models. It focus­ es on his analysis of pentatonicism and tetrachords. In his Folkloristische Tonalitätsstudi­ en, Riemann is generally careful not to commit to a chronology, but supplies instead heuristic links between structural stages. While his explanation of tetrachords as threenote fragments of pentatonic scales with infixed pien is historically doubtful, his study nevertheless allowed him to examine all scalar systems as combinations of these two prin­ ciples. Thus, his study on folk music has been pressed into service to represent a middle ground between strict pentatonicism and modern diatonism. Keywords: Folkloristische Tonalitätsstudien, tonal system, musical structures, melodic angle, melodic analysis, scalar models, pentatonicism, tetrachords, folk music

ONE of the last satisfactions of Hugo Riemann's career was his appointment as director of the Royal Research Institute for Musicology at the University of Leipzig in 1914.1 Two years later Riemann's study Folkloristische Tonalitätsstudien, or “Studies in Folk-Musical Tonality” (1916), appeared as the first volume of the Institute's series of monographs. The question of music from outside the common-practice repertoire had already occupied him for some twenty years; during this time he had published articles on Japanese music, Byzantine chant, and he had arranged a number of “Original Chinese and Japanese songs” for violin and piano. To extend and deepen this foray into “world music” seemed fitting for an important occasion such as the inaugural publication of his research insti­ tute. Page 1 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality The scope of Riemann's study was extraordinarily ambitious—in addition to the Scottish, Irish, Welsh, Scandinavian, and Spanish songs and the Gregorian chant that the subtitle of his book lists, he also discussed examples from Chinese and Japanese music and princi­ ples of ancient Greek music theory. And yet, given that Riemann spent much of his career defending tonality as a natural and universal system, it may seem surprising that late in life Riemann would turn to non-European music. Was he really going to throw his firmly held beliefs overboard? In fact, the motivation for Riemann's study of music from differ­ ent cultures, ironically, was largely to stem the tide of musical ethnography. The “compar­ ative method” of studying music was gaining ground in the early twentieth century— thanks, in no small part, to recent improvements in recording technology, namely the phonograph. In Folkloristische Tonalitätsstudien (p. 141) Riemann railed particularly against this machine and the scientific observations made on the basis of its recordings. He concludes his study with a stern prediction: In light of the role that phonographic recordings play in the young science of mu­ sical ethnography, we must also point out that the transcription of a melody ac­ cording to the recording requires a well-trained musician, but also familiarity with the tonal system to which the melody belongs…. Studies such as the present ones serve in the first place to improve the training of the ear for a fuller understand­ ing of the structure of melodies. Once these studies are deepened, presumably not a lot will be left of the intervals that contradict our musical system, such as 3/4 or 5/4 tones, or the so-called “neutral” thirds that tone psychologists now believe to hear out of phonographic recordings.2 One of the most important centers of comparative musicology was the Berlin school around Carl Stumpf and Erich Moritz von Hornbostel,3 and it is against those scholars that Riemann polemicizes here—though, as was his wont, without mentioning any names. In other words, by looking into new repertoires, Riemann was not contradicting his earli­ er beliefs about the natural basis of the tonal system, but ultimately hoping to reinforce them. In fact, what the comparative musicologists had done was to raise the stakes for Riemann: if he wanted to uphold his claim that his systematic views of music were indeed universal and natural—and not just historically and geographically limited, as compara­ tive musicologists suggested—then he had to tackle music outside the European tonal mainstream head-on to show how his principles still applied. The late Folkloristische Tonalitätsstudien is unique among Riemann's theoretical studies in that it is his only treatise not to start out from the assumption of tonal, triadic har­ monies; instead, it approaches musical structures from a strictly melodic angle. This star­ tling change has moved some commentators to speculate that, had Riemann lived longer, he would have fully reconsidered the harmonic foundations of his musical worldview.4 While his death in 1919 makes a definitive answer virtually impossible, it is worth remem­ bering that such a melodic approach was, on one level, a necessity: if Riemann was to consider monophonic repertoires in their own right, his usual triadic approach—on the basis of the principle of Klangvertretung, the assumption that each pitch was representa­ tive of a triad—would not get him very far. In the past, it is true, he had followed the pop­ Page 2 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality ular tradition of supplying East-Asian melodies with triadic harmonies.5 Here, however, the very possibility of Klangvertretung was in question; only by suspending this principle, at least temporarily, could Riemann counter the challenge of the comparative musicolo­ gists.

Sources and Predecessors In Folkloristische Tonalitätsstudien Riemann took up the challenge from the comparative musicologists by attempting to broaden his own evolutionary prehistory of the diatonic system. The historiographic model he used was the same Whiggish (p. 142) model that had already served him so well in his Geschichte der Musiktheorie and his Handbuch der Musikgeschichte, where the gradual emergence and shaping up of an idea is documented by any historical evidence available. While Riemann was hardly alone in thinking about history in this systematic, teleological way, which was rather typical of his age, Folkloris­ tische Tonalitätsstudien stands out from the many accounts that it echoes by incorporat­ ing some of Riemann's favored models of thinking. It may seem that, in not starting out with harmony, Riemann was ready to jettison the du­ alistic foundations of his musical thought, which he had so rigorously defended. This, however, would probably mean to take the impact of Folkloristische Tonalitätsstudien a little too far. For, as we shall see, the basic principles of mirror symmetry on which his no­ tion of harmonic dualism was based was still firmly in place in the principles guiding Folk­ loristische Tonalitätsstudien. In many ways, in fact, Riemann's desire for symmetry was the most original and personal element he brought to a narrative that otherwise largely followed paths well worn by the time he was writing. Even Riemann's focus on melody was not so much a departure from his earlier thinking as it may seem. Rather, this approach was partly the result of precedent and even neces­ sity: in the history of European writing about non-Western and “folk” music, scale-based theories had predominated to the point of near exclusion of other approaches to the mu­ sic. What is more, while he eschewed harmonic dualism in this study—in favor of what could be dubbed “scalar” dualism—his justification was quite simply that he was here sig­ nificantly concerned with the emergence of the major third, the crucial element that made triadic harmony possible. In this sense, his study of ancient and non-Western music formed a prehistory to the modern major-minor diatonic system, and is implicitly and in parts explicitly conceived as such. As part of this evolutionary model, Riemann's scalar discussions turn out to fit comfortably into his earlier thought. Indeed, they expand directly from that earlier writing. Many of the ideas and details in Folkloristische Tonalitätsstudien go back to Riemann's Handbuch der Musikgeschichte (1904–1913), and before that to articles he had written on Japanese and other non-West­ ern musics (1902 and 1906).6 In these articles and books, Riemann had laid out his belief that the oldest ancient Greek scale was an anhemitonic pentatonic one.7 From this an­ cient anhemitonic manifestation, Riemann asserted, the scale would develop into a more “advanced” heptatonic gamut (and then later develop also hemitonic pentatonic forms). In Page 3 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality other words, the introduction of half steps into the anhemitonic pentatonic ur-scale would ultimately, if slowly, lay the groundwork for modern harmony and tonality. Already in the Handbuch, Riemann had incorporated into his narrative the idea that Greek, Japanese, and Chinese music shared the same path, and he hinted that Scottish and other Celtic and Scandinavian music followed a similar line as well.8 All these claims come center stage in Folkloristische Tonalitätsstudien, where this “prehistory” of universal tonal prin­ ciples itself becomes Riemann's focus. This rhetorical use of multiple parallel examples from different countries to imply that there was a natural, teleological evolution from pentatonicism toward (p. 143) diatonicism and common practice tonality was, however, not only a preexisting tool within Riemann's own writing on non-Western and “ancient” music before Folkloristische Tonalitätsstudien; it was also a common trope going back over a century even when Riemann was writing. Riemann was thus drawing on several established narratives and theories: about penta­ tonicism and nature, about connections between ancient Greek, Chinese, and Scottish music, and about music's inevitable course of development along a path from nature to civilization. The progenitor of such theories in full-grown form was Charles Burney. Near the start of his four-volume General History of Music (1776–1789), Burney had advanced an interpre­ tation—worked out by himself and his friend Thomas Twining—of a crucial passage in (Pseudo)-Plutarch's De Musica. Primarily, Burney tackled Plutarch's claim that the Greek Enharmonic genus was the oldest among the three genera (diatonic, chromatic, enhar­ monic). This statement appeared counterintuitive since the diesis (quarter tone) that de­ fined the enharmonic tetrachord seemed to many like an inherently “late” development in Greek theory. Burney explained that he read Plutarch to mean that before the enharmonic genus came to be defined by its diesis, it was already marked by a “gap” in the notes of the tetrachord.9 He went on to reconstruct a scale based on this version of the enharmon­ ic tetrachord, and called it the “old enharmonic scale.”10 The scale Burney posited was a hemitonic pentatonic scale; in his version it runs (descending) D–B♭–A–F–E–D. For writers in Burney's wake, the most salient feature of Burney's discussion was not so much his re­ construction of this “gapped” scale as the proto-Greek paradigm, but rather the cross-cul­ tural comparisons he brought to bear on it. When Burney presented the Greek scale re­ construction, he immediately followed it with the declaration: “Now this is exactly the old Scots scale in the minor key.”11 Having roped in Scottish music, he asserted that Chinese music used the same scale as well. Alone, these connections might have been dropped as passing dilettantish observations, but Burney provided an explanation that turned out to be irresistibly tantalizing to his readers and followers. He wrote: It is not my intention to insinuate by this that the one nation had its music from the other, or that either [China or Scotland] was indebted to to ancient Greece for its melody; though there is a strong resemblance in all three. The similarity how­ ever, at least proves them all to be more natural than they at first seem to be, as well as more ancient. The Chinese are extremely tenacious of old customs, and equally enemies to innovation with the ancient Aegyptians, which favours the idea Page 4 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality of the high antiquity of this simple music; and as there is reason to believe it very like that of the most ancient Greek melodies, it is not difficult to suppose it to be a species of music that is natural to a people of simple manners during the infancy of civilization and arts among them.12 Burney's connections between Greek, Chinese, and Scottish music—based on the idea that the pentatonic scale was “natural”—were echoed quickly and widely by Orientalist writers and “universal” music historians, though almost all immediately corrected Burney's calculations or otherwise adjusted his theory in order to (p. 144) posit that the shared “natural” scale was an anhemitonic pentatonic scale (that is, D–B–A–F♯–E–D de­ scending, or transposed, the black notes on the piano) rather than the hemitonic version Burney had used as the shared scale.13 Thence followed the increasingly widespread idea that the heptatonic scale, with its two half-step intervals including a leading tone, was a natural extension and development, formed through filling in the gaps of the “universal, primitive” anhemitonic pentatonic scale. Common-practice tonality was the final step in this teleological narrative. It is unclear exactly how much of Burney's discussion Riemann knew and when, or why, he ignored it when he wrote his own interpretation of Plutarch in the first volume of his Handbuch der Musikgeschichte.14 In principle, Burney should have been quite familiar to Riemann, and he would refer to Burney's history vaguely at least in the next volume of his Handbuch.15 Furthermore, in Johann Nikolaus Forkel's Allgemeine Geschichte der Musik, Forkel had quoted Burney's discussion at several pages’ length in order to support parts of Burney's findings and quibble with other aspects of his reading of Plutarch16—and Rie­ mann had certainly read Forkel. Yet in his own discussion of the enharmonic and early Greek scales, he cited neither writer. To be sure, theories of scales developing from an­ hemitonic pentatonicism toward tonal heptatonicism had become so widespread by the mid-nineteenth century that countless writers gave very similar interpretations of Plutarch's enharmonic discussion and noted very similar cross-cultural comparisons with­ out even citing Burney anymore. Sometimes there was a clear path back to Burney via in­ termediate sources. In other cases, such as that of Carl Fortlage,17 the source influences are somewhat more obscure. Helmholtz, for one, did not cite Burney, but his reconstruc­ tion of the old Olympian scale as a hemitonic pentatonic scale was actually closer to Burney's original ideas than those of others at the time who explicitly referred back to Burney.18 Riemann must have been familiar with more of this well-known discourse than he let on when he proceeded to his own dissection of the much-chewed-over passage in Plutarch. In his own account, Riemann explained that by attributing the invention of the enharmon­ ic to Olympos, it might seem Plutarch was claiming that Olympos had been the first to split the half tone into smaller units. However, Riemann maintained, the rest of Plutarch's explanation clearly showed that there had been an older “enharmonic” characterized by gaps in the scale rather than quarter-tone infills within the smaller intervals.19 Not only does Riemann's rhetoric and presentation echo Burney patently,20 but Riemann even dis­ misses the same parenthetical aside in the Plutarch (as extraneous and erroneous) that Page 5 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality Burney had dismissed—both did so in order to reach their conclusions that the old enhar­ monic scale was free of microtonal intervals.21 Yet Riemann did not cite Burney or other predecessors who echoed similar interpretations. Riemann, we know, was a thinker prone to systematic thought, and musings such as those of Burney and many of those who fol­ lowed him may have seemed unsystematic and dilettantish enough to ignore as “real” thought on the issue. Furthermore, Riemann's scale (p. 145) reconstruction differed from Burney's—as we have seen—by positing the old enharmonic scale as an anhemitonic rather than hemitonic pentatonic construction. Those writers who had developed the most systematic explanations of anhemitonic pentatonic scale systems in Greek, Chinese, and British folk music, while openly building on Burney's writing, were Scots, whose Eng­ lish-language work may have remained less known to Riemann.22 In any case, the only predecessors that Riemann did address directly were those whose completely different in­ terpretations he discarded (Fétis), and those whose work he considered “on the right path” (for associating the old enharmonic with the anhemitonic pentatonic), but not quite fully worked out (Fortlage, Bellermann, and Gevaert).23 Riemann also went beyond other German writers in supplementing his reading of Plutarch with a good deal of evidence from other sources. From our current perspective, it is ironic that Riemann's reconstruction itself was every bit as anecdotal and his logic every bit as questionable as Burney's, even if his resulting theories are much more systematized (which perhaps signals an even bigger logical leap than Burney's). For example, it is unclear on what grounds Riemann asserts that the hemitonic pentatonic version of a later Greek period (“deutera archaic period”) could have developed only after a middle era in which the anhemitonic scale had evolved into a full heptatonic scale.24 Riemann's general evolutionary logic from simple to complex is not borne out by the historical record: Plutarch himself had argued that the Olympian en­ harmonic genus was formed by the omission of previously existing notes rather than con­ stituting a true proto-scale.25 Riemann dismisses this element of Plutarch's reasoning, os­ tensibly because other sources indicated to him that the early Greek era was not ad­ vanced enough to have a seven-tone scale yet.26 Such tautological reasoning is typical of the long tradition of using examples of scales developing similarly across different cul­ tures, while ignoring certain data or sources that did not square with the theory. Al­ though Riemann, like Burney and many writers in between, sometimes suggested causal, contextual, or evolutionary links between disparate events, he was often not too worried about the links. On the contrary, he was convinced that the more often an idea flared up in different circumstances the more likely it would be to contain a kernel of truth that would eventually come to the fore.27 In sum, regarding Riemann's relationship to his the­ orizing predecessors, it is hard to know what is suppression, what is ignorance, and what work Riemann simply considered unworthy of comment or acknowledgment. But it re­ mains curious how little the well-read Riemann cited his forbears in outlining his narra­ tive of an implied universal scale development. It is especially striking that Riemann did not call on his precursors for support, consider­ ing the defensive context in which he wrote Folkloristische Tonalitätsstudien: as we re­ call, the whole project was largely conceived as a critical response to the new theories of Page 6 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality the Berlin comparative musicologists, who based their findings on phonograph record­ ings. Given the nature of the new theories against which Riemann was reacting, it might have helped him to cite his many intellectual antecedents rather than trying to stand alone against a modern trend.

(p. 146)

Intellectual Antagonists

What exactly set Riemann going with Folkloristische Tonalitätsstudien? The reason the findings and theories of the Berlin comparative musicologists irked Riemann was that they wreaked havoc with the kinds of musical universals on which Riemann's theories most depended. Historically, there had been two primary ways of asserting natural uni­ versality for musical systems. The oldest way was to assert that the universals of music lay outside the human mind. This idea went back to Pythagorean theory, to the “harmony of the spheres,” and to Rameau's interest in the corps sonore and the overtone series. However, this path had largely been subordinated from the later eighteenth century by attempts to find music's universals in human nature and development. When Rousseau linked the origin of music to human proto-language, he was throwing down a gauntlet not only to Rameau but to a whole long-standing apparatus of music theory and historiogra­ phy. Rousseau was eagerly echoed and followed, with constantly expanding variations, by Herder and many others. It was this conception of universal human mental patterning manifest in music that led to theories such as Burney's about societies passing through predictable early phases in which they would stumble on scales hardwired into the hu­ man mind, and carried through into Darwin's claims about the origins of language in song.28 The acoustic experiments of Helmholtz seemed to swing the pendulum back in the extrahuman direction. At first, Helmholtz and his English translator Alexander Ellis, who also conducted his own musical experiments, must have tempted Riemann with their work, in a positivistic age, on restoring the extrahuman scientific basis of music. To musicians and critics such as Riemann who were invested in nineteenth-century Ger­ man music, however, Helmholtz's and Ellis's results were less tempting than their meth­ ods—for in their manners, both Helmholtz and Ellis concluded that despite increasingly secure understanding of the acoustical and physical properties of sound, human musical activity was somewhat arbitrary, that there was little natural justification for one musical system over another.29 Following Ellis, the Berlin comparative musicologists were out­ wardly skeptical about the existence of universal pentatonic and diatonic scales—ones so engrained in the human brain that they would at some point become the bases for music in any culture. This skepticism seemed to harden as the growing phonographic archive of music collected from all over the world, coupled to more accurate pitch measurements, showed a greater variety of scales and systems than earlier writers had known or ac­ knowledged.30 Any characterization of the Berlin school's work as moving cleanly away from universaliz­ ing explanations, however, would be overly simple, for the music psychologists and com­ parative musicologists such as Stumpf and Hornbostel still hoped and believed they Page 7 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality would find universals on which to base their study of musical development in different cultures. One way they did so was to hypothesize—in the face of the new evidence from around the world—new ways of grounding all music in a natural extrahuman scientific system. (Such a system was von Hornbostel's rather far-fetched theory of “blown fifths.”)31 More flexible and intriguing was Carl (p. 147) Stumpf's idea, set forth in The Be­ ginnings of Music (1909), that acoustical and human physiological bases of music might combine to shape a combination of recurrent features and tastes of musicians around the world while still leaving room for more arbitrary elements. Stumpf hypothesized that the octave, and to a lesser extent intervals such as the fourth and fifth, were naturally used for fusing and carrying voices in group shouts across space, and that these psychophysio-acoustic universals worked their way into a variety of systems with different infill­ ing notes and properties.32 Riemann initially found Stumpf's ideas extremely helpful, but he was troubled by the fact that it implied that his apparently universal triadic harmony was only one arbitrary outgrowth of a more basic situation.33 Riemann's position is unique in its own mix of acoustical and psychological justifications —driven by his joint desire to assert the “rightness” of triadic tonality and of harmonic dualism. In fact, Riemann's reasoning changed over time: his own personal journey showed a move from attempts to justify his music theory in objective extrahuman terms to hard physiological terms to a focus on tone psychology.34 This personal trajectory led to Riemann changing his arguments justifying harmonic dualism, but it also affected his late thinking about the scalar fundamentals of music. By the time he was writing Folk­ loristische Tonalitätsstudien, he had come to regard the phonograph as a false objec­ tivism—a device that recorded sounds as they existed acoustically in particular instances rather than as they should be or as they were perceived, processed, and understood. In the introduction to Folkloristische Tonalitätsstudien, he thus enters into an extended criti­ cism of relying on the phonograph for “evidence” about scales and modes in different cul­ tures—asserting his conclusion that the real “natural” elements in music are psychologi­ cal rather than physical—or at least resulted from a psychological filter though which acoustic phenomena and physiological sound creation and reception passed.35 This al­ lowed him to begin again from the position that, contra the comparative musicologists, the diatonic scale was indeed a musical universal. In general, Riemann was concerned in­ creasingly with protecting what he regarded as a clear, natural, and universal set of rules —which, not at all coincidentally, were specifically the rules that guided the nineteenthcentury German music he prized. (In fact, he framed the entire book by starting out with a claim that “national” and “nationalist” music [read: non-German nationalist music] had run its course, having already given way once again to a broader, universal [read: Ger­ man] current.)36 He was thus defending this nineteenth-century German canon implicitly, even in his work on other musics, by showing how those other musics either contributed to a demonstration of the apparently universal rules on which the music he treasured was based, or deviated from those rules. (And, one might add, spinning out Riemann's way of thinking, if they deviated, so much the worse for those types of music.) All this, ultimately, set him along lines more similar to Burney's goals and those of other pre-Helmholtz writ­ ers who were primarily concerned more with describing and prescribing the course of Page 8 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality music history to conform with their own world views than with researching sounds and sound cultures from around the world for acoustical, comparative, or scientific reasons. We have already considered what Riemann's work shared with these earlier, “nor­ mative” studies: like them, he dwelled on a universalizing narrative of scale development beginning with pentatonicism, and charted a development of scales toward a “modern” (proto-)tonal form. Riemann's narrative, again like those of his predecessors, aspired to being a narrative of all music, teleologically leading toward the rules and prac­ tices of the music he most valued—so he implied that every musical culture had to under­ go the same stages. And he chose the same examples as they had done (Chinese, Greek, and Scottish) to suggest that all cultures had passed through this stage, or in some cases were still passing through it. In this spirit, Riemann spoke of Olympos as “the main repre­ sentative of an epoch…which created melodies with the homely simplicity that we know from ancient Chinese and Japanese, as well as Celtic melodies.”37 For all these reasons, Riemann might have gained by acknowledging his debt to a past discourse when he framed the narrative aspect of Folkloristische Tonalitätsstudien. Yet, for precisely the same reasons, he may also have been moved to hold back—for fear of appearing to (p. 148)

present an argument that was the product of an age before the modern acoustical-scien­ tific rigor of Helmholtz and the Berlin school.

Riemann's Narrative of Scalar Evolution So far, we have been discussing how Riemann responds to a long established discourse. In the body of Folkloristische Tonalitätsstudien itself, Riemann adds his personal stamp to this abstract background. He brings his preferences for symmetrical conceptualization to bear on tetrachordal theory, and he pursues his long-standing quest for the elusive “pure minor mode” as part of his evolutionary narrative. The diatonic scale whose evolution Riemann's study sketches out in the book consists of three major structural elements: 1. Pentatonicism 2. Tetrachords 3. Inserted semitones In order to make his argument as strong as possible—and in order to bolster his univer­ salist agenda—Riemann explores these structural elements in different musical cultures. As for the first two, he examines pentatonic structures in East Asian and Celtic reper­ toires, and tetrachords in ancient Greek music and Gregorian chant. Both are different organizing principles for scales, but Riemann argues that both need to coexist conceptu­ ally to allow the formation of the modern diatonic scale. As we will see, the binding glue between these disparate musical systems—paving the path from one to the other, and ul­ timately on to the modern diatonic scale—is provided with the emergence of the third (and the concomitant scalar interval of the semitone), which in the guise of the leading tone finally makes fully fledged diatonicism possible. Unifying factors (p. 149) such as Page 9 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality chronology or geographical contiguousness do not play a big part in Riemann's considera­ tions: examples drawn from Gregorian antiphons can stand next to snippets of Celtic song.38 What unites these repertoires is that they are all highly developed monophonic systems that help form a prehistory for Riemann's main narrative of Western music. Folk­ loristische Tonalitätsstudien is Riemann's attempt to consider these repertoires in greater depth for their melodic properties. And each repertoire that Riemann draws upon in the construction of his argument delivers further necessary elements that he required to doc­ ument his evolutionary narrative. Riemann begins his observations with a study of anhemitonic pentatonicism. As in earlier theories seeking to find the same pentatonic to diatonic or chromatic path in multiple het­ erogeneous cultures in China, Japan, Greece, and the British Isles, historical accuracy of­ ten fell victim to the urge to project an apparently natural progression onto these differ­ ent music histories. Drawing on Chinese and Japanese derivations of the pentatonic scale, Riemann explains the foundation of anhemitonic pentatonic scales as chains of fifths ex­ tending symmetrically around a central pitch. He gives three examples:

Example 4.1 shows how these chains of fifths are then collapsed into close position— positing the principle of octave equivalence—to create pentatonic scales. While the idea of deriving anhemitonic pentatonic scales from chains of fifths has a long and distin­ guished pedigree, going back to Rameau and Abbé Roussier in the eighteenth century, Riemann gives this story a symmetrical twist: where other writers had based the progres­ sion on the bottom pitch (which acted as a son générateur) and considered this the root of the scale, Riemann turned his attention to the central pitch. In accordance with his dual­ istic principles, Riemann considers this central pitch the “root,” and adds the other scalar elements upward and downward in symmetrical opposition around this central pitch. This symmetrical principle applies both when he presented the pitches as stacked fifths and when he collapsed the pitches into a scale contained in a single octave.

Ex. 4.1. Riemann's symmetrical derivation of an­ hemitonic pentatonic scales, from Folkloristische Tonalitätsstudien, p. 3.

When treating the collection as a scale, Riemann employs his familiar dualistic taxonomy of Roman numbers signifying the minor principle (counting down from the “root”), and Arabic numbers signifying the major principle (counting upward). The circle with the dot ☉ signifies the pitch that Riemann designates as the center of the scale. Riemann introduces two theoretical terms that he adopts from ancient Greek the­ ory. First, Riemann calls the central pitch ☉ of the pentatonic formation mese, after the fixed middle reference pitch of the Greek theoretical tradition. He regards the mese as “playing the part of the tonic.”39 It is noticeable that for the most part Riemann tends to (p. 150)

Page 10 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality avoid the term and quietly employs the symbol ☉ instead—presumably because many of the theoretical implications of the Greek term do not match Riemann's agenda. Second, the central complex of mese plus its upper and lower neighbors Riemann calls, again re­ coining a Greek term, a pyknon, in this case, a “major-third pyknon.” In Greek theory, the pyknon generally refers to the pair of smaller intervals of the tetrachord in the chromatic and enharmonic genera. Riemann's use of the term here refers, rather liberally, to the three central notes in the middle of his pentatonic formations that are separated by whole-tone steps. In other words, pyknon refers to the II–☉–2 complex in example 4.1. The conclusion that Riemann's use of these Greek terms is tendentious is difficult to deny; it is, however, anything but arbitrary: the Greek terms lend gravitas and legitimacy to Riemann's theory. Furthermore, in applying terms from tetrachordal theory to pentatonic structures, he implies an integral connection between those two disparate structural models—and as we shall see below, this point greatly mattered to Riemann. The justification Riemann gives for his decision to depart from the established precedent and assign the tonic (or tonic-like) role to the central pitch is flimsy at best.40 He admits that the only support he has for calling ☉ the most important note is that in his recon­ struction of the oldest Greek enharmonic scale, it is the pitch that would have been desig­ nated mese.41 To support his claim, he presents one melody as an example of a case of a mese in such proto-tonic function, in the sense of a reference point around which the melody circles. It is, however, not a Greek example but a Chinese melody, called “Tsi Tschong.”42 This melody is reproduced in example 4.2. It works as an example “in the ab­ sence of existing Olympian music,”43 as though the Chinese can simply stand in for the ancient Greek—again implying that the pentatonic scale generated a universal system and theory wherever it was used. Riemann observes that the long-established principle of taking the lowest pitch in the generative chain of fifths as the tonic (or “kung,” now work­ ing with Chinese terminology) cannot well be applied to “Tsi Tschong,” as no phrase ends on that pitch (in this case C). Rather, most end on D, that is to say, Riemann's mese. At the same time, however, Riemann has to concede that this melody ends on its lower fifth, or G. We might get the impression that this argument is perhaps not the strongest way to intro­ duce the mese—and Riemann quietly drops the argument after this: the concept of the fi­ nality of the mese plays virtually no part in Riemann's subsequent observations.44 We shall see, however, that this point becomes strategically important to Riemann's argu­ ment: beyond the fact that it provides a theoretical symmetry, Riemann's later explana­ tion of the subsemitonium modi, the leading tone, critically depends on it.

Page 11 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality

Ex. 4.2. Transcription of Chinese Melody, “Tsi Tchong,” from Folkloristische Tonalitätsstudien, pp. 3–4.

Here and elsewhere, Riemann cannot help but add a sideways glance to the (normative) diatonic system, to which all music apparently aspires: in his view, the pentatonic scale is deficient, as it “lacks” the upper and lower thirds (3〉 and III〈), (p. 151) counted outward from the mese. Riemann points out that it is precisely these pitches that will, when chro­ matically altered, enable modulation to the fifth-related key.45 In his explanation of Chi­ nese music theory, Riemann argues that Chinese music fills the wide gap between II and IV and between 2 and 4 with infixes called pien tones. He asserts that these pien tones are rarely employed (usually as changing or neighbor notes),46 and that they do not have fixed intonation.47 Despite these caveats, Riemann employs the concept of pien as though they filled in the thirds in the “deficient” pentatonic scale, and continues to apply the term, culled from its Chinese theoretical context, in all situations to denote the upper or lower third around the mese.48

Ex. 4.3. “Modulations” in pentatonic music, from Folkloristische Tonalitätsstudien, p. 5.

Ex. 4.4. Tonal implications around the pyknon, from Folkloristische Tonalitätsstudien, p. 11. Page 12 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality In Chinese music, he contends, modulations can be observed by following the shift of pyk­ na, even where there are no pien in use. In example 4.3, Riemann observes (p. 152) a shift from the c-scale (or ☉d), to a b♭-scale, and finally to a g-scale (or ☉a). Indeed, the concept of the pyknon within Riemann's symmetrical interval-based conception of pentatonicism allows him to relate the complex back to diatonic procedures. Riemann explains with ref­ erence to example 4.4: “Many melodies with pentatonic tendencies exhibit—for our ears— a noticeable cadential oscillation between the two relative keys to which the central py­ knon is common.”49 Depending on whether the central pyknon appears with the upper or lower fourth, the closing gesture can often be heard as a “sinking down from the major root (II) to the minor fifth (IV).”50 Confusingly, Riemann's verbal explanations here refer to his dualistic model (where the “minor fifth” is the root of the minor mode), while the Roman numbers refer to his new melodic system (i.e., melodic degrees below the mese). Here and elsewhere, we see that Riemann is eager to make sense of musical phenomena in the conceptual framework of dualism, even where triadic shapes play no overt part.51 The sideways glances to diatonic music, and especially to the dualistic minor system, be­ come stronger when Riemann begins discussing the possibility of harmonizing Celtic folk­ songs in pure minor, as Oettingen suggested in 1866: After Carl Fortlage (1847) had pointed out the role of the flat seventh (instead of the leading tone) in Scandinavian melodies, it was especially Arthur von Oettingen who uncovered the nature of pure minor harmony. Even though the full seven-step minor scale, like the seven-step major scale, had developed from pentatonicism by means of inserting the two filling tones (pien), it is nonetheless undeniable that newer music prefers the major scale somewhat—so much so that it has also affect­ ed the minor mode by adding alien elements to it that actually belong to the major mode, with the effect that the specifically minor melody and harmony had almost disappeared from consciousness. It was only the rise of a national Nordic music (Hartmann, Gade, Grieg) that drew attention to the peculiar effects of pure minor again.52 One important side aspect of Riemann's agenda in Folkloristische Tonalitätsstudi­ en was to explore and promote the pure minor mode, which was of course an integral part of his dualistic project. It is likely, as Riemann's allusion to Scandinavian composers suggests, that his interest in the music of the European fringes—Scandinavia, the British (p. 153)

Isles, Spain—was fanned by the hope that the minor mode may have remained in a purer state in these “less civilized” parts.53 With this argument, Riemann hoped to prove, in one fell swoop, not only that the major-minor system had evolved from pentatonic underpin­ nings but also that the two modern modes were equal.

From Pentatonicism to Tetrachords Tetrachords, here in the most general sense as scalar fragments spanning the interval of a fourth, had already formed the tacit background to Riemann's theoretical considera­ tions of pentatonicism: the terms mese and pyknon are drawn from Greek tetrachordal Page 13 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality theory. He next turned his attention more specifically to the tetrachordal principles of modal music, as axioms supposedly shared between ancient Greek music, Gregorian chant, and Northern European folk song. Riemann acknowledged that strict pentatoni­ cism is irreconcilable with strict tetrachordalism, but, nevertheless, “one can gain a good idea of how the trichords of pentatonicism turned into tetrachords of classical antiquity and the Middle Ages by means of the pien.”54 Riemann did not explain how exactly this principle of Chinese music theory, which manages to function as the historical lubricant that provides such smooth passage from Greek antiquity to the European Middle Ages, entered the Western frame of mind. But then again, none of Riemann's unacknowledged sources had done so either.55 It seems that such a question would not even have occurred to Riemann: he felt empowered to mix and match these theoretical concepts because they were merely placeholders for theoretical ideas that he accepted unquestioningly to be universally true. In explaining the tetrachords, too, Riemann took a synchronic systematizing approach that considered their structural properties without any concern for cultural specificity. Based on a suggestion by the philologist August Boeckh, Riemann proposed an applica­ tion of three of the Greek modal names to their characteristic tetrachords:56 Dorian:

E—F G A || B—C D E

(½ + 1 + 1 steps)

Phrygian:

D E—F G || A B—C D

(1 + ½ + 1 steps)

Lydian:

C D E—F || G A B—C

(1 + 1 + ½ steps)

Ex. 4.5. A pyknon can be filled out to a full tetra­ chord by adding a pien, from Folkloristische Tonal­ itätsstudien, p. 36.

Ex. 4.6(a) and (b). The metric placement of the pien influences the tetrachordal structure, from Folkloris­ tische Tonalitätsstudien, pp. 36–37. (p. 154)

The critical difference between these tetrachords resides in the location of their

semitone, and each of the modes listed here can be thought of as built on a double state­ ment of the same tetrachord. Riemann points out that there is no historical precedent for

Page 14 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality calling the tetrachords (as opposed to the whole modes) by these names but argues in fa­ vor of them because of their rigorous and systematic clarity. Riemann explains in some detail how he imagines the transition from pentatonic trichord to tetrachord. In example 4.5, he shows how, depending on whether the pien is added above or below the central pyknon (as 3〉 or as III〈), a Lydian or a Dorian tetrachord will result. In the case of the tetrachord, however, a new musical feature weighs in: the metric posi­ tioning of the pien. Whether the pien occurs on a weak or a strong part of the beat will be of critical importance for Riemann's proto-tonal hearing of the structure. Riemann ex­ plains: if the newly formed Lydian tetrachord from the previous example veers toward B, by sounding it on the strong beat, as shown in example 4.6a, then it does not divert from a tonal sense centered on ☉a. Meanwhile, if it sounds the C (the pien of ☉a) on the strong beat, as shown in example 4.6b, this calls for a reinterpretation of 3〉 as II, that is to say the new ☉ is D, while B is now heard as the new pien (III〉). It is easy to see that this expla­ nation rather begs the question: it presupposes a tonal hierarchy—for it is only in this context that the pien is enabled to act exactly like semitones would in tonal music.

Ex. 4.7. Different pien create different tetrachords, from Folkloristische Tonalitätsstudien, p. 37.

Riemann then, in example 4.7, turns to the case of trichords other than the central py­ knon (i.e., three consecutive scale tones, drawn from an anhemitonic gamut spanning an interval greater than a third, for example E–G–A, where A is the mese). He explains that there are two possible pien that can fill in the gap, resulting in different tetrachords. In this example, with III〉 it will be the Phrygian, with III♮ (p. 155) the Dorian. The emphasis on metric weight is the same as before; the positioning of the pien can induce a quasimodulatory shift to a fifth-related pyknon. Example 4.8 shows a systematic representation of the three tetrachords (i.e., Dorian, Phrygian, Lydian) and their possible derivations from all possible trichordal fragments by addition of the relevant pien tones. It seems that Riemann does not fully trust his own rather exuberant application of the specific Chinese principle of pien to generic pentatoni­ cism resulting in Boeckh's neo-Greek tetrachordal system. For this reason, he turns next to the Guidonian hexachord, in search of further support for his synthesizing maneuver. Riemann explains the hexachord here not only as a combination of the three tetrachords (as shown in example 4.9a), but also as an overlaying of two pentatonic scales with their respective pien tones (as shown in example 4.9b). His point is that tetrachordal and pen­ tatonic theory combine to create proto-diatonicism. Riemann rejoices: “Here again we

Page 15 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality have another bridge from the melody of primordial times to the rarefied system of the middle ages!”57

Ex. 4.8(a)–(c). Riemann shows how trichords can be completed to tetrachords by adding pien, from Folk­ loristische Tonalitätsstudien, p. 38.

Having shown how the interaction of pentatonic and tetrachordal structures can also be applied to the chant repertoire, Riemann proceeds to explain the derivation of the sub­ semitonium modi, the leading tone, in the minor mode. For any evolutionary approach to music this is a problem, as the leading tone in the minor mode must inevitably be figured as an alteration of the natural scale, which requires additional explanation. Riemann's ap­ proach to this issue is oblique, but builds directly on the mechanisms he has laid out pre­ viously.

Ex. 4.9. Riemann explains how the hexachord can be derived (a) from the three tetrachords, and (b) from two pentatonic scales with the appropriate pien tones, from Folkloristische Tonalitätsstudien, p. 39.

Riemann dwells here particularly on the modulatory potential that pykna and pien contain (as we have seen particularly in examples 4.3 and 4.5–4.7). He underlines, howev­ er, that not every sounding of a new pyknon signals a shift of tonal center, as a quick com­ (p. 156)

parison with the modern diatonic scale and its three whole-tone pykna shows (i.e., groups of two consecutive whole tones centered on 2̂, 5̂, and 6̂ of the major scale). Instead, Rie­ mann interprets this analogy in light of the modern harmonic functions, as a shift be­ tween T, S, and D.58 This is an important point in Riemann's argument. We finally seem to have reached a breakthrough, where Riemann feels he has gathered enough theoretical evidence to sup­ port his tacit agenda, to show that latent tonal structures—proto-functional relations—ex­ ist even in monophonic repertoires of other cultures. This might in fact be the reason that Riemann chose to introduce his concept of the pentatonic scale not just once, but in three Page 16 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality fifth-related examples, as we saw in example 4.1 above. The three in combination add up to a suggestive proto-functional scalar ensemble. Modulatory links between them, as we have learned subsequently, would be provided by the strategic employment of pien. Going back to earlier parts of his argument, Riemann explores in greater detail how this tonal analogy applies to non-Western repertoires. In order to explain the derivation of the cadential effect of the leading tone, Riemann needs to revert to an earlier example of pure pentatonicism, the Chinese melody “Tsi Tschong,” which he had already used to ar­ gue that the middle note of the pyknon functioned as the pentatonic “tonic.” A typical closing gesture, Riemann argues, is found in sounding the upper and lower neighbors be­ fore the central pitch, as shown in example 4.10a, drawn from the second measure of ex­ ample 4.2 above. Riemann contends—implausibly—that in strict anhemitonic pentatoni­ cism this is the only way to draw attention to the central pitch. The assertion may not hold water, but it is necessary for Riemann to continue his broader agenda.

Ex. 4.10. Riemann's close analysis of melodic caden­ tial gestures in “Tsi Tchong,” from Folkloristische Tonalitätsstudien, pp. 68–69. (p. 157)

To be able to make his point, Riemann needs to rely on his tendentious argument

of the mese playing the role of “tonic,” which he had ignored in the interim. He gives no evidence of this cadential gesture of circumscribed whole tones in pentatonic music other than “Tsi Tschong” itself, but Riemann needs to emphasize this case to be able to contin­ ue his argument. As far as Riemann is concerned, it seems that there is simply no need for further examples, because he treats this example of Chinese music as nothing more than a demonstration—a structure that can tell us something essential about the relation­ ship between pentatonicism and the modern minor mode in general and that merely hap­ pens to come from Chinese music. From here he can claim, as shown in example 4.10b, that more common cadences using leading tones are just alterations of earlier melodic formulae that came about thanks to the insertion of pien tones into scales. Riemann here conveniently coins the notion of the “minor-third” pyknon, a structure consisting of a three consecutive pitches separated by a semitone and a whole tone, in analogy with the regular pyknon that characterized the pure pentatonic scale to explain these variants. On the basis of this new concept, Rie­ mann goes on to argue that the cadential formulae of example 4.10b should be reheard as implying new tonal centers. That is, if we recognize the central tone in these “minorthird” pykna as the mese, then the same cadential formulae can be heard in a different scalar significance, as expressed in example 4.11. (Note that the second example inverts Page 17 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality the middle tones, which may emphasize the subsemitonium to mese relation.) These new mese are no longer surrounded by two whole-tone steps, but rather by one whole-tone and one semitone step.

Ex. 4.11. Cadential formulae constructed around the “minor third pyknon,” from Folkloristische Tonal­ itätsstudien, p. 69.

Riemann concludes: “The fact that the minor-third pyknon came to the fore instead of the major-third pyknon as the center indicates an emergent understanding (p. 158) of the essence of harmony, a recognition of the third as a part of the sonority, and the separation into major and minor.”59 How and when exactly this breakthrough came to pass, Riemann contends with faux modesty, cannot be reconstructed with any certainty. In fact, he had proposed the year 1200 in his previous historical work.60 Here he cautiously suggests a much earlier date—around 700 B.C.—based on his tendentious readings of Aristoxenus and Plutarch. Japanese music, he suggests, similarly availed itself of the minor-third py­ knon from a very early stage onward. Riemann requires the minor-third pyknon to be able to explain the closing effect of the leading tone, independent of the major diatonic scale. It is for this reason that early on in his treatise he had to argue that the mese was often the closing pitch: for the “minor-third pyknon” in the common-practice tonal system—and probably only there—this contention is true.

Riemann and Melodic Analysis Riemann's evolutionary history of scalar models posits a number of different stages. Be­ ginning with pentatonicism, the rigidity of the system is reduced by the employment of the pien, which allows it to transform into the system of tetrachords. It is noteworthy that these follow a fundamentally different structural principle, as, unlike the pentatonic structures that Riemann examines, tetrachords have no inherent symmetry. In Folkloris­ tische Tonalitätsstudien Riemann is generally careful not to commit to a chronology, but supplies instead heuristic links between structural stages. His explanation of tetrachords as three-note fragments of pentatonic scales with infixed pien is historically doubtful, but it allows him to look at all subsequent scalar systems—hexachords and diatonic scales— as combinations of these two principles. Folk music, as nominally the chief object of his study, has been pressed into service to represent a middle ground between strict penta­ tonicism and modern diatonicism. Of course, it would be wrong to turn to Folkloristische Tonalitätsstudien in the hope of finding new insights into ethnomusicological questions. As we have seen, Riemann had few new facts to add to the study of folk music, he exclusively relied on the field work of Page 18 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality others, and his methods of adapting the findings of others were often tendentious. The polemical occasion that gave rise to the study in the first place, and the defensive stance that Riemann occupied in his argument, means that Folkloristische Tonalitätsstudien can hardly claim to be more than a curiosity in the history of ethnomusicology.61 The polemical aspects of this late study, which are all too easy to dismiss, may overshad­ ow some of its more interesting features. A better angle from which to approach Folkloris­ tische Tonalitätsstudien is from the vantage point of the analytical techniques that Rie­ mann brings to bear on melodic structures. From this angle, Riemann's late work does mark a significant departure from his earlier work, which had always considered melodies in terms of their harmonic implications. Once we take the emphasis away from Riemann's largely indefensible claims con­ cerning non-Western repertoires, a new context opens up in which Folkloristische Tonal­ itätsstudien no longer stands all by itself.62 In the early years of the twentieth century, the analytical study of melody was fast becoming a focal point of theoretical interest again, after a good century of theoretical neglect: Ernst Kurth, for one, effectively presented a theory of melody in his influential Grundlagen des linearen Kontrapunkts (1917), and (p. 159)

Heinrich Schenker was busy writing his two-volume Kontrapunkt and honing his notion of the Urlinie during those years. In fact, it seems that Riemann was quite happy to drop the pretensions to non-Western music after Folkloristische Tonalitätsstudien, and to admit that in current music-theoreti­ cal work the “chief interest is changing from harmony to melody.”63 He was confident that his late work—especially in light of his “theory of the tone imaginations”—had an im­ portant contribution to make to this paradigm shift. The best way to Riemann's Folkloristische Tonalitätsstudien, then, is with an eye on the unique focus on scalar and melodic structure, and the analytical tools it may provide for melodic analysis—within a tonal framework.64 The fact that Riemann chose to tackle a group of particularly challenging repertoires brought out some features of his music-theo­ retical, systematizing thinking that caused him to rethink some of the foundations of his system: Riemann's focus on scale formation means that the other factors which are nor­ mally central to Riemann's musical thought—harmony and meter—take a back seat here, at least temporarily. As we have seen, Riemann never quite forgot the “universal” forces of tonal harmony and metrical position, which in fact formed the backbone of his teleolog­ ical evolutionary trajectory. Folkloristische Tonalitätsstudien thus holds a position that is unique within Riemann's output while also remaining characteristic of that output, in that even this late departure from Riemann's usual analytical practice clearly carries the sig­ nature trait of his music-theoretical work, his deep-seated belief in the explanatory power of symmetries. It is particularly with this trademark feature that he left his mark on the analysis of melody.

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Riemann and Melodic Analysis: Studies in Folk-Musical Tonality

Notes: (1.) See Michael Arntz, Hugo Riemann: Leben, Werk, Wirkung (Cologne: Concerto-Verlag, 1999), 109, and Willibald Gurlitt, “Hugo Riemann (1849–1919),” Veröffentlichungen der Akademie der Wissenschaften und der Literatur, Mainz: Abhandlungen der geistes- und sozialwissenschaftlichen Klasse 25 (1950): 1891. (2.) Hugo Riemann, Folkloristische Tonalitätsstudien (Leipzig: Breitkopf und Härtel, 1916), 112. “Angesichts der Rolle, welche in der jungen Wissenschaft der musikalischen Ethnographie die phonographischen Aufnahmen von Melodien spielen, muß aber auch darauf aufmerksam gemacht werden, daß die Niederschrift einer Melodie nach dem Phonogramm eine Sache ist, die einen tüchtigen Musiker voraussetzt, aber obendrein auch ein Vertrautsein mit dem Tonsystem, dem die Melodie angehört…. Untersuchungen wie die vorliegenden dienen auch in erster Linie der besseren Schulung des Hörens für ein volles Verständnis der Struktur von Melodien. Werden dieselben weitergeführt, so wird vermutlich von den unserm Musiksystem widersprechenden Intervallen von 3/4oder 5/4-Tönen und von den ‘neutralen’ Terzen, die die Tonpsychologen jetzt aus dem Phonogramm heraushören, nicht allzuviel übrig bleiben.” (3.) See Sebastian Klotz, ed., Vom tönenden Wirbel menschlichen Tuns: Erich M. von Hornbostel als Gestaltpsychologe, Archivar und Musikwissenschaftler (Berlin: Schibri, 1998). (4.) Hellmuth Christian Wolff, “Riemann, Hugo,” Die Musik in Geschichte und Gegenwart, ed. Friedrich Blume (Kassel: Bärenreiter, 1949), 11: col. 485. (5.) See Hugo Riemann, Sechs original Japanische und Chinesische Melodien (Leipzig: Breitkopf und Härtel, 1903). (6.) Hugo Riemann, “Über Japanische Musik” Musikalisches Wochenblatt 33 (1902): 209– 210, 229–231, 245–246, 257–259, 273–274, 289–290; see also “Exotische Musik,” Max Hesse's deutscher Musiker-Kalender 21 (1906): 135–137. In this short article, Riemann distances himself from the group of music theorists that seek exotic harmonies to enrich Western composition, and sides with the comparative musicologists of the Berlin school— Stumpf, Abraham, and Hornbostel. (7.) Hugo Riemann, Handbuch der Musikgeschichte (Leipzig: Breitkopf und Härtel, 1904), 1.1: 40–53 and 162. (8.) Riemann, Handbuch, 1.1: 50–51. (9.) Charles Burney, General History of Music (London: Becket, Robson and Robinson, 1776–1789), 1: 34–42. (10.) Ibid., 1: 37. (11.) Ibid. Page 20 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality (12.) Ibid., 1: 41. (13.) See Matthew Gelbart, The Invention of “Folk Music” and “Art Music:” Emerging Cat­ egories from Ossian to Wagner (Cambridge: Cambridge University Press, 2007), 120–152, for a detailed investigation of Burney's claims and influence. (14.) Riemann, Handbuch, 1.1: 43–46. (15.) Riemann, Handbuch, 1.2: 3. (16.) Forkel, Allgemeine Geschichte der Musik (Leipzig: Schwickert, 1788–1801), 1: 335– 337. (17.) Carl Fortlage, Das musikalische System der Griechen in seiner Urgestalt. Aus den Tonleitern des Alypius, zum ersten Male entwickelt (Leipzig: Breitkopf und Härtel, 1847). (18.) Helmholtz even echoed Burney's terminology about the “old enharmonic scale” de­ spite his slightly different derivation of the scale (cited from Hermann von Helmholtz, On the Sensations of Tones as a Physiological Basis for the Theory of Music, trans. Alexander Ellis, 2nd English ed. (London: Longmans, Green, 1885), 257–258, in German Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik (5th ed., Braunschweig: Friedrich Vieweg, 1896). Helmholtz's contemporaries who cited Bur­ ney typically disagreed with Burney's hemitonic reconstruction and found ways to explain why Burney should have reconstructed an anhemitonic scale. (19.) Riemann, Handbuch, 1.1: 43–44. (20.) Like Burney, Riemann is noncommittal about the question of whether Olympos actu­ ally brought an Asian scale to Greece or whether the scale coincidentally developed in dif­ ferent places based on the same natural principles (see Handbuch, 1.1: 50, 162). More specific claims about cultural contact and the Asian origin of Olympos's scale had ap­ peared in Helmholtz's Sensations of Tone, 257–258, 425–426—coupled to typical claims that “in the first stages of the development of music many nations avoided the use of in­ tervals of less than a tone” and hence formed pentatonic scales. The grandest and most eccentric theory of Asiatic origins for both Greek and Celtic scales had come in G. W. Fink's Erste Wanderung der ältesten Tonkunst, als Vorgeschichte der Musik oder als erste Periode derselben (Essen: Bädeker, 1831), 140–168. (21.) This passage is an aside about the spondeion—see the parenthetical remark in (Pseudo-)Plutarch, Plutarch's Moralia in Fifteen Volumes, vol. 14 (Cambridge, MA: Har­ vard University Press, 1967), 377 (from par. 11 of the original). Riemann argues that the passage was marginalia added by a later reader rather than Plutarch proper and thus ig­ nores it (Riemann, Handbuch, 1.1: 44), while Burney had simply claimed that it was “un­ intelligible” and thus omitted it from his discussion (General History, 1: 35, note e). Note that Burney's dismissal of this passage was the locus of Forkel's quibble with Burney's in­ terpretation (Allgemeine Musikgeschichte, 1: 336–337).

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Riemann and Melodic Analysis: Studies in Folk-Musical Tonality (22.) See, for example, George Thomson, “Dissertation Concerning the National Melodies of Scotland,” in Select Melodies of Scotland, Interspersed with those of Ireland and Wales (London and Edinburgh: George Thomson, 1822–1823), 1: 3–19; and Alexander Campbell's preface to the first volume of his collection Albyn's Anthology: A Select Collec­ tion of the Melodies and Vocal Poetry Peculiar to Scotland and the Isles (Edinburgh: Oliv­ er and Boyd, 1816). (23.) Riemann, Handbuch, 1.1: 50. (24.) Ibid. The closest Riemann comes to giving any insight into his claim are some theo­ ries about Japanese organology, which he presents in his article “Über Japanische Musik,” and partially repeats in the Handbuch. (25.) See Plutarch's Moralia, 14: 375. (26.) See Riemann, Handbuch, 1.1: 162–163. (27.) See especially the oft-cited peroration in Riemann's Geschichte der Musiktheorie (Berlin: Max Hesse, 1898), 529. (28.) See Charles Darwin, The Descent of Man, and Selection in Relation to Sex. 2 vols. (London: John Murray, 1871), 1: 56. (29.) See, for example, Ellis, “On the Musical Scales of Various Nations,” Journal of the Society of Arts 33 (1885): 485–527. (30.) See Alexander Rehding, “Wax Cylinder Revolutions,” Musical Quarterly 88.1 (2005): 123–160. (31.) For a detailed critique of the blown-fifths theory, see Kathleen Schlesinger, The Greek Aulos: A Study of Its Mechanism and of Its Relation to the Modal System of Ancient Greek Music (London: Methuen, 1939), 313–350. (32.) See Carl Stumpf, Die Anfänge der Musik (Leipzig: Johann Ambrosius Barth, 1911), esp. 26–31. (First published in 1909 in Internationale Wochenschrift.) (33.) See Alexander Rehding, Hugo Riemann and the Birth of Modern Musical Thought (Cambridge: Cambridge University Press, 2003), 52–53. (34.) This is outlined, for instance, in Robert W. Wason and Elizabeth West Marvin, “Riemann's ‘Ideen zu einer “Lehre von den Tonvorstellungen” ’: An Annotated Transla­ tion,” Journal of Music Theory 36 (1992): 69–116. (35.) Riemann, Folkloristische Tonalitätsstudien, v–vi. “Was man [the comparative musi­ cologists] suchte, waren Anhaltspunkte für eine abweichende Organisation des Hörappa­ rates bei Völkern, die auf niederer Stufe der Musikkultur stehen, und man glaubte solche in den unseren Gewohnheiten widersprechenden Intonationen einzelner Intervalle zu finden (Intervalle von 3/4- oder 5/4-Ganzton, ‘neutrale’ Terzen), wie solche sowohl in Page 22 of 26

Riemann and Melodic Analysis: Studies in Folk-Musical Tonality Phonogrammen als auch auf exotischen Musikinstrumenten sich zu finden schienen. Das ärgerliche Ergebnis dieser Forschungen der vergleichenden Musikwissenschaft war zunächst eine Erschütterung der im Laufe von Jahrtausenden langsam gewordenen Fun­ damente der Musiktheorie. Selbst Hellseher wie Helmholtz wurden wankend in ihrer Überzeugung, daß die Grundlagen des Musikhörens natürlich gegebene Verhältnisse sind, und ließen durchblicken, daß doch vielleicht Musiksysteme nicht naturnotwendig, sondern wenigstens teilweise Ergebnis willkürlicher Konstruktion und Konvention sind…. So stellte sich schließlich heraus, daß die Abhängigkeit unseres Hörorgans von den an dasselbe herantretenden Tonreizen keine absolute, unbegrenzte ist, daß vielmehr beim Musikhören fortgesetzt ein Operieren mit feststehenden Begriffen konstatiert werden muß, ein Beurteilen der musikalischen Geschehnisse nach unser Vorstellen beherrschen­ den Kategorien, in welche die Einzeltonwahrnehmungen sich einordnen, wobei es bis zur strengen Ablehnung der das Ohr treffenden Intonationen als fehlerhaft und unmöglich (unlogisch) kommt. Das Durchbrechen dieser Erkenntnis zwingt aber unweigerlich dazu, an die Stelle einer Lehre von den ‘Tonempfindungen’ eine Lehre von den ‘Tonvorstellun­ gen’ zur Fundamentierung der Musiktheorie und Musikästhetik zu fordern.” (36.) This may be an implied swipe against Georg Capellen, who had recently published Ein neuer exotischer Musikstil (Stuttgart: Carl Grüninger, 1906). Since Capellen's polemic, “Die Unmöglichkeit und Überflüssigkeit der dualistischen Molltheorie Hugo Rie­ manns,” in Neue Zeitschrift für Musik (1901): 529–531, 541–543, 553–555, 569–572, 585– 587, 601–603, 617–619, Riemann and Capellen did not see eye to eye. (37.) Riemann, Handbuch 1.1: 50. “Hauptrepräsentant einer Epoche…welche in jener schlichten Einfacheit Melodien schuf, die wir auch aus uralten chinesischen und japanis­ chen sowie keltischen Weisen kennen.” (38.) Riemann, Folkloristische Tonalitätsstudien, 6. (39.) Ibid. 3. (40.) Erich Fischer and Ambros are mentioned in Folkloristische Tonalitätsstudien, 4, but Riemann is tacitly challenging a long tradition ranging from the later eighteenth century onward. (41.) Riemann, Folkloristische Tonalitätsstudien, 4. (42.) Here he gives no source for the melody, but in fact he had used the same melody, and to the same end, in the Handbuch, 1.1: 52, and in “Über Japanische Musik.” There he gives the sources as “among others in collections by Eyles Irvin and J. Barrow, and in Am­ bros.” He also published a harmonized version of the melody in his Sechs originale chine­ sische und japanische Melodien. (43.) Riemann, Handbuch, 1.1: 52.

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Riemann and Melodic Analysis: Studies in Folk-Musical Tonality (44.) See Riemann, Folkloristische Tonalitätsstudien, especially 12–13: “Schlüsse auf den Zentralton kommen auch noch in schottischen und irischen Liedern vor, haben aber da nach meinem Empfinden nicht die Wirkung eines befriedigenden Abschlusses, sondern die einer dissonanzartigen Spannung.” It is worth pointing out that Riemann here cap­ tures an important feature of Scottish pentatonic music, which often exhibits internal ca­ dences on the degree that Riemann calls mese. (45.) Riemann, Folkloristische Tonalitätsstudien, 5. This notion of modulation only works under the assumption of an a priori diatonicism. Riemann here sidesteps a long debate about modulation within pentatonic scales, including discussions within Chinese theory. (46.) Ibid., 8. (47.) Ibid., 5. (48.) Riemann lists, for instance, examples of different uses of the pien in Celtic melodies, see Folkloristische Tonalitätsstudien, 8–11. (49.) Ibid., 11. “Vielen Melodien mit pentatonischem Einschlag eignet für unser Empfind­ en ein auffallendes Schwanken der Schlüsse zwischen den beiden parallelen Tonarten, denen das Zentral-Pyknon angehört.” (50.) Ibid. (51.) Riemann was not the first to posit that the pentatonic scale could flesh itself out to different diatonic scales depending on which infixes filled the “gaps.” The “modulation” between different pentatonic modes or gamuts via the introduction of infixes as pivot tones was well-established. In the study of Scottish music, it had been discussed in the 1820s and 1830s by several writers; see Louis Necker de Saussure, Voyage en Ecosse et aux Iles Hébrides, 3 vols. (Geneva and Paris: J. J. Paschoud, 1821), 3: 452–454; Fink, Erste Wanderung, 257–259. The Chinese terminology of pien as modulatory notes in cross-cul­ tural comparison with Gaelic or other pentatonic musical systems had been discussed, for instance, in Saussure, Voyage, 3: 456–458. (52.) Riemann, Folkloristische Tonalitätsstudien, 13–14. “Nachdem bereits K. Fortlage (1847) auf die Rolle der Unterganztöne (statt des Unterhalbtones) in der Melodik der Skandinavier hingewiesen, hat besonders A. von Öttingen das Wesen der reinen Mollhar­ monik aufgedeckt. Wenn auch die volle 7-stufige Mollskala ebenso wie die 7-stufige Durskala aus der Pentatonik durch Einfügung der beiden Fülltöne (Pien) sich entwickelt hat, so ist doch nicht in Abrede zu stellen, daß die neuere Musik die Durskala etwas bevorzugt, daß sie auch Moll mit eigentlich demselben fremden, nach Dur gehörigen Ele­ menten so stark versetzt hat, dass die Mollmelodik und -harmonik fast aus dem Bewußt­ sein gekommen war und erst die Vordrängung einer national nordischen Musik (Hart­ mann, Gade, Grieg) wieder auf die eigentümlichen Wirkungen aufmerksam machte, welche dem reinen Moll eignen.”

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Riemann and Melodic Analysis: Studies in Folk-Musical Tonality (53.) Arthur von Oettingen had previously suggested a reharmonization of Beethoven's Scottish folk-song settings as a demonstration of how pure minor might work. Riemann here offers his own version of Beethoven's folk-song settings in pure minor. On Riemann's and Oettingen's engagement with Beethoven's folk-song settings, see Rehding, Hugo Rie­ mann and the Birth of Modern Musical Thought, 173–182. (54.) Riemann, Folkloristische Tonalitätsstudien, 36. (55.) See n. 51 above. (56.) Riemann, Folkloristische Tonalitätsstudien, 35. See August Boeckh, Opera Pindari quae supersunt (Leipzig: August Gottlob Weigel, 1811–1821). (57.) Riemann, Folkloristische Tonalitätsstudien, 39. (58.) Ibid., 68. (59.) Ibid., 69. “Daß aber wirklich ein Kleinterz-Pyknon allmählich statt eines GroßterzPyknons als Zentrum sich dem Verständnis erschloß, deutet auf das Durchbrechen der Erkenntnis des Wesens der Harmonie, auf Bewußtwerden der Terz als Klangbestandteil, auf die Scheidung von Dur und Moll.” (60.) The preface to Riemann's Geschichte der Musiktheorie, 3, is particularly enlighten­ ing here. (61.) This is not to say that Riemann's melodic method was without influence. It was ea­ gerly co-opted for the study of Germanic music that flourished around the same time. See Hans Joachim Moser, Geschichte der deutschen Musik, 3rd ed. (Stuttgart and Berlin: J. G. Cotta, 1923), 1: 19–22. (62.) Riemann refers back to Folkloristische Tonalitätsstudien in his “Neue Beiträge zu einer Lehre von den Tonvorstellungen,” Jahrbuch der Musikbibliothek Peters (1916): 1– 21. This article functions as a continuation and addition of this line of inquiry; it includes a summary of the main argument of Folkloristische Tonalitätsstudien. (63.) Riemann, “Die Phrasierung im Lichte einer Lehre von den Tonvorstellungen,” Zeitschrift für Musikwissenschaft 1 (1918): 26–38. This article is a review of Kurth's Grundlagen des linearen Kontrapunkts—a book that Riemann accepts only insofar as it confirms his ideas of “tone imaginations.” The quotation is originally a comment about Kurth's work, but is one of the few points that Riemann wholeheartedly subscribes to. Kurth's blistering response follows in Zeitschrift für Musikwissenschaft 1 (1918): 176– 182. (64.) A comparison with Riemann's early study, Neue Schule der Melodik (1883), may sug­ gest itself. Despite the title, however, this early work is not so much a treatise on melody as one on tonal counterpoint.

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Riemann and Melodic Analysis: Studies in Folk-Musical Tonality

Matthew Gelbart

Matthew Gelbart is an assistant professor of music in the department of art history and music at Fordham University. His research interests include eighteenth- and nineteenth- century music, how we label and sort the music we listen to, and rock music. He is the author of The Invention of “Folk Music” and “Art Music”: Emerging Categories from Ossian to Wagner. Alexander Rehding

Alexander Rehding teaches music at Harvard University. His interests are in the his­ tory of music theory and in nineteenth and twentieth century music. He is the author of Hugo Riemann and the Birth of Modern Musical Thought, Music and Monumental­ ity, and Beethoven’s Symphony no. 9. He is the editor-in-chief of the Oxford Hand­ books Online series in Music.

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The Problem of Harmonic Dualism: A Translation and Commentary

The Problem of Harmonic Dualism: A Translation and Commentary   Ian Bent The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0005

Abstract and Keywords This article presents a key text in the transition of Riemann's argument from earlier theo­ ries and mature, psychological work. As a reference, the article also forms a link back to historical contexts already explored, and this article invokes figures such as Georg Capellen and Arthur von Oettingen—Riemann's contemporaries. In this article, Riemann is shown to summarize for one last time, the scope and ambitions of his dualistic theory and, implicitly and explicitly, he responds to his critics. As always, Riemann does not merely rework his earlier arguments but designs them in such way they adapt to the cur­ rent situation and cover new ground by answering his critics. Keywords: Georg Capellen, von Oettingen, dualistic theory, Riemann

RIEMANN published “Das Problem des harmonischen Dualismus” in 1905 in the presti­ gious music journal Neue Zeitschrift für Musik. If this journal guaranteed him wide circu­ lation, it had provided less welcome circulation four years earlier to an article by Georg Capellen, “The Impossibility and Superfluousness of Riemann's Dualistic Theory of Mi­ nor” (1901),1 to which Riemann now sought to give the definitive response. In the inter­ im, some of his adherents had expressed misgivings about his theory of harmonic dual­ ism, and had even begun to compromise how they taught it; hence “Das Problem” was al­ so a summons to backsliders among his own ranks, a rallying call to keep the faith. However, he faced a difficult task. His own initial commitment to “the objective existence of undertones” (the title of an 1875 article, in which he argued for the inverse of the over­ tone series as acoustic reality) as the basis of the minor mode had given way finally—af­ ter thirty years of prevaricating and tinkering with the specifics of the hypothesis—to the abandonment of a physics of undertones, which he now saw as a “pseudo-logic.”2 In “Das Problem,” he faces the task of publicly abandoning the undertone series—thereby sacri­ ficing the scientific basis for harmonic dualism—while continuing to claim scientific standing and historical precedent for his theory. Just how much of a problem this is, in a

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The Problem of Harmonic Dualism: A Translation and Commentary context of unwavering faith in the hard sciences, becomes apparent in Capellen's chal­ lenge: (p. 168) How can a theory claim validity under these circumstances that cannot be justi­ fied by natural laws? And how is it possible that Riemann, of all people, who is ea­ ger to erect his system on a scientific foundation, still clings to his undertone hy­ pothesis, despite his acknowledged error?3 Capellen was a firm believer in the science of acoustics, and he clearly understood that Riemann's problem was, and had always been, to justify the minor triad as a consonance equivalent in every respect to the major triad. He effectively threw down a gauntlet to Riemann when he proposed an experiment himself, which is in many ways analogous to Riemann's nocturnal experiment of 1874 that had led to the “objective” undertone hy­ pothesis in the first place, but makes a case for a “monistic” (or, more accurately, “antidu­ alistic”) generation of the minor consonance. In his experiment, Capellen suggested silently holding down the piano keys A2 and C3 and then striking A4–C5–E5. The upper pitches will continue to ring—“as a consonance.”4 This was, for Capellen, the inexorable proof that the minor consonance was indeed possible but it required a double foundation. Capellen, to be sure, based his theory on a different set of assumptions from Riemann—most importantly, Capellen's theory assumed equal temperament. What Capellen's experiment demonstrates above all is the continued se­ ductiveness of a simple experiment and the desire for a “scientific” basis of music theory, which had changed little over the last thirty years. Given this controversy, modern readers of “Das Problem” may be surprised to find “Die Unmöglichkeit” not mentioned at all, and Capellen's name mentioned mostly in passing. In fact, it was a preferred tactic of Riemann's to punish detractors by pointedly ignoring them and effectively silencing their criticisms out of existence.5 Riemann's argument is such that it responds directly to several of Capellen's criticisms, even though Capellen is not cited. Riemann's oblique rebuttal of Capellen's experiment—which refers to a differ­ ent publication by Capellen—is found in a number of absurd suggestions for alternative experiments: if, for instance, we held C2 down on a piano and struck a combination of B♭4 –D5–F♯5–A♭5 (i.e., tones corresponding to the ratio 7 : 9 : 11: 13 relative to C2), we could also justify the “naturalness” of this highly dissonant conglomeration.6 Having ridiculed attempts to justify consonance from an experimental-acoustical perspec­ tive, Riemann delivers the coup de grâce: it is not only the minor consonance that cannot be justified by means of overtones, but also that of the major triad that does not hold wa­ ter from an acoustical perspective. Riemann is in a position to take this line of argument, which amounts to an “emancipation from acoustical phenomena” and which effectively burns the bridges of the scientific appeal that had supported his theory for such a long time, because Riemann in “Das Problem” completes the move from a physical to a psy­ chological foundation, which prepares the ground for his later “Theory of the Imagina­ tions of Tone.”7 Page 2 of 27

The Problem of Harmonic Dualism: A Translation and Commentary To cushion this significant epistemological shift, the argumentative strategy Riemann adopts here primarily serves to minimize differences and emphasize continuities: he re­ visits the intellectual ground covered over the last thirty years of his (p. 169) working life, and he frequently makes reference to historical precedents of dualism, which he traces back all the way to Zarlino, invoking prestigious names such as Goethe, Hauptmann, Oet­ tingen, and his teacher Hermann Lotze in the process. For Riemann, these names form a long and distinguished genealogy, and he claims that “virtually all theorists” since Zarlino have reaffirmed the dualistic principle—at times, even against acoustical arguments ad­ vanced by Helmholtz and other detractors. In place of the acoustical arguments of the “monists”—while, as it were, denying any wrongdoing on his part in the past—Riemann outlines a new psychological theory of “se­ lecting and ordering” on the part of the mind. The perceiving mind is not passive, but rather actively engages in “comparing and associating” tones. However, this still leaves him to justify the equality that he asserts between major and minor triads, and for this, in the fourth installment of the article, he unveils a new theory involving frequencies, string/ tube lengths, and their proportions, whereby major = “growing intensity” (substitute for the overtone series), minor = “accumulating mass” (substitute for the undertone series). Through all of this, he fights vigorously to maintain his view of the minor triad as con­ structed downward, and so to keep his wavering followers on board. Riemann calls his realization to drop acoustics from his argument an “epiphany.” In many ways this rings true: giving up acoustical arguments allowed Riemann to stop worrying about certain problematic aspects of his theory, above all the discrepancies—as Capellen and others mercilessly point out—between the generation of the minor triad and its prac­ tical use in specific part-writing situations.8 The most pressing question is perhaps that of the dualistic “minor root,” which Riemann views in the fifth of the minor triad—even though it is also the chordal tone that can most easily be omitted, as Riemann himself ex­ plains time and again in his harmony tutors. Across this epistemological shift, Riemann is at pains to emphasize the consistency and the continuity of his harmonic thinking throughout, and in a way he is right: very little has changed in the way in which he thinks about harmonic relations; what changes in­ stead is mainly the speculative framework he employs in support of his music-theoretical views. And, with the gradual rise of music psychology, in such thinkers as Carl Stumpf and Ernst Kurth a little later, not to mention such important figures as the Gestalt psy­ chologist Christian von Ehrenfels, it seems that Riemann has his finger on the intellectual pulse of his time. The article's peroration, a rebuttal of his Dutch critic Ary Belinfante, is perhaps the most widely discussed aspect of this article.9 In it, Riemann once again underlines the strong traditional ties that bind and shape his ideas, while underplaying the radical reinterpreta­ tions that some of these traditional concepts have undergone in his hands. One cannot help but feel that Belinfante is held up as a straw man here to avoid having to engage with Capellen's much more aggressive criticisms on similar aspects of Riemann's theory. Page 3 of 27

The Problem of Harmonic Dualism: A Translation and Commentary The article is fascinating not only as a crucial moment in the history of music theory, but also as a study of music-theoretical “politics” in action, as well as for its (p. 170) style and rhetoric. To retain his authority, Riemann adopts a manner of barely controlled patience toward his followers (faintly reminiscent of Bismarck toward an errant state, or even Christ toward his disciples in the Garden of Gethsemane!), and one of ridicule toward his attackers. When the going gets tough, he resorts to long sentences and intricate syntax. However, humorous though Riemann's contortions may be, the article marks a major turning point in his work from physics to the psychology of perception, and lays the foun­ dation upon which Riemann developed his late theory of the “imaginations of tone.” Editors’ note: Throughout the translation, Riemann's (sometimes modified) use of Helmholtz's notation for the registral designation of tones (c’ = middle C, c = viola C, C = cello C) has been converted to ASA (Acoustical Society of America) notation (C4 = middle C, C3 = viola C, and so on). Further, publication information has been provided in the text where Riemann typically provided only titles and dates. —Eds.

The Problem of Harmonic Dualism By Prof. Hugo Riemann I. Introduction The ceaseless demands on my time of projects already in progress have prevented me from complying before now with the urgings of my friends to devote a special study to the question that lies at the very heart of the reforms embodied by my studies in harmonic theory. To be honest, I thought I had already dealt in sufficient detail with the history and scientific, logical and aesthetic basis of harmonic dualism in the closing chapters of my Geschichte der Musiktheorie im 9.–19. Jahrhundert (pp. 369–406, chap. 13 “Zarlino und die Aufdeckung der dualen Natur der Hamonie,” and pp. 450–509, chap. 15 “Musikalis­ che Logik”),10 which appeared in 1898, as well as in my Katechismus der Musikwis­ senschaft (Leipzig: Max Hesse, 1891, especially chap. 2 “Tonkomplexe (“physiologisch— psychologisch)” and chap. 3 “Tonvorstellungen”), and most recently of all in my Elemente der musikalischen Ästhetik (Berlin & Stuttgart: W. Spemann, 1900, pp. 83ff., especially p. 103) to have dispelled objections—the products of addiction to the conventional notions of thoroughbass—to my new formulations of compositional theory. The rapid dissemination of my books (including foreign translations) seemed to justify this assumption. However, since despite this, I keep getting requests—and from the very circle of those who are well-disposed toward my reform of harmony teaching methods—to explain the basic prin­ ciples all together in one comprehensive account, it would indeed appear that there is

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The Problem of Harmonic Dualism: A Translation and Commentary still some obstacle, something that I have not explained sufficiently clearly to avoid mishaps. (p. 171)

This obstacle is the principal tone [Grundton] of the minor chord.

Of those musicians who have kept in touch with recent developments in harmonic theory, and have borne witness to it through their own books or articles, some believe that they banish all difficulties the moment they accept the necessity of interpreting the minor chord as inverted, but that they reconcile this with the “musician's way of thinking,” by using the nomenclature and symbols for the minor chord in terms of its root. Whether they retain the two Oettingen symbols + and ° and designate the A major chord with a+ and the A minor chord with a°, or whether, in place of the ° that Oettingen chose for its sharper distinction between the categories, they adopt the − (= negative) that is the usual opposite of + (= positive) (a− = A minor chord) is, of course, of no consequence. At most one could say that the use of ° in another sense than that introduced by Oettingen and myself invites unnecessary confusion, which can be avoided by choice of another symbol. This is a purely secondary matter. Everyone who invents a new symbol-system naturally counts on the system's being generally adopted. If this occurs, then the different sense in which others have used the same symbols before him quickly falls into oblivion. We will not therefore concern ourselves with the extension of such symbol-systems,11 but merely with the question as to whether the objection that deters one theorist from wholly and un­ reservedly subscribing to the dualistically based theory, and drives another to belated skepticism, and causes him to waver and finally desert it, is valid or not. For this, I shall have to go back some way if I am not to take for granted things that I have written elsewhere. For example, in counting Rameau still among the “monists,”12 Georg Capellen in Die musikalische Akustik als Grundlage der Harmonik und Melodik (Leipzig: Kahnt, 1903), p. 54, demonstrates his complete unfamiliarity with my evidence (Geschichte der Musiktheorie, p. 457) that Rameau had by 1737 (Génération harmonique) definitively abandoned the attack upon Zarlino that he had attempted in his Traité de l’harmonie (1722), and with much-ado joined forces with the dualists. Nor should I, conse­ quently, expect those individuals who now with varying degrees of skill put themselves forward in newspapers and pamphlets as spokesmen against the dual basis of harmony to be any better acquainted with the defense that I, for example, mounted against the core arguments of Helmholtz and Stumpf. Although I have no intention of repeating here what I have written on this subject over the past 30 years, it is imperative nevertheless for me to provide a coherent, unbroken chain of conclusions, if the whole point of this little study is not to be missed.

II. Are Overtones the “Basis” of Consonance? The first attack upon the logical opposition that Gioseffo Zarlino asserted in 1558 in his Istituzioni harmoniche (Book I, chap. 30, and Book III, chap. 31) between the two series

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The Problem of Harmonic Dualism: A Translation and Commentary that mathematically define major and minor consonance:

(p. 172)

—a logical opposition that has been reaffirmed by virtually all theorists since then (Fr. Salinas 1577, Descartes 1615, Mersenne 1636, Rameau [since 1737], Levens 1743, Serre 1753, Tartini 1754, Blainville 1764, Jamard 1769, Vallotti 1779, Sabbatini 1799, Haupt­ mann 1853, A. v. Oettingen 1866, O. Tiersch 1868, H. Riemann 1873, Thürlings 1877, O. Hostinsky 1879, A. Krisper 1882, etc.)—ensued when, after Sauveur had demonstrated (1700) that the sounds of strings comprised an endless series of partial tones, people be­ lieved that they had discovered in this the true cause of consonance. As is well known, Rameau (with or without knowledge of Sauveur's evidence) was the person who made the overtones 3 and 5 (5th above the 8ve, and 3rd above the double-8ve), which he alone as­ serted were audible alongside the fundamental (1), the basis of a new system of harmonic theory whereby the fundamental was treated as the generator (générateur) of the fifth (douzième) and third (dix-septième). We now know, in fact, that to speak of a “generation” of higher tones by the deepest tone is a thoroughly amateurish way of putting it that does not at all correspond to the actual physical-mechanical processes. The fundamental as well as the overtones—and not just the 3rd and 5th, but all the tones corresponding to the natural number-series, as demonstrated by Sauveur, right up to the limit of the auditory

tonal field

are in fact sounded simultaneously upon the impact of the hammer, plucking by the finger or plectrum, attack of the bow, etc.; moreover, each varies in amplitude according to the point of attack, the only rule being that the deepest of the partials is generally the loud­ est. Conversely, several individual tones of the series sounded simultaneously have the power to generate the fundamental (as a combination tone). What is more, on wind in­ struments we cannot speak of the higher tones as arising out of the fundamental; for it is possible to blow [an instrument] in such a way that one of the higher tones becomes ex­ clusively the principal tone [Hauptton] with just its own over-series [Oberreihe] above it, while all the other tones of the series vanish completely. So right away we can totally dis­ count Rameau's son générateur, and we can speak of a series of tones that belong togeth­ er only insofar as the same string or the same column of air vibrating in a pipe can pro­ duce them simultaneously. The tones of this series may well therefore be described as re­ lated to one another; and if the lowest of them is termed the principal tone, then that is adequately accounted for by its usually being the loudest, indeed the one most of­ ten identified by the ear as the sole one, against which the others go undetected. (p. 173)

But even just the attempt to ground the consonance of the major chord in the overtone series [Obertonreihe] demands a completely arbitrary disregard of all the tones above the Page 6 of 27

The Problem of Harmonic Dualism: A Translation and Commentary sixth overtone, since the majority of them do not belong to the major chord of the funda­ mental. The experiments “at the piano” with which G. Capellen (Musikalische Akustik, pp. 12 and elsewhere) prove the “naturalness” of the major chord and the dominant seventh and dominant ninth chords represent a ludicrously elementary knowledge of physics; for on an instrument with strings tuned exactly in the natural relationships of the overtones one would be able to “prove” the naturalness of a large number of other simultaneities

[Zusammenklänge], e.g., the chord 7 : 9 : 11 : 13 :

Admittedly, even Capellen did not dare to infer “consonance” for 5 : 6 : 7 : 9 on the basis of sympathetic resonance under such conditions, only  “naturalness.” But it seems com­ pletely to have escaped him that “naturalness” can equally well be “proven” for the minor chord on the basis of such “experiments,” even without “twin fundamentals,” namely for

the tones 10 : 12 : 15 :

No: there is nothing in this that could be taken seriously. On the contrary, there remains the incontrovertible fact that overtones provide evidence of the close “relationship” be­ tween tones that the ear does not recognize as consonances. Any theory that at all pre­ supposes a relationship between simple frequency proportions and the sensation of con­ sonance will naturally have to come to terms with these tones, for C2: 7 B♭4, 11 F♯5, 13 A♭5, etc. (17, 19, 23, 29, 31, etc.). However, it turns out that in purely physical-acoustical terms nothing whatsoever can be made of them, but only on genuinely musical grounds, proceeding from established musical concepts. On that basis, the ear knows how to cope with these uncongenial incidental tones at least as equably as it does with the third and fifth overtones of a diminished triad, which contradict the sense of the chord: (p. 174)

What a chaos of disconcerting elements (F♯4, A4, C5, D♯5, F♯5, A5)! But these elements cannot prevent the [chord] B2 D3 F3 in C major from being heard as an elliptical G major chord(!) with a seventh. Page 7 of 27

The Problem of Harmonic Dualism: A Translation and Commentary If the derivation of the major consonance from the overtone series is itself already not all that secure, then the “explanation” of the minor consonance by way of the overtones is a downright impossibility. Helmholtz thought he could wriggle out of the situation by using the catchphrase “cloud­ ed consonance”—a white lie of which even Rameau availed himself, but which he soon came to regret,13 and which he retracted. Even Zarlino knew that the minor chord sounds sad (he called it “mesto”); he even ventures to say that it sounds less directly given by Nature, but rejects the thought (Opere, I, p. 221)14 that it might have an “altered” third by comparison with the major chord. If according to Helmholtz the lowering of the third by half a tone merely “clouds” the consonance of the major chord without eliminating it altogether,15 then it beggars comprehension why, for example, a similar alteration applied to the fifth should not likewise have nothing more than a “clouding” effect, and why the 7th and 9th overtones while not clouding the consonance at all (so long as they do not contradict the natural series) are nonetheless classified as dissonance when they are added to the chord. Helmholtz's attempt to explain foundational concepts with such sub­ terfuges has rightly already met with outright rejection by A. v. Oettingen (Harmoniesys­ tem in dualer Entwickelung, Dorpat & Leipzig: W. Gläser, 1866) and Hermann Lotze (Geschichte der Ästhetik in Deutschland, Munich: J. G. Cotta, 1868). But Helmholtz might have learned from Goethe (cf. F. Hiller: Goethes musikalisches Leben, Cologne: M. Du­ Mont-Schauberg, 1883) that it would look no more idiotic to try to derive the major chord by raising the third of the minor chord! A healthy, natural musical instinct cries out for the unconditional recognition of the consonant character of the minor chord, for unre­ served and straightforward coordination with the major chord as one of the two impor­ tant forms of consonance, alongside which there is not, and cannot be, a third option! All explanations that deny the primary and unconditional consonance of the minor chord are and always have been offensive to musicians, and arouse their suspicion. If the minor chord vis-à-vis the major chord were something artificial, derived, secondary, then it would have been obvious, too, that minor was more recent, that it emerged later than ma­ jor, whereas music history bears out precisely the opposite. Hand in hand with the simple derivation of the minor chord from the major chord by way of “clouding,” there comes also in Helmholtz (Lehre von den Tonempfindungen, 4th ed., p. 355)16 the idea of grounding the consonance of the minor chord on twin fundamentals from two overtone series. With this, Helmholtz embarked on a path that was bound to lead to the most dire consequences. His (p. 175) followers along this path were O. Tiersch,17 O. Hostinsky,18 and most recently G. Capellen and A. J. Polak (Über Zeiteinheit in Bezug auf Konsonanz, Harmonie, und Tonalität, Leipzig: Breitkopf und Härtel, 1900, and Über Tonrhythmik und Stimmenführung, Leipzig: Breitkopf und Härtel, 1903 [recte 1902]). The idea that the C-minor chord might combine elements of the C-major chord (C–G), the E♭-major chord (E♭–G), and maybe even also the A♭-major chord (C–E♭) to form an artificial unity contradicts not only the core idea of Rameau's system (consonance as a complex of tones all engendered by the selfsame “son générateur”), but, even more, a categorical demand on the part of the collective musical consciousness, which leaves one Page 8 of 27

The Problem of Harmonic Dualism: A Translation and Commentary uncomprehending that a thinker of the logical ability of a Helmholtz could have put it for­ ward in all seriousness. One doubt over this is certainly dispelled: that consonance cannot be anything other than a combination that leads to a unitary conception. For this reason it can only strike us as risible when Georg Capellen poses as an advocate of harmonic “monism”(!) while at the same time associating himself with Helmholtz and Hostinsky over the reliance of the mi­ nor chord on dual or triple fundamentals. The duality of harmonic relationships, which appears so simply and naturally persuasive for the over-riding separation of major and minor, is brushed aside with a superior smile, only to be resurrected in the separate issue of the grounding of the minor consonance in far more dubious form as the duplicity or even triplicity of fundamentals. Incidentally, the reliance of the minor chord on twin fundamentals actually has its origin in Hauptmann's Die Natur der Harmonik und Metrik, which appeared in 185319 and found its way from there into the work of Helmholtz. In his effort to prove that the upper tone of the minor chord (G in C E♭ G) functions as the center of reference among the three tones, Hauptmann invokes Rameau's idea of the son générateur and ascribes to the C-minor chord the two générateurs C and E♭, so that g becomes a doubly-generated tone. He puts it thus (p. 32): Just as in the major chord one tone (C in C G E) “has the third and fifth” (C–E, C–G), so there comes about in the minor chord one tone (G in C E♭ G) that “is had as third and fifth” (E♭–G, C–G) by the two others.20 A. v. Oettingen's explanation, too, is in fact absolutely identical with that of Hauptmann, since he ascribes to the “phonic overtone,” i.e., the first [common] tone that occurs in the overtone series of the three

tones of the minor chord, the role of the center of the tonal relations:

Thus even Oettingen ultimately upholds grounding the minor consonance in the phenom­ enon of overtones, and even assigns the minor chord three fundamentals in order indi­ rectly to achieve the unifying common tone. He subsequently eliminates the overtone se­ ries, so to speak, and retrospectively obtains the phonic central tone in this manner (p. 31), introducing the complete series of those tones that have the (p. 176) same overtone, using Helmholtz's terminology (p. 76), as the series of harmonic undertones:

All of this does indeed look like an over-riding reliance by the minor consonance on this series, but is in fact nothing of the sort. I openly confess that the pseudo-logic of this un­ dertone series constructed from [several] overtone series had even me fooled for a long time and can still be detected in my earliest writings on harmonic theory. Oettingen him­ self, furthermore, has most specifically (p. 46) pointed to the need for a generalization of

Page 9 of 27

The Problem of Harmonic Dualism: A Translation and Commentary the principle of the relationship of Klänge and stressed that this is “not dependent upon the genuine existence of overtones.” He himself has shown us the way out of this labyrinth, namely by means of his own defini­ tion of dissonance (p. 228), with which he successfully overthrows that of Helmholtz: “Dissonance is the simultaneous occurrence of two or more Klänge (the word Klang to be understood not as an individual tone but in terms of Klang-representation [Klangvertre­ tung]).” Unfortunately, he has overlooked the vicious circle lurking in his derivation of the undertone series from common overtones. On his own terms, the minor chord, too, as the “simultaneous occurrence” of two (or indeed three) Klänge, had inevitably to be a disso­ nance, which in fact he deemed highly appropriate for special cases (for the “secondary triads”), but he nevertheless denied as a general principle (for the main Klänge in minor). His crucial statement (p. 45), “The major chord is tonically consonant and phonically dis­ sonant; the minor chord is phonically consonant and tonically dissonant,” which puts my entire method of harmony teaching in a nutshell (viz. the grounding of the theory of Paral­ lelklänge and Leittonwechselklänge in major and minor), is unquestionably correct; how­ ever, it remains unproven in Oettingen's book. Nowhere is there the evidence for the au­ tonomous status of the minor relationship. The vicious circle by which Oettingen operates shows up most clearly in the sentence (p. 46), “we must confer on the minor chords… their phonically consonant quality. The interpretation that developed originally out of physiologically grounded phenomena now becomes all the clearer in the case of the mi­ nor chord; for the phonic overtone (of the minor chord, e.g., G5 for C3 E♭3 G3) exists in re­ ality, the tonic fundamental (e.g., A♭0 [recte A♭−1] as the tone of which C3 E♭3 and G3 are overtones)21 does not exist…the tonic fundamental is only a virtual Klang, the phonic overtone is a real one.” It is clear that the emancipation of the theory of consonance from real overtones, already perceived by Oettingen as necessary, is in fact totally indispensable if (p. 177) one wishes to escape from such vicious circles. For, as if it were not enough that the phonic overtone G5 for C3 E♭3 G3 already does, through its derivation from the [respective] overtone series of C3 E♭3 (and G3), give the chord two or three real fundamentals, now the tonic funda­ mental A♭0 [recte A♭−1], tone 1, of which all three tones are overtones (10th, 12th, 15th), adds its presence, too, as the fourth entirely real fundamental, even if it is only “virtual” (although in pure intonation it can bring to the interval a disagreeable reality as a combination tone). However elegant as a result the logical opposition may appear in Oettingen, it rests on a fallacy so long as it relies upon the real existence of overtones as essential evidence for the relationships among the tones. The idea of seriously wanting to derive the consonance of the minor chord simply from the overtone series, i.e. disregarding the “tonic” fundamental (C2) [which arises] out of the coexistence of the 3rd, 5th and 15th

Page 10 of 27

The Problem of Harmonic Dualism: A Translation and Commentary tones (G3, E4, B5) must of course be rejected outright, since on the basis of the same evi­

dence the following tone combinations would also have to be rejected:

On the whole, however, the consonance of any combination you care to cite of three, four or even more different tones of the series (needless to say, not on a keyboard in tempered tuning, which proves nothing in this context, but on an instrument yielding the pure rela­ tionships of the natural tones) would emerge as convincingly as the cases that Capellen demonstrated in his amateurish experiments “at the keyboard.” From this we can plainly see, thus, that Stumpf, with his categorical rejection of over­ tones as evidence of consonance, was on the right track. It may sound harsh when Stumpf describes Helmholtz's account of the basis of music theory as an antiquated view—but that is just what it is, and any attempt to revert to it can only revive old white lies and subterfuges. In order to make it easier for us definitively to discard the grounding of har­ mony in overtones (and also in combination tones), let us take an example, viz. the E4–G4 in the following short two-voice phrase:

The first overtones of these two tones bring B5 D6 G♯6 into the collective Klang, but this cannot stop our hearing the E4–G4 as representing the subdominant, viz. the G (p. 178)

minor chord(!), hence corresponding to the four-voice writing:

That such processes of hearing can be explained by means of neither overtones nor com­ bination tones is obvious without more ado. In this case, physics is entirely to be discount­ ed, and only music psychology (music aesthetics) can provide an explanation.

III. Interval Fusion or Klang-Representation? Thus we have Stumpf to thank for the emancipation of music theory from acoustical phe­ nomena. Objectively existing overtones are not the basis of consonance; rather, they are a pointer to a much more general interrelationship that exists between tones by means of the commensurability of the conditions of initiation and continuation of the vibrations producing them in elastic bodies, and of the thereby produced functions of the organs of human hearing that perceive them. Whatever types of transformation of tone vibrations into affections of the central organ­ ism these latter may be, they will surely never be accessible to scientific method; at the very least, there can be absolutely no question today of any precise specification of them. Page 11 of 27

The Problem of Harmonic Dualism: A Translation and Commentary If now even the transition from the mechanisms of the outer ear (the eardrum, the ossi­ cles of the ear), which are well-understood as regards their mechanical dependence upon sound-transmission, to the tone-representations [Tonvorstellungen, i.e., mental represen­ tations of tones], is still obscure at the current state of scientific knowledge (whether Helmholtz's interpretation of the functions of the membrana basilaris is right or not alters matters very little), the dependence of the tone-representations upon tone-vibrations is now beyond doubt, and is so well understood that specific conclusions can be drawn from former to latter. However—and here we stand on the threshold of something important that leads from the physics of hearing, or physical hearing, to musical hearing: the dependence of tone-repre­ sentations upon tone-vibrations is not an absolute, inasmuch as that all the scientifically demonstrable elements of the Klänge striking the ear would have a conditioning influence on the shaping and concatenation of the tone-image, if it were not that it [the depen­ dence] is limited by a selective and ordering activity of the mind that perceives them, which compares to each other the tones that occur (p. 179) successively or simultaneously. For musical hearing, even that of the listener whose ear has not been developed through technical training or habituation, is not just a passive physical process, but rather a psy­ chic activity, a continuous comparing and associating together of successive tones and concurrent simultaneities. That is the only possible explanation for the fact that the par­ ticular acoustical tuning of an interval has absolutely no power to compel the listener to interpret that interval according to the exact quality of that tuning; and moreover that the “musical interrelatedness,” the relationships of the tones to one another, make decisions according to principles of greatest simplicity, of highest possible ‘economy’ of presenta­ tion. The objectively produced Klänge are ultimately only a sort of raw material for musical hearing, a crude base-matter from which the representing mind fashions, in an admitted­ ly restricted but not absolute dependency, the [musical] pattern that affords it delight. The fact that musical hearing is in fact continuously a selecting, order-seeking, consisten­ cy-finding, psychically active [process], and not a physically passive one, is the only possi­ ble explanation of how it contents itself with (tempered) approximations instead of ab­ solutely pure intonations, and how it is in a position to overlook, to completely disregard, incidental tones [Beitöne], often of very considerable strength, that do not lie in the artis­ tic intention but are conditioned by the nature of the Klänge of our musical instruments.22 The recognition that musical hearing is not passive but active, however, at the same time affects the transition from physiological inquiries all the way to the facts about actual mu­ sical hearing. Anyone at all engaged in this issue knows the yawning gulf that lies be­ tween Parts II and III of Helmholtz's Lehre von den Tonempfindungen: in the former, ob­ servations of objective processes in the domain of physical and physiological acoustics; in the latter, a concern with the problems inherent in the processes of musical hearing that play out entirely within the psychological-aesthetic domain. Stumpf's attempt to unify the two separate domains by means of tone-psychology23 cannot yet be said to have succeed­ ed. For the tone judgments that comprise the subject matter of the new science are still Page 12 of 27

The Problem of Harmonic Dualism: A Translation and Commentary as remote from the domain of the actual processes of musical hearing as are the results presented in Parts I and II of Lehre von den Tonempfindungen. Stumpf, in fact all tonepsychologists, have paid far too little attention to the fact that musical hearing, even just in its most primitive elements, is an active, selective process, i.e., for many instances of physical sound it is nothing short of an overlooking, a not-hearing. But without this recog­ nition, without the constant appreciation of this crucial fact, it is not remotely possible to find the connection between physical and physiological acoustics and the beginnings of music! Stumpf made the attempt but gave it up (the continuation of his Tonpsychologie will nev­ er appear [vols. 1–2, 1889, 1892]).24 He moved from investigating single sounds and their distinction to investigating dyads and their greater or lesser fusion, and on that basis con­ structed a scale of degrees of fusion which led gradually from the consonances to disso­ nances and discords. He has not found an overriding distinction between consonance and dissonance, and totally eradicated the distinction between dissonance and (musically meaningless) discord. Thus in sum he has perpetuated through alternative formulations the main errors of Helmholtz's account (p. 180) of the basis of music theory. He has no more found a satisfactory explanation for minor consonance than did Helmholtz, and has come no closer to a concept of chord or to the overriding distinction between major and minor. For, the definition that simultaneities comprising more than two tones are consonant so long as none of the intervals between any constituent pair of tones is a dissonant one, can count only as an attempt at a negative description. It yields a large number of possible combinations which can easily be sorted with the help of our customary concepts into ma­ jor chords and minor chords, but without really understanding why. Incidentally, I acutely embarrassed Stumpf several years ago (Musikerkalender, 1898)25 over a very awkward question, viz. why on a temperately tuned keyboard the augmented triad C E G♯ (A♭) ap­ pears dissonant, although the three pairs of tones: C–E, E–G♯, C–A♭ are understood as consonant. Stumpf's subterfuge that it is impossible for the ear to hear the same tone in the same simultaneity as both G♯ and A♭ can of course be dismissed, since this itself rests on concepts with which, in light of his previous statements, he has no right whatsoever to operate. The fact that the ear willingly attributes consonance to each of the three dyads (despite the tempered tuning!) and yet unhesitatingly categorizes their triadic combination as a dissonance cannot be explained solely by investigating the dyads. I, on the other hand, ever since the appearance of the second volume of Tonpsy­ chologie, have been predicting that Stumpf would never be able to arrive at a satisfactory conclusion so long as he did not move on swiftly from the examination of dyads to ground­ ing harmonic hearing by way of triads. My prophesy that he would never make headway on his current path has been fully vindicated.

Page 13 of 27

The Problem of Harmonic Dualism: A Translation and Commentary Let me not expatiate unduly, but just say, in a word, that the alpha and omega of all music lies decidedly not in hearing tones in terms of intervals (dyads), but rather the contrary: in hearing in terms of triads. The listener of today undoubtedly, and very likely the listen­ er of any era, hears even purely monophonic melody in terms of harmonies (tone-com­ plexes). The only two species of complexes, however, in terms of which individual tones can be heard equally well as two-, three-, and more tone chords, are the major chord and the minor chord (Zarlino, Opere, I, p. 222 “Da questa varietà dipende tutta la diversità e perfettione dell’ harmonie”).26 The major chord and minor chord are not two arbitrary conglomerates of tones existing inter pares alongside however many others; rather, they are the two unique, fundamentally different [entities] in terms of which all other possible combinations are heard. The presupposition for this recognition is of course the close consolidation of all octave tones into the concept of tone in the wider sense, as I estab­ lished it as early as 1877 in my Musikalische Syntaxis,27 and as Stumpf likewise adopted it in his Tonpsychologie in 1892 under the name “expanded meaning.”28 If the concept of the close interrelatedness of all octaves remains valid—and to deny it would be absurd, although its complete explanation or causation is as everybody knows impossible—then all combinations that qualify as consonant, when compressed into the narrowest register, in fact reduce to the two formulas:

(p. 181)

1. Prime with (major) upper third and upper fifth, 2. Prime with (major) under-third and under-fifth. At first sight, of course, nobody would object if that formulation were replaced by this al­ ternative: Prime with (minor or major) upper third and perfect upper fifth to which all the world is accustomed, except that those who give priority to the latter wording will then have to stop talking about grounding harmony in overtones, since every reference of this type immediately calls the minor consonance into question, and makes of it an insoluble problem. But the fact that we hear everything that can occur in music at every single instant in terms of either a major chord or a minor chord speaks unambigu­ ously to the impossibility of there ever being a third category alongside major tonality and minor tonality, that is, the reliance of a melody or chord progression upon a major chord or a minor chord as its central Klang (Rameau's “centre harmonique”).

IV. The True Root [Wurzel] of Harmonic Dualism With this, we come at long last to our central question: Why ought we to think of the con­ struction of the minor chord as the inverse of that of the major chord? If we really do de­ finitively see some way of invoking the overtone series as the basis of consonance, what is there to stop us adopting a twin construction of consonant relationships, in tandem, up­ wards? 1. Prime, major third, fifth.

Page 14 of 27

The Problem of Harmonic Dualism: A Translation and Commentary 2. Prime, minor third, fifth. The first thing to object to in this is that restriction to a single mode of mathematically defined explanation for the two relationships necessary for either chord individually in­ curs complicated numbers, namely:

i.e., the definition in terms of frequencies represents the minor chord as a less simple for­ mulation, while that in terms of string lengths represents the major chord likewise. This simple comparison teaches us that the minor consonance is more correctly derived from the relative size of the sound waves (string lengths, pipe sizes), the major consonance on the other hand is derived from the relative speeds of vibration. If we recall in this respect Lotze's subtle argument (Geschichte der Ästhetik in Deutschland, p. 272) concerning qualitative experience of pitch as a confluence of two quantitative factors: intensity (speed of vibration, number of cycles in a given time-span) and mass [Volumen] (wave-length, overall size of the vibrating elastic body), then it is not (p. 182)

difficult to perceive that the search for the simplest numerical ratios for the two quantita­ tive factors points intrinsically to dualism as the basis for harmony. But perhaps no one has yet arrived at the thought (the realization dawned on me for the first time only [as I worked] through this chain of deductions) that what distinguishes ma­ jor from minor comes down to the essence of major consonance being the simplest ratios in the increase in speed of vibration, and that of minor consonance, by contrast, being the simplest ratios in the enlargement of the vibrating mass (wave-length, length of string, etc.). Thus, put simply, the principle of major can be seen to lie in growing intensity, the principle of minor in accumulating mass. I should like expressly to emphasize that this new definition does in fact correspond best to the character of the two modes. Hauptmann's “downward-dragging weight” of the minor chord can be fully grasped only when one recognizes the growing mass of the deeper tones connected to tone 1, in con­ trast to which the upward-striving, bright, luminous character of the major consonance is best explained in terms of all the tones connected to tone 1 (the prime) owing their origin to the increases in frequency. In a word, then: the principle of major is increase in speed of vibration (ascent toward the next-related tones), the principle of minor is growth in wave-lengths (descent toward the next-related tones). Consequently, the two are numerically best expressed through the same simple number series:

And the invoking of the series of simple fractions ([to express] the logical opposition be­ tween harmonic and arithmetic divisions) that has been customary since Zarlino can Page 15 of 27

The Problem of Harmonic Dualism: A Translation and Commentary henceforth once and for all be discontinued, since it merely serves to obscure the fact of absolutely equal simplicity and originality of the two series. It should further be noted, however, that only polyphony has created the need so obvious in today's musician to think of all simultaneities as constructed from the bottom upward. Indeed, for the first centuries of polyphonic practice (9th–11th centuries) the very oppo­ site way of thinking was customary: in its earlier phase, the “organum” moved mostly be­ neath the main voice, and only with the age of discant did the inverse way of thinking first gradually come about. (Fauxbourdon, too, as late as the 15th century, was thought of in terms of under-thirds.) But for homophonic, i.e., exclusively monophonic music such a concept is not at issue, and even for today it must be stated that singers can hear in their head any interval whatsoever from the top down just as easily as from the bottom up (they can on average sing the former more easily, but there is another reason for that). If nowadays we are (p. 183) accustomed to counting the intervals from the bottom up, this is not in the least because it comes more naturally, and for example the Arabs in the 14th century, and probably still earlier, proceeded inversely; that is, they evolved their scale of intervals (proceeding from the octave to progressively smaller and more complex ones) through the division of the string into twelve equal parts:

To the Greeks, too, the descending order of tone ratios seemed decidedly the more natur­ al one and the only self-evident one. While we are today accustomed to thinking of the ba­ sic scale as the series: the Greeks on the other hand thought of it inversely: Anyone who wants to contest the modesty and naturalness of minor harmony should be very careful not to raise the question of chronology. But even if chronology is left com­ pletely out of it, the sheer contrariness of major and minor relationships is in so many ways so striking that one has to shut one's eyes deliberately not to see it. After we have, with the above, uncovered the true root of harmonic dualism, there re­ mains a problem as to why the concept of consonance in both its forms is limited to the ratios of the fifth and third. Why do the tones corresponding to prime numbers that lie still further away from unity than 5 (7, 11, 13, etc.) not appear also to be connected in the same way to tone 1 and so included within the unity of the Klang? The answer to this, like that to the question of the special status of the octave, can only be to attribute it to a fact of our musical hearing, though perhaps with better reason in this case. It will never be possible to adduce a full explanation as to why the octave of the octave appears to us to be completely the same tone as tone 1, or why also the octave below and any other octave extension upward and downward appears to us as harmonically identi­ cal with tone 1, or why any octave displacements of the fifth or third leave their harmonic quality unchanged, whereas any concatenation of either of the other two intervals, or even any combination of the same, is immediately categorized by the ear as a dissonance, Page 16 of 27

The Problem of Harmonic Dualism: A Translation and Commentary for example, starting with C3, the fifth of the fifth, D4, or the third of the third, G♯3, or the third of the fifth (or fifth of the third), B3. It is perhaps conceivable that such intervals, which allow for a mediated relationship with the prime through division into two simpler intervals, are, just because of this possibility, no longer attracted directly to mediation, but turn the mediating tone into a (p. 184) second prime over against the first prime, and that this independent derivation constitutes the essence of the dissonance (D4 as fifth of G3 and B4 [recte B3] as third of G3 interpolate a second prime, G3, over against the first prime, C3; G♯3 as the third of E3 likewise interpolates the prime E3 over against the prime C3). It is, however, an abiding puzzle that the simplest interval, the first of all intervals, the oc­ tave, is totally immune from being rendered independent in this way: that the fifth of the octave and the third of the octave do not likewise render the octave tone independent. The necessity to recognize the “expanded meaning of tone” as a psychological reality is thus not to be denied; it is a fundamental fact of hearing that must be recognized, and is recognized every time without any attempt to contradict. All the octaves, not only those of the prime but also those of the fifth and third, are thus eliminated from the two series un­ der consideration. All tones whose order numbers in the series constitute products (with the wholesale exception of all the even-numbered ones, which signify only octaves, i.e., are harmonically identical with their halves) are dissonant with the prime (9 = 3 · 3, 15 = 3 · 5, 25 = 5 · 5, 27 = 3 · 3 · 3, etc.). Thus only the prime numbers 7, 11, 13, 17, 19, etc. remain, and for the ear to ignore any of these would naturally demand an explanation. That the tone which corresponds to the ratio of the seventh tone of the series is perhaps really to be understood in musical terms as belonging to the Klang has long been noted, and used for harmonic experiments by, for example, Tartini and Kirnberger. Similar at­ tempts have recently been made also (Claude Debussy, W. Rebikoff) with the eleventh and thirteenth overtones. Nevertheless, I do not hold out any prospect of these attempts lead­ ing to an extension of the concept of consonance, for the simple reason that the forces that have hitherto led to the tones that correspond to these remoter prime numbers of the series being denied any direct relationship with the prime will remain irresistible for the foreseeable future. The basis for this is the possibility that these tones might be interpret­ ed in terms of tones that are differently obtained but which lie very close to them in pitch. The 7th tone of the over-series corresponds closely to the 9th tone of the under-series (the second under-fifth), tone 11 of the over-series to the tone 3 of the under-series (the first under-fifth), tone 13 of the over-series to tone 5 of the under-series (the under-third). Likewise, tone 7 of the under-series [corresponds closely] to tone 9 of the over-series, tone 11 of the under-series to tone 3 of the over-series, and tone 11 [recte 13] of the un­ der-series to tone 5 of the over-series. The seemingly great distance of tone 9 from tone 1 at issue here is in both cases reduced by the fact that it empirically plays its main role not with the tonic but with one of the dominants; for example, C E G B♭ has the feel of a dominant harmony, the tonic chord of which is F major. The tone B♭ appearing with the C major chord is, to be sure, the 9th un­ dertone of C, but only the 3rd [undertone] of F, the true central tone. Likewise, the underseventh, A, appearing with the C minor chord29 corresponds to the 9th overtone of G; but Page 17 of 27

The Problem of Harmonic Dualism: A Translation and Commentary since the chord belongs specially to G minor, this A, which lies two fifths [up] from G, is only the 3rd overtone of the tonic prime D. The two other examples are to be judged cor­ respondingly, but are much easier to understand because they substitute a more distant relationship [on one side] by a closer one on the other side (3 in place of 11, 5 in place of 13). If we now gather the result of our chain of conclusions together summarily, we can say: There are two types of relationship, that of the compatibility [Vereinbarkeit] of tones, and that of the relatedness [Bezogenheit] of tone-complexes to a point of depar­ ture: (p. 185)

(a) the series of simple multiples with respect to frequencies (over-series), (b) the series of simple multiples with respect to string-lengths (under-series). These series do not require verification by way of acoustical phenomena, referral to which, on the contrary, only complicates and confuses the understanding of what are in themselves very simple ratios. The tones of the two series conveniently compress into the closest register through the identity of the octave tones: Just as the consolidation of tones of different registers into the closer unity of the concept of tone ceases at the octave, so, too, is the concept of Klang restricted to the three tones (including whatever octaves) prime, (major) third, and fifth. All more distant tones, corre­ sponding to higher order-numbers, either are subsumed under double relationship via a mediating middle tone ([hence] are dissonant), or are exchanged with tones of the oppo­ site series and are rejected or totally disregarded by the ear as component[s] of the sin­ gle Klang. The next higher, or further, concept beyond that of the tone (including its octaves!) is thus that of the Klang, the only three elements of which are the prime, the third, and the fifth tones. All tones or simultaneities that are at all intelligible as music are heard as repre­ sentatives of such triads and are combined with one another and compared. Outside of all the Klänge constructed from three constitutive tones in this way, there is no reason what­ soever to set up simultaneities of only two tones (intervals) as a special category, since even just single tones are invariably heard in terms of Klänge. The uncertainty as to how to interpret a single tone or even a two-note interval that begins a piece of music in terms of a Klang—uncertainty that will quickly disappear as the piece progresses—affords no reason to posit special categories of interval intermediate between the single tone and the Klang. The consonance of intervals resides in the extent to which they belong to a Klang, and there is no merit whatsoever in distinguishing multiple degrees of consonance for the different intervals that may be formed by the tones constituting a Klang. However, one can of course speak of a Klang as being more clearly or less clearly represented by just one or two of its constituent tones. Ever since the concept of Klang was first recog­

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The Problem of Harmonic Dualism: A Translation and Commentary nized people have spoken of Klang-representation, and for centuries communal musical experience has become accustomed to hearing individual tones as elements of chords. This “way of thinking in terms of Klang-representation,” however, knows no chord that can be “represented” by tones or intervals other than the major chord and the minor chord. For when one, for example, says that in C major F can be heard as a dominant 7th, then that means nothing more than that F is heard as a tone added (p. 186) to the G major chord (G B D F). But nobody would think to suggest hearing G or B in a monophonic melody in C major as a representative of the dominant 7th chord G B D F: only the F itself can operate in that way. Likewise, D in C major can be understood either as the 5th of the G-major chord30 or as the 6th of the F-major chord, thus as a tone added to a chord to which it does not intrinsically belong. These facts have long been known: it would be su­ perfluous to explain them here.31 I raise them only to prove that it comes easily to us to hear individual tones or even “dyads” in terms of complete harmonies comprising prime, third and fifth; and that conversely one absolutely cannot speak of hearing a monophonic melody in terms of dyads (intervals). Therefore, so long as Stumpf and his school do not get beyond researching dyads, they will never enter the realm of musical hearing, which is, after all, sufficiently confirmed through the importance that they attach to beating and incidental tones [Nebentöne] of all types that the musical ear has to disregard.32

V. The Problem of the Fundamental [Grundton] I hope I have been successful, in my argument so far, in demonstrating that if we want to reach an understanding of the dual basis of harmony we must first of all set aside our knowledge of acoustical phenomena, since this, far from providing the key to such under­ standing, is in large measure guaranteed to impede it. I myself have grappled long enough with the spurious evidence of such phenomena, and have spilled much ink, have drawn many an elaborate deduction, that I do not so much have now to retract, as wish to draw attention to as evidence of the contrived convolu­ tions and subterfuges into which one is led when deriving consonance from such phenom­ ena. From the much-sought-after evidence for generation of undertones within the ear through the sound waves that strike the membrana basilaris (in my dissertation, 1873), to the alleged objective existence of these undertones ([Musikalische Syntaxis,] 1877), all the way through to the evidence that, despite multiple generation of each tone of the un­ dertone series by the tone that actually resonates (in accordance with its order-number 2, 3, 4, and multiples), these undertones are annihilated as a result of interference (Kate­ chismus der Akustik, 1891), I wrestled incessantly with acoustical phenomena as the ba­ sis for consonance, until the liberating sentence in Stumpf's Tompsychologie33 at long last revealed—to me at any rate (though not to Stumpf himself)—that we must distance our­ selves from the attempt to find a basis for consonance by way of overtones and combina­ tion tones. Even if the epiphanic force of this crucial sentence had dawned on us with all its implica­ tions, we would still have been faced with the necessity of reckoning with, and having to Page 19 of 27

The Problem of Harmonic Dualism: A Translation and Commentary come to terms with the contiguous tones of these categories, since there is no rationale for disavowing their objective existence. Even if overtones and combination tones are de­ finitively not the cause of consonance, it is still incontrovertible (p. 187) that they so make their presence felt by their often very considerable loudness that one is forced (especially in polyphonic music) to take account of them. This points up that in many instances they still play a not inconsiderable supporting role in musical hearing, and in all cases con­ tribute significantly to the fullness and color of sound. Overtones, especially, can acquire importance for polyphonic composition through their appreciable loudness, which not only makes it impossible for us to disregard them but ac­ tually demands that we take their existence into account for composition [Tonsatz] itself, that one keep them in mind for the regulation of voice leading. In particular, there are two experiences of practical composition which point to overtones as really existing: (1) the hateful effects of so-called false parallels, and (2) the treatment of the third tone of the under-series as a fundamental. First, as regards false octaves and fifths, there are surely not many musicians who will join with the radical reformer Herr Capellen in considering the prohibition of [parallel] fifths and octaves a ridiculous piece of pedantry (Die Freiheit oder Unfreiheit der Töne und Intervalle (Leipzig: Kahnt, 1904), p. 25). I do not deem it necessary to rehearse and counter his arguments here: it is for him to defend them as best he can against others. [Second,] where perfect successive fifths and octaves between real voices are concerned, I take a strict line, and balk at them even when in contrary motion. Be it noted here only that the prohibition of [parallel] octaves and fifths has its basis in the real existence of overtones, because two voices that move (stepwise) in such intervals run the danger of being taken for one [voice] insofar as the tones of the higher voices coincide with over­ tones of those of the lower ones and can be heard as such. Nobody objects to mere mix­ ture-fifths34 or octave doublings (Capellen, however, defends precisely the fifths for which such an interpretation is untenable; these sections of his writing are in my view contrary to art and should be repudiated in the sharpest terms.) The influence of overtones, really existing, on voice leading with respect to the grounding of the minor chord is quite another matter. Those who are in sympathy with harmony teaching as I present it are right in asking: “How is it that in the minor chord it is the (under-)fifth that provides the good bass tone, and that for constructing cadences the only tone properly suited to cadencing is the bass tone?”; and opponents and renegades ask somewhat more shrilly: “How can one denomi­ nate the minor chord after its topmost note when surely its lowest tone must [serve that purpose]?” First, the short answer to the latter, with its abrasive and provocative tone, is that a chord must be denominated by that tone by means of which the others are designated as near­ est relatives (third, fifth, etc.), and that the claim that principal tone (prime) and root [Grundton] are the same under all circumstances still remains to be proven. However, op­ Page 20 of 27

The Problem of Harmonic Dualism: A Translation and Commentary ponents of dualism will not just let it go at that. We will need to go into the question more deeply, not least because the misgivings of the friends [of dualism] must also be dispelled. What is a root [Grundton]? By the terminology customarily found in harmony books, it is generally the lowest tone when the tones of the chord are stacked in thirds. (p. 188) It is perhaps not superfluous to note that this theory of determining the root position of a chord through sorting and stacking of tones in thirds originates with the selfsame great theorist J. Ph. Rameau who himself punched holes in this system of construction by thirds, which was promptly seized and generally adopted by his own and subsequent generations merely because of its crude simplicity—for example, in recognizing the added-sixth chord (accord de sixte ajoutée), viz. the subdominant chord plus 6th (F A C D in C major), the root of which according to Rameau was (if not always, then in many cases) not D (D F A C) but rather F. This same Rameau, however, had earlier, in his first publication (Traité de l’harmonie, 1722), in which he is still a strict “monist” à la Capellen and a fanatic oppo­ nent of Zarlino-style dualism, nonetheless discovered that B D F A in A minor is really a D minor chord with a third sub-posed(!); he does not merely content himself with embrac­ ing the sixte ajoutée for this case, which would have left the D-minor chord its “root,” but invents another third below what would otherwise count as the root! But enough of our historical excursion; anyone who is interested in the history of harmonic dualism and de­ velopment of the theory of the interpretation of chords as Klänge can find the most impor­ tant elements clearly assembled in the third book of my Geschichte der Musiktheorie. Eschewing any further digression, I still have just the central question left to address: Why in the minor chord, in practical multi-voiced composition, does the lower rather than the upper tone of the 5th interval have to assume the role of providing the foundation of the harmony? “Monists” like Capellen and Polak could ask also in all seriousness: Why cannot the 3rd of the minor chord be its root? Now, to my mind the reason is quite simple. All the apparent inconsistencies in the practi­ cal treatment of the minor chord as against its theoretical derivation can be explained in the least forced manner in terms of the objective existence of overtones. To place the third in the bass and somehow to see the layout of the minor chord that corresponds to the numbers 1, 3, 5 of the under-series as the norm:

is forbidden on account of the very strong third overtone of C2 that is present in almost all tone-colors, which runs the danger of hearing a G3 as sounding along with it. Equally well, however, to have E2 as the bass tone would seriously interfere with hearing the chord as of A minor because of the third overtone, B3:

The upshot is thus quite simply that the placement of A1 into the bass produces the layout of the chord that is clearest and that renders the overtones that cause interference least Page 21 of 27

The Problem of Harmonic Dualism: A Translation and Commentary audible. This is because whenever the C4, which is present as a strong (p. 189) chord-tone, is placed in direct proximity to the fifth overtone of A1 (C♯4), it overpowers the latter,

causing it go unheard since it is generally not all that strong:

But the fact that the prime of the minor chord can actually be omitted altogether, and very often is absent, especially at cadences (at least when the dominant-seventh chord strengthens the cadence), can likewise be perfectly satisfactorily accounted for by the strength of the third overtone of A, most particularly when in four-part writing A is given

in three of the voices:

Here we actually have three occurrences of E as third overtone! The fifth overtones of the three as (C♯5, C♯6, C♯4), which are not chord-tones, are paralyzed by the octave- tones [i.e., prime plus overtones 2 and 4] of C4, and go unnoticed. (The musical ear picks up none of the dissonant effect produced by the beating of C against C♯.) Once one has truly divested oneself of the idea of wanting to derive the principles of con­ sonance and dissonance from [acoustical] phenomena, then there is no longer in fact any­ thing problematic about the logical opposition of major and minor. Since it is a fundamen­ tal fact of our musical hearing, yet one that cannot be inferred from the world around us, that we regard octave-tones as more closely unified than any other tones that combine in­ telligibly, so too it is nowhere decreed that the wide-spaced [proportion] 1 : 3 : 5 should be taken as the norm for both types of consonance; on the contrary, it seems obvious when we think of the restricted range of singing voices that the closed position of 4 : 5 : 6

should be regarded as the prototype of both, though with a nod to the unifying 1:

so that the teacher instructing children, particularly beginners in the early stages of har­ mony, has no need whatsoever to speak of acoustical phenomena just so long as he does not have to explain the riddles such as, for example, how a tonic chord that cadences without the upper tone of the fifth-interval (5, I) is perfectly comprehensible. Doubtless my harmony books, all of which still take up battle positions against the old method, contain much that is superfluous to the elementary student of composition, (p. 190)

indeed actually harmful, because confusing. A truly elementary book of harmony along dualistic lines will eschew all speculative theoretical questions as to prime causes and completely avoid aggressive or defensive polemics, restricting itself to prescribing with­ Page 22 of 27

The Problem of Harmonic Dualism: A Translation and Commentary out digression what the student must and must not do. The elementary student does not need reasons, only concise and tightly drawn definitions and stipulations of rules. If he later matures into one who asks Why? then instead of the elementary one he will be ready for a speculative theoretical book, since he will by then be familiar with all the basic drills, and in a better position to understand arguments raising problems. On the basis of long experience of teaching over a quarter of a century, I can with some claim to credibility give an assurance that the denomination of the minor chord after its highest tone is not the slightest source of confusion to the elementary pupil (or even to a child). On the contrary, the very oppositeness inherent in the construction of the two un­ derlying formations makes sense to him immediately, and with every new step he delights in the self-evident nature of the two complementary structures. It is thus entirely correct and pedagogically appropriate to teach major and minor side by side right from the out­ set, and not to dwell first exclusively on major. Any hint of minor being derived in some way from major must be avoided right from the start. It is not wholly clear to me how several of those close to my theory can see in the intro­ duction of function symbols (T D S etc.) something of a regression from the core idea of harmonic dualism. In the Orgaan van de Vereeniging van Muziek-Onderwijzers of 15 Oc­ tober 1904, published by W. Hutschenruyter of Amsterdam, Ary Belinfante in his article “De leer der tonalen functien in conflict met die der polaire tegenstelling” has contended that the teaching of functions actually marks a reneging on the polarity of major and mi­ nor, and regrets that I “have lacked the guts to admit to an honest change of heart.” There is no reason for me here to go into the shaky logic by which Mr. Belinfante imag­ ines himself to have uncovered an inconsistency between function symbols and the dual basis of harmony and tries to expose a change of mind on my part. It is enough for me to give an assurance that I still stand by my original idea, and that the working of function symbols into new editions of earlier books (Handbuch der Harmonielehre,35 Katechismus des Generalbaßspiels,36 Kleine Kompositionslehre37) has in no sense given rise to further text revisions other than the necessary obvious indispensable explanation of the new sym­ bols. If Mr. Belinfante has uncovered inconsistencies, then they are entirely the result of mis­ understandings of my arguments. According to his view (and perhaps those of some other gentlemen), in order to remain “consistent,” I would have had when in A minor to call the D-minor chord the Dominant and the E-minor chord the Subdominant, and consequently use the symbols S and D in minor in the reverse sense from that of major. In so saying, Mr. Belinfante completely overlooks the fact that this selfsame basis would then have ne­ cessitated the use of these signs in the reverse sense in all of my writings on harmonic theory since 1873. He forgets that these names, far from being coined by me, have been in general use ever since Rameau, and that I have maintained them (p. 191) with the same rights and for the same reasons that [I have] the designations [for] major, minor, Parallele, Grundton, and a host of others; and that for that very reason I have, right from the begin­ ning, and even in my most recent writings, introduced the clearly defined designations schlichter Quintklang, Gegenquintklang, Seitenwechselklang, and the whole terminology Page 23 of 27

The Problem of Harmonic Dualism: A Translation and Commentary of Schritte (Gegenquintschritt, etc.), in order to clarify the totally different senses of the dominants in major and in minor. Although one of my private students once assured me that the terminology of the steps no longer gave him any trouble since I introduced the set of symbols for functions, I am per­ fectly sure that the person concerned had discarded only the names, and was far from now regarding the dominants of major and minor as equivalent. That this cannot happen when studying by my methods is expressly insured by the improvements to my symbol system, in which, for example, the prime of the tonic minor is the same tone as the prime of the dominant major. So to speak of “reneging” is nonsense. I take the same view today that I did 30 years ago. All that I have done is to free myself from basing the principles of harmony in acoustical phenomena and laid bare the true roots of dualism, toward which Stumpf was digging except that he covered himself with more and more debris the longer he labored. And so this study comes to an end. I hope I have succeeded in formulating a few sen­ tences that are destined to satisfy the wishes expressed by my friends.

Notes: (1.) Georg Capellen, “Die Unmöglichkeit und Ueberflüssigkeit der dualistischen Molltheo­ rie Riemanns,” Neue Zeitschrift für Musik 68 (1901), 529–531, 541–543, 553–555, 569– 572, 585–587, 601–603, 617–619. This serialized article was later republished in Capellen's Die Zukunft der Musiktheorie (Dualismus oder “Monismus”?) und ihre Ein­ wirkung auf die Praxis (Leipzig: C. F. Kahnt Nachfolger, 1905). (2.) Riemann, “Die objective Existenz der Untertöne in der Schallwelle,” Allgemeine Musikzeitung 2 (1875), 205–206, 213–215. He modified the undertone hypothesis in Handbuch der Akustik (1891), arguing that they exist but are, in fact, inaudible. (3.) Capellen, “Die Unmöglichkeit,” 531. “Wie kann unter diesen Umständen eine Theorie Geltung beanspruchen, die durch die Naturgesetze nicht zu begründen ist? Und wie ist es möglich, dass gerade Riemann, der doch sein System auf wissenschaftlicher Grundlage aufbauen will, trotz seines eingestandenen Irrtums an der Untertonhypothese festhält?” (4.) Ibid., 601. (5.) See Michael Arntz, Hugo Riemann: Leben, Werk, Wirkung (Cologne: Allegro, 1999), and Ludwig Holtmeier's contribution to this volume. (6.) Admittedly, this would work only on a piano that not only follows just intonation but is also capable of accounting for the microtonal deviations of the harmonic series in this range. It seems, however, that in this context such details would merely get in the way of a good polemic.

Page 24 of 27

The Problem of Harmonic Dualism: A Translation and Commentary (7.) Riemann had in fact adumbrated this shift for a long time, most clearly in the article “Die Natur der Harmonik” (1882). A translation of this essay, by Benjamin Steege, is in­ cluded in this volume. (8.) See especially Capellen, “Die Unmöglichkeit,” 641–642. (9.) Henry Klumpenhouwer revisits Belinfante's criticisms in his contribution to this vol­ ume. (10.) [(Leipzig: Max Hesse, 1898), Part III; trans. W. C. Mickelsen as Hugo Riemann's The­ ory of Harmony with a Translation of Riemann's “History of Music Theory,” Book 3 (Lincoln: University of Nebraska Press, 1977).] (11.) I have in mind here, for example, G[eorg] Capellen's new set of symbols (M, L, R [Middle-chord, Left-chord, Right-chord]) in place of my T, S, D, and extensions of this by A. J. Polak and others. (12.) [Monism in this sense is the general theory that there is really only one fundamental kind of thing in the universe. As applied to Rameau, the universe of harmony emanates entirely from one thing: (at first) the first six divisions of the string, (later) the corps sonore.] (13.) [sich schämte: lit. “was ashamed of.”] (14.) [De tutti l’opere del R. M. Gioseffo Zarlino (Venice: Francesco de’ Franceschi, 1588– 1589; facsimile reprint, Hildesheim: Georg Olms, 1968). Riemann refers to a passage from Le Istitutione harmoniche (Venice, 1558), bk. III, ch. 31, 181: “quando si pone la Terza maggiore nella parte grave, l’Harmonia si fà allegra; & quando si pone nella parte acuta, si fà mesta.” Translated by Guy Marco and Claude Palisca as The Art of Counter­ point: Part III of Le Istitutione Harmoniche (New Haven: Yale University Press, 1968), 70: “when the major third is below, the harmony is gay, and when it is above, the harmony is sad.”] (15.) [Riemann refers here to Helmholtz, Die Lehre von den Tonempfindungen, 4th ed. (Braunschweig: Vieweg und Sohn, 1877), 478: “In der Regel finden wir deshalb den Mol­ laccord c–es–g in der modernen Musik so gebraucht, dass c als sein Grundton oder Funda­ mentalbass behandelt ist, und der Accord einen etwas veränderten oder getrübten c-Klang vertritt”; in A. J. Ellis's translation: “Hence in modern music we usually find the minor chord c–e♭g treated as if its root or fundamental bass were c, so that the chord ap­ pears as a somewhat altered and obscured compound tone of c.” On the Sensations of Tone as a Physiological Basis for the Theory of Music (London: Longmans, Green and Co., 1912; New York: Dover, 1954), 294.] (16.) [On the Sensations of Tone, 294–295.] (17.) [Otto Tiersch, Elementarbuch der musikalischen Harmonie- und Modulationslehre, 2nd ed. (Berlin: Robert Oppenheimer, 1888).] Page 25 of 27

The Problem of Harmonic Dualism: A Translation and Commentary (18.) [Ottokar Hostinky, Die Lehre von den musikalischen Klängen (Prague: H. Dominicus, 1879).] (19.) [Die Natur der Harmonik und Metrik: Zur Theorie der Musik (Leipzig: Breitkopf und Härtel, 1853; Eng. trans. W. E. Heathcote as The Nature of Harmony and Meter (London & New York: Swan Sonnenschein, Novello, Ewer, 1888; New York: Da Capo Press, 1989).] (20.) [Hauptmann's discussion of the “having/being” (haben/sein) distinction between ma­ jor and minor occurs on p. 29 of Die Natur (pp. 14–15 of the Heathcote translation).] (21.) [A♭-1 refers to the tone thirteen semitones below A0, the lowest A on a piano. Al­ though Riemann, using modified Helmholtzian notation, twice misidentifies the tonic fun­ damental in the text (2As), he labels it correctly in the diagram (3As). Eds.] (22.) [Beitöne, here translated “incidental tones,” is a catch-all term introduced by Helmholtz to refer to any unintentional sound material or acoustic debris (e.g., combina­ tion tones, summation tones, inharmonic partials, and so on) that are not part of the (pure) Klang. Riemann uses the term Beitöne and its synonym, Nebentöne, interchange­ ably.] (23.) [Tonpsychologie in principle means “music psychology” (by analogy with Tondich­ tung, Tonmalerei, etc.), but is usually associated with research that deals with mental re­ sponse to the stimulus of isolated tones. The distinctions between Tonpsychologie and Musikpsychologie are discussed in detail by Elizabeth West Marvin, “Tonpsychologie and Musikpsychologie: Historical Perspectives on the Study of Music Perception.” Theoria 2 (1987), 59–84.] (24.) [Leipzig: S. Hirzel, 1883; 1890. Square brackets in original.] (25.) [“Was ist Dissonanz?” Max Hesses deutscher Musiker-Kalender 13 (1898), 145–151. Reprinted as “Zur Theorie der Konsonanz und Dissonanz,” in Präludien und Studien: Gesammelte Aufsätze zur Aesthetik, Theorie, und Geschichte der Musik, 3 vols. Leipzig: H. Seemann (1895–1905), 3: 31–46.] (26.) [“On this variety [i.e., arithmetic and harmonic division of the fifth] depend all the diversity and perfection of harmony.” Istitutione harmoniche, 181; The Art of Counter­ point, 69.] (27.) [Leipzig: Breitkopf und Härtel.] (28.) [Stumpf uses the term “Erweiterung” in Tonpsychologie, 179.] (29.) [The prime of the C-minor chord is g (the chord being represented as ºg), and the undertones are measured from that. Likewise at the end of this sentence, the G-minor chord is ºd—and he is placing overtones on this prime presumably because it is the prime of the major dominant, i.e., d+.] (30.) [Riemann's text has “C major chord.”] Page 26 of 27

The Problem of Harmonic Dualism: A Translation and Commentary (31.) Nor can I go into more detail here as to why the positing of double Klänge on the part of the new “monists” (Capellen, Polak) is a backward step. Take a look at my disser­ tation (1873) and my Musikalische Syntaxis (1877), and you will find that I myself grap­ pled laboriously with double Klänge (e.g., C E G♯ as one part A-minor chord, one part Emajor chord; C E G B as C-major chord with one part G-major chord, and so on. The aban­ doning of chords in which two harmonic representations were coordinated, in favor, since my Skizze einer neuen Methode der Harmonielehre (1880), of the absolute subordination of one Klang representation to the other, and its categorizing as (dissonant) degrees with­ in the scale of the overriding harmony, I consider to be one of the most important steps toward understanding that I have made in the entire course of my work. If the term were not in itself so objectionable, then I would doubtless have a claim for the name “monist,” on a different basis than the pseudo-theorists who shun dualism precisely where it is in­ dispensable and categorically demanded by the musical way of thinking, and who adopt it instead, in bits and pieces, for the individual Klang representations, where it defies all logic. (32.) [Concerning Nebentöne, see n. 22] (33.) [Riemann gives no page reference.] (34.) [Presumably referring to “mixture” stops on organs, which involve two or more ranks of pipes producing parallel 5ths, 12ths, etc.] (35.) [Originally as Katechismus der Harmonielehre.] (36.) [Katechismus des Generalbaßspiels (Harmonie-Übungen am Klavier) (1889), later entitled Anleitung zum Generalbaß-Spielen or Handbuch des Generalbaßspiels.] (37.) [This may refer to the Katechismus der Kompositionslehre (Musikalische Formen­ lehre), later entitled Grundriß der Kompositionslehre, to distinguish it from the Große Kompositionslehre (1902–03).]

Ian Bent

Ian Bent is an emeritus professor of music of Columbia University and Honorary Pro­ fessor in the History of Music Theory of Cambridge University, U.K. His publications include Analysis and Music Analysis in the Nineteenth Century; he has served as edi­ tor of Music Theory in the Age of Romanticism, and coeditor of the translations of Schenker's Meisterwerk and Tonwille. He is currently coordinator of the online edi­ tion of all Schenker's correspondence, diaries, and lesson books: Schenker Docu­ ments Online.

Page 27 of 27

Harmonic Dualism as Historical and Structural Imperative

Harmonic Dualism as Historical and Structural Impera­ tive   Henry Klumpenhouwer The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0006

Abstract and Keywords This article aims to provide more sympathetic readings and accounts of harmonic dual­ ism. It makes two claims in particular: first, that the traditional attacks on harmonic dual­ ism that focus on putative structural contradictions in the system are entirely unjustified; and second, that harmonic dualism is a good, legitimate, and useful perspective which can generate enlightening accounts of tonal pieces of music. The argument in this article is part of a broader methodological challenge that questions the means of determining the “correctness” of a music theory. The defense of dualism presented in this article is highly conscious of the modalities of Anglo-American music-theoretical discourse and makes the renewed case for dualism in the context of current argumentative strategies; in the sense, by rehabilitating Riemann's dualism, the article is also holding up the mirror to contemporary music-theoretical practices. There are three parts in this article: the first section focuses on the characteristics of the dualist theory of harmony; the second sec­ tion defends the first claim presented above (that the traditional attacks on harmonic du­ alism that focus on putative structural contradictions in the system are entirely unjusti­ fied); and the last section defend by way of analysis the second claim—that harmonic du­ alism is a good, legitimate, and useful perspective which can generate enlightening ac­ counts of tonal pieces of music. Keywords: harmonic dualism, music theory, dualism, Riemann's dualism, theory of harmony

I. Introduction My goal in the present chapter is fairly straightforward and builds on several earlier ef­ forts to provide more sympathetic readings or accounts of harmonic dualism.1 I make two claims in particular: first, that the traditional attacks on harmonic dualism that focus on putative structural contradictions in the system are entirely unjustified; and second, that harmonic dualism is a good, legitimate, and useful perspective within which we can gen­ erate for ourselves and others enlightening accounts of tonal pieces of music. Page 1 of 25

Harmonic Dualism as Historical and Structural Imperative There are other ways, too, to encourage the reader to warm up to the possibilities of har­ monic dualism. For instance, I could advance along ethnographic lines by arguing that harmonic dualism is an approach taken seriously by certain insiders to nineteenth-centu­ ry Central European musical culture, and that we, as twenty-first-century anglophone out­ siders interested in understanding the tonal literature in question, ought to develop a cor­ respondingly serious interest in the rich diversity of theoretical approaches produced in that musical culture, including the various forms of harmonic dualism it developed. Or I could take a historicist perspective, arguing that in order to understand the musical tradi­ tion in question, we would do well to understand the corresponding traditions of musical thought, especially since they are mutually determinant. One could make that argument without being restrictive, by allowing as interesting gedankenexperiments certain theo­ retical (p. 195) anachronisms, such as analyzing Wagnerian melodies using twelfth-centu­ ry Liège motus theory, and by pointing out that a historicist approach does not exclude the possibility of developing historical arguments or technologies beyond their original articulations. I doubt, however, that such arguments will have much force in the current climate. Con­ temporary American music theorists will not readily accept the claim that they are out­ side of the musical culture they are committed to studying. Furthermore, the dominant strain of music theory is deeply antihistoricist, due, I think, to a belief that structures have no history. Accordingly, I will limit myself to the two claims made in the first paragraph. It is impor­ tant to notice that I plan to defend dualism against attacks on its internal structural con­ sistency, but not against attacks against its violation of the natural laws of acoustics or of human perception or psychology. I will not do so not because I do not believe there are natural laws governing acoustics or human perception or psychology, but rather because I am skeptical that such laws exclude what I take to be dualist thinking and dualist modes of hearing. Furthermore, in a very real sense, it is quite remarkable that music theorists still insist on appealing to Nature (in some form or another) as a means of legitimizing music theoretical arguments about Western tonality, especially after the emergence of a general consciousness of non-Western musics, whose initial appearance in music theory (strictly defined) can be traced back at least as far as Fétis.2 Certainly, part of the prob­ lem for dualism here is that its most visible spokesman, Hugo Riemann, articulated his system along just these very lines.3 Alexander Rehding codifies the animating idea behind critiques of dualism that derive their power by appealing to natural constraints on our musical perception or musical psy­ chology, writing that “[h]armonic dualism, it would seem, has little chance of ever being ‘right’ in [the] instrumentalist sense: as has been pointed out almost without fail, music simply does not work upside down—or, as the saying goes, we do not hear it that way.”4 While the idea that music, on its own, works this way or that way makes only figurative sense, it is certainly true that many of us do not hear triads directed from top to bottom. It is true because we are not trained to conceive and to hear triads organized in this way. The persistence of harmonic dualism (in some form or another) since the beginning of the Page 2 of 25

Harmonic Dualism as Historical and Structural Imperative what might be called the harmonic era, both as a mode of generating the harmonic struc­ tures of tonality and as a mode of organizing the aural experience of them, suggests strongly that it is not a universal law of cognition or perception that we hear triads the way we do. Rather, it is a question of music-technological development, a question with distinctly historical and sociological elements.5 Rehding precedes the cited excerpt above with what strikes me as another very useful summary of a fundamental belief underlying current music theory: “The prevailing view for many contemporary music theorists and analysts is that the decisive criterion is an in­ strumentalist one: on the most basic level, a music theory is considered ‘right’ if it can tell us something about musical practice, or about a musical composition, that in turn en­ hances the listening experience.”6 In fact, my second (p. 196) claim—that harmonic dual­ ism is a good, legitimate, and useful perspective within which we can generate for our­ selves and others enlightening accounts of tonal pieces of music—is made expressly to appeal to this style of logic, and since the standard method for arguing the instrumentali­ ty of a particular theory of music is by way of providing interesting analyses that serve on one hand to exhibit the relevant technical apparatus and on the other hand to prove the theory's suitability (read: truth), we are obligated to provide such analyses if we wish to make a convincing case for dualism, even if we are uncomfortable with the legitimacy of this procedure. The essay will have three parts. First, I will discuss the basic characteristics of a dualist theory of harmony. Next, I will defend my first claim, namely, that the traditional attacks on harmonic dualism that focus on putative structural contradictions in the system are entirely unjustified. Finally, I will defend, by way of analysis, my second claim, namely, that harmonic dualism is a good, legitimate, and useful perspective within which we can generate for ourselves and others enlightening accounts of tonal pieces of music.

II. What Is Harmonic Dualism? There are two characteristic ideas of a dualist view of harmony. 1. Major and minor triads come about as two distinct but equal expressions of a sin­ gle structuring principle, and, as such, together form the whole or totality of harmon­ ic possibilities. 2. Major and minor triads appear as mirror or inverted expressions of one another.7 Some music-theoretical perspectives exhibit only the former. One might call such ap­ proaches “soft dualism.” By articulating dualism in this way, we end up with more dualist approaches to consider, and include harmonic theories that have not traditionally been considered as instances of harmonic dualism. For instance, Rameau's account of the ori­ gins of chords given in Book I, chapter 7 of the Traité, in which he derives major and mi­ nor triads (tous les accords Harmonieux) as the two unique and equally valid products of “multiplying” and “subtracting” a major third and a minor third, is, under my telling, a soft dualist harmonic theory, because it generates major and minor triads as “distinct but Page 3 of 25

Harmonic Dualism as Historical and Structural Imperative equal expressions of a single structuring principle.”8 The account Rameau provides earli­ er in the Traité, however, in which he uses his interval root theory to generate the major triad, and then applies renversement to the order of the constituent thirds to generate a minor triad, is not dualist at all, because the major triad and the minor triad are unequal in derivation.9 Other music theoretical perspectives have both characteristics. One might call such ap­ proaches, which are structured along lines much closer to the traditional definition of du­ alism, “hard dualism.”10 Von Oettingen, Riemann, and Robert (p. 197) Mayrhofer (to name a few) all provide hard dualist theories of chords, even though there are considerable dif­ ferences among them. And there are the traditional forerunners, catalogued, sometimes carelessly, by Riemann himself: Zarlino, Rameau (at times), Tartini, Hauptmann are the most prominent. Without a doubt, the idea of inversion, reversal, or mirroring that characterizes hard du­ alist accounts of triadic structure (and, by entailment, of the tonal system as a whole) makes hard dualism a tough sell in current debates about tonal theory. There is a very long tradition, reflected even in the development of American set theory, of skepticism about the legitimacy of pitch-class inversion (a particular form of structural reversal, or mirroring) on the grounds of perceptual-cognitive limits. It is instructive in this context to compare the final results of this development with Ludwig Bußler's Lexikon der musikalischen Harmonien (Berlin: Carl Habel, 1889). Bußler's dictionary, written, he tells us, to aid students of practical and theoretical harmony, organizes harmonies by way of their total interval content, arranged visually from greater to lesser consonance so as to produce a kind of interval vector. (“Interval” in this case corresponds to dyad class or “in­ terval class.”) For instance, entry 47, the class of harmonies that contains the dominant seventh chord and the half diminished seventh chord, is defined as r5 g3 k3 k3 g2 ü4, (p5 M3 m3 m3 M2 aug 4).11 I reproduce here Bußler's entry 44. 44. p5 M3 m3, unique consonant chord, major and minor triad, c g e, c g e♭ (c a e) = c e g, c e♭ g.12 This description of major and minor triads certainly qualifies as a soft dualist account, be­ cause it presents major and minor triads as “distinct but equal expressions of a single structuring principle.” It is also suggests a certain mode of experiencing and thinking about triads, within which we are to attend to the constituent p5, then the M3, and then the m3. Following this script causes us to understand the relationship between major and minor triad as reversals of one another, by having us concentrate in major triads—using C major as an example—on the dyads c–g, c–e, and e–g and concentrate in minor triads—us­ ing C minor as an example—on the dyads c–g, e♭–g, and c–e♭. However, the two particular pairs of ordering the elements of triads Bußler actually provides—c g e, c g e♭ and c g e, c a e—suggest two different ways to experience and think about the relationship between major and minor triads. In the first pair, c g e and c g e♭, the ordering of the major triad conforms to the ordering suggested by the dyad class catalogue 〈p5, M3, m3〉, namely, c– g, c–e, and g–e, while the ordering of the minor triad, c g e♭, does not. Instead, the order­ Page 4 of 25

Harmonic Dualism as Historical and Structural Imperative ings suggest a correlation between the dyads c–g, c–e, g–e in the major triad and the dyads c–g, c–e♭, g–e♭, respectively, in the minor triad. To my mind, while the characteris­ tic of soft dualism is certainly present, this particular correlation does not suggest a re­ versal or inversion of structure between major to minor triads, but a toggling between the two parallel forms of a single structure: comparing the two orderings, we see c–e and g–e alternate with c–e♭ and e♭–g, respectively. The second pair of orderings, c g e and c a e, however, does attempt to correlate the elements of major and minor triads so as to em­ phasize such a reversal or inversion of structure between major and minor triads: (p. 198) the listing of elements in the major triad, which follows the dyad class ordering 〈p5, M3, m3〉, is reversed in the listing of elements in the minor triad, namely, 〈m3, M3, p5〉. Bußler could also have listed the elements of the minor triad as e a c. That ordering, which con­ forms more obviously to his definition of harmony number 44, would also suggest a rever­ sal or inversion of structure, but along different lines. Either case establishes the condi­ tions for a hard dualist account of triadic structure. There are a number of lessons to draw from this short investigation of Bußler's dictionary. To begin with, we learn that a sense of “reversal or inversion of structure” can emerge in a variety of ways, with or without direct reference to pitch-class inversion, and without appealing to an undertone series (either Helmholtz's or Riemann's). We also learn, and in fact have assumed, that in certain circumstances structural accounts can lead quite natu­ rally to particular modes of organizing our musical experience. Furthermore, we learn that a certain style of experiencing and thinking about chord structures—a style we gen­ erally feel is appropriate to atonal music and within which we feel more or less comfort­ able with the concept of inversion or reversal—emerges organically in connection with late tonal music. The degree to which we feel moved by ethnographic and historical argu­ ments of the kind offered at the beginning of the essay is just the degree to which we will be enlightened by this particular lesson from Bußler's dictionary. We could survey a number of other instances of historical theorizing that engages ideas of reversal/inversion/mirroring between major and minor triads that suggest the character­ istics of hard dualism. A particularly interesting instance of such thinking, which inter­ acts very suggestively with the basic idea behind pitch-class inversion and an axis of sym­ metry, involves the classical use of means in music theory. The calculation of the arith­ metic, harmonic, or geometric mean between two pitches (whether represented as string lengths, frequencies, or string divisions) amounts, among a number other things, to a conceptualization of the relationship between the two pitches by way of a third pitch that one is to understand as holding a middle position between them. Experiencing and think­ ing about intervals in terms of objects, rather than in terms of units of distance, is not at all together unfamiliar to us. For instance, from our systems of scale-based interval names, we are both familiar and comfortable with the practice of representing the inter­ val between two pitches by counting the pitches involved in the scalar segment beginning and ending with the two pitches in question. There have certainly been instances in which the relationship between two pitches has been represented by counting units of measure, but until fairly recently, such schemes have been far less popular. We prefer to call the re­ lationship between C4 and E4, for instance, a third rather than a ditonus. In a theoretical Page 5 of 25

Harmonic Dualism as Historical and Structural Imperative context without scales, one cannot make use of the practice of counting elements in a scalar segment. Nevertheless, the theory of means provides an alternative method of rep­ resenting the relationship between two pitches by way of a mediating pitch. Within the idea that a single mediating pitch can represent the relationship between two pitches, there are a number of different ways (arithmetic, harmonic, and geometric means) to con­ ceptualize the relationship (p. 199) involved. Accordingly, given the context of our usual ways of describing, experiencing, and thinking about intervals, the suggestion that we can describe the interval between two pitches by way of a mediating pitch is hardly for­ eign to us. I do agree that although we recognize the historical role played by the theory of means in generating among other things major and minor triads, we generally do not consider the theory of means as suggesting an interval system (or a system of interval systems).13 Yet the symmetrical character of means is quite evident when in addition to the question, “What is the mean of these two pitches?” we ask, “Of what other pairs of pitches is this pitch the mean?” In that sense, the theory of means strongly implies a particular concep­ tualization of interval that interacts both historically and structurally with the current thinking about inversions. I might be accused at this point of attempting to smuggle the historicist argument back into the picture, when I earlier claimed I would not rely on it. That certainly has hap­ pened. Yet the point here is not (simply) that dualism and in particular the inversional or mirroring element of hard dualism have a long and noble history. Rather, the point is that the idea of structural inversion or mirroring or reversal is not a restrictive or narrow species of musical thinking but a species of thinking with a rich and diverse range of his­ torical expressions. I have to this point discussed only dualist theories of chords. I have said nothing about the characteristics of harmonic dualist approaches as entire systems of tonality. Given our allotment of space, discussions of this sort are not available to us. Yet for my narrowly de­ fined tasks, focusing on chord theory will have the most impact: it is here where objec­ tions emerge. I might list other structural characteristics of harmonic dualisms, such as the absence of scales as fundamental theoretical objects, but the premise of this section is that it is the chord theory that will define the presence and quality of dualism in a giv­ en theoretical approach. I will take this topic up again in the last section of the essay, as I attempt to put together a dualist analysis.

III. The Contradiction between Riemann's Har­ monic Dualism and His Theory of Functions Having just defined harmonic dualism in fairly broad lines, I will now restrict myself to Riemann's harmonic dualism in particular because it provides the correct context for our first claim, namely, “that the traditional attacks on harmonic dualism that focus on puta­ tive structural contradictions in the system are entirely unjustified.” There will be no logi­ Page 6 of 25

Harmonic Dualism as Historical and Structural Imperative cal need to catalog and then refute all such attacks, since almost all are reducible to the idea that there are structural contradictions in harmonic dualism (and in particular, Riemann's), and almost all are flawed in the same way. Accordingly, I will discuss a partic­ ular attack on dualism (and in fact, the most (p. 200) important one historically) and allow the counterargument to serve as a model for other counterarguments. One can reasonably argue that the interaction between harmonic dualism and tonal func­ tion may have been Riemann's central research project. A number of theorists have made just this point. Almost all conclude that Riemann's attempts to make room for both per­ spectives failed utterly from a structural point of view, and that in fact these two aspects of his understanding of tonal music are technologically incompatible. Riemann may have in fact agreed with this judgment at certain later points in his life. His well-known retreat from his early insistence on the objective appearance of undertones in a series of articles published by the Neue Zeitschrift für Musik in early 1905 has caused many (and this may be the view taken by most contemporary music theorists) to consider the matter to have been resolved by Riemann himself in favor of his theory of functions.14 Dutch musicologist Ary Belinfante was among the first to assert in print that there is a structural contradiction between harmonic dualism and tonal functions.15 He argues that the distinction between major things and minor things under harmonic dualism must ex­ presses itself as changes of direction, and that this conflicts with the parallelism that characterizes Riemann's major and minor systems of tonal function. Belinfante's synopsis of Riemann's function labeling system is as follows: 1. Any major triad or a dual minor triad may receive the function label of tonic. 2. To determine the relevant dominant-functioning triad of a tonic Klang, we build a triad (of like kind) using the “structural fifth” of the tonic as the root of the new tri­ ad. Let's assume the tonic is C major: its structural fifth is G, and thus G major will serve as dominant. Now let's assume the tonic is E dual minor: its structural fifth is A, and thus A dual minor will serve as dominant. As Belinfante points out, (2) is not the case in Riemann's work. When E dual minor func­ tions as tonic, Riemann labels B dual minor—not A dual minor—the dominant triad. Belinfante argues that we can only coordinate Riemann's labeling scheme with a monist model of chord structure so that in place of E dual minor we speak of an A-monist-minor triad, whose structural fifth is E. In sum, Riemann's system of functions invokes harmonic monism, and since Riemann has increasingly emphasized his theory of functions at the expense of harmonic dualism is his later writings, we may regard the latter as de facto re­ scinded.16 Example 6.1 clarifies and expands upon the structural claims that underlie Belinfante's critique. The example provides three networks. Function labels appear along the top of each. Example 6.1a displays triads, or Klänge, in C major, using dualist Klangschlüssel notation. The network has two arrows, both extending from the tonic C-major Klang to its Page 7 of 25

Harmonic Dualism as Historical and Structural Imperative subdominant and dominant Klänge. The arrow from the tonic the dominant is labeled by the interval p5 up.

Ex. 6.1. Belinfante's critique of Riemann's dualism

Accordingly, the right half of the display corresponds to Belinfante's assertions about how Riemann thinks major tonic Klänge gives rise to their dominants. The arrow extending from the tonic C major to the subdominant F major is (appropriately) labeled by the inverse interval, p5 down. (p. 201)

Example 6.1b gives the context of C dual minor. The arrow system is strictly identical to that of example 6.1a. However, where example 6.1a has a p5 up, example 6.1b has a p5 down, and where example 6.1a has a p5 down example 6.1b has a p5 up. In Belinfante's view, example 6.1b represents the proper dual minor version of the major tonality given in example 6.1a. If that is true, we are bound to say that F dual minor functions as the dominant of C dual minor, and G dual minor functions as its subdominant. Riemann, however, does not assign T–S–D labels to C dual minor as example 6.1b does. Instead, he assigns them along the lines of the network given in example 6.1c. Comparing examples 6.1c and 6.1a, we see that the arrow system is identical. So are the corresponding intervals. Where example 6.1a has the interval P5 up, so does the network in example 6.1c. That places G dual mi­ nor in the position of C dual minor's dominant and F dual minor as subdominant. Howev­ er, example 6.1c parallels example 6.1a. Monism (not dualism) is constructed around the Page 8 of 25

Harmonic Dualism as Historical and Structural Imperative notion that minor mode structures arise as a parallel (however altered) and not as an in­ version or reversal of major mode structures. Riemann's answer to Belinfante, which appears in his essay “Das Problem,” is decidedly weak. He writes that Belinfante forgets that these names (viz., the function labels Tonic, Subdominant, and Domi­ nant) are not at all mine, but rather have been widely used since Rameau. I have properly retained these terms for the same reasons I have retained the designa­ tions Major, Minor, Parallel, Grundton, and a whole host of others. Accordingly, from the very beginning of my career to my most recent work, I have introduced the designations Quintklang, Gegenquintklang, (p. 202) Seitenwechsel, and the en­ tire terminological system of Schritte and Wechsel. Although one of my students has assured me that he no longer burdens his brain with the these particulars, I know for certain that, while he no longer troubles himself over terminology since I introduced the notational system of Functions, the person concerned is still a long ways from thinking that the Dominant in major tonality and in minor tonality have the same meaning. The particular form of my notational system in which the prime (Grundton) of the dual minor Tonic is the same as the prime (Grundton) of the Dominant makes such a conclusion impossible. Thus a retraction is completely out of the question.17 Riemann's answer to Belinfante fails on several counts. In the first place, he has rearticu­ lated Belinfante's argument so that the question becomes whether or not dominant func­ tion is identical in major tonality and minor tonality. On one hand, that synopsis barely corresponds to Belinfante's line of reasoning. On the other hand, it is just his point. In the second place, while Belinfante is comparing function labeling schemes of Riemann's pure major and pure minor system, Riemann claims that Belinfante ought to be comparing his pure major system and mixed (minor-major) system. The latter, roughly corresponding to the common harmonic minor tonality, is portrayed in example 6.2a. The network labels its arrows with the relevant Riemannian transformation: the arrow extend­ ing from C dual minor to C major is labeled Seitenwechsel (W). This is just the relationship Riemann emphasized in his description of minor tonality: the Grundton of the tonic dual minor triad is the Grundton of the dominant major triad. Com­ paring examples 6.1a and 6.2a, we can appreciate Riemann's argument about the differ­ ence between dominant functions in major and minor. In the former, the dominant Klang, g+, is the Quintklang of the tonic c+. In the latter, the dominant Klang, c+, is the Seiten­ wechselklang of the tonic °c.

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Harmonic Dualism as Historical and Structural Imperative

Ex. 6.2. Primary Klänge of the mixed minor and mixed major systems

Ex. 6.3. Primary Klänge and their transformations

However, pace Riemann, example 6.1a is not the major system we ought to compare to the mixed minor system of example 6.2a. The appropriate dual of example 6.2a is the mixed major system, displayed in example 6.2b. Whereas the mixed minor system of ex­ Page 10 of 25

Harmonic Dualism as Historical and Structural Imperative ample 6.2a varies the pure minor system of example 6.2c by replacing the dominant G dual minor with the dominant c+, the mixed major system of (p. 203) example 6.2b varies the pure major systems of example 6.2a by replacing the subdominant f+ with the sub­ dominant C dual minor. The two systems of example 6.2 do suggest an appropriately in­ verted, hard dualist, relationship. While the arrow system is identical in both networks, the relevant Riemann transformations exchange position from one network to the next. Yet, while Riemann's argument is ineffective, he does provide, by bringing Schritte and Wechsel into the picture, the basis for a reasonable structural refutation of Belinfante's criticism. Example 6.3a provides the context for such an argument. The illustration retains much of example 6.1a, but in place of the intervals p5 up and down, example 6.3a provides the rel­ evant Schritte. These Riemannian transformations are, in fact, the appropriate “intervals” to use for studying relationships between Klänge in a hard dualist context. The arrow extending in example 6.3a from the tonic C major to the dominant G major is labeled Quintschritt (Q). The arrow extending from the tonic C major to the subdominant F major is labeled by its inverse, Gegenquintschritt (-Q). Example 6.3b carries out a corre­ sponding revision of the C dual minor system of example 6.1c, the network that revealed the putative contradiction between dualism and function. Using the relevant Schritte as arrow labels, the figure provides the necessary logic to defend Riemann's theoretical practice from critiques like Belinfante's. Comparing examples 6.3a and 6.3b, we can now observe the appropriate inversion between the Riemann's major system and his dual mi­ nor system in the context of Riemann's function labels. Example 6.3c alters the network of the pure dual minor system in 6.3b, by reversing the direction of its arrows. Consequently, the network replaces Schritte with their inverses: Q for -Q, and -Q for Q. Comparing examples 6.3a and 6.3c, we see just the relationship be­ tween major and dual minor structures Hauptmann uses in reference to internal dualist chord structure. Whereas in major, the formative (p. 204) relationships extend from the Grundton (or here the tonic) to other parts of structure, in dual minor the same relation­ ships extend to the Grundton (or tonic) from other parts of structure: the former instanti­ ates Hauptmann's idea about a pitch-class “having” an interval; the latter instantiates Hauptmann's notion about a pitch-class “being” an interval. Example 6.4 is designed to emphasize the asserted homology between Hauptmann's (and by [qualified] extension, Riemann's) dualist Klang structure, provided on the right side of the diagram, and Riemann's dualist system of Klang functions, provided on the left. The point to bear in mind is that comparison is not meant to collapse the notion of, say, II in Klang structure into the notion of tonal dominant (or Hauptton into tonic), other than to say they play analogous roles in their respective systems (even though Riemann explains functions in just this way in Harmony Simplified). Discussions of a correspondence be­ tween internal Klang structure and Klang relations need to be carried out in the context outlined in example 6.3 (and 6.4). Page 11 of 25

Harmonic Dualism as Historical and Structural Imperative

Ex. 6.4. Klang structure and harmonic functions

The thrust of my defense of Riemann's dualism (in particular) against the specific com­ plaint leveled at it by Belinfante (and Dahlhaus and others) is that the critique defines the conditions for the theoretical consistency of (hard) dualism with monist, not dualist terms of reference, metrics, and expectations. This is essentially the argument made by Suzan­ nah Clark in her study of von Oettingen.18 In addition to Belinfante's argument, we might mention here the argument that if hard dualism were “true” (musically suitable), we would double the fifth in (root position) minor chords and the root in (root position) major chords. We double the root in both, so dualism is false. Yet, why should “double the root” serve as voicing rule in hard dualism, simply because it does under monism? Bearing in mind example 6.4, why not “double the pitch from which the interval p5 up emanates”? Or even more simply, “double chord function I in major and chord function II in minor”? In fact, wouldn’t the reversal from I in major to II in minor be precisely the way inversion­ al/mirroring structuring between major and minor triads should express itself? If my new doubling rules feel engineered, they should. Yet they are no more engineered than “dou­ ble the root.” The lesson we must take away from examining the sort of attacks on hard dualism represented by Belinfante's critique is that if we are to judge the instrumentality of hard (p. 205)

dualisms effectively, either as theoretical systems of tonality or as analytical technologies, we must refrain from dealing with straw-man versions, versions programmed from the outset to fail. And in that connection, we must refrain from projecting monist expecta­ tions for how hard dualisms must operate. At this juncture, the only principle worth bringing to bear on our reflections of hard dualist accounts is the principle of structural mediation, along with its auxiliary concepts of opposition, negation and balance, which in some form or another underwrites hard dualist ways of experiencing and thinking about major and minor chord structure, as well as the dynamics of function theory. At the same time, we must also bear in mind we have no legitimate basis for expecting how such a principle should express itself in the course of analysis.

IV. Dualism in Practice We now turn to the task of analyzing music from the perspective of hard dualism. Doing so does not really require us to spend a great deal of time defining the outlines of a fullblown dualist theory of tonality. We will simply restrict ourselves to the notions of dualism currently in circulation. Nevertheless, we need to remind ourselves of the lessons learned Page 12 of 25

Harmonic Dualism as Historical and Structural Imperative at the end of last section, namely, that we must not judge our dualist account against ei­ ther the values or results of monist theory. Since monism is so deeply established in our theoretical consciousness, this will not be easy. Example 6.5 provides a suitable context for staging the problem of harmonic analysis un­ der dualism. The excerpt is drawn from the first theme of Beethoven's First Symphony. The theme is in Satz form, and the excerpt constitutes its presentation phase or exposi­ tion. The fundamental idea (measures 13–19) (and its transformation [measures 19–25]) also conforms to Satz structure. Both the fundamental idea and its transformation metri­ cally expand a standard four measure unit by two measures. In the following discussion, I shall refer to the fundamental idea as Ged (for Gedanke) and its repetition as Ged´.

Ex. 6.5. Presentation phase of the first theme of Beethoven's First Symphony.

The four-measure unit that begins Ged seems static, even trapped, both melodically and harmonically. In the course of measures 13–15, there are four attempts, with expanding urgency, to push beyond middle C. In this local context, the melodic 〈G, B〉 (the root and the third of the dominant chord) function as a kind of battering ram from below.19 There is a breakthrough at measure 16 as the melody succeeds in surpassing the registral threshold of middle C. However, while the attempts to breach C as a registral limit suc­ ceed, the larger scale attempt to escape C, now regarded as a point of tonal organization rather than as a particular registral barrier, fails. When the arpeggiation in measure 16 reaches G4, G4, the fifth of a C major triad, is transformed (p. 206) into G4, the root of the dominant of C major. The pattern 〈G, B〉 remerges, this time an octave higher. By now the dyad functions, not as an element that attempts to move us beyond C (as a registral barri­ er), but as an element that locks us onto C (as an organizing force). We might also point out that by the end of the four-measure unit, “C” has absorbed the element that initially opposed it. Turning for a moment to the question of metrical balance, we can trace the extension of the standard four-measure unit to the point where the middle C barrier is breached in measure 16. To return the phrase to its “original” metrical balance, we can edit out the last three quarters of measure 16, all of measure 17 and the first quarter of measure 18, so that the anacrusis of the next four-measure unit—the descent from A4—begins on the last quarter of measure 16, initiating Ged´ in measure 17 (instead of measure 19). Under this reading, the 〈G, B〉 dyad that locks us back onto C brings about the extension that fol­ lows, the extension that finally breaks us free from C altogether, not only as a registral Page 13 of 25

Harmonic Dualism as Historical and Structural Imperative limit but also as a harmonic limit. It does so by adapting a strategy that has failed in the preceding measures, namely, by providing the “correct” lower leading tone (the scale the­ oretic name for the third of the dominant triad), a strategy that characterizes the very be­ ginning of the movement. Under this view, the limiting power of C is overcome by replac­ ing it with C♯, one of its chromatic negations, so that it absorbs the function of the pre­ ceding Bs. According to the norms of traditional harmony, the correct analysis of the ex­ tension is I–V7/ii–ii. Turning now to Ged´, we observe that the internal dynamics of Ged´ repeat those of Ged, but transposed to D minor. However, its extension differs considerably. Rather than over­ coming D by negating it chromatically by means of D♯ or D♭, D is maintained as the high­ est sounding pitch of the extension to measure 25. It is A, not D, that is transformed through chromatic displacement by A♭. According to the norms of traditional harmony, the correct analysis of the extension is ii–

.

Clearly, it would be appropriate to take into account the sense of inversion or mir­ roring between the extension of Ged (measures 17–18) and the extension of Ged´ (mea­ sures 23–24). It is worth pointing out here that there is no technical provision at all with­ (p. 207)

in the monist tradition for doing so. We need to augment such an account with technology that will capture this element of the presentation phase. When we do so, we must put constraints on the degree to which some theory of inversion can be absorbed into our ac­ count, even in this local context, so as to preserve the fundamental outlines of monism. In light of these remarks, we might expect that some form of hard dualism can handle the relationship between the extension and its repetition quite well. It is not so clear that it can do as well with the harmonic aspects of the first four measures of the fundamental idea and of its repetition. Let us begin by sketching out a hard dualist account of the two extensions. Example 6.6 isolates the harmonic structures in each, adding the resolutions in brackets. The har­ monies of the first extension appear as alpha. C and C♯ are given in diamond-shaped noteheads. The bass pitches are given as filled noteheads. The constituents of the C-ma­ jor triad (which we will notate as c+) are labeled I, III, II. These are the traditional func­ tions Hauptmann provides for determining triad structure: I and II form a perfect fifth (mod the octave); I and III form a major third (mod the octave). Under many styles of du­ alism, the second structure, A–C♯–E–G, is to be conceived as an aggregate of two triads. But let us set that aside for a moment and isolate the three pitches in the treble clef, C♯– E–G, and let us retain Hauptmann's label assignments from the preceding c+. We can then give an account of the second structure as an alteration of the c+ triad in the follow­ ing way: chromatically alter I (C) so that it forms a diminished fifth with II (G). I derive this style of articulating alterations in triadic structure (which supplements the idea that dissonant structures arise from an aggregation of two Klänge) from Riemann directly.20

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Harmonic Dualism as Historical and Structural Imperative

Ex. 6.6. Harmonies in the extensions of Ged and Ged'.

We now turn to the harmonies of the second extension, given in the example as beta. A and A♭ are given in diamond-shaped noteheads. The bass pitches are given in filled note­ heads. The figure provides a hard dualist explanation of the triad, as a reversal or inver­ sion of the major triad. The constituents of the D minor triad are labeled I, III, II. Again, these are the traditional functions Hauptmann provides for determining triad structure: I and II form a perfect fifth (mod the (p. 208) octave); I and III form a major third (mod the octave), so that A in D–F–A parallels C in C–E–G. Adopting Riemann's system, we shall no­ tate the triad as ˚a. I shall return to this shortly. Under many styles of dualism, the second structure, C–D–F–A♭, is to be conceived as an aggregate of two triads. But let us again set that idea aside for a moment and isolate the three pitches in the treble clef, D–F–A♭, and let us retain Hauptmann's label assignments from the preceding ˚a. We can then give an account of the second structure as an alter­ ation of the ˚a triad in the following way: chromatically alter I (A) so that it forms a dimin­ ished fifth with II (D). Comparing our accounts of the two extensions, we see that the sense of inversion in their relationship turns into a parallelism under our hard dualist account. I think that, general­ ly speaking, this is just the sort of result one would demand of a convincing dualist read­ ing. The particular objection to dualism that emerges in such situations is that, while things work out well in contexts such as ours, what about other aspects of the relation­ ship between the harmonic structure of the fundamental idea and of its repetition? One might be persuaded that a hard dualist account of measures 17–18 and measures 23–24 works reasonably well, but such an account must surely fall flat when we attempt to ex­ tend it to a comparison of the first four measures of Ged and Ged´. I think we have arrived here at a particularly difficult methodological question. How can one reasonably judge that a dualist explanation is providing a useful, enlightening ac­ count rather than a pertinacious distortion? Framed this way, the question is far too unfo­ cused, too abstract, and smuggles in too many related but yet distinct problems to allow an intellectually respectable answer.

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Harmonic Dualism as Historical and Structural Imperative

Ex. 6.7. Ged' revised.

Accordingly, it will helpful to keep our attention on the analytical context at hand, pursu­ ing our dualist analysis a bit further before addressing the larger question. We saw that the inversional or mirror relationship between the extensions of Ged and Ged´ reap­ peared as a parallelism under the hard dualist account. Whether we like to or not, we might admit that if dualist thinking about triadic structure applies at all, it applies in the case of example 6.6. However, we might then point to the four measures of Ged and Ged´ and argue that, in the same way, the dualist chord model certainly fails to apply. In order for such a model to be relevant in the manner it is relevant for measure 18 and measure 24, Ged´ would need to look something like the music in example 6.7. That music, and not the music Beethoven has written, would be open to an appropriately dualist reading. The idea behind example 6.7 is this: for a hard dualist account to succeed analyti­ cally, there must be a sense of reversal or inversion between major things and minor things in the music at hand. The revision given by example 6.7 accomplishes such a rever­ sal. It coordinates the major/minor inversion one finds in the comparison of extensions of Ged and Ged´. The motive that permeates Ged is inverted in example 6.7. In Ged, the mo­ tive extends from II to I (G to C) of the local triad, c+; we can say the same thing about the revised motive in example 6.7: the motive extends from II to I (D to A) of the local tri­ ad, ˚a. (p. 209)

We cannot say that about the motive that appears in Ged´ as Beethoven has written it. Here the motive extends from I to II (A to D). When we expand our analysis to take into account the harmonies in play, we notice the precisely the same problem. In Ged and (Beethoven's) Ged´, the motive projects a harmonic progression from I to V: c+ to g+ in Ged; and ˚a to a+ in Ged´. Both a monist and Riemannian dualist outlook will understand (in their own way) the two progressions functionally as tonic-dominant. By replacing Beethoven's Ged´ with a Ged´ based on the revised motive of example 6.7, we see that in the revised Ged´, the motive projects a harmonic progression from ˚a to ˚d, which both a monist and a Riemannian outlook will understand (in their own way) functionally as a ton­ ic-subdominant progression.

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Harmonic Dualism as Historical and Structural Imperative We might conclude from our little gedankenexperiment that only our revised motive in ex­ ample 6.7 can form the basis of a Ged´ suitable for a hard dualist analysis. Beethoven's motive for Ged´ is suitable for a monist account. Yet there is a great deal of concealed theorizing underwriting this conclusion, theorizing that warrants closer examination. To begin with, we should not confuse an account of a thing with the thing as such. In other words, it is an illusion that the extensions of Ged and Ged´ are dualist while their first four measures are monist. The truth is that the accounts or explanations of the music (and not the music as such) are either dualist or monist.21 The second problem deals with the criteria one employs to determine whether a dualist reading is doing its job as a dual­ ist theory, and as a convincing and interesting account of the music at hand. This is a question we have put off once already, largely because the latter half of the problem seems impossible to answer unless we know to whom the account must be convincing and interesting. So it may be best at this time to reframe the problem, by restricting ourselves to its first half. Hard dualism asserts a reversal or inversion at least between major triads and minor triads, and usually between major things and minor things in general. How or where will this reversal or inversion express itself? What troubles me most about the criticism emerging from the gedankenexperiment of example 6.7 is that it essentially allows the re­ versal to take place only outside of a hard dualist world, and, in particular, within a monist world; and this is just the sort of criticism we have tried to discredit in the preced­ ing section of the essay.

Ex. 6.8. Perfects fifths in Ged and Ged'

Accordingly, as an analytical problematic, harmonic dualism may not in the first instance be a question of poor technology but a question of poor methodology. In other words, the weakness of many criticisms of dualism is that at heart they impose monist methodology to dualist technology. That is precisely what the (p. 210) gedankenexperiment of example 6.7 does. To be sure, it feels absolutely natural to do so, but that is not an indication of its propriety: rather, it is the indication of the degree to which we have naturalized monist methodology. So, how might a dualist methodology differ from a monist methodology? To answer that, let us return to the First Symphony. Example 6.8 isolates the fifths that frame the basic motive in Ged (c+) and Ged´ (˚a): G3/C4 and A3/D4, respectively. The fifths are analyzed dualistically, with Hauptmann-label assignments drawn from example 6.4: in c+, C is as­ signed I, and G, II; in ˚a, A is assigned I, and D, II. Turning first to the Ged triad, we see that example 6.8 supplies an arrow extending from C to G, from I to II of c+, and labels it Page 17 of 25

Harmonic Dualism as Historical and Structural Imperative with the interval “p5 up.” Turning now to the Ged´ triad, we see that example 6.8 sup­ plies two arrows: one, labeled with the interval “p5 up,” extends from D to A, from II to I of ˚a; another, labeled with the interval “p5 down,” extends from A to D, from I to II. Com­ paring the two diagrams, we see how the inversion or reversal between c+ and ˚a ex­ presses itself in the exchange of Hauptmann labels that articulate the interval “p5 up,” or, alternatively, in the transformation of “p5 up” as the label for the arrow extending from I to II in c+ to “p5 down” as the arrow label from I to II in ˚a. That was too easy. Besides, while the music certainly articulates the arrow from I to II in Ged´ (from A to D), it does not do so in Ged. There, the music articulates an arrow from II to I (from G to C), not I to II. Accordingly, the music is not articulating the dualist rela­ tionships asserted in example 6.8. Yet again we see how easily we fall back into our natu­ ralized monist world. The objection just raised still presumes the monist account to be the actual music. But, again, the music, on its own, articulates neither a dualist nor a monist account. When we apply monist thinking to the four measures of Ged and of Ged´, we come away with the idea that Ged and Ged´ are parallel structures. But that is true only because major and minor triads are in fact understood inside of monism as parallel struc­ tures. In the very same way, when we apply dualist thinking to the same music, we come away with the idea that inside of dualism there is a reversal or exchange between Ged and Ged´. But surely there is something of a shell game going on: if we accept the logic of reversal or inversion presented during our comparison of the two measure extensions of Ged and of Ged´ in example 6.6, we cannot do the same now as we compare the first four mea­ sures of Ged and Ged´. Besides, when we consider the harmonic structure of Ged and Ged´ we notice that the polarity shift from Ged to Ged´ is (p. 211) askew: Ged projects ma­ jor, but Ged´ projects harmonic minor, not natural minor, as it should be, especially by du­ alist standards. Furthermore, we can use classic dualist technology to emphasize how poorly Ged and Ged´ serve as reflections or inversion of one another. The progression from c+ to g+ corresponds to Riemann's Quintschritt (Q), while the progression from ˚a to a+ corresponds to Riemann's Seitenwechsel (W), and not Quintschritt or even Gegen­ quintschritt (-Q). I will set the first question aside for the moment, and focus on the second. The observa­ tions are obviously correct. The Riemannian transformations do not correspond at all. But again, this is a methodological not a technological question. I proposed earlier that a hard dualist perspective might understand the dynamics and mechanics of harmonic practice to be centered on mediation, and its auxiliary concepts of opposition, negation and bal­ ance, as its organizing principle. That does not mean that a properly dualist account must find that corresponding aspects of harmonic structure sum to null. It means that in dual­ ist accounts one becomes aware of the processes of negation or opposition, and of the particular musical element(s) that mediate(s) the opposition. In other words, while the unfolding harmonic structure of an excerpt or piece may not be mediated by the tonic or tonic elements or structural sum to null, it is always mediated by some musical element(s). In the present context, the observations made are not fatal to a dualist ac­ Page 18 of 25

Harmonic Dualism as Historical and Structural Imperative count. The corresponding structures of Ged and Ged´ may not sum to null or cancel each other out, but they will still have a mediating element. The determination of what that means musically is not the job of the technology, but rather of how we bring our method­ ological perspective to bear on the context. Another way of thinking about the issue is to understand that the idea of summing to null occurs at the level of the technology, and need not be actualized as such in accounts of particular excerpts or pieces. After having said all that, it is now somewhat embarrassing to point out that in fact there is a possible dualist account within which Ged and Ged´ do, in a rough sense, harmonical­ ly sum to null. We begin by examining the tonality mapped out in Ged´. We pointed out that the progression ˚a–a+ very clearly articulates what is traditionally called D harmonic minor. Example 6.9 provides a run-of-the mill dualist map of the primary triads in D ma­ jor-minor or mixed minor (Hauptmann's Dur-Moll Tonartsystem, the analog of D harmonic minor), including the relevant Klang transformations emanating from ˚a to a+ and ˚d. The “dual” of minor-major or mixed minor is not pure or natural major, within which the domi­ nant and subdominant relate to the tonic by Quintschritt and Gegenquintschritt, respec­ tively. Rather, its dual is major-minor or mixed major. Example 6.10 provides a run-of-the mill dualist map of the primary triads in C minor-major or mixed major (Hauptmann's Moll-Dur), including the relevant Klang transformations emanating from c+ to g+ and ˚c. Comparing examples 6.9 and 6.10, we see the nature of a “dual” relationship between the two tonalities. We could say that characteristic transformations linking the primary triads of one tonality are reversed in the other, along the lines discussed in connection with ex­ ample 6.2. However, Ged does not provide the primary subdominant Klang of C mixed ma­ jor. All we have is the progression c+–g+. We are not in a position to determine whether Ged projects pure C major or C minor-major.

Ex. 6.9. Primary triads, a minor-major.

Ex. 6.10. Primary triads, c major-minor.

Let us take into account the entire Satz exposition, focusing first on the tonal dy­ namics of D mixed minor. Returning to example 6.5, we see that as a tonality, ˚a is initiat­ ed in measure 19, with the appearance of its dominant, a+, and more precisely, its domi­ nant seventh. Earlier, in connection with example 6.6, I pointed out that in Riemann's view, this structure is the aggregation of two triads. a+ is clearly one, and accounts for the presence of A, C♯, and E. G is the representative of a+'s topological opposite, the sub­ dominant triad, ˚d, which belongs to the unfolding of D mixed minor tonality. This concep­ (p. 212)

Page 19 of 25

Harmonic Dualism as Historical and Structural Imperative tion of the seventh chord as a combination of triads needs to be understood in context of tonal function and the effects of mediation. In particular, a sense of tonic emerges when subdominant and dominant are presented either serially or simultaneously.22

Ex. 6.11. Primary and Secondary triads, a pure mi­ nor.

We can also incorporate c+ into the picture. To do so, we could imagine a pure or natural minor in effect, within which c+ functions as the Terzwechselklang of ˚e, the dominant in pure D minor. Example 6.11 provides the topological expression of that tonality. Compar­ ing the upper rank of triads with the rank of triads given in example 6.9, we see that in place of A+, example 6.11 has ˚e, so that the subdominant and dominant triads are the Quintklang and the Gegenquintklang, respectively, of ˚a. The topos makes clear the role of c+ relative to ˚a just asserted. Bearing this in mind, the a+ based structure in measure 18 draws us away from the triadic arrangement in the topos of example 6.11 and to the tri­ adic arrangement in the topos of example 6.9. In sum, the extension of Ged displaces the Terzwechsel of the dominant in pure D minor with the dominant in mixed D minor. Now let us return to the extension of Ged´ in example 6.5. Since we analyzed the extension of Ged in terms of ˚a, the tonality it established, we will carry out the analo­ gous exercise on the extension of Ged´, which establishes C tonality. Again, we have not determined whether pure C major or mixed C major is in effect. (p. 213)

We recall that D mixed minor was initiated by the dominant seventh in measure 18, which we analyzed, following Riemann, as the aggregate of a+ and ˚d. The corresponding struc­ ture in measure 24 is traditionally known as a D half-diminished seventh chord, which we will also analyze, following Riemann, as an aggregate of two triads. Clearly, ˚c is one, and accounts for the presence of C, A♭, and F. D is the representative of ˚c's topological oppo­ site, the dominant triad, which follows immediately in measure 25. Accordingly, we can now find a symmetrical partner for the D-mixed-minor tonality of Ged´ with the C-mixedmajor tonality of Ged, and the development of the first theme, beginning in measure 25.

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Harmonic Dualism as Historical and Structural Imperative And we can do more. We can also incorporate the triad ˚a in measure 24 into the picture. To do so, we could imagine a pure or natural C major tonality in effect, within which ˚a is the Terzwechselklang of f+, the subdominant in pure C major. Example 6.12 provides the topological expression of that tonality. Comparing the upper rank of triads with the rank of triads given in example 6.10, we see that in place of f+, example 6.10 has ˚c, so that the dominant and subdominant triads are the Quintklang and the Gegenquintklang, re­ spectively, of c+. The topos makes clear the role of ˚a relative to c+ just asserted. Bearing this mind, the ˚c based structure in measure 25 pulls us away from the triadic arrange­ ment in the topos of example 6.12 to the triadic arrangement in the topos of example 6.10. In sum, the extension of Ged´ displaces the Terzwechsel of the subdominant in pure C ma­ jor, with the subdominant in mixed C major.

Ex. 6.12. Primary and Secondary triads, c pure ma­ jor.

Comparing the accounts of the extensions of Ged´ and Ged just provided, we see that the Ged account was generated from the Ged´ account by replacing the words ˚a, c+, a+, mixed minor, pure minor, subdominant, and dominant, with the words c+, f+, ˚c, mixed major, pure major, dominant, and subdominant, respectively. The point of all of this is not to prove a required “summing to null,” but to isolate and explore the particular dynamics of negation and opposition, cataloged in the lists just provided, and their affects on the overall harmonic image of the excerpt under review. All of the relationships discussed in the analysis are quite audible when one en­ ters into an experiential world regulated and organized along dualist lines. Within this world, but not outside of it, one may engage in discussions about how well the music ar­ ticulates the observed relationships. Even so, we ought to remember that such talk is con­ tingent on the particular organization of the experiential system. The only necessity in­ volved is the organizational scheme that makes the experiential system possible qua (p. 214)

experience.

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Harmonic Dualism as Historical and Structural Imperative At this point, I break off my analysis. As always, there is much more to point out, but it will not contribute significantly further to the defense of our second claim that “harmonic dualism is a good, legitimate, and useful perspective within which we can generate for ourselves and others enlightening accounts of tonal pieces of music.” This is not to say that I have the feeling of satisfactorily defending the claim either. In the absence of set­ tled-upon standards for evaluating properties such as good, legitimate, useful, enlighten­ ing, the best I can do in this context is to provide a sympathetic demonstration of (hard) dualism in action.23

Notes: (1.) I have in mind Daniel Harrison, Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of Its Precedents (Chicago and London: University of Chicago Press, 1994); Alexander Rehding, Hugo Riemann and the Birth of Modern Musi­ cal Thought (Cambridge: Cambridge University Press, 2003); and Suzannah Clark, “Se­ duced by Notation: Oettingen's Topology of the Major-Minor System,” in Music Theory and Natural Order from the Renaissance to the Early Twentieth Century, ed. S. Clark and A. Rehding, (Cambridge: Cambridge University Press, 2001). (2.) To be clear: I am not complaining about formulating music theory as a science. My concern here is only about attempts to trace the origins of tonal music-theoretical objects to natural phenomena. Projects like Douglas Dempster and Matthew Brown's “Evaluating Musical Analyses and Theories: Five Perspectives,” Journal of Music Theory 34.2 (1990), 247–279, which in the first instance propose to recast the discipline methodologically and epistemologically in the scientific image, differ from the accounts I have in mind here. In his “Letter to the Editor: On the Nature of Music Theory,” Journal of Music Theory 3.1 (1959): 170, Elliot Carter articulates the reservations made here about casting music the­ ory as a natural science, blaming (interestingly) the Yale students of Paul Hindemith. That he supplements his critique with an attack on “historicism,” on the grounds that it has led to a neglect of contemporary composition is equally interesting in the present context. I cite the letter in its entirety. There are many prior or hidden assumptions on which “music theory” is based—I feel that a very thorough examination of these should be undertaken in your mag­ azine. You lean, editorially, rather too heavily on the notion of the “physical” or “natural” basis of music theory as one would expect from the followers of Hin­ demith. I would like to see other points of view given a fair hearing. It seems also that the notion of “historicism,” of the university as the magnifier of the “dead” (not always still alive) past and the consequent neglect of the thinking of the present day and particularly the composing of the present, might be a subject.

(3.) The institutional forces that caused him to do so are strongly related to the institu­ tional forces that cause us to persist in employing Nature as yardstick for the legitimacy of a theory of tonal music. The work of Timothy Lenoir and R. Steven Turner on the for­ Page 22 of 25

Harmonic Dualism as Historical and Structural Imperative mation of scientific disciplines in the nineteenth-century Prussian university offer attrac­ tive models for the study of these forces, which have a great deal to do with the formation of music theory as a discipline and allows one to pose similar questions about the current dynamics of discipline formation. In particular I have in mind Lenoir, Instituting Science: The Cultural Production of Scientific Disciplines (Stanford: Stanford University Press, 1997), and Turner, “The Growth of Professorial Research in Prussia, 1818 to 1848: Caus­ es and Context,” Historical Studies in the Physical Sciences, 3 (1971): 137–182; Turner, “University Reformers and Professorial Scholarship in Germany, 1760–1806,” in Lawrence Stone (ed.), The University in Society, Vol. 2 (Princeton, NJ: Princeton Universi­ ty Press, 1974), 495–531; Turner, “The Prussian Universities and the Concept of Re­ search,” Internationales Archiv für Sozialgeschichte der Deutschen Literatur, 5 (1980): 68–93. Youn Kim's dissertation, “Theories of Musical Hearing, 1863–1931: Helmholtz, Stumpf, Riemann and Kurth in Historical Context” (Ph.D. diss., Columbia University, 2003), is another good source for learning more about such matters. In this context, Aaron Girard's dissertation, “Music Theory in the American Academy” (Ph.D. diss., Har­ vard University, 2007), is of great interest and importance. (4.) Alexander Rehding, Hugo Riemann, 36. (5.) In his diagnosis of the rise and fall of Riemann's harmonic dualism, Rehding makes a similar point. “As soon as we accept that the ‘wrongness’ of harmonic dualism is not an intrinsic quality of the theory,” Rehding writes, “but is brought about by a change of para­ digm, these continual changes, the perpetual reformulation of the foundational elements of Riemann's theories, the undertone hypothesis can in fact be a very useful tool, aiding us in understanding what made Riemann's theories of harmony the success story that they were in the later nineteenth century” (Hugo Riemann, 35). Nicholas Cook's “Episte­ mologies of Music Theory,” in The Cambridge History of Western Music Theory, ed. T. Christensen (Cambridge: Cambridge University Press, 2002), 78–105, presents a much broader exploration of the issue. (6.) Hugo Riemann, 36. (7.) There are a host of other common features, such as a reliance on the triad rather than scale is the fundamental theoretical object, the distinction between primary and sec­ ondary triads, and so on, but here I will need to limit myself to what I see as the most ba­ sic elements of the dualist perspective. (8.) Jean-Philippe Rameau, Traité de l’harmonie réduite à ses principes naturels, ed. Er­ win Jacobi (Rome: American Institute of Musicology, 1967 [1725]), 60. (9.) Rameau, Traité, 42. (10.) Soft and hard dualism are named to recall the two forms of determinism, and not to suggest a preference for one form of dualism or the other.

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Harmonic Dualism as Historical and Structural Imperative (11.) Generally speaking, the entries in the dictionary include discussions of functions and scale degrees the relevant harmonies can conceivably fulfill, along with examples from the tonal literature. (12.) “44. r5 g3 k3, einziger consonirender Accord, Dur- und Molldreiklang, c g e, c g es (c a e) = c e g, c es g” (Bußler, Lexikon, 24). Sperrschrift used for emphasis in the origi­ nal. (13.) Zarlino provides perhaps the most famous instance of generating triads by calculat­ ing the arithmetic and harmonic means of a perfect fifth. Zarlino's generation of major and minor triads certainly qualifies as a hard dualist account. Moreover, accordingly to the present discussion, the construction of major triads and minor triads individually has a sense of mirroring or inversion simply by virtue of involving a mediating pitch. The dif­ ference then (between arithmetic and harmonic means) lies in their different conceptions of the “middle.” And the difference conceptions themselves (arithmetic and harmonic means) stand in an inversional (reciprocal) relationship. As an additional instance of such thinking, we might also mention the important appear­ ance in Schenkerian theory of the concept of Teilung (which serves as the German word for “means”), within which structuring intervals are elaborated through bisection. (14.) “Das Problem des harmonischen Dualismus,” Neue Zeitschrift für Musik 72 (1905): 3–5, 23–26, 43–46, 67–70, translated by Ian Bent as “The Problem of Harmonic Dualism” in chapter 5 of the present volume. While Riemann does reevaluate the role over- and un­ dertone series play in his work, he does not renounce harmonic dualism. We may summa­ rize his line of reasoning as follows: (i.) The more we investigate the relationship between major and minor triads, the more we are impressed by their inversional symmetry. The series 4:5:6 will produce major chords in one domain of sound, but minor in another. Within a single domain 4:5:6 at its reciprocal series 1/4:1/5:1/6 (whose terms he multiplies by 60 to get 15:12:10) will produce major and minor triads. (ii.) A dualist understanding of the relationship between major and minor triads— which is to say, chord theories that view major and minor triads as inversional struc­ tures rather than as varied parallel structures—respects this intuition. Accordingly, in light of his abandonment of undertones, he could now bid his formal farewell to both acoustic and physiological orientations in music-theoretical work in favor of researching and in turn redefining the musical subject. In place of Helmholtz's Tonempfindungen, Riemann now claims the proper object for music-theoretical study is Tonvorstellung. Admittedly, Tonvorstellung is a theme that runs throughout Riemann's writing career, but studying it had always been associated with the styles of research he now seemed to renounce, namely, the physics of sound, and the anatomy of the ear. (15.) “De leer der tonale functieen in conflict met die der polaire tegenstelling,” Orgaan van de Vereeniging van Muziek-Onderwijzers en -Onderwijzeressen IV.9 (1904): 1–2. Page 24 of 25

Harmonic Dualism as Historical and Structural Imperative (16.) Belinfante's argument has been repeated by both Matthew Shirlaw (in his still very useful The Theory of Harmony: An Inquiry into the Natural Principles of Harmony, with an Explanation of the Chief Systems of Harmony from Rameau to the Present Day [New York: Da Capo Press, 1969 (1917)]), and by Carl Dahlhaus in Studies on the Origins of Harmonic Tonality, trans. Robert Gjerdingen (Princeton: Princeton University Press, 1990 [1968]). It is worth noting in Belinfante's critique (and the critique of others) a certain trope detectable in other attacks on harmonic dualism, namely, that dualists are closet monists, and are either lying or confused (to us certainly and perhaps also to themselves). (17.) “Das Problem,” 69–70. (18.) Clark, “Seduced by Notation.” (19.) What I have described here as “increasing urgency” corresponds formally to the continuation or Durchführung of Ged, when Ged is conceived as a local Satz. We note that the same music functions as the Durchführung of the first theme's Satz structure (begin­ ning at measure 25) and the Durchführung of the transition theme (beginning at measure 41). (20.) Riemann, Skizze einer neuen Methode der Harmonielehre (Leipzig: Breitkopf & Här­ tel, 1880), 46–52. Riemann's theories of dissonance are discussed by Edward Gollin in chapter 13 of the present volume. (21.) One can assert a great many things about the relationship between an account of music and music as such, but it is pure mystification to assert they are identical, that our thinking about a thing is identical to the thing. (22.) Playing subdominant and dominant (in either order) in any key, we can easily imag­ ine the appropriate tonic. A dualist explanation of the phenomenon claims that the tonic emerges in our imagination because it mediates the opposition between subdominant and dominant. In other words, tonic relates to subdominant in the way that dominant relates to tonic. (23.) There have been attempts to establish criteria for evaluating analysis. Dempster and Brown (“Evaluating Musical Analyses”), in particular, have provided a useful summary of various positions, including their own.

Henry Klumpenhouwer

Henry Klumpenhouwer is a professor of music at the University of Alberta and for­ mer editor of Music Theory Spectrum. His published work involves the analysis of atonal music, the history of music theory, and analytical methodology.

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Dualistic Forms

Dualistic Forms   Alexander Rehding The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0007

Abstract and Keywords This article explores some of the consequences of Riemann's harmonic dualism that were not pursued by Riemann himself. The question of musical forms is one that Riemann in­ triguingly leaves unresolved—despite the fact that he is considered a formalist theorist, his ideas about musical form remain comparatively underdeveloped. The root of the prob­ lem is assumed to be related to the skepticism with which harmonic dualism is widely re­ garded. In exploring the idea of a dualistically based concept of form, the article takes Riemannian ideas further than Riemann did and (re-)constructs a Riemannian theory of forms based on the concept of harmonic symmetries. Keywords: harmonic dualism, musical forms, Riemannian ideas, harmonic symmetries, theory of forms

THE trio of nineteenth-century harmonic dualists, Moritz Hauptmann, Arthur von Oettin­ gen, and Hugo Riemann, is not generally thought to have contributed much in the way of a theory of musical forms.1 Their major objective was to overhaul music-theoretical spec­ ulation at the chordal level, proposing, each in their own way, that the minor triad should be conceptualized as the polar opposite of the major triad, with the “root” at the fifth.2 With this ambitious project, which amounted to no less than a reconceptualization of one of the cornerstones of musical thought, they were mainly concerned with harmony on the chord-to-chord level, and rarely considered large-scale structures as a whole.3 To complicate matters further, the only dualist prolific enough to consider matters of form in his writings, Hugo Riemann, seemed to insist on the strict temporality of musical form, which is difficult to square with the essentially spatial conception of harmony that invari­ ably underlies dualistic ideas. Witness his contrast between music and architecture, as explained in his Systematische Modulationslehre (1887): It is possible, and to a certain extent self-evident, that an architectonic work of art is initially captured in its totality by a single gaze and only then analytically dis­ sected into its details. An understanding of musical art, by contrast, can only be gained by pursuing the opposite strategy; for even a printed score lying in front of us cannot first be understood as a whole and then in its details, because the read­ Page 1 of 27

Dualistic Forms er (who is, after all, hearing in his imagination) must build up the musical work from small fragments lined up one after the other. These fragments are presented in temporal succession and are combined by the listener's memory into larger components. This understanding of the whole is the consequence of a progressive synthesis.4 Riemann could fall back on the entrenched association between architecture and “frozen music” that had pervaded romantic thought since Schelling and Schopenhauer. Unlike those earlier thinkers, however, Riemann used the comparison to different (p. 219) ends. As he never tired of emphasizing, the act of listening should be thought of not as passive enjoyment but as a logical activity, where the listener had to build up the form of the piece in his mind gradually. This logical activity meant for Riemann in particular that the listener should hear music in full units of metric periods and complete harmonic progres­ sions.5 These very basic points are, unfortunately for subsequent music theorists, the only ones that remain more or less constant in Riemann's intellectual career. Pressured by dire fi­ nancial needs, Riemann was a Vielschreiber, a prolific writer, and the first victim of his necessity to keep publishing was theoretical consistency.6 While it is undeniable that Rie­ mann worked toward one grand project, his views on form appear for the most part scat­ tered, and sometimes only partially thought out. Across his theoretical writings, no fewer than four distinctive approaches to form can be identified, which focus on periodic, the­ matic, harmonic and sectional constructions respectively. From his analytical essays, particularly his rather segmentalized analyses of Beethoven's sonatas, it seems that he understood meter, expressed in ideal types of eight-measure pe­ riods, to be the primary bearer of musical form. These periods, to be sure, are largely predicated upon harmonic rhythm, as William Caplin explores further in his contribution to this volume, but it rarely occurred to Riemann to consider the harmonic relations be­ tween these large-scale units.7 Rather, as Scott Burnham has noted a propos of Riemann's analytical practice, form was for him an open-ended process of eight-measure periods strung together one after the other.8 This would constitute another difference to architec­ ture, or rather the flip side of Riemann's above comparison: while the totality in an archi­ tectonic work of art is clearly demarcated, there was nothing in Riemann's system to limit the series of periods of which musical form was constituted. In this approach the musical totality must remain mysterious. Considering this problem in his theoretical model, it seems surprising that Riemann would also consider musical forms in the traditional Formenlehre approach at the same time, using letter designations of the type A-B-A, without noticing a discrepancy.9 Given Riemann's preoccupation with eight-measure phrases, it is not clear how these standard forms should square with the concatenation of periods. There is little to suggest, beyond common sense, according to what criteria these larger sections should be demarcated, since Riemann's eight-measure units are not cast in a hierarchical relationship to each other. Put differently, Riemann's basic idea of musical form seems to follow two different Page 2 of 27

Dualistic Forms guidelines: up to the eight-measure level, there is a continuous hierarchy with units build­ ing up bottom-to-top from smaller entities, while beyond that, his Formenlehre labels would seem to make sense only when understood as derived top-to-bottom and would therefore seem to be “architectural” rather than “musical.” Between these two notions of musical form there is a gap that appears unbridgeable from within his theoretical system.

Ex. 7.1. Riemann illustrates tonal relations in Beethoven's op. 18 no. 6 on his Tonnetz.

This need not have been this way. In principle Riemann was quite willing to acknowledge that form is also articulated through harmony. In his monumental (p. 220) Große Komposi­ tionslehre, he made an effort to put his Tonnetz to analytical use when he showed that the development section of the first movement Beethoven's B♭-major Quartet op. 18, no. 6, touched on “the keys of G minor, F major, E♭ major, D♭ major, and F major.”10 Riemann was not interested in the scalar implications of this sequential progression, which in a nondualistic space could be seen to outline part of a descending whole-tone scale, before returning to the key of the dominant. Instead, he considered the arrangement of these keys “in their very close proximity to B♭ major,” the center of tonal relations of this move­ ment. Example 7.1 shows how these keys relate to each other and to the central harmony of B♭ major along the lines of Riemann's Tonnetz. This (admittedly rudimentary) analysis presented in Große Kompositionslehre brings the structural role of harmonies into the discussion, but it still results in a fairly amorphous musical form. It seems here that Riemann imagined form to be constituted, somewhat eclectically, from the interaction between harmonic moves across the Tonnetz and largescale A-B-A formal models. These two approaches to form are juxtaposed in Große Kom­ positionslehre, but their interaction is not further clarified. Riemann came closest to tackling the problem of form in Katechismus der Komposition­ slehre of 1889, his only treatise dedicated entirely to the theory of forms: So far, [modulation] was only introduced as a means of filling in the metric scheme, but not as a means of creating form beyond a closed [metric] scheme. But now we shall proceed to a totally different approach, by structuring the order of large-scale forms according to their contents. We no longer regard the metric scheme as any more than a means of creating form on the small scale.11 This decision to base form on harmony in this way resulted in quite radical consequences. (And it is probably no coincidence that it was during the same time that Riemann thought Page 3 of 27

Dualistic Forms most intensely about his notion of harmonic function, which he published only a couple of years later.)12 In this treatise Riemann proceeded to introduce musical forms as four ab­ stract types, irrespective of genre, texture, or function, and actively discouraged his read­ ers from using the conventional generic names, such as rondo or sonata forms.13 All forms were, for Riemann, reducible to the basic pattern of A-B-A (which for him signified “main idea—subsidiary idea—main idea,” or Hauptgedanke—Nebengedanke— Hauptgedanke) in various degrees of elaboration.14 When viewed from this angle, musical forms for Riemann were apparently not so much about distinct formal schemes or types as about an abstract notion of formal process. As he explained in the Musiklexikon, the basis of all musical form was the restoration of an original unity at higher level, by means of the resolution of a contrast or conflict: (p. 221)

No art can do away with form, which is nothing but the combination of the parts of the artwork into a whole. Such a combination is only possible if the diverse ele­ ments are in a deep inner relation to one another. If not, the result is merely an ex­ ternal unity, a succession. The supreme demand for all form, including musical form, is unity. Its aesthetic effect can only fully unfold in the context of opposition, as contrast and contradiction (conflict).15 This organicist approach to form, which owes much to Hauptmann's musical dialectics, allowed Riemann to move beyond the problems that beset his diverse approaches to musi­ cal form. When discussing fugues, for instance, Riemann suggested that there was no es­ sential difference to the other forms: tonal order in fugues could be described in the same way as any other formal types. Because the fugue is, as a rule, a monothematic form, it can even serve to show more clearly the underlying processes that produce contrast and resolution. What ultimately mattered, Riemann maintained, were not themes, but the har­ monic plan: “In other words, such richly elaborated musical works with only one theme (whatever the rhythmic and melodic features, without thematic contrast) reduce the structuring of form to the harmonic element.”16 Riemann was suspiciously silent as to what this “harmonic element” actually entailed. Of course, harmony was for Riemann inextricably bound up with the idea of dualism. And this is, one assumes, also the reason that Riemann was reluctant to put his radical theory of forms into analytical practice, or rather why the more fragmentary approach to form postulated in the later Große Kompositionslehre may seem like a step back. The contro­ versial theory of harmonic dualism, in its purest sense, had rather little to do with har­ monic order as it could be found in most musical works. An obvious problem for harmonic dualism is related to multimovement works, such as Beethoven's Symphony no. 5, which begins in a minor key and ends in the parallel major key: “Such transport from minor to major (‘through darkness into the light’) is of decided­ ly gripping effect and certainly justified, although the unity of key is affected by it.”17 The reason Riemann was so disturbed by an ordinary modal mixture is that for harmonic dual­ ism the reference tone switches between major and minor (in Beethoven's Symphony no. Page 4 of 27

Dualistic Forms 5, from °g to c+, to use Riemann's standard dualist shorthand),18 which would raise the question of synthesis, of the higher unity that Riemann posited for all musical forms. And who would be prepared, save for Riemann, to argue that Beethoven's Fifth Symphony was lacking in unity? It seems that this dualistic notion of resolution and synthesis, so central to Riemann's un­ derstanding of form, would have to be realized in a different way from conventional no­ tions of form. In other words, the mismatch in Riemann's theory of forms—between the abstract notion of synthesis he described in his theories on the (p. 222) one hand, and the concatenation of periods that marked his analyses on the other—was dictated by the con­ straints of applying his very particular principles of harmony to a given repertoire of mu­ sic which did not fully represent his idea of how music ought to work.19 It therefore seems that if we want to explore the notion of dualistic forms we need to ori­ ent ourselves by his theoretical writings more so than his published analyses.20 The first movement of Brahms's late Clarinet Trio op. 114, written in 1890–1891, might serve as a test case of what such a theory might have looked like. While to a certain extent this choice is determined by what can be shown analytically in it, it is not arbitrary. Over the course of Riemann's life, Brahms assumed an ever larger role in the musical pantheon that Riemann promoted in his theoretical and historical works. In this spirit, Katechismus der Kompositionslehre, Riemann's most concentrated treatise on form, was dedicated to “the master Johannes Brahms.”21 An analysis of this kind, within a theoretical framework that was never fully established in its own time, is necessarily a hypothetical and speculative enterprise. The endeavor has a distinct historical tinge to it—it invites us to listen to a piece of music with period ears, as it were, to use analytical tools that are, more often than not, in our day confined to the dustbin of history.22 At the same time, it would be overstating the claims such an analysis can make if we tried to argue for any historical fidelity in this endeavor: we already know that Riemann, or his contemporaries, did not analyze music in this way. This theory of dualistic forms is best understood as the reconstruction of a theory that never was, but could have been. It is perhaps best described as an attempt to take Riemann's theoretical claims seriously, more seriously than he did himself, and to try to put them into analytical practice. For this we need not suspend our experience of music and analysis. Rather, such a reconstruction should serve to enrich our appreciation of the musical work, as do most analytical efforts in our time. The ultimate goal is therefore not an exercise in antiquarian history, but rather in utopian thinking: it aims to show how these “vintage” tools can lead to different, and often startling, concepts of form. But, as always, there is a second side to this analytical project: any potential analytical success also vindicates the premises of the theory. And so, if it becomes possible to assert a “dualistic form,” this raises the question of what this tells us about the work of music under scrutiny. One striking feature of most dualistic theorizing in the nineteenth century was that it had comparatively few musical examples to draw on.23 The pure dualistic ap­ proach has, consequently, been reproached for its insensitivity to musical concerns. What Page 5 of 27

Dualistic Forms would it mean to find a piece whose form can be shown to follow dualistic principles? Could this counter the reproaches against dualism's dubious relation to actual musical works? Could this, indeed, form the starting point of a genuinely dualistic repertory? A word of caution seems in order here. Carrying out an endeavor of this nature is a tightrope walk of theoretical mediation. On the one hand, it is important to remember that the dialogue between theory and practice is more complicated than meets the ear: even where composers and theorists emerged from the same cultural (p. 223) context, as is the case here, we cannot simply assume a one-to-one mapping of compositional and theoretical structures. On the other hand, in this exercise in utopian thinking it is impor­ tant not to feel overly constrained by the desert of the real.24 What matters is to examine carefully the mutual relationship between the repertory under analytical scrutiny and the theory that is brought to bear upon the music. Any theory of dualistic forms will have to be located in this force field between speculative principles and compositional practice.

Brahms's Clarinet Trio op. 114 Let us ignore Riemann's aesthetic difference between architectural and musical forms for the moment and treat Brahms's Clarinet Trio in A minor as though it were a work of ar­ chitecture. Let us briefly survey the formal outline, viewed in the score as a totality. If we take a stroll around the first movement, which in the critical literature is often given short shrift beside the more famous Clarinet Quintet,25 the outer form seems by and large unremarkable.26 The exposition touches, in typical Brahmsian fashion, on three key ar­ eas, moving from A minor to C major, followed by a closing section in E minor. The devel­ opment includes, surprisingly, an intermittent key change to F♯ minor and leads back to A minor. The recapitulation is consequently blurred: the home key is already reached at the retransition, and the first subject is not reprised. The second-subject material is restated in the diatonic lower third relation F major, as if to balance the third relation of the expo­ sition. At the same time, the lower third allows an easy return to the home key: where the exposition closed a major third up from C major to E minor, the same interval in the reca­ pitulation leads back from F major to A minor. A coda concludes the movement. We could do worse than to enter the inside of the movement at the main portal, or the first subject, which opens the work, to marvel at the details and hear the work unfold temporally before us in the manner Riemann suggests in his aesthetics. The opening sub­ ject appears to be a rather clear-cut, if slightly bald, twelve-measure unit, shown in exam­ ple 7.2. The first four measures, played by the cello alone, sustain an A minor harmony, arpeggiating upward and elaborating it downward to comprise the entire natural minor scale.27 The four-measure phrase is metrically guileless, and articulates an iambic pattern at the double-measure level, which is numbered 1–4 in the example, in Riemannian fash­ ion.28 Clarinet and piano enter at measure 4, repeating and varying the opening phrase, but extending it to full eight measures.29 (The Arabic number (2) before measure 6 indi­ cates, in Riemannian fashion, that the previous measure should be understood as a restart of the metrical unit.) The clarinet part brings in some harmonic movement, as it Page 6 of 27

Dualistic Forms shifts from the tonic to the minor subdominant (with its “underseventh,”30 i.e., its sixth scale degree, B), closing on what Riemann would call the tonic Variante (and which in oth­ er harmonic systems usually goes by the English name of “parallel”).

Ex. 7.2. Johannes Brahms, Clarinet Trio, op. 114, I: Allegro, mm. 1–12.

Ex. 7.3. Riemann's “characteristic dissonances” in major (dominant seventh) and minor (subdominant “underseventh”) are inversionally related.

While the subdominant is often considered by “monists,” such as Helmholtz, not to imply a genuine harmonic shift,31 this is not so for the dualists: since the Clarinet Trio is in a minor key, the subdominant does constitute a significant function—as shown in ex­ ample 7.3, it is the precise inversion of the dominant seventh in the major key. For a dual­ ist it may also not be coincidental that at the tutti moment at measure 4, as annotated in example 7.2, all three instruments come in on E in the respective octaves, as E is the du­ alistic primary note of A minor (°e).32 Significantly, from this primary note, the bass line in the piano descends from E to C and then settles on a pedal A, outlining the compo­ nents of the A minor chord in its dualistic order. (p. 224)

This last observation might seem basic, trivial even, but in fact the directionality of chordal dualism has always been a bone of contention. Yet some valuable insights can still be gained from it. David Lewin has reinforced some aspects of dualistic listening strategies, shifting the emphasis slightly from Riemann's rootlike “primary” tone to the notion of a “pivotal” tone of a triad which need not be its root.33 Using short examples from the openings of Brahms's Intermezzo op. 119 no. 1 and The Rite of Spring, he demonstrates how the fifth of a triadic sonority can be the pivotal pitch. Although Lewin, (p. 225)

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Dualistic Forms unlike the dualists, does not distinguish between major and minor sonorities in this case —he uses the opening of the Alla danza tedesca from Beethoven's string quartet op. 130 as an example of a major triad that “works downward” from its fifth—it seems that this listening strategy can be successfully applied as a starting point for this piece. In fact, the large-scale harmonic areas of the exposition in Brahms's op. 114 can be seen as revolving around the pitch E. There are three harmonic areas in the exposition: A minor, C major, and E minor. The fea­ ture of a “double” harmonic move from the principal key area is not unusual in Brahms's sonata forms, and is by and large, as James Webster has pointed out, derived from his study of Schubert's sonata forms.34 This exposition, however, is somewhat unusual in that it tonicizes harmonic areas a third apart. Precedents with the same harmonic outline can be found in only few works that we can expect Brahms to have known: among them Schubert's D minor quartet (d–F–a), and Beethoven's Coriolanus Overture (c–E♭–g), but the way in which synthesis is achieved in the recapitulation in any of these works, is rather different from Brahms's Clarinet Trio. Within Brahms's oeuvre, too, this is the only first movement in a minor key to employ this harmonic procedure. The second symphony may be cited as a reasonably close example in a major key, where the exposition moves from D major to F♯ major and to A major. But the major key sets very different standards: by moving on from the secondary key area, the dominant is reached in the second move, and the conventional polarity of tonic and dominant is reinstated. The same, however, cannot be assumed in a minor context; there it would seem—at least in the abstract—that the closing subject in the minor dominant shoots over the goal and may be regarded as formally redundant.

Ex. 7.4. Klangvertretung in the three key areas of the exposition: E in three different significances.

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Dualistic Forms

Ex. 7.5. Hauptmann's “chain of thirds.”

Ex. 7.6. First and second themes compared.

As example 7.4 shows, each of the tonicized triads of the exposition is grouped around E, which in this way appears “prolonged” throughout the exposition.35 In accordance with Riemann's principle of triad representation or Klangvertretung, the E itself is heard in a different context and changes its significance accordingly from “principal tone” to third to “lower fifth.”36 What is happening here can be (p. 226) represented well by means of Moritz Hauptmann's chain of thirds, as in example 7.5, his distribution of the diatonic scale in intervallic and ultimately chordal units. Tonic value—in Riemann's specific sense as explored by Brian Hyer's contribution in this volume—is transferred between the first and second subjects from the A minor triad to the C major triad. In this case, it is done very subtly, as the thematic substance of the sec­ ond subject is closely oriented toward the first subject. As can be seen in the lower stave of example 7.6, the second subject tonicizes C major, but there is a palpable element of the home key A minor present. In fact, the opening of the subject descends from E along the A minor triad before it replaces the “under-fifth” A emphatically for its diatonic neigh­ bor G. Furthermore, the canon per inversionem between measures 52 and 63, included in example 7.7, is based entirely on the oscillatory power of the subject between A minor and C major, which is not resolved throughout the passage.37 (It is in this feature espe­ cially that the Clarinet Trio reveals itself as the sister piece of Brahms's op. 115 with its harmonically ambivalent opening.) On closer inspection it appears that the tendency to­ ward C major is already dormant in the first subject: as the upper stave of example 7.6 indicates, there is one occasion where a pitch other than a component of the A minor tri­ ad occurs in a metrically strong position, G at the downbeat of measure 2.38 In the melod­ ic shape of both subjects a triad note is literally replaced for its diatonic neighbor: in the first subject G for A, and in the second subject, more obviously, A for G. The harmonic ar­

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Dualistic Forms eas of the first and the second subjects therefore describe little more than a shift of em­ phasis on what can be seen as the basic structure common to both.

Ex. 7.7. Second theme.

Ex. 7.8. Harmonic motion of exposition demonstrat­ ed on Hauptmann's chain of thirds.

Ex. 7.9. Functional shifts in the exposition.

There seems to be altogether fairly little contrast between the first and second subjects. The harmonic similarity is further underlined motivically by the close relation between both themes, as example 7.6 showed. It seems entirely appropriate (p. 227) (p. 228) to ex­ press this close relation, as is shown in example 7.8, and to attach Riemann's label Tp, or Tonikaparallele, to this key area.39 A more decisive harmonic shift occurs only with the move to the minor dominant at mea­ sure 63, reinforced at measure 67, where new thematic material is introduced. Here, for the first time, a reference to the home key of A minor is finally abandoned. Instead, E mi­ nor, its minor dominant, is sounded. On closer inspection, however, a similar subtle link of the two key areas becomes noticeable in the melodic detail of the material: as example Page 10 of 27

Dualistic Forms 7.7 shows, the second subject soon exchanged the C for its semitonal neighbor B at mea­ sure 47. The piano accompaniment, sounding only G and E, supports this harmonic ambi­ guity between C major and E minor: is B a long passing note between C and G, or is C an appoggiatura to B? Similarly, two measures later, the upper neighbors of the dominant seventh chord to C major themselves outline an E minor chord; the sparse piano texture confirms the dominant seventh only a measure later. With a view to the E minor closing theme, the second subject functions as its leading-note change, and should be heard as or Dominantleittonwechsel.40 The total harmonic work of the exposition can be summa­ rized in Riemann's terms of function as shown in example 7.9: the functional representa­ tion shows how the second-subject area relates to its surrounding context, functioning both as Tp and retrospectively as .

The “Triad of Triads”

Ex. 7.10. Hauptmann's “triad of triads” in minor.

The progressive transference of tonic value, from A minor to C major and E minor, that we have observed in the exposition, provides a graphic example of the formal dialectical opposition that Riemann spoke about earlier: is A minor the subdominant of E minor, or is conversely E minor the minor dominant of A minor? (We (p. 229) remember that in a dual­ istic framework, a minor dominant does not constitute a problem but is rather prized as a hallmark of pure minor.) Riemann himself has comparatively little to say about this, as his taxonomy simply assumes the tonic a priori,41 whereas Moritz Hauptmann's dialectical approach to harmony examines the significance of the tonic more closely. We should therefore turn to Hauptmann's model for a moment. Following Hauptmann's dialectical notion of harmony, the closing theme of the sonata has left its state of initial unity, and has “come into opposition or contradiction with itself.”42 We can rephrase our earlier question in Hauptmann's terms: does A minor have a dominant, or is it a dominant—in this case, the lower dominant—of the potential tonic E minor? Hauptmann expressed his dialectical idea of tonality by means of a “triad of triads,” where tonality is established in three stages. What he described in this “triad of triads” is essentially the logical process from an unmediated major triad to its dialectically asserted position as the centre of a tonality.43 (Hauptmann only describes the “triad of triads” for the major system, but his overall symmetrical outlook strongly suggests that an equiva­ lent scheme be valid for the minor system too. Example 7.10 shows a dualistic adaptation into A minor of Hauptmann's diagram in C major.)44 After stating the triad (at I), another fifth-related triad is sounded to challenge the first in its central significance. Since the po­ Page 11 of 27

Dualistic Forms sition between two fifth-related triads is always ambiguous, as I have just argued is the case at the end of the exposition in the Clarinet Trio, it is impossible to decide whether the relation should be heard as a tonic—(minor) dominant or subdominant—tonic. It has come into opposition with itself, which is what is described in the two alternative versions of (II). It is only when the other fifth-related chord is introduced that the first chord can be reinstated as a central triad—now as confirmed tonic. The synthetic task of the tonic, as (III), consists in simultaneously “being” dominant (to its subdominant) and “having” a dominant itself.45

Ex. 7.11. Closing theme “getting stuck” on C♯/A♯.

This ambiguity of function at the end of the exposition, stage (II) in Hauptmann's “triad of triads,” is encapsulated in the cadence that concludes the exposition in measure 82, shown at the end of example 7.11. Unusually enough (but perfectly explicable in the con­ text of this movement), the cadence is plagal. Here the plagal (p. 230) cadence serves as a fleeting reminder of the home key A minor, which makes an appearance there not in its central meaning as the tonic, but merely as the subdominant of its dominant, and under­ lines in this way the dialectical stage of opposition into which the movement has entered.

Undoing the Exposition The development section of the opening movement of the Clarinet Trio is distinctly unBrahmsian. While it has become something of a cliché to say of a Brahms sonata that the development spans the entire piece, beginning at the first measure and ending with the last, the opposite seems to be true here. Where traditionally the (p. 231) development, and particularly the Brahmsian type, is the section that generates most harmonic (and mo­ tivic) interest, this development section is so plain that it is doubtful whether it is appro­ priate to call it a development section at all. It seems in many ways as though the devel­ opment “undoes” the exposition. For what it does is mainly restart the exposition in the dominant key: it abridges the first 33 measures of the movement by simply sounding the Page 12 of 27

Dualistic Forms first subject, simulates the climax of measures 22ff, and cadences on C♯ at measure 96 in the score. When the home key of A minor returns at measure 125, leading into the reca­ pitulation, we seem to have gone full circle: the bass line of the exposition arpeggiated the A minor triad upward (in other words in inverted direction, toward its dualistic son générateur, the fifth), while the development section describes the A major triad down­ ward, also against its imagined direction. The development section is chiefly concerned with establishing this C♯. Although the key signature of the development suggests F♯ minor, the only cadence in this key is on C♯ ma­ jor at measure 96. The urge toward a chromatic C♯ is first noticeable at the end of the ex­ position. As example 7.11 shows, the clarinet takes up the closing theme again but alters it chromatically—from C to C♯—to land on the wrong note at measure 74, A♯ assisted by the piano's C♯, which would resolve into B minor. The phrase cannot be closed, as a con­ sequence of, as it were, the wrong harmonic turn it has taken. Instead, the wrong mea­ sure is repeated obsessively, as if a gramophone needle skipped over a scratch in the record and played a snippet of music over and over again.46 The Riemannian metric enu­ meration, which cannot continue until the next heavy beat occurs, makes this repetitive­ ness particularly clear. In this way, the theme, which in its original guise only comprised five measures, is extended to a long twelve-measure period. (Note that the final measure is assigned the metric value “9” indicating a relatively lighter beat, as tonic resolution is absent from the measure.) The piano accompaniment underscores these “needle skips” with a change of texture, which replaces the arpeggiated figures in the left hand for oc­ tave jumps. Only in measure 77, when the clarinet finally “corrects” itself and reinstates the original A natural and the piano its C natural, can the phrase come to an end in E mi­ nor. The exposition is evidently too early a locus for chromatic alterations.

Ex. 7.12. Harmonic reduction of “short-circuited” circle of fifths in development section.

Within the development section, the C♯ also has a decisive influence on the harmonic events. As example 7.12 shows, the descending circle of fifths starting at measure 105— for Riemann, a passage within which tonality is suspended47—is interrupted at measure 110: C♯, which had already been surrendered in passing, in the dominant seventh chord on D in the previous measure, is taken up again, thus disturbing the chain of seventh chords. The significance of the G major triad is changed with the addition of the C♯ pass­ ing note to sound like a subdominant to an unsounded D major triad. The confused har­ monic progression is thus short-circuited, as example 7.12 shows, and, after briefly re­ turning to E minor at measure 114, further anticipates D major by means of a dominant pedal on A between measures 115 and 117. In analogy to the suppressed C♯ above, it

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Dualistic Forms seems that the development section is too early for D major, which merely gives the lis­ tener a hint of things to come. (p. 232)

Dualistic Synthesis

Ex. 7.13. Riemann's Scheinkonsonanzen.

As often in Brahms's chamber music, the beginning of the recapitulation is blurred: here the return of the first subject is completely suppressed, while a return to the home key of A minor can be registered at measure 125 in the score. The second subject is restated in F major, a perfect fifth below its first occurrence. This procedure is not uncommon—per­ haps the most famous example of a recapitulation in the submediant is found in Beethoven's Waldstein sonata. But as we noted a propos of the exposition, the minor tonality of Brahms's movement makes all the difference: in the Waldstein sonata, the re­ capitulation in A major balances the second subject's occurrence in E major in the exposi­ tion.48 In this way, the harmonic scheme of the sonata substitutes for the conventional fifth relation between the initial and final occurrences of the second-subject area. Riemann's taxonomy of harmonic function, and its notion of apparent consonances (Scheinkonsonanzen), is particularly good at expressing such relations, as shown in exam­ ple 7.13. The upper and the lower third relation, expressed through the “relative” (Paral­ lele) and “leading-tone change” (Leittonwechsel) respectively, demonstrate the relation to the tonic, while encompassing between Tp and the fifth that separates these two occur­ rences. One immediate advantage of such a procedure in this piece is of course the con­ venient recapitulation of the closing theme: no retransition is necessary; the transposition a fifth down means that the closing theme is automatically restated in the home key.

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Dualistic Forms

Ex. 7.14. “Utopian” A-major variant of the third theme. (p. 233)

The main point of this harmonic scheme, however, does not seem to be an expedi­

ent means of achieving closure: the movement does not actually end with the restatement of the closing theme, although all the “work” of the sonata form has already been accom­ plished at that stage.49 What happens instead is that the closing theme is repeated in a changed version in A major. The new version of the closing theme fulfils what the devel­ opment presaged through its C♯. The altered theme, reproduced here in example 7.14, is achieved by reinterpreting the F enharmonically into E♯ as an accented chromatic pass­ ing note to F♯. This very slight change opens the melodic range immensely, and gives the whole passage a radiant, utopian shimmer. Although most accounts of the form of the movement would end here, the piece is not at an end—the dualistic dialectic of the sonata is still incomplete.

Ex. 7.15. Dualistic synthesis along Hauptmann's chain of thirds.

We should remember that the opposition at the base of the dualistic dialectic of this movement does not reside in the conventional opposition between first and second sub­ jects—or indeed, the pertinent harmonic areas—but rather in the articulation of a struc­ tural symmetry in Hauptmann's sense. Hauptmann holds that the stage of “opposition,” of the tonic “splitting up within itself,” must be reconciled by reinstating it in its unambigu­ ous central position, by showing that the tonic both is and has a dominant. The tonic, in other words, must be shown from all sides: as a “dominant” in relation to its subdomi­ nant, and likewise, as the “subdominant” of its dominant. Or, along the chain of thirds, the move to the dominant side achieved in the exposition must be balanced by an equiva­ lent move in the other direction, to complete the harmonic scheme, as shown in example 7.15.

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Dualistic Forms

Ex. 7.16. Symmetrical features in the exposition. (p. 234)

The missing move to the subdominant has indeed been prefigured in the harmon­

ic insinuations toward D major in the development section. The articulation of this key fol­ lows in the cadence at measure 194 and is prepared in measure 190, directly after the Amajor theme of example 7.14 above, which I described as “utopian.” This attribute is meant here in more than a descriptive sense, for neither is A major the tonic of the move­ ment, nor D major its subdominant. Indeed, the dualists were agreed that the minor mode cannot have a major subdominant.50 Rather, the D major cadence must be seen as the tonicization of the subdominant required in Hauptmann's scheme: the “utopian” home key of A major functions as the dominant to D major; D major has a dominant. Numerous allusions to the kind of symmetry that Riemann's aesthetic requires be­ tween the most minute detail and the large-scale structures pervade the surface of the piece. Witness the boxed passages in example 7.16, taken from the exposition, in which the eighth-note figures in measures 15 and 17 melodically mirror each other, and har­ monically emphasize the “upper” and “lower” sonorities of the pedal E in the clarinet and cello parts. Similarly, at the same point in the recapitulation, in example 7.17, the func­ tional opposition is brought out by first outlining a move from A up to D (as tonic to sub­ (p. 235)

dominant), which is then paralleled by a move from A down to E (representing, if not ac­ tually sounding, the tonic and its dominant). The need for a subdominant to balance the harmonic skeleton of the movement has long been foreshadowed since the exposition. The errant cadential gesture at measures 18 and 20, reproduced as part of example 7.16 and indicated with square brackets, both point to­ ward the subdominant region. At that early point in the piece, they seem somewhat un­ motivated and appear to function merely as a suspending device to prepare for the out­ burst in the following measures. However, their effect is carefully calculated: the second

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Dualistic Forms cadence especially is difficult to pin down—do they really indicate plagal motion (in A mi­ nor), or perhaps half-cadential gestures (in D minor)?

Ex. 7.17. Equivalent symmetrical passages in the re­ capitulation.

Part of the unsettling effect of these two short moments is set up by means of metric in­ stability. After the presentation of the first subject until measure 12, the pedal E hovers for five quarter notes before the piano enters with a connecting motive. This is a long enough time to unsettle a sense of metric order (especially given the half-note beat), and it is aggravated by the contradictory onset of the piano motive, which clearly (p. 236) de­ mands a downbeat beginning, at an upbeat moment so that the listener is inevitably a quarter note “out.”51 Although a sense of down and up quarter notes is partly restored by the end of each phrase, it is no longer possible to hear whether the chords of measure 18 are on the first or second half note of the measure, in other words, whether the F major triad possesses greater metrical weight than the A minor triad or vice versa. Also in the following phrase—where the metrical instability is catapulted into three-mea­ sure phrases and triplet rhythms on various levels—the subdominant plays a crucial role. Most pertinent is the strange deceptive cadence at measures 26–27, as shown at the end of example 7.16, where the triads D major (which is nothing but Riemann's “impossible” major-subdominant-in-the-minor-mode)52 and E major resolve into D minor. The function of this interrupted cadence is best understood as a minor tonic Leittonwechsel-Parallele ( ); this is in fact the farthest extent to which Riemann was prepared to take his theory of harmonic function.53 The function label describes that in this metric-harmonic position a tonic function is expected, but the sounding chord has only one pitch in common with the tonic chord. We have moved there by modifying it with two apparent consonances (both Leittonwechsel and Parallele apply successively). It also shows clearly how the harmonic space of the dualists is shaped distinctly differently from Roman numeral space, which would simply declare this chord to be iv.54 This strange moment, of D minor sounding without functioning as subdominant, is indeed closely paralleled by the “impossible” D

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Dualistic Forms major passage that we encountered in the recapitulation, and its lacking fulfillment of the subdominant function. Back in the recapitulation, it is only when the harmony of the D major passage finally changes to a more “real” (and functional) D minor in measure 208 that its subdominant significance is fully articulated: example 7.18 shows how in the following measures, which introduce the coda, the D minor harmony even receives its “characteristic dissonance” that identifies it unambiguously as a subdominant. In this sense, we can hear the recapitulation, in analogy to the exposition, as depicted in example 7.19. Symmetry and dialectical synthesis are finally achieved. Example 7.19 juxtaposes a conventional, “monistic” with a dualistic hearing of the piece: the conven­ tional version would pay more attention to the “synthesis” of the diverse key areas, that is the first and the closing themes returning in A minor/major, as expressed by the upper beam in the graph, whereas for the dualists the downward move from A minor to D minor, expressed in the lower beams, would represent the actual synthesis of the movement.

Ex. 7.18. Coda.

The D minor moment at the coda, which is of no particular interest in a conventional reading, is the moment of dualistic epiphany where everything falls into place. Is it just by accident that the subjugation of the D major into functional tonality coincides with an intrusion in the piano from the first of Brahms's Four Serious Songs op. 121, “Denn es gehet dem Menschen wie dem Vieh,” written shortly after the Clarinet Trio? The underly­ ing sentiment of resignation of this song, the relevant motive of which is reproduced here in example 7.20, contemplating human transitoriness and death—a dystopia if ever there was one—is in every way the negation of the utopian quality conjured up just previously. (p. 237)

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Dualistic Forms

Ex. 7.19. Harmonic content of recapitulation and co­ da. Upper stems show “monistic” form while lower stems show “dualistic” synthesis.

Ex. 7.20. Johannes Brahms, “Denn es gehet dem Menschen wie dem Vieh,” from Four Serious Songs, opening.

It is one of the most fascinating aspects of this movement that the “Denn es gehet” motive had been used as a connecting motive throughout the movement; it is the very motive that created metrical disorder from measure 13 onward. In spite of its promi­ nence, it does not stand out as an intrusion from elsewhere until this very late moment. In all its prior occurrences, the motive had been disguised in one way or another—through figuration in triplets (measure 13, shown in the piano accompaniment of example 7.16), subdivision into breathless two-note motives (measure 35), appearance in various harmo­ nizations, or transformation into the major mode (measure 115). (p. 238)

After the allusion to the Serious Songs, the utopian A major triad has lost its transfigured shimmer; its aura has become hollow. As a variant in the major mode, it lacks the affirma­ tive power of a real tonic. In spite of the repeated plagal cadences between D minor and A major, the movement does not attain full closure—at least not in a tonal universe where major and minor follow different paths. The expected A minor resolution is only granted at the end of the final movement, and there, in like spirit, it is also preceded by a quota­ tion in the piano from one of the Serious Songs: “O Tod, wie bitter bist du.”

Hearing with Dualistic Ears Returning to Riemann's initial conceit regarding the contemplation of an architectural and a musical work of art, we finally know the structures of its detail to project them onto the large-scale form. The very first phrase now reveals itself—in true organicist fashion— as a microcosm of the entire piece: its harmonic skeleton if we return to example 7.2, we can now observe the shift from A minor (°T) to its (p. 239) “utopian” variant A major (T+) with its strong plagal element, which has pronounced repercussions with the structurally important points at the end of the exposition and the coda in its entirety.

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Dualistic Forms And so we can reassemble the entire work in dualistic fashion. The exposition explored the “right” side of the tonic, with its dominant harmonies, while the recapitulation (and coda) complemented this with the “left,” subdominant harmonies. Only after this full ex­ ploration of the tonic, when, following Hauptmann, it has been reaffirmed as the “cen­ tral” sonority in this literal, dualistic sense, can tonality be understood to be established unequivocally. In such dualistic terms, the entire movement can therefore be heard to be laboring toward this end. Genuine dualistic synthesis is not reached until the final phrase, with the end of the coda—and only if we hear it with dualistic ears. But what does it mean, putting it somewhat fancifully, to “hear with dualistic ears”? In phrasing it this way, a strong claim is implied—particularly considering that the first and most powerful criticism of the dualists had always been that we do not hear music in this way.55 What we have been doing here is to assume, contrariwise, that the dualists take their fundamental theoretical position seriously—more so, perhaps, than they themselves dared to do at the time—and to draw out the final consequences of this understanding of harmony in its application to musical form. The theoretical moves in this analysis are all founded on the principles that Riemann and his colleagues posited. It turns out that the dualistic view of form in the case of Brahms's Clarinet Trio opens up a model of synthesis that is otherwise inaccessible. But does the Clarinet Trio need dual­ istic theory (to readjust its position in the chamber-music canon) more than the theory needs the piece (to valorize its claims)? It need not be pointed out that it would be absurd to claim any form of intentionality in this case. We can be almost certain that the Clarinet Trio was not composed with dualistic theorizing in mind. But it might be possible to trian­ gulate theory and repertory in a somewhat more abstract way.56 If we take a step back from theory and repertory, we can see that the same fascination with oppositional symmetry that underlies the basic idea of harmonic dualism also seems to be at play in Brahms's sonata movement.57 From the perspective of oppositional sym­ metries, a number of other musical works of that period—for instance, by Wagner and Liszt and their experiments with symmetrical divisions of the scale—can also be consid­ ered under this dualistic angle. Perhaps the conjunction between dualism and harmoni­ cally derived musical forms is best viewed, ultimately, not so much as a formal model that yields concrete analytical insights—it would probably yield slim pickings if strictly formal­ ized in this way, as explored here in the lonesome example of Brahms's Clarinet Trio. Rather, it would seem most fruitful to consider the utopian project of “dualistic forms” as a reflection of a music-historical mode of thought in the late nineteenth century that af­ fected theory and composition alike. It seems, almost paradoxically, that the more we squint while gazing at the musical work of art, in the way Riemann suggested, the more clearly the dualistic theory of form comes into focus.

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Dualistic Forms (p. 240)

Conclusion

Before we close our eyes and fully succumb to the utopian fantasy of a dualistic theory of forms, however, it might be worth recalling the reality with which we started. Or, put dif­ ferently, remember how we said earlier that no matter how closely related repertoire and theory may appear, they always need to be triangulated via the position from which we view both theory and musical work. In this case, the task of triangulation was consider­ able. We started with the observation that the nineteenth-century dualists were lacking an explicit theory of form, and we hypothetically constructed one for them, which dis­ tinctly injects the interests of early-twenty-first-century Anglo-American music theory into this nineteenth-century enterprise. It is clear from Riemann's writings that he was gestur­ ing toward such a theory, but no matter how hard we squint, this dualistic theory still re­ mains a hypothetical reconstruction or, as we called it earlier, an exercise in utopian thinking: Riemann had to be encouraged to expand his formal interest beyond the eightmeasure unit, and Hauptmann had to be encouraged to be a more consistent dualist than he wanted to be, to help this dualistic theory of forms along the way. It is now time to dis­ entangle ourselves from this complicated constellation, and to consider the implications of the lack of such a fully fledged theory of form. In other words, what does it mean for a group of theorists to be formalists without a con­ cept of form? In fact, for the longest time this paradoxical status was a thorn in the dual­ ists’ side, as they felt vulnerable to attacks in this regard, particularly from the Schenker­ ian camp. Witness Hellmut Federhofer, for instance, who echoed Schenker's own blister­ ing criticisms by arguing that Riemann lacked a cohesive concept of large-scale structure.58 Against these attacks, Elmar Seidel leapt to Riemann's defense, countering that Riemann regarded the first nine measures in slow movement of the Waldstein sonata as “a single gigantic F-major cadential progression (beginning and ending on f+).”59 It is notable how Seidel has to strain Riemann's ideas to parry the charge—and, in all fairness, a reference to one nine-measure phrase is not the most robust counterargument. In fact, there are exceptionally few examples in Riemann's analytical writings that actually go be­ yond the eight-measure level. In other words, the organicist rhetoric that Riemann vigor­ ously engages in his aesthetic writings, as we saw initially, praising the whole as more than the sum of its parts, is not carried out in his analytical endeavors, at least not in any ways that would be immediately apparent. In providing a dualistic theory of form by ex­ emplar, we have gone some steps toward redeeming the organicist promise that Riemann's aesthetics makes. Our own age, however, has gone the other way: what used to be perceived as a weakness of the dualistic outlook on harmony—its fragmentary nature and its indecision over longrange tonal relations—has more recently turned out to be the strength of this approach. It is not by coincidence that the various neo-Riemannian approaches came to the fore at ex­ actly the same time as increasing dissatisfaction with the concept of musical autonomy, and the idealist work concept, became (p. 241) widespread in musicological circles in the 1990s, and classic formalist positions with its emphasis on depth and unity started to be viewed with considerable skepticism. In a disciplinary reflection on the state of music Page 21 of 27

Dualistic Forms analysis Jim Samson noted that, despite this continued attack, neither formalism nor its broader cultural ally, modernism, “have been at all anxious to lie down and die.”60 Instead, the analytical project has had to be redefined. The last few years have seen a number of possible responses: Samson proposes a Barthesian “pleasure of the text” as an apposite defense of formalism; Cohn and Dempster have proposed “plural unities,” while Fink and others have celebrated the musical surface.61 Another response, it seems, is found precisely in Riemann's almost postmodern refusal to decide what exactly consti­ tutes musical form, which in the current intellectual climate can be—and has been— turned into a position of significant force. There is more than a hint of irony in the cir­ cumstance that it was necessary to reconstruct the hypothetical nineteenth-century dual­ istic theory of form, and to think fully through the leads Riemann gave us, to become aware of the lack of one in the twenty-first. The essentially fractured nature of Riemann's theoretical concepts, their blithe indifference to formal totalities, corresponds precisely to the needs of the moment and would seem to make them into the formal theory for our times.

Notes: (1.) I would like to thank Edward Gollin, Suzannah Clark, and especially Brian Hyer, for their generous advice on earlier drafts of this essay. For a recent consideration of Rie­ mann and musical forms, see Scott Burnham, “Form,” in Thomas Christensen, ed., Cam­ bridge History of Western Music Theory (Cambridge: Cambridge University Press, 2002), 880–906. (2.) The extent to which Hauptmann can be considered a dualist is a matter of debate. Pe­ ter Rummenhöller vehemently denies it, as expounded in his “Moritz von Hauptmann, der Begründer einer transzendental-dialektischen Musiktheorie,” Beiträge zur Musiktheorie im neunzehnten Jahrhundert, ed. Martin Vogel (Regensburg: Gustav Bosse, 1966), 11–36. Still, the circumstance that subsequent dualists, above all Riemann, built on his work makes the question as to his true intentions a secondary issue. (3.) See Musikalische Syntaxis (Leipzig: Breitkopf und Härtel, 1877), 104–111, where Rie­ mann considers the key structures of multi-movement works in terms of cadential “the­ ses.” (4.) Hugo Riemann, Systematische Modulationslehre (Hamburg: J. F. Richter, 1887), 2. (5.) For recent elucidations of Riemann's concepts of meter and functional harmony, see Cambridge History of Western Music Theory, 684–691 and 796–800. (6.) See Michael Arntz, Hugo Riemann (1849–1919): Leben, Werk, Wirkung (Cologne: Concerto-Verlag, 1999). (7.) In Große Kompositionslehre (Berlin and Stuttgart: W. Spemann, 1902), 1: 424, Rie­ mann spelled out that he was simply not interested in pursuing “the formal element be­ yond the eight-measure period.” See also Lotte Thaler, Organische Form in der Musikthe­ Page 22 of 27

Dualistic Forms orie des 19. und beginnenden 20. Jahrhunderts (Munich and Salzburg: Emil Katzbichler, 1984), esp. 18–54. (8.) Scott Burnham, Beethoven Hero (Princeton, NJ: Princeton University Press, 1995), 81–88. See also Thaler, Organische Form, 103. (9.) Riemann, Katechismus der Kompositionslehre (Berlin: Max Hesse, 1889), 95, and Große Kompositionslehre, 1: 477–478. (10.) Riemann, Große Kompositionslehre, 1: 480. (11.) Riemann, Katechismus der Kompositionslehre, 91–92. (12.) While it is well known that his theory of function was presented in Vereinfachte Har­ monielehre (1893), which was first published in English as Harmony Simplified, most of the functional terminology and concepts were first introduced two years earlier in an arti­ cle, “Die Neugestaltung der Harmonielehre,” Musikalisches Wochenblatt 22 (1891), 513– 514, 529–531, 541–543. See also Renate Imig, Systeme der Funktionsbezeichnung in den Harmonielehren seit Hugo Riemann (Düsseldorf: Verlag zur Förderung der systematis­ chen Musikwissenschaft, 1970). (13.) Riemann, Katechismus der Kompositionslehre, 97. (14.) Riemann's “first form” corresponds to simple a A-B-A model (two-part song form), the “second form” to a somewhat larger A-B-A model, as in three-part song forms or min­ uet-and-trio, the “third form” to a more elaborate A-B-A model, as found in certain rondo forms, and the “fourth form” to sonata forms. (15.) Riemann, “Formen (musikalische),” in Musik-Lexikon, 4th ed. (Berlin: Max Hesse, 1894), 310–311. The Musik-Lexikon is typically a useful source to follow up Riemann's ever-changing concepts; the fourth edition is chosen here as a standard because it was the first to be published after the important treatises Systematische Modulationslehre (1887) and Katechismus der Kompositionslehre (1889). (16.) Riemann, Katechismus der Kompositionslehre, 186. (17.) Ibid., 189. (18.) See Riemann, Musikalische Syntaxis, 105. (19.) This was not the case in his harmony textbooks: by contrast, they were largely built on abstract harmonic examples, not from examples relating to actual pieces of music. (20.) Thomas Christensen proposes some thoughts toward a formal “dialectic” on the ba­ sis of Riemann's theory of functions, in “The Schichtenlehre of Hugo Riemann,” In Theory Only 6 (1984), 37–44.

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Dualistic Forms (21.) Brahms played an increasing role in Riemann's understanding of “healthy” music in the period of cultural pessimism in the years around 1900. Riemann explores his under­ standing of Brahms particularly in “Die Musik seit Wagners Heimgang: Ein Totentanz (1897),” in Präludien und Studien (Reprint, Hildesheim: Georg Olms, 1967), 2: 33–41, and “Degeneration und Regeneration in der Musik,” Max Hesses deutscher Musikerkalender 23 (1908), 136–138. Margaret Notley has recently examined the role of Riemannian theo­ retical thought on Brahms, including analytical remarks on the Clarinet Trio, in “Plagal Harmony as Other: Asymmetrical Dualism and Instrumental Music by Brahms,” Journal of Musicology 22 (2005): 90–130. (22.) For a historiographic reflection that goes in a similar direction, see also Cristle Collins Judd, “The Dialogue of Past and Present: Approaches to Historical Music Theory,” Intégral 14/15 (2000–2001): 56–63. I also rely on Thomas Christensen, “Music Theory and Its Histories” in Christopher Hatch and David Bernstein, eds., Music Theory and the Ex­ ploration of the Past (Chicago: University of Chicago Press, 1993), 9–39. (23.) Riemann was also a composer, whose oeuvre spans sixty-nine works. As far as I have been able to ascertain, however, not even Riemann's own compositions showed any signs of pure dualism. For a survey of Riemann's compositions, though with little reference to theoretical issues, see Michael Arntz, Hugo Riemann (1849–1919), 183–208. (24.) With apologies to Jean Baudrillard and the makers of The Matrix. (25.) See, for instance, Edwin Evans, Handbook to the Chamber and Orchestral Music of Johannes Brahms (London: New Temple Press, [no year]), 264–270, who scathingly criti­ cizes the work as full of “weaknesses.” The critical rejection that the Clarinet Trio re­ ceived has recently been reviewed in Margaret Notley, “‘Brain-Music’ by Brahms: Toward an Understanding of Sound and Expression in the Allegro of the Clarinet Trio,” American Brahms Society Newsletter 16 (1998): 1–3. (26.) Michael Musgrave summarizes the form of the first movement in The Music of Jo­ hannes Brahms (Oxford: Oxford University Press, 1984), 250–251. (27.) Both Margaret Notley and David Brodbeck comment on the absence of the leading note in this theme, see Notley “Plagal Harmony as Other,” and Brodbeck, “Medium and Meaning: New Aspects of Chamber Music,” in The Cambridge Companion to Brahms, ed. Michael Musgrave (Cambridge: Cambridge University Press, 1999), 128. (28.) See System der musikalischen Metrik und Rhythmik (Leipzig: Breitkopf und Härtel, 1903), 13. (29.) Peter H. Smith and Peter Foster both comment on this theme, noting that the clar­ inet entry, reversing the 1̂ and 5̂ of the theme in the cello, resembles a fugal comes. See Peter Foster, “Brahms, Schenker and the Rules of Composition: Compositional and Theo­ retical Problems in the Clarinet Works” (Ph.D. diss., University of Reading, 1994), and Pe­ ter H. Smith, “Brahms and Subject/Answer Rhetoric,” Music Analysis 20 (2001): 193–236. Page 24 of 27

Dualistic Forms (30.) See “Dissonanz: A. Charakteristische Dissonanz” in Musik-Lexikon, 5th ed. (Berlin: Max Hesse, 1900). The term is not yet included in the fourth edition (1894), 239, al­ though the underlying concept is clearly the same. Most of the relevant entries cited here are also translated in the glossary in my Hugo Riemann and the Birth of Modern Musical Thought (Cambridge: Cambridge University Press, 2003), 186–198. (31.) Hermann von Helmholtz, On the Sensations of Tone, trans. Alexander J. Ellis (Reprint, New York: Dover, 1954), 293. (32.) See “Klang,” Musik-Lexikon, 4th ed. (Berlin: Max Hesse, 1894), 534–536. (33.) David Lewin, “A Formal Theory of Generalized Tonal Functions,” Journal of Music Theory 26 (1982): 23–60. (34.) See James Webster, “Schubert's Sonata Form and Brahms’ First Maturity,” Eigh­ teenth Century Music 2 (1978): 18–35. (35.) On this listening strategy, see also Suzannah Clark's contribution to this volume. (36.) See “Klangvertretung,” Musik-Lexikon, 539. (37.) This would also explain the lack of the leading tone that others have noted. See n. 27 above. (38.) Peter H. Smith also comments on the similarity between the two subjects, but exam­ ines it from a motivic viewpoint, see his “Brahms and the Shifting Barline: Metric Dis­ placement and Formal Process in the Trios with Wind Instruments,” Brahms Studies 3, ed. David Brodbeck (Lincoln: University of Nebraska Press, 2001), 222. (39.) Riemann, Vereinfachte Harmonielehre (London: Augener, 1893), 79–86. (40.) Ibid., 86–92. (41.) Riemann, Musikalische Logik (Leipzig: C. F. Kahnt, 1874), 42. (42.) Moritz Hauptmann, The Nature of Harmony and Metre, trans. W. E. Heathcote (Reprint, New York: Da Capo Press, 1991), 8. (43.) Ibid., 10. (44.) In fact, Hauptmann shies away from this consequence of his theory because he is more attached than Riemann or Oettingen to the major dominant, which for him—unlike his dualist colleagues—is operative in the minor system as well. See ibid., 18–19. Conse­ quently, he completely ducks the question of a “triad of triads” for the minor system. (45.) Ibid., 10–14. (46.) I take the image of the “skipping needle” from Brian Hyer's article “Chopin and the in-F-able,” in Raphael E. Atlas and Michael Cherlin, eds., Musical Transformation and Mu­ Page 25 of 27

Dualistic Forms sical Intuition: Eleven Essays in Honor of David Lewin (Roxbury, MA: Overbird Press, 1994), 147–166, as it seems to describe precisely what goes on here. While in the corre­ sponding place in the recapitulation the passage is not omitted, it is texturally completely integrated and does not stand out as a “scratch on the record.” (47.) Riemann, “Sequenz,” in Musik-Lexikon, 992. (48.) See Suzannah Clark, “Terzverwandtschaft in der Unvollendeten von Schubert und der Waldstein-Sonate von Beethoven—Kennzeichen des neunzehnten Jahrhunderts und theoretisches Problem,” Schubert durch die Brille 20 (1998): 122–130. (49.) See also Scott Burnham, “The Second Nature of Sonata Form,” in Suzannah Clark and Alexander Rehding, eds., Music Theory and Natural Order from the Renaissance to the Early Twentieth Century (Cambridge: Cambridge University Press, 2001), 111–141. (50.) This idea can be found throughout Riemann's work, from Skizze zu einer neuen Methode der Harmonielehre (Leipzig: Breitkopf und Härtel, 1880), 18, onward. Riemann's standard explanation for the prohibition of the major subdominant in the mi­ nor mode (and, correspondingly, the minor dominant in the major mode) is that the dual­ istic roots of the relevant chords would be more than one-fifth apart, in other words no longer directly related. The pure mode of A minor (°e) is formed by °a–°e–°b; to replace the subdominant with the its major variant would result in d+–°e–°b, in other words two chords two-fifths apart, which Riemann no longer considers a relation of the first degree. (51.) Smith, “Brahms and the Shifting Barline,” 191–229, discusses metric issues in the Clarinet Trio in some depth. (52.) In the example I follow Riemann's sneaky way out of this conundrum (as explained in Vereinfachte Harmonielehre) by regarding this impossible subdominant as having a chromatically altered third. In other words, while S+ is not allowed in the minor mode, the identical-sounding SIII〈 is. (53.) See the preface to the 5th ed. of Handbuch der Harmonielehre (Leipzig: Breitkopf und Härtel, 1912), xvi. (54.) See David Lewin, “Amfortas’ Prayer to Titurel and the Role of D in Parsifal: The Tonal Spaces of the Drama and the Enharmonic C♭/B,” Nineteenth Century Music 7 (1984): 336–49. (55.) This criticism of Riemann was made as early as 1878, in Karl von Schafhäutl, “Moll und Dur,” Allgemeine Musikalische Zeitung 13 (1878): cols. 1–137. For an exploration of this criticism, see also Hugo Riemann and the Birth of Modern Musical Thought, chapter 1, and Henry Klumpenhouwer's contribution to this volume. (56.) Again, this notion goes back to Judd's position paper, “The Dialogue of Past and Present,” 63.

Page 26 of 27

Dualistic Forms (57.) See Suzannah Clark, “Seduced by Notation: Arthur von Oettingen and the Topogra­ phy of the Major-Minor System,” in Suzannah Clark and Alexander Rehding, eds., Music Theory and Natural Order, 161–180. (58.) Hellmut Federhofer, “Die Funktionstheorie Hugo Riemanns und die Schichtenlehre Heinrich Schenkers,” in Kongreßbericht Wien (1956): 183–190, and subsequently, Akkord und Stimmführung in den musiktheoretischen Systemen von Hugo Riemann, Ernst Kurth und Heinrich Schenker (Vienna: Veröffentlichungen der Akademie der Wissenschaften, 1977). One of Schenker's most vicious attacks against Riemann's ineptitude at long-range hearing is found in “Beethoven's Sonata in F Minor op. 2 no. 1,” in Der Tonwille, ed. William Drabkin (New York: Oxford University Press, 2005), 2: 92–94. (59.) Elmar Seidel, “Die Harmonielehre Hugo Riemanns,” in Martin Vogel, ed., Beiträge zur Musiktheorie im 19. Jahrhundert (Regenburg: Gustav Bosse, 1966), 89. The analysis in question is found in Hugo Riemann, L. van Beethovens sämtliche Klavier-Solosonaten (Berlin: Max Hesse, 1920), 3: 31. (60.) Jim Samson, “Analysis in Context,” in Nicholas Cook and Mark Everist, eds., Rethink­ ing Music (Oxford: Oxford University Press, 1999), 52. (61.) Ibid., 54; Richard Cohn and Douglas Dempster, “Hierarchical Unity, Plural Unities,” in Katherine Bergeron and Philip Bohlman, eds., Disciplining Music: Musicology and Its Canons (Chicago: University of Chicago Press, 1992), 156–181, and Robert Fink, “Going Flat,” in Cook and Everist, Rethinking Music, 102–137.

Alexander Rehding

Alexander Rehding teaches music at Harvard University. His interests are in the his­ tory of music theory and in nineteenth and twentieth century music. He is the author of Hugo Riemann and the Birth of Modern Musical Thought, Music and Monumental­ ity, and Beethoven’s Symphony no. 9. He is the editor-in-chief of the Oxford Hand­ books Online series in Music.

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Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music   Dmitri Tymoczko The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0008

Abstract and Keywords This article considers the theme of inversional symmetry as it manifests itself in Riemann's theoretical writings and in late-nineteenth-century chromatic music. It exam­ ines the mathematical properties of the concepts of symmetry underlying musical sys­ tems and explores how these symmetrical properties can be brought to bear in innovative ways on musical structures. Section 1 of the article provides a historical background by examining Rameau and his proposed laws of tonal harmony which are invariant under four basic operations: reordering, octave shift, note duplication, and chromatic disposi­ tion. It also discusses Weber's Roman numeral notation which develops and fulfills Rameau's ideas. Section 2 discusses Riemann's “dualism” as an attempt to incorporate in­ version into the Rameau/Weber collection of symmetries. Section 3 examines whether the “second practice” of the nineteenth-century chromaticism involves inversional symmetry. Section 4 provides a Riemannian understanding of dualism and Section 5 illustrates a contrapuntal approach by examining a Brahms intermezzo. Keywords: inversional symmetry, chromatic music, concepts of symmetry, Rameau, tonal harmony, Weber, dual­ ism, Brahms

“THE importance of symmetry in modern physics,” writes Anthony Zee, “cannot be over­ stated.”1 Zee alludes to the fact that some of the most celebrated discoveries in the histo­ ry of science—from Galileo's law of inertia to Einstein's principle of relativity and Dirac's prediction of antimatter—involve the realization that physical laws possess unexpected symmetries. These symmetries allow us to change our description of the world (for in­ stance, by using different numbers to refer to locations in space) without altering the form of our laws. The laws, by virtue of being insensitive to these changes, are symmetri­ cal with respect to them. Though it is not immediately obvious, this notion of symmetry plays a similarly central role in music theory as well. Indeed some of the most important developments in the his­ tory of theory—including Rameau's root functionality, Weber's Roman numerals, and Oettingen's and Riemann's dualism—involve claims that the musical universe possesses Page 1 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music unexpected symmetries. Music theorists, however, are less explicit about the topic than physicists: they typically propose symmetries en passant, by developing notation and ter­ minology that is invariant under the relevant musical transformations. For example, when theorists say that a note is “an F♯,” the description remains true even if the note is trans­ posed by one or more octaves. The term “F♯” thus embodies a symmetry (octave equiva­ lence) by virtue of being insensitive to a musical operation (octave transposition)—much as the laws of (p. 247) Newtonian physics remain the same whether one chooses to de­ scribe oneself as being at rest or in motion with a constant velocity.2 In what follows, I will consider the theme of inversional symmetry as it manifests itself in Riemann's theoretical writings and in late-nineteenth-century chromatic music. Section 1 provides historical background. I begin with Rameau, who proposed that the laws of tonal harmony are invariant under four basic operations: reordering, octave shift, note duplica­ tion, and chromatic transposition. Weber's Roman numeral notation, which develops and fulfills Rameau's ideas, is symmetrical under two additional operations: diatonic transpo­ sition and what I call triadic extension. I argue that traditional tonal syntax does indeed manifest these symmetries, at least to a good first approximation. Section 2 describes Riemann's “dualism” as an attempt to incorporate inversion into the Rameau/Weber col­ lection of symmetries. As many commentators have noted, dualism is unsatisfactory be­ cause traditional tonal syntax is not in fact inversionally symmetric. Section 3 then asks whether the “second practice” of nineteenth-century chromaticism involves inversional symmetry.3 I argue that it does, but only because inversional relationships arise as neces­ sary by-products of a concern with efficient voice leading. Section 4 contrasts this view with a more orthodox, Riemannian understanding of dualism. Finally, section 5 illustrates my contrapuntal approach by analyzing a Brahms intermezzo.

1.  Rameau and Weber Let us begin by defining a “basic musical object”—the atom of music-theoretical dis­ course—as an ordered sequence of pitches.4 Basic musical objects can be ordered in time or by instrument: (C4, E4, G4) can represent an ascending C major arpeggio played by a single instrument or a simultaneous chord in which the first instrument plays C4, the sec­ ond instrument plays E4, and the third G4.5 (Instruments can be labeled arbitrarily: what matters is simply that they are distinguishable somehow.) A progression is an ordered se­ quence of musical objects: thus, (C4, E4, G4) → (C4, F4, A4) is a progression with (C4, E4, G4) as its first object and (C4, F4, A4) as its second. Basic musical objects are uninteresting because they are so particular: until we decide how to group objects into categories, (C4, E4, G4) remains unrelated to (E4, C4, G4).6 Theorists typically classify musical objects by defining musical transformations that leave objects “essentially unchanged.” For example, (C4, E4, G4) can be transformed in three ways without modifying its status as a C major chord: its notes can be reordered, placed into any octave, or duplicated. This process of reordering, octave shift, and note duplica­ tion can be reiterated to produce an endless collection of objects, all equally deserving of Page 2 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music the name “C major”: (C4, E4, G4), (E2, C5, G4, G3), (G2, G4, G7, E2, C6), and so on. “C ma­ jorishness” is therefore determined by an (p. 248) object's pitch-class content, rather than the order or register in which its pitches are stated. We can say the concept “chord” em­ bodies the symmetries of octave equivalence, reordering, and note duplication. Rameau is often considered the first theorist to articulate the modern conception of a chord, determining the harmonic identity of groups of notes by their pitch-class content alone.7 He also classified chords into larger categories, using terms like “major [perfect] chord” and “minor [perfect] chord” to refer to what we would call transpositional set classes. These more general terms are invariant under a larger group of symmetry opera­ tions: we can reorder, shift octaves, duplicate notes, or transpose every note by the same amount, all without changing an object's status as a major chord.8 “Major chordishness” is thus determined not by the specific pitch classes in an object, but by the intervals between them. These intervals, and hence “major chordishness,” are preserved under transposition. Underlying Rameau's classificatory innovations was a third and more far-reaching sug­ gestion: that chords provide the appropriate vocabulary for formulating the basic laws of tonal harmony. Thus, if example 8.1a is an acceptable harmonic progression, then so is ex­ ample 8.1b, since the two passages contain exactly the same series of chords.9 Furthermore, the laws of harmony are on Rameau's view transpositionally invariant: thus if example 8.1a is acceptable in C major, then example 8.1c should be acceptable in G ma­ jor. (The principles of functional harmony, in other words, do not change from key to key.) We can say that the fundamental harmonic laws are invariant under octave shifts, re­ ordering, note duplication, and transposition.

Ex. 8.1. Symmetry in traditional harmonic analysis.

At this point, I should pause to explain that there are actually two different ways in which a symmetry can operate upon a sequence of chords: individual symmetries can be applied independently to the objects in a progression, while uniform symmetries must be applied in the same way to each object. Example 8.1 illustrates. Reordering, octave shift, and note duplication are individual symmetries, and can be applied differently to each chord with­ out changing the progression's fundamental harmonic character.10 (This process may cre­ ate awkward voice leading, but that is a separate matter.) By contrast, transposition is a uniform symmetry: one must use a single transposition when shifting music into a new key. Example 8.1d illustrates (p. 249) the disastrous result of applying transposition indi­ vidually, with the nonfunctional sequence E♭ major →F major → B major bearing little re­ Page 3 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music semblance to the original.11 Clearly, theorists of tonal music want to emphasize the rela­ tion between examples 8.1a–c, all of which are I–IV–V–I progressions; the relationship be­ tween these and example 8.1d, though significant in some twentieth-century contexts, is not important in traditional tonality. Although Rameau invented many features of modern harmonic theory, he did not devise a fully satisfactory notation for progressions. This was accomplished by Gottfried Weber al­ most a hundred years later. Weber's Roman numerals are invariant under (uniform) trans­ position and (individual) reordering, octave shift, and note duplication—precisely as Rameau's theory requires. (Note in particular that the same Roman numerals apply to ex­ amples 8.1a–c, but not example 8.1d.) But Weber's notation also encodes two additional symmetries not intrinsic to Rameau's theory. The first might be called the triadic exten­ sion symmetry: as shown in examples 8.2a–b, it is possible to extend a diatonic “stack of thirds” upward without changing its Roman numeral. This symmetry permits us to group together collections with different pitch class content—for example, to treat {G, B, D}, {G, B, D, F}, and {G, B, D, F, A} as versions of the V chord in C major.12 (Triadic exten­ sion is an individual symmetry, since sevenths can be added to only some of the chords in a progression.) The second symmetry is diatonic transpositional symmetry: as shown in examples 8.2a and 8.2c, it is possible to transpose a passage of music diatonically, be­ tween relative major and minor, without changing its Roman numerals.13 (Since tradition­ al tonality uses only two modes, the action of diatonic transposition is restricted to shifts between relative major and minor.) Like chromatic transposition, diatonic transposition is a uniform symmetry: we would radically transform the sense of a passage if we were to diatonically transpose only some of its chords.

Ex. 8.2. Two additional symmetries. The second pro­ gression relates to the first by triadic extension, while the third relates to the first by diatonic trans­ position.

At first blush, diatonic transpositional symmetry may seem pedestrian. But it is interest­ ing to note that there are broadly tonal styles displaying no such symmetry. For example, in rock, different modes draw on different repertoires of chord progressions: i–VII–vi–VII– i and i–III–vi–VII–i are common in minor, while the analogous major-mode progressions (I– vii°–vi–vii°–I and I–iii–vi–vii°–I) are extremely rare. In some sense, this is to be expected: diatonic transposition, by changing the quality of the triads on each scale degree, changes the sound of the various diatonic chord progressions, and one would not neces­ sarily expect musical syntax to be (p. 250) insensitive to these changes. To my mind, it is somewhat remarkable that classical harmony exhibits such a high degree of symmetry be­ tween major and minor. This symmetry permits the extraordinary expressive effect of pre­ senting “the same” musical material in very different affective contexts. Page 4 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music The Rameau/Weber theory, of course, is merely an approximation to actual tonal practice; a more accurate theory can be obtained by combining Roman numerals with figured-bass notation. This hybrid system, which is nearly universally accepted by American peda­ gogues, allows theorists to make very refined observations about harmonic motion—for instance, that the chord progression V–IV6 is common while the progression V–IV is rare.14 The hybrid system must be further extended with purely contrapuntal principles, such as the emphasis on efficient voice leading and the prohibition of parallel perfect fifths. In this sense, the Rameau/Weber theory describes only some of the conventions of traditional tonal music. In my view this is no flaw: tonality is extraordinarily complex, and we should not ask any one theory to describe it completely. Rather than focus on the inad­ equacies of traditional theory, I prefer to marvel at the fact that it provides an extraordi­ narily efficient description of harmonic patterns found in a wide range of tonal music— even music written by composers trained in the earlier figured-bass tradition.15 Its sim­ plicity and scope surely qualify it as one of the greatest achievements of Western music theory.

2.  Dualism and Traditional Tonality This brings us to the “dualism” of Oettingen and Riemann, which can be understood as an attempt to augment the Rameau/Weber symmetries with (uniform) inversional equiva­ lence.16 Like Rameau and Weber, Riemann articulated his theory by developing analytical terms that are invariant under his favored musical transformations: thus one can often use precisely the same Riemannian description to describe inversionally related pas­ sages. The interesting question is whether traditional tonal practice requires this sort of inversionally symmetrical terminology. Though Riemann's dualistic thinking was guided by dubious forays into metaphysics and acoustics, it can be purified of such connections.17 What is more important, from a mod­ ern perspective, is that inversion and transposition are the only distance-preserving transformations of pitch and pitch-class space (example 8.3a). Since distance is a funda­ mental musical attribute, there are a number of theoretical contexts where it is useful to think dualistically: for example, when cataloguing tertian sonorities,18 or triadic progres­ sions containing common tones,19 or the efficient voice-leading possibilities between set classes of a given type. We will return to this point momentarily.

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Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music

Ex. 8.3. Inversional symmetry in Riemannian analy­ sis. (a) Transposition and inversion preserve the dis­ tances between notes, transforming the nine-semi­ tone interval C–A to the nine-semitone interval E♭–C. (b) Riemann used inversionally invariant terminology to label chord tones, so that inversion sends root to root, third to third, and fifth to fifth. (c) Riemann also labeled chord progressions in an inversionally invari­ ant way, using a single term for both C major → F ma­ jor and its inversion, C minor → G minor.

Riemann, following Oettingen, conceived the minor-key system as the inversion, rather than the diatonic transposition, of major: thus, as shown in example 8.3b, he (p. 251) called C, E, and G the root, third, and fifth of the C major triad, while labeling G, E♭, and C the root, third, and fifth (respectively) of C minor. This “dualistic” terminology is invari­ ant under inversion, which sends the root of a major triad to the Riemannian “root” of a minor triad, the third to the third, and so on (example 8.3b). Riemann also developed an inversionally symmetrical vocabulary for classifying progressions: thus a single Riemann­ ian term, Gegenquintschritt, describes both the progression C major → F major and its in­ version, C minor → G minor (example 8.3c). Though Riemann devised names only for tri­ adic progressions—the so-called Schritte and Wechsel—we can easily extend this idea: let us say that two progressions are dualistically equivalent if they are related by (uniform) transposition or inversion. For example, the progression A♭7 → C major, used in the stan­ dard resolution of the German augmented sixth, is dualistically equivalent to Fø7 → E mi­ nor, the penultimate progression in Tristan.20 This is because the inversion that trans­ forms A♭7 to Fø7 also transforms C major to E minor. In much the same way, the chord progression from C augmented to C diminished seventh (or Caug → C°7) is dualistically equivalent to Caug → C♯°7, since one of the inversions that transforms C°7 to C♯°7 leaves Caug invariant. These definitions allow us to determine whether any two progressions are dualistically equivalent or not.21 So is traditional tonal syntax invariant under inversion? Did Riemann, like Rameau, man­ age to describe a symmetry of traditional tonal chord progressions?

Ex. 8.4. Inversion and diatonic transposition. Page 6 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music Theorists generally agree that the answer is “no.”22 The harmonic patterns in traditional tonal music do not exhibit even approximate signs of inversional invariance: though I–IV– V–I is very common in tonal music, its inversion i–v–iv–i is extremely rare. In this respect, traditional theory is correct in describing major and minor as being related by diatonic transposition rather than by inversion. Example 8.4 demonstrates. We begin with a stan­ dard I–ii56–V–I progression in major. Example 8.4b transposes this pattern downward by two diatonic steps, raising the leading (p. 252) tone G → G♯ in the process; the result is a perfectly well-formed i–

ø

progression in minor. Example 8.4c inverts the har­

monies in example 8.4a around middle C, producing a nonstylistic minor-key i–

–iv–i pro­

gression. The contrast between examples 8.4b and 8.4c illustrates the general point that acceptable tonal progressions in major can usually be transformed into acceptable tonal progressions in minor by way of diatonic transposition, but only rarely by inversion. We can conclude that Riemann's inversionally invariant terminology, though theoretically elegant, is often inconsistent with the actual procedures of traditional tonal music.23 In this sense, it is a theory in search of a repertoire—a speculative description of what might be possible, rather than a faithful description of the music of Riemann's time.24 It is rather remarkable that within a few years of Riemann's death, Schoenberg had devised an entirely new musical language that fully embraced inversional equivalence—a lan­ guage that teems with dualistically equivalent progressions, often resulting from the use of inversionally related twelve-tone rows.25 No doubt Riemann would have rejected Schoenberg's music as violating natural laws of tonality. But from a more distant (and somewhat Whiggish) perspective, we can see twelve-tone music as a vindication of Riemann's speculative music theory—albeit one that Riemann himself would have consid­ ered perverse.

3.  Dualism and Voice Leading The harmonic syntax of traditional tonal harmony does not exhibit even an approximate inversional symmetry, but what about the extended tonality of the late nineteenth centu­ ry? Might it be the case that Riemannian dualism, while not useful for describing the dia­ tonic “first practice” of nineteenth century music, can still tell us something about its chromatic “second practice”? My answer is a qualified “yes.” Dualistic terminology is useful for analyzing chromatic tonality, and this helps explain why Riemann's ideas have been so fruitful in recent music theory. However, I will suggest that contemporary theorists have (p. 253) not produced a fully adequate account of why this is so. In my view, nineteenth-century composers were not explicitly concerned with inversional relationships as such; instead, these relation­ ships appear as necessary by-products of a deeper and more fundamental concern with efficient voice leading.26 Rather than being the syntactic engine that drives the music, in­ version is merely epiphenomenal—the smoke that escapes from the locomotive's chimney, rather than the furnace that makes it go. And though dualism can be useful in analysis,

Page 7 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music this is largely because it is a tool that helps us to comprehend the range of voice-leading possibilities available to nineteenth-century composers. Let me approach these issues by proposing a very simple model of late-nineteenth-centu­ ry tonality, according to which the music combines a diatonic “first practice” inherited from eighteenth-century tonality with a chromatic “second practice” emphasizing effi­ cient voice leading between familiar sonorities.27 This flexible “second practice” sets very few constraints on composers: virtually any voice leading between familiar chords may be used, as long as it is efficient. These chromatic voice leadings serve a variety of musical functions, acting as neighboring chords, passing chords, intensifications of dominants, modulatory shortcuts between distant keys, and so on. (Examples 8.5a–e provide a few representative passages from Schubert, Chopin, and Schumann.) Over the course of the century, one finds a gradual emancipation of the second practice, as chromatic voice lead­ ing—at first sporadic and decorative—controls ever-larger stretches of music.

Ex. 8.5. Efficient voice leading in nineteenth-century music: (a) Schubert, D major piano sonata D. 850/op. 53, I, mm. 11–16; (b) Schubert, Am Meer, m. 1; (c) Chopin, Nocturne Op. 9 no. 1, mm. 23–24; (d) Chopin, Nocturne Op. 9, no.1, mm. 38–39; (e) Schu­ mann, “Chopin,” from Carnival, mm. 11–13.

Ex. 8.6. Semitonal voice leadings between triads.

The interesting point is that this concern for efficient voice leading will necessarily give rise to a wealth of dualistic relationships. For instance, example 8.6 lists the sixteen “semitonal” voice leadings between consonant triads (that is, voice leadings in which no Page 8 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music voice moves by more than a semitone).28 The voice leadings have been grouped into in­ versionally related pairs. They have further been categorized by the retrograde relation­ ship: the voice leading in column 3 is the transposed retrograde of the voice leading in column 1. (Note that the first and fourth columns are related nondualistically, with identi­ cal root motion connecting two major or two minor triads.) It is clear from the table that an interest in semitonal voice leading among consonant triads will necessarily give rise to dualistic relationships; indeed, even a composer who chooses (p. 254) randomly from among the voice leadings in example 8.6 will generate numerous dualistically related pro­ gressions. It turns out that the efficient voice leadings between members of any two set classes can always be grouped into inversionally related pairs. This is because transposition and in­ version are (as discussed above) distance-preserving operations: thus, if a particular pas­ sage of music exhibits efficient voice leading, then we can invert the passage to obtain equally efficient voice leadings (cf. example 8.7, which uses inversion to categorize the voice leadings between half-diminished and dominant-seventh chords). Suppose, then, that a musical style obeys the following three principles: (P1) If a sonority is acceptable then so is its inversion. (P2) Efficient voice leadings are desirable. (P3) Any additional voice-leading prohibitions—such as the prohibition on parallel per­ fect fifths—apply equally to ascending and descending motion. In these styles, if a voice leading is acceptable, then its inversion will also be—which means we should expect a reasonable number of dualistic relations in the music itself.

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Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music

Ex. 8.7. Semitonal voice leading among half-dimin­ ished and dominant seventh chords: (a) voice lead­ ings between two dominant sevenths or two half-di­ minished chords; (b) voice leadings from one type of chord to the other.

Since principles P1–P3 accurately describe the nineteenth century's “second practice,” dualism provides us with a useful tool for cataloging chromatic possibilities. Insofar as we want to develop a systematic grasp of all the efficient voice (p. 255) leading possibilities between familiar sonorities—not just those in examples 8.6 and 8.7, but all the analogous voice leadings between all the familiar tonal chords—then our task will be simplified by cataloging voice leadings on the basis of inversional equivalence: without dualism, we would have to memorize each of the voice leadings in examples 8.6–8.7 separately; but once we understand that they are grouped into dualistically related pairs, we need memo­ rize only half as many. However, dualism is just one of several tools needed here: both the retrograde relationship and what I have elsewhere called individual transpositional equiv­ alence are also useful in this context.29 Together, these concepts allow us to reduce a very large set of voice-leading possibilities to a much smaller set of underlying paradigms. (p. 256)

These ideas derive, ultimately, from Richard Cohn, the first theorist to notice that

dualistic terminology has a natural application to questions about voice leading. In “Maxi­ mally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Pro­ gressions,” Cohn asks, “which equal-tempered harmonies can be connected by singlesemitone voice leading to at least two of their transpositions or inversions?” He answers this contrapuntal question dualistically, noting that two triads can be linked by singlesemitone voice leading if and only if they are related by the neo-Riemannian L or P trans­ formations. The present essay generalizes Cohn's observation by observing that the effi­ cient voice leadings between any two set classes can always be grouped into inversionally related pairs.30 Since inversionally related voice leadings move their voices by the same

Page 10 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music distances, dualism is a natural framework for investigating certain kinds of contrapuntal questions. Following Cohn, then, I conclude that the chromatic voice leadings of the nineteenthcentury's “second practice” do exhibit an important kind of inversional symmetry: we can invert any stepwise (or semitonal, or efficient) chromatic voice leading between tertian sonorities to produce another such voice leading. Since chromatic music exploits virtually all of the efficient voice leadings between familiar chords, we should expect to find nu­ merous dualistic relationships therein. For the same reason, we should expect to find ret­ rograde relationships, as well as instances of individual transpositional equivalence. In­ version, retrograde, and individual transpositional equivalence are important primarily because they help us comprehend the range of options available to nineteenth-century composers—permitting us to group these voice leadings into categories whose members are related in interesting but nonobvious ways. Together, these categories bring a mea­ sure of order to the unruly world of nineteenth-century chromatic possibility.

4.  Harmonic Dualism Let's contrast this “contrapuntal” dualism with a more orthodox form of dualism descend­ ing from Riemann himself. Harmonic dualists reject the suggestion that counterpoint pro­ duces inversional relationships, proposing instead that inversion is explanatorily basic.31 Thus the two dualisms have diametrically opposed understandings of the relative priority of harmony and counterpoint: one conceives of inversional relatedness as a tool for cate­ gorizing voice-leading possibilities, while the other understands inversional relationships as explanatory in their own right.

Ex. 8.8. Lewin and Wagner.

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Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music Consider, in this context, a brief but famous analysis by David Lewin. Lewin observed that the two Wagnerian passages shown in example 8.8 are interestingly related: example 8.8a, the Tarnhelm motive, contains two minor triads (g♯, e) and an open fifth suggesting either b minor or B major. Example 8.8b, from the modulating section of the Valhalla motive, contains major triads on G♭, B♭, and F. The semitonal voice leadings at the beginning of each passage, (G♯, B, D♯) → (G, B, E) and (G♭, B♭, D♭) → (F, B♭, D), (p. 257) are inversional­ ly equivalent; hence from a dualistic perspective, they instantiate the same basic musical schema.32 The second progression in each passage, meanwhile, involves ascending-fifth motion, suggesting tonic → dominant (or subdominant → tonic) motion. Lewin thus con­ structs a single “transformational network” to describe the two passages: the first pro­ gression is described as “LP”—Riemann's Leittonwechsel transformation followed by the neo-Riemannian “parallel” transformation,33 together producing Riemann's Terzschritt; the second progression is labeled SUBD, indicating that the chord moves up by fifth. The analysis, like a good deal of neo-Riemannian theory, is very much in the spirit of Riemann's harmonic dualism. Lewin does not consider the idea that voice leading might help explain the first progressions in examples 8.8a and 8.8b. Nor does he differentiate the first progression in each passage, which arguably arises from efficient voice leading, from the second progression, which is a piece of traditional harmonic syntax. Instead, the “network analysis” in example 8.8d places a neo-Riemannian harmonic label (“LP”) along­ side a more traditional harmonic label (“SUBD”). (The purely harmonic character of this network can be seen from the fact that it applies to any progression from G♯ minor to E minor to B—even registrally disjunct passages such as example 8.8c.) The implication seems to be that the neo-Riemannian LP has a status akin to that of the traditional tonal I–V (or IV–I) progression. Since the harmonic routines of traditional tonality were clearly part of the cognitive framework of nineteenth-century composers, one might read Lewin as suggesting that dualistic harmonic ideas played a similarly important role.34 Certainly, he treats the inversional relationships as significant in themselves, rather than the mere by-products of deeper contrapuntal forces. I am suspicious. From my point of view, analyzing Wagner while ignoring counterpoint is like trying to explain a locomotive's motion on the basis of the shape of (p. 258) the clouds emanating from the smokestack. This is, first, because the contrapuntal view explains facts that the harmonic view does not. From my perspective, it is not at all coincidental that the “Tarnhelm” and “Valhalla” motives exploit major third relationships between ma­ jor and minor triads. Example 8.6 showed that two major triads can be connected by max­ imally efficient voice leading precisely when they are related by major third: voice lead­ ings such as (G♭, B♭, D♭) → (F, B♭, D) move just two notes by one semitone, which is as small as any voice leading between major triads can be. (Inverting, we see that a similar fact holds true of E and G♯ minor triads.) This helps explain why we find so many triadic major-third relationships in pieces as different as Schubert's G major string quartet (movement IV, starting measure 132), Wagner's Ring, and the G minor prelude from Shostakovich's op. 87 Preludes and Fugues. From this point of view, what is most striking about the chromatic progressions in example 8.8 is that they use maximally efficient voice leading between triads, not that they are related by inversion. By contrast, for the har­ Page 12 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music monic dualist, there is nothing particularly distinctive about major-third relationships: the focus of Lewin's analysis is entirely on the relation between examples 8.8a and 8.8b, not on the individual contrapuntal qualities of each example considered in isolation—qualities that in my view help explain the relationship between the passages.35

Ex. 8.9. Chromatic Voice Leading in Tristan.

Second, efficient voice leading potentially explains a wider range of Wagnerian proce­ dures than does harmonic dualism. Example 8.9 presents a number of progressions drawn from Wagner's Tristan, all using efficient chromatic voice leading between familiar tonal sonorities. Harmonic dualism offers no unified explanation of these progressions, nor of their relation to examples 8.8a and 8.8b. (After all, the mere fact that “LP” progressions appear in The Ring gives us no reason to expect that Wagner would elsewhere exploit semitonal voice leadings between seventh chords.) The contrapuntal view thus captures the intuition that there is a single compositional procedure that underlies a wide range of Wagnerian passages.36 Third, contrapuntal dualism offers a simpler and more elegant his­ torical narrative. Composers and theorists have been concerned with efficient voice lead­ ing since the dawn of Western counterpoint. The contrapuntal dualist claims that nine­ teenth-century chromaticism is revolutionary chiefly insofar as it augments traditional tonal syntax with moments of efficient voice leading in chromatic space: thus, triads (p. 259) like E minor and G♯ minor, once thought to be tonally distant, came to be seen as close, since they could be connected by efficient chromatic voice leading. By contrast, it is harder to tell a plausible story that explains how nineteenth-century composers sud­ denly became attracted to dualistic harmonic procedures—particularly since there is so little historical evidence to support this suggestion. These arguments, I suggest, pose a genuine dilemma for proponents of harmonic dual­ ism. If a theorist believes dualistic transformations to be more than by-products of voice leading, then she will need to do more than simply point to sporadic instances of inver­ sional relationships in nineteenth-century music. Instead, she will need to show that these relationships appear especially frequently, or play a significant musical role. If, on the other hand, the theorist does not want to undertake this project, then it is still incumbent upon her to produce some sort of metatheoretical justification for the emphasis on inver­ sion. What is the point of singling out these particular harmonic relationships if we have good reason to think they are mere by-products? The danger is that a too-narrow focus will overemphasize the relations between examples 8.8a and 8.8b, and underemphasize

Page 13 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music the relations among the voice leadings in example 8.6. And this in turn may lead to an im­ poverished perspective on chromatic tonality.

5.  Brahms and the Tristan Chord How does my contrapuntal perspective contribute to analytical practice, if at all? Let's ex­ plore this question by way of a brief analysis of Brahms's Intermezzo, op. 76, no. 4—com­ posed in 1878, thirteen years after the premiere of Tristan. Example 8.10a contains the opening phrase of the Intermezzo's rounded binary form. Example 8.10b summarizes the contrasting middle section, while example 8.10c shows how the opening music is altered in the repeat. The typography reflects my claim that chromatic tonal music involves two distinct systems. Open noteheads represent chords that participate in the first-practice routines of functional tonality—each is assigned a Roman numeral indicating its harmonic function. Closed noteheads refer to chromatic chords whose function is largely contra­ puntal. These have been assigned neither Roman numerals nor neo-Riemannian harmonic labels.

Ex. 8.10. Brahms, Op. 76, no. 4.

Brahms's short piece exemplifies a relatively common nineteenth-century schema, sys­ tematically exploring the voice-leading possibilities of a few characteristic sonorities. Here, the relevant sonorities are the Tristan chord {F, G♯, B, E♭} and the E♭ minor triad. (Note that Brahms's Tristan Chord is the actual Tristan chord, appearing at the correct pitch-class level.)37 The brief piece resolves the Tristan chord in three different ways: to F7 at a1, to A♭ major at a2, and to E♭ major at a3.38 Similarly, the E♭ minor triad resolves to F7 at b1, to G♭7 at b2, and to B♭ major at b3. This chromaticism tends to lead the music in­ Page 14 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music to distant keys: the second resolution of the Tristan chord, at a2, takes us from B♭ major to A♭ major; the dramatic contrapuntal move from G (p. 260) minor to E♭ minor paves the way for a smooth transition into the C♭ major of second phrase; and the return from C♭ major to B♭ major occurs by way of the E♭ minor triad, here acting as iv of B♭ major.39 Harmonically, then, efficient voice leading is a centrifugal force, pulling the music into new tonal territory. It is only at the end of the piece that this force is overcome, as the I– iv–I progression B♭ → e♭ → B♭ tames E♭ minor, returning it to the B♭ major fold. This technique is common in late nineteenth-century music; indeed, the Tristan prelude (and the opera as a whole) could be said to be “about” the various ways of resolving a Tristan chord to the dominant-seventh chord, while Chopin's E-minor prelude can be said to be “about” the various ways of interpolating single-semitone (p. 261) voice leading into a descending-fifth sequence of seventh chords.40 Brahms's piece, like these others, illus­ trates the nineteenth-century principle that any chord can move to virtually any other chord, so long as the two can be connected by efficient chromatic voice leading. But where this description might suggest a kind of unregulated chaos, Brahms is characteris­ tically disciplined: rather than populating the piece willy-nilly with unrelated examples of chromaticism, he returns repeatedly to a few sonorities, demonstrating their capabilities rather like a traveling salesman exhibiting the many functions of an expensive vacuum cleaner. Indeed, there are at least four ways in which Brahms ensures the Intermezzo's coher­ ence. First, the opening phrase features a clear stepwise ascent from F to D, shown by the stems in example 8.10a; in the return, the rising stepwise line is balanced by a chro­ matic linear descent from E♭3 to B♭2 (example 8.10c). Second, the piece is suffused with Brahmsian motivic connections, particularly the double-neighbor B4–D5–C5 from the sec­ ond measure.41 Third, as noted above, the piece returns repeatedly to the same small set of sonorities: not just the Tristan and E♭ minor chords, but also G♭7, which appears both the middle section of the piece and at g3 as a neighbor to B♭. Finally, though the piece is reasonably chromatic, Brahms never lets these centrifugal forces overwhelm the diatonic elements: the music clearly articulates numerous points of tonal arrival and often allows the listener to track the play of diatonic functions. Together, these four factors moderate the “anything goes” radicalism of chromaticism, producing a delicious Brahmsian blend of extravagance and restraint. These observations suggest the more general thought that coherence in nineteenth-cen­ tury music is to be found at the level of specific musical works, and not at the level of general syntactical principles. It is, I think, indisputable that chromatic harmony permits virtually any efficient voice leading between familiar chords. But to say this is not to deny that it takes compositional skill to deploy these options in a musically satisfying way—on the contrary, the more possibilities available to a composer, the harder it is to build logi­ cal musical structures. To understand how nineteenth-century composers constructed in­ telligible pieces, one must therefore look closely at individual works: it is here, and not at the level of universal laws of chromatic tonal syntax, that interesting constraints on musi­ cal coherence are to be found. I suspect that careful analysis of nearly any successful Page 15 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music nineteenth-century music would reveal interesting strategies for harnessing the inherent­ ly destabilizing force of chromatic voice leading—techniques that prevent contrapuntal liberty from devolving into musical anarchy. In section 3, I suggested that we cannot develop a true understanding of chromatic tonali­ ty unless we have a systematic mastery of the voice-leading possibilities between chords. Once we have developed such mastery we will see that Brahms's piece is not a series of idiosyncratic contrapuntal gestures, but a collection of very familiar moves. For example, the voice leading at a1 appears at measures 97–98 of the Tristan prelude (example 8.9e); the voice leading at g1 appears in example 8.8a; the voice leading at g2, which features ♭II7 acting as a tritone substitution for the F7 chord, is very similar to the first voice lead­ ing in Tristan (example 8.9a);42 the voice leading at α3 is very similar to the final voice leading in Tristan (example 8.9g); the voice leading (p. 262) at γ3 opens Schubert's song “Am Meer” (example 8.5b); and so on.43 The piece's various chromatic moves are no more original to Brahms than are its elements of traditional tonal syntax. Indeed, in my view, the efficient chromatic voice leadings constitute the shared syntax of the nineteenthcentury's second practice, just as the shared routines of eighteenth-century tonality con­ stitute its first practice. To understand voice leading is to understand the space in which nineteenth-century composers operated—and is in turn prerequisite for appreciating the often astonishing skill with which they deployed the opportunities available to them.

6.  Conclusion The idea of symmetry thus provides a unifying thread that runs throughout the history of music theory, from Rameau to Weber to Riemann. We could in principle follow this thread into the twentieth century, for example by interpreting Schoenberg as eliminating one of the traditional symmetries: where traditional theorists sometimes consider the order of a group of pitches to be relatively unimportant, twelve-tone rows are defined by their order. Appreciating Schoenberg's music thus involves a two-stage process: not only do we need to sensitize ourselves to the orderings of twelve-tone rows; we also have to desensitize ourselves to their unordered pitch content—since from this point of view, twelve-tone rows are all the same. (Indeed, if one looks at the pitch-class content of moderate spans of music, twelve-tone pieces are remarkably homogenous: rather than modulating from one scale to another, they continually recirculate through the same twelve pitch classes, creating a kind of middle-ground harmonic uniformity.) Whether this twofold reorienta­ tion is psychologically possible or aesthetically desirable is a complex and fascinating question. Unfortunately, a thorough discussion of these ideas is a matter for another time. Instead, let us conclude by reconsidering the ambiguous role of inversional symmetry in tonal music. It is clear that inversion is, in some sense, a genuine symmetry of the musi­ cal universe. Since inversion and transposition are the only distance-preserving opera­ tions on pitch space, there are numerous situations in which it can be useful to think du­ alistically. But while inversion and transposition are equally important mathematically, they are not equally salient psychologically. Many tonal styles take advantage of transpo­ Page 16 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music sitional symmetry, permitting characteristic musical patterns to appear at virtually any pitch level. Individual pieces are often performed in multiple keys, for instance, to accom­ modate different vocal ranges. By contrast, it is hard to think of a robustly tonal style that wholeheartedly embraces inversional symmetry. Nevertheless, it is clear that inversional relationships occur reasonably frequently in chromatic tonal music. Consequently, dualist language can help us describe genuine rela­ tions present in this repertoire—witness Lewin's interesting comparison of examples 8.8a and 8.8b. At the same time, however, it is possible to overemphasize the (p. 263) signifi­ cance of these relationships, since there is good reason to think that they often arise as the by-product of a concern for voice leading. Indeed, it seems likely that a composer or analyst could become expert in chromatic tonality without any explicit awareness of in­ versional symmetry: one would simply have to learn the voice leadings in examples 8.6 and 8.7 (as well as many other analogous voice leadings between other familiar chords) individually, rather than as inversionally related pairs. By contrast, it would be impossible to become an expert tonal composer without understanding transposition in a general and systematic manner. Analyses of nineteenth-century music therefore need to walk a fine line, exploiting dual­ ism for what it can give us, while being careful not to overestimate its role in the music it­ self. I have suggested that the prudent approach is to interpret inversion—like retrograde and individual transpositional invariance—as a tool we use to organize and catalog the musical possibilities available to nineteenth-century composers. To do this is not to deny outright the importance of dualistic theorizing, but it is to reframe its significance some­ what, requiring that analysts adopt a somewhat circumspect attitude toward its claims. For in the language of another great dualist, it is possible that inversional symmetry is a feature of chromaticism as it appears to us, not as it is in itself.

Notes: Thanks to Scott Burnham, Elisabeth Camp, Noam Elkies, Ed Gollin, Dan Harrison, Alex Rehding, and Robert Wason for their help with this article. (1.) Anthony Zee, Quantum Field Theory in a Nutshell (Princeton: Princeton University Press, 2003), 70. (2.) It should be emphasized that this conception of symmetry inheres in our basic nota­ tion and terminology; we are not talking about the manifest symmetry of, say, a palin­ dromic piece. (3.) The essays in William Kinderman and Harald Krebs, eds., The Second Practice of Nineteenth Century Tonality (Lincoln: University of Nebraska Press, 1996), explore the idea that the nineteenth century, like the early seventeenth century, had a “first practice” and a “second practice.” I return to this thought in section 3.

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Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music (4.) Many of the ideas in this section are developed in joint work with Clifton Callender and Ian Quinn. See Clifton Callender, Ian Quinn, and Dmitri Tymoczko, “Generalized Voice Leading Spaces,” Science 320 (2008): 346–348. (5.) I use scientific (i.e., Acoustical Society of America) pitch notation in which middle C is C4; spelling is unimportant. Regular parentheses ( ) denote ordered lists, while curly braces { } denote unordered collections. (6.) Of course, we intuitively consider (C4, E4, G4) to be very similar to (E4, C4, G4), since they are related by reordering. This shows that we instinctively adopt certain musical symmetries even without realizing it. (7.) Various theorists prefigured Rameau with respect to triads. However, Joel Lester credits Rameau with asserting a more general principle that applies to seventh chords as well—and by extension, to harmonies generally. See Lester, “Rameau and Eighteenth-Cen­ tury Harmonic Theory,” in The Cambridge History of Western Music Theory, ed. Thomas Christensen (Cambridge: Cambridge University Press, 2003). (8.) For example, transposition transforms the C major chord (E2, C5, G4, G3) into the D major chord (F♯2, D5, A4, A3). Note that we can use the octave symmetry to shift just some of the notes in an object—for instance, transforming (C4, E4, G4) into (C5, E4, G4)—where­ as we must apply the same transposition to all the notes in an object. (9.) It should be noted that this symmetry is only approximate, since second-inversion tri­ ads have an anomalous status in tonal harmony. (10.) To transform the first chord of example 8.1a into the first chord of example 8.1b, one needs to switch the notes played by soprano and alto, and then transpose the soprano voice up an octave. However, this operation will not transform the second chord of exam­ ple 8.1a into the second chord of example 8.1b. (11.) Here we transpose the first chord in example 8.1c down by four semitones, the sec­ ond down by seven, the third down by three, and the fourth down by seven. (12.) Triadic extension symmetry represents a slight departure from Rameau's ideas: Rameau viewed the chord {D, F, A, C} as both being a D chord with added seventh and an F chord with added sixth. By contrast, the triadic extension principle is typically asso­ ciated with the view that all harmonies are fundamentally “stacks of thirds.” (13.) Chromatic transposition shifts notes by a constant number of semitones; diatonic transposition shifts notes by a constant number of scale steps. Because of this, one may have to change capital Roman numerals to small Roman numerals, and add accidentals to raise the leading tone; I will ignore these details here. (14.) Ian Quinn proposes replacing the standard Roman numeral/figured bass system with an alternative, quasi-Riemannian, system that explicitly represents harmonic functions. See his “Harmonic Function without Primary Triads” (paper presented to the national Page 18 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music meeting of the Society for Music Theory in Boston, 2005). As far as I can tell, Quinn's sys­ tem makes it difficult to express principles like “IV goes to ii but not vice versa” or “roots rarely progress by ascending third.” (15.) See my “Progressions fondamentales, fonctions, degrés, une grammaire de l’harmonie tonale élémentaire,” Musurgia 10.3–4 (2003): 35–64. (16.) A subtle point: it is perhaps more accurate to view Riemann as attempting to relate major keys and minor keys by inversion rather than diatonic transposition. If so, then it is inaccurate to say he wanted to extend the Rameau/Weber symmetry group—instead, he wanted to change it by replacing one symmetry (diatonic transposition) with another (in­ version). This subtlety will not be relevant to the following discussion. (17.) See Alexander Rehding, Hugo Riemann and the Birth of Modern Musical Thought (Cambridge: Cambridge University Press, 2003). Daniel Harrison suggests that Riemann's various derivations of inversional equivalence—some acoustic, some not—are secondary to the principle itself; see his Harmonic Function in Chromatic Music (Chicago: University of Chicago Press, 1994), 259. (18.) The inversion of any tertian sonority is also a tertian sonority. Hence, when we list all the tertian sonorities of a given cardinality, we find they can be grouped into inversion­ ally related pairs. (Some, of course, are their own inversions.) (19.) Suppose we have a chord progression A → B with n common tones. Inversion can be used to produce a second progression, Ix(A) → Ix(B), between sets of the same set class, which also has n common tones. This was well known to Riemann, and is discussed in David Kopp's Chromatic Transformations in Nineteenth-Century Music (Cambridge: Cam­ bridge University Press, 2002), 74. (20.) See my “Scale Theory, Serial Theory, and Voice Leading,” Music Analysis 27.1 (2008): 1–49. (21.) Here I deviate from David Lewin, who interprets Riemann's Schritte and Wechsel as “transformations” or functions that, upon being given a chord as input, return some other chord as output. See his Generalized Musical Intervals and Transformations (New York: Oxford University Press, 2007), and “Some Notes on Analyzing Wagner: ‘The Ring’ and ‘Parsifal,’” 19th-Century Music 16.1 (1992): 49–58. I have instead treated chord progres­ sions as higher order musical objects related by transposition and inversion. My ap­ proach, unlike Lewin's, permits “dualistic” progressions between chords with different symmetries (for example, Caug → C°7 and Caug → C♯°7). No (single-valued) function over chords can capture this sense of “dualistic equivalence,” since it would be necessary to map a single augmented chord to multiple diminished sevenths. (22.) See Carl Dahlhaus, “Über den Begriff der tonalen Funktion.” In Beiträge zur Musik­ theorie des neunzehnten Jahrhunderts, ed. Martin Vogel (Regensburg: Gustav Bosse, 1966), and Studies in the Origin of Harmonic Tonality, trans. Robert O. Gjerdingen (Princeton: Princeton University Press, 1990), Harrison, Harmonic Function (particularly Page 19 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music section 5.4.1), and Rehding, Hugo Riemann, as well as the (nondualistic) maps of allow­ able chord progressions in textbooks such as Stefan Kostka and Dorothy Payne, Tonal Harmony (New York: McGraw-Hill, 2003). Note that even those who defend Riemann against his critics (such as Henry Klumpenhouwer, “Dualist Tonal Space and Transforma­ tion in Nineteenth-Century Musical Thought,” in The Cambridge History of Western Mu­ sic Theory, ed. Thomas Christensen [Cambridge: Cambridge University Press, 2003], 456–476) do not typically assert that there is an inversional symmetry between allowable tonal chord progressions in major and minor. (Martin Scherzinger, with Neville Hoad, “Anton Webern and the Concept of Symmetrical Inversion: A Reconsideration on the Ter­ rain of Gender,” Repercussions 6.2 [1997]: 63–147, may be an exception here.) Instead, they tend to argue that dualistic concepts can be useful even though traditional harmonic practice does not exhibit inversional symmetry. I pursue this strategy in section 3. (23.) These inadequacies may have motivated Riemann's eventual introduction of nondu­ alistic “functional” harmonic labels—a second theoretical system that coexists only some­ what uncomfortably with Riemann's dualism (see Kopp, Chromatic Transformations, and Rehding, Hugo Riemann). However, even function theory does not smoothly account for the diatonic transpositional symmetry between major and minor: Riemann would label the major-mode submediant (vi) as “Tp,” while labeling the minor-mode submediant (VI) as “  ”—even though the submediant triad behaves similarly in the two modes. (24.) Rehding, Hugo Riemann, chapter 3, has cautioned that it is somewhat anachronistic to read Riemann as if he were a contemporary theorist, concerned only with describing the behavior of actual composers: instead, he is (at least in part) a speculative theorist who aimed to provide directions for future compositional work. (25.) Consider, for example, the opening of Schoenberg's op. 33a. If one were to try to an­ alyze the piece without twelve-tone terminology, one might emphasize the retrograded dualistic relationship between the second two chords in the first measure and the first two chords in the second measure. See David W. Bernstein, “Symmetry and Symmetrical Inversion in Turn-of-the-Century Theory and Practice,” in Music Theory and the Explo­ ration of the Past, ed. Christopher Hatch and David W. Bernstein (Chicago: University of Chicago Press, 1993), 377–409, for a more general discussion of symmetry and Schoen­ berg. (26.) By “efficient” voice leading I mean, roughly, “voice leading in which no voice moves very far.” See my “Voice Leadings as Generalized Key Signatures,” Music Theory Online 11.4 (2005), “The Geometry of Musical Chords,” Science 313 (2006): 72–74, and “Scale Theory, Serial Theory, and Voice Leading,” for more mathematical definitions. (27.) I discuss this view in depth in A Geometry of Music (New York: Oxford University Press, 2011). For antecedents of the approach, see Carl Friedrich Weitzmann, Der über­ mässige Dreiklang (Berlin: T. Trautwein, 1853), Ernst Kurth, Romantische Harmonik und ihre Krise in Wagners “Tristan” (Berlin: Max Hesses Verlag, 1920), William Mitchell, “The Study of Chromaticism,” Journal of Music Theory 5 (1962): 2–31, Gregory Proctor, “Tech­ nical Bases of Nineteenth-Century Chromaticism,” Ph.D. diss. (Princeton University, Page 20 of 23

Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music 1978), Richard Cohn, “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions,” Music Analysis 15/1 (1996): 9–40, “Neo-Riemannian Operations, Parsimonious Trichords, and Their ‘Tonnetz’ Representations,” Journal of Mu­ sic Theory 41.1 (1997): 1–66, and “Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective,” Journal of Music Theory 42.2 (1998): 167–180. (28.) I begin each voice leading from C major or C minor. Voices are individuated by regis­ ter: the top note in the first chord moves to the top note in the second; the middle note in the first chord moves to the middle note in the second; and so on. Intuitively, a “voice leading” corresponds a to phrase like “C major moves to E major by holding E constant, moving C down by semitone to B, and G up by semitone to G♯” (see A Geometry of Music or “Scale Theory, Serial Theory, and Voice Leading”). Mathematically, voice leadings are equivalence classes of progressions under (uniform) applications of the reordering and octave-shift symmetries (see Callender, Quinn, and Tymoczko, “Generalized Voice Lead­ ing Spaces”). (29.) As the name suggests, individual transpositional equivalence results from treating transposition as an individual, rather than uniform symmetry. Consider the semitonal voice leading (C4, E4, G4) → (C4, E♭4, G4), shown in the third staff of example 8.6. (The no­ tation (C4, E4, G4) → (C4, E♭4, G4) indicates that C4 moves to C4, E4 to E♭4, and G4 to G4.) By transposing the destination sonority up by semitone, we can obtain a closely related voice leading, (C4, E4, G4) → (C♯4, E4, G♯4), shown on the fourth staff. These two voice leadings each map root to root, third to third, and fifth to fifth, moving their voices by the same distances, up to an additive constant. For more discussion of this idea, see A Geom­ etry of Music and “Scale Theory, Serial Theory and Voice Leading.” (30.) Some of Cohn's writing suggests a more harmonic understanding dualism, of the type we will consider in section 4. For instance, in “As Wonderful as Star Clusters: Instru­ ments for Gazing at Tonality in Schubert,” Nineteenth-Century Music 22.3 (1999): 213– 232, Cohn seems to portray hexatonic key areas as syntactically significant harmonic re­ gions, rather than mere by-products of semitonal motion. It would be interesting to trace the interacting themes of harmony and counterpoint in Cohn's work. (31.) Riemann has often been criticized for ignoring counterpoint. See, for example, Mil­ ton Babbitt, Words about Music (Madison: University of Wisconsin Press, 1987), 136–137. (32.) Recall that the notation (G♯, B, D♯) → (G, B, E) indicates that G♯ moves to G, B moves to B, and D♯ moves to E. Inverting the voice leading (G♯, B, D♯) → (G, B, E) gives us (G♭, B♭, D♭) → (F, B♭, D). For more, see my “Voice Leadings as Generalized Key Signatures,” “The Geometry of Musical Chords,” “Scale Theory, Serial Theory, and Voice Leading,” and A Geometry of Music. (33.) The neo-Riemannian parallel transformation is equivalent to Riemann's Quintwech­ sel, transforming C major into C minor.

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Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music (34.) This conjecture is somewhat speculative, since Lewin does not provide much discus­ sion of the significance of the inversional relationship. I am operating on the assumption that Lewin thought the relationship was important (since he wrote a paper about it) and that he thought it was more than a by-product of contrapuntal forces (since otherwise he would have mentioned voice leading). (35.) It should be noted that any instance of maximally efficient voice leading between two major triads can be related by inversion and (possibly) retrograde to any instance of maximally efficient voice leading between two minor triads—or, to put it another way, the two progressions can be described as exemplifying a single “transformation” drawn from the set {LP, PL, T4, T8}. In the grand scheme of things, then, it is not all that surprising that a single network describes examples 8.8a–b. We would expect the Ring's fifteen hours of music to contain several instances of maximally efficient voice leading between two major or minor triads, followed by an ascending fifth progression; and from these passages we would expect to be able to select pairs that exemplify a single Lewin-style network. (36.) Note also that a contrapuntal approach predicts that triads and seventh chords should in general progress in different ways, since they have different voice-leading capa­ bilities. (In particular, seventh chords are particularly close to their minor-third and tri­ tone transpositions, just as triads are particularly close to their major-third transposi­ tions.) But a purely harmonic approach would give us no reason to expect systematic dif­ ferences between chords of different sizes. Thus we could potentially use statistical analy­ sis to adjudicate between these two explanations. (37.) Lewin explores Brahms's frequent references to the formal and rhetorical proce­ dures of earlier eras. See his “Brahms, His Past, and Modes of Music Theory.” in Brahms Studies: Analytical and Historical Perspectives, ed. George S. Bozarth (Oxford: Oxford University Press, 1990). The appearance of the Tristan chord suggests that Brahms may sometimes have made similar reference to the work of his contemporaries. (38.) Some readers may prefer to analyze the chord at a2 as a diminished seventh chord {B, D, F, A♭} with an E♭ pedal, or as the ninth chord {D, F, A♭, B, E♭}. The important point is that the same sonority occurs at a1, a2, and a3. (39.) Note that the main key areas of B♭ major and C♭ major are largely expressed by their dominants, with the tonic B♭ major chord not arriving until the very end of the piece. This chromatic practice is analyzed at length in Robert Morgan's “Dissonant Pro­ longation: Theoretical and Compositional Precedents,” Journal of Music Theory 20 (1976): 49–91. The large-scale semitonal key relation is also common; see Patrick McCreless, “An Evolutionary Perspective on Nineteenth-Century Semitonal Relations,” in The Second Practice of Nineteenth Century Tonality, eds. William Kinderman and Harald Krebs (Lin­ coln: University of Nebraska Press, 1996).

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Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music (40.) The Tristan prelude is of course also about longing and death. I am engaging in composer's shoptalk here, speaking about how the notes are assembled, rather than ex­ pressive use to which they are put. (41.) Also notable in this regard are the overlapping lines in the contrasting middle sec­ tion. These motivic details are not shown on the harmonic reduction. (42.) Following analytical tradition, I am discarding Tristan's motivic voice exchanges. See A Geometry of Music. (43.) The main pivot-chord modulation in the piece, in which iv of B♭ major becomes iii of C♭ major, is related to the first modulation in Schubert's “Die junge Nonne,” where VI of F minor becomes V of F♯ minor. The harmonic minor scale has two major triads a semi­ tone apart, whereas its inversion, the harmonic major scale, has two minor triads a semi­ tone apart. Schubert exploits the former property and Brahms the latter, with their two modulations being dualistically equivalent.

Dmitri Tymoczko

Dmitri Tymoczko is a composer and music theorist who teaches at Princeton Univer­ sity. His music has been performed by ensembles throughout the country, and he has been the recipient of a Rhodes scholarship, a Guggenheim fellowship, and numerous other awards. His book, A Geometry of Music, has just been published by Oxford Uni­ versity Press; it will be followed shortly by an album of pieces combining classical and jazz ideas.

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From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combinatorial Conception of Interval   Edward Gollin The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0009

Abstract and Keywords This article examines the changing meaning of the Tonnetz in Riemann's writings over the course of his career. It examines how Riemann came to reconcile the literal-acoustical and the spatial-metaphorical views of the Tonnetz—how the Tonnetz as a literal matrix to represent and calculate relative frequencies of the tones in just intonation evolved in his later writings into traversable landscape of tones. This article begins by examining the ta­ ble from Arthur von Oettingen's Harmoniesystem in dualer Entwickelung, which provided not only the graphic model for the table of relations in Riemann's dissertation but also shaped its underlying acoustical conception. The article then examines how two develop­ ments in the following decades were significant for Riemann's conceptual reformation of the Tonnetz. The transition from a literal to a metaphorical understanding of the Tonnetz did not simply mirror Riemann's shift from an acoustical to a psychological view of the foundation of harmony, but rather made possible the transition, providing Riemann a mechanism to mediate between the phenomenal world of musical practice and the un­ bounded noumenal realm of musical meanings. Keywords: Tonnetz, intonation, Arthur von Oettingen, conceptual reformation, Entwickelung, tones, literal-acousti­ cal views, spatial-metaphorical views

EXAMPLE 9.1 presents the opening of the funeral march from Beethoven's Piano Sonata op. 26, a passage Hugo Riemann discusses in his 1880 Skizze einer neuen Methode der Harmonielehre to illustrate the difference between genuine and orthographic enharmonic reinterpretation.1 Riemann observes that the B-major triad in measure 9, a respelling of an actual C♭-major, is a notational change that exists merely “on the outer surface” of the music—a respelling for the convenience of the reader or performer. By contrast, Riemann notes, the “true” enharmonic event in the passage arises through the subsequent har­ monic progression: from the B major in measure 9 (a respelled mediant of the A♭-minor tonic), the passage leads via the parallel B minor in measure 13 (a respelled C♭ minor) to its mediant, D major, in measure 16 (an actual E♭♭ major); the D in measures 16–19 acts

Page 1 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval as leading tone to E♭ (a true F♭♭) in measure 20, which resolves as dominant, not to the departure key of A♭, but rather to the enharmonically related key of B♭♭♭ minor.

Ex. 9.1. Beethoven, funeral march from Sonata op. 26, mm. 1–22.

The enharmonic nonclosure of the passage is visually manifest on the table of tone rela­ tions (Tonverwandtschaftstabelle) or Tonnetz, a potentially boundless, two-dimensional array of tones organized by major thirds and perfect fifths, examples of which are found in the writings of Riemann and others in the late nineteenth and (p. 272) early twentieth centuries.2 Example 9.2a presents the Tonnetz from Riemann's fourth edition MusikLexikon.3 Example 9.2b extracts the open pathway of tones traversed in the Beethoven passage, leading southeast from the initial A♭ tonic to C♭ and E♭♭; E♭♭ as leading tone re­ solves southwest to F♭♭; F♭♭ as dominant resolves one step west to B♭♭♭, a key three steps south of the departure tone.

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From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval

Ex. 9.2. (a) Tonverwandtschaftstabelle from Riemann's Musik-Lexikon, 4th ed., and (b) a pathway traversed by the Beethoven funeral march.

Implicit in example 9.2b is an understanding of the Tonnetz as a metaphorical map or landscape, the pathways through which represent progressions of tones, chords, or keys. This metaphorical view of the Tonnetz has become a familiar trope in neo-Riemannian the­ ory.4 Indeed, harmonic coherence itself, from the neo-Riemannian perspective, is often as­ sociated with the spatial-gestural coherence of progressions as represented on the Ton­ netz. Both ideas find precedent in Riemann's late writings: in his analyses of the Beethoven sonatas, Riemann uses a (p. 273) Tonnetz to discuss and illustrate the enhar­ monic modulation from the tonic A♭ major to E major and back in the slow movement of the Pathétique, op. 13, using the terms Weg (route or pathway) and Pfade (foot path) to describe the course traversed by the music;5 in his Große Kompositionslehre, Riemann ex­ tracts a compact region of the Tonnetz to illustrate the coherent progression of keys in the development of Beethoven's Quartet in B♭ op. 18 no. 6, first movement, observing (p. 274) that the keys all reside in the immediate neighborhood of B♭ (i.e., that they are ad­ jacent to B♭ on the Tonnetz).6 But the metaphorical view of the Tonnetz, particularly as it relates to enharmonic pas­ sages such as Beethoven's, rests on an assumption that the tonal elements of the table are equally tempered, an assumption at odds with the original acoustical function of the Tonnetz as a means to illustrate the derivation and relative frequency of tones in just into­ nation.7 The acoustical function of the Tonnetz is evidenced by the lines (Striche or Kom­ mastriche) placed above or below the letter names on the table to reflect intonational dis­ crepancies among tones in just intonation. Each underline on the table of example 9.2a indicates a syntonic-comma lowering of pitch relative to the like-named tone in the cen­ tral, unlined row; each overline indicates a syntonic-comma raising of pitch relative to the like-named tone in the central, unlined row.8 A consequence of the literal acoustical un­ derstanding of the table—one that is in obvious tension with the spatial analysis of the Page 3 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval Beethoven funeral march in example 9.2b—is that a descent by three major thirds on the just-intoned space of the Tonnetz leads to an arrival tone that is not only nominally dis­ tinct from its departure tone, but one that is intonationally distinct as well: the triple un­ derlined B♭♭♭ is properly intoned an enharmonic diesis—an interval in the ratio 128:125, or approximately 41 cents—higher than the unlined A♭. For Arthur von Oettingen, a strict advocate of just intonation, the acoustical ramifications of example 9.2b posed so great a challenge to his harmonic worldview that he regarded Beethoven's modulation in the funeral march to be a compositional error.9 Riemann, in contrast, saw no fault with the passage, and indeed recognized that enharmonic modula­ tions such as Beethoven's were an inherent feature of the tonal system. Yet the analytical problem that the acoustical view of the Tonnetz posed was nevertheless great enough that Riemann in 1880 could not reconcile the harmonic logic of the passage with its spatial manifestation on the Tonnetz: spatial-harmonic language and imagery are notably absent from Riemann's discussion of the funeral march in the Skizze. Although spatial-harmonic metaphors and Tonnetze do appear in Riemann's early writings, they are typically invoked to discuss harmonic relationships in the abstract, not to describe specific harmonic pro­ gressions in musical practice.10 The present chapter examines how Riemann came to reconcile the literal-acoustical and the spatial-metaphorical views of the Tonnetz—how the Tonnetz evolved from a static ma­ trix of acoustical data in Riemann's early writings to become a traversable map of har­ monic relations in his mature works. The chapter begins by exploring the table from Arthur von Oettingen's Harmoniesystem in dualer Entwickelung (1866), which provided not only the graphic model for the table of relations in Riemann's dissertation, but also shaped its underlying acoustical conception. We then examine how two developments in the subsequent decades were crucial for Riemann's conceptual reformation of the Ton­ netz. First was a general shift away from an acoustical view of the foundations of harmo­ ny, toward the axiomatic perspective of Moritz Hauptmann, who declared the consonance of the major third and perfect fifth as first principles. The emancipation of third and fifth (p. 275) relations from specific defining ratios was significant because it meant that a vari­ ety of actual physical sounds could manifest the generative intervals of the Tonnetz. Second was Riemann's adoption of a symbolic language provided by the mathematician Moritz Wilhelm Drobisch to represent intervals as pathways through the Tonnetz. The symbolic language allowed Riemann to reconceive the Tonnetz, not so much as an array of “things” (i.e., tones with particular intonations), but rather as a network of pathways, combinations of which could bear particular musical meanings. Riemann's changing per­ spective of the table of relations did not simply mirror his general shift from acoustics to psychology as the foundation of harmony. The combinatorial view of interval provided a logically consistent mechanism, structurally independent of any tuning system, that made possible Riemann's relocation of harmonic meaning from the phenomenal-acoustical realm (a aural reckoning of specific frequencies) to the noumenal-psychological realm (a mental reckoning of steps through the Tonnetz).

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From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval

The Table as Matrix: Oettingen and the Acoustical Perspective

Ex. 9.3. The table from Oettingen's Harmoniesystem in dualer Entwickelung.

Example 9.3 reproduces the table of tone relations from Arthur von Oettingen's Har­ moniesystem in dualer Entwickelung.11 Oettingen presents the table in the opening sec­ tion of the work, “Ueber die Tonhöhe. Buchstabentonschrift,” to provide a visual synopsis of tone relations in just intonation. Just intonation was essential to Oettingen's harmonic system and its dual principles of tonicity and phonicity.12 Tonicity refers to the historically well-recognized relationship between tones sharing a common fundamental. The tones C4, E4, and G4, for example, are tonically (p. 276) unified as the fourth, fifth, and sixth par­ tials of the fundamental tone C2. Phonicity, Oettingen's invention, refers to the relation­ ship among tones that are fundamentals to a common upper partial. For example, the tones C4, E♭4, and G4 are phonically unified as fundamentals of the common upper partial G6.13 But whereas tonicity and phonicity stand in a reciprocal logical relationship, the two principles are phenomenologically asymmetric. Because a fundamental tone typically pro­ duces sounding overtones in pure intervallic relationships, tempered major triadic collec­ tions can exist as imperfect images of the pure triads given in the overtone series. Minor triadic collections, by contrast, have no comparable perfect image in the sounding acoustic substrate (such as would be posited by the fictitious theory of undertones). In­ stead, phonicity—to be an objective (i.e., scientifically observable) basis for minor har­ monic relationships—requires that sounding tones be in pure intervallic relationships in order to project a coincident overtone. The pure tuning manifest on the table was conse­ quently necessary to reify the fundamental relationships of Oettingen's harmonic system. Notable in the graphic organization of Oettingen's table is its function as a literal multipli­ cation table. Positive and negative numbers arrayed across the top and left side of the di­ agram represent values of the variables n and m, respectively, in the expression 5m·3n, the integer values of which yield relative vibrational frequencies of tones in pure intona­ tion. Implicit in the expression are the additional factors of 2 needed to keep tones on the table in the same register.14 For example, the entry f in the column n = 3 and row m = −1 has a frequency 5−1 · 33 · 2−2 or 27/5 · 4 = 27/20 relative to the central tone c = 50 · 30 = 1; the entry e̿ in column n = −4 and row m = 2 has a frequency 52 · 3−4 · 22 or 25 · 4/81 = 100/81 relative to c, and so on. Page 5 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval A consequence of the explicit multiplication matrix format of Oettingen's table is that its internal spatial relations are subordinated to its external organization: motions between entries are necessarily mediated by the exponents that define the content of the rows and columns. In his initial exposition of the table, Oettingen observes the geometric form of certain intervals and collections on the table, noting, for instance, that intervals of the mi­ nor third are formed along diagonals directed upward and to the left, that major sevenths are formed along diagonals directed upward and to the right, and that pure triads are manifest by adjacent trios of tones in “right-triangular” formations, above and right for major, below and left for minor triads.15 But in subsequent discussions of intervals, tone collections, and harmonic progressions—both in theory and practice—Oettingen does not make direct reference to their spatial manifestation on the table, but rather, references and illustrates intervals, collections, and progressions through their Buchstabenton­ schriften—the intonationally determined letter names furnished by the table. Further, although Oettingen uses the table to demonstrate the derivation and notation of intervals and tones, he does not use the table directly to describe or explain the related­ ness, behavior, or function of tones in musical context. Rather, discussions of tones and chords refer to the table only indirectly through their Buchstabentonschriften, tokens for the relative frequencies of those tones expressed as factors of 3 and 5. That is, the inter­ vallic transformations among objects on the (p. 277) table constitute transformations on relative frequencies, not on their signifying pitch classes: they involve multiplication or di­ vision of those frequencies by 3, 5, or their powers rather than transpositions in pitch space.16 Although the layout of Oettingen's table privileges relationships that coincide with what would later be understood to be characteristic real or potential motions among tones, the table for Oettingen reflects static relations among tones, tones conceived as to­ kens of unique prime factorizations.17

Ex. 9.4. The table from Riemann's dissertation, “Ue­ ber das musikalische Hören.”

Page 6 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval Example 9.4 reproduces first the table of tone relations in Riemann's writings, from his 1873 dissertation, “Ueber das musikalische Hören.”18 The dissertation sought to synthe­ size Oettingen's duality with Helmholtz's physiological researches, placing Oettingen's dualism upon a physiological foundation. Rather than locating the source of triadic conso­ nance within the purely physical realm (i.e., in the coincidence of sounding overtones or fundamentals), Riemann postulated the dual foundation of triadic consonance to arise from the interaction of sound with the hearing organ. Drawing on Helmholtz's observa­ tion that cilia on the basilar membrane of the inner ear would become sympathetically ex­ cited by pitched sounds at spatial locations corresponding not only to the fundamental frequency and its upper partials, but also at locations corresponding to aliquot frequen­ cies of that fundamental, Riemann hypothesized that undertones, tones whose frequen­ cies were 1/2, 1/3, 1/4, 1/5, and so on, of a given fundamental, although not emitted as part of the periodic air disturbances caused by a vibrating body, were nonetheless real, objective phenomena within our auditory mechanism.19 Although his explanation differed from Oettingen's, the consequence (p. 278) was the same: Riemann defined consonance acoustically, and consonant relations assumed numerical form: . . . consonance is the sounding together of tones (in particular Klänge, since they mostly always bear overtones and undertones) which belong to one and the same Primklang, where we understand by Primklang the complex of a fundamental tone and its overtones of the first order;…we understand by overtones of the first order only 2, 3, 5 and their octaves.20 Relatedness or distance in this view—as in Oettingen's—is measured within the acoustical space of the overtone (or undertone) series. Overtones of the first order are those over­ tones (or undertones) represented by the prime numbers 2, 3, or 5 or their octaves; all others are understood as overtones of overtones. Triads are likewise defined according to prime number ratios representing relative frequencies: major triads are defined as collec­ tions of the form 1 · 2m : 3 · 2n : 5 · 2t; minor triads as collections of the form 1 · 2m : 1/3 ·2n : 1/5 · 2t. Although there are minor notational differences between Oettingen's and Riemann's ta­ bles—Riemann, for example, reverses Oettingen's convention with respect to Striche 21— the table in Riemann's dissertation retains the matrix-like format and conception of Oettingen's. Numbers, not tied to any explicit variable powers, still run across the top, bottom and right side of Riemann's table, just as Oettingen ran the powers of 3 and 5 along the top and side of his. That Riemann's numbers do not include negatives implies either that they represent positive increments along either an overtone or an undertone series, or else that they simply represent fifths and thirds in two directions. Nonetheless, the numbers signify an external organization that determines the inner content of the ta­ ble.

Page 7 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval

Riemann's Changing Perspective: Toward the Table as Map A shift away from an acoustical definition of consonance becomes evident a mere two years after the completion of his dissertation, in Riemann's Die Hülfsmittel der Modula­ tionslehre (1875). Absent from Hülfsmittel is all discussion of acoustics, prime numbers, or the overtone series. Instead, Riemann declares octave equivalence as an axiom, and defines a Klang as simply “the unified statement of several (three) different [i.e., nonoc­ tave-equivalent] tones (triad), which fuse through the unifying, simple relation to one of the tones, the ground- or Hauptton.”22 Riemann then defines the only two possible “sim­ ple relations”: The possible simple relationships between two tones [outside of identity (Einklang, octave), which as a relation belongs not to different, but rather to equivalent tones] are the fifth relation (Quintverhältniß) and third relation (p. 279) (Terzver­ hältniß); all other musically possible relationships (intervals) are combinations or powers of these two.23 Riemann's axiomatic treatment of the third and fifth (and octave) recalls Hauptmann, who declares three directly intelligible intervals in Die Natur der Harmonik und der Metrik (1853): the octave, the fifth, and the (major) third.24 One can see even more clearly the change from a physico-acoustical to an axiomatic perspective in Riemann's subsequent definition of triads. Whereas in the dissertation, triads are defined according to prime number ratios representing relative frequencies, in Hülfsmittel, triads are defined simply as collections in which a tone of reference (the Hauptton) is joined with its upper (major) third and fifth in major or a tone with its lower (major) third and fifth in minor. The Hauptton in Hülfsmittel is not a generator of the other triadic tones. Neither do preor­ dained mathematical ratios define the intervals between the referent Hauptton and the other tones. An even more pronounced shift away from acoustics is evident in Riemann's “Die Natur der Harmonik” (1882), a short treatise that presents a brief teleological history of harmo­ ny from Zarlino to his own theories.25 Riemann therein situates his own work in relation to the writings of Helmholtz, Hauptmann, and Oettingen. Interestingly, Riemann's discus­ sion of prime number ratios and their reciprocals as a means to generate triads is relegat­ ed to the historical discussion of Zarlino. In doing so, Riemann distances himself, by over 300 years, from the very methodologies that were so prominent in his dissertation less than a decade earlier. Riemann's account of his immediate theoretical predecessors places greatest emphasis upon Hauptmann, to whom Riemann attributes the conception of interval that Riemann had essentially presented in Hülfsmittel:

Page 8 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval For Hauptmann, the minor third, the fourth, the sixth and all other intervals do not exist; they are not independently existing, independently significant entities, but rather only products, combinations of the fundamental concepts: octave, fifth and [major] third.26 Composite intervals are not products of frequency ratios, but rather combinations of the axiomatically defined fundamental intervals. Moreover, Riemann explicitly recognizes the noumenal (as opposed to phenomenal acoustical) nature of tonal relations in “Die Natur” in a critique of Helmholtz, observing [t]he error which Helmholtz made is now easy to recognize; he sought to explain from the nature of sounding bodies concepts which could only be explained from the nature of a perceiving mind. Consonance and dissonance are musical con­ cepts, not determinate forms of wave motion.27 With this realization, that musical concepts belonged to a mental realm independent of the acoustical substrate through which they are made manifest, Riemann set the stage for the meaning of intervals on the table to become unmoored from their acoustical founda­ tions. One manifestation of this change is evident in Riemann's labels for the rows and columns of his table. In the tables from his Musik-Lexikon (the fourth edition table is shown in example 9.2a), Riemann lists rows according to Oberterzen and Unterterzen and lists columns according to Oberquinten and Unterquinten relative to the central tone C. In contrast to Oettingen's table, where rows and columns are labeled by exponents that act on relative frequencies, the Terzen and Quinten on Riemann's table are intervals that act on tones themselves, tones which constitute the content of the table. Moreover, by treat­ ing Quintverhältniß and Terzverhältniß as axioms rather than particular ratios, Riemann opened the way for the interpretation of those relations as mental and not physical struc­ tures, and for the interpretation of the Verwandtschaftstabelle as a map of noumenal rela­ tions rather than simply a table of phenomenal properties. (p. 280)

Tonbestimmung, Drobisch, and the Combinato­ rial Conception of Interval Riemann presented the Tonverwandtschaftstabelle in “Ueber das musikalische Hören,” not as part of the text itself, but in an extended footnote concerning “the error that a C♯ should always be higher than a D♭.”28 Riemann sought to demonstrate that the leading tone of some pitch is properly intoned lower than the lowered form of that pitch itself (e.g., that C♯ in the key of D, for example, should be intoned lower than D♭), arguing against performers who would automatically raise sharp pitches or lower flats toward their tones of resolution. Riemann uses the table to explain the precise derivation of the relative frequencies of 133 musically possible tones within an octave.

Page 9 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval In the footnote, Riemann does not explicitly demonstrate how to calculate the intervals between tones using the table. Instead, Riemann simply notes that tones within each row on the table ascend and descend by pure perfect fifth and that tones within columns as­ cend and descend by pure major third, differing in pitch from like-named tones in the central row by syntonic commas. Riemann then provides a table listing 133 distinct inter­ vals formed between C and various tones. A portion of the table is shown as example 9.5. Riemann arranges the list in order of ascending relative pitch, grouping tones according to the [keyboard] key upon which it would be played. The excerpt in example 9.5 includes tones that correspond to C♯/D♭ and D, expressing each as base 2 logarithms and, for the more familiar intervals, as ratios.29 For comparison, Riemann also lists values for inter­ vals among tones in equal temperament, labeled the left of each group, their location marked among tones to the right by dotted lines. The segregation reveals how, for Rie­ mann, the equal-tempered tones are of a different species than the nameable tones, a species that cannot embody musical meanings as do the others, underscoring the impor­ tance Riemann, in the dissertation, places upon intonation as a signifier of musical mean­ ing.

Ex. 9.5. An excerpt of the Tonbestimmungstabelle from Riemann's dissertation.

Using the list, Riemann is able to show that although C♯ as the seventh fifth of C (cis = .094734 = 2187:2048) is higher than D♭ five fifths below C (des = .075190 = (p. 281)

256:243), the reverse is true of the more practically employed third-related forms—C♯ a pure major third above A (cis = .076813 = 135:128) is lower than D♭ a pure major third below F (des = .093111 = 16:15).

Page 10 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval

Ex. 9.6. An excerpt of the Tonbestimmungstabelle from Riemann's Musik-Lexikon, 3rd ed.

The list of tones from the dissertation footnote reappears with minor changes in the first three editions of Riemann's Musik-Lexikon under the article “Tonbestimmung” (tone des­ ignation or determination, the fixing of pitch for tones relative to some reference tone). Example 9.6 presents the section of the Tonbestimmungstabelle from the third edition of the Lexikon (1887) that corresponds to the excerpt in example 9.5. Riemann includes a few more distant tones in the Lexikon table. Comparison of examples 9.5 and 9.6 reveals an E♭♭♭ and a D♭♭ not (p. 282) included in the dissertation table. Riemann also includes tones corresponding to higher partials in the overtone series, marked with asterisks: on example 9.6, the seventeenth overtone is listed as *cis(des). And although Riemann con­ tinues to provide comparative logarithmic entries for tones in equal temperament, he no longer divides the table into twelve groups corresponding to the keys on a piano. Rather, the tables simply list tones consecutively in ascending pitch order.

Page 11 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval

Ex. 9.7. The expanded Tonbestimmungstabelle from Riemann's Musik-Lexikon, 4th ed.

However, beginning with the fourth edition of the Musik-Lexikon (1894), Riemann made a number of significant revisions to the Tonbestimmungstabelle. Example 9.7 presents a section of the revised Tonbestimmungstabelle from the fourth edition Musik-Lexikon. The newer table lists ratios for all, not merely the familiar, intervals (as string lengths rather than as relative frequencies). Intervals are measured not only in base 2 logarithms, but also in base 10 and base

(values of the latter multiplied by 100 yield the familiar

interval measurement in cents). Riemann (p. 283) (p. 284) still presents comparative values for tones in equal temperament, but also gives comparative values for tones in a 53-tone equal-division tuning. But most significant, Riemann includes a “degree of relatedness” (Verwandtschaftsgrad) for all tones relative to the reference tone c, shown in the second column from the left on example 9.7. The degree of relatedness measures the interval between a given tone and the reference tone, reckoned as a combination of perfect fifths, major thirds, and octaves. Riemann, moreover, adopts a symbolic representation for the components of each inter­ val: each ascending fifth is symbolized Q (for Quint), each descending fifth by 1/Q; each ascending major third is symbolized T (for Terz), each descending major third by 1/T; each ascending octave is symbolized O, each descending octave by 1/O. For instance, the first entry on example 9.7, d, has the Verwandtschaftsgrad TO/2Q, meaning to get from c to d, ascend by a major third and an octave, and descend by two perfect fifths. Likewise to get from c to cisis (C𝄪), the next entry on the example, with the Verwandtschaftsgrad 2T6Q/4O, ascend by two major thirds, six perfect fifths, and descend by four octaves. The Verwandtschaftsgrad made explicit the previously implicit connection between the Ton­ verwandtschaftstabelle and the Tonbestimmungstabelle, correlating the degree of har­ monic relatedness among tones with the length of paths between those tones on the Ton­ verwandtschaftstabelle. Page 12 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval The symbolic representation of intervals as combinations of fifths and major thirds was not Riemann's invention, but rather was first introduced in the writings of the Leipzig mathematician Moritz Wilhelm Drobisch (1802–1896), who authored several books and monographs on the mathematics of tuning and temperament. In his 1852 Über musikalis­ che Tonbestimmung und Temperatur, Drobisch introduced the symbols Q and T to repre­ sent the pure intervals of the perfect fifth and major third (3/2 and 5/4, respectively), il­ lustrating how they, in combination with the powers of 2, could be used to express fortytwo distinct intervals in pure intonation.30 For example, Drobisch expresses the whole tone as Q2/2, the major sixth as 2 · T/Q, the lesser augmented fourth as 2 · T2/Q2, the greater limma as Q3 · T/22, and so on. Yet beyond the symbolic notation itself, two aspects of Drobisch's work had particular significance for Riemann's reconception of the Tonnetz: the logarithmic interpretation of those symbols and their potential to represent true vari­ ables, rather than act simply as substitutions for pure intervals.31 In his discussion of pure intonation, Drobisch introduced an additional notational conven­ tion, using the lowercase letters q and t to signify the logarithmic values of the pure fifth and major third (i.e., the logarithmic equivalents of the rational Q and T).32 The logarith­ mic notation allowed Drobisch to express composite intervals, formerly products of Q, T, and 2 and their inverses, now through addition and subtraction of q, t, and units. For ex­ ample, the major second, formerly symbolized by the rational expression Q2/2, could be expressed using logarithmic notation as 2q − 1; the rational expression for the major sixth 2 · T/Q could be represented using logarithmic expression 1 − q + t.33 Curiously, Riemann's Verwandtschaftsgrad represents a hybrid of Drobisch's rational and logarithmic notation. Riemann adopts from Drobisch's rational (p. 285) notation its upper­ case symbology and its use of reciprocal symbols to express descending intervals (e.g., subtracting a major third through multiplication by 1/T). But Riemann reverts to logarith­ mic notation to express symbolic combinations of intervals within the numerator or divi­ sor: implicit in Riemann's notation 3Q is the addition of three fifths (Q + Q + Q) rather than the threefold multiplication of Q (Q · Q · Q = Q3 in Drobisch's notation). Whether the change was inadvertent or intentional, Riemann's quasi-logarithmic notation reflects cer­ tain familiar intuitions about intervals as things extended in Cartesian space, what Lewin calls “transposition-as-characteristic-motion-through-space.”34 That is, it more natural to spatially conceive of an interval such as 3Q as additive motions through the Tonnetz (move east by a fifth, then another, then another) than multiplicatively (multiply by a fifth? then by another? and by another?).35 Even more significant for Riemann's developing music psychology were Drobisch's sym­ bolic conventions as applied in his discussion of equal-division temperaments. The 53tone equal division included by Riemann on the revised Tonbestimmungstabelle was only one of several equal-division tuning systems explored by Drobisch in Über musikalische Tonbestimmung; others included 19-, 31-, 43-, and 74-tone equal divisions of the octave.36 Throughout the discussion, Drobisch uses the symbol q to represent not the logarithmic value of the pure fifth, but rather the general value of a tempered fifth in any given divi­ sion. For example, q = 7/12 or seven semitones in the familiar 12-tone division of the oc­ Page 13 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval tave, but q could also equal 11 parts of an octave in a 19-tone division (q = 11/19), or 24 parts of an octave in a 41-tone division (q = 24/41), or 31 parts of an octave in a 53-tone division (q = 31/53), and so on. Yet regardless of the tuning system in which it is defined, the variably intoned q could nevertheless participate in the same combinatorial intervallic expressions as a purely intoned q. That is, the expression 2q – 1 (two fifths minus an oc­ tave) still formally defines a whole tone, regardless of whether q is interpreted as log2(3/2) or as 31 parts of a 53-tone division of the octave.37 Thus, in Drobisch's treatise, Riemann had a model, a formal mechanism, that could allow him to reconcile the literal and metaphorical understandings of the Tonnetz, since an entity such as q (or in Riemann's case, Q) reckoned as a step along a pathway through the net, could simultane­ ously represent a just-intoned interval and a tempered interval. The symbols not only rei­ fied the individual steps of pathways among tones of the table, but because a symbol such a Q could instantiate a pure as well as a tempered fifth, the table became an instrument to mediate between the outer, tempered world of sounding phenomena, and an inner, noumenal world of distinct and boundless musical meanings. That the mechanism of Drobisch's symbolic language had an increasing resonance for Riemann and his development of a psychological foundation for harmony during the 1890s is suggested by Riemann's entry “Drobisch” in his Musik-Lexikon, and in particular, by a revision made in the fifth edition. After listing a selection of Drobisch's writings on the mathematics of tuning and temperament, Riemann writes in the fourth edition Lexikon (1894), “D[robisch], earlier an advocate of the twelve-tone [pitch] system, in his last writing, adopted in principle the (p. 286) viewpoint of Helmholtz.”38 In the fifth and subsequent editions, the beginning of the entry is largely unchanged, but Riemann rewrites the corresponding sentence to read, “D[robisch], earlier an advocate—based on Herbartian philosophy—of the twelve-tone [pitch] system, recognized in his last writing the importance, in principle, of [the system of] pure intonation.”39 While the overall meaning of the two sentences is largely the same, the differences are telling. Johann Friedrich Herbart (1776–1841), to whom Riemann refers in the revised sentence, was a philosopher who, in the early nineteenth century, proposed a mechanistic theory of the mind.40 Analogously to physical bodies in space, Herbart posited mental “bodies,” ba­ sic units of mental activity, which he designated “Vorstellungen,” the motions and interac­ tions of which, like physical bodies, could be described and predicted mathematically. Moreover, Herbart “argued that because it is our thoughts that determine our behavior, not only can an independent mathematical account be used to predict the existence, mag­ nitude, and duration of mental events but also that physical events in the brain can be thought of as secondary, from a theoretical point of view, to the mental events to which those physical brain events correspond.”41 By deleting the reference to Helmholtz in the “Drobisch” Lexikon article, Riemann deem­ phasized the physical-physiological aspects of Drobisch's work (the aspect, concerning the physical substrate of sound vibrations acting upon the sense organ, that was central to Riemann's dissertation). By invoking Herbart in the revision, Riemann draws attention to the psychological aspect of musical meaning, a change that corresponds to Riemann's Page 14 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval own shift toward a psychological foundation for music theory. That is, Riemann projects onto Drobisch the very ideals he valued in the late 1890s—a rejection of the reliance on acoustical physics and physiology represented by Helmholtz (concerned with “determi­ nate forms of wave motion”), and an embrace of mental mathematics, the activities of a perceiving mind, embodied by Herbart's mathematical psychology.42 The synthesis of Herbart's mechanistic theory of mind and a path-combinatorial concep­ tion of the Tonnetz culminated in Riemann's “Ideen zu einer ‘Lehre von den Tonvorstellun­ gen.’ ”43 Central to Riemann's mature theory of Tonvorstellungen is the Herbartian notion that physical events in the brain (i.e., the result of physiological interactions of the hear­ ing organ with the acoustical substrate) are secondary to the logical activities of the brain —activities that create and impose musical interpretations upon those acoustical signals.44 Those interpretations (e.g., the recognition that a tone is a leading tone, or that a particular minor third dyad is the upper third of a major triad, or that a certain major triad is a dominant) are functions of things-as-perceived, as opposed to raw acoustical da­ ta giving things-in-themselves.45 Tonvorstellung is a mental activity that, although influ­ enced by acoustical data, is largely independent of that data.

Ex. 9.8. The table from Riemann's “Ideen.”

In “Ideen,” Riemann recognized that deviations in intonation had little or no bearing on a listener's understanding of the logic and function of tones in a musical phrase. Instead, Riemann located the functional interpretation of musical tones in the path derivation of those tones on the Tonnetz. Discussing the traditional (p. 287) acoustical distinction be­ tween D as fifth of the dominant in C major and D as the minor third below the subdomi­ nant, Riemann observed the following: Our imagination knows nothing of the intonational difference between d and d, but rather equates both, imagining d as the lower fifth of a and yet at the same time also as the upper fifth of g. This enharmonic identification of acoustical val­ ues differing by syntonic commas is absolutely indispensable for our hearing of music.46 Riemann explicitly recognizes that elements on the Tonnetz are not distinct because of their acoustical properties, but rather that they assume distinct meanings because of the distinct mental pathways through which they are conceived. The Tonnetz of “Ideen,” Page 15 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval shown as example 9.8, is consequently a map not of acoustically determined tones, but of logically conceivable tonal relations reckoned as pathways therein, pathways expressed symbolically through combinations of the fundamental relations Q and T: d and d repre­ sent the same tone acoustically, but have distinct functions or musical meanings in rela­ tion to the tone c, manifest in the distinct pathways 2Q and T/2Q through which they can be derived.47 Evident in Riemann's discussion is that the Striche on the Tonnetz of “Ideen” are no longer signifiers of acoustical discrepancies, but rather signifiers of distinct path deriva­ tions. This change in perspective about the meaning of the Striche is strikingly apparent when one compares a passage from the Musik-Lexikon article “Quinttöne und Terztöne” with its parallel passage in “Ideen.” The passages explain the significance of the table of tone relations for reckoning intervals. In the Lexikon, Riemann emphasizes the actual acoustical values distinguishing intervals among different forms of a like-named tone: The lines (comma-lines) under the letter names indicate a lowering by 80:81 rela­ tive to the same-named tone reached by fifths from [the central tone] c; the lines above the letter names indicate a raising by that same interval. So for (p. 288) ex­ ample, the most closely related f  𝄪 is attained through three [major] third steps and a fifth step and is three commas lower than f  𝄪 in the horizontal row from c (thirteenth fifths).48 Although Riemann recognizes the importance of the pathway of fifths and thirds in the Lexikon, his example is offered to explain the Striche, and to emphasize their importance as acoustical markers. By contrast, Riemann, in an analogous passage in “Ideen” (his ex­ position of the table and its features), calls forth the very same example, but is complete­ ly silent on intonational matters: The table readily furnishes the determination of any interval according to fifth and [major] third steps and discloses for each multiply-determinable tone the simplest, nearest-lying derivation, e.g., [the derivation] for f 𝄪 as Q3T (fifth of the third [ma­ jor] third or third [major] third of the fifth).49 Although the tone to which Riemann refers is the triple-underlined fisis on the table, Rie­ mann does not write or make reference to those underlines in the text, but emphasizes in­ stead the derivation of the tone via a pathway. The Striche of the “Ideen” table are simply markers that distinguish alternatively derivable species (i.e., fisis, fisis, fisis) of the one genus f 𝄪. * * * Riemann's understanding of temperament and enharmonic equivalence on the Tonnetz of “Ideen,” though it bears similarities, should not be confused with the modern (i.e., neoRiemannian) perspective. Neo-Riemannian theory was founded on the assumption of a pitch space comprising twelve equal-tempered pitch classes, an assumption that revealed a number of interesting group-theoretical insights about triadic relations in certain latePage 16 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval nineteenth-century repertoires.50 But the assumption of equal temperament also induced a particular geometrical form upon the Tonnetz. The neo-Riemannian, or conformed, Ton­ netz resides not on a boundless plane, but rather on the surface of a torus, reflecting the acoustical identity of enharmonically distinct tones through the identity of their locations in space.51 Although Riemann recognized the indispensability of enharmonic identifica­ tion for our musical imagination, he did not embrace a neo-Riemannian view of tonal pitch-class space and its ramifications for the Tonnetz. While he could forgive syntoniccomma discrepancies between like-named tones, Riemann in “Ideen” was unwilling to completely give up distinctions between enharmonically related keys such as C♯ and D♭. As if to compensate for the loss of just intonation, Riemann manifests in his late writings an increasing concern for absolute hearing and key character, attributing distinct aesthet­ ic qualities such as darkness or brightness to flat or sharp key areas respectively.52 But neither should Riemann's perspective be regarded teleologically, as an imperfect at­ tempt to approach the neo-Riemannian perspective. Instead, Riemann found a third way. The path-combinatorial conception of interval in “Ideen” allowed Riemann to preserve the appearances of a just-intoned Tonnetz. He recognized that its elements were not an infi­ nite number of acoustically distinct tones, but rather a finite collection of tones that could assume a boundless variety of distinct musical (p. 289) meanings based on their derivation as pathways through the Tonnetz. As neo-Riemannian theory has evolved, so too have its practitioners begun appreciate the analytical value of the path-combinatorial perspective as a means to preserve enharmonic differences among tones, chords and keys.53

Notes: (1.) Leipzig: Breitkopf (1880), 82. (2.) The term Tonverwandtschaftstabelle is used by Riemann. Tongewebe is used by Alfred Jonquière in Grundriss der musikalischen Akustik (Leipzig: Th. Grieben, 1898). The term Tonnetz was apparently introduced by Renate Imig in Systeme der Funktionsbezeichnung in den Harmonielehren seit Hugo Riemann (Düsseldorf: Gesellschaft zu Förderung der systematischen Musikwissenschaft, 1970) and has since been generally adopted in the neo-Riemannian literature. (3.) Leipzig: Max Hesse (1894), 857. The identical diagram appears in all subsequent edi­ tions during Riemann's lifetime. (4.) Neo-Riemannian analytical writings that explore the spatial-harmonic metaphor of the Tonnetz include Richard Cohn's contribution to the present volume; Daniel Harrison, “Nonconformist Notions of Nineteenth-Century Enharmonicism,” Music Analysis 21.2 (2002): 115–160; and David Lewin, “Some Problems and Resources of Music Theory,” Journal of Music Theory Pedagogy 5 (1991): 111–132. Harrison in particular is concerned with enharmonic key relations and their Tonnetz representation in late-nineteenth-centu­ ry musical practice.

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From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval (5.) L. van Beethovens sämtliche Klavier-Solosonaten: Ästhetik und formal-technische Analyse mit historischen Notizen, 3 vols., 2nd ed. (Berlin: Max Hesse, 1919), 2: 24. (6.) Große Kompositionslehre. Erster Band: Der homophone Satz (Melodielehre und Har­ monielehre), (Berlin and Stuttgart: Spemann 1902), 480. “The keys G minor, F major, E♭ major, D♭ major and F major, through which the development of Beethoven's B♭-major Quartet passes, are naturally found in the immediate neighborhood of B♭ major.” (“Die Tonarten G-moll, F-dur, Es-dur, Des-dur, F-dur, welche die Durchführung des Beethoven­ schen B-dur-Quartetts berührte, sind natürlich in der nächsten Nähe von B-dur zu suchen.”) Riemann's illustration of keys is reproduced as example 7.1 in Alexander Rehding's contribution to the present volume. (7.) Just or pure intonation refers to a tuning system in which all thirds and fifths (and consequently all triads) are acoustically pure (i.e., all 3:2 fifths, 5:4 major thirds, 6:5 mi­ nor thirds). (8.) A syntonic comma is the small interval in the ratio 81:80 (approximately 21.5 cents) by which a Pythagorean third (equivalent to two 9:8 whole tones, (9/8)2 = 81/64) exceeds a pure major third (5/4 = 80/64). (9.) Harmoniesystem in dualer Entwickelung (Dorpat: Glaser, 1866), 143. (10.) Notably, in a concluding passage to his Musikalische Syntaxis (Leipzig: Breitkopf und Härtel, 1877), Riemann described the manifold combinations of harmony as path­ ways through a landscape, observing that “one cannot assess the realm of harmony in its entirety by walking across it step by step, but rather, only by flying over it, from a bird's perspective, can one survey it.” (“. . . man kann das Gebiet der Harmonik nicht Schritt für Schritt abgehen, sondern nur überfliegen, aus der Vogelperspektive über­ schauen.” [120]). A still earlier precursor to the Skizze and Musikalische Syntaxis, Riemann's Hülfsmittel der Modulationslehre (Cassel: Luckhardt, 1875), includes a Ton­ netz, but it is used to illustrate key relations in the abstract, not with direct reference to actual musical works or passages. (11.) Harmoniesysteme, 15. (12.) Oettingen's harmonic theories, and in particular his principles or tonicity and phonicity, are discussed in Daniel Harrison, Harmonic Function in Chromatic Music (Chicago: University of Chicago Press, 1994), 242–251. (13.) Riemann illustrates the tonic fundamentals and phonic overtones of major and mi­ nor triads in “Das Problem des harmonischen Dualismus,” translated by Ian Bent in the present volume. (14.) Oettingen uses Helmholtzian registral notation in the Harmoniesystem: C = C2 (cello C), c = C3 (viola C), c´ = C4 (middle C), and so on. Thus, all tones on the table, written in lowercase letters, reside in the small octave (from C3 to B3).

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From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval (15.) Harmoniesystem, 15–16. Oettingen also explicitly illustrates the location of tones in the fully diminished seventh chord using the table (Harmoniesystem, 265). (16.) An example of the difference in the two conceptions is provided by David Lewin in Generalized Musical Intervals and Transformations (New York: Oxford University Press, 2007), 18–19. Lewin considers the interval of the tritone measured as a “chain of musical intuitions,” that is, as a path of intervals in pitch space, in contrast to its “most ‘natural’ mathematical factorization,” as the product of 2–5, 32, and 5. (17.) The periodic table of elements provides an analogy: although elements on the peri­ odic table stand in various relationships with their neighbors (e.g., in numbers and arrangement of electrons), it does not imply transformations (i.e., transmutations) be­ tween those elements. (18.) (Dr. Phil. diss., Göttingen University, 1873. Published Leipzig: Andrä, 1874), 29. The dissertation was republished under the new title, Musikalische Logik: Hauptzüge der physiologischen und psychologischen Begründung unseres Musiksystems (Leipzig: Kahnt, 1874). Riemann introduces the diagram as “a table of relations designed by A. v. Oettingen” (einer von A. v. Oettingen entworfenen Verwandtschaftstabelle. 29). (19.) “Ueber das musikalische Hören,” 12–13. On Riemann's complicated relationship with and ever-changing theories of undertones, see Alexander Rehding's Hugo Riemann and the Birth of Modern Musical Thought (Cambridge: Cambridge University Press, 2003). (20.) “Ueber das musikalische Hören,” 17. “. . . Consonanz ist der Zumammenklang von Tönen (eigentlich Klängen, weil sie meist Obertöne und immer Untertöne mit sich führen), die einem und demselben Primklange angehören, wo wir dann unter Primklang den Complex des Grundtones mit seinen Obertönen erster Ordnung verstehen;…so ver­ stehen wir unter Obertönen erster Ordnung also nur die 2, 3, 5 und ihre Oktaven.” (21.) Riemann retains the conventions of Kommastriche from the dissertation in most of his subsequent writings, reverting to Oettingen's only in his Große Kompositionslehre. Kevin Mooney discusses Riemann's various notational conventions for indicating syntonic comma distinctions in “The ‘Table of Relations’ and Music Psychology in Hugo Riemann's Harmonic Theory,” (Ph.D. diss., Columbia University, 1996), 148. Also unlike Oettingen, Riemann notates all tones on the dissertation table in register relative to the central, mid­ dle C (c´ = C4), continually (and literally) ascending by perfect fifth to the right, and by pure major third upward. Consequently, there is a notated registral shift upward to the northeast (culminating at d𝄪´´´´ = D𝄪7 in the upper right corner) and downward to the southwest (descending to ”E♭♭♭ = E♭♭♭0 in the lower left corner). (22.) Hülfsmittel, 3. “Klang ist die vereinte Angabe mehrerer (dreier) verschiedener Töne (Dreiklang), welche durch einheitliche einfache Beziehung auf einen der Töne, den Grund- oder Hauptton,…verschmelzen.”

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From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval (23.) Hülfsmittel, 3. “Die möglichen einfachen Beziehungen zwischen zwei Tönen sind [außer denen der Identität (Einklang, Octav), welche als Verhältniß nicht verschiedener, sondern gleicher Töne nicht hergehört] das Quintverhältniß und das Terzverhältniß; alle andern musikalisch möglichen Verhältnisse (Intervalle) sind Combinationen oder Poten­ zierung dieser beiden.” (24.) Die Natur der Harmonik und der Metrik, (Leipzig: Breitkopf und Härtel, 1853), 21. “Es giebt drei direkt verständliche Intervalle: I. die Oktav, II. die Quint, III. die (große) Terz.” (25.) Sammlung musikalische Vorträge 4.40 (Leipzig: Breitkopf und Härtel, 1882), 157– 190. Benjamin Steege presents an annotated translation of the work in chapter 2 of the present volume. The translations presented here are mine. (26.) “Die Natur,” 182. “Die kleine Terz, die Quarte, die Sexte und alle anderen Intervalle existieren für Hauptmann nicht; sie sind nicht für sich bestehende, für sich bedeutsame Gebilde, sondern nur Produkte, Kombinationen der Grundbegriffe: Oktav, Quint, und Terz.” (27.) “Die Natur,” 186. “Der Fehler, den Helmholtz gemacht, ist jetzt leicht zu erkennen; er suchte Begriffe aus der Natur der tönenden Körper zu erklären, welche nur aus der Natur percipirenden Geistes erklärt werden können. Konsonanz und Dissonanz sind musikalische Begriffe, nicht aber bestimmte Formen der Schallbewegung.” (28.) “Ueber das musikalische Hören,” 29. “. . . der Irrthum, dass ein Cis jederzeit höher sei als ein Des.” The footnote extends for eleven pages. (29.) The logarithm of any number y to the base a is a number x such that ax = y. For exam­ ple, one can express the interval of the pure major third (5/4 or its decimal equivalent 1.25) in base 2 logarithms as 0.321928 since 20.321928 = 1.25. Logarithmic notation allows one to express intervallic distances linearly: equal logarithmic increments represent equal intervals. Base 2 logarithms for intervals within an octave (whose decimal values range between 1 and 2) are numbers that range between 0 and 1 (because 20 = 1 and 21 = 2). (30.) Über musikalische Tonbestimmung und Temperatur (Leipzig: Wiedmann, 1852), 17. Drobisch uses the notation as well in his Nachträge zur Theorie der Musikalischen Ton­ verhaltnisse (Leipzig: Hirzel, 1855). (31.) Riemann had earlier employed the Q/T notation to substitute for literal acoustical values in just intonation, specifically crediting Drobisch for the symbology, in his article, “Eine musikalische Tagesfrage” (Musikalisches Wochenblatt 13 [1882], 465–466, 477– 479, 489–490, 501–502, 513–515, 529–531, 553–555, 569–570, 593–594, 617–619). The article was later retitled “Das chromatische Tonsystem,” and republished in Präludien und Studien: Gesammelte Aufsätze zur Aesthetik, Theorie, und Geschichte der Musik, 3 vols. Leipzig: H. Seemann (1895–1905), 1: 183–219. Page 20 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval (32.) Über musikalische Tonbestimmung, 28–29. (33.) Recall from note 29 that multiplication (division) by 2 corresponds in base 2 loga­ rithms to addition (subtraction) of 1. (34.) Generalized Musical Intervals, xxxi. (35.) The logarithmic misreading of Drobisch's Q/T notation was already evident in “Eine musikalische Tagesfrage.” (36.) Drobisch discusses equal-division temperaments in chapter IV, “Von den verschiede­ nen Arten der gleichschwebenden Temperatur (Über musikalische Tonbestimmung, 63– 95). Drobisch advocated a 53-tone equal division tuning for its exceptionally close approx­ imation to just intonation. For a modern discussion of equal-division temperaments, see Easley Blackwood, The Structure of Recognizable Diatonic Tunings (Princeton: Princeton University Press, 1986). (37.) Because Drobisch was concerned exclusively with Pythagorean (i.e., fifth-generated) tone systems in his equal division temperaments, intervals in those systems were ex­ pressed exclusively in terms of q. However, there is no conceptual obstacle to including a variable t, defined independently or in terms of q, to form composite intervallic expres­ sions. (38.) “D., früher prinzipieller Verfechter des Zwölfhalbtonsystems, hat sich in der letztern Schrift im Prinzip der Anschauungsweise von Helmholtz angeschlossen.” Musik-Lexikon, 4th ed. (Leipzig: Max Hesse, 1894), 242–243. (39.) “D., früher von der Herbartschen Philosophie aus ein Verfechter des Zwölfhalbton­ systems, erkannte in der letzten Schrift die prinzipielle Bedeutung der ‘reinen Stimmung’ an.” Musik-Lexikon, 5th ed. (Leipzig: Max Hesse, 1900), 272. (40.) Herbart's most important treatise on mathematical psychology is his two-part Psy­ chologie als Wissenschaft: neu gegründet auf Erfahrung, Metaphysik und Mathematik (Königsberg: Unzer, 1824–1825). Herbart's contributions to mathematical psychology are examined in Geert-Jan A. Boudewijnse, David J. Murray, and Christina A. Bandomir, “Herbart's Mathematical Psychology,” History of Psychology 2.3 (1999), 163–193. See al­ so idem, “The Fate of Herbart's Mathematical Psychology,” History of Psychology, 4.2 (2001), 107–132. (41.) Boudewijnse et al., “Herbart's Mathematical Psychology,” 167. (42.) Riemann's Herbartian reading of Drobisch was hardly without basis. Herbart's ideas are central to Drobisch's Der mathematische Psychologie (Leipzig: Leopold Voss, 1850), and Drobisch's study, Ueber die Fortbildung der Philosophie durch Herbart (Leipzig: Voss, 1876), was written for the centennial of Herbart's birth. See also “Moritz Wilhelm Dro­ bisch und mathematische Psychologie: eine kritische Studie,” chapter 2 in Bratsch,

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From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval Deutsche Philosophen: Studien aus dem wissenschaftlichen Leben der Gegenwart (Leipzig: Heitz, 1897), 14–51. (43.) Jahrbuch der Musikbibliothek Peters 21–22 (1914–1915), 1–26. The article is trans­ lated with commentary by Robert Wason and Elizabeth West Marvin, “Riemann's “Ideen zu einer ‘Lehre von den Tonvorstellungen’ ”: An Annotated Translation,” Journal of Music Theory 36.1 (1992): 69–117. The translations presented below are my own. (44.) Concerning the problems of translating Tonvorstellung and the multiple shades of meanings it possesses (e.g., imagination, presentation, a placing before the mind), see the commentary provided in Wason and Marvin's translation of Ideen. For a more extended discussion of Riemann's theory of Tonvorstellungen in historical and philosophical con­ text, see chapter 4 of Mooney, “The ‘Table of Relations.’ ” (45.) These aspects of Tonvorstellung in “Ideen” are explored more deeply in Suzannah Clark's contribution to the present volume. (46.) Unsere Vorstellung weiß nichts von der Stimmungsdifferenz von d̄ [sic] und d, son­ dern setzt beide gleich, stellt d als Unterquint von ā [sic] und doch zugleich auch als Oberquinte von g vor. Diese enharmonische Identifikation der um das syntonische Kom­ ma verschiedenen akustischen Werte ist für unser Musikhören schlechterdings unent­ behrlich. (“Ideen,” 19). (47.) That Riemann drops the Oktavschritte (symbolized O or in the Lexikon) from the symbology of intervallic pathways in “Ideen” further supports the idea that Q and T are not symbols acting in a literal acoustical tone space, but in a metaphorical space of tones (or tone classes) detached from specific registers. (48.) Musiklexikon, 5th ed., (1900), 905. “Die Striche (Kommastriche) unter den Buch­ staben bedeuten die Vertiefung um 80:81 gegen den gleichnamigen, von c aus durch Quintschritte erreichten Ton, die Striche über den Buchstaben die Erhöhung um dasselbe Intervall. So ist z. B. das dem c nächst verwandte fisis (mit 3 Kommastrichen) durch 3 Terzschritte und einen Quintschritt zu erreichen und 3 Komma tiefer als das fisis der hori­ zontalreihe von c (13. Quinte).” The passage is identical in all previous and later editions printed during Riemann's lifetime. (49.) “Ideen,” 20. “Die Tabelle gibt ohne weiteres die Bestimmung jedes Intervalls nach Quint- und Terzschritten an die Hand und verrät für jeden mehrefach bestimmbaren Ton die einfachste nächstliegende Ableitung z. B. für fisis als Q3T (Quint der 3. Terz oder 3. Terz der Quint).” Riemann's reference to the “nearest-lying” derivation of a tone reflects his belief that the mind prefers simpler relationships over more complex ones, a belief he articulated as the “principle of the greatest economy of tonal representation” (“Ideen,” 7). (50.) Some of these are outlined by Richard Cohn, in “An Introduction to Neo-Riemannian Theory: A Survey and Historical Perspective,” Journal of Music Theory 42.2 (1998): 167– 180. Page 22 of 23

From Matrix to Map: Tonbestimmung, the Tonnetz, and Riemann's Combi­ natorial Conception of Interval (51.) The toroidal geometry of the equal-tempered Tonnetz was first recognized by Steven Lubin, “Techniques for the Analysis of Development in Middle-Period Beethoven” (Ph.D. diss., New York University, 1974). (52.) “Ideen,” 23. See also Riemann's analysis of the slow movement of Pathétique Sonata in L. van Beethovens sämtliche Klavier-Solosonaten, 2: 24. (53.) The work of Daniel Harrison in particular comes to mind. See his “Nonconformist Notions.”

Edward Gollin

Edward Gollin is Associate Professor of Music at Williams College.

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On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564)

On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564)   Suzannah Clark The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0010

Abstract and Keywords This article examines figures 1 and 2 in Riemann's 1914–1915 article. First, it examines the different aspect of Riemann's conception of tone: his notion of Klangvertretung, that any tone may project or assume meaning as one of the three elements of a major or minor triad. Second, it explores Riemann's notion of Klangvertretung as outlined in his “Ideen zu einer ‘Lehre on den Tonvorstellungen’”. Third, the article demonstrates the analytical utility of the concept, exploring how attention to the changing triadic-functional identities of tones in three Schubert Lieder offers an enriched view of structural and chromatic third relations in the works. Keywords: tone, Klangvertretung, major triad, minor triad, Schubert, Lieder

IF we plunge straight to the heart of neo-Riemannian theory, it takes us to figure 3 in an article that Riemann wrote in 1914–1915 called “Ideen zu einer ‘Lehre von den Ton­ vorstellungen.’ ”1 This figure, reproduced as example 10.1, consists of three pairs of tri­ ads: C major and C minor on the left, C major and A minor in the middle, and C major and E minor on the right. These correspond respectively to the PRL transformations, as con­ strued by neo-Riemannian theorists.2 I shall rehearse what these transformations are in a minute, but first I shall discuss what Riemann intended to portray through example 10.1.

Ex. 10.1.

By the time Riemann wrote his article in the mid-1910s, he had rejected the physical and physiological foundation for music theory, which had so famously led him to believing in the existence of the undertone series. He now confessed that finding a physical founda­ tion for music theory was a mistaken pursuit. Its replacement would be a theory of the Page 1 of 26

On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) imagination of tone (a “Lehre von den Tonvorstellungen”), an idea he claims he had first postulated but lost sight of in 1873.3 This was to be a theory of the “ ‘logical activity’ of musical hearing.” This new discipline would be akin to the painter who “in advance, gazes inwardly upon a picture that he wishes to paint;” that is, the painter visualizes mentally before (p. 295) painting. Similarly, Riemann argued “a composer hear[s] inwardly and in advance all that he notates afterwards.”4 The key to Riemann's theory of the imagination of tone is “hearing ahead.” But it is not only composers who hear ahead: Riemann argues that performers, score readers, and even page turners should do this too. He adapts the relationship between hearing and no­ tation depending on the agent: where the composer “hears inwardly” or “imagines” and then notates, others start from the notation and then “imagine” the music before sound takes place. That is to say, the performer sees a tone in the notation and imagines it be­ fore playing it; the score reader (who in Riemann's discussion is attending a concert) “is always some distance ahead with his eyes.” Similarly, the page turner, as Riemann em­ phatically puts it, does not so much “read along” as “read ahead” of the performed piece. Riemann argues that a similar sense of anticipation is also possessed by yet another cate­ gory of musician, namely the trained listener, who, if already familiar with a piece, will al­ so imagine a tone before hearing it and according to Riemann will measure mistakes or out-of-tune pitches against his or her expectations. Although such listeners are without notation, there is still a visual aspect to their activity. Brian Hyer has explained this phe­ nomenon in “Reimag(in)ing Riemann.” The listener sees the tone through a visual repre­ sentation of tonal space, such as the image of the Tonnetz.5 So, in other words, the set of triads in example 10.1 arose in a brief pedagogical section of Riemann's article, in which he explores how students can hone their skills in the art of hearing ahead. One of the first things Riemann had to do, however, was to put a limit on his theoretical materials—or to put a limit on what the student might hear. He therefore secured a strictly triadic context for his theory of imagination: “today we hear individual tones and intervals always as representations of triads (major or minor ones) according to available possibilities.”6 Example 10.1 relates to his idea of hearing individual intervals. Riemann observes that the imagination of any perfect fifth or major or minor third may only ever be heard as representing two triads: one major, one minor. Imagining the interval of a perfect fifth yields only one possible major and minor pairing, namely the major/minor tonic, as shown in example 10.1; imagining a major third yields another unique major-minor pairing, namely the relative major and minor; finally, the right-hand pair C major and E minor in example 10.1 shows the unique major/minor pairing when a minor third is imagined.7 Neo-Riemannian theorists have (p. 296) used these observations about the nature of the two common tones and the single displaced tone to distinguish these three sets of triads: P, meaning “parallel,” is the transformation involving the common tones of the perfect fifth (C, G), with a semitone displacement (E to E♭, or vice versa); R, meaning “relative,” is the transformation with the common tones of the major third (C, E) and a displacement Page 2 of 26

On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) of a full tone (G to A, or vice versa); L, meaning “Leittonwechsel,” is the transformation with the common tones of the minor third (E, G) and a semitone displacement (C to B, or vice versa).8 Nowadays, these are generally taken to be the core transformations in neoRiemannian theory—a point to which I shall return. In this essay I wish to take a step back and scrutinize figures 1 and 2 in Riemann's 1914– 1915 article (reproduced here as examples 10.2a and 10.2b). These relate to his theory of the imagination of a single tone—an aspect of Riemann's writings that has so far received no attention from neo-Riemannian theorists. Thus, in the rest of this essay, I shall first re­ hearse what Riemann understood by examples 10.2a and b, and then second, I shall in­ vestigate some of their theoretical implications for neo-Riemannian theory, and finally, I will explore their analytical potential in two songs by Schubert by comparing my Rie­ mannian analysis with a Schenkerian approach.

Ex. 10.2.

So, first of all, what did Riemann intend by the material in example 10.2? Again, it is cru­ cial to note that the examples comprise only major and minor triads. We have already cov­ ered the reason for this, and it is worth repeating Riemann's statement that “today we hear individual tones and intervals always as representations of triads (major or minor ones).” As can be seen in examples 10.2a and 10.2b, the pitch A is treated as 1̂, 5̂, 3̂ (in that order) in major and then minor triads, yielding six possible triads.9 Riemann has the following to say about example 10.2a: “One of the first, most basic exercises of the facul­ ties of the musical imagination would have to be to imagine specifically each individual note in its six possibilities for the representation of a tonal complex.”10 Regarding exam­ ple 10.2b, by contrast, Riemann says that “a single assigned note will be filled out, into a triad by the student by adding the two other notes.”11 So both of these are pedagogical exercises and serve as two sides of the same coin. In one case, the student extracts the common tone from the series of triads (example 10.2a); (p. 297) in the other, he or she starts with the common tone and provides the triads around it (example 10.2b). Note that the student is encouraged to see these triads as a “representation of a tonal complex.” I shall return to this observation later. After presenting these exercises, Riemann went on to muse on the increased musical ap­ preciation and aural dexterity a student would gain from doing them. If hearing a piece for the first time, a listener would know that the sound of a single tone is open to six pos­ sibilities, though Riemann suspects that the most likely possibility to come to mind will be the root of a major chord—an interesting conclusion considering Riemann's propensity for Page 3 of 26

On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) dualism. But if in possession of a score, Riemann points to some visual clues that limit the possible triads down from six—the most obvious being the key signature, which enables the student to narrow down the choice of triads to those suited to the major or minor key denoted by the signature.12 Riemann also points out that a tone in the middle of a piece can be exploited as a kind of riddle, especially if it is “strongly foreign” to the previous harmony; then, he argues, the possibilities are wide open all over again and the “solu­ tion” is only found in the composer's continuation. In such cases, the Riemannian student who is well versed in these exercises will be able (presumably at lightning speed) to rel­ ish in the riddle of possibilities, in the uncertainty of the continuation, and in the pleasure of the actual continuation. Significantly, the kind of hearing that Riemann advocates is very much “in the moment”—a point to which I shall also return when I compare Riemann's theory of tone to Schenker's theory of the Urlinie. Earlier I noted that Riemann understood these six triads as a “representation of a tonal complex.” What are the theoretical ramifications of Riemann's claim? The tonal complex can be mapped onto the Tonnetz, which also features in his 1914–1915 article (see exam­ ple 10.3a). In keeping with Riemann's theory of dualism, the six triads would feature in the Tonnetz as shown in example 10.3b. By contrast, a neo-Riemannian (or fundamental bass) reading places the same triads on the lattice as shown in example 10.3c. Riemann's configuration is naturally symmetrical because there are three major and three minor tri­ ads, and the former are labeled from the bottom while the latter are labeled from the top. In other words, when A serves as a 1̂, it generates the triad above and below it; when it serves as a 5̂, the “roots” of the triads are a fifth below and above it; and when it serves as a 3̂, the “roots” of the triads are also below and above it (this also means that 3̂ is al­ ways a major third away from the “root”). By contrast, in the fundamental bass system, all triads are labeled from the roots at the bottom of the triad and therefore the symmetry of how the pitch sits within the triads is not reflected in the labels. Given that theorists are generally drawn to patterns, they are far more likely to respond positively to the look of example 10.3b than example 10.3c. This may account for the general lack of interest in this collection of triads until neo-Riemannian theorists fashioned them into the PLR cycle, as in Riemann's own configuration in example 10.2a they do not at first sight seem theo­ retically or analytically useful.13

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On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564)

Ex. 10.3.

Even in terms of classic harmonic thinking (that is to say, dispensing with the Tonnetz for the moment and focusing on Roman numerals), they also seem a pretty useless set of tri­ ads. Taking A major as tonic, they form the following chords: I, IV, ♭VI, iv, i, vi (as annotat­ ed in example 10.2a). Note that in this scenario there is no (p. 298) dominant. Of course, it would be possible to create a dominant by instead interpreting the first triad as a domi­ nant, as follows: V, I, ♭III, i, v, iii (also shown in example 10.2a). In the case of the former set of functions, the tone that generates the system is fundamentally a 1̂, whereas in the case of the latter, it is a 5̂. This need not be problematic from a theoretical perspective, but it at least predicts that if a piece of music were constructed entirely around a single pitch and if it were to establish its (major or minor) key around that pitch as 1̂ rather than 5̂, then our (imaginary) piece would lack a dominant. Or, to put this the other way around: if we want to gain the dominant, then A can be interpreted only as 5̂.14 Rather than lament the fact that gaining a dominant severely limits how the common tone can be in­ terpreted, I shall instead explore the joys of how this common-tone theory can throw off the shackles of this apparently fundamental harmony. In other words, without resorting to Riemann's dualism, I shall argue that this complex of triads is theoretically and analytical­ ly useful. However, before getting to some analysis, I have two more points to make. First, in the context of this chapter, the important point to remember about Riemann's two examples is that, in the case of example 10.2a, the student is expected to extract a single common tone from a complex of given harmonies, whereas in (p. 299) example 10.2b he or she generates the complex of harmonies from a single common tone. Trans­ lated into analytical terms, then, Riemann's two examples are useful for demonstrating how a single common tone may be extracted from the fabric of the music (example 10.2a) or, as a kind of opposite, a single common tone may be exposed in the texture in such a Page 5 of 26

On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) way as to suggest that the harmonies turn around it or emanate from it (example 10.2b). This chapter therefore explores further an idea I proposed in an earlier study about Schubert's treatment of pitch in Ganymed (D544), albeit from a Schenkerian perspective. There I suggested that Schubert's treatment of pitch enabled him to expand his harmonic horizons, and I argued that, in Ganymed, the reconstrual of pitch drives the unfolding of the harmonic structure instead of classic progressions or indeed the composing out from an Ursatz. As I demonstrated, Schubert generates harmonic stations by allowing a pitch to serve as a 1̂, 3̂, or 5̂ within a triad, and would also sometimes allow it to serve as a 7̂ in a dominant seventh. Similarly, the pitches in (foreground) voice-leadings that are sugges­ tive of a particular harmonic move, such as 7̂–8̂ or a fifth in the bass, may be reinterpret­ ed to deliver something other than the expected tonic and dominant harmonies.15 Others have also explored Schubert's use of common tones, notably David Kopp, whose transformation system is predicated, as examined further below, on common tones. Kopp has therefore also talked about the change of identity of pitches in Schubert's harmonic unfoldings, and he—rather nicely—includes their use in a greater array of dissonances than I did.16 Other scholars, Diether de la Motte and John Gingerich, have both observed the almost obsessive presence of the pitch G in the cello's melody of the secondary theme in Schubert's Quintet in C Major (D956). During the course of the theme, G serves as 3̂ of E♭, 5̂ of C major and 1̂ of G major.17 These studies (including my own) have tended to focus on common tones when they are obvious in the texture, usually serving as a distinctive melodic feature. As de la Motte ob­ serves, the pitch G is present in the melody for 56 quarter-note beats in 24 measures; this adds up to 14 measures worth of G, or more than half the melody.18 Indeed, as Riemann might put it, the listener “knowing the key of this piece” is likely to expect a high pres­ ence of G in the secondary area of a C-major movement. Schubert, however, translates the expected key into a tone, and his “imagination of tone” places it first as 3̂ of E♭ major —a key not strictly diatonic to C major—then as 5̂ of C major before locating it as 1̂ within G major. Even then, G major appears in a subsidiary harmonic position; the 1̂ in G major is only assertively stated on the arrival of the closing section of the exposition in measure 100. This study will delve further into cases where Schubert's harmonies turn around a single pitch that is showcased melodically or texturally, as suggested by example 10.2b. But tak­ ing my cue from Riemann, I will also scrutinize cases where common tones are hidden and therefore need to be extracted from the harmonic fabric, as suggested by example 10.2a. Additionally, and again following Riemann's cue, I shall look to the key signatures and the harmonic contexts they imply in order to detect how Schubert plays with the imagination of tone. My case studies are Trost (D523), Liedesend (D473), and Gretchens Bitte (D564), all songs written during Schubert's phase of greatest harmonic adventure around the years 1816–1817. The second theoretical point I wish to make before getting to my analyses of these songs has to do with what is arguably the chief debate in neo-Riemannian theory, namely (p. 300)

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On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) which transformations should constitute the core ones. There are three opinions on this, represented mainly by Brian Hyer, Richard Cohn, and David Kopp. In his article “Reimag(in)ing Riemann,” Brian Hyer argued that in order to navigate the Tonnetz (in ex­ ample 10.3a), four transformations are required: PLR + D. Subsequently, Richard Cohn appealed to “the law of the shortest way” and notions of “efficacy” and “parsimony” to ar­ gue that the fewest possible variables should be sought to navigate the Tonnetz. For him, only PLR are necessary. That is to say, Hyer's D transformation, which covers the horizon­ tal axis of the Tonnetz, can be gained by combining LR (to go in the dominant direction) and RL (to go in the subdominant direction). So, for example, an authentic cadence, ac­ cording to Roman numerals, is V–I; according to Hyer, it is D, and according to Cohn, it is RL. In 2002, David Kopp argued against Cohn's use of combinations of R and L to express the dominant, as it gives the false impression that one of the most common harmonic moves is a compound transformation. Although he agreed with Hyer's sentiment for a larger set of transformations, Kopp went further still and developed a common-tone theo­ ry for every possible direct transformation. He ended up with I, D, D−1, F, F−1, M, M−1, m, m−1, R, r, P, and S.19 So, how did he arrive at all these? He wrote out all of the possible triads that share one, two, or all three common tones with C major.20 As it turns out, no neo-Riemannian—other than Kopp himself—has used Kopp's all-inclusive theory of trans­ formations; indeed, the fact that it is all-inclusive is seen as the argument against its use­ fulness. However, as shown in example 10.4, Riemann's set of six triads based on a single common tone offers a solution here: all of Kopp's transformations (except for I: identity) may be found in Riemann's tonal complex. It could be argued that deriving Kopp's trans­ formations from a single common tone, rather than all three pitches of the triad, is more parsimonious, which is a crucial criterion for Cohn and others who have followed suit.21 Moreover, the collection of triads can also be formulated into a cycle, using only the core transformations, as in A+, F♯ –, D+, D–, F+, A–, A+ of example 10.2, using RLPRLP, re­ spectively.22 To my mind, a particularly attractive aspect of Kopp's system of transforma­ tions—especially for the purposes of my analyses—is precisely the fact that it expresses direct relations among all possible transformations within the collection of six triads, and is not seemly confined to articulating an ordered set of maximum common-tone relations. And unlike classic neo-Riemannian theory, which favors fewer transformation types, there are no compound transformational expressions.23 Indeed, the system put forward here— as a revision (of the derivation) of Kopp's transformations—is one that is distinctly Rie­ mannian: as Riemann's career progressed, he was interested in providing a full inventory of Klang relations.24 In this light, I propose that any of Riemann's six triads may be con­ sidered to relate to any other one at least on a theoretical level. How and why they relate as they do in compositional practice is another matter, and it is to some musical analysis that I now turn.

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On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564)

Ex. 10.4.

Let's begin by following Riemann's advice on how to imagine a tone. The key signature of Trost (D523) has four sharps and the first pitch encountered by the singer is a G♯ (see ex­ ample 10.5). In this context, the singer, thinking only of the (p. 301) vocal line, is most like­ ly to imagine the pitch as either 3̂ of E major or 5̂ of C♯ minor. The first surprise, then, is that it in fact serves as a 1̂ of G♯ minor. Pressing onward, we see that Schubert visits B major, then G major, and finally E major in this short 17-measure strophic song. This song has already caught the attention of three scholars, namely Michael Siciliano, Harald Krebs, and Thomas A. Denny, the latter two of whom sought to theorize Schubert's harmonic schemes for songs that begin and end in different keys. Siciliano elegantly shows that all but the final transformations in Trost make their way consistently around the RPL cycle, and he argues that the cycle created by neo-Riemannian relations replaces the lack of a single tonic known to diatonic theory.25 Although Krebs argues a strong case for double tonics as an alternative to monotonality, he analyzes Trost in a single key (ex­ ample 10.6). In part, the song is short enough to be contained in this way, but he also ob­ serves that it does not venture to any keys that are not easily reckoned within an overall E-major Ursatz. The only potentially awkward key is G major, but he shows this to be very much a low middle-ground event. Occurring as it does between two statements of the dominant, he concludes that G major is an “oscillatory progression” between the B major harmonies in measures 6 and 11.26 As Krebs summarizes in a subsequent study, all the conditions are met for the harmonies in Trost to be read as a logical progression in E ma­ jor: there is a large-scale V–I, whereby iii and ♭III serve to embellish the dominant, and the Kopfton 5̂ (B) spans almost the whole song. Interestingly (given Riemann's comments on key signatures), Krebs additionally argues that the key signature clinches his argu­ ment that the song should indeed be regarded as monotonal, in E major.27

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On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564)

Ex. 10.5.

Ex. 10.6.

Ex. 10.7.

Thomas A. Denny responded to Krebs's analysis in his own attempt to theorize Schubert's key relations in songs that begin and end in different keys.28 His approach was starkly different. Instead of looking to voice leading, he looked for patterns in Schubert's har­ monic stations themselves. He came up with three models. The model relevant to Trost is given in example 10.7. Denny explains the concept behind the model as follows: Ia and Ib are two separate but equal-ranking tonics, and the two Xs stand for other keys. Denny noted that generally the Xs tend to be a third away from their respective tonics, and moreover that a rising third relation often incurs a falling one in the second half of such structures. Indeed, Trost is Denny's showcase piece for this model: as the annotations show in example 10.7, the first part of the song rises a third from G♯ (p. 302) (p. 303) minor to B major and the second part descends from G major to E major; as it turns out, the relationship between the two Xs is also a third, though Denny's structural break deemphasizes this feature. In Denny's defense, this structural break is supported by the

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On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) surface of the music in Trost. The shift from B to G major is the most aurally disjunct in the song, aided by the chromatic path (D♯ to D♮) taken by the vocal line. There is an aspect of Schubert's choice of keys that is drawn out by Krebs's use of Schenker that is not explicitly obvious in Denny's method of analysis, namely that the tone B is common to all harmonies in this song. As shown in example 10.8, it serves as 3̂ in G♯ minor, 1̂ in B major, 1̂ of B minor, 3̂ in G major, 1̂ in B major, and 5̂ in E major. Al­ though Krebs's graph shows the Kopfton B as starting in measure 3, the tone is clearly present in the harmony from the beginning of the song. What purpose, then, might a Rie­ mannian reading of this tone serve? To answer this question, we must compare Schenker's imagination of tone with Riemann's.

Ex. 10.8.

An important distinction can be drawn between a Schenkerian and Riemannian concep­ tion of the imagination of tone. Both theorists sought to train the ear, but to do radically different things. Schenker argues the case for a long-term hearing of the Kopfton, in which the pitch is retained mentally as an 8̂ (1̂ ), 5̂, or 3̂ of the triad of the Ursatz. Such graphs in Free Composition as figures 15/1b, 2b–c, 3b–3c2–3, and 5a are helpful in demon­ strating that how the primary tone is retained as 3̂ or 5̂ even as it is supported in the mid­ dle ground by a new harmony. Or—to pick a graph of a work more or less at random—the phenomenon of the mental retention of a pitch is illustrated by figure 130/4b, where the 3̂ is specified above a common tone E even when it belongs to A minor (vi) in the I–vi–I pro­ longation in C major.29 In other words, Schenker's conception of the imagination of a tone aspires to a background hearing that requires the mental retention of an 8̂, 5̂, or 3̂ despite harmonic changes underneath. Or, to demonstrate this through Trost, the primary tone 5̂ is mentally retained (or imagined) as 5̂ of E major, despite the (iii-)V–♮III–V motion that supports it until it begins its descent to 1̂. In a Riemannian conception, the pitch B in Trost should be traced as transforming its identity as each new harmony enters. The point is precisely to grasp the shifts from B̂ as (3̂, −) to (1̂, +) to (1̂, −) to (3̂, +) to (1̂, +) and fi­ nally (5̂, +), as illustrated in example 10.8.30 (p. 304)

A few practical observations may be made about Schubert's presentation of har­

mony in relation to the theoretical derivation of what might be termed Riemann's “single common-tone group” of example 10.2. First, Schubert uses five of the six “aesthetic possi­ bilities,” and clearly construed the available triads around B̂ in order to gain a dominanttonic relationship.31 Second, the tone must be extracted from the surface texture in cer­ tain places and is placed in the ear by the melody in others. Although the vocal line does not include a B̂ in the first phrase (hence it does not feature in Krebs's graph until mea­ sure 3), the second phrase enters early on B̂, as if to anticipate the next new harmony. Meanwhile, the piano has had B̂ in its upper voice all along. As noted earlier, the new key, Page 10 of 26

On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) B major, is stated as V–I in measures 4–6, but the 64 53 formula serves to bring in the B̂ as (1̂, +) immediately, while providing harmonic motion underneath. The vocal line fractures any sense of continuity between B major and the G major of the next phrase because it moves chromatically in measure 6 (these correspond to Denny's Xa || Xb harmonic sta­ tions). By contrast, the piano retains the B̂ in the same voice, albeit an inner voice, in this passage; it shifts from (1̂, +) to (1̂, –) to (3̂, +). The exit from G major back to B major is altogether smoother, as the voice performs the task of repeating the B̂—again entering early, undoubtedly for effect, as the singer sings “nimmer lange weil’ ich hier.” Indeed, during this passage, the voice remains on B̂, changing from (3̂, +) to (1̂, +) to (5̂, +). The piano postlude brings out the B̂ once more, entering early in measure 14 and articulating a downbeat, marked with Schubert's characteristic double emphasis of fp and 〉. At any rate, there is no doubt that the common tone is a hallmark of this song, and that the har­ monies in the song turn on that pitch. Liedesend is one of two songs that George Grove criticized for carrying modulation to an “exaggerated degree” in his dictionary entry on Schubert. In an otherwise sympathetic defense of Schubert's modulatory strategy in the songs (he argues that the key changes are an important means of expressing the text), Grove saw no real justification for the keys in this song: [I]n the short song Liedesend of Mayhofer (Sept 1816), he begins in C minor, and then goes quickly through E♭ into C♭ major. The signature then changes, and we are at once in D major; then C major. Then the signature again changes to that of A♭, in which we remain for fifteen bars. From A♭ it is an easy transition to F minor, but a very sudden one from that again to A minor. Then for the breaking of the harp we are forced into D♭, and immediately, with a further change of signature, into F♯. Then for the King's song, with a fifth change of signature, into B major; and lastly, for the concluding words…a sixth change, with eight bars in E minor, thus ending the song a third higher than it began.32

Ex. 10.9.

It seems Grove may simply have been objecting to the number of times Schubert modu­ lates and possibly to what appears to be a large variety of keys: he lists no fewer than twelve, requiring six key signature changes. According to Grove, the song includes: C mi­ nor, E♭, C♭, D, C, A♭, F minor, A minor, D♭, F♯, B and E minor. What is there to observe about common tones in this set of keys? If the keys are taken at face value and represent­ ed by their tonic triads, then a search for common tones produces the groupings in exam­ ple 10.9 (accidentals apply to individual triads). Each group of four triads (p. 305) is sepa­ rated by a noncommon tone pair of triads: D and C majors, and A minor and D♭ major, re­ Page 11 of 26

On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) spectively. As shown in the example, the first three triads share Ê♭ and the last pair of the group shares Ĝ♭/F̂♯. The next four all revolve around Ĉ. The final group is a series of fifth relations or D transformations, and thus in each case the 1̂ becomes the 5̂ of the next tri­ ad. It will be recalled that Grove generally saw the motivation for key change in the words. In this case, it is almost too easy to offer a cogent explanation for Schubert's key changes: a new key is forged as the narrator depicts, in turn, the king on his throne, the bard playing his harp, the sweet tune he plays, the bard's inability to appease the king, his frustration at his lack of success (the bard breaks his harp), and finally the calm of the king, whose direct speech is then set to the cycle of fifths. Nevertheless, in the passage cited above, the only suggestion Grove makes regarding a hermeneutic motivation for Schubert's choice of harmony is the “breaking of the harp,” which “forces” a new key. Indeed in light of the common-tone links observed in example 10.9 between adjacent harmonies, it seems tempting to say that when the bard breaks his harp, the entry of D♭ “breaks” the com­ mon-tone pattern. However, Grove is not drawn to make any specific comment on the concomitant passage, namely the entry of C major after D major. The main gist of his commentary is to measure how quickly the keys go by and how closely related they are. Therefore he writes that after C minor, Schubert “goes quickly” through E♭ and C♭ major; A♭ “remains” for 15 measures; and, judging from Grove's language, we are as suddenly in D major, as the entry to D♭ is forced. Note also that he draws a distinction between the “easy transition” from A♭ major to F minor and the “very sudden one” from A♭ major to A minor. To be sure, in classic harmonic terms, A♭ major to F minor seems close because F minor is the relative minor of A♭ major, while F minor to A minor seems a distant move because they relate as i–♯iii. However, in common tonality, a move from F minor to A mi­ nor is achieved relatively easily through two semitone displacements around the common tone C. Such a conception still preserves the distinction in the degree of distance ob­ served in the classical model because A♭ major and F minor have two tones in common, while F minor and A minor have only one.

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On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564)

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On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564)

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On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) Ex. 10.10.

Nevertheless, on closer inspection of Schubert's score something is amiss with Grove's analysis, and indeed it has to do with a misreading that emanates from taking in a visual clue (key signature) and narrowing down the imagination of tone (p. 306) to fit the signa­ ture. As noted above, Grove comments on each change of signature. However, after the signature change to no sharps or flats in measure 52, the singer sings an  and Grove imagines it as 1̂ of A minor (example 10.10). This is a perfectly natural impulse. It even fits Riemann's advice on how to imagine a tone in the context of a given signature: one's first port of call should be to construe it as a tonic. An A, appearing after a signature of no sharps or flats, is obviously not going to be part of a C major triad. But Riemann also intended his listeners and score readers to be ready for surprises or “riddles.” Schubert's  is harmonized by an F♯ minor triad. Grove would have done well to observe more care­ fully that the pitch needs to be imagined in the more unlikely scenario of a (3̂, –). Indeed, it forms part of the cadential pattern ii6–V64 53–I in E major. What effect does this correc­ tion have on the common tones found in example 10.9? In substituting E major for Grove's misreading of A minor, the Riemannian student—versed in the extraction of tone, as cultivated by example (p. 307)

(p. 308)

(p. 309)

(p. 310)

(p. 311)

10.2a—will immediately observe that the single tone that unites the harmonies around it is Â♭/Ĝ♯, as illustrated in example 10.9. Indeed, the noncommon tone relationship in Grove's list of keys for the “breaking of the harp” is smoothed over as the Ĝ♯ of E major becomes Â♭ of D♭ major.33 If we now turn to the music (that is to say, to how these harmonies are composed out), we see that Schubert has done little to make the common tones we have been analyzing audi­ ble. Despite the common tones, Grove's perception that this song comprises one contrast­ ing key after another, with the occasional smooth modulation, is accurate. The two first stanzas begin with arpeggiations in their respective home keys (C minor and D major), but in the approach to D major from C♭, the Ĝ♭/F̂♯ common tone is not exploited in any way. The move from C to A♭ majors does at least emphasize the common Ĉ in both the voice and piano. The entry of the fourth stanza is the smoothest in the song, as Grove ob­ served: the move from A♭ major to F minor involves two common tones. Although we saw earlier how the breaking of the harp was less sudden than Grove observed, the exit from D♭ major to F♯ major is not presented as an enharmonic fifth relation on the surface be­ cause the F♯ section does not enter on its tonic. Instead, the pitch F̂♯, which appears in the voice and prominently in the accompaniment, is interpreted not as 1̂ but as 5̂ of F♯'s subdominant. The next main keys (B major and E minor) roll through in dominant-tonic relationships, although during the B major section there is a brief turn to D major; again, nothing is made of the common tone. By and large, then, the keys unfold, but little em­ phasis is placed on the common tone material in the succession of keys. The Riemannian student, carrying out an (aural) common-tone analysis of this song, will necessary rely on the training from example 10.2a to extract tones. Indeed, without the common-tone thread of the keys being emphasized aurally, the keys jar more. This may be one of the Page 15 of 26

On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) aural effects that Grove was responding to when arguing that the modulation was carried out to an exaggerated degree in this song. In my final example, Gretchens Bitte (example 10.11), the common tone is aurally very striking from the very beginning of the song. It is placed in the most salient positions in the vocal line, articulating as it does the beginnings, midpoints, and endings of phrases. Indeed, it governs the harmonic maneuvers for some 33 measures of the song. In short, Gretchens Bitte seems to be a compositional incarnation of Riemann's singing exercise in example 10.2b. As we know by now, Riemann advises the singer to begin by contextualizing a pitch in light of its key signature. The imagination of tone in the case of the vocal line's first D̂♭ is a relatively straightforward affair. Given the six flats in the signature, it is most likely to be either 1̂ of D♭ major or 3̂ of B♭ minor. It turns out to be the latter. The voice enters with a D̂♭ that occupies both the up- and downbeat.

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On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564)

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On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564)

Ex. 10.11.

The first vocal utterance spans the end of measure 14 to the beginning of measure 51, as the harmony traverses tonic to relative major. The vocal line both begins and ends with the D̂♭. The pitch thus starts out as (3̂, –) and becomes a (1̂, +). It also appears at the mid­ point of the passage on the downbeat in measure 4. Here, it is 1̂ of its triad, accompanied as it is by the mediant major. After a brief piano interlude (p. 312) in measures 5–6, the next vocal section opens again with D̂♭ on the downbeat of measure 7, this time prepared by a larger leap than before, namely from an F below. The harmony has returned to the original key of B♭ minor and hence D̂♭ is again (3̂, –). The pitch changes identity as the phrase ensues, ending on yet another D̂♭, which is now (1̂, −).

Ex. 10.12.

Within these ten measures, as example 10.12 shows, Schubert's common tone has ap­ peared in three of Riemann's six triads, namely as 3̂ of B♭ minor, 1̂ of D♭ major, (p. 313) (p. 314) (p. 315) and 1̂ of D♭ minor. Each of these harmonies is tonicized, even if only briefly. One might easily imagine that Schubert could venture next to G♭ major, one of his favorite harmonic spaces (though he prefers to use the key in a tonic-major context). However, his next harmonic turn is to the most unlikely of the remaining candidates among Riemann's six triads; he ventures to A major. The common tone is again placed in the vocal line, although this time as an upbeat to the descending A major triad that starts on the following downbeat (see measures 12–13). Ĉ♯ now serves as (3̂, +). A major is toni­ cized, and in this case remains active for an extended period of time (measures 124–26).

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On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) The remaining two triads G♭/F♯ major and minor, which would complete the full comple­ ment of triads united by D̂♭/Ĉ♯, are sounded in the next section of the song. It also begins with a double iteration of Ĉ♯, both as upbeat and downbeat for “Wohin.” The immediate context in which the pitch appears is a quick shift from A major to F♯ minor harmonies. F♯ minor is strongly implied but never explicitly arrived at measures 284–33, and then an F♯ major chord sounds, as Schubert leads us to B minor in measure 33. The C♯, thus, is first 5̂ of a minor dominant before asserting itself as 5̂ of the major dominant for an authentic cadence in B minor, as shown in example 10.12. The singer of Schubert's song has, in a certain sense, completed the exercise set out by Riemann in example 10.2b. Not all of the triads serve as a tonic, but Riemann never said they had to. As shown in example 10.12, our singer has sung the triads in a different or­ der. She starts always from the same pitch and sings B♭ minor, D♭ major, D♭ minor, A ma­ jor, F♯ minor, and F♯ major. Her Riemannian task is done, but the song is not. Indeed, the fleeting presence of the last two triads, as well as their nontonic function, brings about a new harmonic space for the next portion of the song. Before pressing on to examine the rest of the song, it is worth scrutinizing further the or­ der of presentation of the harmonies around D̂♭/Ĉ♯. As can be seen in example 10.12, in addition to the common tone that has been the focus of our aural attention in the vocal line, there is always another common tone between adjacent harmonies. They unfold therefore in the smoothest possible order, producing the following transformations re­ spectively: R, P, L, R, and P. Note also that they are not construed to produce a dominant anywhere; instead—and undoubtedly because the song sets out in a minor key—the nat­ ural tendency is to construe the triads around the common tone in such a way that gener­ ates a i–III relationship. But Schubert had options in this respect: if F♯ minor were his tonic, then he could have had both III and V. However, in opening up the song with D̂♭ as (3̂, –), and thereafter construing the harmonic motion around this pitch in the melody, Schubert both (p. 316) expanded his tonal palette and eschewed the most fundamental of harmonies, namely the dominant. In this light, the passage in measures 33–35 is interesting. Given the shape of the vocal line, our Riemannian student is liable to assume (especially on looking ahead) that the harmony is likely to be G major, for an authentic cadence into C major. Schubert has something else in mind: he interprets the B̂ and D̂ and 5̂ and 7̂ of an E major harmony, without however treating it like a dominant seventh. Indeed, the music to the words “ich bin, ich bin alleine” does seem to stand alone—or at least apart—from the shape of the harmonies underneath it: the singer seems to be singing a line most suited to G major, while the accompaniment is working to E major. Her subsequent cries, “Ich wein,’ ich wein,’ ich weine,” also sound “alone.” While she repeats her words emphatically, the ac­ companiment wanders off: it neither underscores her apparent 7̂–8̂ in C major (it pro­ vides E major to C major) nor does the accompaniment comply with her simple descend­ ing line at the end of the phrase (it provides an interrupted cadence). That said, while the interrupted cadence is but a gentle departure from the expectations of the vocal line, it does bring about a Ĉ♯, which has been in our ear so prominently from earlier in the song. Page 19 of 26

On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) The piano asserts an accented diminished chord, with Ĉ♯ at the top as again the operative pitch pressing things to D minor. This gesture is, however, an upbeat to another snippet of melody in C major, where the Ĉ is clearly staked out by the voice (measures 38–39). With things apparently settled on C major, the reiteration of measures 354–39 is identical, save for the Ĉ (instead of B̂) upbeat at the end of measure 39. At this point in the song, the voice and piano seem to be fighting each other, rather than, as it were, being in perfect harmony. Indeed, the passages in measures 36–37 and 40–41 where the interrupted cadence is found recall a much earlier passage in the song (mea­ sure 26) when Gretchen sang a descent from 5̂–1̂ in A major to the near-rhyming word “allein.” There—to the words “weisst nur du allein”—the voice and piano resolve the key together. The accompaniment, “knowing” where she is going harmonically, portrays a common understanding in this passage, but later it emphasizes her loneliness.

Ex. 10.13.

One might be easily tempted to make a grand hermeneutic gesture about the fact that the manuscript breaks off at this point; the song is incomplete—or so it at first seems. It ends with a change of key signature in measure 44, introducing four flats. John Reed argued that Schubert lost his way harmonically at this point and concluded that Schubert there­ fore abandoned the composition.34 Any poignant conclusions about Schubert's composi­ tional fragment breaking off like the fragments of Gretchen's heart are quickly deflated if one takes into account more recent suspicions that the rest of this song was lost rather than never composed.35 However, on a more pragmatic note, we can turn to Riemann once more, but this time to “hear ahead” of the notation: if he taught us to hear a tone in light of a key signature, perhaps we can at least use the signature to imagine what the next pitch—and even the next harmony—might have been in the missing portion of this song. In calling upon the exercises in example 10.2, and knowing as we do that the focal pitch of measures 36–43 has been Ĉ, we can imagine the next pitch in the vocal line could (p. 317) also be a Ĉ. From the key signature, it is likely to be either 3̂ of A♭ major or 5̂ of F minor. I vote for a move to A♭ major. Benjamin Britten in his completion of Gretchens Bitte did the same, while N. C. Gatty introduced F minor (with a number of harmonic turns around Ĉ), although he altered the signature to six flats to expedite the return to the opening B♭ minor key.36

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On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) There is a difference between the choices Riemann and Schenker would make if they were imagining the likely harmonic context of this tone. Riemann would entertain either possibility, though given Schubert's predilection for ♭VI, he might vote for A♭ major (that is, Ĉ as (3̂, +)). Schenker, however, would be desperate by now to have a dominant for his bass arpeggiation—not least because after the motley of foreground keys so far in the song, the entry of Ĉ in measure 43 comfortably begins to look like part of an Urlinie descent from D♭. But it needs proper support: the F major harmonies in measures 37–41 do not qualify because they are not (foreground) tonics. So, an arrival of F minor after the key signature would begin to put things right for the bassline. Schenker could even argue that the minor dominant is common in minor-key pieces, and once reached, it invariably seeks the leading note to incur the structural interruption. This would nicely set up a re­ turn to the original key of B♭ minor—and moreover would restore order (I have character­ ized this hypothetical ending in example 10.13). Of course, there is no guarantee that Schubert would have returned to the opening key, as Schenker would undoubtedly prefer him to. By contrast, a Riemannian imagination of the rest of this song could be far more adventurous, allowing the fate of the harmony to lie, as it were, in the hands of the pitch Ĉ. Whatever we might imagine the lost portion of this song to have sounded like, one thing is for certain: a Riemannian and Schenkerian understanding of Schubert's pitch material and choice of keys opens up vastly different possibilities not only for how the music might have continued, but also for how we should hear the music we do have. As we have seen, Riemann's idea of emphasizing the change in identity of a (p. 318) common tone contrasts starkly with the Schenkerian notion of structural common tones, particularly those that belong to the Urlinie and even more particularly, the Kopfton itself. Riemann's and Schenker's theories represent two different aspirations of hearing: Riemann's theory priv­ ileges the moment, where surface key—or even surface triad—is the focus of attention. Schenker's is a large-scale hearing, based in monotonality. The two theorists therefore al­ so represent different conceptions of large-scale tonal structure: Riemann's system allows for a single pitch to anchor a harmonic complex, such that a song or section of a song is not so much “in a key” as “around a pitch”; for Schenker, a single pitch may indeed be prolonged for a long time but it must ultimately move to something else; it must generate counterpoint. For Schenker, the presence of a structural tonic and dominant in the Ursatz is paramount, while in Riemann's conception the need for a large-scale tonic and domi­ nant can easily be obviated. In short, what Riemann hit on in his theory of the “imagina­ tion of tone” is not only a means to engage listeners, performers, score readers, and page turners but also a means by which composers expanded tonal space and went beyond the confines of thinking in terms of root motion.

Notes: (1.) Hugo Riemann, “Ideen zu einer ‘Lehre von den Tonvorstellungen,’ ” Jahrbuch der Musikbibliothek Peters 21/22 (1914–1915): 1–26. References in this chapter will be to the English translation: Hugo Riemann, “Ideas for a Study ‘On the Imagination of Tone,’ ” Page 21 of 26

On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) trans. Robert W. Wason and Elizabeth West Marvin, Journal of Music Theory 36 (1992): 81–117. (2.) The neo-Riemannian designation for P is different from Riemann's. Riemann uses “Parallele” for our relative relationship; our parallel relationship is Riemann's “Variante.” (3.) Riemann, “On the Imagination of Tone,” 82–83. To be sure, Riemann's main theoreti­ cal preoccupation lay in musical hearing (“musikalisches Hören”), although it manifested itself in different ways throughout his career. Riemann's doctoral dissertation was enti­ tled “Ueber das musikalische Hören” (University of Göttingen, 1873). (4.) Riemann, “On the Imagination of Tone,” 83. (5.) Brian Hyer's article, “Reimag(in)ing Riemann,” Journal of Music Theory 39 (1995): 101–138, has been seminal to the development of neo-Riemannian theory, especially with regard to such technical aspects as the mapping of major and minor triads on the Tonnetz and their reconceptualization in a nondualistic, equal tempered space. At the beginning of his article, he scrutinizes the various Riemannian concepts that are obscured by the necessarily more narrow translation of nuanced German terms into the English “imagina­ tion.” As he explains, one important aspect that is lost in translation is Riemann's empha­ sis on the visual aspect of reading ahead; in this context, Riemann uses the word “Ton­ phantasie,” which stems from “phantazein” or “to render visible,” rather than “Tonvorstellung” (p. 103). For an assessment of the importance of Hyer's article to the technical aspects of neo-Riemannian theory, see Richard Cohn, “Introduction to Neo-Rie­ mannian Theory: A Survey and a Historical Perspective,” Journal of Music Theory 42 (1998): 167–180. (6.) Riemann, “On the Imagination of Tone,” 87–88 (my emphasis). (7.) Riemann indicates these intervals using Arabic and Roman numerals for the pitches of major and minor triads respectively, and the triads are reckoned according to his theo­ ry of dualism. Hence in example 10.1, 1 and 5 refers to C and G respectively in the major triad, while I and V refers to G and C respectively in the minor triad; similarly 1 and 3 refers to C and E, while I and III refers to E and C; 3 and 5 refers to E and G, while III and V refers to G and E. Throughout this chapter, I shall refer to the pitches using nondu­ alist designations of the scale degrees, borrowing the Schenkerian caret symbol. The im­ plications of Riemann's conception will, however, be considered at the end of this chapter (see also n. 30 below). (8.) Strictly speaking, Riemann's own explanation of these relationships was based solely on the common tones. See the explanation offered by Brian Hyer and Alexander Rehding, “Riemann, (Karl Wilhelm Julius) Hugo,” The New Grove Dictionary of Music and Musi­ cians, ed. Stanley Sadie, 2nd ed. (London: Macmillan, 2001), 363. (9.) Riemann's dualistic conception is further evident in examples 10.2a and 10.2b as he proceeds in order of decreasing strength of the tone from generating tone to fifth to third, which in the case of the minor therefore yields (in our terms) D minor, A minor, Page 22 of 26

On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) then F♯ minor; if Riemann were thinking in terms of the fundamental bass and the strength of the common tone according to the overtone series, he would have ordered the minor triads as A minor, F♯ minor, and D minor. (10.) Riemann, “On the Imagination of Tone,” 86 (emphasis in original). (11.) Ibid. (12.) Ibid. (13.) The collection was systematized, although not derived from example 10.2a, by Jack Douthett and Peter Steinbach, “Parsimonious Graphs: A Study in Parsimony, Contextual Transformations and Modes of Limited Transposition,” Journal of Music Theory 42.2 (1998): 246–249, and developed in the context of Schubert by Michael Siciliano, “NeoRiemannian Transformations and the Harmony of Franz Schubert” (Ph.D. diss., University of Chicago, 2002). (14.) Note that with this interpretation, we also gain a III and V for the minor key and that this is the only interpretation in which a dominant is gained for the minor key. The is­ sue of whether or not it is problematic from a theoretical perspective to take the common tone in the tonal complex as 5̂ in order to gain the dominant finds a comfortable solution in a neo-Riemannian context: it helps to support Lewin's claim that the D (or DOM) trans­ formation is reckoned as a descending fifth, an idea adopted by both Hyer in “Reimag(in)ing Riemann,” as explained on p. 108, and David Kopp, Chromatic Transfor­ mations in Nineteenth-Century Music (Cambridge: Cambridge University Press, 2002), 169–170. (15.) See my “Schubert, Theory and Analysis,” Music Analysis 21 (2002): 209–243. (16.) The only reason I included just the dominant seventh was that my observations were driven by the harmonies that appeared in Schubert's Ganymed. Kopp has been criticized by Richard Bass for poor reasoning over his choice of which dissonances to include or ex­ clude. As Bass points out, the augmented and diminished triads are out, but the dominant seventh and German augmented sixth are in (the latter because it as the same pitch con­ tent as V7). Bass has a point that some dissonances that are proximate to those included are inexplicably excluded. However, the elegance of Kopp's analysis of, for example, Schubert's Sonata in B♭ Major (D960) lies in the manner in which he exposes the pitch B♭ as the thread that unites not only the harmonic stations of the ABA´ sections of the first thematic statement but also the dissonant harmonies that bring about the return of B♭ major between sections B and A´; one would hope that—had one of the dissonances been, say, a diminished seventh—it would have featured in the system. See Bass, “Review of David Kopp, Chromatic Transformations in Nineteenth-Century Music,” Music Theory On­ line, 10:1 (2004), par. 7. (17.) Diether de la Motte, A Study of Harmony: An Historical Perspective, trans. Jeffrey L. Prater (Dubuque: W. C. Brown, 1991), 218–219, and John M. Gingerich, “Remembrance Page 23 of 26

On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) and Consciousness in Schubert's C-Major Quintet, D 956,” Musical Quarterly 84 (2000): 619–634, here p. 620. (18.) Ibid., 218. (19.) Kopp's labels attached to the transformations are explained in Chromatic Transfor­ mations, 165–176. (20.) Kopp produces a table of how these transformations arise from common tone rela­ tions with the C major triad in Chromatic Transformations, 2. (21.) The importance of parsimony as a criterion for a persuasive theory has been stated on numerous occasions by Richard Cohn. See especially his “Neo-Riemannian Operations, Parsimonious Trichords, and Their Tonnetz Representations,” Journal of Music Theory 41 (1997): 1–66, and “Music Theory's New Pedagogability,” Music Theory Online 4.2 (1998), par. 13. The logical outcome of the principle is Cohn's hexatonic systems, which are out­ lined in “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late Roman­ tic Triadic Progressions,” Music Analysis 15 (1996): 9–40. (22.) See n. 13. (23.) See especially Cohn, “Introduction to Neo-Riemannian Theory.” Cohn argues that the Tonnetz is a “canonical geometry for modelling triadic transformations” (172) and that the D transformation which Hyer added to the basic PLR group is “redundant” be­ cause it can be formed through the combination of R and L (172). (24.) For a synopsis of this progression in Riemann's thought, see Henry Klumpenhouwer, “Dualist Tonal Space and Transformation in Nineteenth-Century Musical Thought,” The Cambridge History of Western Music Theory, ed. Thomas Christensen (Cambridge: Cam­ bridge University Press, 2002), 466. (25.) Michael Siciliano, “Two Neo-Riemannian Analyses,” College Music Symposium 45 (2005): 92, 101, and 105. (26.) Harald Krebs, “Third Relation and Dominant in Late 18th- and 19th-Century Music” (Ph.D. diss., Yale University, 1980), 154; Krebs's graph is in this study as figure III.14, vol. 2, 70. (27.) Krebs, “Alternatives to Monotonality in Early Nineteenth-Century Music,” Journal of Music Theory 25 (1981): 1–16, here, 2–3. The model for Krebs's analysis may be found in Heinrich Schenker, Free Composition (Der freie Satz): Volume III of New Musical Theo­ ries and Fantasies, trans. and ed. Ernst Oster (New York and London: Longman, 1979), sections 244–245 and fig. 110 (d). (28.) Thomas A. Denny, “Directional Tonality in Schubert's Lieder,” in Franz Schubert— Der Fortschrittliche? Analysen—Perspectiven—Fakten, ed. Erich Wolfgang Partsch (Tutz­ ing: Hans Schneider, 1989), 37–53. Page 24 of 26

On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) (29.) Schenker, Free Composition, Figures as cited in the main text above. (30.) A word on my notation: as was mentioned earlier (n. 7), the Arabic and Roman nu­ meral annotations in Riemann's examples common tones distinguish between the same pitch in the context of a major and minor triad. The effect of this is most noticeable in ex­ ample 10.1. Note how each of the two common tones is labeled differently depending on whether they form part of a major or minor triad, as we saw in n. 7. Although, for in­ stance, the C and G ostensibly remain constant between C major and C minor, Riemann seems to suggest they transform from 15 to VI. It is worth exploring whether this is just a product of Riemann's dualism or whether there is some substance to the notion that a tone changes quality and that this could do with being expressed in its label. For Rie­ mann, as is evident from examples 10.1 and 10.2, a single pitch in all its six triadic possi­ bilities can be assigned a different label, thanks indeed to his dualistic conception. From a purely practical (or analytical) perspective, this has a certain advantage: one can write, for instance, that the pitch A is III and this simultaneously reveals the triad it belongs to ˚c♯. From the perspective of perception, it seems right to express that a pitch does change its quality depending on its context. David Kopp has observed this, but turned to Hauptmann to explain how the multiple ap­ pearances of F in Schubert's Die junge Nonne (D828) are “not all the same F” (see Kopp, Chromatic Transformations, 261). He suggests it is advantageous to think of Hauptmann's dialectic labels for components in the triad, and he illustrates how the change in “meaning” of pitches comes about thanks to Hauptmann's dualistic conception of major and minor triads (Kopp, Chromatic Transformations, 58–60). In his dialectical system, I is root (or “unity”), II is fifth (or “opposition”), and III is mediant (or “synthesis”; therefore III is always a major third from the root). Thus, to take one example of how pitches change meaning when common tones exist between triads, in the case of the Leit­ tonwechsel transformation, C major is C = I, G = II and E = III, while E minor is B = I, E = II and G = III. The common tones E and G are III and II in one triad but II and III in the other. In this chapter I seek to specify the quality of the common tone in each triad of Riemann's tonal complex by adapting neo-Riemannian nomenclature familiar from the treatment of triads: (1̂, +) (1̂, –) (3̂, +) (3̂, –) (5̂, +) (5̂, –). Thus, (1̂, +) denotes the root in a major triad and (1̂, –) denotes the root in a minor triad; (3̂, +) denotes the third in a major triad and (3̂, –) the third in a minor triad; (5̂, +) denotes the fifth in a major triad and (5̂, –) denotes the fifth in a minor triad. I also introduce the symbol B̂ in order to denote a pitch, as op­ posed to a triad or key. (31.) Neo-Riemannian cycles—whether LPR, LP (hexatonic), or RP (octatonic)—are in­ tended to show how transformations bypass the traditional tonic-dominant diatonic rela­ tion. Whereas Siciliano emphasizes that Trost follows an almost perfect path through the LPR cycle, thereby lending order to the harmonic structure, I prefer to argue that in this case (unlike the case of Gretchens Bitte that we will examine shortly), Schubert chose the

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On the Imagination of Tone in Schubert's Liedesend (D473), Trost (D523), and Gretchens Bitte (D564) only two tonics from among the six triads that provide a diatonic framework for the song: G♯ minor has its relative major, and E major has its dominant. (32.) George Grove, Grove's Dictionary of Music and Musicians, ed. J. A. Fuller Maitland (London: Macmillan, 1908), 329. (33.) Richard Cohn has noticed that the series of keys (as they are established, rather than as they are identified by Grove) in Liedesend produce a kind of palindromic effect, with third substitutions, around the || mark: C–, C♭+, D+, A+, C+, A♭+ || F–, E+, D♭+, F♯+, B+, E–. After a colloquium that I delivered at Yale University (January 2007), Cohn pointed out to me how these keys turn back on themselves; I am grateful to him for shar­ ing this observation with me. (34.) As John Reed puts it, “The song, what survives of it, is of fine quality, and it is tempt­ ing to speculate about the reasons for Schubert's failing to finish it. The operative quality of his unfinished Faust pieces suggests that he may have cherished an ambition to write an opera based on the drama; but he was not ready for that in 1817, and in the final (C major) cadences one can almost sense the feeling of uncertainty about what happens next.” See Reed, The Schubert Song Companion (New York: Universe Books, 1985), 252. See also Maurice J. E. Brown, who also thought it was unfinished, in Schubert: A Critical Biography (London: Macmillan, 1958), 76. (35.) Walther Dürr (ed.), Neue Schubert-Ausgabe: Lieder, Band 11 (Kassel: Bärenreiter, 1999), 292. (36.) The completion by Benjamin Britten was published under the title Gretchens Bitte: Szene aus Goethes Faust (London: Faber Music, 1998), and N. C. Gatty's completion was published (without commentary or attached article) in Music and Letters 9 (1928): 386– 388.

Suzannah Clark

Suzannah Clark is Gardner Cowles Associate Professor at Harvard University. She previously taught at Oxford University. Her research interests include the history of tonal theory, the analysis and criticism of Schubert's music, as well as the history and analysis of trouvères chansons and thirteenth-century French motets.

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Tonal Pitch Space and the (neo-)Riemannian Tonnetz

Tonal Pitch Space and the (neo-)Riemannian Tonnetz   Richard Cohn The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0011

Abstract and Keywords This article reexamines the Tonnetz as an analytical apparatus. Drawing on Fred Lerdahl's Tonal Pitch Space, which is critical of the homology of tone, chord, and key space posited by the Riemannian Tonnentz, the article proposes a hybrid spatial model that draws on both the Riemannian tradition and Lerdahl's hierarchical model. The prag­ matic solution presented in this article allows for communication, reinterpretation, and relocation between tempered and unbounded spaces of tones, key spaces, and diatonic spaces or regions of triads, through the graphic metaphors of the Tonnetz and its allied spatial models. Apart from providing a pragmatic solution, the article also illustrates the hybrid model in analyses of Schumann, Wagner, and Chopin, with the view of suggesting further explorations and applications of these analytical tools. Keywords: Tonnetz, Fred Lerdahl, Riemannian tradition, Lerdahl's hierarchical model, spatial model, Schumann, Wagner, Chopin

MARTIN Vogel once observed that the Tonnetz “is so deeply anchored in the nature of the matter that it was ‘discovered’ twice”: in 1773 by Leonhard Euler, and again in 1866 by Arthur von Oettingen.1 Vogel's count can be supplemented if we take into account some nineteenth-century graphs that arise from different motivations and display different ori­ entations on the page, but are otherwise equivalent to Euler's. It is further augmented by recent work in music theory and music psychology, which bring at least five further redis­ coveries.2 The following catalog organizes these various Tonnetze according to the inter­ val classes (ICs) that generate their primary axes. IC4 × IC5. The Riemannian Tonnetz (presented, for instance, in his Musik-Lexikon, shown as example 9.2) represents the product of the nonoctave consonant intervals under 5-limit just intonation. The minor third, which exceeds the limit, falls out incidentally along the main diagonal, as the difference between the two principal intervals (in arith­ metic terms, where each interval is expressed in semitones, 7 – 4 = 3). In 1962, Christo­ pher Longuet-Higgins, an acclaimed British theoretical chemist and cognitive scientist

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Tonal Pitch Space and the (neo-)Riemannian Tonnetz who occasionally turned his attention to music, evidently arrived at the same figure inde­ pendently (although perhaps recognition of antecedents was not a priority for him).3 IC3 × IC4. In 1853, Carl Friedrich Weitzmann represented the twelve pitch classes (un­ der equal temperament) as a cross product of the augmented triads and the diminished seventh chords, in other words, the interval cycles of intermediate (p. 323) cardinality.4 The minor and major thirds generate the horizontal and vertical axes, respectively. The perfect fifth, as the sum of the principal intervals, generates the secondary diagonal (4 + 3 = 7). An identical figure was concocted independently by music psychologist Gerald Balzano (1980), at a time when Weitzmann's work was hardly known in North America.5 IC3 × IC5. In the 1760's, François-Guillaume Vial's Genealogical Tree of Harmony presented the twenty-four keys as a product of transposition by minor thirds and perfect fifths, the former interval arising by alternation of parallel and relative scales.6 An ex­ panded version of Vial's array appeared around 1820 in Gottfried Weber's Versuch, one of the most influential music-theory books of the 19th century.7 Major thirds arise residually along the secondary diagonal (9 + 7 = 4, modulo 12). Weber's figure was disseminated without attribution (1954) as Arnold Schoenberg's “Chart of Tonal Regions.”8 A similar figure, wrapped into a torus, appeared in Steven Lubin's 1974 dissertation,9 and in work of Fred Lerdahl (1988, 2001), who defines proximity formally as a product of the acoustic relationship of tonal centers and the degree of scalar intersection.10 Lerdahl's procedure for deriving the figure is sui generis, and he reports that his affiliation of the figure with Weber's (via Schoenberg) was ex post facto. IC3 × IC4 × IC5. Riemann's later writings appropriated a version of the Tonnetz that was introduced in 1879 by the Czech theorist and aesthetician Ottokar Hostinský, and was intended to coequally balance all three consonant interval classes.11 Hostinský, who was familiar with Oettingen's figure, aspired to improve it by balancing the two species of thirds on the two diagonals; in effect, he incrementally expanded the just limit from 5 to 6. Music psychologists Roger Shepard and Carol Krumhansl (1982) introduced a version of Hostinský's graph to represent a set of data derived from experiments in which listen­ er-subjects expressed their intuitions about the relative proximity of pairs of keys.12 Music theorist Brian Hyer (1989) introduced a related graph, consisting of triads rather than pitch classes, and providing a map to be navigated by a set of neo-Riemannian oper­ ations first proposed by David Lewin.13 I subsequently converted Hyer's triadic graph into a pitch-class graph, which, I noted, could be derived by connecting pairs of triads that share two common pitch classes, without any appeal to the acoustic properties of the tri­ ads.14 Just as striking as the sheer number of rediscoveries is that the same graph has been de­ vised in response to a number of distinct premises and urges. The Tonnetze of Euler, Oet­ tingen, Longuet-Higgins, and Hostinský are founded in acoustics: they respond to the sta­ tus of triads as maximal assemblies of consonant intervals. Weber's Tonnetz and mine are set-theoretic, reflecting pitch-class intersection among scales and triads, respectively. Consonance and pc-intersection combine to generate Lerdahl's representation. The Page 2 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz graphs of Weitzmann and Balzano capture group-theoretic properties of a modulo-12 pitch-class system, specifically the interaction of interval cycles. And Krumhansl simply records and organizes the responses of experimental subjects, neutral with respect to any particular interpretation or (p. 324) theoretical agenda. Like water, the Tonnetz is ostensi­ bly a universal solvent, the answer to many independently posed questions. The evident similarity of these graphs, however, masks some distinctions in underlying structure. Those graphs generated by acoustic features assume some form of pure tun­ ing, with nodes syntonically and enharmonically distinct, and implicitly projecting into an infinite plane. Those generated by other principles assume equal temperament, so their nodes reflect syntonic and enharmonic equivalence. Their generating axes are cyclic, and the underlying structure of the graph is a torus if there are two generating axes, a hyper­ torus if there are three.15 Following Daniel Harrison, I shall refer to these interpretations of the Tonnetz as nonconforming and conforming, respectively.16 The similarity of graphic form also masks distinctions in represented content. Some Ton­ netze are populated by pitch classes, others by major and minor triads, still others by keys or tonal regions. Although these three relational levels interact in a rich and com­ plex way, each follows somewhat distinct syntactic principles. Thus, for example, pitch classes B and C are proximate, triads B major and C major are relatively remote, and the keys of B major and C major even more remote. Moreover—to take a phenomenon that chronically confounds harmony students—C minor is closer to G major when we’re speak­ ing of chords, but closer to G minor when we assess distances between keys. Such devia­ tions suggest that perhaps one should be cautious about ascribing too much significance to the evident homology between these graphic structures. Fred Lerdahl's Tonal Pitch Space raises yellow flags of this type. Lerdahl proposes three distinct formulas for measuring proximity among pitch classes, chords, and tonal regions (keys). Because the formulas differ according to the status of the measured objects, so too do the geometries that map the aggregated distances at each of the three levels. Lerdahl's graph of regional space is equivalent to Weber's graph of keys, which, he notes, rotates into a Riemannian Tonnetz. He nonetheless rejects their kinship on the grounds that such a move would “conflate levels of description.” “At a fundamental level…the spaces are dissimilar: the letters in the Tonnetz denote pcs, while in [Weber space] they stand for regions.”17 Lerdahl instead charts the relationship of chord and region through nested arrays, to be considered below. Lerdahl has significant motivations, internal to his theory, for preserving the distinctions among these three levels.18 Outside of that context, however, there are good reasons for conflating pitch-class, chordal, and regional structure, and representing them in a single geometry. Both acoustic consonance and voice-leading parsimony cause the dyadic, tri­ adic, and regional structuring of pitch classes to mutually reflect and reinforce one anoth­ er. These reinforcements are well captured by the Riemannian Tonnetz, provided that that representation is squeezed hard enough to yield all of its interpretive juice. One conse­ quence of this mutual reinforcement is that a Tonnetz user—or, for that matter, a musi­ Page 3 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz cian or listener engaged in the musical experience that the graph is designed to repre­ sent—may enter the (p. 325) structure at any of the three levels, freely shift levels, and en­ dure some indeterminacy for a period of time at a given level of structure. The Tonnetz— or, for that matter, triadic music—therefore has all of the qualities of a “Babylonian” or “robust” structure, one that is derivable from several independent assumptions, and therefore has the capacity to overcome momentary coherence failures at a given level.19 This last quality is particularly significant when regional affiliation is obscured by either Mehrdeutigkeit or (what one might call) Schwebendigkeit—that is, by more or fewer than exactly one plausible assignment of a chord, phrase, or passage to a key. Such situations occur with great frequency in triadic 12-gamut compositions of the type that are charac­ teristically described by “neo-Riemannian theory,” where enharmonic indeterminacy pre­ cludes the determinate assignment of chord to key. Although this practice is prototypical of, and usually attributed to, the music of the late nineteenth century, it is present in much music of Schubert, Chopin, and their contemporaries. Moreover, tonal indetermina­ cy also arises in older and putatively less problematic music, where enharmonicity is not necessarily in play—toccatas and fantasies, development sections, diatonic sequences—or even when chromaticism is not at issue, as in the Wechselwirkung of triads root-related by perfect fifth, what Moritz Hauptmann (1853) conceives of as an unsynthesized dialec­ tic at the triadic level. This essay is motivated by my desire to explore the significant affinities between Lerdahl's research program and the neo-Riemannian program to which many of my own publications have contributed. My ultimate concern is to appropriate aspects of Lerdahl's representational model to sharpen my evolving view of how chromatic and diatonic spaces relate.20 One outcome of this process will be to narrow the gap between neo-Rie­ mannian theory and that of the German Vielschreiber for whom it is named and whose work stimulates this volume.

Pitch Class, Chord, and Region It is easy to demonstrate that the Tonnetz of pitch classes is equivalent to the Tonnetz of triads: the two structures are mutually implicative. Example 11.1 overlays two disconnect­ ed graphs on a segment of the Tonnetz. The nodes of the first graph are pitch classes, with solid edges connecting consonant pc pairs.21 These edges tile the plane into right tri­ angles, each of which is labeled with the name of a major (+) or minor (–) triad. These tri­ adic labels constitute the nodes of the second graph, with broken edges connecting triad pairs that share two pitch classes. Those edges tile the plane into hexagons, each of which is labeled with the name of the pitch class shared by its constituent triads. The la­ bels of each graph are the nodes of the other, and so they implicate each other as geomet­ ric duals.22

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Tonal Pitch Space and the (neo-)Riemannian Tonnetz

Ex. 11.1. (p. 326)

The claim that the Tonnetz of triads is equivalent to the Tonnetz of keys or regions

is a more controversial one. It is supported by an abiding strain of historical theory that maintains or implies that chord and key are two distinct facets, or representational levels, of a single underlying object. That the tonic harmony of a region must be a consonant tri­ ad is held by consensus. But what of the converse proposition, that a consonant triad must function as the tonic harmony of a region? This latter claim is most familiar, in a rel­ atively weak form, in the late writings of Heinrich Schenker, for whom the prototypical (in some of his writings: exclusive) object of prolongation is a consonant triad, and the proto­ typical means of prolongation is via linear operations that engage the remaining compo­ nents of the diatonic scale of which that triad is the tonic. Indeed, in Schenker's view, the principal function of the diatonic collection is to compose out the tonic triad; conversely, scales conceptually collapse at some level into the triads that they exfoliate. Thus, every consonant triad represents a diatonic region in potentia, and the listener is alert to this potential whether or not the composer chooses to actualize it. Schenker's view reflects an attitude that has some historical depth. Consider Johann Philipp Kirnberger's (1771–1780) heuristics on modulation to distant keys. He recom­ mends first attaining the dominant of some closely related key and then “imagining that it is a tonic triad of the main key.” This procedure may then be executed recursively until you achieve a triad that you want to establish as a tonic, at which point Kirnberger rec­ ommends a cadential formula that uses all the members of the diatonic region of which that chord is tonic.23 Accordingly, each triad is a thesis (p. 327) that is provisionally assert­ ed and whose affirmation (or, in Hauptmann's terms, synthesis) engages the remaining members of its diatonic scale. The idea is present in a stronger form in the early writings of Jean-Philippe Rameau (1720s and 1730s), who held that only tonic triads were truly consonant; other apparent triads harbored “supplementary dissonances” that were nei­ ther notationally or acoustically present.24 Adolph Bernhard Marx (1840) held a related view, as described by Nora Engebretsen: “Marx understands every major or minor triad to be ‘borrowed’ from the key in which it is tonic, and he claims that these triads stand in the same relation to one another as the keys they represent.”25 One sees the residue of this attitude in Schenker's (1906) assessment of the opening tutti from Beethoven's G-ma­ jor Piano Concerto: “our feeling gets confused by this continuous change of major tri­ Page 5 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz ads…because we feel tempted, step by step, to impute each one of them the rank of a ton­ ic.”26 A competing strain of historical theory, emerging from eighteenth-century thoroughbass treatises via Gottfried Weber (1821–1824) and François-Joseph Fétis (1844), holds that tonal regions are primarily represented by scales rather than chords. And this strain ac­ cords less well with the view that I am developing here. Scales arise as contiguous and symmetric regions of the pitch-class Tonnetz, but not as primary objects. In some of Riemann's representations, the natural diatonic collection, representing both C major and A minor, appears as an encapsulated region sheltered from the greater chromatic sea. Ex­ ample 11.2 presents a schematic representation, slightly modified to serve my purposes.27 Here the natural (“white-note”) collection appears as a parallelogram, bounded at its ex­ tremes by its two distinctly tuned instances of D, representing the subdominant and domi­ nant extremes of the collection, and bearing to each other the relationship that I shall term a syntonic image. Sliding the parallelogram one station to the west swaps B for B♭, to the east F for F♯, and to the southeast E, A, and B for their flatted versions. According­ ly, modulation to dominant, subdominant, relative, and parallel keys—in other words, those that are closely related on Weber's and Lerdahl's key table—involve either preserv­ ing position on the parallelogram or incrementally translating along one of its two axes. But this net is too coarsely woven. Because each scalar region is oriented to two distinct keys, C major must also be closely related on the horizontal axis not only to F and G ma­ jor but also to D and E minor, and on the diagonal axis not only to C minor but also to E♭ major, A major, and F♯ minor. And these are claims that neither Weber nor Lerdahl wishes to make about regional proximity.

Ex. 11.2.

Is the regional Tonnetz, then, identical to the chordal one? The response depends on our conception of how regions or keys, which are abstractions, are prototypically man­ (p. 328)

ifested in actual sounding structures. Are keys primarily represented by triads, and only secondarily by their scalar exfoliations; or primarily by scales, and only secondarily by the tonic triads selected from their components? This venerable dichotomy would seem to be beyond arbitration on theoretical grounds alone.28 Nonetheless, there are both pragmatic and empirical motivations for viewing regions and chords as similarly structured. The pragmatic motivation, which was alluded to in the introductory section of this essay, con­ cerns the indeterminacy of regional affiliation in many individual segments of triadic mu­ sic: whether a given triad is tonic of its own region, or to which of several competing re­ gions it is diatonically beholden. Mehrdeutigkeit, or what Charles J. Smith calls “function­ al extravagance,” is a hallmark of much nineteenth-century music.29 In these cases, one can either impose a determinate selection from among the plausible options, or disen­ Page 6 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz gage from the process of regional selection until such time as a cadence or some other tonally determining event intrudes. The first of these choices is too often merely arbi­ trary, and thus beyond arbitration; moreover, it requires the imposition of “noise” over the analytical apparatus, potentially masking otherwise significant features or issues, and numbing the analyst's degree of alertness toward them. But the second choice, the deci­ sion not to decide, requires us to suspend our engagement with the analytical enterprise for a while, if our analytical playground is Weber space. And these moments of regional indeterminacy may be just the moments that most draw us in to the musical experience, and where our need for an orienting space is most acute. The empirical motivation for viewing chords and regions as similarly structured is sug­ gested by experimental work of Carol Krumhansl and associates, who asked subjects to rate the “fit” of a presented pitch in relation to an established key. Her correlation of the results for each of the twelve pitch classes across each of the twenty-four tonal regions suggests an indirect measure of regional distance.30 Lerdahl asserts that these results support Weber's conception of regional space,31 but on a different interpretation they pro­ vide stronger support for the Tonnetz conception. From the standpoint of Weber space, the Tonnetz is deficient in respect of its promotion to primary status of keys whose tonic triads are L-related (e.g., C major and E minor), and its relegation to secondary status of modally matched tonics related by D and its inverse (e.g., C and G major, or C and G mi­ nor). Yet the promotion of L is strongly confirmed by Krumhansl's correlations, which rate the L relation as closer than that between the tonic-sharing parallel keys that Weber pro­ motes into his pantheon of closest relations. Moreover, the support for the relegation of D to secondary status is also warranted, but only if the so-related regions are both minor. Averaging the data for major and minor provides strong support for the promotion of L and weak support for the demotion of D and S. It is perhaps on this basis that Krumhansl joins a significant line of historical thinkers about tonality when she suggests that “the distances between (p. 329) keys…will be taken here to be approximately equal to the dis­ tances between tonic triads.”32 In order to begin to assess what is gained and lost by conflating the three levels of struc­ ture, the next three sections of the paper compare a conflationary model, the Tonnetz, with a TPS combined chord-region model, where the levels are separated but coordinat­ ed. An analysis of “Im wunderschönen Monat Mai,” from Schumann's Dichterliebe, intro­ duces the combined-region models to readers unfamiliar with them, and makes some ob­ servations about the capacities of the two modes of representation for the analysis of a piece that is regionally determinate at the local level. We then consider two passages ana­ lyzed in chapter 3 of Tonal Pitch Space, the Faith Proclamation from the Prelude to Wagner's Parsifal, and a phrase from Chopin's E-major Prelude, op. 28 no. 9. In both cas­ es, we consider several alternative TPS combined-region analyses and seek to resolve their differences through a Tonnetz reading. The Wagner passage coordinates the best features of all three models into a hybrid representation, while the Chopin analysis shows the capacity of the Tonnetz to bring out aspects of pitch-class symmetry and Riemannian functional progression. Page 7 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz

Im wunderschönen Monat Mai Example 11.3 presents a segment of Lerdahl's generic combined space, rotated clockwise by 90 degrees in order to facilitate comparison with the Tonnetz. My version frames each local 3 x 3 subarray to clarify its boundaries. Each subarray represents a diatonic region, whose tonic triad is indicated at the center in bold face (uppercase = major, lowercase = minor). Roman numerals represent “diatonic” triads in the standard way, in relation to the tonic at the center of the array. In minor, the seventh degree is borrowed from the parallel major in the case of V and viio; hence not all triads are diatonic in the strict sense. Within each subarray, triadic roots are arranged by generic fifths on the horizontal axis, and by generic thirds on the vertical axis. Both axes are conceived as cyclical, so that each subarray projects an underlying torus. The double representation of III and VI is an artifact of the flattening of circular space. The tonic of each subarray is nested into a larger array that presents regional (“Weber”) space, whose axes are generated by spe­ cific intervallic representatives of the generic intervals that generate the analogous sub­ array axes: perfect fifths horizontally, minor thirds vertically.

Ex. 11.3.

Example 11.4 traces the tonal motion of Dichterliebe's first song on the combined space. An edge with a single arrow represents a one-way harmonic succession, while a doublearrowed edge indicates a round trip. Two parallel edges, without arrows, indicate region­ al reinterpretation of a single harmony, the “pivot chord” of harmony textbooks. The mo­ tion of the song is divided into two graphs in order to counteract the diminishing legibili­ ty resulting from tangles. The motion of example 11.4a (p. 330) is preserved in example 11.4b but recedes into the background. This is one solution to a pragmatic problem that afflicts any two-dimensional representation of musical space, including the Riemannian Tonnetz, and is of no theoretical significance. Music is just that way: efficient traversal of space is not a priority, as it is, say, for shippers of commercial goods. Current technolo­ Page 8 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz gies make better solutions available, but not within the medium of the printed page. Readers may find it useful to imagine (or, if they have the aptitude and applications, bring into realization) a graphic animation, with variations in coloring, intensity, line thickness, and so on, perhaps linked to a sound file that presents the music to the ear as its facsimi­ le unscrolls before the eye. Example 11.4 nicely brings out, among other features, the kaleidoscopic role of the B-mi­ nor triad. We first hear it fleetingly as tonic of its own region, in part by default, in part because of the early sounding of A♯3. We then serially reinterpret it in terms of F♯ minor (half-tonicized), A major (twice tonicized), B minor (finally realized), and D major, a series of regions that is quickly recycled, famously terminating on the still-unrealized dominant of F♯ minor.

Ex. 11.4a.

Page 9 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz

Ex. 11.4b.

One issue is how to best represent the G-minor triad that sets the head of each verse's fi­ nal line. The two edges emerging from the D: vi chord in example 11.4b chart (p. 331) two alternative routes. The longer route connects through the subdominant of an unrealized D minor to a dominant that reorients toward D major. This solution has the virtue of bringing out G minor's mild contextual Fremdheit; but it suppresses what would other­ wise be a virtue of the model, namely its ability to capture the parallel approaches to the four potential tonics. The shorter route substitutes iv♭ in D-major's array of quasi-diatonic chords, echoing the precedent of V♯'s substitution for v in minor regions. Beyond its com­ pactness, this solution has the advantage of avoiding the stylistically dubious implication that this G-minor triad summons a D-minor tonic. (Already with Mozart, minor subdomi­ nants of major tonics are more common than transient (p. 332) unmarked modulations to chromatic regions.) But this solution cannot easily serve as a prototype: when juxtaposed in a single region, modal alternatives of a single Stufe would be forced to elbow each oth­ er for space in that region's subarray.

Page 10 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz

Ex. 11.5.

Example 11.5 presents an alternative representation of the same song, on an unscrolling series of Tonnetze. (Again, the reader might imagine how graphic animation technology might fuse these separate representations into a single, visually modulating space.) The primary objects here are pitch classes, composed into triangular major and minor triads (Klänge). Any sounded triad is eligible to stand as a candidate for regional tonic. The tri­ ad is nominated when horizontally neighboring triads aggregate in such a way that a pitch class is sounded or implied in the vicinity of its syntonic image (as in example 11.2, these images are circled), elected when it is sounded at the center of a region, and sworn in when certain metric and agogic conditions are met. Example 11.5 designates success­ ful candidacies with asterisks, and failed ones with broken edges, terminating in reversed and fragmented arrowheads. Example 11.5a shows Dichterliebe's initial B-minor triad as potential tonic of a di­ atonic region bounded by dual E naturals. This initial hypothesis dissolves at the sound­ ing of G♯5, which falls outside the region and suggests a region bounded by B natural in­ stead. This second suggestion is supported by the sounding of C♯7, half-tonicizing F♯ mi­ nor, and then by E7, leading to a twice realized A major that sets the first couplet of text (p. 333)

(example 11.5b). The initial suggestion is recuperated at the onset of the second couplet, when the syntonic-image potential of E is immediately reasserted by a motion from A ma­ jor to E minor (example 11.5c), a motion that example 11.4 was unable to accommodate into a single region because of the harmonic-minor bias of the chordal pitch space.33 The suggestion is then confirmed by the cadences in B minor and D major at the conclusion of the second couplet (example 11.5d). As an analysis of “Im wunderschönen Monat Mai,” example 11.5 has a few other merits. The conversion of broken lines at 11.5c to solid lines at 11.5d calls attention to dyadic gap filling. The near-filling-in of triadic space, with the arrival of D major, calls attention to the absence of F♯ minor, the only remaining gap in the connected triadic space covered Page 11 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz by this sector of the Tonnetz.34 Example 11.5 also brings out a continuity along the majorthird axis, connecting A♯ to its B♭ enharmonic diesis, and suggesting for them a dual lead­ ing-tone role, the latter summoning the fifth of D major, the former the root (or Riemann­ ian dual under-fifth) of its B-minor relative. Moreover, the graph underlines the unifying role of the B minor triad, as it reappears in different tonal contexts, by locating it at a uni­ tary site in the model. But analytical power is beside the point that I wish to emphasize: with only a thin inter­ pretive overlay, the Tonnetz does the same work as the combined TPS-space graph, show­ ing the interaction of chord and key, while adding the pc level as a bonus. The principal difference between the two models pertains to the status accorded to regional space. For Lerdahl, regional space is a perpetual presence, the necessary envelope within which tri­ ads are cognitively organized. The sounding of a triad commits us to discover a location for it in regional as well as chordal space, and the sounding of a triadic succession com­ mits us to navigate regional super-space at the same moment that we are surfing the chordal subarrays that they contain. By contrast, in the interpretation of the Tonnetz offered here, although the regional location of triads is always cognitively potent, those potentials are mobilized only to the point of organizing the chordal level under special conditions. Those conditions are fulfilled in the Schumann song, but less determinately in the next two analyses.

The Faith Proclamation from Parsifal

Ex. 11.6.

Page 12 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz

Ex. 11.7.

Example 11.6 presents eleven measures from the prelude to Act I of Parsifal. This music, known as the Faith Proclamation, consists of three segments in bar form. The second Stollen transposes the first upward by minor third. The Abgesang, which begins as a fur­ ther minor-third transposition, devolves into an extended phrase that cadences in E♭ mi­ nor. Example 11.7 presents a rotated and cosmetically (p. 334) (p. 335) clarified version of Lerdahl's analysis, which indicates that the first two Stollen execute half cadences in A♭ and C♭ major, and that the Abgesang begins in D major and tonicizes C♯ minor before reaching E♭ minor.35 The graph traces a coherent path downward along a minor-third ax­ is, followed by a sharp departure that reflects the remote quality of C♯ minor, and termi­ nating in a partial recuperation: both column- and row-wise, E♭ minor moves toward tonic without reaching it in either dimension.

Page 13 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz

Ex. 11.8.

Example 11.8 presents a plagal analysis of this same music. Three contextual factors sup­ port this hearing. Globally, plagal cadences play a significant role in the (p. 336) opera's tonal logic and semiotic network.36 Locally, the first Stollen follows a tonicization of E♭ major through a plagal extension of the Dresden Amen. And associatively, the melodic journey—anacrustic tonic, climax on a metrically accented 4̂, inverse arch, metrically ac­ cented cadential 3̂—has already been presented as a motivic topos at the conclusion of the prelude's opening phrase (Spear motive, mm. 4–6). By analogy, the Abgesang begins with a “deceptive” plagal cadence in B♭♭ major (written by Wagner as A, a notational rather than real enharmonicism), lingering for a time in that region but never sounding its tonic. The F♭-major triad serves as a pivot that exits the B♭♭ region and reappears in the vicinity of E♭ minor as the VI of its subdominant. It would be nice to show F♭ major functioning as E♭ minor's “Neapolitan” ♭II, but as noted above, the TPS geometry is unable to establish a direct relation between mixture chords and region­ al tonics. In any case, example 11.8 tells pretty much the same story as example 11.7, but its basis in a single column makes it more compact and coherent. I would, nonetheless, not argue too hard for example 11.8 as the “correct” TPS interpreta­ tion of this passage. The relationship between A♭ major and E♭ major is ultimately under­ determined; for Sechter, it constitutes (following Kant?) a Wechselwirkung, or reciprocal exchange; for Hauptmann, a thesis and antithesis without synthesis. In measures 45–46 of Parsifal, that harmonic underdetermination is topically ensnared with the Wechsel­ wirkung of the G/A♭ semitone, which was first presented in both the diatonic and chro­ matic versions of the Communion music.37 In such situations, it is advantageous to have a conceptual and representational system that does not mandate a determination. Example 11.9 responds to such concerns with a Riemannian Tonnetz. Like example 11.8, it assigns a reorienting role to F♭ major, emphasizing motion through a syntonic seam, rather than pivoting function between distantly related keys. This reorientation may, however, lack Page 14 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz significance, depending on how we conceive the structure of the geometry that underlies the Tonnetz. Instead of moving through the syntonic seam, example 11.9 could have con­ tinued eastward from F♭ major as in example 11.10, which treats D♭ minor as double-sub­ dominant of E♭ minor, the latter key appearing at a location remote from the opening Stollen. Such “plagal drifts” are characteristic of late Wagner in general and Parsifal in particular, and indeed Scott Murphy has emphasized exactly this aspect in an unpub­ lished analysis of the Prelude's Faith music.38 Such an analysis makes a distinctly differ­ ent impression on the eye, but from the standpoint of a “conforming” Tonnetz, this is a distinction without a difference. Both representations lose information, and so the choice between them will be motivated by pragmatic considerations: one option is more com­ pact, while the other is more likely to avoid tangles, and hence is easier to read.

Ex. 11.9.

Ex. 11.10.

Ex. 11.11.

Although example 11.9 has the advantage of echoing the diatonic indeterminacy of the first two-thirds of the passage, it fails to capture the tonal and regional orientation of the Abgesang's final measures. A hybrid model such as example 11.10 provides one possible solution. So long as the regional focus is underdetermined, the model adheres to the Ton­ netz of example 11.9. As the gravitational field of E♭ minor comes into focus, the triads take their position with respect to that emerging (p. 337) tonic. The graph of example 11.10 identifies this reorientation through a double edge that indicates the reinterpreta­ tion of A♭ minor in terms of the emerging diatonic region. There is some awkwardness in adjusting from the pitch-class Tonnetz at the regional level to its chordal dual in the local Page 15 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz subarray. Example 11.11 provides another possibility, transferring the technology devel­ oped in connection with example 11.5. E♭ minor's force is indicated here via the circles around the dual Abs, which mark out the boundaries of the emerging diatonic region, and by the asterisk indicating that triad's realization of its full tonicizing capacity.

Chopin, Prelude op. 28 no. 9, Second Phrase (p. 338)

The final analysis, of Chopin's Prelude in E major op. 28 no. 9, raises similar problems, and offers a similar range of solutions. In addition, it brings the analytical apparatus into proximity with Riemann's own conception of the Tonnetz. The Prelude, which is repro­ duced in example 11.12, contains three phrases whose increasing chromaticism is bal­ anced by a decreasingly dissonant set of surface harmonies. Example 11.13 presents an axis-rotated version of Lerdahl's analysis of the second phrase. Lerdahl hears a symmetri­ cal division of the E octave whose broad motion is captured by a southwestward drift: first to C major, then via an indirect path to a fully realized A♭ major, and finally threading the enharmonic diesis to an E positioned at a remote location. The path from C to A♭ halftonicizes F, by giving C (end of measure 6) a chordal seventh and then inflecting its root upward, forming a leading-tone seventh chord of F (measure 7, beats 2 and 4) that is then reinterpreted in A♭.39 These weak-beat dissonances surround the two strong-beat triads on measure 7 (A major 5/3 and B♭ minor 6/4) and wring the harmonic status out of them via voice exchanges.40 This A major triad, however, is at once the most stable metric event, and the only E-ma­ jor-diatonic event, in the interior of the second phrase, and so one would like to find a role for it. Example 11.14 recuperates this subdominant triad to an alternative TPS-style analysis. Where example 11.13 hears the fourth beat of measure 6 as the unconsummated dominant of F, example 11.14 hears it as an unconsummated “diminished third” of E, a possibility that encourages a hearing of the first two-plus measures of the phrase in rela­ tion to a mixed E major/minor region. (This interpretation responds in part to the resem­ blance between the end of measure 6 and the fermata-roofed chord at measure 23 of the companion E-minor Prelude.) A major is now reinterpreted as ♭II of G♯ major, stimulating an orientation toward that region. (As explained in connection with example 11.8, the combined geometry represents this reorientation indirectly, via the (minor) subdominant.) This suggests that the flat-side spellings are for notational convenience, no enharmonic seam is traversed, and the cadential goal of this phrase lies one major third above the tonic, rather than two major thirds below it. The E major triad that opens the third phrase, in this reading, is “the same” as the opening tonic, rather than its enharmonicdiesis image, as in example 11.13.

Page 16 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz

Ex. 11.12.

Ex. 11.13.

As with the Wagner analysis, I do not want to push example 11.14 too hard as the “cor­ rect” reading of this passage. A major is metrically accented and globally more stable, but C major is agogically accented and locally more stable, being the beneficiary of its own dominant. B♭ in measure 6 is plausibly heard as an A♯, but it enters as a component of a G-minor triad, and resolves as notated. Example 11.15, which traces the triadic motion on the Tonnetz, adopts a more neutral stance with respect to the two readings. Some infor­ mation is lost: dominant seventh/German sixth chords are normalized to their consonant subsets, and diminished seventh chords (p. 339) (p. 340) are elided over. But the graph's compactness allows some significant features of the passage to emerge: the central posi­ tion of its tonic, surrounded in all directions and approximately to the same degree, and Page 17 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz the symmetry of the path, which roughly takes the shape of a book jacket as seen from above.

Ex. 11.14.

Ex. 11.15.

Ex. 11.16.

Page 18 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz Vertically, that symmetry is reflected in a progression from the middle row to the bottom, overcompensating to the top and terminating in the middle. On this dimension, graphic motion reflects chromatic inflection, the transformation of a (p. 341) letter class by adding or subtracting an accidental. The downward motion at the opening reflects the “naturaliz­ ing” of G♯ and D♯ at measure 5; the upward motion at the interior reflects the sharpening of C, E, A, and B in measure 7 and the opening of measure 8. The downward motion at the close captures the elimination of those sharps, including the salient B♯ to B motion at the end of measure 8. The arrows of (p. 342) example 11.16, mimicking the rough flow of the triadic path, capture the most aurally direct of these letter-class transformations. Horizontally, that symmetry is reflected in a palindromic column progression from the middle rightward, overcompensating to the left, reversing the path, and ending in a G♯ (qua A♭) cadence. On this dimension, directed graphic motion reflects directed diatonic step displacement: rightward motion in graphic space corresponds to downward step mo­ tion; leftward motion in graphic space corresponds to upward step motion. The initial rightward motion captures the E → D♯ and G♯ →F ♯ voice-leading as B major displaces E major. The subsequent leftward motion traces F♯ → G as G major displaces B major, D → E and B → C as C major displaces G major, and so forth. Horizontal motion of the graph thus transmits the step-class flow across the phrase: the initial step-class downshift is followed by an upshift through measure 7, a downshift beginning at the downbeat of measure 8 (the climactic overreaching G♯ qua A♭4 is immediately replaced by the D♯ qua E♭4 that is the proper displacement of E♯ qua F4 at the third beat of measure 7), and a step-class up­ shift at the measure 8 cadence. The annotations beneath example 11.17 interpret these directional step motions in terms of Riemann's three functions. The right side of the graph contains the dominant and all chords that plausibly bear dominant function; accordingly, rightward graphic motion cor­ responds to a plagal musical sensation. Conversely, the left side of the graph is affiliated with subdominant function, and leftward motion with an authentic sensation. After an ini­ tial step to dominant, measures 5–6 work their way toward subdominant. The final two measures move back to the dominant side, and then step back to the tonic at the ca­ dence. Example 11.18 suggests that this perspective has a payoff: the three phrases are heard to pursue different pacings through the same functional path.

Page 19 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz

Ex. 11.17.

Ex. 11.18.

The perspective developed in examples 11.17 and 11.18 brings us to Hugo Riemann's doorstep: we are interpreting the master's favorite graphic space in terms of his favorite conceptual categories. Strangely, though, we have not yet earned a position on his (p. 343) overworked Schreibtisch: Riemann navigated the Tonnetz not with functions, but with Schritte and Wechsel.41 It was left to his successors to integrate the Tonnetz with the posthumous reception of Riemann's functional theory in Germany.42 More recently, Daniel Harrison has treated the dominant and subdominant functions in a post-Riemannian dual­ ist framework whose prototypical components, major and minor, are allied with other du­ alized forces: 7̂–1̂ / 6̂–5̂, dominant/subdominant, authentic/plagal, ascending/descending semicadences, and sharp/flat.43 The perspective that we have been cultivating here sug­ gests that all of these antitheses are assimilable into the most fundamental, unassailable duality of all: the two directions of pitch space, or of the linear continuum upon which pitch space is metaphorically mapped.44 From this perspective, triadic motion on the Ton­ netz reduces to three states: neutralized (p. 344) voice-leading, along the diagonal axis that defines transposition by major third; “upward” motion, to the left and above that ax­ is; and “downward” motion, to the right and below that axis.45 One may incline to con­ ceive of this continuum, alternatively, in terms of motion toward the dominant or subdom­ inant. Riemann would reserve such terms for cases where the triads are heard and con­ ceived with respect to a well-defined tonic. In this essay, I have cultivated the notion that such cases are special cases, not the general rule. They come into play only when certain criteria are met. In the general case, directed triadic motion on the Tonnetz is up or down, depending on the voice leading. Only in the specific case do those motions convert

Page 20 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz to the functional terms of subdominant and dominant, authentic and plagal, and so forth.46

A Hybrid Model of Triadic Listening I have argued elsewhere that triads look in two directions: inward toward their acoustic roots and the interpretation of those roots as tonics, and outward toward their position in the chromatic system of twelve pitch classes under equal temperament. They are optimal objects in both directions: in reference to acoustic roots, they are the largest-cardinality consonant collection, while in the chromatic system they bear optimal voice-leading ca­ pacities.47 The historical power of diatonic tonality predisposes the musical mind toward the inward interpretation: the sounding of a triad against a blank ear characteristically motivates one or several hypotheses of a diatonic region in which that triad is embedded, at a first guess as a tonic, at a second guess as representing some other diatonic degree. But subsequent events may survey a panorama of contiguous diatonic regions without ca­ dencing, or make incursions into an unrecuperated chromatic space and penetrate the enharmonic seam, in which case none of our hypotheses is confirmed. In that case, we have recourse to an alternative mode of coherence, optimizing an alternative set of rela­ tions into which triads are capable of entering. Lerdahl's graphic models are capable of mapping both capacities of the triad. Regional clarity is graphed via a chordal subarray, regional modulation via a combined regional/ chordal space, and regional ambiguity or nullification via a regional super-array whose nodes are triads rather than regional tonics. The Tonnetz has a similar set of double ca­ pacities. Regional clarity involves the creation of boundaries that encapsulate the region, partitioning pitch classes and chords into diatonic insiders and chromatic outsiders, re­ gional modulation involves the moving of those partitions, and regional ambiguity or nulli­ fication the dismantling of them.48 As we navigate the regional-qua-chordal super-array or the nearly identical unbounded Tonnetz, we are alert to the possible reassertion of re­ gional discipline on the triadic objects that are engaged. When several triads are sounded in a contiguous horizontal segment of the space, we formulate a hypothesis rooted in the syntactic mechanisms of diatonic tonality. This hypothesis is subject to the same confir­ mation criteria as the initial one. Confirmation reorients our attention from the far(p. 345) flung space toward a well-defined sector of it, and ultimately to a particular loca­ tion in that space, the cadenced triad. Like entering the four walls of our home, or sprout­ ing landing gear and coming within view of a planet, or of an airport. Or getting off the freeway and entering our community, which is well defined, perhaps even gated. On this conception, the Tonnetz and the chord-region graphs resemble a Toyota Prius: one engine for the highway and one for local driving. When one exits the highway of chords qua re­ gions, the chordal arrays and the encapsulated Tonnetz serve as efficient vehicles for cruising the local neighborhoods, with all of their tensions and fraught attractions.49 At this local level, the graphic interfaces of the TPS and Tonnetz models diverge. There are advantages to both graphic protocols. The TPS combined space reifies the regions, Page 21 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz and allows a progression between regions to be charted in a single static representation. The Tonnetz can capture the moving of boundaries only through a series of freeze-frame images, a limitation that can be overcome only by animation capacities unavailable to the print medium. But there are several compensations. The Tonnetz encourages the tracking of pitch classes and dyads independent of their triadic constituency, just as it encourages the tracking of triads independently of regions. Its information converts easily to an inter­ pretation in terms of Riemannian functions. When pitch-class and triadic motion consoli­ dates into hexatonic and octatonic spaces, one simply steers in a different direction on the Tonnetz, without reorganizing the geometry or recalibrating the distances between events. And the Tonnetz has the pragmatic advantage of visual compactness, so long as tangles can be avoided. These distinctions are a matter of feature and inflection, not tonal ontology. The two modes of representation essentially allow the same sorts of claims to be made about diatonic structure and its relation to triadic chromaticism.

Notes: (1.) Martin Vogel, On the Relations of Tone, trans. V. Kisselbach (Bonn: Verlag für system­ atische Musikwissenschaft, 1993), 108. Kevin Mooney, in “The ‘Table of Relations’ and Music Psychology in Hugo Riemann's Harmonic Theory” (Ph.D. diss., Columbia Universi­ ty, 1996), 29–30, shows that Euler anticipated his 1773 graph already in his 1739 Tenta­ men. Edward Gollin shows that Oettingen appropriated the graph from the 1858 disserta­ tion of E. A. Naumann; see “Some Further Notes on the History of the Tonnetz,” Theoria 13 (2006): 99–111. (2.) The figure is probably somewhat higher; Roger Shepard lists a handful of additional sources, including one from a physics publication, in “Structural Relations of Musical Pitch,” in Diana Deutsch. ed., The Psychology of Music (New York: Academic Press, 1982), 378. (3.) Christopher Longuet-Higgins, “Letter to a Musical Friend,” Music Review 23 (1962): 244–248. (4.) Carl Friedrich Weitzmann, Der übermäßige Dreiklang (Berlin: T. Trautwein, 1853), 22; trans. Janna K. Saslaw, “Two monographs by Carl Friedrich Weitzmann. I: The Augmented Triad (1853),” Theory and Practice 29 (2004): 133–228. (5.) Gerald Balzano, “The Group Theoretic Description of 12-Fold and Microtonal Pitch Systems,” Computer Music Journal 4.4 (Winter 1980): 72. (6.) Joel Lester, Compositional Theory in the Eighteenth Century (Cambridge, MA: Har­ vard University Press, 1992), 229–230. (7.) Gottfried Weber, Theory of Musical Composition, trans. James F. Warner (Boston: Wilkins, Carter, 1846), 320. (8.) Arnold Schoenberg, Structural Functions of Harmony, rev. ed. (New York: W. W. Nor­ ton, 1969), 20. Page 22 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz (9.) Steven Lubin, “Techniques for the Analysis of Development in Middle-Period Beethoven.” Ph.D. diss., New York University, 1974. (10.) Fred Lerdahl, “Tonal Pitch Space,” Music Perception 5.3 (1988): 315–349; idem., Tonal Pitch Space (New York: Oxford University Press, 2001), 64–65. (11.) Ottokar Hostinský, Die Lehre von den musikalischen Klängen: Ein Beitrag zur aes­ thetischen Begründung der Harmonielehre (Prague: H. Dominicus, 1879). (12.) Shepard, “Structural Relations”; Carol Krumhansl and Edward Kessler, “Tracing the Dynamic Changes in Perceived Tonal Organization in a Spatial Representation of Musical Keys,” Psychological Review 89 (1982): 334–368; Krumhansl, Cognitive Foundations of Musical Pitch (New York: Oxford University Press, 1990), 45–48. (13.) Brian Hyer, “Tonal Intuitions in Tristan und Isolde” (Ph.D., diss., Yale University, 1989); idem, “Reimag(in)ing Riemann,” Journal of Music Theory 39.1 (1995): 101–138. Lewin's operations are identical to three of the twelve Wechsel first identified by Riemann in Skizze einer Neue Methode der Harmonielehren (Leipzig: Breitkopf und Härtel, 1880). See Mooney, “The Table,” 236. (14.) Richard Cohn, “Neo-Riemannian Operations, Parsimonious Trichords, and Their Ton­ netz Representations,” Journal of Music Theory 41.1 (1997): 1–66. (15.) This underlying structure is retrospectively implicit in the graphs of Weber and Weitzmann, but has been explicit in all of the cited publications of the last forty years. (16.) Daniel Harrison, “Nonconformist Notions of Nineteenth-Century Enharmonicism,” Music Analysis 21.2 (2002): 115–160. (17.) Lerdahl, Tonal Pitch Space, 45, 84. (18.) Elsewhere I argue, however, that the chord/region distinction causes internal prob­ lems for Lerdahl's theory as well. See my “Review of Tonal Pitch Space,” Music Theory Spectrum 29.1 (2007): 101–114. (19.) William C. Wimsatt, “Robustness, Reliability, and Overdetermination,” in M. Brewer and B. Collins, ed., Scientific Inquiry in the Social Sciences (San Francisco: Jossey-Bass, 1981), 124–163. Wimsatt attributes the Babylonian ascription to Richard P. Feynman, The Character of Physical Law (Cambridge, MA: MIT Press, 1965). (20.) For a different approach to the relationship between tonal pitch space and neo-Rie­ mannian theory, see Michael Spitzer, “The Metaphor of Musical Space,” Musicae Scienti­ ae 7 (2003–2004): 101–118. (21.) Lerdahl defines pc-proximity as chromatic distance (Tonal Pitch Space, 49). Defining it in terms of consonance, though, is more consistent with his practice elsewhere of corre­ lating proximity with “goodness of fit.” Krumhansl's subjects judged consonant pcs to fit a

Page 23 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz tonic better than those that are semitonally proximate. This point is further elaborated in my review of Tonal Pitch Space. (22.) Jack Douthett and Peter Steinbach, “Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition,” Journal of Music Theo­ ry 42.2 (1998): 241–263; Edward Gollin, “Representations of Space and Conceptions of Distance in Transformational Theories” (Ph.D. diss., Harvard University, 2000), 238 n. 24. (23.) Johann Philipp Kirnberger, The Art of Strict Musical Composition, trans. David Beach and Jürgen Thym (New Haven: Yale University Press, 1982), 141–142. (24.) Thomas Christensen, Rameau and Music Theory in the Enlightenment (Cambridge: Cambridge University Press, 1993), 131. (25.) Nora Engebretsen “The Chaos of Possibilities: Combinatorial Group Theory in Nine­ teenth-Century German Harmony Treatises” (Ph.D. diss., State University of New York at Buffalo, 2002), 70. (26.) Heinrich Schenker, Harmony, trans. Elisabeth Mann Borgese (Chicago: University of Chicago Press, 1954), 254. (27.) Candace Brower's “Paradoxes of Pitch Space,” Music Analysis 27, no. 1 (2008): 51– 106, also explores the chromatic Tonnetz as an expansion of an encapsulated diatonic one. (28.) Carl Dahlhaus provides a sketch of the positions as part of a wide-ranging compari­ son of Riemann's and Fétis's views of tonality, in Studies on the Origin of Harmonic Tonali­ ty, trans. Robert Gjerdingen (Princeton: Princeton University Press, 1990), 9–12. (29.) Charles J. Smith, “The Functional Extravagance of Chromatic Chords,” Music Theory Spectrum 8 (1986): 94–139. On Mehrdeutigkeit, see Janna K. Saslaw and James P. Walsh, “Musical Invariance as a Cognitive Structure: ‘Multiple Meaning’ in the Early Nineteenth Century,” in Ian Bent, ed., Music Theory in the Age of Romanticism (Cambridge Universi­ ty Press, 1996), 211–232. (30.) These results are summarized in Krumhansl, Cognitive Foundations, chapter 2. (31.) Tonal Pitch Space, 82. (32.) Carol Krumhansl, “Perceived Triad Distance: Evidence Supporting the Psychological Reality of neo-Riemannian Transformations,” Journal of Music Theory 42.2 (1998): 267. (33.) Example 11.4a could have accommodated this possibility by replacing viio with VII in the b-minor subarray, as with the minor subdominant of D major in example 11.4b. (34.) Completions of this type are introduced, in a different context, in David Lewin, “Notes on the Opening of the F♯ Minor Fugue from WTC 1,” Journal of Music Theory 42.2 (1998): 235–238. Page 24 of 26

Tonal Pitch Space and the (neo-)Riemannian Tonnetz (35.) Tonal Pitch Space, 127. (36.) See David Lewin, “Amfortas's Prayer to Titurel, and the Role of D in Parsifal: The Tonal Spaces of the Drama and the Enharmonic C♭/B,” 19th-Century Music 7.3 (1984): 336–349; reprinted in idem, Studies in Music with Text (Oxford: Oxford University Press, 2006), 183–200. (37.) On the underdetermination of the diatonic Liebesmahl, see Lewin, “Amfortas's Prayer;” on the chromatic versions, see Richard Cohn, “Hexatonic Poles and the Uncanny in Parsifal,” Opera Quarterly 22.2 (2006): 230–248. (38.) Scott Murphy, “Wayward Faith: Divergence and Reconciliation of Melodic Sequence and Harmonic Cycle in Some Measures from the Prelude of Wagner's Parsifal,” talk pre­ sented at the annual meeting of the Society for Music Theory, Philadelphia, November 2001. On plagal drifts in Wagner, see Hyer, “Tonal Intuitions,” 218–226, and Lewin, “Amfortas's Prayer.” (39.) It is unclear why Lerdahl interprets C7 with respect to both unrealized F-modes. F minor provides a context for B♭ minor 6.4 at beat 3, but that event lacks harmonic status and hence should have no impact on the regional analysis. Example 11.12 offers a simpli­ fication (“Cohn's alternative”) that bypasses F minor. (40.) Lerdahl's hearing follows Edward Aldwell and Carl Schachter, Harmony and Voice Leading, 2nd ed. (New York: Harcourt Brace, 1989), 539. (41.) Alexander Rehding, Hugo Riemann and the Birth of Modern Musical Thought (Cambridge: Cambridge University Press, 2003), 59. For a different view, see Mooney, “The ‘Table of Relations,’” 99. (42.) See Renate Imig, Systeme der Funktionsbezeichnung in den Harmonielehren seit Hugo Riemann (Düsseldorf: Gesellschaft zur Förderung der systematischen Musikwis­ senschaft, 1970). (43.) Daniel Harrison, Harmonic Function in Chromatic Music (Chicago: University of Chicago Press, 1994), 27. (44.) Our musical culture conceives of that continuum in terms of up and down; others do so in terms of young/old, small/large, or sharp/dull. See Lawrence Zbikowski, Conceptual­ izing Music: Cognitive Structure, Theory, and Analysis (New York: Oxford University Press, 2002), 67–68. (45.) Although the invitation to conceive the up/down pitch continuum in terms of north­ west/southeast may initially seem perverse, our bodily interactions with musical instru­ ments show us to be malleable in this regard. For a pianist, up is right; for a cellist, up is down; and for a flutist, up is left. I suspect that for most musicians, kinesis precedes lo­ gos as the foundation of internalized musical geometry.

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Tonal Pitch Space and the (neo-)Riemannian Tonnetz (46.) I elaborate this point extensively in Chapters 5 and 6 of Richard Cohn, Audacious Eu­ phony: Chromatic Harmony and the Triad's Second Nature (New York: Oxford University Press, forthcoming 2011). (47.) Richard Cohn, “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions,” Music Analysis 15.1 (1996): 9–40; idem, “Neo-Rie­ mannian Operations.” The argument is considerably elaborated in my forthcoming book, Audacious Euphony, and in Dmitri Tymoczko, The Geometry of music (New York: Oxford University Press, 2011). (48.) The multiple coherences of the Tonnetz were first identified in Brian Hyer's disserta­ tion (“Tonal Intuitions,” 214–215), and his language bears reproduction here. His “group” is geometrically represented by a conforming Tonnetz, a hypertorus with syntonic and en­ harmonic images fused: The group itself…does not have a tonic triad. The group…disperses the presence normally accorded to the signified tonic throughout its structure…. The assertion of a tonic can be said to warp the symmetrical surface of the lattice. It imposes a sense of perspective on the surface, a point of view from which all the other major and minor triads appear to be near by, more or less distant, or beyond the hori­ zon…. The effect of hearing any given major or minor triad as a tonic is also to de­ circularize the whole representation of the group…. The moment we hear any ma­ jor or minor triad as a tonic we immediately measure all relevant…relations with respect to that particular triad. The relations we do hear with respect to the tonic, moreover, are measured in modal intervals…. When it becomes strained to hear relations between triads with respect to a given tonic triad, then we in fact no longer hear that triad as a tonic. At that moment…the circularized form of the lat­ tice comes back into play.

(49.) Chapter 8 of Audacious Euphony presents further discussion of the convertible Ton­ netz.

Richard Cohn

Richard Cohn is Battell Professor of Music Theory at Yale University and editor of the Oxford Studies in Music Theory series. His book on triadic progressions in nine­ teenth-century music is forthcoming from Oxford University Press, and a book on geometric modeling of metric states is in preparation.

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Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Har­ monieschritte   Nora Engebretsen The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0012

Abstract and Keywords This article examines Riemann's discussion of the Harmonieschritte within the Skizze, lo­ cating that discussion within a nineteenth-century combinatorial tradition shaped by Riemann's conception of key. In this article, Riemann's Harmonieschritte is examined from three neo-Riemann standpoints. The first section offers a short introduction to the Harmonieschritte and examines neo-Riemann theory's embrace of these relationships, emphasizing the conflation of functional relations and root-interval transformations that this embrace has entailed. The second section discusses the development of Riemann's system of Harmonieschritte despite of neo-Riemannian theorists's acknowledgement of the system's susceptibility to interpretation as a group of transformations on the conso­ nant triads. The third section focuses on tonal coherence, with particular interest on Riemann's recognition that Harmonieschritte might portend the sort of harmonic practice embraced by neo-Riemannians if left unchecked. The article concludes with a translation of Riemann's Systematik der Harmonieschritte, the summary of the complete “chromatic” family of triadic relations from the Skizze. Keywords: Harmonieschritte, Skizze, conception of key, neo-Riemann theory, functional relations, root-interval transformations, tonal coherence

THE development of a neo-Riemannian practice, derived from relationships central to Hugo Riemann's conception of harmonic functions, has contributed to a broad reap­ praisal of Riemann's theories and has given rise, in particular, to a revival of his system of Harmonieschritte. Recent interest in Riemann's Harmonieschritte, a wide-ranging taxono­ my of root-interval relations between chords that existed alongside and supplemented his functional theory, stems primarily from neo-Riemannian theorists’ recognition of an affini­ ty between their transformational systems and Riemann's root-interval system (commonly referred to in neo-Riemannian writings as the Schritt/Wechsel system or Schritt/Wechsel group).1 Richard Cohn has remarked that “the S/W system anticipates the triadic trans­ formations of Lewin and Hyer in spirit as well as in substance” insofar as its capacity to account for triadic relationships independent of key anticipates the fully chromatic orien­ Page 1 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte tation of the neo-Riemannian approach and its dually conceived, directed-root-interval re­ lations can be interpreted as (p. 352) being “essentially transformational.”2 From this per­ spective, the system of Harmonieschritte in effect constitutes Riemann's adumbration of the neo-Riemannian perspective. The neo-Riemannian appropriation of the Harmonieschritte is well founded in many re­ spects, and yet the characterization of the Harmonieschritte as a group of transforma­ tions akin to neo-Riemannian transformations somewhat distorts Riemann's conception of these relationships. Riemann's Harmonieschritte, like the PLR family of operations, do, as Cohn notes, reflect a dualist model of relationships between consonant triads and de­ scribe these relationships without reference to key. Other correspondences can be cited, such as the systems’ similar combinatorial underpinnings, to be discussed below. In spite of these significant resemblances, however, the framing of Riemann's Harmonieschritte as an anticipation of a neo-Riemannian perspective is at odds with Riemann's ideas about these relationships and their bearing on matters of coherence, in particular: although the Harmonieschritte describe relationships independent of key, Riemann did not necessarily hold these relationships to confer a strong sense of coherence independent of key—essen­ tially the position assumed by neo-Riemannian theorists, who have located the source of a nonfunctional tonal coherence in the combinatorial logic of a handful of triadic transfor­ mations.3 On the contrary, Riemann clearly recognized the threat to tonality posed by such an interpretation of the Harmonieschritte, and indeed one of the outgrowths of the renewed interest in the system of Harmonieschritte has been a reappraisal of its relation to Riemann's theory of harmonic functions, which would seem to impose certain con­ straints on harmonic succession.4 This essay examines Riemann's Harmonieschritte from three standpoints strongly in­ formed by neo-Riemannian theory. The first section provides a brief introduction to the Harmonieschritte and surveys neo-Riemannian theory's embrace of these relationships, emphasizing the conflation of functional relations and root-interval transformations that this embrace has entailed. The second section explores the development of Riemann's system of Harmonieschritte in light of neo-Riemannian theorists’ recognition of the system's susceptibility to interpretation as a group of transformations on the consonant triads. The discussion situates Riemann's treatment of triadic relationships within a com­ binatorial tradition, extending back into the mid-nineteenth century, in which all relation­ ships between consonant triads are understood through reference to the composition of a few directly intelligible relationships and in which emphasis is placed on these generative relations as bearers of coherence. The third section builds on this discussion of tonal co­ herence, focusing on Riemann's recognition that Harmonieschritte might portend the sort of harmonic practice embraced by neo-Riemannians if left unchecked. The essay is fol­ lowed by a translation of Riemann's Systematik der Harmonieschritte, the most thorough exposition of the Harmonieschritte in the Skizze einer neuen Methode der Harmonielehre (1880) and the source most frequently cited in neo-Riemannian discussions of his Schritt/ Wechsel system.

Page 2 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte

The Harmonieschritte and Neo-Riemannian Appropriations (p. 353)

The group-theoretic potential of Riemann's Harmonieschritte and their relevance to neoRiemannian practice were first identified by Henry Klumpenhouwer, who presented the Schritt/Wechsel system as a corrective to flaws he perceived in the transformational sys­ tems developed by Lewin and Hyer.5 Klumpenhouwer specifically objected to Lewin's and Hyer's mixture of dual transformations, such as the parallel, relative and leittonwechsel transformations (transformations whose effect depends upon the quality of the triad to which they are applied), with transformations based on nondualist, fundamental-bass re­ lationships, such as the dominant and subdominant transformations (transformations whose effect remains constant regardless of triad quality), noting that their combinations often yield somewhat counterintuitive results. The compound leittonwechsel–then–rela­ tive transformation, for example, is equivalent to the subdominant transformation when applied to a major triad, but is equivalent to the dominant transformation when applied to a minor triad.6 Klumpenhouwer proposed that a fully dualized system of Schritte (transpositions) and Wechsel (exchanges or inversions) drawn from Riemann's 1880 treatise Skizze einer neuen Methode der Harmonielehre, rather than from the later function theory, might serve well as “a less conflicted group of transformations.”7 The table in example 12.1 summarizes Riemann's presentation of the root-interval relations between diatonic triads from which Klumpenhouwer departs to generate the complete Schritt/Wechsel group act­ ing on the twenty-four major and minor triads.8 Although the Schritt/Wechsel system is tonic-blind and ultimately accommodates relationships between any consonant triads, Riemann's initial presentation in the Skizze emphasizes relationships found within the context of a given major or minor key, admitting the minor subdominant in major and the major dominant in minor. Here each diatonic relationship is illustrated both in the context of C major and A minor. Major triads are identified by their root followed by a plus sign (“c+” for C major); minor triads are given in dual notation (that is, the A-minor triad is identified as “oe”). Each relationship is identified as being a Schritt or a Wechsel, where a Schritt preserves mode and a Wechsel reverses mode. Each relationship is further defined by the root interval spanned and the direction of the root-interval motion. The modifier “schlicht” (plain) refers to root motion in the direction of the initial triad's generation (up for major triads, down for minor triads); the prefix “gegen” (contrary) generally refers to motion opposite that of the initial triad's generation.9 Relationships are assumed to be plain unless otherwise specified. Finally, the far right column, to which we will return in the next section, shows the implied derivation of each relationship in terms of the gener­ ating Quintschritt (Q), Seitenwechsel (⊕) and Terzwechsel (T), though this is not an aspect of Riemann's presentation much emphasized by Klumpenhouwer.

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Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte

Ex. 12.1. Diatonic root-interval realtions in the Sk­ izze (1880). (p. 354)

Further combinations of these diatonic relationships yield the remaining transfor­

mations in the group, which are given in full in the appendices of Klumpenhouwer's arti­ cle and are in certain respects equivalent to the roster of relationships given in the table in example 12.2, the Systematik der Harmonieschritte, or listing of twenty-five “potential­ ly intelligible” relationships offered by Riemann himself in a later section of the Skizze.10 (Klumpenhouwer's list adds an identity element and omits the Gegentritonusschritt and Doppelterzwechsel, which are redundant under equal temperament.) Here again, the col­ umn headed “Implied Derivation” should be ignored for the moment, as should that head­ ed “Directly Intelligible?” Notice that in this presentation, Riemann illustrates each pro­ gression in relation to a C-major and an A-minor triad, rather than in the context of the keys of C major and A minor, underscoring the points that these relationships are not de­ pendent upon key and that a major or minor triad can be related to any of the remaining twenty-three consonant triads via some Schritt or Wechsel. While Klumpenhouwer advocated the fully dualized Schritt/Wechsel system as an alterna­ tive to transformational systems derived from Riemann's function theory, his Schritt/ Wechsel group has gained little traction as an analytical tool. Rather, the shift toward a fully dualized perspective has come about through the privileging of the parsimonious PLR transformations and their compounds in the works of Richard Cohn, which has been accompanied by an abandonment of vestiges of the functional tonic-subdominant-domi­ nant framework to which these relationships were originally subordinated in Riemann's theories. The Schritt/Wechsel group in turn has accrued status on this basis of its isomor­ phism with this PLR group—that is, on the basis of its standing as a structural analogue of this PLR group—and on the basis of the historical precedent it provides for the neo-Rie­ mannian PLR group within Riemann's own writings.11

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Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte

Ex. 12.2. Riemann's Systematik der Harmoni­ eschritte from the Skizze (1880).

Viewed in this light, the neo-Riemannian appropriation of the Harmonieschritte highlights a widely recognized slippage inherent in their appropriations of Riemann's functional re­ lationships. Whereas Riemann's functional labels identify individual chords’ meanings in relation to the tonic, the neo-Riemannian abandonment of (p. 355) tonic reference leaves its “functional” PLR transformations to serve much the same purpose that the Harmoni­ eschritte serve in Riemann's theory, namely to provide tonic-blind descriptions of harmon­ ic successions. Moreover, as David Kopp observes in the context of a discussion of Lewin's and Hyer's adaptations of Riemann's functional relationships, including the rela­ tive and leittonwechsel relationships, “By construing the concepts as progression types rather than as identity processes associating chords, they revise the concepts of relative chord and leittonwechsel chord to represent root-intervals…Thus what would appear to be a synthesis of the functional and root-interval approaches reveals itself to be a pure root-interval system.”12 In other words, while Cohn's P, L, and R transformations ostensi­ bly derive from and retain names associated with Riemann's function theory, they exem­ plify the dually conceived, directed motion associated with the Harmonieschritte, and like the Harmonieschritte describe only local relationships between chords, saying nothing about the meaning of those chords within a key.13 To say that the PLR transformations embody the same notion of relationship as the Har­ monieschritte, or to say that the PLR and Schritt/Wechsel groups are isomorphic, is not to say that they are identical. While a PLR family analog exists for each Schritt and Wechsel, and while the groups do share the same structure, each system carries its own assump­ tions about harmonic relationships. Kopp, for instance, contrasts Riemann's system of root-interval relationships, which he characterizes as (p. 356) “all-inclusive,” with Lewin's and Hyer's—and by extension Cohn's—transformational systems, which he describes as being “limited to common-tone relationships.”14 That is, under the Schritt/Wechsel system, any triadic succession can be accounted for as an instance of some unitary rela­ Page 5 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte tionship, whereas under Cohn's LPR system successions preserving only a single tone or involving no common-tone connection are accounted for through compounds of the group's fundamental, generative transformations L, P, and R—meaning that they are un­ derstood not directly, but through reference to a series of stepwise displacements and changes of mode. Thus, while a progression from a C-major triad to a G-major triad is un­ derstood directly, as a simple Quintschritt, by Riemann, under the LPR model it is under­ stood less directly, through reference to an L-then-R compound involving, to some extent, imagined motion through a mediating E-minor triad. It can be and has been argued that these compound transformations represent more than the sum of their parts and have an intrinsic intelligibility, a position taken by Hyer when, in discussing the coherence of Wagner's Schlafakkorde, he asserts that “LP gains a cer­ tain conceptual independence…when it enters into an algebraic group, becoming a sin­ gle, discrete transformation rather than a process that combines two more primitive transformations, L and P.”15 Even if we admit the direct intelligibility of LP, however, the resulting LP intervals continue to reflect a notion of harmonic distance informed by the group's parsimonious generators—a notion of distance not directly evident in a listing of discrete Harmonieschritte. The idea that groups of the same structure can be presented in different ways, reflecting different analytical or theoretical priorities—such as a desire to emphasize root-interval motion or voice-leading connections, or diatonic versus chro­ matic third relationships—will be explored in the next section, which will focus in particu­ lar on generated groups and the connections that exist between their fundamental, gener­ ating transformations and the conceptions of tonal proximity and coherence they imply.

Combinatoriality and the Generation of the Harmonieschritte Interest in generated neo-Riemannian groups, such as the PLR group, has drawn atten­ tion not only to the Harmonieschritte, but also to the role combinatoriality played in later nineteenth- and early-twentieth-century harmonic theories, including Riemann's. Brian Hyer's and Richard Cohn's explorations of the algebra and group structures associated with their chosen neo-Riemannian transformations highlight the combinatorial potential of a few maximally proximate, directly intelligible relationships, and Cohn in particular has endorsed a conception of tonal organization under which these generators serve as the bearers of tonal coherence.16 This combinatorial perspective and its emphasis on group generators has led to (p. 357) reexaminations of Riemann's Harmonieschritte focused not only on exhaustive listings of root-interval relationships, such as the roster presented by Klumpenhouwer, but also on the combinatorial details of Riemann's discus­ sions and the notions of harmonic proximity and coherence underlying these lists. In short, the neo-Riemannian revival of the Harmonieschritte has entailed a renewal or “re­ generation” of Riemann's Schritt/Wechsel systems, as well as a reappraisal of earlier har­ monic theories with respect to their combinatorial potential.

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Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte Riemann was not the first theorist to advocate this combinatorial approach to harmonic relations, which will be shown to underlie the Harmonieschritte. From roughly the midnineteenth century onward, a number of theorists—including Hauptmann, Oettingen, and Hostinský—displayed a keen interest in identifying some basis, independent of key, for tonal coherence at the level of chord-to-chord succession. Two competing perspectives emerged, one favoring common-tone connections as a basis and another favoring acousti­ cally proximate root relations, but adherents of both proceeded along a path familiar from neo-Riemannian practice, initially defining collections of directly intelligible, tonally co­ herent relationships between triads, and then exploring the combinatorial potential of these relationships. Not surprisingly, given this focus on generative relationships and their interactions, the nineteenth-century theorists’ findings prove to be well modeled by combinatorial group theory, which describes groups in terms of “group presentations,” essentially listings of a collection of generating elements and a set of rules governing the ways in which those generators will combine to yield remaining group elements.17 Somewhat more formally, a presentation describes a group in terms of a system of “generators”—a collection of group elements from which all other group elements can be derived as compounds—and a set of “defining relators” on those generators. Relators place constraints on the ways in which the generators compose, most often by showing how certain combinations of the generators align with the group's identity element, and a set of defining relators on some given generators is a set of relators from which all other relators on those generators can be derived. The familiar LPR group, for example, might be given by the presentation 〈L, P, R; L2, P2, R2, (LP)3, (PR)4, and (LR)12〉, in which the set of defining relators (shown to the right of the semicolon) align the group's unary and binary generators with the identity el­ ement. Combinatorial group theory accords with nineteenth-century treatments of triadic rela­ tions in several additional respects. Notions of harmonic proximity inherent in nine­ teenth-century discussions of directly intelligible relationships and of the “mediated” or “elided” successions associated with their compounds were often realized in spatial schema, such as Oettingen's well-known table, which in turn supported intuitive realiza­ tions of the connections between pathways across a musical space, the composition of generators to form strings or what are known as “words in the generators” (compounds, such as LP), and the notion of intervallic distance captured by the concept of “word length” (the number of generator symbols used to form a word—for example, the binary compound “LP” is a word of length two).18 (p. 358) Group presentations easily accommo­ date both finite and infinite groups—an important point in that under just intonation, fa­ vored by many nineteenth-century theorists, the compounding of generators is not con­ strained as under equal temperament. And, perhaps most significantly, the combinatorial approach lends itself particularly well to the examination of a nascent, rather than fully formed, group theoretic perspective, in that its emphasis on the construction of a group from its generators corresponds to nineteenth-century theorists’ interest in building out­ ward from local relations, rather than requiring any recognition of or interest in the uni­ versal actions of these implicit groups on a particular set of triads. In keeping with this Page 7 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte concentration on “building out,” generators figure much more prominently in nineteenthcentury theory than do defining relators. The appeal to concepts central to combinatorial group theory also links Riemann and his predecessors to a more broadly defined combinatorial tradition including the roughly con­ temporaneous emergence of “combinatorial space” recently explored by Catherine Nolan.19 Nolan's study focuses on applications of combinatorial techniques to pitch-class relationships within the finite, twelve-note, equal-tempered system—rather than to the ex­ amination of relationships among potentially infinite collections of consonant triads—and specifically surveys the exploration of various partitions and groupings of pitch classes, which led theorists to exhaustive enumerations of all subsets within the system and tax­ onomies of related equivalence classes based on transposition. Formal connections be­ tween the combinatorial practices underlying the identification of these pitch-class rela­ tions and the development of a combinatorial approach to harmonic relations can be forged, for example by emphasizing the notions of permutation and partition in relation to group actions on the triads: the familiar neo-Riemannian groups effect permutations of the twenty-four consonant triads and partition the triads in various, potentially musically interesting ways.20 This perspective involves a restriction to groups acting on the finite universe of the equal-tempered triads—a restriction acceptable to Riemann, but not to all nineteenth-century theorists—and also requires an awareness of the group as a whole, a perspective evident to some degree in Riemann's complete enumerations of the Harmoni­ eschritte, but not in all nineteenth-century discussions of triadic relations. More general­ ly, we might note that these combinatorial traditions both reflect a widespread preoccu­ pation with algebraic and geometric formulations of musical space and what Nolan de­ scribes as “an important shift in music-theoretical thought” involving the application of combinatorial techniques “not as compositional devices, but as a means to illuminate pre­ compositional relations among the fundamental elements” of pitch classes, triads, and in­ tervals.21 Returning to the matter of combinatoriality and the generation of Riemann's Harmoni­ eschritte, although the neo-Riemannian literature associates the appearance of the finite Schritt/Wechsel group with the 1880 Skizze, Riemann's implicit formulation of this group in fact dates to an earlier treatise, his 1875 Die Hülfsmittel der Modulation.22 Because the format and terminology of the presentation in the Skizze would become the norm for Riemann's subsequent discussions of the (p. 359) Harmonieschritte, however, the current consideration of Riemann's understanding of the Harmonieschritte will focus on the pre­ sentation in the Skizze and reference Die Hülfsmittel only where relevant. Musikalische Syntaxis (1877), which appeared between Die Hülfsmittel and the Skizze, includes a less complete listing of Schritt and Wechsel relations but is nevertheless interesting where matters of proximity and intelligibility are concerned. Like other nineteenth-century theorists, Riemann did not explicitly invoke the terminolo­ gy of combinatorial group theory in his discussions of triadic relations, but instead can been seen to have arrived at an implicit formulation of the Schritt/Wechsel group through an exploration of the combinatorial potential of a few highly proximate, directly intelligi­ Page 8 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte ble relationships. As noted earlier, in conjunction with the rosters of Schritte and Wechsel given in examples 12.1 and 12.2, Riemann offers two distinct presentations of the Har­ monieschritte in the Skizze. The initial presentation, summarized in example 12.1, is in­ complete in the sense that it lists only those relationships that obtain between triads found in a diatonic context (including relationships involving both the major and minor forms of the subdominant in major keys and of the dominant in minor keys). This is the presentation that Klumpenhouwer expanded to produce the complete Schritt/Wechsel group. In a subsequent discussion of harmonic relationships, Riemann himself presents a fully chromatic version of the system of Harmonieschritte, given in example 12.2, compa­ rable in most respects to that produced by Klumpenhouwer and also to Riemann's own earlier presentation of the Harmonieschritte in Die Hülfsmittel. What was not noted above, but is evident from careful comparison of the columns headed “Implied Deriva­ tion,” found toward the right of each table, is that these two presentation involve two dif­ ferent sets of generative relations—represented by Q, ⊕ and T in the first case and Q, ⊕ and T in the second (the difference being T versus T, representing the diatonic Terzwech­ sel and chromatic Terzschritt, respectively)—reflecting two different conceptions of har­ monic proximity and ultimately of tonality. The generator set associated with Riemann's complete, fully chromatic presentation of the Harmonieschritte most accurately expresses his own view of tonality, for which his ini­ tial, diatonically oriented presentation in a sense provides historical context. Although the inclusion of the diatonic presentation was almost certainly a response to pedagogical exi­ gencies not raised in his earlier, speculative works—Riemann's desire to present a readily accessible Harmonielehre led him to defer any substantive discussion of his own views on tonality until his survey of diatonic root-interval relations had been completed—this dia­ tonic presentation in effect recapitulates Oettingen's (1866) root-interval system, upon which Riemann's own system of Harmonieschritte was based.23 That said, a slight caveat might be in order, in that Oettingen's interests lay in the extrapolation of a wide range of diatonic and chromatic relationships from a set of diatonically based generators, whereas Riemann's interests (in this context, at least) lay in defining tonality through reference to root relationships involving motion through the acoustically privileged intervals of the perfect fifth and major third. In his survey of the fourteen root-interval relationships commonly found between diatonic triads, Riemann first describes the root-interval relationships found between the primary triads within a key and then turns to relationships involving the key's secondary (p. 360)

triads.24 Like Oettingen, Riemann casts the Quintschritt, the Terzwechsel, and the Seiten­ wechsel as fundamental relationships from which all others can be generated. Riemann begins his discussion of individual Harmonieschritte by deriving major and minor key sys­ tems from their tonic triads through reference to the Quintschritt, its inverse, the Gegen­ quintschritt, and the Seitenwechsel—represented by the symbols Q, Q−1, and ⊕ in exam­ ple 12.1.25 The Quintschritt and Gegenquintschritt pair a tonic with its upper and lower dominants of the same quality; the Seitenwechsel pairs a major tonic with its minor sub­ dominant and a minor tonic with its major dominant. Riemann accounts for the three re­ maining relationships between primary triads as compounds of the generative Page 9 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte Quintschritt, Gegenquintschritt, and Seitenwechsel relationships: the Ganztonschritt arises as the product of two successive Quintschritte (Q2), the Quintwechsel as the prod­ uct of a Quintschritt followed by a Seitenwechsel (Q⊕), and the Gegenquintwechsel as the product of a Gegenquintschritt followed by a Seitenwechsel (Q−1⊕). Riemann next introduces the secondary triads, which he characterizes as standing in the Terzwechsel (T) relation to a key's primary triads—as the Terzwechselklang, the Terzwechselklang of the Quintklang, and the Terzwechselklang of the Gegenquintklang. Two aspects of this characterization merit comment in light of later developments in Riemann's theories. First, the representation of the secondary triads as Terzwechselk­ länge of the primarily triads suggests a conceptual equivalence of sorts between the Terzwechsel and relative relationship to be introduced as part of Riemann's later func­ tional theory, which in turn suggests that the slippage between root-interval and function­ al transformation discussed earlier is evident even in Riemann's own works. Second, Rie­ mann here employs the Schritt/Wechsel terminology both to describe relationships be­ tween triads and to identify individual triads. An even greater reliance on Schritt/Wechsel-based labeling of triads is found in the earlier Musikalische Syntaxis, as well as in the final chap­ ter of the Skizze, where Riemann's advocacy of his chromatic conception of tonality (to be discussed below) leads him to emphasize each triad's meaning within a given key as a product of its root-interval relationship to the tonic. In Riemann's later theory, his system of function labels assumes the duty of chordal identification, leaving the Schritt/Wechsel system to express chordal relationships, the latter being the domain of transformational approach implicit in Riemann's system of Harmonieschritte but explicit in neo-Riemann­ ian theory.26 The Terzwechsel is the last generative relationship to be presented. The remainder of Riemann's preliminary discussion of Harmonieschritte addresses relationships involving secondary triads, and Riemann's accounts of these relationships reveal a combinatorial perspective centered on the Quintschritt and Gegenquintschritt, the Seitenwechsel, and the Terzwechsel. This combinatorial attitude emerges most clearly through his labeling of triads. Riemann observes, for example, that a (p. 361) Kleinterz-Wechsel relationship ob­ tains between the tonic and the Terzwechselklang of the Gegenquintklang, from which we can easily infer that the Kleinterz-Wechsel may be understood as the compound Gegen­ quintschritt-then-Terzwechsel (Q-1T). The implied derivations for the remaining diatonic relationships are given in example 12.1. Though combinatorial in conception, Riemann's survey of the diatonic Harmonieschritte does not imply a presentation of the Schritt/Wechsel group on the generators Q, T, and ⊕.27 Certainly, these generators can and do compose so as to yield a complete a group on the twenty-four consonant triads, but this group exceeds the bounds Riemann himself has imposed on his initial discussion and he does not pursue the matter. When he does even­ tually discuss a fully chromatic system of Harmonieschritte later in the Skizze, the diaton­ ic Terzwechsel is abandoned as a generator and replaced with the chromatic Terzschritt.

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Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte As Riemann turns to his own views on tonality in the final chapter of the Skizze, he reca­ pitulates a position set forth in his earlier treatises, Die Hülfsmittel and Musikalische Syn­ taxis. Rejecting the standard, scale-based conception of tonality, Riemann argues that tonality should instead be understood as a system of acoustically defined relationships around a tonal center, and that diatonic and chromatic chords alike may stand in direct relation to that tonal center, so long as the primacy of that tonic—as the chord in relation to which all other chords receive their meaning and as the ultimate goal of cadential mo­ tion—remains undisputed. Accordingly, Riemann's choice of generative relationships is not constrained by diatonicism, and the relationships he favors as generators are simply those involving the most acoustically immediate root-interval relationships: the Seiten­ wechsel, which involves no root motion, is the most fundamental relationship, followed by the Quintschritt and the Terzschritt.28 Rather than departing from a survey of direct relationships, as in his presentation of dia­ tonic Harmonieschritte, Riemann proceeds immediately with his presentation of the com­ plete roster of twenty-five “potentially intelligible” relationships shown in example 12.2. On the basis of the root intervals involved, Riemann groups these twenty-five relation­ ships into seven categories of Harmonieschritte (given in the leftmost column of the ta­ ble), which are ordered to reflect the acoustic immediacy of the root intervals entailed. Apart from the first category, the Seitenwechsel, each category comprises four relation­ ships—a Schritt, a Gegenschritt, a Wechsel, and a Gegenwechsel—spanning a shared root interval. Riemann breaks this pattern only once, when he replaces the expected Gegentri­ tonuswechsel with the Doppelterzwechsel, which does not share the root motion by tri­ tone that seemingly defines the final category. As noted above, each relationship is illus­ trated with respect to a C-major triad and to an A-minor triad. As in Riemann's presentation of the diatonic Harmonischritte, the combinatorial deriva­ tions shown in example 12.2 follow, for the most part, from descriptions of the individual relationships (the formerly direct Terzwechsel is notably described as the relationship from a tonic to the contrary chord [gegentheiligen Klange] of its third, pointing to the de­ rivation T⊕), or in some cases from (p. 362) Riemann's demonstrations of the required me­ diations of indirect successions (the Gegenquintwechsel from C major to B♭ minor [c+ to o f] becomes intelligible through reference to a mediating F-major or F-minor triad [f+ or o c]—pointing to a derivation as either Q−1⊕ or ⊕Q).29 The combinatorial approach thus revealed does this time support a group presentation: 〈⊕, Q, T; ⊕2, (Q⊕)2, (T⊕)2, and QT = TQ〉.30

On Intelligibility, Derivation, and Harmonic De­ generacy After introducing all twenty-five “potentially intelligible” relationships comprised by the Systematik der Harmonieschritte, Riemann builds upon preliminary comments he has made regarding their relative intelligibility, compiling a roster of the sixteen relationships he considers to be directly intelligible (though not all equally so). These sixteen relation­ Page 11 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte ships are identified in the rightmost column of example 12.2. The roster clarifies the sta­ tus of a few borderline cases—the Gegenterzwechsel and Gegenkleinterzwechsel, which Riemann originally described as “difficult to understand,” are admitted as directly intelli­ gible, while the Gegenganztonwechsel, originally described as “not easily understand­ able,” is not—but Riemann provides little insight into the reasoning behind these classifi­ cations. The sixteen Harmonieschritte include, but obviously are not limited to, the gener­ ative Seitenwechsel, Quintschritt, and Terzschritt. A notion of derivational distance, ex­ pressed in “word length,” provides some guidance, albeit not much: direct relations in­ clude all words of length one, over half of those of length two, under half of those of length three, and none of those of length four. The immediacy of the underlying root-in­ terval relationship likewise seems to play some role, but again not a defining one: with the exception of the Gegenquintwechsel, all Harmonieschritte within the first four cate­ gories are directly intelligible, while only one-third of those in the remaining three cate­ gories are. Riemann's acceptance of an increasingly broad range of relationships as directly intelligi­ ble initially seems to call into question the privileged status of the group generators. Clearly, however, Riemann does not view all sixteen of the direct relationships as genera­ tive: we know this from the preceding combinatorial presentation of the group. Rather, in light of Riemann's assertion that diatonic and chromatic chords alike may stand in direct relation to a tonal center, so long as the primacy of that tonic is not disturbed, this regis­ ter of intelligible relations is intended to provide guidance as to which relationships will and will not threaten to undermine a tonic. This shifting understanding of intelligibility neither requires Riemann to abandon, nor bars him from assuming, the position that co­ herence is borne by the generators, but the weakening of the correlation between deriva­ tional simplicity (p. 363) and harmonic proximity or intelligibility does limit the effective­ ness of intelligibility as a constraint on motion within the Schritt/Wechsel system, a con­ straint Riemann finds very much wanting. Just as Riemann was not the first theorist to advocate a combinatorial approach to triadic relations, he was not the first to recognize the potential for harmonic chaos—what Oettin­ gen termed the “chaos of possibilities”—that such an approach might usher in.31 Indeed, as Klumpenhouwer notes, in cultivating the system of Harmonieschritte, “Ultimately, Riemann's purpose is to provide a thorough enough lexicon of relations so that any two klangs could find a relevant transformation.”32 Yet, the notion that “any chord can follow any other chord” realizes precisely what Alexander Rehding describes as “Riemann's hor­ ror vision” of a theory that allows everything and explains nothing.33 That Riemann appears to have felt this predicament more acutely than other nineteenthcentury theorists working in the same combinatorial tradition was due at least in part to the notion of tonality he espoused. Whereas Hauptmann and Oettingen, for instance, cast their diatonically conceived generators as bearers of tonal coherence but also invoked key-based relationships in limiting the composition of these generators to produce a high­ ly compatible two-tiered system of local and global coherence,34 Riemann's rejection of a key-based conception of tonality leaves him with only an acoustically determined model of Page 12 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte coherence at the chord-to-chord level—albeit one that is perhaps weakened by the liber­ ties he takes in assessing the intelligibility of relationships—and the desire for some glob­ al principle of musical logic. At the conclusion of Musikalische Syntaxis, Riemann outlines his predicament, noting both his embrace of a full range of harmonic relationships and his desire for a prescrip­ tive theory capable of accommodating them: Thank God the combinations [of harmonies] are inexhaustible in number, and one cannot explore the area of harmony in its entirety by walking across it step by step but only by flying over it and surveying it from a bird's eye view. It is sufficient, however, to recognize the chief paths through this magnificent Garden of Eden, which Heaven has left us after the Fall; everybody may then find new side paths for himself leading to ever new perspectives on regions never entered before. It all depends on [. . .] the logical laws of musical listening and thinking; since mod­ ern practice has broken the old laws, the student of composition has to become aware of new and higher laws according to which to create and to judge the cre­ ations of the masters: only in this way is it possible to counter the tendency of our modern theorists and practitioners toward formlessness and capriciousness.35 Riemann would revisit this final point, recast as the need to answer degeneracy, anarchy, and decadence in modern music with clearly articulated, prescriptive boundaries, some thirty years later in the article “Degeneration und Regeneration in der Musik” (1908), lo­ cating the potential for musical regeneration in the logic inherent in Brahmsian classi­ cism.36 Whether the classically conceived cadential paradigms of Riemann's own function theory, introduced in Vereinfachte Harmonielehre (1893), were prescriptive enough to im­ pose meaningful limits on the relationships described by the Harmonieschritte is debat­ able, at best. Indeed, Carl Dahlhaus and subsequent authors have observed that while Riemann's function theory does address relationships of chords to a tonal center, it pro­ vides little in the way of normative rules of progression and, as such, falls short as a “logi­ cal” system.37 As Daniel Harrison notes, in Riemann's theory, tonic, subdominant, and dominant functions were “allowed to roam freely over the musical surface with only the barest syntactic constraint.”38 Kopp's similar assessment recognizes both the potential flexibility of Riemann's function theory—an asset in approaching the harmonic variety (p. 364)

presented by the harmonic “Garden of Eden”—and also its similarity to the system of Har­ monieschritte as a descriptive, rather than prescriptive, system: “Progressions do tend to follow general cadential formulas in the long run, but the particulars of chord-to-chord progression are only lightly constrained, while the possibilities of their combination, as apparent from Riemann's analyses, are manifold.”39 Function theory may not have imposed significant limits on successions, but it neverthe­ less provided Riemann with some means of addressing their tonal coherence. Whereas the Harmonieschritte accounted for coherence at the chord-to-chord level—coherence tied to (though not exclusive to) inherited notions of the immediacy of group generators Page 13 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte and of a general correlation of derivational simplicity with intelligibility—the functional system accounted for tonal coherence by fixing chords’ individual meanings within a key.40 Following the introduction of his theory of tonal functions, however, Riemann's ex­ planations of the intelligibility of individual Schritte and Wechsel focused increasingly on tonal coherence rather than on derivation. Subsequent editions of the Skizze, revised and retitled the Handbuch der Harmonielehre, presented familiar summaries of all possible Schritt/Wechsel relationships in their Sys­ tematik der Harmonieschritte, but added examples of functional contexts in which each of the relationships shown in example 12.2, as well as some others, would be intelligible. In the 1906 Handbuch, for example, Riemann describes the potential intelligibility of the Gegentritonuswechsel—a relationship not even included on the roster given in the Skizze —as follows: “The Gegentritonuswechsel is hardly likely to be made intelligible; its sim­ plest interpretation in major and minor would be as ( )” (that is, in C major or mi­ nor, as a D♭-minor triad moving to a D-major triad).41 In addition to demonstrating the functional system's ability to provide justification for a wide range of successions—sug­ gesting that almost any chord can follow any other if given some appropriate tonal setting —these kinds of explanations reflect Riemann's increasing reliance on tonal rather than derivational coherence. Riemann declares that “the true guide through the labyrinth of possible harmonic successions is no longer the nomenclature of the Harmonieschritte but rather that of function.”42 In his article initiating the neo-Riemannian appropriation of the Harmonieschritte, Klumpenhouwer declines to find fault with Hyer and Lewin for (p. 365) having misrepre­ sented Riemann's ideas, instead offering an avowal of his own presentist biases: “it could be fairly argued that…it is I who will be misrepresenting Riemann. My own distortions (as far as I can tell) involve projecting a particular ideational structure on Riemann's trans­ formations…and then appropriating the transformations for analytical purposes other than those discussed by Riemann.”43 Setting aside the legitimate question as to whether Riemann intended the Harmonieschritte as “essentially transformational” or merely “rela­ tional,”44 as well as the matter of the explicit appeal to the ideational structure of group theory, Klumpenhouwer's disclaimer that he is “appropriating the transformations for an­ alytical purposes other than those discussed by Riemann” points toward a more funda­ mental distinction between Riemann's view of the Harmonieschritte and a neo-Riemann­ ian view of the Harmonieschritte. To a degree, the neo-Riemannian rendering of the Harmonieschritte as the Schritte/Wech­ sel group realizes Riemann's “horror vision” of his system's degenerative potential. If the logic of the group as a whole stands as the source of harmonic coherence, as Hyer has suggested, it is true that “any chord can follow any other.”45 But the structural relation­ ships inherent in the group are, of course, not the end point of neo-Riemannian analysis. Neo-Riemannian analyses aim to reveal underlying order by showing how these structur­ al relationships are drawn into configurations specific to the works or passages being ex­ amined. Klumpenhouwer, for instance, continues his disclaimer by noting, “Following Lewin and Hyer, I will be interested in extending the use of Riemann's transformations to Page 14 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte seek out repetitions of harmonic patterns, which are then presented as motivic,” and Cohn has developed an analytical practice centered on networks of relationships within harmonic spaces ordered by generative cycles. Riemann himself was, of course, interested in networks of relationships—between chords placed in succession and of chords to a tonic—and had worked out almost fully the combi­ natorial “logic” underlying the Schritt/Wechsel group. What sets neo-Riemannian theory apart from Riemann's theories, in terms of their treatment of the system of Schritte and Wechsel, is the abandonment of tonal reference. While the Harmonieschritte were defined independent of key and conveyed information about the harmonic coherence of succes­ sions—here the neo-Riemannian appropriation remains true to the original—in practice they were subjugated to Riemann's views on tonality and tonal coherence. The signifi­ cance of this is not merely that Riemann's focus on tonal relationships prevented him (to take a very presentist view of the matter) from conceiving of the Harmonieschritte in for­ mal terms as a group of transformations acting on equally weighted triads, or that the overlay of harmonic function limited the range of possible successions (indeed, it exclud­ ed few, if any), but rather that Riemann's tonal orientation led him to back away from the approach to harmonic coherence, correlating derivational simplicity with harmonic prox­ imity, inherited through the combinatorial tradition. While root-interval relations contin­ ued to inform Riemann's characterizations of the strength and immediacy of chord-tochord successions, once his functional theories were in place questions about the intelli­ gibility of specific successions depended (p. 366) at least as much on tonal context as on these root-interval relationships. Under the neo-Riemannian approach, by comparison, the combinatorial structure of the group is the sole source of coherence. Kopp gets at much the same point being made here when he notes, “The reorientation of focus from goal-directedness to structural coherence becomes a defining aspect of latter-day Rie­ mannian theory.”46 This difference in perspective is, of course, a product of historical po­ sition. Whereas Riemann felt compelled to strike a balance between the harmonic poten­ tial well described by his system of root-interval relationships and the fin-de-siècle fear of a degenerating harmonic practice, neo-Riemannians freely embrace Riemann's descrip­ tive system, untroubled by the prescriptive constraints he found lacking. In effect, the neo-Riemannian revival of the Harmonieschritte amounts not merely to a re­ covery of Riemann's Schritt/Wechsel terminology and of his theories’ dualist aspects, but a “spiritual” renewal of sorts, not resolving Riemann's concerns about the lack of a pre­ scriptive, tonal element, but rather embracing the flexibility his system provides for rep­ resenting different notions of coherence implicit in repertoire, rather than explicit in a re­ ceived theoretical model. What appears to be a degeneration of Riemann's tonally con­ ceived Schritt/Wechsel system proves, in this sense, a regeneration.

Riemann's Systematik der Harmonieschritte [The “Systematik der Harmonieschritte”—a systematic exposition or taxonomy of the root-interval relations—appears as section 38 of Riemann's Skizze einer neuen Methode Page 15 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte der Harmonielehre (1880). Throughout the translation that follows, the names of root-in­ terval progression types and related chord names will be left in the original German to avoid awkward neologisms. A brief explanation of Riemann's terminology can be found at the outset of the preceding essay. Likewise, familiar terms such as “Klang,” “Unterklang,” “Oberklang,” “Untertonart,” and so forth are generally not translated. I would like to thank Christopher Williams for our conversations about an early draft of this translation. Any errors or infelicities that remain are, of course, my own.] If we renounce the old concept of key in this way, we are by no means underestimating the value of the traditional scales. These remain, as always, of the greatest significance as types of melodic motion.47 But what, now, is the nature of the modern key? Does it dis­ solve into the indefinable with the setting aside of the scale? No. On the contrary, it is more precisely definable than before; its realm is wider and yet its borders are redrawn more sharply. In a word: key is nothing more than the meaning of a Klang as tonic; the key is left as soon as this meaning changes, which can occur without the introduction of chromatically altered notes.48 The modern name for key in this sense is: tonality [Tonalität]. We are in C-major so long as the c+ Klang forms the center of our harmonic conception, so long as it appears as the sole (p. 367) cadential goal and all other Klänge receive their characteristic effect and meaning through their connection with this tonic; likewise, the e-Untertonart reigns so long as oe is understood as the tonic. It was previously indicated that the Gegenquintklang has an entirely different position and meaning for the harmony of the key than the schlichte Quintklang (§9); we now come to describe, to name and to show the compositional meaning of the understandable species of Harmonieschritte according to the degree of relationship to the tonic of the Klang thereby reached. We found a large number of these already in the harmony adhering to the scales of the major and minor keys, [and] still more through the introduction of the Seitenwechselklang and the Terzwechselklänge. The possible, intelligible successions are: I. The Seitenwechsel, the relation of the tonic to its contrary [gegentheilige] Klang, that is a) from major to minor, c+–oc. b) from minor to major, oe–e+. The Schritt is directly intelligible and has cadential force returning to the tonic. II. Quintschritte, they are 2) the schlichter Quintschritt, that is the relation of the tonic to the Klang of the same kind (that is, [to the Klang] consonant according to the same Klang-princi­ ple—major or minor) of its fifth: a) between Oberklänge: c+–g+ (+Tonic–+Oberdominante; cf. §5ff).

Page 16 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte b) between Unterklänge: oe–oa (oTonic–oUnterdominante; §7). 3) the Gegenquintschritt (§9), the relation of the tonic to the like-mode Klang of its Gegenquint (that is, in view of the construction of [the tonic's] Klang, the fifth lying on the opposite side of the tonic: f ¬ c: e. g.): a) between Oberklänge: c+–f+ (+Tonic–+Unterdominante). b) between Unterklänge: oe–ob (oTonic–oOberdominante). 4) The Quintwechsel (§11), the relation of the tonic to the contrary Klang of its fifth. The Schritt is directly intelligible and occurs, for example, between the Gegenquintklang and Seitenwechselklang, but can just as well occur from the tonic itself, without indicating a modulation; from major to minor: c+–og; from minor to major: oe–a+ (the meaning is the same). 5) The Gegenquintwechsel (§12), the relation of the tonic to the contrary Klang of the Gegenquint. The two Klänge are no longer directly related; their under­ standing requires the mediation of [another] one lying between and closely re­ lated to both, i.e., either the Gegenquinklang or Seitenwechselklang (the Seiten­ wechselklang is related to the Gegenquintwechselklang by the schlichter Quintschritt, whereas the Gegenquintklang is related to the Gegenquintwech­ selklang by the Seitenwechsel). In terms of harmonic structure, the necessity of the progression's mediation by the specified Klänge indicates the expected con­ tinuation; even though the Gegenquintwechsel is not adequate for a (p. 368) mod­ ulation, it does, in any case, introduce the Gegenquintklang or Seitenwechselk­ lang with greater emphasis:

a) from major to minor: c+–of, with f+ or oc mediating:

b) from minor to major: oe–b+, with ob or e+ mediating:

III. Terzschritte, they are 6) the schlichter Terzschritt (§19), the relation of the tonic to the like-mode Klang of its third. The Schritt is directly intelligible and, like all directly intelligi­ ble Schritte, has cadential force if it occurs going back to the tonic (see example 108.6 below).49 As, however, scale-based harmony has accustomed us particular­ ly to the Quintschritte and Seitenwechsel, a Seitenwechsel follows the Terz­ schritt well, and as a result the Terzwechselklang (the tonic of the relative key) is reached: a) between Oberklänge: c+–e+;

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Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte b) between Unterklänge: oe–oc:

7) The Gegenterzschritt, the relation of the tonic to the like-mode Klang of the Gegenterz (a♭ 〈—c. e. g; a. c. e—〉g ♯). The Schritt is likewise directly intelligible and has cadential force when occurring in reverse (a plagal cadence, like the fall back from the Gegenquintklang to tonic); our habituation to scale-based harmo­ ny has, however, made it an easy means of modulation into the key of the Quin­ twechsel or of the Gegenkleinterzschritt: (p. 369) a) between Oberklänge: c+–a♭+; b) between Unterklänge: oe–og ♯:

8) The Terzwechsel, the relation of the tonic to the contrary Klang of its third. The ease of intelligibility of this Schritt has already been addressed (§14); the re­ turn has cadential force (see example 109.8).50 Its form is: a) from major to minor: c+–oe. b) from minor to major: oe–c+. 9) The Gegenterzwechsel, the relation of the tonic to the contrary Klang of the Gegenterz. The Schritt is scarcely comprehensible, especially since, due to habit­ uation to scale-based harmony, we are already inclined to see the schlichte Terz­ schritte and Gegenterzschritte as something peculiar. Intelligibility is found through the Gegenterzklang, which is, as a consequence, therefore expected: a) from major to minor: c+–oa♭, with a♭+ mediating. b) from minor to major: oe–g ♯+, with og ♯ mediating.

Page 18 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte The Schritt exhibits an enharmonic connection, [Ligatur] e

f ♭ and c

 b ♯,

respectively:

All the same, the Gegenterzwechselklang can also be introduced closing to the tonic, since one note is held, two [voices] make Leitton­ schritte and the bass makes a Terzschritt from root to root. Cf. example 108.9.51 (p. 370)

IV. Kleinterzschritte. It has already been pointed out, in the note to §15, that the Kleinterzschritte are actually to be thought of as Sextenschritte and correspond to the passage from the third to fifth partial of a major or minor Klang.52 The intelligibil­ ity of the interval of the minor third (major sixth) is therefore admittedly not entirely direct; neither of the two tones is the overtone or undertone of the other. Interpreted as representatives of the same Klang, for example, e. g as the representative of the c+- or ob-Klang, however, the two blend completely in the same manner as, for exam­ ple, c. g or c. e as representatives of the c+-Klang and as e. b and g. b as representa­ tives of the ob-Klang. It is probably more correct to consider c. g directly intelligible not because g is an overtone of c or c is an undertone of g, but because both are rep­ resentatives of the same Klang (c+ or og). With this deduction, I will have allayed the doubts cast on [my conception] of the minor third by Dr. Ottokar Hostinský (Die Lehre von den musikalischen Klängen, Prague 1879, p. IV). The Kleinterzschritte are: 10) The schlichter Kleinterzschritt (§20), the relation of the tonic to the schlichter Terzklang of the Gegenquintklang, or more simply: the relation of the tonic to the like-mode Klang of its major sixth. The Schritt is directly intelligible and has cadential power in retrograde (see example 108.1053), but, due to our habituation to scale-based harmony, can be used to good effect in modulations, resulting especially in a Seitenwechsel to the Kleinterzwechselklang, which then appears to be introduced with some emphasis: a) between Oberklänge: c+–a+. b) between Unterklänge: oe–og.

11) The Gegen-Kleinterzschritt, the relation of the tonic to the like-mode Klang of its minor third or, more correctly, to the like-mode Klang of its Gegensexte. The Schritt is directly intelligible and has cadential force going back [to tonic] (see example 108.1154), but is well used for modulations: a) between Oberklänge: c+–e♭+.

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Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte b) between Unterklänge: oe–oc ♯.

12) The Kleinterzwechsel, the relation of the tonic to the contrary Klang of its major sixth (or rather, Unterkleinterz). The Schritt is still directly intelligi­ ble, but it best passes to the Gegenquintklang by means of the Terzwechsel: a) from major to minor: c+–oa. b) from minor to major: oe–g+. (p. 371)

13) The Gegenkleinterzwechsel, the relation of the tonic to the contrary Klang of its minor third (Gegensexte), [is] a scarcely comprehensible Schritt, which nev­ ertheless has cadential force when occurring back to tonic, probably because three voices are able to carry out Halbtonschritte (one Leittonschritt and two chromatic) and the bass makes a Terzschritt: a) from major to minor: c+–oe♭. b) from minor to major: oe–c ♯+.

It goes

without saying that the Schritt is usually used for modulation; the Gegen­ kleinterzklang (e♭+, oc ♯) follows most comfortably. V. Ganztonschritte. The Ganztonschritte, whose harmonic relationships, as Dop­ pelquintschritte, are not directly intelligible because the root interval is dissonant and therefore only indirectly understandable. Comprehension is aided by a mediat­ ing Klang closely related to both Klänge: 14) The schlichter Ganztonschritt, the relation of the tonic to the like-mode Klang of its second fifth: c. e. g–d, d–a. c. e, was discussed in detail above (§10).55 It can never close directly to the second Klang, but causes the skippedover, like-mode Klang (the schlichter Quintklang) to be expected: a) between Oberklänge: c+–d+, with g+ mediating.

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Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte

b) between Unterklänge: oe–od, with oa mediating.

15) The Gegenganztonschritt. The relation of the tonic to the like-mode Klang of its second Gegenquint (b♭–f–c. e. g, a. c. e–b–f♯) likewise leads to the skipped-over Klang (the Gegenquintklang): a) between Oberklänge: c+–b♭+, with f+ mediating. (p. 372)

b) between Unterklänge: oe–of♯, with ob mediating.

16) The Ganztonwechsel (cf. §17), the relation of the tonic to the contrary Klang of its second fifth, is never understood as such, but is always confused with the Terzwechselklang of the Gegenganztonklang. That is, wherever the Schritt appears, one of the two Klänge is understood as the schlichter Quintklang of a skipped-over Klang and the other is understood as the Kleinterzwechselklang [of that same skipped-over Klang]. a) from major to minor: c+–od (d understood as the third of b♭, the second Unterquint of c, therefore leading to the skipped-over oa or f+), b) from minor to major: oe–d+ (d understood as the Unterterz of f♯, the sec­ ond Oberquint of e, therefore leading to the skipped-over g+ or ob):

17) The Gegenganztonwechsel, the relation of the tonic to the contrary Klang of the second Gegenquint, is not easily understood but is of good effect (particular­ ly with the introduction of both sevenths, as a result of which it gets a double common-tone connection); it leads to a skipped-over Klang: a) from major to minor: c+–ob♭. b) from minor to major: od–f♯+.

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Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte VI. The Halbtonschritte. The Halbtonschritte occupy a peculiar position, particular­ ly the schlichter and Gegenleittonschritt. The indirect relationship of the roots (the Leittonschritt is a combined Quintterzschritt c–g–b; f–a–e) leads one to suspect that the succession of Klänge may also not be directly (p. 373) intelligible, but practice proves the opposite. The Leitklänge are suitable, actually quite excellent, for direct confrontation with the tonic and have closing force to it in much the same way as the Quintklänge and Terzklänge. This strange fact can be explained only as follows: that the Leittonschritte of all the voices, indeed the “leading relationship” [Leitverhältniss] of the entire chord is regarded in a similar way as the Leittonschritt from the third of the schlichter Quintklang to the tonic. The Leittonschritt is the smallest, still unques­ tionably intelligible melodic step, for chromatic steps are all too often mistaken for Leittonschritte and owe their good effect precisely to this confusion. For the same reason, the Leittonschritt is also the smallest step that can be sung with certainty by the singer. The Leittonschritt undoubtedly owes its meaning, as leading toward clo­ sure, to this proximity in pitch; it is after all so close to the tonic, which forms the center of the tonal conception, that it already makes trouble for the singer endeavor­ ing to successfully resist passing over into it [tonic]. It is the reverse of this conception if the harmony rises or falls a half step, [but] the Klang reached as a result is not the tonic—on the contrary the tonic is being moved away from—and yet [this new Klang] creates no expectation of a skipped-over Klang, rather the effect is a conclusive one. The then-prevailing feeling is that the key has not been left at all, but has only been shifted, moved. In such cases, the spelling in the sense of a chromatic Secundschritt may correspond most perfectly to the sensa­ tion of the event. The chromatic shifts are no empty folly; rather with them the mas­ ters achieve the most beautiful effects. For example, the sudden introduction of the a-major chord instead of the a♭-major chord has a magical effect, like a sudden flash of lightning—all at once we are transported out of the depths of the flat keys into the bright heights of the sharp keys. The occurrence is then similar to when in a chord the fifth, the root or the third is altered; the chromatic Secundschritt is an alteration of the entire chord. The individual species of Halbtonschritte are: 18) The rising Halbtonschritt. Regarded as a Leittonschritt, it is a) in the major sense (c+–d♭+) a Gegenleittonschritt. b) in the minor sense (oe–of) a schlichter Leittonschritt. If in the major sense the Klang reached (d♭+) is intended as tonic, then the Schritt is a retrograde schlichter Schritt and has cadential force; if that is not the case, then it has an opposing [gegensätzlich] effect, like the Gegen­ quintklang, Gegenterzklang, etc., and will best lead to a skipped-over Klang

or the schlichter Quintklang follows it directly, leading back to the tonic:

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Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte Likewise, in the minor sense the Schritt has cadential force if the Klang reached (of ) was intended as tonic; it is then a retrograde Gegenleit­ tonschritt (we have, of course, two kinds of leading tones, that in the minor sense—from above—and that in the major sense—toward above; both are valid in either mode). If the second Klang is viewed according to its harmon­ ic relationship, as an indirect relative it leads to a direct relative, or the (p. 374)

tonic's Seitenwechselklang follows it, leading back to tonic:

Regarded as a chromatic Schritt, the rising Halbtonschritt always involves an upward step, both in its major meaning and in its minor meaning, be­ cause this effect is not dependent upon connections of harmonic relation­ ship, but rather upon the difference in pitch (melodic connection). More­ over, frequently with Halbtonschritte spelling is arbitrarily chosen accord­ ing to the convenience of the accidentals. To discover the right one is not entirely easy and is often a matter of keen musical instinct. 19) The falling Halbtonschritt. This is in all respects the opposite of the rising one. Regarded as a Leittonschritt, it is a) in the major sense (c+–b+) a schlichter Leittonschritt. b) in the minor sense (oe–od♯) a Gegenleittonschritt. If, in the major sense, the Klang reached (b+) is intended as tonic, then the Schritt is a retrograde Gegenleittonschritt and has cadential force; if the ini­ tial chord was the tonic, then the Klang reached will lead to a skipped-over

one, which leads back to the tonic:

Likewise, in the minor sense the Schritt has cadential force if the second chord (od♯) was intended as tonic; it is then a retrograde schlichter Leitton­ schritt. If the initial chord was tonic, then the Leitklang will best move back to the tonic through the Seitenwechselklang:

(p. 375)

20) The Leittonwechsel (cf. §16) is an extraordinarily easy to understand

Schritt and comes out of the scale belonging to the tonic; in retrograde it has ca­ dential power: a) from major to minor: c+–ob, Page 23 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte b) from minor to major: oe–f+. 21) The Gegenleittonwechsel is, in spite of the distant harmonic relationship, of good effect, particularly if the sevenths of both Klänge are introduced: a) from major to minor: c+–od♭. b) from minor to major: oe–d♯+.

A coherent sense of tonality is disturbed by such Schritte, as it is scarcely possible to return directly to the tonic. In general, the introduction of more distant Schritte must be handled very cautiously and judiciously. The digres­ sion from the tonic's closer relations must not appear unmotivated. These sorts of Harmonieschritte are therefore actually more a means of modula­ tion than part of tonal harmony. VII. Tritonusschritte. As notorious as the “horror tritoni” is in the history of counterpoint, more recent times not only have grown fond of the tritone as a simultaneity and put up with it as a melodic interval [Stimmenschritt]—they have even learned to value as extremely effective a Harmonieschritt that is, in consideration of the roots, a Tritonusschritt. Not Wagner and Liszt, no, Mozart and especially Beethoven had already naturalized this Schritt without any masking by dissonances. The only species of this Schritt that does not occur (at least is not known to me to occur) is the Gegen-Tritonuswechsel; the others are very frequent: 22) The schlichter Tritonusschritt is the relation of the tonic to the Terzklang of the second fifth (c+–g–d–f ♯+; oe–a–d–obf); the Schritt reaches out a great dis­ tance and will therefore lead to its skipped-over Klänge: a) between Oberklänge c+–f ♯+, b) between Unterklänge oe–ob♭. Usually it serves modulation, particularly modulation back from the con­

trary side of the tonic.

(p. 376)

23) The Gegentritonusschritt is the relation of the tonic to the Gegen­

terzklang of the second Gegenquint (c+–f–b♭–g♭+; oe–b–f ♯–oa♯). Its effect is just as good as that of the schlichter Tritonusschritt; of course it leads the modula­ tion to the opposite side: a) between Oberklänge: c+–g♭+,

Page 24 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte b) between Unterklänge: oe–oa♯.

24) The Tritonuswechsel (§18) is found within the scale and is quite well under­ stood; taken from the tonic it either initiates a modulation or emphatically intro­ duces a Klang from the schlicht side: a) from major to minor: c+–of ♯, b) from minor to major: oe–b♭+.

The last, to my knowledge, occurring Harmonieschritt is a Doppelterzschritt, namely 25) The Doppelterzwechsel: a) from major to minor: c+–og ♯. b) from minor to major: oe–a♭+.

The Schritt has a common tone (the third), which makes it considerably eas­ ier to understand.

Notes: (1.) Kevin Mooney, in “The ‘Table of Relations’ and Music Psychology in Hugo Riemann's Harmonic Theory” (Ph.D. diss., Columbia University, 1996), 243–247 and appendix 2, presents a formalization of what he terms the “SW-system.” References to both the “Schritt/Wechsel system” and the “Schritt/Wechsel group” appear throughout the special issue of the Journal of Music Theory (42.2, Fall 1998) devoted to neo-Riemannian theory. (2.) Richard Cohn, “Introduction to Neo-Riemannian Theory: A Survey and Historical Per­ spective,” Journal of Music Theory 42.2 (1998): 174. Cohn refers to David Lewin, General­ ized Musical Intervals and Transformations (New Haven: Yale University Press, 1987; reprint New York: Oxford University Press, 2007) and Brian Hyer, “Tonal Intuitions in ‘Tristan und Isolde,’” (Ph.D. diss., Yale University, 1989).

Page 25 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte (3.) See especially Brian Hyer, “Reimag(in)ing Riemann,” Journal of Music Theory 39.1 (1995): 115, 130; and Richard Cohn, “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions,” Music Analysis 15.1 (1996): 9–40; al­ so idem, “Neo-Riemannian Operations, Parsimonious Trichords, and Their Tonnetz Representations,” Journal of Music Theory 41.1 (1997): 1–66. (4.) This idea is central to the work of David Kopp, in particular his Chromatic Transfor­ mations in Nineteenth-Century Music (Cambridge: Cambridge University Press, 2002). (5.) Henry Klumpenhouwer, “Some Remarks on the Use of Riemann Transformations,” Music Theory Online 0.9 (1994). (6.) Klumpenhouwer, “Some Remarks,” paragraphs 3–4. (7.) Klumpenhouwer, “Some Remarks,” par. 7. (8.) The table summarizes the discussion of diatonic root-interval relations found in Riemann's Skizze einer neuen Methode der Harmonielehre (Leipzig: Breitkopf und Här­ tel, 1880), chapters 2 and 3. The table originally appeared in my “The Chaos of Possibili­ ties: Combinatorial Group Theory in Nineteenth-Century German Harmony Treatises” (Ph.D. diss., State University of New York at Buffalo, 2002), 240. Klumpenhouwer's extension of the diatonic Schritt/Wechsel system into the Schritt/Wech­ sel group is in keeping with a second, “chromatic” presentation of the Schritt/Wechsel system later in the Skizze, to be discussed below. (9.) Klumpenhouwer (“Some Remarks,” par. 22) notes the exception of the Kleinterz­ schritt, commenting “one need provisionally to take kleinterz as signifying a major sixth.” (10.) Riemann Skizze, section 38. This table originally appeared in my “Chaos,” 245. (11.) See Julian Hook, “Uniform Triadic Transformations,” Journal of Music Theory 46.1/2 (2002): 57–126, especially sections 2.2–2.8. See also Cohn, “Introduction to Neo-Rie­ mannian Theory,” Edward Gollin, “Some Aspects of Three-Dimensional Tonnetze,” Journal of Music Theory 42.2 (1998): 195–206, and John Clough, “A Rudimentary Geometric Mod­ el for Contextual Transposition and Inversion,” Journal of Music Theory 42.2 (1998): 297– 306. (12.) Kopp, Chromatic Transformations, 154; emphasis added. (13.) Ibid., 159. (14.) Ibid., 154–155. (15.) Hyer, “Reimag(in)ing Riemann,” 115. Cohn (“Neo-Riemannian Operations,” 58–59) identifies this epistemological problem, which he describes as one of a “path/goal duali­ ty,” as an open question meriting further consideration. For discussion of the issue, see my “Chaos,” 20–26.

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Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte (16.) Hyer, “Reimag(in)ing Riemann,” and “Tonal Intuitions in ‘Tristan und Isolde’ ”; Cohn, “Neo-Riemannian Operations.” See the related discussions in Edward Gollin, “Represen­ tations of Space and Conceptions of Distance in Transformational Music Theories” (Ph.D. diss., Harvard University, 2000), 275–276, and Engebretsen, “Chaos,” 23–25. (17.) Nineteenth-century theorists did not, however, explicitly engage combinatorial group theory, which was emerging roughly contemporaneously (Engebretsen, “Chaos,” section 5.2). On group presentations, see Magnus, Karrass, and Solitar, Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations (New York: John Wiley & Sons, 1966). Both Gollin, “Representations,” and Engebretsen, “Chaos,” re­ ly upon a group-combinatorial perspective, Gollin as part of a survey of musical spaces in eighteenth- and nineteenth-century theories and Engebretsen as part of a response to a questions posed in Cohn, “Neo-Riemannian Operations,” about nascent group-theoretic content in nineteenth-century harmonic theory. Though a combinatorial perspective per­ vades neo-Riemannian discussions of groups, Gollin deserves credit for introducing the terminology and formal apparatus of combinatorial group theory into the neo-Riemannian discourse. (18.) Gollin, “Representations,” explores these notions in some detail. See especially his section 2.3. (19.) Catherine Nolan, “Music Theory and Mathematics,” in The Cambridge History of Western Music Theory, Thomas Christensen, ed. (Cambridge: Cambridge University Press, 2002), 272–304. (20.) For example, each of the relators in presentation of the LPR above effects a parti­ tion of the consonant triads: L2, P2, and R2 effect pair-wise partitions, (LP)3 yields Cohn's hexatonic cycles, and so on. (21.) Catherine Nolan, “Combinatorial Space in Nineteenth- and Early Twentieth-Century Music Theory,” Music Theory Spectrum 25.2 (2003): 238. For a discussion of composition­ al uses of combinatorial techniques, see Nolan, “Music Theory and Mathematics.” (22.) Discussed in Gollin, “Representations,” 217–224. (23.) From a systemic perspective, Riemann's scheme of root-interval classifications re­ fines Oettingen's original root-interval system in one very significant respect. Whereas Oettingen had classified relationships between consonant triads by indicating whether the relationship was between chords of the same or opposite qualities and by identifying the interval between the triads’ roots, Riemann treats the interval between the triads’ roots as a directed interval, gauged with respect to the direction of the initial triad's gen­ eration. Among its other implications, this refinement leads to the dual definition of both transposition and inversion relations (this is, in effect, the very refinement that Klumpen­ houwer proposes to Lewin's and Hyer's systems in advocating the Schritt/Wechsel system) and yields one-to-one mappings between triads.

Page 27 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte (24.) Riemann distinguishes between root-interval relationships involving primary triads exclusively and those involving one or more secondary triads. The distinction drawn in ex­ ample 12.1 between relationships involving primary and secondary triads, versus those involving only secondary triads, is implicit in his discussion but is recognized explicitly in David Kopp, “A Comprehensive Theory of Mediant Relations in Mid-Nineteenth-Century Music” (Ph.D. diss., Brandeis University, 1995), 115. (25.) Gollin (“Representations,” 230) presents a similar listing of derivations. The symbol­ ic representations of generators given here are not found in the Skizze, but they do have a precedent in the symbology of Riemann's Die Hülfsmittel and play a role in his later es­ say, “Ideen zu einer ‘Lehre von den Tonvorstellungen,’ ” as discussed by Gollin and in En­ gebretsen, “Chaos.” (26.) On chordal identity versus relationship, see Daniel Harrison, Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of Its Precedents (Chicago: University of Chicago Press, 1994), 281–282, and Kopp, “A Comprehensive Theory,” 181– 184. (27.) See Gollin, “Representations,” 230. (28.) Riemann does not present these relationships as generators per se, but their status is clear from the symbology associated with the earlier presentation in Die Hülfsmittel (see note 25) and can be inferred readily from the discussion of individual relationships found in the Skizze. Riemann was not alone in his treatment of the Terzschritt as a gener­ ator: Hostinský, in Die Lehre von den musikalischen Klängen (Prague: H. Dominicus, 1879), also considered it to be a fundamental relationship, but considered the Terzwech­ sel to be one as well. (29.) Gollin (“Representations,” 233) presents a listing of derivations like that found in ex­ ample 12.2. In cases where more than one derivation is suggested, I have followed Riemann's convention of representing compound Wechsel relations in the form Schritt-then-Wechsel. (30.) Gollin (“Representations,” 234) offers a slightly different presentation of the same group. Justification of the chosen relators is omitted here, but can be found in Engebret­ sen, “Chaos,” 223. (31.) Arthur von Oettingen, Harmoniesystem in dualer Entwickelung (Dorpat: W. Gläser, 1866), 156. (32.) Henry Klumpenhouwer, “Dualist Tonal Space and Transformation in Nineteenth-Cen­ tury Musical Thought,” in The Cambridge History of Western Music Theory, ed. Thomas Christensen, 456–476 (Cambridge: Cambridge University Press, 2002), 466. (33.) Alexander Rehding, Hugo Riemann and the Birth of Modern Musical Thought (Cambridge: Cambridge University Press, 2003), 51.

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Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte (34.) On Hauptmann's and Oettingen's use of key relations to constrain the composition of their chosen generators, see Engebretsen, “Chaos,” sections 2.3 and 3.3. (35.) Riemann, Musikalische Syntaxis (Leipzig: Breitkopf und Härtel, 1877), 120; quoted in Rehding, Hugo Riemann, 105 (translation Rehding's). (36.) Riemann, “Degeneration und Regeneration in der Musik,” Max Hesses deutscher Musikerkalender 23 (1908): 136–138. Reprinted in ‘Die Konfusion in der Musik’: Felix Draesekes Kampfschrift von 1906 und die Folgen, ed. Susanne Shigihara (Bonn: Gudrun Schröder, 1990), 245–249. See also Rehding, Hugo Riemann, 138. (37.) See Carl Dahlhaus, “Terminologisches zum Begriff der harmonischen Funktion,” Die Musikforschung 28.2 (1975): 197–202. Riemann's quest for a prescriptive musical logic is discussed in Rehding, Hugo Riemann, in particular 42–45 and chapter 3. See also Harri­ son, Harmonic Function chapter 6; and Kopp, Chromatic Transformations, chapter 4 and especially section 6.1. (38.) Harrison, Harmonic Function, 279–280. (39.) Kopp, Chromatic Transformations, 136–137. (40.) This distinction between harmonic coherence and tonal coherence is drawn sharply in Kopp, Chromatic Transformations, 99, 137, and 150. (41.) Riemann, Handbuch der Harmonielehre, 4th ed. (Leipzig: Breitkopf und Härtel, 1906), 134. (42.) Ibid., 135. (43.) Klumpenhouwer, “Some Remarks,” par. 6. (44.) This question is answered in different ways by different authors. Cohn (“Introduc­ tion to Neo-Riemannian Theory,” 174), citing Mooney, indicates that a transformational approach is anticipated in the writings of both Oettingen and Riemann, whereas Kopp (Chromatic Transformations) presents Riemann's Schritt/Wechsel system as a classifica­ tion scheme describing static relationships between triads. For those in agreement with Kopp, the transformational overlay then becomes part of the imported “ideational struc­ ture” to which Klumpenhouwer refers. (45.) Hyer, “Tonal Intuitions,” 129–130. (46.) Kopp, Chromatic Transformations, 137. (47.) In the preceding section of the Skizze (section 37), Riemann has discussed the short­ comings of the diatonic scale as the schema for modern tonality, noting, for example, that acceptance of a direct relationship between the tonic and its Terzklänge—e+ and a♭+ in c major—undermines the diatonic scale, suggesting instead the scale c. d. e♭. e. f. g. g ♯. a♭. a. b. c. Rather than advocating the enharmonic-chromatic or some other scale as an alter­ Page 29 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte native schema, Riemann characterizes scales as “nothing more than a connection of the tones of the tonic triad through passing tones.” If scales are “really nothing more than arpeggiations of a singe chord with passing tones,” different scales simply reflect differ­ ent types of melodic motion between elements of the tonic triad. (48.) The emphasis is Riemann's. Throughout this translation, text emphasized through increased spacing (Sperrschrift) has been italicized. (49.) Riemann refers his reader to example 108 b, but surely intends example 108.6, where the progressions c+–e+–c+ and oe–oc–oe appear as part of a compilation of cadential progressions. Because example 108 is not part of section 38, the Systematik der Harmoni­ eschritte, it is not reproduced here. Riemann will refer to several other examples not in section 38; these, too, will be described but not reproduced. (50.) In section 14, Riemann introduces the Terzwechsel in a diatonic context, emphasiz­ ing its association of secondary Klänge [Nebenklänge] with the primary Klänge [Hauptk­ länge] and pointing to the smooth voice-leading connections involved (Klänge related by the Terzwechsel share two common tones, while the third voice moves a Ganztonschritt and the bass moves between the major root and the minor fifth). In example 109, Riemann demonstrates the improved voice-leading and/or greater intelli­ gibility that results from the addition of a minor seventh in many Harmonieschritte. Example 109.8, in particular, shows the progressions c+–oeVII (adding f ♯ beneath the oe triad) and oe–c+7 (adding b♭ above the c+ triad). (51.) Example 108.9 shows the progressions c+–oa♭–c+ and oe–g ♯+–oe in four voices, illus­ trating the voice-leading connections just described. (52.) Section 15 of the Skizze includes a rather extensive note addressing this issue of mi­ nor-third relationships as major-sixth relationships, a portion of which follows: The minor third is related only indirectly to the Hauptton, namely through the con­ nection of a third tone, that is c+ is related to a through f +, c belongs to the f  + -Klang as its fifth and a as its third, c–a is thus the passage from the fifth to the third. We can also seek mediation with respect to oe; in oe, c is the third and a the fifth; then c–a is thus a passage from third to fifth. In both cases the Schritt c–a is a rising one, if we think of the overtone series for the Durklang and of its inver­ sion, the undertone series, for the Mollklang:

that is, a actually appears as a großer Sextenschritt, in the overtone series as the passage from the third partial to the fifth, in the undertone series as the passage from the fifth partial to the third. If we see in the ascending that which is suitable to major [and] in the descending that which is suitable to minor, then from c+ we must designate the Schritt c–a as schlicht [and] thus regard c+–oa and g+–oe as schlichter Kleinterz-Wechsel (or Sextenwechsel); correspondingly, the passage Page 30 of 31

Neo-Riemannian Perspectives on the Harmonieschritte, with a Translation of Riemann's Systematik der Harmonieschritte from a to c or from e to g is, in the oe-Tonart, to be considered schlicht, so that here o a–c+ and oe–g+ also appear as schlichter Kleinterz-Wechsel. (53.) Example 108.10 shows the progressions c+–a+–c+ and oe–og–oe in four voices as part of a roster of cadential progressions. (54.) Example 108.11 illustrates the cadential progressions c+–e♭+–c+ and oe–oc ♯ –oe in four voices. (55.) This earlier discussion focused on voice-leading issues encountered with this pro­ gression (the potential for forbidden parallels and melodic tritones).

Nora Engebretsen

Nora Engebretsen , an associate professor of music theory at Bowling Green State University, holds a Ph.D. in music theory from the State University of New York (SUNY) at Buffalo. Her research interests include chromatic harmony, transforma­ tional theory, and the history of theory. Her work has appeared in Music Theory Spectrum, Theoria, the Journal of Music Theory Pedagogy, and collections published by the University of Rochester Press and Stockholm University Press.

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On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen

On a Transformational Curiosity in Riemann's Schema­ tisirung der Dissonanzen   Edward Gollin The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0013

Abstract and Keywords This article explores the less examined aspect of Riemann's Skizze, the Schematirisung der Dissonanzen. The article focuses in particular on Riemann's category of Doppelklänge, by examining the conceptual origin of the category in the Skizze and its relation to Riemann's harmonic theories. Doppelklänge are dissonant chords that arise through a combination of two consonant triads in a coordinate relationship. While the Doppelklänge category is short-lived, it nevertheless fascinating for its relation to Riemann's taxonomy of harmonic relations. Specifically, Riemann classifies Doppelklänge according to the Har­ monieschritte that relate their component Klänge, reflecting within those dissonant chordal structures the same relations that organize harmonic progression in music. The article then examines how an explicitly transformational reinterpretation of Riemann's Doppelklänge can offer analytic insights into tonal, post-tonal and transitional musical repertoires such as by Chopin, William Grant Still, and Ravel. The article ends by provid­ ing an appendix that presents a translation of Schematirisung der Dissonanzen. Keywords: Skizze, Doppelklänge, Harmonieschritte, Klänge, dissonant chords, Chopin, William Grant Still, Ravel

THE present chapter explores Riemann's seldom-discussed Schematisirung der Disso­ nanzen, a taxonomy of dissonant chord types presented in his 1880 Skizze einer neuen Methode der Harmonielehre. The chapter focuses in particular on Riemann's category of Doppelklänge, dissonant chords that arise through the combination or union of two con­ sonant triads, examining the conceptual origin of the category in the Skizze and its rela­ tion to Riemann's harmonic theories. The Dopppelklang category, though short-lived (ap­ pearing only in the Skizze), is nevertheless fascinating for its relation to Riemann's taxon­ omy of harmonic relations. Specifically, Riemann classifies Doppelklänge according the Harmonieschritte that relate their component Klänge, reflecting within those dissonant chordal structures the same relations that organize harmonic progression in music. The chapter then suggests how an explicit transformational interpretation and extension of Riemann's theory can offer analytical insights in works by Chopin, William Grant Still,

Page 1 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen and Ravel. An appendix to the chapter presents a translation of the complete Schema­ tisirung der Dissonanzen. Although the Skizze is generally associated with Riemann's system of Harmonieschritte, his catalog of dualistically conceived triadic relationships, more than half the treatise (chapters 4 through 7) is dedicated to the derivation and explication of dissonant chord types. The Schematisirung der Dissonanzen appears at the end of chapter 6, and presents in outline form a summary of the dissonant chord (p. 383) types discussed in the treatise up to that point.1 The Schematisirung forms an integral part of Riemann's overall Har­ monielehre, functioning as a complement to the Systematik der Harmonieschritte.2 Within the regulative framework of Riemann's harmonic theory, the taxonomy of the Schematisirung illustrates the derivation of multifarious dissonant chord forms as modifi­ cations of consonant Klänge, which in turn may participate in the normative progressions outlined in the Systematik der Harmonieschritte.3 Dissonance, for Riemann, is defined in opposition to the unity of triadic consonance, and the two main categories of dissonant chords in the Schematisirung correspond to the two primary means by which triadic unity can be disrupted: through extension or through al­ teration. Dissonances of class I, those that arise through extension, are discussed in the fourth and fifth chapters of the Skizze, and involve adding a sixth, seventh, or ninth to a consonant Klang. Dissonances of class II, those that arise through alteration, are dis­ cussed in the sixth chapter, and involve the chromatic inflection of the prime, third or fifth of a consonant Klang. In dual fashion, Riemann presents each generic dissonance type in inversionally related major and minor forms. For example, the natural seventh chord, Riemann's first chord type in class I, exists in a major form, derived through the addition of a minor seventh above the Hauptton of a major Klang (e.g., G B D | F), and a minor form, derived through the addition of a minor seventh added below the Hauptton of a minor Klang (e.g., B | D F A). Altered chords are similarly conceived dualistically: the di­ minished fifth chord involves the lowering of the upper fifth of a major Klang (e.g., C E G becomes C E G♭) or raising the lower fifth of a minor Klang (e.g., A C E becomes A♯ C E). To the two main categories of dissonance in the Schematisirung, Riemann proposes a third: dissonances that arise through the combination or simultaneous presentation of two consonant Klänge. The seeds of the Doppelklang idea appear in the opening para­ graph of the Skizze: There are two kinds of consonant chords, namely major chords and minor chords. A major chord is composed of a Hauptton, its upper fifth and upper third; a minor chord [is composed] of a Hauptton, under fifth and under third. These three tones —Hauptton, fifth and [major] third—are the essential components of each conso­ nant chord. Yet only upper fifth and upper third, or under fifth and under third, to­ gether with the Hauptton, can form a consonant chord: not upper fifth and under third, or under fifth and upper third. An assemblage of this last kind rather always produces a dissonant chord. We name the joining of Hauptton, upper third, and upper fifth, instead of major chord, Oberklang; that of Hauptton, under third, and Page 2 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen under fifth, instead of minor chord, Unterklang. So we can say: Oberklang and Un­ terklang are opposed to one another, and do not merge into a unity, even if they share the same Hauptton; their combination is instead a Doppelklang, a Klang-du­ ality (Klangzweiheit), as is the combination of Klänge that do not have the same Hauptton.4

Ex. 13.1. Augmented triads and Seitenwechsel-disso­ nances within (a) the C-major mixed-mode tone scheme and (b) the a-minor mixed-mode tone scheme.

Riemann expands upon the notion of Klangzweiheit at the opening of chapter 6 in his dis­ cussion of the augmented triad.5 Riemann illustrates the derivation of the augmented tri­ ad using Hauptmann's mixed-mode tone schemes. Examples 13.1a (p. 384) and 13.1b present, respectively, Hauptmann's C-major mixed-mode tone scheme (his Molldur system), and Hauptmann's a-minor mixed-mode tone scheme (his Durmoll system).6 Brackets above each scheme illustrate the location of the augmented triads thereupon: in C major, the augmented triad A♭–C–E arises through the union of the lower third of the tonic triad and the upper third of the minor subdominant; in a minor, the augmented triad C–E–G♯ arises through the union of the lower third of the major dominant and the upper third of the tonic. The augmented triad, Riemann observes, is consequently the incom­ plete manifestation of the simultaneous presentation of the two mode-opposed Klänge that share the same Hauptton, which Riemann names the Seitenwechsel dissonance after the Harmonieschritt that relates the two Klänge. On example 13.1, the full Seitenwechsel dissonances are shown with brackets below the schemes. Using his Klangschlussel notation, Riemann symbolizes the Seitenwechsel dissonance by affixing symbols for both major and minor triads (“˚” and “+”) to the letter name of Hauptton: in the C-major mixed-mode scheme, ˚c+; in the a-minor mixed-mode scheme, ˚e+. Riemann then pro­

Page 3 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen vides a notational shorthand, overlaying the major and minor symbols, yielding the new symbol 𝆴 to represent the augmented triad or the complete Doppelklang.7 Riemann suggests further that other sonorities could be similarly construed. Brackets on example 13.2 illustrate Riemann's derivation of Doppelklänge through the union of fifthrelated triads within pure major and minor tone schemes: f+c designates the dissonance formed by the union of a major tonic and its lower fifth in the C-major scheme, c+g designates the major tonic and its upper fifth; e˚b designates the dissonance formed by the union of a minor tonic and its upper fifth in the pure a-minor scheme, a˚e designates the minor tonic and its lower fifth. Riemann names the sonorities of example 13.2 dissonances of the Quintschritt; as with the Seitenwechsel dissonance, Riemann names the Doppelklang dissonance after the Harmonieschritt that governs the relationship be­ tween its component Klänge.

Ex. 13.2. Dissonances of the Quintschritt within (a) the pure C-major tone scheme and (b) the pure a-mi­ nor tone scheme.

But Riemann embraces the notion of Doppelklänge with caution. One concern is practical: Riemann recognized that in four-voice settings, expression of the complete five-tone Dop­ pelklänge of examples 13.1 and 13.2 is not possible, and he suggests that one could more justifiably notate elliptical versions of a Seitenwechsel dissonance as a single Klang with an added minor sixth (Class I.4 in the Schematisirung). (p. 385) Riemann illustrates how the four-tone subset F A♭ C E from the Seitenwechsel dissonance in example 13.1a could be more simply notated as an f-minor Klang with added minor under sixth, cVI〈; or how the collection C E G♯ B, a subset of the Seitenwechsel dissonance in example 13.1b, could be notated as an E-major Klang with added over sixth, e6〉.8 Riemann's illustrations under­ score a mechanical (or statistical) understanding of harmonic identity, and Riemann sug­ gests that the Doppelklang notation might be appropriate only in cases where there is am­ Page 4 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen biguity of over which Klang is predominant—predominance here equated with a prepon­ derance (or majority) of tones. More fundamentally, Riemann expresses doubt about the viability of the co-ordinate (ver­ sus subordinate) conception of Klänge implied by the Doppelklang notation, suggesting that the Doppelklang category might exist “more in theory rather than in practice.”9 Riemann notes that the tone F in the seventh chord G B D | F, understood in the context of the C-major tone scheme, may be understood as a component of an F-major chord (a subdominant tone in the scheme). But Riemann observes that the tone does not project the functional identity of the F-major triad, but instead simply disturbs the consonance of the G-major Klang: F is subordinate to, not coordinated with, the G-major Klang. Similar­ ly, Riemann argues that an augmented triad, even in cases where there is no fifth, can be understood in the sense of a single consonant Klang with the minor sixth presented in place of the fifth.10 But despite his concerns, Riemann nevertheless retains the notion of Doppelklang dissonances as a third category in the Schematisirung, expanding the category beyond Seitenwechsel-related and Quintschritt-related chord forms to include ten distinct chordal dissonances that range from tetrachords to hexachords. The dissonances are largely cir­ cumscribed by the triads available within the pure and mixed-mode tone schemes (and concomitantly, by the Harmonieschritte that relate triads within those schemes).11 The one exception is the dissonance of the Gegenganztonwechsel, which joins the minor sub­ dominant triad with the major dominant of the dominant, providing a Doppelklang derivation of the augmented sixth chord. The Doppelklang conception of dissonance could only have arisen at that moment in the development of Riemann's harmonic theories when his recognition of triadic consonance (a theory of consonance based on the Klang) had superseded the acoustical view, but be­ fore his concept of harmonic function had sufficiently coalesced. (p. 386) Indeed the sup­ position of sonorities that project multiple, coordinate Klänge was fundamentally at odds with Riemann's evolving notion of function: although it is possible to conceive of a collec­ tion that projects the tones of two distinct Klänge, it is not possible in Riemann's function theory to conceive of a collection that simultaneously projects two distinct functions—for example, to conceive of a collection as simultaneously being a tonic and a dominant—in the same local key. Riemann's Doppelklang theory was consequently purged from the second edition of the Skizze, renamed the Handbuch der Harmonielehre.12 In the second edition Handbuch, Riemann retains the same basic chapter structure, but removes the Schematisirung der Dissonanzen from chapter 6, presenting instead a new section, the Übersicht der Disso­ nanzen (Overview of Dissonances), at the end of the seventh chapter. The Übersicht retains the summary of extended and altered chords (categories I and II) but replaces the Doppelklänge category with a new category III, suspension chords.13 In later editions of the Handbuch, even the Übersicht der Dissonanzen was discontinued: as Riemann ex­ panded his Dissonanzlehre to include an ever greater palette of melodic dissonances, the Page 5 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen attempt to create a comprehensive listing of dissonant chord types was abandoned in fa­ vor of simply demonstrating harmonic figuration through the Klangschüssel notation.14

Some Analytical and Transformational Exten­ sions of Riemann's Doppelklang Theory Riemann provides no examples of Doppelklang dissonances in musical practice (although examples drawn from musical practice are largely absent throughout the Skizze). Yet even if Riemann had been disposed to demonstrate his theories empirically, he would like­ ly not have found many true common-practice examples of dissonances formed by distinct Klänge in coordinate, rather than subordinate, relationships. If, however, one detaches Riemann's theory of Doppelklänge from its regulative function within the context of his Harmonielehre, the theory invites a number of analytical and theoretical extensions. Most notably, the conceptual organization of Riemann's Doppelklang theory by his system of Harmonieschritte naturally suggests a transformational interpretation. Transformational music theory offers a means to illustrate how intervals internal to a collection can be pro­ jected externally, as the transformational relationships between elements in a work. Like­ wise, because the same relations that define harmonic succession in Riemann's harmonic system also define the distinct categories of Doppelklang dissonances (i.e., define their in­ ternal structure), Riemann's Doppelklänge offer a means to relate dissonant sonorities in a work to the salient transformations among those sonorities.

Ex. 13.3. Leittonwechsel manifest in Chopin, Ballade op. 38, (a) relating phrases in the pastorale section, mm. 17–19 (b) the Leittonwechselklang as tonic sub­ stitute at m. 34, (c) the Leittonwechselklang-disso­ nance in the Presto con fuoco, m. 47. 

Page 6 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen Example 13.3 explores the transformational perspective as it informs passages from Chopin's Ballade op. 38. Chopin's Ballade has been much discussed as an example of tonal pairing or directional tonality, a work that ends in a key different than that in which it begins.15 Specifically, the Ballade begins in F major and closes in the Leittonwechsel-re­ lated key of a minor, oscillating between the two over the course the piece. In particular, pastorale-like homophonic passages in F, shown in example 13.3a, are contrasted with the stormy Presto con fuoco passages in a, shown in example 13.3c. The Leittonwechsel relationship is manifest between successive phrases within the pastorale sections as well. In example 13.3a, Leittonwechsel defines both the direct relation between F-major and aminor triads in measure 18, and the relation between phrases in F and a. Example 13.3b illustrates a phrase ending parallel to that of measure 17, resolving not to the F-major tonic, but to a-minor, the tonic Leittonwechselklang. The opening of the Presto con fuoco at measure 47, shown as example 13.3c, is curious in light of the salient Leittonwechsel relationships throughout the work. Although the passage indisputably arpeggiates the lo­ cal a-minor tonic with an added f6̂ (in Riemann's notation, eVII〉, compare 3f in the Schematisirung), recognizing the chord as a Leittonwechsel dissonance provides a poten­ tially more unifying view of the (p. 388) passage—it posits the same transformation, mani­ (p. 387)

fest externally in the salient relationship between keys, phrases and triads, as an internal relationship organizing components of the dissonant sonority that opens the Presto con fuoco. Although Riemann circumscribes the range of possible Doppelklänge based on the Haupt­ mannian notion of closed key schemes, one could pursue the obvious extension of his sys­ tem, forming set classes from the eighteen distinct binary combinations of triads under Schritte and Wechsel. Table 13.1 lists the eighteen combinations of triads possible under Riemann's Harmonieschritte. The table replaces Riemann's cumbersome German nomen­ clature with a simpler notation to represent the contextual relations of Riemann's system: S0, S1, S2,…, S11 represent the twelve Schritte that map mode-identical triads 0, 1, 2,…, 11 semitones in the direction of chord components (up in the case of major, down in the case of minor); W0, W1, W2,…, W11 represent the twelve Wechsel that map any triad to the mode-opposed triad that lies 0, 1, 2,…, 11 semitones apart in the direction of chord com­ ponents.16 Thus Quintschritt = S7, Seitenwechsel = W0, Terzwechsel = W4, Gegenganzton­ wechsel = W10, and so on. The table presents illustrative triadic combinations for each re­ lationship (relative to C-major and a-minor triads), as well as the set class of the resulting collections. The rightmost column lists the corresponding entry for the ten Doppelklänge in the Schematisirung.17 However, more interesting than the taxonomy itself are the possibilities that arise from a transformational interpretation of Schritte and Wechsel as generators of sets. Such a transformational perspective not only allows one to describe the static content of a collec­ tion, but also allows one to discuss how internal relations in a set (p. 389) may interact in a dynamic way with external collections and events, to describe how collections and events progress in a work, particularly in post-tonal contexts.

Page 7 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen Table 13.1 The Distinct Doppelklänge Formed through Binary Combination of Schritte and Wechsel. Transformation

Triadic Combinations

S0

(Identity—no combinations)

S1 = S11

C + D♭; a + g♯

[013478]

S2 = S10

C + D; a + g

[023579]

S 3 = S9

C + E♭; a + f ♯

[01469]

S 4 = S8

C + E; a + f

[01458]

S 5 = S7

C + F; a + e

[01358]

S6

C + F♯; a + d♯

[013679]

W0

C + f; a + E

[01348]

W1

C + f ♯; a + D♯

[014679]

W2

C + g; a + D

[02469]

W3

C + g♯; a + C♯

[014589]

W4

C + a; a + C

[0358]

cf. 23

W5

C + b♭; a + B

[013689]

cf. 21

W6

C + b; a + B♭

[013578]

cf. 27

W7

C + c; a + A

[0347]

W8

C + c♯; a + A♭

[01478]

W9

C + d; a + G

[024579]

cf. 26

W10

C + e♭; a + F♯

[013579]

cf. 22

W11

C + e; a + F

[0158]

cf. 24

Page 8 of 20

Set Class

cf. 20

cf. 19

cf. 18

cf. 25

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen Example 13.4a presents the opening eight measures of William Grant Still's “Cloud Cra­ dles” from his seven Traceries (1939). The work as a whole moves between sections that express various hexatonic and octatonic collections, often exploring mirror-symmetric and “triadic” partitions of those collections. In the opening four measures, for example, a hexatonic collection is partitioned by the pianist's two hands into G♭-major and d-minor triads (the “vagrant” C of measure 2 will be discussed shortly). The reference here to tri­ ads does not imply tonal functionality, nor does it suggest any sort of polytonality. Rather, the triads and various seventh chords in the context of the work are things one does with one's hands when playing the piece, and consequently the intervals they project are sim­ ply part of the larger sound fabric. In this context, viewing triads as generative collec­ tions under certain contextual operations seems appropriate.

Ex. 13.4. (a) William Grant Still, “Cloud Cradles,” from Traceries (1939), mm. 1–8; (b) a network of re­ lationships in mm. 1–6; (c) a network of relationships in mm. 5–8.

On example 13.4b, I illustrate how the opening hexatonic collection may be viewed as the combination of a G♭-major triad and its W3-associate (Riemann might have named this the Gegenkleinterzwechseldissonanz). Under W3, G♭, the tone to which B♭ and D♭ are re­ ferred, is mapped 15 (= 3) semitones up to A, the tone to which F and D (p. 390) are re­ ferred in mirror-symmetric fashion. The appearance of C in the right hand of measure 2— an added under-sixth of the d-minor triad from the Riemannian perspective—disturbs this symmetrical balance. Balance is restored by the appearance at the downbeat of measure 5 of E♭, the over-sixth of G♭ and W3 associate of C. The internal transformation W3 thus impels the music toward the E♭ of measure 5, which in turn participates in the W3 partition of a different hexatonic collection: an enharmonically spelled B-major triad in the left hand is joined with g-minor in the right. Just as Riemann recognized that many dissonances could admit a variety of interpreta­ tions, so too might we view differently the kernels of the hexatonic collections in mea­ sures 1 and 5, regarding the central four pitches sounded on the first two eighths of each Page 9 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen measure as Major-minor triads on B♭ and E♭. I illustrate on example 13.4c how one could thus posit W7 as the operative contextual relation that joins E♭-major and e♭-minor triads in measure 5. W7, the internal relation, is in turn projected outward, as the relation that impels “B-major” in the left hand of measures 5 and 6, onto the arpeggiated b-minor bridge of measure 7. Certain observations we have made about “Cloud Cradles” could also have been made without recourse to contextual operations or triadic cells. Reference to an I3 inversional axis (i.e., the axis about which mapped pcs sum to 3 in a C = 0 labeling system) could as well have explained the motion to E♭ as the I3 associate of C. But neither I3 nor any fixedreference operation can express as clearly as W3 what is transformationally the same about measures 1–4 and 5–6: both arise from the partition of hexatonic collections via the same contextual operation, W3.18 As a final illustration of how a transformational reinterpretation of Riemann's Doppelk­ lang theory offers a means to mediate between tonal and post-tonal analytical perspec­ tives, we examine the opening of the Forlane from Ravel's Tombeau de Couperin. Exam­ ple 13.5a presents the rondo theme in measures 1–5. The passage is curious for its deft juxtaposition of tonal and post-tonal idioms: though it seems to subvert its presumed eminor tonic thoroughly in its first three measures, it clearly defines and projects an e-mi­ nor tonic on the approach to the downbeat of measure 5. The tonal aspects of the passage run deeper than the foreground ii7–v♭3–i cadence at the phrase ending. Example 13.5b illustrates how the compound bass line projects an upper-line stepwise descent from 5̂ in e-minor, over a leaping 4̂–2̂–5̂–1̂ in a lower line. However, the bass-line sketch of example 13.5b, while reinforcing the final cadence, says little about the harmonic content of the first three measures, content that resists interpretation within the tonal world of e-minor. Example 13.5c illustrates an aspect of that harmonic content, extracting and labeling five distinct [0148] tetrachords presented as strong-beat simultaneities in measures 1–3 (tetrachord 4 will be discussed shortly). While the example demonstrates the theme's generally high degree of set class consistency, it gives no account of how the [0148]-tetra­ chords interact with the tonal features of the passage—in particular the perfect fifth A–E sounded as a pedal below tetrachords 2 and 3.

Ex. 13.5. (a) Maurice Ravel, rondo theme from the Forlane of Tombeau de Couperin, mm. 1–5; (b) a lin­ ear sketch of the compound bass line; (c) five [0148] tetrachords in mm. 1–3.

Page 10 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen We begin by asserting that the passage is in e-minor. The closing e-minor tonic of mea­ sure 5 that begins the repeat of the four-bar phrase is the paradigm of which the opening [0148] tetrachord is a dissonance-beclouded image: the opening is an (p. 391) e-minor tri­ ad with an added D♯.19 It is possible to read the opening [0148] tetrachord as an incom­ plete Seitenwechsel-dissonance, a dissonance joining the tonic e-minor with the dominant B-major triads about their common Hauptton B, omitting the F♯.20 However, a different view of the opening tetrachord can be gleaned by observing not simply its pitch content, but the opening transformational gesture in which it participates. The motion from tetra­ chord 1 to 2 is an example of a parsimonious transformation that David Lewin has called a DOUTH2 relation: two tones are fixed—in this case E and D♯—and two tones move by semitone to yield a new set of the same class.21 The motion, an inversion of the [0148] about the semitone D♯/E, is manifest by a real exchange of place and function of the tones E and D♯ in the two chords. The tonic E4 of chord 1 is given to the right hand while D♯ participates in the augmented triad of the left. In chord 2, D♯5 is given to the right hand while E participates in the left hand's augmented triad. The inversion of the [0148] about its semitone is a contextual inversion analogous to W3 with respect to triads. That is, W3 maps the embedded e-minor triad of tetrachord 1 to the embedded G♯-major triad of tetrachord 2. W3 thus defines the relationship between tetrachords 1 and 2, but also rep­ resents an internal relationship within tetrachords themselves: it is the interval between D♯ and E that preserves the dyad within the opening chords. In this sense, we may justifi­ ably regard the [0148] tetrachords as W3 dissonances.

Ex. 13.6. A network relating [0148] tetrachords in the rondo theme of the Forlane.

While I can hear the characteristic triadic motion between hexatonic poles, e-minor and G♯-major, embedded within the opening measure, I admit that my triadic hearing there­ after breaks down.22 What I can hear, however, is an implicit DOUTH2 relation between tetrachord 4, a nonliteral-[0148], and tetrachord 5, an explicit [0148]. That is, I hear an implied motion from {A, C♯, E♯, G♯} to {A, C, E, G♯}, predicated on hearing an implied C♯ Page 11 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen in tetrachord 4. This inference is aided, however, by the (p. 392) presence of an actual C♯4 both before and after the downbeat of measure 3. In other words, understanding tetra­ chord 4 as an implied [0148] allows one to posit the same contextual transformation that was an agent of departure from e-minor—inversion of an [0148] about its semitone—as an agent of return to tetrachord 5. An interpretation of tetrachord 5 as a (misspelled) a-minor triad with added leading tone, by analogy to the opening e-minor, is not difficult to hear, particularly because it picks up, and makes sense of, the open fifth, A–E, in measure 2. That is, tetrachord 5 fills in the empty subdominant fifth that had been subposed below tetrachord 2. Example 13.6 consolidates these observations within a single network that makes sense as well of tetra­ chord 3. The operation W3 is here understood to be equivalent to the clumsier expression “invert an [0148] about its semitone.” I have labeled the upper two nodes subdominant and tonic to reflect the functional identity of their embedded triads. The lower corner nodes, as W3 conjugates of the upper nodes, have been labeled “W3 Subdom” and “W3 Tonic” for purely formal reasons—I am not asserting that one necessarily hears either chord as a substitute for the tonic or subdominant, or that one necessarily be aware of their embedded major-triad subset. What is curious, however, is that the pathway from tetrachord 2, the putative “tonic,” to tetrachord 4 a putative “subdominant,” is mediated by tetrachord 3, a putative “submediant.” That is, tetrachord 3 is the minus-3-transpose of tetrachord 2 and the 4-transpose of tetrachord 4—it forms a “third divider.”

Ex. 13.7. W3 relating e-minor and G♯-major mixedmode tone schemes.

Page 12 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen

Ex. 13.8. Ravel, Forlane from Tombeau de Couperin, (a) mm. 9–11 and (b) mm. 16–18.

A different perspective on the same structural relations can be gleaned by extending the contextual inversion among [0148]s to the larger context of tone schemes. The upper half of example 13.7 illustrates a Riemannian mixed-mode key scheme for e-minor; the lower half of example 13.7 presents the Riemannian G♯-major tone scheme that results from the inversion of the e-minor scheme about E and D♯.23 The example illustrates how the e-mi­ nor tonic triad with added leading tone maps via W3 to G♯-major with added f6. One can also see the analogous transpositional pathway, T-3-then-T-4, that leads from the G♯ to C♯ via E♯, the actual submediant in the G♯-major scheme. The view of our contextual inver­ sion acting on the e-minor key scheme intersects curiously well with larger harmonic events in the Forlane. Example 13.8a shows how at measure 10, in a passage analogous to measure 2, the subdominant A–E fifth is replaced by its W3-subdominant analogue, C♯– G♯. Further, a half cadence on a G♯-major triad (an implied dominant to c♯-minor?) at measure 18, shown on example 13.8b, is the first large point of repose in the work. The Riemannian notion of Doppelklänge provided us a means to view the salient tetrachordal collections in the opening of the Forlane as unified wholes. Moreover, unlike a purely set-theoretical perspective, it allowed us to relate those collections directly to Riemannian functional tone schemes, which in turn imparted functional interpretations to the collections. (p. 393)

Appendix: A Translation of Riemann's Schema­ tisirung der Dissonanzen [In the following, I have used the archaic “greater” and “lesser” to translate Riemann's groß and klein applied to chordal intervals (e.g., translating Große Septimenaccorde and Kleine Sextaccorde as “greater seventh chords” and “lesser sixth chords,” respectively) rather than the more idiomatic English “major” and “minor” in order to distinguish Riemann's use of the terms dur and moll as adjectives applied to the mode of the underly­ ing Klang within a dissonant chord. For example, Riemann's terminology distinguishes “greater ninth chords” (Große Nonenaccorde) as a generic (p. 394) category of ninth Page 13 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen chord from its specific manifestation as either a major ninth chord (Durnonenaccord, a major triad with major over-ninth) or its dual minor ninth chord (Mollnonenaccord, a mi­ nor triad with major under-ninth).] 30. Schematic overview of dissonances. Looking over the various kinds of dissonant chords presented thus far, we can by-and-large distinguish two main classes: I. when a fourth (or fourth and fifth) tone is introduced to a major or a minor chord, the added tone disrupts the consonance. II. when one of the three tones of a Klang is chromatically altered and thereby be­ comes a tendency tone; in this case the altered tone is dissonant. Within the first class one can distinguish the following particular types: I. 1) (Natural) seventh chords, namely: a)Major seventh chords, e.g., g7 = G . B . D | F. b)Minor seventh chords, e.g., aVII = B | D . F . A. 2) (Greater) sixth chords, namely: c) Major sixth chords, e.g., f 6 = F . A . C | D. d) Minor sixth chords, e.g., eVI = G | A . C . E. 3) Greater seventh chords, namely: e) Major chords with a major seventh, e.g., c7〈 = C . E . G | B. f) Minor chords with a major seventh, e.g., eVII〉 = F | A . C . E. 4) Lesser Sixth Chords, namely: g) Major chords with a minor sixth, e.g., c6〉 = C . E . G | A♭. h) Minor chords with a minor sixth, e.g., eVI〈 = E | F . A♭ . C. 5) Augmented sixth chords, namely: i) Major chords with an augmented sixth, e.g., c6〈 = C . E . G | A♯. k) Minor chords with an augmented sixth, e.g., eVI〉 = G♭ | A . C . E. In addition to these four-tone dissonances in the first class are the following five-tone dis­ sonances (with two dissonant tones—seventh and ninth): 6) (Greater) ninth chords, namely: l) Major ninth chords, e.g., g9 = G . B . D | F . A. m) Minor ninth chords, e.g., aIX = G . B | D . F . A. 7) Lesser ninth chords, namely: n) Major seventh chord with minor ninth, e.g., [g9〉 =] G . B . D | F . A♭. o) Minor seventh chord with minor ninth, e.g., [aIX〈 =] G♯ . B | D . F . A

Page 14 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen Further, there are the four-tone elliptical constructions, which one could name third-fifthseventh-ninth chords, or third-ninth chords:24 6a) (Greater ninth chords without the Hauptton, compare 1) p) g9⃥ = B . D | F . A. q) aIX ⃥ = G . B | D . F. and 7a) (Lesser ninth chords without the Hauptton, the diminished seventh chords of figured-bass terminology): r) g9〉 ⃥ = B . D | F . A♭. IX〈 s) a⃥ = G♯ . B | D . F (p. 395)

And finally, the three-tone elliptical constructions:

1a) (Seventh chords without the Hauptton; one could designate these as thirdfifth-seventh chords or third-seventh chords): t) g7⃥ = D | F . A. u) aVII ⃥ = B | D . F. In the second class (altered chords) belong the following: II.  8) Augmented fifth chords: v) Major chords with augmented fifth, e.g., c5〈 = C . E (G) G♯. w) Minor chords with augmented fifth, e.g., eV〉 = A♭ (A) C . E. 9) Diminished fifth chords: x) Major chords with diminished fifth, e.g., c5〉 = C . E . G♭ (G), particularly as G♭ . C . E. y) Minor chords with diminished fifth, e.g., eV〈 = (A) A♯ . C . E, particularly as C . E . A♯. 10) Augmented prime chords: z) Major chords with raised prime, e.g., c1〈 = (C) C♯ . E . G. aa) Minor chords with lowered prime, e.g., eI〉 = A . C . E♭ (E). 11) Diminished prime chords: bb) Major chords with lowered prime, e.g., g1〉 = G♭ (G) B . D. cc) Minor chords with raised prime, e.g., aI〈 = D . F (A) A♯. 12) Augmented third chords: dd) Major chords with raised third, e.g., c3〈 = C . (E) E♯ . G, particularly as (E) G . C . E♯. ee) Minor chords with lowered third, e.g., eIII〉 = A . C♭ (C) E, particularly as C♭ . E . A (C).

Page 15 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen 13) Diminished third chords: ff) Major chords with lowered third, e.g., c3〉 = C . E♭ (E) G, particularly as E . G . C . E♭. gg) Minor chords with raised third, e.g., eIII〈 = A (C) C♯ . E, particularly as C♯ . E . A . C. 14) Seventh chords with augmented fifth: hh) Major seventh chord with augmented fifth, e.g., c75〈 = C . E . G♯ . B♭, particularly as B♭ . C . E . G♯. ii) Minor seventh chord with augmented fifth, e.g., aVIIV〉 = B . D♭ . F . A, particularly as D♭ . F . A . B. 15) Seventh chords with diminished fifth: kk) Major seventh chord with diminished fifth, e.g., c75〉 = C . E . G♭ . B♭, particularly as G♭ . B♭ . C . E. ll) Minor seventh chord with diminished fifth, e.g., aVIIV〈 = B . D♯ . F . A, particularly as F . A . B . D♯. 16) Seventh chord with augmented prime: mm) Major seventh chord with augmented prime 71〈 = C♯ . E . G . B♭. (com­ pare r) nn) Minor seventh chord with augmented prime aVII ⃥ I〉 = B . D . F . A♭. (com­ pare s) (p. 396)

17) Greater major ninth chord with augmented fifth:25

oo) c95〉 = C . E . G♯ . B♭ . D, for example:

If still other types of altered chords are encountered here or there in the works of modern and contemporary composers, they are easy to classify using the terminology developed here. One could distinguish a third class of chords, of which we have thus far only explicitly dis­ cussed a few cases: chords understood in terms of two Klänge. The conception of any dis­ sonance is possible in terms of two (or three) Klänge; even the natural seventh chord, if one considers the triadic derivation of its components, can be regarded as a Doppelklang. More commonly, one of the Klänge represented in such a conception will dominate, so that the notation in terms of a single Klang is more appropriate. Only for the following dissonances might Doppelklang notation be appropriate to express that they comprise two Klänge [that exist] in a coordinate relationship. III. 18) The Seitenwechsel-dissonance (compare 4 and 8):

Page 16 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen 19) The dissonance of the Quintschritt:

20) The dissonance of the Ganztonschritt:

21) The dissonance of the Gegenquintwechsel: vv) c/g = G . B . D | F . A♭ . C. 22) The dissonance of the Gegenganztonwechsel (compare 15): ww) c/d = (F) A♭ . C | D . F♯ (A). Only exceptionally would the following be notated as Doppelklänge (perhaps where they would be used to modulate to the relative key): 23) The dissonance of the Terzwechsel (compare 2):

(p. 397)

24) The dissonance of the Leittonwechsel (compare 3):

25) The dissonance of the Ganztonwechsel (compare 6):

26) The dissonance of the Kleinterzwechsel:

27) The dissonance of the Tritonuswechsel:

In four-part composition, most of these constructions (those with five or six tones) are meaningless, since they can only be expressed elliptically. Yet the possibility for their no­ tation in terms of two Klänge is important for their theoretical explanation and perhaps for the role it could play, in much the way harmonic notation through chord roots (as with Gottfried Weber, Richter, etc.) exists alongside figured-bass notation. For practical pur­ poses, however, I prefer to employ notation in the sense of a single Klang, because for the most part it corresponds more directly to a musical thought process for understanding harmonic succession.

Page 17 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen

Notes: (1.) Skizze einer neuen Methode der Harmonielehre (Leipzig: Breitkopf und Härtel, 1880), 52–57. (2.) Skizze, 69–81. The Systematik der Harmonieschritte is discussed and translated by Nora Engebretsen in chapter 12 of the present volume. (3.) Alexander Rehding has eloquently discussed the regulative function of Riemann's har­ monic theories and their focus upon the normative features of tonal music as a means to define and prescribe the way tonal music “ought to be.” See Rehding's Hugo Riemann and the Birth of Modern Musical Thought (Cambridge: Cambridge University Press, 2003), in particular, chapter 2. The relation of Riemann's Dissonanzlehre to his system of Harmoni­ eschritte is generally overlooked in the neo-Riemannian literature, which tends to regard the Harmonieschritte as transformations acting solely on the family of consonant triads (i.e. members of sc [037]). (4.) Skizze, 1. “Es giebt nur zwei Arten consonanter Accorde, nämlich Duraccorde und Mollaccorde. Ein Duraccord besteht aus einem Haupttone, seiner Oberquint und Obert­ erz, ein Mollaccord aus Hauptton, Unterquint und Unterterz; diese drei Töne: Hauptton, Quintton und Terzton sind die wesentlichen Bestandtheile jedes consonanten Accordes, doch können nur Oberquint und Oberterz, oder Unterquint und Unterterz mit dem Haupt­ tone zusammen einen consonanten Accord bilden, nicht aber Oberquint und Unterterz oder Unterquint und Oberterz, eine Verbindung letzterer Art ergiebt vielmehr immer einen dissonanten Accord. Nennen wir die Verbindung von Hauptton, Oberterz und Oberquint statt Duraccord: Oberklang, und die von Hauptton, Unterterz und Unterquint statt Mollaccord: Unterklang, so können wir sagen: Oberklang und Unterklang wider­ sprechen einander, verschmelzen nicht zur Einheit, auch wenn der Hauptton derselbe ist; ihre Verbindung ist so gut ein Doppelklang, eine Klangzweiheit, wie die Verbindung von Klängen, die nicht denselben Hauptton haben.” (5.) Skizze, 44. (6.) The Durmolltonartsysteme and Molldurtonartsysteme are discussed and illustrated in Hauptmann's Die Lehre von der Harmonik (Leipzig: Breitkopf und Härtel, 1868), 18–22. (7.) Riemann introduces the “𝆴” symbol in Die Hülfsmittel der Modulationslehre (Cassel: Luckhardt, 1875), not to represent a Doppelklang, but to symbolize the Seitenwechsel relation itself (there named the antinomic Wechsel), 8. (8.) Skizze, 45. (9.) Skizze, 46. “Die Doppelklänge existiren überhaupt mehr in Theorie als in der Praxis.” (10.) Skizze, 46.

Page 18 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen (11.) On Riemann's two-tiered conception of Harmonieschritte in the Skizze—those that relate triads within a closed key scheme and those that relate all (including chromatic) triads—see my “Representations of Space and Conceptions of Distance in Transformation­ al Music Theories” (Ph.D. diss., Harvard University, 2000), esp. chapter 5; and Nora Engebretsen's contribution to the present volume. (12.) Handbuch der Harmonielehre. Zweite, vermehrte Auflage der “Skizze einer neuen Methode der Harmonielehre” (Leipzig: Breitkopf und Härtel, 1887). (13.) Riemann had discussed suspensions as a fourth category of dissonances in chapter 7 of the Skizze, but considered them separately from dissonances of categories I–III be­ cause of their melodic origin. (14.) The Doppelkläng idea was revived and extended in the writings of Hermann Erpf. See in particular his Studien zur Harmonie- und Klangtechnik der neueren Musik (Leipzig: Breitkopf und Härtel, 1927). Erpf does not combine Klänge based on the Har­ monieschritte, but rather based on their functional categories: tonic-plus-dominant Klänge, Doppeldominantklänge, and so on. (15.) Harald Krebs discusses tonal pairing in Chopin's Ballade op. 38 in “Alternatives to Monotonality in Early Nineteenth Century Music,” Journal of Music Theory 25.1 (1981): 1–16. Tonal pairing is a central topic in the essay collection The Second Practice of Nine­ teenth-Century Tonality, ed. William Kinderman and Harald Krebs (Lincoln: University of Nebraska Press, 1996); Chopin's Ballade op. 38 is discussed in contributions by Jim Sam­ son (“Chopin's Alternatives to Monotonality: A Historical Perspective,” 34–44) and Kevin Korsyn (“Directional Tonality and Intertextuality: Brahms's Quintet op. 88 and Chopin's Ballade op. 38,” 45–83). (16.) I introduce the Sx/Wy  notation in “Some Aspects of Three-Dimensional Tonnetze,” Journal of Music Theory 42.2 (1998): 195–206. (17.) The eight relations, W1, W3, W7, W8, S1, S3, S4, and S6, that produce distinct combi­ nations of Klänge are omitted by Riemann. (18.) The use of operations to both construct and relate sets is reminiscent of Richard Cohn's work on transpositional combination (TC). See, for instance, his “Inversional Sym­ metry and Transpositional Combination in Bartók,” Music Theory Spectrum 10 (1988): 19–42. TC is a binary operation that makes use of the pitch-class/interval isomorphism to combine pc set classes. For example, one can generate the hexatonic collection [014589] via TC through the combination of set classes [048] and [01], either by starting with an augmented triad building and a new augmented triad “one semitone higher,” or by start­ ing with a chromatic dyad and building two new chromatic dyads “an augmented triad higher,” that is, 4 and 8 semitones higher. (19.) That the work is in e appears to be supported by external evidence: Ravel tran­ scribed an E-major Forlane from Couperin's Concerts royaux as a model for his Forlane. See Arbie Orenstein, “Some Unpublished Music and Letters by Maurice Ravel” Music Fo­ Page 19 of 20

On a Transformational Curiosity in Riemann's Schematisirung der Disso­ nanzen rum 3 (1973): 328–331. On intertextual aspects of the Tombeau de Couperin and its For­ lane, see Carolyn Abbate, “Outside Ravel's Tomb,” Journal of the American Musicological Society 52.3 (1999): 465–530. Dissonant tonics are the topic of a talk by Daniel Harrison, “Dissonant Tonics and Post-Tonal Tonality,” delivered at the 2002 meeting of the New York State Music Theory Society. (20.) Riemann would certainly reject such a reading, given the rhetorical position of the chord and its embedding of the complete tonic triad; he would likely view the collection instead as a tonic minor sixth chord, b〈6. (21.) Lewin defines and discusses the DOUTH2 relation in “Cohn Functions,” Journal of Music Theory 40.2 (1996): 181–216. (22..) “Hexatonic poles” refer to pairs of mode-opposed triads within a hexatonic collec­ tion that share no common tones. See, for instance, Richard Cohn, “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions,” Mu­ sic Analysis 15.1 (1996): 9–40; also idem, “Uncanny Resemblances: Tonal Signification in the Freudian Age,” Journal of the American Musicological Society 57.2 (2004): 285–324. (23.) Riemann illustrates his composite key schemes, featuring both raised and lowered forms of 6̂ in major, and raised and lowered forms of 7̂ in the Skizze, 18. The Riemannian composite e-minor key scheme is particularly fitting for the Ravel passage, given its use of both raised degree 7 at the opening and its characteristic lowered degree 7 at the ca­ dence. (24.) [Riemann's terminology (Terzquintseptnonenaccorde, Terznonenaccorde, and so on) should not be confused with figured-bass chord names, in which numerals refer to inter­ vals measured above the bass (e.g., six-four chords). Rather, Riemann's third, ninth, et alia refer to elements of the Klang, reckoned in dual fashion.] (25.) [On whole-tone dominants in neo-Riemannian and historical contexts, see the sec­ ond section of Daniel Harrison's contribution to the present volume, and in particular n. 42.]

Edward Gollin

Edward Gollin is Associate Professor of Music at Williams College.

Page 20 of 20

Chromaticism and the Question of Tonality

Chromaticism and the Question of Tonality   David Kopp The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0014

Abstract and Keywords This article examines the role of key and function as a component of Riemann's relational harmonic system. It is argued in this article that while the neo-Riemannian abstraction of Riemann's Harmonieschritte offer certain insights into the nature of chromatic relations in the nineteenth-century music, it has also resulted in a view of harmonic relations un­ comfortably divorced and separated from the tonal and functional contexts in which they were conceived. In addition to examining the role of key and function as component of Riemann's relational harmonic system, and chromaticism, the article also suggests how neo-Riemannian analysis can benefit by reconnecting Riemannian harmonic relations to the functional tonal contexts in which they arose, illustrating the recovered and renewed nineteenth-century perspective with analyses of music by Beethoven, Schubert, and Wolf. Keywords: key, function, relational harmonic system, chromaticism, neo-Riemannian analysis, Riemannian har­ monic relations, functional tonal contexts, Beethoven, Schubert, Wolf

I. Introduction One of the ironies of the neo-Riemannian enterprise is its furtherance of a point of view that runs counter to one of Riemann's central claims. Where Riemann sought to expand the theoretical boundaries of tonality to accommodate broad aspects of nineteenth-centu­ ry chromaticism, this modern model, developed a century after his and in his name, has drawn these boundaries tightly back. To explain chromaticism in nineteenth-century mu­ sic that poses challenges to more traditional modes of analysis, neo-Riemannian theory posits mechanisms of structural coherence to which harmonic fifth relations and even the presence of a tonic are extraneous. The foundational work of David Lewin as developed by Richard Cohn and others asserts basic triadic operations that, while resembling tonal relationships and invoking aspects of classical dualism, combine systematically to form progressions defined extratonally.1 Transformational processes tied to symmetrical divi­ sions of the octave are defined in a neutral environment of minimal voice-leading opera­ tions without reference to notions of key or tonic. Constructs such as hexatonic and octa­ tonic systems supplant more tonally referent concepts of chromatic third relations and di­ Page 1 of 16

Chromaticism and the Question of Tonality minished seventh chords; model progression types tend to the sequential rather than the cadential. In this view, since extratonal transformational processes can be identified in chromatic passages where diatonically based functional interpretations fall short, one might suspect the music under analysis must have nontonal content. It is becoming com­ monplace to encounter references to nontonal (or what one might even call “atonal”) pas­ sages in earlier as well as later nineteenth-century music whose structures lend them­ selves to modeling in terms of the principal (p. 401) neo-Riemannian transformations.2 However, there are other possible ways to understand music like this. For if there are chromatic passages in Beethoven and Schubert, for example, that make intuitive sense to us in relation to the whole, but which the usual tonal-analytic means do not explain well, must we conclude that the music in these passages is not tonal? Perhaps we should try in­ stead to understand how it can be that this music is tonal, especially since it is written in a style that was considered at the time to be thoroughly tonal (and, as at all times, sub­ ject to experiment), and in a milieu that significantly predates the introduction of modern notions of nontonality and “atonality.”3 Consequently, we may also explore how relevant transformational processes may be interpreted within an expanded sphere of tonal rela­ tions.4 In the entry “Tonalität” in his Musik-Lexikon, first published in 1882, Hugo Riemann de­ fined the concept expressly to embrace chromatic as well as diatonic relations to a tonic, in contrast to Tonart, the familiar diatonic conception of key. Example 14.1 presents his now-familiar illustration of a progression exemplifying Tonalität. The example involves a root-position cadence juxtaposing the tonic and its two major-third chromatic mediants, rather than its two dominants.5 The cadence's presentation highlights the similarities in common-tone content and voice-leading between the mediants and their diatonic counter­ parts. Much later, in a 1917 publication, Riemann was to attribute not only tonal meaning but independent functional status to these chromatic mediants.6 How ironic, then, that the relation between these three chords, cited by Riemann as a fundamental example of tonality in chromatic music, has come to serve as a fundamental example of nontonal tri­ adic relations in neo-Riemannian theory, when reinterpreted as the product of the hexa­ tonic cycle.

Ex. 14.1. Riemann, Musik-Lexicon: Chromatic ca­ dence exemplifying Tonalität.

Page 2 of 16

Chromaticism and the Question of Tonality In what sense may the hexatonic cycle be taken as nontonal in a tonal context? Tonality, as Brian Hyer has noted, is a concept that may be understood in a number of related but separate meanings.7 In one of its principal meanings, according to Hyer, tonality is delim­ ited through the association of its familiar elements. Two of these elements, already men­ tioned above, are relevant here. First, the hexatonic cycle, and relations between hexa­ tonic cycles, are defined without reference to a tonic; moreover, the cycles’ symmetrical nature precludes any determination of tonic from their internal configuration. In this sense the cycle operates beyond tonal organization, although a tonic orientation could in principle be imposed from without. Second, the hexatonic cycle, along with the larger neo-Riemannian system predicated on parsimonious voice-leading, is defined in the ab­ sence of, and separate from, diatonic fifth relations.8 This more restricted sense, resonat­ ing with (p. 402) Schenkerian perspectives, would likely hold hexatonically organized mu­ sic to be nontonal even in the presence of a tonic, as in Riemann's example.9 To whatever extent each of these meanings is implicit in the neo-Riemannian viewpoint, an opposing argument in favor of tonal interpretation should address them both. Following are three analytic examples to advance the tonal idea.

II. Analyses

Ex. 14.2. Harmonic reduction, Beethoven, “Emper­ or” concerto, III, mm. 112–246.

The interior section of the third movement of the Emperor Concerto, shown in harmonic reduction in example 14.2, contains as its centerpiece a series of harmonic moves typical of those neo-Riemannian theory might designate as extratonal, operating outside the key. My claim, to the contrary, is that a continual awareness of tonic and of key is part and parcel of this extraordinary passage. The section overall has three parts: a rising-fifths se­ quence of tonicized minor triads; a descending major-third cycle of major triads; and a descending whole-tone sequence of minor triads, culminating in the return to tonic E♭ major. The major-third cycle (neo-Riemannian PL cycle) is situated at the formal point of greatest distance from tonic influence, and its aesthetic impression is not unlike an in­ creasingly dreamy free fall. Moving directly from C major to A♭ major, and again by the same means to F♭ major (enharmonically notated as E major by Beethoven, but, no differ­ ent from the previous progression, clearly a major third below A♭ major in the ear), the music appears, in its immediate context, detached from tonic reference. There is no di­ rect tonic connection at either end of the third cycle: C major is not reached as a chro­ matic mediant in relation to tonic E♭, nor is F♭ major treated as the Neapolitan chord. However, this does not mean that the tonic exerts no influence on the passage. By this relatively late point in the concerto, we have heard a great deal of E♭ major-centered mu­ Page 3 of 16

Chromaticism and the Question of Tonality sic, along with the striking contrast of the chromatic-third-related C♭ major of the second movement (enharmonically notated as B major).10 This cumulative memory cannot help but orient and ground the harmonic ear as the section unfolds. First comes the rising-fifth passage, ascending from E♭ through B♭ and F to C, its increasing harmonic tension and swift centrifugal trajectory mitigated slightly by the mi­ nor tonicizations. While the C minor goal could potentially be oriented back toward the tonic as the relative minor, Beethoven instead slips E♭ up to E♮, firmly planting C major three fifths above the tonic through this parallel transformation, and positioning the mu­ sic toward its next path, the descending major-third cycle. The local ascent by fifth, exu­ berant forte character of the passage, long-range harmonic background, and final chro­ matic push dislodging the tonic scale degree upward all combine to give this C major an unmistakable brightness and marked profile to the sharp side of the tonic. This effect is accentuated by the comparatively longer duration of C major in relation to the previous harmonies. (p. 403)

Next is the first major-third drop, into A♭ major. This type of chromatic mediant relation by descending major third is arguably the strongest, most natural-sounding of the four chromatic mediant types, since it alone contains a leading-tone progression to the goal chord. It also preserves the tonic of the first chord as common tone into the second.11 These two attributes, which it shares with the dominant-tonic relation, give this mediant progression its particular cadential quality. Here, the arrival to subdominant A♭ major al­ so restores the tonic pitch-class, E♭; thus the direct mediant shift brings the music per­ ceptibly back into the purview of tonic harmony, reinforced by the amiable character of the solo. At the second major-third drop, identical to the first, we enter the depths of F♭ major, well to the flat side of the tonic, beyond the (enharmonic) C♭ major of the second movement. At this extraordinary moment time nearly stands still: the melody locked in place, pianissimo, pulse receding, the soloist almost disappearing behind the brass pedal, strings subsequently withdrawing behind the soloist to pianississimo. E♭ has again moved up by semitone, but diatonically now, as leading tone, accentuating the distant position of F♭ major in relation to the tonic and the music surrounding it. The musical embodiment of the enharmonic transformation of F♭ major into the notated E major, preparing the se­ quential descent back to tonic E♭, can be identified in this fadeaway to nearly nothing, fol­ lowed by the piano's reemergence, energetic and forte, at the cadential arrival of mea­ sure 200. After this, the minor-to-major parallel transformation that initiated the majorthird cycle is now reversed at the cycle's end, unexpectedly producing E minor at mea­ sure 212, no longer tied to the major-third descent and demarcating a new subsection. Thus this extraordinary series of events framed in a descending major-third cycle derives a necessary portion of its meaning from the heard relation of each element to the tonic, even if those relations are not direct. Positioned by a rising-fifth series, C major hovers brightly above the tonic; A♭ major falls within the tonic sphere; and F♭ major dips deeply below. In this sense the music is profoundly tonal. Just because the elements of the local progression are capable of being described by a mechanism independent of a tonic, we need not conclude that the progression itself is not tonal, particularly if an alternative ex­ Page 4 of 16

Chromaticism and the Question of Tonality planation that links its elements both to each other and to their larger surroundings by tonal means provides a comprehensive account of the music. Moving forward, E minor yields to A minor, a tritone distant from the tonic and of opposite mode. This progression initiates a second series of minor-triad tonicizations, each unit spanning two descending fifths compared to the single ascending fifths of the earlier sequence, doubling the harmonic distance covered by the same number of individ­ ual progressions (refer to example 14.2).12 Here the two-chord units of tonicization define a descending whole-tone sequential process (A minor, G minor, F minor, E♭ minor briefly implied), that, not unlike the descending major-third cycle, could potentially be seen as organized without reference to the tonic, related instead to equal division of the octave and a certifiably nontonal harmonic system. But we do not think of progressions like this “Emperor” sequence as nontonal, since we are able to relate them to a tonal process (e.g., the composing out of a tonally bounded intervallic pattern). Here also the descend­ ing-fifth path terminates at a dominant pedal leading to the tonic, whose major mode negates any nontonal impression of whole-tone sequential continuation in minor, and in­ corporates the preceding two tonicizations as diatonic elements. Moreover, the whole(p. 404)

tone cycle does not readily lend itself to neo-Riemannian modeling for a number of rea­ sons.13 Ultimately, then, while the major-third cycle of measures 138–208 connects less directly to the tonic than do the sequences that surround it, it is no less tonal, and its wide harmonic orbit corresponds appropriately to its central position in a developmental section. The first movement of Schubert's D major piano sonata, D. 850, contains an unusual, highly chromatic passage near the opening, shown in example 14.3, that appears to breach the key at an unusually early point. After an initial four-bar phrase straightfor­ wardly establishing tonic D major, a consequent phrase, already intimating chromaticism by beginning in the parallel minor, abruptly bends harmony through a path that seems to strain its connection with the tonic. Without implying any new tonal center, it projects a broad space defined by chromatic mediants, in contrast to the more circumscribed dia­ tonic space of the first phrase. From subdominant G minor in measure 7, the music spi­ rals first through an insistent, cadential-sounding F major 46 at measure 8, then into a surprising C♯ major harmony at measure 12, before landing back in diatonic territory on dominant A major at measure 14 on its way to the tonic in measure 16.14 Conventional harmonic analysis falls short in explaining the coherence of this particular progression; F major and C♯ major, along with A major, define a descending major-third cycle outside typical cadential and sequential process. A linear-reductive analysis of the passage, shown in example 14.4, works better, revealing a structural ascending upper-voice line consisting of composed-out diatonic intervals of a perfect fifth and fourth, framed within the octave compass of the ascending D minor scale, with the bass traveling along in par­ allel and similar motion. F major appears at the pivotal 5̂, extending melodically to natur­ al 7̂, after which C♯ major accompanies the leading tone, with the chord's jarring pres­ ence accentuating the note's alternative identity as a chromatic passing tone. Although this analysis offers a thorough means for contextualizing these harmonies within a con­ ventional framework, the potent, dramatic triadic juxtapositions in this passage merit an Page 5 of 16

Chromaticism and the Question of Tonality explanation truer to their harmonic nature than as chromatic elaborations subsumed within a diatonic infrastructure. This suggests a transformational approach.

Ex. 14.3. Schubert, piano sonata in D major, D. 850, I, opening section, mm. 1–16.

Ex. 14.4. Voice leading sketch, Schubert, piano sonata D. 850, I, mm. 5–16.

Ex. 14.5. Transformation graph, principal harmonic arrivals in the Schubert example. (M / PL etc. = Kopp / Cohn transformations; solid lines = direct re­ lations; dashed lines = indirect relations).

While the long-range connection between G minor and A major is clear in the con­ text of tonic D, the intervening harmonies could be said to pass through a nontonal chro­ matic phase. As shown in example 14.5, this interlude may be modeled as a transforma­ tional process: a descending major-third cycle played out in neo-Riemannian hexatonic space, linking to the tonal processes at its borders. But a different interpretation of the same transformational analysis aids in understanding this passage as fully tonal, every­ (p. 405)

thing in relation to tonic D, without resorting to diatonically based (p. 406) explanations such as the elaborated linear process just described or a succession of altered scale-de­ gree harmonies related by the system of fifths. The individual chords can be best under­ Page 6 of 16

Chromaticism and the Question of Tonality stood tonally if the functionality of chromatic mediants, for which I have advocated else­ where, is allowed as part of a system of chromatic or common-tone tonality.15 In nine­ teenth-century style, chromatic mediants may occupy a stable position in harmonic space distinct from their diatonic counterparts, defined less by scale degree than by the constel­ lation of intervallic relations between roots, voice-leading content, and qualitative har­ monic change, in ways strongly analogous to, but separate from, functional fifth relations. As mentioned above, Riemann himself formally defined a bona fide concept of mediant function toward the end of his career. One good way to assert the mediants’ functional identities is to name them according to the intervallic and harmonic relations to the tonic —above or below, and to the sharp side or the flat side—which give them their distinctive qualities, rather than assigning them altered Roman numerals designed for diatonic chords. These specific names for the four chromatic mediants are shown in table 14.1, along with their root relations to a major tonic. The associated keys of the chromatic me­ diants are also shown toward the end of this essay in example 14.9.16 (Also, in the net­ work analyses of examples 14.5, 14.7 and 14.8, pairs of transformational expressions are shown. The first expression in each pair comes from the system I have proposed, which retains transformations for functional dominant and mediant relations.17 The second ex­ pression in each pair is the familiar neo-Riemannian formula.) In the passage under consideration, F major relates comfortably enough to a D major-mi­ nor chromatic tonality, although Schubert's chromatic approach through a common-tone diminished seventh chord gives the chord a harmonically more distant sound resonant with its status as the upper flat mediant of the major tonic. C♯ major makes sense after the fact, as the upper sharp mediant of the dominant which follows, especially as its root persists as the leading tone of the home key. On the other hand, the progression between F major and C♯ major stands at the greatest remove from the tonic; the juxtaposition of its members might still be thought to rupture the tonal flow. But we are accustomed to tonal successions whose members relate to their neighbors but not directly to each other —involving secondary dominants, for instance. In chromatic tonality this progression is analogous. It can be understood as a passage between the functions just described, while the arrival upon C♯ major may additionally derive some of its meaning from the paradox of its close linear proximity (p. 407) to, and significantly greater harmonic distance from, tonic D major. Moreover, the motion from F major to C♯ major is itself a chromatic medi­ ant relation, providing a locally coherent connection between chords similar to those that connect them individually to the key. Thereby C♯, locally the lower flat mediant of F ma­ jor, is reinterpreted as the upper sharp mediant of A major. Here, then, as in the Beethoven example, what could be dismissed as a nontonal descending major-third cycle embedded in the music surrounding it—more jarring in this case—may be much more deeply and profitably analyzed not only as a cycle, but also as a succession of chords each having a separate and distinct harmonic meaning in relation to the key. Table 14.1. The four chromatic mediant relations to a major tonic. Chromatic mediant type Page 7 of 16

Root relation to tonic

Chord in C major

Chromaticism and the Question of Tonality Upper sharp mediant (USM)

M3 up

E major

Upper flat mediant (UFM)

m3 up

E♭ major

Lower sharp mediant (LSM)

m3 down

A major

Lower flat mediant (LFM)

M3 down

A♭ major

A final, more complex example of a tonally referent major-third cycle is Hugo Wolf's song Und schläfst du, mein Mädchen, from vol. 4 of his Spanisches Liederbuch, shown as ex­ ample 14.6. This late nineteenth-century Lied of three stanzas and one minute's duration contains only eleven chords with nine distinct identities, but their nature and manner of succession creates a complex web of harmonic interrelation difficult to represent using conventional models. Some progressions seem almost randomly connected. The music ap­ pears to lack tonal coherence, above and beyond a major-third circle in its middle that re­ sembles a neo-Riemannian nontonal episode. The form of the song itself implies a loosely sequential rather than cadential basis. Explanation of tonal meaning in this song requires a combination of approaches and an expanded view of tonal relations. The key of the song is in doubt until the end of the final stanza; until that point the music sounds neither like it is in a particular key, nor like it is not in a particular key. Numerous unexpected juxta­ positions of chords take place within an environment in which direct, indirect, and im­ plied progressions all contribute to a singular sense of tonal meaning. The poem, based on an early sixteenth-century Spanish text, depicts a man riding at night on horseback, calling anxiously to his beloved to wake and steal away with him, so quickly that she should not even put on her shoes, on a journey to faraway lands. The song's tonal struc­ ture reflects the poem's sense of urgency, uncertainty, and excitement.

Page 8 of 16

Chromaticism and the Question of Tonality

Ex. 14.6. Wolf, “Und schläfst du, mein Mädchen,” from Spanisches Liederbuch, vol. 4 (poem by Gil Vi­ cente, c. 1500, trans. from Spanish).

The first stanza opens with an alternation between an open fifth on G and a half-dimin­ ished seventh chord on D, while the melody fixates on D, their common tone. Despite G's missing third, the progression most strongly suggests a half cadence in C minor, strength­ ened by the brief melodic appearance of G's seventh, F, in measure 4, and the phrase end­ ing on the G sonority in measure 6. This expectation is thwarted, however, at the begin­ ning of the next phrase, first by the vocal entrance on B♭, which clashes with the implied B♮ of the previous harmony, then by the E♭ major triad, which stands (p. 408) (p. 409) (p. 410) in a chromatic-third relation to the cadence rather than the fifth relation of ex­ pected C minor. Thus the G sonority resolves as upper sharp mediant, a functional alter­ native, rather than as dominant. Perception of tonic and key in this passage is ambiguous Page 9 of 16

Chromaticism and the Question of Tonality at best: is it C minor, favoring the progression at the beginning; E♭ major, favoring the ar­ rival at the end, with the persistent melodic D acting as leading tone; or perhaps G phry­ gian, suggested through repetition and emphasis as well as initial presentation? All of these possibilities collectively lend tonal meaning to the music. The half-diminished chord on D, in particular, accrues a heady mix of potential identities: iiø7 of C; dominant of phry­ gian G; then leading tone seventh in relation to the arrival at E♭, attenuated by the inter­ vening G sonority. Example 14.7a illustrates this nexus of implied and realized relation­ ships through a transformational network graph. Direct relationships are shown by solid lines, indirect relationships by dashed lines, and implied relationships by dotted lines. As with the Schubert analysis, alternative transformation types are shown.18

Ex. 14.7. Transformation graphs for each stanza of Wolf, Und schläfst du, mein Mädchen, ➀ mm. 1–10; ➁ mm. 11 -20; ➂ mm. 21–37.Solid lines = direct rela­ tionsDashed lines = indirect relationsDotted lines = implied relations*Bass transformation type RES 6-3a (3, 5)

The second stanza, depicted in example 14.7b, begins with a literal repetition of the open­ ing phrase transposed down a semitone, invoking the sequential principle, but lacking a secure framework, since it remains unclear what the starting point represents. Just as measures 1–6 implied tonic C, the half-cadential progression of measures 10–14 implies tonic C♭. The potential for this resolution, in favor of the sequentially determined D major (now also implied), gains weight at the end of measure 14, where the voice breaks the se­ quence, singing G♭ rather than the sequentially expected BD. This anticipation of domi­ nant resolution quickly dissipates as well, though, as the vocal line descends a semitone to F, resulting in harmonic arrival to a B♭ major triad at measure 15, a semitone lower than expected. This development, however, triggers associations with earlier music: B♭ major's chromatic mediant relationship with G♭ major, up a major third, is the inversion of the downward chromatic mediant relationship formed by G and E♭ at the end of the previ­ ous stanza. Where the G sonority resolved locally as upper sharp mediant, G♭ major re­ Page 10 of 16

Chromaticism and the Question of Tonality solves in opposite fashion as lower flat mediant. The quickly rising melodic line, in com­ parison with the descending line at measures 7–10, mirrors this inverse relationship. B♭, as the immediate goal of the half cadence, also activates an implied relation with the opening D♭ø7, parallel to the first stanza. But where that relation was functional, this dis­ placed relation is transformational, a non-neo-Riemannian type as defined by Richard Bass.19 Furthermore, in their shared prominence as arrival points, B♭ is in fifth relation with E♭, indicated in example 14.8, which traces relations between stanzas. This formal balance is disturbed by an abrupt repositioning, almost a rupture, to a D major triad at measure 17, as if it were an alternative resolution of the G♭ major triad as upper sharp mediant, realizing the implicit sequential possibility that was suppressed two measures earlier. At the level of the stanza, D major is the expected goal of the strict sequence, granting B♭ the character of an interruption as well as a resolution. Locally, this chain creates an ascending major-third cycle—refer to example 14.7b—which could be inter­ preted as a nontonal transformational process organizing the bulk of this stanza, especial­ ly given a context lacking a firm tonic. The resultant effect of traveling a great harmonic distance resonates with the uprooting from home and impending journey to distant (p. 411) places described in the text. But these arrival points also have strong internal tonal and structural associations, several of which have just been documented, as evident in example 14.8 from the web of interconnections engaging the cycle. Additionally, just as B♭ major relates to the earlier arrival on E♭, D major refers back to the song's opening progression and its insistent melodic D. The accompaniment's Lydian fourth, however, precludes a revival of G as a tonal focus.

Ex. 14.8. Transformation graph for relations within and between stanzas in Und schläfst du, mein Mäd­ chen Black lines: relations between stanzas Gray lines: relations within stanzas (from Fig, 4) LN = lower neighbor relation (p. 412)

The third stanza, shown in example 14.7c, resumes the sequence at first, its half

cadence pointing to arrival at the B♭ major just superseded at measure 17, with a poten­ tial sequential goal of D♭. This time through, however, the sequence is less strict. First, the melody immediately reaches up to a dominant ninth rather than down to a seventh; then, in measure 24, the harmonic sequence is broken after the third bar, with B♭ appearing a measure early in the dominant position rather than in measure 25 as tonic. In the song's only authentic cadence, the chord resolves to an unambiguous tonic E♭ major triad, where the music plays itself out until the end. In retrospect, as shown in ex­ ample 14.8, both arrivals of the previous stanza, disjunct at the time to each other, can be Page 11 of 16

Chromaticism and the Question of Tonality now understood to relate to the newly revealed tonic. B♭ major, transient at the time, is its dominant, and D major, a strong local arrival, is its lower neighbor, reminiscent of the po­ sition of Schubert's C♯ major. In this way, E♭ major can be seen to organize aspects of the entire song, although this does not necessarily mean that it is always present as its key. Rather, it exerts an influence while the music veers between various harmonic states as it makes its way toward its arrival point. Whereas Beethoven's chromatic mediant cycle ex­ ists at the periphery of its movement's tonal structure, and Schubert's cycle acts to en­ hance and heighten the cadential energy of its thematic area, Wolf's cycle of major thirds participates in a pivotal way in the song's structure, each triad a focal point in the projec­ tion of a network of tonal relations leading to the song's culmination. In all three cases, though, these chromatic elements exhibit a complex yet tangible and fully identifiable re­ lation to their tonic. Designating the structures of these extraordinary passages as the nontonal products of voice-leading operations would do an injustice to their content.

(p. 413)

III. Postscript

Ex. 14.9. Key-flips on the Tonnetz producing the four chromatic mediants Solid lines: initial C major key Dashed lines: resultant chromatic mediant keys Upper left: Upper flat mediant (E♭ major) Lower left: Lower flat mediant (A♭ major) Upper right: Upper sharp mediant (E major) Lower right: Lower sharp mediant (A major)

As a postscript to this essay, consider the projection of the diatonic set of the key onto an adaptation of Riemann's classic Tonnetz from his “Ideen zu einer ‘Lehre von den Ton­ vorstellungen.’ ”20 Changes of key may be represented by directed shifts of the template on the matrix. But what if one were to flip the key template along its longer edges, in the manner of the neo-Riemannian triadic flips producing the LPR transformational set, which change the orientation of the triadic triangle and always produce a change of mode? Ex­ ample 14.9 displays the results of 180-degree flips along both long edges of the template in its two alternative forms. They are exactly the keys of the four chromatic mediants, in close and direct relation to the tonic key. Flips of the template having D at the lower right Page 12 of 16

Chromaticism and the Question of Tonality of the key yield the upper mediants, while flips of the template having D at the upper left yield the lower mediants, since the Tonnetz tends upward to the right. Upward flips yield the sharp mediants, while downward flips yield the flat mediants, since the Tonnetz tends sharpward toward the top.21 The diagrams also show the even distribution of shared (common) and nonshared tones in these relationships, either three-to-four or four-tothree, a key element in their impression of distance-yet-connectedness. This intrinsic property of the Tonnetz helps to validate the essential and unmediated harmonic nature of the chromatic mediants within the tonal system, particularly the smoothness (p. 414) asso­ ciated with juxtaposing mediant-related keys. This is in contrast to the neo-Riemannian hexatonic model, which generates chromatic mediants as the compound products of two different successive diatonic voice-leading operations at the chordal level. While exam­ ples of true hexatonic organization certainly exist in the literature, it seems questionable to assert hexatonicism as the source and identity of the common chromatic third relations and many of the major-third circles characteristic of a wide range of nineteenth-century music, given the other ways in which these relations may be convincingly shown to func­ tion. A final observation: we call music pentatonic, whole tone, diatonic, or octatonic when it takes place largely within individual instances or related groups of those sets. Music or­ ganized by chromatic third relations, however, does not normally remain within the hexa­ tonic set defined by its structural triads, but (other than strict triadic sequences) tends to be as locally diatonic or chromatic as its style and surroundings. Thus, in this context, “hexatonic” seems like a misnomer. Harmonic organization in music such as this is, I think, better understood as one aspect of a greater chromatic tonality, akin to Riemann's Tonalität.

Notes: (1.) Lewin's collection of common-tone transformations includes both tonally referent functional types (DOM, SUB, MED), which preserve mode, and locally referent types (P, R, L, and SLIDE), which change mode. Taken together, these transformations form a het­ erogeneous and internally inconsistent system. The (PRL) transformations, originally pre­ sented as contextual inversion operations by Lewin, were recast as minimal voice-leading operations by Cohn to form the basis of a more homogeneous, tightly drawn, and consis­ tent system. Although referred to as “neo-Riemannian,” the system draws on only a part of Riemann's overall tonal conception. See David Lewin, “A Formal Theory of Generalized Tonal Functions,” Journal of Music Theory 26.1 (1982): 23–60; Lewin, “Amfortas’ Prayer to Titurel and the Role of D in Parsifal: The Tonal Spaces and the Drama of the Enharmonic C♭/B,” 19th Century Music 7.3 (1984): 336–349; Richard Cohn, “Neo-Riemannian Opera­ tions, Parsimonious Trichords, and Their Tonnetz Representations,” Journal of Music The­ ory 42.1, (1997): 1–66; David Kopp, Chromatic Transformations in Nineteenth-Century Music (Cambridge: Cambridge University Press, 2002), 142–164. (2.) See, for instance, n. 9 below, as well as contributions by Steven Rings and Robert Cook in the present volume. Page 13 of 16

Chromaticism and the Question of Tonality (3.) We may perhaps consider the neo-Riemannian view as exemplifying, in some degree, the influence of what Thomas Christensen has called “presentism” in history of theory as well as analysis. Presentism involves, among other things, the inclination to focus selec­ tively on those aspects of a historical theory or theories which have relevance to a con­ temporary approach (hence assigning the name “neo-Riemannian” to a theory with nonRiemannian attributes), and to find evidence in earlier music or music theory of a concept not articulated until later (nontonality). Thomas Christensen, “Music Theory and Its His­ tories,” in Music Theory and the Exploration of the Past, ed. David Bernstein and Christo­ pher Hatch (Chicago: University of Chicago Press, 1993), 9–39. (4.) In his 2002 treatise Tonal Pitch Space, Fred Lerdahl also proposes models for chro­ matic systems of hexatonic and octatonic organization (among others) that are indepen­ dent of the diatonic system. Unlike the neo-Riemannian conception, in which hexatonic and octatonic organization are essentially different from diatonic organization, Lerdahl's systems are all constructed on similar principles and exhibit similar fundamental proper­ ties, which by definition renders them all tonal. This allows for analyses of music which travels alternately through diatonic and chromatic tonal spaces. Lerdahl, Tonal Pitch Space (New York: Oxford University Press, 2001), 249–343. (5.) Hugo Riemann, Musik-Lexicon (Leipzig: Max Hesse, 1882), 923. Discussed in Kopp, Chromatic Transformations, 80. See also Alexander Rehding, Hugo Riemann and the Birth of Modern Musical Thought (Cambridge: Cambridge University Press, 2003), 48–49. (6.) Riemann, Handbuch der Harmonielehre, 6th ed. (Leipzig: Breitkopf und Härtel, 1917), xvii. I have discussed this in detail in Kopp, Chromatic Transformations, 99–102. (7.) Brian Hyer, “Tonality,” in The Cambridge History of Western Music Theory, ed. Thomas Christensen (Cambridge: Cambridge University Press, 2002), 726–752. (8.) Fifth relations can, of course, be formulated in terms of PLR transformations, but only as compound expressions. As tonal phenomena, they have not been a focus of neo-Rie­ mannian theory and analysis. (9.) Other transformational approaches that expand relationships beyond the basic PLR repertory to include seventh chords also set their subject apart from music based on fifth relations. Recent work by Graham Hunt, for example, proposes an apples-and-oranges an­ alytic method for highly chromatic music of Wagner, in which long stretches of conven­ tionally tonal music represented by Schenkerian graphs are interrupted by complex transformational episodes (often involving incremental voice-leading) represented by net­ work graphs. Schenkerian process is suspended during these episodes and resumes at their termination. Hunt stops short of calling these episodes nontonal, though, despite the article's name. See Graham G. Hunt, “When Chromaticism and Diatonicism Collide: A Fu­ sion of Neo-Riemannian and Tonal Analysis Applied to Wagner's Motives,” Journal of Schenkerian Studies 2 (2007): 1–32.

Page 14 of 16

Chromaticism and the Question of Tonality (10.) The key of C♭ major is enharmonically notated as B major by Beethoven, but its con­ textual identity is clearly defined by its interaction with E♭ major at the movement's boundaries. The Adagio un poco mosso links to the first movement by a direct chromatic third relation by descending major third, with tonic E♭ as common-tone bridge initiating the melody. It ends with an isolated tonic bass note in the horns which, reinterpreted as lowered 6̂, descends by semitone to dominant B♭, initiating the transition to the finale. Thus the tonic of the second movement is directly heard to be a major third, not a dimin­ ished fourth, below the tonics of the surrounding movements. (11.) For a detailed discussion of the properties of the individual chromatic mediants, see Kopp, Chromatic Transformations, 15–17. (12.) The interpolation of E minor at measure 212 is intriguing; were the measure to con­ tinue in E major instead, it would more directly introduce A minor and the ensuing se­ quence of tonicizations. Imagining the familiar sound of measures 212–213 in major may seem uncomfortable to us, but its plausibility demonstrates that E minor is not harmoni­ cally necessary. Rather, E minor's presence is important for marking a boundary with the previous section and establishing the new character of the sequence. (13.) Two reasons are primary. First, the succession is comprised of step progressions having no common tones and consequently greater transformational distance than chro­ matic third-relations. Second, the succession commonly breaks down into equal parts (fifth progressions, either with or without mode change) which as neo-Riemannian trans­ formational components are themselves compound and would generate a cycle exceeding the octave, while the hexatonic and octatonic cycles can be generated from transforma­ tionally simple, unequal parts (parallel-mode and diatonic-third shifts) within a single oc­ tave. (14.) For a compelling precedent for a discussion of a harmonically weighty 46 in a chro­ matic context, see Gregory Proctor's discussion of the third movement of the same Schu­ bert sonata, in “Technical Bases of Nineteenth-Century Chromatic Tonality: A Study in Chromaticism” (Ph.D. diss., Princeton University, 1978), 140. (15.) Kopp, Chromatic Transformations, 5–8. (16.) This classification scheme is introduced and described in Kopp, Chromatic Transfor­ mations, 8–15. (17.) Kopp, Chromatic Transformations, 165–191. (18.) The seventh chords in example 14.7 are considered to be triads for transformational purposes, in part to show the functional connections, although neither transformational system used here incorporates seventh chords. Transformational relationships involving seventh chords have been the subject of further investigation, including Edward Gollin, “Some Aspects of Three-Dimensional Tonnetze,” Journal of Music Theory 42.2 (1998): 195–206; Adrian Childs, “Moving Beyond Neo-Riemannian Triads: Exploring a Transfor­ mational Model for Seventh Chords,” Journal of Music Theory 42.2 (1998): 181–193; and Page 15 of 16

Chromaticism and the Question of Tonality Richard Bass, “Enharmonic Position Finding and the Resolution of Seventh Chords in Chromatic Music,” Music Theory Spectrum 29.1 (2007): 73–100. (19.) Bass, “Enharmonic Position Finding,” 87. (20.) Riemann, “Ideen zu einer ‘Lehre von den Tonvorstellungen,’ ” Jahrbuch der Musik­ bibliothek Peters 21–22 (1914–15): 1–26; the Tonnetz appears on p. 20. Riemann's keytemplate incorporates both contiguous occurrences of pitch class D, the symmetrical axis of the diatonic set whether arranged in scalar steps or in fifths, at symmetrically opposite positions at the ends of the set. For a probing study of this aspect of the diatonic set see Norman Carey and David Clampitt, “Aspects of Well-Formed Scales,” Music Theory Spec­ trum 11.2 (1999): 187–206. For present purposes, dual versions of the template, each in­ cluding only a single D within the diatonic set, are used. (21.) Flipping the template on its short edges produces, interestingly, different nondiaton­ ic cyclic sets: an incomplete octatonic set in one direction, and a hexatonic set with re­ dundant member in the other.

David Kopp

David Kopp is an associate professor in the department of composition and theory at the Boston University School of Music. He is the author of Chromatic Transforma­ tions in Nineteenth-Century Music and articles in the Journal of Music Theory and Music Theory Online, among other publications. As a pianist, he has recorded for the New World Records, CRI, ARTBSN, and Arsis labels.

Page 16 of 16

Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter

Criteria for Analysis: Perspectives on Riemann's Ma­ ture Theory of Meter   William E. Caplin The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0015

Abstract and Keywords This article focuses on Riemann's theories of rhythm and meter. It specifically aims to clarify the criteria that Riemann uses in justifying his metrical analyses by examining his theories from two general perspectives. The first perspective assumes that musical events are understood to receive their metrical interpretation—that is, which events are deemed metrically accented and which are metrically unaccented—according to the mechanics of notation associated with that theory, such as time signatures and bar lines. This perspec­ tive is termed notated meter. The second perspective assumes on the contrary that the musical events themselves can express their own metrical interpretation independent of the notation. That is, the interaction of certain musical parameters (such as duration, mo­ tivic contour, duration) can engender a sense of meter in a listener who is unaware of how the music may be actually notated. This perspective is termed expressed meter. While Riemann failed to realize the full potential of his own skepticism on the status of notation, his attempt to account for the origin of accent on the basis of musical content alone remains a significant achievement in the history of metrical theory. Keywords: theories of rhythm, meter, notated meter, expressed meter, notation, metrical theory

THOUGH Hugo Riemann's reputation as a music theorist is based largely on his theory of harmonic functions, his contributions to the theory of musical rhythm and meter are no less significant. Indeed, Riemann was preoccupied with issues of musical temporality throughout his career, proposing along the way a variety of theoretical formulations and analytical models. By the late 1890s, his views on meter had become relatively fixed, and this “mature theory,” as it may be called, found expression in a wide range of publica­ tions.1 Although his earlier views are of considerable interest, his mature theory of meter has exerted a greater impact, both positive and negative, on the subsequent history of theory. Like his theories of harmony, his theories of rhythm and meter still dominate in Germany and Northern Europe, while they continue to be regarded with suspicion in most Anglo-American academic circles.

Page 1 of 21

Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter A number of important studies have already laid out the basics of Riemann's metrical the­ ory,2 yet some aspects of it remain open to scrutiny. In particular, the various ways in which Riemann understands how metrical accentuation comes into being—whether it en­ sues directly from a preconceived model or whether it arises naturally out of the musical materials themselves—call for further study and interpretation. The present essay seeks to clarify the criteria that Riemann uses to justify his metrical analyses by examining his theories from two general (p. 420) perspectives, ones from which any theory of meter can be formulated and applied analytically. From the first perspective, musical events are un­ derstood to receive their metrical interpretation—that is, which events are deemed metri­ cally accented (or strong) and which are metrically unaccented (or weak)—according to the mechanics of notation associated with that theory, such as time signatures and bar lines. Such a perspective can thus be termed a notated meter. A second perspective as­ sumes, on the contrary, that the musical events themselves can express, so to speak, their own metrical interpretation independent of the notation. That is, the interaction of cer­ tain musical parameters (such as pitch, duration, motivic contour) can engender a sense of the meter in a listener who is unaware of how the music may actually be notated, a sit­ uation that arises in many listening contexts. This perspective can be termed expressed meter. At first consideration, Riemann's mature theory of meter, as represented at various times in his writings by the kinds of models shown in example 15.1 seems to function as a notat­ ed meter. Though different in format, four essential features are common to each model. First, they present a hierarchical organization comprising four levels of structure—the level of the beat, the measure, the two-measure half phrase, and the four-measure phrase. By proposing a metrical interpretation for levels residing beyond the confines of the notated measure, indeed as high as a full eight-measure period, Riemann brings to a culmination the general nineteenth-century tendency to view higher level rhythm as hy­ permetrical.3 Second, each level of this temporal hierarchy contains a series of regularly alternating accented and unaccented events. In this respect, Riemann's mature model conforms to traditional approaches. Third, every rhythmic grouping of the events at each level is understood to be “end-accented.” Unlike most earlier views, in which the accent is regarded as the beginning of a structural unit (such as the measure), Riemann dogmati­ cally holds that in all cases, and at every level of metrical organization, the accented event is an end, a goal of musical motion. His model of musical meter has therefore been appropriately termed an Auftakttheorie (“theory of upbeat”). The aesthetic principle un­ derlying this view is that of “active hearing.”4 According to Riemann, we do not passively relate an unaccent to the accent that precedes it, but rather we actively direct our atten­ tion to the accent that follows it; we hear toward a goal, not away from a starting point. Fourth, and finally, Riemann establishes his model as preexisting, as a theoretical a priori. The musical content receives its metrical interpretation from the model, which compen­ sates for the lack of notational symbols for meter at levels above that of the measure. In short, the model can be considered a kind of notated meter, such that the model itself provides the metrical interpretation for higher levels of structure.

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Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter

Ex. 15.1a. From Elementar-Schulbuch, 180.

Ex. 15.1b. From “Neue Beiträge zu einer Lehre von den Tonvorstellungen,” 11.

Ex. 15.1c. From Große Kompositionslehre, 1: 27–28.

A closer examination reveals, however, that Riemann's metrical theory can also be under­ stood from an expressed-meter perspective. Indeed, Carl Dahlhaus has argued that Riemann's metrical analyses are grounded in specific pitch relationships: “The harmonic criteria upon which Riemann bases the distinction between heavy and light measures are never explicitly articulated or grounded by him. They can, however, be reconstructed through an analysis of his analyses. Riemann is still a systematizer, even when he is silent.”5 Though Dahlhaus is surely correct in identifying principles of (p. 421) harmony that lie at the root of Riemannian metrics, an investigation of Riemann's writings reveals that he is far from silent on the criteria he uses for analyses. In a number of remarks scat­ tered throughout his mature writings, he justifies his metrical readings by appealing to the actual content of the music under consideration. In doing so, he invokes five princi­ ples: (1) Harmoniewirkung (“effect of harmony”), (2) motivic imitation, (3) durational ac­ cent, (4) tonal accent, and (5) Schlußwirkung (“effect of cadence”).6 In an effort to inter­ pret Riemann's own account of his system, I examine each of these principles in order to consider the extent to which they reflect an (p. 422) expressed-meter perspective. I fur­ ther attempt to determine whether theorists would likely hold these criteria as valid and persuasive today.

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Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter

Harmoniewirkung The first extensive treatment of Riemann's main principle of harmonic-metric interaction appears in the first volume of Grundriß der Kompositionslehre: “First of all, it can be gen­ erally stated that the more accented a note is, the more one expects it to have a change of harmony; in other words, the moments of time upon which new harmonies preferably en­ ter are the strong points of the motive, group [of motives], and phrases.”7 Har­ moniewirkung thus involves a change of harmony and the relationship of the new harmo­ ny to accent. In Grundriß der Kompositionslehre, the principle is formulated from a notat­ ed-meter point of view: structural units are identified as accented or unaccented prior to a consideration of their harmonic content. Indeed, the heading of the chapter in which Harmoniewirkung is treated, “Die Stellung der Harmonie im Satzbau” (“The Placement of the Harmony in the Structure of the Composition”) directly refers to a preexistent frame­ work within which the actual music is set. In a later work, Vademecum der Phrasierung, Riemann reconsiders the relationship of content to structure in a new light. In particular, he addresses the problem of how performers can determine the correct metrical interpre­ tation of the music so that they can properly convey this understanding to listeners (who may not have the notation before them). Simply following the composer's notation is not always a sure guide for the performer: The differentiation of accented and unaccented notes is not dependent upon the caprice of the composer, but rather already lies in the nature of the musical ideas themselves, and it is only a question of characterizing this differentiation in the notation. In what, then, does the essence of the various weights of the tones con­ sist? 8 In this important passage, Riemann explicitly calls for the formulation of a metrical theo­ ry from an expressed-meter point of view. Indeed, this is perhaps the first time in the his­ tory of music theory that the need to determine the way in which accents and unaccents “lie in the nature of the musical ideas themselves” is so consciously articulated. Riemann answers his own question about the “essence of the various weights of the tones” by re­ ferring to Harmoniewirkung: Since we do not want to introduce here detailed aesthetic and theoretic discus­ sions, it can just be stated short and to the point that the accented beats are, in general, the bearers of Harmoniewirkungen, that the composer must set the bar lines accordingly, and that the listener can also recognize the various weights of the tones from the harmonic content. Assuming that a listener writes down by ear a melody that is unknown to him before, then the places where the harmony changes…will reveal themselves as the most important, as those that are entitled to characterization in the written copy [through bar lines].9 With respect to the examples that follow this passage, some of which are shown in example 15.2, Riemann notes that not every chord change marks a strong beat (e.g., the chord marked with N. B. in example 15.2a). Yet the examples “document to some extent (p. 423)

Page 4 of 21

Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter the dependence of the choice of meter and time signature on the motion of the harmony.”10

Ex. 15.2. From Vademecum der Phrasierung, 42–43; (a) Beethoven, Piano Sonata in F, Op. 2, No. 1, ii, mm. 1–4; (b) Beethoven, Piano Sonata in E, Op. 14, No. 1, i, mm. 1–4; (c) Beethoven, Piano Sonata in G minor, Op. 49, No. 1, i, mm. 1–4; (d) Beethoven, Pi­ ano Sonata in B-flat, Op. 22, iv, mm. 1–4; (e) Beethoven, Piano Sonata in E-flat, Op. 31, No. 3, iii, mm. 1–4.

In example 15.2a, b, and c, the change of harmony creates what I have termed initial ac­ cents, following the lead of Moritz Hauptmann.11 Such accents arise when we can identi­ fy two levels of motion, such that the onset of events at one level creates the sense of ac­ cented events at the next lower level of motion. Here, the onset of a new harmony at the level of the measure creates an initial accent at the level of the beats within the measure. The placement of the bar lines in these cases corresponds to these initial accents. In ex­ ample 15.2d and e, the change of harmony in measures 2 and 4 also creates initial ac­ cents at the level of the beat. But in these two (p. 424) examples, which feature the har­ monic progression T–D–D–T, another set of structural levels comes into play as well. If the tonic harmony of measure 4 is continued into measure 5 (which is the case in both ex­ amples), then the Harmoniewirkung operates at the level of the double measure: mea­ sures 2 and 4 thus become initial accents in relation to measures 3 and 5. Here, then, Harmoniewirkung creates initial accents at two levels of structure. Riemann also uses Harmoniewirkung to explain meter at the level of the measure even though the conditions for initial accents, as I have defined them, are not present. For in­ stance, he considers that the tonic–subdominant progression in the full cadence T–S–D–T of example 15.2b can be analyzed as unaccented–accented because of Harmoniewirkung: “The beginning whole (i.e., the first measure) becomes understandable as unaccented Page 5 of 21

Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter through the change of harmony in the second measure, and the cadence T S D T comes plainly to a close in the fourth measure.”12 The second measure of the passage, however, is not a genuine initial accent: the change of harmony there does not introduce a higher level event (at the double-measure level) because measure 3 also brings a new harmony, the dominant. The accent in measure 2 could be considered an initial accent only if Rie­ mann could explain how S and D together comprise one harmonic event (which, within his theory of harmonic functions, would be nonsense). It would seem that for Riemann, Harmoniewirkung creates accent merely on the basis of a change in harmony. When these accents actually reside at a level lower than that of the harmonic change itself, then they can be explained according to the expressed-meter principle of initial accentuation; when the accents are identified as arising at the same level as that of the change of harmony, then they cannot be so justified. In those cases, Riemann reveals a misunderstanding about the hierarchical conditions necessary for har­ monic change to create accent. In example 15.2d and e, the accents associated with Harmoniewirkung happen to conform exactly to his Auftakttheorie. In other cases, though, Riemann has to ignore some changes of harmony in order to preserve the regular alternation of accents and unaccents as­ sumed in his notated-meter model. For example, the change from S to D at measure 3 of example 15.2b is, as already discussed, merely disregarded without comment. Another in­ teresting case concerns the progression T–T–D–T in example 15.3, where the motion from T to D would seem to make measure 3 accented according to the principle of Har­ moniewirkung. Riemann's explanation for why an accent does not occur at this place is significant: Among the simple presentations of the tonic, we could also consider the frequent structure that does not actually progress to a dominant, but that just makes a ret­ rogression from such [a dominant] to the tonic, so that only a passing dominant, so to speak, is inserted between two tonics that are brought on relatively accented beats.13

Ex. 15.3(a)–(b) From Grundriß der Komposition­ slehre, 72; (a) Beethoven, Overture to Fidelio, mm. 49–52; (b) Kuhlau, Op. 20, No. 3, ii, mm. 1–4.

A similar explanation for the same progression is given in the Handbuch der Harmonieund Modulationslehre: “If a foreign harmony enters between two relatively accented beats of the same harmony (e.g., the accented beat of the first (p. 425) and second mea­

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Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter sure), this foreign harmony does not produce a complete effect, but appears only as pass­ ing.”14 In both explanations, the progression T–D does not represent a change of harmony; rather, the dominant is considered as merely ornamental. When the T–D progression oc­ curs between measures 1 and 2 (as in example 15.2a, c, d, and e), Harmoniewirkung can be identified, but when the same progression takes place between measures 2 and 3 (ex­ ample 15.3), the dominant is passing. The harmonic content of both progressions is the same; the only difference is the placement of the harmonies within the phrase. Thus, Rie­ mann returns here to a purely notated-meter account of Harmoniewirkung. Rather than allowing the manifest change of harmony to indicate the location of accents, the a priori scheme itself determines the harmonic change. Although Riemann employs Har­ moniewirkung from an expressed-meter point of view in Vademecum der Phrasierung, he abandons his position as soon as an analysis from that perspective does not conform to his notated-meter Auftakttheorie.

Ex. 15.4. Beethoven, Piano Sonata in C minor, Op. 10, No. 1, i, mm. 1–31; from Hugo Riemann, L. van Beethovens sämtliche Klavier-Solosonaten, 3 vols. (Berlin: M. Hesse, 1917–19), 1: 279–80.

In some cases, however, Riemann's implicit appeal to Harmoniewirkung results in analy­ ses that stand in a more flexible relation to his model. In example 15.4, the change from tonic to dominant at measure 4 conforms to the model as does the change back to tonic at measure 8. (It is unclear, however, how we are to understand measures 2 and 6 to be strong in relation to measures 1 and 5.) At the final measure of period II (last measure of system 5, measure 22 of the movement), Riemann analyzes a structural elision (“8 = 1”), whereby the cadence of period II (which closes the main theme) projects measure “8” of the model (according to the principle of Schlußwirkung to be discussed below) while at the same time beginning period III (the closing section) on a measure “1.” This nine-mea­ Page 7 of 21

Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter sure period opens with the harmonic pattern T–T–D–D. According to the principle of Har­ moniewirkung, the third measure of the period (measure 24 of the movement) would be accented; so, too, would be the return to tonic at the fifth measure (measure 26). But such an analysis would not conform to the model. As a result, Riemann indicates that the change to dominant at measure 24 is a “2” (rather than a “3”) by extending the interpre­ tation of “1” over both measures 22 and 23. Riemann is thus able to employ the principle of Harmoniewirkung in a way that agrees with his model, though in order to do so, he must break with a mechanical counting of the measures.15 (p. 426)

Melodic Imitation

Ex. 15.5. Beethoven, “Leonore Overture,” No. 2, mm. 57–65; from Vademecum der Phrasierung, 50.

Although Riemann places primary emphasis on Harmoniewirkung as a metrical determi­ nant, he admits that in some situations, harmony “leaves us in the lurch”: “In all cases where the harmony remains the same for a longer series of measures,…it is clearly neces­ sary to establish the various weights of the measures in another way.”16 Where Har­ moniewirkung is not forthcoming, Riemann turns to melodic content for the expression of “statement” (Aufstellung) and “response” (Antwort) that are his (p. 427) main metaphors for unaccent and accent respectively: “It is easy to recognize that the relationship of statement and response is expressed not only, but above all, in the return of the same or similar melodic phrases, in the imitation of the motive.”17 Riemann illustrates his principle that melodic imitation creates accent with the opening of Beethoven's “Leonore Overture,” no. 2 (example 15.5). He notes that the second phrase is metrically stronger than the first because of a repetition of melodic contour. Furthermore, within each phrase, he claims that a simple change in direction suffices to express melodic imitation: “It has indeed long been recognized that inversion is a form of imitation.”18 Thus the second half of each phrase in example 15.5 reverses the melodic di­ rection, thereby imparting greater metrical weight to the double measures 3–4 and 7–8. Since Riemann considers that one “real” measure in this example consists of two “notat­ ed” measures, the real measures 2 and 4 are accented in relation to measures 1 and 3, an analysis that fully conforms to his notated-meter model. Armed with the criteria of Har­ moniewirkung and melodic imitation, Riemann confidently asserts that he can explain the metrical structure of most musical phrases: “Both of these factors—the melodic contours and the harmony—will be found sufficient at least in the great majority of cases for a cer­ tain determination of the metrical weight.”19

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Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter Riemann may be certain of his analyses, but, as Dahlhaus points out, the premise that melodic imitation creates accent is open to serious question: “In general, motivic repeti­ tion (not unlike harmonic repetition) gives rise to a copy, which is less weighty than the model.”20 Indeed, if one accepts the principle of initial accentuation, then it is contradic­ tory to assume that a “second” of something is necessarily accented in relation to a “first.” Nevertheless, Riemann's fundamental aesthetic of “active hearing” leads him to just such an experience of musical meter.

Durational Accent

Ex. 15.6. From Musikalische Dynamik, 31–32.

Ex. 15.7. From Große Kompositionslehre, 1: 23.

In an early treatise, Musikalische Dynamik und Agogik,21 Riemann introduces the concept of “agogic accent,” in which an event receives accentuation through a performed, minute extension of its durational value. Such an accent is not, however, based on any durational differentiation notated by the composer. The note that the (p. 428) performer elongates may well be notated with the same rhythmical value as the preceding or following notes, as in example 15.6, where, within a succession of steady eighth notes, agogic accents (in­ dicated by the carets) can be used by the performer to differentiate a 3/4 meter from a 6/8 one. In his mature theory, Riemann appeals to another process of accent formation, one that is rooted in a manifest difference in the duration of the events, differences that the composer specifically notates. These accents generally arise from the proportions 2:1 or 3:1, as in example 15.7, taken from the Große Kompositionslehre. As Riemann notes, “The ear instinctively attributes the greater weight to the tones that are distinguished by longer duration; that is, the ear assumes a time signature that places these notes directly behind the bar line.”22

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Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter In Vademecum der Phrasierung, he specifically uses durational differentiation as a criteri­ on for metrical analysis. The aesthetic effect of relatively long notes is to bring musical motion to a standstill, and such a cessation is most appropriate for the end of a thematic idea:

Ex. 15.8. Beethoven, Piano Sonata in E-flat, Op. 27, No. 1, i, mm. 1–4; from Vademecum der Phrasierung, 57–58.

Therefore, the longer values generally fall consistently on the accented beats…. After we have once understood that the accented beat is articulative in itself, this combination [of long and short] appears as a quite natural and obvious means of assisting the articulating force of the accented beats; since the accented beats (p. 429) are not always immediately recognizable as such, this assistance is often enough necessary.23 Riemann then appeals to durational differentiation to support his metrical analysis of the beginning of Beethoven's Piano Sonata in E-flat, op. 27, no. 1 (example 15.8): “The as­ sumption that the bar lines really stand correctly [in staff a] forces one to interpret the motive as [in staff b], hence, with incessantly hindering long values in the upbeat and with ‘appended motives’ on all of the accented measures.”24 By changing the bar lines to indicate his idea of the correct metrical organization, Riemann now makes the longer val­ ues correspond to the metrical accents, as shown in staff c.25

Ex. 15.9.

Ex. 15.10.

Though Riemann's appeal to durational accents in this example might seem unproblemat­ ic (even if his ultimate metrical interpretation and renotating may not convince today's Page 10 of 21

Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter listeners), the relationship of short to long in this example raises further complications. Looking back at example 15.7, we can observe that the alternation of short and long events occurs at a single level of metrical motion: at the level of the quarter note (for al­ ternating half notes and quarter notes) or at the level of the eighth note (for the dotted quarters and eighths). And it would be hard for listeners to hear an implied metrical in­ terpretation that places bar lines before the short note values. But the situation in exam­ ple 15.8 is more complex. Consider the more abstract situation of example 15.9. As shown by the upward-stemmed notes, the single half note can relate to the prior and subsequent quarter notes at the level of quarter-note motion and thus acquire a durational accent, as in interpretation (a). But at the level of half-note motion, the single half note also relates to the combined quarter notes, shown as a bracketed half note in the downward stemmed line of notes; the situation here would not generate any durational differentiation. Thus at the half-note level, both interpretations (b) and (c) are reasonable. In other words, it is not so evident that the metrical setting in line b of example 15.10 is necessarily so much more compelling than that of line a. To be sure, there may be a residual effect of the dura­ tional differentiation at the level of the quarter note that slightly (p. 430) allows us to fa­ vor line b, but that effect is hardly as striking as that which arises in the case of the un­ ambiguous durational differentiation of example 15.7. That many listeners today would not readily accept Riemann's rebarring of the Beethoven Sonata (staff c of example 15.8) suggests that the force of the durational accents identified by Riemann is perhaps not as strong as he claims.

Tonal Accent

Ex. 15.11.

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Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter

Ex. 15.12.

Another criterion for metrical analysis involves the relationship of harmonic function and meter. In an earlier study, I have shown that a handful of theorists from Rameau to Rie­ mann link tonic harmony and metrical accentuation in various ways—ranging from Hauptmann's highly abstract “correspondence” of tonic and accent to Vogler's rigid rule requiring all tonic harmonies to be placed on strong metrical positions.26 Early in his ca­ reer, Riemann forcefully articulated a position associating tonic harmony and metrical ac­ cent, a criterion that can now be termed tonal accent.27 Somewhat later, when focusing on his theory of “dynamic shading” in Musikalische Dynamik, he seemed to reverse posi­ tions and associated dominant harmony (as opposed to tonic) with the “metrical climax,” the moment of greatest intensification within a metrical motive (see example 15.11).28 Given his earlier interest in such harmonic–metric relationships, it is surprising then to discover that he largely ignores this issue within his mature writings, focusing instead on Harmoniewirkung and Schlußwirkung (“effect of cadence,” to be discussed shortly). The only statement suggesting the existence of tonal accents is found in Grundriß der Kompo­ sitionslehre: (p. 431) If we now seek out the natural relationship between harmonic motives and the metrical elements of form, it follows as most simple and obvious that a positive and negative harmonic motive stand in symmetry with each other; that is, a first member brings the turning away from the tonic, the second member brings the re­ turn to the tonic.29 In describing a “positive” and “negative” development, Riemann seems to be expressing a view similar to that found in his earlier theory of dynamic shading, where the dominant harmony receives the greatest intensification (see again example 15.11). But Riemann's mention of a first and second “member” (Glied) is a reference to the Auftakt model of no­ tated meter in his mature theory, and the relationship of accentuation and harmony is ac­ tually different from that presented in Musikalische Dynamik when, as shown in example 15.12, two levels of metrical structure are taken into account. At the lower level, each “member” is made up of two harmonies: T–D and D–T. The higher level consists of a sin­ Page 12 of 21

Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter gle member embracing the two lower level ones. According to the Harmoniewirkung, and in correspondence with his a priori model, the second part of each lower level member is accented in relation to the first. At the next level up, the second complete member (D–T) can be seen as a tonal accent in relation to the first member (T–D) because, according to the model, the actual point of accent of the higher level corresponds with the accent of the lower level. In this way, the (second) tonic harmony, not the dominant, is directly asso­ ciated with the higher level accent. Thus a rudimentary notion of tonal accent can be seen to play a role in certain situations involving the relatively common distribution of harmonies shown in example 15.12. Indeed, many classical themes open with just this al­ ternation of tonic and dominant harmonies, such as those in example 15.2d and e. And to the criterion of Harmoniewirkung already offered (to explain the metrical analysis at the level of the measure) can now be added a criterion of tonal accent to explain the metrical analysis at the level of the double measure. Insofar as Riemann associates a return to the tonic with the metrical strong point, he presents only a partial concept of tonal accent, for he never finds the opening tonic of a phrase to be metrically strong. The motion from T to D at the beginning of a progres­ (p. 432)

sion does not express the pattern “accent–unaccent” because of tonal differentiation, but rather, it forms the reverse pattern, “unaccent–accent,” because of Harmoniewirkung. That Riemann does not recognize the existence of initiating tonal accents is, of course, consistent with his general aesthetic principle, in which the process of hearing is always directed toward a goal. In his mature theory, the tonal accent is linked exclusively to the end of a musical idea, which is frequently a point of cadence. Since Riemann often refers to a Schlußwirkung (“effect of cadence”), this concept must now be examined in some de­ tail.

Schlusswirkung One of Riemann's most frequent rationales for renotating the bar lines of a musical work (in order to reflect what he deems to be its true metrical interpretation) is that the Sch­ lußwirkungen must fall on accented positions. An understanding of the relationship that Riemann draws between “cadence” and “accent” is complicated, because the very notion of cadence traditionally includes a variety of factors—harmonic, melodic, rhythmic, and metric. In order to avoid a logical circularity, it is necessary to determine whether or not a cadence can exist independent of a determinate metrical position. If so, then at least it is possible that cadence can be used as a criterion of expressed meter. The most complete definition of Schlußwirkung is found in the Elementar-Schulbuch der Harmonielehre: The harmonies that enter on the beginning of the fourth and eighth measures make Schlußwirkungen [cadential effects] or Schluß-like effects, according to the extent to which the chords are suitable for the cadence or not. In the strictest sense, only the tonic is suitable for the cadence…. But a kind of Schlußwirkung Page 13 of 21

Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter can also be made by all other harmonies that fall on these accented beats, thus the dominant as well, assuming that it enters as a consonant chord.”30 Here, the identification of Schlußwirkung is dependent directly upon a notated meter (the a priori schemes of example 15.1, above). At the same time, however, a definite harmonic component, a chord that is “suitable for the cadence,” is also required for the effect. And although Riemann gives preference to the tonic harmony, he nevertheless concedes that other triads can also create “a kind of cadence.” Riemann presents a similar definition of the cadence in Grundriß der Kompositionslehre: “If the ‘answers’ given here [D–T or S–T] are grouped symmetrically with their corre­ sponding ‘statements,’ they are entitled to the name cadence.”31 (p. 433) The symmetrical grouping refers again to Riemann's notated-meter model. Like the explanation in Elemen­ tar-Schulbuch, the cadence is considered to possess two necessary components—harmon­ ic content and metrical position: “Thus in addition to symmetry, the harmonic retrogres­ sion [e.g., D returning to T] belongs to an actual cadence: both factors assume and mutu­ ally support each other.”32 But Riemann then qualifies this mutual relationship: As we will see, however, there is the possibility of breaking through the symmetry with the help of the harmony and of forcing Schlußwirkungen where the metrical prerequisites are lacking; in such cases, the tonality must naturally be expressed especially powerfully, and the assistance of motivic imitation can hardly be dis­ pensed with.33 Under some circumstances, then, meter is not a necessary condition for Schlußwirkun­ gen. By allowing the cadence to be defined in terms of harmony, melody, and rhythm (the reference to “motivic imitation” incorporates these last two parameters) and by making it independent of an a priori metrical placement, Riemann lays the foundation for the use of Schlußwirkung as a criterion for expressed meter. In Vademecum der Phrasierung, for ex­ ample, he regards the location of the Schlußwirkungen in the slow movement of Mozart's Piano Sonata in G, K. 283, as decisive for the correct metrical interpretation (example 15.13 gives the opening measures): Therefore, the critical question becomes: does the beginning tonic or the following dominant have the greater weight in the theme? The course of the theme through twelve full common-time (𝄴) measures shows that all of the Schlußwirkungen fall in the middle of the measure, and that by choosing his notation in 𝄴 rather than in 2/4, Mozart did not leave out the bar lines of the unaccented measures (1st, 3rd, 5th, and 7th) as would have been correct, but rather he left out those of the ac­ cented ones (2nd, 4th, 6th, and 8th).34

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Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter

Ex. 15.13. Mozart, Piano Sonata in G, K. 283, ii, mm. 1–2.

The first Schlußwirkung is a half cadence with the dominant harmony on the third beat of measure 2. To be sure, we sense at this point a kind of closure, articulated most strongly by the motion of the bass voice, but whether such a sense of cadence is in itself sufficient to express accent remains open to question. It is not clear that any genuine “expressed meter” criteria can justify reading the second half of measure 2 as metrically stronger than the first half: no initial, durational, or tonal (p. 434) accents can be readily identified here. In effect, Riemann defines the boundaries of a musical idea (e.g., a motive, phrase, period) and then assumes that the end of the idea is metrically strong. That is, he finds the mere fact of being a conclusion of an idea (an “answer” to a “statement,” to use his favorite metaphors) to be a sufficient condition for expressing accent. Such an assump­ tion, of course, runs entirely counter to that underlying the idea of initial accentuation, in which the fact of being a beginning results in accent creation. Since most higher level structures comprise units of two parts (and this fact find its expression in Riemann's no­ tated-meter model), it is not logically possible to analyze as accented both the first and the second part; indeed, it seems that a theorist must choose between the principle of ac­ cent of initiation or Schlußwirkung.35 Riemann's choice is clear—in his mature theory, he never explicitly discusses an accent associated with the beginning of a structural unit. To be sure, those analyses based on Harmoniewirkung are often explainable in terms of ini­ tial accentuation (as discussed earlier), but Riemann himself says nothing about any initi­ ating quality that is responsible for accent creation. Thus while it may be questionable whether accent can indeed be expressed by Schlußwirkung, Riemann's acceptance of that idea is consistent with a general rejection of initial accentuation and fully consistent with his fundamental aesthetic orientation.

Notated Meter versus Expressed Meter

Ex. 15.14. Beethoven, Piano Sonata in G, Op. 31, No. 1, ii, mm. 1–4; from Vademecum der Phrasierung, 50.

The various criteria for metrical analyses to which Riemann explicitly appeals have now been presented and evaluated. The relationship between these criteria and the Auftakt model that lies at the heart of Riemann's metrical theory has also been discussed but now deserves further consideration. As Dahlhaus points out, it is unlikely that metrical analy­ ses based exclusively on musical content could fully conform to any a priori scheme: ei­ Page 15 of 21

Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter ther the model of regularly alternating accents and unaccents must be abandoned, or the metrical implications of the musical content must be ignored.36 Of the two possibilities, Riemann clearly follows the second. But there is perhaps a third way to understand how Riemann operates with both expressed-meter and notated-meter viewpoints. By present­ ing such a wide range of expressed-meter criteria, Riemann can in almost all cases find some justification for an analysis that agrees with his notated model. For example, at the opening of the second movement of Beethoven's Piano Sonata in G, op. 31, no. 1 (example 15.14), Riemann notes that the harmonic progression T–T–D–D implies an accent at the change of harmony (Harmoniewirkung) in measure 3.37 But Riemann is also able to ap­ peal to the notion of melodic imitation by claiming that the melody changes direction from the first to the second measures and from the third to the fourth: “But is there not, then, a contrast of melodic content within the individual motives? Now to be sure, not in the form of a strict imitation, but in the free inversion: the ‘sinking down’ of the melody is contrasted as an answer to the ‘rising up.’ ”38 With his criterion of melodic imitation, Rie­ mann thus justifies an (p. 435) analysis that locates accents in measures 2 and 4 in spite of the manifest Harmoniewirkung in measure 3. In this way, the analysis now corresponds to the Auftakt model. As a general method, Riemann seems to analyze the eight-measure period by first locat­ ing an accent at the cadential points, the Schlußwirkungen. Then, he simply counts back­ ward assigning alternating accents and unaccents. The accented measures can in almost every case be justified by one of his expressed-meter criteria: Harmoniewirkung, melodic imitation, durational accentuation, or tonal accentuation. If an “unaccented” measure can also be seen as accented in terms of one of these principles (such as the Har­ moniewirkung at the third measure of example 15.14), that fact is simply overlooked in fa­ vor of his a priori model. The validity of this procedure depends, however, on the extent to which his expressed-meter criteria conform to most competent listeners’ perception of metrical accent. If some of his principles are found unsatisfactory in accounting for ex­ pressed meter (and indeed, the capability of melodic imitation and Schlußwirkung to cre­ ate accent has been called into question), then Riemann must ultimately be seen as a dog­ matist who holds true to his preconceived notions in the face of contradictory evidence from the musical content. Dogmatist or not, Riemann is perhaps nonetheless the first theorist to appreciate fully the need to explain how accents arise from the music itself. To be sure, his frequent renotat­ ing of compositions leads to numerous misinterpretations, yet his justification of this practice is based on the sound principle that the notation alone cannot determine meter. And although Riemann may have failed to realize the full potential of his own skepticism regarding the status of notation by adopting in the end another notated-meter model, his attempt to account for the origin of accent on the basis of musical content alone remains a significant achievement in the history of metrical theory. Whereas earlier theorists had articulated with varying degrees of precision some conceptions of expressed meter, Rie­ mann is the first to formulate a wide variety of principles that he then applies analytically in order to explain our perception of metrical phenomena. Page 16 of 21

Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter

Notes: (1.) Representative works include Große Kompositionslehre, 3 vols. (Berlin and Stuttgart: W. Spemann, 1902–1903); System der musikalischen Rhythmik und Metrik (Leipzig: Bre­ itkopf und Härtel, 1903); Grundriß der Kompositionslehre, 3rd rev. ed. (Leipzig: Max Hesse, 1905); Vademecum der Phrasierung, 2nd rev. ed. (Berlin: Max Hesse, 1906); Hand­ buch der Harmonie- and Modulationslehre, 3rd ed. (Berlin: Max Hesse, 1906); ElementarSchulbuch der Harmonielehre (Berlin: Max Hesse, 1906); “Neue Beiträge zu einer Lehre von den Tonverstellungen,” Jahrbuch der Musikbibliothek Peters 23 (1916), 1–21. (2.) General studies of Riemann's theories of rhythm and meter include Howard Elbert Smithers, “Theories of Rhythm in the Nineteenth and Twentieth Centuries with a Contri­ bution to the Theory of Rhythm for the Study of Twentieth-Century Music” (Ph.D. Diss., Cornell University, 1960), chap. 5; Ernst Apfel and Carl Dahlhaus, Studien zur Theorie und Geschichte der musikalischen Rhythmik und Metrik, 2 vols. (Munich: Emil Katzbich­ ler, 1974) (most of this is book is authored by Apfel; Dahlhaus contributed two chapters, one of which, “Zur Kritik des Riemannschen Systems,” directly relates to Riemann's theo­ ries); Wilhelm Seidel, Über Rhythmustheorien der Neuzeit (Bern: Francke, 1975), chap. 5; Ivan F. Waldbauer, “Riemann's Periodization Revisited and Revised,” Journal of Music Theory 33.2 (1989): 333–392. (3.) See William E. Caplin, “Theories of Musical Rhythm in the Eighteenth and Nineteenth Centuries,” in The Cambridge History of Western Music Theory, ed. Thomas Christensen (Cambridge: Cambridge University Press, 2002), 675, 682–683. (4.) Hugo Riemann, “Ideas for a Study ‘On the Imagination of Tone,’ ” trans. Robert W. Wason and Elizabeth West Marvin, Journal of Music Theory 36.1 (1992): 81. “A guiding principle that extends throughout my music-theoretic and music-aesthetic work…is that music listening is not merely a passive processing of sound effects in the ear but, on the contrary, a highly developed manifestation of the logical functions of the human intellect.” (“Daß das Musikhören nicht nur ein passives Erleiden von Schallwirkungen im Hörorgan sondern vielmehr eine hochgradig entwickelte Betätigung von logischen Funk­ tionen des menschlichen Geistes ist, zieht sich als leitender Gedanke durch meine sämtlichen musiktheoretischen und musikästhetischen Arbeiten.”) (5.) Dahlhaus, “Zur Kritik,” in Apfel and Dahlhaus, Studien, 1: 185. “Die harmonischen Kriterien, auf die sich Riemann stützte, um schwere von leichten Takten zu unterschei­ den, sind von ihm niemals explizit ausgesprochen und begründet worden. Sie sind jedoch —durch Analyse von Analysen—rekonstruierbar. Riemann ist noch dort Systematiker, wo er es verschweigt.” (6.) Riemann's remarks on Harmoniewirkung and melodic imitation have already been ex­ amined by Apfel (in Apfel and Dahlhaus, Studien, 1: 58–68). Though Apfel considers ways in which these criteria interact with metrical accentuation, his concern lies more in their connection to musical phrasing, articulation, and form.

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Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter (7.) Riemann, Grundriß der Kompositionslehre, 1: 51. “Zunächst ist allgemein zu konsta­ tieren, daß, je schwerer ein Wert ist, desto mehr er einen Wechsel der Harmonie er­ warten läßt, mit andern Worten: die Zeitmomente, auf welche vorzugsweise neue Har­ monien eintreten, sind die Schwerpunkte der Motive, Gruppen, und Halbsätze.” For the sake of clarity, Riemann's copious use of emphasis has been greatly reduced in my Eng­ lish translations; the original emphasis is retained in the German text. (8.) Riemann, Vademecum der Phrasierung, 42. “Die Unterscheidung der schweren und leichten Werte hängt nicht von der Willkür des Komponisten ab, sondern liegt bereits in der Natur der musikalischen Ideen selbst, und es handelt sich bei der Niederschrift nur darum, sie richtig zu kennzeichnen. Worin besteht nun aber das Wesen des verschiede­ nen Gewichts der Töne?” (9.) Ibid. “Da wir uns hier nicht in umständliche ästhetische und theoretische Erörterun­ gen einlassen wollen, sei nur kurz und bündig festgestellt, daß schwere Zeiten im allge­ meinen Träger von Harmoniewirkungen sind, daß der Komponist hiernach die Taktstriche zu setzen hat und daß daher auch der Hörer aus dem harmonischen Sachverhalte heraus das verschiedene Gewicht der Töne erkennen kann. Angenommen ein Hörer schreibt eine ihm vorher unbekannte Melodie nach dem Gehör auf, so werden die Stellen, wo die Har­ monie wechselt…sich ihm als die wichtigeren offenbaren, als diejenigen, welche Anspruch auf Auszeichnung in der Niederschrift haben.” (10.) Ibid., 43. “… dokumentieren…einigermaßen die Abhängigkeit der Taktwahl and Tak­ tbezeichnung von der Harmoniebewegung.” (11.) See William E. Caplin, “Der Akzent des Anfangs: Zur Theorie des musikalischen Tak­ tes,” Zeitschrift für Musiktheorie 9.1 (1978): 17–28, and “Moritz Hauptmann and the The­ ory of Suspensions,” Journal of Music Theory 28.2 (1984): 251–269. (12.) Riemann, Vademecum der Phrasierung, 49. “Die beginnende Ganze (d. h. also der erste Takt) wird durch den Harmoniewechsel auf den zweiten Takt als leichte ver­ ständlich und die Kadenz T S D T kommt glatt auf den vierten Takt zum Abschluß.” (13.) Riemann, Grundriß der Kompositionslehre, 1: 72. “Zu den einfachen Hinstellungen der Tonika dürfen wir auch jene besonders häufigen Bildungen rechnen, welche zu einer Dominante nicht eigentlich fortschreiten, sondern nur einen Rückgang von einer solchen zur Tonika machen, so daß zwischen zwei auf relativ schwere Zeiten gebrachte Toniken sich eine gleichsam nur durchgehende Dominante auf die leichte Zeit einschiebt.” (14.) Riemann, Handbuch der Harmonie- und Modulationslehre, 214. “. . . tritt zwischen zwei relativ schwere Zeiten mit derselben Harmonie, z. B. die schwere Zeit des ersten und zweiten Taktes, eine fremde Harmonie, so wirkt dieselbe nicht voll, sondern er­ scheint nur durchgehend.” (15.) A mechanical counting would also have worked if the cadence of measure 22 were considered to be an “8” exclusively; then the following measure could have been seen as a “1,” and the rest of the analysis would have conformed to his a priori model as a matter Page 18 of 21

Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter of course. But Riemann understands that measure 22 is the real beginning of the new unit and correctly identifies an elision at that point; as a result, he is then forced to re­ gard measure 23 as an extension of “1,” as just discussed. (16.) Riemann, Vademecum der Phrasierung, 50–51. “In allen Fällen, wo für längere Tak­ treihen die Harmonie dieselbe bleibt,…liegt die Notwendigkeit, das verschiedene Gewicht der Takte anderweitig zu begründen, klar vor.” (17.) Ibid., 51. “… so ist leicht zu erkennen, daß das Verhältnis von Aufstellung and Antwort sich nicht nur auch, sondern sogar in allererster Linie in der Wiederkehr gleich­ er oder ähnlicher Melodiewendungen, in der Nachahmung der Motive aussprechen wird.” (18.) Ibid. “Daß die Umkehrung eine Form der Nachahmung ist, weiß man ja schon lange.” (19.) Ibid., 52. “Diese beiden Faktoren, die melodischen Konturen und die Harmonie wer­ den wenigstens in der großen Mehrzahl der Fälle mit Sicherheit zur Bestimmung des metrischen Gewichts ausreichend befunden werden.” (20.) Dahlhaus, “Zur Kritik,” 1: 188. “Durch motivische Repetition entsteht—nicht anders als durch harmonische—im allgemeinen ein Nachbild, das weniger gewichtig als das Modell erscheint.” (21.) Riemann, Musikalische Dynamik und Agogik (Hamburg: D. Rahter, 1884). (22.) Riemann, Große Kompositionslehre, 1: 23. “Unwillkürlich weist das Ohr den durch längere Dauer ausgezeichneten Tönen auch das größere Gewicht zu, d. h. nimmt eine Taktart an, welche dieselben direkt hinter den Taktstrich stellt.” (23.) Riemann, Vademecum der Phrasierung, 54. “Deshalb fallen im allgemeinen die Län­ gen stets auf relativ schwere Zeiten…. Nachdem wir einmal erkannt haben, daß die schwere Zeit an sich gliedert, erscheinen diese Kombinationen als ganz natürliche und naheliegende Mittel der Unterstützung der gliedernden Kraft der schweren Zeiten; denn da die schweren Zeiten nicht immer ohne weiteres als solche erkennbar sind, so ist diese Unterstützung oft nötig genug.” (24.) Ibid. 58. “Die Annahme, daß die Taktstriche wirklich richtig ständen, zwänge, die Motive so zu deuten:…also fortgesetzt mit hemmenden Längen im Auftakt und mit langen Anschlußmotiven bei allen schweren Takten….” By “appended motives,” Riemann would seem to be referring to the bracketed motive “F–F–B♭” (in measure 2 of example 15.8b) and the corresponding motive “B♭–B♭–E♭” (measure 4). (25.) The indication of

(that is, a cadential six-four) at the beginning of the renotated

first measure is undoubtedly meant to refer to a Harmoniewirkung at this point; however, Riemann misreads the bass that is implied here, which is not B♭ (to be sure, the lowest sounding note), but rather E♭, whose appearance at the beginning of measure 3 clearly establishes what is implied at measure 1. Thus, the first change of harmony (from T to D) Page 19 of 21

Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter is found at the beginning of the (originally notated) second measure. Ironically, this more accurate interpretation of the bass would allow the original notation to create a Har­ moniewirkung on the downbeat of the second measure, according to Riemann's more typ­ ical metrical interpretations, for that moment would bring the first real change from T to D. (26.) William Caplin, “Tonal Function and Metrical Accent: A Historical Perspective,” Mu­ sic Theory Spectrum 5 (1983): 1–14. (27.) Ibid., 9–11. See also Kevin Mooney, “Hugo Riemann's Debut as a Music Theorist,” Journal of Music Theory 44.1 (2000): 81–99. (28.) Caplin, “Tonal Function,” 11–13. (29.) Riemann, Grundriß der Kompositionslehre, 1: 64. “Suchen wir nun die natürlichen Beziehungen zwischen den harmonischen Motiven und den metrischen Elementen der Formgebung auf, so ergiebt sich als die einfachste und selbstverständliche, daß ein posi­ tives und ein negatives Harmoniemotiv zu einander in Symmetrie treten, d. h. daß ein er­ stes Glied die Wegwendung von der Tonika und ein zweites die Rückkehr zu ihr bringt.” (30.) Riemann, Elementar-Schulbuch der Harmonielehre, 181. “Die auf den Anfang des vierten und achten Taktes eintretenden Harmonien machen daher Schlußwirkungen oder schlußartige Wirkungen, jenachdem sie schlußfähige Akkorde sind oder nicht. Sch­ lußfähig ist im engeren Sinne nur eine Tonika…. Eine Art von Schlußwirkung machen aber auch alle andern Harmonien, die auf diese schwersten Zeitwerte fallen, also auch die Dominanten, vorausgesetzt nur, daß sie als konsonante Akkorde…eintreten.” (31.) Riemann, Grundriß der Kompositionslehre, 1: 65–66. “Werden die hier aufgestellten Antworten symmetrisch zu den zugehörigen Aufstellungen gruppiert, so haben sie Anspruch auf den Namen Schluß.” (32.) Ibid., 1: 68. “Zum wirklichen Schluß gehört also außer der Symmetrie der harmonis­ che Rückgang; beide Faktoren setzten einander voraus und heben sich gegenseitig.” (33.) Ibid. “Doch ist, wie wir sehen werden, auch die Möglichkeit da, mit Hilfe der Har­ monie die Symmetrie zu durchbrechen und Schlußwirkungen zu erzwingen, wo die metrischen Vorbedingungen dafür fehlen: in solchen Fällen muß dann natürlich die Tonart besonders scharf ausgeprägt sein, und auch die Mitwirkung motivischer Imitation ist kaum zu entbehren.” (34.) Riemann, Vademecum der Phrasierung, 56. “Die Frage spitzt sich daher dahin zu: hat in dem Thema die beginnende Tonika oder aber die ihr folgende Dominante das größere Gewicht? Der Verlauf des Themas durch zwölf ganze Takte 𝄴 beweist dadurch, daß alle Schlußwirkungen auf die Taktmitte fallen, und daß Mozart bei der Wahl der Notierung in 𝄴 statt 2/4 nicht, wie es korrekt gewesen wäre, die Takstriche der leichten (1., 3., 5., 7.), sondern die der schweren Takte (2., 4., 6., 8.) fortgelassen hat.” The nota­

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Criteria for Analysis: Perspectives on Riemann's Mature Theory of Meter tional practice described here by Riemann was understood by many eighteenth-century theorists as compound meter. (35.) One might cite Edward T. Cone, however, as one theorist who seems to want it both ways. His model of thematic structure, which is effectively tripartite, recognizes the mo­ ments of both beginning and ending to be accented in relation to the middle of a theme; see Musical Form and Musical Performance (New York: W. W. Norton, 1968), 26–27. (36.) Dahlhaus, “Zur Kritik,” 1: 185. (37.) Riemann, Vademecum der Phrasierung, 51. “Bei [example 15.14] könnte man noch wegen des Harmoniewechsels annehmen, daß der dritte Takt als der schwerer zu verste­ hen wäre.” (38.) Ibid. “Aber ist denn nicht auch innerhalb des einzelnen Motivs noch eine Gegenüber­ stellung melodischen Inhalts nachweisbar? Nun, in der Form strikter Nachahmung zwar nicht, aber in der freier Umkehrung: dem Hinauftreten wird das Zurücksinken als Antwort gegenübergestellt.”

William E. Caplin

William E. Caplin is James McGill Professor of Music Theory in the Schulich School of Music, McGill University. His 1998 book Classical Form: A Theory of Formal Func­ tions for the Instrumental Music of Haydn, Mozart, and Beethoven won the 1999 Wal­ lace Berry Book Award from the Society for Music Theory. In addition to his work on musical form, he has published on the history of harmonic and rhythmic theory in the eighteen th and nineteenth centuries, including a chapter in the Cambridge History of Western Music Theory.

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Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas

Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas   Scott Burnham The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0016

Abstract and Keywords This article examines Riemann's most sustained treatment of harmony and meter, as found in his analyses of the Beethoven piano sonatas. In this article, Riemann's analyses are used as a basis, not only to elucidate on how Riemann's theories work but also to demonstrate how they can illuminate Beethoven's music. Focusing on three sonatas of Beethoven's Op. 31, the article reveals the circumstances in which Riemann does not have the luxury or freedom to make abstract or systematic pronouncements, but is in­ stead confronted with the concrete situation of having to make sense of a musical compo­ sition. In this article, the focus is on Riemann's rhythmic analysis. It aims to answer how Riemann uncovers periodic logic in Beethoven's labile phrase rhythm, and what this logic means for the examination of Beethoven's compositional trajectory. Keywords: harmony, meter, Beethoven piano sonatas, periodic logic, labile phrase rhythm, Beethoven, sonatas

My phrasing editions as well as my textbooks on phrasing…fulfill the same pur­ pose as commentary on difficultly understood literary works: to show how one reads between the lines, how one advances from primitive sight-reading of what's on the page to actual understanding of its sense.1

For many years, Hugo Riemann sought to read between the lines of canonic music, confi­ dent that his interrelated theories of rhythmic and harmonic function could unveil the of­ ten occluded workings of a cogent musical logic. This enterprise culminated in the threevolume analysis of Beethoven's piano sonatas, whose final volume was published in 1919, the year of Riemann's death. Wilhelm Seidel speculates that the Beethoven analyses con­ stitute Riemann's testament: His three-volume book on Beethoven's piano sonatas is one last attempt—now in the form of a monumental work—to draw attention to that element through which he knew that the artistic character of music was determined in the zenith of its history, and [to draw attention] to the metric security and aesthetic mastery of Page 1 of 20

Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas every last detail [of Beethoven's sonatas]. Riemann's work on Beethoven's piano sonatas was his testament. He marked it as such: under the last analysis of the third volume he wrote his name.2 In his compulsion to analyze every bar of every sonata, Riemann indeed wished to show “how a musical art in full possession of its possibilities can be represented.”3 But he was also indulging a growing sense of the Beethoven piano sonatas as a coherent cycle, one that is best treated in toto. Not only do the sonatas—like the string quartets—sprawl across Beethoven's three “style periods,” but they seem to tell their own story of Beethoven's development, a story made more intimate by the commonly expressed sense of the sonatas as Beethoven's hands-on musical diary. Individual sonatas are understood as notable events occurring within a well-charted narrative, and most critical assess­ ments assign at least part of any given sonata's significance to its placement within the entire cycle. For example, who ever discusses the two-movement Op. 54 without noting its placement between the Waldstein and the Appassionata? It gains interest from this placement, as an intriguing dale between two imposing hills. Or consider what Op. 2, 1 and Op. 111 gain from being the first and last of the sonatas. Critics who narrate the (p. 441)

course of the sonatas often detect rhythms of expansion and contraction, intensification and decompression. For example, Op. 22 is almost always described as a watershed con­ solidation of the inherited idea of a sonata, such that Beethoven now felt free to experi­ ment formally in Op. 26 and in the fantasia-like Op. 27 sonatas. The prevalence of sets— groupings of sonatas that relate in complementary ways to each other—mark the first ten years of Beethoven's sonata production: Op. 2, Op. 10, Op. 14, Op. 27, Op. 31, Op. 49. His subsequent production of single sonatas only is most often read as a sign of artistic matu­ ration: each work is now too individuated and coherently self-reliant to brook inclusion within any collective smaller than the great cycle itself. The set of three sonatas in Op. 31 is usually treated as marking a turning point, a “new way,” in the cycle and in Beethoven's overall style. Op. 31 is also the final single-opus set in the sonatas (Op. 49, 1 and 2 were composed earlier). The novelty of this set, as well as its coherence as a set, is often localized to the highly characteristic openings of each sonata. As Lewis Lockwood observes, “The openings of all three speak a new language, each presenting a new and original mode of entry into a large sonata-form movement.”4 These three openings and the expositions that follow offer a useful opportunity to gauge the rewards and challenges of Riemann's method. To further enhance the focus here, I will restrict my observations primarily to Riemann's rhythmic analysis. How does Rie­ mann uncover periodic logic in Beethoven's increasingly labile phrase rhythm, and what can this logic mean for us as we continue to negotiate this rich turn in Beethoven's com­ positional trajectory?

Downbeats Riemann's stated modus operandi makes the assumption that one can read between the lines, that the musical text does not offer every last clue regarding its proper reading Page 2 of 20

Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas (and to pretend that it does is to indulge in primitive reading). (p. 442) Additional com­ mentary is needed to penetrate to the true sense of the music. Riemann offers a bar-bybar elucidation, whose symbols—phrase markings, harmonic functions, periodic functions —elaborate a single-staff reduction of the score. As Carl Dahlhaus observed, “For Rie­ mann, musical analysis was the addition of a script of signification to the sounding script of notation.”5 Riemann might well have characterized his method as “reading the lines,” because his analytical interpretations divide the temporal flow of the music into cogent units that are not unlike lines of classical poetry that possess a fixed number of feet and some invariant metrical functions. A line for Riemann is any concrete realization of his prototypical eight-measure period; he thus refers to these delineated sections of music as periods. Riemann's prototype is an abstraction based on what he believes to be the first fact of musical rhythm: the fundamental impulse of upbeat to downbeat. As can be seen in example 15.1 (chapter 15, p. 421) the upbeat/downbeat impulse perme­ ates Riemann's construction at several levels: the last four measures answer the first four; the two-measure pairs 3–4 and 7–8 answer the two-measure pairs 1–2 and 5–6; each even measure answers each odd measure: 2 answers 1, 4 answers 3, 6 answers 5, and 8 answers 7. In actual realizations of Riemann's prototype, even-numbered measures (“schwere Zeit­ en”) usually carry harmonic impact—these are the places where the harmony changes.6 Example 16.1 shows an unproblematic example from the first Op. 31 sonata. The arrow at the outset indicates the presence of an upbeat before the upbeat-functioning first mea­ sure. Note that Riemann marks only the even-numbered strong measures. In keeping with his theory of harmonic function, every harmony is marked as some type of T, D, or S function (tonic, dominant, subdominant—I will not take the time to elucidate these sym­ bols except in those cases where it bears on his assignment of rhythmic functions).

Ex. 16.1. Op. 31, 1, Rondo theme, Period I.

Beethoven's rondo theme works well as a realization of Riemann's prototype. Each evennumbered downbeat is a harmonic resolution, and it is easy to hear the various answer­ ing downbeats of the prototype, at three different levels: 2 answers 1; 3–4 answers 1–2; 5–8 answers 1–4. (p. 443)

In line with its construction around resolving downbeats, Riemann's prototypical

period is end-oriented. Among other things, this means that the final 8 function is an in­ variant feature. Every Riemann period must contain an 8 function, just as every tree must have a trunk (structuralist linguists would refer to Riemann's period as a left-branching Page 3 of 20

Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas structure—everything branches off from the final term). One can never lop off the trunk: a period left incomplete will be some version of 5–8, never 1–4. And in keeping with this end-oriented construction, local upbeats are more dispensable than local downbeats. There can be downbeats without upbeats, but no upbeats without downbeats.7 Ends of periods rarely seem problematic in Riemann's analyses, but his insistence on even-numbered measures as strong downbeats leads at times to readings that would strike us as counterintuitive at first. He will never allow the first measure of a period to be strong. If the sounding first measure is strong, then he posits a 1 function that pre­ cedes it as an imaginary upbeat. This kind of move can be conceptualized as an elision and is not necessarily problematic. But Riemann is constrained at times to number the rest of his period in odd ways, in order to preserve the first sounding measure as a 2 function rather than a 1 function. Take the second theme of Op. 31, 3 (example 16.2). This would sound to most of us like a straightforward 8-measure theme, governed by a parallel and symmetrical construction. But because it commences with a strong measure and there is no change of harmony in the next measure, Riemann is constrained to designate the opening measure as a 2. The choice seems sound as we make our way through the rest of the theme, because the points of harmonic change in the theme will then fall on 4, 6, and 8. But an awkward bump ensues at the end of the phrase, as Riemann has to in­ clude two 8s. It's not at all unusual to have multiple 8s in the same period, as we shall see. But to have this occur in a theme that sounds so straightforward in terms of its con­ struction is somewhat jarring.

Ex. 16.2. Op. 31, 3, Second theme, Period V.

There is an interesting adjustment of harmonic function at the point of the duplicated 8 function. Riemann marks his first 8 as a T, though it is actually a Tp (i.e., (p. 444) he marks it as a tonic B♭ harmony, but it is a G minor harmony, a “tonic parallel” in Riemann's system). He makes the same adjustment in the recapitulated version of this theme. It is hard to imagine that this could be an oversight; a likelier scenario is that Riemann's T is more theoretical than empirical at this point in the period. By reverting to the main token of the T family of harmonies, he can more explicitly justify the consecutive 8 functions: both bring on the resolving tonic. Another slight adjustment happens with the 5 function, which Riemann designates as a back-relating 5. Here the fifth measure re­ lates back to 4 rather than acting as an upbeat to 6. This breaks with the usual sense of 4

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Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas as an end to a unit, followed by a new four-measure unit. Riemann's analysis of this theme thus lays out as 2–3–4–5/6–7–8–8a.8 Both these adjustments encourage hearing the entire theme in two-measure units, only in down-up configurations rather than up-down (2–3, 4–5, 6–7, 8–8a). Such a strategy is plausible to the ear, since this theme clearly falls into two-measure groups, whose bound­ aries are drawn by harmonic change and bassline. It is also forecast by the 8–8a pair that precedes it, in the measures of dominant prolongation in which the bassline comically drops several stories downstairs (this can be seen in example 16.4 below, as the end of Riemann's Period IV). More important, the opening 2 designation gets at the effect of starting the bass not on the root but on the third of B♭. This creates an “in medias res” opening that immediately moves the theme into a cadential progression spanning the en­ tire eight measures, with a significant bass ascent every two measures. To follow that overemphasized dominant F not with a strong resolving tonic in the bass but with a third in the bass is to isolate the “falling downstairs” moment even more decisively (more comi­ cally?). The resolution to tonic takes place not down in the bass but up in the melody, cre­ ating an emblematic registral transfer in a movement that is very much about wide-rang­ ing registral play. Meanwhile, the bass itself begins a cadential drive that is more animat­ ed (less flat-footed) for starting on the third scale degree of B♭ rather than its root. Riemann's 2 function begins to capture the way the bass jumps ahead here. And taking in this theme's bass motion as a cadential progression that begins in medias res makes us aware of a connection to the nature of the first theme, how it too gradually creates a ca­ dential progression without a strong opening tonic. The B-major second theme from the first movement of Op. 31, 1, shown in example 16.3, presents a similar challenge to Riemann's periodic functions. It too consists of two-mea­ sure pairs that start strong and thus cannot be numbered 1 through 8. And the second group of four measures begins exactly as the first four measures, which creates an addi­ tional complication for Riemann.

Ex. 16.3. Op. 31, 1, Second theme, Period V.

The oddness of his solution prompted him to gloss it in his prose introduction to the analysis of this movement. His logic is this: because the 5th measure of this theme sounds like a return and resolution, it is a strong measure, in fact an 8; thus the measure it an­ swers to (first measure of the theme) must be a 4, because 8s answer 4s. So far so good, but Riemann then goes on to reinterpret this 8 as a 5 (in order to end on an 8). This then makes the first pair of measures a 4–5, and the answering pair of measures an 8 = 5–6. Thus 4–5 becomes 5–6, which constitutes a radical reinterpretation.9 Riemann is willing Page 5 of 20

Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas to live with this inconsistency. What does he gain? Two noncongruent 8 functions in the same theme. And what does this emphasize? For (p. 445) one thing, it helps us hear the theme as a four-measure utterance, doubled up and heavy on closural harmony. This is al­ most the opposite design from the second theme in Op. 31, 3. There the theme took eight measures to complete its progression; here it goes through two cadential cycles. And the superfluity of 8 functions will prove to have further analytical payoff when we look at the opening of the movement as well as the rest of its exposition. These examples demonstrate how Riemann's eight-measure periodic designations mix empirical and theoretical evidence, as well as what types of analytical decisions he will make to preserve his most fundamental precepts. At times he seems to apply his func­ tions in an inconsistent, ad hoc manner, as in the case of the Op. 31, 3 theme, where he allows a Tp to function as a closing T, or in the Op. 31, 1 theme, where he allows a 4–5 pair to be recast as a 5–6 pair. But Riemann's designations almost always reveal some­ thing interesting about the themes in question. Puzzling out the motivations for his func­ tional designations forces one to ask different kinds of questions of these themes, pre­ venting us from simply classifying them as straightforward 4 + 4 constructions. At the very least, Riemann's prototype offers a consistent set of criteria about cadence and har­ monic weighting. When some given phrase does not match up with his prototypical tem­ plate, the points of deviation can lead to a more nuanced sense of what is afoot in that phrase.

Reading between the Lines

Ex. 16.4. Op. 31, 3, Period IV.

At the heart of Riemann's analytical enterprise is the ability to distinguish the essential stations of each period from material that is inserted or appended. Harmonic progression represents the evolving musical argument and thus helps determine the primary stations of his prototypical construction; patches of harmonic stasis serve as confirmation or filler within that construction.10 Mastering this distinction and applying it to the act of parsing by period is what allows Riemann to process the highly differentiated musical flow of a Beethoven as the realization of so many (p. 446) eight-measure periods. This kind of ana­ lytical distinction forms the most palpable bridge between background prototype and foreground realization. Page 6 of 20

Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas We can illustrate this distinction initially with a clear-cut example from Op. 31, 3, in ex­ ample 16.4. Here several pairs of measures (3–4 and 5–6) are repeated in a higher regis­ ter and marked as 3a–4a and 5a–6a; they obviously serve to extend the fundamental peri­ od without furthering the harmonic argument. Riemann also indicates an added 8 func­ tion at the end (8a). Here the register is dropped dramatically, reversing the tendency of the earlier insertions. The resulting period goes something like this: 1–2, 3–4 (3–4), 5–6 (5–6), 7–8 (8). But this is a very transparent example; ultimately, Riemann has a more complex—and consequential—differentiation in mind. For Riemann the historian, increased differentia­ tion is the distinctive mark of the modern theme, which he claims was ushered in by Ger­ manic composers such as Fasch and Stamitz. In the chapter of his composition treatise entitled “Die thematische Arbeit in den grösseren Formen der Instrumentalmusik,” Rie­ mann characterizes the nature of the modern theme as “the union of a greater number of different motives into a greater unified configuration. The modern theme is no longer on­ ly a melodic fragment but is much more an entire melody, more or less completed within itself.”11 He goes on to declare that Allegro themes are the first best place to distinguish the new style from the old.12 And one cannot read the modern theme, or even recognize its modernity, without being able to make Riemann's crucial analytical distinction be­ tween primary and subordinate. Because an opening Allegro theme is thus a fertile place to register this new style, and because the opening themes of Beethoven's Op. 31 sonatas have long been noted for their characteristic variety, it will be of some interest to see how Riemann chooses to analyze them. All three arguably contain “a number of different motives” placed within a larger configuration.

Ex. 16.5. Op. 31, 2, Period I.

The most staggering contrast in the Op. 31 set is the now mythical pairing of Largo and Allegro at the outset of the D-minor “Tempest” Sonata. Riemann stitches (p. 447) the two together as a single period, though one that does without its first two measures. Instead, he designates the Largo as a 3–4 pair, and also as a Vorhang, or curtain. The notion of a Vorhang that is then swept into the first full period of the movement both honors the extreme contrast but also encourages us to hear the entire opening as a single harmonic process.

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Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas The first sonata of the Op. 31 set also begins with contrast. Here a two-measure running figure precedes a blocky chordal theme that makes a running joke out of the rhythmic misalignment of the hands. Example 16.6 shows the first two periods of Riemann's analy­ sis. In his prose commentary, Riemann again figures the contrasting opening measures as a Vorhang, though this Vorhang fulfills periodic functions 6 through 8. Period I then con­ tinues afresh with functions 1 through 8. The combination of Vorhang and full period is then repeated in Period II. Because of Riemann's reading of the Vorhang measures, by the time we get to the end of Period II, we have heard no less than seven 8 functions. As in his analysis of the second theme of this movement, we are again being made aware of a superfluity of 8 functions (and there will be more to come . . .).

Ex. 16.6. Op. 31, 1, Periods I and II.

Despite this interesting feature, Riemann is much more eager to discuss the striking tonal gambit of the opening section: the fact that the second period begins a step lower than the first, on F. He hears this as a progressive move that foreshadows the openings of Op. 53 and 57 (mutatis mutandis), and he speculates that Beethoven was advancing on the slight tonal divagation at the outset of Op. 28 (by trumping that opening's fleeting flatseventh scale degree—as part of a V7/IV—with an abrupt move into the key of the flat-sev­ enth). Moreover, the tonal divagations of the first two periods add up to an interesting kind of story. In the first period, an unequivocal, even stolid, tonic G is destabilized with its vi, which is then retrospectively reinterpreted as an S function (E-minor Tp of G be­ comes E-minor Sp of D) and spun into a cadence in the dominant key of D. Thus the tonic G becomes Subdominant of D (T = S).13 Then, in a surprising harmonic jump, the opening eleven measures repeat, now transposed into F major. This new tonic F then follows the same harmonic route as in the first period and becomes the subdominant of C. Finally, C is reinterpreted (p. 448) as the subdominant of G. This last move takes place within an ex­ tension of Riemann's second period (5a through 8a), and sounds as a larger-level version Page 8 of 20

Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas (a trumping version) of the same T = S move, whose ultimate cadence is repeated three times for emphasis (8a, 8b, 8c). For Riemann, this opening scena is a good example of Beethoven's progressive sense of tonality. The composer can actually ground his tonic key more securely by setting up threats to its reign: the key of G now comes on as something that sweeps the competition away, as an earned tonality rather than an uncontested as­ sertion. What Riemann does not point out in this dialectical reading of the opening, is that Beethoven's tonal gambit also guarantees a steady proliferation of 8 functions. The Vorhang's peremptory tonic cadence is answered by a cadential progression in the domi­ nant; this is then answered by a Vorhang cadence in the subdominant of the subdominant, followed by a full cadence in the subdominant, which is followed in turn by a threefold full cadence in the tonic. This series of emphatic 8 functions is developing into a comic tic of the movement.

Ex. 16.7. Op. 31, 3, Period I.

Ex. 16.8. Mannheim sigh in a theme of Stamitz.

Ex. 16.9. Riemann's interpretation of the opening motive of Op. 31, 3.

Riemann does not posit a Vorhang for the opening of Op. 31, 3, and in some ways this opening, shown in example 16.7, is the most straightforward. Following (p. 449) the har­ monic changes, one hears that 4, 6, and 8 fit perfectly; the problem comes with 2. Unlike the second theme of this movement, here Riemann seems willing to let a repeating mea­ sure function as a 2. To support this reading, he refers to the opening motive, which he hears as a Mannheim sigh. In accordance with the use of the Mannheim sigh by Stamitz, as illustrated in example 16.8, Riemann suggests that Beethoven's opening motive be heard as shown in example 16.9.

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Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas

Ex. 16.10. Op. 31, 3, Period III.

Riemann explains that the sigh figure in Stamitz's theme is to be understood as phrased in version b (under the staff) rather than version a (over the staff). Example 16.9 shows how he then extends this kind of reading to Beethoven's opening. The long note and sub­ sequent quarter-note rest is meant to be heard as a long upbeat into the downbeat recur­ rence of the sigh in measure 2. Yet one strains to hear it this way: to do so would be to cut the sigh off at the sixteenth note, with the effect of a hiccup. The Stamitz sigh is fash­ ioned out of even eighth notes; all bets are off with Beethoven's dotted sixteenth. In fact, this figure sounds more like a birdcall than a sigh (consider as well the effect of its itera­ tion). The sigh figure seems almost a red herring, fished up from Riemann's never flag­ ging sense of the historical importance of the Mannheim school. He makes a stronger jus­ tification for his reading of the first period by invoking what happens in the third period (example 16.10). Here the (p. 450) harmony does indeed change on 2 (“the harmonic pro­ gression comes into the motive itself” as Riemann puts it). Thus he reads the opening pe­ riod with foreknowledge of what comes later. Riemann also notes the “remarkable” off-tonic opening of Period I: “the tonality is not be­ trayed right away by the first chord, as per usual practice, but is led to with a discursive cadential progression [umständliche Kadenz] that fills an entire 8-measure period.”14 Riemann maps each of the idiosyncratic openings of Op. 31 into coherent periods, as in­ stances of internally differentiated, thoroughly modern themes. The ability to differenti­ ate between primary and subsidiary material also proves indispensable for his under­ standing of the ways that such initial themes grow into large-scale forms: The inner necessity with which thematic configurations unfold out of each other in the imagination, differentiate from each other, and are grouped into larger propor­ tions, rests precisely on this clear distinction between the essential and the sub­ sidiary, between that which is firmly formed and that which is loosely accommo­ dated, between the essential musical action and the moments of idle lingering in­ serted between the main phases of that action.15 What counts as logical necessity for Riemann is his sense that every measure, even if part of a more loosely formed interpolation, carries a periodic function, a function that would Page 10 of 20

Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas be missed if the analyst were incapable of making the distinction between invariant for­ mation and variant accessory. To be able to entertain such a distinction is thus to be able to parse entire movements by a Beethoven into periods. Each period is a carrier of logical necessity—and if the entire movement can (p. 451) be shown to consist of such periods, then it is a process characterized by logical necessity. And not only does Riemann hold this analytical distinction—and the inner logic it reveals —to be the indispensable key to understanding both the modern theme and modern largescale form, but he also claims that it can help elucidate Beethoven's developmental progress throughout the course of his piano sonatas. In defending the chronological arrangement of his analyses, Riemann observes that such an arrangement can serve as a “life sketch of the master, one that makes visible the imposing growth and strengthening of his artistic powers. Surely it will be instructive to recognize that what is characteristic of the maturing master is not a rending of form, but on the contrary, an ever firmer posi­ tioning of the actual structure, around which all accessory material is arranged.”16 To see what Riemann might mean by “ever firmer positioning of the actual structure,” we need to look at some of the more expansive periods in his analyses of the Op. 31 first move­ ments.

Expansion and Closure Period III in Riemann's analysis of Op. 31, 1, shown in example 16.11, is a running pas­ sage of sixteen measures that takes up the figure from the first two measures. In keeping with the increasing number of 8 functions so far in the movement, this period trumps the four 8s of Period II with five 8s in its drive to rest on the dominant. Within this passage, Riemann notes an important “about-face” [Umwenden] at measure 36, in which an 8 function is heard to function retrospectively as a 7 function. This is the point at which the passage shifts from a tonic center of gravity to a dominant center of gravity, and Riemann urges the performer to underline this shift with crescendo and al­ largando. Each harmonic goal now has its own 8 function: measure 36 is the 8 for the ton­ ic, and the very next measure is the first of four 8-functioning measures on the dominant. By the time the music gets to Riemann's 8b, the 7–8 upbeat/downbeat pair has been firm­ ly reestablished. At this point, the remainder of the period adds some back-relating 9s, such that the 8 functions now have echoes. The resulting analysis offers a very nuanced sense of a passage that could easily be mistaken for an undifferentiated effusion.17

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Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas

Ex. 16.11. Op. 31, 1, Period III.

Ex. 16.12. Op. 31, 1, Period IX.

Going further into the movement, one can track with Riemann a progressive addition of 8s as the exposition continues. The next station is the emphasis on the dominant of B, with five 8s; then comes the B-major second theme, which we have already observed to make two separate motions to an 8 function within the same eight measures. The move to B minor entails a short period that (again) finds an 8 within four measures (Riemann's Pe­ riod VI). And the final period of the exposition, which also acts as an epilogue for the pre­ ceding period, presents a much-iterated melodic figure that first plays out as a two-mea­ sure 7–8 and is then sped up into a (p. 452) series of one-measure 8s. Counting the 8a right before Period IX, Riemann charts a total of nine 8 functions at the end of this exposi­ tion. Tracking Riemann's 8 function reveals a previously neglected comic tendency of this exposition: an over-the-top proliferation of closural figures. For a cartoonish sight-gag version of this effect, imagine the tonic arrival in measure 98 as a door that slams shut, only to pop back open on the following root-position dominant, slam shut again on the en­ suing tonic, and so on. That slamming sound has been in the air ever since the first 8 in measure 3, and it is echoed not only in all the 8s to follow but also every time a (p. 453)

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Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas sixteenth-note upbeat in one hand hits a downbeat in the other. The movement lives on the comic premise of the peremptory downbeat.18 If Op. 31, 1 represents the comedy of closures, the first movements of the other two sonatas furnish realizations of the 8 function that are at once more expansive and more dramatic. Period VI in Op. 31, 3, shown in example 16.13, extends for thirty-three mea­ sures, unfolding like a gigantic run-on sentence. Here the in medias res energy of Beethoven's second theme (discussed above in regard to its onset in Period V) remains in play all the way to the end of the exposition. At every point where one might reasonably expect it to close off, it flares up again, first in a series of trills spawned by the trill on the chord seventh of the local dominant (the wrong trill for a strong cadence), then in an ascending group of sixteenth-note arpeggio figures, then in a dramatically slowed and reiterated arpeggio on a six-four, then in a long trill on the second scale degree (the right trill for a cadence) that finally resolves onto a conclusive 8, which is yet followed by a quiet epilogue from 6 to 8 via what Riemann calls a Triole (insertion of three measures for two between strong downbeats). Starting from measure 65, the functions of the added measures make for a telescoping elongation, always mov­ ing a step further from the 8 (note the boldface numbers): 7–8, 7–8, 6–7–8, 5–6, 5–6, 5–6, 4–5–6–7–8, (6–7–8). The second theme is thus heard to touch off a flurry of excited clatter that motors around through the registers via chained trills and prolix arpeggios, delaying the definitive 8 for as long as possible. This process can be heard as humorous, though it is quite unlike the slaphappy comedy of the final expository period of Op. 31, 1, where every other card drawn from the deck is an 8.

Ex. 16.13. Op. 31, 3, Period VI.

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Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas The most dramatically consequential extensions are found in the Tempest sonata. First notice the way, in example 16.14, that Period II reaches a climactic 6 function (the

of

measure 13) and then furiously churns away on the now stalled 6, building up to a tremendous discharge of energy at measure 21. (p. 454) This kind of proliferation of the cadential antepenultimate turns out to be one of the primary means Beethoven will use to effect dramatic delays of a conclusive cadence. (p. 455)

It works because the antepenultimate (often a cadential

) offers an unequivocal sense

that the resolution is right around the corner, such that a continuing delay of that resolu­ tion can be highly dramatic. And because the 6 is a downbeat function, it can be reiterat­ ed more plausibly in Riemann's phrase syntax. Stalling on the penultimate dominant would have a different effect, because it would not be able to build up a charge in the same way.19 The dominant is better at discharging energy, like a switch thrown to com­ plete the circuit. If Riemann's period were laid out as an electrical circuit, the 6 would be a capacitor, building up current until the 7 leads it into a grounding 8. As in Op. 31, 3, the second theme of the Tempest moves in two broad periods all the way to the end of the exposition. This is shown in example 16.15. From the arrival of the sec­ ond theme in measure 41, Beethoven's bassline mostly prolongs the dominant E. It touch­ es down briefly on a root-position A minor at Riemann's 8c, squirms away at the elided beginning of Period V, and is finally brought home to tonic at Riemann's 8g and confirmed in 8h. The action from the first 8 of Period V is as follows: 8, 7–8, 7–8, 7–8, 7–8, 7–8, 8, 8, 7–8. This string of 8s comes close to the amount of 8s featured at the end of the Op. 31, 1 exposition. But the prolonged dominant in the bass makes the repeated figuration and harmonic oscillation between tonic and dominant less about comic repetition (remember the door that wouldn’t stay closed) and more about an implacable process that keeps blowing through proffered 8s like a storm that will not move off.

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Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas

Ex. 16.14. Op. 31, 2, Period II.

Massive expansions like these may help support Riemann's claim to show how Beethoven attained an “ever firmer positioning of the actual structure,” if by “actual structure” he means the primary underlying functions of his prototypical (p. 456) period. The most im­ portant of these is the 8 function, and we can hear how Beethoven's expansions make this function that much more momentous, whether as the high voltage 8 that discharges the gathered energy of an iterated 6, or as the series of 8s that bring a process of great mo­ mentum back to the ground. The stations of the prototype, especially the 6 and 8, are made to carry and distribute the weight of ever larger spans of music. As Riemann sug­ gests, far from tearing this underlying form asunder, Beethoven gets it to support and channel the weightiest, most extended, most consequential musical processes. It is more firmly positioned than ever.

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Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas

Ex. 16.15a. Op. 31, 2, Periods IV and V.

Ex. 16.15b.

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Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas Riemann's crucial analytical distinction between essential and accessory provides him with an analytical interface between his theoretical prototype and actual music. With it he can show what Tovey was later to describe as “the inexhaustible expansive and con­ tractile power of Beethoven's phrase-rhythm.”20 In particular, Riemann's end-oriented prototype excels in profiling the many ways that Beethoven harnesses (p. 458) and ampli­ fies the closural energy of a period, and—by extension—the closural energy of the classi­ cal style. (p. 457)

And yet for Riemann himself, the stakes were even higher than this. The great thrust of his Lebenswerk was to discover and refine a system of musical logic. Riemann hyposta­ tized the jump from upbeat to downbeat as the fundamental spark of music's temporal en­ ergy and thus of his dynamic prototype for the eight-measure period. Downbeats function as points of resolution at many levels within that prototype: at the beat, the measure, every two measures, every four measures, every eight measures. This creates a function­ al rhythmic logic that works along with the functional harmonic logic of T, S, and D. Every measure has a function within the logic of the period, and every harmony has a function within the logic of the T–S–D–T cadential progression. Riemann viewed this logical system as nothing less than natural law revealed by history. In the foreword to his Beethoven analyses, Riemann unveils his Grail for what would be the last time: (p. 459) The continued operation of artistic fantasy in both productive and receptive modes with naturally given categories that have come into existence historically is a fact of our spiritual life whose significance cannot be overestimated. These categories rid artistic creation of every vestige of caprice and make it into a logically neces­ sary imperative. To demonstrate the absurdity of those who despise form and rule, who trumpet the unfettered caprice of artistic production, is the most distin­ guished goal of my labors.21 The greater task undertaken by the Beethoven analyses is thus to rescue musical art from the arbitrary and secure its place as a product of nature revealed through history. And Riemann spares no pains: not a detail is allowed to go astray in Beethoven's determined, determinable, works. Standing at the end of the long nineteenth century, an age that mus­ tered tenacious faith in the determining forces of both history and natural law, Riemann's is easily the most developed, or at least the most discursive, expression of that belief as applied to a body of music. However quixotic such an agenda may sound today, however obsessed Riemann might ap­ pear to us in his need to label every measure in Beethoven with a function, and however inconsistent he must appear to us in his wavering criteria for deriving all those functional designations, a certain flair pour suivre la piste arises whenever we stop to pick up the thread of any of his analyses. The mechanics of Riemann's prototypical period stir our own analytical preoccupations about phrase rhythm in the classical style. To observe

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Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas Hugo Riemann reading between the lines of these piano sonatas is to discover yet anoth­ er way to gauge the transfiguration of that style in the hands of a Beethoven.

Notes: (1.) Hugo Riemann, Vademecum der Phrasierung, 2nd rev. ed. (Berlin, M. Hesse, 1906), 15. “Die Phrasierungsausgaben sowohl wie die Lehrbücher der Phrasierung haben de­ shalb nur denselben Zweck wie die Kommentare schwerverständlicher Dichtungen: die Wege zu weisen, wie man zwischen den Zeilen liest, wie man vom primitiven Ablesen dessen, was da steht, zum wirklichen Verständnis des Sinnes vordringt.” (2.) Wilhelm Seidel, “Riemann und Beethoven,” Hugo Riemann (1849–1919): Musikwis­ senschaftler mit Universalanspruch, ed. Tatjana Böhme-Mehner and Klaus Mehner (Cologne: Böhlau Verlag, 2001), 151. “Sein dreibändiges Buch über die Klaviersonaten Beethovens ist der Versuch, abermals, ein letztes Mal—nun in Gestalt eines monumental­ en Werkes—auf das Moment aufmerksam zu machen, durch das er den Kunstcharakter der Musik im Zenit ihrer Geschichte bestimmt wusste, auf die metrische Gefasstheit und ästhetische Bewältigung eines jeden, auch des kleinsten Details. Das Werk über Beethovens Klaviersonaten war Riemanns Vermächtnis. Er hat es gezeichnet: Er hat unter die letzte Analyse des letzten, dritten Bandes seinen Namen geschrieben.” (3.) Ibid., 150. (4.) Lewis Lockwood, Beethoven: The Music and the Life (New York: Norton, 2003), 137. (5.) Carl Dahlhaus, Geschichte der Musiktheorie: Die Musiktheorie des achtzehnten und neunzehnten Jahrhunderts (Darmstadt: Wissenschaftliche Buchgesellschaft, 1981), 1: 30. Cited in (and translated by) Alexander Rehding, “Riemann's Functions—Beethoven's Function,” (Paper presented at “Tonality in Perspective” conference, London 2008). (6.) Riemann, Vademecum, 42. “[s]chwere Zeiten [sind] im allgemeinen Träger von Har­ moniewirkungen.” (7.) On occasion, Riemann will allow two consecutive 5s or two consecutive 7s, but the first will “back relate” to either the strong 4 or strong 6, while the second will revert to type (as upbeat to 6 or 8). (8.) There's another analytical possibility, used by Riemann on other occasions: one could posit a so-called Takttriole extending from 6 to 8. This is an ad hoc device that expands the normative two-bar space between functional downbeats to a three-bar space. I be­ lieve that Riemann deploys the Takttriole only between 6 and 8, where it functions as a kind of extended wind-up to the cadential 8. In the case of the present theme, such a reading would take the F in the bass as prolonged for three bars (i.e. decorated by upper and lower neighbors, but prolonging the V7 sonority until the cadence in 8a). It may be the case, however, that the prevailing two-bar patterning of this theme feels too strong to Riemann to break with the insertion of a Triole. Page 18 of 20

Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas (9.) Riemann, L. van Beethovens sämtliche Klavier-Solosonaten: Aesthetische und formaltechnische Analyse mit historischen Notizen, 3rd ed. (Berlin: M. Hesse, 1920), 2: 324–325. The B-major theme marks another case in which a Takttriole could be applied. For exam­ ple, why not make the first bar a 2 and then use a Takttriole to make the entire theme re­ solve on an 8? This would also rid us of the inconsistency of reinterpreting a 4–5 as a 5–6. Instead of Riemann's 4–5–6–7/8=5–6–7–8 we would have 2–3–4–5/6–Triole–8. On the other hand, doing so would disturb the even-number subdivisions of the theme as a 4 [2+2] + 4 [2+2] construction (the second half would be [3 + 1]). (10.) Riemann, Große Kompositionslehre (Berlin: W. Spemann 1902), 1: 425. “Die wichtig­ ste Ergänzung der neuen Fassung der Formenlehre bildet nun aber die durchgeführte Un­ terscheidung von eigentlich den Aufbau konstituierenden, entwicklenden Partien und von Einschaltungen.” Emphasis in original. (11.) “. . . Vereinigung einer grösseren Anzahl verschiedener Motive zu einem grösseren Einheitsgebilde. Das moderne Thema ist nicht mehr nur ein Melodiefragment, sondern vielmehr eine in sich mehr oder minder abgeschlossene ganze Melodie.” Emphasis is originally Riemann's. Riemann, Große Kompositionslehre, 414. (12.) Ibid., 415. (13.) The modulation to the surprising key of the second theme, B major, is also the result of a T=S transaction. In this case, Tp (e minor in key of G) is reinterpreted as minor S of B minor, whose prolonged dominant resolves at the last moment to B major. (14.) Riemann, Beethovens Klavier-Solosonaten, 2: 426–427. “Merkwürdig ist dieser An­ fang auch darum, weil er die Tonart der Sonate nicht, wie gewöhnlich, gleich durch den Anfangsakkord verrät, sondern mit einer umständlichen Kadenz, die eine ganze achttak­ tige Periode füllt, zu ihr hinleitet . . .” (15.) Riemann, Große Kompositionslehre, 425. “Die innere Notwendigkeit, mit welcher sich die thematischen Bildungen in der Phantasie auseinander entwickeln, gegeneinander differenzieren und in grösseren Proportionen sich gruppieren, beruht eben auf solchen deutlichen Unterschieden des Wesentlichen und des Beiwerks, des fest Geformten und des loser Gefügten, des eigentlichen musikalischen Geschehens und der zwischen dessen Hauptphasen sich einschaltenden Momente beschaulichen Verweilens.” (16.) “…Lebensskizze des Meisters, welche das imposante Erstarken und Wachsen seiner künstlerischen Potenz ersichtlich macht. Gewiss wird es von Nutzen sein, zu erkennen, dass nicht ein Zerbrechen der Form, sondern vielmehr im Gegenteil ein immer festeres Hinstellen des eigentlichen Gerüstes, um welches das Beiwerk sich ansetzt, für den reifenden Meister charakteristisch ist.” I have relied to some extent on Alexander Rehding's translation of a portion of this quotation, from his paper, “Riemann's Functions —Beethoven's Function.”

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Reading between the Lines: Hugo Riemann and Beethoven's Op. 31 Piano Sonatas (17.) Riemann renumbers this same passage when it reappears in the coda. The 8–9 pairs become 4–5 (back-relating) pairs; he apparently makes this adjustment because he needs to house the next utterance within the same period (though he might have just as easily included those next bars into the succeeding period). Though this adjustment seems an inconsistency, it at least shows that 4–back5 is analogous to 8–9: both situations provide a back-relating echo to a strong downbeat function. (18.) The “peremptory downbeat” could also be described as an upbeat pulled up short. The comedy of the pianist's misaligned hands is only the most local expression of this kind of situation. (19.) The prolonged dominant of a retransition is a different animal. Riemann usually gives such a dominant its own period. (20.) TOVEY, “Sonata Forms,” Musical Articles from the Encyclopaedia Britannica (Oxford: Oxford University Press, 1949), 220. (21.) “Das fortgesetzte Operieren der produktiven wie der rezeptiven künstlerischen Phantasie mit natürlich gegebenen und historisch gewordenen Kategorien, welche das Kunstschaffen jeglicher Willkür entkleiden und es zu einem logisch notwendigen Müssen machen, ist eine Tatsache unseres Seelenlebens, die gar nicht ihrer Bedeutung nach überschätzt werden kann. Die Form- und Regelverächter, die Verkünder einer schranken­ losen Willkür des Kunstschaffens ad absurdum zu führen, ist der vornehmste Zweck meiner Arbeit.” Riemann, Beethovens Klavier-Solosonaten, Vol. 1 (1919) Vorwort (unnum­ bered page).

Scott Burnham

Scott Burnham is Scheide Professor of Music History at Princeton University. He is the author of Beethoven Hero, translator of A. B. Marx, Musical Form in the Age of Beethoven, and coeditor of Beethoven and His World. Forthcoming writings include “Late Styles,” for Rethinking Schumann, and “Intimacy and Impersonality in Late Beethoven,” for New Paths.

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Metric Freedoms in Brahms's Songs: A Translation and Commentary

Metric Freedoms in Brahms's Songs: A Translation and Commentary   Paul Berry The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0017

Abstract and Keywords This article focuses on Riemann's thoughts on the temporal features of music, found in his article “Metric Freedoms in Brahms's Songs”. Riemann's essay is notable, first, be­ cause it presents an almost contemporary account of Brahms's music. Second, it presents what is for Riemann a rare analytical account involving vocal music—a repertoire Rie­ mann generally considers as inferior to absolute music. And third, its dissimilarity to Riemann's more systematic theories of rhythm and meter makes his essay interesting. In this article, the aim is to demonstrate using two examples, how in an individual case, one can clarify musical structure through detailed analysis hence opening up new sources of appreciation. Two universally known songs are used as samples in this article and these are: “Immer leiser wird mein Schlummer,” op. 105/2, and “Schwalbe, sag'mir an,” op. 107/3. In addition, the article also reveals Riemann as a sensitive analyst, responding to issues of declamation, harmony, and notated meter, and offers insights not simply into the music but into Riemann's theoretical and analytical methodologies as well. Keywords: music, Brahms's music, vocal music, musical structure, harmony, notated meter

Metric Freedoms in Brahms's Songs Hugo Rie­ mann in Leipzig The history of song at the turn of the seventeenth century tells of a striking transforma­ tion in the relationship of musical setting (i.e., melodic structure) to the meter of the po­ em set to music. Previously, composers strictly preserved the poem's rhythmic framework in the melodic progression (at least in dance songs, villanelles, etc., which, though admit­ tedly polyphonic, are set note against note), and therefore a succession of identically con­ structed stanzas could be sung without variation to the same music, as still occurs today in the common strophic song. Now, however, Giulio Caccini introduced the sophisticated innovation that made him world-renowned and that, today, differentiates the higher art song from the strophic song, namely, scrupulous responsiveness to the accentual de­ Page 1 of 23

Metric Freedoms in Brahms's Songs: A Translation and Commentary mands and expressive meaning of individual words—parameters that, of course, remain the same across multiple stanzas only in exception cases. He therefore introduced that which differentiates a song by Schumann, for example, from a student song. But he went still further by incorporating all kinds of elongations or accelerations, such as an artful singer knows how to (p. 463) apply in performance to even a simply constructed song, into the musical notation, not with performance indications like ritenuto, stringendo, and the like, which were not in use at the time, but rather with alteration of note values. On this point Caccini himself states, in the preface to the second volume of the Nuove musiche, 1614: “Questa mia maniera di cantar solo, la quale io scrivo giustamente come si canta.”1 In fact, according to our current concepts of meter and declamation, Caccini's manner of notation would entail repeated metric shifts. Nevertheless, he notates almost all of his melodies entirely without metric shifts throughout, with an initial : that is, just as recita­ tive in Italian opera has been notated for more than 200 years, without regard for the po­ sition of verbal accents within the measure. Naturally, the consequence of this notational style was that nowadays Caccini's declamation has been considered abysmal or worse, so that one could hardly comprehend what actually gave him his initial reputation. Since, however, Caccini set himself precisely the task of declaiming better than his predeces­ sors, and since his contemporaries praised him to the skies for his accomplishments in that arena, the blame lay not with him but with the critics of our time, who no longer un­ derstood his intentions. Detailed evidence that this truly is the case appears in the recent­ ly published fourth part of my Handbuch der Musikgeschichte (volume II, part 2: The Age of Figured Bass). A wholly analogous situation to Caccini's has befallen Johannes Brahms, except that he has been misunderstood by a considerable proportion of the musically inclined not after 100 years, but during his own lifetime. Brahms, too, considers it his overriding objective to do justice to the poem in fullest measure; he, too, writes exactly what should be sung, and although he naturally avails himself of performance indications adopted in the inter­ vening years since Caccini, instances are by no means uncommon in which, for example, he expresses a ritardando using longer note values in conflict with the prevailing metric organization. Thus he, too, has not escaped the unfavorable verdict on the part of less in­ sightful listeners, readers, or singers, that he frequently makes errors in declamation. This is certainly a severe allegation against the master whom universal opinion considers the greatest proponent of the Lied after Schumann. His reputation is indeed so firm, and a great many of his songs have sung themselves to sleep in so many hearts,2 that it can seem downright superfluous to argue on his behalf as if for a neglected genius. But when faced with the abstruseness of the formal structure of many Brahms songs whose sub­ limely artful effect is beyond question, it would be evidence of incapacity if modern music theory were to surrender and concede that it was stymied by such songs. In other words: despite all appearances to the contrary, whatever truly functions compellingly and seems artistically well-motivated must be based upon inner regularity, and whatever can be so securely comprehended by intuition must ultimately admit of an explanation.

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Metric Freedoms in Brahms's Songs: A Translation and Commentary The history of music has already provided manifold opportunities to recognize that our continually dance-like, ever-ticking meter is no absolute necessity, for instance in the lyri­ cal strains of antiquity with their lines that shift so often from duple to triple time. Like­ wise, frequent metric shifts speak an intelligible language in the recitative and arias of composers such as Lully, and Franz Liszt in particular resisted rigidly continuous (p. 464) meter in every conceivable circumstance. And yet, to conclude that perhaps the age of precisely ordered meter is over, that except in dance music no strict time need prevail, would be a fatal error. All of the freer forms are to be measured, today as in all times, ac­ cording to the standard of the strictly regular; they obtain their individual quality and their individual effect through their deviation from the norm. The downright abolishment of normative concepts is of no use to anyone—specifically, in the arena of the Lied, nei­ ther to the singer and listener nor to the composer. Therefore, as long as one's power of comprehension does not suffice to recognize the well-proportioned design and immanent logic in Brahms's works, even if only intuitively with the senses, it is more honorable for one to admit that Brahms's structure is discomfiting, that one cannot attain a full appreci­ ation thereof. But one would do better to avoid pronouncing such foolish criticisms as that of poor declamation and, instead, humbly acknowledge that the perceived deficiency inheres not in Brahms, but in one's own abilities. Of course, I cannot hope to clear away all the difficulties that interfere with an immediate understanding of Brahms's intricate ideas all at once in a short essay. I propose nothing more than to demonstrate using two examples how, in an individual case, one can clarify musical structure through detailed analysis and thereby open up new sources of appreci­ ation. Admittedly, anyone who feels compelled to deny the pleasurable values of investi­ gating formal relationships will acquire little taste for my exposition.3 I am selecting two universally known songs: “Immer leiser wird mein Schlummer,” op. 105/2, and “Schwalbe, sag’ mir an,” op. 107/3.4 Brahms-connoisseurs may ask in astonish­ ment what needs to be explained regarding these transparent songs. Granted, they are not among those accused of poor declamation; nevertheless, we will soon see that they contain various issues to be elucidated. For the time being, my foremost concern is to re­ veal the affinity between Brahms's and Caccini's free disposition of metric proportions. However, I will spare the reader parallel passages in Caccini's music; those who take a deeper interest in the question will know to find such passages elsewhere. To begin with, in the opening of op. 105/2 it is striking that the first two melodic lines each take up three measures despite their eight-syllable normal verse and the uniform

continuation of the dotted rhythm:

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Metric Freedoms in Brahms's Songs: A Translation and Commentary Whoever depends completely upon Brahms's barlines will experience a slight unease pro­ duced by the elongation of the rhyming syllables and the long rest. He will be inclined to understand the accompanimental imitation of the conclusion of the first line as an (p. 465) intensification, a repetition of the focal point of the second measure such as Brahms in­ deed loves; but then the great difficulty arises of apprehending the rhythmic organization of the continuation, since after the second line the vocal line takes up the cadential for­ mula instead of the accompaniment, but as a continuation rather than a static echo. By contrast, whoever is not looking at the score will hear only the two identically construct­ ed melodic lines with their points of emphasis defined by the rhymes and will never real­ ize that the two lines each consist of three measures. One should not overlook Brahms's allabreve mark (𝄵). The half note, not the quarter, should be felt as the beat, albeit a slow beat in virtue of the tempo indication, “langsam.” The harmonic motion admittedly places a new chord in conjunction with the first barline, but in such a metric position this chord gives the word “leiser” almost too much accentuation, and in fact the smooth, stepwise motion of the vocal line disguises a single C♯-minor chord that remains undisturbed until the line-ending rhyme. Whoever knows the song hears a significant shift in harmony only at the rhyme, and thus perceives the entire structure not as 𝄵, but as

:

Since the continuation begins with a repetition of the close of the second melodic line (g  ♯, e ♯, f ♯), cut time replaces Page 4 of 23

Metric Freedoms in Brahms's Songs: A Translation and Commentary inconspicuously, and the consequent proceeds in the most natural fashion. An echo of the consequent's cadential formula results in a (p. 466) repetition of the eighth measure. (The rhyme scheme is mutilated by the “über mir” instead of “über mich,” which must certain­ ly be correct, since “bitterlich” rhymes with it later on.)5 It is marvelous how Brahms fills the entire consequent of this first period with the five syllables “zitternd über mich.” Fur­

thermore, Hermann Lingg's stanzaic form gave him no easy task. Compare the following:

Evidently Brahms (or Lingg?) transposed lines 4 and 5 of the first or second stanza, which is a decided improvement in regard to the sense of the poem but disrupts the correspon­ dence in rhyme scheme between the two stanzas. Also worth noting is that the final line of the first stanza comprises nine syllables, while that of the second stanza only five. (Repetitions, for example “Komm, [o komm,] o komme bald,” are suppressed above.) Now let us next observe to what degree the setting of the second stanza confirms our ex­ planation of the structure of the first period. The melodic progression is strongly diver­ gent. The words “Ja, ich werde sterben müssen” are transferred to the higher register of the second line of the first strophe; in fact, ultimately the melodies of the first and second lines are simply interchanged, quite certainly reflecting the subtle consideration that the intensification of the second line would substitute a jealous emotion in place of poignant resignation. The words “Wenn ich bleich und kalt” are displaced by a half note so that the primary accent falls on “bleich,” not on “kalt.” In other respects the period proceeds com­

pletely analogously and has the same cadence as in the first strophe:

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Metric Freedoms in Brahms's Songs: A Translation and Commentary The first phrase of the second period is set entirely identically in both strophes. The aforementioned transposition of the two poetic lines is barely noticeable (admittedly the impure rhyme “Tür—dir” is somewhat exposed because the lines again follow one an­ other directly):6 (p. 467)

The identical structure of these two melodic lines (despite the missing rhyme connection) clearly places them in parallel with lines 1–2 of the first period, even though there the lines were longer and the meter . The consequent of the second period, too, is comprehensible in both strophes as a paral­ lel construction to the consequent of the first period, albeit widely elongated in accor­ dance with the text, which, especially in the first strophe, contains so many more sylla­ bles. I believe I do not err in assuming that the consequent must be interpreted with whole notes as the beat; justification thereof is provided by the harmonies prolonged through two measures of in the second strophe.7 Segmentation into multiple consequents is prohibited in both stro­ phes by the bold harmonic progression based on the succession of multiple motions by minor third, a sequential structure of the most daring kind during which no cadential ef­ fect can occur. Because the subsequent triad consistently occurs in second inversion, each element in the sequence creates the illusion that a cadential motion is being initiat­ ed, namely toward the relative [Parallele] of the parallel [Variante] of the first chord:

Or, on the other hand, another possible interpretation is that a plagal cadence is made on the chord of the Neapolitan sixth relative to the principal key (C♯ minor):

(p. 468)

but instead of being continued tonally, this formula is repeated, descending se­

quentially, so that the complete progression in the first strophe is nothing but:

The analogous portion of the second strophe is far more freely constructed insofar as it does not follow the return path back to the principal key, but rather approaches a ca­ dence in D♭ major through three consecutive ascending minor third motions (E, G, B♭ D♭). Page 6 of 23

Metric Freedoms in Brahms's Songs: A Translation and Commentary D♭ major must be taken as enharmonically identical to C♯ major and as the parallel to the principal key (a major-mode ending). Yet one might arguably assume that Brahms select­ ed this enraptured detour into the utterly remote region of B♭-related keys, which project merely the illusion of tonal coherence (for this is all that enharmonic relations entail in such cases), with especially poetic intent. Moreover, the ascending minor third motive in the vocal line is disposed (all three times in each of the two strophes) such that it begins its ascent from the tone that is chromatically altered in the next harmony, so that accord­ ing to vulgar terminology a cross-related progression emerges (more accurately an inner

metamorphosis of thrilling effect). The progression in the second strophe looks like this:

and thus avoids the mitigatingly explanatory resolutions of the chords. The strong deviations in melodic progression in relation to the first strophe are so well-motivated by the text that no further commentary is necessary. I will simply allude to the suppression of the value of a whole note at the first “Komm,’ o komm.” Perhaps this is related to the shortening of the poetic line (one measure of instead of two measures of ). The consequent of the second period of both strophes looks like this: (p. 469)

Brahms's notation, which is otherwise throughout, employs two measures of time at only one place, namely at the second “weine bitterlich,” where he was compelled to indicate

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Metric Freedoms in Brahms's Songs: A Translation and Commentary in the middle of a measure.8 Whether my metric arrangement is an aid for the singer and player is perhaps debatable; but it is indispensable in exposing the periodic structure in its great regularity.— Circumstances are similar in the Schwalbenlied, except that the meter of the initial vocal motive:9

determined the notated meter for the entire song even though the effective metric pro­ portions change more mercurially here than in op. 105/2.10 The consistent cheerfulness of mood made it possible to set the two stanzas of the poem completely identically, render­ ing continuous notation barely necessary. To begin, for me the meter of the prelude (which returns as an interlude between the two strophes and as a postlude completed by a full cadence) is deeply problematic. This charming ritornello has nothing to do with the melodic material of the song; instead, it stands independently be­ tween the sung stanzas like the prelude/postlude in Beethoven's Bagatelle op. 126/6 or, still more similarly, the streaming triplet theme that occurs between the three presenta­ tions of the main theme. In fact, its meter is not but 𝄴, or rather 𝄵. Here I include only the melody:

The last quarter rest in the score is a Luftpause which I have therefore placed above the double barline; it simply delays the beginning of the vocal line (for which the ancients had the symbol 𝄐). It will be difficult for anyone to deny that one may not pro­ ceed from the prelude to the song in strict tempo; every intelligent accompanist will hold back somewhat in the final measure of the prelude. The falsity of the original notation easily induces a misguided presupposition of well-rounded completion that the prelude (p. 470)

does not really possess:

But only the antecedent of the first period of the song adheres to the meter that now ensues; the consequent flits quite clearly over to in virtue of its motivic construction:

The seven measures in Page 8 of 23

Metric Freedoms in Brahms's Songs: A Translation and Commentary in the original notation thus quite unnecessarily conceal the structural simplicity of the consequent as well as the prelude. Now, however, the real difficulties begin. That the en­ suing interlude foreshadows the next measures of the vocal line is beyond doubt, but in both strophes the verbal stresses reveal that the barlines may not entirely fulfill their

obligations:

The harmonic motion is the deciding factor. Although often enough with Brahms one en­ counters harmonies that begin on weak beats as (syncopated) anticipations and are held over into strong beats, no reason to hypothesize so complex a formal structure exists in

this case. The common listener hears:

that is, a single measure of occurs initially at the transition, and the penultimate measure, with its prescribed ritar­ dando, comprises only two instead of three quarters. One should probably deny that this new structure presents an independent period because the text is too much simply the conclusion and practical application of the previous lines, the harmony provides only a single cadence initiated repeatedly, and the vocal line, too, does not develop with suffi­ (p. 471)

cient independence:

In effect the melody remains fixed on c♯ until the definite descent in the final measure, and the entire section is a single consequent which, beginning after the closing quarter note on “gebaut,” blithely ignores that close and commences with the sixth measure, while emphatically strengthening the latter twice with an elaborated upbeat (5–6). It mat­ ters little whether one prefers to regard the c♯ at the conclusion of the first period as an incomplete measure (only ) or, instead, together with the ensuing three quarters, as a measure of . In any event, one might seriously consider whether the following far freer analysis dis­ closes the underlying sense of the passage:

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Metric Freedoms in Brahms's Songs: A Translation and Commentary Thus, if I outline the entire song together with both stanzas of the text:

No doubt one is in agreement that in many of his songs Brahms proceeds very freely with respect to metric organization (not, of course, in those that depend on a folk­ song-based foundation). But if one truly is in agreement, one should not be content to es­ tablish that Brahms's measures are not proper measures; rather, one should attempt to explore thoroughly what lies hidden beneath the misleading notation. With Brahms it is no different than with Heinrich Albert, for example, and other composers of the seven­ teenth century. When Kretzschmar, in his Geschichte des neuen deutschen Liedes (vol. I, page 28), simply states that “the age of mensuration knows no strong or weak beats, but places emphasis according to the verbal accentuation,”11 he thereby promulgates a highly problematic confusion regarding fundamental rhythmic principles. As is well known, the entire theory of strict composition is based upon the continuous differentiation of strong (p. 472)

and weak beats, which the theorists of the thirteenth century know just as well, and for­ mulate just as precisely, as we do today. Yet the era in which notation relinquished the ca­ pability of using time signatures to differentiate strong and weak beats with consistent accuracy is exactly the era in which so-called mensuration (more properly diverse mensuration, that is, the varied significance of the same series of rhythmic values de­ pending upon the meter signature) became obsolete. If that era places emphasis accord­ ing to the verbal accentuation in contradiction to a meter signature (the true meaning of which is often, unfortunately, simply misunderstood), this means for us today no more than that one can uncover the true metric relationships according to the verbal accentua­ Page 10 of 23

Metric Freedoms in Brahms's Songs: A Translation and Commentary tion. To do so is certainly not easy, and Kretzschmar fails altogether in his attempt to transcribe H. Albert's (see ibid., page 26):12

into “modern form”:

instead of the only correct version:

The advent of the barline (previously familiar only in tablature) in mensural music around 1600 at first had a highly dubious effect: this mechanical means of aligning whatever be­ longs together in a clearly arranged layout, though providing (p. 473) an excellent service for accompanying organists in their assembly of scores for many-voiced works, also di­ verted attention away from rhythmic proportions. It was at least a half century before this evil influence was overcome and, where triple meter obtained, a meter signature that stipulated as much was consistently applied (for a long while in conjunction with the sign 𝄴, as is well known—thus, , , , etc). This evidence may indicate to those less familiar with ancient music that our own age still occasionally comes up against very curious obstacles to the proper appreciation of such music, and that in suffering the effects of the atrophied state of rhythmic theory in musical education, Brahms was not alone.

Commentary The closing couplet from Goethe's sonnet Natur und Kunst could stand as a motto for “Metric Freedoms in Brahms's Songs,” the most detailed and penetrating music-analytic study that Hugo Riemann devoted exclusively to the music of Johannes Brahms.13 Published in 1912 in the Berlin periodical Die Musik, the article was overshadowed from the first by Riemann's contemporaneous Handbuch der Musikgeschichte (the second part of volume II was completed in 1912, the third and final part in 1913). Aside from a recent, largely critical response from the Viennese musicologist Matthias Schmidt,14 “Metric Freedoms” is virtually unknown today and has never been made available in English translation. Yet the problem of metric ambiguity in Brahms's songs exposed fascinating intersections between historical and theoretic concerns within Riemann's broader outlook and provoked one of his most resonant articulations of the role of norms and deviations in music analysis. Moreover, although the article's analyses of individual songs are open to justifiable and familiar criticisms with respect to their treatment of meter and phrase Page 11 of 23

Metric Freedoms in Brahms's Songs: A Translation and Commentary structure, intriguing interpretive insights emerge as well, both in Riemann's arguments themselves and in the subtle inconsistencies in analytic technique that Brahms's songs elicited. In der Beschränkung zeigt sich erst der Meister,

Within constraints the master first arises,

Und das Gesetz nur kann uns Freiheit geben.

And law alone can give us freedom.

“Metric Freedoms” can be divided into three main parts. First, Riemann establishes met­ ric organization as his central concern and places the metric practice of Brahms's songs into a sweeping historical context that stretches back to the beginning of the seventeenth century. In the process, he provides a concise defense of normative comparison as the guiding principle of music analysis. Next, the bulk of the article provides comprehensive accounts of meter, phrase structure, (p. 474) and harmony in two of Brahms's best-known late songs, Immer leiser wird mein Schlummer, op. 105/2, and Das Mädchen spricht, op. 107/3.15 These analyses are as much demonstrations of method as explications of the indi­ vidual songs at hand. Finally, Riemann closes with a brief return to broader historical con­ cerns coupled with a polemical attack on contemporary practices of transcription from early seventeenth-century sources, particularly the work of Hermann Kretzschmar. In combination, these three sections articulate multiple, partially overlapping goals for the article as a whole. Indispensable to Riemann's project but almost inevitably alienating to twenty-first-centu­ ry readers are the scope of his argument and historical breadth of his evidence. The arti­ cle begins and ends not with Brahms's songs, but with the monodies of Giulio Caccini and the Arien of Heinrich Albert. Framing the article with references to solo vocal music from the early Baroque serves at least two distinct purposes. On a pragmatic level, emphasiz­ ing the music-historical importance of Caccini's melodic style and text-setting principles enables Riemann to promote the latest volume in his major scholarly project of the early 1910s: the introduction to “Metric Freedoms” is sprinkled with teasers, both subtle and overt, for “the recently published fourth part of my Handbuch der Musikgeschichte,” which opens with nearly thirty pages of detailed historical and music-analytic commen­ tary on Caccini's music and devotes substantial attention to Albert as well.16 At the same time, evaluating Brahms's songs in the context of early monody also lends credence to a central tenet of Riemann's music-analytic method. Having presented both Caccini and Brahms as misunderstood by conventional listeners on the basis of the text declamation implied by traditional metric notation, Riemann employs this transhistorical similarity as implicit support for his (now notoriously) normative approach to metric organization: “All of the freer forms are to be measured, today as in all times, according to the standard of the strictly regular; they obtain their individual quality and their individual effect through

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Metric Freedoms in Brahms's Songs: A Translation and Commentary their deviation from the norm.”17 Historical synchronicity thus justifies normative analy­ sis and an explicitly conservative musical aesthetic. In turn, Riemann's conclusion relates the introduction's prescriptions for analytic method to his general theory of rhythm and meter, drawing once more upon evidence from a vast music-historical continuum. Indeed, his brief discussion of “the continuous differentiation of strong and weak beats” relies upon adumbrations of Auftaktigkeit from as early as the thirteenth century. “Metric Freedoms” closes, however, where it began, in detailed criti­ cal engagement with the problem of transcription from early seventeenth-century sources in which both meter signature and measure lines contradict proper patterns of text decla­ mation. Riemann's ostensible purpose in his conclusion is to counteract the widely shared but, from his perspective, erroneous assumption that mensural music simply follows the accentual patterns of its text rather than systematically differentiating strong beats from weak. Yet his ultimate target is more specific: he explicitly attacks Hermann Kretzschmar's Geschichte des neuen deutschen Liedes as a prominent source of “confu­ sion regarding fundamental rhythmic principles,” reproduces and criticizes (p. 475) Kretzschmar's re-barred transcription of the opening vocal phrase of an Arie by Albert, and proposes an alternative version. Ironically, a misprint in Kretzschmar's reproduction of Albert's original notation leads Riemann's proposed alternative astray.18 Even neglect­ ing the misprint and its effects, the alternative version is radical in its disregard for pro­ portionate rhythmic values in favor of metric continuity, but an accurate transcription of Albert's Arie was almost surely not Riemann's primary motivation in singling out Kret­ zschmar for polemical assault. The two men had recently engaged in public debate re­ garding the foundational goals of musicological inquiry itself,19 which must have made the opportunity to disparage Kretzschmar's latest book difficult to ignore. If Riemann's introduction and conclusion locate the thesis of his article firmly in the con­ text of his overarching music-theoretic concerns and promote his perspective at the ex­ pense of an academic rival, his analyses of Immer leiser and Das Mädchen spricht provide focused demonstrations of the applicability and implications of his rhythmic, formal, and harmonic theories. The two analyses proceed analogously, beginning from small-scale metric disruptions in the initial measures of their respective songs and gradually present­ ing integrated interpretations of meter, phrase structure, and salient harmonic detail that illuminate each song as a coherent whole. Both songs are explicated in terms of their de­ viations from a strophic norm. Immer leiser consists of two highly varied strophes in which only four measures of music return unchanged in the same position relative to large-scale formal boundaries, whereas Das Mädchen spricht is almost literally strophic (only slight alterations in the figuration of the piano accompaniment prevented Brahms from simply using repeat signs). Nevertheless, in both cases, Riemann not only compares successive strophes explicitly, but also represents their melodies and pivotal harmonies atop one another in graphic notation. This technique is borrowed directly from his ap­ proach to Caccini's vocal music in the Handbuch der Musikgeschichte, where it is applied progressively to a succession of two monodies in which poetic stanzas are set with in­ creasing degrees of musical variation.20 For the straightforwardly strophic Das Mädchen spricht, Riemann's analytic premise is inevitable and its implications trivial, but in the Page 13 of 23

Metric Freedoms in Brahms's Songs: A Translation and Commentary case of Immer leiser, systematic evaluation according to a strophic norm opens up an in­ terpretive space in which Brahms's compositional choices can be related productively to the text. To today's readers, Riemann's prose is perhaps most convincing in those por­ tions of his treatment of Immer leiser that focus on poetic motivations for harmonic and melodic deviations from strict strophic form. Indeed, given the preponderance of modi­ fied strophic songs in Brahms's output from the 1880s, “Metric Freedoms” arguably pro­ vides a fresh point of departure for further scholarship. It is when one turns from large-scale form to small-scale metric, harmonic, and textual or­ ganization that Brahms's songs present significant obstacles for Riemann's method and, occasionally, provoke revealing inconsistencies or misreadings. This is not to say that Riemann's normative approach to meter and phrase rhythm is consistently ill-suited to the songs at hand. In fact, in confronting the first period of each song, “Metric Freedoms” emerges largely triumphant, arguing plausibly on the (p. 476) basis of rhythmic and har­ monic evidence that the vocal line's initial phrase can be parsed in both songs as consist­ ing of eight downbeats, the first four in triple meter and the next four in duple. While Rie­ mann himself does not comment on this striking similarity between two otherwise strong­ ly divergent songs, his interpretation accounts in both cases for subtle but unsettling gaps between the acts of hearing the songs and reading the scores as notated. One can at least begin to surmise how the theorist would have approached metric organization in works like Brahms's Auf dem Kirchhofe, op. 105/4, which alternates explicitly between triple and duple meter in each of its two strophes. Elsewhere in each song, however, Riemann's unswerving commitment to the eight-measure period leads to fascinating but highly abstract analyses. For instance, at the conclusion of his truly virtuosic account of the final vocal cadence of Immer leiser, one is forced to wonder—along with countless readers of his analyses of works by Bach, Beethoven, and others—how passages from a single song can be reconceived in , , , and finally without stretching the concept of meter beyond the limits of audibility. Moreover, Riemann's analytic figures are sometimes revealing in their unspoken alter­ ations of Brahms's original compositional fabric. His reproductions of the melodic line of the prelude to Das Mädchen spricht dutifully include Brahms's final quarter rest, placing it in brackets above the last bar line, but tacitly shift the position of a crescendo to coin­ cide with the newly positioned downbeat generated by reconceiving the passage in instead of . Along similar lines, the penultimate quarter rest in the reduction of the postlude to the same song is entirely spurious but analytically convenient: its appearance delays the final tonic triad until the downbeat of the final measure in the newly notated time (see Example 17.17). Misreadings such as these may frustrate twenty-first-century readers, but they preserve instructive traces of their author's habits of mind. A similar mixture of plausible analysis and creatively retrofitted evidence marks Riemann's ap­ proach to harmony and voice-leading. His interpretations of the first three beats of Immer Page 14 of 23

Metric Freedoms in Brahms's Songs: A Translation and Commentary leiser and measures 11–16 of Das Mädchen spricht both argue convincingly on behalf of a static, underlying harmonic or contrapuntal structure projected across a changing mo­ tivic surface. Schenkerian or otherwise prolongationally oriented readers should bear in mind, however, that the perception of such underlying structures is not an end in itself, but instead a key component in Riemann's subsequent metric interpretations of the pas­ sages in question.21 In “Metric Freedoms,” methodological pluralism is nearly always used as a tool in the service of music-analytic orthodoxy. At two points in Immer leiser, Brahms's harmonic and contrapuntal practice forces Rie­ mann outside of purely theoretic application and into metaphorical description. Measures 15–20 and 42–47 consist of chromatic sequences featuring harmonic motion by descend­ ing whole tone and ascending minor third, respectively. Riemann explains: The ascending minor third motive in the vocal line is disposed (all three times in each of the two strophes) such that it begins its ascent from the tone that is chro­ matically altered in the next harmony, so that according to vulgar (p. 477) terminol­ ogy a cross-related progression emerges (more accurately an inner metamorpho­ sis of thrilling effect).22 The effect he so admires exploits a principle of voice-leading expounded, among other places, in his own Handbuch der Harmonielehre, published twenty-five years prior in 1887. In describing the voice-leading proper to a Quintwechsel (the shift from major to parallel minor or vice versa), the Handbuch explains: “It is important, however, that the transfer to the chromatically altered tone occurs in the same voice, or at least in the same octave register.”23 The sequential progressions in Immer leiser combine a Quintwechsel with a simultaneous Parallele or Leittonwechsel, making the vocal line's leap away from the chromatically altered tone all the more striking. As it happens, Brahms himself owned a copy of Riemann's Handbuch der Harmonielehre, which is preserved in the archives of the Gesellschaft der Musikfreunde in Vienna. Aside from annotations pointing out printer's errors, the copy is completely devoid of marks in pencil or ink, with a single ex­ ception: in the passage above, Brahms underlined “the same voice” (derselben Stimme) in pencil and placed an exclamation point in the margin. Given when the volume was pub­ lished, this annotation must have been made at least a year after Immer leiser was com­ posed in August 1886.24 Nevertheless, the mark registers Brahms's interest in the specif­ ic voice-leading patterns associated with chromatic progressions and hints that Riemann's analytical approach toward his sequences might have resonated with his own notions of musical syntax. Ultimately, however, neither Riemann nor Brahms's own acquaintances could come to terms with the composer's harmonic practice in the second of the song's chromatic se­ quences, which consists of three successive triads in second inversion. In “Metric Free­ doms,” Riemann responds to Brahms's thoroughly unorthodox voice-leading with another tacit correction, interpolating root position triads between the second inversion triads in his harmonic reduction of the passage in an effort to mitigate its blatant parallel octaves (see Example 17.8). Brahms's friends were less tactful. Having encountered the song in Page 15 of 23

Metric Freedoms in Brahms's Songs: A Translation and Commentary manuscript nearly two years before its publication, Elisabeth von Herzogenberg confront­ ed the composer with her impressions in a letter dated December 2, 1886: Are the successive six-four chords in the C♯-minor song really all right with you, especially in the second version at the end—G major, B♭ major, D♭ major right af­ ter one another and blatant six-four chords; have you ever actually done anything like that before? I truly don’t know an equally dreadful place in all of your music, and I imagine you yourself are still seeking another expression for the impulsive yearning of that passage. What you want is certainly clear, but how you want it is not as beautiful as Brahms otherwise is, and something in me positively says “Ow!” there. And it's too bad about the so beautifully fading, twilit, tender song which suddenly boxes one's ears.25 Brahms's reply hinted that he would consider her criticisms,26 although the published version of the song demonstrates his eventual decision to proceed with his original plan. With respect to the more radical of the two chromatic sequences, then, Riemann's unspo­ ken revisions to Brahms's harmonic practice extend a tradition of negative reception that began among the composer's own friends. More important to Riemann's argument, however, is another strain of criticism di­ rected toward Brahms's songs from both inside and outside his circle: the accusation of poor text declamation. Shortly after the composer's death, partisans of Wolf were joined by defectors from among Brahms's acquaintances in attacking the musical scansion of many of his best-known songs. Hermann Levi, for instance, pointed out the purportedly flawed text declamation of five songs, including the consistently popular Wie bist du meine Königin, op. 32/9, in a series of letters to a mutual friend, the great baritone and Lieder-singer Julius Stockhausen, in 1899.27 “Metric Freedoms” is ostensibly designed to counteract precisely such negative assessments of Brahms's text accentuation. Yet the ar­ ticle actually recapitulates the problem of scansion in Brahms's songs rather than ad­ dressing it head on. Although Riemann claims that neither of his two chosen analytic case studies has been accused of poor declamation, he often relies upon the assumed percep­ tion of questionable verbal accentuation to support his arguments in favor of reconsider­ ing Brahms's metric organization. In fact, questionable declamation provides one of the two primary methods of relating music and text in “Metric Freedoms.” The other method begins with textual structure conceived entirely independently from its musical setting. (p. 478)

Here, in venturing outside the purview of music analysis altogether and entering the realm of poetics, Riemann proposes what is undoubtedly his most inflexible and easily re­ futed reading of either words or music. Extrapolating from his strictly strophic model of song composition, he argues that the two stanzas of Hermann Lingg's text for Immer leis­ er must have originally employed identical rhyme schemes and that composer or poet must have subsequently changed a pronoun and revised the order of lines in either the first or the second stanza of the poem. Leaving aside the fact that Riemann's substitute pronoun borders on grammatical nonsense, comparison with the poem in the version in which Brahms encountered it thoroughly vitiates his reading. (The texts of Immer leiser

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Metric Freedoms in Brahms's Songs: A Translation and Commentary wird mein Schlummer and Das Mädchen spricht are provided in an appendix in the ver­ sions with which Brahms was familiar, along with my translations.) Taken to its logical extreme, then, the normative analytic approach advocated throughout “Metric Freedoms” ends in absurdity. Nevertheless, even with respect to poetic structure, documentary evidence highlights tantalizing affinities between Riemann's analytic method and Brahms's compositional priorities. Preserved in the Stadtbibliothek in Vienna are three of the composer's handwritten collections of song texts. The final page of one of these collections presents the text to Immer leiser in ink, copied with Brahms's customary accuracy from the published version of the poem.28 Like other poems in his collections, the handwritten text also bears witness to retrospective editing. In addition to adding a large X in blue pencil (indicating, as in many other instances, that he had completed a setting of the poem in question), Brahms amended the text itself in gray pencil, crossing out the second “e” in the final words of lines 11 and 13. “Wehen” and “sehen” thus be­ come “wehn” and “sehn.” He also linked and highlighted his revisions by drawing a verti­ cal stroke in the right hand margin alongside lines 11–13, placing question marks at ei­ ther end of the stroke and the letter “u” in between (standing for “und”), and prefacing the marginalia (p. 479) with the symbol “NB.” The cumulative effect of these annotations is an equivocal but powerful reminder to replace the final syllables of lines 11 and 13 with silent contractions, as in the published version of the song. The revision renders the first six lines of the poem's two stanzas precisely equivalent in terms of their accentuation pat­ terns. In other words, part of Brahms's engagement with Lingg's poem consisted in rec­ ognizing and rectifying a potential lack of rhythmic symmetry between its two stanzas. His song itself entailed a process of normative structural comparison no less deliberate than Riemann's, if not as sweeping in its implications. In the end, “Metric Freedoms” occupies an unusual position among Riemann's analytic studies. Read skeptically as a collection of related music-analytic and historical truth claims, the theorist's arguments are susceptible to a wide array of familiar and often welldeserved objections. But understood sympathetically as the record of a paradigmatic con­ frontation between method and material, the article reveals unique and intriguing reso­ nances between Riemann's analytic approach and Brahms's compositional practice. In text-interpretive stance as well as harmonic and metric details, Brahms's late songs pro­ vided compositional constraints peculiarly appropriate to Riemann's problematic but mas­ terful blend of normative analysis and broad-based historical comparison. The results are fascinating, frustrating, and ultimately rewarding.

Notes on the Translation and Acknowledge­ ments Following Riemann's own principles of transcription from early sources, my translation adheres as strictly as possible to the large-scale sentence structure of the original article, while altering small-scale organization where necessary for the sake of clarity. The words Lied and Strophe are each used in several senses in “Metric Freedoms.” I have translated Page 17 of 23

Metric Freedoms in Brahms's Songs: A Translation and Commentary Lied as “song” except where it refers to the genre of the nineteenth-century German art song, in which case I have retained the German; throughout the translation, the word “strophe” refers exclusively to repeated or varied musical units, while the word “stanza” is used strictly with respect to versification. Riemann's article avoids footnotes entirely in favor of in-text citations. Thus, all parenthetical and bracketed insertions in the transla­ tion are Riemann's own, and my explanatory and editorial remarks are confined to foot­ notes. Like most German prose from the nineteenth and early twentieth centuries, arti­ cles in Die Musik indicated special emphasis in Sperrschrift, by adjusting the amount of space between individual letters of important words; I have preserved such emphases in the translation through the use of italics. Riemann's article further highlights many names and titles, no doubt to assist the reader scanning the text for key words; the use of these emphases has been carefully adjusted to modern-day standards. My (p. 480) thanks are due to Professor Otto Biba of the Gesellschaft der Musikfreunde in Wien and the staff of the Handschriftenabteilung at the Wiener Stadtbibliothek for their kind assistance in locating relevant materials.

Appendix: Brahms's Song Texts The following song texts appear here in the versions in which Brahms encountered them, transcribed directly from their original publications and translated as literally as possible within the broad confines of their respective meters. Brahms found “Immer leiser wird mein Schlummer” under the title “Lied” in Hermann Lingg's “Vermischte Gedichte” (“Miscellaneous Poems”), a collection of individually titled poems which ap­ peared, in versions of varying length, in all editions of Lingg's Gedichte. Multiple editions of the Gedichte reproduce the poem identically, and no copy of the publication survives among the remnants of Brahms's library in the archives of the Gesellschaft der Musikfre­ unde in Vienna, making it impossible to determine definitively which edition was his source.29 Immer leiser wird mein Schlummer,

Ever lighter grows my slumber,

Nur wie Schleier liegt mein Kummer

Only sorrow lies above me

Zitternd über mir.

Trembling, like a veil.

Oft im Traume hör’ ich dich

Often in my dreams I hear you

Rufen draus vor meiner Thür,

Calling from outside my door,

Niemand wacht und öffnet dir,

No one wakes and opens for you,

Ich erwach’ und weine bitterlich.

I awaken weeping bitterly.

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Metric Freedoms in Brahms's Songs: A Translation and Commentary Ja, ich werde sterben müssen,

Yes, it's true I’ll have to die,

Eine andre wirst du küssen,

You will kiss another woman

Wenn ich bleich und kalt,

When I’m pale and cold,

Eh die Maienlüfte wehen,

Ere the May-time breeze is blowing,

Eh die Drossel singt im Wald;

Ere the thrush sings in the wood;

Willst du mich noch einmal sehen,

If you want once more to see me,

Komm, o komme bald!

Come, o come soon!

Brahms's source for “Das Mädchen spricht” was Otto Friedrich Gruppe's “Gedichte” of 1835, which presents the poem as the fifth of eleven untitled texts that appeared under the heading “Das Mädchen spricht.”30 Brahms evidently considered Gruppe's collective ti­ tle an appropriate clarification of tone and poetic speaker for the particular poem at hand, and used it as the title for his song. His copy of Gruppe's Gedichte survives in the archives of the Gesellschaft der Musikfreunde; it bears no annotations in pencil or ink. (p. 481)

Schwalbe, sag mir an,

Swallow, let me know,

Ist's dein alter Mann

Is it your old husband

Mit dem du's Nest gebaut,

With whom you make your nest,

Oder hast du jüngst erst

Or have you just now

Dich ihm vertraut?

Given him your heart?

Sag,’ was zwitschert ihr,

Tell me what you twitter,

Sag,’ was flüstert ihr

Tell me what you whisper

Des Morgens so vertraut?

So privately this morn?

Gelt, du bist wohl auch noch

Ha! You, too, have not long

Nicht lange Braut?

Been a bride?

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Metric Freedoms in Brahms's Songs: A Translation and Commentary

Notes: (1.) [Giulio Caccini, Nuove musiche e nuova maniera di scriverle (Florence: Pignoni, 1614), 2. Emphasis in Riemann's original. Caccini's original publication includes a comma after “giustamente,” and may be rendered in English as follows: “This my style of solo singing, which I write exactly as it is sung.”] (2.) [Riemann's choice of verb, sich einsingen, reflects the high value placed upon Brahms's lullabies, most famously the Wiegenlied, op. 49/4, and Geistliches Wiegenlied, op. 91/2, in both contemporary and posthumous evaluation of his solo songs.] (3.) [Riemann's term, Lustwerte, carries the connotations of a philosophical category of value; one might also translate it as “pleasure values” or even “hedonistic values.”] (4.) [Here and for most of his article, Riemann avoids the title under which op. 107/3 was published (“Das Mädchen spricht”) in favor of the first line of the poem, which he some­ times reduces to “Schwalbenlied.”] (5.) [Here and in the following paragraph, Riemann's suggested emendations of Lingg's text are erroneous. In addition, throughout his article Riemann's musical and textual ex­ amples diverge from Linng's poem in subtler and, one assumes, less intentional ways, in­ cluding the consistent but spurious substitution of “schwebt” for “liegt” in line 2. For a transcription and translation of the text in its original published version, see the Appen­ dix.] (6.) [The distinction between pure and impure rhymes was a special preoccupation in German poetics of the early 20th century. A mid-century redaction of the principal con­ cerns involved can be found in Wolfgang Kayser, Kleine deutsche Versschule (Bern: Francke, 1946; 25th ed., Tübingen: Francke, 1995), 83–87.] (7.) [See measures 43–44 and 45–46.] (8.) [That is, in the middle of one of Riemann's reconceived measures. Brahms's change of meter occurs on the downbeat of measure 23 in the published score.] (9.) [Riemann's musical examples throughout his analysis of op. 107/3 show that he was familiar with the song not in its original key of A major but in the F♯-major transposition for low voice, which was first published in October 1888. See Margit McCorkle, Johannes Brahms: Thematisch-bibliographisches Werkverzeichnis, ed. in collaboration with Donald M. McCorkle (München: Henle, 1984), 434.] (10.) [Riemann's sentence refers to op. 105/2 as the “Mädchenlied”; in order to avoid con­ fusion with “Das Mädchen spricht,” I have substituted the opus number instead.] (11.) [Kretzschmar, Geschichte des neuen deutschen Liedes, I. Teil: von Albert bis Zelter (Leipzig: Breitkopf und Härtel, 1911), 28: “Die Mensuralzeit kennt keine guten und schlechten Taktteilen, sondern betont nach den Wortakzenten.” Emphasis in the original.] Page 20 of 23

Metric Freedoms in Brahms's Songs: A Translation and Commentary (12.) [The stem on the second note in the third measure is spurious. The error stems from Kretzschmar's Geschichte des neuen deutschen Liedes, 26; compare to Hans Joachim Moser, ed., Heinrich Albert: Arien I, Denkmäler deutscher Tonkunst, vol. 12, rev. ed. (Wiesbaden: Breitkopf und Härtel, 1958), 15.] (13.) Hugo Riemann, “Die Taktfreiheiten in Brahms’ Lieder,” Die Musik 12.1 (October 1912): 10–21. Two listening guides to the Third and Fourth Symphonies, completed short­ ly before Brahms's death, are the only other analytic studies in which Riemann focused exclusively on Brahms's works. As befits their intended audiences, these listening guides identify primary thematic material and trace its recurrences throughout the large-scale formal outlines of each movement. Completed before Riemann had fully developed his mature approach to meter and harmony, neither guide engages his music-theoretic as­ sumptions or addresses harmony and phrase structure in detail comparable to that of “Freedoms of Meter.” First published separately in the late 1890s, they were brought to­ gether in August Morin, Johannes Brahms: Erläuterung seiner bedeutendsten Werke von C. Beyer, R. Heuberger, Prof. J. Knorr, Dr. H. Riemann, Prof. J. Sittard, K. Söhle und Musikdir. G. H. Witte. Nebst einer Darstellung seines Lebensganges mit besonderer Berücksichtigung seiner Werke (Frankfurt am Main: Bechhold, n.d.), 97–132; Morin's vol­ ume was published just after Brahms's death, in either 1897 or 1898. Both of Riemann's listening guides were later reprinted in Meisterführer Nr. 3. Johannes Brahms: Sym­ phonien und andere Orchesterwerke. Erläutert von I. Knorr, H. Riemann, J. Sittard nebst einer Einleitung von A. Morin (Berlin: Schlesinger; Vienna: Haslinger, n.d.), 60–95, with minimal editorial changes, the most obvious being the removal of a sentence expressing hope for a “most longingly awaited fifth symphony”; compare to Morin, Johannes Brahms, 132. Susan Gillespie has provided an English translation of the guide to the Fourth Sym­ phony in Kenneth Hull, ed., Johannes Brahms Symphony No. 4 in E Minor Op. 98: Authori­ tative Score, Background, Context, Criticism, Analysis (New York: Norton, 2000), 200– 213. (14.) Matthias Schmidt, “Syntax und System: Brahms’ Taktbehandlung in der Kritik Hugo Riemanns,” Studien zur Musikwissenschaft: Beihefte der Denkmäler der Tonkunst in Österreich 48 (2002): 413–438. (15.) Throughout most of “Metric Freedoms,” Riemann avoids Brahms's title for Op. 107/3 in favor of the first line of the poem: “Schwalbe, sag’ mir an,” sometimes shortened to “Schwalbenlied.” (16.) Riemann, “Die Taktfreiheiten,” 10; Handbuch der Musikgeschichte 2, 2 (Leipzig: Breitkopf und Härtel, 1912), 1–29 and 330–337. (17.) Riemann, “Die Taktfreiheiten,” 11: “Alle freieren Bildungen werden heute wie zu allen Zeiten am Maße der streng regulären zu messen sein und erhalten eben durch ihre Abweichung von den schematischen ihren Sonderwert und ihre Sonderwirkung.”

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Metric Freedoms in Brahms's Songs: A Translation and Commentary (18.) Compare Hermann Kretzschmar, Geschichte des neuen deutschen Liedes, I. Teil: von Albert bis Zelter (Leipzig: Breitkopf und Härtel, 1911), 26, and Riemann, “Die Taktfrei­ heiten,” 21, with Hans Joachim Moser, ed., Heinrich Albert: Arien I, Denkmäler deutscher Tonkunst, vol. 12, rev. ed. (Wiesbaden: Breitkopf und Härtel, 1958), 15. The misprint con­ sists of a stem that erroneously transforms the second note of the third measure of origi­ nal notation from a breve into a long. Kretzschmar himself completed the introduction for the original edition of the Denkmäler publication of Albert's works in 1903. Like his later volume on the history of German song, Kretzschmar's introduction singles out Albert's Mein Kind, dich müssen Leute lieben as a demonstration of his transcription technique and philosophy, but the introduction reproduces the original rhythmic notation accurately and provides a different transcription into modern notation. See Moser, Heinrich Albert: Arien I, xix. (19.) See Achim Heidenreich, “ ‘Die Ungeheuerlichkeit dieser Art Hermeneutik’: Ein Dis­ put zwischen Hugo Riemann und Hermann Kretzschmar,” in Tatjana Böhme-Mehner and Klaus Mehner, eds., Hugo Riemann (1849–1919): Musikwissenschaftler mit Univer­ salanspruch (Cologne: Böhlau, 2001), 153–157. (20.) Riemann, Handbuch der Musikgeschichte 2, 2: 20–28. (21.) On the other hand, the role of Riemann's quasi-prolongational interpretations in his broader metrical account of the songs is not in itself sufficient cause to dismiss those in­ terpretations. In the only recent response to “Metric Freedoms,” Matthias Schmidt criti­ cizes Riemann's reading of the first three beats of Immer leiser, claiming that instead of maintaining tonic harmony throughout, the passage projects a shift to the subdominant in measure 1 (Schmidt, “Syntax und System,” 428). I find Schmidt's account thoroughly un­ convincing, but the mere fact that one can reasonably disagree demonstrates the contin­ ued relevance of Riemann's study. For a more promising alternative to Riemann's ac­ count, see David Beach, “The Functions of the Six-Four Chord in Tonal Music,” Journal of Music Theory 11 (1967): 2–31; Immer leiser is addressed on pp. 20–21. (22.) Riemann, “Die Taktfreiheiten,” 16: “Ist doch auch das Kleinterzmotiv der Singstimme (je dreimal in jeder der beiden Strophen) so geartet, daß es sich von dem Tone nach oben wegwendet, der in der folgenden Harmonie chromatisch verändert ist, so daß also nach vulgärer Terminologie eine querständige Führung (richtiger eine innerliche Umwandlung von packender Wirkung) entsteht.” (23.) Riemann, Handbuch der Harmonielehre: 2. vermehrte Auflage der Skizze einer neuen Methode der Harmonielehre (Leipzig: Breitkopf und Härtel, 1887), 41: “Es ist aber von Wichtigkeit, dass der Übergang in den chromatisch veränderten Ton in derselben Stimme geschieht, oder wenigstens in derselben Oktavlage.” Emphasis in the original. (24.) For details regarding the date of composition for Immer leiser, see Margit McCorkle, Johannes Brahms: Thematisch-bibliographisches Werkverzeichnis, ed. in collaboration with Donald M. McCorkle (Munich: Henle, 1984), 426.

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Metric Freedoms in Brahms's Songs: A Translation and Commentary (25.) Johannes Brahms and Elisabet and Heinrich von Herzogenberg, Johannes Brahms im Briefwechsel mit Heinrich und Elisabet von Herzogenberg, ed. Max Kalbeck (Berlin: Deutsche Brahms-Gesellschaft, 1907), 2: 132–133: “Sind Ihnen wirklich die auf einander folgenden Quart-Sext-Akkorde im cis moll-Liede recht, besonders in der zweiten Fassung am Schluß, G dur, B dur, Des dur hintereinander und lauter Quart-Sext-Akkorde; haben Sie so etwas eigentlich je sonst gemacht? Ich weiß gar keine ähnliche grausame Stelle in Ihrer ganzen Musik und bilde mir ein, Sie suchen noch selber nach einem anderen Aus­ druck für die treibende Sehnsucht jener Stelle; was Sie wollen, ist gewiß klar; aber wie Sie's wollen, ist nicht so schön, wie sonst Brahms ist, und etwas in mir sagt förmlich Au! dabei; und wie schade um das so schön dahindämmernde weiche Lied, das einem plöt­ zlich Ohrfeigen erteilt.” Emphases in the original. (26.) Brahms/Herzogenberg, Briefwechsel, 2: 135. (27.) Julia Wirth (née Stockhausen), Julius Stockhausen, der Sänger des deutschen Liedes: Nach Dokumenten seiner Zeit (Frankfurt am Main: Englert und Schlosser, 1927), 482– 485. (28.) The collection in question is catalogued as H. I. N. 55730; the copy of Lingg's text occurs on folio 30r. (29.) Margit McCorkle pinpoints the third edition (published in 1857) as Brahms's source; see Werkverzeichnis, 426. Since McCorkle provides no direct evidence supporting her as­ sertion, one must simply trust that she had access to a volume whose whereabouts are now unknown. Eduard Hanslick seems to have brought the poem in question to Brahms's attention (see Brahms/Herzogenberg, Briefwechsel, 2: 135n); perhaps he lent Brahms a copy of Lingg's Gedichte as well. In any case, the transcription here follows the poem as found in Hermann Lingg, Gedichte. Dritte vermehrte Auflage (Stuttgart and Augsburg: Cotta, 1857), 56. The poem is reproduced identically in both the second and fourth edi­ tions (published by Cotta in 1855 and 1860, respectively). (30.) See Otto Friedrich Gruppe, Gedichte (Berlin: Reimer, 1835), 55.

Paul Berry

Paul Berry is an assistant professor (adjunct) of music history at the Yale School of Music. His current work centers on historical, critical, and analytic approaches to nineteenth-century chamber music and song, particularly that of Brahms, Schubert, and Schumann. Related focuses include rhetorical studies, connections between bi­ ography and historiography, and theorizing and contextualizing the kinesthetics of performance.

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Riemannian Analytical Values, Paleo- and Neo-

Riemannian Analytical Values, Paleo- and Neo-   Steven Rings The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0018

Abstract and Keywords This article considers Riemann's analysis of Schubert's triadic but highly chromatic Gbmajor Impromptu. This article compares Riemann's own analysis of the work with neoRiemannian view inspired by the writings of Richard Cohn, assessing the differences in analytical methodology and technology, and locating those differences with the divergent ideologies of the two approaches. In this article, the central focus is not on the analytical technologies, but rather on the assumptions and values that underlie the distinct analyti­ cal perspectives. It focuses on the analytical values, with a focus on the methodological and ethical contrasts between these two approaches. The article ends by considering the ways in which a technical rapprochement between the theories might open ethical hori­ zons and provide new ways to value music through Riemann-inspired analytical activity. Keywords: Schubert, Gb-major Impromptu, Richard Cohn, analytical methodology, technology, methodological contrasts, ethical contrasts

I

Ex. 18.1. Schubert, Impromptu in G♭, D. 899, no. 3, mm. 78–82.

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Riemannian Analytical Values, Paleo- and NeoThe passage in example 18.1—from the coda of Schubert's Impromptu in G♭—provides a useful point of departure. The tonal bottom seems to drop out of the music here: in just over two measures, we progress from the tonic G♭ major, through B minor, to G minor.1 A menacing bass trill on C in measure 79 announces the imminent arrival of the latter—we hear the G-minor chord coming before it sounds. (Such a menacing trill could hardly be preparing us for G major.) Miraculously, this forecasting does not lessen the shock of the chord when it actu­ ally arrives. (p. 488)

The G-minor chord of course admits of a tonal interpretation: it is the minor Neapolitan, enharmonically respelled. The chord nevertheless emanates a surplus of harmonic ener­ gy, overflowing the bounds of such a familiar tonal category. This surplus registers not on­ ly sonically, but also in the notation. Spelled “correctly,” the chord would be A𝄫 minor; as a tonic, its key signature would have 14 flats (that is, double flats on every diatonic pitch). Schubert has already begun from a point of flatward extremity: given his six-flat signa­ ture, any motion flatward will exert pressure on the notation.2 In the passage in question, the flatward pressure is so great that it forces an enharmonic snap in the music, creating visual fissures on the page where the six-flat signature is cancelled in measure 79 and then reinstated halfway through measure 80. The reinstatement coincides with a ffz augmented-sixth chord, which effortfully hauls the music back from its G-minor nadir, leading to a confirming cadential progression in G♭. The passage is a great intensification of a gesture Schubert has traced throughout the piece, beginning with the first phrase: a bass descent in thirds from the tonic into sub­ dominant regions, with a return by ascent at the last minute, under dominant energy. The descent in example 18.1, however, presses so far in the subdominant direction that it has the character of a tonal crisis or trauma, the intensity of which registers visually on the page, in the fissured notation. We can indeed hold the G♭ tonic in our ears throughout the passage—thus retaining the minor-Neapolitan hearing—but it takes some effort to do so. If we listen while looking at Schubert's fractured score—perhaps while playing the piece —we may be encouraged to give up that effort altogether, opening our ears to the chord's extratonal surplus. How we respond to such a passage analytically says much about what we value in music —and in musical analysis (the two are not necessarily the same). Given its harmonic com­ plexities, Schubert's passage provides an especially fruitful context for exploring some of the divergent values inherent in (echt-)Riemannian and neo-Riemannian approaches to harmonic analysis. The various technical differences between Riemann's harmonic theory (in its many iterations) and neo-Riemannian theory (in its many iterations) are, by now, relatively well known.3 Less attention has been paid, however, to the theories’ strikingly different attitudes toward the analytical act itself, including the different ways they seem to value music (in both senses: “cherish music” and “invest music with value”). Such dif­ ferences are, it need hardly be said, products of the theories’ highly distinct historical, ideological, and cultural moments. Page 2 of 27

Riemannian Analytical Values, Paleo- and NeoIn what follows, I will take an initial step toward mapping some of these divergences in value (and uncovering some unexpected points of contact), taking Schubert's G♭ Im­ promptu as a point of reference. Section II compares a model (p. 489) neo-Riemannian analysis of the passage—based on the work of Richard Cohn—to Riemann's own analyti­ cal comments about the piece, which bookend his career, appearing first in the early Musikalische Syntaxis (1877) and then in the sixth edition of the Handbuch der Har­ monielehre (1917). Section III then explores the methodological and ethical contrasts be­ tween the two approaches in depth, tracing aspects of the intellectual and ideological contexts in which they arose. The chapter concludes in section IV by considering some ways in which a technical rapprochement between the theories might open our ethical horizons, providing new ways in which we can value music through Riemann-inspired an­ alytical activity.

II It seems safe to say that the music in example 18.1 would catch the ear of any neo-Rie­ mannian analyst, perhaps even providing the first point of analytical entry into the piece. (One thinks here of the many analytical forays into Parsifal that have begun not at the work's outset, but with the most chromatically distorted version of the Grail motive, very near the end of act III.) Neo-Riemannians have often explored such passages by turning attention away from the traditional categories of tonal harmony and toward voice-leading efficiency, in an effort to detect pattern and regularity where there might otherwise ap­ pear to be tonal strain or disorder.4 By invoking enharmonic equivalence, such approach­ es further sidestep enharmonic complexities such as those discussed above. Example 18.2 sketches aspects of the Schubert passage from this perspective. The grand staff at (b) shows a reduction of the passage. The single staff at (a) extracts Klänge from the music.5 A key at the bottom of the example explains the noteheads in (a), which indicate whether the note in question is a common tone from the previous chord or has moved by ic1 or ic2. A quick scan of the noteheads reveals that every chord maintains at least one common tone with its predecessor; furthermore, motion by ic1 predominates. The annotations above staff (a) tally the results of the total voice-leading between the chords. DVLS is Richard Cohn's “directed voice-leading sum.”6 It measures the directed voice-leading motion between chords, distinguishing between “up” and “down.”7 Thus, the first entry in the row, +2, indicates total voice-leading of two semitones “up” from G♭+ to the B–: the two filled noteheads in the B– Klang indicate the two voices that have moved up by semitone from G♭+. The –2 that follows indicates total voice-leading of two semitones “down” from B– to D+, as one voice descends by whole tone. And so on. A clear pattern emerges: DVLS values alternate between +2 and –2 until the Ed+ (= D+) chord of measure 80 proceeds to the G♭+ of measure 81, yielding a DVLS value of 0: here two voices move by semitone, but in opposite directions, canceling each other out. This is the very moment at which the ffz augmented-sixth chord wrenches the music back to a ca­

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Riemannian Analytical Values, Paleo- and Neodential progression in G♭. The “wrenching” registers here in the contrary motion of DVLS = 0, which ends the +2/–2 tailspin.

Ex. 18.2. Harmonic reduction of Example 18.1, with voice-leading analysis.

The row below DVLS is labeled AVLS for “absolute voice-leading sum.” This mea­ surement takes no account of the direction of the voice-leading, instead measuring only (p. 490)

the absolute distance traversed in interval classes, registering what Joseph Straus calls the total voice-leading “work” or “exertion” of the progression.8 Again, there is a clear pattern: AVLS is 2 for all entries until the cadential oscillation between G♭+ and D♭+, where it increments to 3. This reading notes a continuity in the progression from E𝄫+ to G♭+, which traverses the same absolute voice-leading distance as all of the preceding progressions. The wayward chromatic successions of the first part of the phrase thus all show AVLS = 2, while the key-reaffirming cadential tag in G♭ projects AVLS = 3. The prevalence of 2s in the AVLS row suggests a particular voice-leading space, which Cohn calls a “Weitzmann region.” All of the chords in such a region relate to one another by AVLS = 2.9 Example 18.3(a) shows the Weitzmann region containing G♭+.10 The solid, undirected edges circling the perimeter of the network indicate the transforma­ tions that relate adjacent triads within the system: N and R. The former is Cohn's transfor­ mational label for Weitzmann's nebenverwandt relation;11 the latter is the familiar neoRiemannian relative. Dashed edges indicate transformations between nonadjacent triads: Klänge “two apart” on the cycle are related by PL or LP, and those opposite one another are related by Lewin's SLIDE. These five transformations—N, R, LP, PL, and SLIDE—are the only neo-Riemannian transformations (out of 24) for which AVLS = 2.12

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Riemannian Analytical Values, Paleo- and Neo-

Ex. 18.3. (a) Weitzmann region including G♭+; (b) passes through this region in measures 78–81. (p. 491)

All of the Klänge in example 18.2(a), with the sole exception of the dominant D♭+,

reside in this Weitzmann region. Example 18.3(b) shows the passage's progression through the region, up to the G♭+ cadential six-four in measure 81. The progression be­ gins with G♭+ at 12 o’clock and proceeds clockwise around the outer edge until it reach­ es G– at six o’clock. Along the way, an LP arrow leads from B– to G–, indicating that the D+ chord that intervenes in the second half of measure 79 plays a passing role between the two harmonies on the downbeats. On the “return trip” counterclockwise from G–back to G♭+, a similar LP arrow leads from D+ to G♭+; this is the “wrenching” LP motion asso­ ciated with the resolution of the augmented sixth to the cadential six-four. (The return trip bypasses B– altogether.) After returning to G♭+, the music leaves this Weitzmann re­ gion to engage in the confirming cadential progression via AVLS = 3.

Ex. 18.4. T, S, and D Weitzmann regions showing harmonic events in the B section.

Other passages in the Impromptu also trace out significant portions of a single Weitz­ mann region. Most notable among these is the other highly “purple” patch in the piece— the sojourn to C♭ major and E♭ major within the B section (measures 32–53). This is mapped in example 18.4, which presents the three Weitzmann regions containing G♭+, C♭+, and D♭+, labeling them T, S, and D in a manner analogous to Richard Cohn's labels for hexatonic systems in his analysis of Schubert's B♭ Sonata.13 The example first shows the move to E♭– and its dominant at the opening of the B section (measures 25–31); these harmonies still reside in the tonic region.14 Then, at measure 32, there is a move to the Page 5 of 27

Riemannian Analytical Values, Paleo- and Neosubdominant region's C♭+, via an interregional L transform of E♭–. The first “purple” mo­ tion traced in the subdominant region is the alternation between C♭+ and F♭– in measure 35. The dashed edges then link these chords to the next local tonic in the section, the E♭+ that enters in measure 48, which leads to the transitional A♭– in measures 52–53. From here the music progresses to the dominant D♭+, which is highlighted within its network on the right-hand side of the example. Notably, no other node is “lit up” within the domi­ nant network—in fact, none of the other Klänge in the dominant region sound prominently anywhere in the Impromptu. (p. 492) This is indicative of the way in which the Impromptu thoroughly explores the subdominant side of G♭, but not its dominant side.15 To be sure, there are several infelicities in the analysis in example 18.4. For one, and most obviously, the analysis leaves out many harmonies within the B section that do not fit into its regions, most notably all of the stormy diminished-seventh-based music in mea­ sures 40–45 and some local dominants. Furthermore, it leaves the question of the rela­ tionship between voice-leading efficiency and harmonic function (T, D, and S) somewhat undertheorized, conflating the two in a way that causes the distinction between chord and key to break down (e.g., by treating syntactical harmonies in the same way that it treats tonicized harmonies). Similarly, the separation of the tonic, dominant, and subdom­ inant triads into separate regions seems to do violence to their local syntactic connected­ ness at the level of the phrase. These are familiar problems in certain strands of neo-Rie­ mannian analysis. Yet, despite these shortcomings, the analysis provides a suggestive heuristic for tracing the piece's voice-leading activity, showing the ways in which it navi­ gates via AVLS = 2 in its most ear-catching passages. This neo-Riemannian reading—only the beginning of a fuller analysis—has proceeded as such readings often do, beginning with the most chromatically exceptional moment in the piece and moving outward from there to construct a broader interpretation of the move­ ment. Cohn takes a similar approach, for example, in his analysis of Schubert's B♭ Sonata, beginning by observing the hexatonic-polar relationship between the work's B♭+ tonic and its F♯– secondary key area, then seeking out other hexatonic-polar progressions (such as the one at the transition into the development section), and ultimately constructing a hexatonic analysis of the entire movement. The contrast with Riemann's own analytical practice could hardly be more stark. We can see it by consulting his analysis of the G♭ Impromptu, published in (p. 493) Musikalische Syntaxis in 1877. This is not only Riemann's first published analysis, but also one of his longest, at seven pages. Not until the Beethoven piano sonata analyses from the end of his career would he publish further analyses of comparable scope and technical detail. His primary concern is the Impromptu's phrase structure and its demonstration of the principles of harmonic syntax that he develops in the book, based on an arcane terminolo­ gy developed from Oettingen.16 Riemann's descriptions are often minutely detailed: he devotes an entire paragraph to the first phrase, and two paragraphs to measures 17–24. And about the phrase in example 18.1 he says…almost nothing. The passage gets only a very brief passing mention in his main prose, but it is not singled out; it is simply listed as Page 6 of 27

Riemannian Analytical Values, Paleo- and Neoone of several progressions in the coda: “The final consolidation of the primary key through progressions [Thesen] to C minor, G major, A♭ minor, and G major has a wholly excellent effect.”17 Riemann is referring here to the music in measures 74–81; note that he analyzes the piece in G major.18 The progression to A♭ minor—the minor Neapolitan in measure 80—is given no special emphasis: it is merely the goal of one of the four Thesen that Riemann mentions. Even more strikingly, Riemann cites these progressions as play­ ing a role in the “consolidation of the tonic” [Festigung der Haupttonalität]. This is in vivid contrast to the comments at the head of this chapter, in which I suggested that the music in example 18.1 can be heard to lead to a tonal crisis, creating a harmonic surplus that overflows the tonal frame. Riemann, by contrast, hears in these measures nothing more than a final confirmation of the global tonic, a confirmation that, moreover, has an “excellent effect” [vorzüglicher Wirkung]. It is hard to tell exactly what aspects of the passage Riemann finds vorzüglich, but whatever they are, they seem to have little to do with any undermining of the tonal order. His language instead suggests a celebration of the piece's confirmation of eternal tonal laws—its exemplary establishment and reinforce­ ment of a Haupttonalität. Thus, the passage that received so much attention in the neoRiemannian account, serving as the starting point from which all other observations radi­ ated, is little more than a footnote for Riemann, a negligible chromatic ripple on the sur­ face of an exemplary tonal masterwork. Indeed, Riemann frames his analysis in just this way, describing the Impromptu as a mod­ el citizen of the tonal realm, a “formally rather clearly structured composition.”19 He praises the piece's orderly construction: “The whole is a masterpiece as regards not only melodic form and metric structure, but especially as regards the ordering of its progres­ sions [Thesenordnung]. And over all of it reigns the tonality of G major, the principal key.”20 The second sentence makes clear Riemann's firm commitment to monotonality. (Modern readers will be struck by the pre-echoes of Schenker and Schoenberg.) Ten years later he voiced a similar sentiment in a more general context in his Systematische Modulationslehre: One is constantly struck by the controlling force [Geltung] of the main tonic, even during the boldest and most wide-ranging modulations. When we find ourselves at the end of the path, looking back, we know that we have learned how to trace ever wider circles around the unshakeable center.21 Though he is not specifically discussing the Impromptu here, this passage clearly applies to Riemann's understanding of the piece, in which a single tonic not only controls (p. 494)

the whole, but does so in model fashion. The piece's harmonic excursions do not weaken the tonic, but instead contribute to its greater glory, concentrically expanding its domin­ ion. (Again, the Schoenbergian resonance is striking.) There is only one other passing reference to the minor Neapolitan of measure 80 in Riemann's extensive discussion of the piece. In a tabular overview of the piece's harmonic progressions, he writes beneath the chord symbols for measures 79–80: “NB. Modulation to the antilogic antinomic third-key: g+—ºes.”22 The Oettingen-inspired terminology sim­ Page 7 of 27

Riemannian Analytical Values, Paleo- and Neoply means a progression from G major to the minor harmony whose dual root is a major third below—that is, a progression from a G over-triad to an E♭ under-triad (i.e., A♭ mi­ nor). Again, it is not entirely clear just what we are to “note well” about the passage. The annotation could be taken for an exclamation of surprise, and perhaps admiration, at Schubert's harmonic audacity: “Note well: A remarkable progression!” But the invocation of the arcane theoretical nomenclature might also suggest something quite different. We are to note not simply a striking progression, but the fact that the music proceeds to the antilogic antinomic third-key. This yields a very different sentiment: “Note well: My theo­ ry even has a name for this chord.” Read in this latter sense, the statement seems to betray an anxiety, an attempt to contain the harmonic extravagance of the moment within the rational bounds of the theory.23 It suggests a desire to demonstrate that no part of the Impromptu eludes the theory's ex­ planatory reach: all of its harmonic maneuvers are easily contained and rationalized with­ in the theory's bounds. Riemann himself explicitly thematizes the notion of spatialized boundaries to harmonic possibility in the book's closing pages. Here he suggests that he has mapped out a spatialized realm of tonal order, comparing it to a harmonic Garden of Eden: Thank God the combinations [of harmonies] are inexhaustible in number, and one cannot explore the area of harmony in its entirety by walking across it step by step but only by flying over it and surveying it from a bird's-eye view. It is sufficient, however, to recognize the chief paths through this magnificent Garden of Eden, which Heaven has left us after the Fall; everybody may then find new side paths for himself leading to ever new perspectives on regions never entered before.24 Schubert's progression would seem to represent one of the exotic, “new side paths” with­ in this realm, off of the beaten track of the Hauptwege, but nevertheless admissible. Yet, as Alexander Rehding has noted, Riemann's passage belies a profound worry: he presents the Garden of Eden as universal and transhistorical, but his language “implies at the same time a premonition—conscious or not—of its actual, contingent nature…the whole theory is built on a feeling of angst, a Spenglerian feeling that the end of an age—the end of German music—is imminent.”25 The harmonies on the borders of the Garden are thus fraught with peril, and perhaps temptation. After all, the invocations of the Garden and the Fall vividly suggest the (p. 495) possibility of harmonic sin. To sin against the tonal or­ der could bring about permanent banishment from the Garden—that is, banishment from the realm of tonal order into the atonal wilds. Did Riemann have doubts as to whether Schubert's minor Neapolitan might represent just such a harmonic sin? He may have. For we find him preoccupied with the chord forty years later, when revising his Handbuch der Harmonielehre for its sixth edition. His fore­ word to this edition of the Harmonielehre contains the last additions to the theory of func­ tions, which he had first introduced in 1893's Vereinfachte Harmonielehre. Rehding ob­ serves that the theory of functions represented a way to control and corral the overly per­ missive possibilities for harmonic progression in some of Riemann's earlier harmonic the­ Page 8 of 27

Riemannian Analytical Values, Paleo- and Neoories, including that in Musikalische Syntaxis, thus better fortifying the boundary around the Garden of Eden.26 It thus makes sense that he would return to the Schubert chord to make sure its energy was contained within his new system. In the foreword to the Handbuch, Riemann adds symbols for direct third relations, as well as a symbol for the modal Variante of any function—a v after a function symbol, which simply switches the triad's mode.27 The single example Riemann adduces for the new symbol is the minor Neapolitan from Schubert's Impromptu, which he now analyzes as : the variant of the leading-tone change of the minor subdominant.28 Even with the new symbol, the chord clearly puts a strain on Riemann's functional system, as it requires three alterations to the initial S function, which are made visible in the three accretions to the S symbol: 1. Major S to minor (º); 2. Leading-tone change of that (〉); 3. Variant of that (v). As Rehding has noted, Riemann otherwise seemed wary about admitting multiple alter­ ations to a function symbol.29 And indeed, his wariness is apparent here: he calls the chord “exceptional” (an Ausnahmserscheinung) and otherwise uses the v symbol only rarely in his later analyses. Nevertheless, the sense of Riemann's symbol is quite clear, and it is not that far from the way in which the chord would be analyzed in a modern American theory classroom using Roman numerals. Example 18.5 compares a Roman-numeral analysis of the passage at (a) with a full Riemannian-functional reading at (b) and the neo-Riemannian reading at (c). In order to make the progression's correspondence to the tonal readings more legible, I have adjusted the enharmonic spelling.

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Riemannian Analytical Values, Paleo- and Neo-

Ex. 18.5. Three harmonic readings of mm. 78–81: (a) Roman-numeral; (b) Riemannian; (c) Neo-Riemann­ ian.

The Roman-numeral interpretation constructs the chord on the downbeat of measure 80 as an alteration of a harmony built on the (lowered) second scale degree, while Riemann constructs it as a modification of subdominant function. There are important conceptual distinctions between the two theoretical concepts,30 but overall, the readings at (a) and (b) are quite similar: both trace a descent from tonic, via applied dominants,31 to an exot­ ic, “subdominant-side” harmony, before pulling back to the dominant at the downbeat of measure 81. Both analyses show the greatest amount of cognitive work at the downbeat of measure 80 (though Riemann's (p. 496) augmented sixth chord shows considerable in­ terpretive exertion as well). The neo-Riemannian analysis, by contrast, simply shows the first six chords unified by voice-leading motion of AVLS = 2, all unfolding within the tonic Weitzmann region. It is only at the resolution of the G♭+ cadential six-four to D♭+ that the tonic region is left for the dominant one. The infelicities of the neo-Riemannian account from a functional perspective leap into view here: it captures none of the phrase-level functional distinctions in the first six chords, and further is unable to read the G♭+ caden­ tial six-four as functionally distinct from the opening G♭+ tonic. The functional labels for the Weitzmann regions should thus indeed be understood as a secondary overlay—an in­ formal tonal label applied to a theoretical system whose essential logic is not tonal. Notably, the Riemannian reading at (b) casts the progression as syntactically normative, following the preferred paradigm for cadential motion: T–S–D–(T). The subdominant re­ gion is expanded considerably by modifications, but this does not obscure the phrase's overall syntactic sense. The analysis further reveals a clear syntactical resemblance to the opening progression of the piece (in measures 1–3), which Riemann would also ana­ lyze as departing from a tonic, passing through modifications of S, and then arriving at D. Thus, the passage that I read at the opening of the chapter as representing a tonal crisis or trauma, a reading that led to neo-Riemannian exploration of its extratonal logic, is re­ Page 10 of 27

Riemannian Analytical Values, Paleo- and Neofashioned here as a model tonal phrase, one that vividly demonstrates the efficacy of Riemann's functional principles in a chromatic context. The dangerous chord of measure 80 has now been fully contained within the boundaries of Riemann's Garden of Eden.

(p. 497)

III

This returns us to the broader question of value. The contrasts are as obvious as they are vivid. Riemann analytically constructs chromatic passages so that they show conformance to his tonal theories, which he portrays as universal laws.32 The neo-Riemannian analyst, by contrast, constructs chromatic passages so that they appear tonally “disunified,” and thus require nontonal explanation. Riemann's theory thus places a high value on order and conformance to putative universals of tonal harmony, while neo-Riemannian theory, it would seem, values crisis and disruption of that order.33 We can better understand this sharp divergence if we briefly survey the intellectual and ideological contexts that nurtured Riemann's theory, on the one hand, and neo-Riemann­ ian theories on the other. Riemann's context has been masterfully reconstructed by Alexander Rehding, so I will merely summarize his argument here. Rehding characterizes Riemann as seeking to define a universal “classicism”34 that transcends history and thus acts as a brake against further historical change in music, clearly demarcating the bound­ aries beyond which music should not progress. For Riemann, music theory had an ethical responsibility to set limits for composers, acting as “a bastion against historical change.”35 Appeals to the burgeoning natural sciences allowed Riemann to provide “hard” support for his claims of universality,36 while institutional and pedagogical factors played a role as well, as he sought to develop a harmony pedagogy that would displace Roman-numeral based Weberian approaches, thus allowing him to influence future musi­ cians directly, instructing them in the laws and limits of musical possibility. The result was a conservative theory shot through with a “relentless normativity.”37 In short, Riemann sought, through this theory, to stem the tide of historical change in music, which seemed to him (rightly, it turns out) to be perilously close to transgressing the boundaries of the tonal Garden of Eden.38 By the time of the American revival of interest in Riemann's theoretical ideas in the 1980s and 1990s, that transgression had of course occurred long ago. Indeed, the crossing of music over the atonal threshold was one of the primary factors leading to the disciplinary consolidation of music theory in the American academy: theorists had taken advantage of the challenges posed by posttonal music to argue for the institutional necessity of music theory as a research discipline. Atonality, which before had been a looming threat to Rie­ mann, to be resisted at all costs, now enjoyed great institutional privilege and prestige, especially among theorists in the second half of the twentieth century. That neo-Riemann­ ian theorists would value tonal crisis far differently from Riemann should thus come as no surprise. If a given passage by, say, Wagner was perceived to veer perilously close to tonal incoherence, it could now be embraced analytically using the technologies of atonal

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Riemannian Analytical Values, Paleo- and Neotheory, thus inheriting the institutional values associated with avant-garde atonal musics through a sort of ethical transitive property. This is only part of the story, however. In the new, institutionalized theory of the American academy, analysis became an end in itself—a means of engaging (p. 498) deeply with indi­ vidual works via various technical hermeneutic genres, often with a liberal humanist fo­ cus on interpreting the telling compositional idiosyncrasy. Such a practice carries with it a strong element of Romantic ideology, with the individual work valued for its originality and uniqueness, as a product of genius. Neo-Riemannian theory clearly participates in this ideology, with its almost exclusive focus on the compositionally extraordinary. (The chromatic Grail motive is a classic case.) This is in stark contrast to Riemann's own prac­ tice, in which analysis serves first and foremost to illustrate and validate his theory. His analytical emphasis is not on what makes a work remarkable or individual, but on the ways in which it exemplifies the normative, law-like aspects of his theory. As Rehding ob­ serves, regarding Riemann's analysis of the “Waldstein” Sonata: [I]t seems that Riemann is not interested in the special features of [the sonata's] opening. Rather—it would appear—he plays down the particularity of this opening in favor of its general features. While we have come to appreciate the first few bars of the “Waldstein” sonata as a paradigm of Beethoven's harmonic boldness, Riemann's analysis of this passage is actually a demonstration of its ordinariness.39 Riemann's ability to find the ordinary within the compositionally extraordinary extends to all aspects of his theory, from harmony, to rhythm, phrase structure, and form. For exam­ ple, he says of the first movement of Beethoven's op. 130: “Correctly interpreted, the movement offers no cause to speak of disruption and formal difficulty, but instead clearly shows the normal framework of sonata form.”40 This breathtakingly matter-of-fact assess­ ment—turning one of Beethoven's most fissured movements into a self-evident and un­ problematic sonata-form—dissonates not only with modernist (Romantic) theory and analysis, which would seek to explore the structural particularities that make the move­ ment unique, but also with more recent critical musicology. Daniel Chua, for example, characterizes the same movement as nothing less than a “direct assault” on the listener: “the audience is simply thrown into confusion by a disarticulated syntax, by a language so violent and contradictory that to analyze the disunity is to be more obvious than ‘poststructuralist.’”41 Obvious to us, perhaps, in our postmodern age, but not to Riemann, whose aesthetic system allowed no place for disunity or disruption in a masterwork. Styles of critical musicological thought such as Chua's are not irrelevant to this study. For it is not a coincidence that neo-Riemannian theory rose to prominence around the same time as the disciplinary upheaval caused by the New Musicology. The valorization of dis­ unity, crisis, fragmentation, and heterogeneity in that literature finds a curious and dis­ torted echo in neo-Riemannian theory. Indeed, in his introductory article to the Journal of Music Theory issue dedicated to neo-Riemannian theory, Richard Cohn explicitly situates neo-Riemannian analytical approaches with respect to “an evolving post-structuralist crit­ Page 12 of 27

Riemannian Analytical Values, Paleo- and Neoical practice,” suggesting a community of purpose as regards claims of tonal disunity.42 There are further parallels. Neo-Riemannian theory focuses on the very literature privi­ leged by the New Musicologists—nineteenth-century opera and concert music—turning its (p. 499) attention to many of the very same “problem” passages that would capture the attention of the postmodern critic. The neo-Riemannian literature's emphasis on “de-cen­ tered” harmonic spaces also resonates with such critical practices (though, more cynical­ ly, one might simply recognize here a co-optation of jargon).43 More substantively, the em­ phasis on analytical pluralism in many transformational approaches, with roots in Lewin's methodological writings, squares well with postmodern interpretive practices. Finally, certain recurring hermeneutic tropes in the neo-Riemannian literature—such as Cohn's explorations of the harmonic uncanny44—are clearly indebted to the New-Musicological spirit. Neo-Riemannian theory thus appears to pull off a seemingly impossible double play, benefiting from two sets of prestigious but conflicting institutional values at once (both of which would be anathema to Riemann): those associated with the atonal avantgarde on the one hand, and with postmodern critical practices on the other. But, as Cohn makes clear, any commitment to the latter is trumped by more familiar mu­ sic-theoretical concerns: Both paradigms [neo-Riemannian theory and post-structuralist music criticism] recognize the potential for tonal disunity in music that uses classical harmonies, and accordingly resist shoehorning all chromatic triadic music into the framework of diatonic tonality. For the post-structuralist, the recognition of tonal disunity leads immediately to an ascription of disunity tout court, and from there to a clus­ ter of cognate terms…: “unstructured,” “incoherent,” “indeterminate,” “coloris­ tic,” disjunct,” “arbitrary,” or “aimless.” The recognition of tonal disunity could in­ stead lead to a question: “if this music is not fully coherent according to the princi­ ples of diatonic tonality, by what other principles might it cohere?”45 Thus, despite nods to postmodern sensibilities, the most time-honored value of modernist music theory remains firmly intact: the demonstration of coherence through formalism. And here we find common ground with Riemann himself. For is the “coherence” of the neo-Riemannian analyst really that far removed from the “logic” or “syntax” of Riemann? Despite some obvious differences in philosophical underpinnings, both projects are un­ derwritten by a drive toward systematization and logical rigor; a penchant for elegant, symmetrical theoretical structures; and a desire above all to detect order in complex mu­ sic, containing harmonic extravagances in controlled, rational spaces. These values, it would appear, are pan-Riemannian. But neo-Riemannian theory contains methodological tensions not present in Riemann. The high value it places on both disruption and coherence leads to a peculiar sort of hy­ bridity or double focus. Surely a prime reason for the success of neo-Riemannian theory is the fact that it allows analysts to dwell on the most remarkable sounding passages in a chromatic work, those moments when the tonal fabric is stretched or torn. But it is not the remarkable sound of those passages that is analyzed; it is their coherence. One thus Page 13 of 27

Riemannian Analytical Values, Paleo- and Neobegins to wonder what the relationship is between the sound and the analysis. Is the “co­ herence” that the method detects responsible for what it is that makes these disorienting passages so aurally captivating? Or are the two unrelated? In other words, do we value the analysis for the same reasons that we value the music? This question could surely be made of any style of systematic music analysis. But it has a special urgency in neo-Riemannian practice, as critical responses to the theory at­ test. Charles Fisk, for example, states that Cohn's analysis of Schubert's B♭ Sonata runs the risk of “making even the most extraordinary progressions in Schubert seem ordinary —or at least, in some respects, normative.”46 (The echoes of Rehding's interpretation of Riemann are striking, suggesting further pan-Riemannian similarities.) Fisk is concerned that Cohn's theory does not do justice to the sound of Schubert's music, making the aural­ ly arresting seem theoretically commonplace. Cohn responds by invoking what amounts to a music-theoretic fact/value distinction, arguing that theoretical categories do not nec­ essarily correlate with sonic affects in a simple one-to-one fashion, even in traditional the­ ory.47 Cohn's writing on the uncanny effects of hexatonic-polar progressions is an elo­ quent testament to this. (p. 500)

Nevertheless, Fisk's criticism is hard to dismiss. The Schubert passage in example 18.1 sounds extraordinary, but the analysis of examples 18.2–4 does not tell us about that, in­ stead revealing order and pattern in its voice-leading. We are thus left to wonder just what it is that this music does to us after it enters our ears, why it thrills and captivates us. Reflecting on this matter, we might come to value Riemann's original theory a bit more. For Riemann throughout his career intended his theories to provide an answer to the question “Wie hören wir Musik?” That the question was always framed in normative terms (“How should we hear music?”) and that the answers were therefore tinged with a sense of prescriptive “ought” does not diminish greatly the value of his approach in this regard. Riemann's focus was indeed on what happens to the music after it enters our ears, and despite his many theoretical and rhetorical excesses and detours, some of his ideas hit the mark so successfully that they remain with us, in some form or another, to this day. Chief among these is of course the idea of tonal function, whose influence is still felt, not only in Germany's Musikhochschulen, but also in many strands of Anglo-Ameri­ can Roman-numeral-based harmony, even in some Schenker pedagogy (however obfuscat­ ed the debt to Riemann may be).48 In the concluding section of the chapter, I will thus ex­ plore one way in which Riemann's functional ideas can be reanimated in a transforma­ tional context, thus shedding some light on the remarkable sonic effect of Schubert's pas­ sage, and narrowing the neo-Riemannian fact/value gap, if only slightly.

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Riemannian Analytical Values, Paleo- and Neo-

IV

Ex. 18.6. LRP map with G♭+ tonic at center.

Riemann's functions model the relationship of harmonies to the tonic, either directly or via one of its two dominants.49 One way to interpret his function symbols is thus not as la­ bels for chords, but as descriptions of the actions that listeners perform as they interpret sounding harmonies with respect to the tonal center. In this understanding, to hear a chord as a subdominant is to perform the (p. 501) subdominant operation (S), directing awareness from the sounding chord to the tonic via S. The S-ness of the perception re­ sides not in the sounding harmony (the raw acoustic signal) but in the action whereby the listener relates it to the tonic.50 I have elsewhere referred to this action of directing awareness toward the tonic as “tonal intention.”51 We can trace such Riemannian intentional acts on various species of Tonnetz. Example 18.6 shows one such space that is useful in exploring Schubert's Impromptu: a dual of the familiar neo-Riemannian Tonnetz that Michael Siciliano calls the “LRP map” and Douthett and Steinbach call the “Chicken-Wire Torus.”52 The edges represent the three canonical neo-Riemannian operators (P, L, and R), as shown by the key to the left. The network is rotated 90º from its usual presentation in neo-Riemannian studies (and in historical Ton­ netze), so that fifth-related triads are on the vertical axis, capturing the familiar metaphor of dominants residing “above” a specified tonic, and subdominants “below.” Dominant (D) and subdominant (S) arrows can be added to this vertical dimension as necessary, to indi­ cate direct functional relationships, while the P, L, and R dimensions allow for the model­ ing of Riemann's Parallele (R), Leittonwechsel (L), and Variante (P) functional modifica­ tions.53 If enharmonically conformed, the network wraps around into a torus; it is arranged on the page here so that G♭+—the tonic of Schubert's Impromptu, shown with a double border—has a central position.

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Riemannian Analytical Values, Paleo- and Neo-

Ex. 18.7. (a) Intentional paths traced by two differ­ ent subdominants in the Impromptu; (b) Modal vec­ tors within the functional space.

Example 18.7(a) demonstrates the intentional paths traced by two modified subdominants in the Impromptu: the E♭– in measure 2, the second chord in the piece; and the A𝄫– (= G–) minor Neapolitan of example 18.1, measure 80. The relevant Klänge are indicated with crosshatching, while solid dark arrows show the intentional paths traced by their Rie­ mannian interpretations. The E♭– chord is (p. 502) interpreted as a subdominant leadingtone change, or ,suggesting an intentional path of LS back to the G♭+ tonic, passing through the subdominant C♭+ along the way. Note that if the same Klang were interpret­ ed as a tonic parallel, Tp, it would trace out a different intentional path: directly back to the G♭+ tonic via R. Riemann's different interpretations of the same sounding chord thus traverse different paths in the space, making clear that, in the present interpretation, the function describes not the chord, but the path whereby it is related back to the tonic. The minor Neapolitan, A𝄫–, is analyzed following Riemann's interpretation of it in the sixth edition of the Handbuch as , that is, the Variante of the leading-tone change of the minor subdominant. This suggests a considerably more complex intentional path back to the tonic: PLPS. The initial P models Riemann's Variante (v); that is, it traces our interpre­ tation of the chord as a Variante of the “proper” (p. 503) minor subdominant leading-tone change, as we mentally interpret A𝄫– as related to A𝄫+ via P. The remaining transforma­ tions indicate similar interpretive activity, L corresponding to the leading-tone change (〉) and the second P to the modal alteration of the subdominant (º). I stated above that “it takes some effort” to “hold the G♭ tonic in our ears throughout the passage—thus retain­ ing the minor-Neapolitan hearing.” That effort registers in the PLPS intentional path— specifically, in the concatenation of four interpretive moves it suggests. While AVLS in ex­ ample 18.2 expressed the exertion involved in voice-leading one chord to the next, the

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Riemannian Analytical Values, Paleo- and Neopresent system expresses the exertion involved in interpreting each chord tonally with re­ spect to G♭ via Riemannian categories. We can make vivid that exertion by imagining the intentional arrows in 18.7(a) traveling through a substance or medium—a medium that offers a certain resistance, limiting how far we can travel in tracing a harmony back to a tonic. The resistance might cause us to give up altogether in our effort to interpret the A𝄫– chord with relation to G♭+, thus breaking the chain of arrows departing from A𝄫– in 18.7(a). Or we may reach G♭+, but on­ ly weakly, our tonal intentions exhausted in the effort. Our sense of G♭+ as tonic would thus be considerably attenuated, its hold on the music in our ears now precarious. But the arrows are not only a measure of exertion. They also give us a way to think about the tonal quality of a given harmonic function—its characteristic sonic affect (and effect). A subdominant sounds different from a dominant, after all, as do the many modifications of these harmonies from one another. These qualitative differences are often described metaphorically by reference to color (dark harmonies, bright harmonies, and the like). Such differences in sound are not a product of the raw acoustic signal, but a product of the way in which we relate a given sounding harmony to a tonic. (After all, the same chord can be a subdominant in one context and a tonic or dominant in another.) That is, the affect arises only after the music “enters our ears,” setting our tonal-interpretive ac­ tivity in motion. The quality can thus be understood to inhere in the path of arrows we trace from the sounding chord back to the tonic. Example 18.7(b) explores aspects of the resulting affective or coloristic regions in the space. As it shows, the “brightest,” sharpward regions are up and to the left from the ton­ ic, while the “darkest,” flatward regions are down and to the right. These coordinates provide a rough sense of the color that will accrue to a chord as it is related back to the tonic from a given region of the network. The farther in any given direction a chord re­ sides, the more intensely will it acquire that color. The A𝄫– chord resides deep in the dark­ est quadrant of the space, tracking back to G♭+ from the farthest flatward reaches and acquiring an extremely dark tinge in the process. It is as though with each interpretive/in­ tentional act—each darkened arrow—the harmony accrues a new layer of color, or better, of shading. (The word evokes Kurth, whose ideas are highly pertinent here.) These accre­ tions of interpretive shading are what give the G–/A𝄫– chord its remarkable chiaroscuro quality; one thinks of the multiple layers of paint around the dark perimeter of a Rem­ brandt portrait. Again, though the harmonic color seems to infuse the sounding medium (p. 504) itself, it is not a property solely of the raw acoustic signal (i.e., the minor triad). We can experience this by clearing our ears of the harmonic context of the Impromptu and simply playing a G-minor chord alone, hearing it as a tonic. The effect is one of strip­ ping away the layers of shading that are present when we hear the chord in the context of example 18.1, as though we have stripped away the many layers of Rembrandt's deep browns, revealing the blank canvas of the minor triad beneath. In this sense, the chord of measure 80 has not less tonal character than the more traditional tonal har­ monies in example 18.1, but more.

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Riemannian Analytical Values, Paleo- and NeoExample 18.8 integrates the two chords from example 18.7(a) into their respective pro­ gressions. Dashed arrows show the chord-to-chord progressions within the passages; these are numbered to indicate their order. Solid arrows show the intentional path back from each chord to the tonic. Example 18.8(a) analyzes the opening phrase up to the G♭+ chord of measure 4; 18.8(b) analyzes the phrase in example 18.1 (measures 78–82). The opening phrase remains largely vertical within the space of 18.8(a), moving first down in­ to the subdominant region, and then balancing this with a motion to the dominant. Note that this reading takes advantage of the toroidal possibilities of the space, reinterpreting the A♭– chord in a manner analogous to Rameau's double emploi. Example 18.8(b), by contrast, spreads out horizontally—and chromatically—across the network, dipping into its darkest corner. Note the intentional interpretation of the E𝄫+ chord: It is heard not as the relative (or Riemannian Parallele) of the previous C♭– chord (Riemannian ºSp). In­ stead, it is heard as the dominant of the upcoming A𝄫–, consistent with the Riemannian analysis in example 18.5(b). This is the chord that contains the “menacing bass trill on C,” which announces the imminent arrival of the A𝄫–/G– chord, which we can thus hear com­ ing before it sounds. The sense that we can “hear it coming” is reflected in the PD arrow chain that departs from E𝄫+ to the right and down, directing our attention toward the coming A𝄫– via its dominant. The “menace” of the chord resides partly in the fact that it is pointing us further into the darkest regions of the space—further to the right and down­ ward. As I noted above, both progressions trace a similar T–S–D–T progression, via altered sub­ dominants; this similar trajectory is evident visually on the two examples, as the progres­ sions move first to the subdominant side, below the tonic, then return to the tonic from above, “under dominant energy.” Unlike the progression in 18.8(a), however, the flip to the dominant side in 18.8(b) does not occur through a Rameauian reinterpretation. In­ stead, there is simply a snap from one extreme of the network to another, as the A𝄫– chord moves to “A♭+” via arrow 4.54 The snap from one edge of the space to the other co­ incides with Schubert's ffz dynamics and the second key-signature fissure, as the aug­ mented-sixth “effortfully hauls the music back from its G-minor [= A𝄫–] nadir.”

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Riemannian Analytical Values, Paleo- and Neo-

Ex. 18.8. (a) Analysis of mm. 1–4; (b) Analysis of mm. 78–82.

This reading values the music in example 18.1 by tracing something of a middle path be­ tween the Riemannian and neo-Riemannian positions sketched above. As in neo-Riemann­ ian accounts, I have sought here to valorize the remarkable in this passage; as in Rie­ mann, I have done so by situating the harmonies within a tonal space. (p. 505) But I have not invoked that space in order to contain the music—at least, that was not my intent. In­ stead, I wished to show the ways in which the Impromptu's harmonies are invested with qualitative intensity via their tonal context and the ways in which we as listeners partici­ pate in generating that qualitative intensity. The not-so-implicit claim is that the tonal context is responsible for the extraordinary sound: the latter results from the ways in which the music moves toward the outer, benighted realms of G♭ major, or better, the ways in which we work to interpret its harmonies from those realms back to the tonic. This restores a sense of extremity—both harmonic and emotional/expressive—to the mu­ sic.55 This extremity is also captured by (p. 506) the fragility of the intentional path from A𝄫– back to G♭+, reflecting the danger we face of losing cognitive contact with the music's rational center. The analysis seeks to narrow the fact/value gap between the sounds we cherish and the analyses we construct. However successful or unsuccessful it is in that effort, it is clear that the gap is not closed. No formal model can capture all aspects of our musical experi­ ence, even when we limit ourselves to one parameter, such as harmony. For myself, I find that the picture in example 18.8(b) turns what had been a flickering and contingent expe­ rience into something more fixed and stable, even overdetermined. In my prose I have sought to mitigate this, reinvesting the formal model with some sense of fragility. But this represents an intervention from outside the space of the formal theory, suggesting the continued persistence of the fact/value split. The present approach has a signal merit, however, in that it gives the “value” side of the equation specific hooks to attach to in the Page 19 of 27

Riemannian Analytical Values, Paleo- and Neoformal model, allowing our evanescent aural sensations to interact with the model in sug­ gestive ways. We are lucky in that our historical position allows us, unlike Riemann, to relish those mo­ ments in which chromatic works threaten to overflow the rational bounds of our tonal theories, or in fact do overflow those bounds. As I suggested above, the relishing—indeed, valuing—of those moments seems to be one of the defining traits of the neo-Riemannian habitus. Yet, in the desire to detect coherence at all costs one notes a continued reluc­ tance to step over the next threshold, to relish that unruly part of musical experience that resists formal containment. Perhaps this is only a matter of time, however—the neo-Rie­ mannian turn has introduced a new flexibility into tonal analytical thought, moving us to­ ward a highly salutary methodological self-awareness and interpretive pluralism. This shift may ultimately lead us to relinquish coherence (and its implicit sense of rational con­ tainment) as music theory's ethical lodestar, allowing us to employ our analytical methods freely in the exploration and construction of manifold musical experiences, without feel­ ing the need to claim comprehensiveness for any one of them. For surely the best way we can value music is to acknowledge that it will always exceed the manicured gardens of our theories.

Notes: (1.) The cited chords occur on the downbeats of measures 78, 79, and 80. The passage ex­ pands a previous gesture in measures 74–77, which had already traversed part of this path, from G♭ major to C♭ minor and back. As Charles Fisk has noted, the B/C♭ minor har­ mony in both passages recalls the B-minor middle section in the previous Impromptu, in E♭. Charles Fisk, Returning Cycles: Contexts for the Interpretation of Schubert's Im­ promptus and Last Sonatas (Berkeley: University of California Press, 2001), 118. For fur­ ther discussion of the E♭ Impromptu, see my “Perspectives on Tonality and Transforma­ tion in Schubert's Impromptu in E♭, D. 899, no. 2,” Journal of Schenkerian Studies 2 (2007): 33–63; on the intertextual resonances of Schubert's B-minor harmonies, see 47 n. 26 in the latter article. (2.) On notational “pressure” forcing enharmonic shifts, see Daniel Harrison, “Noncon­ formist Notions of Nineteenth-Century Enharmonicism,” Music Analysis 21.2 (July 2002): 140–142. On the historical importance of the six-flat signature in Schubert's Impromptu, see Hugh MacDonald, “[Six-Flat Key Signature, 9/8],” 19th-Century Music 11.3 (Spring 1988): 221–237. MacDonald calls the Impromptu “a breakthrough toward a new concept of the key” of G♭ (p. 225). The Impromptu was first published by Haslinger in 1857 in G major; Hugo Riemann seems to have based his discussions of the piece—about which, more below—on Haslinger's (or a later) G-major edition. (3.) For an overview of some of the most salient technical differences between paleo- and neo-Riemannian theories, see David Kopp, Chromatic Transformations in Nineteenth-Cen­ tury Music (Cambridge: Cambridge University Press, 2002), 150–151.

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Riemannian Analytical Values, Paleo- and Neo(4.) For example, Richard Cohn states that neo-Riemannian theory seeks to answer the question, “If this music is not fully coherent according to the principles of diatonic tonali­ ty, by what other principles might it cohere?” Cohn, “Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective,” Journal of Music Theory 42.2 (Fall 1998): 169. (5.) I mean Klang here—and throughout this chapter—in the familiar neo-Riemannian sense of major and minor triads (not in Riemann's sense of a dualistic emanation of over­ tones and undertones from a single pitch). The Klänge in Example 18.2(a) omit chordal sevenths and the one augmented sixth. The omissions would need to be addressed in a broader transformational analysis, but they are not consequential here. For a transforma­ tional model that integrates members of SC 3–11 and 4–27 see Julian Hook, “Cross-Type Transformations and the Path-Consistency Condition,” Music Theory Spectrum 29.1 (Spring 2007): 1–39. (6.) Richard Cohn, “Square Dances with Cubes,” Journal of Music Theory 42.2 (Fall 1998): 283–296. (7.) The scare quotes make clear that DVLS values obtain in pitch-class space, in which the concepts of “up” and “down” are traditionally considered problematic. I have never­ theless retained those words in the text for their intuitive immediacy. I have also replaced the directed pitch-class intervals of Cohn's DVLS with positive and negative integers, for the same reason. These numbers should be understood as substitutes for their mod-12 equivalents. (Recently, Clifton Callender, Ian Quinn, and Dmitri Tymoczko have recuperat­ ed the notions of “up” and “down” in pitch-class space; I do not, however, rely on their formalism here.) (8.) Joseph Straus, “Uniformity, Balance, and Smoothness in Atonal Voice Leading,” Music Theory Spectrum 25.2 (2003): 321–322; see especially n. 39. AVLS is the same as Straus's “total displacement” and Cohn's “voice-leading efficiency” or VLE (“Square Dances,” 284). (9.) This lends a consistency of voice-leading distance to a Weitzmann region that is not present in a hexatonic cycle, in which AVLS values range from 1 to 3. On Weitzmann re­ gions, see Richard Cohn, “Weitzmann's Regions, My Cycles, and Douthett's Dancing Cubes,” Music Theory Spectrum 22.1 (Spring 2000): 89–103. See also “Square Dances,” 290–295. Cohn's Weitzmann regions arise from a transformational interpretation of ideas in Carl Friedrich Weitzmann's pamphlet Der übermäßige Dreiklang (Berlin: T. Trautwein, 1853). All of the triads in a Weitzmann region share two tones with a single augmented triad. Interestingly, several augmented triads appear prominently on the surface of Schubert's Impromptu—{D♭, F, A♮} in measures 4 and 58; {G♭, B♭, D♮} in measure 24; and {C♭, E♭, G♮} in measure 73. The sense that the piece tends toward an augmented-tri­ ad sound world—especially in moments of transition—is suggestive, but I would not push the idea too hard: all of these chords operate in ways far more traditional than the Liszt­ ian possibilities that Weitzmann had in mind. Page 21 of 27

Riemannian Analytical Values, Paleo- and Neo(10.) In this and later networks, undirected edges are a shorthand for a symmetrical pair of arrows (or more colloquially, double-headed arrows). I have drawn the region in a hexagonal format analogous to Cohn's images of his hexatonic systems in “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progres­ sions,” Music Analysis 15.1 (1996): 9–40, and in “Weitzmann's Regions,” 95, ex. 7. This approach essentially reverses Cohn's graph-theoretic priorities in “Weitzmann's Regions,” in which it is the hexatonic cycles that are drawn cyclically, with Weitzmann regions join­ ing them via a mediating augmented triad. (11.) “Weitzmann's Regions,” 92 and 98. N inverts a triad about its Riemannian (NB) root, for example, . In more familiar Anglo-American tonal terms, N maps a major tri­ ad to its minor subdominant (and back) or a minor triad to its major dominant (and back). Weitzmann's nebenverwandt relation is formally the same as Riemann's Seitenwechsel and Oettingen's Wechsel, which also exchange triads that share the same dual root. (12.) Along with the identity operation E, these five operations form a tidy dihedral group of order 6. On SLIDE, see David Lewin, Generalized Musical Intervals and Transforma­ tions, reprint ed. (New York: Oxford University Press, 2007), 178. (13.) Richard Cohn, “As Wonderful as Star Clusters: Instruments for Gazing at Tonality in Schubert,” 19th-Century Music 22.3 (Spring 1999): 213–232. (14.) For Riemann, E♭– in the key of G♭ major could act as either a tonic (Tp) or a sub­ dominant ( ). We will explore the differences between Riemann's function theory and neo-Riemannian regional analyses such as that in example 18.4 below. (15.) Riemann himself notes the piece's exploration of subdominant key areas at the ex­ pense of dominant ones, using the fact as an argument for the dualistic equality of the two dominants. Hugo Riemann, Musikalische Syntaxis: Grundriß einer harmonischen Satzbildungslehre (Leipzig: Breitkopf und Härtel, 1877), 71. (16.) Michael Kevin Mooney provides a lucid and thorough overview of Riemann's Musikalische Syntaxis, including its Oettingen-inspired terminology and the analysis of the Schubert Impromptu, in his dissertation “The ‘Table of Relations’ and Music Psycholo­ gy in Hugo Riemann's Harmonic Theory” (Ph.D. diss., Columbia University, 1996), 162– 175. (17.) Riemann, Musikalische Syntaxis, p. 69. “Die schließliche Festigung der Haupttonal­ ität durch die Thesen von ºg, g+, ºes und g+ ist von ganz vorzüglicher Wirkung.” Riemann uses the word These (a holdover from his earlier Hauptmann-inspired work in “Musikalis­ che Logik”) throughout the book to refer to motions away from, or back to, the tonic, via its upper and lower dominants. It comes to mean little more than “progression,” and that is how I have translated it above. (18.) As observed in n. 2, the Impromptu was first published in 1857 in G major, only twenty years before Riemann's book. It is curious, however, that Riemann continued to re­ Page 22 of 27

Riemannian Analytical Values, Paleo- and Neofer to the piece in its G-major version even in the sixth edition of his Handbuch der Har­ monielehre, published in 1917. The volume of the Schubert Alte Gesamtausgabe includ­ ing the present Impromptu in the correct key of G♭ had been published in 1888, and the correct key was surely well known to Riemann by 1917. It is tempting to speculate that he retained the G-major version not only for pedagogical clarity, but also as it does not ex­ hibit the same notational disruption as the G♭ version does: there is no shift in key signa­ ture in measures 79–80 in the Haslinger edition, thus removing any notational sign of tonal disruption and visually clarifying the minor-Neapolitan hearing of the harmony in measure 80. (19.) Riemann, Musikalische Syntaxis, 65. “eine formell recht übersichtlich gegliederte Komposition.” (20.) Ibid., 69. “Das Ganze ist ein Meisterstück sowohl hinsichtlich der Melodiebildung als der metrischen Struktur, besonders aber in Hinblick auf die Thesenordnung. Den bei weitem größten Theil beherrscht die Tonalität von g+, die Haupttonart.” (21.) Hugo Riemann, Systematische Modulationslehre (Hamburg: J. F. Richter, 1887), 202. “Immer wieder drängt sich uns die Geltung der Haupttonalität auch während der kühn­ sten und weitestausholenden Modulation auf. Wenn wir daher nun am Schluss auf den Weg, den wir zurückgelegt, zurückblicken, erkennen wir, dass wir nun gelernt haben, im­ mer weitere Kreise um das unverrückbare Centrum zu beschreiben.” (22.) Riemann, Musikalische Syntaxis, 69. “NB. Ausweichung nach der antilogen antin­ omen Terztonart g+—ºes.” (23.) Riemann's anxiety about chromatic progressions such as this one is evident earlier in the book, when he twice stresses that one should treat such progressions (to antilogicantinomic third chords, among others) with the greatest caution (Vorsicht). Ibid., 19–21. He observes (p. 21) that such progressions are best used only at the end of a piece, after the tonic has been established securely, as, presumably, in Schubert's Impromptu. In his valuable discussion of Riemann's analysis of the Impromptu, Michael Kevin Mooney ob­ serves that the antilogic-antinomic Terzklang is one of the most distant harmonies from the tonic, when measured by a metric Riemann provides in his slightly later Skizze einer neuen Methode der Harmonielehre of 1880. See Mooney, “The ‘Table of Relations,’” 240– 241. Notably, if one were to take seriously Schubert's enharmonic spelling of the chord in its original key—as a G-minor triad in G♭ major—it would represent the most distant har­ mony from the tonic on Riemann's scale, the Doppelterzwechselklang (g♭+—ºd♮). (24.) Ibid., 120, trans. Alexander Rehding, Hugo Riemann and the Birth of Modern Musi­ cal Thought (Cambridge: Cambridge University Press, 2003), 105. “Die Kombinationen sind Gott sei dank unerschöpflich an Zahl und man kann das Gebiet der Harmonik nicht Schritt für Schritt abgehen, sondern nur überfliegen, aus der Vogelperspektive über­ schauen. Es genügt aber, die Hauptwege durch diesen herrlichen Garten Eden, den uns

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Riemannian Analytical Values, Paleo- and Neoder Himmel nach dem Falle gelassen, zu erkennen; jeder mag dann selbst weitere Seit­ enpfade zu immer neuen Durchblicken in nie betretene Reviere finden.” (25.) Rehding, Hugo Riemann, 111–112. (26.) Ibid., 51–59, 105, and 114. (27.) Renate Imig observes that the Variante is in fact inconsistent with Riemann's dualist theory, in which major and minor thirds are not merely exchangeable within a triad, but are instead generated in opposed, dualist directions. Renate Imig, Systeme der Funktions­ bezeichnung in den Harmonielehren seit Hugo Riemann (Düsseldorf: Gesellschaft der Förderung der systematischen Musikwissenschaft, 1970), 51–52. (28.) Hugo Riemann, Handbuch der Harmonielehre, 6th ed. (Leipzig: Breitkopf und Här­ tel, 1917), xvii. (29.) Rehding, Hugo Riemann, 58 n. 52 and 168. (30.) On the conceptual differences between Stufen and functions (for example, the differ­ ence between a IV Stufe and an S function), see David Lewin, “Music Theory, Phenomenol­ ogy, and Modes of Perception,” Music Perception 3.4 (Summer 1986): 342–343; and Brian Hyer, “Tonal Intuitions in Tristan und Isolde” (Ph.D. diss., Yale University, 1989), 105. (31.) The Ds in parentheses in the Riemannian reading indicate applied dominants of the following harmonies. Note that the preferred Riemannian reading of the German sixth is as an altered applied dominant. The slash through the D indicates that the root is omit­ ted, while the 〉 symbol after an Arabic numeral indicates chromatic lowering. (32.) On universality in Riemann's thought, see Rehding, Hugo Riemann, chapter 4. Riemann's analytical practice cuts both ways, of course: he also uses his theory to demon­ strate the alleged violation of his universal laws in composers like Berlioz. See ibid., 152– 156. (33.) It is important not to monumentalize neo-Riemannian theory as a single practice— there are notable instances of transformational approaches to chromatic harmony that do not fit this description. David Lewin, for example, never speaks of “disunity” (or “unity,” for that matter) in his harmonic-transformational writings; both he and David Kopp fur­ ther employ transformational approaches to explore specifically tonal characteristics of chromatic passages (as I have in my work). I base my discussion here on the influential species of neo-Riemannian analysis made popular by Cohn, within which the values out­ lined above have been remarkably consistent. (34.) Riemann's “universal” classicism is of course simply Viennese classicism (the roots of which he repeatedly traces back to the Mannheim symphonists, especially Stamitz). On the nationalist motivations behind this project, most explicit in the 1890s, see Rehding, Hugo Riemann, chapter 4. (35.) Ibid., 110. Page 24 of 27

Riemannian Analytical Values, Paleo- and Neo(36.) In addition to Rehding, see Scott Burnham, “Method and Motivation in Hugo Riemann's History of Harmonic Theory,” Music Theory Spectrum 14.1 (Spring 1992): 1– 14. (37.) Rehding, Hugo Riemann, 9. (38.) In addition to the Garden of Eden passage, the idea of spatialized boundaries to compositional possibility emerges vividly at the end of Riemann's history of nineteenthcentury music, published in 1901. After a negative assessment of Richard Strauss, he writes, “But one hopes that this trend [toward program music] has reached a boundary with Strauss, at which it must turn back.” Hugo Riemann, Geschichte der Musik seit Beethoven (1800–1900) (Berlin: W. Spemann, 1901), 759. “Doch steht zu hoffen, daß diese Richtung mit Strauß an einer Grenze angekommen ist, die zur Umkehr zwingt.” (39.) Rehding, Hugo Riemann, 39. (40.) Hugo Riemann, Beethovens Streichquartette erläutert von Hugo Riemann (Berlin: Schlesinger, 1910), 129. “Verständig interpretiert giebt der Satz keinerlei Anlass, von Zerissenheit und schwerverständlichem Aufbau zu sprechen, zeigt vielmehr deutlich das normale Gerüst der Sonatenform.” (41.) Daniel Chua, The “Galitzin” Quartets of Beethoven (Princeton: Princeton University Press, 1995), 201. (42.) Cohn, “Introduction,” 167. (43.) The rise of neo-Riemannian theory can be read in one sense as a savvy disciplinary response to the challenge of New Musicology, in which the analytical tools of a discredit­ ed high-modernist canon are turned toward new interpretive ends in the very repertory prized by critical musicologists, with certain buzzwords retooled along the way. (44.) Richard Cohn, “Uncanny Resemblances: Harmonic Signification in the Freudian Age,” Journal of the American Musicological Society 57.2 (Fall 2004): 285–323. (45.) Cohn, “Introduction,” 169. (46.) Charles Fisk, Comment & Chronicle, 19th-Century Music 23.3 (Spring 2000): 301. (47.) Richard Cohn, Comment & Chronicle, 19th-Century Music 23.3 (Spring 2000): 303. (48.) Allen Cadwallader and David Gagné, for example, introduce the functional cate­ gories T, int (for “intermediate”), and D in their Schenker textbook without once mention­ ing Riemann by name. Allen Cadwallader and David Gagné, Analysis of Tonal Music: A Schenkerian Approach, 2nd ed. (New York: Oxford University Press, 2006). (49.) As Riemann put it in the fifth edition of his Musik-Lexikon: “Functions…describe… the various significances that chords possess, depending on their position [with respect]

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Riemannian Analytical Values, Paleo- and Neoto the tonic.” Translated in Rehding, Hugo Riemann, 188. See also Kopp, Chromatic Transformations, 99. (50.) This discussion raises a perennial question in Riemannian exegetics: Are his func­ tions labels for chords, relations, or syntactic categories? The present study opts for the second choice, interpreting functions as symbols for relational paths from the sounding chord to the tonic, following Lewin (Generalized Musical Intervals and Transformations, 177) and Hyer (“Tonal Intuitions,” 99–107). Cogent discussions of the function-as-chord versus function-as-category/relation problem may be found in Mooney, “The ‘Table of Tonal Relations,’” 102–108; Daniel Harrison, Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of Its Precedents (Chicago: University of Chica­ go Press, 1994), 266–276; and Rehding, Hugo Riemann, 61 and 78–79. (51.) Steven Rings, Tonality and Transformation (New York: Oxford University Press, 2011). (52.) Michael Siciliano, “Neo-Riemannian Transformations and the Harmony of Franz Schubert” (Ph.D. diss., University of Chicago, 2002); Jack Douthett and Peter Steinbach, “Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition,” Journal of Music Theory 42.2 (Fall 1998): 246–249. (53.) I maintain the neo-Riemannian transformational letters here (R, P, and L) for famil­ iarity, even though they create some dissonance with Riemann's own terms. A formal note: once a D or S is added to the network, its underlying graph is downgraded from “path consistent” to “universally realizable,” per Julian Hook's terminology in “Cross-Type Transformations and the Path-Consistency Condition,” 29. (54.) The latter represents the German-sixth chord, which, as we noted, Riemann would interpret as an applied dominant; the quotes around the A♭+ node in the example indi­ cate that the alterations to the chord have significantly obscured its triadic basis. (55.) It is hard to gain a sense of such extremity without a tonic center from which to measure such things, as in many neo-Riemannian “de-centered” spaces. The present ap­ proach thus restores the “distortions” created by a tonic that Brian Hyer—in an influen­ tial move—explicitly eliminated from his renewed, de-centered Tonnetz in “Reimag(in)ing Riemann,” Journal of Music Theory 39/1 (Spring 1995): 127–128.

Steven Rings

Steven Rings is an associate professor of music and the humanities at the University of Chicago. His research focuses on transformational theory, phenomenology, popular music, and voice. Before turning his attention to music theory, he was active as a concert classical guitarist in the United States and Portugal. His book Tonality and Transformation (Oxford 2011) won the Emerging Scholar Award from the Society for Music Theory.

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit   Robert C. Cook The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0019

Abstract and Keywords This article provides an analysis of Cesär Franck's Le chasseur maudit, which serves as an extended and elegant reflection on the potential and limitations of various analytical frameworks. This article is placed with respect to notions of chromatic music, specifically on the idea that chromaticism poses analytical difficulties that Riemannian and neo-Rie­ mannian perspectives are particularly well suited to address. After discussing the work from both linear and functional perspectives, and examining the conceptual problems that attend each, the article then demonstrates how a contextual, neo-Riemannian view can capture the work's important gestures, and offers a balance between a desire to un­ derstand the work as a reflection of an orderly and coherent relational system, and the need to engage the aural experience of the music. Keywords: Cesär Franck, Le chasseur maudit, chromatic music, Riemannian perspectives, neo-Riemannian per­ spectives

A Rhenish count rides out to hunt on a Sunday, violating the sanctity of the Sabbath. Sud­ denly, riding beside him appear two other knights: a bright, fair one to the right, and a dark, ominous one to the left. They are the proverbial angel and devil perched on one's shoulder, personifications of the count's good and bad consciences. This incident and its consequences are the program for César Franck's symphonic poem Le chasseur maudit (“The Accursed Huntsman”), which closely follows the original ballade (ca. 1778) by Got­ tfried August Bürger.1

Ex. 19.1a. Franck, Le chasseur maudit, mm. 114– 117.

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit The count's ghostly good conscience pleads with him to break off his blasphemous recre­ ation: (p. 513) Schlecht stimmet deines Hornes Klang,

(“The sound of your horn harmonizes poorly,”

Sprach der zur Rechten, sanften Muts,

The one on the right said gently,

Zu Feierglock und Chorgesang.

“With the solemn bells and hymns.

Kehr um! Erjagt dir heut nichts Guts.

Turn back! You win nothing good for yourself today.

Laß dich den guten Engel warnen,

Heed the warning of the good Angel,

Und nicht vom Bösen dich umgar­

And do not bind yourself to the Devil!”)

2

nen!

“Harmonizes poorly”? Franck's evocation of galloping horses and the count's strident horn is, ironically, an aggressive fanfare firmly in G minor. Example 19.1a shows a portion of this passage. The good knight's admonition, with its reference to the sacred music of a solemn high mass, is triadic, but by contrast to the fanfare, chromatic and modulatory. This music appears in example 19.1b. The passage leaves the fanfare's G minor through a 5–6 exchange, then pairs of triads related by major third lead to a tonicization of F minor. After repeated confirmation of F minor, parallel dominant seventh chords on F and F♯ lead to music in B minor. Throughout, the voice-leading is generally by chromatic semi­ tone. In this chapter, we seek to understand how example 19.1b, like the good knight's plea, could be sehr sanft—not stretching diatonic expectations very far, at least from measure to measure—and yet lead to unsatisfying descriptions under traditional analytical meth­ ods. We shall explore theoretical questions raised when we interpret the passage first in a contrapuntal way and then in a functional way. Sorting through the resulting observa­ tions, I shall then recommend a system of neo-Riemannian transformations, one that mod­ els more directly our intuitions about the passage and its context.

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit

Difficulty

Ex. 19.1b. Franck, Le chasseur maudit, mm. 129– 140.

That one might find the ways example 19.1b is chromatic remarkable speaks to the com­ mon attitudes toward chromatic music: that it is unstable, diffuse, and requires much ex­ planation, often with reference to some deeper diatonic structure. We can imagine that pairs of metaphors like stable/unstable, clear/unclear, and ordered/disordered lie at a deep level in theoretical thought about harmony and chromaticism (p. 514) well back into the nineteenth century, even in the writings of theorists seeking to legitimize chromatic practices.3 In response to historical characterizations of chromatic techniques as obfusca­ tory, destabilizing, or even morally questionable, we develop new models that expand, generalize, or replace theoretical categories in order to elucidate, anchor, and discipline intuitions about chromatic music. Still, prominent among our responses is to begin with the idea that chromatic music is difficult, and then to ask why our analytical languages cannot verbalize any logic—or better, grasp the intuitive sense—made by chromatic mu­ sic. We want to make sense of the music; we feel that such must be possible. But if the possible statements bring us close, we are not close enough. Framed in terms of existing analytical languages, the argument goes, the music is difficult, elusive, even disorderly.4

“Hugo-Riemannian” and Neo-Riemannian ideas We shall see below that a functional interpretation of example 19.1b in the manner of Hugo Riemann circa 1890 is unsatisfying, but Riemannian ideas will exert some (p. 515) influence on our transformational interpretation. Since the 1980s, theorists aiming to cut through “difficulty” in their accounts of chromatic harmony have found two aspects of Riemann's harmonic theories useful in structuring their conceptual frameworks: (1) the

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit decoupling of categories of harmonic effect from specific sonorities, and (2) the listening subject—listening as mental activity, as an exercise of agency.5 Within this Riemannian framework, neo-Riemannian theorists generalize the first aspect to remove the impediment of an acoustic tonic, and generalize the second aspect to regis­ ter apprehension of motivic and voice-leading paths. An intuitive, Gestalt recognition of closure, described in mathematical terms, takes the place of—or shares place with—clo­ sure with reference to key. For the capability of triads to communicate function with respect to a tonic, Brian Hyer substitutes the capability of functional transformations between triads to communicate tonal coherence independent of a governing diatonic collection. Writing of Wagner's Sch­ lafakkorde music from Die Walküre, Hyer says that the progression of major triads E–A♭– C–E “prolongs [E major's] tonal significance as a tonic.”6 Though each pair of triads in the progression is locally intelligible as “a move to the parallel major of the mediant,” as an LP (Leittonwechsel–parallel) step, in diatonic terms all three pairs together compose out no single tonic. Instead, the coherence of the prolongation derives from the algebraic closure of the LP cycle: “the algebraic group imparts an immediate intelligibility to trans­ formational relations between harmonies, however remote from one another those har­ monies might appear to be. In this sense, tonal coherence does not require a piece to elaborate a single prolonged tonic, but rather that we regard relations between har­ monies as being tonal…..”7 Hyer's model strongly implies a Riemannian active listening subject in pursuit of tonal intelligibility, coherence in a harmonic domain structured by consonant triads as Klänge. There is no reason this must be the case. Richard Cohn's hexatonic system models incre­ mental smooth voice-leading routines that distinguish triads from all but a few other types of pitch-class collections. These routines are familiar from diatonic tonal practice, and they structure the set of consonant triads identically to Hyer's algebraic group, but Cohn focuses explicitly on them as relations between pitch-class sets. Similarity to tonal relations, relations between Klänge, is secondary.8 To some readers, the reliance on group algebra in neo-Riemannian theory shortchanges musical intuitions and practices in favor of mathematics. But this is to miss the point of the algebra as one instrument employed when interpreting music in the transformational mode. For much music composed from the later seventeenth to the late nineteenth centu­ ry, the rich, thick, tangled, historically contingent, and problematic-but-demonstrably-effi­ cacious “common practice” language suffices clearly to set, at the very least, the terms for disagreement among musicians. With a relatively thin layer of a priori theoretical, practical, and historical soil in which to cultivate an interpretation of music from the late nineteenth century on, however, the algebraic models and locutions provide at least some of the necessary phenomenological compost.9

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit (p. 516)

Contextuality

Our approach in this chapter is neo-Riemannian in the sense just outlined. The music in example 19.1b is not difficult, elusive, or disorderly. Contrapuntal and functional analyses of the passage, as we shall see presently, clearly verbalize pertinent tonal intuitions. The problem with these statements is that for all their richness—perhaps too rich for the size of serving—they seem to miss the point of the music. They do not promote the contextual features from which we conceive of the musical objects that are important. They do not closely follow the topography of the space these objects inhabit. I propose self-conscious­ ly to hear this music as contextual in Milton Babbitt's sense of the word. I propose we hear this music “defining very many of its own premises within itself, even while the modes of connection, the harmonic aspects of it, the contrapuntal aspects, the rhythmic aspects [are] still pretty much defined within just a generally shared communal frame­ work of tonality.”10 A great part of the difficulty traditional tonal theories have with chromaticism is what Leslie Blasius has called (with reference specifically to Schenker), the “masked” episte­ mology of music theory.11 I take Blasius to mean that, among the kinds of things there are to know about music and the kinds of ways there are to do that knowing, there is much that is tacitly assumed, not subjected to critical scrutiny, or conceived of in such ethical terms that it is exempt from question. Transformational theory is particularly suited to addressing the problems of chromatic music because the approach encourages one to draw analytical categories from intuitive responses to music heard and played, and then to extend intuition, through some logical formalization, toward a model of a domain of possible relations through which some particular music moves.12 At the same time, cen­ tral to Lewin's theoretical project (which, in a general sense, is coextensive with transfor­ mational theory itself), from the first musings on the “transformational outlook,”13 is awareness of the underlying and possibly mutually exclusive metaphors informing theo­ retical concepts and their use in analysis. This concern is conceptually prior to any specif­ ic musical situation. Thus Lewin's project is, in contrast to Schenker's and those of others (Lewin himself looks as far back as Zarlino), epistemologically open, laying bare the con­ tents of its analytical categories and acknowledging the limits of their application.14 I want to worry this issue a bit more here because, toward the end of the chapter, we will need to address questions about what our various interpretations tell us about Le chas­ seur and why the contextual approach is warranted. Insistence on a diatonic tonal center is certainly not among the masked epistemological elements of traditional theories. In­ deed, beneath the nineteenth-century concerns voiced in moral or ethical terms, there are clear technical concerns that the expansion of chromatic techniques in music of the middle and late nineteenth century separates (p. 517) one's experience of the music from received practice and the received axioms of music theory. The dominant-tonic cadence may no longer define the boundaries and levels of a tonal hierarchy; the categories of consonance and dissonance, particularly in a contrapuntal context, may lose significance as resolutions cease to be obligatory. Not only are the pillars of diatonic theory destabi­ lized, their conceptual priority is threatened, and with it, the epistemological order: the Page 5 of 39

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit aural effect of particular progressions, their motivic or thematic character, and their par­ ticipation in a contextually defined syntax—rather than the extent to which they support a diatonic tonal center—are factors determining their status as structurally fundamental.15 In response, as Steven Rings has recently pointed out, some theorists performing trans­ formational interpretations of chromatic music have set aside the apperception of tonality or demoted it to a rhetorical or local status rather than a structural one. Rings reads these moves as assertions of tonal disunity and connects them with issues of “structural immanence.”16 Schenkerian theory and transformational theory are epistemologically complementary, according to Rings, because the former is makes claims of immanence while the latter makes esthesic claims. But, Rings writes, “To tie transformational ac­ counts to immanent claims about musical structure [by using such accounts to find other­ wise missing coherence] is, in my view, to sever them from the analytical strategies under which they flourish.”17 I think neither that the distinction is as sharp nor that the danger is as imminent. Rings himself notes that though the “discursive tradition” of Schenkerian theory (he could easily have said fundamental bass theory or functional theory) promotes immanence, Schenkerian analyses can certainly be read as esthesic statements.18 The dif­ ference between traditional tonal approaches and transformational approaches is, again, one of epistemological openness. The Schenkerian attitude is epistemologically closed in part because it denies—explicitly or implicitly—the existence of knowledge, or kinds of knowledge, about some music that are not expressed in the interpretation. The transfor­ mational attitude explicitly acknowledges its own provisionality in its very formalism.19 By virtue of its aim for epistemological self-awareness on the part of the analyst, transforma­ tional analysis ideally aims neither to refute nor replace more traditional modes of analy­ sis. Lewin notes that a “perception” of a musical moment in his model of a phenomenolog­ ical analytical method includes not simply the auditory input of a performance, but the in­ fluences of one's theoretical and historical context as well.20 Thus, when I propose a con­ textual interpretation of example 19.1b below, I do so not simply because the tonal inter­ pretations we do first are inadequate. Rather, our very engagement with Schenkerian contrapuntal thinking and Riemannian functional thinking drives our interest in alterna­ tive modes of explanation for the chromatic features of the passage. The contrapuntal and functional hearings are part of the analytical experience, along with the contextual hear­ ing, not merely missteps on the path, or straw men to be knocked down.

Contrapuntal and Functional Interpreta­ tions (p. 518)

As we listen to the music in example 19.1b, there are three features that we notice imme­ diately. First is the sequence in measures 129–134 and its characteristic leading-tone res­ olutions across the bar lines. Absent the F♯ in the bass of measure 133, we very clearly hear the sequence shown in example 19.2.21 The parenthetical accidentals in the third measure of example 19.2, if played, would preserve the major mode of the previous mea­ sures. Though the major-third pairs that make up this sequence are not diatonic, the Page 6 of 39

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit smooth voice-leading and the leading-tone resolutions in the top voice of Franck's version (example 19.1b) are so gentle that we are hardly aware that tonal ground rules have changed when the passage settles into the strictly diatonic cadential figures tonicizing F minor in measures 134–139.22 The second thing we notice is the mode switch from the major of the first two iterations of the sequence (measures 129–132) to the minor of the last (measures 133–134). This and the stepwise descent from C5 to F4 (measures 134–136) drains momentum from the music, leaving it coasting in F minor. Had Franck continued the sequence of major-third pairs as he began, the passage would have returned to the opening E♭ major. Why, we might ask, would he make this switch? He could easily move to the B minor with which the excerpt closes from E♭ major; after all, the passage opens with the progression E♭ ma­ jor–B major. The third salient feature is the pair of parallel dominant seventh chords on F and F♯ in measure 139. They move by so quickly, however, that our general impression of the pas­ sage as smooth and orderly remains undisturbed.

Ex. 19.2. Model sequence for Franck, Le chasseur, mm. 129–134.

To these impressions I would add a fourth, the significance of which will become clearer when we study more of Le chasseur below: the music preceding example 19.1b is in G mi­ nor (it is the complete fanfare music, of which example 19.1a is the theme), and the music following example 19.1b is a transposition of the G-minor music to B minor. Thus the structural function of the passage is to make a large-scale major-third pair with a se­ quence of local ones. The undiatonic character of these progressions and the associated common notion of chromaticism as “disorderly” are at odds with our hearing of the pas­ sage as a whole. This is not to say that this music is diatonic; rather, its easy character does not reflect its harmonic complexity. Furthermore, that one can recognize the first six of the twelve measures (p. 519) as a sequence submerges their complexity beneath a wash of convention: we are accustomed to granting the interior of a sequence tonal license, at­ tending to the relation obtaining between its endpoints.23 Thus the deployment of the chromatic relations themselves draws attention away from their chromaticism. One would like to make an interpretation that accounts for this structure without burying the serene orderliness of the surface.

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit

A Contrapuntal Interpretation

Ex. 19.3. Franck, Le chasseur, mm. 129–140, (a) foreground and (b, c) two possible middleground sketches.

Example 19.3a shows a foreground sketch of the passage, and it recognizes a far more complex situation than was apparent at first.24 The sequence of major-third pairs comes through in the upper-voice arpeggiations, but the sequential character appears to be (p. 520) nothing more than a surface feature. Instead, the sketch asks us to hear the first two major-third pairs performing an octave transfer of D4 to D5, elaborated by E♭ as a neighbor note. Under the arpeggiation of E♭ major in measure 129, the E♭ neighbor note shifts enharmonically to D♯. The shift avoids a diminished third with the following C♯, which becomes a lower neighbor to D resolved an octave higher. In this hearing, the third major-third relation between A minor and F minor loses status in favor of the minor-third relation between D major and F minor. The upper voice of the sketch highlights this new relationship first by the arpeggiated minor third between D5 and F5, and second through a reversal of the octave transfer, bringing F5 down to F4. There are significant issues for the interpretation of the passage on which this sketch re­ mains equivocal, as indicated by the frequent question marks beside annotations, and the lack of indication, either by stem length or slurs, of the relationship between F♯ and F♮ in

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit the bass. Certainly both the F and F♯ dominant seventh chords cannot exist at the same structural level. Examples 19.3b and 3c offer two possible middle-ground interpretations of the fore­ ground sketch. Example 19.3b accepts the arrival on and expansion of F minor in mea­ sures 134–139 as clues to its structural importance. In this interpretation, F♮ in the bass is a lower neighbor to G, while F♯ is first a chromatic passing note between G and F♮, and second the root of a local dominant of B minor. An important consequence of interpreting F♯ this way is the disappearance of V6 in measure 132. The upper-voice D, then, is no longer the resolution of a neighbor figure on 5̂, but becomes an inner voice. Instead, E♭ is a diatonic passing note from the initial D to F, just as its counterpart in measure 139 is a passing note returning from F to D. The end of example 19.3b suggests that B minor takes the place of an expected B♭ major, and thus that F–B♮ in the bass is not an augmented fourth but a chromatic inflection of the perfect fourth F–B♭. In other words, the governing tonic remains G minor and the expected descent in the upper voice remains the same; on­ ly the local tonicizing action of the F♯7 in measure 139 inflects III to ♯III. Example 19.3c offers a different hearing. We should hear F♯ in the bass as a lower neigh­ bor to G, and F♮ as a chromatic lower neighbor to F♯. E♭ accomplishes its purpose early as an upper neighbor to D; the final E♭ in measure 139 is a chromatic inflection of E♮, which itself is an upper neighbor to D. F♮, then, in both the upper and lower voices, is a chro­ matic event. In the lower voice, as noted, it is a neighbor note; in the upper voice, it is a chromatic arpeggiation from D, the root of V. Finally, example 19.3c asserts that F♯7 in measure 139 is an inflection of V6. Example 19.3c has in its favor the way it emphasizes smooth semitonal counterpoint across the passage. It encourages us to hear the third of G minor, B♭, moving up to B♮, the root of B minor; it encourages us to hear the root of G minor moving down to F♯, the fifth of B minor; and it encourages us to hear the common-tone D in the upper voice. Example 19.3c also picks up on one aspect of the major-third pairs we recognize as important in this music: if we consider the G minor and B minor endpoints to be bound together mo­ tivically, F♯ in the bass serves both as the agent of departure and the agent of return to the G-minor/B-minor “home base.” By contrast, example 19.3b privileges F♮ in the bass and obscures our appreciation of (p. 521) the semitonal voice-leading. Still, it does res­ onate with the intuition that the relaxation into F minor in measures 134–138 is a signifi­ cant event. We need not choose between examples 19.3b and 3c, or among them and other possible interpretations. They serve to illustrate how, if we presume a strictly diatonic background in G minor, the possible explanations for its elaboration in part through the music of ex­ ample 19.1b and other similar passages in the piece furnish an inviting salon for analyti­ cal debate on the relationship of chromatic diminutions to their diatonic underpinnings. Does this not, however, miss some intuitively important features of the passage? What of the sequential triad pairs? What of the shift to minor in measure 133–134?

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit We shall find it difficult to answer these questions from our contrapuntal perspective be­ cause several crucial surface chords belong “to the sphere of counterpoint.”25 Thus not only the B- and F♯-major triads of measures 130–131 (see example 19.1b), but perhaps even V6 in measure 132 disappear as independent entities. In their place is a complex voice-leading structure linking the Stufen G and B minor that but dimly reflects the music we hear.

A Functional Interpretation Example 19.4 shows a functional interpretation of the passage. Each of the triad pairs, in­ cluding A minor–F minor (measures 133–134) is an expression of the relationship D–T, that is, dominant–tonic. So too is the large-scale pair consisting of the G-minor tonic of the preceding music and the new tonic B minor in measure 140, though in reverse: T–D. The dependence of this reading on the leading-tone resolutions in the upper voice is re­ flected in the enharmonic reinterpretation of B♭ (measures 129–130) from upper fifth in relation to E♭ to upper seventh—in other words, A♯—in relation to B♮. Riemann's ap­ proach to dissonance explains how we can hear both a half-diminished-seventh chord and an A-minor triad in measure 133. Coming from D major as dominant in measure 132, we understand F♯ ø 7 as V9 without the root, ; in relation to the following F minor, however, and attending to the motivic parallel of the tune with measures 129 and 131, we hear A minor with an under-seventh DpVII. The interpretation falters, however, in measures 139–140, where it lacks much utility in explaining the simple parallel motion from F7 into F♯7 and on to B minor. While we might have little difficulty understanding F7 as the dominant of B♭ major, (D7)[Tp], given the phenomenal weight of the hundred-some bars in G minor preceding the passage in ques­ tion, it is not clear what dimension of aural skills should be brought to bear reinterpret­ ing this dominant of an absent relative major as a subdominant seventh of the subdominant's Leittonwechselklang, or (S7). Indeed, this secondary subdominant has an added minor seventh, encouraging us to hear it as a dominant.

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit

Ex. 19.4. Franck, Le chasseur, mm. 129–140, func­ tion analysis.

If we attempt to simplify the function of F7 in relation to B minor, we dive into waters even more murky. Given the salience of leading-tone resolutions in the passage, it makes sense to consider how F♮ might be reinterpreted as E♯, leading to F♯. To (p. 522) so con­ sider is to hear F7 as the dominant of the dominant. It would be a drastically altered chord, however, with a missing root (there is no C♯), an added ninth (E♭ becomes D♯), and a raised fifth and seventh (A and C become G and B♯, respectively). If we are in the busi­ ness of supposing roots, we might avoid enharmonicism and simply treat F7 as a D-minor chord, the relative major of B minor, Tp, with a lowered third (F♮), a seventh, and a ninth, but missing the root. Neither of the alternative readings suggested seems particularly rel­ evant to our listening experience.

Conceptual Problems in Both Approaches From the contrapuntal perspective taken in example 19.3, we were unable to disentangle the sequential triad pairs, the coincident leading-tone resolutions, and the underlying chromatic but smooth and intelligible semitonal voice-leading from the concept of the Stufe, under which only some of the chords have true triad status, while the remaining ones are voice-leading simultaneities. This is not to say that such a tonal ontology is ulti­ mately flawed or analytically useless—it is clear that musical understanding is aided by viewing certain events in certain musical contexts as elaborative. Rather, it means that in a fairly straightforward passage from the late (p. 523) nineteenth century, featuring typi­ cal (for the period) major-third relations and using no unprepared or unusual dissonance, the insights available from this perspective do little to build firm analytical support for those aspects of the music that are intuitively important. The functional approach in example 19.4 deals better with the major-third pairs, but the ultimate requirement that local functions be directed to the global tonic produce not only the difficult functional snarl in measures 139–140, but also a rather uninformative “back­ ground” interpretation of motion from essentially tonic function to essentially dominant Page 11 of 39

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit function, or T–D. This shift takes place almost immediately, and we remain under D throughout. To an extent, the Schenkerian sketches of example 19.3 are much more rep­ resentative of our phenomenological response: “return” is a common metaphor for mo­ tion in music, and the return to F♯ (albeit sudden) in measure 139, the return of the G-mi­ nor theme in B minor in measures 140ff, and the return of D in the upper voice are strongly salient features in example 19.1b. Example 19.4 does little to support this experi­ ence. Examples 19.3 and 19.4 do support some aspects of the intuitions expressed above in suggestive ways. Example 19.3 shows how voice-leading by semitone, most immediately apparent in the music as the leading tones in the top voice resolve across the bar lines, governs virtually the entire texture. Both examples 19.3b and 19.3c, though the latter more strongly, point to the importance of F in the bass as a sort of pivot around which the harmony moves away from and back to harmonies that include F♯. Example 19.4 preserves few of these local details, but draws functional parallels between the majorthird pairs of the sequence and the endpoints of the passage, G minor and B minor. These contributions to our understanding of the music will be useful in the construction of a transformational model below.

Contextual Interpretation Our contextual neo-Riemannian approach will roughly follow a trajectory from direct re­ flection upon our experience with the music and the traditional interpretations above, to selection of salient events and gestures to serve as models for a family of relations, and then through a process of formalization and generalization toward a model of the musical space through which the piece moves (and we move with it).26 At first, we will concen­ trate our efforts on the first two stages, proceeding as intuitively as possible, considering in what clear, direct fashion we may represent salient attributes of the music in example 19.1b. Later we will see how our interpretation plays out across a larger portion of Le chasseur maudit.

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit

Ex. 19.5. Franck, Le chasseur, mm. 129–140, net­ work of basic motion.

We begin by considering the large-scale motion of the passage. The passage grows out of music in G minor, and ends in music of B minor. As we noted earlier, these endpoints form another major-third pair, embracing the major-third pairs within the passage. One way to depict the salience of the major-third pairs in this (p. 524) music is to treat them as a sin­ gle object. Shortly, we will define more precisely how major-third pairs can be a unit; for the moment, we shall simply enclose labels for the triads in each pair in a single node. Ex­ ample 19.5 does this for the major-third pair forming the endpoints of the passage. The circling arrow reflects our intuition of departure from and return to the major-third pair {G-, B-}. From our contrapuntal analysis, we came to consider F♯ as the agent of departure and re­ turn. Hence the network in example 19.6a shows us leaving G- by way of F♯+, and then coming full circle to B- from F♯+. The sense of leaving a place may sometimes be matched by a sense of returning through constant travel in one direction; in other words, by moving roughly in a circle.27 This sort of motion is what example 19.6a captures. At Page 13 of 39

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit other times, the sense of returning is one of backtracking, returning over the same ter­ rain by which one left. If we imagine the bass notes G and B beneath the endpoints of the motion in example 19.1b as belonging together—in other words, as members of the majorthird pair {G-, B-}—then we may imagine further that the F♯3 between G3 and B3 in the bass of example 19.1b marks a turning point, where we reverse our descending motion from G and ascend to B. Example 19.6b captures this idea, depicting the motion of the music is from G- and back to B-, rather than the unidirectional motion of example 19.6a.

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit

Ex. 19.6. Franck, Le chasseur, mm. 129–140, effect of “return” as circular and as reversal of direction.

Example 19.7 fleshes out our representation by adding additional major-third pairs. We shall add even more below—indeed, the two empty nodes are there to foreshadow the eventual shape of the model—but for the moment, economy will be helpful. By arranging some nodes inside the larger circle and some outside, we distinguish between major-third pairs of major triads and of minor triads. Though below we shall distribute the nodes more evenly on the page, here we wish to respond (p. 525) to our earlier interpretation of the sequence as being disrupted by the shift to minor in measures 133–134.

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit

Ex. 19.7. System of several major-third pairs.

Recall that if the sequence were to continue in major, we would navigate the network in the manner of example 19.8a. Beginning with E♭+ and B+ (corresponding to measures 129–130 of example 19.1b), we would move counterclockwise to F♯+ and D+,t just as Franck's music does (measures 131–132), but then to A+ and F+ instead of A- and F-. The next would be C+ and A♭+, before returning to E♭+. Indeed, Franck might have written the passage just this way, then taken the (E♭+, B+) motion of measures 129–130 as a model and moved the upper voice through B♭-or-A♯ to B as the root of B minor. Instead, the passage takes us through the network in example 19.8b. After F♯+ and D+, we cut across to A- and F-, then back to the F+ the sequence leads us to expect. Our counter­ clockwise motion has been arrested, however, and we move back the other way to F♯+. Note that the example distinguishes between (p. 526) moves between nodes of the same mode and moves between nodes of opposite mode using heavy arrows for the former and

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit lighter arrows for the latter. Example 19.8b responds to our tentative intuitions of return that grew out of the contrapuntal analysis above.

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit Ex. 19.8. Franck, Le chasseur, mm. 129–140 net­ work interpreting (a) the model sequence of Example 19.2, (b) mm. 129–139, and (c) the preceding G-mi­ nor and following B-minor music.

Example 19.8c completes this first attempt at a contextual interpretation by adding the node containing G- and B-. As it stands, the model depicts in gestural terms our intuitions that major-third pairs are motivically important and that the (p. 527) mid-passage switch to minor has some significance. To complete the model, we first need to find a way to ac­ count for the importance of semitonal voice-leading, brought first to our attention at first by the leading-tone resolutions across the bar lines in the sequence (example 19.1b, mea­ sures 129–134), and later highlighted by our contrapuntal interpretation (particularly ex­ ample 19.3c). We should also define a group of transformations through which to formal­ ize our intuitions. This group should treat major-third pairs as identical, distinguish be­ tween transformations that preserve mode and transformations that switch mode, and be sensitive to semitonal voice-leading.

Ex. 19.9. Franck, Le chasseur, mm. 129–134, semi­ tonal voice leading in the tune.

The salience of leading-tone resolutions in the tune of example 19.1b was among the first features of the passage we noted above. Example 19.9 examines in detail the voice-lead­ ing of the tune during the major-third sequence. Recall that our contrapuntal (p. 528) interpretation made a strong case for the importance of semitonal voice-leading between G minor and B minor: G moves down to F♯ in the bass, B♭ moves up to B♮ in the tenor, and D is a common tone in the upper parts. Every major-third pair in the sequence has a simi­ lar voice-leading pattern: one voice moves down by semitone, another up by a semitone, and the third holds as a common tone. We can represent these voice-leading intervals with integers modulo 12. Thus, a semitone down is 11; a semitone up is 1, and the com­ mon tone, of course, is 0. Now, if we sum these voice-leading intervals, we get a single in­ teger that represents the total distance spanned by the voice-leading between the two tri­ ads in question. The sum for each major-third pair is 0 because 11 + 1 = 12 = 0mod12. Cohn calls these numbers “directed voice-leading sums.”28 They fulfill much the same function as the terms, “parallel,” “oblique,” and “contrary” with regard to voice-leading, though with a higher degree of specificity and less of the appeal to traditions of contra­ puntal practice.29 Table 19.1 summarizes all directed voice-leading sums from G-. In order to distinguish between transformations that move between triads of the same mode, and those that move between triads of opposite mode, we shall call one sort X and the other Y. X will stand for mode-switching transformations: X as in “eXchanging major for minor or vice versa.” Y will then stand for mode-preserving transformations, in large part because the letters x and y are often associated with one another in colloquial dis­ Page 19 of 39

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit course. We may then use the voice-leading sums described above to distinguish between different Xs and different Ys. Example 19.10 revises the previous network of triad pairs in this fashion, matching directed voice-leading sums from table 19.1 with the appropriate X (mode exchange) or Y (mode preservation). Any given triad may participate in two possi­ ble major-third pairs: for example {G-, B-} and {G-, E♭-} are the two possibilities for G-. The given triad's two major-third partners themselves make a major-third pair: for exam­ ple {B-, E♭-}. Thus our nodes of major-third pairs, to be complete, should contain not two but three triads apiece. The X and Y transformations form an algebraic group equivalent to one devised by Lewin to study operations among pitch classes in octatonic collections and related to one de­ vised by Lewin to study passages from Schoenberg's String Trio, op. 45.30 X1 is the trans­ formation that exchanges a triad for another of opposite mode by directed voice-leading summing to 1, G- to E♭+, for example. X1 is its own inverse, so it is also the directed voice-leading transformation that takes E♭+ to G-. (p. 529) In order to move the other di­ rection around the network, we need X10, which exchanges a triad for another of oppo­ site mode by directed voice-leading summing to 10, G- to F♯+, for example. Table 19.1. Directed Voice-Leading Sums from G− Triad relation

Directed voice-leading sum

(G−, G−)

0

(G−, B−)

0

(G−, E♭−)

0

(G−, E♭+)

1

(G−, G+)

1

(G−, B+)

1

(G−, E−)

3

(G−, A♭−)

3

(G−, C−)

3

(G−, C+)

4

(G−, E+)

4

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit (G−, A♭+)

4

(G−, D♭−)

6

(G−, F−)

6

(G−, A−)

6

(G−, A+)

7

(G−, D♭+)

7

(G−, F+)

7

(G−, B♭−)

9

(G−, D−)

9

(G−, FS−)

9

(G−, FS+)

10

(G−, B♭+)

10

(G−, D+)

10

If X1 and X10 are in the group, then so must be (X1, X10) and (X10, X1). The former moves from one triad to another of the same mode two nodes clockwise in the network. The latter moves from one triad to another of the same mode two nodes counterclockwise in the network. To preserve the distinction between transformations that switch mode and transformations that preserve mode, we define Y3 = (X1, X10) and Y9 = (X10, X1). Continuing in this same manner, we would find the remaining mode-switching transfor­ mations X4 and X7 and the remaining mode-preserving transformation Y6. Transforma­ tions within nodes—between triads of major-third pairs—are Yo.31

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit

Ex. 19.10. Network of consonant triads under the X/ Y group of transformations.

Ex. 19.11. Franck, Le chasseur, mm. 129–140, trans­ formational interpretation.

With a contextual model of triadic space in Le chasseur complete, we may revisit our net­ work interpretation. Example 19.11 revises example 19.8c to account for the more com­ plete transformational model. (The reader may wish to pause here and thumb example 19.1b, referring to it as necessary through the following narration, (p. 530) according to the parenthetical cues.) After moving away clockwise from G- to E♭+ via X1 to begin the passage (measure 129), we begin what appears to be a counterclockwise tour of the sys­ tem by Y3 (measures 129–134). When we reach F+ (measure 139), however, our motion reverses, moving by Y9 to F♯+ (measure 139), then back by X10 to B- (measure 140).

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit If we were to have continued our Y3 tour of the network, we would have returned to E♭+, from which we might have moved either to B-, as suggested above (in example 19.8a) or to the original G- by X1, the same way we began. Instead, our course reverses both its path around the network and the direction from which we reenter the “home” node. The mechanism of this reversal is X4, taking D+ to A- (measures 132–133), fol­ lowed by X1, taking F- to F+ (measures 138–139). This series of transformations effects its own local reversal, causing us to approach F+, which is the next logical stop on the Y3 tour (had it not been interrupted by A– in measure 133), from the “wrong” direction (clockwise, instead of counterclockwise). Thus the shift to minor, through X4, arrests our circumnavigation of triadic space. (p. 531)

Let us recall once again the words of warning to the huntsman: his good conscience says, “Turn back!…And do not bind yourself to the Devil!” Under our model, Y transformations will just keep spinning around the network in the same direction unless stopped by an X transformation. In example 19.11, X4, which steps ahead of the oncoming Y3 sequence, turning it back the way it came, is all that prevents the music from circling back and be­ ginning again (and, perhaps, again, and again…. ).

Incompleteness To summarize these observations, we recall first that problems with our contrapuntal and functional interpretations above arose to a great extent from each approach's difficulty in negotiating between two different concepts of a triad: as an object occurring on the musi­ cal surface, and as a token for some higher level category, be it Stufe or function. Though each interpretation informed our contextual model, our contextual interpretation suc­ ceeded particularly in modeling important features of our earliest intuitions. This result follows not only from careful musicianship, but more generally from the conceptual order­ liness and epistemological transparency of transformational theories in general. Concep­ tual order, as a theoretical desideratum, is strongly supported by the use of algebraic groups to focus analytical attention on a clearly defined set of musical relations. Episte­ mological transparency ascribes high value first to the phenomenally salient in music, those events and relations that emerge as important or fundamental to the understanding of a piece, as we experience it, and second to the network of all possible such relations, while demanding wariness of theoretical assumptions that are not necessarily consonant with intuition or the music in question. An analytical approach built on such principles will produce incomplete analyses because the building of such a model requires that a significant amount of the musical experience be excluded. Indeed, we must remember that our Schenkerian and Riemannian hearings are as much part of the analytical picture we paint of example 19.1b as the transforma­ tional analysis. Unlike earlier analytical approaches, (p. 532) however, (such as Schenker's) in which that which does not fit the model is dispatched as either unimpor­ tant or antithetical to true Art, transformational theory is built on the premises that an analysis seeks to answer particular questions in a particular musical context (not surpris­ Page 23 of 39

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit ing, given its origins in atonal analysis), that in a given piece there may be multiple, even contradictory contexts, that the analyst gives a performance of a piece (in a sense, at a certain level of abstraction), which can only ever be a partial realization of the piece. Inevitable analytical incompleteness is suited to the chromatic-yet-triadic tonal music of the late nineteenth century particularly because we sense that this music often works in the seams between traditional diatonic hierarchies and a more fluid syntax of (only) local­ ly intelligible chord-to-chord relations.32 Also, the ontological status of triads and triad re­ lations, already at issue in tonal theory, can be particularly thorny in chromatic music; transformational theory can help to sort out, though not necessarily choose, various posi­ tions.

Le chasseur maudit From the interpretation depicted in example 19.11 and the accompanying commentary, one could reasonably ask, “Does the music of Le chasseur maudit ever complete a tour of the system?” One could also ask, “Does the count heed the warnings and turn back?” I wish neither to engage in the “trite, simple-minded, unprofitable” leitmotivic identifica­ tion of musical figure with literary character or object33 nor to enter the debate on pro­ gram music. Seeking language suitable for the context of Le chasseur, and recognizing the conceptual sparseness of our transformational model, I turn to Bürger's tale for perti­ nent figurative language. We should be clear on the role of Bürger's poem in relation to Franck's program and our hearing of the piece: we are not concerned here with the narrative power, or lack of same, of music, nor are we concerned with program music as a generic context for Le chasseur. Our account of Le chasseur does not seek justification in the program. Rather, we use the figural language of the tale to “tell the story of the music,” to motivate (ani­ mate, instigate) the technical language. We use both technical and figurative languages to interpret music. Even for Schenker (the goat for all that is wrong with music analysis), figurative description attempts to fill the interpretive gaps where musical experience exceeds the power and scope of technical description.34 By the same token, technical description “serves to amplify and substanti­ ate” figurative description.35 But even this dichotomy may be too simplistic: figurative de­ scriptions are a story, “the incorrigible claims of [which] interweave with the exposition of the empirical data [technical description] to make the fabric of the account,” an account in which scientistic truth is replaced with “plausibility.”36 Indeed, these sorts of stories are necessarily part of satisfactory analyses.37 Speaking of the structure of the piece in terms of the story engages our hearing of the music—a temporal phenomenon—with some immediacy. The riding of the count, the warnings of his good conscience and his renewed galloping after them, and his even­ tual damnation highlight our rhythmic response to the shifting diatonic and chromatic (p. 533)

materials of the music. We do not assert that the music retells Bürger's tale. Instead, we Page 24 of 39

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit tell the story of the music through Bürger's tale. At bottom, it matters little for our pur­ poses precisely how Franck intended to depict Bürger's poem or to follow his own pro­ gram, and indeed it matters little whether we consider the program or the transforma­ tional model as conceptually prior in our own engagement with the piece; when we follow Franck's program through the piece we adopt it as an analytical model, just like our transformational model.38 In Bürger's poem, the warning by the count's good conscience quoted at the beginning of this chapter comes near the beginning of the hunt. After the count's first refusal to listen, they ride furiously across the landscape. As they go they meet three of the count's sub­ jects: a farmer, a shepherd, and a hermit, each of whom pleads with his lord to repent. Immediately after each commoner addresses the count, the right-hand riding companion (the count's good conscience), springs forward to plead with the count once more. Bürger makes a political point here about the good German peasants and morally lax nobility.39 The sower of seeds, or vineyard worker, the shepherd, and the practitioner of quiet prayer in the wilderness are also important Christian images in a poem about blasphemy. Franck's use of the story concentrates on the alternation between riding and pleading, rather than on the progression through various parts of the count's domain. The synopsis of the story published in the first edition of the score reads: Hallo! Hallo! The hunt rushes through the fields [of wheat], the moors, the mead­ ows—“Stop, count, I beg you, listen to the pious hymns”—“No”…Hallo! Hallo! —“Stop, count, I beseech you; beware …. “—“No,” and the cavalcade hurtles on like a whirlwind. Suddenly the count is alone; his horse will advance no more; he blows on his horn, and the horn sounds no more…a lugubrious voice, implacably pronounces the curse: “Sacrilege,” it says, “Run forever in hell.”40 The first plea corresponds to example 19.1b, the second to a transposition of the same music in measures 192–203. Various guises of the riding music alternate with these two passages. The second paragraph of the synopsis corresponds to a large portion of the piece (measures 273–346) during which the music moves slowly from a suddenly calm (“Molto lento”) and quiet B minor to the count's final, accursed, eternal ride in G minor (“Allegro molto,” and in ever-increasing tempi through the closing “Quasi presto”). The riding music is generally diatonic and tonal stable, while the good knight's pleas and the sudden quiet before damnation are chromatic and in nearly constant motion. Thus we may imagine the motion of the piece through our system as gestures illustrating the warnings to the count and the consequences of ignoring those warnings. (p. 534)

Following the music of example 19.1b, the horn call and riding music of measures

76–128, which was in G minor (example 19.1a shows the principal motivic material for this section), returns in B minor (measures 140–191). The differences between the G-mi­ nor and B-minor versions of the riding music are few and almost entirely in the orchestra­ tion. The music of example 19.1b returns as well, transposed up a major third, in mea­ Page 25 of 39

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit sures 192–203. At measure 203, however, the riding music returns in E♭ major rather than E♭ minor, as we might expect from the preceding 127 bars. We will not bother with a transformational depiction of the B-minor section and the transition that follows. The net­ work would be identical to that shown in example 19.11 because measures 140–202 con­ stitute a large-scale major-third pair with measures 76–139. Example 19.12 picks up the score at measure 203. The third presentation of the riding music, now in E♭ major, breaks off suddenly after four bars (measures 203–206). A simi­ larly abbreviated recollection of example 19.1b follows (measures 207–210), leading to a second short riding-music fragment in F major (measures 211–212). Once again Franck recalls the chromatic transition (measures 213–222), but this time doing diatonic duty as an elaboration of V in G minor. The last measure of example 19.12 (measure 223) begins a reprise of the G-minor riding music. Example 19.13 plots the music of example 19.12 as paths through our system of nodes. The arrow labeled “X1 to measure 203” represents a move from music structured by G minor and B minor to the E♭-major music beginning example 19.12. In transformational terms, measures 203–223 constitute an expanded reprise of measures 129–140, a rela­ tionship we can see by comparing example 19.13 with example 19.11. The earlier inter­ pretation traced a Y3 path from E♭+ and B+ to F♯+ and D+, before moving by X4 and X1 to F+ (which would have been, as we noted above, the next step on the Y3 tour). The move back to the {G-, B-, E♭-} node in example 19.11 is then twofold: Y9 then X10. Exam­ ple 19.13 skips directly across the system from E♭+ to F+ by Y6, then returns to G- (in the same node as B-) by Y9 then X10. The shape of the two collections of gestures is the same, moving about the left- and lower left-hand portion of the system, never completing a circumnavigation. Having reached G minor again, a reprise of the riding music works toward B minor (mea­ sures 223–272), including a pause to approach F♯ as V (measures 247–272). Upon reach­ ing B minor, the music begins a long, occasionally interrupted path through tonal space that eventually engages those portions of our system hitherto avoided. Because this sec­ tion of the piece is rather long at seventy-four measures (measures 273–346), the abbrevi­ ated score shown in example 19.14 must suffice. A double bar indicates missing measures that repeat or otherwise continue the material since the last double bar. The reader may wish to keep a thumb at example 19.14 and refer to it methodically while studying the fol­ lowing narration of the passage.

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit

Ex. 19.12. Franck, Le chasseur, mm. 203–223.

Three ascents by minor-third transposition, or Y9 transformation, divide the passage: measures 273–294, B minor to F minor; measures 305–307, E♭ minor to A minor; and measures 312–328, F♯ minor to C minor.41 The constituent triads and transformations are labeled in example 19.14. Versions of a 5–6 neighboring figure (p. 535) in the brass or winds, and reminiscent of the riding fanfare (cf. example 19.1a), distinguish the ascents from the rest of the passage. These figures are marked in brackets in example 19.14. (The arrows beneath the staff in example 19.14 interpret the passage in our transformational system and will facilitate comparison with example 19.15.)

Ex. 19.13. Franck, Le chasseur, mm. 203–223, trans­ formational interpretation.

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit The first ascent from B minor through D minor to F minor breaks at measure 294. A new melodic figure in the low brass at measure 295 (shown in example 19.14 (p. 536) as a tenor voice) begins a descent by semitone root motion from F minor to E♭ minor (measures 303–304). This intrusion of step-wise root motion in a third-dominated context is reminis­ cent of measures 203–223, where the motion was by whole step in the opposite direction: E♭–F–G. Following the arrival on E♭ minor in measure 303, a new minor-third ascent be­ gins in measure 305, this time breaking off after arriving on A minor in measure 307. The full

cadence in F♯ minor in measures 308–310 is remarkable not simply be­

cause it breaks the chromatic sequence but because it does so in a diatonic manner out of character with its surroundings. We have heard F♯ in a remarkable role before. Our con­ trapuntal interpretation in example 19.3c nominated F♯—lower neighbor to G, third of D major, and root of the dominant of B minor—the crucial event in the bass line. Our trans­ formational interpretation in example 19.11 depicted F♯ major as the musical expression of the command “Turn back!” The present music recalls this role by turning back from A minor through C♯ major to F♯ minor. The third ascent soon begins from this F♯ minor—as if the ascent from E♭ minor needed to step back, catch its breath, and try again—this time passing through A minor to C minor in measure 328.

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit Ex. 19.14. Franck, Le chasseur, mm. 273–346, ab­ breviated score.

At the arrival of C minor, the dotted fanfare figure returns in the horns and the tempo quickens. The chromatic transposition scheme is more complex: major-third root motion supports the motivic 5–6 figure, and transposition of each major-third (p. 537) (p. 538) triad pair by a whole tone moves from C-minor/A♭-minor (measures 328– 329) through D-minor/B♭-minor (measures 330–331) to E-minor/C-minor (measures 332– (p. 539)

333).

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(p. 540)

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit

Ex. 19.15. Franck, Le chasseur, transformational in­ terpretation of (a) mm. 273–304, (b) mm. 305–310, (c) mm. 312–346.

Beneath C minor in measures 334–335, the bass ascends to G. Rather than support a ca­ dential six-four chord in the following music, G supports a diminished seventh chord Page 32 of 39

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit (spelled first with a C♯ root, then E) in measures 336–339. Neighboring G in the bass of measures 336–337 we hear F♯ and F, in measures 338–339, only F♯ (together the last echoes of the possibility of redemption promised in example 19.1b?). The goal here is E♭ major in first inversion (measures 340–343), not, as one might expect, a long dominant. The return to G minor in measure 344–346, where literally all hell breaks loose, is thus not diatonically tonal. It is a reversal of the 5–6 (p. 541) (p. 542) exchange that brought us into the first call to repent in measure 129. It also closes a chromatic circuit of triadic space. Examples 19.15a–c interpret the music of example 19.14. The network in example 19.15a depicts the first minor-third ascent (measures 273–294) and the following descent by semitone in measures 295–304. As before, heavy arrows mark mode-preserving Yn transformations and lighter arrows mark mode-exchanging Xn transformations. Though the transposition scheme focuses attention on the Y9 and Y6 transformations—that is, on familiar motion back across the system instead of around it—modal inflections in mea­ sures 295–304 move through all of the unused nodes around the top of the system. E♭ minor in measures 303–305 stands roughly at the midpoint of the passage and is the one occupant of the “home” node containing G- and B- as yet unheard. From a dramatic perspective, E♭-'s appearance looses the demons with whom the count must ride for eter­ nity, in punishment for his blasphemy. From a transformational perspective, it opens the way for a complete circumnavigation of the system. In this sense E♭- has tonic effect, and the second minor-third ascent in measures 305–307 and the F♯-minor cadence in mea­ sures 308–310 are recapitulatory.42 Example 19.15b illustrates. The minor-third ascent moves by Y9 to the node occupied by A-, the node that includes F- and at which the music of example 19.1b made a full cadence before turning back through F♯+. Here, the full ca­ dence in F♯ minor turns back the motion in a similar way. Example 19.15c interprets the remaining music of example 19.14, from the third minorthird ascent beginning on F♯ minor in measure 312 to the return of G minor in measure 346. The ascent moves by Y9 to the node occupied by C-. From there the music of mea­ sures 328–333 dash back across the system by Y6 and return. The final steps through E♭+ to G- complete the circumnavigation of the system suggested—or warned against—in ex­ ample 19.1b. The hunter is damned.

Conclusion The neo-Riemannian interpretation of Franck's Le chasseur maudit presented here re­ sponds to the recognition that chromatic music may be syntactically regular but not ac­ cording to diatonic tonal customs and that the organizing force of tonic harmony—in oth­ er words, the interpretive efficacy of granting conceptual priority to the composing-out of a single harmony—is diminished in this music, in some cases challenged by the compos­ ing-out of a network of sonorities.43 It also adopts the perspective—borne of (hermeneuti­ cally aware, preferably) post-posttonal interpretive practice—on the contextual formal role of motivic features. If neo-Riemannian approaches are hardly Riemannian in their ea­ Page 33 of 39

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit gerness to find other agents of coherence than the tonic Klang, they are vigorously Rie­ mannian in their attempts to account systematically for the experience of the engaged lis­ tener.

Notes: (1.) Franck printed a synopsis of the story in the first edition of the score. I shall explore that program and its relationship to the structure of the piece at the end of this chapter. (43.) For a wide-ranging study of major-third cycles as expressive and structural features, see Matthew Bribitzer-Stull, “The A♭–C–E Complex: The Origin and Function of Chromatic Major Third Collections in Nineteenth-Century Music,” Music Theory Spectrum 28.2 (2006): 167–190. (2.) Bürger, “Der wilde Jäger,” in Bürgers Werke in einem Band, selected and intro. Lore Kaim-Kloock and Siegfried Streller (Berlin: Aufbau-Verlag, 1973), 69. (3.) François-Joseph Fétis, Moritz Hauptmann, and Hugo Riemann all warned of the en­ tropic dangers in progressions that use distantly related chords, irregular resolutions of dissonant chords, and frequent enharmonic reinterpretation. Fétis writes that pervasive and formulaic use of chromatic leading tones and enharmonic reinterpretation, permit­ ting constant and distant modulations, is a “degradation” of the art, and that, in the luxu­ rious orchestration of his day, “their effect is more often to produce exhaustion of the mind and of the organs than to satisfy them.” (“[L]eur effet est de produire plus souvent la fatigue de l’esprit et des organes que de les satisfaire”), Traité complet de la théorie et de la pratique de l’harmonie contenant la doctrine de la science et de l’art, 4th ed. [Paris: Brandus, 1849], 200.) Moritz Hauptmann is concerned with the ontological status of en­ harmonic relations because such relations treat as identical notes the inward natures of which are different and unrelated. Those progressions requiring enharmonic reinterpreta­ tions cannot stand beside “those which depend upon an organic union.” They are “tainted with untruth” and have no “natural life.” (The Nature of Harmony and Metre, 2nd ed., trans. W. E. Heathcote [London: Sonnenschein, 1893; reprint, New York: Da Capo Press, 1991], 167). Interestingly, Hauptmann's principal practical concern with regard to enhar­ monicism is its effect on vocal performance, misleading the singer toward almost certain intonation problems (ibid., 168). Riemann, in a more practical fashion, notes that enhar­ monic reinterpretation of diminished seventh chords is “deceptive [trügerisch]” and should be used sparingly (Skizze einer neuen Methode der Harmonielehre, Leipzig: Bre­ itkopf und Härtel, 1880, 83). Each of these three theorists admits altered chords and enharmonic reinterpretation to the canon of compositional practice—whether such techniques are allowed is not at issue. They worry, rather, of the disordering effect of chromaticism, which they frame in ethical terms. The nature of the worry is clear from the bodily focus of Hauptmann and Fétis. For the former, enharmonicism misleads the singer, whose body is the instrument; for the lat­ ter, enharmonicism debilitates the listener's mind. While early-twenty-first-century musi­ cians are less likely to question the morality of chromatic music, we do, as Daniel Harri­ Page 34 of 39

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit son reminds us, “conceive of chromatic music as representing the dissolution of tradition­ al tonality instead of, say, its culmination” (Harmonic Function in Chromatic Music: A Re­ newed Dualist Theory and an Account of Its Precedents [Chicago: University of Chicago Press, 1994], 9). (4.) See, for example, William E. Benjamin, “Interlocking Diatonic Collections as a Source of Chromaticism in Late Nineteenth-Century Music.” In Theory Only 1.11–12 (1975): 31– 51; Charles Smith, “The Functional Extravagance of Chromatic Chords,” Music Theory Spectrum 8 (1986), 94–139; Harrison, Harmonic Function; and Richard Cohn, “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progres­ sions,” Music Analysis 15.1 (1996): 9–40. All four are aware that the principal difficulty lies less with the music and more with the appropriateness of the languages we use to in­ terpret the music. Gregory Proctor addresses this issue directly in his review of The Se­ cond Practice of Nineteenth-Century Tonality, ed. William Kinderman and Harald Krebs, Music Theory Spectrum 21.1 (1999): 138–139, and Harrison alludes to it in the descrip­ tion of his workshop at the 2006 Mannes Institute for Advanced Studies in Music Theory (supplied in personal communication). (5.) Harrison, Harmonic Function, 11 and 262. See also David Lewin, “Amfortas's Prayer to Titurel and the Role of D in Parsifal: The Tonal Spaces of the Drama and the Enharmon­ ic C♭/B,” 19th-Century Music 7.3 (1984): 336–349; reprinted in Studies in Music with Text (Oxford: Oxford University Press, 2006), 183–200. Lewin imagines intersecting but inde­ pendent tonal spaces structured by Stufen and Funktionen, thus distinguishing between scalar-acoustic and categorical-spatial harmonic effects. (6.) Hyer, “Reimag(in)ing Riemann,” Journal of Music Theory 39.1 (1995): 101–138; see in particular 115. (7.) Ibid, 130. (8.) Cohn, “Maximally Smooth Cycles,” 12–13. (9.) The gardening metaphor is intentional. What Lewin refers to as language in his inter­ pretation of Schubert's Morgengruß is a historically and culturally rich medium, thick with mutually engaging and mutually exclusive concepts of genre, encrusted with diverse practices of composition and performance. (See “Music Theory, Phenomenology, and Modes of Perception,” in Studies in Music with Text [Oxford: Oxford University Press, 2006], 65–67.) Nonetheless, most musicians would likely agree with more statements about harmony and voice-leading in a Schubert song than would disagree, though they might disagree vigorously about the import of such statements. However stained and grit­ ty our culture may leave our discourse, it is the soil in which that discourse can grow. Mu­ sic composed since the late nineteenth century—and especially since the early twentieth century—has yet to find same rich interpretive soil that nourishes our understanding of the concert canon from Bach to Beethoven.

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit (10.) Babbitt, Words about Music: The Madison Lectures, ed. Stephen Dembski and Joseph N. Straus (Madison: University of Wisconsin Press, 1987), 167. Babbitt is talking about Schoenberg's String Quartet no. 1, op. 7, a relatively “difficult” composition to be sure; Le chasseur certainly defines fewer of its own premises. Still, the perspective will be useful. See also Lewin, “Music Theory, Phenomenology, and Modes of Perception,” 60–67 and 86–87; and Dora A. Hanninen, “A Theory of Recontextualization in Music: Analyzing Phenomenal Transformations of Repetition,” Music Theory Spectrum 25.1 (2003): 59–97; see especially 64–74. Both Lewin and Hanninen define “context” in a straightforwardly temporal way as a musical segment—a note, a chord, a measure, a phrase, and so on—but necessarily a segment that can be recognized meaningfully as such through some “language” (Lewin) or “structural criteria” (Hanninen). I think, though, that Babbitt is talking about music in which available languages/criteria are inadequate, which is as much to say that a priori bases for constructing “contexts” in Lewin's and Hanninen's senses are unreliable, or at least less reliable than usual. (11.) Leslie David Blasius, Schenker's Argument and the Claims of Music Theory (Cambridge: Cambridge University Press, 1996), xvi. (12.) On transformational analyses and the musical spaces through which they move, see Lewin, “Transformational Techniques in Atonal and Other Music Theories,” Perspectives of New Music 21.1–2 (Fall–Winter 1982/Spring–Summer 1983): 312–371; see in particular 335–336. (13.) Lewin, “Forte's Interval Vector, My Interval Function, and Regener's Common-Note Function,” Journal of Music Theory 21.2 (1977): 194–237; see in particular 227–235. (14.) Lewin, “Some Problems and Resources of Music Theory,” Journal of Music Theory Pedagogy 5.2 (1991), 111–32. See also Henry Klumpenhouwer, “In Order to Stay Asleep as Observers: The Nature and Origins of Anti-Cartesianism in Lewin's Generalized Musi­ cal Intervals and Transformations,” Music Theory Spectrum 28.2 (2006): 277–289, in par­ ticular 285. (15.) Carl Dahlhaus, Between Romanticism and Modernism: Four Studies in the Music of the Later Nineteenth Century, trans. Mary Whittall (Berkeley: University of California Press, 1980), 56 and 66. (16.) Steven Rings, “Tonality and Transformation” (Ph.D. diss., Yale University, 2006), 33– 36. For one seminal example that Rings does not cite, see Cohn, “Maximally Smooth Cy­ cles,” 13. Cohn says he will demonstrate properties of “voice-leading potential of motion between triads, [which] may be characterised in group-theoretic terms without any ap­ peal to tonal centres, diatonic collections, harmonic roots and the like.” As with Hyer (see note 6 above), I think Cohn means to suggest not that we abandon tonal intuitions, but that in chromatic music these intuitions may not lead to interpretations that account for entire compositions. (17.) Rings, “Tonality,” 35. Page 36 of 39

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit (18.) Ibid., 28. This is a major part of Robert Snarrenberg's thesis in Schenker's Interpre­ tive Practice (Cambridge: Cambridge University Press, 1997), though he does not say it this way. (19.) The transformational analyst is free to pretend this provisionality does not exist, of course, but the pretense does not make the model less provisional. (20.) “Music Theory,” 59–67. (21.) There are three good reasons to ignore the bass F♯ in measure 133. First, were the bass to have moved to E in measure 134, it would either have been forced into parallel octaves with the soprano going into measure 134, or forced to move by skip to either C or A♯. Neither of these options fits the character of the passage. Second, motivic parallelism encourages hearing measure 133 as similar to measures 129 and 131, in which case F♯ is some sort of alteration or added note to the “real” harmony. Third, the notion that a chord might have an added sixth (if the root is A) or an underseventh (if the prime note is E) as a common dissonance has a good pedigree in the works of Rameau and Riemann. (22.) The E♮–F figures in measures 136–139, despite the rhythmic compression arising from hemiola in measures 138–139, recall the leading-tone figures of preceding mea­ sures, further discouraging attention to the earlier chromaticism. (23.) The comments of Fétis in his Traité, 26–27, regarding the weakening effect se­ quences have on tonal centeredness are well known. Also see Ernst Kurth, Romantische Harmonik und ihre Krise in Wagners “Tristan,” 3rd ed. (Berlin: Hesse, 1923), 333–335, trans. by Lee A. Rothfarb in Ernst Kurth: Selected Writings (Cambridge: Cambridge Uni­ versity Press, 1991), 135–137. (24.) I refer to the interpretation offered in example 19.3 as “contrapuntal” instead of “Schenkerian” to underscore the attitude from which I want to proceed. I do not intend to encounter and elucidate the work as a subject in the world of the tones, though that is a beautiful, human way to engage music. Rather, I want to use heuristically what we have learned from Schenker about counterpoint and the effects of Stufen on it to scout possible paths through the tonal space of Example example 19.1b. (25.) Oswald Jonas, “Introduction” to Schenker, Harmony, trans. Elizabeth Mann Borgese (Chicago: University of Chicago Press, 1954; reprint, 1980), ix. (26.) Henry Klumpenhouwer, “Remarks on American Neo-Riemannian Theory,” Tijdschrift voor Muziektheorie 5.3 (2000), 155–169, in particular 157; Robert C. Cook, “Parsimony and Extravagance,” Journal of Music Theory 49.1 (2005), 109–140, in particular 122. (27.) On this sort of conceptualization of musical motion and space, see Lawrence Zbikowski, “Large-Scale Rhythm and Systems of Grouping” (Ph.D. diss., Yale University, 1991); Janna Saslaw, “Forces, Containers, and Paths: The Role of Body-Derived Image Schemas in the Conceptualization of Music,” Journal of Music Theory 40.2 (1996): 217– 243; and Candace Brower, “A Cognitive Theory of Musical Meaning,” Journal of Music Page 37 of 39

Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit Theory 44.2 (2000): 323–379. To my knowledge, no one has yet explored the activity of transformational analysis from the perspective of embodied metaphor. (28.) Richard Cohn, “Square Dances with Cubes,” Journal of Music Theory 42.2 (1998): 283–296. (29.) The terminological shift is appropriate: we are considering the ways Le chasseur defines its own context. (30.) Lewin, Generalized Musical Intervals and Transformations (New Haven: Yale Univer­ sity Press, 1987), 251–253; “Generalized Interval Systems for Babbitt's Lists, and for Schoenberg's String Trio,” Music Theory Spectrum 17.1 (1995): 81–118. Edward Gollin uses Lewin's group in “Some Unusual Transformations in Bartók's ‘Minor Seconds, Major Sevenths,’” Intégral 12 (1998): 25–51. Michael Siciliano develops a similar group to refine representation of neo-Riemannian transformations in “Neo-Riemannian Transformations and the Harmony of Franz Schubert” (Ph.D. diss., University of Chicago, 2002), 40–50. See also Robert D. Morris, “Set Groups, Complementation, and Mappings among PitchClass Sets,” Journal of Music Theory 26.1 (1982): 101–144. Some of Morris's operations are related to those of Lewin's group. For a discussion of this relationship with an extend­ ed example, see Cook, “Transformational Approaches to Romantic Harmony and the Late Works of César Franck” (Ph.D. diss., University of Chicago, 2001), 105–107. (31.) The use of the X/Y group to model voice-leading appears in Cohn, “Square Dances.” Cohn alludes (296, note 4) to the early relationship of the present work to his article. In 1997, Cohn and I were working separately on further implications of voice-leading pat­ terns in the hexatonic system. In unpublished work, I presented a revision of the system that accounted for the difference between intra- and intercyclic relations. Cohn suggest­ ed that I adopt the system of SUM classes instead, which I did. I then applied a form of the X/Y group to the SUM classes. After some joint revision, the group appeared in a con­ sistent form in Cook, “Voice Leading, a Non-Commutative Group, and the Double Reprise in Franck's Piano Quintet” (presented at the annual meeting of the Society for Music The­ ory, Chapel Hill, NC, December 1998), and Cohn, “Square Dances.” (32.) Dahlhaus, Between Romanticism and Modernism, 66. (33.) Carolyn Abbate, “What the Sorcerer Said,” 19th-Century Music 12.3 (Spring 1989): 221–230. (34.) Snarrenberg, Schenker's Interpretive Practice, 5. (35.) Fred Everett Maus, “Music as Drama,” Music Theory Spectrum 10 (1988): 56–73; es­ pecially 63. (36.) Marion A. Guck, “Rehabilitating the Incorrigible,” in Theory, Analysis and Meaning in Music, ed. Anthony Pople (Cambridge: Cambridge University Press, 1994), 57–73, in particular 67 and 72.

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Tonal Interpretation, Transformational Models, and the Chromatic Calls to Repent in Franck's Le chasseur maudit (37.) Maus, “Music as Drama,” 73; Guck, “Analytical Fictions,” Music Theory Spectrum 16.2 (1994): 217–230, see in particular 218. (38.) Zbikowski, “The Blossoms of ‘Trockne Blumen’: Music and Text in the Early Nine­ teenth Century,” Music Analysis 18.3 (1999): 307–345, on conceptual blending in Lieder, works over similar territory in the sense that the text of a song is not simply considered in juxtaposition to the music but as sharing some sort of conceptual background with the music. Thus, text and music analyze each other. (39.) William A. Little, Gottfried August Bürger, Twayne's World Authors Series 270 (New York: Twayne, 1974), 107–109. The oppressed peasant addressing the negligent, hunthappy nobleman is the subject of Bürger's “Der Bauer an seinen durchlauchtigen Tyran­ nen,” written at the same time as “Der wilde Jäger.” See Douglas G. Baird, “Essential Mo­ tif-Patterns in the Lyrical Poems and Ballads of Gottfried August Bürger” (Ph.D. diss., Rut­ gers University, 1982), 142–143. Lore Kaim-Kloock addresses the issue of class conflict in both poems in Gottfried August Bürger: Zum Problem der Volkstümlichkeit in der Lyrik, Germanistische Studien (Berlin: Rütten und Loening, 1963), 205–213, though firmly from the perspective of East German socialism. (40.) “Hallo! Hallo! La chasse s’élance par les blés, les landes, les prairies—Arrête, comte, je t’en prie, écoute les chants pieux—Non…Hallo! Hallo!—Arrête, comte, je t’en supplie; prends garde…. —Non, et la chevauchée se précipite comme un tourbillon. Soudain le comte est seul; son cheval ne veut plus avancer; il souffle dans son cor; et le cor ne résonne plus…une voix lugubre, implacable le maudit: Sacrilège, dit-elle sois éter­ nellement couru par l’enfer.” Franck, Le chasseur maudit, ed. André Coeuroy (London: Eulenberg, 1973), ii. This synopsis was published in the first edition of the score, pub­ lished in 1884, according to the foreword to the present edition by Roger Fiske (iii). (41.) The passage is a rich example of chromatic transposition as a voice-leading tech­ nique in Proctor's sense. See “Technical Bases of Nineteenth-Century Chromatic Tonality: A Study in Chromaticism” (Ph.D. diss., Princeton University, 1978), 159–170. See also Patrick McCreless, “An Evolutionary Perspective on Nineteenth-Century Semitonal Rela­ tions,” in The Second Practice of Nineteenth-Century Tonality, ed. William Kinderman and Harald Krebs (Lincoln: University of Nebraska Press, 1996), 87–113. (42.) On “tonic effect,” see the discussion above of Hyer, “Reimag(in)ing Riemann.”

Robert C. Cook

Robert C. Cook teaches music theory at the University of Iowa. His interests include chromaticism, contextual music, and languages and practices of analysis. He was ed­ ucated at the University of Chicago and Northwestern University.

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Three Short Essays on Neo-Riemannian Theory

Three Short Essays on Neo-Riemannian Theory   Daniel Harrison The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music, Music Theory Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.013.0020

Abstract and Keywords This article presents a three-section discussion, exploring the specific interrelated themes and questions central to the transformational and neo-Riemannian enterprise. The first section discusses the natures of musical objects and relations within the transformational worldview. It asks what happens when tones and chords are imagines not as objects but as transformations. The second section delves further into the object/transformation di­ chotomy. It explores the structural and functional differences among dissonant and conso­ nant trichords in a particular nonatonic cycle. It also explores how the voice-leading func­ tional and set-theoretical implications of the cycle might be engaged by a transformation­ al perspective as a means to impart “sensuous distinctions”. The last section examines the analytical implications of the first two sections, by examining Vaughan Williams's neomodal triadic Fantasia on a Theme by Thomas Tallis. Keywords: neo-Riemannian enterprise, tones, chords, transformations, object/transformation dichotomy, dissonant trichords, consonant trichords, nonatonic cycle

I. The New Riemann: Same as the Old Rie­ mann? A Meditation on Transformational Mu­ sic Theories Surely one of the main benefits of neo-Riemannian theory is the animation it brings to re­ lationships that have long been thought to be static. David Lewin's celebrated figure 0.1 from Generalized Musical Intervals and Transformations, reproduced here as example 20.1, is accompanied by an inspired gloss: Instead of regarding the arrow…as a measurement of extension between points s and t observed passively “out there” in a Cartesian res extensa, one can regard the situation actively, like a singer, player, or composer, thinking: “I am at s; what characteristic transformation do I perform to arrive at t?”1

Page 1 of 31

Three Short Essays on Neo-Riemannian Theory

Ex. 20.1.

The attraction of this language is that, among other things, it honors long-held and cherished notions of music being something “in motion.” As Victor Zuckerkandl puts it, “Musical contexts are motion contexts, kinetic contexts. Tones are elements of a musi­ cal context because and in so far as they are conveyors of a motion that goes through them and beyond them. When we hear music, what we hear above all is motion.”2 (p. 549)

The promise of example 20.1, then, is the promise of a music theory that can model musi­ cal motion. The figure, as well as Lewin's gloss, also supports the currently fashionable position that musical analysis is a product of an engaged, musical body, not of some ob­ jective and disembodied oracle regarding music “out there.” As Lewin puts it, “[the trans­ formational] attitude is by and large the attitude of someone inside the music, as idealized dancer and/or singer. No external observer (analyst, listener) is needed.”3 It is easy to move from Lewin's image of two points separated by an arrow to an image of two points connected by an arrow—which is to say that it is easy to change the gloss from speculating about what has to be done in order to move from s to t to speculating about what has to be done to move s itself to t. This change is evident in some transformational analytic discourse, such as the following: Within each voice, the underlying transformation can be heard as motivating the movement from note to note. When the first chord moves to the second, for exam­ ple, I sends each note in the first chord onto a corresponding note in the second. The inversion of G onto C♯ and D onto F♯ can be heard to push the F♯ onto D, as each note is urged onward by the behavior of the other two.4

Page 2 of 31

Three Short Essays on Neo-Riemannian Theory

Ex. 20.2.

The author, Joseph Straus, takes advantage of the (metaphorical) technical term “onto,” which denotes a kind of mapping, to animate the image of notes being “sent,” “urged,” and “pushed” on their way. In this reformulation, example 20.1 is reimagined to be some­ thing like example 20.2, a crude representation of a series of strobe-lighted exposures capturing s being transformed “into” t in the sense that it “becomes” t. This version is even more attractive from the point of view of musical motion, since it further subjectifies the activity. That is to say, it actually changes s and t entirely from static to dynamic points, from things one is at to things with which one moves. (p. 550)

At this point, Zuckerkandl reminds us to rein in our thinking:

What is it that moves in a melody? It will be answered: the tones. But is a tone something that can move? What moves is objects, things—and have we not shown that the tones of music are precisely not that, are not like things, are not like ob­ jects, and have no reference to things and objects? And now are they suddenly to do what only things do—to move?5 In other words, how can the tone s move to the tone t and still be s? As soon as s starts, say, to be “pushed,” does it not stop being the pitch s and become something else— pushed-s, or “s on its way to t”? As Zuckerkandl puts it, “if we attempt actually to connect tone with tone…taking real motion as our model, the result is the familiar screeching glis­ sade of the siren.”6 I do not intend at this point to explore further the general issue of musical motion, prefer­ ring instead to deal with some implications of a highly mobilized theory of chordal rela­ tions—the current state of neo-Riemannian theory—that the previous discussion has sug­ gested are problematic.7

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Three Short Essays on Neo-Riemannian Theory First, let us be clear about the metaphorical status of musical motion—but also about the fact that this status is not the real source of the problem. Roger Scruton explains: Music is the intentional object of an experience that only rational beings can have, and only through the exercise of imagination. To describe it we must have re­ course to metaphor, not because music resides in an analogy with other things, but because the metaphor describes exactly what we hear, when we hear sounds of music.8 So, if the modeling of metaphor is inescapable, then the issues at hand are how much power gets invested in a particular metaphor, and to what extent does that power drain other, perhaps equally useful and important metaphors. In transformational theories, and in neo-Riemannian theory in particular, the attention paid to the metaphor of motion af­ fects the status of the moving things in two seemingly contradictory ways. The first effect is that objects lose stability as things-in-themselves and either become things-that-move (in the example 20.2 sense) or are reduced solely to things-in-transfor­ mational-relations (in the example 20.1 sense). To the extent that this loss is compensated for by the power of node-arrow transformation networks, in which the contents of nodes (the objects) can be variously constituted in order to illustrate analytically powerful iso­ graphies, it is perhaps a fair trade. But the loss can be felt when the differences between various kinds of musical objects are disregarded or downplayed in order to assert isogra­ phy or some other desirable algebraic relation. A mild example was encountered above, in Straus's description of a transformational relationship between two chords: the rela­ tionship as described was actually between individual pitch classes (“inversion of G onto C♯ and D onto F♯,” etc.), subsuming the concept of chord under that of pitch-class set, a move that undermines the status of “chord” as an analytic entity. While this may not be a (p. 551) problem for Straus's immediate purposes, which involve investigating voice-lead­ ing in an atonal context, it is a move that should not be made unwittingly. A more signifi­ cant example is found in the assumption that musical dynamic markings, mf, p, f, and so on, might be amenable to the same kind of transformations that pitch classes are subject­ ed to (as in, for example, total serialist work). This whimsy, which attributes the same kind of properties to both dynamics and pitch classes, subverts “natural” properties of dy­ namics as musical objects—that they are infinitely rather than discretely scaled, for exam­ ple.9 So, the first effect, which leads directly to the construction of example 20.2, is the evacu­ ation of the concept of musical object itself. It becomes an absence of sorts, an empty cir­ cle in a node-arrow graph, a thing in such constant becoming that it has no being. The second, and apparently opposite effect is overdefinition of the transformational ob­ ject. In the preceding brief discussion of musical dynamics, the essential maneuver was giving dynamic markings the same transformational opportunities as pitch classes by defining them rigidly as points instead of as potentially overlapping bands.10 For a trans­ formational theory to operate, it needs homogenous and well-defined things upon which to act. (The act of well definition, paradoxically, need not be anything other than the com­ Page 4 of 31

Three Short Essays on Neo-Riemannian Theory plete listing of transformational relations in which the thing participates, which under­ scores the evacuation of the object itself mentioned above.) If an algebraic group struc­ ture is used to model some relationship between various objects, those objects should all be of the same essential type—pitch classes, dynamics, tempi, time points, and so on, or consistently defined ordered tuples of such types in order for the model to make sense. It is not appropriate for group theory to model transformations between heterogeneous ob­ jects—pitch classes and tempi, timbres and time points, and so on—or at least to model them without recourse to ontological strong-arm tactics.11 The statement “Transforma­ tion X of pc 0 results in p” is a mere formal statement needing a real-world justification. The effects of objects too well defined were clearly felt in the early stages of neo-Rie­ mannian research, where the operations had purchase on triads but not on seventh chords. This resulted in some analytic difficulties, which surfaced during the meetings of the initial working group on neo-Riemannian theory (1993), when, for example, Richard Cohn's analysis of a passage from Parsifal (Act I “Amfortasklage,” at 1369) had to omit “subposed” bass tones that formed seventh chords. Adrian Childs used this very example to point out the need for a transformational model for seventh chords, which he then con­ structed.12 Yet the models for triads and seventh chords were essentially separate, and could communicate only by means of other mapping technology, such as Clifton Callender's “split” relations.13 In addition, other chordal objects request a patch into the developing relational grid—certainly, for late-nineteenth-century music at least, ninth chords, and perhaps even “higher” extended tertian chords: elevenths, thirteenths, and so on. Undoubtedly, linking the various well-defined chordal objects can be done success­ fully, but might (p. 552) the resulting unified theory be massive and overbuilt relative to the musical experience and metaphorical structure of musical motion it models? In the end, transformational theory in general requires a separation of object and activity, of what something is and what is done to it—with “what something is” sometimes being defined solely by “what is done to it.” Objects are inert and without tendency, and all ac­ tivity and meaning are supplied by transformations applied to them. From this far van­ tage point, transformational theory appears to model the metaphor of musical motion by constructing a ventriloquist's dummy; it only appears to be alive, but is in fact a construc­ tion of lifeless parts that are made to move by some external force.14 At this point, a wide look at Riemann's various descriptions of harmonic relations is use­ ful, for we can see with some newfound sympathy why there seem to be so many overlap­ ping components: the Schritt/Wechsel system,15 the L/P/R system that apparently sup­ planted it, and the independent concepts of T, D, and S. All of these, as Lewin pointed out, are inherently transformational in potential: An even more basic problem for Riemann was that he never quite worked through in his own mind the transformational character of his theories. He did not quite ever realize that he was conceiving “dominant”…as something one does to a Klang, to obtain another Klang…. [He was led] to conceive “dominant” and the like

Page 5 of 31

Three Short Essays on Neo-Riemannian Theory as labels for Klangs in a key, rather than as labels for transformations that gener­ ate Klangs . . .16 Despite Lewin's explicit illustration of (a piece of) the T, D, and S system, only the S/W and L/P/R systems have been significantly adapted to neo-Riemannian theory; T, D, and S carry freight that is costly to handle, as Brian Hyer noted: Rather than regarding the parallel, relative, leittonwechsel, and dominant as rela­ tions between harmonies, Riemann hypostatizes them, transforming abstract im­ material relations into concrete substances, or tonal “functions.” There is, in other words, an enormous ontological difference between thinking of the dominant as a relation and thinking of it as an actual chord…. For Riemann, the dominant is a “being,” something a triad is, rather than a “doing,” something one does to a triad. After commenting on his own transformational reworking of Riemann's theories, he con­ tinues: Our use of Riemann's harmonic theories…attests to the enormous theoretical pow­ er of regarding a transformation as a musical “doing.” Yet I think we can also ac­ knowledge that certain harmonic configurations do seem to insist on an intrinsic relation or affiliation with a referential tonic, even if that affiliation is of our own imaginative making. It is this sense of musical “being” that Riemann's own musiccritical practice answers to.17 Implicit in these quotations is the idea that transformational theory cannot deal well with “being,” a point explored above. But T, D, and S are fundamentally about being—T especially. Elsewhere, Hyer has noted that “in discussing the tonic, (p. 553) musicians most often resort to metaphors of presence, designating the tonic as a ‘centre,’ a ‘home,’ etc.”18 By having presence, the tonic gains substance, quality, and an intrinsic power.19 Dominant and subdominant, which require a concept of tonic, are similarly constituted as substantial presences. The reason that T, D, and S cannot easily be brought under trans­ formational concepts is because they suggest that the ventriloquist's dummy, like Pinoc­ chio, is somehow alive. What does this mean, and is it a good thing? While the substances of T, D, and S impose a distorting gravitational field upon pure transformational space, they do restore a sensu­ ous dimension to the hearing and experience of tonal music, a dimension that Ernst Kurth recognized as the necessary counterweight to the energetic dimension so well modeled by neo-Riemannian theory.20 Sensuousness in music involves objects that have mass and substance and that, moreover, ask to be savored, appreciated, and even caressed by one's own voice. (I’m thinking of “humming along” here.) With some experience in analytical listening, these objects also acquire energetic properties of T, D, and S, which I have de­ fined elsewhere as attitudes, orientations, and moods that these objects can adopt.21 To be sure, these properties may “arise more from the language we use to talk (and think) about music than from the music itself. [T, D, and S] are not immanent in the music, but rather occur there as a result of our critical activities.”22 Yet, as Scruton noted above, mu­ Page 6 of 31

Three Short Essays on Neo-Riemannian Theory sic is an intentional object, and as such requires these kinds of critical activities, if we are to experience it as music. The reader may have noticed a move in the previous paragraph that collapsed the distinc­ tion between object and activity. By endowing objects with attitudes and moods, I thereby allowed them tendencies and urges—which is to say, their own motive power. This move in effect makes musical motion the product of sensuous actors, placing the origins of such motion somehow within these actors, out of sight, and thereby out of the reach of trans­ formational theory as it currently stands. This complicates theorizing greatly, since en­ dowing tonal objects with these kinds of functional “attitudes” involves perhaps even more problematic ontology than that pertaining to isolated and pure musical motion it­ self. Still, I have written a book working out the implications of hearing, in a “renewed” Riemannian way, musical activity as the property of sensuous-functional musical objects, so I am familiar with the pitfalls. But I am also familiar with the rewards. In the following essay, I attempt a modest reconciliation between these contending modes of Riemannian thought.

II. Some Hypotheses about Tonic and Antitonic Trichords At the 2000 “Musical Intersections” conference in Toronto, Matthew Santa presented “Hexatonic and Nonatonic Systems in Late Nineteenth-Century Music,” in which he creat­ ed a cyclic system made up of consonant triads and “dominant (p. 554) seventh chords.”23 (The reason for the scare quotes will be given below.) The system uses nine pitch classes, and hence was termed nonatonic. Example 20.3a reproduces Santa's example 4b, a threevoice realization of the nonatonic cycle beginning on C major. Note that [026] trichordal subsets represent the complete [0258] seventh-chord set class. This is not a controversial substitution; the omission of the chordal fifth is common and sanctioned in elementary part writing since it does not affect the character and function of the chord. Example 20.3b offers Santa's four-voice realization of the nonatonic cycle using a bass line consist­ ing of alternating root and previously missing chordal fifths, forming a whole-tone scale.24

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Three Short Essays on Neo-Riemannian Theory

Ex. 20.3.

I was struck by the pleasing neatness of this result. But I was also unexpectedly struck with a suspicion of this neatness, a sense that something had been scrubbed too thor­ oughly. My suspicion focused on the fit of the whole-tone scale with the three-voice nona­ tonic cycle; there seemed to me a possibility for greater looseness, for an alternative scalar path that, while not strictly adhering to the requirement that the dominant seventh chords be complete, made nonetheless the same functional point. What I imagined was example 20.3c. Here, the whole-tone scale is replaced by the chromatic, with the domi­ nant sevenths now acquiring not just “the (p. 555) fifth,” but “possible fifths.” The root names of the seventh chords are replaced by Dom, a generic label for dominant function, with arrows pointing toward their tonics. The chromatic chordal fifths, as I wrote above, do not alter the functional attitudes of the [026] trichords; they instead strengthen these attitudes by imparting a greater energetic charge to the resulting chords (which exempli­ fy what Kurth called the “intensive alteration style”). I will return briefly to these ener­ getic chords later in the essay. Example 20.3c suggests three things. First, it casts the three-voice realization of 20.3a as functionally sufficient and the particular four-voiced realization of 20.3b as somewhat overbuilt. Second, it points up a practical consequence of having objects be “too well de­ fined” in the sense that I wrote about in the previous essay; the assumption of “dominant seventh” required that it be conceived as an entity fundamentally similar to the triad: a chord whose members could be defined as pitch classes; but—and this is the third thing— example 20.3c points out that the seventh-chordal manifestation of dominant function need not be thought of as structurally similar to the triad, but could rather be constituted as having three pitch classes and one pitch-class band.25 For the moment, I want to focus on the proposition that example 20.3a is functionally suf­ ficient, that the [026] trichord is not only a minimally adequate representation of the dominant seventh chord, but also the fundamental dominant discord. This idea was first suggested by Rameau, who in discussing the perfect cadence focused exclusively on Page 8 of 31

Three Short Essays on Neo-Riemannian Theory those notes forming the [026] subset of the dominante-tonique, calling them dominant (the “2” in the set), major dissonance (“6”), and minor dissonance (“0”).26 The chordal fifth, while shown and voice-led in the accompanying examples, he mentioned not at all. Further, Rameau maintained that nontonic chords were, by nature, seventh chords, so that even a simple dominant triad was, as Riemann would later put it in a slightly differ­ ent context, simply a feigning consonance, a representative of the fundamental dominant discord. In a similar but less controversial way, Rameau proposed that the tonic chord was properly consonant, going so far as to imply that any true consonant chord was, by definition, a tonic. While this style of thinking is somewhat antique, it does lead to some interesting intersections of harmonic function and neo-Riemannian ideas concerning voice-leading parsimony.

Ex. 20.4.

Example 20.4 arranges the chord labels of example 20.3a around a circle, which Santa, following Richard Cohn, termed the “Northern” system (because it is one of four nonaton­ ic systems).27 Santa's discussion focused on transpositional opportunities within the sys­ tem, measured by the number of clockwise moves from one chord to another. Thus T1(C+) = E♭7, T2(C+) = A♭+, and so on. One result of handling chordal relations in this way is the avoidance of traditional if awkward functional language to describe nonstan­ dard juxtapositions of, say, B7 and C+ (e.g., V7–VI in the [putative] key of E minor). But in bypassing such awkwardness, it also fails to engage productive functional relations. While it may seem at first that the only directly intelligible adjacency relationship is be­ tween a triad and its “dominant seventh” one station counterclockwise, analysis of the voice-leading behaviors (p. 556) among the members of the family system suggests that no two members are estranged.

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Three Short Essays on Neo-Riemannian Theory Example 20.5 shows discharge functions, which are resolutions from the [026] dominant discords to the [037] concords. Three kinds of discharges are possible in the system. The first is the normal root motion by descending fifth, labeled Da. The second, Db, has a de­ scending minor-third root motion, and the third, Dc, an ascending minor-second motion.28 The harmonic functional relationships, however, are better understood by disassembling the constituent motions into their component scale-degree behaviors, which are shown in table 20.1. It is stipulated that that all discharges tonicize the triad of resolution (symbol­ ized by →T), so that scale degrees receiving discharge are reckoned as members of the tonic triad (→1̂/8̂, 3̂, or 5̂).29 The scale-degree composition of the [026] discord can then be determined retroactively. The ordering of the rows within each discharge function is made according to the functional power of the scale-degree motions, with the most pow­ erful listed in the first row and the least in the third.30 All of the motions involve function­ al mixtures of D and S—even Da, which involves the traditional dominant seventh, al­ though its S component is the weakest of the three. Because all the [026]s are interpreted as functional mixtures, I will hereafter substitute the adjective “antitonic” for “dominant” when describing them. All three antitonic discharges are similar in a number of respects, although their differ­ ences are telling. Db shares Da's semitonal voice-leading converging upon 1̂ and 3̂ and its common-tone 5̂. Its S, however, is the chief functional attitude. Dc discharges its D components semitonally, like Da and Db, but its S discharges by whole step. Note that while all three antitonic discharges involve different sets of scale-degree assemblies, the assemblies all consist of two Ds and an S, and, further, that the sum of the respective di­ rected voice-leading motions is 0, according to methods discussed by Richard Cohn.31 The 0 voice-leading displacement within the system creates a balance and even a certain sense of aptness that marks all the discharges, even the unconventional Db and Dc. We shall return to these voice-leading issues later.

Ex. 20.5.

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Three Short Essays on Neo-Riemannian Theory Not shown in example 20.5 are the inverse of the antitonic discharge functions, which, by analogy, are tonic charging functions. These can easily be read by reversing the direction of the arrows in both example 20.5 and table 20.1. (Thus: Da−1, Db−1, Dc−1.) Combining discharging and charging functions allows for a mobilized family in which tonicized concords communicate with each other through antitonic discords. As men­ tioned above, only one communication, Da and its inverse, is canonically recognized in tonal theory. But the others are not unknown and useless; example 20.6 shows a wellknown passage from Brahms's Requiem involving a Db discharge/charge pattern. The ulti­ mate goal of the passage (not shown in the example) is F major, reached in measure 45. Thus, the activity in measures 37–42 might be considered dilatory, especially the “failed” cadence concluding the excerpt. Yet the Db functions do not promote stasis or other symptoms of biding time, instead maintaining and even strengthening the energetic flow of the passage.32 As for the Dc function, a well-known example is found at measures 16– 17 of the Tristan Prelude (and passim).33 (p. 557)

Table 20.1. Da

Db

Dc

7̂→8̂ D→T

f2̂→1̂ S→T

7̂→8̂ D→T

5̂→5̂ D→T

♯2̂→3̂ D→T

6̂→5̂ S→T

4̂→3̂ S→T

5̂→5̂ D→T

♯2̂→3̂ D→T

Ex. 20.6. Brahms, Ein deutsches Requiem, I, mm. 35–42. (p. 558)

So far, we have constructed channels of communication between antitonic dis­

cords and tonic concords, but not among discords and concords themselves. Neo-Rie­ mannian transformations obviously apply to the latter, which instance 〈PL〉/〈LP〉 cycles on the [037] trichord. In terms of possible T, D, and S sensuous-functional relationships, these cycles can take on different attitudes depending upon the musical and analytical context. If one triad has structural superiority and the others are subordinate, relations among them can be analyzed by what I have elsewhere termed linking analysis.34 If each triad receives the same kind of harmonic accent, accumulative analysis aptly describes the relationships. Example 20.7, adapted from an earlier article, shows these two types of analysis in a 〈PL〉 cycle starting from an E-major triad.35 The top staff shows the linking analysis, taking E major as T. The other “triads” are then conceived as functional mix­ Page 11 of 31

Three Short Essays on Neo-Riemannian Theory tures of various D and S components, as well as modal flavorings of T. Movement among and between the [037] elements is thus always in reference to the governing T. The bot­ tom two staves show two possible functional accumulations: retrospective, in which the second of the two triad pairs receives a tonicizing discharge from the first; and prospec­ tive, in which the tonicized first charges the second. In both cases, enharmonic/functional reinterpretation takes place, symbolized by the turn symbol and the respelling of the chords. In all staves, shaped notes indicate directional tendencies. The article uses these function-analysis possibilities to interpret the coda of Liszt's E♭ piano concerto, arguing that the coda's typically strong subdominant bias allows us to hear the arrival of each numbered chord in example 20.7 (transposed down a chromatic semitone to the key of E♭) as a tonicized S. The accumulation of these subdominants then discharges through a final plagal cadence to tonic (measures 491–492).

Ex. 20.7.

Ex. 20.8.

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Three Short Essays on Neo-Riemannian Theory We now come to constructing channels of communication between the [026] antitonic dis­ cords. Interestingly, they, too, instance 〈PL〉/〈LP〉 cycles. Example 20.8, taken from an arti­ cle by Richard Cohn, shows the intervals of displacement (p, ī, r) involved in P, L, and R transformations in normal, 12-pc chromatic space, with conjunct (i.e., parsimonious) in­ tervals within the space marked in boxes.36 These intervals are conceived as one or two unit intervals of the space; so, for (p. 559) (p. 560) chromatic space, the intervals are one half step (= 1 or 11 unit interval) or a whole step (= 2 or 10 unit intervals). The original purpose of example 20.8 was to show that of all the conventional trichords, only [037] en­ joyed set-class preservation under parsimonious pitch-class displacement resulting from P, L, or R operations; the others, to varying degrees, required some disjunct displacement in order to preserve set class.37 But of all the unparsimonious trichords within chromatic space, [026] is the least unparsimonious—which is to say that it alone has two parsimo­ nious displacements, while all other trichords have one or zero. Should we be interested in somehow construing the [026] trichord as fully parsimonious, the solution lies in re­ moving it from chromatic and placing it in whole-tone space, where the unit interval is the whole step, and where parsimony is thus one or two whole steps. Reading example 20.8 in this way, we would find whole-tone space parsimony expressed in displacements by 2/10 and 4/8 (but not in 1/11 or 3/9, which do not connect to pitch-class points in the space).38 The appearance of whole-tone space here should be reassuringly familiar to neo-Riemannian theorists, since it pops up with curious frequency in various neo-Rie­ mannian theoretical artifacts; “hexatonic” systems, for example, can be profitably exam­ ined by means of whole-tone lenses, as can Santa's “complete” nonatonic system of exam­ ple 20.3b, which involved a whole-tone scale component. Further, even if whole-tone space is left behind as an amusing transformational “gee whiz” in order to explore the sensuous-functional aspects of the 〈PL〉/〈LP〉 cycle on [026], it reappears unexpectedly as a means to understand one path of twentieth-century harmonic development in tonal music. Part of the unexpectedness is that there seems so little of [026] to explore in the function­ al dimension. As we saw in example 20.7, there are many possible functional relations among the [037] triads in the system, depending on whether one or more of the triads is understood as Wahr- or Scheinkonsonanzen. The top staff of example 20.7, for instance, assumed a fixed T (E+), and read the other triads in the system as (following Kurth) neighbor-note insertions. The bottom two staves assumed a movable T and showed vari­ ous accumulations. In the case of the [026] trichord, however, there is no question of it functioning as a T in tonal music without substantial contextual support, let alone a Scheinkonsonanz.39 It is an antitonic entity that tonicizes upon discharge. But what func­ tional change might an 〈LP〉 transformation effect? None, really. We have seen that any [026] discord can successfully discharge to any [037] via Da–c. Thus, moving from one [026] in the system to another can, at most, accumulate antitonic charge as the various S and D components activate or deactivate.40 As a result of this all-purpose antitonicism, the [026]s of the system combine harmonically in functionally useful ways.

Page 13 of 31

Three Short Essays on Neo-Riemannian Theory

Ex. 20.9.

Example 20.9 shows paired combinations of [026] trichords from the family under discus­ sion, voiced as normally spaced chords. The two trichords from the system are distin­ guished by open and filled noteheads.41 The resulting [02468] pentachords are antitonic discords of mass and energy greater than the sum of their constituent [026] elements, be­ cause they embed four other forms of [026] besides the originating pair. (One easily seen example is found in the treble staff (p. 561) of each progression, made up of one open and two filled noteheads—an “inverted” [026] that connects to the consonant triad in –2 voiceleading displacement instead of the 0 displacement of the Da–c resolutions.) A more de­ tailed disassembly of these other, “secondary” forms of [026] will be undertaken in con­ nection with example 20.12. Aside from bringing up suggestive harmonic and voice-leading issues, which I will explore a bit further later on, example 20.9 also connects to a currently inactive area of tonal the­ ory concerned with the development of “extended” sonorities; Horace Alden Miller, in his 1930 treatise on “modern harmonic problems,” devoted an entire chapter to “whole-tone dominant” structures of the kind as shown in example 20.9. (I must also pay tribute to A. Eaglefield Hull, whose example 120 from his 1915 work inspired the layout of the current graphic.)42 These chords are now largely understood as unitary pitch-class sets (or as “jazz” chords, depending upon the repertory). Yet it seems fruitful to conceive of them— and other similar structures—as agglomerations of [026] antitonic trichords that could be potentially useful in some extended tonal context. Indeed, in this light, the usefulness of example 20.3c, in which I replaced Santa's whole-tone scale with a chromatic one, de­ pends upon the fact that the “new” notes of my chromatic scale create other forms of the [026] trichord with the given trichord. That is, the raised and lowered fifths in altered V7 chords—which form the upper and lower bound of the “pitch-class band” mentioned earli­ er—create additional forms of [026] in the chord. And it is the proliferation of these [026] trichords (p. 562) that endows the “altered” dominant tetrachords with their characteris­ Page 14 of 31

Three Short Essays on Neo-Riemannian Theory tic functional energy gain. If both altered fifths are present in the chord, the [02468] pen­ tachord results, various forms of which we have seen in example 20.9. (This last para­ graph has been uncomfortably sketchy, since my main points lie elsewhere, but the possi­ bility of reviving an interesting and useful branch of harmonic theory that seemed to dead-end after the work of Persichetti and Ulehla required at least a passing mention.)43 At this point, we should see how the discharge family appears (in example 20.10) with its functions fully marked. (Inverses of functions, i.e., Da−1, Db−1, Dc−1, and 〈PL〉, can be read backward from the arrowheads.) To recapitulate, the various transformational functions in the system interact with the sensuous functions in the following way: 1. 〈PL〉/〈LP〉 on the [037] concord: linking and accumulative possibilities, some of which are shown in example 20.7. 2. 〈PL〉/〈LP〉 on the [026] discord: accumulative antitonic charge, either serially or, as in example 20.9, 3. Da–c/(Da–c)−1 on the [026]«[037] pair: (dis)charge of tonal function as shown in ta­ ble 20.1.

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Three Short Essays on Neo-Riemannian Theory

Ex. 20.10.

This heterogeneity of harmonic function and set-class type can be more strongly linked to a neo-Riemannian system by adapting the approach taken by Richard Cohn, which views voice-leading displacement as foundational.44 Adapting Cohn's definition 6, we can consti­ tute

as the class of [037] and [026] trichords X such that SUM(X) = Q (i.e., the sum of

pitch classes in trichord X is Q mod 12). Santa's “northern system” is thus class

 (i.e.,

the sum of the pitch classes in every trichord in the system is 11 mod 12). At this point, we can graft the antitonic [026] discords as dotted-box extensions onto Cohn's sum-class system (his figure 3), the results of which are shown in example 20.11.45

Page 16 of 31

Three Short Essays on Neo-Riemannian Theory

Ex. 20.11. (p. 563)

In addition to the dotted-box graftings, the example also adds interior arrows

showing paths of communication between class

and other classes based on harmonic

combinations of antitonic trichords shown earlier in connection with example 20.9. To un­ derstand how these paths are made, we construct example 20.12, which shows all the “secondary” antitonic discords that fall out of combinations of two primary discords, us­ ing only the first two “measures” of example 20.9 as a data source.46 (The other “mea­ sures” work out similarly.) The sum-class of the secondary discords is shown underneath. Note that two of these discords belong to class   antitonic pentachord made up of two class chords belonging to class er from class    

; generalizing, we may note that any

antitonic trichords also contains two tri­

, and, as the example indicates, one from class

and anoth­

. This situation is symbolized by the interior arrows, the paths of commu­

nication, in example 20.11 above.47 Note the extra-thick, “wide” pathway to flects the twofold embedding of secondary

, which re­

trichords. Combining those two 

trichords recreates the pitch-class content of the original antitonic pentachord, which is to say that the pentachord can serve as a pivot chord since it can be built from two anti­ tonic trichords from either  charge in

or

; a progression could charge in

and, pivoting, dis­

, with 0 voice-leading displacement in both. The same situation applies, mu­

tatis mutandis, to all other sum classes. Again, this could be a resource in some extended tonal context.

Page 17 of 31

Three Short Essays on Neo-Riemannian Theory

Ex. 20.12.

The basic idea of this essay has been to explore a system whose objects are both similar in fundamental structural properties (trichords) and heterogeneous in functional proper­ ties (tonic concords and antitonic discords). Santa's work stressed similarities, allowing for sensuously indifferent transpositional movements among elements of his systems; my work here has attempted to reinscribe the sensuous differences among the objects, which necessitated more complicated apparatus of 〈PL〉/〈LP〉, (p. 564) Da–c, and sum classes. Along the way, I have been tempted constantly to explore formal, historical, and speculative im­ plications, which I have managed either to confine within footnotes or to devote hurried paragraphs in the main text. The result is admittedly rather sketchy and awaits appropri­ ate infilling, but other work now beckons.

III. Remarks on Fantasia on a Theme by Thomas Tallis by Ralph Vaughan Williams The first essay of this chapter addressed the conflict between a sensuous view of harmon­ ic relations, embodied in function theory, and the highly energetic view that nourishes current neo-Riemannian theory. The second sketched a way of reconciling these two modes of thought. This third essay attempts to combine the two analytically, though along a different front from that of the second essay. In much transitional tonal music of a century ago (as, indeed, in traditional, commonpractice tonal music), the presence of functional tonics—at any level of structure in which they can be heard—creates what Brian Hyer imagined as a gravitational distortion of equally tempered Tonnetz space.48 Neo-Riemannian theory seems happiest operating in an apparently tonic-free, zero-gravity state, as its analytic products are transformational labels that have no sensuous-functional significance. Thus, passages of consonant triads that offer great resistance to Stufen interpretations or that otherwise require torturous la­ beling in some system—in other words, passages in which functional tonics are absent or only weakly effective—respond well to neo-Riemannian analysis and “gain intrinsic intelli­ Page 18 of 31

Three Short Essays on Neo-Riemannian Theory gibility from the algebraic structure of the group.”49 But under conditions of tonal gravity, (p. 565) neo-Riemannian analysis takes on a certain dreamlike quality. It works uncon­ strained by the laws of gravity or of nature—just as we do when in our dreams we fly, or converse with friends and family members from long ago, or find ourselves magically transported from place to place. Yet such analysis may also be a window into a composi­ tional unconscious, an occasion for insight and new perspective, and a source of thera­ peutic information to an analyst. But, like dreams, it does not touch a substantial and con­ sequential reality, which can be addressed only while awake and, in obedience to gravity, with feet on the ground. Even so, there are pieces that have functional gravitation, but of a kind that is unusual and even dreamlike in a way. A representative work is the Fantasia on a Theme by Thomas Tallis by Ralph Vaughan Williams, a renowned composition that heads a list of twentieth-century works that mix the sounds of old church modes with the conventions of late-tonal compositional rhetoric, producing something new that is also vaguely antique. The Tallis theme that Vaughan Williams used as the basis of his piece is shown in example 20.13 along with some minimal analytic overlay, which I will explain shortly. The “third mode” of its title is, of course, Phrygian, and Tallis, being an excellent composer in the strict style, knew how to handle expertly this comparatively difficult mode, even in a sim­ ple homophonic setting like this one. I am not now going to drop the bombshell that neo-Riemannian theory is the analytic key for a late-sixteenth-century modal piece, which is a dud, I should think. I will show, how­ ever, that such theory is appropriate to illuminate the circumstances in which Vaughan Williams found himself when he decided to appropriate the Tallis tune. Vaughan Williams heard this piece as a fine example of functional Tudor music, but he also heard it as an early-twentieth-century composer of tonal music, characteristically on the lookout for new resources. It is likely that Vaughan Williams's interest in the chord usage of this piece—for that is what stands out most prominently from its homophonic texture—centers on the clear sep­ aration of the abstract and systematic role of the various triads based on major/minor tonality from their particular structural and rhetorical roles in the piece. Put more con­ cretely, he recognized that the G-major chords can sound like dominants of a key on C yet are clearly acting as tonics of a mode on G. In the same way, the C-minor chords, which should by rights be tonics, sound like subdominants. It is possible to hear tonal functions in this environment, but these functions are no longer attached to the traditional majorminor system that gave rise to them. In an important study on the functional extravagance of chromatic chords, Charles Smith introduced the term “modal chromaticism” to cover this very effect. Modal chromaticism is a useful construct

Page 19 of 31

Three Short Essays on Neo-Riemannian Theory when we find a fragment of a conventional progression that has been forced into a context that contradicts its apparent functional allegiance and imposes another functional interpretation, usually around a different tonic. The effect of noticing such a functional transformation is that we hear both of these contradictory func­ tional interpretations, the systematically implicit and the contextual, resonating, sometimes quite uneasily, through the passage.50

Ex. 20.13.

Helpfully for us, Smith sketched a preliminary outline of a “Phrygian” progression system, reproduced as example 20.14. In this system, the 𝄬2̂ is what I would term the (p. 566)

agent of the pivotal (or “dominant”) function. The preliminary outline is full of question marks, which underscore its speculative nature, which was unavoidable since no latenineteenth-century composer was writing—nor any theorist analyzing—consistent Phry­ gian-system pieces.

Page 20 of 31

Three Short Essays on Neo-Riemannian Theory

Ex. 20.14.

Returning to the Tallis piece, example 20.13, let us attend to the simple functional labels, which are mostly those used by mainstream European Riemannists. The top-row (p. 567)

labels reflect systemic function—the sense that a key-profile algorithm would propose Cminor as a strong candidate for background key. I have boxed two progressions where a C-minor background comes through the strongest. The bottom row, whose functional la­ bels are circled, show the rhetorical and structural functions of chords in this Phrygian piece; Smith's 𝄬2̂ pivotal function is added to the mainstream labels. This analysis recog­ nizes that G is the tonal center and that it is frequently confirmed through a plagal ca­ dence from C. Also of note here is the constant fluctuation between major and minor tri­ ads over the G final. While this problematizes to some extent the hearing of G as domi­ nant of C, it also contributes mightily to the unease that Smith mentioned by creating a modally indistinct but shimmering tonal center. My point is this: because of the bifurcation of systemic implication and contextual behav­ ior, the motive power of the chords here is screwed up. That is to say, the functional atti­ tudes and moods of the objects are confused, and we can confidently rely on neither their character nor their good behavior. Functional analysis must pretend that everything is okay and that behavior can be normalized or otherwise explained analytically. But that is not the substantial and consequential reality of this piece. In this environment, while the chordal objects are not lifeless, they are tonally unmotivat­ ed. The question arises: How can Vaughan Williams make his chords move? I am not ask­ ing about the Tallis, which moves under its own, pre-common-practice power. The issue is how Vaughan Williams can conserve and enrich the environment created by the Tallis tune while composing a piece of twentieth-century music fully aware of a harmonic com­ mon practice that arose after Tallis's time.

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Three Short Essays on Neo-Riemannian Theory As to conservation, Vaughan Williams maintained a harmonic link between his work and the Tallis by (1) restricting the chordal vocabulary of the fantasia to major and minor triads as much as possible. This restriction was a profound discipline for a postWagnerian composer writing in 1910, but it is responsible for successfully working in and from the Tallis. As we will see, it is also responsible for some striking harmonic progres­ sions that Tallis certainly wouldn’t have composed. (2) Vaughan Williams orchestrated his piece for two string choirs, a larger and a smaller, from which a solo quartet also was drawn. This timbral restriction is another gesture of respect for the Tallis original, which was a similarly restricted choral work. (3) Vaughan Williams treated his choirs much as Tallis might have treated a thirty-voice ensemble (or as he did with his famous forty-voice Spem in Alia) by using them sometimes in tutti and sometimes in antiphonal combina­ tions. (p. 568)

The first narrative arc of the piece consists of an introduction, a statement of the theme, a figurational variation of the theme, and then a “modern” exploration of the Tallis environ­ ment. The postintroduction section can be understood as a kind of bar form, with two Tallis Stollen followed by a Vaughan Williams's Abgesang, an analysis that harmonizes and magnifies the bar form of the Tallis piece itself. I will direct my analytic comments first to the introduction and then to the Abgesang, the two places where Vaughan Williams takes precedence over Tallis. The introduction begins with the chord and counterpoint progression shown in example 20.15: introduction through rehearsal A+3, the first “measure” of the example. Vaughan Williams signals a number of things in this progression. The contrary-motion counterpoint is pure Tallis: 8–6–3–8–6. The major-minor shimmering that characterizes Tallis's treat­ ment of the modal final, G, is also reenacted here in the relationship of the first and third chords, emphasized by the plus and minus signs (+ for major, and – for minor). Note also the mixed 2̂ associated with the modal shimmering: a non-Phrygian 2̂ (A♮ in the bass) fol­ lows the non-Phrygian G-major triad, and a Phrygian 2̂ follows the Phrygian G-minor triad (the A♭ 63 chord). These backward-looking touches are balanced by a thoroughly modern chord-root pro­ gression, noted between the staves, that is in neither a key nor a mode. It floats in a very light, G-ish tonal gravity before bumping unceremoniously into a surprising G♭-major tri­ ad, in the contrapuntally surprising 64 position to boot. The outer-voice counterpoint, still pure from Tallis's point of view, also shows some modern touches: space between the voices closes at a consistent rate of 4 semitones per chord change—until the last, that surprising G♭, which closes by only 3 semitones from the previous chord. This outer-voice counterpoint is the main constructive feature of the opening, since mode is already shim­ mering and tonal center, while attempting to materialize as G in the first few chords, is blown away by the G♭ at the end. What creates the particular chord qualities and roots? I submit that these are induced “from without” and that a neo-Riemannian transformation­ al apparatus is useful to show this.

Page 22 of 31

Three Short Essays on Neo-Riemannian Theory

Ex. 20.15.

In the first analytic “measure” of example 20.15, relationships are shown using L/P/R transformational labels between the chords on the upper staff. I use this particular set to reflect the remoteness of one triad compared to its immediate neighbor, an attribute I hear as expressively primary here. That is, in such light tonal (p. 569) (p. 570) gravity the overall impression can be one of “each chord for itself,” or what Kurth termed an “ab­ solute progression.”51 Each chord proposes itself, however tentatively, insecurely, or im­ probably as T, with each succeeding chord supplanting the claims of the previous one. L/ P/R transformations are uniquely suited to this environment because they derive from procedures of chordal “alteration,” a situation in which the sounding aspect of a chord is changed (transformed) without changing some underlying meaning. For example, as orig­ inally used by Riemann, a label like describes a tonic-functioned chord altered by the Leit­ tonwechsel transformation, yet the chord retains its T-ness despite the swapping in of the leading tone for the root. Riemann himself was reluctant to recognize multiple alterations —for example, the relative of the Leittonwechsel—lest the claim that a constant underly­ ing tonal function was still effective despite considerable surface deformation be taken as merely (and laughably) notional instead of actually (and productively) hearable. Neo-Rie­ mannian theory, on the other hand, is predicated on the idea that a whole set of triads can be manufactured by variously altering a single prototype. In this regard, however, the fact that a C-major triad when transformed by LR becomes a G-major triad is noncongruent with the observation that the two stand in relationship of, say, T and D (or that LR “is the same as” some operation X that transforms a tonic into a dominant);52 for LR claims that if C was construed as T, then G also expresses T; G is an “altered” C and retains an essen­ tial C-ness. As Hyer recognized, “from the transformational potential of a single triad, the group as a whole disperses the functional ‘significance’ of that triad among the harmonic consonances woven together to form its algebraic fabric; there is no one triad that forms a tonic for the group as a whole.”53 In the end, L/P/R neo-Riemannian analysis stands quite apart from a sensuous-functional analysis; it is rather the manufacturing record of Page 23 of 31

Three Short Essays on Neo-Riemannian Theory how chord X, possessing attribute A, was molded into chord Y, still possessing attribute A, at least in theory. In the string of absolute progressions here, attribute A is a vague, ten­ tative, and insecure T. The L/P/R system shows relationships according to root motions by third and modal change; T, D, and S are based on root relations by perfect fifth. The opening of the Tallis Fantasia is composed using root changes by second, which suggests the penetration of melodic-contrapuntal values—located in previous discussion between outer voices—fur­ ther into the harmonic dimension. The ultimate expression of melodic-harmonic preemi­ nence, however, is not in the first analytic measure of example 20.15 but in the second, where root-position major triads are planed in a kind of organum. As a result, the analytic technique there shifts from L/P/R to conceptually simpler pitch transposition Tx, with x being semitone-interval units; while L/P/R can “work” there, it models the situation there less naturally and accurately than pitch transposition. (Each alteration from one chord to the next requires four operations, such as LRPR, etc., which makes them all four-letter words—a rather discouraging analytic portent.) Back in the first measure, where L/P/R is suggested by the absolute quality of the chord changes, operations can be disciplined by using G-Phrygian as (p. 571) prototype and tool­ ing the individual chords accordingly. That is, given a choice of transformations that take one chord to another, the “best” one is that which distorts a G-Phrygian scale the least, which, in other words, uses the fewest notes from outside the prototype scale. It is for this reason that the first and fourth operations, both of which transform a major triad into another major triad a whole step lower, are analyzed differently. Example 20.16 explains. The top staff shows the first transformation, from a G-major to an F-major triad, which can be described by three L/P/R synonyms. G major contains B♮, a harmonically “perfect­ ing” variant that characterizes Tallis's own treatment of Phrygian, as we have seen; while a departure from prototype, it is but a mild one. A♮, as noted above, also departs from prototype. The first L/P/R word, PRLR, restores B♭ as its first operation and holds off in­ troducing A♮ until the last moment, while the other two fail to temper B♮ at all and, in one case, produce A♮ earlier than PRLR. Moreover, the other two introduce non-Phrygian pitches, E♮ and F♯, both of which eventually precipitate out after working the appropriate transformations. In terms of faithfulness to the prototype, then, PRLR is the superior pro­ cedure. In contrast, the lower staff of example 20.16 shows that PRLR is the least faithful in working the last chord change, from A♭ to G♭, while LRPR is most, producing the nonPhrygian deformations, G♭ and D♭, only at the end of the process.

Ex. 20.16.

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Three Short Essays on Neo-Riemannian Theory Moving into the second analytic measure, the previously unheard syllable PL is sounded as a transformational word and introduces the second section of the introduction. (None of the previous words had this syllable, associating Ps with Rs and Rs with Ls.) This word is accompanied by an articulation and change of texture, with the hint of D♭ acting like C♯ leading tone to D. This section stands on dominant D, a repeated P operation shimmering its mode: minor tune-fragments (bracketed as –) alternate with major-chord responses (bracketed as +). At the conclusion of the section, Vaughan Williams uses the ascending upper tetrachord of the mode to span D to G. This gesture is taken from the opening of Tallis's third line, where the systemic function (but not contextual function) was shown to discharge dominant to tonic. (Consult example 20.13, beginning of third system.) After the presentation of the Tallis tune and its varied reprise, Vaughan Williams was faced with the problem of extricating himself from Tallis's sixteenth-century Phrygian constraints, which comes to a head a rehearsal E. (Example 20.15: rehearsal E ff.) Vaugh­ an Williams lingers over a fragment from the final cadential gesture of the Tallis, all of whose chords fit properly into the G Phrygian mode (“measure” 1). This fragment is then echoed at quite a distance both tonal (via RPR between the chords beginning the seg­ (p. 572)

ment) and dynamic, a distance that seems responsible for the distortion in the echo that denatures the Phrygian mode (“measure” 2). After this transition, Vaughan Williams re­ turns to the technique of call-and-response used in the introduction (“measures” 3–5). Here, however, the harmonic effects are more striking while the outer-voice counter­ points of the responses revert to the pristine standard of the opening gesture. The motion to and from the hexatonic pole, F♯–d–F♯, which accompanies the upper-neighbor motive around the reciting note, is a signal response of the twentieth century to the sixteenth. At the next venture into this figure (“measure” 4), the upper-neighbor motive is transformed into a whole step and, by not returning to C♯, strives to pass toward E, attained only after considerable boundary turbulence between tonal centers, manifested by colliding dynam­ ic and harmonic masses in “measure” 5: fortissimo major chords separated from pianissi­ mo minor ones by the distancing transformation RPR that was so effective at the begin­ ning of this section. The middle section of the piece inhabits the home pitch class of Phrygian, E. But Vaughan Williams, true to his tonal traditions, ends the piece back in G, a tonal center he chose, I suspect, for the resonant low C the cellos and basses could use for the penultimate chord. The return passage, sketched in example 20.15: U–6, is also through boundary turbu­ lence. The transformations PLP and RP, which were prominent features of the material back around rehearsal E, are reapplied here at the beginning of the trip journey to G. Un­ like the earlier boundary passage, the thoroughgoing minorizing of triads here so cloud the interior that the dynamic collisions seem to be happening in a fog; the arrival of the G major chord is particularly gratifying as a result. In general, examples 20.15 and 20.16 attest to the competence of neo-Riemannian and other transformational labels in contexts where sensuous-functional chord labels have trouble making sense, where, in other words, tonal flow is strongly manipulated and dis­ orienting. Such contexts, I suggest, correlate in particular to Kurthian absolute progres­ Page 25 of 31

Three Short Essays on Neo-Riemannian Theory sions rather than to generalized instances of sensuous-functional stress. (For example, re­ lations among tonic and antitonic objects examined in the second essay do exhibit stress and, from a narrow view of “harmony” that is frequently prefaced by the adjective “dia­ tonic,” may suggest a transformational palliative. Yet I maintained both there and in other writings that sensuous-functional relations are hardly so brittle as to break from an effect like B7 progressing to C+ in [putative] E minor, or some other inventive if nonstandard relation.) In other words, when individual chords in a succession of chords propose them­ selves (however meekly or ineffectually) as local tonics, and when (p. 573) such proposals pile up and accumulate, there is no longer any sensuous-functional transactions and dis­ charges, no circulation of T, D, and S. Yet there is still measurable chord change, which can be adequately modeled by L/P/R or other sensuously indifferent transformational sys­ tems (such as pitch transposition Tx or a “Uniform Triadic Transformation” system as out­ lined by Julian Hook). And now a strategic retraction: in the opening paragraphs of this essay, I opened up space for transformational analysis in cases when “functional tonics are absent or only weakly effective” evincing a “tonic-free, zero-gravity state.” It is now clear that I identify this state not as being absent of tonics but rather as overcrowded with them. Still, the ef­ fect upon analysis is the same—a loss of orientation that prevents sensuous-functional transactions from taking place.54 To have attempted this distinction earlier might have struck readers as fussy, so I thought it better to set down a heuristic position and then let the Vaughan Williams composition undercut it. My purpose in these three essays has been, obviously, threefold: (1) to deflate immoder­ ate enthusiasms for transformational approaches as cure-alls for various music theoreti­ cal and analytical anxieties;55 (2) to cross such approaches with others in the hopes of producing vigorous analytic hybrids; and (3) to identify cases in which transformation theory may indeed be the best first approach. That the three are related yet stand inde­ pendently—that they, in other words, create an absolute progression of ideas—is a happy accident, and hopefully a useful one.

Notes: (1.) David Lewin, Generalized Musical Intervals and Transformations (New York: Oxford University Press, 2006), xxxi. (2.) Victor Zuckerkandl, Sound and Symbol: Music and the External World, trans. W. R. Trask, vol. XLIV, Bollingen Series (Princeton: Princeton University Press, 1956), 76. (3.) Lewin, Generalized Musical Intervals and Transformations, 159. (4.) Straus, “Voice Leading in Atonal Music,” in Music Theory in Concept and Practice, ed. David W. Beach, James M. Baker, and Jonathan W. Bernard (Rochester: University of Rochester Press, 1997), 243–244.

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Three Short Essays on Neo-Riemannian Theory (5.) Zuckerkandl, Sound and Symbol, 82. Zuckerkandl devotes an entire chapter to “The Paradox of Tonal Motion,” (75–87). And this is within an entire section devoted to motion (75–148). Roger Scruton, in The Aesthetics of Music (Oxford: Oxford University Press, 1997), 49–54, highlights the problematic features of both musical motion and musical space, concluding that “musical space, and musical movement, are not even analogous to space and movement in the physical world” (51). (6.) Zuckerkandl, Sound and Symbol, 83. (7.) A useful exploration of the problems in the metaphor of musical motion is presented by Judy Lochhead, “The Metaphor of Musical Motion: Is There an Alternative?,” Theory and Practice 14–15 (1989–1990): 83–103; see especially 84–91. Robert Gjerdingen, “Ap­ parent Motion in Music?,” Music Perception 11.4 (1994): 335–370, deals extensively with the perceptual and cognitive issues involved. (8.) Scruton, The Aesthetics of Music, 96. (9.) Pitches, though generally conceived as discrete points, can on occasion participate in infinite, continuous scaling: glissando, portamento, and so on. The equivalent in music dy­ namics is, of course, the crescendo/descrescendo. (10.) The definition of dynamics as points is required in computer applications such as no­ tation or sequencer programs, where, for instance, is assigned a particular value (say, in terms of MIDI key velocity). In contrast, the human treatment of dynamics regards them as overlapping bandwidths within which are a large number of acceptable and “true” re­ alizations. The same is true of tempo when indicated by the usual Italian terms—but not when nonoverlapping max and min points of these terms are mapped onto a metronome face. (11.) That is, it is not possible to do so without getting rid of some or most commonsense definitional attributes of the heterogeneous objects—to make them, in other words, ho­ mogenous objects. For an extended study of this issue involving pitch and time, see Justin London, “Some Non-Isomorphisms between Pitch and Time,” Journal of Music Theory 46 (2002): 127–151. (12.) Adrian Childs, “Moving Beyond Neo-Riemannian Triads: Exploring a Transformation­ al Model for Seventh Chords,” Journal of Music Theory 42.2 (1998): 181–193. (13.) Clifton Callender, “Voice-Leading Parsimony in the Music of Alexander Scriabin,” Journal of Music Theory 42.2 (1998): 219–331. A similar observation has been made by Ju­ lian Hook, “Uniform Triadic Transformations” Journal of Music Theory 46 (2002): 57–126; see in particular 58. (14.) One might argue (with considerable effect, perhaps) that modeling the appearance of motion is consonant with the metaphor of musical motion itself, which, as Zuckerkandl reminded us, is only apparent motion. Still, when Lewin writes that because Riemann's “dominants just sit around, not going anywhere, [which] makes his musical analyses sub­ Page 27 of 31

Three Short Essays on Neo-Riemannian Theory ject to inertia and lifelessness” (Lewin, Generalized Musical Intervals and Transforma­ tions, 177), I can note only that I espy the same potential in neo-Riemannian analytic work, albeit in a different way—hence, the title of this essay. (15.) Described first in Riemann, Skizze einer neuen Methode der Harmonielehre (Leipzig: Breitkopf & Härtel, 1880); expanded in Henry Klumpenhouwer, “Some Remarks on the Use of Riemann Transformations,” Music Theory Online 0.9 (1994); discussed in connection with other systems in Hook, “Uniform Triadic Transformations.” (16.) Lewin, Generalized Musical Intervals and Transformations, 177. If I have succeeded in drawing the reader's attention away from the dummy and to the ventriloquist, then the phrase “something one does to a Klang” in the quotation above should be seen in spot­ light. Who, exactly, is doing the doing? Composer? Listener? Analyst? A combination of thereof? The vagueness and misdirection here is essential to the ventriloquist's act. (17.) Hyer, “Reimag(in)ing Riemann,” Journal of Music Theory 39.1 (1995): 101–138; quo­ tation on p. 128. (18.) Hyer, “‘Sighing Branches’: Prosopoeia in Rameau's Pigmalion,” Music Analysis 13.1 (1994): 7–20; quotation on p. 15. (19.) Hyer discusses the “disruptive” and “damaging” power of the “substantive pres­ ence” of a tonic on the transformational lattice in “Reimag(in)ing Riemann,” 127–128. (20.) See Lee A. Rothfarb, Ernst Kurth as Theorist and Analyst (Philadelphia: University of Pennsylvania Press, 1988). For instance, Rothfarb notes that “according to Kurth harmo­ ny exhibits two properties, sensuousness and energy. Their mixture, in various propor­ tions, determines the finished musical product” (115). See also pp. 152ff. (21.) Harrison, Harmonic Function in Chromatic Music, 36–42. (22.) Hyer, “‘Sighing Branches,’” 40. (23.) The conference was a joint meeting of fourteen scholarly societies. Santa's paper was read at session “3–46 (SMT)” on Friday, November 3. The paper was subsequently published as “Nonatonic Systems and the Parsimonious Interpretation of Dominant-Tonic Progressions,” Theory and Practice 28 (2003): 1–28. (24.) Scott Murphy pointed out to me that this progression also appears (transposed to start on a G-major triad) in Nicolai Rimsky-Korsakov, Practical Manual of Harmony, trans. J. Achron (New York: Carl Fischer, [1886] 1930), 117. (25.) Since Helmholtz's day at least, the consonant triad also was potentially thought of as a collection of pitch classes and pitch-class bands, with the “fifth” being constituted by pitch classes and the “third” by a pitch-class band (i.e., minor or major third above the fundamental). See Paul Hindemith, The Craft of Musical Composition (New York: Associat­ ed Music, 1942), I: 72, for a Helmholtz-derived presentation of this idea. This way of thinking can be applied with great effect, I believe, to certain harmonic theories that fea­ Page 28 of 31

Three Short Essays on Neo-Riemannian Theory ture movable chordal members—jazz theory being an outstanding example. It is also of potential use in dealing with Kurth's ideas of harmonic shading, coloration, and identifica­ tion. I reluctantly concede that exploration of this interesting idea must await another pa­ per. (26.) Jean-Philippe Rameau, Treatise on Harmony, trans. Philipp Gossett (New York: Dover, 1971), 65–67. See also Thomas Christensen, Rameau and Musical Thought in the Enlightenment (Cambridge: Cambridge University Press, 1993), 115–116. The crude setnotation shorthand must be understood as representing a recto form of the set, in other words, a tritone with an attached interior major second on the bottom. (27.) Cohn applies compass labels to hexatonic systems in “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions,” Music Analysis 15.1 (1996): 9–40. (28.) In terms of Santa's transpositional descriptions, for any [026] = X: Da = T1(X); Db = T3(X); and Dc = T5(X). (29.) Musical context, of course, determines the strength of these tonicizations, as will be illustrated briefly in connection with example 20.6 and n. 32 below. (30.) The rationale for these orderings, as well as its origins in the theories of Rameau and Riemann, is discussed in Harrison, Harmonic Function in Chromatic Music, 90–126. (31.) “Square Dances with Cubes,” Journal of Music Theory 42.2 (1998): 283–295. (32.) More should be said about context here: the Db discharges from C7 do not really sta­ bilize the A+ chords as tonics because of the oscillation between C7 and A+ as well as the mobile, first-inversion voicing of A+. A more nuanced hearing of the A+ describes it as a tonicized dominant, with a C♯/ 7̂ that eventually resolves to D/ 8̂ in the last measure of the example, after the C7 attempts yet fails to complete a Da discharge onto F+ (with 7 4 suspension). (33.) I analyze the passage in ways consonant with the present discussion in Harmonic Function in Chromatic Music, 153–156. (34.) Harrison, Harmonic Function in Chromatic Music, 134–153. (35.) Harrison, “Nonconformist Notions of Nineteenth-Century Enharmonicism,” Music Analysis 21.2 (2002): 115–160. (36.) “Neo-Riemannian Operations, Parsimonious Trichords, and Their Tonnetz Representations,” Journal of Music Theory 41.1 (1997): 1–66. (37.) Cohn, in “Neo-Riemannian Operations,” 62 n. 8, notes that “other aspects of [the fig­ ure] are intriguing and suggestive.” He draws particular attention to mod-3 congruence of some of the trichords.

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Three Short Essays on Neo-Riemannian Theory (38.) Of course, there are only three trichordal set-classes in whole-tone space, [024], [026], and [048], so the whole-tone parsimony prize is contested in a particularly weak field. (39.) I have sketched out some contextual structures in “Dissonant Tonics and Post-Tonal Tonality,” paper read at Music Theory Society of New York State, April 2002, in New York City. (40.) Of course, enharmonic reinterpretation of the various [026] discords can occur with a Db resolution. The energetic aspects of enharmonicism are discussed in Harrison, “Non­ conformist Notions.” (41.) The example is not meant to show pristine voice-leading, since there are parallel oc­ taves, but rather the resolution of the transpositionally combined trichords. (42.) See Horace A. Miller, New Harmonic Devices: A Treatise on Modern Harmonic Prob­ lems (Philadelphia: Oliver Ditson, 1930), 107–118; and A. Eaglefield Hull, Modern Harmo­ ny: Its Explanation and Application (London: Augener, 1915), 59. (43.) The relevant works are Vincent Persichetti, Twentieth-Century Harmony: Creative Aspects and Practice (New York: W. W. Norton, 1961), and Ludmila Ulehla, Contemporary Harmony; Romanticism through the Twelve-Tone Row (New York: Free Press, 1966). (44.) Cohn, “Square Dances.” (45.) The Ø symbol represents an inversus form of [026] = (046): a tritone with embedded major second attached to the upper member of the tritone. There is an interesting study, for which I have done only the barest preliminaries, involv­ ing parsimonious 0-sum bijective functions from a given trichord. Using {0,4,7} as a test case, the following set classes can be reached: [037], [026], and [027] three times each; [016] and [014] two times, and one instance of [013]. Based on this result, admitting [025] into the system of [037] and [026] seems the next move. This example should be compared to Cohn's figure 9, which shows the “tetrachordal (4–27) system of sum class­ es,” grouping the tetrachords into six classes of four chords each, with the chord roots being related by minor thirds. (46.) That is, the top system shows the secondary discords from measure 1 of example 20.9, and the bottom shows those from measure 2. (47.) The labels X4 and Y6 are the sum-class transformations that take

to

and

, re­

spectively. See Cohn “Square Dances,” 288–289. (48.) Hyer, “Reimag(in)ing Riemann,” 128–129. (49.) Hyer, “Reimag(in)ing Riemann,” 115, in reference to an analysis of the Schlafakko­ rde from Die Walküre.

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Three Short Essays on Neo-Riemannian Theory (50.) Charles J. Smith, “The Functional Extravagance of Chromatic Chords,” Music Theory Spectrum 8 (1986), 94–139; the quotation is on page 129. (51.) See Rothfarb, Ernst Kurth as Theorist and Analyst, chapter 7. (52.) A similar point is made by Henry Klumpenhouwer in “Some Remarks on the Use of Riemann Transformations.” (53.) Hyer, “Reimag(in)ing Riemann,” 127. (54.) Related problems of having “too much tonality” are discussed in Harrison “Noncon­ formist Notions,” 126–128. (55.) In this matter, it shares common ground with Philip Lambert, “Isographies and Some Klumpenhouwer Networks They Involve,” Music Theory Spectrum 24.2 (2002): 165–195, Justin London, “Some Non-Isomorphisms,” and Shaughn O’Donnell, “Klumpen­ houwer Networks, Isography, and the Molecular Metaphor,” Intégral 12 (1998): 53–80.

Daniel Harrison

Daniel Harrison is the Allen Forte Professor of Music Theory at Yale University, where he is also chair of the department of music. He is the author of Harmonic Function in Chromatic Music, and has published on tonal-music topics in Journal of Music Theory, Music Theory Spectrum, Musical Quarterly, Theory and Practice, and Music Analysis, among other venues.

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Glossary The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011

(p. 578)

(p. 579)

Subject: M usic Online Publication Date: Sep 2012

Glossary

Achttaktigkeit. Analogous to his idealized notions of harmonic progression, Riemann posits an idealized metric configuration comprising eight-measure units. Auftaktigkeit. Refers to Riemann's supposition of a universal iambic configuration in the metric domain of musical phrasing. Even where it is not notated, Riemann understands musical phrases to begin with an implicit upbeat. Cadence. Riemann posited two idealized successions of functions: in major, T–S–D–T; in minor, ˚T–˚D–˚S–˚T (see Dualism). Dualism. At a most basic level, dualism reflects the structural equality of major and minor systems: in a dualistic framework, the minor triad is recognized as the mirror-symmetric image of the major. The concept became problematic in the nineteenth century when theorists sought acoustical justification for dualism. For example, Riemann sought to locate the source of minor triadic generation in a supposed undertone series, the mirrorsymmetric dual of the overtone series, which historically was considered the source of generation for the major triad. Even though Riemann's specific explanation for dualism changed over the course of his writings, dualism remained an essential and foundational component of his harmonic thought. Function. Riemann distinguishes three functions: tonic (T), dominant (D), and subdominant (S). However, it remains unclear whether Riemann construed functions as chords or categories. Understood as chords, each function is manifest by primary forms (I, V, and IV) and modified forms (see Scheinkonsonanz). Understood as categories, the functions reify stages in idealized harmonic successions (see Cadence). Harmonieschritte. Riemann's dualist system to categorize particular harmonic successions between two chords according to the directed intervals between chordal Haupttöne (see Hauptton). Riemann divided Harmonieschritte into Schritte that relate like-mode triads, and Wechsel that relate mode-opposed triads. Riemann further modifies Schritte and Wechsel according to the direction of the interval between Haupttöne relative to their mode: Schlichte (simple) relations progress in the direction of generation of chord components relative to the Hauptton of the first SOFIA P - LES JEUX SONT FAITS "Produced by Shuka4Beats" (outgoing) chord, Gegen (contrary) relations progress in the direction opposite to that of chord components. For example, the schlichter Quintschritt relates any major triad to the major triad a perfect fifth higher (e.g., c +–g+, the directed relation from C, the Hauptton of the C-major triad, upward to G, the Hauptton of the G-major triad)

or any minor triad to the minor triad a perfect fifth below (e.g., ˚e–˚a, the directed

relation from E, the Hauptton of an A-minor triad, to A, the Hauptton of a D-minor triad). For Riemann, “schlicht” is an unmarked term, understood when generally omitted: “Schlichter Quintschritt” is more simply referred to as “Quintschritt.” The Gegenquintschritt relates any major triad to the major triad a perfect fifth lower (e.g., c +–f+) or any minor triad to the minor triad a perfect fifth higher (e.g., ˚a–˚e). The (schlichter) Terzwechsel relates any major triad to (p. 580)

the minor triad whose Hauptton lies a major third higher (e.g., c +–˚e) or any minor triad to the major triad whose Hauptton lies a major third lower (e.g., ˚e–c +). The Gegenterzwechsel relates any major triad to the minor triad whose Hauptton lies a major third lower (e.g., c +–˚af) or any minor triad to the major triad whose Hauptton lies a major third higher (e.g., ˚a♭–c +). Hauptton (principal or referential tone). In his dualist harmonic conception, Riemann posits a Hauptton, distinct from a triad root, specifically that tone in a triad to which the other tones refer as major third and perfect fifth. In major, the Hauptton corresponds to the triadic root (C in a C-major triad); in minor the Hauptton corresponds to the triadic fifth (E in an A-minor triad) (see Klangschlüssel, Dualism, Klang). Klang. Klang is multifaceted concept in Riemann's writings. At the most general level, Klang refers to the acoustical signal of a sounding body. Riemann, however, posits a more specific meaning, distinct from, but not always clearly differentiated from, the triad. In line with his dualistic conception, a Klang is an abstraction comprising a Hauptton surrounded by its upper and lower triadic components (major third and perfect fifth). A consonant, sounding Klang (i.e., a triad), however, manifests only one side of the idealized abstraction (Hauptton and upper components in major, Hauptton and lower components in minor). Klangschlüssel. Riemann's system of notation, in which triads are labeled according to their Haupttöne, with symbols affixed to indicate mode. For example, c + represents a C-major triad (Hauptton C with affixed “+” for major); ˚e represents A-minor (Hauptton E with affixed “˚” for minor). Modifications to any triad can be indicated with numerals and symbols affixed to the letter names. Arabic numerals represent tones at intervals above a major-mode Hauptton; Roman numerals represent tones at intervals below a minor Hauptton. Thus, c 6 indicates a C-major triad with added sixth (C–E–G–A); aVII indicates a d-minor triad with added under-seventh (B–D–F–A); d5〉 indicates a D-major triad with a chromatically lowered fifth (A♭–D–Fs), and so on. Klangvertretung. Refers to the ability of individual tones and of dyads to project triadic and functional identities. For example, the tone C can assume the identity of the root of a C-major or C-minor triad, the third of an A♭major or A-minor triad, or fifth of an F-major or F-minor triad. Similarly, an A–C dyad can represent either the upper third of an F-major triad or the lower third of an A-minor triad. Scheinkonsonanz (apparent consonance). A collection that appears to be a triad but that for Riemann represents an elliptical manifestation of a dissonant collection. (p. 581) For example, Riemann could construe an A-minor triad in the context of C major either as an elision of a dissonant configuration, F–A–C–E, in which the Hauptton F is suppressed and replaced by the E, or the dissonant configuration C–E–G–A, in which the chordal fifth, G, is suppressed and replaced by the A. In Riemann's function theory (see Function), apparent consonances form the basis for functional substitution. In the given example, the A-minor triad as ellipsis of F–A–C–E would be construed as a subdominant Leittonwechselklang (i.e., an apparent triad that results from the substitution of the triadic Hauptton by the leading tone); alternately, the A-minor triad as ellipsis of C–E–G–A, would be construed as SOFIA P - LES JEUX SONT FAITS "Produced by Shuka4Beats" a tonic Parallelklang (i.e., an apparent triad that results from substitution of the triadic fifth with sixth).

Tonvorstellung. The imagination or mental representation of a tone. Tonvorstellung became a central concept in Riemann's late writing when he located the foundation of tonal relations within psychology. This relieved him of the obligation to argue from an acoustical perspective. Transformation. As a contemporary music-theoretical concept, transformation has a twofold meaning. As a general term, transformation refers to a perspective that places emphasis on the relationships between musical objects, rather than upon the objects themselves (in contrast to a set-theoretical perspective). More specifically, transformations refer to mathematical functions that map elements of a set to themselves in a manner that is both one-to-one and onto. Musically, the set may contain chords, tones, keys, and so on. Voice-leading parsimony. Refers to relations between triads or other pitch collections—in the literal or underlying voice-leading—that minimize moving voices and voice-leading distances between them. For example, a C-major harmony can be voice-led parsimoniously to an E-minor harmony: the tones E and G can remain fixed and C can move by semitone to B to complete the transformation. The Tristan progression is an example of a parsimonious tetrachordal relationship: B and Gs remain fixed; F and Ds move by semitone to E and D♮, respectively.

Selected Bibliography

Selected Bibliography   The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195321333.004.0001

(p. 582)

(p. 583)

Selected Bibliography

Apfel, Ernst, and Carl Dahlhaus. Studien zur Theorie und Geschichte der musikalischen Rhythmik und Metrik, 2 vols. Munich: Emil Katzbichler, 1974. Arntz, Michael. Hugo Riemann (1849–1919): Leben, Werk und Wirkung. Cologne: Allegro, 1999. Babbitt, Milton. Words about Music: The Madison Lectures. Edited by Stephen Dembski and Joseph N. Straus. Madison: University of Wisconsin Press, 1987. Balzano, Gerald. “The Group Theoretic Description of 12-Fold and Microtonal Pitch Sys­ tems.” Computer Music Journal 4 (1980): 66–84. Bass, Richard. “Enharmonic Position Finding and the Resolution of Seventh Chords in Chromatic Music.” Music Theory Spectrum 29.1 (2007): 73–100. ——— . “Review of David Kopp, Chromatic Transformations in Nineteenth-Century Music.” Music Theory Online 10.1 (2004). Belinfante, Ary. “De leer der tonale functien in conflict met die der polaire tegenstelling.” Orgaan van de Vereeniging van Muziek-Onderwijzers en -Onderwijzeressen 4.9 (1904): 1– 2. Benjamin, William. “Interlocking Diatonic Collections as a Source of Chromaticism in Late Nineteenth-Century Music.” In Theory Only 1.11–12 (1975): 31–51. Bernstein, David. The Harmonic Theory of Georg Capellen. Ann Arbor, MI: UMI Press, 1986. ——— . “Symmetry and Symmetrical Inversion in Turn-of-the-Century Theory and Prac­ tice.” In Music Theory and the Exploration of the Past, ed. Christopher Hatch and David W. Bernstein, 377–409. Chicago: University of Chicago Press, 1993. Blackwood, Easely. The Structure of Recognizable Diatonic Tunings. Princeton: Princeton University Press, 1986. Page 1 of 20

Selected Bibliography Blasius, Leslie David. Schenker's Argument and the Claims of Music Theory. Cambridge Studies in Music Theory and Analysis 9. Cambridge: Cambridge University Press, 1996. Böhme-Mehner, Tatjana, and Klaus Mehner, eds. Hugo Riemann (1849–1919): Musikwis­ senschaftler mit Universalanspruch. Cologne: Böhlau Verlag, 2001. Bribitzer-Stull, Matthew. “The A♭–C–E Complex: The Origin and Function of Chromatic Major Third Collections in Nineteenth-Century Music.” Music Theory Spectrum 28.2 (2006): 167–190. Brower, Candace. “A Cognitive Theory of Musical Meaning.” Journal of Music Theory 44 (2000): 323–379. ——— . “Paradoxes of Pitch Space.” Music Analysis 27.1 (2008): 51–106. Budday, Wolfgang. Harmonielehre Wiener Klassik: Theorie-Satztechnik-Werkanalyse. Stuttgart: Berthold & Schwerdtner, 2002. Burnham, Scott. Beethoven Hero. Princeton, NJ: Princeton University Press, 1995. ——— . “Form.” In The Cambridge History of Western Music Theory, ed. Thomas Chris­ tensen, 880–906. Cambridge: Cambridge University Press, 2002. ——— . “Method and Motivation in Hugo Riemann's History of Harmonic Theory.” Music Theory Spectrum 14.1 (1992): 1–14. ——— . “The Second Nature of Sonata Form.” In Music Theory and Natural Order from the Renaissance to the Early Twentieth Century, ed. Suzannah Clark and Alexander Rehding, 111–141. Cambridge: Cambridge University Press, 2001. (p. 584)

Bußler, Ludwig. Lexikon der musikalischen Harmonien. Berlin: Carl Habel, 1889. Callender, Clifton. “Voice-Leading Parsimony in the Music of Alexander Scriabin.” Journal of Music Theory 42.2 (1998): 219–233. Callender, Clifton, Ian Quinn, and Dmitri Tymoczko. “Generalized Voice Leading Spaces.” Science 320 (2008): 346–348. Capellen, Georg. Die Abhängigkeitsverhältnisse in der Musik. Leipzig: C. F. Kahnt, 1904. ——— . Die Freiheit oder Unfreiheit der Töne und Intervalle. Leipzig: Kahnt, 1904. ——— . Die musikalische Akustik als Grundlage der Harmonik und Melodik. Leipzig: Kah­ nt, 1903. ——— . “Die Unmöglichkeit und Überflüssigkeit der dualistischen Molltheorie Riemanns.” Neue Zeitschrift für Musik 68 (1901): 529–531, 541–543, 553–555, 569–572, 585–587, 601–603, 617–619.

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Selected Bibliography ——— . Die Zukunft der Musiktheorie (Dualismus oder “Monismus”?) und ihre Einwirkung auf die Praxis. Leipzig: Kahnt, 1905. ——— . Ein neuer exotischer Musikstil. Stuttgart: Carl Grüninger, 1906. ——— . Fortschrittliche Harmonie- und Melodielehre. Leipzig: Breitkopf und Härtel, 1908. Caplin, William E. “Der Akzent des Anfangs: Zur Theorie des musikalischen Taktes.” Zeitschrift für Musiktheorie 9.1 (1978): 17–28. ——— . “Moritz Hauptmann and the Theory of Suspensions.” Journal of Music Theory 28.2 (1984): 251–69. ——— . “Theories of Musical Rhythm in the Eighteenth and Nineteenth Centuries.” In The Cambridge History of Western Music Theory, ed. Thomas Christensen, 657–694. Cam­ bridge: Cambridge University Press, 2002. ——— . “Tonal Function and Metrical Accent: A Historical Perspective.” Music Theory Spectrum 5 (1983): 1–14. Carter, Elliot. “Letter to the Editor: On the Nature of Music Theory.” Journal of Music Theory 3.1 (1959): 170. Childs, Adrian P. “Moving Beyond Neo-Riemannian Triads: Exploring a Transformational Model for Seventh Chords.” Journal of Music Theory 42.2 (1998): 181–193. Christensen, Thomas. “Music Theory and Its Histories.” In Music Theory and the Explo­ ration of the Past, ed. Christopher Hatch and David W. Bernstein, 9–39. Chicago: Univer­ sity of Chicago Press, 1993. ——— . Rameau and Musical Thought in the Enlightenment. Cambridge: Cambridge Uni­ versity Press, 1993. ——— . “The Schichtenlehre of Hugo Riemann.” In Theory Only 6 (1982): 37–44. Clark, Suzannah. “Schubert, Theory and Analysis.” Music Analysis 21 (2002): 209–243. ——— . “Seduced by Notation: Oettingen's Topology of the Major-Minor System.” In Music Theory and Natural Order, ed. Suzannah Clark and Alexander Rehding, 161–180. Cam­ bridge: Cambridge University Press, 2001. ——— . “Terzverwandtschaft in der Unvollendeten von Schubert und der Waldstein-Sonate von Beethoven—Kennzeichen des neunzehnten Jahrhunderts und theoretisches Problem.” Schubert durch die Brille 20 (1998): 122–130. Clough, John. “A Rudimentary Geometric Model for Contextual Transposition and Inver­ sion.” Journal of Music Theory 42.2 (1998): 297–306.

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Selected Bibliography Cohn, Richard. “As Wonderful as Star Clusters: Instruments for Gazing at Tonality in Schubert.” Nineteenth-Century Music 22.3 (1999): 213–232. (p. 585)

——— . “Hexatonic Poles in Parsifal.” Opera Quarterly 22.2 (2006): 230–48.

——— . “An Introduction to Neo-Riemannian Theory: A Survey and Historical Perspec­ tive.” Journal of Music Theory 42.2 (1998): 167–180. ——— . “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions.” Music Analysis 15.1 (1996): 9–40. ——— . “Music Theory's New Pedagogability.” Music Theory Online 4.2 (1998). ——— . “Neo-Riemannian Operations, Parsimonious Trichords, and Their Tonnetz Representations.” Journal of Music Theory 41.1 (1997): 1–66. ——— . “Review of Tonal Pitch Space.” Music Theory Spectrum 29.1 (2007): 101–114. ——— . “Square Dances with Cubes.” Journal of Music Theory 42.2 (1998): 283–295. ——— . “Uncanny Resemblances: Harmonic Signification in the Freudian Age.” Journal of the American Musicological Society 57.2 (2004): 285–323. ——— . “Weitzmann's Regions, My Cycles, and Douthett's Dancing Cubes.” Music Theory Spectrum 22.1 (2000), 89–103. Cohn, Richard, and Douglas Dempster. “Hierarchical Unity, Plural Unities.” In Disciplin­ ing Music: Musicology and Its Canons, ed. Katherine Bergeron and Philip Bohlman, 156– 181. Chicago: University of Chicago Press, 1992. Cook, Nicholas. “Epistemologies of Music Theory.” In The Cambridge History of Western Music Theory, ed. Thomas Christensen, 78–105. Cambridge: Cambridge University Press, 2000. Cook, Robert C. “Parsimony and Extravagance.” Journal of Music Theory 49.1 (2005): 109–140. ——— . “Transformational Approaches to Romantic Harmony and the Late Works of César Franck.” Ph.D. diss., University of Chicago, 2001. ——— . “Voice Leading, a Non-Commutative Group, and the Double Reprise in Franck's Piano Quintet.” Paper presented at the annual meeting of the Society for Music Theory, 1998. Cramer, Albert. “Music for the Future: Sounds of Early-Twentieth-Century Psychology and Language in Works of Schoenberg, Webern, and Berg, 1908 to the First World War.” Ph.D. diss., University of Pennsylvania, 1997.

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Selected Bibliography Dahlhaus, Carl. Between Romanticism and Modernism: Four Studies in the Music of the Later Nineteenth Century. Translated by Mary Whittall. California Studies in 19th-Centu­ ry Music 1. Berkeley: University of California Press, 1980. ——— . Die Geschichte der Musiktheorie im 18. und 19. Jahrhundert, Zweiter Teil. Darm­ stadt: Wissenschaftliche Buchgesellschaft, 1989. ——— . Studies on the Origins of Harmonic Tonality. Translated by Robert Gjerdingen. Princeton: Princeton University Press, 1990. ——— . “Terminologisches zum Begriff der harmonischen Funktion.” Die Musikforschung 28.2 (1975): 197–202. ——— . “Über den Begriff der tonalen Funktion.” In Beiträge zur Musiktheorie des 19. Jahrhunderts, ed. Martin Vogel, 93–102. Regensburg: Gustav Bosse, 1966. ——— . “Über einige theoretische Voraussetzungen der musikalischen Analyse.” In Bericht über den 1. Internationalen Kongreß für Musiktheorie, ed. Peter Rummenhöller, Friedrich Christoph Reininghaus, and Habakuk Traber, 153–176. Stuttgart: Ichtys, 1972. de la Motte, Diether. A Study of Harmony: An Historical Perspective. Translated by Jeffrey L. Prater. Dubuque: W. C. Brown, 1991. De la Motte-Haber, Helga. “Musikalische Logik: Über das System von Hugo Riemann.” In Musiktheorie, ed. Helga de la Motte-Haber and Oliver Schwab-Felisch, 203–223. Laaber: Laaber, 2004. Dempster, Douglas, and Matthew Brown. “Evaluating Musical Analyses and Theo­ ries: Five Perspectives.” Journal of Music Theory 34.2 (1990): 247–279. (p. 586)

Diergarten, Felix. “Riemann-Rezeption und Reformpädagogik: Der Musiktheoretiker Jo­ hannes Schreyer.” Zeitschrift der Gesellschaft für Musiktheorie, Bd. 2.1 (2005). Douthett, Jack, and Peter Steinbach. “Parsimonious Graphs: A Study in Parsimony, Con­ textual Transformations, and Modes of Limited Transposition.” Journal of Music Theory 42.2 (1998): 241–263. Drobisch, Moritz W. Nachträge zur Theorie der musikalischen Tonverhältnisse. Leipzig: Hirzel, 1855. ——— . Über die mathematische Bestimmung der musikalischen Intervalle. Leipzig: Wied­ mann, 1846. ——— . Über musikalische Tonbestimmung und Temperatur. Leipzig: Wiedmann, 1852. Ellis, Alexander. “On the Musical Scales of Various Nations.” Journal of the Society of Arts 33 (1885): 485–527.

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Selected Bibliography Engebretsen, Nora. “The Chaos of Possibilities: Combinatorial Group Theory in Nine­ teenth-Century German Harmony Treatises.” Ph.D. diss., State University of New York at Buffalo, 2002. Ergo, Emil. “Die Taktlehre der Tonkunst.” Zeitschrift der IMG (1911): 180–188. ——— . Richard Wagner's Harmonik und Melodik: Ein Beitrag zur Wagnerschen Harmonik. Leipzig: Breitkopf und Härtel, 1914. Ernst, Anselm, and Wolfgang Rüdiger. “Reformtendenzen in der Musikpädagogik (1900– 1933): Resümee und Ausblick auf die Gegenwart.” In Visionen und Aufbrüche: Zur Krise der modernen Musik 1908–1933, ed. Günther Metz, 375–80. Kassel: G. Bosse, 1994. Erpf, Hermann. Studien zur Harmonie- und Klangtechnik der neueren Musik. Leipzig: Breitkopf und Härtel, 1927. Federhofer, Hellmut. Akkord und Stimmführung in den musiktheoretischen Systemen von Hugo Riemann, Ernst Kurth und Heinrich Schenker. Vienna: Veröffentlichungen der Akademie der Wissenschaften, 1977. ——— . “Die Funktionstheorie Hugo Riemanns und die Schichtenlehre Heinrich Schenkers.” In Bericht über den internationalen musikwissenschaftlichen Kongreß Wien, 1956, ed. Erich Schenk, 183–190. Graz-Cologne: Böhlau, 1958. Fétis, François-Joseph. Traité complet de la théorie et de la pratique de l’harmonie con­ tenant la doctrine de la science et de l’art. 4th ed. Paris: Brandus, 1849. Fink, Robert. “Going Flat.” In Rethinking Music, ed. Nicholas Cook and Mark Everist, 102–137. Oxford: Oxford University Press, 1999. Fortlage, Carl. Das musikalische System der Griechen in seiner Urgestalt. Aus den Ton­ leitern des Alypius, zum ersten Male entwickelt. Leipzig: Breitkopf und Härtel, 1847. Foucault, Michel. The Order of Things. New York: Vintage Books, 1970. Gabriel, Gottfried. “Frege, Lotze, and the Continental Roots of Early Analytic Philosophy.” In From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy, ed. Erich H. Reck, 39–51. Oxford: Oxford University Press, 2002. Gelbart, Matthew. Invention of Folk Music and Art Music. Cambridge: Cambridge Univer­ sity Press, 2007. Girard, Aaron. “Music Theory in the American Academy.” Ph.D. diss., Harvard University, 2007. Gjerdingen, Robert O. “Apparent Motion in Music?” Music Perception 11.4 (1994): 335– 370.

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Selected Bibliography Gollin, Edward. “Representations of Space and Conceptions of Distance in Transforma­ tional Theories.” Ph.D. diss., Harvard University, 2000. ——— . “Review of Hugo Riemann (1849–1919): Leben, Werk und Wirkung.” Jour­ nal of the American Musical Society 56.1 (2003): 192–198. (p. 587)

——— . “Some Aspects of Three-Dimensional Tonnetze.” Journal of Music Theory 42.2 (1998): 195–206. ——— . “Some Further Notes on the History of the Tonnetz.” Theoria 13 (2006): 99–111. ——— . “Some Unusual Transformations in Bartók's ‘Minor Seconds, Major Sevenths.’ ” Intégral 12 (1998): 25–51. Grabner, Hermann. Die Funktionstheorie Hugo Riemanns und ihre Bedeutung für die praktische Analyse. Munich: O. Halbreiter, 1923. ——— . Lehrbuch der musikalischen Analyse. Leipzig: C. F. Kahnt, 1926. Guck, Marion A. “Analytical Fictions.” Music Theory Spectrum 16.2 (1994): 217–230. ——— . “Rehabilitating the Incorrigible.” In Theory, Analysis and Meaning in Music, ed. Anthony Pople, 57–73. Cambridge: Cambridge University Press, 1994. Gurlitt, Willibald. “Hugo Riemann (1849–1919).” Veröffentlichungen der Akademie der Wissenschaften und der Literatur, Mainz: Abhandlungen der geistes- und sozialwis­ senschaftlichen Klasse 25 (1950): 1865–1901. Halm, August. Harmonielehre. Leipzig: Breitkopf und Härtel, 1900. Hanninen, Dora A. “A Theory of Recontextualization in Music: Analyzing Phenomenal Transformations of Repetition.” Music Theory Spectrum 25.1 (2003): 59–97. Harrison, Daniel. “Dissonant Tonics and Post-Tonal Tonality.” Paper read at the meeting of the Music Theory Society of New York State, 2002. ——— . Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Ac­ count of Its Precedents. Chicago: University of Chicago Press, 1994. ——— . “Nonconformist Notions of Nineteenth-Century Enharmonicism.” Music Analysis 21.2 (2002): 115–160. ——— . “Supplement to the Theory of Augmented Sixth Chords.” Music Theory Spectrum 17.2 (1995): 170–195. Hartmann, Günther. Karg-Elerts Harmonologik: Vorstufen und Stellungnahmen. Bonn: Or­ pheus, 1999. Hasel, Johann Emmerich. Die Grundsätze des Harmoniesystems. Vienna: Kratochwill, 1892. Page 7 of 20

Selected Bibliography Hauptmann, Moritz. Die Lehre von der Harmonik. Leipzig: Breitkopf und Härtel, 1868. ——— . Die Natur der Harmonik und der Metrik. Leipzig: Breitkopf und Härtel, 1853. Translated by W. E. Heathcote as The Nature of Harmony and Metre (Reprint, New York: Da Capo Press, 1991). Heidenreich, Achim. “‘Die Ungeheuerlichkeit dieser Art Hermeneutik’: Ein Disput zwis­ chen Hugo Riemann und Hermann Kretzschmar.” In Hugo Riemann (1849–1919): Musik­ wissenschaftler mit Universalanspruch, ed. Tatjana Böhme-Mehner and Klaus Mehner, 153–157. Cologne: Böhlau, 2001. Helmholtz, Hermann. Die Lehre von den Tonempfindungen. 4th ed. Braunschweig: Vieweg und Sohn, 1877. Translated by A. J. Ellis as On the Sensations of Tone as a Physio­ logical Basis for the Theory of Music (Reprint, New York: Dover, 1954). ——— . “The Facts in Perception.” [1878]. In Science and Culture: Popular and Philosophi­ cal Essays, ed. David Cahan, 342–80. Chicago: University of Chicago Press, 1995. ——— . “On the Interaction of the Natural Forces.” [1854]. In Science and Culture: Popu­ lar and Philosophical Essays, ed. David Cahan, 96–126. Chicago: University of Chicago Press, 1995. ——— . “On the Physiological Causes of Harmony in Music.” [1857]. In Science and Cul­ ture: Popular and Philosophical Essays, ed. David Cahan, 46–75. Chicago: University of Chicago Press, 1995. (p. 588)

Hindemith, Paul. Unterweisung im Tonsatz. Mainz: Schott, 1937.

Holtmeier, Ludwig. “Der Tristanakkord und die Neue Funktionstheorie.” Musiktheorie 17 (2002): 361–365. ——— . “From ‘Musiktheorie’ to ‘Tonsatz’: National Socialism and German Music Theory after 1945.” Music Analysis 2/3 (2004): 245–266. ——— . “Grundzüge der Riemann-Rezeption.” In Musiktheorie, ed. Helga de la MotteHaber and Oliver Schwab-Felisch, 230–262. Laaber: Laaber, 2005. ——— . “Heinichen, Rameau and the Italian Thoroughbass Tradition: Concepts of Tonality and Chord in the Rule of the Octave.” Journal of Music Theory 51.1 (2007): 5–49. ——— . “Ist die Funktionslehre am Ende?” Tijdschrift voor Muziektheorie 5 (1999): 72–77. Hook, Julian. “Cross-Type Transformations and the Path-Consistency Condition.” Music Theory Spectrum 29.1 (2007): 1–39. ——— . “Uniform Triadic Transformations.” Journal of Music Theory 46.1/2 (2002): 57– 126.

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Selected Bibliography Hostinský, Ottokar. Die Lehre von den musikalischen Klängen. Prague: H. Dominicus, 1879. Hull, A. Eaglefield. Modern Harmony: Its Explanation and Application. London: Augener, 1915. Hunt, Graham G. “When Chromaticism and Diatonicism Collide: A Fusion of Neo-Rie­ mannian and Tonal Analysis Applied to Wagner's Motives.” Journal of Schenkerian Studies 2 (1997): 1–32. Hyer, Brian. “Chopin and the in-F-able.” In Musical Transformation and Musical Intuition: Eleven Essays in Honor of David Lewin, ed. Raphael E. Atlas and Michael Cherlin, 147– 166. Roxbury, MA: Overbird Press, 1994. ——— . “Reimag(in)ing Riemann.” Journal of Music Theory 39.1 (1995): 101–138. ——— . “ ‘Sighing Branches’: Prosopopoeia in Rameau's Pigmalion.” Music Analysis 13.1 (1994): 7–20. ——— . “Tonal Intuitions in Tristan und Isolde.” Ph.D., diss., Yale University, 1989. ——— . “Tonality.” In The Cambridge History of Western Music Theory, ed. Thomas Chris­ tensen, 726–752. Cambridge: Cambridge University Press, 2002. Imig, Renate. Systeme der Funktionsbezeichnung in den Harmonielehren seit Hugo Rie­ mann. Düsseldorf: Verlag zur Förderung der systematischen Musikwissenschaft, 1970. Jonquière, Alfred. Grundriss der musikalischen Akustik. Leipzig: Th. Grieben, 1898. Judd, Cristle Collins. “The Dialogue of Past and Present: Approaches to Historical Music Theory.” Intégral 14/15 (2000–2001): 56–63. Kim, Youn. “Theories of Musical Hearing, 1863–1931: Helmholtz, Stumpf, Riemann and Kurth in Historical Context.” Ph.D. diss., Columbia University, 2003. Kinderman, William, and Harald Krebs, eds. The Second Practice of Nineteenth Century Tonality. Lincoln: University of Nebraska Press, 1996. Kirnberger, Johann Philipp. The Art of Strict Musical Composition. Translated by David Beach and Jürgen Thym. New Haven: Yale University Press, 1982. Kirsch, Ernst. Wesen und Aufbau der Lehre von den harmonischen Funktionen: Ein Beitrag zur Theorie der Relationen der musikalische Harmonie. Leipzig: Breitkopf und Härtel, 1928. Klumpenhouwer, Henry. “Dualist Tonal Space and Transformation in Nineteenth-Century Musical Thought.” In The Cambridge History of Western Music Theory, ed. Thomas Chris­ tensen, 456–476. Cambridge: Cambridge University Press, 2002.

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Selected Bibliography ——— . “In Order to Stay Asleep as Observers: The Nature and Origins of Anti-Cartesian­ ism in Lewin's Generalized Musical Intervals and Transformations.” Music Theory Spec­ trum 28.2 (2006): 277–289. ——— . “Remarks on American Neo-Riemannian Theory.” Tijdschrift voor Muziek­ theorie 5 (2000): 155–169. (p. 589)

——— . “Some Remarks on the Use of Riemann Transformations.” Music Theory Online 0.9 (1994). Koch, Heinrich. Musikalisches Lexikon. 1802. Facsimile reprint, Kassel: Bärenreiter, 2001. Kopp, David. Chromatic Transformations in Nineteenth-Century Music. Cambridge: Cam­ bridge University Press, 2002. ——— . “A Comprehensive Theory of Mediant Relations in Mid-Nineteenth-Century Mu­ sic.” Ph.D. diss., Brandeis University, 1995. Kraushaar, Otto. Der accordliche Gegensatz und die Begründung der Skala. Kassel: C. Luckhardt, 1852. Krebs, Harald. “Alternatives to Monotonality in Early Nineteenth-Century Music.” Journal of Music Theory 25 (1981): 1–16. ——— . “Third Relation and Dominant in Late 18th- and 19th-Century Music.” Ph.D. diss., Yale University, 1980. Krehl, Stephan. Allgemeine Musiklehre. Leipzig: Göschen, 1904. ——— . Harmonielehre [Tonalitätslehre]. Leipzig: Vereinigung wissenschaftlicher Ver­ leger, 1922. Krones, Hartmut. “Hugo Riemanns Überlegungen zu Phrasierung und Artikulation.” In Hugo Riemann (1849–1919): Musikwissenschaftler mit Universalanspruch, ed. Tatjana Böhme-Mehner and Klaus Mehner, 93–115. Cologne: Böhlau, 2001. Krumhansl, Carol. Cognitive Foundations of Musical Pitch. New York: Oxford University Press, 1990. ——— . “Perceived Triad Distance: Evidence Supporting the Psychological Reality of NeoRiemannian Transformations.” Journal of Music Theory 42.2 (1998): 265–281. Krumhansl, Carol, and Edward Kessler. “Tracing the Dynamic Changes in Perceived Tonal Organization in a Spatial Representation of Musical Keys.” Psychological Review 89 (1982): 334–368. Kurth, Ernst. Bruckner, 2 vols. Berlin: M. Hesse, 1925.

Page 10 of 20

Selected Bibliography ——— . Die Voraussetzungen der theoretischen Harmonik und der tonalen Darstel­ lungssysteme. Bern: Drechsel, 1913. ——— . Romantische Harmonik und ihre Krise in Wagners “Tristan.” Berlin: M. Hesse, 1920. Lambert, Philip. “Isographies and Some Klumpenhouwer Networks They Involve.” Music Theory Spectrum 24.2 (2002): 165–195. Lenoir, Timothy. Instituting Science: The Cultural Production of Scientific Disciplines. Stanford: Stanford University Press, 1997. ——— . The Strategy of Life: Teleology and Mechanics in Nineteenth-Century German Bi­ ology. Chicago: University of Chicago Press, 1982. Lerdahl, Fred. “Tonal Pitch Space.” Music Perception 5.3 (1988): 315–349. ——— . Tonal Pitch Space. New York: Oxford University Press, 2001. Lester, Joel. “Rameau and Eighteenth-Century Harmonic Theory.” In The Cambridge His­ tory of Western Music Theory, ed. Thomas Christensen, 753–777. Cambridge: Cambridge University Press, 2002. Lewin, David. “Amfortas's Prayer to Titurel and the Role of D in Parsifal: The Tonal Spaces of the Drama and the Enharmonic C♭/B.” 19th-Century Music 7.3 (1984): 336–349. Reprinted in Lewin, Studies in Music with Text, 183–200. New York: Oxford University Press, 2006. ——— . “Brahms, His Past, and Modes of Music Theory.” In Brahms Studies: Analytical and Historical Perspectives, ed. George S. Bozarth, 13–27. Oxford: Oxford University Press, 1990. ——— . “A Formal Theory of Generalized Tonal Functions.” Journal of Music Theo­ ry 26.1 (1982): 23–60. (p. 590)

——— . “Forte's Interval Vector, My Interval Function, and Regener's Common-Note Func­ tion.” Journal of Music Theory 21 (1977): 194–237. ——— . “Generalized Interval Systems for Babbitt's Lists, and for Schoenberg's String Trio.” Music Theory Spectrum 17.1 (1995): 81–118. ——— . Generalized Musical Intervals and Transformations. New Haven: Yale University Press, 1987. Reprint, New York: Oxford University Press, 2007. ——— . “Music Theory, Phenomenology, and Modes of Perception.” Music Perception 3 (Summer 1986): 327–392. Reprinted in Lewin, Studies in Music with Text, 53–108. New York: Oxford University Press, 2006.

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Selected Bibliography ——— . “Notes on the Opening of the Fs minor Fugue from WTC 1.” Journal of Music The­ ory 42.2 (1998): 235–239. ——— . “Some Notes on Analyzing Wagner: The Ring and Parsifal.” 19th-Century Music, 16.1 (1992): 49–58. ——— . “Some Problems and Resources of Music Theory.” Journal of Music Theory Peda­ gogy 5 (1991): 111–132. ——— . “Transformational Techniques in Atonal and Other Music Theories.” Perspectives of New Music 21.1–2 (Fall–Winter 1982/Spring–Summer 1983): 312–371. ——— . “Two Interesting Passages in Rameau's Traité de l’harmonie.” In Theory Only 4.3 (1978): 3–11. Lochhead, Judith. “The Metaphor of Musical Motion: Is There an Alternative?” Theory and Practice 14/15 (1989–1990): 83–103. London, Justin. “Some Non-Isomorphisms between Pitch and Time.” Journal of Music The­ ory 46.1/2 (2002): 127–51. Longuet-Higgins, Christopher. “Letter to a Musical Friend.” Music Review 23 (1962): 244–248. Lotze, Hermann. Geschichte der Ästhetik in Deutschland. Munich: J. G. Cotta, 1868. Louis, Rudolf. Anton Bruckner. Berlin: G. Müller, 1904. ——— . Der Widerspruch in der Musik: Bausteine zu einer Ästhetik der Tonkunst auf real­ dialektischer Grundlage. Leipzig: Breitkopf und Härtel, 1893. ——— . “Unsere Harmonielehre.” Süddeutsche Monatshefte 3.2 (1906): 430–437. ——— . “Zu Hugo Riemanns Besprechung der Louis-Thuilleschen Harmonielehre.” Süd­ deutsche Monatshefte 4.1 (1907): 614–620. Louis, Rudolf, and Ludwig Thuille. Harmonielehre. 4th ed. Stuttgart: C. Grüninger, 1913. Maler, Wilhelm. Beitrag zur durmolltonalen Harmonielehre. Leipzig: F. E. C. Leukert, 1931. Marvin, Elizabeth West. “Tonpsychologie and Musikpsychologie: Historical Perspectives on the Study of Music Perception.” Theoria 2 (1987): 59–84. Maus, Fred Everett. “Music as Drama.” Music Theory Spectrum 10 (1988): 56–73. Mayrberger, Carl. “Die Harmonik Richard Wagner's an den Leitmotiven des Vorspiels zu Tristan und Isolde erläutert.” Bayreuther Blätter 4 (1881): 169–180.

Page 12 of 20

Selected Bibliography ——— . Lehrbuch der musikalischen Harmonik. Pressburg/Leipzig: Gustav Heckenast, 1878. McCreless, Patrick. “An Evolutionary Perspective on Nineteenth-Century Semitonal Rela­ tions.” In The Second Practice of Nineteenth-Century Tonality, ed. William Kinderman and Harald Krebs, 87–113. Lincoln: University of Nebraska Press, 1996. Mehner, Klaus. “Hugo Riemanns ‘Ideen zu einer Lehrer von den Tonvorstellungen.’” In Hugo Riemann (1849–1919): Musikwissenschaftler mit Universalansprach, ed. Tatjana Böhme-Mehner and Klaus Mehner, 49–57. Cologne: Böhlau Verlag, 2001. Mickelsen, W. C. Hugo Riemann's Theory of Harmony with a Translation of Riemann's “History of Music Theory,” Book 3. Lincoln: University of Nebraska Press, 1977. (p. 591)

Miller, Horace Alden. New Harmonic Devices: A Treatise on Modern Harmonic Problems. Philadelphia: Oliver Ditson, 1930. Mooney, Michael Kevin. “Musical Logic: A Contribution to the Theory of Music.” Journal of Music Theory 44.1 (2000): 100–126. ——— . “Riemann's Debut as a Music Theorist.” Journal of Music Theory 44.1 (2000): 81– 99. ——— . “The ‘Table of Relations’ and Music Psychology in Hugo Riemann's Harmonic The­ ory.” Ph.D. diss., Columbia University, 1996. Morris, Robert D. “Set Groups, Complementation, and Mappings among Pitch-Class Sets.” Journal of Music Theory 26 (1982): 101–144. Nolan, Catherine. “Combinatorial Space in Nineteenth- and Early Twentieth-Century Mu­ sic Theory.” Music Theory Spectrum, 25.2 (2003): 205–241. ——— . “Music Theory and Mathematics.” In The Cambridge History of Western Music Theory, ed. Thomas Christensen, 272–304. Cambridge: Cambridge University Press, 2002. Nowak, Adolf. “Wandlungen des Begriffs ‘musikalische Logik’ bei Hugo Riemann.” In Hugo Riemann (1849–1919): Musikwissenschaftler mit Universalansprach, ed. Tatjana Böhme-Mehner and Klaus Mehner, 38–43. Cologne: Böhlau Verlag, 2001. O’Donnell, Shaugn. “Klumpenhouwer Networks, Isography, and the Molecular Metaphor.” Intégral 12 (1998): 53–80. Oettingen, Arthur von. Harmoniesystem in dualer Entwickelung. Dorpat: Glaser, 1866. Pearce, Trevor. “Tonal Functions and Active Synthesis: Hugo Riemann, German Psycholo­ gy, and Kantian Epistemology.” Intégral 22 (2008): 81–116. Page 13 of 20

Selected Bibliography Persichetti, Vincent. Twentieth-Century Harmony: Creative Aspects and Practice. New York: W. W. Norton, 1961. Ploeger, Roland. “Zum Problem Monismus-Dualismus.” In Studien zur systematischen Musiktheorie, 2nd ed. Eutin: Petersen-Mickelsen, 2002. Polak, A. J. Über Tonrhythmik und Stimmenführung. Leipzig: Breitkopf und Härtel, 1902. ——— . Über Zeiteinheit in Bezug auf Konsonanz, Harmonie, und Tonalität. Leipzig: Bre­ itkopf und Härtel, 1900. Proctor, Gregory. “Review of The Second Practice of Nineteenth-Century Tonality.” Music Theory Spectrum 21.1 (1999): 131–39. ——— . “Technical Bases of Nineteenth-Century Chromatic Tonality: A Study in Chromati­ cism.” Ph.D. diss., Princeton University, 1978. Rameau, Jean Philippe. Génération harmonique. Paris: Chez Prault fils, 1737. ——— . Traité de l’harmonie. Paris, 1722. Translated by Philip Gossett as Treatise on Har­ mony (New York: Dover, 1971). Rehding, Alexander. Hugo Riemann and the Birth of Modern Musical Thought. Cam­ bridge: Cambridge University Press, 2003. ——— . “Review of Hugo Riemann (1849–1919): Leben, Werk und Wirkung.” Music Theory Spectrum 24.2 (2002): 283–293. ——— . “Wax Cylinder Revolutions.” Musical Quarterly 88.1 (2005): 123–160. Richter, Ernst Friedrich. Lehrbuch der Harmonie. Leipzig: Breitkopf und Härtel, 1853. Riemann, Hugo (pseudonym Hugibert Ries). “Musikalische Logik” [1872]. In Präludien und Studien (Reprint, Hildesheim: Georg Olms, 1967), 3: 1–22. Riemann, Hugo. Beethovens Streichquartette erläutert von Hugo Riemann. Berlin: Sch­ lesinger, 1910. ——— . “Das Problem des harmonischen Dualismus.” Neue Zeitschrift für Musik 72 (1905): 3–5, 23–26, 43–46, 67–70. (p. 592)

——— . “Degeneration und Regeneration in der Musik.” Max Hesses deutscher Musik­ erkalender 23 (1908): 136–138. Reprinted in ‘Die Konfusion in der Musik’: Felix Drae­ sekes Kampfschrift von 1906 und die Folgen, ed. Susanne Shigihara, 245–249. Bonn: Gu­ drun Schröder, 1990. ——— . Die Hülfsmittel der Modulationslehre. Cassel: Luckhardt, 1875. ——— . “Die Musik seit Wagners Heimgang: Ein Totentanz.” [1897]. In Präludien und Stu­ dien (Reprint, Hildesheim: Georg Olms, 1967), 2: 33–41. Page 14 of 20

Selected Bibliography ——— . “Die Natur der Harmonik.” Sammlung musikalische Vorträge 4.40. Leipzig: Bre­ itkopf und Härtel (1882): 157–190. ——— . “Die Neugestaltung der Harmonielehre.” Musikalisches Wochenblatt 22 (1891): 513–514, 529–531, 541–543. ——— . “Die objective Existenz der Untertöne in der Schallwelle.” Allgemeine Musikzeitung 2 (1875): 205–206, 213–215. ——— . “Die Phrasierung im Lichte einer Lehre von den Tonvorstellungen.” Zeitschrift für Musikwissenschaft 1 (1918): 26–38. ——— . “Die Taktfreiheiten in Brahms’ Liedern.” Die Musik 12.1 (October 1912): 10–21. ——— . “Eine musikalische Tagesfrage.” Musikalisches Wochenblatt 13 (1882): 465–466, 477–479, 489–490, 501–502, 513–515, 529–531, 553–555, 569–570, 593–594, 617–619. Reprinted as “Das chromatische Tonsystem.” In Präludien und Studien (Reprint, Hildesheim: Georg Olms, 1967), 1: 183–219. ——— . “Eine neue Harmonielehre: Harmonielehre von Rudolf Louis u. Ludwig Thuille.” Süddeutsche Monatshefte 4.1 (1907): 500–504. ——— . Elementar-Schulbuch der Harmonielehre. Berlin: Max Hesse, 1906. ——— . Elemente der musikalischen Ästhetik. Berlin and Stuttgart: W. Spemann, 1900. ——— . “Exotische Musik.” Max Hesses Musikerkalender 21 (1906): 135–137. ——— . Folkloristische Tonalitätsstudien. Leipzig: Breitkopf und Härtel, 1916. ——— . Geschichte der Musik seit Beethoven (1800–1900). Berlin: W. Spemann, 1901. ——— . Geschichte der Musiktheorie im IX.–XIX. Jahrhundert. Leipzig: Max Hesse, 1898. ——— . Geschichte der Notenschrift. Leipzig: Breitkopf & Härtel, 1878. ——— . Große Kompositionslehre, 3 vols. Berlin and Stuttgart: W. Spemann, 1902–03. ——— . Grundriß der Kompositionslehre, 3rd rev. ed. Leipzig: Max Hesse, 1905. ——— . Grundriß der Musikwissenschaft. 4th ed. Leipzig: Quelle und Meyer, 1908. ——— . Handbuch der Harmonielehre (Zweite, vermehrte Auflage der “Skizze einer neuen Methode der Harmonielehre). Leipzig: Breitkopf und Härtel, 1887. Subsequent editions in 1898, 1906, 1912, 1917. ——— . Handbuch der Musikgeschichte. Leipzig: Breitkopf und Härtel, 1904. ——— . “Hie Wagner! Hie Schumann!” [1880]. In Präludien und Studien (Reprint, Hildesheim: Georg Olms, 1967), 3: 204–214. Page 15 of 20

Selected Bibliography ——— . “Ideen zu einer ‘Lehre von den Tonvorstellungen.’” Jahrbuch der Musikbibliothek Peters 21–22 (1914–1915): 1–26. ——— . Katechismus der Akustik. Berlin: Max Hesse, 1891. ——— . Katechismus der Kompositionslehre. Berlin: Max Hesse, 1889. ——— . Katechismus der Musikwissenschaft. Berlin: Max Hesse, 1891. ——— . L. van Beethovens sämtliche Klavier-Solosonaten: Ästhetik und formal-technische Analyse mit historischen Notizen, 2nd ed. Berlin: Hesse, 1919. ——— . Musikalische Dynamik und Agogik. Hamburg: D. Rahter, 1884. ——— . Musikalische Logik: Hauptzüge der physiologischen und psychologischen Begrün­ dung unseres Musiksystems. Leipzig: Kahnt, 1874. ——— . Musikalische Syntaxis: Grundriß einer harmonischen Satzbildungslehre. Leipzig: Breitkopf und Härtel, 1877. (p. 593)

——— . Musik-Lexikon. Berlin: Max Hesse, 1882. Subsequent editions in 1884, 1887, 1894, 1900, 1905, 1909, 1916. ——— . “Neue Beiträge zu einer Lehre von den Tonvorstellungen.” Jahrbuch der Musik­ bibliothek Peters (1916): 1–21. ——— . Neue Schule der Melodik. Hamburg: K. Grädener, 1883. ——— . Sechs original Japanische und Chinesische Melodien. Leipzig: Breitkopf und Här­ tel, 1903. ——— . Skizze einer neuen Methode der Harmonielehre. Leipzig: Breitkopf und Härtel, 1880. ——— . Systematische Modulationslehre als Grundlage der musikalischen Formenlehre. Hamburg: Richter, 1887. ——— . System der musikalischen Rhythmik und Metrik. Leipzig: Breitkopf und Härtel, 1903. ——— . “Über Japanische Musik.” Musikalisches Wochenblatt 33 (1902): 209–210, 229– 231, 245–246, 257–259, 273–274, 289–290. ——— . “Ueber das musikalische Hören.” Dr. phil. diss., Göttingen University, 1873. Pub­ lished Leipzig: Andrä, 1874. ——— . Vademecum der Phrasierung, 2nd rev. ed. Berlin: Max Hesse, 1906.

Page 16 of 20

Selected Bibliography ——— . Vereinfachte Harmonielehre; oder, Die Lehre von den tonalen Funktionen der Akkorde. London: Augener, 1893. Translated by Henry Bewerunge as Harmony Simpli­ fied, or The Theory of Tonal Functions of Chords (London: Augener, 1895). ——— . “Was ist Dissonanz?” Max Hesses deutscher Musiker-Kalender 13 (1898): 145– 151. Reprinted as “Zur Theorie der Konsonanz und Dissonanz.” In Präludien und Studien (Reprint, Hildesheim: Georg Olms, 1967), 3: 31–46. Rimsky-Korsakov, N. Practical Manual of Harmony. Translated by J. Achron. New York: Carl Fischer, 1930. Rings, Steven. “Perspectives on Tonality and Transformation in Schubert's Impromptu in E♭, D. 899, no. 2.” Journal of Schenkerian Studies 2 (2007): 33–63. ——— . “Tonality and Transformation.” Ph.D. diss., Yale University, 2006. Rothfarb, Lee Allen. Ernst Kurth as Theorist and Analyst. Philadelphia: University of Pennsylvania Press, 1988. Rummenhöller, Peter. “Die fluktuierende Theoriebegriff Riemanns.” In Hugo Riemann (1849–1919): Musikwissenschaftler mit Universalansprach, ed. Tatjana Böhme-Mehner and Klaus Mehner, 31–36. Cologne: Böhlau Verlag, 2001. ——— . “Moritz von Hauptmann, der Begründer einer transzendental-dialektischen Musiktheorie.” In Beiträge zur Musiktheorie im neunzehnten Jahrhundert, ed. Martin Vo­ gel, 11–36. Regensburg: Gustav Bosse, 1966. Santa, Matthew. “Nonatonic Systems and the Parsimonious Interpretation of DominantTonic Progressions.” Theory and Practice 28 (2003): 1–28. Saslaw, Janna. “The Role of Body-Derived Image Schemas in the Conceptualization of Mu­ sic.” Journal of Music Theory 40 (1996): 217–243. Saslaw, Janna K., and James P. Walsh. “Musical Invariance as a Cognitive Structure: ‘Mul­ tiple Meaning’ in the Early Nineteenth Century.” In Music Theory in the Age of Romanti­ cism, ed. Ian Bent, 211–232. Cambridge: Cambridge University Press, 1996. Schenker, Heinrich. Harmony. Edited by Oswald Jonas. Translated by Elisabeth Mann Borgese. Chicago: University of Chicago Press, 1954. Originally published as Har­ monielehre (Stuttgart: Cotta, 1906). Schillings, Max. “Besprechung der Harmonielehre von Rudolf Louis und Ludwig Thuille.” Die Musik 23 (1906–1907): 365–369. Schinköth, Thomas, ed. Sigfrid Karg-Elert und seine Leipziger Schüler. Hamburg: von Bockel, 1999.

Page 17 of 20

Selected Bibliography Schmidt, Matthias. “Syntax und System: Brahms’ Taktbehandlung in der Kritik Hugo Riemanns.” Studien zur Musikwissenschaft: Beihefte der Denkmäler der Tonkunst in Österreich 48 (2002): 413–438. (p. 594)

Schmitz, Eugen. Harmonielehre als Theorie, Ästhetik und Geschichte der musikalischen Harmonik. Kempten/Munich: J. Kösel, 1911. Schoenberg, Arnold. Harmonielehre. 3rd ed. Vienna: Universal Edition, 1922. ——— . Structural Functions of Harmony Rev. ed. New York: W. W. Norton, 1969. Schreyer, Johannes. Harmonielehre: Völlig umgearbeitete Ausgabe der Schrift ‘Von Bach bis Wagner.’ Dresden: Holze und Pahl, 1905. ——— . Von Bach bis Wagner: Ein Beitrag zur Psychologie des Musikhörens. Dresden: Holze und Pahl, 1903. Scruton, Roger. The Aesthetics of Music. Oxford: Oxford University Press, 1997. Sechter, Simon. Die Grundsätze der musikalischen Komposition: Erste Abtheilung: Die richtige Folge der Grundharmonien oder vom Fundamentalbass und dessen Umkehrun­ gen und Stellvertretern. Leipzig: Breitkopf und Härtel, 1853. Seebeck, August. “Ueber Klirrtöne.” Annalen der Physik und Chemie, n.s., 10 (1837): 539–547. Seidel, Elmar. “Die Harmonielehre Hugo Riemanns.” In Beiträge zur Musiktheorie im 19. Jahrhundert, ed. Martin Vogel, 39–92. Regensburg: Gustav Bosse, 1966. Seidel, Wilhelm. “Riemann und Beethoven.” In Hugo Riemann (1849–1919): Musikwis­ senschaftler mit Universalanspruch, ed. Tatjana Böhme-Mehner and Klaus Mehner, 139– 151. Cologne: Böhlau Verlag, 2001. ——— . Über Rhythmustheorien der Neuzeit. Bern: Francke, 1975. Shirlaw, Matthew. The Theory of Harmony: An Inquiry into the Natural Principles of Har­ mony, with an Explanation of the Chief Systems of Harmony from Rameau to the Present Day. Reprint, New York: Da Capo Press, 1969. Siciliano, Michael. “Neo-Riemannian Transformations and the Harmony of Franz Schu­ bert.” Ph.D. diss., University of Chicago, 2002. ——— . “Two Neo-Riemannian Analyses.” College Music Symposium 45 (2005): 81–107. Smith, Charles J. “The Functional Extravagance of Chromatic Chords.” Music Theory Spectrum 8 (1986): 94–139. Snarrenberg, Robert. Schenker's Interpretive Practice. Cambridge Studies in Music Theo­ ry and Analysis 11. Cambridge: Cambridge University Press, 1997. Page 18 of 20

Selected Bibliography Spitzer, Michael. “The Metaphor of Musical Space.” Musicae Scientiae 7 (2003–2004): 101–118. Steege, Benjamin. “Material Ears: Hermann von Helmholtz, Attention, and Modern Aural­ ity.” Ph.D. diss., Harvard University, 2007. Straus, Joseph N. “Voice Leading in Atonal Music.” In Music Theory in Concept and Prac­ tice, ed. David Beach, James M. Baker, and Jonathan W. Bernard, 237–274. Rochester: University of Rochester Press, 1997. Stumpf, Carl. Tonpsychologie, 2 vols. Leipzig: S. Hirzel, 1883, 1890. Thaler, Lotte. Organische Form in der Musiktheorie des 19. und beginnenden 20. Jahrhunderts. Munich and Salzburg: Emil Katzbichler, 1984. Tiersch, Otto. Elementarbuch der musikalischen Harmonie- und Modulationslehre. 2nd ed. Berlin: Robert Oppenheimer, 1888. Turner, R. Steven. “The Growth of Professorial Research in Prussia, 1818 to 1848: Causes and Context.” Historical Studies in the Physical Sciences 3 (1971): 137–182. ——— . “The Prussian Universities and the Concept of Research.” Internationales Archiv für Sozialgeschichte der Deutschen Literatur 5 (1980): 68–93. ——— . “University Reformers and Professorial Scholarship in Germany, 1760– 1806.” In The University in Society, ed. Lawrence Stone, 2: 495–531. Princeton: Princeton University Press, 1974. (p. 595)

Tymoczko, Dmitri. “The Geometry of Musical Chords.” Science 313 (2006): 72–74. ——— . “Key Signatures as Voice-Leadings.” Music Theory Online 11.4 (2005). ——— . “Progressions fondamentales, fonctions, degrés, une grammaire de l’harmonie tonale élémentaire.” Musurgia 10.3–4 (2003): 35–64. ——— . “Scale Theory, Serial Theory, and Voice Leading.” Music Analysis 27.1 (2008): 1– 49. Ulehla, Ludmila. Contemporary Harmony: Romanticism through the Twelve-Tone Row. New York: Free Press, 1966. van Eycken, Heinrich. Harmonielehre. Leipzig: Hofmeister, 1911. Vogel, Martin. On the Relations of Tone. Translated by V. Kisselbach. Bonn: Verlag für sys­ tematische Musikwissenschaft, 1993. Waldbauer, Ivan F. “Riemann's Periodization Revisited and Revised.” Journal of Music Theory 33.2 (1989): 333–392.

Page 19 of 20

Selected Bibliography Wason, Robert. Viennese Harmonic Theory from Albrechtsberger to Schenker and Schoenberg. Ann Arbor, MI: UMI Press, 1985. Wason, Robert, and Elizabeth West Marvin. “Riemann's ‘Ideen zu einer “Lehre von den Tonvorstellungen” ’: An Annotated Translation.” Journal of Music Theory 36.1 (1992): 69– 117. Weber, Gottfried. Theory of Musical Composition. Translated by James F. Warner. Boston: Wilkins and Carter, 1846. Weitzmann, Carl Friedrich. Der übermäßige Dreiklang. Berlin: T. Trautwein, 1853. Trans­ lated by Janna K. Saslaw in “Two Monographs by Carl Friedrich Weitzmann. I: The Aug­ mented Triad (1853),” Theory and Practice 29 (2004): 133–228. Wetzel, Hermann. Elementartheorie der Musik: Einführung in die Theorie der Melodik, Harmonik, Rhythmik und der musikalischen Formen- und Vortragslehre. Leipzig: Bre­ itkopf und Härtel, 1911. Zbikowski, Lawrence. “The Blossoms of ‘Trockne Blumen’: Music and Text in the Early Nineteenth Century.” Music Analysis 18 (1999): 307–345. ——— . Conceptualizing Music: Cognitive Structure, Theory, and Analysis. New York: Ox­ ford University Press, 2002. ——— . “Large-Scale Rhythm and Systems of Grouping.” Ph.D. diss., Yale University, 1991. Ziehn, Bernhard. “Der Weise aus Großmehlra.” Allgemeine Musik-Zeitung 17 (1890): 355– 561. Zuckerkandl, Victor. Sound and Symbol: Music and the External World. Translated by W. R. Trask. Bollingen Series, vol. XLIV. Princeton: Princeton University Press, 1956.

Page 20 of 20

Index

Index   The Oxford Handbook of Neo-Riemannian Music Theories Edited by Edward Gollin and Alexander Rehding Print Publication Date: Dec 2011 Subject: Music Online Publication Date: Sep 2012

(p. 596)

(p. 597)

Index

A absolute progression, 486, 570–573 abstract algebra, 248, 288, 356–362, 365, 515, 527–531, 550–551 acoustics, musical, 57–59, 65, 71, 167–182, 274, 323 Albert, Heinrich, 472, 474–475 analysis empirical and theoretical approaches, 5, 65, 445, 497 incompleteness in, 531–532 melodic, 141–159 neo-Riemannian, 365, 400–401, 487–506, 512–542, 564–573 phrase and metrical structure in, 441–459, 464–472 post-tonal, 389–393, 497, 532 post-post-tonal, 542 reductive, 15–32, 402, 450, 457 role in music pedagogy, 13–33, 40 ancient Greek music theory, 142–150, 153–156, 183 antitonic, 556–564 Apfel, Ernst, 436n6 apparent consonance, 232, 580–581 in Louis, 11, 35–36 in Riemann, 83, 102, 104 in Schreyer, 19 Arabic-Persian music theory, 66–67, 89n25, 90n26, 183 Arntz, Michael, 3–4, 112 Auftaktigkeit, 417,579. See also phrase and metrical structure augmented triad, 180, 383–385, 507n9 B Babbitt, Milton, 516, 544n10 Bach, Johann Sebastian Fugue in A minor from Well-Tempered Clavier, 18–19, 20–21 Two-Part Fugue in C minor, 15 Baldini, Guglielmo, 268 Balzano, Gerald, 323 Page 1 of 17

Index Bass, Richard, 319n16, 410 Beethoven, Ludwig van, 219, 417–418, 440–459 Bagatelle op. 126/6, 469 Concerto for Piano in E-flat major “Emperor”, 402–404, 412 Concerto for Piano in G major, 327 Coriolan Overture, 225 Fidelio Overture, 425 Leonore Overture, 427 Quartet in B-flat major op. 130, 225, 498 Quartet in B-flat major op. 18/6, 220, 273–274 Sonata No. 1 in F major op. 2/1, 423 Sonata No. 5 in C minor op. 10/1, 425–426 Sonata No. 8 in C minor “Pathétique” op. 13, 273 Sonata No. 9 in E major op. 14/1, 423 Sonata No. 11 in B-flat major op. 22, 423 Sonata No. 12 in A-flat major op. 26, 271–274 Sonata No. 13 in E-flat major, op. 27/1, 429–430 Sonata No. 16 in G major op. 31/1, 434–435, 442–445, 447–448, 451–453 Sonata No. 17 in D minor “Tempest” op. 31/2, 446–447, 453–458 Sonata No. 18 in E-flat major op. 31/3, 423, 443–444, 446, 448–451, 453–455 Sonata No. 19 in G minor op. 49/1, 423 Sonata No. 21 in E major “Waldstein” op. 53,  82, 498 Sonata No. 28 in A major op. 101, 110–111 Symphony No. 1 in C major, 205–213, 232 Symphony No. 5 in C minor, 27, 28–29, 30–31, 221 Belinfante, Ary, 7, 166, 169, 200–205, 216n16 Bent, Ian, 166 Berry, Paul, 418 Blasius, Leslie, 516 Brahms, Johannes, 463–464, 472, 476–478 “Auf dem Kirchhofe”, 476 Clarinet Trio, 222–239 Das Mädchen spricht (“Schwalbe, sag’ mir an”), 464, 469–471, 475–476, 480–481 Ein deutsches Requiem, 557–558 Four Serious Songs op. 121 “Denn es gehet”, 236–238 “Immer leiser wird mein Schlummer”,  464–469, 475–480 Intermezzo op. 76/4, 259–262 Intermezzo op. 199/1, 225 “Wie bist du meine Königin”, 478 Bribitzer-Stull, Matthew, 547 Bürger, Gottfried August, 512–513, 532–533 Burney, Charles, 143–145 Burnham, Scott, 219, 418 Bußler, Ludwig, 197–198 (p. 598) C Caccini, Giulio, 462–463, 474–475 Callender, Clifton, 551 Capellen, Georg, 2, 6, 166, 171 Page 2 of 17

Index critique of dualism, 167–170, 173, 175 Caplin, William, 219, 418 Childs, Adrian, 551 Chinese music, 148–158 theory of, 148–150 “Tsi Tschong”, 150–151, 156–157 Chopin, Frederic Ballade op. 38, 387–388 Nocturne op. 9/1, 253 Prelude op. 28 in E minor, 260 Prelude op. 28 in E major, 337–344 chromaticism, 36–39, 513–517, 522–523, 542 Brahmsian, 468, 476–477 cross-relations, 477 extended/“second practice” tonality, 252–262, 263n3 key-displacement and, 373, 410 Louis and, 36–39 modal, 565–573 modulation and, 10–11, 344, 373 Riemann and, 361, 400–402, 509n23 Schenkerian theory and, 317, 517 Schubertian, 299–317, 488, 492 sequences and, 333–337, 402–404, 407–412, 468, 476–477, 513, 518–519 third progressions and, 85, 258, 401–414, 468, 476–477, 523–524, 534–536 Vaughan-Williamsian, 486, 564–573 Wagnerian, 256–258, 336, 415n9 Clark, Suzannah, 204, 270 Chua, Daniel, 498 coherence in neo-Riemannian analysis, 272, 400, 485, 497–500, 542 in Nineteenth century music, 261–262 in Riemann, 352, 359, 362–366, 441, 450, 468, 485 incoherence and disunity, 488, 498–500, 506, 513–514, 517, 543n3 tonal, 356–357, 400–402, 407 Tonnetz and, 324–325, 344–345, 348n48 Cohn, Richard, 241, 256, 266n30, 270, 300, 321n33, 351, 398n18, 492, 498, 546n31, 551, 558– 560 combination tone, 75–76, 78, 172, 186 combinatoriality, 280–289, 356–362, 366 comma, syntonic, 81–82, 274, 280, 289n8, 324, 327 common tone relationships, 83, 300, 356, 406 and Tonvorstellung/- vertretung, 296–299, 581 in Riemann, 102–104, 296–297 Cone, Edward T., 439n35 consonance and dissonance characteristic dissonance, 95, 97 in harmonic dualism, 83–85, 171–186, 383 in pre-Rameauvian music theory, 66–67 Page 3 of 17

Index in Rameau, 327 in Riemann, 92, 104–105, 383–386, 393–397 triad and, 323–324, 517, 556 Cook, Robert, 486 D Dahlhaus, Carl, 114, 204, 442, 420–421, 427, 434 critique of Riemannian function, 94, 101, 122–123, 364 d’Alembert, Jean-Baptiste, 71–73 Darwin, Charles, 146 degeneration, 363–364, 543n3 Dempster, Douglas, 241 Denny, Thomas, 301–304 diatonicism relation with chromaticism, 253, 257–262, 361, 572 transposition in, 249–252, 264n13 Diergarten, Felix, 14, 30 diminished triad, 173–174 dissonant sonorities, 382–397 as simultaneous chords, 350, 383–386, 394–397 function and, 385–386 taxonomy of, 388, 396–397 transformations among, 386–393 dominant in functional-logical form, 118–119, 130–131 in Hauptmann, 229–230 in neo-Riemannian theory, 300, 319n14, 554–564, 566 in Rameau, 72, 83 in Riemann, 84, 94, 98–102, 128, 430 secondary, 9–11, 38, 406, 509n31 Doppelklänge. See dissonant sonorities Drobisch, Moriz Wilhelm, 61, 269, 275, 284–286 dualism. See harmonic dualism dyads, 180, 197–198, 333 E Ehrenfels, Christian von, 169 Ellis, Alexander, 146 Engebretsen, Nora, 327, 350 enharmonicism, 488–489, 543n3 in Riemann, 180, 271–274, 280–282, 287, 468 seams and snaps, 488, 505–506 Tonnetz and, 324 epistemology, 124–125, 135n50, 377n15, 516–517, 531 Ergo, Emil, on contrapuntal harmony, 30–32 on dualism, 4–5 on modulation, 9–11 Erpf, Hermann, 39–40, 398n14 Euler, Leonhard, 269, 322–323 Page 4 of 17

Index F Federhofer, Hellmut, 240 Fétis, François-Joseph, 94–95, 195, 327, 543n3 (p. 599) Fichte, Johann Gottlieb, 100 figured bass, 68–70, 87, 230 Fisk, Charles, 500, 506n1 Fogliani, Ludovico, 67 Forkel, Johann Nikolaus, 144 form, musical, 218–223, 463–464, 493, 506. See also harmonic dualist analysis—musical form and Fortlage, Carl, 144, 152 Foucault, Michel, 68 Franck, César Le Chasseur Maudit, 512–514, 518–542 Frege, Gottlob, 2, 112–131 concepts in, 120, 127–128 mathematical functions in, 114–119 on sense and meaning, 121–124 on thought and idea, 124–129 function theory. See harmonic function fundamental bass theory, 6, 38, 71–73, 95–96, 297 fugue, 16, 18–19, 20–21, 111, 221 G Gelbart, Matthew, 2 Gingerich, John, 299 Glarean, Heinrich, 68 Goethe, Johann Wolfgang von, 174, 473 Gollin, Edward, 270, 350 Grabner, Herrmann, 14, 40–41 große Cadenz, 99–101, 107, 496 group theory. See abstract algebra Grove, George, 304–311 H Halm, August, 13 Hanninen, Dora, 544n10 harmony alteration of, 72–73, 84–86, 383, 385, 394–396, 555, 562, 570–571 Riemann tone relations as forms of, 102–103, 366–367 ambiguity and extravagance, 325, 328, 494, 499, 565, 572–573 chord and key, 324–329 counterpoint and, 27–32, 36–39, 68, 519–521, 565 cycles, 321n31, 322–323, 357, 365, 401, 553–554 PL3, 321n31, 401, 403–405, 410–412, 558–564 PLR2, 300, 315 PR4, 321n31, 468, 534–536 RN3, 490–491 form and, 219–223 hierarchical levels of, 15–21, 517 Page 5 of 17

Index intelligibility of, 80–81, 99, 101, 112, 127, 356–376, 515, 522, 532, 555. See also coherence parenthetical, 20–32, 37 phrase and meter and, 420–426 qualia of, 247–248, 491, 499, 503–505, 553, 564 Schwebende, 325, 363–364 sensuous aspect of, 6, 360, 486, 552–553, 558–560, 570–572 sin and, 363, 495, 512–514 third stacking, 74, 249, 264n12, 551–552, 561 transformational accounts of, 102, 355–356 harmonic dualism, 5–6, 147, 165–166, 196–199, 579 analysis and, 196, 205–241 before Riemann, 171–178 critiques of, 4, 6, 199–205 present day, 195–197, 257–259 Riemann's contemporaries, 33–34, 167–169 defense of, 7–8, 168–191, 194–214, 265n22 development of, 73–83 dissonance and, 383, 385–386 function and, 96–97, 107, 190–191, 199–205, 223–239, 250–252 fundamental bass and, 186–190 higher tertian chords in, 184–186, 383, 394–395 major-minor polarity, 7, 12, 69, 75, 78–80 mathematical means and, 198–199 melody and, 142, 152–153, 180 minor scale/triad and, 33–34, 83, 171–191, 580 mode of listening, 165–170, 195, 225, 238–241 monism and, 6, 12, 171, 188, 192n12, 199–214, 224, 236 musical form and, 218–241 pedagogy of, 190–191, 195 repertoire for, 239–240 residues of, 11–13 “soft” and “hard” forms, 196–199 symmetry in, 246–247, 251–262 voice-leading and, 253–262 harmonic function, 1–2, 92–112, 118–120, 123, 127, 579 as appearance/chord/object, 100–102, 118–119, 550, 552–553 as concept/idea/representation, 84–129 as disposition/behavior, 108–109 as intention, 500–502, 552–553 as mathematical function, 118 chord succession and, 107–109, 250–256, 364, 579 critiques of, 122–123, 265n23, 364 dissonance and, 97, 385–386 dualism and, 199–205, 223–239, 250–252 limits of, 236, 364, 566–567 meanings of, 92–93, 98–102, 111–112, 122 notation of, 7, 109–112, 130–131, 511n50. See also Klangschlüssel phrase rhythm and, 424–426, 430–434, 442, 458 Page 6 of 17

Index propagation of theory, 5, 12, 500 relation to scale degree, 122–123 taxonomies of, 9, 11–15, 109–112 (p. 600) tonal space and, 323–329, 333, 336–337, 343–344 tonality and, 109, 361, 365–366, 430, 443–444 harmonic ratios arithmetic and harmonic division, 67, 74–75 consonance and dissonance, 172–173, 181–186 differentiation of adjacent pitches and, 81 in post-Rameauvian music theory, 74–77, 80–81, 269 in pre-Rameauvian music theory, 66–68, 70 prime factorization of, 80, 183–184, 278–279 Tonnetz and, 275–288 harmonic progression, 107–109, 247–250, 352 by fifth, 81, 360, 401 in Riemann, 367–368 by second, 371–375, 415n13, 570–571 by third, 73, 82, 256–258, 301–304, 356, 476–477, 515, 518, 523–524 in Riemann, 360, 368–370, 380n52, 468, 494–495 by tritone, 361, 375–377, 404 in dualist systems, 81, 85, 225–226, 232, 352 in non-Classical music, 249 Rameau and, 73, 95–96 voice-leading efficiency and, 103–104, 253–262, 477 Harmonieschritte, 251, 351–376, 579–580 background of, 349–350, 358–359 dissonant configurations and, 382–386, 394–397 function and, 107–109, 202–204, 210–214, 352, 360 taxonomies of, 353–356, 366–376 See also tone relations Harrison, Daniel, 101, 264n17, 343–344, 364, 486, 507n2 Hauptklänge, 92, 94–96, 104–106, 200–204, 343, 380n50, 491–492 Hauptmann, Moritz, 66, 80–82, 86, 175, 182, 207, 274, 279, 363, 423, 543n3 musical dialectics of, 203–204, 225–230, 233, 325, 336 tonal space in 81, 228–229 Helmholtz, Herrmann von, 33, 57–64, 80, 86, 144, 171 on chords, 82 on consonance and dissonance, 62–63, 78–79, 174–177, 277, 279 on tone representation, 60–61 on tone sensation, 78 Henried, Richard, 8 Herbart, Johann Friedrich, 268 Herder, Johann Gottfried, 146 hermeneutic edaphology, 515, 544n9 Herzogenberg, Elisabeth von, 477 hexatonicism, 413–414, 414n4, 491, 515, 560 hex-collection, 389–393, 398n18 hex-cycle, 401, 403–405, 410–412, 507n9, 547n43, 558 hex-pole, 391, 399n22, 492, 499 Page 7 of 17

Index historicism in music theory, 65–86, 194–195, 214n2, 462–464, 472–475 Holtmeier, Ludwig, 1, 137n5 Hook, Julian, 507n5, 573 Hornbostel, Erich Moritz von, 141, 146 Hostinský, Ottokar, 323, 357 Hull, A. Eaglefield, 561 humor in music, 444, 448, 452–455 Hunt, Graham, 415n9 Hyer, Brian, 295, 300, 318n5, 323, 348n48, 353, 401, 511n55, 515, 552–553, 564, 570 hypermeter, 420–421 I imagination of tone. See Tonvorstellung interval, 80, 295–296 genera and, 66, 143–145, 150 transformational conception of, 198–199, 285, 386, 548–549 octave, 180, 183–185, 247–249, 520 intonation and temperament, 177, 191n1, 270, 274–289, 322–323, 358 inversion. See symmetry J Jameson, Frederic, 56, 69 Journal of Music Theory, 498 K Karg-Elert, Sigfrid, 8 Kayser, Hans, 8 Kirnberger, Johann, 74, 86, 326 Kirsch, Ernst, 114 Klang, 1, 73, 82–86, 363, 366, 382–386 conceived by Helmholtz, 82 conceived by Oettingen, 82–83, 176 defining, 90n29, 278, 507n5, 580 over and under-Klänge, 83, 92, 366–376 Klangschlüssel, 14–15, 19, 130, 200, 386, 580. See also harmonic function—notation of Klangvertretung. See triad representation Klumpenhouwer, Henry, 166, 353–354, 359, 364–365 Knecht, Justin Heinrich, 74 Kopp, David, 299, 300, 350, 355, 507n3 Krebs, Harald, 301–304 Krehl, Stephan, 7–8 Kretzschmar, August Ferdinand Hermann, 418, 472, 474–475 Krumhansl, Carol, 323–324, 328–329 Kuhlau, Friedrich, Sonatina Op. 20/3, 425 Kurth, Ernst, 159, 169, 503, 553, 555, 560, 570, 572 and Riemann reception, 6, 11, 14, 39 (p. 601) L Lerdahl, Fred, 270, 324–345, 414n4 Levi, Hermann, 478 Lewin, David, 123, 225, 285, 323, 353, 400, 516–517, 528, 544, 548–549, 552–553, 574n16 Page 8 of 17

Index on Wagner, 256–258, 262–263 Lingg, Hermann, 466, 478 Liszt, Franz, 463–464 Consolation No. 1, 30 Piano Concerto in E-flat Major, 558 Valse impromptu 27, 29 Lockwood, Lewis, 441 Longuet-Higgens, Christopher, 322–323 Lotze, Rudolf Hermann, 62, 79, 85, 169, 174, 182 Louis, Rudolf, 2, 13, 32–39 “conceptual dissonance” of, 35–38 critique of Riemann, 32–34 intellectual background of, 33–34 “intermediate harmony” of, 36–37 scholarly aims of, 32–33 M “major chordishness”, 247–248 Maler, Wilhelm, 40–41 Mannheim school, 449 Marx, Adolf Bernhard, 81–82, 327 mathematical function, 112–118, 130, 357–358. See also harmonic function Matrix, The, 243n24, 274 Mayrberger, Carl, 30 Mayrhofer, Robert, 46n52, 197 McCorkle, Margit, 483n–28 Medieval music theory, 155 Mendelssohn, Felix Song without Words op. 52/2, 19–21, 24–26 Mersenne, Marin, 71 meter. See phrase and metrical structure Miller, Horace Alden, 561 minor triad, 33–34, 171–191 Hauptmann and, 80 Oettingen and, 77 Rameau and, 71–72, 172, 174 Riemann and, 83, 181–191 Zarlino and, 68, 171–172, 174, 180 See also harmonic dualism minor scale harmonic minor, 202, 267n43 natural minor, 143, 148, 152–158, 404 modulation, 326–327, 513, 543n3 expressive, 304–305, 316, 373 relationship to secondary dominants and, 9–11 Riemann on, 85, 109–111, 156, 366–367, 373–375, 447–448 monism. See harmonic dualism—monism and monotonality. See tonality—monotonality Mooney, Michael Kevin, 508n16 Moser, Hans Joachim, 8 Page 9 of 17

Index Motte, Diether de la, 299 Mozart, Wolfgang Amadeus, Piano Sonata in G major KV. 283, 433 String Quartet in C major “Dissonance” KV. 465, 15, 17–18 Munich School, 2 Murphy, Scott, 336, 575 musical form, 166, 218–241 musical logic, 128, 229, 365, 450–451, 458–459, 496, 499 in Riemann, 99–101, 219 relation to mathematical logic, 113, 125–126, 130–131 musical motion, 78, 100, 129, 247, 250, 277, 279, 285–286, 353–355, 366, 533, 548–553 musical universals, 5, 140–141, 145, 148–149, 497 musico-passeriformal topoi, 449 musicology, comparative, 2, 140–148 N National Socialism, 13, 41 Naumann, Carl Ernst, 269–270 Nebenklänge, 104–106 neo-Riemannian theory, 485–486 analysis and, 365, 400–401, 487–506, 542, 565–573 coherence in, 344, 365–366, 497–500, 515, 531–532 diversity of, 485, 510n33 harmonic function and, 127, 138n123, 492, 496, 517, 570 harmonic indeterminacy and, 325, 351, 400–402, 488, 564 New Musicology and, 240–241, 485, 498–500, 510n43 origins of, 497–498, 561 paleo-Riemannian theory and, 486, 488–489, 497, 500–504, 514–515 tensions of, 485, 492, 499–500, 550–553, 574n16 Tonnetz and, 270, 272, 288–289, 295, 329–345, 501–506 transformation theory and, 351–352, 515–517, 548–553 transformations of, 295–296, 300, 350, 351–360, 364–366, 414n1 Twentieth century music and, 389–393, 564–573 Neue Zeitschrift für Musik, 167, 200 “New Empiricism”, 13, 33 Nolan, Catherine, 358 nonatonic collection, 486, 554–555 non-Western musics, 140–159 O octatonic collection, 321n31, 389, 400, 413, 414n4, 528 octave equivalence, 66, 69, 180, 183–185, 246–248, 278 (p. 602) Oettingen, Arthur von, 60, 76–79, 82, 86, 152–153, 166, 171, 174–177, 218, 359 harmonic progression and, 349, 363 tonal space in, 82, 274–278, 280, 322–323, 357 organicism, 220–221, 240 overtones and overtone series, 78, 171–181, 188–189, 275–276 arithmetic division, 67 multiplication tones, 76–77, 175–178, 275–276 Rameau and, 70, 73–74 Page 10 of 17

Index P parsimony. See voice-leading—efficiency Pearce, Trevor, 135n50 Peirce, Charles Sanders, 101 pentatonic collection, 413 Riemann and, 142–158, 163n51 tonicity and, 150–152, 156–158 Perischetti, Vincent, 562 phrase and metrical structure, 5, 99–100, 154, 219, 417–418, 419–435, 440–459 accent, cadential, 418, 432–435 accent, durational, 427–430 accent, tonal, 430–432 downbeats and upbeats, 417–418, 420, 441–445, 458, 472 melodic imitation and, 426–427, 433–434 notated and expressed meter, 420, 422, 431, 434–435, 472 period structure and, 442–443, 445–447, 450, 457–458 text setting and, 462–480 Phrygian mode, 565–568 physiology, 57–60, 65, 78–79, 83, 147, 165 plagal drift, 335–336, 559 Plutarch, 143–145 psychology, 2, 57–58, 65, 82–85, 126–128, 147, 178, 186, 193n23, 262 See also harmonic dualism —mode of listening Pythagoras, 66, 144 Q Quinn, Ian, 264n14 R Rameau, Jean-Philippe, 70–71, 86, 95–96, 146, 149, 248–250, 555 double emploi, 504 fundamental bass theory of, 71–73, 188 minor triad and, 71–72, 171 Ravel, Maurice Tombeau de Couperin “Forlane”, 390–393 reductionism, 92, 102, 109 Reed, John, 316, 321n34 reformism in German music theory, 8, 13, 40–41, 43n15, 55, 170 Rehding, Alexander, 2, 55–56, 94–95, 100, 127, 166, 195, 215n5, 397n3, 494–495, 497–498 rhetorical ventriloquism, 552, 574n16 Riemann, Hugo analysis and, 109–112, 158–159, 272–274, 418, 440–459, 488, 492–494 Beethoven and, 417–418, 440–441, 445–446, 450–451, 457–459, 493 Brahms and, 222, 242n21, 363, 418, 463–464, 472–474, 477, 482n13 composer, as, 242n23 contributions to practical music theory, 8–13 East Asian musics and, 148–152, 154–158 Frege and, 112–114 German musical canon and, 147, 440–441, 446, 472, 510n34 harmonic function and, 92–112, 118–120, 127–131, 361, 366, 430, 443–444 Page 11 of 17

Index Harmonieschritte and, 107–109, 349–350, 351–356, 360–376 history of music theory, on, 62, 65–86, 171–175, 183 immediate influence of, 4–8, 240 intellectual sources of, 101–102, 143–147 melodic analysis in, 140–141, 150–153, 158–159, 426–427, 446 minor scale/triad, origins of, 82–83, 153–155, 167–169, 249–250 modernity and, 55–56, 64–65, 146–147, 446, 450–451, 459 musical form and, 218–233, 238–241 non-Western repertoires and, 140–159 normativity in, 148, 365–366, 383, 397n3, 464, 474, 496–498 notational systems of, 109–112, 149, 442 pedagogy and, 169, 187–191, 296–297 personality and rhetorical style of, 3–5, 144–147, 168–170 phrasing and metrical structure and, 99–100, 154, 219, 417–418, 419–435, 440–459, 462–479, 579 posthumous reception of, 3–41, 43n23, 419, 473 psychological justification for dualism, 181–191 science and technology and, 56–59, 65, 140–141, 497 tone relations of, 102–109, 351–366, 552 tone representation and, 61–64, 178–179 Tonnetz and, 271–289 undertones, argument for, 76–78, 165, 182–183 Riemann, Hugo, writings of Beethovens sämtliche Klavier-Solosonaten, 111, 440–459 “Degeneration und Regeneration in der Musik”, 363 Elemente der musikalischen Ästhetik, 170 Elementar-Schulbuch der Harmonielehre, 433 Fokloristische Tonalitätsstudien, 140–159 (p. 603) background of, 141–148 evaluation of, 158–159 Geschichte der Musiktheorie, 142, 170, 188 Große Kompositionslehre, 220–221, 273–274, 428 Grundriß der Kompositionslehre, 422–424, 430–433 Handbuch der Harmonielehre, 4, 32, 94, 107–108, 424–425, 489, 495 Handbuch der Musikgeschichte, 144, 364, 463, 473–475 Die Hülfsmittel der Modulationslehre, 278, 349, 358, 360 “Ideen zu einer ‘Lehre von den Tonvorstellungen’”, 128, 286–288, 294–296, 412 Katechismus der Musikwissenschaft, 170, 186, 220–222 “Metric Freedoms in Brahms’ Songs”, 462–473 background of, 473–479 Musikalische Dynamik und Agogik, 427–428, 431 “Musikalische Logik”, 99–101, 107 Musikalische Syntaxis, 61, 86, 71, 180, 186, 349, 360–363, 489, 493 Musiklexikon, 4–5, 93, 94–95, 105, 221, 401 function in, 99 tonal space in, 272, 280–287, 323 “Die Natur der Harmonik” 2, 55–86, 98–99, 127–128, 279 background of, 55–65 Page 12 of 17

Index Neue Schule der Melodik, 32 “Die Neugesetaltung der Harmonielehre”, 242 “Das Problem des harmonischen Dualismus”, 166, 167–191, 201–202 background of, 167–170 Skizze einer neuen Methode der Harmonielehre, 86, 107, 271–274, 349 “Schematisirung der Dissonanzen” in,  382–386, 393–397 “Systematik der Harmonieschritte” in, 350, 353–359, 362–376 Systematische Modulationslehre, 218, 417, 493–494 “Über das musikalische Hören”, 277–278, 280–281 Vademecum der Phrasierung, 422–424, 428, 433–435 Vereinfachte Harmonielehre, 495 function in, 92–94, 96–97, 99, 102, 112–113, 127, 204, 364 Harmonieschritte in, 107–108 Richter, Ernst Friedrich, 4, 9–10 Rings, Steven, 486, 517 Rousseau, Jean-Jacques, 146 Roussier, Abbé Pierre-Joseph, 149 Royal Research Institute for Musicology, 140 Rummenhöller, Peter, 101–102, 241n2 S Santa, Matthew, 553–555, 560–563 Sauveur, Joseph, 71, 172 scale degree theory, 1, 123, 495–497, 509n30, 564 Scandinavian musics, 152–153 Scheinkonsonanzen. See apparent consonance Schenker, Heinrich, 159, 326–327, 493 imagination of tone of, 303, 317–318 relationship to function theorists, 32 Schillings, Max, 36–37 Schmidt, Matthias, 473, 483n21 Schoenberg, Arnold, 63, 252, 262, 493, 528 Schreyer, Johannes, 2, 4, 13–32 and harmonic parenthesis, 20–32 “apparent harmonies” of, 15–16, 19 reductionist analyses of, 15–32 Schritte and Wechsel, S/W group. See Harmonieschritte Schubert, Franz Am Meer, 253, 262 Ganymed, 299 Gretchens Bitte, 311–317 Impromptu in G-flat major, 487–506, 508n18 Liedesend, 304–311 Piano Sonata in D major op. 53, 253, 404–407, 412 Quartet no. 14 in D minor, 225 Quintet in C Major, 299 Trost, 300–304 Schumann, Robert Carnival “Chopin”, 253 Page 13 of 17

Index Dichterliebe “Im wunderschönen Monat Mai”, 329–333 Scruton, Roger, 550, 553 Sechter, Simon, 35–36, 38, 336 Seidel, Elmar, 240, 440 sensation and perception, 60–64, 90n36 seventh chords, 197, 251, 254–262, 383, 385, 394, 507n5, 553–555 Shepard, Roger, 323 Siciliano, Michael, 301, 501 Smith, Charles J., 328, 565–567 Smith, Peter H., 243n38 sounding body, 57, 65, 83 Still, William Grant Traceries “Cloud Cradles”, 389–390 Stockhausen, Julius, 478 subdominant, 500–501 in Rameau, 72, 83, 95 in Riemann, 234, 244n50, 353, 360, 495 minor, 331–332, 384, 362 Steege, Benjamin, 2, 88n13, 192n7, 291n25 Straus, Joseph, 549, 551 Strauss, Richard, 510n38 Stravinsky, Igor The Rite of Spring, 225 Stufen. See scale degree theory Stumpf, Carl, 141, 146–147, 169, 171, 177–180, 186 symmetry, 142, 166, 198–199, 262–263, 297, 499 cadential, 432–433 form and, 234–236, 239 individual and uniform, 248–249, 255, 266n29 (p. 604) inversional, 103, 246–247, 250–263, 353 pentatonic scalar, 148–152 pitch collections bearing, 389, 400–401 tonal space and, 338–343 varieties of, 246–255 See also harmonic dualism T Tallis, Thomas, 565–568 Tarnhelm progression, 256–258 Tartini, Giuseppe, 68, 74–76, 86 tetrachords, 143, 148, 153–158, 390–392, 562 theo-Riemannian theory, 363, 495, 496 thirds, progression by. See harmonic progression—by thirds Thuille, Ludwig, 4, 50n116 tonal function. See harmonic function tonal luminosity, 373, 403, 502–503, 568–572 tonality chromatic, 360, 400–402, 403, 468, 565 monotonality, 493–494 Riemann on, 84–85, 221, 359–367, 401–402, 406, 448, 493–494 Page 14 of 17

Index non-monotonality, 387, 407 origins of, 141–158 post-tonality and, 390–393, 568–570 scale and, 326–327, 361, 366, 412–414 tonal complex, 296–297 tonal design, 317–318, 402–412 and dualism, 220, 223–239 double tonic complexes, 301 tonal gravity, 96, 451, 565 tone relations, 102–109, 269–270, 355 by thirds and fifths (Q/T), 274–279, 284–288, 359–360, 367–370, 404, 468, 477 compound/unary relationships, 356–357, 415n8, 568–572 cross-type, 251, 254–255, 507n5, 551–552 dualist and non-dualist, 353 key and, 328–329, 344, 351–352, 357, 400–401, 488–491 PLR group, 294–296, 300, 350, 352, 354–355, 501, 570–571See also harmonic function; Klang; Harmonieschritte; transformational operations tone representation. See Tonvorstellung tonal space. See also transformation networks; Tonnetz distance/proximity and, 250, 258–259, 278, 305, 326–328, 346n21, 413, 568 Garden of Eden, as, 363, 494, 496 in Lerdahl, 324–325, 329, 333, 344–345 in Riemann, 271–289 functional continuum, 105–107 functional representation, 96–97 temporality and, 218–219 whole-tone space, 540, 560 tonic, 97–98, 150–152, 552–553, 573 and antitonic, 556–564 in Hauptmann, 229–230, 233 in Riemann, 361, 366, 430, 443–444 Tonnetz analysis with, 271–274, 329–345, 501–506 as acoustic matrix, 274–283, 323 as map of relations, 272–274, 285–289, 323 chord, pitch, and region in, 325–329, 412–414 geometry of, 288, 324, 329, 501 in Oettingen, 82, 270, 322 in Riemann, 220, 322, 237, 280–288 pre-Riemannian developments of, 269–278, 322–323 tone representation with, 295–298 varieties of, 323–325, 501 Tonverwandtschaften. See tone relations Tonverwandtschaftstabelle. See Tonnetz Tonvorstellung, 6, 9, 34, 60–64, 165, 168–170, 294–318, 515 defining/translating, 90n36, 104–105, 138n128, 292n43, 295, 580 Helmholtz on, 60–61 Riemann on, 83–85, 102, 128–129, 178–179, 185–186, 286–287, 294–297, 317–318 Page 15 of 17

Index tone reinterpretation and, 296–297, 301–317 Tonnetz and, 286–289, 297–298 Tovey, Francis, 457 transformation, 247, 529, 548–553, 580 contextuality in, 349–350, 353, 516–517, 527–531, 542 Riemann's conception of, 102, 127, 285, 378n23 Schritte/Wechsel vs. tone relations, 107–109, 349–350, 353–366 stance of, 388–389, 404, 502, 515–517, 548–550, 574n16 theories, 386, 516, 531, 552 tonality and, 485–486, 500–501, 515, 552 transformation networks, 550, 563–564 analytical, 256–257, 406, 410–412, 490, 491, 524–531, 536–541 function and, 201–204, 212–214, 411, 492, 496, 502 post-tonal, 389–393, 406 transformational operations D/DOM, 300, 328 DOUTH2, 391 H/hexatonic pole, 391, 399n22, 492, 499, 572 IDENT, 354, 357, 388 L/leittonwechsel, 11–13, 102–106, 328, 387–388, 570 N/nebenverwandt, 490–491, 508n11 P/parallel, 102–106, 495, 403, 571–572 Q/T (Quintschritt/Terzwechsel), 274–279, 284–288, 353, 359–360, 367–370, 404, 468, 477 R/relative, 9, 11–13, 102–106 S/slide, 266n29, 361, 369–370, 376, 488, 490–491, 503–504, 508n12, 509n23, 524, 529 (p. 605) S/SUBDOM, 500–501 Tn/transposition, 246–249, 255, 266n29, 285, 392, 398n18, 571, 573 Wn/Doppelklänge-generators, 388–391 X/Y transformations, 528–529, 534, 542, 546n31, 562–564 triad, ontology of, 344, 531–532, 548–553 triadic atonality. See tone relations—independent of key triadic inversion, 106–107 fundamental bass and, 69–72, 187–189, 467–468, 477 structure of major and minor triads, 181–186, 197, 204 triad representation, 102–103, 270, 318, 580 trichords, 555–564 Triole, 453, 456, 460n8 Tristan chord, 259–262 Tymoczko, Dmitri, 166, 507n7 U Ulehla, Ludmila, 562 undertones and undertone series, 7, 167–168, 182–183, 200, 277 empirical case for, 77, 186 harmonic division, 67 in Oetttingen, 77, 175–178 in Rameau, 71 V Vallotti, Francesco Antonio, 73–74 Page 16 of 17

Index Vaughan Williams, Ralph Fantasia on a Theme by Thomas Tallis, 564–573 Verwandtschaftstabelle. See tonal space; Tonnetz Vial, François-Guillaume, 323 Vogel, Martin, 269, 322 Vogler, Abbé Georg Joseph, 430 voice-leading, 1, 166, 500 absolute and directed, 489–491, 507n7, 528–529, 556, 562–564 efficiency, 253–262, 300, 324, 330, 354, 400, 489, 555, 558–560, 581 in Riemann, 103–104, 477 maximal smoothness, 256, 258, 267n35 voice exchange, 29–30 Vorhang, 447–448 W Wagner, Richard, 10–11, 375 Parsifal, 121–122, 333–337, 489, 498, 551 Rheingold, Das 256–258 Tannhäuser, 30–32, 121–122 Tristan und Isolde, 258–261, 557 Walküre, Die, 356, 515 Wason, Robert, 13, 30 Weber, Gottfried, 249–250, 323, 327–328 Webster, James, 225 Wechsel. See Harmonieschritte Weiner, Joan, 129 Weitzmann, Carl Friedrich, 322–323, 507n9 Weitzmann region, 490–492, 496, 507n9 Wolf, Hugo Spanisches Liederbuch “Und schläfst du, mein Mädchen”, 407–412 X X and Y transformations, 528–529, 534, 542, 546n31, 562–564 Z Zarlino, Gioseffo, 67–70, 74–75, 169, 171, 180, 216n13 Zee, Anthony, 246 Ziehn, Bernhard, 3 Zuckerkandl, Victor, 549–550, 573

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