Audacious Euphony: Chromaticism and the Triad’s Second Nature 019977269X, 9780199772698

Citation preview

Audacious Euphony

OXFORD STUDIES IN MUSIC THEORY Series Editor Richard Cohn

Studies in Music with Text, David Lewin Music as Discourse: Semiotic Adventures in Romantic Music, Kofi Agawu Playing with Meter: Metric Manipulations in Haydn and Mozart’s Chamber Music for Strings, Danuta Mirka Songs in Motion: Rhythm and Meter in the German Lied, Yonatan Malin A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice, Dmitri Tymoczko In the Process of Becoming: Analytic and Philosophical Perspectives on Form in Early Nineteenth-Century Music, Janet Schmalfeldt Tonality and Transformation, Steven Rings Audacious Euphony: Chromaticism and the Triad’s Second Nature, Richard Cohn

Audacious Euphony Chromaticism and the Triad’s Second Nature Richard C ohn

1

1 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 © 2012 by Oxford University Press 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 Cohn, Richard Lawrence, 1955Audacious euphony : chromaticism and the consonant triad’s second nature / Richard Cohn. p. cm. — (Oxford studies in music theory) ISBN 978-0-19-977269-8 (hardback : alk. paper) — ISBN 978-0-19-983282-8 (companion website) 1. Harmony. 2. Triads (Music) I. Title. II. Series. MT50.C736 2011 781.2΄5—dc22 2011008754

Publication of this book was supported by the Otto Kinkeldey Endowment of the American Musicological Society. 135798642 Printed in the United States of America on acid-free paper

in memoriam John Clough

David Lewin

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CONTENTS

Introduction About the Companion Web Site

ix xviii

1 Mapping the Triadic Universe Three Ways to Calculate Triadic Distance Triads in Chromatic Space Remarks on Syntax and Maps 2 Hexatonic Cycles A Minimal-Work Model of the Triadic Universe The Hexatonic Trance Contrary Motion and Balance Hexatonic Progressions, Tonnetz Representations, and Triadic Transformations Near Evenness, Minimal Voice Leading, and the Central Role of Augmented Triads Remarks on Dualism Triadic Structure Generates Pan-Triadic Syntax Triads Are Homophonous Diamorphs

1 1 8 13 17 17 20 24

3 Reciprocity The Historical Emergence of Augmented Triads Consonance/Dissonance Reciprocity Two Early-Century Examples: Beethoven and Schubert Three Late-Century Examples: Liszt, Rimsky-Korsakov, Fauré Reciprocity in Weitzmann’s Der Ubermässige Dreiklang 4 Weitzmann Regions The Structure of a Weitzmann Region Weitzmann Transformations and N/R Cycles Remarks on the Tonnetz Historical Origins of Weitzmann Regions The Double-Agent Complex Expanded N/R Chains Weitzmann Regions without Sequences: Wagner and Strauss

43 43 46 48 49 56 59 59 61 65 67 72 76 78

25 33 37 39 40

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 Contents 5 A Unified Model of Triadic Voice-Leading Space How Hexatonic and Weitzmann Regions Interact Chromatic Sequences Transformational Substitutions Voice-Leading Zones Remarks on Disjunction and Entropy 6 Navigating the Triadic Universe: Three Compositional Scripts Neighborhoods and Pitch Retention Loops Departure → Return Scripts Continuous Upshifts 7 Dissonance Four Eighteenth-Century Approaches to Dissonance Reduction to a Triadic Subset Hexatonic Poles in Parsifal The Tristan Genus as Nearly Even Tetrachord Circumnavigating the Tristan-Genus Universe Scriabin’s Mystic Species and Generalized Weitzmann Regions 8 Syntactic Interaction and the Convertible Tonnetz Some Previous Proposals The Diatonic Tonnetz Horizontal Extensions Vertical Extensions The Convertible Tonnetz Two Analytical Vignettes: Wagner and Brahms

83 83 89 95 102 106 111 113 121 131 139 139 142 145 148 159 166 169 169 175 179 184 186 189

9 Double Syntax and the Soft Revolution A Summary Example from Schubert Double Syntax and Its Skeptics Code Switching and Double Determination Cognitive Opacity The Soft Revolution On Musical Overdetermination

195 195 199 201 203 205 208

Glossary Bibliography

211 215

Index

229

I N T R O D U C T IO N

The admittedly audacious but effective and euphonious progression shown [above] defies definition in terms of an older doctrine of key. But . . . it consists only of closely related chords contrasted with the tonic triad. —Hugo Riemann, s.v. “Tonalität,” Musik-Lexicon, 1909

Two questions arise. First, what notion of harmony underlies Hugo Riemann’s judgment that these chords are closely related? Current textbooks have inherited the eighteenth-century formulation that triads are closely related if their eponymous scales are identical to within one degree of difference: they share at least six out of seven tones. These harmonies don’t come close to qualifying: their associated scales share three tones out of seven. Riemann’s conception of harmonic distance is evidently rather different from our own. How can we construct and represent that conception? Second, if the triads are closely related, why does Riemann call the progression “audacious”? Close relations are unmarked, well formed, normal—not the stuff of audacity. These questions lead to different kinds of responses. The first is susceptible in principle to a systematic inquiry. If we can establish that Riemann and his contemporaries calculated harmonic distance in a consistent way, even if distinct from the way we do so, then we might have a chance to understand the basis for his judgment. The second, because it identifies a paradox, is an invitation to an interpretation. Audacious Euphony reconstructs conceptions of triadic distance that were proper to nineteenth-century harmonic thought but have since been stripped from music theory’s inheritance. What effect do these alternative conceptions have on our understanding of how the nineteenth-century ear understood harmonic relations and how nineteenth-century composers crafted strategies and made choices that both appealed to and molded that ear? How did these alternative conceptions, and the strategies and choices they motivated, contribute to the charismatic, entraining, and sublime qualities that we still hear in many compositions of that era? Harmonic theorists of the nineteenth century provided a partial response to these questions. This book expands that sketch into a fully realized proposal using

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 Introduction conceptual, technical, and representational resources unavailable a century ago. Its principal thesis is that major and minor triads are not only the familiar acoustic consonances of eighteenth-century classical theory. They also have a second property, equally rare and equally generative of syntax.1 That property underlies an alternative method for measuring harmonic proximity, leading naturally to an account of the triadic universe and to its representation through an interrelated set of maps. This book also explores the paradox identified in Riemann’s formulation, a paradox that is writ large across the music of the long nineteenth century. Chromatic progressions of triads excited the Romantic imagination not because they conformed to expectations about triadic behavior and succession but because they confounded them. In song, opera, and programmatic music of the period, such progressions are often explicitly affiliated with altered or heightened realities: Schubert’s magic harp, Wagner’s magic sleep, Rimsky-Korsakov’s magic and exotic kingdoms, Liszt’s mountain-top meditations (Kurth 1923; Taruskin 1985). Even in nonprogrammatic instrumental music, they are capable of evoking the strange, magical, and inscrutable (Hoffmann 1989 [1813–14], 318; Cohn 2004). To address how chromatic progressions summon those responses, we will need to explore their position on a field animated by opposing forces: of classical and romantic syntax, of normativity and aberration. How do the old and the new syntaxes coexist and interact in nineteenth-century music, and what are the implications of this coexistence for our understanding of musical cognition and of the transformation of musical syntax across the long nineteenth century? In order to cultivate an unfamiliar conception of the harmonic universe, it will be useful to suspend some overlearned habits. Because some species of classical diatonic theory is the first, and in many cases the only, music theory that we are exposed to, its status as theory, a way of looking, easily hardens into nature: the way that music is. In response, I will occasionally invite readers to “forget” that major and minor triads are acoustic consonances generated from roots that occupy a position in a diatonic scale with respect to some tonic. This is a cultivated and strategic denial, motivated by the desire to prevent our default habits from flooding back in and drowning those properties and relations that I wish to foreground. The move is perhaps akin to one that physicians make when they bracket off the aesthetic qualities of the bodies that they are examining—qualities that might interest them in other contexts. Those qualities are not relevant to the diagnosis or treatment of some ailment or condition, and to focus on them would be a distraction from the task before them. As we explore the relations of the two syntaxes in the final chapters, readers will increasingly find themselves on familiar ground, although their perspective on that ground will have been altered in some meaningful way. That, at least, is my hope and intention. In establishing the viability and utility of the alternative syntax whose proposal is at the core of this book, Audacious Euphony draws on four different kinds of evidence. The syntactic model will be shown to be theoretically powerful, in the 1. Syntax is a problematic term whose chief virtue is that it is better than its alternatives. It nonetheless requires clarification, which I provide in the final section of chapter 1.

Introduction sense that it generalizes in surprising and productive ways, beyond the consonant triads for which it was designed. It will be shown to be analytically productive, in the sense that it leads to new ways of hearing, conceiving, and representing music composed by a broad cross section of nineteenth-century composers. It will be shown to be historically appropriate, in the sense already indicated: its approach emerges from indigenously nineteenth-century ways of thinking and writing about musical materials and their relations, although it is often couched in language that those historical figures would find alien. And it will be shown to be historically productive, affording a new way to address the transformation of musical style across the long nineteenth century. My hope is that each of these four types of evidence will bear its own intrinsic and independent value for readers, so that, for example, readers who do not connect with my analyses, who find my historical readings tendentious, or for whom abstract modes of thinking about music are distasteful, might nonetheless find value elsewhere in the book. My aim has been to write a book that is as robust, multidimensional, and overdetermined as its topic.

Genesis and Relation to Prior Work In 1990, a passage from Schubert’s Eᅈ Piano Trio called my attention to a voiceleading property of consonant triads. (The passage is partly reproduced as figure 2.8 in chapter 2.) The property was special; it was shared by no other three-note chords, and the few chords of other sizes that shared it nontrivially included the diatonic and pentatonic collections, which are equally privileged within the European musical tradition, and within others as well. After seeking it in chords and scales in universes with more or fewer than twelve tones, I began to formulate some conjectures about this property and its impact on historical repertories. Informal correspondence led John Clough to convene a small conference in Buffalo during the summer of 1993, where Jack Douthett and David Lewin refined and generalized my conjectures and suggested a scope for them beyond the limited domain within which I had initially envisioned them (Douthett 1993; Lewin 1996; Douthett and Steinbach 1998). These unanticipated applications to living repertories (rather than hypothetical ones that might be constructed ad hoc) helped build my confidence in the significance of my observations and the conjectures that they had prompted. A second wave of reinforcements arrived in 1995–96, during a residency at the University of Chicago’s Franke Institute of the Humanities. Reading in and around nineteenth-century harmonic theory, aided by the recent dissertations of David Kopp (1995) and Michael Kevin Mooney (1996), I discovered affinities with my developing conception. An article of Larry Todd’s (1988) led me to Carl Friedrich Weitzmann’s short monograph on The Augmented Triad (1853), which provided the final building block for a model of triadic space that I presented at a second Buffalo conference in 1997, the proceedings of which became a special issue of the Journal of Music Theory (vol. 42, no. 2, 1998).

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 Introduction In the fourteen years since, the ideas emerging from the two Buffalo conferences have been further generalized, merged with other theoretical concerns, and adapted to various analytical and historical purposes, in dissertations (and subsequent publications) of David Clampitt (1997), Edward Gollin (2000), Robert Cook (2001), Nora Engebretsen (2002), Michael Siciliano (2002), Julian Hook (2002), So-Yung Ahn (2003), and Steven Rings (2006). Those ideas are also absorbed into, and significantly transformed by, the writings of Dmitri Tymoczko, which have appeared in a steady stream since 2005 and culminated in the publication of A Geometry of Music (2011). The theoretical power of that emerging work, and the pedagogical gifts of its author, prompted me to divert the flow of my own developing book away from some of its early theoretical ambitions, the better to focus on analytical and historical questions pertinent to its music-historical moment. With so many horses out of the barn and reproducing, I have two distinct sets of motivations for bringing this material together into a book now. First, I have aimed to make my ideas about nineteenth-century music accessible to readers uncomfortable with or uninterested in the technical literature in scholarly journals. The core of the exposition assumes no knowledge beyond a semester of undergraduate harmony. I intend that this book be read not only by music theorists and their students but also by music historians and psychologists, performers of nineteenth-century repertories, composers who find creative vitality in triads and seventh chords, and anyone else who has the appropriate background and temperament to work through a close argument about musical properties and relational systems and close readings of particular compositions. I have avoided, to the extent possible, the mathematical modes of discourse that have dominated much of the technical literature on this topic. Most of the book will ask readers to be comfortable with counting small quantities, representing them as numbers, and adding or subtracting their values; for example, one apple over here and two apples over there make three apples in all. Some parts of the book use modulo twelve arithmetic, which you have been performing all your life, although perhaps beneath your threshold of awareness. If you worked from nine this morning until six this afternoon, how long was your shift? If you board the train at eleven, and the journey lasts five hours, what time do you arrive? If you know the answers spontaneously, or can calculate them quickly, then you are performing arithmetic modulo twelve. My hope is that this level of mathematical rigor will not drive away readers who are inclined to head for the hills when a scientistic odor wafts about. For those who are comfortable with such discourses, references to the technical literature are furnished. At the ends of chapters 2, 5, and 7, the technical bar rises. Second, I intend this book as a fresh contribution to music theory and analysis. Although readers familiar with my papers on chromatic harmony will recognize traces of many of them, almost every sentence and analysis presented here is new. My earlier publications have offered a kaleidoscope of concepts and analytical perspectives, all radiating from my initial observation about the consonant triad’s special voice-leading properties and their relationship to nineteenthcentury triadic chromaticism. A principal concern of Audacious Euphony is to

Introduction

 xiii

show that these perspectives constitute paths through a unified field, reflecting an integrated vision. A second concern is to work out a model of the relationship between classical and romantic harmony and to pursue its historical implications, building on a program that I first sketched at the end of my initial publication on the topic (Cohn 1996). Several readers, evidently having overlooked that sketch, have interpreted me as claiming no such relationship (Fisk 2000; Lerdahl 2001; Kopp 2002; Horton 2004; Damschroder 2010), and I am intent to put those responses to rest. A related aim is to stake out positions with respect to some technical topics of particular interest to music theorists. These discussions are mostly confined to subsections titled “Remarks.” Readers who find these concerns parochial may skip these passages without loss of continuity. A final goal is to convince readers that these assumptions, concepts, and representational modes have the capacity to illuminate aspects of musical artworks that they may cherish. Prior work, of mine and others, has focused mostly on refining and generalizing the model, rather than pursuing its implications for understanding particular compositions and the responses they provoke in musicians and listeners (Hook 2002; Rings 2007). The core chapters include analyses of passages of music, and sometimes entire compositions, by Schubert, Chopin, Liszt, Wagner, Brahms, Rimsky-Korsakov, Dvořák, Bruckner, Fauré, and Richard Strauss. I hope that these analyses will be satisfying on their own terms and that they might provide models for readers to explore other compositions from this glorious repertory that continues to occupy a prominent position in concert halls, opera houses, and mp3 players. My prior articles on chromatic harmonic flew under the banner of neoRiemannianism, and I owe it to readers to explain why I avoid that term in the present book. The first of my two distinct motivations is that it gives too much credit to Hugo Riemann. It was David Lewin’s reading of Riemann’s harmonic writings that constituted the originary moment for this branch of theorizing, and the insights that he fashioned from that reading fuel my approach. But I have come to believe that, with respect to the nineteenth-century ideas most at the heart of this book, Riemann was functioning more as transmitter than generator (Cohn 1998a). My second motivation is that the domain of neo-Riemannian theory has never been very stable (Hook 2002), and I am not comfortable with all of the views that have been attributed to it or with all of the practices that have been performed under its name. For example, Tymoczko (2009a) claims that a commitment to harmonic dualism is at the heart of neo-Riemannian theory. On this basis, I am no neo-Riemannian. At the same time, a strain of neo-Riemannianism has arisen whose focus is on the application of transformational labels to harmonic progressions. Although these labels are descriptively useful, they do not in themselves lead to an understanding of triadic syntax any more than Roman numeral or set-class designations alone constitute explanatorily adequate accounts of compositions from the tonal and atonal repertories. I view these labels as a bridge to a first approximation; what lies on the other side of the bridge, if one is lucky, is an understanding of how the moves designated by these labels behave as part of a compositional system. It is the exploration, representation, and application of that system that is the primary aim of this book.

xiv

 Introduction Although neo-Riemannian properly labels an approach toward hearing and representing musical relations, the term has also been applied to the musical relations themselves; thus, a tonally indeterminate progression of triads, or a composition filled with the same, is said to be neo-Riemannian. The term I favor for the compositions or passages that invite such modes of audition, conceptualization, and representation is pan-triadic, first used by Evan Copley (1991). The term is inspired by its obverse, pan-diatonic (Slonimsky 1937): both terms designate music that uses fundamental materials of tonality in tonally indeterminate ways, one by using diatonic scales without triads, and the other by using triads without diatonic scales.

Organization of Chapters Figure 0.1 models the organization of the book. The nine chapters form three groups of three: premises and theses, model construction and analysis, extensions and implications. This symmetry is perturbed, however, by the beginning of chapter 2, where significant aspects of the model are put into place. The form of the book in some respects mirrors an aspect of its thesis: that it is the consonant triad’s perturbed symmetry that motivates those triadic progressions most characteristic of the nineteenth century. Chapter 1 probes the limits of the classical view of harmonic distance, presents some alternative approaches cultivated by nineteenth-century theorists, and suggests that those approaches point the direction toward a model of nineteenthcentury triadic syntax that operates independently of classical principles. The first half of chapter 2 introduces a preliminary model of the triadic universe, based on the idea that triadic distance is conditioned by voice-leading proximity among consonant triads, and shows how that model captures a number of characteristic passages that extend chronologically from Mozart and Haydn of the 1780s to Wagner and Brahms of the 1880s. The central role of major thirds and augmented triads in the model leads to a proposal for why chromatic major third relations are affiliated throughout the nineteenth century with altered and destabilized psychological states, the supernatural and the uncanny, as Richard Taruskin and others have observed.

Figure 0.1. The organization of this book.

Introduction

 xv

In the middle of chapter 2, the arc of model construction is suspended in order to advance the major theoretical claim of the book, which follows from elements already in play. That claim is that pan-triadic syntax follows from an internal property that consonant triads hold independently of the acoustic properties that are claimed to generate classical syntax. One aspect of that claim, that augmented triads are central to pan-triadic syntax even when they are absent from the musical surface, may be counterintuitive, especially to readers whose understanding of harmony is unilaterally shaped by textbook accounts of harmonic tonality. Chapter 3 aims to defuse any skepticism that is aroused, by clarifying the nature of the assertion and providing historical and analytical support for it. Following ideas introduced by François-Joseph Fétis in the 1840s and refined by Ernst Kurth around 1920, I recognize that repetition, conceived broadly, has the power to offset the monolithy of diatonic tonality. Some ideas from Carl Friedrich Weitzmann’s 1853 treatise on augmented triads, introduced at the end of chapter 3, serve as the basis for a second preliminary model of the triadic universe, based on the voice-leading proximity of each consonant triad to one of the four augmented triads, and partitioning the triads into four Weitzmann regions. Chapter 4 traces the historical growth of Weitzmann regions from basic diatonic routines; shows how Schubert, Chopin, and others use four-chord subsets of these regions, forming what I call a double-agent complex, to tease the enharmonic seam; and analyzes some passages of Schubert, Liszt, Wagner, and Richard Strauss that plunge through that seam in order to explore the full chromatic extent of a Weitzmann region. The opening of chapter 5 demonstrates that the two preliminary models stand in figure–ground relationship and that a comprehensive model of the triadic universe results from their synthesis. Chapters 5 and 6 explore various modes of geometric representation and take the model for an analytical ride across some familiar and lovely musical terrain. Chapter 5 focuses on sequential passages and their transformations, with examples from Chopin, Liszt, Brahms, and Bruckner. Chapter 6 documents three strategies for exploring triadic space using less strictly patterned surfaces and analyzes complete short compositions and large sections of longer ones, including Schubert’s “Der Doppelgänger,” “Auf dem Flusse,” and Bᅈ major Piano Sonata, Brahms’s Second Symphony, and Dvořák’s “New World” Symphony. A major focus of these chapters, following an implication of Daniel Harrison’s work (1994), is to position Riemann’s harmonic dualism as a particular instance of a more fundamental phenomenon, the relationship between upward and downward melodic motion. Chapters 7 and 8 take up two issues that have been bracketed off until the model could be constructed on its own terms. Chapter 7 proposes several ways to integrate dissonant harmonies into the model, with a particular analytic focus on Chopin’s piano music and Wagner’s late operas. The second half of the chapter constructs a model of seventh chords by analogy to the pan-triadic model, leading to a general version of the model whose breadth is suggested by its applicability to Alexander Scriabin’s treatment of mystic chords. Chapter 8 responds to a question implicitly teased but explicitly skirted throughout the preceding chapters: how does pan-triadic syntax interact, in alternation or overlay, with the familiar

xvi

 Introduction diatonic syntax of classical tonality? After reviewing the approaches of prior theorists, including David Lewin, Brian Hyer, Steven Rings, and Candace Brower, as well as those offered in my own publications, I present a new proposal and analyze passages from Liszt, Wagner, and Brahms. My proposal asks readers to accept the evidently controversial proposition that good music can be fashioned from multiple syntaxes, and that listeners are cognitively capable of integrating them. Chapter 9 supports that proposition’s epistemological value and uses it to approach the question of how the harmonic practice of Mozart mutated into that of Webern. It is this material that provides the basis for the claim that the model is historically productive. The analysis of a passage from Schubert’s C major Symphony presented at the beginning of that chapter summarizes the approach developed in this book and might serve as an effective initial exposure for readers who want to get a sense of its central ideas.

Acknowledgments First thanks go to Heather, whose love calmed and focused me, and whose generosity sustained and nourished me, through many years of work on this project. The encouragement of David Lewin and John Clough, to whose memories this book is dedicated, endured in me long after they passed away, within months of each other, in 2003. David’s ideas were foundational, and the power of his ideas is reflected in the number of citations to them in my text. John’s interest in my ideas led him, on three occasions, to gather music theorists to Buffalo to grapple with their implications. I will be forever grateful for the honor, as well as for the ideas and perspectives that I took away from those sessions. Jack Douthett’s early extensions of my ideas are amply represented in this book, and his skill at communicating complex insights about musical systems has been a continuing inspiration. I have also benefited from vigorous discussions with Robert Cook and Michael Siciliano, who worked closely with me at a crucial point in the development of the ideas presented here. I am grateful to William Benjamin and an anonymous reader for energetic readings of an initial partial draft; their impact was immense. Prof. Benjamin also contributed an insightful reading of a later complete draft. Dmitri Tymoczko provided a generous end-stage reading of the entire manuscript, corrected a number of errors, and helped clarify some questions concerning the relationship of my work to his. Steven Rings was an invaluable consultant during the final stages of the project. Specific passages in the book benefited from friendly communications from Robert Bowers, Candace Brower, Scott Burnham, Deborah Burton, Adrian Childs, Thomas Christensen, David Clampitt, Suzannah Clark, Cliff Eisen, Daniel Goldberg, Floyd Grave, Berthold Hoeckner, Christoph Hust, Stefano Mengozzi, Tahirih Motazedian, Ian Quinn, Ramon Satyendra, and Judith Schwartz. I regret that I am unable to honor by name other students and colleagues whose responses to my ideas left traces that detached from their sources.

Introduction

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I am pleased to acknowledge the dedication and skill of Andrew Maillet, who produced the graphics for the book and for the accompanying Web site, and of Christopher Brody, a meticulous bibliographer, indexer, reference checker, and proofreader. Their work was partly funded by a grant from the Otto Kinkeldey Fund of the American Musicological Society, for whose assistance I am grateful. At an early stage, I benefited from institutional support of the Franke Institute of the Humanities at the University of Chicago, and from Philip Gossett and Janel Mueller, both former deans of the Humanities Division there. Toshiyuki Shimada and Brian Robinson kindly provided access to the recordings archive of the Yale Symphony Orchestra and granted permission to use selections to accompany the animated graphics on the companion Web site. Patrick McCreless performed the piano excerpts that appear there and accompanied singer Paul Berry in the performance of “Der Doppelgänger.” I am grateful to both of them, and to Mateusz Zechowski, who recorded and edited the sound files. Final thanks go to Heather’s mother, Meryl, whose loving gift of time made it possible for me to finish on schedule; to my mother, Dorrit, a model writer and teacher and an intrepid theorist; and to my daughter, Sylvia, who has tickled my soul every day for the last seven years.

A B O U T T H E C OM PA N IO N W E B SI T E

www.oup.com/us/audaciouseuphony The Web site contains scores for those passages that are subjects of sustained analysis, and whose length exceeds a single page. These scores are in pdf format and may be downloaded to be viewed on screen or printed. Every effort has been made to render the notes and rhythms as accurately as necessary for study purposes, but the scores lack dynamics, articulations, and other performance indications. Scores are not provided for passages discussed only briefly. Most of these scores are widely available and also are available online, for instance, via the International Music Score Library Project. The Web site also contains recorded performances of some passages, coordinated with graphic animations that correspond to static graphics in the book. Recordings are limited by their availability under fair-use statutes of copyright law. The orchestral excerpts are performed by the Yale Symphony Orchestra, under the direction of Toshiyuko Shimada. Patrick McCreless performed the excerpts for piano, and Paul Berry sang “Der Doppelgänger.” References in the form “Web figure x.y” direct readers to these Web resources, accompanied by the symbol . User name: Music5

Password: Book1745

C HA P T E R

One

Mapping the Triadic Universe

Three Ways to Calculate Triadic Distance It is self-evident that those keys whose scales have most notes in common are most closely related. —Johann Phillip Kirnberger, The Art of Strict Musical Composition, 1771 It is as though a hidden, sympathetic bond often connected the most remotely separated keys, and as though under certain circumstances an insuperable idiosyncrasy separated even the most closely related keys. —E. T. A. Hoffmann, Kreisleriana, 1814

In the age of Mozart, distance between keys is linear and easily calculated. In the age of Beethoven, the matter is more complicated, although Hoffmann (writing as Kreisler) is unprepared to say why. As key proximity became more complicated in the age of Beethoven, so too did the calculation of distances between the triadic harmonies of which keys are composed.1 This is because for theorists of the time, triadic relations tracked those of their eponymous keys. Jean-Philippe Rameau proclaimed in 1722 that “every note that supports a perfect chord should be considered a tonic” (Dahlhaus 1990 [1967], 28). Adolph Bernhard Marx (1841–47) “understands every consonant triad to be ‘borrowed’ from the key in which it is the tonic, and he claims that these triads stand in the same relation to one another as the keys they represent” (Engebretsen 2002, 70). Hugo Riemann (1897, 86) wrote pithily that “key relation is nothing other than the relation of their two tonic triads.” And Heinrich Schenker collapsed the distinction altogether, regarding “keys” as triads under prolongation (Schachter 1987). 1. Throughout this book, “triad,” in its unmodified form, refers restrictively to the twenty-four consonant triads. Particular triads are identified in the standard manner, by root and mode. Roots for minor triads are sometimes in lower case. In the figures and tables, major and minor are often abbreviated as plus and minus signs, respectively; thus, C+ stands for C major, and c– for c minor.

1

2

 Audacious Euphony Figure 1.1. Schubert, Sonata in Bᅈ major, D. 960, 1st mvt., mm. 217–56.

To introduce the porous boundary between chord and key, as well as the complicated proximity judgments at both levels of relation, consider figure 1.1, which opens the first-movement recapitulation of Schubert’s Bᅈ major Piano Sonata (D. 960, 1828). The first theme, which ends at m. 233, and its counterstatement, which begins at m. 254, are separated by three spans that respectively prolong Gᅈ major, fᅊ minor, and A major. Each span is locally diatonic. That is, within each local span’s own context, the role of each note and chord is specified, consistently and without ambiguity, by any of the protocols (e.g., Roman numerals, Schenkerian graphs, Riemannian functions) that represent the detailed inner workings of diatonic tonality. From a global perspective, too, the passage is normatively tonal: it begins and ends in the tonic. Everything that we have observed so far points to the conclusion that the music of figure 1.1 adheres to the syntactic principles of classical diatonic tonality. It would be premature, however, to conclude that the passage is determinately tonal in all of its aspects. We have yet to consider how the local keys (or, from a different perspective, the triads prolonged by the local spans) relate to one another and how they work together to express the global tonic of Bᅈ major. If we are unable to do so, we just have a bunch of tubs floating around on their own bottoms. Each vessel is internally coherent and occupies a space bounded by the Bᅈ shores. But in relation to one another, their relation is random, for all we know. And that is no way to express a tonality. We can’t just goᇳBᅈ major, Cough, Wheeze, Honk, Bᅈ majorᇴand pretend that we have made coherent music in Bᅈ major (Straus 1987). If a tonal theory is to meet its claims of explanatory adequacy, it needs to be able to specify the role, with respect to tonic, of the harmonies that

CHAPTER 1

Mapping the Triadic Universe

separate the bounding tonics. The fact that each harmony may also be a tonic of its own local context in no way relieves it of that responsibility, any more than my role in my own home relieves me of my role in the community. Let us locate each triad, in turn, with respect to Bᅈ major: [1] The opening triad, Bᅈ major, is rooted on the tonic axiomatically. [2] Gᅈ major is rooted on the flatted 6th scale degree of Bᅈ major, as notated. We know this because we identify the cantus Bᅈ of m. 235 with the cantus at the previous cadence, which we identified as tonic in [1] above, and we hear the bass as a consonant third beneath it. [3] “fᅊ minor” is a notational surrogate for gᅈ minor, rooted on the flatted 6th degree of Bᅈ major. We know this because we identify the bass pitch at m. 239 with the bass pitch in the measures just preceding, which we described as Gᅈ in [2] above. [4] “A major” is a notational surrogate for Bᅈᅈ major, rooted on the flatted 1st degree. We know this because we identify the bass pitch at m. 241 as octave-related to the cantus pitch in the preceding measure, which was a consonant third above the bass pitch that we identified as Gᅈ in [3] above, and because the bass proceeds from “Fᅊ” to “A” through three steps of a scale in m. 240. [5] “Bᅈ major” is a surrogate for Cᅈᅈ major, rooted on the doubly flatted 2nd degree. We know this because we identify the cantus “D” at m. 255 as a notational surrogate for Eᅈᅈ, the proper tonic of the previous dominant seventh, which was rooted on Bᅈᅈ, as identified in [4] above; and we hear the bass of m. 255 as a consonant major third below that Eᅈᅈ. But our syllogisms have led us astray! No amount of logical sophistry can dislodge us from the conviction that the final chord of the progression represents the tonic degree, not the doubly flatted second. There must be an error to repair. Perhaps we can find it by retroengineering the analysis: [5] The final chord is Bᅈ major, axiomatically; and so [4] its immediate predecessor is rooted on its leading tone, qua dominant of its mediant, just as notated; and so [3] the chord just prior, a minor third below the leading tone, represents the fifth degree, Fᅊ, again just as notated; and so [2] despite its notation, “Gᅈ major” represents Fᅊ major; and so [1] the cadence at m. 233 is on Aᅊ major. The problem remains unrepaired. We have backed ourselves into another corner, on the opposite side of the room. Fortunately, there are still some options to explore. Taking them in reverse order: [4] Perhaps the roots of the last two chords are separated by a chromatic rather than a diatonic semitone? [3] Perhaps the root of the third chord lies an augmented second beneath that of its successor? (But then the bass of the latter represents a different scale degree than the soprano of the former; moreover, the stepwise approach in the bass signifies that the consecutive roots are not related by step.) [2] Perhaps the Fᅊ and the Gᅈ really do represent different degrees, just as Schubert notated it? (This is implausible prima facie: if you sing the bass while you perform the passage, nothing will persuade

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 Audacious Euphony you to fracture the sustained pitch Gᅈ2 into two noncommunicating entities.) [1] Perhaps the root of the second chord represents Fᅊ despite its notation? (But then, the soprano Bᅈ4 represents a different scale degree than it did a moment ago.) Are any of these perceptions plausible? What might motivate one to make a case for any one of them, other than the desire to preserve the initial premise, which is that the passage is determinately tonal in all of its aspects? If none of these questions can be answered in the affirmative, then we can only conclude that the passage is not entirely determined by the logic of classical diatonic tonality. This conclusion is independent of how we choose to regard the status of the entities that are progressing, that is, whether they are placed under the auspices of harmonic or modulatory or linear-prolongational syntax. We can corroborate this conclusion by means of a simple measure of diatonic coherence: how many pairs of triads (not limiting to those presented in immediate succession) share membership in at least one diatonic collection?2 In a typical diatonic passage in major mode, a randomly selected group of four distinct triads share membership in a single diatonic collection; ipso facto, so do the six pairs that they form. In a passage with a single applied dominant chord, four or five of the six pairs coexist in some diatonic collection (although not a single unified one). In figure 1.1, only a single pair, A major/fᅊ minor, shares membership in some diatonic collection. This is, of course, very low on the spectrum of possibilities: of the 33,649 (= 23 choose 5) sextets of distinct triads that include Bᅈ major, only eight contain fewer (= zero) common diatonic memberships.3 From a diatonic standpoint, this progression is among the most entropic. To the extent that Schubert is employing the logic of diatonicism here, it is in a negative sense: it is present in its absence. We might then conclude that Schubert is being disjunctive, irrational, or arbitrary. To do so would place us in good company (Clark 2011a). Some critics of Schubert’s time “described harmonic indirection as a kind of aimless wandering towards extraneous goals, which injected a quality of randomness and lack of plan into the music” (Shamgar 1989, 530–31). The more progressive of them placed high aesthetic value on tonal ruptures and disjunctions, connecting their inexplicability to the mysterious and sublime qualities so valued in the Romantic imagination (e.g., Hoffmann 1989 [1813–14], 131–36). A related view became the inheritance of historical musicology in its poststructural phase, for which ruptures constitute traces of ideological, sociocultural, and psychological formations that are otherwise occluded by the passage of historical time.4 2. I regard this measure as more suggestive than definitive. A more useful metric might additionally track the number of pairs that share membership in some harmonic minor scale, although that introduces other problems; for example, is {Aᅈ, B, Eᅈ} a diatonic chord in c minor? 3. These eight include Bᅈ major together with one chord from each of the following pairs: {G major, e minor}; {E major, cᅊ minor}; {Dᅈ major, bᅈ minor}. 4. Examples include Kramer 1986, 233; Subotnik 1987; Abbate 1991; and McClary 1994, 223. Carolyn Abbate’s analysis of a scene from Die Walküre in her Unsung Voices (1991) is a particularly fertile garden for such tropic varietals; it refers to “harmonic irrationality and incongruence” (189), “unstructured harmonic improvisation” (192), “cannot be heard as a logical harmonic progression” (194), “disjunctive gap” (194), “no progress, no development . . . a repeated succession of discontinuous chords” (199).

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But there is another available interpretation: perhaps diatonic distance is not the best metric for the situation at hand. In a treatise published in 1796, Francesco Galeazzi estimated the relationship between C major and d minor triads as “very irregular and poor” (irregolarissimo e pessimo), even though each has diatonic status when the other is tonic (Galeazzi 1796, 264).5 Yet he values the relation between C major and E major as “regular and good” (regolare e buono), although there is no diatonic collection that includes them both. Why does he judge the diatonic progression less normative than the chromatic one? Because the latter has a common tone that the former lacks. He considers the relation between C major and e minor to be “even better” (migliore) than the previous two. Because the relation is diatonic? No—because the chords share two common tones (see Galeazzi 1796, 263–64). Diatonic collections play no role in the model of triadic proximity that underlies Galeazzi’s judgments. Nor, for that matter, do harmonic roots have any role to play (although they are implicitly present to the extent that they furnish labels). Galeazzi’s judgments are based on properties and relations that are independent of those identified by classical theory, such as acoustic consonance and diatonic inclusion. If we are willing to suffer the anachronism and the scientistic odor, we can express Galeazzi’s implicit conception in the language of modern mathematical set theory: triadic proximity correlates with cardinality of pitch-class intersection. Galeazzi’s association of harmonic proximity with common-tone preservation recurs consistently in music theory treatises throughout the nineteenth century. K. C. F. Krause asserted in 1827 that “the most closely related consonant triads are those that have two notes in common with the given triad, then follow those with one note in common with the given chord” (qtd. in Engebretsen 2002, 69 n. 1). Nora Engebretsen notes that Krause “presents his view without any fanfare, in a manner suggesting that this is the standard approach” (69). Ten years later, Marx offered the opinion that co-occurrence of triads in a diatonic collection counted as a “superficial unity” but that “a more distinct tie exists in the connecting notes which each of our chords has in common with its neighbors” (Marx 1841–47, qtd. in Engebretsen 2002, 69). Two influential treatises from midcentury, by Moritz Hauptmann and Hermann Helmholtz, were equally dedicated to the commontone basis of harmonic proximity, even though their epistemological bases (respectively, in idealist philosophy and scientific empiricism) were diametrically opposed (Hauptmann 1888 [1853], 45; Helmholtz 1885 [1877], 292). In the final decades of the century, the common-tone view of the harmonic Verwandschaft began to lose ground to a renewed interest in acoustic generation and consonant root relations (Engebretsen 2008). But the two methods are nonetheless frequently seated side by side. For example, Tchaikovsky’s 1872 Guide to the Practical Study of Harmony distinguishes between “inner” relations, based on root distance on the circle of fifths, and “external” connections based on common tones (Tchaikovsky 1976 [1872], 11–13; compare Riemann 1897, 85ff.).

5. For an annotated translation of Galeazzi’s treatise, see Burton and Harwood (forthcoming).

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 Audacious Euphony Table 1.1(a). Number of common tones between each triadic pair in figure 1.1

Bᅈ major Gᅈ major fᅊ minor

Gᅈ major

fᅊ minor

A major

1

0 2

0 1 2

Applying this criterion to the Schubert passage gives a rather different picture of its coherence. For example, the fᅊ minor triad, which is the most difficult of the four to integrate into a Bᅈ major tonal framework, shares two common tones, the maximum possible, with both its predecessor, Gᅈ major, and its successor, A major. Table 1.1(a) counts the number of common tones between each pair of triads in the progression, disregarding their order of presentation. The total of six common tones positions the progression toward the upper end of the range for quartets of triads, which extends from zero to nine, and well above the average, which is just below four.6 In counting common-tone connections in a particular passage, we have implicitly assumed that voice leading is idealized.7 In most compositions, tones freely transfer registers, and multioctave tone doublings liberally appear and disappear. We say that two triads have a common tone even when, in a particular setting, those tones appear one or more octaves apart. Identity of tones, then, is independent of the particular register in which those tones appear. When we speak of common tones, then, we are adopting a conception of tone that is allied with pitch class rather than pitch. There is nothing special about idealized voice leading; music theory teachers and scholars assume it every day of their working lives. It is so familiar, indeed, that it takes a special effort to acknowledge it. Idealized voice leading is also assumed by a related method for calculating the distance between triads, which attends not only to the number of moving voices but also to the absolute distance of motion.8 We define a unit of voice-leading work as the motion of one voice by one semitone. The initial Schubert progression, Bᅈ major → Gᅈ major, requires two units of work: the voices containing F and D both move by semitone (up and down, respectively), while the voice containing Bᅈ stays put. The progression Gᅈ major → fᅊ minor involves only a single unit of work, Bᅈ to A (assuming no surcharge for enharmonic exchanges). And the progression from fᅊ minor to A major involves two units of work, Fᅊ → E. Table 1.1(b) calculates the work for the six pairs of triads in the Schubert progression. Summing the values in the table, the progression as a whole involves fourteen units of work.

6. An example of the maximum is {C major, a minor, e minor, c minor}. An example of the minimum is {G major, eᅈ minor, Dᅈ major, a minor}; see figure 5.25(b) in chapter 5. 7. Proctor 1978 attributes the term to Godfrey Winham. 8. Not all theorists agree that voice leading should be idealized when voice-leading measurements are assessed. Tymoczko (2005, 2009c, 2011b) presents an argument in favor of measuring voice leading along paths in circular pitch class space, distinguishing between upward and downward motions. My own views are flexible on this matter, in accordance with the position taken in Rings 2011, 51–54.

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Table 1.1(b). Number of semitonal displacements (“voice-leading work”) between each triadic pair in figure 1.1

Bᅈ major Gᅈ major fᅊ minor

Gᅈ major

fᅊ minor

A major

2

3 1

3 3 2

This is on the lower end: for a set of four triads, the minimal total work is ten, the maximum twenty-eight.9 The assumptions underlying this method of calculating triadic proximity are even more venerable than Galeazzi’s. Already in the early fourteenth century, Marchettus of Padua was articulating a “closest approach” preference for semitonal voice leading (Schubert 2002, 506).10 Gioseffo Zarlino wrote that “when from the third we wish to arrive at the unison . . . the third should always be minor—this being closer” (1968 [1558], 79). Early-nineteenth-century theorists cultivated melodic fluency as an alternative to fundamental bass progression (Engebretsen 2002), and at the turn of the twentieth century, Georg Capellen proposed that triadic connections are based on a combination of common tones and semitonal motions (Bernstein 1986, 142). Maximum common tone retention and minimal voice-leading work are so closely related to each other that one might be tempted to think of them as equivalent. They are conflated, for example, in the “law of least motion,” which decrees that voices should move by minimal intervals, holding common tones in the same voice.11 This principle has the status of a robust prescription if one takes the classical view that voice leading is secondary to harmony. If one first selects a pair of chords and then considers how most economically to join them, maximum common-tone preservation entails minimal voice-leading work. If, however, these metrics serve as a primary determinant for selecting harmonies, rather than as a criterion invoked only after the harmonies have been selected, then they do not yield identical judgments about triadic proximity (Cook 2005, Tymoczko 2009b). In some cases, voice-leading work makes a finer set of distinctions than does common-tone retention, since the former spreads its results across six distinct values whereas the latter returns only three (see figure 4.7 in chapter 4). For example, in figure 1.1, fᅊ minor shares two common tones with both the preceding Gᅈ major and the subsequent A major. Yet the moving voice travels by semitone in the first case, whole tone in the second, a distinction that disappears when one is merely counting common tones. In other cases the two metrics make 9. An example of the minimum is {Bᅈ major, bᅈ minor, Gᅈ major, fᅊ minor} (see chapter 2). The maximum is fulfilled by {G major, eᅈ minor, Dᅈ major, a minor}, which is the minimum of note 6. 10. Dahlhaus (1990 [1967], 335 n. 7) speculates on even earlier origin. 11. The law of least motion was erroneously deposited in Arnold Schoenberg’s theoretical account, but, like so many of the other treasures banked there (the chart of regions from Weber, the emancipation of dissonance from Weitzmann), it was siphoned from the accounts of predecessors. The “law” was a staple of thoroughbass theory and debuted no later than Charles Masson’s 1694 treatise.

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 Audacious Euphony contradictory claims about relative proximity. Is C major closer to g minor or to gᅊ minor? The former preserves a common tone where the latter has none, and it is thus closer on one criterion. But the latter involves three units of voice-leading work ({C, B} = 1; {E, Dᅊ} = 1; {G, Gᅊ} = 1), whereas the former involves four ({C, Bᅈ = 2; {E, D} = 2; {G, G} = 0), producing a proximity judgment that contradicts the previous case. In summary, we have reviewed three distinct metrics, each of which formalizes a different set of intuitions about triadic proximity. The classical metric evaluates triadic proximity in terms of mutual membership in diatonic collections and interprets figure 1.1 as very disjunct. The same passage is interpreted by Galeazzi’s common-tone metric to be fairly conjunct, and by the voice-leading metric to be very conjunct. The diatonic collection, which plays a central role in the first metric, has no privilege whatsoever in the remaining two. In the voice-leading metric, which most successfully captures the intuition that the triads in Schubert’s progression inhabit a similar neighborhood, it is the chromatic collection that explicitly comes forward as the template against which distance is assessed.

Triads in Chromatic Space To view consonant triads against the background of chromatic space is to decline to interpret them in terms of the number of diatonic degrees that separate their root from some tonic. This choice cuts against the multiple denominations of classical tonal theory and their pedagogical offshoots, which all teach that chromatic harmonies are primarily to be understood as transformations of some underlying diatonic one. The idea that the diatonic collection conceptually precedes and regulates the interpretation of the chromatic one, already implicit in the names of notes, their position on the staff, and the system of key signatures, became canonized with respect to classical tonality in the early nineteenth century, at roughly the same historical moment that musical education became institutionalized in conservatories, analysis evolved into its own discipline, a theory of tonality began to congeal under that name, and Roman numerals became the default first-level descriptors for triads (Wason 1985, 53). That idea has proven hardy indeed, as can be confirmed with reference to any English-language harmony textbook. The diatonic view of chromaticism has prevailed for good reason. Triads and diatonic scales together constitute the foundational organizing materials of classical tonality. Although the diverse traditions of classical theory assign consonant triads and diatonic scales different values in relation to each other (Dahlhaus 1990 [1967]), they all agree that it is through their coordination that major and minor keys are established. For an acculturated listener, a major or minor triad, sounded in isolation and without prior context, signals the tonic status of its root by default. In a process first described by Gottfried Weber (1846 [1817–21]), a listener spontaneously imagines an isolated triad housed within a diatonic collection, signifying a tonic that bears its name.

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Yet there is a difference between a default interpretation and a necessary one. The tonic status of a triad requires confirmation, weakly through the remaining tones of its associated diatonic collection; more strongly by arranging those tones into a local cadence; more strongly yet by repeating that cadence, perhaps with supplementary rhetorical packaging, at the end of the movement or composition. Such a confirmation is by no means inevitable. Sometimes an initial triad comes to be understood as representing a nontonic degree (Kirnberger 1982 [1771–76], 45; Aldwell and Schachter 1989, 135). And sometimes it initiates a succession of triads, any of which could claim tonic status under appropriate conditions (Kirnberger 1982, 114; Schenker 1954 [1906], 254). The more tones put in play, the less likely their alignment with respect to a diatonic collection that organizes their position and role with respect to some tonic. Until such a collection emerges and is cadentially crowned, the triadic progressions are diatonically indeterminate. Once essential enharmonic relations arise, indeterminacy evolves into contradiction. By essential enharmonicism, I am excluding those notated enharmonic conversions that arise as artifacts of notational pragmatics, as might happen if a composer presumes that a performer is more comfortable reading a signature of four sharps than eight flats. What distinguishes essential enharmonicism is that the composer has no choice but to convert between sharps and flats in order to retain global diatonic logic (Schenker 1954 [1906], 333–34). In such cases, the exact point where the composer notates the conversion is a pragmatic matter without significance; in a phenomenological sense, such a conversion happens everywhere and nowhere, which is tantamount to saying that it is distributed evenly across all of the possible moments when it could occur (Proctor 1978, 177; Telesco 1998; Harrison 2002a). When enharmonically paired pitch classes are juxtaposed directly, the ear cannot avoid identifying them as the product of a single tone. As we saw with reference to figure 1.1, the splitting of a tone’s scale-degree constituency can have a ripple effect, destabilizing the diatonic collection and the tonic that is claimed to anchor it. The recognition that triadic music is not always fully determined by the principles of diatonic tonality is by no means a new one. As already noted, earlynineteenth-century critics intuited that contemporary music defied familiar logic (although they disagreed as to whether this was a good thing). The impulse to systematize these intuitions was first acted on in the writings of François-Joseph Fétis, a Belgian music critic working in Paris, initially in a series of articles from 1832, eventually in his 1844 harmony treatise. Fétis divided tonality into four stylistic species, each representing a stage of historical development, and each defined by its own syntactic principles and affective properties. The most progressive of the four species, omnitonality, is distinguished by a proliferation of enharmonic relations that indicate a “multiplicity, or even the universality of the keys” (Fétis 2008 [1844], 190), a process that Fétis predicted would lead to “the total destruction of the scale in certain cases, and the beginnings of an acoustic division of the musical scale into twelve equal semitones” (Berry 2004, 257, quoting Fétis 1832). For Fétis, the objects of omnitonality are chromatically intensified dissonant harmonies, rather than the consonant triads that concern us in the present study. It is rather in

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 Audacious Euphony the historically earliest of Fétis’s four species, unitonality, that one finds tonally indeterminate chromatic successions of triads, as in some music of Marenzio and Gesualdo from the turn of the seventeenth century. Such successions fail to define a key because their constituent triads do not communicate with each other: “No attraction is evident, because every perfect chord is a harmony of repose” (Fétis 2008 [1844], 163). If every chord is a potential tonic, then no chord can fulfill that potential by functioning as one. Each tub is on its own bottom, bobbing around the sea independently of the others. Although each of Fétis’s four tonal species arises at a particular historical moment, the later species do not supplant the earlier ones. According to his historical model, they are cumulative; once available, the best composers know how to combine them in a single work of art (Berry 2004, 255). It is evident, then, that Fétis conceives of classical tonality (“transitonality”) as a category whose constituent elements are not integral “pieces”—compositions or complete movements— but rather musical moments. On Fétis’s view, the faculty of (transi)tonal listening is capable of spontaneous suspension and reengagement without notice or fuss, like a carpenter exchanging a screwdriver for a hammer. He recognizes a similar dynamic in a purely diatonic environment, as when a sequence arises midphrase. At the moment that the sequence is recognized, the “law of tonality” is placed in abeyance, as our cognition is submitted to a “law of uniformity.” “The mind, absorbed in the contemplation of the progressive series, momentarily loses the feeling of tonality, and regains it only at the final cadence, where the normal order is reestablished” (Fétis 2008 [1844], 27). The idea of simultaneously accessible tonal schemata was developed specifically with relation to pan-triadic progressions seventy-five years later by Ernst Kurth, who was raised in an era of rampant, fully ramified omnitonal chromaticism that Fétis could only divine. Kurth’s Romantische Harmonik und ihre Krise in Wagners “Tristan,” initially published in 1920, proposed that many chromatic progressions, particularly those that involved root relations by third, introduced rifts, wedges, and fissures into the fabric of tonality. The identity and function of these chords are found in their internal structure and in their local connections to their immediate antecedents and successors. When concatenated with sufficient intensity and persistence, such absolute progressions bring about “the total disruption of the original embracing tonal unity” (Kurth 1991, 120). Kurth discovered an agent of tonal disruption in chromatic sequences, which, like Fétis’s diatonic ones, are governed by the logic of repetition. Such progressions are “extratonal” in the sense that their relation to the tonal pillars that bound them on either side is not tonally determined. After Kurth’s 1920 treatise, it became a commonplace of German musicology that neither the appearance of consonant triads nor their framing by occasional cadential progressions was sufficient to justify the judgment that their syntax was governed by the principles of classical tonality; other factors were necessary in addition (Adorno 1964; Kunze 1970; Dahlhaus 1980a [1974]; Motte 1976). Among the adherents of this view were Theodor Adorno and Carl Dahlhaus, both of whom eventually acquired a significant readership in North America, one result of which was that Kurth’s views immigrated into the arena of American musicology

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 11

(e.g., Newcomb 1981; Meyer 1989, 302; Agawu 1989, 27; Abbate 1991, 192). Similar views can also be found, perhaps surprisingly, in the writings of American Schenkerians, who otherwise are committed to the vision of the masterwork as organically unified by Ursatz emanations that function uniformly at all compositional levels. These included Adele Katz, for whom the Magic Sleep music from Die Walküre “lack[s] . . . tonal implication” (1945, 213); William J. Mitchell, who noted that a triadic circle of fifths “can be arrested at any point or it can just as easily go on in perpetuity” (1962, 9); and Felix Salzer and Carl Schachter, who wrote that “we register the equal intervallic progressions without referring them to a supposed diatonic original. This temporary lack of a diatonic frame of reference creates, as it were, a suspension of tonal gravity” (1969, 215). The dissemination of this view has not, however, dislodged a broadly shared commitment to the notion that the chromatic triadic progressions characteristic of the nineteenth century are determined by their position with respect to some tonal center. This commitment is evident not just in the profusion of inflected Roman numerals or function symbols that dominate the textbook teaching of nineteenth-century harmony on both sides of the Atlantic. It also dominates various branches of research, whether based in Roman-numeral/fundamental bass traditions (Lerdahl 2001), Schenkerian/linear approaches (Darcy 1993; Brown 2005), Riemannian functions (Harrison 1994), or Lewinian transformations (Kopp 2002). Although these denominations interpret triadic harmony according to quite different sets of assumptions, and express those interpretations using distinct modes of representation, they all share a base in the late-eighteenth-century classical harmonium, from which they reach out to lay claim to the chromatic triadic music of the nineteenth century. I can think of three reasons that analysts of nineteenth-century triadic music have continued to dance to a modified eighteenth-century beat, despite the many stumbles induced by the terrain. First is the promiscuity of triadic descriptive categories, combined with the illusion that to describe is to explain. Roman numerals are flexible enough to furnish a first-level description of almost any triad in almost any key (Dahlhaus 1980a [1974], 68; Hyer 1989, 229–30). Many Roman numeral practices are satisfied, moreover, with finding a local tonic for each harmony, without any demand that local tonics be reconciled to each other and to a global tonic. Riemannian functions likewise are catchall categories, such that “a student of Riemann’s system can analyze virtually any chord into any one of the three functions should the occasion demand” (Harrison 1994, 284). Schenkerian approaches allow chromatic triads to degrade into coordinated linear spans (Benjamin 1976; Smith 1986), which serve as carpets under which to sweep enharmonic paradoxes. A second reason for the continued resistance to alternative views of triadic chromaticism is that it requires an embrace of some form of double syntax. Most nineteenth-century passages that can be seen to juxtapose triads according to nonclassical principles exist in close proximity to other behaviors that are normal under classical diatonic tonality. The Schubert excerpt with which we began (figure 1.1) is not atypical: while the local spans are classically tonal, the middleground tonics adhere to a different logic. To analyze such a composition requires

12

 Audacious Euphony not only that we navigate, sometimes in rapid alternation, between two or more syntaxes, as Fétis imagined listeners moving between his four kinds of tonality. It also requires the capacity to simultaneously process two distinct sets of syntactic principles that unscroll at different speeds. Can music of high aesthetic value really partake of two systemic modes of organization, shuttle between them quasi instantaneously, and even overlay them? Are our musical brains wired in such a way that we have the capacity to shift between these syntaxes as if at the click of a switch, or to multitask between them? If the responses of several prominent music scholars are representative, it seems that there is a strong motivation to reject any such idea on a priori grounds, which is to say, independently of the details of the proposal under which a double syntax program might be carried out (Dahlhaus 1990 [1967], 111; Smith 1986, 109; Lerdahl 2001, 85). Chapter 9 considers and responds to this line of objection; readers who share this prima facie skepticism may wish to teleport there before proceeding with the linear exposition. The final reason pertains to the absence of a fully ramified alternative. We are inclined to come out from under familiar technologies only when we are prepared to substitute for them an alternative that is plausible, coherent, and productive. To acknowledge that chromatic progressions of triads might be based in some syntactic principles other than those of diatonic tonality is to clear a space, but that is not the same thing as building a house. One needs to be able to say something about what that syntax is, not just what it is not. To say that “Beethoven’s third period seemed destined to shake the absolutist regime of the main tonality for the first time” (Draeseke 1987 [1861], 315) or that some Wagnerian progressions “stand . . . in certain opposition to tonal unity” (Kurth 1923, 249; my translation) constitutes a necessary first step. To allow for the existence of a “countersyntax” that stands in “dialectical” relation to classical tonality (Kramer 1986) constitutes a significant second one. But to posit the terms of that countersyntax, it is necessary to do more than substitute a Latin adjective for its Greek equivalent (as occurs whenever a writer feels that they have scratched an explanatory itch when they have attributed a chromatic harmony to a “coloristic” effect),12 or refer to linear processes without being prepared to specify anything beyond pointing to lots of semitones (e.g., Dahlhaus 1980a [1974], Agawu 1989). In addition, one wants to know what principles underlie the syntax, how it operates, how its analyses are represented. Are its claims consistent, well formed, and free of internal contradiction? How is the syntax motivated by the lexicon; that is, what properties do triads possess that qualify them for the job that (the syntax claims) nineteenth-century composers put them up to perform? What sorts of problems does this syntax help solve? Does it generate analyses that reflect some aspect, however obliquely and abstractly, of a musician’s or listener’s experience? Does it lead us to notice interesting things about a score, or about its relationship to other scores, that would have otherwise escaped attention? Does it help us think differently about historical problems of genre, style, evolution, 12. See Tischler 1964, 233; Rosen 1980, 245; Kramer 1986, 203; Todd 1988, 94; Meyer 1989, 299; Ratner 1992, 113; Somer 1995, 219; and Taruskin 2005, 69. “This chromaticism has a coloristic effect” has roughly the propositional status and explanatory value of “this box is so heavy because it weighs a lot.” David Kopp (1995, 345) makes a similar point.

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and the like, or about the relationship between music and the historical conditions of the individual, society, or culture that produced it? This book responds to these questions by adapting a conceptual framework erected between 1955 and 1980 by the field of atonal pitch-class theory, whose great achievement was to develop a systematic approach for exploring the properties, potentials, and interrelations of chords (“sets”) within the chromatic universe. Atonal theorists of that era were not much interested in consonant triads, as their analytic interests were focused on a repertory whose principal phonological constraint was, on some accounts, their absence (Boulez 1971 [1963]; Forte 1972; but see Straus 1990). Reciprocally, music scholars of that era who were open to the cultivation of alternative approaches to nineteenth-century triadic music were alienated from American atonal theory because of geography, the serendipities of disciplinary configuration, or the low priority that cold-war theorists placed on disciplinary outreach. In exploring the properties and potentials of consonant triads using a method adapted from atonal theory, I hope to defuse the suspicion that “applying analytical techniques derived from contemporary music” to lateRomantic repertory is “menial and easily accomplished” (Dahlhaus 1989 [1980], 381–82), or an act of desperation (Harrison 1994, 2).

Remarks on Syntax and Maps Syntax is a central term in the study of natural language, and not all of the meanings that it accumulates there can be transferred into music. Syntax is that branch of linguistics that studies how words and their constituent particles combine to form coherent sentences, independently (in principle) of how those sentences represent concepts and states, or motivate actions, in the world. I use syntax in this book in three different ways. First, syntax contrasts with phonology and lexicon, which respectively treat the internal structure of atomistic units and their firstlevel bundling into units of signification or reference. Because music, under ordinary conditions, lacks the referential dimension of language, phonology and lexicon come close to fusing: a lexicon is a list of available sounds (chord, scales, sets), and phonology provides a principled account of what properties make those sounds available for use. Second, syntax is the study of the ordering of events as they sequentially unfold in time: how triads “progress” in a moment-by-moment sense, and perhaps also in a middleground sense where such interpretations are appropriate. Third, and most important for present purposes, I use syntax in the same sense that the Greeks used harmonia, the “means of codifying the relationship between those notes that constituted the framework of the tonal system” (Dahlhaus 1980b, 175). This broader domain is roughly equivalent to what Roger Sessions (1950, 33) designated as “the relationships between tones, and . . . the organization which the ear deduces . . . from those relationships” and what David Lewin (1969, 61) characterized as the way that “sound [is] conceptually structured, categorically prior to any one specific piece.” It is at this most abstract level that we can also refer to Fétis’s four types of tonality as evincing distinct syntaxes,

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 Audacious Euphony or think of his laws of tonality and uniformity as manifesting distinct syntactic principles. They are distinct in the sense that they generate different orderings of the harmonies, doublings of chords, and expectations about dissonance treatment. But they are also distinct in the sense that they evoke different modes of musical cognition. In this third sense, musical syntax has long benefited from geometric and graphical representation. Geometric models of pitch space have been in use for some 1,500 years (Popovic 1992; Westergaard 1996). During the eighteenth and nineteenth centuries, they were frequently applied to relations among keys, and later among chords. American music theory of the postwar era generally favored algebraic models, for their compactness (a significant consideration in print media) and their strong generalizing capacity. A surge in geometric models began in the 1980s, and has intensified in the last decade, in part due to the increasing accessibility of graphics software and the economies afforded by electronic space. Geometric models can thrive as effective modes of exploration and communication only if the phenomenon being modeled meets certain structural and psychological conditions. Structural problems arise if there are more conceptual dimensions than are available in the physical medium. In linear space, one dimension is great, two’s fine, three’s the limit, and four blows the mind. In cyclic space, even a second dimension introduces falsifications and distortions, like the Bering Strait problem familiar from Eurocentric world maps. Moreover, Euclid’s logic often collides with that of the psyche. The symmetry of spatial distance may lack psychological salience for someone walking uphill, and the triangle inequality prohibition is violated whenever two miles walked in intense conversation feels shorter than one mile alone on a sore ankle. Fortunately for my project, in the case of triadic distance measurements these problems are kept to a minimum. The cyclic structure of chromatic space will create some Bering Strait problems, but we will find that these are easily negotiable with the help of a supplementary “legend” that guides interpretation of the map. The supreme advantage afforded by musical maps is their capacity to reflect judgments about the psychological proximity of musical objects or states (Popovic 1992). Elementally, such judgments come in binary form (“these two notes sound close, those two sound distant”) that lead naturally to comparison (‘“these notes sound closer together than those”). When structural and psychological conditions align, a map has the capacity to draw together a family of pairwise distance assessments. Such a map then acquires the capacity to capture syntactic judgments, which might take the form of conjunct versus disjunct, normal versus unusual, or acceptable versus unacceptable. Moreover, it earns the potential to aid in the exploration of semantic predicates, such as “betweenness” (there is a gap that one expects will be filled), “orientation” (we can chart our distance from and direction with respect to some “home”), or “momentum” (there is a pattern whose continuation we anticipate). Like a geographical map, a good representation of musical space does not merely sit there, as a static structure. It acts as a stage upon which imaginative performances are mounted, thus serving the same function as a geographical map for a child with a toy car, or for a medieval monk tracking a crusade (Connolly 1999).

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A musical map can illuminate compositional decisions as selections from a finite menu. It can move composers to ask, “How many ways are there to connect these two chords?” “What chord stands halfway between these two chords?” “How can I form a cycle, with some desired number of elements, that begins and ends at the same element?” or “If I’m at A, what state B should ensue, if I want to mimic the gesture that carried Q to R?” It invites analysts to ask, “Is this a step or a leap?” “Is this connection the most direct one?” “Are these two paths parallel?” or “Is this path an embellishment of that one?” Questions of this type are concerned with corpuswide ideals, norms, and limitations, as well as with “motivic” elements that shape and individuate a particular composition in dialogue with those norms and ideals. These are the musical equivalents of langue and parole, language and utterance, the topics most central to the syntactic study of natural language. Steven Rings (2006) suspects me of using coherence, in related contexts, as a fourth-order stalking horse for the universalization of nineteenth-century German aesthetic ideology, by way of the intermediate terms “unity,” “autonomy of the artwork,” and so forth, and he and others may suspect that syntax just heaps another shell or two of derivatives on top. I am not committed to either italicized term and would by happy for readers simply to substitute some other, or perhaps some neutral, term (e.g., X-factor) in their place. I do think that there is some profit in acknowledging that, among communities, some musical phenomena “go down easy” and some “go down hard.” Asyntactic and incoherent signify the neighborhood of aesthetic responses that might alternatively take the form “doesn’t make sense,” “sounds weird,” “sounds erratic,” “doesn’t fit,” “sounds random,” “sounds awful,” “I don’t get it,” “I wasn’t expecting that,” “that’s not normal,” “not immediately intelligible.” Someone with a historical sense might posit those same responses through comparison, similar to the way that Forkel responded to that odd passage from C. P. E. Bach’s f minor Piano Sonata (Kramer 2008, 11), or that some Viennese critics heard Schubert’s modulations (Shamgar 1989), or that the first European heard South Asian music or the first Indian heard European music. Although I don’t care what term is used to make the distinction, I am pretty sure that there is a distinction to be made. I would even go so far as to suggest that that distinction is universal, on the hypothesis that, for individuals or communities or cultures, there are things that make sense and things that don’t, things that go down easy and things that go down hard, things that are familiar and things that are foreign, and so forth. An anthropologist might make this distinction with the term emic (fits the world-view of the folks who live there), an intellectual historian with episteme (fits the worldview of the folks who lived then), and a linguist with syntactic (has potential meaning within that linguistic community). What I mean by “syntactic,” then, is the musical equivalent of all of those.

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C HA P T E R

Two

Hexatonic Cycles

Chapter 1 proposed that triads could be related by voice leading, independently of roots, diatonic collections, and other central premises of classical theory. This chapter pursues that proposal, considering two triads to be closely related if they share two common tones and their remaining tones are separated by semitone. Motion between them thus involves a single unit of work. Positioning each triad beside its closest relations produces a preliminary map of the triadic universe. The map serves some analytical purposes, which are explored in this chapter. Because it is not fully connected, it will be supplemented with other relations developed in chapters 4 and 5. The simplicity of the model is a pedagogical advantage, as it presents a circumscribed environment in which to develop some central concepts, terms, and modes of representation that are used throughout the book. The model highlights the central role of what is traditionally called the chromatic major-third relation, although that relation is theorized here without reference to harmonic roots. It draws attention to the contrary-motion property that is inherent in and exclusive to triadic pairs in that relation. That property, I argue, underlies the association of chromatic major-third relations with supernatural phenomena and altered states of consciousness in the early nineteenth century. Finally, the model is sufficient to provide preliminary support for the central theoretical claim of this study: that the capacity for minimal voice leading between chords of a single type is a special property of consonant triads, resulting from their status as minimal perturbations of perfectly even augmented triads. The consequences of that claim are the focus of the final sections of this chapter.

A Minimal-Work Model of the Triadic Universe We will say that two triads are in the minimal-work relation if motion between them involves the displacement of a single voice by semitone. According to this

17

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 Audacious Euphony definition, each triad is in the minimal-work relation to two triads of the opposite mode. Each major triad is in the minimal-work relation with its parallel minor and with the minor triad whose root lies four semitones above it. For example, C major is in the specified relation with c minor and with e minor. Reciprocally, each minor triad is in the specified relation with its parallel major and with the major triad whose root lies four semitones below it. For example, c minor can reach both C major and Aᅈ major by a single semitonal displacement. Each consonant triad is thus situated in a chain of alternating major and minor triads. C major is flanked by c minor and e minor, producing the three-element chain {c minor, C major, e minor}. That trio is nested within a five-element chain, {Aᅈ major, {c minor, C major, e minor}, E major}. That quintet is, in turn, nested within a seven-element chain, {aᅈ minor, {Aᅈ major, {c minor, C major, e minor}, E major}, gᅊ minor}. The external elements of that septet are identified under enharmonic equivalence, and so the outer elements are “glued together,” converting the seven-element chain into a six-element cycle. Figure 2.1 presents that cycle in its upper quadrant, along with three other cycles, each germinated in an analogous way from other triadic seeds. Together, the four cycles partition the twenty-four consonant triads.1 Connections among the cycles will emerge as a central project of chapters 5 and 6. This chapter focuses on the internal workings of the individual cycles. Because they relate to each other by transposition, internal features proper to one are proper to all. Two triads are situated in the same cycle if they have the same roots, or if their roots are four semitones apart. Classical theory presents various languages for identifying the collections of co-cyclic triadic roots. We might say that they form

Figure 2.1. A graph of the twenty-four triads under single semitonal displacement, producing the four hexatonic cycles.

1. Derek Waller called attention to these cycles in a 1978 article in the Mathematical Gazette, which anticipated some aspects of my presentation in Cohn 1996.

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an augmented triad. If we are concerned with projecting neutrality about enharmonic choices, we might say that the roots form a symmetric division of the octave into three equal parts. In circumnavigating the cycle in either direction, the progression consists of paired triads transposed by major third, or (again more neutrally) by an interval of four semitones. We might then be inclined to refer to it as a T4 or T8 cycle, where “T” refers to transposition and the subscripted number indicates the size of the transposition, measured in semitones. We might be inclined to view this progression in terms of a chromatic sequence whose generating interval is the major third or minor sixth, or whose generating transpositions are T4 and T8. All of these characterizations are more or less interchangeable, and I shall treat them as equivalent. It has become standard to view the cyclic progressions portrayed in figure 2.1 in terms of “third relations” and thereby to throw them into the same pot with progressions that transpose by a series of minor thirds. This conflation, which is encouraged both by the labeling conventions for intervals and by the appearance of both types of cycles at roughly the same historical moment, obscures a fundamental distinction: cycles generated by major thirds exhibit balanced voice leading, alternating between up and down, whereas those generated by minor thirds lead their voices in a uniform direction. For example, in transposing by major third from C major to E major, under idealized voice leading one voice moves up (G → Gᅊ) and one moves down (C → B), while the third voice, E, holds its place. In transposing by minor third from C major to Eᅈ major, again under idealized voice leading, both moving voices move down (C → Bᅈ and E → Eᅈ) while G holds its place. The latter case is the norm. Under least-motion voice leading, recursive transposition by any interval other than major third generates uniformly directed voice leading. It is transposition by four or eight semitones that is special: these alone generate transposition cycles whose voice leading is balanced.2 A significant entailment of balanced voice leading is contrary motion: for every voice that rises, another falls by the same magnitude. Contrary motion among triads of the same mode thus inheres exclusively to triads that are transpositionally related by major third (Cohn 1996). This entailment, which is at the heart of the approach developed in this book, has gone unrecognized by the many scholars who have studied third relations during the last thirty years.3 I will argue, in this chapter and beyond, that the contrary motion of major-third relations underlies both their central role in the syntax of pan-triadic progressions and their association with the semiotics of the supernatural. The reason that the major third has this special status is that it divides the octave into as many equal parts as the triad has notes. One job of this chapter is to show why.

2. Balanced voice leading corresponds to balls atop an arch. Some released balls will fall east, some west, and the behavior of one ball does not predict the behavior of its successor. Uniform voice leading corresponds to the situation of balls on a tilted plane. If one ball falls east, they all fall east; one can predict the behavior of all from the behavior of any one. 3. These include Proctor 1978; Krebs 1980; Taruskin 1985, 1996, 2005; Bailey 1985; Cinnamon 1986; Aldwell and Schachter 1989; Agawu 1989; Kraus 1990; Rosen 1995; Somer 1995; Kopp 2002; and Bribitzer-Stull 2006. Tymoczko 2011b is a recent and notable exception.

20

 Audacious Euphony Figure 2.2. Voice leading through a hexatonic cycle. Arrowheads indicate the direction of semitonal displacement between adjacent triads.

The Hexatonic Trance Figure 2.2 progresses clockwise about one of the cycles from figure 2.1, in a format that emphasizes the behavior of the individual voices. The repeat sign suggests a continuously cyclic process. (The features that we identify in this progression will also be present in its retrograde, as voice-leading features are independent of cyclic direction.) Arrowheads indicate the location and direction of semitonal displacements. Each individual voice holds its pitch for three “beats” and then displaces to a new pitch that is likewise sustained for three beats before returning to the original pitch. The entire cycle engages only six pitches, two for each of its three voices. Accordingly, the progression is referred to as a hexatonic cycle. Ordered linearly within an octave, the six tones form a hexatonic scale, alternating semitone and minor third.4 The triple periodicity within each individual voice interlocks with the duple periodicity resulting from their combination. This duple periodicity has both a directly perceptible aspect and a more abstract structural one. The direct aspect emerges in the periodic alternation of upward and downward motion, from one “beat” to the next. The more abstract aspect emerges from the relations between the three voices, each of which executes its displacements two beats later and four semitones lower than its predecessor (assuming octave equivalence). The voices thus combine to form a hocket canon, a structure familiar to music historians from the caccia of late-medieval polyphony (Bukofzer 1940). The interlocking of duple and triple periodicities, induced respectively within and between the individual voices, forms a 3:2 phasing (also referred to variously as polyrhythm, grouping dissonance, hemiola, and cross pattern). (Readers may find it useful at this point to kinetically engrave and aurally entrain these periodicities by playing through the repeating hexatonic cycle with one hand at a keyboard, or by arpeggiating the successive triads on a single-line instrument.) Figure 2.3 juxtaposes nonadjacent triads of the same hexatonic cycle. Next-adjacent triads are of the same mode; either major (a) or minor (b). 4. Jazz players know the same pattern as the augmented scale, because the collection combines two adjacent augmented triads. It has also been referred to variously as the Ode to Napoleon collection, Miracle hexachord, Liszt model, source-set E, 1:3 collection, and set class 6–20.

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Figure 2.3. Voice leading between nonadjacent triads within a hexatonic cycle.

Diametric triads, three positions apart, are of opposite mode (c). The number of semitonal displacements in each progression is equivalent to the cyclic distance of its constituents. In figure 2.3, (a) and (b) show the double displacements that occur in major-to-major and minor-to-minor juxtapositions, and (c) shows the triple displacement between nonadjacent triads of opposite mode. All three progressions involve balanced voice leading, in the sense that at least one voice moves in each direction. These progressions can be seen to compress the serial alternation of up and down in figure 2.2, transforming the directional alternation into a simultaneous contrary motion. In music of the nineteenth century, and throughout the history of music for film, the progressions illustrated in figure 2.3 frequently depict sublime, supernatural, or exotic phenomena. In “Nacht und Traüme,” Schubert slips directly from B major to G major to depict a nocturnal fixation on evanescent dreams (Schachter 1983b). In the Ring, Wagner uses the progression at figure 2.3(b) to portray the Tarnhelm, which makes its wearer invisible. Rimsky-Korsakov used the same progression, transposed through the minor triads of a hexatonic cycle, to depict the reclusive Antar adrift on the sands of the Sahara desert (see figure 3.6 in chapter 3). Wagner subsequently used the same cycle in Parsifal to depict the sorcerer Klingsor, another recluse in the Arabian wasteland. The diametric progression at figure 2.3(c) depicts a range of uncanny phenomena. The many examples cited in Cohn 2004 include Kundry’s de-souling (Parsifal), the dead Siegfried shaking his fist (Götterdämmerung), Scarpia’s murder (Tosca), Aase’s arrival at St. Peter’s Gates (Peer Gynt), Strauss’s Salomé singing to the severed head of Jochanaan, and Schoenberg’s self-portrait in death (String Trio, Op. 45). In a chapter titled “Music and Trance,” Richard Taruskin (2005) recognizes the semantic charge of chromatic progressions by major third but implies that their link to altered and uncanny states is conventional, relying on what Swiss linguist Ferdinand de Saussure referred to as the arbitrary bond between signifier and signified.5 Although convention is certainly an element of this semiotic system, there is something else at work: the affective power of the progressions in figure 2.3 derives from a paradoxical characteristic that is inherent to them, when they are 5. Taruskin’s association of chromatic progressions by major third with trance, the uncanny, etc., seems to be limited to the major mode, in which all of his examples occur. In minor, the submediant is “naturally” flat, and so ᅈVI and “flat submediant” are not meaningful designations.

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 Audacious Euphony heard against the expectations of classical diatonic tonality. By default, the classically conditioned ear interprets a relation between two tones as diatonic rather than chromatic (Agmon 1986, 185; Temperley 2001, 128). The empty ear filling with music interprets a two-semitone interval as a major second rather than a diminished third, a three-semitone interval as a minor third rather than an augmented second, and so forth. (A canny composer cultivates various resources for reversing these defaults, just as an engineer can raise objects in physical space; but they nonetheless hold in an “everything else equal” context.) This same principle dictates that a single semitone be heard, again ceteris paribus, as a change of diatonic degree, rather than as a chromatic inflection of an invariant degree. Applied to a perfect fifth whose voices move outward by semitone, as in the outer voices of the three progressions at figure 2.3, the default principle dictates that the interval between them increase by two diatonic degrees, producing a diminished seventh. Yet when heard in their own insular context, the default principle dictates that two tones separated by an interval of nine semitones express a major sixth.6 The principle thus produces contradictory information: on the first application, the nine-semitone interval is dissonant; on the second, it is consonant. In the attempt to reconcile these interpretations, the ear is caught in a liminal space, where the binary distinction between consonance and dissonance is eroded. Such breakdowns in the division between otherwise securely demarcated categories, prototypically the boundary between reality and illusion, or life and death, are a mark of the psychological uncanny.7 The capacity of chromatic root progressions by major third to signify altered and unstable mental states is thus based not on mere convention but on a homology between the signifying progression and the signified affect. In Peircean terms, the progressions in figure 2.3 are icons rather than symbols of altered or destabilized mental states. Consider, for example, the Tarnhelm progression (figure 2.4), whose first two chords match those of figure 2.3(b). Warren Darcy (1993, 170) writes of the “aural sense of . . . eerie power” and of “radical disjunction. . . . the motif seems almost to have fallen in from another world.” For Carolyn Abbate (2006, qtd. in Parly 2009, 166), that world is subterranean, “as if excavated from primeval time.” Nila Parly (2009, 167) attributes this effect to the open B/Fᅊ fifth, an archaic symbol that evokes an “air of something ‘uncanny,’” by virtue of incongruity, when embedded into Wagner’s progressive harmonic language. 6. Riemann (1890, 38) wrote that, in these cases, one “will more or less always feel the inclination . . . and indeed with good reason,” to hear one of the triads as a dissonance, spelled as a consonance only for convenience. Kopp 1995, 141ff., contains a translation and exegesis. Prout (1903, 256) calls them “false triads.” See also Louis and Thuille 1982 [1913], 409–10, and Hull 1915, 42. Louis and Thuille note that Liszt frequently spells mediant triads as dissonances. Lendvai (1988, I: 60) observes a specific instance in Verdi’s Otello. 7. Although the analogy may initially seem far-fetched, it taps into a significant history of theorizing about consonance and dissonance. Heinrich Schenker considered consonances to uniquely possess life-generative capacities, in the form of the capacity for prolongation; in one passage, he refers to the tonic triad as the maternal womb. He also considered dissonant harmonies to be false and illusory. Thus, to confound consonance with dissonance was literally to confound musical reality with musical illusion. Similar passages can be found in writings of his contemporaries, Ernst Kurth and Alfred Lorenz. See Cohn 2004. I return to this theme in chapter 7, in connection with Parsifal.

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Figure 2.4. The Tarnhelm progression from Wagner’s Ring.

In his initial notation of the progression, Wagner struggled with enharmonic decisions (Darcy 1993, 168–69). He initially sketched the first chord as aᅈ minor. He led cantus Eᅈ to Fᅈ, as the diatonic logic suggests, and then retracted it in favor of E, as part of an e minor triad that supplies the G leading tone. The third presentation of that chord, though, is rewritten as an fᅈ minor triad. The approach pursued here suggests that the indeterminate enharmonics and the uncanny and disjunctive semantics stem from the same source. The melodic Eᅈ ought to move diatonically to Fᅈ; the bass Aᅈ ought to move diatonically to (tenor) G; and the interval between bass and melody ought to be a diatonic and consonant major sixth. One of these imperatives must be discarded so that the other two can survive. But the effect is overdetermined. Hexatonic progressions are also able to depict the world of noumena by virtue of their tonal multistability. Each Tarnhelm triad contains the other’s leading tone and hence signifies the other as tonic, like the Escher hands drawing each other’s sleeves. A hexatonic cycle abjures the cadential resources of classical tonality, such as fifth-related roots, dissonances, and diatonic coordination. A composer can suggest one of its constituent triads as a tonic “factitiously by virtue of its recurrence” (Taruskin 1996, I: 259) but can secure it only by recruiting external syntactic routines. The six triads are equally likely recipients of rhetorical or cadential benefaction; the progression itself is neutral with respect to its potential tonics. In this way, hexatonic progressions resemble the so-called “standard” bell pattern (variously called bembé and gagokoe) of West Africa or the Caribbean, whose temporal “tonic” (i.e., downbeat) can occur at several points along the cycle. The cycle itself is neutral with respect to these bestowals and is in this sense multistable (Pressing 1983). The analogy that I have just ventured takes us into a speculative terrain that merits light treading, because it is counterintuitive from a historical standpoint. Yet a small cluster of features line up suggestively in support of pursuing it at least around one or two bends in the road, before beating a hasty retreat back to the main path. We observed, in connection with figure 2.2, that a hexatonic cycle embeds a hocket canon that projects a 3:2 phasing. This particular combination of attributes is likewise identified with African musical traditions associated with spirit possession and trance. Rouget 1985 suggests that such concerns are not entirely remote from Europe. In the post-Enlightenment world, these concerns bore scientific trappings. Dr. Franz Mesmer’s theories of animal magnetism and universal fluids, as well as his therapeutic practice of hypnosis, held considerable interest among artists and intellectuals in Biedermeier Vienna; Schubert’s sustained encounter with them is documented in Feurzeig 1997. One can imagine, then, that a composer of the early nineteenth century might be interested in exploring ways to depict states or sensations associated with hypnosis, as an

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 Audacious Euphony uncanny state liminally perched at the juncture between reality and illusion, or life and death, and might find in hexatonic progressions a number of homologous attributes.

Contrary Motion and Balance How is it that contrary motion is a special case, when it is universally acknowledged as a prescriptive norm in voice-leading theory and pedagogy? The resolution to this peculiarity points to an overlooked circumstance that turns out to be fundamental to our understanding of pan-triadic harmony. When three-voice triads (i.e., lacking tone doublings) are connected by idealized voice leading, contrary motion is indeed a special case: it arises only between triads whose roots are related by major third and hence share membership in a hexatonic cycle. Any given triad (e.g., C major) can be juxtaposed with eighteen other triads that lie outside of its cycle. The progression to one of these triads (in the given case, A minor) involves motion in a single voice, to which the categories of paired motion do not apply. The progression to each of the remaining seventeen triads involves similar or parallel motion.8 The bidirectional voice leading in a hexatonic cycle, whether successive, as in the case of incremental motion through the cycle, or simultaneous, as in the case of motion between nonadjacent triads, invites an interpretation in terms of internal points of symmetry. In figure 2.2, each triad has a point of balance whose location is roughly constant throughout the progression, despite local fluctuations. This even balance results from the quasi-uniform size of the intervals of which the triad is composed, which fall within a narrow compass of from three to five semitones. The particular point of balance depends, of course, on the registral ordering of the voices, which is arbitrarily selected in figure 2.2. But the general feature of balance would remain under any other ordering of the pitches, provided that the ambitus remained within an octave. (Readers comfortable with representations of pitch-class space will recognize that the center of symmetry is more properly shown as a vector cutting through a cycle, but the basic point holds in pitch space as well.) As the triads move incrementally through the hexatonic cycle, the semitonal alternation of up and down causes the center of balance to toggle back and forth between two pitches separated by one-third of a semitone. Circumnavigation of a hexatonic cycle thus produces neither upward nor downward motion through the pitch spectrum. Like a walker or a waterfall, the incessant 8. This observation has a significant entailment for one aspect of the history of harmonic tonality. Latemedieval contrapuntal theory privileged small melodic intervals within voices and contrary motion between them. Under triadic tonality, these two principles come into conflict. Any three-voice progression between diatonic triads must violate one or the other of them, if it involves two moving voices. This conflict may have been a stimulant to collective creativity, a problem whose solution incidentally introduced a number of varietals that ultimately enriched the compositional soil of harmonic tonality: among them, the addition of a fourth voice that either doubled a triadic pitch class or introduced a dissonance that resolved in contrary motion to the other voices.

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local fluctuations are underlain by a global stasis. This stasis, however, coalesces around the prolongation not of a tonic, in any standard construal of the term, but rather of a zone of voice-leading space (or voice-leading zone). What is meant by the italicized term is taken up in chapter 5, after several more elements that support it are put in place.

Hexatonic Progressions, Tonnetz Representations, and Triadic Transformations Having observed hexatonic cycles in the laboratory, we are now in a position to study them in the field. Figures 2.5 through 2.8 present four of the earliest hexatonic cycles, from four decades of Viennese music. It is characteristic of these chronologically early examples that tonicizing dissonances buffer the stations of the cycle, which are presented at a thinly veiled layer of the middleground. Like the Schubert excerpt examined in chapter 1, each of these excerpts instantiates what Ramon Satyendra (1992) calls layered tonality: classically tonal at both the most global and the most local level but chromatic at the level of the local modulations (also see McCreless 1996, 102). The diatonic indeterminacy resides in the succession of tonics, as symptomized by the essential enharmonic transformation that a hexatonic cycle requires. Figure 2.5 presents a passage from the final movement of Mozart’s Symphony in Eᅈ major, K. 543 (1788). The excerpt initiates the developmental core of a monothematic sonata-form movement. Beginning in Aᅈ major with a partial quotation of the principal theme, the music proceeds, via sostenuto wind chords, through a gᅊ minor triad to E major at m. 115. A series of violin/cello stretti, based on the

Figure 2.5. Mozart, Symphony in Eᅈ major, K. 543, finale, mm. 109–26.

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 Audacious Euphony Figure 2.6. Haydn, Symphony no. 98, finale, mm. 148–98.

one-measure head, unfolds in the subsequent measures, connecting through e minor and C major to c minor at m. 123, and then intensifies through a diatonic sequence that ultimately prolongs c minor. Figure 2.6 presents a passage from a Haydn symphony composed four years later; the passage likewise constitutes the developmental core of a final movement. The development begins with a series of four-measure thematic segments that tonicize Aᅈ major, its submediant f minor, and its subdominant Dᅈ major, the latter tonicization precipitating a sudden caesura. For four measures, the first violin paws at the rising-third incipit of the principal theme, finally grabbing hold of a Sturm und Drang arpeggiation of Cᅊ minor, the enharmonic parallel of the immediately previous tonic. Gᅊ → A leads to a thematic statement in A major, from which vantage point we can reinterpret f minor → Dᅈ major, initially heard in Aᅈ major diatonic space, as the first step in a journey through a hexatonic cycle that is completed by an a minor arpeggiation (m. 178) that ultimately leads to F major at m. 190. The Mozart and Haydn passages are harmonically open and texturally diverse, as befits a developmental core. In the following two passages, similar progressions are harmonically closed, unfold at a leisurely pace, and involve block transpositions of lyrical thematic material. Figure 2.7 presents the sixteen-measure period in the center of the ternary-form Adagio of Beethoven’s “Spring” Sonata, Op. 24 (1802).9 After a cadence in Bᅈ major, the antecedent phrase arises from a modal mutation to bᅈ minor and leads after eight measures to a cadence in its submediant, Gᅈ major. The consequent phrase likewise begins with a modal mutation to 9. Schenker models this passage in Free Composition, figure 100.6(b), commenting only that “here we have a descending register transfer by means of three major thirds” (1979 [1935], 82). Proctor 1978, 175–76, provides an astute commentary.

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Figure 2.7. Beethoven, Sonata for Violin and Piano, Op. 24, 2nd mvt., mm. 38–54.

fᅊ minor and leads to a tonicization of its submediant D major, but this tonal motion is transacted in half the time. This acceleration leaves a balance of four measures, which are filled by yet a third transposition of the same progression, leading from d minor to Bᅈ major for the start of the final section of the movement. Figure 2.8 presents the first of three phrases from the conclusion of the immense initial movement of Schubert’s Piano Trio in Eᅈ major (1828). The unmelodied accompaniment (or “vamp”) mutates a major chord to its parallel minor, leading to an eight-measure period that modulates down a major third to the latter’s submediant. The phrase given in figure 2.8 is followed twice by its exact transposition, beginning first in B major and then in G major, the latter modulating back to the closing Eᅈ major tonic. Figure 2.9 models these four hexatonic passages in a graphic format that brings out some features obscured by the cyclic graphs of figure 2.1. Points represent individual tones, rather than the triads formed by their combination, and edges connect tones that form consonant intervals. This graphic format is a fragment of the Tonnetz (“tonal network” in German), a planar figure that coordinates axes representing the consonant interval classes. In the version that will be used

Figure 2.8. Schubert, Piano Trio in Eᅈ major, Op. 100, 1st mvt., mm. 584–595.

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Figure 2.9. Tonnetz models of figures 2.5–2.8.

throughout this book, perfect fifths rise from left to right along the horizontal axis, minor thirds rise from northwest to southeast, and major thirds from southwest to northeast.10 What appears as clockwise motion in figure 2.1 is converted in figure 2.9 to downward motion through a strip whose external boundaries form augmented triads. Each strip’s interior is tiled into triangles, representing consonant triads. Major triads extend upward, and minor triads subtend downward, from their shared perfect-fifth edge. Internal edges represent shared dyads. The motion of each passage through the strip is modeled by an arrow. Numbers inside the triangles reference bar numbers of the score. These planar graphs suffer from a Bering Strait flaw. Each tone and edge at the top of a strip reappears at its bottom, masking identities and distorting distances. Were these identities honored by “gluing together,” the strip would convert to a cylinder. Our failure to so honor them is a concession to the dimensional limits of the printed page. These limitations are worth tolerating because of the many advantages that the triangularly tiled planar representations afford. A few of those advantages will become apparent in this chapter; many more will accumulate as we penetrate more deeply into the heart of the model.

10. The Tonnetz was first presented by Leonhard Euler in 1739. It was revived by German harmonic theorists in the second half of the nineteenth century and was independently reinvented numerous times, and for numerous reasons, by late-twentieth-century music theorists, psychologists, and dilettantes interloping from other academic fields. The angled format was introduced by Ottakar Hostinský in 1879 and adopted by Hugo Riemann in his later publications. For more on the history of the Tonnetz, see Vogel 1993 [1975], Mooney 1996, Gollin 2006, and Cohn 2011. One innovation adopted here, following an idea of Daniel Harrison (2002b), is the double labeling of nodes that correspond to enharmonic exchanges in the score being modeled (e.g., the Aᅈ/Gᅊ along the left border of the Mozart Tonnetz in figure 2.9). This makes it easier to identify triads and track their progressions.

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Figure 2.10. A progression on a Tonnetz strip, as incremental moves through a hexatonic cycle.

Figure 2.10 extracts one of these graphs for closer study. It breaks the single arrow into a series of local ones that indicate individual pitch-class displacements. Downward arrows, indicating chromatic-semitone descents, alternate with diagonal ones, indicating diatonic-semitone ascents. To distinguish these two species of local progression, we draw on a tradition initiated by Arthur von Oettingen (1866), who identified several exchange operations (Wechsel) that connect opposite-mode triads. These include a leading-tone exchange (Leittonwechsel) that connects opposite-mode triads that share a minor-third dyad, as exemplified by the diagonal arrows in figure 2.10. Oettingen’s exchange operations were developed by Riemann (1880), and some of them were revived and formalized a century later by David Lewin (1982, 1987).11 Following Brian Hyer’s 1989 adaptation from Lewin, I will use the letter P (parallel major/minor) to indicate the motion between triads that share two common tones and a common root, and L (Leittonwechsel) to indicate triads that share two common tones and whose roots are a major third apart. Both operations are involutions, which is to say that they “undo themselves”: two consecutive applications produce an identity. Table 2.1 summarizes the information about these two transformations. Figure 2.11 presents Tonnetz models of four additional passages, which traverse hexatonic cycles in a manner more characteristic of the nineteenth century. Figure 2.11(a) models the passage presented at figure 2.12, from the first movement of Brahms’s Concerto for Violin and Cello, Op. 102 (1887). The passage resembles those studied above in connection with Figures 2.5 through 2.9, but the diatonic buffers have been removed. Each station along the path is approached 11. The history of transformations, on their own and as they relate to the Tonnetz, is documented in Klumpenhouwer 1994, Mooney 1996, Gollin 2000, Kopp 2002, and Engebretsen 2002. The exchange operations were theorized as contextual inversions in Clough 1998.

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 Audacious Euphony Table 2.1. Incremental hexatonic transformations Name

Symbol Root motion

Parallel Leittonwechsel

P L

Common dyad

Semitonal Planar angle species on Tonnetz

No change Perfect fifth Chromatic 0° Major third Minor third Diatonic 120°

directly from its predecessor, and as a result the passage sounds only the six tones of the cycle’s associated hexatonic scale. Figure 2.11(b) represents the second sentence of Liszt’s Consolation no. 3 (1840s), for which a score is available at Web score 8.3 . The passage presents upward motion through the strip, equivalent to counterclockwise cyclic motion, for the first time.12 A second new feature is the omission of one member of the cycle, cᅊ minor, resulting in the direct juxtaposition of two triads of the same mode (see figure 2.3(a)). Although the name for this transformation, LP, suggests a compound transformation, with intermediate cᅊ minor deleted, I prefer to think of the transformation (and its inverse, PL) as a unitary Gestalt whose name happens to have two syllables.13

Figure 2.11. Four hexatonic passages from the nineteenth century. 12. For analyses, see Lewin 1967, Gollin 2000, Kopp 2002, Santa 2003, and chapter 8 of this book. 13. Kopp (2002, 159) suggests that LP is a compound operation by virtue of its compound name. Gollin 2000 (6–12) argues that this need not be so; any compound transformation, such as “retrograde inversion,” can be furnished with a unary name (he suggests “George”). Although there is heuristic value in the compound name, there is no necessary significance to it. The same is true in natural language, where words like breakfast and handicap autonomously accrue and shed meanings apart from their compound origins.

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Figure 2.12. Brahms, Concerto for Violin and Cello, Op. 102, 1st mvt., mm. 270–79.

The final two passages modeled in figure 2.11 are presented at figure 2.13: the opening of the Sanctus movement from Schubert’s Eᅈ major Mass (1828) and a chromatic Grail distortion from Wagner’s Parsifal (1882).14 Both passages begin with Eᅈ major → b minor, diametrically across a hexatonic cycle. Two intermediate triads on the hexatonic cycle are omitted, and all three voices move simultaneously by semitone, as in figure 2.3(c). I will describe this relation as a hexatonic pole, and the corresponding transformation with the label H (after Cook 1994). Like all mode-switching operations, H is an involution that “undoes itself ” when performed twice consecutively. Both passages also continue by indirectly connecting b minor → eᅈ minor, but they do so in different ways. Wagner interpolates G major, which shares two

Figure 2.13. Two similar hexatonic progressions.

14. For a consideration of the Schubert passage, see Salzer and Schachter 1969 (215–18). Concerning the Grail distortion, see Lewin 1984 and 1992, Clampitt 1998, Lerdahl 2001, Cohn 2006, Lerdahl and Krumhansl 2007, Brower 2008, and Rings 2011.

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 Audacious Euphony common tones with its predecessor and none with its successor. As a result, the closing G major → eᅈ minor transposes the opening H progression, Eᅈ major → b minor. Schubert interpolates g minor, which shares one tone with both of its neighbors, dividing the four units of voice-leading work evenly between the two progressions. The progression into and out of g minor transposes a minor triad downward by major third, meriting the label LP. That label is familiar from the Liszt Consolation (figure 2.11(b)), where it identified the transposition of a major triad upward by a major third. Here we arrive at a circumstance that has come to be viewed by some as the Achilles heel of triadic transformational theory. In furnishing g minor → eᅈ minor, which terminates the Schubert progression, with the same label as A major → Dᅈ major, which terminates the Liszt one, we are implicitly claiming that the two progressions are in some sense equivalent. Three sorts of objections have been raised against this claim. First, the claim is seen to violate the intuition that progressions, in order to be considered equivalent, ought to move roots not only by the same magnitude but also in the same direction (Kopp 2002; Tymoczko 2009b). Second, the claim is seen to result solely from the way that the transformational logic plays out. The LP operations, and their PL inverses, become the boorish in-laws that need to be tolerated if we want to marry into the otherwise attractive and well-behaved kinship system. Third, the claim seems to entail a commitment to the harmonic dualism under whose banner it first arose in the writing of Oettingen and Riemann: the notion that major and minor triads are generated by equal but opposite metaphysical or physical forces. Such a commitment would be embarrassing on a priori grounds, since the metaphysics is obsolete and the physics apocryphal (Harrison 1994; Rehding 2003). All three objections are neutralized by an appeal to voice leading, a dimension of experience that is, in principle, independent of root motion. What C major → E major shares with c minor → aᅈ minor is the behavior of each individual voice: the G voice moves up by semitone, the C voice moves down by semitone, and the remaining voice holds constant. More generally, any LP operation sends its perfect-fifth dyad to a major sixth (or perfect fourth to minor third), and any PL operation does the reverse. Consider figure 2.14, which juxtaposes two passages from Richard Strauss’s “Frühling” of 1949. A score is available at Web score 4.19 . After an orchestral alternation between c minor and aᅈ minor, the singer enters on the wave of that same progression at m. 5. The second stanza opens on a C major triad that moves directly to E major in 64 position. The two passages feature major-third transpositions but in opposite directions. What they share is their voice leading: both progressions lead C down by semitone and G up by semitone while keeping their third voice invariant. It is this voice-leading equivalence that the label LP captures.15 Although there is no appeal to harmonic dualism here, there is, nonetheless, a more benign melodic dualism lurking about in the wings, whose implications are treated below.

15. See also Lewin 1992, figure 3; for a similar case involving PL, see Cohn 1999.

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Figure 2.14. Two LP transformations in Strauss’s “Frühling” (Four Last Songs).

Near Evenness, Minimal Voice Leading, and the Central Role of Augmented Triads Our work with the Tonnetz strips (figures 2.9 and 2.11) has focused on the relations between the triangles at their interior. In a move that will have significant repercussions for the remainder of this book, we now attend to the augmented triads that bound the strips. The role that they play, in the hexatonic passages that those strips represent, is not immediately apparent. In none of the passages does an augmented triad appear as a surface harmony. One of the augmented triads, on the left boundary of the strip, does have a certain salience in the bass register, where its tones slowly unfold one by one. We might be inclined, by habit, to say that this augmented triad is arpeggiated, or even prolonged. But we may not be prepared to take on board some implications of such a claim. Are the tones of the augmented triad fused into a corps sonore, which generates the passage by distributing its components across time and sprouting triads from each one? The invocation of arpeggiation and prolongation in this context has a metaphorical component whose heuristic value has been a site of heated controversy among Schenkerian theorists since the 1950s.16 The notion that augmented triads are “prolonged” is particularly problematic in the current context, since it suggests that the smooth voice leading of these passages is a by-product. If we consider semitonal voice leading as primary, then such a conception places the tail where the head should be. It is the semitonal voice leading that stands at the core. The disjunct bass is merely running about town making calls where its services are needed (see Schachter 1983b, 75). Moreover, even if we are comfortable with assigning fundamental status to the augmented triad at the left boundary of each strip, what is the role of its partner at its right boundary? In proposing an equal role for the two augmented triads that bound a Tonnetz strip, we arrive at a major theoretical claim of this book: when triadic progressions are pursuing the logic of smooth voice leading rather than that of acoustic 16. For a nuanced discussion of this issue, see Proctor 1978, 157ff.

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 Audacious Euphony consonance, augmented triads play a central role in their syntax, even when occluded from the music’s surface and hence not directly accessible to perception. By virtue of their status as perfectly even trisections of the octave, augmented triads are the invisible axes about which pan-triadic progressions spin. Consonant triads acquire their distinctive voice-leading features in chromatic space by virtue of their status as minimal perturbations of the perfectly even augmented triads. The crucial property that consonant triads bear is one that Dmitri Tymoczko has named near evenness.17 Major triads are nearly even because they can be formed from an augmented triad by a single semitonal displacement downward; minor triads, conversely, are formed by a semitonal displacement upward. A consonant triad is thus like a wheel that is dented just enough to affect a wobble but not so much that it is knocked off its rotation. It is this property that makes possible the minimal voice leading that hexatonic progressions feature. Why does a nearly even chord bear the capacity to connect to an equivalently structured chord by minimal voice leading? Why does it uniquely bear that capacity? More generally, what is the connection between degree of evenness (proper to a chord’s internal structure) and voice-leading magnitude (proper to its external relations)? We can begin to explore these questions by inspecting figure 2.15(a), a circle that intersects the vertices of an equilateral triangle. The vertices are labeled with the tones of an augmented triad. Each vertex is flanked by two black circles, labeled in two ways: with the tone that semitonally displaces a member of the augmented triad, and with the major or minor triad that results when that tone is combined with those of the remaining fixed vertices.18 Figure 2.15(b) shows how a sample triad, C major, is generated from the augmented triad by the displacement Gᅊ → G. Our interest is in exploring the behavior of this triad as it is subjected to those shape-preserving transformations that involve minimal change. When the triangle is transformed in such a way that two vertices are preserved, how far is the third vertex displaced? The shape-preserving transformations are of two types: rotation, equivalent to pitch-class transposition; and reflection, equivalent to pitch-class inversion. As the triangle in figure 2.15(b) is scalene, it cannot be rotated in such a way that two of its vertices are invariant. (This corresponds to the impossibility of transposing a consonant triad such that two tones are preserved.) Reflection, though, is another matter: to preserve two vertices, exchange their position by reflecting them about an axis halfway between them. As there are three pairs of vertices, there are three axes around which such a reflection is possible. They are positioned as broken lines in the three components of figure 2.16. In each component, the C major triangle is presented in half-tones, and its inversion about the axis is presented in full tones. A double-headed arrow indicates the exchange of the two common tones. A single-headed arrow indicates 17. The connection between smooth voice leading and near evenness was initially suggested to me by John Clough in 1993, mentioned in Cohn 1996, 39n40, and elaborated in Cohn 1997, although using different terms. Tymoczko 2011b (14, 61, 85–93) positions the connection between near evenness and voice leading within a very broad framework with many concrete applications. 18. Figure 2.15(a) resembles some graphs that appear in the teaching materials of jazz guitarist Pat Martino (Capuzzo 2006). See also Siciliano 2005a.

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Figure 2.15. Derivation of consonant from augmented triads via single semitonal displacement.

the path from the remaining tone of the C major triad to the tone that replaces it as a result of inversion about that axis. That path’s magnitude is double the distance that separates the tone from the axis. This is, of course, how inversion/reflection works: the farther the distance of some object from the axis, the farther that object is projected by inversion about it. •

In figure 2.16(a), the axis (represented as a broken line intersecting the center of the circle) is positioned halfway between E and G, so that those two tones reflect into each other (as indicated by the double-headed arrow). Under reflection about that same axis, the remaining tone, C, is replaced by B (as indicated by the single-headed arrow). Interpreted in pitch-class space, this reflection models the application of the L transformation on C major to produce e minor.

Figure 2.16. Double common tone transformation of C major, depicted as inversion about an axis that holds two of its tones invariant.

36

 Audacious Euphony •

•

In figure 2.16(b), the axis mediates C and G, which invert into each other. The remaining tone, E, reflects into Eᅈ. This reflection models P as it transforms C major to c minor. In figure 2.16(c), the axis mediates C and E, which invert into each other. The remaining tone, G, reflects into A. As this transformation involves two units of work—small, but not minimal—it has not arisen in this chapter; we shall study it in chapter 4.

For comparison, figure 2.17 shows the behavior under reflection of two other chord types, which are, respectively, more and less even than the C major triad of figure 2.16. Both chords share the CE dyad with C major; what defines their degree of evenness is the position of the third voice. In figure 2.17(a) that third voice is subject to a “null” perturbation, remaining on Gᅊ and creating a perfectly even augmented triad. At (b), the third voice is perturbed by three units, moving to B and creating a relatively uneven [015]-type trichord. In the perfectly even case, the axis that reflects C and E into each other also reflects Gᅊ into itself. Consequently, the perfectly even augmented triad cannot be distinguished from its reflection, and the transformation is a phantom. In the uneven case, reflection about the same axis slings F a tritone away, to B. This exercise demonstrates that, in order to create a small but recognizable displacement of a single voice under inversion, the trichord must be as even as possible, but not perfectly even. What is true of nearly even trichords in a chromatic space of twelve tones is equally true of nearly even chords of any cardinality in a chromatic space of any size.19 I mention here briefly several other cases of near evenness that are familiar to music theorists, or that we will have occasion to refer to in chapter 7.

Figure 2.17. Double common tone transformations of two dissonant chords.

19. The general principles are established, in different ways, in Douthett 2008 and in Tymoczko 2011b.

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• •

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Dominant and half-diminished seventh chords, as minimal displacements of perfectly symmetric fully diminished seventh chords, voice lead to each other more smoothly than any other tetrachord type. Mystic chords, as minimal displacements of whole tone collections, have a similar capacity among hexachords. Diatonic collections, which are maximally but not perfectly even (Clough and Douthett 1991), communicate with each other via minimal displacement, a circumstance that underlies the system of key signatures at the heart of musical notation and theory of modulation. The same is true of their pentatonic complements (Kopp 1997). Diatonic triads, which are maximally but not perfectly even in seven-tone diatonic space, communicate with each other by single stepwise motion, a circumstance that underlies the system of diatonic third-relations central to some theories of harmonic function (e.g., Agmon 1991, 1995).

For the more circumscribed case that concerns us in this chapter, the central point is this: Via single semitonal displacement, each major triad communicates with two minor triads, and each minor triad with two major triads, precisely because major and minor triads are nearly even. One can thus draw a direct connection between near evenness and the unique ability of triads to participate in hexatonic chains and cycles. The role of near evenness, with respect to the participation of triads in hexatonic progressions, is thus analogous to that of consonance, with respect to their participation in diatonic ones. This suggests that the role of the augmented triad in the former case is analogous to that of the harmonic series in the latter: it is the concealed noumenon that gives rise to the revealed phenomenon. We might then find value in the claim that augmented triads generate pan-triadic syntax by way of nearly even trichords, just as many theorists have found value in the claim that the harmonic series generates diatonic syntax by way of consonant triads. This conjecture is given close scrutiny in chapter 3, which focuses more closely on the relationship between consonant and augmented triads in the music and the music theory of the nineteenth century.

Remarks on Dualism The above discussion places us in a position to refine our understanding of the way that the system of triadic transformations rest on a dualistic basis. David Kopp (2002, 155) frames an extensive discussion of my early work on this topic under the subtitle “A Dualist Transformation System” and suggests without comment that the dualism is a limitation. Dmitri Tymoczko (2009a) makes a more sweeping claim: that the system of triadic transformations is “fundamentally dualist” and hence ripe for wholesale rejection. It is hardly necessary to explain why attributions of dualism are a priori problematic, since the term is identified with the

38

 Audacious Euphony discredited harmonic dualism of Oettingen and Riemann under which the triadic transformations were initially conceived. There are two points to make in response. The first is that concepts may be detached from the framework in which they were initially conceived, in principle. This point will be familiar to music theorists, who identify Schenkerian prolongations and progressions without fear that they will be suspected of subscribing to the ideas of German racial superiority in support of which those ideas were evidently conceived. The second point is that relations that may appear to be fundamentally dualist may arise as epiphenomena of other relations. This is the central argument of Tymoczko 2011a: “Nineteenth-century composers were not explicitly concerned with inversional relationships as such; instead, these relationships appear as necessary by-products of a deeper and more fundamental concern with efficient voice leading (253).” The same argument applies to twenty-first-century theorists. Major and minor triads constitute a fundamental and coherent class of objects not because they are related to each other by inversion about an axis. They are related because they share the property of near evenness, and degree of evenness is invariant under inversion. Their inversional relation is a consequence of the capacity of the tones of an augmented triad to be semitonally perturbed in two directions, up and down. What is true of the objects is true of their transformations: any hexatonic transformation introduced in this chapter can be defined with respect to its structure as a minimally displaced augmented triad. For example, L can be defined as the transformation that acts upon the tone that is further from the displaced tone, moving the former a semitone closer to the latter; and P as the transformation that acts upon the tone that is nearer to the displaced tone, moving the former a semitone further from the latter. Similar formulations can be used to characterize any of the remaining transformations introduced in this chapter and, indeed, any of the transformations and transformation classes introduced in chapters 4, 5, and 7.20 In adopting this position, we are not washing our hands of harmonic dualism, which Henry Klumpenhouwer characterizes as “music-theoretical work that accept[s] the absolute structural equality of major and minor triads as objects derived from a single, unitary process that structurally contains the potential for twofold, or binary, articulation” (2002, 459). Instead, we are viewing harmonic dualism as the product of a more fundamental melodic dualism, which posits that melodic motion proceeds in two opposite directions, which we figure in our culture as “up” and “down,”21 and that there exists a sense in which it is productive to grant equivalent status to directed motions of equivalent magnitude but not equivalent direction. This dualism is so familiar as to be transparent; we invoke 20. To be sure, these formulations are cumbersome, so I often use inversion with reference to the triadic transformations. But, as I stressed in note 13 above, heuristics should not be confused with ontology. One can refer to inversional relations without believing that they are essential, just as one can refer to a sunset while still believing that the sun’s position is fixed and the earth’s is variable. 21. In other cultures it is figured as old/young, sharp/flat, skinny/plump, etc. See Zbikowski 2002, 67–68.

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it implicitly whenever we refer to arpeggiation, passing or neighboring motion, or refer to an interval without specifying its direction. On this position, the unitary process to which Klumpenhouwer refers is not a harmonic one, as it was for nineteenth-century German theorists, but rather a melodic one: the single semitonal displacement of the augmented triad. The binary articulation involves neither overtones/undertones nor having/being, but rather an acknowledgment that such displacements may proceed either up or down.

Triadic Structure Generates Pan-Triadic Syntax Material may suggest what process it should be run through (content suggests form), and processes may suggest what sort of material should be run through them (form suggests content). —Steve Reich, Writings about Music

One of the desirable qualities of a theory is the ability to demonstrate a relationship between the internal properties of an object and its function within a system. A successful model of triadic music ought to give a coherent account not only of triadic behavior but also of why composers have selected triads to do the behaving. Accordingly, one of the most enduring features of classical tonal theory is its capacity to generate syntax from the phonological properties of its constituent objects. Remarkably, the conviction that phonological consonance generates syntactic proximity is held by consensus across the many denominations of classical theory, which conceive and represent their subject in distinct and often competing ways. Riemann’s functions, Piston’s Roman numerals, Schoenberg’s structural functions, Schenker’s Ursätze, and Lerdahl’s pitch-space grids say different things about tonal syntax, but the acoustic properties of major and minor triads are foundational to each. If triads are nothing but quintessentially consonant objects, why should they be asked to generate a syntax that is not predicated on their consonance? The good composer listens to the musical object, identifies its properties and tendencies, and recognizes the transformations that will extract the dynamic life from the object’s interior, just as the good sculptor recognizes and extracts the form latent in the stone or the fallen log. It is a poor composer who runs any old object through any old machine and calls the result “art.” A passage from Daniel Harrison’s “Three Essays on Neo-Riemannian Theory,” written in 2001 but only recently published (2011), gives some sense of what is at stake: Transformational theory in general requires a separation of object and activity, of what something is and what is done with it. . . . Objects are inert and without tendency, and all activity and meaning are supplied by transformations applied to them. From this far vantage 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 construction of lifeless parts that are made to move by some external force. (552)

40

 Audacious Euphony Although transformational theory, in its broadest outlines, may suffer from Harrison’s gruesome vision, that branch of transformational theory that takes consonant triads for its objects, and subjects those objects to transformations that minimize voice leading, is immune from it, by virtue of the work we have done in this chapter. Establishing near evenness as a unique feature of consonant triads places us in a position to see that the structure of triads, as objects, is intimately related to their function, as participants in hexatonic (and, more broadly, pan-triadic) syntax.

Triads Are Homophonous Diamorphs Identifying the triad as an optimal voice-leading structure by virtue of its near evenness does not detract from its status as an optimal acoustic structure by virtue of its consonance. What it suggests is that the triad is a homophonous diamorph: one sound, two forms.22 There are two distinct, independent reasons for selecting major and minor triads as primary structures on which to build a musical syntax.23 Even in some alternative universe where major and minor triads were acoustically dissonant, there would still be a musical motivation for inventing them and basing a musical system upon their properties. Dmitri Tymoczko (2011b, 64) makes a similar point with a Deist parable: Suppose God asked you, at the dawn of time, to choose the chords that humanity would use in its music. There are two different choices you might make. You might say, “. . . I’d like some nearly even chords that allow us to combine efficient voice leading and harmonic consistency. . . .” And God would hand you a suitcase containing nearly even chords, including the perfect fifth, the major triad, and dominant-seventh chord. On the other hand, you might say “. . . I’d really like to hear chords that sound good—chords whose intrinsic consonance will put a smile on my face.” In this case, God would hand you a suitcase containing . . . the perfect fifth, 22. The term is appropriated from linguistic theories of code switching (Muysken 2000, 123), whose connections with music are explored in chapter 9. 23. The assertion that near evenness is independent of consonance is complicated by the strong correlation between the two properties. Tymoczko points out that nearly even chords in the twelve-note universe are among the most consonant of their cardinality (2011b, 14), and that “for small chords, maximal consonance implies near evenness” (61). The implication goes only one way—it is not the case that near evenness implies maximal consonance. This is true whether one assesses consonance by an overtone method or by an interval vector method. With an overtone method, the maximally consonant chord of cardinality n is the one that most closely approximates the first n odd partials of some generating fundamental. On this standard, the [02469] major ninth chord is the maximally consonant pentachord; but it is less even than the [02479] “usual” pentatonic. With an interval vector method, the maximally consonant chord of a given cardinality is the one whose interval vector entries for classes 3, 4, and 5 sum to the highest value. On this standard, the [0347] split third is the maximally consonant tetrachord, but it is less even than the nearly even [0258] dominant/ half-diminished chord. The divergence between the two properties becomes more acute in larger “microtonal” universes, where the nearly even dyad moves away from the perfect fourth and closer to the tritone.

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the major triad, the dominant-seventh chord. In other words, he would hand you the very same chords, no matter which choice you made.

The general phenomenon described here turns up frequently in the natural and social worlds, where it is referred to as overdetermination, robustness, or Babylonianism. These terms characterize “the use of multiple means of determination to ‘triangulate’ on the existence and character of a common phenomenon, object, or result” (Wimsatt 1981, 125).24 These might include “using different assumptions, models, or axiomatizations to derive the same result or theorem” (127). Human physiology presents many easily accessible examples of overdetermination. Mouths are for eating, talking, and breathing; ears serve auditory and vestibular functions; male urethrae channel both excretory and reproductive fluids. Organs transform seamlessly between the different functions that they fulfill. In humans, these organs are housed within a body that also includes an organ responsible for achieving and articulating awareness of the world within, without, and beyond. And yet these transformations evade recognition by that organ, under ordinary circumstances. Organs fulfill their overdetermined potentials well beneath the threshold of awareness. But music is different, as its active production and passive experience necessarily involve, in some measure, the participation of a conscious, aware mental faculty. How, then, does triadic music execute the transformation between the multiple potentials of its constituent objects? This question will emerge as explicitly central in the closing chapters of this book. In the meantime, our concern will be directed toward refining the model of triadic syntax introduced in this chapter, which is predicated exclusively on fulfilling the triad’s syntactic potential as a nearly even object. We will nonetheless be unable to forget that the triad is also something else that we have long known it to be. We will preserve that memory in our terms of reference: I will continue to refer to them collectively as consonant triads, even though we are more interested in their extensionally identical but intensionally distinct status as nearly even trichords. I will continue to name them individually by their roots, even though roots have no theoretical status in the theory of pantriadic syntax. (Heuristics ≠ ontology; see notes 13 and 20.) And, in discussions of particular passages, I shall continue to casually invoke all manner of overlearned ascriptions and categories, in ways that contain unspoken implications about how composers move between or overlay diatonic and pan-triadic syntax, the nearly even trichord with the consonant triad. What I defer, until chapters 8 and 9, is the explicit theorization of this process. Our docket is otherwise full, as we seek to refine the notion of how the nearly even trichord generates pan-triadic syntax on its own terms, independently of diatonic tonality. 24. Wimsatt 1981 traces the origin of the tradition to Aristotle, “who valued having multiple explanations of a phenomenon” (125). Freud’s Interpretation of Dreams (1900) is the moment when overdetermination emerges to prominence in modern thought. Feynman 1965, 46, referred to this approach to knowledge as “Babylonian.”

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C HA P T E R

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The Historical Emergence of Augmented Triads Chapter 2 proposed that pan-triadic progressions, exemplified by hexatonic cycles, arise from the status of consonant triads as minimal perturbations of the perfectly even augmented triad. Some readers might worry that too much weight is being placed on a relatively slender shoot. When an augmented triad appears in music before 1830, its behavior is normally well regulated and unobtrusive, tucked into the middle of a phrase rather than exposed at its boundaries, passed through quickly and lacking metric accent. In an 1853 monograph titled The Augmented Triad, Carl Friedrich Weitzmann portrayed his protagonist as a serf, scurrying in and out the rear entrance, occasionally showing his face but never intruding on the conversation in the salon. After agitating on behalf of “granting [the augmented triad] an abiding place in the kingdom of tones,” Weitzmann “gave its further fate over to our enlightened composers” (Weitzmann 2004 [1853], 144, 224; my translations). As if in response, some late-nineteenth-century composers featured augmented triads as motivic emblems in individual compositions: Brahms in his Alto Rhapsody (Forte 1983) and Wagner in Siegfried Idyll (Anson-Cartwright 1996) and his musical portraits of the Valkyries and of Amfortas. Liszt and Wolf acted more boldly, incorporating augmented triads into their normative sonic core.1 It is nonetheless difficult to make a case that augmented triads ever achieved a normative, unmarked status. Charles Moomaw’s 1985 dissertation is the most comprehensive Englishlanguage source concerning the augmented triad’s origins and early history. Moomaw locates the chord in France as early as 1636, typically when the fifth of a dominant triad is displaced up a diatonic semitone (Moomaw 1985, 251). He also reports that figured bass treatises consistently instruct that the +5 figure be 1. On augmented triads in Liszt, see Forte 1987, Todd 1988, and Satyendra 1992. Hantz 1982 analyzes the augmented triads in Liszt’s “Blume und Duft” in a way that particularly relates to the approach developed here. On augmented triads in Wolf, see McKinney 1993.

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44

 Audacious Euphony rendered with a seventh or ninth above the bass even when the latter is not explicitly ciphered (128). Georg Andreas Sorge was evidently the first to recognize the augmented triad as a primary harmony, although initially in 1745 he did so with great reluctance: “The best thing about this harsh harmony, if one may speak of it as one, is that it seldom appears” (Sorge 1980 [1745], 440). In 1760, Sorge upgraded its status incrementally, observing that the augmented triad is tolerable when it results from a chromatic passing tone that connects fifth-related major triads (Moomaw 1985, 323). It is in such passing contexts, bisecting a whole step, that the augmented triad most characteristically and frequently occurs in music of the late eighteenth and early nineteenth centuries. During the 1770s, French theorists began to accept the augmented triad as a fundamental sonority, bearing a distinctive character, and even a capacity to support accretions (Gessele 1994, 84–86). This acceptance becomes evident in a remarkable D major Minuet that has been attributed to Mozart.2 The short composition contains seven augmented triads, of which only the first behaves in the manner sanctioned by contemporaneous treatises. The remaining six dissonances are anomalously accented in three independent ways: each initiates a phrase, occurs on a metric downbeat, and is marked sforzando. Howard Boatwright astutely observed that “each augmented chord has a different melodic origin and a different harmonic function” (1966, 30) and concluded that the sonority has motivic value, in and of itself, rather than as a diminutional accretion to some other formation (also see Sobaskie 1987). Abbé Georg Vogler’s 1802 Handbuch zur Harmonielehre was the first treatise to explore the augmented triad’s potential for enharmonic reinterpretation. Writing that the augmented triad “appears to consist of three similar major thirds,” Vogler claimed that its proper roost was the third scale degree of harmonic minor and that “each III chord in minor . . . can be multiply interpreted as a III chord in three different keys” (1802, 103, 109; my translations). Vogler illustrated this potential for Mehrdeutigkeit (multiple meaning) through the progression given here as figure 3.1, whose anacrusis/downbeat combinations form a hexatonic cycle. The third beat of each measure hosts some spelling of the CEGᅊ augmented triad, acting successively as dominant of each triad on the following beat.

Figure 3.1. From Georg Vogler’s Handbuch zur Harmonie (1802).

2. The Köchel number is K. 355/576b. The attribution, from an 1801 publication, is suspicious on internal grounds (Oster 1966) and has never been corroborated. In any case, no evidence exists as to date of composition (Cliff Eisen, e-mail correspondence with the author, 2007).

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Figure 3.2. Schubert, Symphony no. 2, 4th mvt., mm. 300–12.

Vogler’s progression serves as a template for the passage presented in linear reduction at figure 3.2, from the finale of Schubert’s Second Symphony in Bᅈ major (1812).3 The core of the development consists of three transpositionally identical phrases that divide the octave by major thirds, beginning and ending with the F major that terminates the exposition. Immediately preceding each tonic is a locally appropriate spelling of the CEGᅊ augmented triad. Prolongation does not capture the relation of these chords to one another, because the chord functions as a local dominant to a series of well-articulated tonics. Nor is motivic association quite adequate; it is too static to capture the phenomenology of the passage. More than waving their hands and crying, “Remember me, here I am again,” these augmented triads are also saying, “You thought I was this; well think again, ’cause I can be that too.” If prolongation is at work in this passage in any form, then its more plausible object is the FACᅊ triad that unites the three tonics. In graphs of similar passages by Beethoven and Wolf, Heinrich Schenker implied that he understood the arpeggiated augmented triad as the prolonged displacement, by chromatic neighbor, of a major triad.4 In figure 3.2, that triad would be F major, which appears as a tonic at the end of the exposition, returns at m. 352, and ultimately acquires a retransitional seventh (m. 392). When Dᅈ major is tonicized at m. 312, C is displaced by a Dᅈ neighbor, which continues to be locally supported (qua Cᅊ) when A major is tonicized at m. 332, and only returns to C at m. 352, when F major is retonicized. If, as Schenker implies in analogous passages, FADᅈ is the prolonged harmony from m. 312 to m. 347, then it follows that both Dᅈ major and A major are subordinated to a controlling dissonance. Yet the score contains no vertical slice or contiguous patch, even an egregiously gerrymandered one, to be circled and labeled as a “controlling harmony.” In passages such as these, then, the augmented triad is

3. Seidel (1963) draws attention to this passage. Wason (1985, 19) speculates on a possible lineage from Vogler to Schubert. Vogler was a peripatetic, ambitious, and charismatic personality who lived in Vienna from 1802 to 1805 and later taught composition to such prominent figures as Carl Maria von Weber, Gottfried Weber, and Meyerbeer (Grave and Grave 1987). 4. See Schenker’s analyses of passages from Beethoven’s “Appassionata” and “Spring” Sonatas and Wolf ’s “Ständchen” (2005 [1924], 41–64; 1979 [1935]: fig. 100.6). Many scholars (e.g., Slatin 1967; Morgan 1976; Proctor 1978; Stein 1985) have observed that his treatment of middleground equal divisions cannot be reconciled to his pronouncements elsewhere that only consonant harmonies are susceptible to composing out. What is of primary interest here is that Schenker found dissonant prolongations aurally and conceptually plausible, even if they “prolong” an idea that dissonates with the fundaments of tonality.

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 Audacious Euphony not directly available to perception. Its status, as a collocation of bass pitches or triadic roots or local tonics, is virtual and liminal. This analysis suggests that the relationship between consonant and dissonant harmonies is not diodic. Consonant harmonies provide the context in which dissonant harmonies can operate, as a rule. But, as Robert P. Morgan showed in “Dissonant Prolongation” (1976), there are situations where these priorities are reversed, and consonant triads subordinate to dissonant ones, not only locally but across spans of significant duration. The relation between consonance and dissonance, then, is fluid in principle. The potential for this fluidity opens up a compositional dynamic, where a terrain of fixed relations is transformed into a site for negotiation. Consonant and augmented triads gain the potential, in principle, for reciprocity.

Consonance/Dissonance Reciprocity The nineteenth century was familiar with reciprocity as a general cultural condition. Kant developed it in his influential Critique of Pure Reason (1982 [1787]) as his third analogy of experience. The term was imported into music theory by Simon Sechter (1853–54), who noted (following Kirnberger 75 years earlier) that, lacking further context, two fifth-related triads are tonally indeterminate.5 C serves as dominant to F, which serves as subdominant to C, triggering a recursive circle whose resolution requires external intervention (see Lewin 2006, 64). A similar situation arises in the case of diatonic third relations, whether relative major/ minor (C major/a minor) or Leittonwechsel (C major/e minor). Both of these species focus their tonal indeterminacy at a single melodic fulcrum, a whole step in the first case and a semitone in the second. The potential indeterminacy of the former case is well documented, particularly with respect to Schumann (e.g., Rosen 1995, 674). That of the latter case is encoded into its German name. The modern conception of leading tone is restricted to the relationship between a tonic and its semitonal lower neighbor. For German theorists of the middle of the nineteenth century, this relationship captured only one half of a duality: Leitton applies equally to the relationship of dominant and its semitonal upper neighbor. 6 Accordingly, when C major and e minor are juxtaposed, the attraction of C to B ^ ^ ^ ^ (as 6–5 in e minor) is as strong as that of B to C (as 7–8 in C major). The semitonal relation thus projects an unstable force field that pulls simultaneously in both directions. Carl Friedrich Weitzmann identified a third type of reciprocity that shared aspects of those identified above: the relationship between a minor triad and its major dominant, which he regarded as equivalent to that of a major triad and 5. Kirnberger (1982 [1771–76], 44–45). See also Hauptmann 1888 [1853]. Such bilateralism is also characteristic of the sixteenth-century view (Dahlhaus 1990 [1967], 241). 6. The conception originates in dualist thinking but was sensible enough that it was taken up by theorists with no commitment to dualism, such as Louis and Thuille 1982 [1913], Kurth 1923, and Lorenz 1933. Harrison 1994 provides an excellent elaboration on these matters.

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its minor subdominant. The reciprocal leading tone energies are divided between the thirds, which Daniel Harrison (1994) calls the agents: the upward-pressing E, borrowed from f minor’s parallel major, and the downward-pressing Aᅈ, borrowed from C major’s parallel minor.7 Although this triadic relation plays a central role in the writings of many theorists after 1850, it never achieved a stable name. I shall refer it to using Weitzmann’s term, nebenverwandt, which Janna Saslaw translates as “adjacency relation” (Weitzmann 2004 [1853]). Of particular relevance for present purposes is Arthur von Oettingen’s name for the same triadic pairing: reciprocal, a German/English cognate (Mooney 1996, 56). The situations examined so far involve a relation between two consonant triads. Such relations are bilateral in principle, since no consonant triad is more stable than any other absent a particular context. The reciprocity that we identified with regard to figure 3.2, however, is of a different kind, as it involves the relationship between a consonant and dissonant triad. The scale is inherently out of balance and can only be leveled through the application of external forces. In the crudest cases, such as Schubert’s “Die Stadt”, a dissonant harmony achieves a quasi stability by squatting like a brute and appropriating the rhetorical garments normally reserved for consonances (first, last, loudest, longest; see Harrison 1994, 75ff.). In contrast to such ad hoc solutions, François-Joseph Fétis recognized a way to override the forces of tonality by cultivating more systematic resources, which he referred to under the terms uniformity and symmetry. In a passage quoted in chapter 1, Fétis described the experience of a diatonic sequence in phenomenological terms: “the succession and . . . movement fix the attention of the mind, which holds on to the form so strongly that any irregularity of tonality is not noticed. . . . The mind, absorbed in the contemplation of the progressive series, momentarily loses the feeling of tonality. . . . The attention of the musical sense is diverted from the feeling of tonality by symmetry of movement and succession” (2008 [1844], 27, 30). Fétis writes that a sequence levels the distinction between consonance and dissonance. A diminished fifth no longer requires resolution; in this context, its behavior is indistinguishable from that of the perfect fifth. In a diatonic sequence, the law of uniformity is kept in check by the prior commitment to the diatonic scale. Although each pattern iteration replicates the generic intervals of its predecessor, its specific intervals are channeled within the banks of the diatonic scale. The forces that Fétis identifies become more fully unleashed in chromatic sequences, such as the hexatonic cycles explored in chapter 2 or the Schubertian third-divisions represented by figure 3.2. In these cases, the law of uniformity has a monolithic force, and the rapid turnover of chromatic pitch classes ruptures any ability of the diatonic collection to hold a focus on a particular global tonic. The binary distinction between diatonic and chromatic sequences is a particular manifestation of a more general dynamic that arises in many passages that we would not consider to be sequential per se. Whenever a motivic fragment migrates across a series of transpositional levels, or a fugal point of imitation is 7. Smith 2006 identifies several Brahms compositions that thematize this reciprocity as an ambiguity.

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 Audacious Euphony replicated on a different degree of the scale, the absolute sizes of the intervals may conform to the locally governing scale, or they may be preserved at the expense of subverting or even rupturing that government. The pressures toward uniformity may be confined within the diatonic channel or may jump those banks and lay down their own channels. Later theorists formulate this same duality in terms of diatony versus repetition (Schenker 1954 [1906]) and magnetism versus inertia (Larson 1994). To see how this duality manifests in the relation between consonant and augmented triads, consider the following classroom situation. Two students are presented with a melodic gesture from C up to E and asked to replicate that gesture beginning on E. One responds with E up to Gᅊ, projecting an augmented triad; the second with E up to Gᅉ, projecting a consonant one. Both responses are correct, but one interprets replication as raw uniformity; the other, as tempered to the diatonic collection. Gᅉ and Gᅊ displace each other across the melodic fulcrum upon which the diatonality/uniformity tension is balanced, in the same way that the same two tones constitute the modal fulcrum in the case of an E tonic, or the melodic fulcrum in a Leittonwechsel relation between c minor and Aᅈ major, or one of two such fulcrums in the nebenverwandt relation between C major and f minor.

Two Early-Century Examples: Beethoven and Schubert Composers of the early nineteenth century sometimes treated this melodic fulcrum as a site for motivic play. Consider the initial movement of Beethoven’s f minor Piano Sonata (Op. 57, “Appassionata”).8 A secondary theme in Aᅈ major (m. 35) has a consequent phrase that mutates to aᅈ minor (m. 42) and remains in that key until the end of the exposition at m. 65, featuring Eᅈ/Fᅈ motivic play throughout those measures.9 The motive is raised to a higher power in the development, which begins in aᅈ minor, renotated as gᅊ minor, and progresses to E major at m. 67, saliently featuring the motion from Dᅊ to E on successive downbeats. Motion continues around the hexatonic cycle, to e minor (m. 79); skipping over C major, whose status as global dominant requires it be reserved for a later moment; and proceeding directly to c minor (m. 83) and Aᅈ major (m. 87). The entire passage prolongs Aᅈ major by displacing its fifth Eᅈ to its augmented fifth E and then restoring it. Similar motivic play of the dominant and its upper neighbor is evident in the first movement of Schubert’s A major Piano Sonata, D. 959. A score of the exposition and development is available at Web score 3.3 . The dominant reached at m. 28 of the exposition is prolonged for more than one hundred measures through two extended expansions, each initiated by a chromatic sequence that arpeggiates 8. The analysis offered here is based on Proctor 1978, 173–74. See also Bribitzer-Stull 2006, 179–80. 9. These echo the Fᅈ/Eᅈ play at m. 23 (bass) and mm. 26 and 29 (treble), which are in turn echoes of the Dᅈ/C emanations that conclude the initial f minor theme at mm. 10–15.

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downward through the stations of an E augmented triad. Both arpeggiations involve B → C displacements that, upon resolution, trigger significant motivic reverberations. The initial major-third cycle, sketched at figure 3.3(a), culminates at m. 39 when C is displaced back to B in the bass. The bass then isolates and works over the B → C displacement throughout the subsequent extension of the B major local dominant (figure 3.3(b)). Similar bass motivic play occurs locally at the G major fantasy drift (mm. 65–68, figure 3.3(c)), more structurally at the reanimation of the major-third division (mm. 82–91, figure 3.3(d)), and prior to the final stabilization of E major, where a C major sforzando (m. 103) is not recuperated until a medial caesura eight measures later. The apotheosis of this motive occurs in the development section, whose opening measures (figure 3.3(e)) have been the subject of much marvel by performer/ critics. Charles Fisk describes it in the following evocative terms: The new theme articulates itself as a fantastical ten-measure period: its first phrase [mm. 131–35] slips away from C major into B major, while its second [mm. 136–40] slips just as magically back up to C. An even more ethereal variant of the same phrase pair immediately follows [mm. 141–50], its sixteenths now spun out into gossamer webs. For these two periods, the music simply oscillates between C and B, achieving what [Charles] Rosen characterizes as a stasis with a “physical effect . . . like nothing in music before.” (2001, 216, quoting Rosen 1980, 287)

The oscillation identified by Fisk persists, indeed, through the remainder of the developmental core, even after escaping the “poised, transfixed stasis” (Brendel 1991, 126) of its opening musette. The subsequent ten-measure period (mm. 151–60) modulates from C major to b minor and back. The C → B melodic arc is then carried by the phrases of the final extended period (mm. 161–80), which approach a retransitional E major first from c minor, its hexatonic pole, and then from a minor, its minor subdominant. The phrase pairings throughout the developmental core suggest that B acts as lower neighbor to structural C (Jonas 1982 [1934], 92). The key that jointly provides a context for both harmonies is e minor (see Schenker 1954 [1906], p. 226; Hauptmann 1888 [1853], 159–60), whose shadow control is indicated by the phrygian approach to its dominant (mm. 134, 144) and the deceptive return to its submediant (mm. 139, 149). E minor in turn substitutes for the E major that frames the development, which opens with a melodic motion from B4 to C5 (m. 129 bis) and concludes with its reversal (m. 179). B thus performs the role of lower neighbor to its own upper neighbor, magnifying the melodic fulcrum introduced with the equal divisions of the exposition.

Three Late-Century Examples: Liszt, Rimsky-Korsakov, Fauré The give and take between consonant and augmented triads becomes foregrounded in a number of compositions from the second half of the nineteenth century.

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 Audacious Euphony Figure 3.3. Excerpts from Schubert, Sonata in A major, D. 959, 1st mvt.

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Figure 3.4. Liszt, A Faust Symphony, 1st mvt., mm. 1–22.

The opening of Liszt’s Faust Symphony, completed in 1854, famously and explicitly inverts the values classically accorded these sonorities (figure 3.4). The passage consists of two slow rotations through three segments of material (marked A, B, and C in the example), each of which extends approximately four measures. Augmented triads dominate the surface. Moreover, with the exception of mm. 1 and 13, the pitch-class pool for the entire passage draws exclusively on the CEGᅊ and FACᅊ augmented triads, which combine to form a hexatonic collection. Of particular interest are the four boxed figures, whose staggered downward motion tropes a suspension figure, but with consonance and dissonance inverted with respect to formal function (but not metric location): the position of preparation and resolution is occupied by dissonant augmented triads; that of the suspension, by consonant minor triads (Morgan 1976, 60). It seems likely that Rimsky-Korsakov had the opening of Faust in his ear when he wrote the opening measures of his Symphony no. 2 (1868), subtitled Antar (figure 3.5). Antar, like Faust, begins with two slow rotations through a series of three texturally differentiated segments, each approximately four measures long. The second rotation transposes the first segment by a minor sixth (down in Antar, up in Faust); the final two segments are then transposed upward by major third. As both compositions combine two augmented triads into a hexatonic collection, their second rotations recirculate the same tones as their respective antecedents.

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 Audacious Euphony Figure 3.5. Rimsky-Korsakov, Symphony no. 2 (Antar), 1st mvt., mm. 1–24.

Looking back two decades later at the kuchist movement of which he had been principal in the 1860s, Rimsky wrote that “Liszt was extreme, so was Berlioz, so was Wagner. And so were we” (Taruskin 1996, I: 70). The opening of Antar suggests that Rimsky was understating his capacity to nuance that extremism to artistic ends. Whereas Faust’s opening overturns the asymmetric consonance/ dissonance binary with one swipe of the hand, Antar’s balances an exquisitely fine point between its terms. Although Antar is stricter than Faust in its hexatonicism—the passage contains not a single pitch foreign to the collection— its augmented triads are less apparent. The segment labeled A presents three minor triads, and the segment labeled C selects one of them for prolongation. The segment labeled B, by contrast, prolongs the FACᅊ augmented triad, embellishing two of its components with an escape tone that very tentatively suggests a reconstitution of one of the minor triads. Whereas figure 3.5 symmetrically segments the opening of Antar into six units, based on thematic and textural rotation, figure 3.6 asymmetrically partitions the same music on the basis of harmonic content, splitting the A material into two segments and fusing the B and C material into a single one. In the first rotation,

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Figure 3.6. Analysis of the opening of Antar.

the A material consists of a box and an oval that respectively enclose fᅊ minor → d minor and bᅈ minor → fᅊ minor. The remaining material echoes and expands the initial fᅊ minor → d minor progression, interpolating an augmented triad between them. The role of the augmented triad, on this interpretation, is to connect the two more stable consonant triads that flank it, grossly distending a progression that would have otherwise been at home in the eighteenth century. In effect, the augmented triad staggers the simultaneous semitonal motions of the opening progression: first Fᅊ → F in the bassus and then, four measures later, Cᅊ → D in the cantus.10 In transposing its predecessor downward by minor sixth, the second rotation inverts the function of the two triadic pairings. The bᅈ minor → fᅊ minor at the interior of the previous rotation is now positioned at the head, and it is this unit that is subsequently expanded through the same passing augmented triad. Conversely, the fᅊ minor → d minor that dominates the first rotation is tucked into the interior of the second one. As a result, the series of four minor triads that opens the composition, fᅊ minor → d minor → bᅈ minor → fᅊ minor, is expanded in the progression from one rotation head to the next (fᅊ minor → d minor, m. 1; bᅈ minor → fᅊ minor, m. 13) and also in the progression from one expansion (fᅊ minor → Faug → d minor, mm. 4–12) to the next (bᅈ minor → Aaug → fᅊ minor, mm. 16–24). Where figure 3.6 presents the augmented triad as prolonging a motion between its flanking consonant triads, figure 3.7 inverts those roles. The opening gesture in the first bassoon (= cantus) is a hexatonic spiral that, on the basis of parallelism, suggests three semitonal pairs: Cᅊ → D, A → Bᅈ, F → Fᅊ. Assuming that we are inclined to hear parallel passages in parallel ways, we are encouraged to hear the melodic gesture as unfolding an augmented triad.11 But which one? As the metric grid does not lock in until m. 4, it is unclear whether the first or second component of each pair is the accented one, and hence whether it is the FACᅊ or BᅈDFᅊ 10. On staggered semitones in Liszt, see Satyendra 1992, 102–3. A more complete interpretation would acknowledge the tentativeness of d minor at m. 8. D falls back to Cᅊ throughout the segment labeled C, at the same time as A escapes to Bᅈ, suggesting a bᅈ minor triad and delaying the ultimate consolidation of d minor until m. 11, when the sustained Bᅈ finally resolves to A. 11. The “parallel passages in parallel ways” dictum was stated by Gottfried Weber (1846 [1817–21], 365) as “What the ear has once heard in a certain passage, it will not only expect again, on the recurrence of the same passage, but will sometimes even perceive beforehand,” and reappears prominently in Lerdahl and Jackendoff 1983.

54

 Audacious Euphony Figure 3.7. Alternative analysis of the opening of Antar.

augmented triad that is unfolded. The second bassoon (= bass) presents a similar but complementary problem. It also unfolds the hexatonic collection in semitonal pairs: Fᅊ → F, D → Cᅊ, Bᅈ → A. Here, too, the floating metrics defeat any assignment of priority to a component of each pair, and hence to one of the two augmented triads. Moreover, simultaneous tones in the cantus and bassus belong to different augmented triads. Even if our ears locked into a particular metric orientation, they would be receiving conflicting information from the outer voices. Only the inner voice of the opening segment, sounded by the timbrally distinct horn, has a clear commitment to one of the augmented triads: it sounds A → F → Dᅈ → A. This ever-so-slight tipping of the balance, in an otherwise austere equilibrium, is subtly confirmed by the pitches that are held invariant in the first three measures as their respective triads are registrally redistributed. The segment labeled B in figure 3.5 provides clarity, first to the bassus and then the cantus. In the bass, the arrival of pedal F2 at m. 4 stakes down the metric grid clearly for the first time, conferring the accent of the Fᅊ → F onto its second term. Applying this information in retrospect to the opening gesture causes us to hear the second bassoon line in terms of Fᅊ → F, D → Cᅊ, Bᅈ → A, emphasizing the same augmented triad sounded in the horn. The subsequent escape-tone figures in the cantus similarly disambiguate the hexatonic spiral of the previous measures, by tracing the same melodic course an octave lower. What was metrically flat and amorphous in the first segment becomes shaped in the second gesture, clearly thrusting the accentual weight onto the first term of each pair: Cᅊ → D, A → Bᅈ. The timbral continuity of the bassoon helps to forge this connection and to project this weighting retrospectively onto the cantus of mm. 1–3, which now is interpreted in terms of Cᅊ → D, A → Bᅈ, F → Fᅊ. This analysis of the second segment leads us to hear the opening segment, in each of its three melodic parts, as projecting FACᅊ, even though its constituent tones are not sounded simultaneously before their prolongation at mm. 4–8. Through this lens, each of the minor triads sounded in the opening segment results from displacement of a component of the augmented triad. This same hearing then extends naturally to the third segment of each rotation, which alternates between two minor triads, in 63 and 64 inversion, respectively, neither of which projects convincingly as an object of prolongation. The passage excerpted as figure 3.8, from Gabriel Fauré’s Requiem Mass of 1877, provides an instructive comparison. Like much of the d minor Introit/Kyrie

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Figure 3.8. Fauré, Requiem Mass, Introit, mm. 50–61.

movement from which it is drawn, this passage is animated by the tonic’s relationship to an F major triad that consistently functions as dominant. The melodic fulcrum of this relationship is the Cᅊ that mediates between C and D, and hence the harmonic fulcrum is the FACᅊ augmented triad that appears at m. 60, just prior to the cadence. This same augmented triad, spelled variously, also appears three times in the interior of the phrase, each time as the second component of a two-chord unit whose first component is a consonant triad: F major at m. 52, fᅊ minor at m. 54, and bᅈ minor at m. 56. Other features confirm the two-bar groupings throughout the passage: with the exception of the final cadential pairing, the first measure of each pair descends a melodic fourth on its final beat, and the second measure of each pair supports an A4 reciting tone. Our preference for analyzing parallel passages in parallel ways presents us with a choice similar to the one that we faced in our analysis of Antar: the Exaudi’s harmony is structured either by the connections between the initial, consonant measure of each pair or by those of its terminal, dissonant ones. The first of these options does not present a very coherent species of diatonic tonality: F major is embellished by fᅊ minor and bᅈ minor before resolving as dominant of d minor. bᅈ minor is easily reconciled as the minor subdominant of F major. What remains intractable is the fᅊ minor triad. Perhaps it functions as iii of a D major that otherwise has no presence in the passage (or elsewhere in the movement)? This feels a little desperate and, moreover, does not address the enharmonic metamorphosis of Cᅊ into Dᅈ as fᅊ minor is displaced by bᅈ minor at m. 56. The second option understands this bouquet of harmonies in terms of the augmented triad to which each one leads. This alternative places FACᅊ at the conceptual center of the passage, assigning it the role of a switching station through which the various consonant triads are threaded. We are aware that the augmented triad plays this role because Fauré shows us, by leading each chord in and out of the switching station, thereby isolating each semitonal displacement. There is no consistent diatonic explanation that accounts for the simultaneous presence of this particular group of triads in a single phrase. What draws them together is their shared status as single semitonal displacements of FACᅊ.

56

 Audacious Euphony The same can be said of the opening measures of Antar, where the same collection of minor triads is no more tonicizing than in Fauré (van den Toorn 1995, 127–28). This is so even though their mutual relationship to the FACᅊ augmented triad only unfolds slowly, across the entire introduction. The augmented triad can function as a switching station whether it has the presence of chronological mediator, as in the Exaudi, or chronological consequent, as in Antar, or no role at all, as in many of the pieces examined in chapter 4. The center of a circle is equally orienting to a set of dancers, whether marked by a pole, a hole, or the imagination of the dancers.12

Reciprocity in Weitzmann’s Der übermässige Dreiklang Carl Friedrich Weitzmann was the first theorist to recognize the compositional dynamic documented here. Weitzmann’s 1853 monograph on The Augmented Triad tells three genesis stories about its protagonist, each of which involve the verb entstehen or its nominal equivalent, Entstehung. Saslaw translates the verb as “arise,” and the noun as “origin.” An alternative translation, generate/generation, emphasizes organicist implications that may or may not be nested within Weitzmann’s conception. Because such an implication is not guaranteed, Saslaw has done well to avoid them; but nor is it precluded, and it will serve my interest to pursue it. The first story occurs in his chapter 2, “Preparation, Origin [Entstehung], and Introduction of the Augmented Triad,” which offers “a primer as to how this strange chord could come to life [in Leben treten könne], prepared through major and minor triads and their inversions” (Weitzmann 2004 [1853], 166; my translation). The primer presents sixteen ways to connect a consonant triad to an augmented one via semitonal voice leading. The second story occurs in chapter 6, “Natural Origin [Entstehung] of the Augmented Triad Most Important to Each Key.” Weitzmann combines f minor and C major triads into a pentachord, FAᅈCEG, from whose interior he extracts the augmented triad: F[AᅈCE]G. “From the connection of these two nebenverwandt chords arises [entsteht] the augmented triad most important to the two keys represented by them” (184–85). Weitzmann explains that even though E is foreign to the key of f minor, and Aᅈ to the key of C major, each arises as that key’s most important neighbor tone.13 These two tales relate to each other as specific to general. The first account concerns how an augmented triad comes into being at a particular moment in a particular composition. The second deals with the augmented triad’s position in a musical system,

12. The absence of the perfectly even chord about which the nearly even ones circulate is a theme of Tymoczko 2011b, which shows that it is productive to think of pentatonic and diatonic collections as circulating about perfectly even, and thus microtonal, collections. See also Douthett 2008. 13. Saslaw translates Nebenton as “secondary tone,” emphasizing that these neighbor tones are chromatic to the respective keys.

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apart from its particular instantiations. The first is in the sense of “Isaac was born of Abraham”; the second, in the sense of “invention is born of necessity.” In chapter 7, Weitzmann explores the augmented triad’s Mehrdeutigkeit, in the sense that interested Vogler fifty years earlier. He notes that once enharmonic variants are taken into account, the AᅈCE triad also arises in four other keys, besides the f minor and C major already explored: “So we find the augmented triad AᅈCE and its enharmonic equivalents in the nebenverwandt keys F minor and C major, further in the relative keys of each, in Aᅈ major and A minor, finally in the nebenverwandt keys of the latter, in Dᅈ minor and E major” (186–87). Although Weitzmann’s ordering has transformational implications that we will consider in chapter 4, for him that ordering evidently held no value except as an aid to memory. On a subsequent page, he lists the same six keys in the format reproduced here as the first block of table 3.1, writing that CEGᅊ and its enharmonic equivalents “can appear as the most important [augmented triad] of the following keys listed under them” (188–89). The remaining eighteen triads are grouped into three analogous clusters, each headed by an augmented triad and listing the keys in which it is “the most important.” Having created this list, Weitzmann’s discourse begins to project a subtle inversion. Until now, he has viewed the augmented triad as a serf in the employ of the particular consonant triad from which it arises. But now, having observed that each augmented triad has multiple patrons, he begins to wonder what life is like from its point of view. “The closest relatives of an augmented triad,” he writes, “are thus the major triads on its bass tone, third, and fifth, [plus] the minor triads to whose roots each of [the augmented triad’s] three voices forms the leading tone. . . . Its more distant relatives are the minor versions of the just-designated major chords and vice versa” (188–89). Several chapters later,

Table 3.1. Weitzmann’s grouping of the consonant triads as displacements of augmented triads I. {C, E, Gᅊ} (and its enharmonic transformations) 1. C major 2. E major 4. a minor 5. cᅊ minor

3. Aᅈ major 6. f minor

II. {Dᅈ, F, A} (and its enharmonic transformations) 1. Dᅈ major 2. F major 4. bᅈ minor 5. d minor

3. A major 6. fᅊ minor

III. {D, Fᅊ, Aᅊ} (and its enharmonic transformations) 1. D major 2. Gᅈ major 4. b minor 3. eᅈ minor

3. Bᅈ major 4. g minor

IV. {Eᅈ, G, B} (and its enharmonic transformations) 1. Eᅈ major 2. G major 4. c minor 5. e minor

3. B major 6. gᅊ minor

58

 Audacious Euphony Figure 3.9. From Weitzmann’s Der übermässige Dreiklang. Upper- and lower-case letters are the roots of major and minor triads respectively.

Weitzmann graphically portrays these relationships in a diagram that is reproduced here in translation as figure 3.9. Each augmented triad is presented at the center of a cluster of consonant triads; major and minor triads are indicated by large- and small-case roots, respectively. And here is where Weitzmann’s third genesis tale involving the augmented triad can be found: From the following augmented triads . . . arise [entstehe] the [consonant] triads indicated by the letters next to them. . . . The chords placed immediately next to the augmented triad are attained through the half-step progression of one of their voices; the [chords] further away [are attained] through the half-step progression of two of their voices. (202–5)

With this, Weitzmann turns back the flow of his second genesis narrative. At the systematic level, it is the augmented triads that are the sources, and the consonant triads the products. The first genesis narrative nonetheless remains intact. Immediately following the passage just quoted, Weitzmann presents seven full pages of examples, comprehensively enumerating the ways that an augmented triad can resolve. It is always the dissonance that is resolving to the consonance, never the other way around. In a moment-to-moment sense, the relation of consonant triad to dissonant augmented triad continues to be diodic. But in a systematic sense, Weitzmann is able to entertain the possibility that the relation is reciprocal. These passages from Weitzmann’s treatise are so rich in implication that they guide the work presented in the next three chapters of this book. Chapter 4 considers the internal structure of the six-triad pools that are clustered in table 3.1, from the standpoint of the Tonnetz graphics and triadic transformations introduced in chapter 2. Chapter 5 uses figure 3.9 as a stage from which to extend the Tonnetz, and its attendant transformations, so that it breaks out of the augmented triad boundaries that confined them in chapter 2. Chapter 6 uses that extended universe as a playing board, or map, upon which to present pan-triadic analyses of extended passages from the Romantic repertoires, and to assess and categorize those passages on the basis of the voice-leading strategies that they execute.

C HA P T E R

Four

Weitzmann Regions

This chapter explores a second preliminary model of the triadic universe, based on table 3.1’s six-triad groups, which we will refer to as Weitzmann regions. The initial model studied in chapter 2, based on minimal-work voice leading, also features six-triad groups, the hexatonic cycles. Both models partition the twenty-four triads into eight triplets, each of which contains three major-third–related triads of the same species, and each of which is paired with a triplet featuring triads of the opposite species. Where hexatonic and Weitzmann regions differ is in how those triplets are paired. Hexatonic regions pair major triads with minor triads built on the same root, or on roots four semitones away. Weitzmann regions pair triads whose roots lie an odd number of semitones apart. The first half of this chapter studies the internal structure of Weitzmann regions, focusing on their voice-leading properties, graphic representations, and the transformations that connect the triads within a region. The second half samples the diverse ways that composers of the long nineteenth century, from C. P. E. Bach in 1763 to Richard Strauss in 1949, first dipped their paddles into sectors of a Weitzmann region and ultimately ran their canoes directly down its rapids.

The Structure of a Weitzmann Region The Weitzmann regions have a quite different internal structure from the hexatonic regions studied in chapter 2. The six triads of a hexatonic region naturally fall into a cycle, an inherently graded space in which triadic distance correlates with voice-leading work. On that same basis, a Weitzmann region is a flat, uniform space: all of its triadic pairs stand exactly two voice-leading units apart. The triads can be cyclically ordered in several ways, but there is no ordering that is more natural than the others, from the standpoint of voice leading. Figure 4.1 uses two graphic metaphors to suggest the voice-leading structure of a Weitzmann region. The image at (a) is of a water bug with augmented-triad 59

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Figure 4.1. Two portraits of the structure of a Weitzmann region. body and six consonant-triad feet, the three on each side representing the modally matched subregions. All journeys from one foot to another are routed through the body. Some pairs of feet are located on a single side, others directly opposite, and others across the body at an angle. Each of these three pair types requires a twostage connection: one from the source foot into the body, and one from the body to the destination foot. Figure 4.1(b) reimagines the water bug as an oval hallway with three rooms on each side. This metaphor encourages one to inhabit the space and travel through it. Moving between rooms, one might stay on the same side, or move directly across, or move across at an angle. Each of these three trajectories can be experienced in various ways. A firefighter would likely execute each of them with a single kinetic impulse and experience each as a single Gestalt of essentially equivalent magnitude, despite small differences. A professor might experience the hallway as a marked intermediate point that articulates the journey into two stages (“I came into the hallway for a reason. . . . Oh, yeah, to go to class”). A retiree might be inclined to linger in the hallway for a while and see who else might be passing through. In the same way, the augmented triad may or may not be marked as such: some music will linger there, some will invite passing notice of its features, and some will rush quickly through the passageway without registering any impression of it. Figure 4.2 sets this abstract structure to music, modeling the voice leading between C major and the five other triads in its region. The first stage of each route is the same: G → Gᅊ/Aᅈ converts C major to C augmented. What distinguishes them from each other is what happens at the second stage: which tone moves by semitone, and in which direction. The progressions at (a) and (b), connecting two major triads, are equivalent to connections between legs on a single side of the water bug. The two stages have identical magnitude but opposite direction, preserving the center of balance discussed in chapter 2. Although these progressions are coextensive with the LP and PL transformations through a hexatonic cycle, it is not clear that these labels are pertinent, as they suggest that each motion is implicitly routed through a minor triad, not an augmented one. I will nonetheless retain the labels despite their isolation from their etymological source, since ordinary language works that way as a matter of course. (We say that this book is written in the English language even though no tongue is involved in producing it.)

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Figure 4.2. Voice leading from C major to the remaining five triads of its Weitzmann region.

The progression at (c), C major → a minor, resembles a straight line across the body. Here the two stages are consolidated in a single voice that moves by whole step, rather than divided among two voices moving by semitone as in all of the other intraregional progressions. The progressions at (d) and (e) connect C major to the two remaining minor triads in the region, f minor and cᅊ minor. As in a hexatonic region, the three cross-modal progressions shift the bug’s center of balance from one side to the other. The distance of that shift is by two units, more unsettling than the analogous hexatonic transformations (L, P, and H), which shift that center by a single unit.

Weitzmann Transformations and N/R Cycles Although the triads of a Weitzmann region have no natural cyclic ordering on the basis of voice-leading proximity, they do fall quasi naturally into a cycle on the basis of their historical origins in classical syntactic routines. This ordering was already suggested by Weitzmann in his initial, heuristically motivated description of the regions (quoted on p. 57 of this book). The chain suggested by that account is presented as figure 4.3. Diachronically, the account radiates outward from C major and f minor and terminates at the boundary triads, E major and dᅈ minor. The latter two form a direct connection that links the chain into a cycle. This is not a connection that Weitzmann indicates. It does not suit his purposes to do so, since his interest is in the contents of a region rather than its internal structure, and he has already generated all six triads without this final link in the cycle. Although Weitzmann conceived of nebenverwandt and relative as bilateral relations between keys, his conception easily converts into a set of idealized voice-leading actions on triads, each of which is associated with a triadic transformation.1 The nebenverwandt transformation (abbreviated N) takes any 1. I noted in chapter 1 that triads and tonics are interchangeable in much nineteenth-century theory. The transformational conception, although not explicitly present in Weitzmann’s 1853 monograph,

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Figure 4.3. A graphic depiction of the N/R chain described in Weitzmann 2004 [1853], extended to a cycle. triad to its nebenverwandt chord, which Weitzmann defines as the inversion about the root of a major triad, or about the fifth of a minor triad.2 The relative transformation (abbreviated R) takes a consonant triad to its relative major or minor. Figure 4.3 accordingly can be characterized as an N/R cycle, analogous to the L/P alternation that generates the hexatonic cycle. Complete N/R cycles appear in at least four passages from Schubert’s instrumental music, always traveling in the “authentic” direction (clockwise on the figure 4.3 cycle). Figure 4.4, from the first movement of his Fourth Symphony of 1816, is the chronologically earliest of the four, and the only one to restrict its lexicon to consonant triads.3 Each major triad is prolonged as a local tonic, which each minor triad serves as subdominant. In the remaining three passages, these

Figure 4.4. Schubert, Symphony no. 4, 1st mvt., mm. 86–106.

is not egregiously anachronistic: Oettingen 1866 interpreted the nebenverwandt in explicitly transformational terms (Mooney 1996, 71). 2. Weitzmann’s teacher, Moritz Hauptmann, defined this relation as the inversion about the tone that participates in both the major third and the perfect fifth of its triad (1888 [1853]). Morris 1998 proposes the label L′ (L-inverse) for this transformation, whose semitonally moving and stationary voices swap those of L. 3. Langlé 1797 contains, among a number of four-voice synthetic examples, a progression that embeds an N/R cycle (p. 73, no. 26).

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Figure 4.5. Schubert, Symphony no. 9, 1st mvt., mm. 304–15.

functional roles are exchanged: minor triads are local tonics, and the major triads that precede them are embedded within dominant sevenths. These passages include the famous trombone solo from the first movement of the Ninth Symphony, partly presented at figure 4.5. The trombones first carry solo material at mm. 197–219 of the exposition, through a partial N/R chain. The passage excerpted here, from near the end of the development, presents the first of two passes through a complete N/R cycle. The addition of the sevenths puts all twelve pitch classes in play, enabling Schubert to present complementary whole-tone scales in the treble register, chaining pungent minor-second suspensions with minor-third resolutions (Krebs 1980, 87).4 Liszt’s appropriation of the progression in the first movement of his Faust Symphony of 1854 (Cohn 2000) resembles figure 4.4: it moves in the authentic direction, employs no chordal dissonances, and accents the major triads as local tonics. In several other compositions, Liszt runs the cycle in the reverse direction, exploring its plagal capacity: minor subdominants move to their major tonics, and minor tonics, to their major dominants.5

4. Whole-tone scales are also evident in the two remaining Schubert passages, from the Octet (Taruskin 1996, I: 261) and the c minor Piano Sonata, presented below as figure 4.17(c). See also Borodin’s Second Symphony, 1st mvt., mm. 41ff. 5. “O Lieb, so lang du lieben kannst” of 1845, transcribed as the Liebestraum no. 3 for piano, and the piano etude “Vision” of 1852 (see Ahn 2003, 69).

64

 Audacious Euphony In focusing on the combination of N and R, we have neglected the Weitzmann region’s remaining mode-reversing relation, C major → cᅊ minor at figure 4.2(e). Like most juxtapositions of chromatically related triads, this one lacks a canonical name. Sigfrid Karg-Elert (1930) called this the same-third (Terzgleich) progression, and David Lewin (1987) proposed Slide, both names of which (following Kopp 1995, 289) suggest letter S.6 This direct pairing is not classically normative, because S-related triads share no membership in a diatonic scale. In the eighteenth century, S-related triads are less likely to be juxtaposed directly than to substitute for one another, as alternative modal expressions of a mediant or submediant degrees. It is only with Schubert that the progression achieves prominence, both as an indirect relationship between tonics and as a direct juxtaposition between triads.7 S-progressions become more frequent and fluent in Liszt, Wagner, and Prokofiev and have become a staple of film music (Lehman 2010). The Slide transformation nonetheless retains a maverick status among the Weitzmann transformations, similar to H within the hexatonic group. Figure 4.6(a) portrays a Weitzmann region on the Tonnetz. The structure resembles a stalk with six leaves. The thickening of the stalk enhances the presence of the augmented triad as an entity, no longer the mere boundary of chapter 2. Figure 4.6(b) superimposes transformational arrows that model the five progressions studied in connection with figure 4.2, as they act on C major. Four of the arrows pass through a vertex that contains the tone common to the connected triads. The fifth arrow, labeled R, passes across an edge that connects the two tones common to C major and a minor. Here we encounter one of the benefits of treating the augmented-triad axis as an object rather than as a boundary. The latter interpretation conveys the impression that R involves less distance than the other Weitzmann-internal relations (Tymoczko 2009b). Indeed, it suggests that R covers the same distance as L and P, the other common-tone maximizers. From the standpoint of voice-leading magnitude (“work”), this is an illusion. R is a more distant relation than L and P, just as surely as a whole step is larger than a semitone, and is neither closer nor farther than the other four relations internal to a Weitzmann region. Considering the augmented-triad stalk as an object dispels that illusion. This status can be rendered unmistakably clear by expanding the stalk of figure 4.6(a) into the gray parallelogram of figure 4.6(c). Like the consonant triads, the augmented triad is a two-dimensional polygon with boundaries and area conveying its status as a surface in its own right, rather than just a boundary between regions. All voice-leading distances are now “true.” The image is consistent with the hallway metaphor of figure 4.1(b), and with the notion of a

6. Karg-Elert’s treatment of the common-third progression has had a particular influence on harmonic theory in Russia, as documented in Segall 2011. 7. Direct juxtapositions occur in the Characteristic Allegro for Piano Four Hands (Lebensstürme), m. 260 (Rings 2006, 202), and mediated by an augmented triad at the opening of the Sanctus from the Aᅈ major Mass. Prominent structural S-pairings occur in Lebensstürme, the opening two movements of the Bᅈ major Piano Sonata, and the second moment of the String Quintet. Rings’s (2006) analysis of Lebensstürme engages many of the issues treated here.

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Figure 4.6. Three depictions of a Weitzmann region on the Tonnetz. Weitzmann region as a free-commerce zone rather than one that is governed by a natural internal order.

Remarks on the Tonnetz These multiple depictions of the augmented triad represent a tension that has been worked over repeatedly in the last chapter and a half. On the uninflected Tonnetz of Arthur von Oettingen and Hugo Riemann, the major-third axis is simply another boundary that, like those generated by the remaining consonant intervals, contributes to the definition of triads. The area that it attains in figure 4.6(c) suggests its capacity to represent a musical object that might be prolonged for a significant span of compositional time. The ambiguous status of R in figure 4.6 manifests this tension and clarifies what is at stake. On the common-tone basis on which much nineteenth-century harmonic theory rests, R joins its common-tone– maximizing brethren, L and P, at the first rank. The uninflected Tonnetz captures

66

 Audacious Euphony this view, bestowing equivalent status on the three axes generated by the consonant intervals.8 By contrast, from the viewpoint of voice-leading work, measured in semitonal units, L and P alone are the primary triadic relations; R joins the remaining Weitzmann transformations at the second rank. The heavily adapted Tonnetz of figure 4.6(c) captures this second view. A compromise position is suggested by the mild inflection of the major-third axis of figure 4.6(a), which is susceptible to either interpretation. This flexibility may or may not be a good thing, depending on one’s interpretive values and goals. There are good reasons to prefer voice-leading measures to common-tone– based measures and, accordingly, to ship the Tonnetz off to the museum, as Tymoczko 2009b recommends. Figure 4.7 models the distribution of the twentythree types of triadic pairings (i.e., connecting some fixed triad to each of the remaining triads) under three measurements: (a) number of common tones; (b) Tonnetz distance, measured as edge-traversals that represent double-commontone retention; and (c) voice-leading work. Method (c) creates the most categories and most evenly distributes the twenty-three progressions among the available categories. Voice-leading distance is also superior to Tonnetz distance because it provides more intuitively satisfactory results, in those cases where their results disagree. Tymoczko (2009c, 2010) notes that modally matched fifths such as C major → F major are closer on the Tonnetz than are nebenverwandt relations, such as C major → f minor, even though the latter progression involves less voice-leading work. He notes further that these discrepancies become more acute when one moves from the specific case of triadic relations in a twelve-note universe to generalized chordal relations in an n-note universe. In part for these reasons, he uses fused-triad graphs such as figure 2.1, where points represent chords rather than their pitch-class constituents. There are six reasons why I will nonetheless continue to cultivate the Tonnetz as a primary (but not exclusive) mode for representing the triadic universe. First, when the augmented-triad axis is interpreted as an object as in figure 4.6(c), distances on the Tonnetz align with those on the fused-triad graphs. The discrepancies that Tymoczko identifies are thus neutralized, and the Tonnetz can be treated as a true map of voice-leading proximity. Second, although elsewhere

Figure 4.7. Triadic distributions for three distance measures. 8. I adopted that view in Cohn 1997, and it has been developed since by a number of theorists, such as Capuzzo 2004, Goldenberg 2007, and McCreless 2007.

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I have explored nineteenth-century triadic practice against a general backdrop of hypothetical possibilities (Cohn 1997), the primary aims of the present work are more historical and analytical. The limitations of the Tonnetz in the general case are not as limiting for my specific purposes. Third, the “unconformed” (or flat) Tonnetz fixes the directional axes, facilitating comparisons between passages and compositions. Although fused-triad graphs can also be fixed in this way (Tymoczko 2011b, 416–17), their directional axes lose legibility because some of their components recede into a third dimension. Fourth, chapter 6 shows that there are advantages to maintaining individual tones as primary objects, rather than prepackaging them into triads. The Tonnetz can track continuities among individual pitch classes, yielding analytic information that would otherwise be difficult to recover, and provides locations for other pitch-class combinations that arise in nineteenthcentury music. These include consonant dyads and dissonant seventh chords that would otherwise need to be referred to a consonant triad via expansion or contraction, forcing interpretations that might be underdetermined or even arbitrary. Fifth, the Tonnetz maintains contact with a certain mode of historical thinking, recovery of which is part of the project of this book. Finally, because the Tonnetz’s historical origins are associated with ideas about tuning, it presents an opportunity to explore the interplay of the triad’s two forms, and hence of the two syntaxes to which they respectively give rise. This exploration is undertaken in chapter 8. There will nonetheless be circumstances where the fused-triad graphs prove useful. They have their own more recent history, with which we will also want to maintain contact. Their elimination of the atomic pitch-class level makes them trimmer and more geometrically flexible. As shown in chapter 2, they wrap more easily into a cycle. Accordingly, fused-triad graphs are substituted when surfaces are composed exclusively of triads, and when focusing the eye on cyclic closure is more central to the analysis than tracking particular pitch classes or voices.

Historical Origins of Weitzmann Regions I am aware of no eighteenth-century compositions that explore all six triads of a Weitzmann region. Its origins can nonetheless be traced to classical routines, notably involving a i–III–Vᅊ complex. In the most typical case, relative major divides the motion from minor tonic to major dominant. (Particularly in the eighteenth century, such progressions are more likely to occur as structural pillars—middlegrounds—rather than as direct successions. See Schmalfeldt 2011, chap. 8.) Expressed in transformational language, and as depicted in figure 4.8, N is executed by combining R with LP. In a more direct variation, the order of the final two triads can be exchanged. Minor tonic proceeds to its major dominant via N; after a fermata, an LP disjunction, with characteristically paradoxical contrary motion, brings new music in the relative major, characteristically at a faster tempo (LaRue 2001 [1957]). That the three chords form an unordered space during the later part of the eighteenth century is underscored by C. P. E. Bach’s assertion that

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Figure 4.8. Minor tonic, major mediant, and major dominant, as a subset of a Weitzmann region. direct motion from C major to E major can be executed by anyone “who knows that E is the dominant of a and that a minor is very closely related to C major” (Kramer 1985, 552). There are several ways to expand this core progression by adding a fourth triadic member of the Weitzmann region. In a famously perplexing passage from C. P. E. Bach’s f minor Piano Sonata of 1763, Aᅈ major (III) is connected to C major (Vᅉ) through an anomalous Fᅈ major, the tonic’s Slide relation. Although the contemporaneous critic Johann Nikolaus Forkel found no beauty in this passage, he nonetheless defended it on the grounds that it was written by a great composer and ipso facto must contain something of virtue (Kramer 2008, 11–12). Forkel and his contemporaries were likely intrigued and undone by the diatonic indeterminacy of this interpolated chord, which can be understood as Fᅈ major only in retrospect and as E major only in prospect. Once enharmonic distinctions are neutralized in chromatic space, the progression is understood in terms of two parallel PL motions that substitute for the single LP of figure 4.8.9 Figure 4.9 depicts these local progressions as downward motions on the Tonnetz and recuperates dominant to its proper location by means of a broken arrow that recognizes the cyclical structure underlying the planar representation. A more common way to expand the i–III–Vᅊ core is to juxtapose the relative major with its own minor subdominant (ivᅈ of III). Figure 4.10 shows the basic complex of relations. C minor is flanked by its major dominant and relative major. The latter in turn is flanked by its minor subdominant. The transformational symbols indicate that the inner two chords are R-connected, the outer adjacencies are N-connected, and the progression is a contiguous segment of an N/R cycle (figure 4.3). The remote outer triads are connected by S, but the identity of their common third is obscured by the enharmonic distinction indicated by the broken arrow. The Bᅉ/Cᅈ enharmony is essential: no renotation will bring them into orthographic conformance without sacrificing locally consonant and diatonic connections, each of which is fully salient to perception. The relation of these two

9. Recall from chapter 2 that PL and LP progress two stations around a six-station hexatonic cycle, but in opposite directions. A four-station journey in one direction reaches the same destination as a two-station journey in the other; hence, PL × 2 = LP.

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Figure 4.9. Tonnetz model of C. P. E. Bach, Sonata in f minor, H. 173 (Wq. 57/6), 1st mvt., development.

Figure 4.10. A double-agent complex about C minor/Eᅈ major.

tones creates an opportunity for a composer to gently tease the enharmonic seam or, if so inclined, to plunge directly through it. Two Chopin compositions from 1830 illustrate the two options. Figure 4.11(a) presents mm. 8–16 of the first Mazurka (Op. 6 no. 1, fᅊ minor). After tonicizing the relative major at m. 12, Chopin leads three times to its subdominant: twice as a minor triad and once as major. The replacement of Fᅊ for Fᅉ prevents the latter from coming into direct contact with its enharmonic equivalent Eᅊ when the global dominant arrives at the end of that measure.10 Figure 4.11(b) presents the opening phrase of the first Nocturne (Op. 9 no. 1, bᅈ minor). After attaining the relative major at m. 5, Chopin again leads three times to its subdominant, twice as a minor triad and once as a major. Here, though, the order is reversed: Dᅈ’s major subdominant precedes the twice-iterated minor version. The direct motion from gᅈ minor to F major enacts a Slide transformation that transforms Bᅈᅈ3 10. The enharmonic duality of Eᅊ and F (circled in the score) inspired a significant set of analytic musings in Hyer 1994. I cannot, however, follow Hyer in hearing d minor as a local tonic, because of the Bᅉs, and because his suggestion is inconsistent with the relative stability of the later and lower member of motivic semitones (F → E, Fᅊ → Eᅊ) throughout the Mazurka. Similar progressions arise in the finale of Chopin’s b minor Piano Sonata (compare the Bᅈ at m. 12 with the Aᅊ at m. 16) and in his f minor Etude Op. 25 no. 2, where accented Fᅈ in the tenor register at mm. 16–17 anticipates the E4 when dominant arrives on the downbeat of m. 19.

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 Audacious Euphony Figure 4.11. Two early Chopin examples.

directly into A3 in the same voice (as indicated by the arrow); the two notated pitches are metrically accented and analogously positioned in the arpeggiated figuration.11 What is entered as the third of a minor triad, pressing downward by semitone, exits as the third of a major triad, pressing upward by semitone. Figure 4.12 presents Tonnetz models of both phrases. In the Mazurka (a), the motion to d minor at the bottom of the figure is undone, and relative major proceeds to dominant. The minor subdominant is thus a sideshow that delays the 11. Compare Smith 1986, 123–24. I consider the Eᅈ that is sounded with the gᅈ minor triads at both mm. 6 and 7 to be a dissonant under-seventh that transforms at m. 8 into a dissonant over-seventh. Justification of this position is deferred until chapter 7.

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Figure 4.12. Tonnetz models of figure 4.11.

structural motion to the goal dominant. In the Nocturne (b), the S transformation is modeled on the Tonnetz as a downward motion, enharmonically reinterpreting the common tone and landing on the dominant at the bottom of the strip. The identification of that dominant with the one adjacent to the tonic at the top of the strip is indicated by the broken line at the right side of the figure.

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 Audacious Euphony The Double-Agent Complex Developing an idea that was prevalent at the turn of the twentieth century, Daniel Harrison (1994) has suggested that the melodic energy invested in thirds bestows agency upon their constituent harmonies: triadic major thirds strive upward, bearing dominant energy, while triadic minor thirds press downward with subdominant “attitude.” These appellative tendencies are intensified when the thirds are chromatic inflections that displace their diatonic equivalents. In the Chopin Nocturne, the downward agency of Bᅈᅈ is converted instantly into the upward agency of A. I find that I need to sing the pitch continuously in order to convince myself that the two pitches are identified; and even then, it feels like a trick. The redirection of an energized tone through enharmonic transformation is one of Vogler’s (1802) principal genera of Mehrdeutigkeit. In the most familiar case, the root of a diminished seventh chord (FᅊACEᅈ) is reinterpreted as a seventh (ACEᅈGᅈ), redirecting the energy from g minor to its relative major, Bᅈ major. In an analysis of two passages from the first movement of Beethoven’s Fifth Symphony, Lewin recognized the rhetorical potential of this redirection and vested it inside an anthropomorphic parable: [In the exposition] you enter . . . wearing your Fᅊ cloak, as leading tone to G; but you abruptly hurl the cloak away and reveal yourself in a suit underneath as Gᅈ, upper neighbor to F. . . . [At the parallel moment of the recapitulation] everyone is waiting for you throw off your Fᅊ cloak and reveal yourself as Gᅈ. You throw off your Fᅊ cloak all right, but now you are wearing an Fᅊ suit beneath it! You resolve as a leading-tone to G. (2006, 107)

Lewin’s story is a Cold-War allegory, a musical version of a trope that arises in Fleming, LeCarré, Fowles, and dozens of dime-store novels, movies, and television scripts from the era from the end of the war to the fall of the wall. It is the story of the double agent. All is not what it seems. Or perhaps it is what it seems, in which case it is not what it seemed not to seem. And so forth, in a recursive fractal explosion, an eternal ricochet, a toggle that is turtles all the way down. Some double agents are gnarl-visaged trench-coat lurkers, and others sparkle and glisten in honeyed tones and décolletage. The two Chopin passages suggest that musical double agency is found not only in the dissonant harmonies of Vogler and Lewin but in acoustically perfect beauties as well. Both passages present the components of the double-agent complex in a canonical order, and in a canonical tonal context. Consideration of three further pairs of excerpts will demonstrate that nineteenth-century composers found the complex remarkably malleable, both in the ordering of its elements and in the tonal environments in which they operate. The first pair, consisting of Wagner’s Tarnhelm motive (1853; figure 2.4) and the opening phrase of Liszt’s Il Penseroso (c. 1840; figure 4.13) from volume 2 of the Années de pèlerinage, demonstrates the permutational possibilities of the double-agent complex. The Tarnhelm swaps the order of two of Chopin’s

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Figure 4.13. Liszt, Il Penseroso, mm. 1–9.

components: the relative major follows rather than precedes its minor subdominant. This produces a direct LP progression from gᅊ minor to e minor, creating the uncanny effect discussed in chapter 2, and temporarily destabilizing gᅊ minor’s status as tonic. E minor’s tonic potential is suggested by the opening melodic gesture and by the half-cadential rhetoric of the B open fifth. Wagner often exploits this potential by adding an under-seventh F to the initial triad and resolving it as an augmented sixth. B major also accrues potential tonic value, which is activated on those occasions when the second phrase is omitted. At the same time, the salience of the underlying i–III–Vᅊ core is sufficient to prompt a tonally well-formed hearing in gᅊ minor (Darcy 1993, 169–70).12 The opening phrase of Il Penseroso omits the relative major altogether, only to restore it as the goal of its second phrase. The composition opens with an LP progression, as in the Tarnhelm (which it precedes by a decade); after acquiring an Fᅊ under-seventh, a minor proceeds directly to dominant Gᅊ major through a Slide operation that reinterprets C as Bᅊ (as in the Chopin Nocturne).13 The subdominant potential of a minor is exercised in the composition’s second phrase, which ends by tonicizing the relative major, represented as an open fifth as in the Tarnhelm. (In the event, Liszt fills the open fifth as e minor at the beginning of the third phrase.) ^

12. The passages that fit this model problematize Kevin Swinden’s assertion (2005, 261) that ᅈ1 always ^ ^ notationally stands in for the leading tone. Swinden acknowledges that although “bass line (ᅈ)6–(ᅈ)3 . . . may support such structures, I have yet to find convincing examples to demonstrate this pattern.” He cites as typical the Tarnhelm progression (see figure 2.4 in chapter 2 of this book), ^ “where ᅈ1 was treated as a leading tone (F ) in the key of Gᅊ minor.” Yet the subsequent ^ ^ progression of e minor to a BFᅊ dyad presents an instance of what he was unable to find: a 6 → 3 bass ^ ^ in gᅊ minor, supporting a ᅈ1 → (ᅈ)7 soprano. 13. Rings 2011, 78, has some pertinent comments on the affective paradox that this respelling represents.

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 Audacious Euphony Figure 4.14. Two passages from Schubert’s Impromptus, Op. 90.

The double-agent complex is as tonally adaptable as it is permutationally malleable. In each of the examples examined so far, tonic has been situated on the interior minor triad of the four-triad complex. (It is interior with respect to the layout of figure 4.10, where the R-related triads are interior and the S-related ones exterior.) Figure 4.14 presents excerpts that tonicize exterior members of the complex. The excerpts are related by source and by formal function: they begin the codas of Schubert’s second (Eᅈ) and third (Gᅈ) Impromptus (D. 899, Op. 90, 1828), compositions that have been substantively linked in other ways (Fisk 2001, 46–47). The Eᅈ Impromptu is a sectionalized ternary composition, both of whose principal sections terminate with a scalar eᅈ minor leading to a marcato Gᅈ major (figure 4.14(a)). The four measures that begin the b minor interior section are replicated at the beginning of the coda. Upon its initial presentation, this phrase

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Figure 4.15. Tonnetz models of figure 4.14.

elicited a V → i consequent in b minor. In the coda, the i → V antecedent in b minor is answered by V → i in eᅈ minor. The upward transposition by major third across the caesura, transforming dominant Fᅊ major to dominant Bᅈ major via LP, is undone by a downward transposition four measures later, transforming tonic eᅈ minor to tonic b minor, also via LP (figure 4.15(a); compare Rings 2007, 60). The double agency of Eᅈᅈ qua D is quite piquant in the measure following the half cadence, as indicated by the arrows in figure 4.14(a). The third Impromptu’s coda enters its double-agent complex from the opposite side, with Gᅈ-major tonic playing the role of exterior major triad (figure 4.15(b)). The coda begins by transforming Gᅈ major into a local dominant of cᅈ minor, exactly as in the Second Impromptu (figure 4.14(b)). Cᅈ minor proceeds to its relative major, Eᅈᅈ major, which functions as a German sixth chord, immediately restoring Gᅈ major as tonic. (The trio of triads is as at figure 4.8, but here a major tonic arpeggiates down to its subdominant, instead of a minor triad arpeggiating up to its dominant.) The second phrase of the coda retraces the same path but goes a step farther: D major acts as dominant to a fleetingly tonicized g minor, transposing the earlier Gᅈ → cᅈ and completing the four-triad complex.14 Figure 4.14 suggests the tonal adaptability of the double-agent complex, which can in principle center around any of its four triadic constituents.15 The final two examples, presented at figure 4.16, demonstrate that the double-agent complex can also be present where none of its constituent chords serves as tonic. Figure 4.16(a) presents some measures from the first song of Schumann’s Dichterliebe, “Im wunderschönen Monat Mai.” The complex is centered on local tonics b minor and D major, 14. This passage is analyzed in Riemann 1877. 15. Only three of the four possibilities are exemplified here. The fourth possibility, oriented toward the interior major triad, is represented by the development section of Mozart’s Piano Sonata in Bᅈ, K. 333, which makes much of the enharmonic relation between Gᅈ qua subdominant agent and Fᅊ qua dominant agent of an unrealized g minor tonic.

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 Audacious Euphony Figure 4.16. Two nontonic examples of a double-agent complex.

neither of which is a candidate for global tonic. The same complex appears, untransposed and in the same register, at mm. 3–5 of “Ich will meine Seele tauschen,” the fifth song of Dichterliebe, which is in b minor (Komar 1971, 75). Brahms’s song “Von ewiger Liebe” (Op. 43 no. 1), also in b minor, presents an even more tonally remote version of the double-agent complex. After the first couplet tonicizes the mediant, the second couplet, presented as figure 4.16(b), begins with its modal variant, d minor. A double-agent complex sets the couple’s journey through the smokeless darkness, “Nirgend noch Licht und nirgend noch Rauch.” The parallel halves of the text are set by a parallel set of motions from a tonic to its major dominant: first d minor → A major, and then fᅊ minor → Cᅊ major, each harmony extending for one measure. The four-measure unit thus connects the tonic’s minorized mediant, whose F agent discharges downward, to the dominant of its dominant, whose Eᅊ agent presses upward (see Karg-Elert 1930, 271).

Expanded N/R Chains Although the four-element double-agent complex attained a life of its own in the nineteenth century, it also began to expand in the direction of the fully ramified

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Figure 4.17. Three expanded N/R chains.

six-element N/R cycle, and complete Weitzmann region, that we find already in Schubert (see figures 4.4 and 4.5). Figure 4.17 models two chromatic excerpts from late-eighteenth-century pieces, both of which embed a double-agent complex within a five-triad N/R chain. As it happens, both progressions feature the same set of triads, presented in the same order, although the progressions are embedded in compositions whose tonics are a tritone apart! The first excerpt is from Jiří Benda’s Piano Sonata in a minor (1783), of which Rey Longyear and Kate Covington (1988) write that “in the eighteen measures of the second theme-group the composer starts in C major (III), pauses on a half cadence in E minor, then goes to such remote keys as C minor and Aᅈ major before the sudden modal mutation into C major for the closing theme-group” (454). Figure 4.17(a) models the passage beginning from the B7 caesura, whose resolution to e minor occurs in 63 position, precipitating a series of harmonies over a G pedal, and connecting e minor to its Slide-related Eᅈ major through an alternation of N and R. The entire

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 Audacious Euphony N/R chain-segment connects B major to Eᅈ major, neither of which acquires an easy role in the overall C major prolongation. Figure 4.17(b), adapted from an analysis of James Webster (1991, 321), models a passage from the opening Adagio of Haydn’s Symphony no. 99 (1793) in Eᅈ major. Haydn comes upon B major by renotating the upper neighbor of Eᅈ’s dominant. The five-chord progression issuing from B major is the same as Benda’s, although the final connection between C minor and tonic Eᅈ major is divided by a retrogression to a prolonged G major, acting as the former’s back-relating dominant. The culmination of this process of expansion can be seen at figure 4.17(c), from Schubert’s Piano Sonata in c minor (1828), where the sequential progression of Haydn’s Symphony is compressed and extended to cover the entire six-triad Weitzmann region as a fully ramified N/R chain, such as that presented earlier in connection with figures 4.4 and 4.5. In sketching the expansion from the four-chord double-agent complex, which might merely tease the enharmonic seam, to the six-chord N/R chain, which necessarily plunges through it, we have purchased a view onto an incremental historical process that moves from the circumscribed realm of classically determinate diatonic tonality into the symmetric chromatic universe of twelve equally tempered tones. This opens a file drawer into which material will be inserted in the coming chapters. In the final chapter of this book, I shall attempt to organize those contents into a historical framework.

Weitzmann Regions without Sequences: Wagner and Strauss The flat-terrain status of a Weitzmann region dictates that not all of its progressions will be sequential or cyclic. One of many possible nonsequential paths is presented in figure 4.18, a passage from act 1 of Parsifal. The passage is locally in b minor, which in the larger context represents the minor dominant within a tonally Figure 4.18. Wagner, Parsifal, act 1, mm. 404–13.

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closed section in e minor. Beginning at m. 405, a locally subdominant e minor spins into its Weitzmann region, sounding its minor triads in succession, followed by its major ones. An augmented sixth is attached to the final G major and resolves to the Fᅊ major dominant, reintegrating into the b minor orbit, preparing an e minor cadence some seven measures later.

Figure 4.19. Strauss, “Frühling” (Four Last Songs), mm. 4–33.

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 Audacious Euphony We conclude with an excerpt at the near end of the long nineteenth century, composed nearly two centuries after C. P. E. Bach’s sonata. The opening stanza of “Frühling,” from Richard Strauss’s Orchesterlieder (1949), is organized around a Weitzmann region, although Strauss’s characteristically restless harmonic practice requires the analyst to wrestle the interpretation to the ground a bit. Figure 4.19 provides a sparse score of the opening thirty-three measures, and figure 4.20 provides a series of analytical snapshots. The score is available at Web score 4.19 . The alternation between c minor and aᅈ minor (considered earlier in connection with figure 2.14) moves at m. 10 to the latter’s relative major, B major, combining LP and R to form S, and exhausting half of the Weitzmann region (figure 4.20(a)). After visiting several harmonies outside the region (A major and Bᅈ major, not shown), the first stanza concludes with a melismatic setting of “Vogelsang” (birdsong) that uses the region’s remaining components, cadencing on Eᅈ major (figure 4.20(b)). Here the previous transformational progression is reversed: a broad S-gesture, connecting e minor to Eᅈ major, is bisected by G major, the former’s relative major, so that here S is formed by combining R with PL. In figure 4.20, (a) and (b), summarizing the harmonic motion so far, make evident one additional feature: the six bass pitches form a whole-tone collection (Kaplan 1994), permuting the whole-tone scale used by Schubert in his C major Symphony and Octet (see figure 4.5). The relative major now achieved, an orchestral interlude progresses to dominant using a double-agent juxtaposition much like the opening of Chopin’s bᅈ minor Nocturne (see figure 4.11(b)). After a return trip to tonic, Eᅈ major proceeds to its minor subdominant, at m. 24. The latter progresses immediately to the cadential dominant, threading the seam and juxtaposing double agents Cᅈ and B (figure 4.20(c)). The cadential dominant discharges onto C major, moving into a different Weitzmann region for the start of the second verse at m. 29. From a transformational standpoint, this second region is established in the same way as its predecessor: with an S gesture connecting C major to cᅊ minor, routed through the latter’s Relative major, E major (figure 4.20(d)). Figure 4.21 summarizes the first thirty-three measures of “Frühling” on the Tonnetz. The crooks overlaying the figure indicate the appearances of the R + LP = S motive, as it is presented at the opening of the first stanza, retrograded

Figure 4.20. Sparse-score model of “Frühling”.

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Figure 4.21. Tonnetz model of Frühling. at the close of the first stanza, fragmented into its component transformations in the interlude, and inverted at the opening of the second stanza. The figure presents the image of a continuous cascade along the triads of the region structured by GBEᅈ, including two enharmonic transformations, followed by a shift to the region centered on CEAᅈ. With this observation, we have ruptured the boundaries of the individual Weitzmann region and started to explore the ways that adjacent regions connect and collaborate. In doing so, we have trespassed into territory to be explored in chapters 5 and 6.

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C HA P T E R

Five

A Unified Model of Triadic Voice-Leading Space

How Hexatonic and Weitzmann Regions Interact We have studied the internal structures of the hexatonic and Weitzmann regions and characteristic nineteenth-century passages that remain within one of them. The challenge now is to traverse the boundaries of the individual regions, in order to view the larger triadic universe as a connected system under efficient voice leading. The four hexatonic regions, and their constituent transformations, are closed, mutually disconnected systems (in a mathematical sense, they are groups), as are the four Weitzmann regions. On their own, the regions are static, like a right and left leg hopping in place. In collaboration, Weitzmann and hexatonic transformations gain the power of perambulation, unifying the triadic universe. Producing a connected universe from its constituent parts makes good on the aphorism “any chord can go to any other chord.” Yet it does so without unbridling the twenty-four triads into an aleatoric torrent.1 The connected universe affords a method of evaluating the voice-leading distance between any pair of consonant triads, of recognizing patterns of motion, and of judging the coherence of progressions on the basis of their voice-leading properties. Once we understand the mechanisms of coordination, and the anatomy of the larger universe that ensues, we will be in a position to explore progressions that are unidirectional, moving up or down through registral space, or clockwise or counterclockwise through pitch-class space. The point of entry is a figure from Weitzmann’s Der übermässige Dreiklang, presented as figure 3.9 and reproduced here with overlay as figure 5.1. Weitzmann created this figure in order to portray the first- and second-order relations of each of the four augmented triads. In the first tier of triads, highlighted in this version of the figure, are those that can be reached by displacing one of the augmented triad’s tones by one semitone; together, these six triads constitute its 1. The aphorism is attributed variously to Weitzmann, Liszt, and Reger; the metaphorical gumbo is cooked to a recipe supplied by Harrison 1994, 1–7.

83

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 Audacious Euphony Figure 5.1. Figure 3.9 with Weitzmann regions overlaid. Upper- and lower-case letters are the roots of major and minor triads respectively.

Weitzmann region. The second tier contains those triads that can be reached by displacing two of its tones in the same direction. Although Weitzmann has nothing further to say about this figure, several aspects of it merit our attention. First, those consonant triads that displace the augmented triad downward are consistently positioned to its left, and conversely, those that displace its tones upward are positioned to its right, conferring a directional consistency: within each quadrant of the figure, rightward motion in graphic space is associated with upshifting in pitch space.2 Second, each cluster of triads appears twice in the figure: those that appear to the right of a given augmented triad are identical in content to those that appear to the left of its rightward neighbor, and vice versa. Pruning the redundancies leaves only those twenty-four triads highlighted by the overlay. Third, each group of consonant triads that separates adjacent augmented ones forms a hexatonic region (although its six components are not arranged into a cycle). Fourth, the entire figure is implicitly cyclic: the same consonant triads appear at its left and right margins. Taken together, these observations imply figure 5.2. Its cardinal points contain the four augmented triads, each of which is connected to the six triads of its Weitzmann region: minor triads to its clockwise side, major to its counterclockwise side. Left-to-right ordering in figure 5.1 corresponds to clockwise ordering in figure 5.2. Intersecting the Weitzmann regions are the hexatonic regions, shown as textured pools. The figure suggests that the Weitzmann and hexatonic regions are in a figure–ground relation. For the hexatonic mariner, the Weitzmann regions form bridges. For the Weitzmann landlubber, the hexatonic regions are rivers to cross. To circumnavigate the entire complex, one need only locate an appropriately amphibious vehicle. Figure 5.3 gives structure to the hexatonic pools by connecting their constituent triads into the familiar hexatonic cycles. The cosmetic distortions of the hexatonic cycles align modally matched triads on the same side of the figure, in proximity to the augmented triad that they displace. Figure 5.3 is a version of Cube Dance, a graph created by Jack Douthett in 1992 and published in Douthett and Steinbach 1998. Cube Dance is a “true” model of voice-leading distance between triads: “Every distance can be interpreted as representing voice-leading size” (Tymoczko 2009b, 271).3 Directional motion on the graph correlates consistently 2. Upshift and downshift, from Lewin 1998, refer to directed voice leading in pitch or pitch-class space. An ordered pair of chords upshifts if, in the transition from one to the other, more voices move up than down. The same pair downshifts when the order is reversed. The terms may apply either to register-specific pitch sets, such asᇳC4, E4, G4ᇴ→ᇳC4, F4, A4ᇴ , or to pitch-class sets under idealized voice leading, such as C major → F major. 3. This observation first appeared in Tymoczko 2006, which positions Cube Dance as a contiguous sector of the continuous space representing all three-chords. See also Tymoczko 2011b, chap. 3.

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Figure 5.2. Four Weitzmann water bugs in union with four hexatonic pools.

with melodic direction: clockwise and counterclockwise motion denote upshifting and downshifting, respectively. Since Cube Dance includes the four hexatonic cycles and the four Weitzmann regions as contiguous subgraphs, it models all of the passages that are internal to them, as studied in chapters 2 and 4, respectively. But it also supplies, for the first time, a way to model progressions that move between regions, which is the usual case. To get a preliminary sense of how Cube Dance models a composition that crosses boundaries between regions, consider the Adagio opening of the Overture to Schubert’s Die Zauberharfe from 1821. Schubert later used the same composition as the Overture to Rosamunde, under which name it is normally performed today. The score is available at Web score 5.4 . After an eight-measure fanfare, the oboe sounds an antecedent phrase in c minor, followed by a consequent in its relative major. After an Eᅈ major cadence, Schubert pulsates on this chord and then drops G to Gᅈ and pulsates further on eᅈ minor. The bass now slips down through D (forming a transient augmented triad) to Dᅈ, supporting a cadential 64 that resolves in classical fashion to Gᅈ major. After a tonally closed period in Gᅈ major, the same sequence of events occurs twice in transposition: pulsations on Gᅈ major and fᅊ minor, transient motion through FACᅊ, cadence in A major; pulsations on A major and a minor, transient motion through AᅈCE, cadential 64 in C major. In the event, the latter initiates a deceptive motion to Aᅈ major, and a standing-on-the-dominant, before proceeding with a C major Allegro.

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Figure 5.3. Jack Douthett’s Cube Dance. The four hexatonic cycles are portrayed by the circuits of unbroken lines; the four Weitzmann regions by the broken-line “water bugs.”

Figure 5.4 charts the path of prolonged harmonies on Cube Dance, together with the transient augmented triads. The figure brings out the modulatory progression’s semitonal logic, its arc of motion away from and then back toward an origin, and its constant downshifting. Figure 5.5 shows the same passage on a Tonnetz that expands and unites the segments presented in chapters 2 and 4. Like those earlier graphs, perfect fifths rise to the right, major thirds rise toward the northeast, minor thirds “rise” toward the southeast, and major and minor triads are represented, respectively, by up-tipped and down-tipped triangles. The figure conjoins adjacent hexatonic systems at their shared major third axes. Figure 5.6 presents a complementary conception of how this Tonnetz is assembled, from the standpoint of the four Weitzmann regions. If the major-third axes are interpreted as boundaries, then the Tonnetz is a relation graph of the twenty-four triads under maximal pitch-class intersection (adjacent triads share two tones). If interpreted as locations in their own right, as in figure 4.6(c), each axis represents an augmented triad, and the Tonnetz becomes, like Cube Dance, a model of the twenty-four consonant and four augmented triads under single semitonal displacement.

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Figure 5.4. Schubert, Overture to Die Zauberharfe, opening measures, portrayed on Cube Dance.

Figure 5.5. Schubert, Overture to Die Zauberharfe, portrayed on the Tonnetz.

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Figure 5.6. Connected Tonnetz assembled from the four Weitzmann regions. Broken edges indicate where corresponding points are glued together.

Under this second interpretation, the Tonnetz shares the characteristic that Tymoczko (2009c) demonstrated for Cube Dance: “Every distance can be interpreted as representing voice leading size.” Unlike Cube Dance, the Tonnetz suffers from a Bering Strait distortion, and in two distinct dimensions: locations at the left side of the graph replicate those at the right, and locations at the bottom of the graph replicate those at the top. Were we freed from the dimensional constraints of the printed page, these distortions could be corrected by gluing together pitch-class identities in one dimension to create a cylinder and then in a second dimension to wrap the cylinder into the shape of a doughnut, or what topologists call a torus. Although we can avoid these distortions by reverting to Cube Dance, we would lose some analytically valuable information and historical connections, as discussed in chapter 4. The progression from Schubert’s overture is depicted in figure 5.5 as a direct motion along the main diagonal. As such, it resembles the hexatonic progressions that move along the opposite diagonal. This resemblance, however, occludes a distinction emphasized in chapter 2. Vectors that parallel the major-third axis preserve a constant center of balance, as upshifting and downshifting voices cancel out. Any vector orthogonal to that axis represents voice leading that either cascades or escalates. One might imagine figure 5.5 as a staircase seen from far above. Each hexatonic strip is a tread, and each augmented triad is a riser. From our distant vantage point, step height is equalized, and we must imagine which directions correspond to up and down. The downward direction of the vector in figure 5.5 is consistent with the downward direction of Schubert’s voice leading. The rightward direction of that vector, however, is problematic from that same standpoint. Passage of time is associated with left-to-right motion by virtue of European orthographic convention, with clockwise motion by virtue of clock conventions, and with upward motion by virtue of organismal physiology. As a result of cognitive blending beneath the

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threshold of awareness, upshifting in pitch space is spontaneously identified with rightward motion across the page, as in figure 5.1, and with clockwise motion about a circle, as in Cube Dance. The Tonnetz confounds those intuitions, by identifying rightward motion with downshifting in pitch or pitch-class space. To reengage those default intuitions, it is tempting to recast the Tonnetz by inverting it around its vertical axis. I resist this temptation in order to retain the orientation familiar to veteran Tonnetz surfers. It will also serve the project of the final chapters, when historical issues come to the fore.4 Cube Dance and the Tonnetz present us with two essentially equivalent modes for representing triadic progressions from the standpoint of their voice-leading properties. Each graph has its own heuristic advantages. The Tonnetz is more compact, shows individual pitch classes rather than fusing them into triads, preserves connections with historical modes of thought, is more amenable to tracing common tones, and consistently correlates planar directions with transformations. Cube Dance is a more direct model of true voice-leading distance, both because it does not require an exercise of imagination across the Bering Strait and because it gives the augmented triads explicit locations. We shall move back and forth between these two models in response to the characteristics of the particular composition at hand.

Chromatic Sequences In chapters 2 and 4 we studied sequences that transpose by a series of major thirds. These sequences exhibit balanced voice leading and are internal to a hexatonic or Weitzmann region. We are now in a position to study those sequences that move parsimoniously between adjacent regions, in some cases circumnavigating the triadic universe. The transpositional values that generate those sequences represent an odd number of semitones. Transposition by semitone (T1, T11) engages all three voices in parallel motion. The remaining transpositions (T3, T5, T7, and T9) feature two voices moving in similar motion, one by semitone and the other by whole step. Thus, all odd-value transpositions involve three units of semitonal work, ranking them next on the scale of parsimony, after the hexatonic and Weitzmann transformations already studied. In incrementing from two to three units of voice-leading work, we have crossed over a threshold. Just as a three-semitone motion in a single voice is characteristically conceived as a leap that implicitly combines and elides across two distinct 4. The historical orientation arises because the designers of early Tonnetze were concerned with acoustic ratios rather than melodic motion. Euler, Oettingen, and Riemann conceived C → B as a rising major seventh, conjoining a 3:2 fifth and a 5:4 major third, and it is this upward motion that projects from left to right on their Tonnetze, in accordance with European orthographics and organicist/ teleological conditioning. Interpreting C → B as a melodic motion reverses the directional flow, so that upshifting now travels the Tonnetz from right to left. Incidentally, this suggests that these historical theorists thought of voice leading in idealized terms, rather than in terms of the pitch-class paths emphasized in Tymoczko’s work.

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Minor-third (T±3) sequences The minor third transpositions in Die Zauberharfe result from the alternation of R (from the Weitzmann group) and P (from the hexatonic group). When modeled

Table 5.1. Combination table of W- and H-group operations R

N

S

L

T±5

T±1

T±3

P

T±3

T±5

T±1

H

T±1

T±3

T±5

Each number indicates the number of semitones that a triad is transposed when the H-group operation at the head of its row is combined with the W-group operation at the head of its column. The transposition may be up or down, depending on the order of the operations and on whether the initial triad is major or minor.

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Figure 5.7. Schubert, Drei Klavierstücke, D. 946, no. 2, mm. 46–52.

on the Tonnetz (figure 5.5), the bounding axes are generated by minor thirds, forming diminished seventh chords that combine to form an octatonic collection. The P/R cycle thus can be considered as a primordial triadic generator for octatonic systems, analogous to the role of L and P with respect to a hexatonic system.5 A minor-third sequence is also formed by two other combinations of H- and W-group operations: N with H, and S with L. Since both pairings involve one rare operation, they occur less frequently. Figure 5.7 illustrates a T3 sequence formed by an N/H chain, from a late Schubert piece published posthumously as the second of the Drei Klavierstücke. In this passage, the classically normative N transformation occurs within each segment, and the maverick H occurs across phrasing boundaries. In later compositions, these functions are swapped: Liszt’s Malediction for piano and string orchestra (seven measures after rehearsal B), RimskyKorsakov’s orchestral tone poem Skazka (see Taruskin 1996, I: 271), and Bruckner’s choral motet “Ecce Sacerdos” (from rehearsal 2) all feature H within phrases, and N between them. The alternation of S and L generates an extended downshift sequence (offset by upward registral transfers) in Liszt’s Grande fantaisie symphonique for piano and orchestra (“Lélio Fantasy,” 1834; figure 5.8). The opening of the Sanctus from Schubert’s Aᅈ major Mass follows a similar progression in the upshifting direction (with falling bass), although the insertion of a back-relating dominant after S makes the voice leading less efficient than in the Liszt passage presented here.

5. This generative parallelism is but one reflector of a broader structural parallelism that relates hexatonic and octatonic organizations of chromatic space. See Cohn 1991 and 1997, Lerdahl 2001, 258, and Tymoczko 2011b, 125. Siciliano 2005b and Goldenberg 2007 present further examples of octatonic R/P chains, in Schubert and Smetana, respectively.

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Perfect-fifth (T±5) sequences Chromatic sequences that modulate by perfect fourth or fifth become common quite early, due to the classically privileged position of that interval. Lowinsky 1989 [1967] calls attention to a number of hexachord fantasias that upshift by perfect fourth, the earliest of which originate in Elizabethan-era England. The most common T5 sequence results from the alternation of L with R. It is found in a circle diagram of Werckmeister (1698), where it served as a map of modulations between keys (tonics) rather than direct progressions between chords. It was recognized as a more local harmonic map by Georg Vogler in 1778 (Brower 2008, 99n41) and Honoré de Langlé in 1797 (Damschroder 2008, 80). Carl Stein recommended that piano students learn to play the entire twenty-four-triad cycle, “as it forms the groundwork on which may be constructed an almost infinite number of passages and variations,” noting the progression’s symmetry and its “double union of intervals—two of them always remaining undisturbed” (1888, 38).6 Eventually, the L/R chain became the central component of Moritz Hauptmann’s influential conception of tonal space (1888 [1853]). Although the cycle is too long to be circumnavigated in a single gesture, a remarkable passage from the Scherzo of Beethoven’s Ninth Symphony comes close. A reduced score is available at Web score 5.9 . The progression begins on C major and traverses nineteen triads before terminating at A major, completing more than two tours of Cube Dance (Cohn 1991, 1992, 1997). On the Tonnetz, it requires more space than can legibly fit on a page, moving from right to left along the horizontal axis. Figure 5.9 presents an example of more modest dimensions and more leisurely tempo, from the first movement of Brahms’s D major Symphony, Op. 73. The passage presents the liquidating consequent of the movement’s fᅊ minor theme, in preparation for the arrival of the dominant 6. Cohn 1997 attributes the quote directly to Logier 1827, but it was introduced by Stein in his 1888 adaptation of Logier’s text for an American readership.

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Figure 5.9. Brahms, Symphony no. 2, 1st mvt., mm. 102–18.

(mm. 106–18). It exits the fᅊ minor diatonic system at Gᅉ. The subsequent Cᅉ and Fᅉ bring us into another realm entirely, as chromatic alterations of root and fifth. By this point, we are in free fall in the bass, and “free rise” in the upper voices, without any way of predicting the duration of this sequential process. Figure 5.10 depicts this progression on the Tonnetz as a leftward drift (see also Web animation 5.10 ). The sequence terminates at its ninth station, d minor. Figure 5.11 traces the same passage on Cube Dance, starting and ending at 4:00 and traveling clockwise. The representation suggests that d minor brings a measure of closure to an R/L chain initiated from fᅊ minor, by virtue of the return to the original radius. A second way to effect a sequence by perfect fourths is through the alternation of N and P. Figure 5.12 indicates that the progression is equally effective in both directions. For Daniel Harrison (1994, 33–34), this pair of progressions typifies a set of dualisms that includes major/minor, plagal/authentic, dominant/subdominant, and upward/downward leading-tone energy. Thus, in the authentic direction of progression (a), minor tonics are preceded by their major dominants, whose agents discharge upward; in the plagal direction of progression (b), major tonics are preceded by their minor subdominants, whose agents discharge downward.

Figure 5.10. Tonnetz model of figure 5.9. An animation, with recorded performance, is at Web animation 5.10.

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Figure 5.11. Cube Dance model of Brahms excerpt (figure 5.9). Both directions are well represented in the literature; Harrison points to passages from J. S. Bach’s g minor Organ Fantasy and Schubert’s d minor String Quartet as exemplary.

Semitone (T±1) sequences The most common, and chronologically earliest, option involves the alternation of N and L. The ascending progression at figure 5.13(a) appears more frequently; one Figure 5.12. T5 and T7 sequences generated by N/P alternation.

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Figure 5.13. T1 and T11 sequences generated by N/L alternation.

example, from among many, is discussed below (see figure 5.21). Although the descending progression is less frequent, it was used by Rameau in one of the most famously expressive passages in the history of pre-Classical opera (Rehding 2005).7 More rare is the progression at figure 5.14, excerpted from Liszt’s Il Penseroso, which connects Gᅈ major to D major through a series of semitonal descents, first lowering the third of each major triad and then lowering the root and fifth of each minor triad. Like the Brahms passage illustrated in figure 5.11, the entire S/P chain executes a complete tour around Cube Dance, elliptically connecting two PL-related harmonies that stand directly adjacent in the same hexatonic and Weitzmann region.8

Transformational Substitutions A composer who pursues a chromatic sequence beyond several stages, and lacks a programmatic motivation for doing so, risks being perceived as mechanical, lacking invention, and so forth. Some of the excerpts that have arisen in this exposition preserve their stake to more aesthetically positive qualities by varying the pace Figure 5.14. T11 sequence generated by P/S alternation in Liszt’s Il Penseroso, mm. 17–20.

7. Langlé 1797, 82–83, includes synthetic models of both progressions among his tours de l’harmonie. Ahn 2003, 105–7, shows that both directions are represented in Liszt’s “Wilde Jagd” (Études d’exécution transcendante, 1852). 8. Gollin 2000, 306, identifies a similar progression in Prokofiev’s War and Peace.

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 Audacious Euphony with which the stations of the sequence are delivered. But a composer who wishes to circumnavigate the pan-triadic universe can introduce variation in the progression of triads, breaking the sequential pattern while maintaining the voice-leading trajectory and its pacing. This can be done by substituting, for any triad in a sequence, one of its major-third transpositions.9 Describing the same phenomenon in transformational terms, we can say that transformations within the same group can freely substitute for one another without perturbing the voice-leading trajectory. Represented on Cube Dance, the directional trajectory continues but traverses different nodes. Represented on the Tonnetz, vectors maintain their angle, but crooks displace their position on the plane. Beyond introducing variety to the transformational and transpositional palette, these substitutions can also provide a shortcut to a tonal goal that would be remote were the sequence to strictly run its course. If an H/W-group pairing induces a chromatic sequence that transposes by semitone or fifth, then the triadic chain must traverse all twenty-four triads before closing back to its triad of origin. Major-third substitution allows a goal triad to move forward in the chain by eight or sixteen stations, expediting its arrival without interrupting or reversing the voice-leading trajectory of the passage. Figure 5.15 presents an excerpt from the Die Erlöseten des Herrn fugue, from Brahms’s German Requiem, as a clear introduction to transformational substitutions. The initial fourfold setting of “wird weg” progresses from bᅈ minor to B major (an enharmonic proxy for Cᅈ major) through an upshifting L/R chain, each chord progressing to its diatonic submediant. At “müssen,” N substitutes for R, taking B major to its minor subdominant, e minor, rather than its submediant, gᅊ minor. The L/R chain immediately resumes, carried in the voices by a stretto of the fugal incipit, and is carried through six further stations, terminating at Bᅈ major, the movement’s tonic. The connection from bᅈ minor to Bᅈ major, through an unimpeded L/R chain, would require seventeen stations; the N-for-R substitution allows the target to arrive after nine moves. Figure 5.16 depicts the progression as a leftward motion on the Tonnetz, interrupted by a diagonal jog that transfers files, marking the substitution of e minor for gᅊ minor, and N for R (see also Web animation 5.16 ).10 The fault lines, at a 45° angle, show the orthogonal cut against the major-third regions, upshifting across all four major third alleys and restoring the original. In Chopin’s g minor Ballade, Op. 23 (1835), a mediant chain runs in the opposite direction and requires two N-for-R substitutions to achieve tonal closure. The extended Eᅈ major prolongation in the middle of the Ballade is dominated by the earworm for which the composition is famous, sounded in Eᅈ major at m. 68, in the tritone-related A major at m. 106, and again in Eᅈ major at m. 167. In the abstract, an antipodal progression of this type can be conceived as a reversal, or as the continuation of a trajectory that leads around the back side of the chromatic 9. For further discussion and examples, see Cohn 1998b. Tymoczko 2011b, 87–88, 282–83, describes the same phenomenon using a different geometric layout, which resembles Cube Dance but has an additional virtue: by projecting into a third dimension, Tymoczko’s continuous space consistently affiliates root interval with direction along a single axis. 10. For a different view of this passage, see Brown, Dempster, and Headlam 1997, 173–75.

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Figure 5.15. Brahms, Ein deutsches Requiem, 2nd mvt., mm. 261–71.

universe, executing a two-stage cycle of tritone transpositions. The latter conception is more pertinent in this case: although not transpositionally related, the two tritone modulations downshift along parallel tonal trajectories. Initial Eᅈ major proceeds through a downshifting L/R chain, each triad progressing to its diatonic mediant (mm. 90–93). The chain’s terminus, d minor, is the minor subdominant of A major, which it reaches after an extended dominant. A similar L/R chain connects A major to gᅊ minor (mm. 115–24), although its internal terms are

Figure 5.16. Tonnetz model of figure 5.15. An animation, with recorded performance, is at Web animation 5.16.

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Figure 5.17. Tonnetz model of Chopin Ballade, Op. 23, mm. 68–167. permuted, with E major preceding cᅊ minor rather than following it. gᅊ minor is accorded an under-seventh at m. 124, attaining subdominant function in Eᅈ major, to which it leads after a prolonged Bᅈ dominant. Figure 5.17 sketches the progression as a rightward motion on the Tonnetz but with two northeast-jogging N-for-R substitutions, the first taking d minor to A major rather than the expected F major, and the second performing the same function at a tritone transposition. The two file transfers move the tonic forward by sixteen stations in the L/R chain while preserving its overall downshift trajectory.11 N-for-R substitutions also guide the opening of the development from Bruckner’s Third Symphony, which connects f minor to a minor through two transpositionally related phrases, each modulating up a whole step. Each phrase begins with a slowly evolving series of soundsheet arpeggios that upshift across four stations of an L/R chain, and ends with an N substitution that carries a major dominant to a minor tonic. Described in this way, the transformational progression should be identical to that of the Chopin Ballade (although retrograded on the Tonnetz, since it begins with a minor rather than major triad). Yet the phrases are transpositionally related by whole step rather than tritone, because a PL transformation (outfitted as motion from a major tonic to its German sixth) is interpolated prior to the N substitution. As shown in figure 5.18, PL’s balanced voice leading causes the progression to linger within a Weitzmann region (Gᅈ major → D major → g minor), putting the upshifting juggernaut on pause while a second file transfer is executed, to supplement the one provided by the N substitution. As with the Chopin Ballade, the combined phrases execute a complete traversal of the four major-third alleys (this time moving from right to left). A fourth passage based on an L/R chain, from Chopin’s f minor Fantasy, Op. 49, presents a variation on the phenomenon we have been studying. A reduction is given as figure 5.19; the score is available at Web score 5.19 . The transition from the opening Lento to the agitato core arpeggiates upward through a series of overtone-distributed triads whose highest pitches are circled by inverse cambiatas. An antecedent phrase downshifts through six links of an L/R chain, connecting f minor to Bᅈ major. The goal triad acquires a seventh and resolves across the fermata to its nebenverwandt, eᅈ minor, from which a consequent phrase 11. Schenker 1954 [1906], 300, ventures a similar reading. For discussion and an alternative reading, see Brown, Dempster, and Headlam 1997, 167–69.

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Figure 5.18. Tonnetz model of Bruckner, Symphony no. 3, 1st mvt., opening of the development.

L/R-chains back to tonic f minor. Like the previous two passages, N is interpolated within an L/R chain. What is different about this passage is that the interpolated N functions not to continue the downshift trajectory but rather to temporarily reverse it: Bᅈ major → eᅈ minor is an upshift amidst a cascade of downshifts. This circumstance results from N’s interpolation not between two H-group progressions, as in the previous examples, but rather between two operations from its own W-group. Consequently, the rapid turnover of H- and W-regions, characteristic of all of the chains we are studying in this section, is slowed by a momentary tarrying within a single W-region, via the R/N-chain segment that connects g minor to Gᅈ major. Figure 5.20 represents the passage on a Tonnetz (see also Web animation 5.20 ). As in the previous three examples, there is a horizontal migration by five alleys, interrupted by a file transfer that expedites the return of the tonic. What is new here is the sharp angle of the jog, representing a temporary retreat before further advance. It is also worth noting here, anticipating a topic of chapter 6, that the internal six triads of the progression constitute and exhaust a pitch retention neighborhood, represented by the hexagon about Bᅈ.

Figure 5.19. Chopin, Fantasie, Op. 49, mm. 43–68.

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Figure 5.20. Tonnetz model of figure 5.19. An animation, with recorded performance, is at Web animation 5.20.

One final example, from Liszt’s Grande fantaisie symphonique for piano and orchestra (“Lélio Fantasy”) of 1834, demonstrates that H/W substitutions are not limited to L/R chains.12 Measures 379–429 feature an oboe melody that is transposed successively upward by semitone. Figure 5.21 presents a synopsis of the underlying harmonies, which execute a segment of an N/L chain, transposing from Dᅈ major to E major. The latter’s nebenverwandt, a minor, proceeds not to the projected F major via L but rather to C major via R. As with the Chopin Fantasy, this amounts to a temporary retreat, briefly colonizing a Weitzmann region. From C major, the N/L chain now proceeds on its course, arriving not at the projected F major but rather on the tonic Dᅈ major. Figure 5.22 summarizes the progression on the Tonnetz. Having examined these five passages, this is a good moment to stand back and evaluate the Tonnetz as a mode of representation. Its directional vectors reflect the tight patterning of chromatic sequence, as well as the disruption of those sequences when substitutions occur. The superposition of the 45° fault lines encourages the tracking of voice-leading trajectories. But a limitation of the Tonnetz is beginning to come into view. The Tonnetz makes a two-dimensional presentation of an underlying phenomenon that, under certain conditions, inhabits three or more dimensions. Those conditions are met when we accept enharmonic equivalence, as we must in the case of chromatic sequences. At that point, the Tonnetz disguises cyclic closure. Each of the four passages studied in this section proceeds through exactly five hexatonic regions, each represented by an augmented-triad-bounded alley, and exactly five Weitzmann regions, each represented by an augmented-triad

Figure 5.21. Liszt, Grande fantaisie symphonique über Themen aus Berlioz’ “Lélio,” mm. 379–439.

12. See also Brahms’s “Walpurgisnacht.”

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Figure 5.22. Tonnetz model of figure 5.21.

stalk. Each type of region is represented by four species, one per augmented triad, and so a progression through five regions closes a cycle. But this cyclic closure is not represented on the Tonnetz. This is the domain where Cube Dance excels. Figure 5.23 gives a Cube Dance representation of the passage from the Liszt Fantasy. The graph shows the passage’s consistent clockwise = upshift trajectory, the temporary downshift to C major, and the recontinuation as C major progresses toward cyclic closure at the return of Dᅈ major. What the graph fails to honor is the sequential nature of the upshift progression. The progression could have selected one triad at random from each trio of T4-related triads that constitute the intersection of an H- and W-region; the cyclic progression would have looked just as orderly. This suggests that Cube Dance, with its cyclic explicitness, is superior to the Tonnetz, to the extent that our interest focuses on tracking the voice-leading trajectory of a triadic passage, at the expense of the particular progression of triads that realizes that trajectory. The decision between the two modes of representation is analogous to the more familiar one of choosing to track a pitch succession in pitch or pitch-class space. The former tracks register, the particular choice of an octave location for the presentation of a pitch class, whereas the latter makes cyclic positioning explicit. Both have their uses for particular purposes, and one would not wish to dispense with either of them. The remaining work of this chapter, and that of chapter 6, focuses more on voice-leading trajectories, cyclic closure, and other sorts of patterns that occur against a circularized enharmonic backdrop and less on the particular triads that are selected to carry out these trajectories and patterns. Our first task will be to retune our theoretical and representational apparatus, to make it as sensitive as possible to these new considerations as they come to the fore.

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Figure 5.23. Cube Dance model of figure 5.21.

Voice-Leading Zones I now make explicit two intertwined ideas that have floated close to the surface in the analytical work of the previous section. The first idea is about triadic objects: underlying the claim that T4-related triads can freely substitute for each other in the fulfillment of a voice-leading trajectory is the idea that these triads are equivalent, that they belong to the same equivalence class of objects.13 The second, corresponding idea is about actions on those objects: underlying the claim that the three members of the hexatonic group of transformations, and correspondingly, the members of the Weitzmann group, can freely substitute for each other in the fulfillment of a voice-leading trajectory is the idea that these transformations are equivalent, that they too belong to the same equivalence class of transformations. It is important to proceed very carefully here, so as not to convey the wrong idea. When we say that two objects or transformations are equivalent, we are 13. The equivalence of T4-related triads was first advanced by the French theorist Camille Durutte in his Esthétique musicale: Technie ou lois générales du système harmonique of 1855, whose relevance to this work was brought to my attention by Levenberg 2008. The same triplets arise in Carol Krumhansl’s research (1990, 43), where their clustering arises through dimensional scaling of data concerning how listeners assess proximity between keys. See Krumhansl 1998, 257, for a discussion of how those data might apply to triads.

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saying that they are so with respect to some well-defined context, not with respect to every conceivable context. For example, we might say that all of the even numbers are equivalent with respect to their ability to divide integrally in half. This is not to say that I would be just as happy with two dollars as with two million, or would walk just as well with ten legs as with two. Analogously, when we say that C major and E major are equivalent, we are saying that they are so with respect their capacity to function in a certain voice-leading context, not that they are equally good final chords for the Jupiter Symphony. Recognition of the equivalence status of these relations leads naturally to a model of voice-leading zones,14 equivalence classes of trichords with the capacity to share a center of balance, in the sense that the latter term was cultivated in chapter 2. Consonant triads share a voice-leading zone if they are transpositionally related by major third. On the Tonnetz, zone-equivalent triads occupy the same hexatonic strip, and their triangles are equivalently oriented (up-tipped or downtipped). On Cube Dance, zone-equivalent triads are positioned on the same radius. The twenty-four consonant triads thus partition into eight zones, coextensive with the “triplets” referenced at the beginning of chapter 4. The augmented triads occupy the remaining four zones. Thus, the objects on Cube Dance occupy twelve distinct voice-leading zones. The number twelve is fortuitous, tapping as it does into well-consolidated intuitions about the reckoning of time. We can take advantage of these intuitions by labeling the twelve voice-leading zones as the stations of the clock. Figure 5.24 does so, by superimposing a clock face over Cube Dance. The clock face is altered in several insignificant ways. Adopting a convention familiar to music theorists, zero substitutes for twelve at the top of the figure. Multiples of three, adjacent to the augmented triads, are presented in a special typeface. And all twelve numbers are underlined, a convention that distinguishes zone labels from numbers as they might be used for other purposes in the exposition. Naming is its own virtue, since it allows for unambiguous reference. But the assignment of particular labels to particular zones here is not arbitrary; rather, it identifies several ways that the structure of the zones is isomorphic to the structure of the numbers modulo 12 (and, transitively, to the structure of the pitch classes). Those isomorphisms will be useful, to the extent that they encourage certain realizations about structure to emerge as by-products of our intuitions about numbers (or the pitch class universe). The assignment of these particular numbers to these particular sets of triads is useful in two different ways, the first of which is self-evident, the second perhaps less intuitive on first encounter. First, with a single exception, assigning numbers to voice-leading zones allows voice-leading distance to be modeled as subtraction, modulo 12.15 Consider, for example, the voice-leading distance between d minor and G major, located in zones 4 and 8, respectively. Identifying their zones is sufficient to identify the voice-leading distance between them, by subtracting modulo 12 and grabbing the 14. Cohn 1998b develops the same model under the term sum classes. 15. The exception is the hexatonic pole. Its maverick status is related to its contrary motion, a property that it alone possesses, among relations between triads in different zones.

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Figure 5.24. The twelve voice-leading zones, depicted on a clock face. Each zone contains the triad(s) on its radius. The pitch classes of each triad sum, modulo 12, to the number indicated on its radius.

absolute value. As |8 – 4| = |4 – 8| = 4, modulo 12, motion between the two triads, in either direction, requires four units of voice-leading work. We can verify this easily at a piano, or on a staff, or by counting edges on Cube Dance. But the isomorphism of voice-leading zones and the numbers modulo 12, together with our intuitions about the structure of numbers, allows us to short-circuit all of these procedures. Selecting now any of the eight remaining pairs of triads from these same voice-leading zones, we know that they are separated by four voice-leading units. What is proper to one pair is ipso facto true of the other eight. Behold the beautiful cognitive economy of equivalence! Second, and perhaps counterintuitively, the number assigned to each triad is the sum of its pitch classes, modulo 12 (Cohn 1998b; Tymoczko 2011b, 89).16 Knowing this allows us to assign zone labels to triads without looking them up on a table or a diagram. For example, d minor ({DFA} = {2, 5, 9}) belongs to zone 4 by virtue of 2 + 5 + 9 = 16 = 4modulo 12; similarly, G major ({GBD} = {7, 11, 2}) belongs 16. Pitch-class sums were first explored, in a completely different context, by Babbitt 2003, 299 (initially published 1972). Their pertinence here was first suggested to me by Jack Douthett.

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to zone 8 by virtue of 7 + 11 + 2 = 20 = 8modulo 12. Readers who worry that these sums require tedious calculation or brute-force memorization will be gratified to learn that they can be determined in a musically engaging way with low cognitive overhead. Mentally fix the C augmented triad at 0, piggybacking on the C = 0 convention of atonal theory. Then secure the remaining augmented triads, in ascending order, to the ascending multiples of three. Each of the remaining eight zones is now adjacent to one of these four cardinal points. To ascertain a particular consonant triad’s zone, one need only determine which augmented triad it displaces, and in which direction. If you have made it this far in this book, you should by now be familiar with thinking of consonant triads in this way. D minor is at zone 4 because it upshifts FACᅊ at zone 3; G major is at zone 8 because it downshifts GBEᅈ at position 9. Having established a system of classifying and labeling consonant triads on the basis of their positions within a voice-leading system, the next step is to establish corresponding equivalences for the transformations that connect them. We define three such transformation classes: an H class, consisting of the three members of the hexatonic group (L, P, and H); a W class, consisting of the three members of the Weitzmann group (R, N, and S); and an E class, consisting of three transformations that map triads within their own zone (LP, PL, and identity operation E).17 An H-class operation maps a consonant triad in zone X into the unique zone X ± 1, and a W-class operation maps a consonant triad in zone Y H into the unique zone Y ± 2. Thus, we write 1 ← → 2 to summarize a range of propositions that include the following: • • • •

To get from a triad in zone 1 to some triad in zone 2 requires some H-group transformation. To get from a triad in zone 2 to some triad in zone 1 requires some H-group transformation. Any H-group transformation applied to a triad in zone 1 produces a triad in zone 2. And Any H-group transformation applied to a triad in zone 2 produces a triad in zone 1.

W Similarly, we interpret 2 ← → 4 as making corresponding claims about the role of the W-group transformations, in moving between triads in zones 2 and 4, and so forth. With the H- and W-group transformation classes as basic elements, we can establish classes of compound transformations that connect triads from nonadjacent zones. Triads that are transpositionally related by an odd value, three voiceleading units apart, are connected by a compound HW or WH transformation; in effect, this is the generalization that underlies all of the work with sequences and 17. H and W are a species of exchange operations, which are introduced in Lewin 1987, appendix B, and developed in Lewin 1995 and Cook 2001.

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 Audacious Euphony transformational substitutions in this chapter. Triads that are separated by four units of work, as in the {d minor, G major} example considered above, are connected by HWH; by five units of work, by WHW; and finally, triads that are the maximum of six units apart (transposition by major second or tritone) are connected either by HWHW or by its inverse, WHWH. The final sentence of the preceding paragraph consolidates a large number of specific instances: the three-term compounds cover twenty-seven specific sets of transformations, and the four-term compounds cover eighty-one. Thus the generalizing power is along the lines of a claim about numbers such as odd + odd = even, which consolidates in one statement an infinite variety of propositions covered by it. For readers mystified by this mode of discourse, be assured that the remainder of the book proceeds independently of these abstractions.

Remarks on Disjunction and Entropy The model proposed here is premised on the proposition that conjunct voice leading is privileged in nineteenth-century music. Yet such a proposition is risky, if assigned universal value. The inversion of asymmetric (“violent”) binaries is among the cardinal features of European society and culture after the French revolution. If nineteenth-century composers accorded privilege to smooth voice leading, they also accorded privilege to the contravention of privilege. One need not search far for examples. Since the Carolingian era, the asymmetry of consonance and dissonance constituted an absolute hierarchy; as we have seen in connection with the opening of the Faust Symphony (figure 3.4), by the middle of the nineteenth century it became possible to stabilize a dissonant chord and destabilize its consonant neighbors. In classical tonality, a dissonant chord requires resolution to a key; in Tristan und Isolde, “the various would-be ‘keys’ come to sound . . . as an agglomeration of phenomena accessory to the Chord-as-Ding-an-sich” (Lewin 2006, 220). Since the sixteenth century, the chromatic is an ornament to the diatonic; in the nineteenth century, the diatonic often becomes a subset of the chromatic. It would be naive to fantasize that the privileges accorded to melodic conjunction are immune from analogous reversals. Moreover, one of the central tropes of Romantic aesthetics is a fascination with the remote, unattainable, and inexplicable. Already quite early in the century, composers were moved to find ways to depict these tropes through music (Kramer 1994; Hoeckner 1997). A late manifestation of this concern is revealed in a letter of Gustav Mahler, concerning the C major finale of his First Symphony: Again and again, the music had fallen from brief glimpses of light into the darkest depths of despair. Now, an enduring, triumphal victory had to be won. As I discovered after considerable vain groping, this could be achieved by modulating from one key to the key a whole tone above (from C major to D major, the principal key of the movement). . . . My D chord . . . had to sound as though it had fallen from heaven, as though it had come from another world. (qtd. in Buhler 1996, 127)

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The passage turns our initial assumption on its head. Mahler was privileging not conjunction but its opposite. Fortunately, a pan-triadic model that privileges conjunct voice leading brings with it, as a residual benefit, the ability to model disjunction as well. When we were exploring the hexatonic and Weitzmann regions as autonomous entities, the best emblems of disjunction were the maverick transformations H and S, which represent diametric opposition within the normative hexatonic and N/R cycles. With the drawing together of these components in this chapter, other possibilities emerge. Imagine that you are a triadic composer who wishes to jump as far away from C major as possible in a single stroke. But, unlike Mahler, you have at your disposal a model of the triadic universe as a metric space under voice-leading proximity, in the form of figure 5.24. Locate C major at zone 11, note zone 5’s diametric relation, and consider the options: transpose either by tritone or by major second. No need to vainly grope. Imagine now that you wish to embellish this progression with the addition of two further triads. Your first impulse might be to select from zones 2 and 8, midway between 5 and 11. The outcome is a set of four modally matched triads whose roots are drawn from the four distinct augmented triads (and hence form one of the all-combinatorial tetrachords). But perhaps modal uniformity lacks sufficient lexical diversity for your purposes. You would prefer to mix the two triadic species in equal proportion while still maintaining a sense of distance. One set of options is illustrated in figure 5.25: two zone-diametric triads are combined with their hexatonic poles. These progressions uniquely feature minimal pitch-class intersection: the four triads have no pitch-class overlap, and together they use all twelve tones. In the excerpt at figure 5.25(a), from a J. S. Bach Prelude (Dahlhaus 1967b, 87), each major triad is followed by its hexatonic pole; the pair is then transposed down by a zone-diametric whole step. The four-chord complex rotates

Figure 5.25. Two examples of triadic disjunction.

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 Audacious Euphony through voice-leading zones 1, 8, 7, and 2, in that order; its transposition in the following measure rotates through the remaining voice-leading zones. Bach’s fourchord complex became a trope of disjunction in the long nineteenth century: examples occur in such diverse venues as Mozart’s c minor Fantasy, Mendelssohn’s Variations sérieuses, Chopin’s G major Nocturne (mm. 25–27), Schumann’s d minor Piano Trio, Wagner’s Parsifal, Debussy’s Prelude to the Afternoon of a Faun, Elgar’s King Olaf, and Hugo Distler’s motet Fürwahr, er trag unsere Krankheit. Under permutation, the same complex occurs in Chopin’s G major Nocturne (mm. 1–12), Liszt’s Die Legende vom heiligen Stanislaus, Strauss’s Salome, and Kodály’s Mountain Nights.18 In the excerpt presented as figure 5.25(b), from Parsifal, a pair of H-related triads is transposed down by the other zone-diametric interval, a tritone. Together, these progressions exhaust (to within transposition and reordering) the possible ways of distributing the pitch-class aggregate among four distinct consonant triads. All of the examples explored so far have an aspect of sequential patterning. If we want to forgo consistency as well as conjunction, then we want to project not maximal disjunction but maximal variety along the conjunct/disjunct spectrum. Here we can take advantage of the homology between figure 5.24 and the universe of twelve tones, about which we know a good deal. If we consider the voice-leading distance between two triads in terms of an interval, then we are in a position to define the total interval content of an ensemble of triads and to represent that information in an array. To achieve maximum entropy, the triadic pairs should be distributed as evenly as possible among the six intervals. The optimal way to do this is to find an ensemble of triads whose voice-leading zones are homologous to one of the all-interval tetrachords, of which there are two types, prime form [0146] and [0137]. Both tetrachords are octatonic subsets, and the eight voice-leading zones occupied by the consonant triads have an octatonic structure. Thus, there exist triadic quartets whose constituent pairs are maximally distributed across the six interval classes. Taking inspiration from an aspect of Elliott Carter’s compositional practice, Robert Morris (1990) has shown that any nonintersecting union of a tritone with a minor third creates an all-interval tetrachord. Piping this observation through our homology, we learn that a maximally entropic ensemble of four triads consists of two zone-diametric (T2- or T6-related) triads of one mode, combined with two triads of the opposite mode that are related to each other by an odd transposition (and thus three zones apart, as established earlier in this chapter). Figure 5.26 presents two options, one for each species of all-interval tetrachord. Figure 5.26(a) models mm. 8–12 of the finale of Beethoven’s “Tempest” Sonata, which moves through aᇳiv, ᅈII, V6, iᇴprogression in d minor. The voice leading zones, {g = 7, Eᅈ = 8, A = 2, d = 4}, correspond to prime form [0146], one of the all-interval tetrachords. Figure 5.26(b) models the opening measures of Götterdämmerung, which toggles between eᅈ minor and a series of foils: first Cᅈ major, then dᅈ minor, and finally D major, with dominant seventh, in a quotation of the Fate motive. 18. Sources that identify and analyze these passages include Hull 1915, Karg-Elert 1930, Lendvai 1983, Cohn 1996, and Cohn 2004.

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Figure 5.26. Two “all-interval” quartets of triads. (a) is aᇳi, iv, ᅈII, V6ᇴprogression in d minor. (b) alternates eᅈ minor with Cᅈ major, then dᅈ minor, and finally D major (with a seventh), in a quotation of the Fate motive.

These voice leading zones, {eᅈ = 7, Cᅈ = 8, dᅈ = 1, D = 5}, correspond to prime form [0137], the other all-interval tetrachord. Arrowed lines connect order-adjacent triads, and the remaining connections are indicated by broken lines. Disjunction and entropy are the province of poststructural approaches to nineteenth-century analysis, and it is the practitioners of those approaches who are best qualified to weave those appraisals into significant critical arguments. The comments provided here suggest, perhaps surprisingly, that atonal pitch-class theory has the capacity to stimulate, undergird, or nuance those attributions, when its conceptual framework is analytically directed toward the model of the triadic universe proposed in these pages.

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C HA P T E R

Six

Navigating the Triadic Universe: Three Compositional Scripts

Having established the voice-leading structure of the triadic universe, explored ways to represent that structure, and identified vehicles for navigating it, we are now in a position to investigate how nineteenth-century composers created compositional strategies in response to it. In the compositions explored in chapter 5, varied repetition was the guiding principle, generating chromatic sequences and deforming them via transformation-class substitutions. In the compositions analyzed in this chapter, repetition plays a less systematic role. Compositional material is still repeated, varied, and transformed, but the periodicity, flow, contrast, and trajectory of events acquire the flexibility associated with sonata, symphony, and Lied, genres that furnish most of the excerpts investigated here. The analyses of this chapter are organized around three scripts. Some compositions establish neighborhoods and enact pitch retention loops. They visit a small set of adjacent voice-leading zones, and create continuity through retention of one or more focal tones. Some progress through a cycle. The analytical focus is on Schubert’s song “Der Doppelgänger” and the first-movement development of Brahms’s Second Symphony. Other compositions execute a classical departure → return trope, using novel resources to define a route, a destination, and the nature of reversal. In one variation familiar from the eighteenth century, the return overshoots the origin, requiring a final recuperation. I shall analyze two Schubert songs, the well-known “Auf dem Flusse” and the little-known “Liedesend’” as well as two instrumental compositions: the well-known first movement of Schubert’s Bᅈ major Piano Sonata, and a little-known Organ Kyrie of Franz Liszt. The final script, continuous upshift, is also a variation on a classical trope, in nuce the Mannheim Rocket, in elaboration the Sturm und Drang developmental core. After reviewing some sequential models from Beethoven, I examine the upshifting development of Dvořák’s “New World” Symphony, which treats thematic materials more freely. These analytical vignettes make no claim to completeness. My goal for them has been to establish that pan-triadic voice-leading models can make contributions to interpretation. But they do not constitute interpretation in and 111

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Neighborhoods and Pitch Retention Loops Example 6.1 reproduces a Tonschema (tonal scheme) from Czech theorist Ottokar Hostinský’s 1879 Die Lehre von den musikalischen Klängen. The figure is a variation on Riemann’s Tonnetz; its diagonal positioning of major and minor thirds is echoed in the Tonnetze presented in this book. Hostinský grows the tonal network from the C circled at its center. Radiating from C are six edges that connect to the six tones with which it is consonant. Six further edges, forming a hexagon, bind those six tones directly to each other by consonances. The hexagons and radii together form six triangles, representing the six triads that include C (as root, third, or fifth of a major or minor triad). Following Igor Popovic (1992, 101), we will call this complex of tones and triads C’s neighborhood. Each of these six tones generates its own neighborhood, whose bounding hexagon overlaps with the one surrounding C, producing a third tier of tones, initiating a process that radiates outward as far as one pleases to imagine. It is not difficult to imagine what might have stimulated a late-century theorist to explore the structure of pitch-class neighborhoods. Much music of the nineteenth century is concerned with kaleidoscopic pan-triadic harmonizations of a static pitch. This concern is well documented in the case of Italian opera, which William Rothstein (2008) posits as historically antecedent to central European developments.1 German Lieder that exploit this technique to obvious programmatic effect include Schubert’s “Zügenglöcklein,” D. 871 (1826), and Peter Cornelius’s “Ein Ton,” Op. 3, no. 3 (1854).2 Schubert’s late Schwanengesang in b minor, “Der Doppelgänger,” presents an example of an entire composition based on a neighborhood centered on Fᅊ (Saslaw and Walsh 1996, 231). A score of the song is available at Web score 6.2 . The song opens with the serial presentation of four tones of that region,ᇳB, Aᅊ, D, Cᅊᇴ , in dyadic partnership with a drone Fᅊ. This segment develops into an ostinato, sometimes substituting A for Aᅊ as a fifth tone of that neighborhood.

Figure 6.1. Hostinský’s Tonnetz. 1. Webster 1991, 15–17, indicates another antecedent in Haydn’s “Farewell” Symphony (1772). Siciliano 2002 and Clark 2011a both devote considerable attention to pitch retention schemes in the music of Schubert. 2. I am grateful to Christoph Hust for calling my attention to these songs. See also Siciliano 2005b regarding Schubert’s “Trost.”

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 Audacious Euphony Each dyad is susceptible to triadic completion in two distinct ways, and the song engages all of these possibilities (Code 1995; Kurth 1997). The song’s dramatic wave crests at the completion of Fᅊ’s pitch-class neighborhood, when Dᅊ is introduced at m. 47 as the root of a minor triad. When that Dᅊ is presented in conjunction with B, as components of a B major triad, Fᅊ’s neighborhood is completed at the dyadic and triadic level as well, marking the end of the song. Figure 6.2 presents this narrative as a series of cumulative snapshots. Web animation 6.2 unites them as a continuous presentation. Figure 6.2(a) shows the situation at m. 1. By m. 4 (figure 6.2(b)), four of the tones in Fᅊ’s neighborhood have been presented, each as a component of a dyad. At m. 7 (c), the singer arpeggiates b minor, filling the BD dyad. At mm. 10–11 (d), A is sounded for the first time, as a component of fᅊ minor and D major. At m. 12, a fourth triad, Fᅊ major, completes the first couplet. At m. 46 (e), the neighborhood’s final tone, Dᅊ, is sounded, as part of a dᅊ minor triad. At the final cadence (f), BDᅊ is sounded within a B major triad, completing the neighborhood’s dyadic and triadic “aggregates.” Figure 6.2(g) sketches some events omitted from the above narrative, which transpire mostly to the left (subdominant) flank of the central hexagon. (1) At the end of the first two couplets (mm. 12–14 and 22–24), E is outposted as a lone representative of b minor’s subdominant region; its appearance locally is as the seventh of the dominant.3 (2) E returns at the end of the third couplet (mm. 32–33) as a component of a CE dyad, again disconnected from the central hexagon, and again sounding against dominant pitch classes as part of a French sixth chord. (3) At the end of the fourth couplet (m. 42), E returns as part of a complete C major triad, still disconnected from the Fᅊ neighborhood, and sounding against a single dominant pitch, Aᅊ, as part of a German sixth chord. (4) The climactic dᅊ minor alternates with its nebenverwandt at mm. 48–50, briefly cultivating new territory for the first time on the dominant (northeast) side of the central complex. (5) The enharmonic reinterpretation of C at m. 51, as part of a G major German sixth, causes a return to the subdominant region and begins to close the triadic gap that separates the C major complex from the central neighborhood. (6) This gap is completely filled with the sounding of an e minor triad, initially over a dominant pedal point at m. 54, ultimately over a tonic pedal, as part of the final plagal motion at m. 61. These analytical remarks, in addition to illustrating pitch retention strategies via neighborhoods, also showcase three virtues of the Tonnetz as a mode of description and analytical representation. First, by representing triads as assemblies of atomistic pitch-class components, the Tonnetz adapts well to surfaces that feature dyads and individual tones. Rather than compelling us to decide whether the initial dyad signifies b minor or B major, it simply documents its presence and invites us to infer its plausible completions. Second, the Tonnetz is neutral with respect not only to triadic constituency but also to tonal (key) centricity. Although, as we 3. The idea that a subdominant pitch represents a subdominant function, even when attached to a dominant triad, is a central element of Riemann’s theory of function and its development in Harrison 1994.

Figure 6.2. Stroboscopic Tonnetz portrait of “Der Doppelgänger.” An animation, with recorded performance, is at Web animation 6.2.

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 Audacious Euphony shall see in chapter 8, it can be adapted to express tonal interpretations, it is not compelled to do so. Thus, for example, we need not submit to arbitration the question of whether the final harmonies of “Der Doppelgänger” move from minor subdominant to B major tonic (L. Kramer 1986; Kurth 1997) or to major dominant from e minor tonic (R. Kramer 1994; Code 1995; Schwarz 1997) or form some liminal blend of the two (Saslaw and Walsh 1996). Finally, as an interpretative consequence of these two levels of descriptive neutrality, the Tonnetz facilitates tracking the accumulation of dyads and pitch classes, encouraging us to observe how they might form systematic aggregations independently of their triadic constituency. In the case of “Der Doppelgänger,” for example, we were able to watch Fᅊ’s neighborhood build, and come to completion, at the level of individual tones, dyads, and triads and to trace the filling of a problematic gap opened up by the outlying E first introducted at m. 12. Our discussion so far has not presupposed any particular ordering of a neighborhood’s six triads. But the triadic adjacencies on Hostinský’s Tonschema visually emphasize the relationships between those triads that hold two tones in common, suggesting a canonical ordering that makes a cyclical tour of the neighborhood. The transformational vehicles for such a tour are the three commontone maximizers, P, R, and L, placed in any order and then repeated in the same order. Taking advantage of an acronymous fortuity, we will call such a tour a pitch retention loop. Figure 6.3 models the cantabile section of Verdi’s “Ah sì, ben mio” (Il Trovatore, act 3). The first of the cantabile’s three quatrains modulates from f minor to Aᅈ major. The second quatrain begins in aᅈ minor, sounds Fᅈ major through its consequent phrase, and repeats the text of that consequent over dᅈ minor. The final quatrain prolongs Dᅈ major. A final L would return the initial triad; instead, this transformation indicates the progress of the cantabile across its entire span.

Figure 6.3. Tonnetz model of Manrico’s aria from Verdi’s Il Trovatore.

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A more ambitious example of the structuring force of pitch retention loops occurs in the first movement of Brahms’s Second Symphony. The final part of the development, which begins at m. 246 and elides into a characteristically underarticulated recapitulation fifty-six measures later, is initiated by the reentry of the timpani, which has been absent since the middle of the exposition. A score of the extended passage is available at Web score 6.4 . As is normal for a symphony in D major, the timpani sounds only two pitches during the retransition: tonic D at its beginning and near its end, and dominant A through its middle. The neat conventionality of this description masks an anomaly: with one underemphasized exception, these pitches serve not as roots of tonic and dominant triads (or their dissonant extensions) but rather as thirds and fifths that disperse inventively across their respective neighborhoods. The passage proceeds in a consistent four-bar hypermeter, articulated by harmonic changes and by the periodic rotation of three thematic/textural modules whose first appearances are transcribed in figure 6.4. A is a dyadic figure derived from the opening theme, sounding heterophonically and in double hemiola throughout the orchestra. B is a metrically malleable violin theme accompanied by tremolo strings. The lyrical, metrically stable C is the only module to feature significant internal harmonic progressions. The triple rotation of these modules establishes a twelve-bar hypermeter as well, which is sustained even after the rotation is abandoned. Five twelve-bar hypermeasures are indicated in figure 6.5, which presents the timpani in metric reduction. Double bars indicate the boundaries of four-bar hypermeasures. (The timpani’s characteristic rhythm is given for each hypermeasure, so that the durational values do not always add up to equal units as in regular score notation.) My discussion takes this reduction quasi-literally: I use “measure” (in quotes) to refer to what is notated as four bars of the score, reserving “hypermeasure” for Brahms’s twelve-bar units. The first two hypermeasures cycle through the

Figure 6.4. Brahms, Symphony no. 2, 1st mvt., mm. 246–57, with three themes designated.

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 Audacious Euphony Figure 6.5. Brahms, Symphony no. 2, 1st mvt. thematic rotations in mm. 246–302.

three modules in order. The third hypermeasure disrupts this pattern, omitting the initial dyadic segment and filling out the hypermeasure by repeating the final lyrical module, B → C → C. The fourth hypermeasure is dominated by the previously absent A module, performed by the entire orchestra for its first two “measures” and by the brass alone for the final one. The boundaries of the fifth hypermeasure are ambiguous, as indicated by the overlap in figure 6.5. Continuing to project a twelve-bar periodicity, one expects a new hypermeasure at m. 294. This expectation is rewarded by the timpani roll that has initiated each previous hypermeasure, and is further supported by the timpani’s return to the tonic after an absence of twenty-four measures. On this hearing, the fifth hypermeasure dissolves one “measure” early, with the arrival of the recapitulation at m. 302. There is, however, motivation to hear this fifth hypermeasure as a complete twelve-bar unit, beginning already at m. 290. The downbeat of that measure is a “loud rest,” marked especially by the absence of the timpani, which has sounded at every other “downbeat” of the retransition but is tacit throughout this “measure.” Four other factors reinforce the accentual status of m. 290. A rootposition tonic chord sounds for the first time since m. 50 (and indeed will not sound again until m. 477!). The B module, absent since m. 273, returns in the violin and continues into the subsequent “measure,” so that m. 294 sounds as continuation rather than a new event. The subito piano at m. 290 creates an accent of change, and the counterpointing of two melodic modules creates a textural accent. These factors conspire to suggest an equivocal reading and to assign a transitional role to the “measure” beginning at m. 290. These considerations create a context for analyzing the harmonic design of these measures, which revolve kaleidoscopically about the timpani pedals without privileging the tonic and dominant triads that they traditionally imply. One can get an immediate sense of how Brahms withholds the harmonic definition that D → A → D might otherwise signal in a D major context by examining the

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harmonies that occur when the timpanist smacks a new drum. The initial D, at m. 246, is the fifth of a G major triad and, moreover, one that initially sounds like dominant of C by virtue of its phrygian approach. The switch to the A drum at m. 262 occurs, ironically, at the moment that D sounds as a harmonic root. When the timpani returns to D at m. 294, it is a diminished fifth above the Gᅊ sounding in the bass.4 Figure 6.6(a) models mm. 246–70 on a Tonnetz that combines two overlapping hexagons. The first twelve-bar hypermeasure charts a progression around part of the D retention loop, G major → g minor → Bᅈ major.5 The harmonically mobile C module hurries through d minor to D major at m. 257 (shown by dotted arrows) before returning to Bᅈ major at the beginning of the second twelve-bar unit (258). The following “measure” (262) revisits d minor but over an A in the timpani. Preparing now to exit the D neighborhood, we can note that four of its triads have been prolonged for a full four-bar “measure.” Tonic D major has appeared briefly in the anacrustic m. 257, and its relative minor not at all. The initial progression d minor → F major that enters the A neighborhood at the onset of the C module (m. 266) is a transposition of the corresponding moment twelve measures earlier (g minor → Bᅈ major), as is the continuation a minor →

Figure 6.6. Tonnetz models of Brahms Symphony no. 2, 1st mvt. 4. In this respect, the entry of the timpani on D at m. 294 associates with the first timpani entrance of the movement, at m. 32, where a solo D tremolo, initially understood to opaquely represent an eagerly anticipated tonic, becomes absorbed into a diminished seventh chord in the low brass, with Gᅊ as the lowest pitch of all. See Schachter 1983a. 5. The augmented triad at m. 252, and at analogous points in the B module, is omitted for visual clarity but may easily be imagined along lines sketched out in connection with figure 4.6(c).

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 Audacious Euphony A major (267–69). This transpositional logic projects a return to F major at the opening of the third hypermeasure, corresponding to the return to Bᅈ major at m. 258. Instead, Brahms continues a counterclockwise tour of the A neighborhood, establishing fᅊ minor at m. 270. To this point, the motion through the loop is entirely “stepwise.” One further step around the loop completes the tour of the A neighborhood and, at the same moment, returns to the tonic triad. Brahms forestalls this outcome by reversing course, as shown at figure 6.6(b): he toggles between fᅊ minor and A major for six measures (mm. 274–279) and returns to F major at m. 281. The fourth hypermeasure, beginning at m. 282, is dominated by the dyadic A module that went missing in the previous rotation. In its two previous occurrences, the major-third dyad in the strings sounded as part of a major triad completed by the winds. At m. 282, only the FA dyad is sounded. This dyad is initially heard to represent F major, the harmony of the previous measure. But the remaining events in this rotational unit complicate this interpretation. At m. 286 the same music repeats, but F is replaced by Fᅊ. If we hear FA as root and third of F major, then we have motivation to hear FᅊA as root and third of Fᅊ minor, by virtue of the parallelism. But a Slide relation from F major to fᅊ minor is not consistent with the double common-tone retention that characterizes the rest of the passage. Moreover, at m. 290, FᅊA is completed by D. If we heard fᅊ minor at m. 286, then we hear an implicit Leittonwechsel, with D emerging from a phantom Cᅊ. But perhaps it is more plausible to retrospectively interpret FᅊA as completing a D major that was already implicitly present. And if so, perhaps we have a motivation, on analogous grounds, to retrospectively interpret FA in term of d minor, if not when it is first sounded at m. 282 then perhaps at some point in the middle of its span. These questions are ultimately not subject to arbitration. But, as we noted in connection with our discussion of “Der Doppelgänger,” one of the virtues of the Tonnetz is that it assigns a determinate but triadically neutral location to a dyad, at the edge that separates the two viable triadic interpretations. Accordingly, we can avoid arbitration entirely, in favor of a Solomonic option that splits the difference. The graphic solution, which I have adopted in figure 6.6(b), is to open up the dyadic edge into an oval, so that it becomes a location in its own right, not just a boundary between two adjacent triangles. Although the arrival of D major at m. 290 marks the return of the tonic, the release of the A pedal in the timpani, and the completion of the A neighborhood, its rhetorical force is compromised by the subito piano and the absent timpani, which has attacked every previous hyperdownbeat. Accelerating the periodicity of harmonic change, Brahms now completes the tour of the D retention loop that had been aborted at m. 262, traversing b minor (with Gᅊ under-seventh) at m. 294 and arriving back at G major (with E added sixth, 296), from which the retransition had originated in m. 246. After the G major → g minor at m. 298 (with added under-seventh) reprises the progression with which the retransition began, Brahms concludes with a plagal nebenverwandt from minor subdominant to tonic (in 64 position) for the beginning of the recapitulation. Figure 6.7 provides a synopsis of the sixty measures (see also Web animation 6.7 ). Larger dots mark harmonies that arrive on hypermetric downbeats, most

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Figure 6.7. Synoptic Tonnetz model of Brahms Symphony no. 2, 1st mvt., mm. 246–302. An animation, with recorded performance, is at Web animation 6.7. of which are prolonged for four measures. Harmonies that arrive elsewhere in the hypermeasure are shown selectively, with smaller dots. The tour sounds every triad in the double neighborhood, and no triads outside of it. The figure communicates the comprehensive logic of the entire passage as a counterclockwise tour of the space, from subdominant to subdominant, with some back-filling that extends the retention loop around the dominant A and delays the premature return to the tonic region.

Departure → Return Scripts The path → reversal sequence is a fundamental template of musical form. Its classical prototype moves from tonic to dominant and back. Tonics and dominants can be realized in many ways, not all of which will project an equally strong sense of reversal. But if both departure and return realize least-motion voice leading, then each upper voice is structured as a palindrome. The sense of departure → return is particularly strong for a performer, for whom the complementarity of downshifting and upshifting involves a kinetic aspect.6 The situation just described is not specific to the relationship between a tonic and its dominant. Reversing the order of any series of harmonies reverses their individual voices. The complementarity of departure and return thus is reflected in a complementarity of upshifting and downshifting, as those terms apply both to the individual voices and to their aggregation into triads. Once fifth relations lose their prototypical status, the specific case of I → V → I is absorbed into the general case of downshift/upshift complementarity. This suggests that a departure → return script can be fulfilled by any triadic progression whose voices move in a uniform direction. A return script is fulfilled not only by a literal reversal of that triadic progression but also more abstractly by a progression that moves through the same voice-leading zones in reverse order, freely substituting same-group transformations and same-sum triads. 6. Depending on the mode of performance, this kinesis involves a shift of bow angle, hand position, throat physiology, embouchure, or some combination thereof.

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 Audacious Euphony To illustrate, compare the opening of “Auf dem Flusse,” the seventh song from Schubert’s Winterreise, with the beginning of its final stanza. A score of the opening and closing stanzas of the song are available at Web score 6.8 . The first quatrain, “Der du so lustig rauchtest . . . ,” is set as a period whose antecedent leads from e minor to a half-cadence, and whose consequent prolongs and fully cadences in the tonally distant dᅊ minor. The antecedent contains only the e minor tonic and its N-related major dominant. Across the phrase boundary, the root of B major drops a semitone to the fifth of dᅊ minor, effecting a leading-tone exchange. Because N and L represent different transformation classes (Weitzmann and Hexatonic respectively), their alternation produces a unidirectional downshifting, from zone 10 to 5. This motion is reversed at the cadence, when Aᅊ major upshifts to dᅊ minor. (This upshift is idealized; as is standard at the full classical cadence, voices move down, forgoing common-tone preservation and voice-leading parsimony in order to effect melodic resolution.) This reversal triggers an unraveling of the previous transformations: the leadingtone exchange is undone at the midpoint of the two-measure interlude, and the opening nebenverwandt is reversed at the onset of the second stanza, which repeats the music of the first. The final stanza, “Mein Herz, in diesem Bache” is delivered at the rate of one couplet per period, rather than one quatrain as previously, and the entire stanza is given two complete settings. Consequently, this single quatrain of text is accorded four full periods, as many as the four previous quatrains combined. The initial setting of the first couplet at mm. 41–47 replicates that of the opening quatrain, cadencing in dᅊ minor. Schubert connected the initial stanza to its successor through a two-measure interlude (mm. 12–13), prolonging dᅊ minor for a measure and then converting it to a dominant through Aᅊ → B. At m. 48, Schubert compresses the interlude into a single measure: one beat of dᅊ minor and one beat of semitonal ascent to the dominant of the new tonic. But Schubert assigns the mid-measure semitonal ascent to the “wrong” voice:” Fᅊ instead of the Aᅊ of m. 12. Accordingly, dᅊ minor leads not to B major via L but rather to Dᅊ major via P. And when that chord acts as a dominant, it leads not to tonic e minor but rather to gᅊ minor. As at the end of the first stanza, Schubert has upshifted through zone 8 to the home zone 10. But the P-for-L substitution causes the triadic representatives of those zones to transpose upward by major third. It is the job of the second couplet, then, to undo this diversion. It does so by directly juxtaposing the dominants of gᅊ minor and e minor, paving the way for the tonic cadence at the end of the final stanza’s first full setting. Figure 6.8 presents a paradigmatic analysis (Agawu 2009) that facilitates comparison of these progressions. The first line shows what the two settings have in common, in an abstract sense: both use W → H → W to downshift zones 10 → 5 and the same combination to upshift back to zone 10. (Recall that W and H, respectively, signify Weitzmann and hexatonic classes of transformations, not the transformations themselves.) The first and second stanzas (“I, II”) realize both legs of this journey in terms of N → L → N. The first couplet of the fifth stanza (“V:1”) realizes the departure leg in the same way but diverts the course of the return leg through a P-for-L substitution, symbolized here by a diagonal arrow that throws

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Figure 6.8. Voice-leading zones in Schubert’s “Auf dem Flusse.”

the progression off its horizontal course, and requires rectification through the rectilinear N → PL → N combination (“mm. 50–55”). The second setting of the final stanza, beginning at m. 55, likewise sets each couplet to its own period, and each of these spins a new variation on the downshift/upshift script. Schubert accelerates the harmonic rhythm, and hence the rate of downshifting, so that zone 5 is reached already by the half-cadence of each fourmeasure antecedent, rather than just before the full cadence of each eight-measure period as previously. We will take the final period (mm. 64–72) first, as presenting the simpler case. Whereas at m. 48 Schubert diverted the harmonic flow through an H-class substitution during the upshift leg, here he does the same during the downshift, substituting for the anticipated L the third and final constituent of the H-class group, the “shocking” hexatonic pole (H) (Newcomb 1986, 164). Thus, B major proceeds to g minor rather than to dᅊ minor, and its nebenverwandt proceeds to D major rather than to Aᅊ major as in the first two stanzas. As at m. 48, a substitution for L has created a transposition upward by major third but this time at the end of an antecedent phrase rather than at its beginning. In the setting of the previous period (mm. 55–62), the antecedent phrase is identical, except that G major substitutes for g minor. The rare juxtaposition of

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 Audacious Euphony two major triads (mm. 56–57) prolongs zone 8 for an additional measure and causes the omission of zone 7 altogether. The half cadence is followed, as in the earlier stanzas, by a leading-tone exchange, taking the root of D major to the fifth of fᅊ minor and driving us into the as yet uncharted depths of voice-leading zone 4 and then fleetingly to 2 (Cᅊ major) in preparation for the cadence. The twomeasure interlude reclaims tonic e minor on schedule by the unparsimonious, but conventionally diatonic, upshift from fᅊ minor to B major. Leaving aside the tonally stable E major third and fourth stanzas, the chromatic sections of “Auf dem Flusse” all involve a departure → return script, with each departure realized as a downshift from e minor and each return as an upshift to that tonic or its zonal affiliate, gᅊ minor. Scanning down a column of chords in figure 6.8 brings to light substitutional relations within voice-leading zones, all involving transposition by major third. Scanning down a column of transformations, we see that all of those chordal substitutions result from transformational substitutions within the Hexatonic class: P and H both stand in for L. The Weitzmann-class transformations are perpetually represented, without substitution, by the ubiquitous nebenverwandt. Richard Kramer has written that “the modulatory adventure of Auf dem Flusse is extreme. No other song in Winterreise is endowed with a tonal graph anywhere near as complex” (1994, 159). Encountering this characterization without knowledge of the song, one might imagine that “Auf dem Flusse” is deeply entangled in the enharmonic ambiguities characteristic of Schubert’s most tonally adventurous compositions. Against this expectation, it comes as a surprise to realize that “Auf dem Flusse” achieves the status that Kramer assigns it without ever approaching, much less threading, an enharmonic seam. Except for the diatonically conventional E major setting of the third and fourth stanzas, “Auf dem Flusse” lives entirely “downstream” from its tonic. The song is nebenverwandt-saturated: every minor triad is followed or preceded by its major dominant, and with the single exception of G major (m. 58), every major triad is adjacent to some minor tonic to which it is dominant. The subdominant, accordingly, is absent to a remarkable degree: aside from a tonicized fᅊ minor at m. 62, whose subdominant status is realized only in retrospect, only eight chords have subdominant value, and none of these has a duration of more than a single beat. The previous paragraph brings to the fore the potential affiliation of downshifting and upshifting, respectively, with dominant and subdominant. To what extent does the voice-leading dualism of down and up interact with the Riemannian dualism of subdominant and dominant? It is tempting, for example, to hear gᅊ minor at m. 50 as a tonic substitution, the Leittonwechsel of the tonic’s modal variant. By extension, its Dᅊ major predecessor functions as dominant, and by further extension, so too do the other tonicized leading tones at mm. 12 and 21, as David Lewin suggests (2006, 115). Projecting the logic, Aᅊ major represents doubledominant Fᅊ major, as the modal variant of its Leittonwechsel, and so too does the fᅊ minor tonicized at m. 62. This leads to a hypothesis that these functional assignments extend to entire zones, so that 10 always represents tonic, 7 and 8 dominant, and 5 double-dominant. But this proposition immediately attracts counterexamples. Lewin, for example, hears gᅊ minor (m. 50) as the relative of the dominant,

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rather than as the tonic representative suggested by the hypothesis. And the parallelism, with respect to phrase positioning and text, of G major (m. 58) and g minor (m. 66) with e minor at m. 45 inclines us to hear G in terms of tonic function, not the dominant of our hypothesis. The hypothesis that chords whose roots share an augmented triad always share harmonic function, which I advanced in Cohn 1999 and critique in chapter 8, is as Procrustean as Ernő Lendvai’s theory of axis tonality (1971), which makes an analogous claim about roots that share a diminished seventh chord. One of the benefits of voice-leading zones is that they supervene upon such functional hearings, suggesting but not mandating them. Downshifting and upshifting may be interpreted in terms of plagal and authentic motions, respectively. But if they make claims that are counterintuitive or uncomfortable, the functional scaffold can be shed, leaving the zones to carry the weight of the structural claims. One advantage of this view of dualism is that it is immune to dismissal on the grounds that it is based on the metaphysical or spurious physical claims discussed in chapter 2. It merely explores the ramifications of the following aggregation of circumstances: that pitch motion exists on a linear continuum, which our culture metaphorically projects onto a vertical axis, as higher and lower; that pitch-class motion exists on a cyclical continuum; thus, that motion along either continuum can occur in two opposite directions; and that those motions have experiential correlates in the aural perception of sound and the kinetics of its production. This is a dualism that even the most skeptical realist can abide!7 These considerations come to the fore in the first movement of Schubert’s Bᅈ major Piano Sonata, whose entire score is available at Web score 6.9 . In its general outlines, the tonal structure of the movement conforms to classical principles: an exposition that establishes tonic, proceeds through secondary thematic material, and concludes in the dominant; a development that concludes with a retransitional dominant; and an ordered recapitulation of the thematic material of the exposition, tonally adjusted to conclude in the tonic. This skeleton, though, supports a number of tonal events that are anomalous from a classical standpoint. Intervening between the first theme and its perorational counterstatement is a passage that prolongs Gᅈ major. The second theme is presented in fᅊ minor and tonicizes A major before arriving at the dominant, the four paragraphs of the development depart respectively from cᅊ minor, A major, Dᅈ major, and d minor, and the recapitulation of the second theme is given in b minor and D major. Of the eight tonic triads just listed, only one (d minor) is diatonic to Bᅈ major, and only two (Gᅈ major, Dᅈ major) are diatonic to bᅈ minor, and hence derivable via firstorder modal mixture. The remaining five keys are diatonically indeterminate and involve some degree of enharmonic paradox along lines already explored in chapter 1, in connection with a segment of this movement. 7. The upshift/downshift duality is also strongly related to the duality of strong and weak harmonic progression, a distinction made by Arnold Schoenberg and developed by Nicholas Meeus (2000). Strong progressions, whose roots descend by third or fifth, or ascend by second, all upshift under idealized voice leading. The only complication here is the equivocal voice-leading status of modally matched chords whose roots are a major second apart (IV → V in major; ii → iii in minor), which involve six units of voice leading in either direction.

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 Audacious Euphony Figure 6.9. Three models of Schubert’s Sonata in Bᅈ major, D. 960, 1st mvt.

Figure 6.9 presents the main tonal events of the movement, together with three analytic underlays.8 The first of these analyses assigns function according to the hexatonic constituency of a triad. Triads in Bᅈ major’s hexatonic system are assigned tonic function; similarly, subdominant and dominant function are bestowed on triads hexatonically associated with Eᅈ and F major, respectively. The structural transition from tonic to dominant, on this reading, occurs at m. 58, when A major displaces fᅊ minor. (Measures 68–71 are heard as transiently revisiting tonic amidst an overriding dominant prolongation; see Webster 1978–79.) The first three paragraphs of the development begin with three hexatonic associates of the dominant, cᅊ minor, A major, and Dᅈ major, respectively. The analysis interprets the rhetorically punctuated and extensively prolonged d minor as a premature tonic return, overriding the spineless and opiated dominant seventh that precedes the recapitulation. The recapitulation is entirely in the tonic, except for a 8. This first analysis is equivalent to one proposed in Cohn 1999; the second two analyses develop observations made near the end of that article, using the technologies and representational modes introduced in chapter 5.

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transient tonicization of A major, representing dominant, and the sounding of the fᅊ theme in b minor, representing subdominant. The large-scale functional scheme is consistent with the Classical sonata, including the recapitulatory subdominant overshoot characteristic of many of Schubert’s first movements. Only two elements, both associated with d minor, are anomalous: the transient revisitation of tonic after dominant has been established in the exposition, and the premature arrival of a tonic representative at the end of the development. The principal thematic materials of the movement emphasize major thirds that in combination outline a series of augmented triads (Cohn 1999, 224–25). If we affiliate these augmented triads with the consonant triads that they displace, then we hear the functional progression in a different way. The structural motion to dominant occurs already at the fᅊ minor second theme (m. 48); it need not wait for the A major cadence ten measures later. The retransitional d minor at m. 171 is similarly reevaluated: its FA major third affiliates that triad with dominant, not tonic. These motivic considerations suggest that the harmonies be reinterpreted according to their Weitzmann-region constituency, leading to my second analysis. This analysis conforms more closely to the classical ideal for coordinating themes and structural harmonies: dominant arrives with the second theme and is prolonged for the remainder of the exposition and the entire development (except for its cᅊ minor opening, which uniquely inhabits the distant region of the double dominant). Tonic returns at the beginning of the recapitulation and maintains its functional hold throughout, with the exception of some quite transient tonicizations. According to this interpretation, the b minor theme in the recapitulation represents tonic, by virtue of the common D third. Using voice-leading zones, the third analysis creates a hybrid of the first two. The exposition is interpreted as a two-phase descent in voice-leading space. The fᅊ minor second theme displaces Bᅈ major and its Gᅈ surrogate downward by one voice-leading degree, 5 → 4. The F major third theme then displaces fᅊ minor downward by two voice-leading degrees, 4 → 2. The nadir of this downshift is reached with cᅊ minor at the opening of the development. The trajectory is reversed when A major displaces cᅊ minor, restoring zone 2. The motion through d minor to Bᅈ major (retransition, recapitulation) represents a compensatory upshift through zones 4 and 5. According to this interpretation, the b minor theme in the recapitulation constitutes an overshoot to sum class 7. We have the option of interpreting this voice-leading scheme in terms of Riemannian functions; indeed, Schubert gives us many reasons for doing so. Zones 5 and 2 may be assigned tonic and dominant value, respectively; zone 4 may be interpreted as occupying an intermediate space between the two functions, and zones 1 and 7 may be heard as overshooting toward the double-dominant and subdominant sides, respectively. But we need not make this move if aspects of it make us uncomfortable. If we are attracted to functions primarily because they encourage us to bundle musical events into coherent trajectories, such as the departure → overshoot → return scheme documented here, then they are dispensable. The zones provide another, more theoretically neutral way of documenting those same trajectories. We may not like that neutrality. We may like the added value that the language of functions provides, and may be prepared to

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 Audacious Euphony Figure 6.10. Liszt, Missa pro organo lectarum, Kyrie.

articulate its nature. But we are not stuck with the functional apparatus if we do not find added value in it or, worse, if we find that it bears implicit baggage that detracts value. To focus the point, we now analyze two pan-triadic compositions that execute a departure → return scheme but where functional ascriptions sit far less comfortably than in Schubert’s Piano Sonata. Figure 6.10 presents a brief wordless setting of the Kyrie from a Liszt Organ Mass of 1879. Each of its three phrases, corresponding to the Kyrie → Christe → Kyrie sequence of the Latin Mass ordinary, contain three parallel segments, corresponding to the standard threefold repetition of each textual unit, followed by a fourth, presumably wordless, cadential segment. The reduced score notates each of these twelve segments as a measure. The composition is entirely triadic and begins and ends in Bᅈ major. Aside from the two interior cadences, there is little to suggest diatonic scale degrees or Riemannian functions. There is nonetheless an exquisite patterning, the door to which is opened by an observation of Ramon Satyendra (1992, 98–101): leaving aside the first two neighboring cadential gestures, each individual voice constitutes a palindrome whose axis is the double bar about which the key signature converts from flats to sharps. Yet, because the firing order of the voices is staggered, the palindrome evident in each voice individually does not result in a palindromic structure for the triads that those voices comprise. Figure 6.11 normalizes the voice leading in order to illustrate the trajectory of the individual voices. The arcs suggest which triads would be identical, if the

Figure 6.11. Palindromic model of figure 6.10.

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triadic palindrome were strict. The deviations occur in the third and fourth triads, where Dᅈ major → f minor is mirrored by cᅊ minor → A major, its retrograded transposition down a diminished fourth. The palindrome thus is proper not to the progression of the triads themselves but to that of voice-leading zones that they represent. The progression of zones illustrates what is already evident to the organist’s hand: that all voice leading shifts downward in the first two phrases and upward in the final phrase. Figure 6.12 displays this departure → return scheme on Cube Dance. Schubert’s 1816 song “Liedesend’” provides a second, more complex example of a departure → return scheme realized as a palindrome of voice-leading zones. A score of the entire song is available at Web score 6.12 . What is at issue here is not the local harmonic progressions, which define tonics using standard diatonic conventions, but rather the progression of tonics, which George Grove selected as an example of Schubert’s inscrutable modulatory practice (Clark 2011b). The song is sectionalized and episodic: each of the nine strophes has its own mood and tempo, and six of them terminate with a fermata. The modulatory scheme is evidently no less episodic. Seven of the nine strophes are preceded by a change of key signature, and the song closes in a key that is first introduced only at the beginning the poem’s final couplet. Suzannah Clark’s recent analysis (2011b, 304–11) suggests that the song is tonally structured by a series of common-tone

Figure 6.12. Cube Dance model of figure 6.10.

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 Audacious Euphony Figure 6.13. Voice-leading zones in Schubert’s “Liedesend.”

modulations. This observation can be extended by charting the progression of voice-leading zones. Figure 6.13 presents the tonal structure, as a palindromic motion through voice-leading zones. The nine strophes are indicated by Roman numerals and represented by their beginning and ending tonics. Where successive tonics lack common tones, they are bridged over by a prolonged dominant that shares a tone with both. Minor tonics arise three times: at the song’s beginning, conclusion, and palindromic center. Connecting the c minor beginning to the f minor center is a series of major triads that cascade from zone 8 down to 11; connecting the center to the e minor conclusion is a series of major triads that upshift through the same zones in reverse order. The zonal palindrome is ruffled only by the D major conclusion of the eighth stanza, which introduces a local downshift against the prevailing upshift. This perturbation defers the arrival at the goal zone 10, while reprising and ultimately reversing, at the start of the song’s final stanza, the exact triadic succession that occurred at the conclusion of its initial one. Figure 6.14 plots the same design on a sparse Tonnetz, painted over with major-thirdgenerated fault lines. Broken lines indicate a new tonic that is asserted rather than approached through a modulatory process. Asterisks indicate tonally closed verses. The graph highlights the downshift/upshift scheme, as well as the common-tone continuities that Clark observes. Like the surface harmonies of Liszt’s Kyrie, the tonics in “Liedesend’” do not convey a strong sense of harmonic function. Any attempt to organize the tonics about either the opening c minor or the closing e minor would be tendentious to an extreme degree, an act of desperation in support of a monotonality that is being

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Figure 6.14. Sparse Tonnetz model of Schubert’s “Liedesend’.” clung to at all costs. The palindromic voice leading suggests that the modulatory scheme need not be deemed arbitrary even if its logic is not monotonal.

Continuous Upshifts Chapter 5 studied sequential passages that lead in a uniform direction through a series of adjacent voice-leading zones. Our concern here is with passages that do so without benefit of patterned repetition. I document continuous upshifting in two quite distinct environments: as a slow unfolding of structural chords, mostly tonics, across Schumann’s song cycles, and as a characteristically energized surge through the development section of first movements of Beethoven and Dvořák. Fred Lerdahl (2001, 138) plots the key sequence of Robert Schumann’s cycle of sixteen songs, Dichterliebe, as a coherent path through Weber’s graph of regions.9 The key signatures of songs 1–8 decrement from three to zero sharps, monotonically every two songs were numbers 4 and 5 swapped. The scalar downshift accelerates in songs 9–11, which increment one flat per song. Beginning with the reversal from three flats to two at song 12, the pattern in key-signature space is fractured. Lerdahl notes that the keys of the final songs nonetheless fulfill the projected path by substituting parallel or minor-third related keys, which are privileged in Weber’s space and Lerdahl’s. The continuous downshifting in scalar space through the first eleven songs of Dichterliebe is realized in triadic space as a phased upshifting that continues straight through to the end of the cycle.10 Figure 6.15 traces the progression of the 9. Much ink has been spilled concerning whether the cycle is “organically unified,” and on what grounds. Ferris 2000, 26–38, summarizes positions; Perrey 2002 and Hoeckner 2006 are more recent contributions. 10. Komar 1971, 78, observes a pattern of upward displacements between adjacent tonics.

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Figure 6.15. Tonnetz model of the tonal plan for Schumann’s Dichterliebe. The numbers, corresponding to the ordinal position of each song in the sixteen-song cycle, are positioned at their beginning and ending triads. Arrows lead the eye at points where the right-to-left is perturbed or where a song ends in a different key than it began. boundary harmonies, defined as those that begin and end songs. Boundary harmonies are tonics except when a song begins and/or ends with a dominant prolongation (songs 1 and 9) or when its conclusion modally inflects the opening tonic (songs 9 and 16). Numbers refer to order position of the sixteen songs. Arrows guide the eye when a song references multiple structural harmonies or when the prevailing flow is reversed. Dichterliebe executes two revolutions about Cube Dance. The initial point of cyclic renewal occurs at the beginning of song 9, the exact halfway point in the cycle, where a prolongation of A7 matches that of Cᅊ7 in song 1. In this respect, the first nine songs of Dichterliebe echo Schumann’s Op. 24 Liederkreis, also setting poems of Heine, whose ninth and final song completes a similar upshifting rotation about voice-leading space (Hoeckner 2001). This suggests dividing the set into equal halves, a possibility supported by a circumstance of compositional genesis: Schumann initially composed Dichterliebe as a twenty-song set but withdrew the original pairs at order positions 5–6 and 15–16, the median points of each hypothesized half (Hallmark 1975). Table 6.1 documents the progression of the songs through the two rotations about the voice-leading cycle, with the withdrawn songs restored (shaded in the table). The pace of upshifting through the first rotation does not precisely match that of the second, but some parallels are worth noticing. Each half upshifts 2 → 8 and then introduces a unique downshift (onset of songs 5 and 12). The final four songs (in each group of eight) upshift 7 → 1. In the initial conception, the final five songs (in each group of ten) upshift from zone 5. E. T. A. Hoffmann’s Kreisleriana, with which Schumann was famously absorbed, provides a possible source for his interest in the patterned upshifting and zonal completion that guides the key sequences of his two Heine cycles. In the vignette titled “Kreisler’s Musico-Poetic Club,” Kreisler slowly improvises a series of ten triads (some with sevenths) and reports the sensations and images that each one evokes (Hoffmann 1989 [1813–14], 131–36). Kreisler’s progression begins with a downshift Aᅈ major → aᅈ minor and ends with the zone-identical downshift C major → c minor. The six interior harmonies execute an upshift rotation about voice-leading space, connecting E major → a minor → F major → Bᅈ major → Eᅈ major → G major.

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Table 6.1. Voice-leading zones in Schumann’s Dichterliebe Songs 1–8

Songs 9–16

Song no.

Triad(s)

Zone(s)

Zone(s) Triad(s)

Song no.

1

Cᅊ+

2

2, 4, 5

A+, d–, D+

2

A+

2

7

g–

10

3

D+

5

8

Eᅈ+

11

4

G+

8

5

Bᅈ+

12

a

Eᅈ+

8

7, 8

g–, G+

A

b

g–, D+

7, 5

5

Bᅈ+

B

5

b–

7

7

eᅈ–

13

6

e–

10

8

B+

14

7 8

C+ a–

11 1

11 1, 2

E+ Cᅊ–, Dᅈ+

15 16

9

The published songs, numbered 1–16, are supplemented by four that Schumann composed for the cycle but omitted, which are inserted here in the initial position that he intended them and labeled a, b, A, B. Triads are those prolonged at the beginnings and endings of songs, whether or not they are local tonics.

The interior of a Classical sonata-form development, which Caplin 1998 calls its core and Hepokoski and Darcy 2006 its central action zone, characteristically projects tonal instability and mounting tension, frequently through one or more sequences that sustain an upshift trajectory. Figure 6.16 models two upshifting cores from first movements of middle-period Beethoven sonatas. The first, from the “Tempest” Sonata (d minor, Op. 31 no. 2), contains an intensely compressed upshift bracketed by two briefer downshifts (indicated by lines beneath the reduced score). Ametric strummed arpeggios precede that core, leading to Fᅊ major. The propulsive core begins with a “lights out” downshift from Fᅊ major to fᅊ minor.11 Like Kreisler’s slow and dreamy improvisation, the core of the “Tempest” development begins and ends with two zone-identical downshifts, fᅊ minor → Cᅊ major (99–106) and d minor → A major (117–21), flanking an upshifting passage that orbits voice-leading space. A related script is executed over a broader span in the development of the “Waldstein” Sonata, Op. 53 (figure 6.16(b)).12 The principal theme in F major (m. 90) begins a downshift that culminates four measures later at D7, as dominant of g minor. A T5 chain from m. 96, at a four-measure pace, accelerates eightfold at m. 104, prolonging bᅈ minor and Gᅈ major. After a brief downshift, the sequence 11. A 2009 collection of essays on the “Tempest” Sonata edited by Pieter Bergé, Jeroen D’hoe, and William E. Caplin contains the following characterizations of the fᅊ minor triad at m. 99: “electrifying, wrenching shock, jolt” (Burnham, 42–44), “surprising harshness” (Burstein, 70), “brutally negat[ing] explosion” (Hatten, 170), “punched with a vengeance” (Hepokoski, 201), “shocking” (Kinderman, 220). The critical consensus is as stunning as the moment itself. 12. The Tonnetz interpretation is based in part on Lubin 1974, 109ff.

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continues at its four-measure pace. Beginning at m. 120, the upshift stalls in a Weitzmann region, connecting Bᅈ major → b minor through an N/R chain and reaching the retransitional dominant at m. 130. These two passages suggest that extended upshifts are ingredients of the adrenal concoction with which Beethoven so often injected his developments. The opening movement of Dvořák’s e minor Symphony (1895), “From the New World,” contains a development section with an even more relentless upshifting trajectory, carried by a free circulation of thematic materials that are not organized into sequential repetitions as in the classical prototype. This ninety-six-measure development is the longest continuous segment of music analyzed in this book and, as such, constitutes a fittingly weighty close to the six-chapter arc that constitutes the core presentation of pan-triadic syntax. A score of the entire development is available at Web score 6.17 . As a backdrop to our study of this development, I begin with some observations about the entire movement. One of its peculiarities is its avoidance of the dominant. B is never tonicized, as major or minor. On those rare occasions when cadential dominants appear, their characteristic effect is undercut by weak bass degrees (mm. 237, 269), deceptive resolution (m. 408), or brief duration (mm. 56, 431). The sole exception is the two-measure caesura that connects the initial Adagio to the exposition (mm. 22–23). Throughout the movement, the off-tonic role is appropriated by four upper mediant triads (Kopp’s term, 2002), presented in upshifting order.

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Figure 6.17. Dvořák, Symphony no. 9, 1st mvt., tonal arc of the second theme.

Figure 6.17 tracks the second theme as it appears in exposition and recapitulation. It first appears in g minor (m. 91), and then in G major at m. 137, the key in which the pastoral third theme closes the exposition. The corresponding measures of the recapitulation transpose this music up by chromatic semitone. Aᅈ major, a notational proxy for Gᅊ, is the hexatonic pole of the tonic, an extraordinary key in which to close a recapitulation. In figure 6.18, the keys of the second theme climb the Tonnetz, through the gray band. This progression can be captured as a climb in voice-leading space, 7 → 8 → 10 → 11, as part of a P/S chain. The beginning of the coda takes this onward to 2 (A major and then F major) before the latter moves as a Neapolitan to the fleeting structural dominant at m. 408. The relaxed upshift through the movement’s pastoral breathing points is mirrored by an intense upshift across its breathless development section, tracing

Figure 6.18. Tonnetz analysis of figure 6.17.

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 Audacious Euphony Table 6.2. Triads and voice-leading zones of the development of Dvořák’s “New World” Symphony, organized around the four thematic entries Measure 189 209 233 257

Theme 3 3 1 1

Triads

Zones

E+ F+, fᅊ– eᅈ–, e– fᅊ–, A+, Bᅈ+, B+, e–

11 2, 4 7, 10 4, 2, 5, 8, 10

two complete rotations from the G major of the end of exposition to the e minor of the beginning of the recapitulation. The entire development is in a clear four-bar hypermeter. Hortatory thematic incipits structure the development into four paragraphs, the first two dominated by the pastoral third theme and the final two by its principal theme. These two themes are linked by their #MRRR#M rhythm, associated with the Hungarian choriambus, and are related by contour inversion. Table 6.2 indicates the harmonic platforms from which these choriambic volleys are launched. The series of initial stations, E major → F major → eᅈ minor → fᅊ minor, does not present a very coherent image of purposeful tonal progression. Interpreting these triads and their successors as representatives of voice-leading zones, one begins to get a sense of the prevailing upshift winds that blow across these plains. Figure 6.19 segments the development into four-bar units. Ovals enclose hypermeasures that do not represent voice-leading zones; either a diminished seventh chord is prolonged, as at m. 213, or there is rapid planing between dissonances, as at m. 253. The first paragraph presents E major (zone 11), then moves to cᅊ minor (1), and progresses through its hexatonic region to F major (2), from which the second paragraph is initiated and proceeds to fᅊ minor (4). Voice-leading zone 5, represented by a Bᅈ dominant seventh, ushers in the third paragraph. The third paragraph, which is articulated at m. 233 with the return of the principal theme, executes an extended N/L chain that carries straight through to the start of the fourth paragraph at m. 257. A series of semitonally ascending minor tonics appear every eight measures: eᅈ minor (233) → e minor (241) → f minor (249) → fᅊ minor (257). With a single exception, the four-bar median of each eightbar span is marked by the arrival of a dominant seventh chord of the subsequent minor tonic. The exception occurs in the third span, which alone lacks a thematic incipit. The Dᅈ major dominant projected for m. 253 arrives two bars early. This buys Dvořák four bars in which to further accelerate the harmonic rhythm in anticipation of the fourth paragraph at m. 257, where thematic activity resumes. Dvořák fills these four measures by synopsizing the harmonic course of the prior twenty measures, planing the series of minor triads on eᅈ minor, e minor, and f minor, and accompanying each with an under-seventh (see Bass 2001, 51). The fᅊ minor with which the fourth paragraph begins is rhetorically marked not only by the return of the principal theme but also by virtue of fulfilling two trajectories right on time: a slow one initiated from m. 233 and following an eight-bar periodicity, and a fast one initiated from m. 255 and following a one-bar schedule. The final

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Figure 6.19. Dvořák, Symphony no. 9, 1st mvt., voice-leading zones in the development section (mm. 177–273).

paragraph then reprises the first part of the N/L chain with one addition, A major, and one deletion, eᅈ minor. Figure 6.20(a) presents the entire development of the “New World” Symphony on the Tonnetz. An animated version of this figure, synchronized with a recorded performance, is presented at Web animation 6.20 . Starting at the right edge, the initial gesture, covering the precore and the onset of the first paragraph, travels along an octatonic R/P chain. The second arrow, moving southwest from cᅊ minor, indicates the hexatonic progression that connects the first paragraph to the opening of the second. A disjunctive move to fᅊ minor, at the end of the second paragraph, commences an intensive southwest slide along the N/L chain that dominates the third paragraph. A broken arrow emerging from the graph’s southwest fringe repositions fᅊ minor back at the top of the slide, to prevent a drift outside the frame but also to emphasize how the fourth paragraph reprises the opening of the third. Figure 6.20(b) extracts the right side of the figure and overlays a different interpretation on the same data. The first paragraph is interpreted here as executing a four-station tour about an E retention loop. The second paragraph is initiated

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Figure 6.20. Dvořák, Symphony no. 9, 1st mvt., Tonnetz model of the development. Animation, with recorded performance, is at Web animation 6.20.

at the moment that E is displaced to F. Bridging the two paragraphs is a second pitch retention loop about A, beginning at m. 201 (A major) and extending to the diminished seventh chord at m. 227. This reading draws attention to the relatively slow pace of pitch-class turnover and upshift pacing through the opening two paragraphs, as compared with the accelerated upshifting of the final two paragraphs. The two components of figure 6.20 constitute two “performances” upon a single pattern of black dots. Both are “note accurate,” but they shape the data in different ways and draw breaths at different articulation points of the score. The two graphs illustrate the interpretive leeway available to the pan-triadic analyst, using the representational “instrument” of the Tonnetz. To what extent do these acts of “performance” relate directly to musical performances, as traditionally construed? To what extent are they metaphoric and suggestive, inviting imaginative translation? That is something that only performers can determine; I merely suggest here that both possibilities are available for contemplation.

C HA P T E R

Seven

Dissonance

Four Eighteenth-Century Approaches to Dissonance Aside from the augmented triad, this book has largely ignored dissonant harmonies, or swept them out of the frame with a series of deferrals. Readers would be justified in suspecting the waving of hands and in wondering whether the neglect of seventh chords, and other larger-cardinality harmonies, is simply a flaw in pantriadic theory. Even if classical models strain in response to the highly chromatic repertory of late Romanticism, they do have the virtue of supplying a first-level descriptor for most dissonant harmonies, and of making claims about their behavioral tendencies. Why should such models be exchanged for one that appears to write dissonant harmonies out of existence? Now is the moment when I hope to reward readers for their patience concerning these matters. I propose here not a unified theory of chordal dissonance but rather a set of responses that extend pan-triadic theory in several orthogonal directions. This flexibility is consistent with the precedent of classical theory. Jean-Philippe Rameau, the theorist for whom dissonant harmonies first arose as categorical possibilities, generated them through a series of ad hoc, even mutually incompatible, methods (Christensen 1993, 98–100). Rameau’s various approaches can be classified into four strategies—deletion, reduction, substitution, and combination—all of which endured in the nineteenth century and three of which survive into present-day harmonic theory and pedagogy. It is important to distinguish here between a strategy, as a general set of assumptions, and the several methods by which it might be implemented. The methods that I propose are distinct from Rameau’s, which in turn are distinct from the familiar methods of modern harmony texts. In the Traité of 1722, Rameau principally generated dissonant harmonies from a consonant triad by adding one or more thirds above its fifth or below its root. The task for the analyst is to identify the consonant triad en route toward locating its root. This procedure is simple if the triadic subset is unique, as with dominant and 139

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 Audacious Euphony half-diminished sevenths and minor ninths. But with diminished seventh chords, which include no consonant triads, or with major and minor sevenths or major ninths, which include several, the process does not yield determinate solutions, and Rameau addresses them through a set of ad hoc responses to be considered shortly. This problem was obviated around 1800, when diminished and augmented triads were added to the roster of basic sonorities (Damschroder 2008, 17–19). Any stack of thirds could now be generated upward from a root, and any seventh or ninth chord could be unambiguously reduced to its triadic base, be it dissonant or consonant. What was retained through this change of method was the strategy of reduction to a subset, a strategy that continues to underlie the ubiquitous practice of labeling dissonant harmonies as extensions of the lowest three members of their third-stack. In his ad hoc responses to harmonies lacking a unique root, Rameau put two other strategies into play. The Traité interpreted the diminished seventh chord as resulting from the displacement of the root of a dominant seventh chord, thereby appropriating to harmonic theory a substitution strategy familiar from theories of counterpoint and of rhetorically based melodic analysis (Christensen 1993, 100). Substitutional strategies were subsequently invoked in Johann Philipp Kirnberger’s notion of the accidental seventh chord, whose dissonant tone substitutes for its consonant resolution; by François-Joseph Fétis (2008 [1844]), who considers chromatic passing and neighboring chords to be substitutes for, and intensifiers of, diatonic ones; and in harmonic pedagogy influenced by Heinrich Schenker.1 Rameau advanced an alternative approach to diminished seventh chords in his 1737 Génération harmonique, where he generated them from the combination of two triads, the major dominant and minor subdominant, each of which is represented by only two of its constituent tones. A similar strategy is evident in his 1760 Code pratique, which viewed major and minor seventh chords as combining two complete triads. Although generation of dissonant harmonies as the combination of consonant ones is only occasionally sighted in contemporary harmonic theory (e.g., Straus 1982; Harrison 2002a), it was quite common in the nineteenth century: it underlay Moritz Hauptmann’s and Hermann von Helmholtz’s analyses of major and minor sevenths, Helmholtz’s analysis of minor triads as clouded consonances representing three distinct major triads, Hugo Riemann’s theories of dissonant harmony (Gollin 2011), Georg Capellen’s 1908 theory of hybridization (Doppelklang), and Schenker’s analysis of the French sixth chord.2 The fourth historical strategy, deletion, is implicit in eighteenth-century modulatory theory, which interprets one level of a composition’s structure in terms of a succession of keys that stand in one-to-one relationship with the eponymous consonant triads. Thus, a succession of keys lends itself to interpretation (if not yet representation) as a succession of individual harmonies. This conception becomes 1. For example, Aldwell and Schachter 1989, Gauldin 2004. Schenker, in his later writings, denied harmonic status to substitutional dissonances, but modern Schenkerians have conceded this point, perhaps in deference to the universal and evidently irreversible harmony centricity of music theory pedagogy in the academy. 2. Hauptmann 1888 [1853], 67; Helmholtz 1885 [1877], 294, 341; Gollin 2011; Bernstein 1993, 89; Schenker 1954 [1906], 278.

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fully implicit in conjunction with Rameau’s theory of imputed dissonance, which held that only tonic triads are consonant, all others bearing an acoustically suppressed sixth or seventh (Christensen 1993, 129). The implication is that a triadic progression is coextensive with its progression of tonics. If one wishes to model a composition as a succession of consonant triads, one’s job is not to assign triadic constituencies to dissonant harmonies, whether their dissonance is acoustically realized or virtually imputed, but rather to delete them from the analysis. Rameau’s views on the relationship of modulation and consonant succession were adopted, although inflected somewhat differently, by Riemann (Mickelsen 1977, 66; Gollin 2000, 236). Schenker adopted a different version of the deletion strategy by stipulating that consonances alone are capable of prolongation. Unlike Rameau or Riemann, it was immaterial to Schenker whether a consonant triad carried a local status of tonic, as he rejected the notion of local tonics altogether (Schachter 1987). What is significant is Schenker’s treatment of nonadjacent consonant triads as if directly adjacent, by suppressing the dissonant harmonies between them. More significant yet is Schenker’s translation of this strategy into a mode of representation that resembled a musical score: his graphs represent nonadjacent consonant triads as register-specific pitches that participate in linear strands, or voices, whose coordinated behavior is available for study by analogy with the behavior of voices in a direct succession of consonances such as one might find in a part-song or chorale. The four strategies play different roles in this chapter. I shall say little about deletion, which tacitly underlies many analyses already presented and to which Anglophone readers are habituated by Schenkerian practice. A second strategy that has already been in evidence is the reduction to a subset, which I have implicitly invoked whenever I have considered a dominant seventh chord to represent a major triad, ignoring its seventh, or, more controversially, considered a halfdiminished seventh chord as a minor triad, ignoring its putative root. The second part of this chapter develops this way of thinking about dissonances that have a unique consonant subset. The third part develops the substitution strategy in a way that inverts Rameau’s method. Taking a cue from Benjamin Boretz’s analysis of the Tristan Prelude, it interprets dominant sevenths, as well as their half-diminished inversions, as semitonal displacements of fully diminished seventh chords. The two chord types constitute a system of nearly even tetrachords, and motion between them benefits from the voice-leading efficiency that accrues to such a system. Many, but not all, of the terms, concepts, and modes of representation that have been developed for triads can therefore be adapted for the tetrachordal case. Although the fourth strategy, combination, has not figured heavily in my thinking, it is worth taking a moment here to briefly sketch some ways that it might be integrated it into the theory developed here. The analytic output of that theory has been communicated, to a large degree, by means of the Tonnetz, which represents triads and their progressions in compact, efficient, and legible ways. Those qualities do not, for the most part, transfer to dissonant harmonies, most of which are represented on the standard Tonnetz by shapes that are irregular, sprawling, or even disconnected. Edward Gollin (1998) solves this problem by extending

142

 Audacious Euphony Figure 7.1. Franz Schreker, Kammersymphonie, at rehearsal 13.

the Tonnetz into a third dimension, according the interval of the minor seventh its own axis, and representing dominant and half-diminished seventh chords as tetrahedra. This ingenious solution is limited, as a mode of exploration and communication, by the dimensional constraints of the printed page. There is, however, a genus of dissonances that do benefit from compact and determinate locations on the Tonnetz. These are the tetrachords that combine two edge-adjacent triads, forming a parallelogram from their two triangles. Of the three dissonances generated in this manner, two are familiar, and their generation by combination was identified by Rameau 1760 and Hauptmann 1853. The chord of the major seventh is formed by combining two complete L-related triads to form a parallelogram bisected by their shared minor-third edge. Similarly, the chord of the minor seventh is formed by R-related triads straddling a major-third edge. The third harmony, which does not arise to prominence until the twentieth century, is the “split third” [0347]-type tetrachord formed by the union of two P-related triads. This is the “alpha” sonority that Lendvai 1971 locates prominently in the music of Bartók. To briefly suggest its descriptive potential, figure 7.1(a) presents a brief progression from Franz Schreker’s Kammersymphonie of 1916 (see Harrison 1994, 22). The Cᅊ split third fills a gap that is left along the major third axis by the juxtaposition of the two previous triads, the hexatonic poles A major and f minor, with both of which it shares two common tones. Figure 7.1(b) indicates this hybrid chord as a parallelogram.

Reduction to a Triadic Subset The strategy and the method are Rameau’s: a dissonant harmony reduces to a consonant triad when the latter is uniquely contained in the former. Figure 7.2 shows

Figure 7.2. Six chords with uniquely embedded consonant triads.

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how this method assigns analyses to six chords, presented in inversionally related pairs. Standard third-stacking theory roots all six chords on B, but only the first chord of each pair is so rooted according to the approach pursued here. Concerning the rare sonorities presented here as (d) and (f), there is little received opinion against which the analysis proposed here can deeply cut. But the analysis of (b) as d minor chord with added sixth or under-seventh, rather than as a halfdiminished seventh chord rooted on B, is a hard sell in a musical culture habituated to the view that root generation by stacked thirds is bestowed by nature, if only implicitly via the standard names for chords. Carl Dahlhaus writes that “the notion that B is the ‘characteristic dissonance’ of the subdominant seventh chord b-d-f-a is frustrated by musical reality, in which it is evident that a is the chord’s actual dissonance” (1990 [1967], 56), and Harrison 2011 suggests that the recourse to subposed dissonances in such cases exemplifies “analytic difficulties,” presumably by virtue of the prima facie untenability of the concept. Although the analysis of figure 7.2(b) as d minor was first suggested by Rameau, it is most frequently associated with harmonic dualism, as developed by Arthur von Oettingen and Riemann in the second half of the nineteenth century. For Riemann, the DFA chord is generated downward from A. When B is added to that chord, it functions as a characteristically dissonant under-seventh (Rehding 2003, 94–96). The interpretation of the half-diminished seventh chord as a minor triad with under-seventh or added sixth was hardy enough to survive well into the twentieth century, even among theorists otherwise uncommitted to a dualist theory of harmony (see chapter 3, note 6). Ultimately, that interpretation was overcome by a renewed third-stacking monolithy, to the point that its peremptory dismissal became self-evident. Recent empirical work has motivated a revival of Riemann’s proposal, grounding it in perceptual experiments rather than physical or metaphysical speculation (Parncutt 1989, 149). A consideration of some passages from Wagner will suggest that late-nineteenthcentury dualists might have been partly motivated by an empiricism of a different sort. Figure 7.3(a) presents the final cadence from Tristan und Isolde, connecting a minor subdominant to a major tonic. On its final beat, the subdominant receives an added major sixth in the highest voice, temporarily creating a “ø7” sonority. But the “seventh” of that chord, B, does not resolve, nor does it feel any pressure to do so. It is the putative root, Cᅊ, that is transient: as an under-seventh, it passes stepwise upward to the third of the resolution chord, just as a dominant seventh typically behaves, but in the reverse direction. The minor-to-major plagal progression with rising under-seventh, which is a mirror image of the major-to-minor authentic progression with falling seventh, has its origins in the sixteenth century and serves as a final cadence in Chopin, in Brahms, and in classic film music, among other places.3 A dismissal of this Cᅊ as naught but a passing tone courts the danger of missing something, in an opera that derives so much expressive bang from the 3. Thomas Tallis’s Third Tune for the Psalter of Archbishop Parker (1557, familiar from the Vaughan Williams Fantasy) concludes with such a progression: a minor → E major, with passing cantus Fᅊ. A similar progression is evident a century later in Monteverdi’s Incoronazione di Poppea (act 1, scene 12, cadence of the second quatrain).

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 Audacious Euphony Figure 7.3. Under-sevenths in Wagner.

syntactic possibilities of the ø7 chord. Act 3 of Tristan (figure 7.3(b)) opens with a bᅈ minor triad to which is appended a G. There is little temptation to hear this as a passing formation, as the G does not evidently issue from F, and it endures for three very slow beats. Figure 7.3(c) shows a similar example from Parsifal. Here the foreign tone (again a G) is heard as an outlier by virtue of registral distribution and orchestration: bᅈ minor sounds in the brass, whereas G is registrally and timbrally isolated in the timpani and pizzicato strings. Moreover, by Parsifal’s third act, we have often heard the horn music, affiliated with the title character. Characteristically that motive is fast, loud, major, and consonant in keeping with its brash protagonist. Here it is slow, muted, minor, and muddied by dissonance, to suggest Parsifal’s desultory wandering. Wagner often treats minor triads and their ø7 supersets as interchangeable in his late music. In figure 7.3, (d) and (e) present two versions of the Communion theme from Parsifal, the first triadic, the second with a timbrally isolated underseventh. Wagner’s treatment of the Tarnhelm motive in the Ring tetralogy is analogous. In its initial and most characteristic form (figure 2.4) it juxtaposes gᅊ minor and e minor, but Wagner frequently adds an under-seventh to the initial chord (Rothfarb 1988, 154; Lewin 1992). The final two examples, figure 7.3, (f) and (g), juxtapose the beginnings of two parallel phrases from Parsifal; but for their initial bass pitch, the phrases are identical to within transposition. Each of these

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examples suggests that Wagner’s musical reality stretched beyond the limits that frustrated Dahlhaus’s putative listener.

Hexatonic Poles in Parsifal I now bolster the claim advanced in the previous section by exploring the case of hexatonic poles in Parsifal.4 The opera’s first chromatic progression occurs at m. 28 of the Prelude, when Aᅈ major → e minor deforms the diatonic theme associated with the ritual of Holy Communion. Its last chromatic progression occurs eighteen measures before its final measure, when Kundry is “de-souled” to a hexatonic distortion of the music associated with the Holy Grail (similar to figure 5.25(b)). Hexatonic poles similarly bookend the second half of act 2. Kundry opens that scene by twice singing Parsifal’s name, once arpeggiating cᅈ minor (m. 739), the second time its hexatonic pole, Eᅈ major (m. 751). She closes her role in the scene by twice cursing Parsifal to a life of perpetual wandering; her two iterations of “Irre” are set to d minor and Gᅈ major. Twelve measures later, Klingsor hurls the spear in bᅈ minor and Parsifal catches it in D major. Hexatonic poles also play a role in the large-scale tonal structure of each of Parsifal’s three acts. The first act begins in Aᅈ major, and returns to that key for the Communion service near the end of the act. Its hexatonic pole, e minor, opens the Amfortasklage, the psychological climax of the act. The second act is tonally closed in b minor; its pole, Eᅈ major, is the principal key of the flower maidens’ music. The third act begins in bᅈ minor, reaches D major at the Good Friday meadows music (m. 676), and returns to bᅈ minor with the choral music that leads into the final scene (m. 862). That final scene begins in e minor (m. 918), and culminates in the Aᅈ major that opens the shrine (m. 1088) and closes the opera. The four structural keys of act 3 retrograde the Grail-theme distortion at figure 5.25(b). Hexatonic poles can thus be said to have motivic value in Parsifal, in the dual sense of unifying the opera and marking its individuality. This abstract idea of motive is more consistent with twentieth-century uses of the term by such theorists as Schenker, Rudolf Réti, and Hermann Keller (surveyed in Cook 1987; Dunsby 2002) than with the well-known nineteenth-century Leitmotif tradition of Wagnerian analysis, where motives depend for their identity on melodic, rhythmic, and/or timbral features. This motivic network is expanded considerably by taking into consideration hexatonic poles whose triadic constituents are uniquely embedded into dissonant formations. Consider the two passages excerpted in figure 7.4, the first opening Amfortas’s lament, the second associated with Parsifal’s memory of his mother. Both passages involve hexatonic poles, with their characteristic contrary semitonal motion in three voices. In the Amfortasklage, dominant Gᅊ major is accompanied by an over-seventh. In the Herzeleide motive, subdominant cᅊ minor is 4. An expanded version of this Parsifal analysis is found in Cohn 2006; some of its ideas originate in Cook 1994.

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 Audacious Euphony Figure 7.4. Hexatonic poles in Parsifal, with supplementary dissonance.

accompanied by an under-seventh. A stacked-third approach to harmonic structure might grant the hexatonic pole in the first case and admit it as a member of the motivic network documented above but would deny it in the second case on the basis that the root progression is different. And what is the reward for seeing a new motivic connection? Music theorists treasure them in themselves, because they allow us to see a composition as a tightly knit and unified web. But to stop there is to invite the charge of complicity in (or worse, fetishization of) the ideology of unity. There it is again, and there it is again, and again . . . who cares? There’s a good response to this question, but it is rarely made explicit: the value of motivic connections is not only what they are but what they allow the analyst to do. To see a connection is to open a door; one still needs to walk through it. In order to do so, we place our observations about hexatonic poles into an interpretative framework that engages the core of the Parsifal story. The opera’s two principal Leitmotive, one associated with a ritual (Communion) and the other with a solid object (Grail), are theologically related through the blood of the savior, which the latter contains and the former symbolically transmits into the Christian body. The central problem of the opera is that Amfortas is unable to perform the Communion ritual because the process of ingesting the blood inflicts agony on his wounded body beyond his ability to bear it. Object and ritual are thus bound together with fluid and sensation into a causal chain: the Grail is the source, the Communion ritual the agent, the blood transmission the action, and the agony of sin the result. Admitting that consonant triads may be embedded into dissonances enables us to see that Wagner uses hexatonic poles in Parsifal not only to deform the Communion and the Grail but also to depict the other two components of this schematic knot, the blood and the agony. Figure 7.5(a) sets the first textual reference to the pain of Amfortas. At the inception of the text (“The pain soon returned, but more intensely”), V 24 in d minor (over a tonic pedal) is displaced by iiø7 in c minor. Each dissonant chord embeds a unique consonant triad, A major and

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Figure 7.5. The pain and the blood in act I of Parsifal.

f minor, whose connection executes the semitonal displacements characteristic of the hexatonic pole. The same two triads are concealed in the music of figure 7.5(b), which occurs at the first mention of the Holy blood in act 1 and each of the three times that the blood is referred to in act 3. The four-stranded knot is drawn most tightly in the final scene of the first act, when the Grail is uncovered and the Communion ritual is performed. Amfortas, “shot through with pain,” imagines the savior’s blood surging out of the chalice, entering his body through his wound, flowing through his heart, and mingling with his own blood “defiled by shame.” The music most characteristic of Amfortas’s lament, the Sündenqual (“pain torment,” figure 7.3, (f) and (g)), receives its first explicit verbal association just after Amfortas enters the scene, to choral singing of “Den sündigen Welten, mit tausend Schmerzen, wie einst sein Blut geflossen” (“as once his blood flowed, for the sinful world of a thousand pains”). Figure 7.6 analyzes the Sündenqual progression as a sequence of hexatonic poles concealed behind a persistent descending fifth motion in the bass that feigns a red-herring tonality (see Lendvai 1988, I: 142–43). The dominant and ø7 chords paired in each leg of the sequence are those of the “pain” progression at figure 7.5(a). The four components of Parsifal’s central problem are thus bound by a single musical signifier, the H progression. Far from arbitrary, this signifier relies on a homology between the structures of hexatonic poles and of the circumstances that

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 Audacious Euphony Figure 7.6. Parsifal, act 1, mm. 1369–71 (“torment of sin” motive, P/V score, p. 83).

they portray. At many levels, Parsifal straddles the border between life and death. Christian doctrine of transubstantiation holds the blood to be alive, even though its organic source, the redeemer, has been dead for a millennium. This is the logic of the uncanny, as theorized by Ernst Jentsch and Freud: what ought to be dead is unaccountably living. This same logic is played out in four of the opera’s six named characters. Amfortas teeters on the brink of death, and ancient Titurel inhabits a tomb. Kundry, shielded from death by a curse, is older yet; Wagner describes her death as the de-souling (Entseelung) of a zombie. And Klingsor conjures botanical abundance from a desert wasteland: from death springs life. A homologous logic is played out in the hexatonic pole progression, with the life/death duality mapped onto consonance and dissonance. When a consonant triad progresses to its hexatonic pole, its root is displaced down to the raised seventh degree, while its fifth is displaced upward to the flatted sixth degree. The resulting interval ought to be a dissonant diminished seventh. But if we perceive this new chord as a triad, then we are perceiving the resulting interval as a consonant major sixth. What ought to be dissonant is unaccountably consonant, in a dynamic that I sketched in chapter 2 (see pp. 21–22). Alfred Lorenz writes, of figure 5.25(b), that “during the lingering on the notes that are initially understood as dissonant, the chord cleanses itself, without any motion, into the most radiant beauty” (1933, 89). There is much more that could be said about Parsifal from this standpoint (see Cohn 2006), but my primary purpose here is not to interpret that opera. The topic is pertinent to this chapter insofar as it illustrates the analytic and interpretative profit that can be made available by viewing dissonant harmonies in terms of their consonant-triadic subsets. This analytic move broadens the motivic network, enriching our understanding of how Parsifal’s musical relations organize and interpret its web of extramusical symbols. And these results, to the extent that they satisfy our desire to interpret this most complex of nineteenth-century masterpieces, help to justify the approach toward analysis of dissonant harmonies advocated in this section.

The Tristan Genus as Nearly Even Tetrachord Reduction to a consonant subset, however, hardly provides a universal solution for all dissonances that contain a unique consonant subset. Consider the common

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Figure 7.7. A low-work tetrachordal progression embedding a high-work triadic progression.

classical prototype at figure 7.7(a). Taken on its own terms, the progression involves only two units of voice-leading work: two upper voices descend by semitone; two lower ones are stationary. The progression between its constituent triads at figure 7.7(b), however, involves five units of work, the maximum possible for two triads of opposite species. The “supplementary” dissonances smooth the voice leading of a progression that is otherwise disjunct. These considerations provide a motivation for seeking an alternative approach, one that takes dissonant harmonies as they are, rather than reducing them to something simpler and more familiar. This section responds by advancing a model of dissonant harmony based on an analogy with the triadic case, rather than an extension of it.5 Nineteenth-century music contains many compositions, or extended passages, whose surfaces are dominated not by consonant triads but rather by dominant (“V7”) and half-diminished (“ø7”) seventh chords, or their enharmonic equivalents. Figure 7.8 presents three sequential templates. Figure 7.8(a), which involves only V7 chords, arises already in the eighteenth century. Embellished by chromaticized voice exchanges, the progression becomes known as the “omnibus” or “devil’s mill” (Wason 1985; Telesco 1998). Figure 7.8(b) and (c), which alternate V7 and ø7 chords, also serve as occasional late-eighteenth century templates and are frequently elaborated in music of Chopin (Tymoczko 2011b, 284–92).6 Each of the three sequences transposes by an odd value and features double semitonal motion against two stationary voices. But there is a difference: the first sequence features contrary motion; the remaining two, similar motion. Thus, figure 7.8(a), which transposes V7 chords by minor thirds, shares some of the anomalous features that accrue when triads are transposed by major thirds (Tymoczko 2011b, 97). As we shall soon see, this is related to the structure of V7 chords: the minor third has this special status because it divides the octave into as many equal parts as a seventh chord has tones. (This formulation is italicized to emphasize its parallelism to the one on p. 19 of this book.) Not all passages that harness these parsimonious features are sequential. In Tristan und Isolde, Wagner famously blew open the gates to a compositional field cohabited by V7 and ø7 chords whose interactions were disciplined neither 5. This analogy was initially reported in Cohn 1996 (40n39). Its details were initially worked out in five papers presented at Buffalo conferences in 1993 and 1997 (Lewin 1996; Callender 1998; Childs 1998; Gollin 1998; Douthett and Steinbach 1998) and are adapted and transformed in broadly synthetic writings of Jack Douthett (2008) and Dmitri Tymoczko (2006, 2011b). The work presented in this section is indebted to all of these writings. 6. Tymoczko (2011b, 284–93) presents a number of examples of seventh chords in T11 and T5 sequences, as well as in T2 and T8 sequences with which they share properties.

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 Audacious Euphony Figure 7.8. Three sequential templates for Tristan-genus seventh chords.

by tonality nor by patterned sequential repetition. Succeeding generations of composers romped on this Tristan field for a half-century and more, using semitonal voice leading to connect ø7 chords directly to each other, and to V7 chords, in a manner that defied diatonic/tonal coherence more frequently than not.7 The coast of Cornwall is littered with the detritus of ill-equipped music-theoretic vessels that perished in search of a suitable tonic at which to moor.8 After touring the wreckage, Benjamin Boretz approached those shores with a different conveyance. Figure 7.9 presents the opening six chords of the Prelude in three pairs, each of which consists of a ø7-type chord linked to a dominant seventh. Of the first pair, Boretz writes that “these two chords . . . share a common relation to . . . D–F–Gᅊ–B; each . . . contains just three of its four pitch classes, with one pitch ‘contrapuntally’ displaced by a semitone; for only the Dᅊ ‘spoils’ the first chord of m. 2, and when it ‘resolves’ to D, the F of the complex is ‘displaced’ to E”

Figure 7.9. Prelude to Tristan und Isolde, opening measures. Each filled notehead is a “spoiler” that lies outside of the majority diminished seventh chord.

7. Bass 2001 provides a number of examples of parsimonious motion among ø7 chords in early-twentieth-century music. 8. The history of Tristan analyses is told in Motte 1976, Wason 1985, Bailey 1985, and Nattiez 1990.

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(1972, 163).9 Boretz’s observation is striking in light of its affinity to two distinct episodes in the history of music theory. Like Rameau in the Traité, Boretz claims that fully diminished and dominant seventh chords stand in a substitutional relation. But Boretz reverses Rameau’s generative arrow. Whereas Rameau values the dominant seventh as progenitor by virtue of its relative consonance, Boretz bestows primacy on the diminished seventh by virtue of its status as an equal division of the octave. The second affinity is with Carl Friedrich Weitzmann’s Der übermässige Dreiklang: Boretz’s reversal, for the case of tetrachords, is analogous to Weitzmann’s reversal for the case of major and augmented triads, as discussed at the end of chapter 3. Nor does the analogy end there. The perfectly even diminished seventh chord, like the perfectly even augmented triad, may be displaced upward as well as downward, yielding a second species of nearly even tetrachord: the half-diminished as well as dominant seventh chord. Together, these two species of chords, which I shall designate the Tristan genus, forms a parsimonious voice-leading system analogous to the pan-triadic system.10 The following exposition develops the system of nearly even tetrachords by analogy with the trichordal case, up to the limits of that analogy’s productive power. Table 7.1, which aligns analogous terms from the two domains, can serve as a guide to this development, as it unfolds in the pages ahead. Because the analogy relies on generalization, the exposition is more technical and abstract than most other parts of this book. Boretz’s observations about the remaining chord pairs in the opening of the Tristan Prelude already mark out some of the lines along which this analogy develops. The description that he provides for chords (1) and (2) (with reference to figure 7.9) applies with equal force to chords (3) and (4). The only change is in the identity of the tones that “spoil” the BDFGᅊ chord: Dᅊ and E in the first pair, and Fᅊ and G in the second. The same description likewise applies to chord (5), where C is the spoiler. But it does not apply to chord (6), whose B root is the single remnant of the former complex and now plays its own role as spoiler of a new 9. Boretz’s 1970 dissertation, “Metavariations,” was published in installments in volumes 8–11 of Perspectives of New Music. The Tristan analysis appears in volume 11, no. 1 (1972), to which my page references apply. Metavariations was republished by Open Space Press in 1995. Related views of the Tristan Prelude are elaborated in Bass 2001 and Chafe 2005, and of related passages from the opera in Lerdahl 2001, 302. See also Morgan 1976, on the Prelude to act 3 of Parsifal. Only Chafe acknowledges, en passant, the powerful precedent of Boretz’s analysis. 10. This is not the Tristan constellation of hexachords developed in Soderberg 1998. The Tristan genus is equivalent to set-class 4–27, prime form [0258], a system of classification that honors inversional equivalence. That equivalence relationship is a by-product here, as in the triadic case discussed in chapter 2 (see p. 38); what is primary is the relationship of single semitonal displacement that this group of chords holds with respect to the equal division of its cardinality. In using traditional names for seventh chords, I do not wish to take on board the functional agency that classical theory bestows on them. Nor do I wish to deny the force of this agency, which is metaphorically figured variously as energy, appellation, attraction, magnetism, charge, desire, and so forth, but only to neutralize it at this stage of theoretical development. It is not a necessary feature of Tristan-genus chords, to the same degree, in each and every one of its manifestations in the universe of compositions that use them. For the same reason, despite my reservations about the universal value of third-stacking, I will continue to take advantage of its ability to furnish familiar and unambiguous chord labels.

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 Audacious Euphony Table 7.1. Comparison of nearly even chords of cardinality three (triads) and of cardinality four (Tristan genus) Genus

Species, n = 3

Species, n = 4

(1)

Perfectly even n-note chord Augmented triad

Diminished seventh chord

(2)

Downward displacement of (1)

Major triad

Dominant seventh chord

(3)

Upward displacement of (1)

Minor triad

Half-diminished seventh chord

(4)

Union of (2) and (3)

Consonant triads

Tristan genus

(5)

Set of displacements of an instance of (1)

Weitzmann region

Boretz region

(6)

Geometric representation of (5)

Weitzmann water bug Boretz spider

(7)

Bridges between adjacent (5)’s

Hexatonic region

Octatonic region

(8)

voice-leading zones

1 2 4 5 7 8 10 11

1 3 5 7 9 11

(9)

Transformations within (5)

(10)

(a) One voice by whole step R

R*

(b) Two voices by semitone N, S

S3(2), S3(4), S6

Transformations within (7) (a) n – 2 voices by semitone L, P

(11)

S2, S4, S5

(b) n voices by semitone (“maverick”)

H

Octatonic pole

Voice-leading map of unified system

Cube Dance Tonnetz

4-Cube Trio 3-D Tonnetz (Gollin 1998)

diminished seventh chord complex centered on ACDᅊFᅊ. Boretz interprets the shift between majority diminished seventh chord complexes as the first of several interregional modulations that, he argues, provides the Prelude with its harmonic scaffold (169).

Boretz regions The triadic case presents a model for fashioning Boretz’s observations into a systematic framework for exploring voice-leading relations among members of the Tristan genus. More than a century earlier, Weitzmann placed each nearly even

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(consonant) triad adjacent to the perfectly even augmented triad that it minimally displaces. The six consonant triads that cluster about an augmented triad, by virtue of their mutual adjacency to it, are second-order adjacent to each other. Chapters 3 and 4 explore how those clusters form a tight voice-leading system and propose that the augmented triad to which they are adjacent asserts a virtual power over that system even when absent from the musical surface. By analogy, a consequence of Boretz’s observation that eight members of the Tristan genus minimally displace a fully diminished seventh chord is that those eight chords are second-order adjacent to each other. Thus they, too, form a tight voice-leading system even when, as in the Tristan Prelude the diminished seventh chord is absent from the surface.11 As the first five chords of the Tristan Prelude are single semitonal displacements of a single equal division, BDFGᅊ, we will say that they share membership in a Boretz region, a tetrachordal analogue to the Weitzmann regions of triads. This region contains eight members: the four ø7 chords that displace B, D, F, or Gᅊ upward, and the four V7 chords that displace one of those tones downward. The two remaining diminished seventh chords have their own Boretz region, each with a cluster of eight Tristan-genus chords. Table 7.2 classifies the members of the Tristan genus by Boretz region (compare with table 3.1).

Boretz spiders Figure 7.10 models voice leading within the third Boretz region, which furnishes the initial five chords of the Tristan Prelude.12 As with the Weitzmann water bug Table 7.2. The contents of the three Boretz regions, modeled after Weitzmann’s grouping of his four triadic regions (table 3.1) I. {C, Eᅈ, Fᅊ, A} and its enharmonic transformations 1. Ddom7

2. Fdom7

3. Aᅈdom7

4. Bdom7

5. Eᅈø7

6. Fᅊø7

7. Aø7

8. Cø7

II. {Cᅊ, E, G, Bᅈ} and its enharmonic transformations 1. Cdom7

2. Eᅈdom7

3. Fᅊdom7

4. Adom7

5. Cᅊø7

6. Eø7

7. Gø7

8. Bᅈø7

III. {D, F, Aᅈ, B} and its enharmonic transformations 1. Cᅊdom7 5. Dø7

2. Edom7 6. Fø7

3. Gdom7 7. Gᅊø7

4. Bᅈdom7 8. Bø7

11. In Tristan, the virtual presence of the diminished seventh chord is corroborated by Wagner’s initial sketch of the Prelude (Bailey 1985, 131). Moreover, diminished seventh chords often replace Tristan-genus chords in contexts where the latter normally appear, as in m. 68 of the Prelude (Mitchell 1967, 190–91). 12. Tymoczko 2011b, 371, has a formally identical graph but with the additional feature that axes correspond to motion in the three musical voices.

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Figure 7.10. A Boretz spider. (figure 4.1), any two constituent chords are separated by two units of voice leading. When tetrachords are of the same species, as in figure 7.8(a), voices move in contrary motion, as the resolution of the “spoiler” into the diminished seventh body is offset by the displacement of a different voice out of that body, on the same flank. When they are of opposite species, the spoiler resolves into the body and a new spoiler emerges on the other side. The semitones are either distributed between two different voices, as in chords (1) → (2) of figure 7.9, or concentrated into a single voice moving by whole step, as in chords (2) → (3). Back-and-forth motion across a Boretz spider, as across a Weitzmann water bug, toggles downshift with upshift, balancing between adjacent zones in voiceleading space. Figure 7.11 presents the Tristan progression’s idealized voice leading. Arrows indicate direction of motion. The first five chords alternately upshift and downshift, moving back and forth across the Boretz spider and balancing between adjacent zones in voice-leading space. Zone labels beneath the score,

Figure 7.11. Idealized voice leading of the Tristan Prelude opening, with transformations labeled above and voice-leading zones indicated below.

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derived via the summing protocol presented in chapter 5, indicate an alternation between zones 3 and 1, balancing about the BDFGᅊ diminished seventh chord, whose corresponding integers sum to 2.13

Boretz-group transformations The transformational labels above the score in figure 7.11 are adapted from Childs 1998 and Douthett and Steinbach 1998. “S” here stands for similar motion (it no longer refers to Slide, as in the triadic case). The two superscripts classify the interval of the stationary and moving dyad, respectively. In the initial progression, the stationary BGᅊ and moving FDᅊ belong to interval classes 3 and 2, respectively, hence the S3(2) label. R*, like triadic R, moves a single voice by whole step. In addition to the three cross-species Boretz transformations used in this excerpt, there is a fourth, S6, which holds the tritone dyad and moves the interval class 5 dyad; we will encounter this in figure 7.14 below.

Octatonic regions Regions that combine the half-diminished seventh chords of one system with the same-rooted dominant seventh chords of the neighboring system form bridges between adjacent Boretz regions. As the eight chords of this bridging region draw their tones from an octatonic scale, they are referred to as octatonic regions. Their role with respect to the Boretz regions is exactly analogous to the role of the hexatonic regions with respect to the Weitzmann regions, although their individual transformations lack the minimal work of the hexatonic transformations L and P. The progression from chord (5) → (6) (in figure 7.9) is one of four octatonic transformations that can bridge adjacent Boretz regions.

Octatonic-group transformations Adrian Childs (1998) dubs the particular transformation from chord (5) to chord (6) an octatonic pole because it plays the same maverick role as the hexatonic pole progression does in triadic space. Sigfrid Karg-Elert’s 1930 treatise identified their affinity, calling them both Kollektivwechsel, or “collective exchange” (285). Figure 7.12 documents the analogy: both progressions combine inversionally related species; involve upshifting in all-but-one voice, offset by downshifting in the remaining voice; and use all of the tones of their eponymous scales.14 13. The labels for zones acquire meaning only within the limited context of the tetrachordal system of voice leading. Some of the labels are identical to those used for voice-leading zones in triadic space. These are “false friends”: the triadic and tetrachordal systems are not in communication with each other. 14. Lendvai (1983, 510; 1988, 139) identifies prominent octatonic poles in the Ring and in Boris Godunov. Cohn 1996, 26–28, observes that hexatonic and octatonic poles are directly juxtaposed in the first movement of César Franck’s Piano Quintet. See also Cook 2005.

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 Audacious Euphony Figure 7.12. Hexatonic and octatonic poles.

Figure 7.13. The four octatonic-bridge transformations from Dø7.

The three remaining octatonic transformations, presented in figure 7.13, share the feature of double semitone voice leading with the Boretz-region transformations. Adapting from Childs’s labeling protocol, these S transformations bear only a single superscript, indicating the interval classification of the stationary dyad. The octatonic transformations lack the minimal-work feature of their hexatonic analogues because their chords contain more tones. After the held dyad is accounted for, there are two voices, not one, that are left to migrate between equal divisions, tipping the majority status from one to the other.

Brünnhilde’s Immolation Lewin 1996 observes that the opening music of Brünnhilde’s Immolation (Götterdämmerung, act 3) follows a voice-leading script similar to the opening of the Tristan Prelude. Figure 7.14 presents the harmonies, beginning with the tempo change (p. 318 of the Schirmer piano/vocal score) and extending through the first ten measures of Brünnhilde’s aria. As with the Tristan opening, the progression alternates V7 and ø7 chords, here occupying the Boretz region centered on EGAᅊCᅊ.

Figure 7.14. Brunnhilde’s Immolation, from Götterdämmerung, act 3.

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Figure 7.15. Three Boretz spiders in union with three octatonic pools. The octatonic bridge into a new region involves two chords rooted on Cᅊ and, unlike the Tristan excerpt, sustains the double semitone voice leading of its predecessors. The consecutive upshifts, Eᅈ7 → Cᅊø7→ Cᅊ7, enter a new voice-leading zone. As in Tristan, this marks the moment when the seventh chords hearken to their appellative properties after ignoring them for some measures.

4-Cube Trio Figure 7.15 portrays the entire system of interlocking Boretz and octatonic regions. As in the triadic case (figure 5.2), the spiders and pools are subject to figure– ground reversals. The Tristan and Götterdämmerung passages foreground Boretzregion membership, but one can also conceive of passages where octatonic regions act as the primary macroharmonies, and the Boretz spiders constitute interregional bridges between them. Because the octatonic transformations involve two voices moving by semitone in parallel motion, they jump voice-leading zones, leaving a gap that is not present in the triadic case. That gap can be bridged using either minor seventh or French sixth chords. In figure 7.16, known as 4-Cube Trio, both of these chord types appear simultaneously as octatonic bridges at the 12:00, 4:00, and 8:00 positions.15 15. Douthett initially presented this figure in 1993 under the name Power Towers. Douthett and Steinbach (1998, 256) reassigned that name to a version of the graph lacking the French sixth chords. In a footnote (262, n. 12), Douthett and Steinbach attach 4-Cube Trio to a prose description

158

 Audacious Euphony Triangular nodes indicate minor seventh chords, and stars indicate French sixth chords. As both fulfill the bridging function independently, either can be removed without disconnecting the graph. The labels for these chords are omitted in figure 7.16 to avoid clutter but may be inferred for each node from its incident Tristan-genus chords, which occupy positions associated with the six odd voice-leading sums. The set of nodes is completed by the spiders’ heads at 2:00, 6:00, and 10:00, representing the three diminished seventh chords.

Figure 7.16. Jack Douthett’s 4-Cube Trio.

of the original figure. (The graph links three four-dimensional cubes, or tesseracts.) Dmitri Tymoczko (2011b, 106) independently rediscovered a version of 4-Cube Trio, which emerged as a subgraph of his continuous four-dimensional voice-leading space. By virtue of that status, Tymoczko shows that 4-Cube Trio, like Cube Dance, is an accurate model of voice leading among its included chords (in the sense that the most direct edge distances between chords represent their most efficient voice leadings). Peck and Douthett 2011 describes other features of 4-Cube Trio and clarifies in what sense it is cubic.

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Circumnavigating the Tristan-Genus Universe 4-Cube Trio, like Cube Dance, is a true model of voice leading (Tymoczko 2011b, 106). Voice-leading distance between any two chords is calculated by determining the shortest path between them and counting the edges. 4-Cube Trio also serves as a space for circumnavigation of the universe of nearly even tetrachords. As in the triadic case, the simplest path is through transpositional sequences that perpetually upshift or downshift. If the sequence uses only members of the Tristan genus, then only odd zones of voice-leading space are engaged. The diminished seventh, minor seventh, and French sixth chords in the even zones have virtual status, in the same sense that the augmented triads have virtual status in sequences of consonant triads. As with the triadic case, not all transpositional values induce uniform voice leading. If two Tristan-genus chords are transpositionally related by minor third or tritone—that is, by exactly those intervals that are internal to a diminished seventh chord—then the voice leading is balanced, featuring contrary motion, as in the examples from the Tristan Prelude and Brünnhilde’s Immolation (figures 7.8(a), 7.9, and 7.14). It is the remaining transpositional values, involving intervals absent from a diminished seventh chord, that produce sequences with circumnavigatory powers. Table 7.3 presents a composition table of the Boretz and the octatonic transformations. Each transpositional value is produced by combining the Boretz transformation to its left with the octatonic transformation heading its column. As with table 5.1, the direction of transposition is determined by the order in which the transformations are applied and by the species of the transformed chord. Each of the four transposition values is produced by four distinct transformation pairs. Table 7.3 thus documents sixteen distinct transformation pairs. When a particular pair is selected, and its components presented in alternation, a Tristan-genus sequence tours the odd voice-leading zones and circumnavigates 4-Cube Trio.

Table 7.3. Combination table for the four Boretz-region and four octatonic-region operations, modeled after Table 5.1 S2

S4

S5

Octatonic pole

S3(2)

T±1

T±5

T±4

T±2

S

T±5

T±1

T±2

T±4

S6

T±2

T±4

T±1

T±5

R*

T±4

T±2

T±5

T±1

3(4)

Each transposition is the product of the Boretz transformation at the head of the row and the octatonic transformation at the head of the column. The transposition may be up or down, depending on the order of the operations and on whether the initial triad is dominant or half-diminished.

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 Audacious Euphony Figure 7.17. Roulade from Chopin’s Prelude, Op. 45.

In figure 7.8, (b) and (c) exemplify two of these sixteen possibilities: the T11 sequence is generated by S2 from the octatonic group and S3(2) from the Boretz group; the T5 sequence substitutes S3(4) as its Boretz-region transformation. To give some sense of the variety of transformation combinations available even within a single transpositional value, compare figure 7.8(b) with figure 7.17, an extended downshifting roulade in Chopin’s Op. 45 Prelude that circumnavigates 4-Cube Trio multiple times by alternating S4 and S3(4) (Childs 1998). As in the triadic case, class substitutions can introduce variety while still maintaining an overall voice-leading trajectory. Figure 7.18 illustrates with a passage from Chopin’s f minor Mazurka, Op. 68 no. 4.16 Tymoczko (2011b, 284–86) shows that the initial two phrases of this Mazurka execute one of many possible realizations of a script that connects a series of dominant seventh chords, each a semitone lower than its predecessor (T11), through intermediate ø7 chords. He also shows, in a companion analysis, that the e minor Prelude, Op. 28 no. 4, similarly connects a series of dominant seventh chords transpositionally related by descending perfect fifth (T5) and conjectures that the former piece is “a virtual rewriting of the latter.” The passage at figure 7.18, which precedes the reprise of the Mazurka’s initial phrase, supports that conjecture. The final harmonies in mm. 33–37 form a T5 chain of dominant seventh chords, as in the e minor Prelude. Locally, the progression alternates S2 and S3(4), realizing the template given earlier as figure 7.8(c). Leading into m. 38, S4 substitutes for S2, so that Aø7 sounds in place of projected Eᅈø7. This substitution triggers a conversion of the T5 sequence to one generated by T11, Figure 7.18. Chopin, Mazurka, Op. 68 no. 4, mm. 33–40, with voice-leading zones and Tristan-genus transformations indicated.

16. Any claims about this remarkable piece must be tempered by the knowledge that its text is notoriously unstable. Its editions, which disagree in many details, have been patched together from a nasty set of scratchings from Chopin’s pen. See Kallberg 1985.

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now realizing the figure 7.8(b) template. The semitonal transpositions continue until Dᅈ7, the German sixth chord in f minor, is reached, at which point the reprise begins. As with the transformational substitutions in chapter 5 (beginning with figure 5.15), the substitutions do not affect the voice-leading trajectory, which continues to downshift at a uniform pace throughout the passage. Zone 1 plays a central articulating role in this six-measure passage and, indeed, throughout the Mazurka. It is initially represented by G7, functioning as f minor’s double dominant, in m. 33. The return of zone 1, represented by Bᅈ7 at the third beat of m. 36, prompts an acceleration of surface rhythm, as diminished seventh chords are interpolated on the second beat of each measure beginning at m. 37. The next return of zone 1, represented by Dᅈ7 on the third beat of m. 39, breaks the T11 sequence in order to reprise the Mazurka’s initial phrase. In that reprise (not shown), zone 1 returns immediately in the form of a G7 chord, which triggers the T11 sequential descent that Tymoczko traces in that phrase. That descent is aborted with the appearance of yet another zone 1 representative, the E7 at the third beat of measure 5. In the consequent phrase, that same chord triggers an extended dream fantasy in A major (mm. 12–18), the sole moment in the Mazurka when a single triad, f minor’s hexatonic pole, is stabilized for an extended period. The interpolated diminished seventh chords at mm. 37–39 (figure 7.18) show how even-numbered stations can be engaged in the midst of a passage otherwise monopolized by Tristan-genus chords, and bridge gaps in the cumulative voiceleading descent. Even-numbered stations are integrated more comprehensively in the much-studied e minor Prelude, whose score is presented as Web score 7.19 . The Prelude is a slowly unfolding parallel period both of whose twelve-measure phrases are constructed of four-measure segments. Figure 7.19 models the first seventeen measures in a 4:1 reduction that represents each measure as a beat, and each segment as a measure. The first phrase reaches its structural subdominant at m. 9 and then prolongs dominant for three measures. The second phrase roughly inverts those proportions. Subdominant arrives already at the end of its fourth measure followed by eight measures of dominant prolongation prior to the final cadence at m. 25. What has caught the interest of literally dozens of publishing analysts, and many more teachers of analysis, are the connections between the opening tonics of each phrase and their structural subdominants. The voice leading of both connecting gestures is exclusively semitonal and distributed among the four voices so as to combine into species familiar from the classical harmonic bestiary: dominant sevenths, minor sevenths, half- and fully diminished sevenths, and French sixths. It is this feature that prompts Tymoczko (2011b, 286) to speculate that the Prelude and Mazurka realize the same abstract script. Framing these chromatic downshifts are progressions featuring whole steps. These “leaps” in chromatic space, indicated by diagonal lines in figure 7.19, mark the moments when tonic first loses and then regains its focus. The chromatic linear progressions are well formed from a Schenkerian standpoint, and accordingly, it is attractive to interpret the entire passage as completely secured by its diatonic frame and unilaterally oriented toward e minor. On Carl Schachter’s reading (1988), the events internal to the chromatic downshift bear no harmonic value and colonize no Stufen. Taking a more harmonic approach,

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Fred Lerdahl hears the downshifting tritones summoning a series of alternative tonics, none of which materializes (2001, 109; see also Tymoczko 2011b, 287 n. 21). Figure 7.20 presents a more complete 4:1 reduction of the opening phrase, omitting only the upper neighbors in the cantus. Diagonal lines indicate tritones. Those tones disposed to leading tone status are selected for annotation. The tritones initially summon tonics distributed along a line of fifths: first E and then A and D. Each leading tone defies the summons and continues downward in a triumph of gravity and inertia over magnetism (using terms cultivated by Larson 1994). Near the beginning of the second segment, an FᅊC tritone summoning G, the next projected tonic, sustains into the following measure. There it is joined by another tritone, DᅊA, which potentially interrupts the line of fifths, circling back toward the initial E minor tonic. Like their predecessors, first Dᅊ and then Fᅊ resist the call, descending instead to Dᅉ and Fᅉ, and the leading-tone-seeking energy is transferred to another diminished seventh two measures later, at the end of the second segment. Here finally, the call to resolution is heeded: as if awakened from its omnitonal haze by the whole-step motion in the bass, the Gᅊ remembers its monotonal responsibilities, and its rise to A triggers the reorientation to tonic. Attention to these leading-tone potentials leads easily to assignment of harmonic status for each chord, as dominants of unrealized tonics. Like the linear analysis advanced by Schachter, the harmonic one is well formed, provided that one allows tonality to structure one’s hearing even in the absence of sounded local tonics. Robert Gauldin (2004, 715) suggests a third hearing that combines aspects of the previous two, as a linear span harmonically supported not only at the boundaries but also by an interior structural pillar: E7 on the downbeat of m. 4, whose bass Figure 7.20. Chopin, Prelude, Op. 28 no. 4, mm. 1–9, with tritones indicated and leading tones identified.

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Figure 7.21. Chopin, Prelude, Op. 28 no. 4, mm. 1–9, adapted from Gauldin 1997.

E is a plausible component in the Bassbrechung of the tonic triad, and whose GᅊD tritone reappears at the end of the second segment.17 Figure 7.21, adapted from Gauldin, presents this hearing as a well-formed linear graph. The first four-measure segment converts the e-minor tonic to a V7 of the subdominant, and the second segment prolongs that dissonant harmony by opening up a chasm of arpeggiation, and filling it chromatically within each individual voice. The harmonic stasis of the second segment suggests figure 7.22 as an underlying model of the first phrase, suturing the end of the first segment to that of the second. The second phrase, modeled at figure 7.23, amply rewards this hypothesis. Its opening segment retraces the course of the first phrase but arrives at E7 a halfmeasure early. Chopin uses the two netted beats to break the constraints that the Prelude has husbanded with such consistent dedication. While the melody leaps a diminished seventh to the upper octave on beat 3, the lower three voices jump a minor third, a Gulliver stride in the Lilliputian world of this Prelude. The location where the chasm of arpeggiation is hurdled corresponds to that in the first phrase where the hypothesized jump of figure 7.22 was declined, in favor of the fourmeasure bridge that traverses the chromatic span. Figure 7.24 presents the first phrase in 4:1 reduction and addresses the details and pacing of its downshift, which covers a total of fifteen semitones: four in each Figure 7.22. Chopin Prelude, first phrase, with m. 4 sutured to m. 8.

17. Schachter 1995, 150, considers the possibility that the E at m. 4 is structural but rejects it since its dominant seventh resolves locally to a dissonance.

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lower voice and three in the cantus. An initial segment, leading from B7 at m. 2 into E7 on the downbeat of m. 4, descends by four cumulative semitones in the span of two measures.18 A second, quicker segment, extending from the third beat of m. 4 to the following downbeat, drops five further semitones in the span of a single measure, omitting the odd voice-leading zones. The final segment slows the pace of downshifting, requiring three measures to descend the four units from zone 8 to zone 4 before accelerating in m. 8 in preparation for the tonal reorientation at the following downbeat. Figure 7.25 wraps the circumnavigatory path about 4-Cube Trio (see also Web animation 7.25 ).19 The path intersects itself in the motion from 3 → 2, represented as Bø7 → Bo7. The two instances of this progression mark the two incremental accelerations of the outer phases, in preparation respectively for the arrival of the structural V7 of iv and its resolution to iv6 five measures later. The consequent

Figure 7.24. Chopin Prelude, first phrase, with downshift segmented into three phases.

1st segment

2nd segment

3rd segment

18. I follow the tradition of considering the tenor E as a suspension that stands in for Dᅊ. A more literal reading of the harmony would add EFᅊAB as a harmony at zone 6. 19. This figure closely tracks Tymoczko 2011b, 287–89, using a simplified geometry.

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Figure 7.25. Chopin Prelude modeled on 4-Cube Trio. Animation, with recorded performance, is at Web animation 7.25.

phrase is also represented on figure 7.25; divergences from its antecedent are indicated by broken lines. This phrase, too, begins with an incrementally downshifting segment to an e minor seventh chord at zone 0, with some changes in details due to a permuted firing order, followed by a leap to EGAᅊCᅊ at zone 10. After the voices of that chord are exchanged, further leaps to zones 7 and 5 prematurely terminate the downshifting segment of the consequent phrase, as both of the chords represented by these zones discharge standard tonal responsibilities with respect to e minor. The focus of this analysis of the Chopin Prelude, as with those that pertained exclusively to progressions limited to chords of the Tristan genus, has been on cultivating ways of documenting and exploring motion among standard dissonant harmonies, when those motions do not hearken to the call of the tonal forces to which those harmonies are normally subjected under the terms of classical tonality. But in each of these compositions, the summons is eventually heard, and dissonant sonorities revert to their inborn behaviors, sometimes retrospectively projecting those behaviors onto their predecessors. This suggests that the denizens of 4-Cube Trio, like those of Cube Dance, are overdetermined creatures, homophonous diamorphs through which composers and listeners can rapidly and

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 Audacious Euphony radically exchange syntactic expectations. Ways to conceive and represent these syntactic exchanges are taken up in chapter 8, in the tetrachordal as well as the triadic case.

Scriabin’s Mystic Species and Generalized Weitzmann Regions The analogy between triads and Tristan-genus chords is so rich and multileveled because the two are specific manifestations of a general phenomenon: the voiceleading efficiency that exists between nearly even chords of uniform size. The following paragraphs sketch the general case, necessarily resorting to an abstract mode of discourse, including a number of technical terms that are italicized at point of presentation. Given some nonprime universe of nq tones (n and q are integers, n > 2, q > 1), there exist q perfectly equal divisions of chord size n.20 Each perfectly even division serves as the core for 2n nearly even chords: n upshifters that result from upward semitonal perturbation, and n downshifters that result from downward perturbation. Each such region plays a role corresponding to that of a Weitzmann region (for q = 4, n = 3) or a Boretz region (for q = 3, n = 4); I shall refer to it as a generalized Weitzmann region (GWR). Any two chords that share a GWR are exactly two voice-leading units apart, whether related to each other by transposition or inversion. Two GWRs are adjacent if their cores are transpositionally related by semitone. If so, one GWR of the pair will be lower, and the other higher. Given two adjacent GWRs, the n upshifters of the lower GWR and the n downshifters of the higher GWR combine to form a bridging region, which plays a role corresponding to that of the hexatonic regions (for q = 4, n = 3) or octatonic regions (for q = 3, n = 4). Each nearly even chord is connected to n – 1 opposite-mode members of its bridging region by semitonal displacement in n – 2 voices (the other two voices remaining fixed) and to the remaining opposite-mode member of its bridging region by displacing n – 1 voices in one direction and the remaining voice in the opposite direction (generalized H, or Karg-Elert’s collective exchange; see Cook 2005, 131). The union of each chord with its collective exchange is equal to the union of the two adjacent cores of their respective GWRs. Earlier I noted that for every chord size there is a unique nearly even chord species, with superior voice-leading properties, but that, even among this parsimonious company, the potential of the nearly even trichord holds a particularly privileged status. The general formula just given shows why this is so. For every perfectly even division, there is a nearly even chord species whose members 20. I place to the side those nearly even chords whose size is prime relative to that of the chromatic system, for example, the diatonic and pentatonic scales within a 12-tone chromatic universe, or the triad within a 7-tone diatonic universe. Nearly even chords of this type are prime generated (Pressing 1983; Lewin 1996) and maximally even (Clough and Douthett 1991) and stand near the center of what Tymoczko calls type-2 lattices (2011b, 107–12). Because they do not displace perfectly even chords, they do not participate in the GWRs described in this section.

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partition into GWRs. Intra-regional progressions always involve two units of voice leading, no matter the size of the chord or the universe. Accordingly, there is nothing special about the 3-in-12 case. It is in the motion within the bridging regions that variety is located. Leaving aside the collective exchange as a special case, in all other motions through a bridging region, the number of voices is two less than the number of tones in the chord. Thus, from the standpoint of minimal voice leading, the ideal situation occurs when n = 3. It is only then that n – 2 = 1, that is, that a bridging motion involves only a single unit of voice-leading work. In this trichordal case, the bridges between two Weitzmann regions are the hexatonic transformations of chapter 2, among which are the minimal-work L and P transformations. In the tetrachordal case studied earlier in this chapter, where n = 4, the octatonic bridges between Boretz regions (S2, S4, S5) involve n – 2 = 2 units of voice-leading work, and hence no minimal-work transformations. As the cardinality of the nearly even species grows further, so too does the number of moving voices, and the bridging transformations become increasingly less efficient. Consider the nearly even six-tone chord species, Forte-class 6–34 with prime form [013579], whose members minimally perturb one of the two whole-tone scales. Both of its GWRs consist of six downward perturbations that produce Wozzeck chords (featured in the Berg opera; see Perle 1967) and six upward perturbations that produce mystic chords (featured in many late compositions of Scriabin). Motion between any pair, selected from these twelve, involves two units of voice-leading work, just as in a Weitzmann or Boretz region. Figure 7.26 models the opening of Scriabin’s Feuillet d’album, Op. 58 (1910), which James Baker (1986, 129) calls “a study of the properties of 6–34, with which Scriabin was then preoccupied.”21 The score of the opening measures is available at Web score 7.26 . Figure 7.26(a) presents the three transpositions of the mystic chord as they appear at the opening of the first three phrases. Thin lines indicate pitch-class prolongations through registral displacement; the thicker crosses indicate the progressive voice leading from one chord to the next. The first chord would be a whole-tone collection but for the substitution of Dᅊ for D in the “alto” voice. This substitution is recuperated in the second chord by Eᅈᅈ in the bass but is offset by E → F, which takes Dᅊ’s place as the alto. In the third chord, F is recuperated to E in the bass voice, while a new spoiler, Cᅊ, appears in the alto. Scriabin makes these idealized voice leadings salient through registral transfers and motivic play (see figure 7.26(b)). At m. 4, the Dᅊ5 spoiler and soon-to-bespoiled E4 of the first chord exchange registers, triggering a semitonal motion that resolves Dᅊ back into the source whole-tone collection and then spoils that collection by E5 → F in anticipation of the second mystic chord. A similar tenor/alto voice exchange occurs between the spoiler F5 and the soon-to-spoiled C/Bᅊ at mm. 8–9. In this case, the spoiling motion is delayed until after the appearance of the remaining voices on the downbeat of m. 9, and the motion from Bᅊ to Cᅊ 21. See also Pople 1989, who charts the interplay between the mystic and whole-tone collections. For related approaches to Scriabin’s late music, see Reise 1983 and especially Callender 1998.

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transforms a whole-tone collection on beat 2 to a mystic chord on beat 3. In each of the four semitonal transitions, the direction of semitonal motion echoes the direction of registral displacement. Both of the progressions illustrated here are in and out of a single equal division and thus are analogous to transformations within a Weitzmann or Boretz system. Bridging progressions, which would travel between two systems, would involve a transfer from one majority whole-tone scale to the other, and would be far less parsimonious, because of the number of tones included in each chord. Two tones, the spoiler and a soon-to-be-spoiler, would be held invariant, and each of the remaining four voices would be displaced by semitone, migrating from one whole-tone scale to the other. For example, the first mystic chord in Feuillet could be inverted into a Wozzeck chord by holding C and Dᅊ invariant and raising the remaining pitches by semitone. Whether or not some past composer has made, or some future composer could make, interesting music with such routines is an open question. Even were the question closed with a resounding no, the exploration is still worthwhile for the light that it sheds on the special capacities of triads, as selected from all possible nearly even chord types.

C HA P T E R

Eight

Syntactic Interaction and the Convertible Tonnetz

Everywhere, Romanticism exploits the ability to hear one and the same phenomenon in two and more ways; it is fond of this coexistence and its indefiniteness. —Ernst Kurth, Romantische Harmonik und ihre Krise in Wagners “Tristan”

The pan-triadic model developed in this book has left classical tonality to the side, for strategic reasons detailed in the introduction. Yet because the two syntaxes are so deeply intertwined in most nineteenth-century compositions, the terms and concepts of classical tonality have surreptitiously crept into virtually every analysis in the book. If the syntaxes operate independently of one another, then the challenge is to model their intertwining without collapsing them into each other. As Steven Rings has emphasized in a recent assessment, such a model must do more than dump transformational labels into the same cage with Roman numerals or Schenkerian representations; it needs to study how they get along within those confines. Otherwise, he writes, “the theoretical divide is thus reified in analytical practice, resulting in a curiously bifocal view of chromatic harmony, one in which the triad seems not so much ‘overdetermined’ as dichotomous” (2007, 34). In this chapter I review some ways that recent theorists of chromatic harmony, including David Lewin, Brian Hyer, myself, Steven Rings, and Candace Brower, have met this challenge, before developing a model that builds on aspects of that earlier work but overcomes some of its limitations.

Some Previous Proposals Figure 8.1 presents two models of syntactic interaction, both crafted in response to late piano compositions of Schubert. Figure 8.1(a) is a version of a figure from my 1999 analysis of the first movement of the Bᅈ Sonata. In figure 6.9 of this book, I presented three interpretations of that movement; this figure is associated with the first of those. The figure positions the four hexatonic cycles as cross sections of 169

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(b) adapted from Rings 2007

Figure 8.1. Two models of syntactic interaction.

a cylinder (Cohn 1999) and connects each triad horizontally to its perfect-fifth transpositions. The two ends of the cylinder glue together with a one-third twist, forming two continuous circles of fifths, composed of major and minor triads, respectively. I suggest there that the cycle that includes Bᅈ major has tonic function and that the fifth relations above and below bear dominant and subdominant function, respectively. Figure 8.1(b), adapted from Steven Rings’s 2007 analysis of the Eᅈ Impromptu for Piano, D. 899 no. 2, positions each triad at the intersection of a

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“vertical tonal-functional” and a “horizontal hexatonic” axis. His model is implicitly cylindrical and hence isomorphic to mine. What distinguish his figure from mine are the arrows along the vertical axis, accompanied by transformational labels adapted from David Lewin (1982, 1987). The arrows are single-headed because, unlike the Hexatonic- and Weitzmann-group transformations used in this book, these transformations only work in a single direction; they are not their own inverses. DOM is equivalent to T5; Lewin writes that the source chord “becomes the dominant of ” the target chord (1987, 176). (According to this definition, G major is not the dominant of C minor, even though musicians frequently refer to that relation in those terms.) SUBD, its inverse, is equivalent to T7 = T–5; the source chord “becomes the subdominant of ” the target chord (177). Rings emphasizes that these transformations capture not only the syntactic flow but also the semantic charge that is associated with classical tonality. This charge roughly stands in for what musicians elsewhere capture through such terms as summoning, leading, gravity, magnetism, attraction, desire, and so forth. Rings writes that the dimensions “mark out a unified space in which we can map progressions that exploit both the triad’s tonal-gravitational properties and its triadic-transformational potential, without privileging one at the expense of the other” (52). Three considerations limit the capacity of figure 8.1 for generalization beyond the compositions to which they initially responded. The first is their orientation to a hexatonic configuration of chromatic space, which limits their applicability to a repertory that often organizes its chromatic progressions along a minor-third/ octatonic axis.1 A second limitation is that Rings arrows implicitly commit to the presence of tonal-gravitational forces whenever motion occurs along the fifths axis, excluding the possibility that those forces are slackened in the case of sequences, whether diatonic as François-Joseph Fétis argued, or chromatic as in the L/R chains studied in chapter 5. That possibility can only be honored if the Rings arrows function as a default component that can be deactivated under appropriate circumstances, rather than as an inherent property of the model. Finally, both models relegate third relations to the chromatic axis and thus have difficulty capturing the common tonal procedure of dividing a diatonic fifth progression into two constituent thirds. Lewin proposed dividing DOM into two MED transformations, each taking the root down a diatonic third; he writes that the source chord “becomes the mediant” of the target chord (1987, 176). Inversely, SUBD, the motion from a subdominant to its tonic, divides into two SUBM transformations, each taking the root up a diatonic third. To integrate these transformations on figure 8.1(b), the mediants would need to be placed halfway between two DOM- or SUBD-related triads, as in figure 8.2. 1. Minor third relations are not significant in the sonata movement, but they occur prominently in the Impromptu. Rings responds by shearing the figure, derigging major third L relations and rigging up minor third R ones in their place. Rings ingeniously gets this contraption to carry interpretive hay by asserting that the moment of realignment in the model coincides with a moment of disjunction in the Impromptu, reflecting (or creating?) a semantic bang. He thereby converts an awkwardness in the model into a profit in the interpretation, in the manner of Lawrence Kramer’s hermeneutic windows (1990). But the particular case resists generalization. Where such semantic compensations are lacking, the deficit in the model will remain a deficit in the net reckoning.

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Figure 8.2. Rings model with mediants interpolated.

This solution introduces an interpretive complication: c minor → Aᅈ major projects both eastward into a hexatonic cycle and southward toward subdominant f minor.2 Depending on one’s analytical aims, this complication need not be a disadvantage, and indeed might even be an advantage if there were an analytical motivation to conceive of these two projections of c minor → Aᅈ major as representing a crisp phenomenological distinction. Lewin 1992 asserts just such a distinction in an analysis of Parsifal, claiming that one hears c minor → Aᅈ major and its transpositions in terms of L when the progression is affiliated with Amfortas’s suffering, but otherwise in terms of MED. But this offers little guidance in the general case. More commonly, the distinction will lack a difference, forcing an arbitrary choice, or will involve an ambiguity that might benefit from cultivation rather than premature resolution.3 To indicate the kinds of difficulties that can arise from such a crisp distinction, consider figure 8.3, which presents a synopsis of the first forty-three measures of Liszt’s Consolation no. 3 in Dᅈ major. A complete score is given at Web score 8.3 , and a recording is embedded in Web animation 8.9 . The music consists of two sentences, both of whose presentation phrases cadence in f minor. The initial sentence classically continues through ii6 to a perfect cadence in Dᅈ major, executing an expanded cadential progression (Caplin 1998, 61). The second sentence continues instead to a cadence in A minor, after which a second continuation returns to Dᅈ major, completing a major-third division (see figure 2.11(b)). 2. An additional problem is that the figure ceases to function as a product network, with all the technical advantages that such a status entails (Lewin 1987, 206; Rings 2007, 52). In order for the graph to maintain its status as a product network, the newly added mediant chords would need to form L and P relations with their neighbors along the rows. Instead, they participate in R/S cycles, creating a transformational path through a Weitzmann region rather than a hexatonic one. 3. Lewin concedes as much when he suggests that a diatonic hearing of Amfortas’s pain figure is “latently possible” (1992, 55n4).

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Figure 8.3. Synopsis of Liszt Consolation no. 3. See Web score 8.3 for a complete score.

From the standpoint of a final-state hearing, the double positioning of f minor on figure 8.2 is well motivated. The initial sentence moves northward, engaging f minor as a mediant on the way to a dominant. After returning to tonic at its initial position, the second sentence moves westward, engaging f minor as a Leittonwechsel that initiates a hexatonic journey. Yet this conception ignores a significant aspect of in-time experience. Arriving at the second f minor cadence, one has no reason to be aware of having embarked on a westward journey through chromatic space. Indeed, the principle of “parallel passages in parallel ways” (see chapter 3, note 11) suggests rather a retracing of the northward path toward dominant, as at m. 7. The continuation phrase forces a retrospective reevaluation of that position; we realize that we were migrating leftward, not upward. This reevaluation depends on identifying f minor on the vertical axis of figure 8.2 with its associate on the horizontal axis. But the model presents us with no means for establishing that identity: the two f minors occupy different positions, and our phenomenological journey from one to the other involves a magical wormhole for which the model has no explicit account. An influential paragraph from a 1984 article by Lewin will help identify the problem and suggest a solution. 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 simply the old objects and relations dressed up in new packages with new labels; they are essentially different objects and relations, embedded in an essentially different geometry. That is so even if in some contexts the two spaces may coexist locally without apparent conflict; in this way the surface

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Is the geometry that Lewin envisions compatible with my cylinder and Rings’s grid? Standing at a triad, one inhabits two distinct spaces, represented by the intersecting axes, “without apparent conflict.” Yet Lewin’s thrice-iterated conjunction of “objects and relations” suggests that the intersection of the spaces includes not only triads but also relations that pair them. He imagines the intersecting space as a surface rather than a set of discrete points. Paths intersect not only at points where they cross but also at segments where they merge. The grids of figure 8.1 show the triadic objects coexisting without apparent conflict, but the forced assignment of diatonic third relations to one axis or the other in figure 8.2 precludes the possibility that a triadic relation can coexist simultaneously in two spaces. Brian Hyer’s 1989 dissertation developed a geometry capable of simultaneously modeling both of Lewin’s spaces, while situating each object and relation in a unique location. Hyer positions each triad as a point and connects it to its L, P, and R associate, as well as directly to its modally matched fifth. This Tonnetz models chromatic space by identifying (“gluing”) enharmonically and syntonically equivalent points at opposite ends of the plane. Each such dimensional folding individually creates a cylinder like figure 8.1(a). As Hyer phrased it in a subsequent article, “the [transformational] group as a whole disperses the functional ‘significance’ of [a single] 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” (1995, 127). Hyer’s Tonnetz, however, has the capacity to change shape in response to how the listener hears the relations among its objects. If the triads are heard to collaborate in the definition of some tonic, then the glue loses its bond. “To assert a given triad as a tonic . . . forces us to imagine transformational relations with regard to the tonic, and to calculate them in scale degrees rather than generic semitones, in effect decircularizing [the Tonnetz], extending its [axes] in all directions” (1995, 127). Converting from a circular to a planar geometry “impos[es] a sense of perspective on the surrounding terrain, a point of view from which all the other triads appear to be near, more or less remote, or over the horizon” (127–28). Inversely, “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 lattice comes back into play” (Hyer 1989, 215).5 Hyer’s convertible Tonnetz is ordered up to Lewin’s blueprint in almost every respect. Each triad occupies a unique position, as does each direct triadic relation. 4. The italicized passages in the original 1984 publication were romanized in the 2006 reprint (194). The relevance of this passage to the present situation hinges on the interpretation of “Riemannian tonal space,” whose domain of reference was mobile in Lewin’s writings of the 1980s. It is nonetheless clear that the Riemann/scale degree distinction has strong affinities with binary relations that Lewin elsewhere cultivates in terms of chromatic/diatonic and atonal/tonal. 5. The distinction between the circular and planar interpretations is equivalent to the conforming/ nonconforming distinction in Harrison 2002a.

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The interpretation of each relation is contingent on whether the space is closed or open. The structure of the space cannot be inferred from the standpoint of a single triad, or even of a direct relation such as the Dᅈ major and f minor of figure 8.3. That structure is cylindrical when the space is closed, exactly as in Lewin’s metaphor. Only one detail is astray: where Lewin envisions a Möbius strip, Hyer constructs a plane. To bring the vision to full realization requires us to imagine Hyer’s plane closed into a loop, with a half-twist. Candace Brower’s 2008 consideration of the Tonnetz as a model of diatonic space provides a motivation for exercising our geometric imagination in exactly this way.

The Diatonic Tonnetz Rather than accepting the Tonnetz as a space given all at once, Brower generates it in stages from a seed node. The stages roughly retrace the development of pitch systems across historical time (Brower 2008, 71) and encourage us to explore with some precision the interplay of forces that precipitate and shape this development. Brower recognizes a centrifugal force that operates on purely tuned triads, transposing them by purely tuned intervals. This is balanced by a centripetal force that relies on our psychophysiological propensity to ignore pitch differences beneath a threshold of just-noticeability, treats similarity as equivalence, and induces systemic closure through symmetric completion (68). Brower’s tale is thus a variation on Plato’s allegory of musical forces whose reconciliation provides a model for the ideal civic society (McClain 1978). Although motion from a tone to its incremental approximation masquerades as stasis under our notational system, it can be conceived in terms of voice leading through a comma. Accordingly, Brower’s distinction between pure and flexible tuning can be recast in terms of a tension that has been central to this book from the outset, between acoustic consonance and melodic proximity. Accordingly, in retracing the stages of Tonnetz generation, the following exposition periodically pauses to evaluate the respective contributions of consonance maximization and voice-leading minimization. Although at times these forces collude so deeply as to be indistinguishable, we will occasionally find seams that allow them to be pried apart. At the first stage, shown at figure 8.4(a), a purely tuned C major is transposed by its most acoustically powerful interval, yielding F major to the left and G major to the right (Brower 2008, 71).6 Because the interval of transposition is internal to the structure of the triad, where it appears only once, neighboring triads share a single point. Alone among the initial tones of the triad, E is unshared at this stage of development. Yet E stands in the same fifth relation with two neighbors on its 6. This derivation of the diatonic scale is appropriated from Hauptmann and Riemann; see Harrison 1994, 44–45. Brower’s Tonnetz rotates mine by 90°, to bring absolute direction in correspondence with the intuition that dominants are above their tonics. I have not followed her in this, not only because of historical precedent but also because her orientation causes upshift voice leading to flow down the page, and vice versa.

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Figure 8.4. Genesis of a diatonic Tonnetz.

rank, A to its left and B to its right. Honoring this acoustic potential leads to the closure of two local triangles and creates a trapezoid about the periphery of the seven-tone figure, as shown at figure 8.4(b). The relation of the two minor triads inverts the relation of adjacent major triads, suggesting the possibility of replicating the entire three-triad complex on the minor side. This suggestion can be fulfilled by adding either b minor to the right or d minor to the left (figure 8.4(c)). The latter solution is preferred because it evidently replicates a tone already present, thereby creating a first stage of systematic closure. From the standpoint of pure intonation, the closure is an illusion: if each edge represents a justly tuned interval, then the two D tones differ by a syntonic comma. Honoring the comma was a cardinal priority for many theorists of the mid-nineteenth century and, indeed, continues to be so in some musical microcultures (Perlman 1994; Duffin 2007). Yet there is a strong incentive to accept some nonidentity as a quasi identity, since to refuse to do so at every opportunity is to invite a boundless proliferation. Accepting this identity creates a first stage of systemic closure, completing the portrait of the diatonic collection as a parallelogram whose interior is tiled into six consonant triads, as at figure 8.4(d). I will use the term syntonic image to characterize the relation between the circled tones occupying its remote corners. The parallelogram encapsulates its contents into a microecological hothouse, activating the internal dynamics signified by Rings arrows. Under Renaissance modality, the six encapsulated triads are in principle equally stable, and any of them can serve as finalis. Under classical tonality, only the central triads, sheltered from the parallelogram’s unstable corners, enjoy this privilege. In a process elegantly described by Gottfried Weber (1846 [1817–21]) as

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Figure 8.5. T → S → D → T on the Tonnetz.

“Attunement of the Ear to a Key,” and recently revived in a more overtly psychological framework by Fred Lerdahl (2001, 193ff.), the blank ear takes a triad as a metonym for a diatonic system that remains a hypothesized default, subject to confirmation. Hearing a consonant triad, one provisionally situates it inside its appropriate parallelogram. Standard confirmation comes in the form of a tonic → subdominant → dominant → tonic (abbreviated T → S → D → T) progression represented on the parallelogram by two leftward arrows, one initial and one terminal, linked by a longer rightward arrow connecting its two sides (Brower 2008, 71). There are two distinct ways to interpret the motion of the arrows in figure 8.5. On the planar Tonnetz, subdominant and dominant balance about tonic, and the rightward arrow represents an overcompensation. Riemann imagined the subdominant as a “stretched bow which slings the arrow beyond the mark” (Riemann 1893, qtd. in Rings 2007, 50). The slung arrow may be interpreted to represent the “vault” across the syntonic comma. Yet Riemann took equal temperament as an empirical given (Rehding 2003, 50), and so that vault was executed not in the acoustic signal but rather in the imagination.7 Once the acoustic identity of the two D tones is recognized, it is a small step to honor their unification in conception, leading to a more conjunct relation between subdominant and dominant. This conjunction is represented on the Tonnetz by a geometric manipulation that is difficult to represent or imagine in two dimensions but can be reproduced in three dimensions with appropriate implements. First cut out the parallelogram of figure 8.4(d). Then loop it and glue together its remote corners. D is now at the center of two orthogonal line segments that share points of terminus: a sevenpoint coil of fifths, FCGDAEB, and a three-point line of minor thirds, FDB. The second stage of closure involves accepting the pseudo-fifth BF as if it were perfect, creating a direct connection between its components. The result is that the coiling line of fifths becomes a circuit. Since the coiling process positioned B two ranks above F on the FDB line of minor thirds, they can be brought into direct contact only by crooking that line at its center, fixing the DF edge, and folding the BD edge so that B comes alongside F. B now intersects the coil of fifths as it is projected beyond its F terminus. The new edge that connects B and F locally closes a triangle that represents the diminished BDF triad and globally closes the chain of triangles into a loop, representing the seven diatonic triads arranged into a continuous series of diatonic mediants (Agmon 1995). Because the DF edge is fixed and the BD edge inverts, every intermediate point of the parallelogram downfolds 7. On the relation of the ideal and the material in Riemann’s treatment of intonation, see Harrison 1994, 237, and 2002a, 136.

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8. Compare chapter 2, p. 37 of this book.

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Figure 8.6. Two voice leadings for T → S → D → T.

upshifting figure 8.6(b), which reconstructs the double emploi of Jean-Philippe Rameau. There is little reason to doubt the deep connection between eighteenth-century diatonic tonality and the acoustic capacities of the triad. Nonetheless, there exist segments of diatonic music that force acoustic properties into remission and tame the semantic power of Rings’s arrow. What alerted Fétis in the 1840s to the special nature of these sequential moments was the behavior of the pseudo-fifth, whose appellative powers are neutralized by the “symmetry of movement and succession” (2008 [1844], 30). The specific identities of intervals dissolve into their generic categories, and the contours of the Möbius strip disindividuate into a strangely shaped, greased racing oval. Although what remits Fétis’s “law of tonality” is his “law of uniformity,” there is another, silent partner in the success of these progressions: the nearly even status of the diatonic triad, which ensures that any two of its species are step connected (Agmon 1991). The diatonic Tonnetz is already a robust space, dually determined by the properties of the overdetermined diatonic triad that generates it. Here the potential reciprocity of acoustics and voice leading, and their analytic inseparability, can already be glimpsed even before the limits of the parallelogram are breached.

Horizontal Extensions Incremental extensions: Modulation to closely related keys The encapsulation of the diatonic Tonnetz as a Möbius strip depends on the acceptance not only of the syntonic comma as a perfect unison but also of the pseudofifth as a perfect fifth. No amount of tempering will cure the latter case; one is asked to simply accept it axiomatically. Renaissance musicians often opted instead to extend the chain of perfect fifths outside the parallelogram. The result is a historical expansion of the diatonic Tonnetz along the fifth axis, at first incremental, ultimately compounded in a process whose documentation was a lifelong project of Edward Lowinsky (1946, 1989 [1967]; see also Brower 2008, 83). By the classical

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 Audacious Euphony era, the tracks, in all their potentially infinite extension, were a thing made rather than a thing in the making. Although eighteenth-century composers had the option of traveling the length and breadth of the extended line of fifths, they usually declined to do so. A narrower band of fifth space, closed by the pseudo-fifth, was better suited to the cultivated microecology of the hothouse and the aesthetic and semantic attributes of classical tonality that arose within it. A limited degree of mobility along those tracks nonetheless suited the ends of classical tonality, as it allows the diatonic capsule to be embedded into a larger “modulatory” shell that replicates its properties and dynamic features on a broader scale. Sharps to the right and flats to the left temporarily displace their diatonic letter-name equivalents, shifting the parallelogram along a horizontal track. The principle of close relation dictates that such a shift be limited to a single degree on either side of the tonic. With each shift comes a new pair of syntonic images, a new pair of potential tonics, and a reversal of Rings arrows and the syntactic and semantic forces they represent. These shifts might be characterized in terms of secondary (applied) chords, tonicizations, or modulations, depending on their duration, the presence or absence of cadential signifiers, and the proclivities of the characterizer. In order to demonstrate the plotting of fundamental diatonic progressions and close modulations on the Tonnetz, figure 8.7 analyzes the opening sentence of Liszt’s Consolation no. 3, as a series of snapshots of a listener’s awareness as the music unfolds. These images are fused in Web animation 8.9, an animated version with embedded sound file . Figure 8.7(a) depicts the opening three measures, which arpeggiate a Dᅈ major triad in the manner of a nocturne. In addition to presenting the triad as a triangle, the figure also presents the diatonic collection in which any acculturated listener imagines it contained, a collection of tones inferred but as yet unsounded. Figure 8.7(b) represents the four-chord basic idea of mm. 3–6, in terms of the T → S → D → T paradigm introduced in connection with figure 8.5. North American harmony pedagogy would likely analyze this same progression in terms of T → [D] → D → T, positioning the {G, Bᅈ, Dᅈ, F} chord to the right of the dominant. The subdominant interpretation offered here follows the embedded-triad protocol introduced in connection with figure 7.3, maintaining the Dᅈ and F common tones in a single location rather than displacing them a syntonic comma to the southeast. Only the G, treated as a supplementary under-seventh, is located to the dominant side, dislocated from the remaining tones of the chord on the plane. This interpretation is consistent with the Riemannian approach to dissonant harmonies, treating them as a mixture of dominant and subdominant components. Figure 8.7(b) places G within a star and connects it by a dotted line to the triad that it supplements. (This is an ad hoc solution that functions well here but will create clutter in many other cases.) The initial displacement of tonic thus shifts in both directions on the Tonnetz: leftward at the level of the triad (Aᅈ → Bᅈ) and rightward at the level of the diatonic collection (Gᅉ for the hypothesized Gᅈ of figure 8.7(a)). Both involve an upshift in pitch-class space. The subsequent motion to the dominant seventh chord involves a compensation at both levels. The harmony moves to the right, or dominant, side of tonic, as indicated by the overshifting arrow. At the

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Figure 8.7. Liszt Consolation no. 3, first sentence. See Web score 8.3 for a score. Animation, with recorded performance, is at Web animation 8.9. (a)–(b) The basic idea, tonicizing Dᅈ major. The starred tones are supplementary dissonances for the triads to which they are attached. (c) Repetition, modulating to f minor. (d) Continuation from f minor to Dᅈ: ii6. (e) Cadence.

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 Audacious Euphony same time, the dissonance, Gᅈ, starred at the left edge of the figure and connected to the Aᅈ major that it supplements, replaces the Gᅉ that supplemented the prior harmony. Thus, the collection shifts left as the harmony shifts right, both motions involving a downshift in pitch-class space. The cadential tonic at m. 6 is then reached via triangulation from both the dominant and subdominant side. Although both dissonant chords internal to the basic idea individually bear dominant and subdominant elements, in combination they balance more strongly to the dominant side, marking out territory to be pursued when the basic idea is repeated. Figure 8.7(c) models the subsequent tonicization of f minor, fulfilling the subdominant potential of the basic idea’s initial dissonant harmony. The supplementary dissonances here are not starred, to enhance the legibility of the figure, but are easily imagined. The cadence in f minor shifts the syntonic images rightward, from Eᅈ to Bᅈ. The initial dissonance on bᅈ minor comes into relief here, as the only diatonic triad that bears subdominant function in both keys. The figure also includes an initial vertical extension to the rank above, in order to acknowledge the honorary-diatonic status of E as f minor’s leading tone and as the third of its major dominant, substituting in both capacities for Eᅈ. Such vertical extensions are theorized in the next section of this chapter and exemplified by the second sentence of Liszt’s Consolation. Figure 8.7(d) presents the beginning of the first sentence’s continuation phrase at mm. 12–13, where f minor is transformed into a diminished seventh chord over a common F bass, which presses toward eᅈ minor qua diatonic ii6. The diminished seventh chord presents a particular challenge to the graphic apparatus, as it mixes subdominant and dominant tones that are not adjacent on the plane.9 Again, I have adopted an ad hoc solution, forming a quadrilateral bounded by DF above the tonic and AᅈCᅈ on the left edge, and inserting a black dot to represent the temporary occupation of that space. Figure 8.7(e) presents the expanded cadential S → D → T progression. As an analytic statement about Liszt’s Consolation, figure 8.7 is far from ideal. Its four components cumulatively consume considerable space in order to make some fairly rudimentary claims about music well understood using other representational modalities that are more familiar, more economical (in the case of chord and function labels), and less abstract (in the case of Schenkerian graphs). The Tonnetz representations do have some added value, allowing one to track the role of common tones in forging moment-to-moment connections, redefine a single triad in multiple tonal contexts, and observe the interaction of dominant and subdominant regions (or, more neutrally, upshift and downshift voice leading). But these considerations hardly justify the investment in initial pedagogical overhead and ongoing production costs. A more significant motivation will emerge as we study the interaction of these diatonic Tonnetz representations with their expansion into chromatic pan-triadic space.

9. The analysis of a diminished seventh chord as mixture of subdominant and dominant derives from Rameau (see p. 140 of this book). Its Tonnetz depiction is inspired by Oettingen 1866, 265. See also Harrison 1994, 65–69.

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Recursive extensions along the L/R chain The familiar processes described in the preceding section are subject to recursive application, precipitating more ambitious excursions away from the global tonic. A pair of shifts in a single direction already exits the system of closely related keys, tonicizing a chord that is chromatic with respect to the initial tonic. Sequential tonicization of fifth-related, modally matched tonics, as in many classical development sections, compounds this process. The moving window begins to resemble a local train, migrating incrementally in a uniform direction and locking temporarily into a series of adjacent stations. The track on which this journey takes place is the L/R chain studied in connection with figure 5.9. Sequential migrations along the L/R track can also occur in the absence of cadential activity to lock in the series of stations. In such cases, the window slides transiently through a series of keys, each closely related to its neighbors in the chain. An early example is the Adagio from J. S. Bach’s d minor keyboard Toccata (BWV 913, pre-1708; a score is available at Web score 8.8 ). After establishing g minor as tonic, the Toccata moves incrementally leftward, substituting flats for naturals in key signature order until six flats are in play. It then reverses course, replacing the flats with naturals in the order that the latter were cast out. This rightward motion overshoots, replacing all six flats with their corresponding naturals, ultimately terminating at d minor’s dominant in preparation for the final fugue. Each added flat or natural signifies a new minor tonic, but none are cadentially confirmed. Some approaches are abandoned at the cadential 64 ; others, at its dominant resolution. Still others make it as far as an imperfect cadence, which is immediately undone; the sensation is of a housefly landing and immediately taking off again. In such situations, the incrementally migrating window is a surveillance engine that slows to inspect each station but alights at none. In still other passages, a series of mediant-related triads acts as a slippery slope for an extended excursion into chromatic space. The prototype is mm. 159–71 of the Scherzo from Beethoven’s Ninth Symphony (see p. 92 of this book, and Web score 5.9 ). As the triads fly by, they have the potential to be gathered into bouquets of six and constituted as a diatonic space. But after several hypothesized tonics fail to materialize, we give ourselves over to the momentum of the journey. Once the pattern is broken, our tonic-seeking radar might reawaken, stimulating us to gather up the most recent few triads and kindle a cadential fire that activates their tonal potential in retrospect. I imagine this a bit like an express-train traveler who stops wondering what it might be like to disembark at each local station and focuses on the somatic sensations of the ride, until the slowing of the wheels reengages images of the external world. Here is where the Tonnetz earns its value despite the extra weight it requires us to port about. Exiting the initial diatonic space, the parallelogram slides incrementally left or right. Flying past multiple diatonic spaces, its diagonal edges ease open, and the horizontal track converts to an alley of indefinite extension. Rings arrows retract into the body of the moving vehicle. Approaching a new diatonic station, doors flip closed, Rings arrows reemerge, and we psychologically reorient

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Vertical Extensions The L/R chain cannot capture one chromatic extension that arose at an early historical stage, that flared up prematurely in connection with figure 8.7(c), and that is familiar to any harmony student from an early stage of tutelage: the substitution of the semitone below tonic for the whole tone, when the latter is proper to the diatonic scale. Figure 8.8(a) locates this substitution on the Tonnetz, with respect to A minor: Gᅉ is replaced by the Gᅊ directly above it on the dominant side of tonic, converting the minor dominant east of tonic to the northeast-positioned major dominant, the nebenverwandt of the minor tonic. A dual echo that initially arose in the sixteenth century (see chapter 7, note 3) and became increasingly prevalent between 1725 and 1825 involves the substitution of the semitone above the fifth scale degree for the whole tone, again when the latter is proper to the diatonic scale. Figure 8.8(b) locates this dual substitution with respect to C major: Aᅉ is replaced by Aᅈ directly beneath it on the subdominant side of tonic, converting the major subdominant west of tonic to the southwest-positioned minor subdominant, the nebenverwandt of the major tonic. Since Schenker’s time, it has been standard to explain the phenomena portrayed in figure 8.8 by invoking modal “borrowing” or “mixture.” One says that, lacking a semitonal discharge to tonic, minor borrows that capacity from parallel ^ major. Lacking a semitonal discharge to 5, major borrows an asset from minor’s private stash. This lovely allegory of civic reciprocity in the late-Platonic republic has three limitations, qua explanation: it is presumptuous, logically unparsimonious, and ahistorical. It presumes that, from the white-note perspective, there already exist fully ramified three-sharp and three-flat scales. But one searches in vain for those collections in figure 8.8, as one does in early polyphony. It is unparsimonious because it borrows three tones, only to discard two. Why order a

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Figure 8.8. Diatonic collection with nebenverwandt substitutions. three-course dinner when the tone for which you hunger is available à la carte? The substitutions in figure 8.8(a) are sufficiently explained by a voice-leading principle articulated by Marchettus of Padua in the early fourteenth century without any reference to scales: a semitone may substitute for a whole tone. As I showed in chapter 2, in the general case the displacement of a single tone by a single unit requires an exchange of chord type, and thus a violation of harmonic consistency. It is only when the chord is nearly even and its cardinality is odd that such a semitonal substitution preserves chord type. It was only in the fifteenth century, a century after Marchettus, that harmonic consistency became a compositional desideratum and that the Gᅉ → Gᅊ substitution at the pitch-class level became staged as an e minor → E major substitution at the triadic level. In our genesis story, this is the moment when the near-evenness of the consonant triad becomes crucial, and its acoustic properties become epiphenomenal by-products. The e minor → E major substitution reduces voice-leading work while maintaining harmonic consistency, a circumstance enabled by the triad’s near evenness in chromatic space. True, the major-for-minor substitution is also an acoustic gain, as it replaces a minor triad with a more purely tuned major triad.10 But this gain is more than offset by an immense loss at the level of the scale: the Gᅊ enters into a howlingly dissonant relation with all of the natural-scale tones with which it is not directly connected on the Tonnetz. It is in the rectification of this 10. Local acoustic considerations nonetheless have considerable historical force in motivating majorfor-minor substitutions, independently of any gain in voice-leading parsimony. This is most clear in the case of the tièrce de Picardie, and its nineteenth-century expansion as the aspera → astra plot prototype. Moreover, although e minor → E major and F major → f minor register equivalent voiceleading gains, the former arises at an earlier historical moment, a circumstance that can only be explained in terms of an acoustic bonus.

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The Convertible Tonnetz Such diachronic fantasies are of little consequence to composers and listeners of the classical and later eras. For them as for us, the universe is given all at once, as a set of opportunities to be taken or declined, depending on composerly temperament and the perceived tastes and tolerances of patrons and listeners. The Tonnetz becomes a black-box theater, whose walls can be erected, dismantled, and moved without fuss. One is free to establish an encapsulated diatonic space, activating the Rings arrows with their associated tensions and attractions. One may also navigate purposefully along one of its bounded hexatonic or octatonic alleys, or over a set of connected diatonic spaces without alighting, or by establishing residence in the encapsulated space of a pitch retention loop. One may perambulate more freely, meandering from triad to triad across shared edges. Adding a supplementary dissonance to any triad, one may transfer into an orthogonal space consisting of Tristan-genus objects, and wander that space as if oblivious to the semantic values of classical tonality and the syntactic obligations that they impose. Yet as we navigate the unbounded Tonnetz, we are alert to the possible reassertion of classical syntax on the triadic and Tristan-genus objects that are traversed. 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. Confirmation reorients our attention from the far-flung space toward a well-defined sector of it, and ultimately to a particular location in that space, the cadenced triad. The process might be likened to entering the four walls of our home, or sprouting landing gear and coming within view of a planet or an airport, or exiting the freeway and entering a fortified village or gated community. This last image suggests that the Tonnetz resembles a hybrid automobile: one engine for the highway of chords, and one for touring the encapsulated neighborhoods, with all of their tensions and fraught attractions. At any moment, one may brake on a subdominant, reverse into a dominant seventh along the curb, pull forward to a tonic, and kill the engine. Figure 8.9 analyzes the second sentence of Liszt’s Consolation as a sequence of six stroboscopic snapshots, arranged chronologically from top to bottom. (Web animation 8.9 merges them into a single animation, with sound file embedded .) The first image, labeled “first sentence,” presents a synopsis of the composition’s first twenty-three measures, treating f minor as the mediant degree within a

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Figure 8.9. Liszt Consolation no. 3, second sentence: a stroboscopic Tonnetz portrait. Animation, with recorded performance, is at Web animation 8.9.

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 Audacious Euphony well-ordered Dᅈ major region. The second image synopsizes the presentation phrase of the second sentence. This time, as in figure 8.7(b), f minor is treated as a modulatory goal, sliding the diatonic capsule to the right and engaging a new set of syntonic images at Bᅈ. What is new in figure 8.9 is the sprouting of a hexatonic alley, indicating the future trajectory of the phrase. The remaining components of figure 8.9 show the progressive ascent on the hexatonic elevator. The triad sublimates its earlier role as acoustic generator, becoming the optimizer of voice-leading parsimony. Yet these triads are not only objects in pan-triadic voice-leading space. Each one is tonicized through the resources of classical tonality. F minor becomes F major at m. 31 through their shared dominant seventh, and F major becomes a minor at m. 35 by interpreting the former as bearing subdominant function and sounding the latter’s dominant. Every tonicized triad thus carries a dual role, as a voice-leading object in Dᅈ hexatonic space and as an acoustic object in its own local diatonic space. Every event in mm. 31–35, for example, has a determinate role within the diatonic pitch space of a minor. Within that context, there exists no functional or enharmonic ambiguity. All of the modal degrees of a minor are present. A minor serves as a field of attraction for its surrounding pitch classes, bestowing leading-tone charges upon Gᅊ, F, and so forth. We are fully comfortable with mm. 31–35 as a diatonically determinate microcosm whose perceptions are organized in relation to a minor. What we do not understand is the diatonic function of that tonic itself, in relation to the ^ ^ larger Dᅈ major macrocosm: whether it is rooted on its 5 or 6; whether it functions as subdominant by virtue of carrying the subdominant agent or as dominant by virtue of carrying the dominant agent; whether it is consonant by virtue of its local tonic status or dissonant by virtue of carrying the two agents characteristic of a diminished seventh chord (see p. 22). A minor, like F major before it and A major after it, is a diatonically organized, sealed capsule that bobs on the pan-triadic sea. Its interior is an ecological microsystem within which the fraught tensions and attractions, gravities and magnetisms of tonality are active. On its exterior, it is tethered to the broader universe by different forces and relations. Figure 8.9 thus presents a more complex picture of the passage than our earlier consideration of the passage in connection with figure 2.11(b). There, we considered the same music as passing through a hexatonic cycle whose stations are “buffered” by diatonic detritus. We are now in a position to give the latter their due. Each station of the hexatonic cycle brings along its own entourage of objects, internal syntactic relations, and external semantic effusions, all representing and expressing the principles of classical tonality. Most of the analyses offered throughout this book might plausibly benefit from reconsideration in light of the hybrid model of the convertible Tonnetz offered in this chapter. In all of them, familiar tonal terms have made unmarked cameo appearances, as so many faces in the crowd of words. Readers should now be in a position to translate these informal remarks, cast as they are in the pidgin dialect that we use when we talk about classical tonality—that familiar merger of language and perspectives of Schenker, Rameau, Riemann, Fétis, Kurth, Tovey, Rosen, Krumhansl, Hepokoski . . .—into a more formal apparatus along the lines of figure 8.9.

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Such a hybrid apparatus is even relevant in those passages where pan-triadic syntax does its best to cleanse our ears of the tonal semiotic. Even at its most pantriadic, a composition can immediately invoke classical syntax, as Liszt does at the end of each presentation phrase of his Organ Kyrie (see figure 6.10). Every major and minor triad, whether bobbing on its isolated tub or swept into the currents of parsimonious voice leading and sequential replication, is capable of diatonic encapsulation. Our filled ears are as subliminally alert to this potential as the blank ear that attunes a triad to a key. Although the convertible Tonnetz is specific to the nearly even trichord, the process of convertibility that it models is also pertinent to that other nearly even chord-class that doubles as a participant in classical syntax: the Tristan genus. These affinities are already implicit in several formulations from chapter 7: in Brünnhilde’s Immolation, the Cᅊ7 chords “hearken to their appellative properties after ignoring them for some measures” (see p. 157 of this book). Similarly, the Gᅊ in Chopin’s e minor Prelude, “awakened from its omnitonal haze by the whole-step motion in the bass, . . . remembers its monotonal responsibilities” (see p. 162 of this book).11 The dimensional limitations of the Tonnetz prevent us from proceeding by direct translation from the trichordal to the tetrachordal case. But this is a heuristic barrier, not an ontological one. If it were possible to assign members of the Tristan genus distinct locations on the Tonnetz, or if their position on Edward Gollin’s tetrahedron (1998) were legible on the planar page, then their participation in a voice-leading system, and their potential to revert to diatonic syntax, could be traced as in the triadic case. In order to imagine this participation, we need only contemplate the triadic case and boot it through the metaphorical conduit.

Two Analytical Vignettes: Wagner and Brahms Figure 8.9 interlocks and overlays diatonic and pan-triadic syntaxes in the projection of an essentially sequential motion through a hexatonic cycle. This chapter concludes with two analytic vignettes that navigate more diverse, less patterned routes through pan-triadic space, and use the convertible Tonnetz to show how those navigations interact with classical diatonic syntax.

The Faith Proclamation from Parsifal Figure 8.10 presents eleven measures from the Prelude to act 1 of Parsifal. This music, known as the Faith Proclamation, is structured as a sentence. The basic idea, beginning on Aᅈ major, is transposed up a minor third, beginning on Cᅈ major. Lewin (1987, 161) and Lerdahl (2001, 127) both hear the presentation 11. See also the treatment of the Tristan chord in Lewin 2006, which views the chord perched on the cusp of a double syntax consistent with the model developed here.

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 Audacious Euphony Figure 8.10. The Faith Proclamation, Parsifal, act 1 Prelude, mm. 45–55.

segments as half-cadential motions from tonic to dominant. But there are three contextual factors that support hearing them as motions from a local subdominant to its tonic. Globally, plagal cadences play a significant role in the opera’s tonal logic and semiotic network (Lewin 1984). Locally, the basic idea follows a tonicization of Eᅈ major through a plagal extension of the Dresden Amen. And associatively, the melodic journey—anacrustic tonic, climax on a metrically ^ ^ accented 4, inverse-arch, metrically accented cadential 3—has already been presented as a motivic topos at the conclusion of the Prelude’s opening phrase (Spear motive, mm. 4–6). I would, nonetheless, not argue too hard for a plagal hearing as “correct.” The relationship between Aᅈ major and Eᅈ major is ultimately underdetermined, a Hauptmannian antithesis without synthesis, such as discussed in chapter 3. In such situations, it is advantageous to have a conceptual and representational system that does not require a determination. The Tonnetz of figure 8.11 provides such an analysis. The continuation phrase, itself a nested sentence, begins up another minor third, on Eᅈᅈ major, and devolves into a plagal drift, indicated on figure 8.11 as a motion through an L/R chain.12 The extent of that drift is not predictable; this is one of those triadic chains that “can be arrested at any point or . . . can just as easily go on in perpetuity” (Mitchell 1962, 9). The first sign of an impending diatonic encapsulation occurs in the leftward motion from eᅈ minor to Cᅈ major, which stanches the perpetual downshift that has pervaded the entire sentence. The reverse hemiola, which elongates the counting pulse, reinforces the sense that 12. Wagner writes the chord as D major for pragmatic notational reasons; as there is no motion through the enharmonic seam, the entire passage could have been written using flats, as in figure 8.11.

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Figure 8.11. A Tonnetz analysis of figure 8.10.

some force is being applied to stop the flow. The appearance of the first accented dissonance in the passage, an Fø7 acting as subdominant with characteristic dominant dissonance, clinches the sense of eᅈ minor’s impending encapsulation. “Stopping the flow” has a particular meaning in Parsifal, an opera whose central event is the healing of Amfortas’s bleeding wound. The force that stops the flow is figured here as eᅈ minor. The hermeneutic interpretation can be extended if we notice the role of eᅈ minor in the presentation phrase, indicated by the gray triangle in figure 8.11. Each of its three double-common-tone associates is sounded in turn; the music perambulates about eᅈ minor without knowing how to respond to it, just as the goose-headed Parsifal cluelessly gawked at the Communion service in act 1. Only after an evidently aimless journey does Parsifal stumble back upon that eᅈ minor thing in act 3, recognize its value, and use it to stanch the plagal drift. All of this, of course, is foreshadowing; following the eᅈ minor cadence, the plagal flow is temporarily reversed by a Cᅈ major sounding of the Grail theme at m. 56 but then devolves into an even more extended plagal drift that extends all the way to the chromatically tortured reprise of the Communion theme at m. 79 (Murphy 2001).

The finale of Brahms’s First Symphony Figure 8.12 presents a reduced score of the opening of the recapitulation of the finale of Brahms’s First Symphony. (An orchestral score is available at Web score 8.12 .) The principal theme is presented by the violins in C major, diverted into a chromatic episode at m. 204, and presented again in a modified C major counterstatement at m. 220. Both phrases of the chromatic episode begin with partial thematic statements in the winds, in Eᅈ major (m. 204) and in B major (m. 212), and are followed by four-measure liquidations in the horns and pizzicato strings. The tonal domiciles of these four thematic statements are traced on figure 8.13(a) by a series of black dots, ordered from top to bottom on the Tonnetz. For clarity, the graphs encapsulate only the C major regions, but the reader should imagine the Eᅈ major and B major thematic statements defined by similar parallelograms, as they are in Web animation 8.13 .

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 Audacious Euphony Figure 8.12. Brahms, Symphony no. 1, 4th mvt., mm. 186–221.

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Figure 8.13. Two Tonnetz analyses of figure 8.12. Animation, with recorded performance, is at Web animation 8.13. The C major region of mm. 186–200 is inflected by only two brief and easily contextualized chromatic tones. The flatward inflections that appear beginning in m. 201 coalesce around Eᅈ major at m. 204, projecting an R/P chain on the southeast diagonal that is taken to the next step by eᅈ minor at m. 207. The horn fifths at m. 208 initially project Bᅈ major, as eᅈ minor’s nebenverwandt to the northeast, but the Dᅈ at mid-measure defines a southward move from hypothesized Bᅈ major to realized bᅈ minor. After three measures of prolongation, bᅈ minor is connected to Cᅈ major by a westward motion along an L/R chain. Each chord of this chain bears latent potential to blossom into a diatonic region, but only the final one exercises that capacity, by virtue of its positioning at a hypermetric downbeat. The liquidation of the B major thematic statement, beginning at m. 215, is a nearly exact transposition of the Eᅈ major liquidation eight measures earlier. The parallel shapes on figure 8.13(a) bring these out nicely. The only significant difference is that the fᅊ minor triad that begins the pizzicato unison passage is

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 Audacious Euphony abbreviated, affording the westward L/R chain a bonus measure with which to overshoot G major, so that tonic C major rather than dominant G major arrives at the next hypermetric downbeat. Because figure 8.13(a) crawls fairly low to the ground, its explanatory power is somewhat attenuated. One stumbles through the darkness, eventually blundering into the light of C major, blinking and disoriented. Figure 8.13(b) proposes an alternative hearing that remains oriented to C major throughout, organizing the progression of harmonies about an L/P (hexatonic) chain that unfolds at the middleground. Eᅈ major and B major are the most salient stations of that chain, by virtue of the thematic statements at mm. 204 and 212, respectively.13 Eᅈ minor, which arises at m. 207 through the substitution of Gᅈ for G and returns at m. 211, is prolonged in the interim through a retention loop about Bᅈ, before progressing to Cᅈ-qua-B major when the retained Bᅈ discharges as a leading tone. B minor arises through a similar substitution, is prolonged through a retention loop around Fᅊ, and proceeds to G, the actual dominant, which in the event is overshot on the way to tonic. From the standpoint of C major, the entire chromatic episode can be seen to fulfill a dominant function, with the entire alley acquiring the function of its most diatonic member. By extension, the pitch retention loops visit the region of the double dominant. But the work carried out in chapter 6 suggests that such an interpretation is not mandatory, if all it does is provide a language with which to express the intuition that the passage executes a departure–return scheme. The voice-leading zones provide an alternative means for capturing the same intuition. They suggest that the motion from C major to eᅈ minor represents a downshift, that the motion through the hexatonic alley balances along the fulcrum, that the retention loops temporarily move to a lower voice-leading zone, and that only in the final overshoot from G major to C major is the initial downshift reversed. In this Brahms excerpt, as in the Faith Proclamation, diatonic encapsulations emerge from triadic drifts along the L/R chain. In the Wagner case, the forces that halted the rightward drift and erected the diatonic walls included temporary reversal of the drift, accretion of supplementary dissonances, change in harmonic rhythm, and the flanking of a triadic goal by its syntonic images. Brahms invokes rather different resources to halt his leftward drift. There are no supplementary dissonances, no reversals, and no syntonic images. His walls are erected instead through the sounding of a diatonic and triadic theme on a hypermetric downbeat. Where Wagner methodically stakes his tent using structural means, Brahms simply plants his feet using what Harrison (1994, 81) refers to as rhetorical ones.

13. The hexatonic connection of Eᅈ major to G major, along the same path as figure 8.13(b), arises in the final section of Brahms’s Alto Rhapsody, a c minor composition from six years earlier that also transforms to C major in its final section.

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A Summary Example from Schubert We end, as we began, with an excerpt from Schubert’s instrumental music, whose consideration will bring together the central analytic concerns of this book and provide a focus for the central topic of this chapter. The passage, from the scherzo of the C major Symphony, begins at m. 93 with a waltz fragment that is presented first in the dominant and then in the tonic and ends with a dominant prolongation that precedes the reprise of the Scherzo at m. 153 (figure 9.1). Its interior, from m. 109, forms a chromatic tunnel that recedes from C major in stages, loses contact with it entirely as the hypermeter broadens to six-bar units at m. 117, and then gradually reattunes to that key as four-bar units are restored at m. 141. The initial chromatic events are the Aᅈ and Dᅈ neighbors that sound in mm. 109 and 110. These receive subsequent harmonic support respectively from f minor at m. 113 and Dᅈ major four bars later. To a monotonal ear, these tones are products of phrygian mixture. To one with more modulatory flexibility they represent a tonicization of f minor, the minor subdominant and nebenverwandt. The honorary diatonic status of the nebenverwandt relation in chordal space papers over what is already a fairly significant journey through key space, from zero flats to four. Exiting the tunnel, C major comes back into view when A major (m. 129) proceeds to d minor (135). When the latter assumes its diatonic role as a supertonic, the former is heard as its applied dominant, also evincing a nebenverwandt. Approached from either portal, the cᅊ minor triad at m. 123 is inscrutable. Entering from Dᅈ major, it is heard as dᅈ minor, modally inflecting a chord that already presses the limits of modal mixture in C major.1 Exiting through A major, 1. Here we encounter a peculiarity in the theory of modal mixture. Phrygian occupies the extreme end of modal space, involving four flatwise substitutions on major. The other extreme, lydian, requires four sharpwise substitutions on minor. The current textbook sanctioning of phrygian but not

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 Audacious Euphony Figure 9.1. Schubert, C major Symphony, 3rd mvt., mm. 101–53.

it is heard as cᅊ minor, a local diatonic mediant. But there exists no enharmonic construal of the triad in m. 123 that renders it diatonic in either C major, or in the locally tonicized f minor and d minor that directly flank it, or even in the parallel modes of any of those three keys. lydian mixture rests on Schenker’s whims and tastes a century ago rather than that of systematic consistency and completeness.

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Riemann’s function theory, because it is relatively free of diatonic constraints, is helpful here, providing a well-formed and plausible reading of cᅊ minor as the Leittonwechsel of A major, the dominant of the supertonic.2 One benefit of this approach is that it calls attention to a symmetry in the passage: cᅊ minor’s protensive relation to an emerging supertonic inverts Dᅈ major’s retensive relation to a receding tonic. Transformations express that symmetry: C major is transposed up a semitone to Dᅈ major through the transformational sequenceᇳN, Lᇴ . cᅊ minor is transposed to d minor by the same transformations applied in reverse order,ᇳL, Nᇴ . The Tonnetz at figure 9.2(a) provides a geometric expression of that same symmetry. An initial southwest arrow exits the C major capsule along an NL axis, connecting to Dᅈ major. A second arrow along a parallel axis charts the reentry into the C major capsule, connecting from cᅊ minor to d minor, which discharges its supertonic function according the syntactic demands of life within the capsule. Riemannian theory is thus able to assign every harmony in the passage a function with respect to the C major tonic. But there still remains the problem of how to hear the enharmonic Dᅈ major → cᅊ minor junction across the barline that precedes m. 123. The dotted arrow that connects them in figure 9.2(a), which seems to travel a great distance, belies the aural perception that these two triads are proximate, just as Schubert’s conversion from sharps to flats confuses the visual perception that the triads share individual tones. Honoring those perceptions, while retaining fidelity to some classical theory of tonality, will never remove the problem; it will only displace it to some further outpost. If Dᅈ at m. 117 becomes “Cᅊ” at 123, which becomes “Cᅊ” at 129, then “Cᅊ” at 129 is heard as Dᅈ and cannot function as the leading tone of the supertonic at m. 135. Conversely, if Gᅊ at m. 123 is inherited from “Aᅈ” at m. 117, which is what becomes of “Aᅈ” at m. 113, then “Aᅈ” at m. 113 is heard as Gᅊ and cannot function as the flatted submediant of the tonic triad at m. 109. This circle can never be squared by diatonic logic. For chromatic triadic progressions that move through the enharmonic seam, seven-tone diatonic space provides no adequate stick for measuring distances. The solution comes easily once we relinquish the diatonic gauge for a chromatic one. Cᅊ and Dᅈ become notational variants of a single object. Like the Morning Star and the Evening Star, they achieve identity by intension as well as extension, by sense as well as by reference. Equivalence of root distance then follows automatically: the unified Cᅊ/Dᅈ object is heard as equally distant from the F that precedes it and the A that follows it. What is compromised in this move is the syntactic position and semantic force of that tone. From a protensive phenomenological perspective, it can be heard as subdominant agent of f minor, and from a retensive one as dominant agent of d minor. But, because there is no reason to prefer one of these vistas over the other, the sound notationally indicated by Cᅊ at m. 129 is essentially neither Cᅊ nor Dᅈ. The ability of that sound to coexist at once in both diatonic spaces results from its position in chromatic space, equidistant 2. Schubert also transposes a minor tonic down by semitone in “Auf dem Flusse,” studied in chapter 6. McCreless (1996, 88–89) identifies a similar modulation in Beethoven’s Trio, Op. 1 no. 3.

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 Audacious Euphony from F and A. Although the chord still retains its forked syntactic and appellative residues in diatonic space, what comes to the fore as syntactic driver is the minimal change relation between adjacent triads: each motion requires only a single semitonal motion. Once attuned to the minimal change at the center of the passage, that attunement disperses toward its exterior. We might then hear C major → f minor (m. 109) and A major → d minor (m. 135) as double semitonal upshifts. (This perception in no way overrides or otherwise conflicts with our simultaneous perception of their classical relation to C, measured in diatonic fifths.) This might then lead us to notice that these two double upshifts are bridged by a cumulative semitonal upshift, as f minor progresses through the hexatonic alley to A major. Pressing one stage further on both flanks, to G major → C major at m. 105 and d minor → G major at m. 141, we might now notice their participation in the same upshift, although taking larger strides in voice-leading space. Figure 9.2(b) records this second hearing on the Tonnetz. No diatonic region is encapsulated. All triadic locations are governed by common-tone relations.

Figure 9.2. Two Tonnetz analyses of figure 9.1.

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A wave of voice-leading efficiency crests at the hexatonic alley, whose balanced voice leading temporarily suspends the pervasive upshift across the entire passage and then recedes in the rearview mirror. The voice-leading perspective of figure 9.2(b) is inverse to the monotonal one of figure 9.2(a): the one focuses as the other blurs. What we initially conceived as a tunneling into and out of diatonic/ tonal darkness becomes, from the standpoint of voice leading, a transient emergence into the hexatonic light. The central claim of this book is that these two incommensurate ways of measuring triadic distance emerge respectively from two independent properties of consonant triads. Listening to the passage with reference to C major, as it disappears into chromatic obscurity, crosses the enharmonic crevice, and reemerges back into diatonic clarity, we are hearing in terms of a system of distance relations governed by the degree to which harmonic roots are consonantly related to each other and to C major. That system is predicated on the membership of C major and its satellites in a family of objects, the consonant triads, that uniquely optimize acoustic consonance. Listening to the same passage attuned to voice leading, we hear an elimination of chordal dissonance and the elimination of whole steps in individual voices (m. 109), then a reduction from two moving voices to one (m. 117), a central passage that both maximizes common-tone retention and minimizes voice-leading sizes (mm. 123–40), and an increase in voice-leading work synchronized with a restoration of chordal dissonance (m. 141). Attuning to these processes, we are hearing trichords as participants in a system predicated on their nearly even status, as minimal perturbations of the augmented triad that divides the octave into three equal parts.

Double Syntax and Its Skeptics My characterization of sense-making in response to the music of figure 9.1 relies on an assumption of double syntax that some scholars have found implausible. Objections to that assumption come in two forms: an ontological/aesthetic one that ascribes immanent properties to compositions, and an epistemological/cognitive one that describes mental capacities in response to them. The ontological objection, characteristic especially of American music theory before 1990, is that music of high aesthetic value is organically unified and hence cannot be generated from multiple sources. This objection is sometimes framed with an appeal to the alleged unity of natural language. For example, Charles J. Smith, responding to work of Gregory Proctor (1978) that had elements that could be characterized in terms of double syntax, wrote that Proctor’s approach “suggests the separateness of chromatic and diatonic tonalities, since it treats them as virtually distinct languages. I cannot believe that any chromatic master conceived of his musical terrain as so partitioned; I hear no grinding of gears as one area is left and the other entered” (Smith 1986, 109). Claims of double syntax sit uneasily with postwar music theory’s commitment to the idealist notion that a good composition resembles an organism in its indivisibility. Although that metaphor has become epistemically

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 Audacious Euphony and historically bracketed in the last twenty years, and to a certain extent has receded altogether from the explicit discourse of American theorists, I suspect that it nonetheless continues to subliminally work on the instinctive sensibilities of music theorists. Smith’s objection can be reframed so as to transfer its weight from immanent properties to human listening capacities. Thus, Carl Dahlhaus writes that “the supposition that the musical hearing switches between tonal and non-tonal apprehension during a composition or a phrase would be problematic” (1967, 100–1; my translation). Although directed toward an earlier historical moment when “nontonal apprehension” is of a quite different nature than that which concerns us here, Dahlhaus’s admonition is framed as a universal claim about human capacities. A more explicit psychological turn is evident in a similar formulation of Fred Lerdahl, who also shares Smith’s perception of a seamless unity in the music itself: “The argument that listeners switch between two systems . . . is implausible as a psychological position. It is problematic in particular for late tonal music, which moves smoothly between diatonicism and chromaticism even within a single phrase” (2001, 85).3 Neither Dahlhaus nor Lerdahl presents evidence in support of his judgment, implicitly relying instead on its appeal to some species of common sense. That species may nonetheless be broadly enough shared that it is able to bear the weight of the objection. Dahlhaus, Smith, and Lerdahl evidently voice their objections to double syntax independently of one another and converge on those objections at three distinct historical moments, from three distinctive eras and intellectual traditions. This convergence suggests that the double syntax hypothesis touches a broad cultural nerve, clashing not only with idiosyncratic habits of thought sedimented within the microecology of some sequestered musicological terrarium but also with ideas that are deeply lodged in the experiences and sensibilities of the late-twentieth-century Euro-American music scholar and composer. Accordingly, it seems likely that the responses of Dahlhaus, Smith, and Lerdahl represent those of a much broader population of music scholars, who perhaps share the impulse to reject the claims advanced in this book on the grounds that they are untenable on a priori grounds, which is to say, independently of any evidence that might be brought to bear in support of them. My response to these objections takes three forms. First, the door that Smith attempted to close with his appeal to the autonomy of natural languages I will reopen through an appeal to multilingualism, a branch of linguistics developed in the quarter-century since Smith lodged his objection. Second, by reflecting on some quotidian modes of experience, and speculating on the mental operations that underlie them, I seek to establish that there are other sorts of common sense than those to which Dahlhaus and Lerdahl implicitly appeal when they dismiss double syntax on the basis of some tacit theory of cognitive capacities. Finally, and most speculatively, I suggest that double syntax provides a framework through 3. Lerdahl’s diatonicism/chromaticism dichotomy comes in the context of a critique of neo-Riemannian theory and thus is equivalent to both the classical/pan-triadic distinction of this book and to Dahlhaus’s tonal/non-tonal distinction.

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which to achieve a perspective on some aspects of the historical developments that connected the First and Second Viennese Schools, such that we can see how the one in some sense “became” the other. It thus suggests an approach to a problem that is often swept under the rug for lack of alternatives.

Code Switching and Double Determination The recent evidence from linguistics provides ample motivation for questioning Smith’s assumption, characteristic of its era, that natural languages are rigidly discrete, autonomous systems, compartmentalized and segregated in the minds of speakers and hearers. It turns out that most people have the ability “to alternate between different linguistic varieties at the drop of a hat, and to make use of several simultaneously when it suits them to do so” (Gardner-Chloros 2009, 177). Although this ability might seem a special case or odd trick within the demographic pool that stocks the Society for Music Theory with members (i.e., Anglophones above the poverty line), for most outside that pool it is an uncultivated norm of linguistic behavior from birth. Penelope Gardner-Chloros observes that “if you add together people who live in multilingual areas of the world . . . ; people who speak a regional language or dialect on top of a national language . . . ; and migrants and their descendants . . . , you are left with small islands of monolingualism in a multilingual sea” (2009, 5–7). Linguists refer to language juxtaposition as code switching, a term that is used by consensual inertia even though it is widely acknowledged to be misleading in several dimensions (Gardner-Chloros 2009, 11–12). The fault lines along which competing theoretical models of code switching fracture may have familiar resonances for music theorists. The “classical” model developed in the early 1990s maintains that interacting languages are asymmetrically related: one is the “matrix” into which the lexical items of the other are inserted (Winford 2009). More recent evidence suggests that many code switchers hold equivalent knowledge of the two languages, shuttling between them as equals. A second question pertains to the distinctiveness of the languages when placed into dialogue with each other. The early view is that the juxtaposed languages have autonomous status and thus that the “switch” between them crisply snaps off at a precisely locatable juncture between words. Katherine Woolard polemically responds that “alternation between rigidly discrete systems [is] a mythic ground of analysis that is as empirically and theoretically ill-founded as the idealized speaker/hearer” (1998, 6), and a considerable body of evidence has accumulated in support of the complexity of interlinguistic transition (Gardner-Chloros 2009, 45). Of particular relevance to the approach presented here is the phenomenon that Pieter Muysken identifies as congruent lexicalization. Where language populations overlap, share a significant geographic boundary, or associate through trade, colonization, or migration patterns, there often exist words that have similar form and meaning in both languages. These cognate terms, or homophonous diamorphs, are thought to “cause, or at least facilitate, a codeswitch from one

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 Audacious Euphony language to the other” (Broersma and de Bot 2006, 2; see also Muysken 2000, 123). Their function thus resembles that of pivot chords in theories of modulation: a dual-language zone intercedes between language 1 and language 2, so that “it is very possible to know that a definite codeswitch has occurred even when we are unable to say at what point it occurred” (Woolard 1998, 7–8). The topic becomes acutely relevant to music when viewed in terms of the mental processes of the individual language user, equivalent to the level at which both Dahlhaus and Lerdahl register their skepticism. On a hot summer day in Alsace, Gardner-Chloros (2009, 45) recorded an office worker complaining on the phone about a malfunctioning air-handling system. “The whole extract was delivered at high speed and the fifteen or so switches [between French and Alsatian] in no way interrupt the flow.” Moreover, exchanges of this type tend to occur beneath the horizon of consciousness: “Speakers are on the whole not very aware of their code-switching behaviour, and tend to be surprised at their own performance if you play it back to them” (121). The behavior of listeners has been less studied; although some neural researchers have begun to measure electrical activity in response to bilingual speech, the evidence is as yet preliminary (140). Yet it seems safe to infer that, like any other competent language user, a “code switcher” would spontaneously adjust her elocutionary pace to the perceived tolerances of her listener. If the Alsatian office worker is code switching rapidly in an effective conversation, it is likely that her interlocutor is receiving her switches with as few hitches as she has in creating them. Evidence from bilingualism, then, suggests that the human brain has ample potential for coprocessing distinct languages within compressed time spans, without cognitive hiccups. That potential is amplified when the interacting languages possess intersecting lexical units, which serve as bridges that facilitate the exchanges between the constituent languages. Such a unit performs distinct roles in the distinct languages, and functions as a seam through which a speaker slips from one language to another, bringing along her appropriately acculturated interlocutor without stumble. It is overdetermined, functioning both with respect to the language that precedes it and that which follows. Moreover, the entire process is cognitively opaque, transacted beneath the horizon of awareness.4 This evidence does not in itself mandate any conclusions about music, which is no language. Although the overlap is significant enough to have fueled 2,500 years of metaphorical cross-appropriation, there are also significant blocks in the metaphorical pipeline. In this connection, it is often noted that music is a form of art practiced by specialists, not of universally practiced discourse. An aesthete might then assert that the existence of some quotidian ability or tendency may have no implications for our understanding of the ineffably sublime masterwork. Such claims come less easily, however, if one’s concerns are with music as received rather than produced. Recent work at the music/language boundary suggests that 4. There would seem to be a number of other ways that code-switching research could stimulate the thinking of musical scholars, particularly given the intensive interest in hybridity among ethnomusicologists and in cross-national encounters among historical musicologists. Evidently, only Mark Slobin (1992) has mined this field; the theoretical and empirical explosion in bilingual studies suggests that a return visit might be productive.

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“as cognitive and neural systems, music and language are closely related” (Patel 2008, 417). If one has the impulse to dismiss the double syntax hypothesis on the basis of hypothesized cognitive capacities and proclivities in response to acoustically delivered data, the evidence from bilingualism provides a strong motivation for reevaluating that impulse and the hypotheses that underlie it.

Cognitive Opacity There is reason, moreover, to believe that overdetermination is cognitively opaque not only in the limited case of language but also more broadly in our quotidian interactions with the world about us. Consider, for example, how we conceive our interactions with garments, which perform (“optimize”) at least three distinct functions: they protect the body (from natural elements for survival), conceal it (from socially undesirable exhibiting), and adorn it (to signal social position and in support of sexual selection). Usually a dress or sweater performs multiple functions simultaneously, conflating them in our awareness. But one can imagine situations that isolate these functions from each other. Alone in your remote wood-heated cabin in December, you pull on a rugged old sweater. You have no need to conceal or adorn your body; there is no social function to be performed. You are simply protecting your body from the cold. Asleep in the city on a warm June night, you are woken by your dyspeptic dog; you throw on an oversized T-shirt and grab his leash. You have no need for allure, prestige, or warmth. You simply want to conceal your body from voyeurs. Back at the cabin in August, you anticipate an intimate evening by slipping into an item recently ordered from an apparel catalogue. You have no need to conceal your body from your partner or to protect it from the elements. You simply want to project an allure. In each instance, you go through the same sequence of actions and experience the same set of sensations: open the drawer, locate the item, check the label for orientation, raise arms, locate headhole, feel the garment slip down your torso, smooth, straighten, tuck. As you proceed through this familiar series of actions and experiences, you may feel no motivation to keep these functions mentally compartmentalized, to introspect on which you are fulfilling or in what combination. Dressing may feel pretty much like a unitary act. Although we may use the same object to fulfill quite distinct, nonoverlapping functions, in our minds the functions can merge into a single set of subroutines. We move between them without “grinding the gears.” The clothing analogy may lack force in one respect: when double syntax is claimed for music, the functions are not merely swapped, one for the other. Something potentially more disjunctive occurs: there is a sense in which their terms exchange function. When the triads at the boundaries of the Schubert Scherzo (figure 9.1) are diatonically encapsulated by C major and ordered according to eighteenth-century norms, they are referred to C for their meaning. Equivalently, depending on theoretical tradition, one might say that each triad is heard in terms of, is oriented toward, functions with respect to,

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 Audacious Euphony is generated/controlled by, or is hierarchically subordinate to C. I will use the expression “triad (3) → tonic (1)” to generalize across each of these expressions as a neutral party; the parenthesized numbers refer to cardinality, to prevent confusion between tonic-as-tone and tonic-as-triad. The formulation lends itself to recursive expansion. When C is tonic, the triadic tones orient the diatonic ones, which in turn orient the chromatic ones, leading to the expanded expression chromatic (12+) → diatonic (7) → triad (3) → tonic (1).5 By contrast, when the triads at the interior of that same passage are released from the diatonic capsule and the enharmonic seam breached, they are referred to the entire chromatic collection for their meaning. Each individual tone, including C, now “takes its meaning from” (functions with respect to, orients toward, etc.) the triad of which it is a constituent, which in turn “takes its meaning from” its nearly even status with respect to the chromatic collection. Accordingly, the arrows flip; we now have chromatic (12) ← triad (3) ← tone (1). It is as if the arrow that initially points from chromatic (12+) to tonic (1) is attached to the system at the point of the triad (3), where it rests on a spring-loaded pivot and is locked into position by the diatonic (7). When the diatonic lock is released, the spring uncoils and the direction of the arrow instantly reverses. It requires some diatonic reconstitution to reload the spring, and some cadential labor to lock it back into its initial position, pointing away from the chromatic (12) toward the tonic (1). The metaphor of arrow reversal helps explain why double syntax encounters such a deep vein of resistance in the musical case. Its advocates are not merely suggesting that listeners exchange modes of organizing relations “on the fly.” They are suggesting that all of the relations of subordination, causality, and orientation undergo a reversal. It is as if at one moment the person wears the garment and at the next the garment wears the person. A second round of introspection suggests, however, that causal inversion also has a potential for cognitive opacity. On Tuesday, you drive your car to town for a tune-up. On Wednesday, you drive your car to town so that you can get your hair cut. In the first case, you take the car. If the car could get to the shop without you, you would gladly send it. In the second case, the car takes you. If the barber were near enough, you would leave the car in the garage. Comparing these two situations, the arrow of causality is reversed. Yet, from a cognitive standpoint, one hardly notices the difference. Find your keys, unlock the car, warm it up, buckle the seat belt, down with the handbrake, down with the clutch, car into reverse, open garage door—it’s all the same. These reversals are insidious; they float below the threshold of consciousness unless one exerts some effort through analysis. Do you want money to get power, or power to get money? Do you seek wealth to buy a yacht, or a yacht to demonstrate your wealth? Do you smile because you are happy, or to elicit a mirroring response that will make you happy? Even if these introspective musings identify some intersubjective tendencies, as I suspect they do, one is not perforce compelled to accept their application 5. Compare the cone in Krumhansl 1990, 128, replicated in Lerdahl 2001, 46. The reference to “12+” chromatic elements indicates that, when enharmonic distinctions are honored, there is no crisp cutoff in the number of distinct chromatic pitch classes.

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 Audacious Euphony is generated/controlled by, or is hierarchically subordinate to C. I will use the expression “triad (3) → tonic (1)” to generalize across each of these expressions as a neutral party; the parenthesized numbers refer to cardinality, to prevent confusion between tonic-as-tone and tonic-as-triad. The formulation lends itself to recursive expansion. When C is tonic, the triadic tones orient the diatonic ones, which in turn orient the chromatic ones, leading to the expanded expression chromatic (12+) → diatonic (7) → triad (3) → tonic (1).5 By contrast, when the triads at the interior of that same passage are released from the diatonic capsule and the enharmonic seam breached, they are referred to the entire chromatic collection for their meaning. Each individual tone, including C, now “takes its meaning from” (functions with respect to, orients toward, etc.) the triad of which it is a constituent, which in turn “takes its meaning from” its nearly even status with respect to the chromatic collection. Accordingly, the arrows flip; we now have chromatic (12) ← triad (3) ← tone (1). It is as if the arrow that initially points from chromatic (12+) to tonic (1) is attached to the system at the point of the triad (3), where it rests on a spring-loaded pivot and is locked into position by the diatonic (7). When the diatonic lock is released, the spring uncoils and the direction of the arrow instantly reverses. It requires some diatonic reconstitution to reload the spring, and some cadential labor to lock it back into its initial position, pointing away from the chromatic (12) toward the tonic (1). The metaphor of arrow reversal helps explain why double syntax encounters such a deep vein of resistance in the musical case. Its advocates are not merely suggesting that listeners exchange modes of organizing relations “on the fly.” They are suggesting that all of the relations of subordination, causality, and orientation undergo a reversal. It is as if at one moment the person wears the garment and at the next the garment wears the person. A second round of introspection suggests, however, that causal inversion also has a potential for cognitive opacity. On Tuesday, you drive your car to town for a tune-up. On Wednesday, you drive your car to town so that you can get your hair cut. In the first case, you take the car. If the car could get to the shop without you, you would gladly send it. In the second case, the car takes you. If the barber were near enough, you would leave the car in the garage. Comparing these two situations, the arrow of causality is reversed. Yet, from a cognitive standpoint, one hardly notices the difference. Find your keys, unlock the car, warm it up, buckle the seat belt, down with the handbrake, down with the clutch, car into reverse, open garage door—it’s all the same. These reversals are insidious; they float below the threshold of consciousness unless one exerts some effort through analysis. Do you want money to get power, or power to get money? Do you seek wealth to buy a yacht, or a yacht to demonstrate your wealth? Do you smile because you are happy, or to elicit a mirroring response that will make you happy? Even if these introspective musings identify some intersubjective tendencies, as I suspect they do, one is not perforce compelled to accept their application 5. Compare the cone in Krumhansl 1990, 128, replicated in Lerdahl 2001, 46. The reference to “12+” chromatic elements indicates that, when enharmonic distinctions are honored, there is no crisp cutoff in the number of distinct chromatic pitch classes.

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to music. We are still free to insist, as a metaphysical priority, that music is different, that our responses to it are isolated not just from language but from all other mental functions, such that whatever we say about the functioning of the brain does not apply to its functioning in response to music. But such a reversal flies in the face of evidence that causal reversal is cognitively opaque in music, too. At the opening of Parsifal, G is a neighbor to Aᅈ; soon enough, we recognize that Aᅈ has become a neighbor to G. Eᅈ major is dominant of Aᅈ; soon enough, we recognize that Aᅈ has become the subdominant of Eᅈ. Arsis and thesis flip, so that strong beats become weak and vice versa. A cunning composer or crafty performer knows how to take advantage of the opacity of causal reversal, to flip the arrow of causality without rocking a listener’s boat. If musical cognition is not entirely remote from other sorts of mental processing (Zbikowski 2002), then we cannot dismiss the possibility that switching between musical syntaxes might transpire beneath our horizon of awareness, especially when the switching is routed through an overdetermined object such as a triad or a dominant seventh chord.

The Soft Revolution In this music of Schubert, so gorgeous and so tonally secure to the foregrounded ear, the position of the tonic as functional center is placed into question for the first time, like a soft revolution. —Diether de la Motte, Harmonielehre

In the age of Mozart, musical pitches are primarily organized by diatonic tonality. In the age of Webern, there exists some species of music whose tones are organized in some other way. How does music that is heard to be organized by diatonic tonality become music that is heard to be organized in some other way? Either that “other way” is conjured tabula rasa from what was absent from diatonic tonality, or it is conjured partly by what was present in it. The first possibility is no possibility at all; it is inconsistent with everything else we know about historical process and human cognition. That leaves the second possibility: that there is some aspect of diatonic tonality that was reshaped, recontextualized, developed in some direction that had been hitherto inconceivable (compare Morgan 1998, 3–4). Where is the thread that unravels the garment and then gets rewoven? Not, evidently, in the sound combinations, which are complementary: acoustic, rootindexing harmonies in the age of Mozart; anything but, in the age of Webern. Nor, evidently, in the relations between sounds: in the former, harmonies gain their systemic function in their relations to tonics; in the latter, harmonies gain their systemic function according to their properties and potentials within the chromatic system. Neither lexicon nor syntax furnishes a lever that can trigger a phased process of change, such as would be consistent with the evolutionary model implied by the second possibility. The burden of historical continuity can be distributed onto the backs of motive, rhythm, and form. But the primacy of

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 Audacious Euphony pitch renders such a move artificial: motivic relations rely on similarity and equivalence judgments that take shape in part within a set of systematic assumptions; rhythm is heard in the context of a meter that is established in part by rate of harmonic change; formal articulation in part depends on what counts as closure and repetition. All such considerations in some measure presuppose some prior organization of pitch relations. A theory of double syntax suggests a way out of this impasse. It allows that at the same time that triads are root-indexing consonances, they have the capacity to serve as something else entirely: nearly even chords participating in a system of tight voice leading. Under cover of a unified lexicon, a composer learns to hear a different set of syntactic possibilities, and to train listeners to that hearing. This step can be seen as the initial, evidently innocuous stage in a process that eventuates in the music that arises and characteristically represents the terminal end of the long European century. One way of conceiving of this process, following Patrick McCreless’s appropriation of evolutionary theory, is to say that the triad is preadapted for this second role. In a passage that McCreless (1996, 108–10) quotes from evolutionary biologist Stephen Jay Gould, preadaptation “asserts that a structure can change its function radically without altering its form as much. We can bridge the limbo of intermediate stages by arguing for a retention of old functions while new ones are developing” (Gould 1977, 108). For example, fish jaws “were well designed for their respiratory role; they had been selected for this alone and ‘knew’ nothing of any future function. In hindsight, the bones were admirably preadapted to become jaws. The intricate device was already assembled, but it was being used for breathing, not eating” (108). McCreless suggests (1996, 110) that classical tonality is preadapted to chromatic space, but it is not clear from his account what aspect of classical tonality assumes the role of the fish’s jaw. That burden falls variously on “structural semitone harmonic relations” (110), directed linear motion through chromatic space (98–99), and sequential transposition in chromatic space, on the basis that each initially arises in passages that are fully reconcilable to the seven-tone diatonic and is subsequently reused by composers in ways that can only be understood within the twelve-tone chromatic. Yet none of these phenomena is ever intrinsic to a diatonic system, in the way that the jaw is proper to the fish. They arise only when a diatonic system is supplemented by chromatic tones, at which point the evolutionary train has already left the station. The work presented in this book suggests a different candidate: it is the consonant triad, fully native to a nonchromaticized diatonic space, that plays the crucial role. Triads are preadapted to play the role of voice-leading optimizers by virtue of their near evenness, which manifests in both diatonic 7-space and chromatic 12-space. Triads first assume that role in the diatonic sequence, when, as described by François-Joseph Fétis (2008 [1844]), they cease to exploit the orienting, or “summoning,” capacity of that scale’s tritone and instead exercise their capacity, described in Agmon 1991, for intertriadic stepwise voice leading within a quasiflat diatonic space. Moving through a Baroque ritornello, from presentation to Fortspinnung to cadence, the arrow of orientation first flips outward, then back in,

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as triads move with ease between their dual natures in diatonic space. This dynamic is replicated and intensified in chromatic space, which presents a new habitus in which intertriadic voice leading proves capable of even greater parsimony. Moving through the Schubert Scherzo of figure 9.1, the arrow of orientation undergoes a similar journey, only this time without need for sequential repetition to sustain it. Further evolutionary stages can be provisionally proposed, at the risk of bleaching the nuance out of a complex set of musical transformations through a few broad strokes. At the second stage, it is the syntax, in the form of a prioritization of voice-leading efficiency, that is firmly planted, while the lexicon is exchanged. If consonant triads can serve as objects in a system of relations defined by common-tone preservation and voice-leading work, so too can other nearly even chords, the members of the Tristan genus, beginning with the omnibus progressions that arose already in the eighteenth century (Telesco 1998), and eventuating in late Wagner. We are now at one further level of remove on the historical continuum: the syntactic arrow has been reversed, and a hitherto secondary member of the lexicon has been promoted to primary status. A third step involves another lexical transformation, but this time at the scalar rather than the chordal level. The most characteristic harmonic progressions to exploit the syntax for which the triad was preadapted are the L/P-chaining major third cycle, and the R/P-chaining minor third cycle. These progressions respectively create six-tone (hexatonic) and eight-tone (octatonic) scales that take their place as substitutes for the seven-tone diatonic gamut. An alternative route to the octatonic scale is through the parsimonious organization of Tristan-genus seventh chords. Although arising initially as secondary by-products of harmonic progressions that exploit the capacity for efficient voice leading among nearly even trichords and tetrachords, eventually these scales become things in themselves.6 Individual tones now relate directly to these new scales, without the intervention of nearly even trichords or tetrachords. Thus arises the scalar tonality associated with the tradition that extends from Liszt through Rimsky-Korsakov to the earlytwentieth-century Russian and French composers, recently explored in chapter 9 of Tymoczko 2011b. The triad can now be replaced by other dissonant formations, octatonic and hexatonic subsets. At the fourth and final stage, hexatonic and octatonic scales are absorbed into a broader atonality. Chords and scales are transformed into the sets and set classes of Second-Viennese-School atonality, 6. Richard Taruskin (1985, 1996, 2005) also emphasizes that octatonic collections initially arise as by-products of other procedures of diatonically contextualized chromaticism. Taruskin’s claim that Liszt is the first composer to lay out the octatonic scale in a single voice and to “categorically express” circles of minor thirds as harmonic progressions (2005, 429) is supportable only on an idiosyncratic reading of “categorical.” Theorist Honoré de Langlé (1797) included a circle of minor thirds with octatonic bass among his sequential tours de l’harmonie. Thomas Christensen (1992, 9) suggests an even earlier precursor from mid-seventeenth-century Spain, although it is more likely a modulatory blueprint than a directly sounded progression. The finale of Beethoven’s “Acht Variationen über Tändeln und Scherzen” of 1799 presents a complete series of minor-third transpositions (Seidel 1963, 201). See also figure 5.4 in chapter 5, from an 1821 composition of Schubert’s. The passage from the Scherzo of Beethoven’s Ninth Symphony (1822), referred to on p. 88 and presented as Web score 5.9, lays out octatonic collections in each of its constituent voices, including the melodically prominent top voice, as a by-product (Cohn 1991). Blum 1986 raises similar questions about Taruskin’s claim.

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 Audacious Euphony which take their meaning directly, like the consonant triad at stage 1, in relation to the chromatic 12-gamut. In each of the latter stages, the arrow of orientation remains sprung, pointing from smaller-cardinality entities toward larger ones. All of the stages are in that sense “atonal,” although only gradually do they begin to take on the sonic properties that we associate with the prototypical atonality of Schoenberg and Webern. What distinguishes the later stages from the earlier ones is the number of degrees of remove from diatonic tonality. At the moment that diatonic scales, triads, and Tristan-genus seventh chords lose their privileged lexical positions, the ontogenetic road of return becomes bumpy. Once composers begin to use [0148]-type chords in a hexatonic context, or [0146]-type chords in an octatonic one, the imposition of a diatonic cadence often sounds like the grinding of gears that Smith describes. The lexical substitutions of the later stages are relatively easy to effect. It is the syntactic substitution at the first stage that provides the key that unlocks the capsule of diatonic tonality. The central argument of this book is that the consonant triad has two natures, and that those two natures lead to two syntaxes. The central argument of this chapter is that the juxtaposition of syntaxes is not as problematic as is sometimes thought, particularly when smoothed over by lexical continuity. The central suggestion of this penultimate section is that lexical continuity can bridge over a syntactic impasse along the historical axis, as well as within a phrase of Schubert.

On Musical Overdetermination Since no later than the early eighteenth century, Western musicians and listeners have, as a first-level default, interpreted musical events with respect to a tonic that is expressed through the coordination of consonant triads and diatonic collections. Independently, composers cultivated a taste for economical voice leading and harmonic consistency and sought harmonic structures that could respond to them both simultaneously. What they were seeking was sitting right in front of their ears, preadapted and ready to serve. The simultaneous optimization of efficient voice leading and harmonic consistency led composers to discover new syntactic possibilities in the familiar sounds of diatonic tonality and provided a forum for exploring the parallactic clash between old and new ways of construing musical distance, at a historical moment when old and new were being placed into acute tension in every domain of European culture. The triad’s double determination can be seen as a fortuitous curiosity, a quirk, a lucky draw to a straight flush. And yet, the triad is not the only homophonous diamorph in Western music, nor is the nineteenth century the only moment when preadaptation contributed to fundamental syntactic change. For it turns out that the diatonic scale, the triad’s necessary partner in the production of classical tonality, is also doubly determined. There are many plausible reasons why the diatonic collection became privileged in the Carolingian era, but it was surely not because

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of its position within an equally tempered chromatic space first theorized five centuries later and first realized three further centuries down the road. Yet once that space arose, diatonic collections had properties that proved both unique and functionally powerful: “The major scale is a maximal structure possessing [the property of unique multiplicity] in the usual equal-tempered division of the octave” (Babbitt 2003[1965]).7 Although the diatonic scale evidently originated at a moment when modulation was inconceivable, it was preadapted to serve as a modulatory vehicle capable of achieving maximum variety and depth. Thus, the diatonic scale’s overdetermination played a central role in the new modulatory system of early-eighteenth-century harmonic tonality, just as the consonant triad’s overdetermination played a central role in the early nineteenth century’s discovery of an alternative to it. The story has one further chapter. The double determination of the threeelement triad and the seven-element scale are matched by a double determination of the twelve-element equal-tempered chromatic gamut.8 The chromatic collection arose in stages as musicians rejected the tritone as a surrogate for the perfect fifth. Projecting further along the chain of just fifths, the next opportunity for systemic closure arises at the twelfth generation. Acceptance of the octave-adjusted Pythagorean comma as a suitable pseudounison halts the proliferation tones, leaving a chromatic gamut of twelve tones. The chromatic collection thus has twelve elements through an arithmetic fortuity: it is when m = 12 that 3m suitably approximates 2n for some value of n. Once temperament distributes the twelve tones equally about the cyclic octave, another property of that number comes forward as a stylistic determinant: twelve is an abundant number, divisible by 2, 3, 4, and 6, that is, by every nonunit interval besides its just-fifth generator. The system of twelve tones is thus preadapted for equal division. The projection of just fifths “knew nothing” of the abundant divisibility of the number 12. Had the eleventh or thirteenth powers of 3 been suitable approximations of some power of 2, we would have inherited a chromatic system incapable of equal division: no whole-tone scales, no augmented triads, no diminished seventh chords. Accordingly, the richness of nineteenth-century chromatic syntax results from overdetermination and preadaptation at two distinct levels of structure. (One might say, with only a slight abuse of language, that its double determination is itself doubly determined.) One level is the one emphasized throughout this book, which is also a central theme of Tymoczko’s Geometry of Music: the acoustically consonant triads are preadapted to serve as optimal voice-leading objects in chromatic space by virtue of their near evenness. But the second level of 7. By “unique multiplicity,” Babbitt refers to the number of instances of each interval class: six perfect fourths, five whole steps, four minor thirds, and so forth, what Gamer 1967 refers to as a deep scale. Common-tone retention under transposition is determined by the multiplicity of the interval of transposition. Consequently, a scale with maximum intervallic multiplicity has a maximum diversity of common tones under its various transpositions. For any particular diatonic collection, there is another diatonic collection that shares n tones with it, where n ranges from 2 to 6. Among seven-tone “scales,” only the chromatic heptachord shares that potential. 8. For a more detailed account, see Balzano 1980.

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 Audacious Euphony overdetermination, the preadaptation of the fifth-generated series of pitch classes to the multiple even divisions of chromatic space, is just as essential. Had the series of just fifths achieved pseudoclosure at a prime number of generations, then the chromatic system would lack perfectly even divisions. Without perfectly even divisions, the 3-in-12 nearly even consonant triads would behave like 7-in-12 diatonic collections in chromatic space, or 3-in-7 diatonic triads in diatonic-scalar space. Without perfectly even trichords to minimally perturb, the consonant triads would group neither into hexatonic systems nor Weitzmann regions. Without diminished seventh chords, the Tristan-genus tetrachords would group into neither octatonic systems nor Boretz regions. Under incremental voice leading, all nearly even chords of a given cardinality would form a single exhaustive cycle, rather than partition into equivalent co-cyclic regions. The munificent economy of Tymoczko’s distributive god is thus spread across multiple sectors of the musical firmament. Through their serendipitous coordination, the several layers of overdetermination generate much of the structural intricacy and affective multiplicity of much of the music at the heart of the repertory of concert halls and operatic stages. The tale of this book accordingly spins a variation on a tale twice told elsewhere. The script that unifies these variants is the primal tension and collaboration between physical forces of the external world, on the one hand, and relations internal to the musical chord or scale, on the other. To understand these forces, as they play out in music, requires a structural conception of musical materials and relations, such as is characteristic of (but not limited to) what we think of as atonality. It requires a conception of the cognitive mechanisms associated with the homophonous diamorph, to model how the grammars come into contact and thread one into the other. And it requires a theory of classical diatonic tonality, but that alone is not sufficient.

G L O S S A RY

The following technical terms used in this book either are not common to standard music-theory pedagogy or are used here in nonstandard ways. A term used in the definitions is italicized if defined elsewhere in the glossary. Augmented triad. A perfectly even three-note chord. Balanced voice leading. Under idealized voice leading, at least one voice moves up and another voice moves down. Boretz region. A collection of eight Tristan-genus chords—four half-diminished seventh chords and four dominant seventh chords—each of which is a single semitonal displacement of the same diminished seventh chord. Chain. A series of triads derived by alternating two distinct triadic transformations. Consonant triad. A nearly even three-tone chord, either major or minor (equiv. harmonic triad; in German, Klang). Cube Dance. A connected graph whose nodes are the twenty-four consonant triads and the four augmented triads, and whose edges represent minimal voice-leading work, or single semitonal displacement. Diminished seventh chord. A perfectly even four-note chord. Double syntax hypothesis. The hypothesis that the mind is capable of organizing musical patterns, simultaneously or in immediate succession, in two distinct and incompatible ways. Downshift voice leading. A motion between two chords, under idealized voice leading, where all moving voices proceed downward. 4-Cube Trio. A connected graph whose nodes are the twenty-four Tristan-genus chords, the four diminished seventh chords, the six French sixth chords, and the twelve minor seventh chords, and whose edges represent single semitonal displacement. Generalized Weitzmann region (GWR). A collection of all of the nearly even chords related to a single perfectly even chord by a single semitonal displacement. Weitzmann regions and Boretz regions are examples. Generating interval. An interval that forms a chord by recursive application. In a whole-tone scale, the whole tone is the generating interval; in the diatonic scale, the perfect fourth or fifth is the generating interval. Hexatonic cycle. An arrangement of six consonant triads such that each is adjacent to those two triads to which it relates by single semitonal displacement. 211

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 Glossary H(exatonic)-group transformation. Any transformation that maps a consonant triad within its hexatonic region and outside of its Weitzmann region. The three H-group transformations are the Leittonwechsel (L), the parallel (P), and the hexatonic pole (H). Hexatonic pole (H). A relationship between two triads that share membership in a hexatonic cycle but that share no tones in common; or the transformation that produces one of these triads from the other. Hexatonic region. An unordered collection of consonant triads containing the six members of a hexatonic cycle. Hexatonic scale. The union of two adjacent augmented triads. In scalar order, the step intervals alternate semitones and minor thirds. Homophonous diamorph. A sound that has meanings or functions in two distinct systems. Idealized voice leading. The ordered dyads between two chords when their tones are paired one to one and the voice-leading work is as small as possible. The idealized voice leading from C major to F major is (C, C), (E, F), (G, A), no matter how those chords are registrally realized or how the actual instruments or singers move between their constituent tones. Leittonwechsel (L) (English: leading-tone exchange). The relation between two consonant triads that share a minor-third dyad, for example, C major and e minor; or the transformation that produces one of these triads from the other. Nearly even. A collection of tones that are distributed as evenly as possible without being perfectly even. Nebenverwandt (N) (English: next related). The relation between a major triad and the minor triad whose root lies a perfect fourth above it, for example, C major and f minor; or the transformation that produces one of these triads from the other. Neighborhood. A collection of six consonant triads that share a single tone. Octatonic region. A collection of eight Tristan-genus chords—four half-diminished seventh and four dominant seventh chords—that draw their tones from the same octatonic scale. Octatonic scale (or collection). The union of two distinct diminished seventh chords. In scalar order, the step intervals alternate semitones and whole steps. Pan-triadic. Any composition, or segment thereof, that consists exclusively or predominately of major or minor triads without determining a tonal center. Parallel (P). The relation between two consonant triads that share a root, for example, C major and c minor; or the transformation that produces one of these triads from the other. Perfectly even. A collection of tones that divides the octave into equal parts. Pitch retention loop. A cyclic ordering of a neighborhood such that each pair of adjacent triads shares two tones. Relative (R). The relation between two consonant triads that share a major-third dyad, for example, C major and a minor; or the transformation that produces one of these triads from the other.

Glossary

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Single semitonal displacement. A relation between two chords of cardinality n that share n – 1 common tones and whose remaining tones are a semitone apart; or, the motion between two such chords. Slide (S). The relation between two consonant triads that share a common third, for example, C major and cᅊ minor; or the transformation that produces one of these triads from the other. Syntonic image. The relation between two tones that are separated by a syntonic comma in just intonation. Typically one of these tones is two 3:2 perfect fifths (minus an octave) above the tonic (3/2 × 3/2 × 1/2 = 9/8) and the other is a 4:3 perfect fourth minus a 6:5 minor third above the tonic (4/3 ÷ 6/5 = 10/9). The difference between the tones is 9/8 ÷ 10/9 = 81/80, slightly larger than one-fifth of an equally tempered semitone. Tonnetz. An graph whose nodes are pitch classes and whose edges represent consonant intervals. Tristan genus. The union of the twelve dominant and twelve half-diminished seventh chords. Upshift voice leading. A motion between two chords, under idealized voice leading, where all moving voices proceed upward. Voice-leading work. The sum of the magnitude of all moving voices of two chords connected by idealized voice leading. Voice-leading zone. A collection of chords whose pitch classes sum to a constant value. Consonant triads share a voice leading zone if they are transpositionally related by major third, in which case they share membership in both a hexatonic region and a Weitzmann region. Tristan-genus chords share a voice leading zone if they are transpositionally related by minor third or tritone, in which they share membership in both an octatonic region and a Boretz region. W(eitzmann)-group transformations. Any transformation that maps a consonant triad within its Weitzmann region and outside of its hexatonic region. The three W-group transformations are the relative (R), nebenverwandt (N), and slide (S). Weitzmann region. A collection of six consonant triads, three major and three minor, each of which is a single semitonal displacement of the same augmented triad.

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B I B L IO G R A P H Y

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INDEX

Piano Trio in c minor, Op. 1 no. 3, 197n2 Sonata for Violin and Piano in F major, Op. 24 (“Spring”), 2nd mvt., 26–28, 45n4 Symphony no. 5 in c minor, Op. 67, 1st mvt., 72 Symphony no. 9 in d minor, Op. 125, 2nd mvt., 92, 183 Benda, Jiří Piano Sonata in a minor, 1st mvt., 77–78 Berg, Alban Wozzeck, 167 Bering Strait problem, 14, 28, 88–89 Blum, Stephen, 207n6 Boatwright, Howard, 44 Boretz, Benjamin, 141, 150–51 Boretz-group transformation, 155, 159 Boretz region, 141, 152–56, 159 Boretz spider, 153–55 Bot, Kees de, 201–2 Brahms, Johannes Alto Rhapsody, Op. 53, 43 Concerto for Violin and Cello in a minor, Op. 102, 1st mvt., 29–31 Deutsches Requiem, Ein, Op. 45, 2nd mvt., 96–97 Symphony no. 1 in c minor, Op. 68, 4th mvt., 191–94 Symphony no. 2 in D major, Op. 73, 1st mvt., 92–94, 117–21 “Von ewiger Liebe,” Op. 43 no. 1, 76 “Walpurgisnacht,” Op. 75 no. 4, 100n12 Brendel, Alfred, 49 Bribitzer-Stull, Matthew, 48n8

Abbate, Carolyn, 4n4, 22 Adorno, Theodor, 10 agent, 47, 68–78, 93, 188, 197 Agmon, Eytan, 22, 37, 177, 179, 206 Ahn, So-Yung, 95n7 all-interval tetrachord, 108–9 Aristotle, 41n24 arrow reversal, 204–5 augmented triad, 19, 33–37, 43–46, 56–58, 64–65, 83–86 Babbitt, Milton, 104n16, 184, 209 Bach, C. P. E. and relatedness of C major and E major, 67–68 Piano Sonata in f minor, H. 173 (Wq. 57/6), 1st mvt., 68–69 Bach, J. S. Fantasy and Fugue for Organ in g minor, BWV 542, 94 Prelude in a minor, BWV 889 (Well-Tempered Clavier, vol. 2), 107–8 Toccata in d minor, BWV 913, 183 Baker, James M., 167 balanced voice leading, 19–21, 159 Balzano, Gerald, 209n8 Bass, Richard, 136, 150n7, 151n9 Beethoven, Ludwig van Acht Variationen über “Tändeln und Scherzen” (WoO 76), 207n6 Piano Sonata in d minor, Op. 31 no. 2 (“Tempest”), 1st mvt., 133–34; 3rd mvt., 108–9 Piano Sonata in C major, Op. 53 (“Waldstein”), 1st mvt., 133–34 Piano Sonata in f minor, Op. 57 (“Appassionata”), 1st mvt., 45n4, 48

229

230

 Index bridging region, 84, 155–56, 166–67 Broersma, Mirjam, 201–2 Brower, Candace, 175–78, 184 Brown, Matthew, 11, 96n10, 98n11 Bruckner, Anton “Ecce Sacerdos,” 91 Symphony no. 3 in d minor, 1st mvt., 98–99 Callender, Clifton, 167n21 Capellen, Georg, 7, 140 Caplin, William E., 133 Capuzzo, Guy, 34n18, 66n8 Chafe, Eric, 151n9 chain minor-third (T±3), 90–92 perfect-fifth, (T±5), 92–94, 160–61 semitone (T±1), 94–95, 160–61 see also L/R chain; N/H chain; N/L chain; N/P chain; N/R chain; S/L chain; S/P chain; sequence, chromatic Childs, Adrian P., 155, 160 Chopin, Frédéric Ballade in g minor, Op. 23, 96–98 Etude in f minor, Op. 25 no. 2, 69n10 Fantasy in f minor, Op. 49, 98–100 Mazurka in fᅊ minor, Op. 6 no. 1, 69–71 Mazurka in f minor, Op. 68 no. 4, 160–61 Nocturne in bᅈ minor, Op. 9 no. 1, 69–71 Nocturne in G major, Op. 37 no. 2, 108 Piano Sonata in b minor, Op. 58, 4th mvt., 69n10 Prelude in e minor, Op. 28 no. 4, 160–66 Prelude in cᅊminor, Op. 45, 160 Christensen, Thomas, 139–41, 207n6 Clark, Suzannah, 113n1, 129–30 Clough, John, xi, 29n11, 34n17 code switching, 201–3 cognitive opacity, 202–5 coherence, 15 collective exchange, 155–56, 166–67 combination, see dissonance, strategies for analyzing common tones, 5–8, 17, 29, 34–36, 129–30, 209n7; see also pitch retention loop

consonant triad, 1n1 contrary motion, 19–24, 103n15, 149, 154, 159 Cook, Robert, 31, 105n17, 145n4 Copley, R. Evan, xiv Cornelius, Peter “Ein Ton,” Op. 3 no. 3, 113 Covington, Kate R., 77 Cube Dance, 84–89, 101, 103–4, 152 Dahlhaus, Carl, 7n10, 10, 13, 143, 200 Darcy, Warren, 22, 133 Debussy, Claude Prelude to the Afternoon of a Faun, 108 deletion, see dissonance, strategies for analyzing Dempster, Douglas, 96n10, 98n11 determination, double, see syntax, double developmental core, 25–26, 45, 49, 133–34 diatonic encapsulation, see encapsulation, diatonic diminished seventh chord as dissonant sonority, 140, 182 as perfectly even chord, 150–55 disjunction, 4, 22, 106–9 dissonance, strategies for analyzing combination, 140, 141–42 deletion, 140–41 reduction, 140–43 substitution, 140–41, 150–51 Distler, Hugo Fürwahr, er trag unsere Krankheit, 108 DOM, 170–71 double-agent complex, 68–78 double syntax, see syntax, double Douthett, Jack and near evenness, 36n19 and Cube Dance, 84, 86 and Tristan-genus chords, 149n5, 155, 157–58n15, 158 role in developing pan-triadic theory, xi downshift voice leading, 84–85, 121, 154–55, 160–66 Draeseke, Felix, 12 dualism harmonic, xiii, 32, 37–38, 46n6, 143 melodic, 32, 37–38, 93, 124–25

Index Durutte, Camille, 102n13 Dvořák, Antonín Symphony no. 9 in e minor, Op. 95 (“From the New World”), 1st mvt., 134–38 efficient voice leading, see voice leading work, minimal Elgar, Edward King Olaf, 108 encapsulation, diatonic, 176–78, 189, 190–94 Engebretsen, Nora, 5 enharmonicism, 9, 11, 18, 23, 68–69, 72, 174–75 essential, 9 entropy, 106–9 equivalence class, 102–3 Euler, Leonhard, 28n10, 89n4 evolutionary theory, 205–7 Fauré, Gabriel Requiem, Introit, 54–56 Fétis, François-Joseph and chromaticism, 140 and species of tonality, 9–10, 13–14 and summoning, 206 uniformity and symmetry in, 47, 171, 179 Feurzeig, Lisa, 23 Feynman, Richard P., 41n24 Fisk, Charles, 49, 73 4-Cube Trio, 152, 157–59 Franck, César Piano Quintet in f minor, 155n14 Freud, Sigmund, 41n24, 148 function, harmonic, 11, 114n3, 124–28, 170–71 fused-triad graph, 66–67 Galeazzi, Francesco, 5 Gamer, Carlton, 209n7 Gardner-Chloros, Penelope, 201–2 Gauldin, Robert, 162–63 generalized Weitzmann region (GWR), 166–67 geometric models, 14, 173–75 Goldenberg, Yosef, 66n8, 91n5 Gollin, Edward, 30n13, 95n8, 140–42, 189

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Gould, Stephen Jay, 206 Grieg, Edvard Peer Gynt, 21 GWR, see generalized Weitzmann region H, see hexatonic pole Hantz, Edwin, 43n1 Harrison, Daniel and atonal analysis of nineteenthcentury music, 13 and harmonic progression, 83n1 and dualism, xv, 46n6, 93–94 and Riemann’s treatment of intonation, 177n7 and leading-tone agency, 47, 72 and rhetoric, 47, 194 and Riemannian function, 11, 114n3 and subposed dissonances, 143 and the Tonnetz, 28n10, 174n5 and transformational theory, 39 Hauptmann, Moritz, 5, 62n2, 92, 140, 142, 175n6 Haydn, Franz Joseph Symphony no. 98 in Bᅈ major, 4th mvt., 26, 28 Symphony no. 99 in Eᅈ major, 1st mvt., 78 Headlam, Dave, 96n10, 98n11 Helmholtz, Hermann von, 5, 140 Hepokoski, James, 133, 133n11 hexatonic cycle, 17–24, 84–86, 169–70 in Beethoven, Piano Sonata in f minor, Op. 57 (“Appassionata”), 1st mvt., 48 in Beethoven, Sonata for Violin and Piano in F major, Op. 24 (“Spring”), 2nd mvt., 26–27 in Brahms, Concerto for Violin and Cello in a minor, Op. 102, 1st mvt., 29–31 in Brahms, Symphony no. 1 in c minor, Op. 68, 4th mvt., 193–94 in Haydn, Symphony no. 98 in Bᅈ major, 4th mvt., 26 in Liszt, Consolation no. 3 in Dᅈ major, 30, 186-88 in Mozart, Symphony in Eᅈ major, K. 543, 4th mvt., 25–26 in Schubert, Piano Trio in Eᅈ major, Op. 100, 1st mvt., 27

232

 Index hexatonic cycle, (continued) in Schubert, Mass in Eᅈ major, D. 950, Sanctus, 31–32 in Wagner, Parsifal, 31–32 hexatonic-group transformations, 90–91, 105–6 hexatonic pole (H), 31, 103n15, 107–8 in Bach, J. S., Prelude in a minor, BWV 889 (Well-Tempered Clavier, vol. 2), 107–8 in Chopin, Mazurka in f minor, Op. 68 no. 4, 161 in Dvořák, Symphony no. 9 in e minor, Op. 95 (“From the New World”), 1st mvt., 135 in Schubert, Piano Sonata in A major, D. 959, 1st mvt., 49 in Schubert, “Auf dem Flusse” (Winterreise, D. 911), 123 in Schubert, Mass in Eᅈ major, D. 950, Sanctus, 31–32 in Wagner, Parsifal, 31–32, 107–8, 145–48 hexatonic region, see hexatonic cycle hexatonic scale, 20, 207 hexatonic transformations, see hexatonicgroup transformations Hoffmann, E. T. A., 1, 132 homophonous diamorph, 40, 165–66, 201–2 Hostinský, Otakar, 28n10, 113, 116 Hyer, Brian, 29, 69n10, 174–75 hypermeter, 117–21, 136 idealized voice leading, 6, 19, 89n4 Jackendoff, Ray, 53n11 Jentsch, Ernst, 148 Kant, Immanuel, 46 Karg-Elert, Sigfrid, 64, 155, 166 Katz, Adele, 11 Kirnberger, Johann Philipp, 1, 46, 140 Klumpenhouwer, Henry, 38–39 Kodály, Zoltán Mountain Nights, 108 Kollektivwechsel, see collective exchange Komar, Arthur, 131n10 Kopp, David, xi, 11, 12n12, 30n13, 37, 64 Kramer, Lawrence, 4n4, 171n1

Kramer, Richard, 68, 124 Krause, K. C. F., 5 Krumhansl, Carol, 102n13, 204n5 Kurth, Ernst, 10–11, 12, 22n7, 46n6, 169 L, see Leittonwechsel Langlé, Honoré de, 62n3, 92, 95n7, 207n6 Larson, Steve, 162 Leittonwechsel (L), 29–30, 38, 65–66 Lendvai, Ernő, 22n6, 125, 142, 155n14 Lerdahl, Fred and Chopin’s e minor Prelude, 162 and diatonic space, 177 and double syntax, 200 and hexatonic versus octatonic space, 91n5 and structural parallelism, 53n11 and the Faith Proclamation from Parsifal, 189–90 and the key sequence of Dichterliebe, 131 and Tristan und Isolde, 151n9 Lewin, David and classical versus Wagnerian tonality, 106 and directed voice leading, 84n2 and enharmonic transformation, 72 and exchange operations, 105n17 and harmonic function, 124–25 and Riemannian versus scale-degree space, 173–74, 178 and syntax in music, 13 and triadic transformations, 29, 64, 171–72, 189–90 and Tristan-genus chords, 156, 189n11 role in developing pan-triadic theory, xi, xiii Liszt, Franz “Blume und Duft,” 43 Consolation no. 3 in Dᅈ major, 30, 172–73, 180–82, 186–88 Faust Symphony, 1st mvt., 51, 63 Grande fantaisie symphonique über Themen aus Berlioz’ “Lélio” (“Lélio Fantasy”), 91–92, 100–102 Legende vom heiligen Stanislaus, Die, 108

Index Liebestraum no. 3 (“O Lieb, so lang du lieben kannst”), 63n5 Malediction for piano and string orchestra, 91 Missa pro organum lectarum, Kyrie, 128–29, 189 Penseroso, Il (Années de pèlerinage, vol. 2), 72–73, 95 Transcendental Etude no. 6, “Vision,” 63n5 Transcendental Etude no. 8, “Wilde Jagd,” 95n7 Longyear, Rey M., 77 Lorenz, Alfred, 22n7, 46n6, 148 Louis, Rudolf, 22n6, 46n6 Lowinsky, Edward E., 92, 179 L/P chain, see hexatonic cycle L/R chain, 92, 183–84 in Bach, J. S., Toccata in d minor, BWV 913, 183 in Beethoven, Symphony no. 9 in d minor, Op. 125, 2nd mvt., 92, 183 in Brahms, Ein deutsches Requiem, Op. 45, 2nd mvt., 96–97 in Brahms, Symphony no. 2 in D major, Op. 73, 1st mvt., 92–94 in Brahms, Symphony no. 1 in c minor, Op. 68, 4th mvt., 193–94 in Bruckner, Symphony no. 3 in d minor, 1st mvt., 98–99 in Chopin, Ballade in g minor, Op. 23, 96–98 in Chopin, Fantasy in f minor, Op. 49, 98–99 in Wagner, Parsifal, 190 Lubin, Steven, 133n12 Mahler, Gustav Symphony no. 1 in D major, 4th mvt., 106–7 Marchettus of Padua, 7, 185 Martino, Pat, 34n18 Marx, Adolph Bernhard, 1, 5 Masson, Charles, 7n11 maximal evenness, 37 McClary, Susan, 4n4, 112 McCreless, Patrick, 66n8, 197n2, 206 McKinney, Timothy R., 43n1 MED, 171–72 Meeus, Nicholas, 125n7

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Mehrdeutigkeit, 44, 57, 72 Mendelssohn, Felix Variations sérieuses, Op. 54, 108 meter, 53–54; see also hypermeter; rhetoric minimal perturbation, see near evenness Mitchell, William J., 11, 153n11, 190 mixture, modal, 184, 195 Möbius strip, 173–74, 178–79 Monteverdi, Claudio Incoronazione di Poppea, L’, 143n3 Moomaw, Charles Jay, 43–44 Mooney, Michael Kevin, xi Morgan, Robert P., 46, 151n9, 205 Morris, Robert D., 62n2, 108 Motte, Diether de la, 205 Mozart, Wolfgang Amadeus Fantasy in c minor, K. 475, 108 Minuet in D major, K. 355/576b (attrib.), 44 Symphony in Eᅈ major, K. 543, 4th mvt., 25–26, 28 Mussorgsky, Modest Boris Godunov, 155n14 Muysken, Pieter, 201 mystic chord, 166–67 N, see nebenverwandt near evenness, 34, 166–67, 204 of consonant triads, 34–37, 40, 179, 184–85, 207–8, 210 of Tristan-genus chords, 151–52, 189 relation to acoustic consonance, 40, 40n23 relation to efficient voice leading, 34–37, 40 nearly even, see near evenness nebenverwandt (N), 47–48, 61–62, 122–24, 184–85 neighborhood, 113–14, 116 neo-Riemannian theory, xiii–xiv Newcomb, Anthony, 123 N/H chain, 91 N/L chain, 94–95 in Dvořák, Symphony no. 9 in e minor, Op. 95 (“From the New World”), 1st mvt., 136–37 in Liszt, Grande fantaisie symphonique über Themen aus Berlioz’ “Lélio” (“Lélio Fantasy”), 100

234

 Index N/P chain, 93–94 N/R chain, 61–62, 76–78 in Beethoven, Piano Sonata in C major, Op. 53 (“Waldstein”), 1st mvt., 134 in Benda, Piano Sonata in a minor, 1st mvt., 77–78 in Chopin, Fantasy in f minor, Op. 49, 99 in Haydn, Symphony no. 99 in Eᅈ major, 1st mvt., 77–78 in Liszt, Faust Symphony, 1st mvt., 63 in Schubert, Piano Sonata in c minor, D. 958, 4th mvt., 77–78 in Schubert, Symphony no. 4 in c minor, D. 417, 1st mvt., 62 in Schubert, Symphony no. 9 in C major, D. 944, 1st mvt., 63 octatonic pole, see collective exchange octatonic region, 155, 157, 159 octatonic-group transformations, 155–56, 159 octatonic scale, 155, 207 octatonic transformations, see octatonicgroup transformations Oettingen, Arthur von, 29, 32, 38, 47, 62n1, 65, 89n4, 143, 182n9 omnibus progression, 149, 207 overdetermination, 40–41, 165–66, 179, 202–3, 208–10 P, see parallel palindrome, 121, 128–31 pan-triadic, xiv paradigmatic analysis, 122 parallel (P), 29–30, 38, 65–66 Parly, Nila, 22 parsimony, voice-leading, see voiceleading work, minimal Patel, Aniruddh, 203 Peck, Robert, 158n15 pitch retention loop, 116–21 Plato, 175 Popovic, Igor, 113 poststructuralism, 4, 109 preadaptation, 206–210 Proctor, Gregory, 6n7, 26n9, 33n16, 48n8, 199

Prokofiev, Sergei War and Peace, 95n8 prolongation, 22n7, 33, 45–46, 141 Prout, Ebenezer, 22n6 Puccini, Giacomo Tosca, 21 R, see relative R*, 155 Rameau, Jean-Philippe, 1, 95, 139–43, 151, 179, 182n9 reciprocity, 46–48 reduction, see dissonance, strategies for analyzing Rehding, Alexander, 95, 177 Reise, Jay, 167n21 relative (R), 62 rhetoric, 47, 194 Riemann, Hugo and dissonance, 140, 141, 180 and dualism, 32, 38, 143 and enharmonicism, 22n6 and harmonic distance, ix, 1, 5 and harmonic function, 11, 114n3, 124, 127–28, 177, 197 and intonation, 89, 177 and neo-Riemannianism, xiii and the Tonnetz, 28n10, 65, 113, 175n6 and Wechsel, 29 Riemannian tonal space, 173–74, Rimsky-Korsakov, Nikolai Antar (Symphony no. 2), 1st mvt., 21, 51–54, 56 Skazka, 91 Rings arrows, 170–71, 176–83 Rings, Steven, 6n8, 15, 64n7, 73n13, 169–71 Rosen, Charles, 49 Rothstein, William, 113 Rouget, Gilbert, 23 S (similar-motion transformations on Tristan-genus chords), 155 S2, 160, 167 S3(2), 155, 160 S3(4), 155, 160 S4, 160, 167 S5, 167 S6, 155

Index Salzer, Felix, 11 Saslaw, Janna, 47, 56, 56n13 Satyendra, Ramon, 25, 43n1, 53n10, 128 Schachter, Carl, 11, 21, 119n4, 161, 163n17 Schenker, Heinrich and consonance versus dissonance, 22n7, 45 and equal division of the octave, 26n9 and keys versus chords, 1 and mediant relationships, 98n11 and mode mixture, 184, 196–97n1 approaches to dissonance, 140, 141 Schmalfeldt, Janet, 67 Schoenberg, Arnold and strong versus weak harmonic progressions, 125n7 and the law of least motion, 7n11 String Trio, Op. 45, 21 Schreker, Franz Kammersymphonie, 142 Schubert, Franz “Auf dem Flusse” (Winterreise, D. 911), 122–25, 197n2 Characteristic Allegro for Piano Four Hands (“Lebensstürme”), D. 947, 64n7 “Doppelgänger, Der” (Schwanengesang, D. 957), 113–16 Impromptu in Eᅈ, Op. 90 (D. 899) no. 2, 74–75, 170–71 Impromptu in Gᅈ, Op. 90 (D. 899) no. 3, 74–75 Klavierstuck in Eᅈ major, D. 946 no. 2 (Drei Klavierstücke), 91 “Liedesend’,” D. 473, 129–31 Mass in Aᅈ major, D. 678, Sanctus, 64n7, 91 Mass in Eᅈ major, D. 950, Sanctus, 30–32 “Nacht und Träume,” D. 827, 21 Piano Sonata in c minor, D. 958, 4th mvt., 77–78 Piano Sonata in A major, D. 959, 1st mvt., 48–50 Piano Sonata in Bᅈ major, D. 960, 1st mvt., 2–4, 6–8, 64n7, 125–27, 169–70 Piano Trio in Eᅈ major, Op. 100, 1st mvt., xi, 27–28

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Rosamunde Overture, see Zauberharfe, Die, Overture “Stadt, Die” (Schwanengesang, D. 957), 47 String Quartet in d minor, D. 810 (“Death and the Maiden”), 94 String Quintet in C major, D. 956, 2nd mvt., 64n7 Symphony no. 2 in Bᅈ major, D. 125, 4th mvt., 45 Symphony no. 4 in c minor, D. 417, 1st mvt., 62–63 Symphony no. 9 in C major, D. 944, 1st mvt., 63; 3rd mvt., 195–99 “Trost,” D. 523, 113n2 Zauberharfe, Die, Overture (Rosamunde Overture), 85–88 “Zügenglöcklein,” D. 871, 113 Schumann, Robert Dichterliebe, Op. 48, 131–33; no. 1, “Im wunderschönen Monat Mai,” 75–76; no. 5, “Ich will meine Seele tauschen,” 76 Piano Trio no. 1 in d minor, Op. 63, 108 Scriabin, Alexander Feuillet d’album, Op. 58, 167–68 Sechter, Simon, 46 Seidel, Elmar, 45n3, 207n6 semantics, 14, 21–22, 171, 171n, 178–80, 184, 186, 188 sequence diatonic sequence, 10, 47, 207 chromatic, 10, 19, 47, 89–90 see also chain Sessions, Roger, 13 Shamgar, Beth, 4 Siciliano, Michael, 34n18, 113n1 single semitonal displacement, 18, 34–35, 37, 55, 151n10, 153; see also Leittonwechsel; parallel S/L chain, 91–92 slide (S), 64 Slobin, Mark, 202n4 Slonimsky, Nicolas, xiv Smith, Charles J., 199–201 Smith, Peter H., 47n7 Soderberg, Stephen, 151n10 Sorge, Georg Andreas, 44

236

 Index S/P chain, 95 in Dvořák, Symphony no. 9 in e minor, Op. 95 (“From the New World”), 135 in Liszt, Il Penseroso, 95 spoiler tone, 150–54, 167–68 Stein, Carl, 92 Steinbach, Peter, 84, 155, 157–58n15 Strauss, Richard “Frühling” (Four Last Songs), 32–33, 79–81 Salome, 21, 108 SUBD, 170–71 SUBM, 171–72 substitution, see dissonance, strategies for analyzing substitution, transformational, see transformational substitution sum, pitch-class, 104–5; see also voice-leading zone Swinden, Kevin J., 73n12 syntax, 12–15, 39–40 double, 11–12, 169–72, 195–201, 205–8 syntonic comma, 175–77 syntonic image, 176–78 T (transposition), see chain Tallis, Thomas Psalm tune no. 3 for Psalter of Archbishop Parker, 143n3 Tarnhelm, 21–23, 72–73 Taruskin, Richard, xv, 21, 23, 207n6 Tchaikovsky, Peter Ilich, 5 third relations, 19, 171 Thuille, Ludwig, 22n6, 46n6 Todd, R. Larry, xi, 43n1 Tonnetz, 27–30 and Cube Dance, 86–89, 100–101 and diatonic progressions, 175–79 comparison to other geometrical models, 65–67 convertible (circular versus planar), 174–75 flexibility of, 114–16 spatial orientation of, 89, 175n6 three-dimensional, 141–42 torus, 88 transformation, 29–32, 38, 39, 90, 155–56, 159; see also transformational substitution

compound, 30, 60–65, equivalence among, 102–6 transformational subsitution, 95–96, 121–24, 160–61 Tristan genus, 148–52 Tymoczko, Dmitri and dualism, xiii, 37–38 and measurements of voice-leading distance, 6n8, 7, 66, 84, 88, 159 and near evenness, 34, 36n19, 40n23, 56n12, 166n20 and octatonic triadic systems, 91n5 and representation of transformational substitution, 96n9 and scalar tonality, 207 and triads as homophonous diamorphs, 40, 209–10 and Tristan-genus chords, 149nn5–6, 153n12, 158n15, 160–61, 164n19 and voice-leading direction, 19n3, 32, 89n4 uncanny, 21–24 uniformity, law of, 10, 47–48 upshift voice leading, 84–85, 121, 131–38, 154–55, 166 Verdi, Giuseppe “Ah sí, ben mio” (Il Trovatore), 116 Vogler, Georg Joseph, 44–45, 72, 92 voice-leading work, 6–8 minimal, 6–8, 17–19, 33–37, 38, 151, 166–67 voice-leading zone, 102–6, 154–55, 159 Wagner, Richard Götterdämmerung, 21, 108–9, 156, Parsifal, 21, 30–32, 43, 78–79, 107–8, 144–48, 172, 189–91, 194 Ring des Nibelungen, Der, 21, 22–23, 72–73, 155n14 Siegfried Idyll, 43 Tristan und Isolde, 143–44, 149–55 Walküre, Die, 4n4, 11, 43 Waller, Derek, 18n1 Wason, Robert W., 45n3 Weber, Gottfried, 7n11, 8, 53n11, 131, 176–77 Webster, James, 78, 113n1

Index Weitzmann, Carl Friedrich and the emancipation of dissonance, 7n11 and the relationship between consonant and augmented triads, 56–58, 83–84, 151–53 metaphorical presentation of the augmented triad by, 43 nebenverwandt in, 46–47 triads versus tonics in, 61

Weitzmann-group transformation, 90–91, 105–6 Weitzmann region, 59–61 graphical representation, 64–65 Werckmeister, Andreas, 92 Wolf, Hugo “Das Ständchen,” 45n4 Zarlino, Gioseffo, 7 zone-diametric triads, 107–8

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