Single-Voice Transformations : A Model for Parsimonious Voice Leading [1 ed.] 9781443818940, 9781443818605

Citation preview

Single-Voice Transformations

Single-Voice Transformations: A Model for Parsimonious Voice Leading

By

Brandon Derfler

Single-Voice Transformations: A Model for Parsimonious Voice Leading, by Brandon Derfler This book first published 2010 Cambridge Scholars Publishing 12 Back Chapman Street, Newcastle upon Tyne, NE6 2XX, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2010 by Brandon Derfler All rights for this book reserved. No part of this book 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 the copyright owner. ISBN (10): 1-4438-1860-7, ISBN (13): 978-1-4438-1860-5

TABLE OF CONTENTS Page List of Figures

vii

List of Tables

ix

Acknowledgments

x

Chapter 1: Introduction A Brief Preliminary Remark: Conventions Used

1 8

Chapter 2: Some Transformational Models of Voice-Leading Space

11

Chapter 3: The Single-Voice-Transformation Model Qualities Privileged in Voice-Leading Models Callender 2004 and Tymoczko 2006 Direct-Product Transformations Splits and Fuses SST-Succession Classes Split-Succession and Fuse-Succession Classes

32 32 35 39 47 55 62

Chapter 4: Graphical Representations of Parsimonious Voice-Leading Spaces 66 Graphs of Voice-Leading Transformations on PCs and PC Sets of the Same Type 66 Graphs of Voice Leading Transformations between PC Sets of Different Types 72 Graphs of Optimal Offset between SC Types of the Same Cardinality 76 Graphs of Optimal Offset between SC Types of Differing Cardinality 91 Chapter 5: The SST Model and Neo-Riemannian Theory A Brief Overview of Tetrachordal Neo-Riemannian Studies Relating SST-Succession Classes to Neo-Riemannian Operations A Comparison of Neo-Riemannian Labeling Systems “Neo-Riemannian” SST-Succession Classes and the Inclusional Principle

104 105 125 125 131

Chapter 6: Analyses Tonal Analyses

138 140

vi

Table of Contents

Chopin, Prelude in E Minor, Op. 28 no. 4 Scriabin, Etude in C-Sharp Minor, Op. 42 no. 5 Atonal Analyses Webern, Cantata No. 2, Op. 31: I Lansky, Modal Fantasy, “Prelude” Post-Atonal Analyses Adams, Nixon in China Conclusion

140 147 151 151 154 158 160 171

Appendix A: Summary of [SSTi]-Relations Between Trichordal and Tetrachordal Equivalence-Class Pair-Successions 173 Number of Distinct [SSTi]-Relatable Set Classes [B] in a Pair-Succession ¢[A],[B]²SSTi 173 Equivalence-Class Pair-Successions Relatable through Some Single [SSTi] (Including Split- and Fuse-Succession Classes) 174 Appendix B: Chart of [SSTi] Subscripts Relating Trichordal and Tetrachordal Equivalence-Class Pair-Successions 177 Equivalence Classes [B] Relatable to [A] by Single [SSTi]s as Part of an SST-Succession Class 177 Equivalence Classes [B] Relatable to [A] by Single [SSTi]s as Part of an SST-Succession Class or Fuse-Succession Class 178 Notes

180

References

189

Index

195

LIST OF FIGURES Figure Number

Page

2-1. Voice leading of a [0146]-type tetrachord 2-2. Voice leading where voice and line coincide 2-3. Voice leading where voices do not coincide with lines 2-4a. Wagner, Prelude to Tristan und Isolde, mm. 1–3 2-4b. Voice-leading via T7I transformation on a member of SC [0258] 2-5. Musical representation of a voice-leading rule (from Cope 2002) 2-6. Uniformity, “fuzzy” transpositions between {F,F,B} and {G,B(,D} 2-7. Balance in “fuzzy” inversions between {F,F,B} and {G,B(,D} 2-8. Smoothness in all voice-leadings between {F,F,B} and {G,B(,D} 3-1. Voice-leading which privileges pitch-class smoothness 3-2. Fundamental region of voice-leading space, trichordal SC types 3-3. The T3/S3 orbifold (from Tymoczko 2006) 3-4. A split transformation T10/T10 + 3 (after O’Donnell 1997) 3-5. Constant min-m-voice transformations: a) for splits; b) for fuses 3-6. Variable m-voice transformations: a) for splits; b) for fuses 3-7. Constant max-m-voice transformations: a) for splits; b) for fuses 3-8. A fuse function SPLIT5–1(SST(3–1)(5–1)(¢0,3,4,7,8²)) = ¢0,3,3,7² 3-9. Continuation of voice leading in four voices after a fuse 4-1. The Riemann Tonnetz 4-2. A “traditional” Tonnetz (from Gollin 1998) 4-3. Voice leading from set inversion operations L, P, R, and R′ 4-4. A geometric dual graph of the Tonnetz (“chicken-wire torus”) 4-5. Parsimonious voice leading between members of SC [0258] 4-6. “Cube dance” showing parsimonious voice leading between triads 4-7. Graphs of strongly parsimonious 7th-chord voice-leading space 4-8. Voice-leading space with optimal offset between trichordal SCs 4-9. [SSTi]s between trichordal equivalence-class pair-successions 4-10. Voice-leading space of optimal offset between tetrachordal SCs 4-11. 3D voice-leading with optimal offset between tetrachordal SCs 4-12. [SSTi]s between tetrachordal equivalence-class pair-successions 4-13. Additional parallel paths between tetrachordal pair-successions 4-14. Cohn’s tetrahedral graph of tetrachordal voice-leading space 4-15. Graph of parsimonious tri- and tetrachordal voice-leading space 4-16. Sectional layers in the trichordal split-succession-class graph 4-17. Layers of the tri-tetrachordal split-succession-class graph 4-18. Trichordal SST-succession classes and split-succession classes 4-19. Graph of trichordal split-succession classes

11 20 20 21 21 23 25 26 26 34 37 38 47 49 50 50 53 54 67 67 68 70 71 72 74 77 79 83 84 86 88 90 92 93 95 96 97

viii

List of Figures

4-20a. Graph of Figure 4-18, identifying [SSTi] or [SSTj] relations 4-20b. Rear view of Figure 4-20a 5-1. P2 transformations between members of [0258] in an octatonic set 5-2. Partial ambiguity associated with voice leading operation C3(2) 5-3. A Tonnetz for SC [0258] showing flips about central tetrachord 5-4. S2, S3, and S4 transformations; hypothetical S3 transformations 5-5. Comparison of Kopp’s M function to an SST-succession class 5-6. N-like SST-succession classes where [A] = [0358] 6-1. SST analysis of Chopin, Prelude in E minor, Op. 28 no. 4 6-2. SST-succession-class analysis of Chopin, Prelude in E minor 6-3. Chopin, Prelude in E minor with extended neo-Riemannian labels 6-4. Formal plan of Scriabin, Etude in C-sharp minor, Op. 42 no. 5 6-5. Scriabin, Etude in C-sharp minor, Op. 42 no. 5, measures 30–42 6-6. SST and SST-succession-class analyses of Scriabin, mm. 31–6 6-7. Webern, Cantata No. 2, Op. 31, I: row forms 6-8. Webern, Cantata No. 2, Op. 31, I: measures 1–5 (reduction) 6-9. Webern, Cantata No. 2, Op. 31, I: measures 11–2 (reduction) 6-10. Paul Lansky, Modal Fantasy, “Prelude,” mm. 1–4 6-11. SVT analysis of Lansky, Modal Fantasy, “Prelude,” mm. 1–4 6-12. SST-succession-class analysis of Modal Fantasy, “Prelude” 6-13. Adams, Nixon in China, Act I, Scene 3, mm. 390–404 analysis 6-14. Adams, P2,0 cycles 6-15. Adams, Nixon in China, Act I, Scene 1, mm. 487–558 analysis 6-16. Adams, Nixon in China, Act III, mm. 748–809, reduction 6-17. Adams, Nixon in China, Act III, mm. 791–6 (piano–vocal score)

99 101 112 113 116 118 130 134 141 143 144 147 148 150 152 153 154 155 156 157 161 162 164 169 170

LIST OF TABLES Table Number 3-1. List of all Tn/TnI representatives of SCs [02] and [03] 5-1. Baker’s tables of semitonal transformations (from Baker 2003) 5-2. Label comparison for [SSTi]-related SST-succession classes 5-3. Label comparison for composite tertian SST-succession classes 5-4. Comparison of labels for tertian split-succession classes 5-5. P-, L-, and R-like tetrachordal SST-succession classes 5-6. Other neo-Riemannian-like tetrachordal SST-succession classes 5-7. Inclusion relations between penta- and tetrachordal set classes

Page 56 122 126 127 128 132 133 136

ACKNOWLEDGMENTS The author wishes to express sincere gratitude to the individuals who helped bring this study to fruition. John Rahn, as dissertation advisor, was always willing to meet with me to discuss the progress of this work, and his help in formalizing the mathematical aspects of the topic proved to be invaluable. His suggestions that I pursue certain intriguing avenues of investigation proved to be fruitful, and led directly to the formulation of several new sections. Committee member Jonathan Bernard was always encouraging and stimulated new ways of thinking about the concepts involved in transformational theory, some of which made their way into the final work. David Kopp succeeded in kindling in me an interest in neo-Riemannian theory; his tutelage was indispensable in pointing me toward literature on the topic. Sincere thanks must also be extended to Joseph N. Straus, Scott Baker, Amy Shimbo, and Michael Callahan for providing me with copies of their unpublished papers. My deepest gratitude goes to Tenia Danielle Ackles, without whose seemingly never-ending support I would never have been able to complete this work. Licenses and Permissions: Adrian Childs, “Moving Beyond Neo-Riemannian Triads: Exploring a Transformational Model for Seventh Chords,” Journal of Music Theory 42/ 2. © 1998, Yale University. All rights reserved. Used by permission of the publisher, Duke University Press. Anton Webern, 2. KANTATE für Sopran, Bass, gemischten Chor und Orchester op. 31. © 1951 by Universal Edition, Wien, English version Copyright 1955 by Universal Edition A.G., Wien/UE 11885. © Renewed. All Rights Reserved. Used by permission of European American Music Distributors LLC, U.S. and Canadian agent for Universal Edition, Wien. Dmitri Tymoczko, “The Geometry of Musical Chords,” Science 313/5783. Reprinted with permission from AAAS. Edward Gollin, “Some Aspects of Three-Dimensional Tonnetze,” Journal of Music Theory 42/2. © 1998, Yale University. All rights reserved. Used by permission of the publisher, Duke University Press.

Acknowledgments

xi

Jack Douthett & Peter Steinbach, “Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition,” Journal of Music Theory 42/2. © 1998, Yale University. All rights reserved. Used by permission of the publisher, Duke University Press. John Adams, Fearful Symmetries. © 1989 by Hendon Music, Inc., a Boosey & Hawkes Company. Copyright for all countries. All rights reserved. Reprinted by permission of Boosey & Hawkes, Inc. John Adams, Nixon in China. Lyrics by Alice Goodman. © 1987 by Hendon Music, Inc., a Boosey & Hawkes Company. Copyright for all countries. All rights reserved. Reprinted by permission of Boosey & Hawkes, Inc. Robert D. Morris, “Voice-Leading Spaces,” Music Theory Spectrum 20/2. © 1998 by University of California Press. Reprinted with permission from The University of California Press.

CHAPTER 1 INTRODUCTION It is likely that no music-theoretic study in recent times has had a greater impact on subsequent research than David Lewin’s 1987 book Generalized Musical Intervals and Transformations (hereafter abbreviated GMIT). Among the many possible applications of Lewin’s transformational theory, two have been particularly fruitful and have been developed by Lewin and others into major branches of modern music theory: transformational theories, applied to voice leading in pioneering studies by Lewin, John Roeder, Joseph Straus, and Henry Klumpenhouwer, and so-called “neoRiemannian” theory, stemming from chapter 8 of Lewin’s book and extended by Brian Hyer, Richard Cohn, Lewin himself, and others. Transformational theories of voice leading have relied almost exclusively on the Tn/TnI dihedral group of operations on verticalities to map pcs from one chord to another. One particular offshoot of transformational voiceleading theory of interest to many theorists (including Lewin) involves the study of network isographies (“Klumpenhouwer networks” or “K-nets”). With their emphases on chordal and network transformations and the fractallike recursive self-similarities found in K-nets, studies of this kind have tended to relegate voice leading to a position of lesser prominence, seemingly viewed as not much more than an incidental by-product of chordal mappings. Additionally—as will be seen—the very notion of “voice leading” itself is problematic when one attempts to make a distinction between, say, voice and line on the one hand, or between manifest and projected voice leading, to borrow Lewin’s terms (1998, 18), on the other hand. Neo-Riemannian theory at times seems to suffer from the same malady that plagues set transformational theory: much more emphasis has been given to the group properties of transformations on the Riemannian triadic Klänge (and, more recently, seventh chords) than to the actual voice leading that results from the transformations. In both areas of study, theorists have focused much more closely on the operations that map one pc set onto another, to the detriment of exploring the contrapuntal paths used to get there. One aspect of certain types of voice leading—namely, smoothness or parsimony—has been explored in neo-Riemannian studies (particularly by Richard Cohn), and forms an important bridge between chordtransformation theories and theories of counterpoint and line. A third, unrelated branch of musical explanation that enjoys great currency can be traced to the writings of Heinrich Schenker and his disciples.

2

Chapter One

While Schenkerian analysis is non-transformational, not being founded on a mathematical basis as Lewin’s system is,1 it offers a much more intimate look at voice leading, at counterpoint, and at the linear aspect of music than do the branches of theory described above. While I do not propose a type of “unified field theory” that reconciles the Schenkerian and the transformational approaches to voice leading, in this study I employ a transformational approach in a manner that takes into account the voice-leading connections between chords to a greater degree than do the transformational voice-leading and neo-Riemannian models. As will be seen, there is nothing particularly Schenkerian about my approach, other than a heightened sensitivity to the horizontal element in progressions between vertical sonorities. As some proponents of neo-Riemannian theory privilege voice-leading smoothness, in this dissertation I too will focus much of the discussion on voice-leading parsimony, which has been a concept basic to linear writing in non-monophonic music from the advent of polyphony until the present day. This principle of smooth voice leading as the “norm,” with disjunct motion serving as the exception (to provide contrast), has been recognized by theorists and composers for centuries and is perhaps best described by Schoenberg (quoting Bruckner) as “the law of the shortest way” (Schoenberg 1978, 39). Key to my approach is the single-voice transformation (SVT), whose archetype is the single-semitone transformation (SST). This function on a chord transposes one pc by T1 while the remaining pcs map onto themselves. Multiple iterations of this transformation or its inverse through the composition of functions produces larger voice-leading intervals. As a great deal of music privileges smooth voice leading over disjunct voice leading, the SST—whether iterated singly or doubly—serves as an effective model for better than half of the voice-leading connections in most music (i.e., in music in which half- and whole-step voice leading predominates). For such music in which parsimony is the predominant voice-leading paradigm, the occasional skip can be modeled by the single-fifth transformation (SFT) and its multiples. This function on a chord transposes one pc by T7 while the remaining pcs map onto themselves. As with the SST, multiple iterations of this function or its inverse—as well as its combination with an SST—produce different voice-leading intervals. The SST system and the SFT system, both based on transposition intervals that generate the complete ETS 12 pitch-class universe, are isomorphic. Some readers will recognize the single voice transformation’s resemblance to Allen Forte’s “unary transformation,” in which one pc set mutates to another by the change of a single element of that set (Forte 1988). In order to limit the scope of this study, for the most part I will be discussing the SST, first focusing on parsimonious transformations in the abstract and subse-

Introduction

3

quently examining music in which the voice leading can for the most part be modeled by SSTs. A “parsimonious transformation” may be defined as one or more SSTs acting on a chord, with a maximum voice-leading displacement of one whole step per chord member (“voice”). Thus to be considered parsimonious, a voice-leading mapping may not consist of more than two consecutive applications of the SST affecting the same voice. This limit at first appears related to Robert Morris’s artificially imposed constraints on total voice leading (Morris 1998, 178). There really is no limit to the number of iterations of SSTs and SFTs that can be applied to a chord member; the whole-step artificial limit is only meant to define the term “voice-leading parsimony” and in no way compromises the integrity of the SVT system were it to be removed. Music in which the voice leading can be modeled by a combination of parsimonious and non-parsimonious transformations is the norm through all historical periods, and the works analyzed in this study are no exception to this norm.2 The structure of the present study is as follows: As a preface to discussion of my model of voice leading, it is helpful to summarize some of the work of theorists in modeling transformational voice-leading space. An overview of transformational models of atonal voice leading is presented in chapter 2, examining studies by Klumpenhouwer, Straus, O’Donnell, Lewin, Cope, Jurkowski, Callahan, and Roeder. Most of these models propose variations on the theme of voice leading via Tn/TnI operations mapping pc set to pc set of the same set-class type. While these studies are fascinating and at times quite compelling—and, in the case of Klumpenhouwer, have opened up new areas of theoretical inquiry—they all, in varying degrees, suffer from one or both of two principal shortcomings. First, and most significantly, in most of these studies, voice-leading transformations can only be mapped between pc sets of the same set-class type. This problem has recently been partially addressed in articles by Joseph Straus through his concepts of “near-transposition” and “near-inversion” (Straus 1997, 268) which he later calls “fuzzy transposition” and “fuzzy inversion” (Straus 2003, 318).3 Second, and perhaps more relevant from a practical standpoint, is the problem of musical voices—as a result of pitch-class mapping via transposition and inversion operations—actually being heard (or at least musically intuited). The fact that many of the paths traced through this type of “pitch-class counterpoint”4 correspond with neither concepts of voice-leading parsimony nor (more importantly) what most would intuit as individual voices—made audible in actual music through factors such as register, instrumentation and pitch/ pitch-class interval distance—naturally causes one to question the analytical applicability of such proposed voice-leading connections. Integral to a discussion of such an important topic is an understanding of the distinction

4

Chapter One

between voice and line, and part of chapter 2 emphasizes some of the finer points of this often difficult differentiation. In Chapter 3 the single voice transformations SST and SFT are presented and formally defined as ordered m-tuple elements ¢g1,g2,...,gm² of the direct product group of mod-12 transposition operations G1 + G2 + ... + Gm acting on a chord C of cardinality m, with chord members ci also ordered as an mtuple: ¢c1,c2,...,cm².5 The chief differences between my voice-leading model and those discussed in chapter 2 are presented in this chapter and include a number of characteristics which I will briefly outline here. Previous transformational voice-leading theories have relied on the Tn/TnI group of operations to map pc set to pc set. My model employs only the group of transposition operations (Tn) to effect mappings between m-tuples. This has the net result of describing voice-leading operations in a way that is more naturally intuited by listeners and (possibly in the majority of cases) as conceived by composers. Another principal difference between methodologies is the capability of SST and SFT operations and their compositions to allow for mappings between chords belonging to different set-class types. The desirability of this result is self-evident. The strengths and limitations of the direct-product voice-leading model are weighed in chapter 3. For example, the problem of multiple iterations of single-voice transformations is discussed: at what point does the conglomeration of “moves” become excessive? Another one of the model’s shortcomings is its difficulty in convincingly explaining music in which there is a fair amount of voice leading by ic 3 or 4. On the flip side, however, one of the strengths of my model is its use of ordered m-tuples—optionally indexed by a unique set-class type—as the domain and range of voice-leading functions. Through simple addition, one can apply one or more single-voice transformations to an m-tuple and easily calculate the resultant m-tuple. By extension, SST-succession classes can be created to generalize parsimonious voice-leading relations between Tn/TnItypes on a more abstract level. The generalization to SST-succession classes builds upon recent work by Clifton Callender (2004) and Dmitri Tymoczko (2006). Through the use of ordered m-tuples and generalized SST-succession class types, the registral order of the chord members is unimportant, unlike some previous transformational voice-leading theories based on voice permutation. In those theories, the permutation of the voice positions drives the voice leading. If pitch-interval size between adjacent voices in a chord is a key factor in a transformational voice-leading model (and it often is), permutation of the registral order positions (“chord inversion” in the parlance of tonal harmony) completely changes the interval size; thus a different operation must be used for chords of the same set-class type appearing in different registral configurations.

Introduction

5

Another problem which has plagued transformational theory since its inception involves mappings between sets of different cardinalities. In chapter 3 I discuss the manner in which theorists have coped with this issue, which in the pitch/pitch-class realm has involved the use of splits and fuses, terms which seem to have originated in a neo-Riemannian study by Clifton Callender (1998, 224). My concept of splits and fuses tends to differ somewhat from the “traditional” definition, and is perhaps the first to formalize these types of operations mathematically. Generalization of these concepts to split-succession classes and fuse-succession classes rounds out the chapter. Chapter 4 deals with network graphs of parsimonious voice-leading systems. A number of these graphs have been constructed—in two- and threedimensional form—of voice-leading space involving triads and seventh chords of limited set-class types. For example, the familiar 2D Riemannian Tonnetz lattice models L, P, and R relations between major and minor triads ([037]). More recently Robert Morris (1998) has constructed a generalized “Tonnetz space descriptor” which adjusts the ic values on each axis of a Tonnetz to allow for lattice models of all trichordal set-class types, not all of which preserve parsimonious voice leading. Parsimonious voice-leading relations between all tetrachordal set-class types have been graphed by Straus in 2D (2003, 339) and by Straus (2005) and Richard Cohn (2003) in 3D. In chapter 4 I present two distinct network graphs of parsimonious voiceleading space: the tetrachordal SST-succession-class-space graph which is a 3D version of Straus’s “optimal offset” chart, and the trichordal/tetrachordal split-succession-class-space (“Christmas tree”) graph which is a 3D model showing all split-succession-class relations between trichordal and tetrachordal SC types. As it is difficult to reproduce 3D renderings on the 2D printed page, links are provided to an internet URL which contains CAD files of the network graphs, along with a link to the free software used for viewing them. The purpose of this is to provide a hands-on experience in which readers can zoom in and out of the graph and pan in all three dimensions to view the structure from any angle. The website also contains short .avi video clips recorded with a virtual camera “flying” in different trajectories through the graphs. For those disinclined to use the CAD software, 2D snapshots of the graphs appear as figures in the body of the dissertation. Both graphs are “ball-and-stick” networks, with set-class types being represented by spheres and cubes, and the associated SST- or split-relation subscripts by the connecting “sticks.” In several cases, different SSTsuccession-class types may comprise identical pair-related set classes. In cases such as these we can speak of duplicate paths of distinct SST-relations between m-tuple elements of the ordered pair of (identical) set classes. Duplicate SST-succession-class paths are most frequently associated with

6

Chapter One

symmetrical set classes. One well known—and extreme—example of this concept is the collection of SST-succession classes that relate [0369] to [0258], where there are eight distinct voice-leading path types between mtuple elements of the set classes.6 The tetrachordal network graph, while simpler than the trichordal/ tetrachordal network graph, has one clear advantage over the latter: it shows a clear correspondence between SST-succession classes affecting “order positions” of the set-class integers and the X–Y–Z axes of the Cartesian coordinate system. It will also be seen that set class [0148] occupies a unique position in the tetrachordal graph, and reasons for this are discussed in the chapter. The trichordal/tetrachordal graph comprises nested cones of setclass types sharing certain properties, and displays split-succession-class connections between trichordal and tetrachordal SC types. At first glance, this second graph appears to resemble Cohn’s tetrahedral network graph (2003), but it is actually quite different; these differences are discussed in the chapter. The study of single-voice transformations acting on m-tuple members of an important subset of trichordal and tetrachordal set classes—namely, the “neo-Riemannian” set classes—forms the basis for chapter 5. There is, in the literature, disagreement about exactly which set class types’ members can be operated on by neo-Riemannian transformations. The original essays defining neo-Riemannian theory (originating with Riemann himself) only admitted the major and minor triads as Klänge which could be subjected to Schritten and Wechseln such as the Dominantschritt and Leittonwechsel operations. In the late 1990s a number of theorists began to examine parsimonious voice-leading transformations between seventh chords, particularly between dominant seventh/half-diminished seventh chords of set class [0258].7 In this study, the “neo-Riemannian” subset is defined as set class [037] and all tetrachordal SC types that include it. This collection includes the standard major, major-minor, minor, and half-diminished seventh chords as well as several atonal tetrachords, but does not include the fully diminished seventh chord or the augmented or diminished triads (parsimonious voice-leading involving these latter chords is discussed in chapter 4). The first part of chapter 5 provides a brief overview of the literature on neo-Riemannian theory, limited to those studies that examine 1) transformations on chords of cardinality 4 or 2) transformations that increase or decrease cardinality between chords. Work by Lewin, Cohn, Callender, Childs, Douthett and Steinbach, Gollin, Hook, and others is discussed. It will be seen that most of these studies do not directly address the relationship between neo-Riemannian operations and voice leading (the principal exception is Childs). As most of these studies regard neo-Riemannian functions such as L, P, and R as contextually-defined inversion operations (“cio”s,

Introduction

7

after Kochavi 1998), they run into the same twin problems that plague the authors discussed in chapter 2: difficulty in achieving mappings between chords belonging to different set-class types, and musically non-intuitive voice leading (if voice leading is even a consideration). By eliminating these inversion operations from the group of transformations on members of the neo-Riemannian SC collection, voice leading comes to the fore and becomes more of a determinant of chordal succession than would be suggested by cioprivileging approaches. Interestingly, only Gollin (and to a lesser extent, Shimbo) approach neo-Riemannian transformations through Riemann-like dualist systems, even if they do not explicitly and formally describe the voice-leading that results from their chord mappings. Although the present study eschews tonal and dualist tonal theories for an atonal transformational approach, by its abstracting the ordered m-tuples operated upon by the single-voice transformations to SST-related Tn/TnI types, it bears some resemblance to the Riemannian dualist system. For example, the set {0,4,7} shares the same Tn/TnI type as its Riemannian dual, {5,8,0}, namely, [037]. Applying the SST operation SST2–1 to ¢0,4,7² is equivalent to the “traditional” P operation, in practical terms mapping a C-major triad to a C-minor triad. However, the SST operation SST2 would need to be applied to ¢5,8,0² to map an F-minor triad to an F-major triad. Both of these SST-related chord successions are representatives of the SST-succession class, ¢[037],[037]² SST2. Also in chapter 5, the multitude of labeling systems used by the various authors to describe the different “neo-Riemannian” functions on trichords and tetrachords are examined. These are compared with the conventions of the labeling system used in this dissertation and with each other; all are summarized in Tables 5-2 through 5-4. Chapter 6 presents analyses of music using the single-voice-transformation model. I have deliberately chosen music composed in several distinct time periods and styles to show the near-universal applicability of the SVT model to non-monophonic works from the Western art-music tradition. The selections can be grouped in three broad categories: 1. Tonal music, specifically Romantic-era compositions more highly saturated with chromaticism than earlier styles. Pieces by Chopin (Prelude Op. 28, no. 4 in E minor), and Scriabin (Etude Op. 42, no. 5 in C-sharp minor) are analyzed; 2. Atonal music, represented here by analyses of the first movement of Webern’s Second Cantata, Op. 31, and the opening of Paul Lansky’s Modal Fantasy; and 3. Music by contemporary composers dating from the last quarter of the twentieth century which incorporates elements of atonal as well as nonfunctional triadic music and which, for lack of a better term, I will refer

8

Chapter One

to as “post-atonal” music. Shorter excerpts from pieces by John Adams are first examined, followed two lengthier analyses of selections from Nixon in China. This last category requires some additional explanation. I have undertaken analyses of several pieces from the post-atonal repertoire for a couple of reasons. First, this literature has, until quite recently, been underrepresented in music analysis, and particularly in transformational analyses, which have instead focused on the “classic” serial literature (atonal voiceleading analyses) and on late-Romantic pieces (neo-Riemannian analyses). Another reason for the choice of this particular literature is its suitability for analysis which emphasizes parsimonious voice leading. Much of this literature fits—however tightly or loosely—under the rubric of “minimalist” music, in which smooth voice leading is, if not privileged, at least more readily apparent to the ear than in atonal music from the same time period. The dissertation concludes with a summary of topics covered in the study and posits suggestions for avenues of future research. This is followed by appendices which list tables of trichordal and tetrachordal SST- and splitsuccession classes.

A Brief Preliminary Remark: Conventions Used My notational conventions for m-tuples, sets, and set classes in this study for the most part follow those in Rahn (1980) with one minor exception: I have eliminated the commas separating pc integers in both Tn- and Tn/TnItype set-class designations to follow an increasingly common practice in the literature. The absence of the commas should not create confusion if we use “t” for 10 and “e” for 11. Thus a Tn-type for the whole-tone hexachord would read (02468t) and the Tn/TnI-type would read [02468t]. Another reason for eliminating the commas in the Tn-type is to avoid confusion with a common mathematical convention for an m-tuple. Following standard practice, commas will be retained between elements of m-tuples (enclosed in angle brackets, thus ¢a1,a2,...,am²) and sets (enclosed in curly braces, thus {a1,a2,...,am}). There is slight disagreement in the music-theoretic literature about the extent to which the following terms differ: transformation, operation, function. The terms are largely synonymous in the mathematical community (Hook 2002, 192; 197) but Lewin assigns subtly different shades of meaning to each in GMIT. Lewin’s definition of a function follows the standard mathematical convention: Let S and S′ be sets of objects. The Cartesian product S × S′ is the set of all ordered pairs ¢s,s′ ² such that s is a member of S and s′ is a member of S′ .

Introduction

9

A function (or mapping) from S into S′ is a subset f of S × S′ which has this property: Given any s in S, there is exactly one pair ¢s,s′ ² within the set f which has the given s as the first entry of the pair. We say that s′ , in this situation, is the value of the function f for the argument s; we shall write f(s) = s′ (Lewin 1987, 1).

Lewin goes on to define a transformation on S as a function from a set S into S itself. An operation on S is a bijective function (1987, 3). It is interesting to note that by Lewin’s definition, an operation is not necessarily a transformation (if the function maps set elements from S to S′ ). Fortunately, in most of this study the set S equals the twelve equal-tempered pitch classes, and functions on this set map elements of the set onto elements of the same set injectively. Thus Lewin’s distinctions are largely a moot point here: I will use the terms “function,” “mapping,” and “transformation” interchangeably throughout the text. As the SVT system involves bijective functions, the term “operation” can and will also be used in subsequent chapters to describe voice-leading transformations. Lewin’s definitions for semigroups and groups follow the standard mathematical definitions: A semigroup is an ordered pair ¢X,•² comprising a set X and an associative binary composition • on X (Lewin 1987, 5). A group is a semigroup with identity in which every element has an inverse (6). In this study for the most part I will be limiting the discussion to the group G of transpositions modulo 12. Finally, chapter 4 presents a number of graphical representations of pitches and relations between them (as in, for example, the Riemannian Tonnetz), of sets of pitches (chords) and transformations that relate them, and of SC types and parsimonious voice-leading paths between them. These representations have been called anything from models to maps, graphs, and networks. While in a general sense these terms are fairly interchangeable (and I will ask the reader’s forgiveness if I occasionally slip into this practice), each term carries its own subtle shade of meaning. The term “model” is perhaps best reserved for theoretical systems devised to explain aspects of pitch and pitch-class relations in compositional space, and will not be used here to describe transformational diagrams. Straus (2003) prefers to use the term “map” for his diagrams, but to me this term is too close to the mathematical term “mapping” for comfort. This may explain his choice to use the term “graph” (of voice-leading space) for the same type of diagram in 2005. However, “graph” itself is a loaded word, which after Lewin’s usage implies a node-and-arrow diagram without content assigned to the nodes.8 Lewin himself uses the terms “transformation network” or “network graph” for his node-and-arrow diagrams; the term network graph seems perfectly

10

Chapter One

reasonable to describe transformational “ball-and-stick” representations of voice leadings and SST-succession classes between m-tuples and paired set classes, respectively.

CHAPTER 2 SOME TRANSFORMATIONAL MODELS OF VOICE-LEADING SPACE While the study of voice leading in tonal music has a long and venerable historical pedigree (with Schenker as the most visible proponent), the study of voice leading in non-tonal music has only recently been an area of serious theoretical inquiry. Although a rudimentary model of voice leading existed as early as the 1940s in the writings of Joseph Schillinger, it was not really until the 1990s that scholars turned their attention to the possibilities of transformational theory applied to voice leading in atonal music. The term “transformational” is key here, describing a model based on concepts of David Lewin as set forth in GMIT and earlier works. Transformational theories are one of three types of models for atonal voice leading, as recognized by Joseph Straus (1997, 237), the other two being “prolongational” and “associational” models. Prolongational voice-leading theories owe much to Schenkerian analysis and are usually considered to be modifications to his theory to allow for the analysis of non-tonal music. Associational models attempt to explain the voice leading of specific musical lines as “projections” of vertical harmonies or set-class types that are prominent at the musical surface (241). As my approach is transformational, in this chapter I will give an overview of some of the transformational models of atonal voice leading beginning with Henry Klumpenhouwer’s 1991 dissertation and continuing to recent studies.

Fig. 2-1: Voice leading of a [0146]-type tetrachord, as a result of a permutation/transposition function (after Klumpenhouwer 1991).

12

Chapter Two

Inspired by Lewin’s GMIT, Klumpenhouwer’s dissertation, “A Generalized Model of Voice-Leading for Atonal Music” is perhaps the locus classicus of subsequent approaches to atonal voice leading in that it posits transposition and inversion operations on pc sets as the driving forces behind its transformational method. The model is basically a permutational one, where stacked pcs, ordered by register, “lead” to verticalities through permutation cycles affecting the ordered registers. For example, the permutation cycle (sbt) maps the pc in the “soprano” register position to that in the “bass” position in the subsequent verticality; the “bass” pc maps to the pc occupying the “tenor” position, and the “tenor” pc maps to the pc occupying the “soprano” position. Given a four-register texture, the “alto” pc maps to its own register position (not necessarily to the same pc). These register positions can also be (and are later) re-labeled 1, 2, 3, or 4, from highest to lowest. When a register position is associated with specific pcs, a registral permutation, paired with a transformation belonging to the Tn/TnI group of TTOs, acts upon the pcs and maps them to a new verticality of the same set class. The operation is expressed as an ordered pair ¢A,X² where A = a registral permutation and X = a TTO. For example, ¢(143),T9² acting on the collection {2,C}, {3,A}, {4,C}, {1,G} produces {2,A}, {1,G(}, {3,B(}, {4,E} (see Figure 2-1). Klumpenhouwer later fashions a quotient GIS which makes all transformations with the same permutational cycle congruent, regardless of TTO employed. The construction of the quotient GIS is the first of several steps which gradually lead the topic of inquiry away from voice leading per se, toward the more abstract and generalized topic of network isomorphisms. These later chapters of the dissertation, in fact, develop Klumpenhouwer’s bestknown contribution to transformational theory: not a permutational model of voice leading, but rather the Klumpenhouwer network (or K-net). Klumpenhouwer posits the K-net as a way of relating chords belonging to different SCs by comparing their internal structure, noting where there is a “skew” in moving from one set type to another. He also spends considerable time studying networks of networks, noting the “possibility of recursive structuring” between the internal structure of networks and network isomorphisms (see his chapter 6).1 So much time, in fact, is spent on K-nets that one starts to wonder whether the dissertation is accurately titled. Certainly the author begins by constructing a system for atonal voice leading, but the model becomes so generalized and remains at such a macro-level for so long that voice leading as a topic is largely abandoned by the fifth of nine chapters. In the end, his dissertation becomes more a celebration (or fetishization?) of network isographies than an exploration of voice leading in atonal music, and as such it should be regarded as an important contribution to theories of isomor-

Some Transformational Models of Voice-Leading Space

13

phisms in music. Its value as a theory of voice leading lies chiefly with its proposal of a transformational Tn- and TnI-based model for voice mapping between pc sets that became the prototype for several subsequent studies. These later studies inherit from the dissertation a number of issues that raise questions about the efficacy of the Tn- and TnI-based transformational model in convincingly portraying voice leading in a way that is commensurate with one’s logical and audiological intuitions about the music in question. One of the principal issues or problems attached to any Tn- and TnIbased analytical approach (not just limited to models of transformational voice leading) is that of relating objects of different class-types, in this case chords of different set-class types. Klumpenhouwer provides a roundabout way of relating chords belonging to different SCs through his K-nets, but the network isographies are stronger between certain pairs of chords than others, and in any case the voice leading between any two chords is never directly shown. In fact, in the early chapters of the dissertation he is compelled to skip certain chords in his analyses, chords that do not belong to the SC type of the particular verticality undergoing a voice-leading transformation. It should also be pointed out that the registral (vertical) arrangement of chord members is of paramount importance in Klumpenhouwer’s model. This is an absolute registral ordering and does not allow for voice crossing as frequently occurs in the more abstract concept of a musical “voice.” Once a registral permutation has occurred, a transformed “line” is assigned a new order number based on its new registral position; thus the concept of a quasiindependent voice cannot be maintained, since it would be destroyed with each mapping. Without taking this absolute ordering into account, Klumpenhouwer’s model does not work. Again, it will be seen that this issue continues to arise in later voice-leading studies. A recent paper by Michael Callahan (2006) reviews some previously noted parallels between Klumpenhouwer networks and George Perle’s cyclic sets and then investigates sum-and-difference motion along a network graph comprised of two separate K-net cycles. The first of these is the “strongisography class cycle” in which the K-net interval and one of the K-net sums increase or decrease in tandem by one while the remaining sum decreases or increases equally in opposition to them. Any three-pc K-net strongisography cycle will contain twelve steps. The second K-net cycle is the “semi-transpositional cycle” (after Perle) which maintains the same K-net interval and one of the sums while increasing or deceasing the remaining sum by one. Like the strong-isography cycle, the semi-transpositional cycle contains twelve steps. Callahan later constructs a 3D network graph of threemember K-nets that includes “axial-isography cycles,” which are used to relate K-nets which share no transposition- or inversion-related intervalclasses.

14

Chapter Two

Callahan’s contention is that one can reinterpret K-nets as pc sets and show the semitonal voice leading between the sets as a combination of “moves” along the network graph. His claim that “the treatment of these relationships in terms of semitonal motion allows us to avoid the potential objection that sums-of-sums and differences-of-sums may be abstract and too far removed from the musical surface” seems to fall short simply because K-nets themselves, upon which his network graphs are constructed, are already a significant theoretical abstraction. Additionally, Callahan does not express motion along any of the axes of his network graphs in terms of a mathematical transformation, instead choosing to show parsimonious relations in his graphs without identifying specific transformations—either musically intuitive (such as an L, P, or R operation) or purely abstract—leading from one K-net/pc set to another. Thus constituted, Callahan’s study perhaps does not precisely belong in a chapter devoted to transformational models of voice-leading space, but it does show a heightened sensitivity to independent voice-leading paths and does not rely upon in toto set-class transposition and inversion as Klumpenhouwer’s and many others’ approaches do (as summarized later in this chapter). It is also notable for its being the only study that I am aware of that combines the privileging of voice-leading parsimony (à la Richard Cohn and the neo-Riemannians) with Lewin’s research on K-nets. Callahan’s study is not without precedent, however. Edward Jurkowski’s dissertation, “A Theory of Harmonic Structure and Voice Leading in Atonal Music” (1998), also relates non-TTO-related sets through special K-nets which he calls “Interval-Difference Networks,” or I-DIFF Networks. Operations within these networks are on intervals and ics rather than on pitches and pcs; this gives Jurkowski the capability of discarding inversion operations entirely and using only transpositions. The I-DIFF Networks are composed of dyadic subsets of verticalities, and like Klumpenhouwer, rely upon reorderings (permutations) of registral positions in order to create the pitchclass counterpoint of transformational lines. Though an intriguing investigation of a new direction for the study of K-nets, Jurkowski’s method is somewhat hampered in the same way Klumpenhouwer’s and Callahan’s are: all rely heavily on a very abstract concept, the K-net, which—though elegant theoretically—is so far removed from the musical surface and from an actual listener’s experience that there is little, if any, correlation between the intrachordal K-net transformations and the perceived voice leading. Even the models which propose set transpositions/inversions as the driving force behind the mappings of the individual voices (see the discussion of Straus, O’Donnell, and Lewin, below) seem to conform more closely to one’s listening experience than do those which first require a K-net transformation before proceeding to a TTO on a set.

Some Transformational Models of Voice-Leading Space

15

Drawing on Klumpenhouwer’s transformational approach and on earlier, related work by John Roeder (1984 and 1994, discussed later in this chapter), Joseph Straus (1997) also uses a Tn- and TnI-based model to describe voice leading between pc sets. In Straus’s system, voice-leading connections are still limited to pc sets of the same SC type. Like Klumpenhouwer’s model, the destination tones of individual voices comprising a chord are determined by transposition or inversion operations on the set. Straus, however, abandons the permutational aspects of the earlier study, instead tracing individual voice paths by connective lines in diagrams placed under the score (see for example Figure 2-2). In addition to showing the chord-tochord voice-leading connections effected by set transposition and inversion operations, Straus shows the overall Tn or TnI level traced by summing all the chord-to-chord transformations, from the first to the last chord of a musical passage. By examining the total sum of voice-leading operations on small-, medium-, and large-scale levels, Straus is able to show transformational patterns that otherwise would not be readily apparent. These patterns suggest intriguing but inexact parallels to foreground, middleground, and background structures in Schenkerian analyses. However, Straus’s most important contribution lies in his proposal of (as yet unformalized) concepts of “near-transposition” and “near-inversion.”2 Despite this tantalizing suggestion, unfortunately the model still suffers from the frustrating by-product of pc-set transposition- and inversion-based transformational systems: the resultant “voice leading” usually runs counter to what is musically perceived/intuited. Reconciling audible musical lines with the more abstract concept of voices is not a problem unique to Straus’s study; on the contrary, it has been an ongoing challenge for most theorists attempting to construct logical models of voice leading. I will have more to say on this later in this chapter. An important contribution to the theory of transformational voice leading is found in Shaugn O’Donnell’s dissertation, “Transformational Voice Leading in Atonal Music” (1997). O’Donnell’s principal contribution lies not so much in his voice-leading model or system, which is cast in the Klumpenhouwer and Straus set transposition- and inversion-driven mold, but rather in his insightful criticism of the very limitations of this type of model. Among the most severe deficiencies he cites are those mentioned earlier. For one thing, voice leading may occur only between chords of the same SC type, and these are often widely separated in the actual music: The most significant limitation of a transformational voice-leading model based solely on transposition and inversion is the self-contained nature of the set-class equivalence classes. . . . The transformational interpretation of the traditional operations, that is, transposition and inversion as dynamic processes rather than static relationships, is severely weakened in such

16

Chapter Two

circumstances. . . . It is absolutely essential that a successful voice-leading model be capable of describing complete passages, because there must be some aural continuity in these musical gestures or we would not reify them with the term “passages.” (16–7)

O’Donnell offers more perceptive criticism of the Tn- and TnI-based voice-leading model than any of his predecessors, and promises to expand “the transformational machinery” in his dissertation to compensate for the system’s deficiencies. He is only somewhat successful in achieving this promise, however. His solutions for the most part seem to be relatively minor fixes, subtle “tweaks” to the system that patch up only the most egregious deficiencies. O’Donnell recognizes—perhaps more than any other author on this subject—the rôle that our auditory faculties and mental pre-conditionings play in shaping our perception of voice leading. “I want voices that most richly interact with my experience of the musical surface. In other words, I prefer voices that are reinforced by one or more surface lines, such as registral or instrumental lines” (64). Much of the dissertation focuses on the attempt to choose, from multiple possible analyses of a piece’s voice leading, a reading that most closely approximates a hypothetical listener’s experience. This is not always successful, as a purely perceptual interpretation of the voice leading frequently comes in conflict with more abstract constructs, as it does, for example, with voice-leading paths determined by Tn and TnI operations on pc sets or with Klumpenhouwer network isography.3 In trying to “have it both ways”—tracing the voice leading on a perceptual level while at the same time justifying voice leading as a result of strict transformational processes, O’Donnell runs into a problematic result familiar to many: a jarring disjunction between a perfectly logical theory and a musical surface that refuses to pigeonhole itself by readily conforming to that theory. As a way of coping with the problem of transposition and inversion between members of different set classes, O’Donnell endorses the idea of the “singleton transformation” (26–9). In this type of transformation, all chordal elements but one map by the same operation. The remaining element (voice), in mapping by some other operation, violates the integrity of the composite mapping, keeping it from being an exact transposition or inversion. There are several precedents for this type of transformation. Forte’s “unary transformation” (Forte 1988) maps pc sets of different set types by the transposition of only one of its members. In “Transformational Techniques in Atonal and Other Music Theories” (1982–3) David Lewin proposes an “if-only transformation,” which is basically the reverse of the unary transformation in that all members of a pc set but one participate in a TTO. (The remaining singleton maps to itself.) Straus’s “near-transposition” and “near-inversion” (1997, 268–70) are similarly conceived. This “new” type of

Some Transformational Models of Voice-Leading Space

17

transformation allows for the modeling of a greater number of voice-leading connections between proximate pc sets than would simple, straightforward set transposition and inversion. O’Donnell then ups the ante by allowing two distinct Tn or TnI (or even wedge—“W”) operations to occur simultaneously. This “split transformation”4 likewise adds another option for explaining voice leading between chords of different SCs, and serves fairly well in analyses of passages from Scriabin (98ff) and Babbitt (154–5). However, even this broader palette of transformations produces analyses faintly tinged with rose-colored methodological hues, as the voice-leading paths traced by the mappings are often still musically non-intuitive or inaudible, especially when resulting from an inversion operation. Partly to defend his split transformations, O’Donnell writes, “Recent transformational models tend to parse pc sets into subsets to explain transformations among sets belonging to different SCs” (48). I believe that O’Donnell is on to something important here. An even greater number of simultaneous transformations on chords will be necessary if one is to have the freedom to connect a series of chords from any and all of the different SCs, let alone to chords of differing cardinalities. As the number of concurrent transformations acting on a pc set increases to encompass more than half the cardinality of the set, it would seem to be wise to re-evaluate what exactly these specialized transformations are acting upon, and whether it would be propitious to begin thinking less about transformations on sets or even subsets, and more about transformations on individual pitch classes. In summary, O’Donnell fulfils his promise to expand “the transformational machinery” originally conceived by Klumpenhouwer and Straus, but even with this expanded toolbox he achieves only mixed results in terms of offering insight into the voice leading of any given musical passage. Ultimately his most important contribution to the field of atonal voice leading studies consists of some important criticisms of Tn/TnI pc-set transformational models in general. Lewin’s important JMT article, “Some Ideas about Voice-Leading Between Pcsets” (1998), also touches upon transformations of the type discussed earlier, in which all but one member of a pc set participate in a straight transposition. If the nonparticipating set member’s transposition differs as little as possible from the transposition of the other set members (in ETS 12 systems by a semitone) the voice leading is called maximally uniform.5 Lewin assigns a “pseudo transposition number” to the transposition which is equal to Tn where n = the transposition number of the majority of the set members, and also designates an “offset number” which equals the deviation from true transposition by the total (absolute) number of nonconforming semitones (33). In his article, Lewin also raises an important issue touching on the

18

Chapter Two

“retrogradability” of voice-leading functions, namely that only bijective functions have an inverse function. To allow for “inverses” (retrogradable voice leading) for non-bijective voice-leading functions between pc sets, Lewin dispenses with the label “function” altogether in favor of the more general term relation. A relation R between two sets X and Y is a collection of ordered pairs ¢x,y² where the x of each ¢x,y² is a member of X and the y of ¢x,y² is a member of Y. R is a subset of the Cartesian product X + Y. Functions are simply relations R in which every member of X appears once on the left, within some member-pair of R, and only once (Lewin 1998, 64).6 Lewin’s choice to “settle for” the mathematically weaker relations to describe voice leading transformations in the last section of his article, instead of the functions that he had so carefully prepared in the previous sections of his paper, is very curious.7 Even the non-bijective voice leadings can still be described in terms of functions; they simply do not have unique inverses. Although functions can be injective, surjective, or both, they can also be neither. As an example, consider Lewin’s example of voice leading between the sets X = {D,F,A} and Y = {E,G,B}. The closest voice leading between pitch classes would result from a function V where V(D) = E, V(F) = E, and V(A) = G. Although the voice-leading transformation V is neither surjective (the element B of Y did not occur as a result of the function V) nor injective (elements D and F of X both mapped to the same element of Y) it still satisfies the definition of a function as described at the end of the previous paragraph. If all functions were unidirectional this would not be a problem, but what if we wanted to “undo” this voice leading by moving set element E of Y “back” to both D and F of set X? A move like this would defy the criteria for a function, as the image of the argument E under this inverse transformation V–1 would not be uniquely determined. The simpler term “relation” must be used instead to describe such a transformation. It is for this reason—to mathematically justify retrogradable voice leading—that Lewin discards voice-leading functions in the last section of his paper, even though he had very carefully constructed a number of specific types of voice-leading functions in the previous section.8 I assume—although it is nowhere explicitly stated—that Lewin is suggesting that functions should be used to model voice leading except in cases that require retrogradable mappings, or to put it in musical terms, in cases of voice doubling or of increasing/decreasing the number of voices in a homophonic texture. In these cases, the best one can do, mathematically, is to revert to the simpler relations. Why Lewin does not simply use relations to describe all voice-leading moves, ditching the more narrowly restrictive functions to allow for more voice-leading flexibility, is open to conjecture. There is another alternative, wherein Lewin could have used bijective functions for all voice-leading moves, dispensing with relations entirely. This

Some Transformational Models of Voice-Leading Space

19

scenario involves the use of (ordered) n-tuples as the objects operated upon, and not unordered sets. This approach will be described in the discussion of Roeder (1994) at the end of this chapter, and in greater detail in chapter 3. There is one other topic discussed by Lewin in his “Some Ideas” article that had gone largely unexamined in the previous voice-leading studies. To introduce this topic, we must first specify exactly what a voice is. John Roeder (1994) defines a voice as a succession of pitches in the same registral order position between two simultaneities. Like Klumpenhouwer before him, Roeder’s approach to voice leading9 must assign considerable weight to pitch register and order position within a chord for his transformational approach to work. Roeder’s term “voice” receives a much stricter definition—necessarily to avoid part crossings in his system—than that which is usually associated with the term. In some ways his definition of the word “voice” more closely resembles the confusingly similar term “line.” The unfortunate conflation of the these two terms has a long history in music theory. Recently Joseph Straus has attempted to clear up some of the confusion by providing definitions for the two concepts (1997, 241–2; 2003, 306–11). Straus’s definition of voice is not nearly the same as Roeder’s: a voice arises from the path traced by a set member as it maps onto different pitch classes through transposition and inversion transformations. Straus’s voice resembles Roeder’s in that it is “operational and systematic” but it differs in that it is not limited to a specific registral position when moving from verticality to verticality. A musical line, on the other hand, is a contextual pitch-class path not based on transformational mappings but is based rather on register and instrumentation and also—to a large degree—on musical perception and intuition. This study adopts Straus’s definitions for and distinctions between the terms “voice” and “line.”10 A voice and a line may be one and the same in certain passages of music. To demonstrate, Straus examines measures 2–6 of the fifth of Webern’s Five Movements for String Quartet, reproduced here as Figure 2-2. With the exception of the ’cello part, all voices move by straight transposition: by T8 between the first two chords, by T7 between the third and fourth chords, and by T2 between chords four and five (the mapping of individual pitch classes is traced by the connective lines in the diagram below the excerpt). Other aspects of the music reinforce the voice leading suggested by the transformational analysis. It is immediately apparent that the registral lines exactly reproduce the transposition-driven voice leading (i.e., the transformations produce no registral crossing) and—upon consulting the score—each instrumental part also exactly follows the path of the transformational voice leading. Here, then, we may say the voices and the lines coincide. Figure 2-3 reproduces another example from Straus (2003) which demonstrates voice leading in the second of Webern’s Five Movements. As in

20

Chapter Two

1

Vn. 1, 2 Va.

Vc.

Ł ¹ −−ŁŁŁ Ł

Š 42 Ý 2 Ł Ł 4

 Ł ²Ł

T4

 Ł Ł

3

4

Ł ¹ −ŁŁ Ł −Ł

 Ł ²Ł

¹ −ŁŁ Ł −Ł Ł  Ł Ł

5

6

Ł ¹ ²²−ŁŁ Ł Ł ¹ −ŁŁ Ł ² −Ł Ł Ł

 ¦Ł

¦Ł

 Ł ²Ł

¹  Ł ²Łý

T9 1

D G E B E

(a)

2

−Ł ¹ −² ŁŁŁ Ł

8 8 8 8 8

2

3

B E B F C

C E D A E

4 7

5 2

G B A E B

7 7 7 7

2

2 2 2

T7

T8

A C B F C

T2 T9

Fig. 2-2: Transposition-driven voice leading where voice and line coincide in Webern, Five Movements for String Quartet, op. 5, no. 5, mm. 2–6 (from Straus 2003)

Va.

Vn. 2 Vc.

(b)

Š 00 Ł

T4

Łý Ý 0 ¹ ¦ ŁŁ ýý 0 −Ł ý

Ł

q Ł Ł ²Ł 1

¹  ¦Ł Ł ² Ł −Ł

q ŁŁ Ł Ł ¹ ¹ ¦ ŁŁ ½ Ł −Ł  w

¹ −¦ ŁŁ ýý Łý

 Ł ¹ ²Ł Ł Ł 2

½ A E G

 Ł ¦Ł −¦ ŁŁ Ł

Ł ²Ł

Ł 3

¼

4

¹ ¦ ŁŁ ýý ²Łý G B C

4 4

T4

Fig. 2-3: Transposition-driven voice leading where voices do not coincide with lines in Webern, Five Movements for String Quartet, op. 5, no. 2, mm. 1–3 (from Straus 2003)

Figure 2-2, a single transposition operation—in this case, T4—maps all pitch classes in the “accompanimental” sonority from chord two to chord three. The transposition operation drives the voice leading here, but unlike Figure 2-2, the lines do not correspond with the voices. The twin factors of instrumentation and register lead one to hear this passage differently than the transformational reading would suggest. To be more explicit, one hears the pitch A3 dropping slightly in register to G3 and played by the same instrument—the second violin—rather than progressing to the low C2 in the

Some Transformational Models of Voice-Leading Space

21

‘cello. This combination of registral and instrumental continuity produces a strong impression of a musical line, an impression that is at odds with the voice leading suggested by the transformational operation T4. A similar sense of line is produced by the register and instrumentation of the doublestopped ’cello pitches.

Fig. 2-4a: Wagner, Prelude to Tristan und Isolde, mm. 1–3, piano reduction

To give another example of some of the problems inherent in interpreting voice-leading purely as the result of twelve-tone operations on chords belonging to one particular set class, let us consider the opening bars of a very well-known piece (Figure 2-4a). If we reduce the passage to the two principal verticalities and show the voice leading that would result if we applied a T7I mapping to the first sonority, we would obtain Figure 2-4b.

Fig. 2-4b: Voice leading obtained via a T7I transformation on a representative of SC [0258]

Some would argue—and not entirely without reason—that music so

22

Chapter Two

closely associated with the tonal tradition is too closely bound to that tradition to be profitably examined using non-tonal analytical techniques. But Wagner’s music—and Tristan in particular—uses “tonal” materials in distinctly unorthodox, “non-functional” ways that share some similarities with early atonal music. Of course transposition and inversion operations do play a role in tonal music, especially in the realm of melody, but also increasingly in harmonic constructs in the late nineteenth and early twentieth centuries. Thus it should be quite appropriate to analyze the Tristan Prelude using both tonal and non-tonal approaches.11 Regardless of whether Wagner should be modeled using TTOs, one of the principal problems associated with applying transposition- and inversion-driven voice-leading transformations to pc sets is readily apparent in this figure: the supposed “voice leadings” do not precisely correlate with the audible musical lines, as can be seen in the “bass” and “alto” voices. Chord 1’s D4, played by the English horn, is perceived by the listener as leading directly to the D94 in chord 2, with no change in timbre and minimal motion in pitch space (by the directed interval –1).12 A T7I voice-leading interpretation, on the other hand, shows the D4 mapping by the much larger directed interval –11 to the E3 with a change in timbre from English horn to bassoon occurring over the barline. A similar pair of readings can be obtained by tracing either the inversion-driven voice leading or the audible line, each of which commences with the pitch F3 in the bassoon. The choice of this excerpt was deliberate and perhaps a bit extreme; my intent was to trace transformational voice-leading in a very familiar “tonal” piece to demonstrate what is not always evident in non-tonal and less familiar music where the musical lines are sometimes more difficult to perceive: the disjunction between the pitch-class counterpoint of transformationally driven voices and the perceived lines. This same disjunction is present whether the set-class transformational apparatus is applied to Beethoven, Schoenberg, Stravinsky, or Gubaidulina. It just may not be as readily apparent in more recent music as it is in the Wagner example. Returning from this digression to Lewin’s JMT article (1998), it should be noted that it is the importance of hearing musical lines that leads Lewin to attempt to formally describe aspects of lines that make them cohere, that allow us to recognize them as such and to distinguish them from other, simultaneously presented lines. Although he does not neglect discussing the importance of timbre in helping to define lines, Lewin turns much of his attention to describing registral proximity as a primary component of an audible musical line. In particular, the concept of “maximally close voice leading” is posited as a desirable quality in a line, where a pitch in a verticality leads to the closest possible pitch of the next harmony (1998, 17–8). When this literally happens in a passage of music, the motion is said to manifest the maximally close voice leading. When it doesn’t literally happen, but

Some Transformational Models of Voice-Leading Space

23

the motion is displaced by an octave or octave multiple (i.e., maximally close voice leading between pitch classes rather than between pitches) the voice leading is said to be projected. We will frequently return to this distinction between voice and line, as well as to the distinction between “manifest” and “projected” voice leading in the course of this study, particularly in discussions of parsimonious voice leading.

Fig. 2-5: Musical representation of voice-leading rule (((0 0) 2 1) ((3 3) 2 2) ((5 1) –2 3) ((8 5) –1 4)) (from Cope 2002)

We turn our attention now to some publications which will be seen to have considerable bearing on the SVT model proposed in chapter 3. A study that succeeds in breaking away from the transposition-and-inversion transformational voice-leading paradigm is David Cope’s “Computer Analysis and Composition Using Atonal Voice-Leading Techniques” (2002). Although Cope’s article largely focuses on the creation of computer music using voice-leading “rules” (algorithms), the rules themselves are of interest as a detailed way of describing voice-leading motion independent of setclass transformations. Figure 2-5 is taken from Cope’s Example 4 and shows in music notation how his program ALICE (Algorithmically Integrated Composing Environment) would interpret the rule (((0 0) 2 1) ((3 3) 2 2) ((5 1) –2 3) ((8 5) –1 4)). In Cope’s algorithm, each 4-member rule part (e.g., ((0 0) 2 1)) represents one channel (or “part,” here congruent with Straus’s “line”). The channel number is indicated by the last number in each rule part. The head sub-list (e.g., (0 0) in the first rule part) represents the consecutive pcs in that particular channel, labeled in such a way that the normal form for

24

Chapter Two

the set type of each chord starts on 0 (i.e., so that it corresponds to the prime form of the set-class type). The number immediately following the sub-list represents the directed voice-leading interval (in half steps) between the two chords for that particular voice. One of the most attractive features of Cope’s voice-leading rules is its particularity in describing every parameter of the voice-leading space, from channel (register/part) number to directed voice-leading interval, and even to listing the SC types of the origin and target chords. Ironically, it is this very particularity that is also the algorithms’ greatest drawback. There is simply too much information presented in a computer-friendly but human-no-sofriendly format. A line of code works well in ALICE, but it takes some time and practice for a flesh-and-blood person to extract the information that a member of set class [0358] moves to a member of set class [0135] from the rule (((0 0) 2 1) ((3 3) 2 2) ((5 1) –2 3) ((8 5) –1 4)). While Cope’s algorithm makes it possible for a human to divine the precise voice-leading intervals between chord members, the rule says nothing about the precise set members created by the directed motion. In this instance, the directed motions 2, 2, –2, and –1 are applied to members of the set {0,3,5,8} to produce the set {2,5,3,7}—rearranged in normal form to produce {2,3,5,7}—but nowhere is this stated explicitly.13 Additionally, the nomenclature of Cope’s rules differs considerably from music-theoretic “norms” in that it lacks familiar symbols such as Tn, and—perhaps more significantly—in its avoidance of standard mathematical orthography where a function maps a domain to a range.14 Cope’s compositional algorithms represent one attempt at describing voice leading between chords belonging to different set-class types, an activity that—as we have seen previously—can be quite problematic when approached via the transformational machinery of strict Tn and TnI operations on pitch-class sets. Recently, Joseph Straus has proposed a different “fix” to this problem by expanding upon the concepts of “near-transposition” and “near-inversion” hinted at in his 1997 article. Straus (2003) measures uniformity and balance in operations between chords of different set-class types and even between chords of differing cardinalities (if the difference is ± 1). Uniformity is the extent to which pitch-class voices move by the same intervallic distance in a “fuzzy transposition,” where the transposition interval is not uniform between all voices. Balance is a related concept, the extent to which pitch-class voices move by the same index number in a “fuzzy inversion,” where the index number is not uniform for all voices undergoing an inversion. The extent of difference is called offset by Straus, the amount by which all the voices would have to be adjusted to bring them all into conformity with each other, measured in semitones from the convergence point, the Tn or TnI level from which there is the least possible offset (2003, 311–18).

Some Transformational Models of Voice-Leading Space

25

To demonstrate uniformity and balance, Straus examines all the possible voice-leading connections between two sets, {F,F,B} and {G,B(,D}. The following examples (Figures 2-6 and 2-7) show the 2 +3! = 12 distinct mappings between these three-member sets involving transposition and inversion.

ð Š ²ð

ð −ð

ð ²ð

ð −ð

ð ²ð

ð −ð

ð ²ð

ð −ð

ð ²ð

ð −ð

ð ²ð

ð −ð

Ý ð

ð

ð

ð

ð

ð

ð

ð

ð

ð

ð

ð

G B D

F 5 F 8 B8

G B D

F 9 F 1 B 11

G B D

F 5 F 1 B 3

G B D

F 9 F4 B8

G B D

F 2 F8 B 11

G B D

F 2 F 4 B 3 *T3 (2)

*T8 (3)

*T11 (4)

*T3 (4)

*T8 (5)

*T11 (6)

Fig. 2-6: Voice-leading uniformity in all possible “fuzzy” transpositions between {F,F,B} and {G,B(,D} (from Straus 2003)

Figure 2-6 shows the degree of uniformity of voice leading as a result of “fuzzy” transpositions between the two sets (the asterisk before the Tn symbol differentiates this special type of transposition from a “straight” transposition). In each voice-leading diagram, the solid line represents the base convergence point and the dashed lines represent the offset voices. Each number associated with a line measures the number of semitones the initial pitch class is transposed in pitch-class space. The number in parentheses below each diagram is a sum of the voice-leading offset. Thus the first voiceleading diagram in Figure 2-6 represents the fuzzy transposition with the greatest degree of uniformity. Figure 2-7 depicts an analogous situation involving “fuzzy inversion” between the trichords. Here the degree of balance—the deviation of the index number from the convergence point—is measured. The first instance, in which two of the voices map by inversional index number 1, and the third by inversional index number 3, involves a deviation from the baseline convergence point I1 of only two index increments (signified by the parenthetical “2” under the voice-leading diagram) and is thus the most “balanced” of the six inversional mappings.15 However, it should also be pointed out here that one could call into question the idea that a greater difference in the inversional index correlates with a larger perceived “distance” in the voice mapping.

26

Chapter Two

ð Š ²ð

ð −ð

ð ²ð

ð −ð

ð ²ð

ð −ð

ð ²ð

ð −ð

ð ²ð

ð −ð

ð ²ð

ð −ð

Ý ð

ð

ð

ð

ð

ð

ð

ð

ð

ð

ð

ð

G B D

F 7 F4 B6

G B D

F 0 F8 B 9

G B D

F 0 F4 B 1

G B D

F 3 F8 B6

G B D

F 7 F1 B 9

G B D

F 3 F 1 B1 *I1 (2)

*I6 (3)

*I9 (4)

*I1 (4)

*I6 (5)

*I9 (6)

Fig. 2-7: Voice-leading balance in all possible “fuzzy” inversions between {F,F,B} and {G,B(,D} (from Straus 2003)

Fig. 2-8: Smoothness in all possible voice-leadings between {F,F,B} and {G,B(,D} (from Straus 2003)

Another way in which voice leading can be analyzed is in term of its degree of smoothness, or parsimony. Many recent studies, particularly in neo-Riemannian theory, consider voice-leading smoothness to be a desirable quality, intentionally cultivated and privileged in certain musical styles; but—as Straus points out—smoothness often appears to be at odds with uniformity and balance. Figure 2-8 revisits the six voice-leading possibilities between the {F,F,B} and {G,B(,D} chords and measures the total displacement in semitones between the chord members connected by lines in the diagrams.16 It should be noted here that the voice-leading displacement is measured by an unordered pitch-class interval between dyads. This shows the efficiency of each progression, with the smoothest voice-leading configuration appearing on the left of the example and the least smooth on the right. When one locates the left-most, smoothest voice-leading mapping of Figure 2-8 in Figures 2-6 and 2-7, it becomes apparent that voice-leading parsimony can be incompatible with the qualities of uniformity and balance that are so important to Straus’s set-class-transformational model of voice leading. In Figure 2-6, the smoothest voice-leading path is only the third-

Some Transformational Models of Voice-Leading Space

27

most uniform transpositional path (with an offset of 4), and in Figure 2-7 it happens to be the least balanced inversional path, with an offset of 6. Straus rightly recognizes the historical importance of smoothness as a desired voice-leading quality and devotes approximately the second half of his article to it. He introduces the concept of optimal offset, which is the closest possible distance between any two set classes of the same cardinality, measured in terms of total displacement (2003, 335–40). It is possible to generalize optimal offset distances between different set-class types, and Straus produces a series of charts and spatial maps (network graphs) which show these possibilities for all trichord classes and tetrachord classes. The spatial maps in particular are an efficient medium for showing the smoothest possible voice-leading paths between set-class types of the same cardinality, but Straus does not go so far as to create charts and network graphs showing optimal offset between set-class types of differing cardinalities. Instead, “one needs to imagine the 220 set classes hovering in a multi-dimensional space and linked together by the bonds of parsimonious voice leading” (2003, 340). This is easier said than done, of course, whether in one’s imagination or realized as a 3D lattice in 2D space (on paper, on computer monitor, etc.). Creating such a lattice would be a very labor-intensive task and would result in a hopelessly convoluted nest once all 220 set classes are connected by optimal offset links. It is only properly done, strictly speaking, in five-dimensional space, to account for all set classes of cardinality 6 or less (with the higher-order set classes of cardinality n appearing in an isomorphic 5D space as the set-class complements to those set classes of cardinality (12 – n)). If we limit our lattice to set classes belonging to the same cardinality and also include set classes that differ in cardinality by plus or minus one (but not both!) our task is somewhat simpler. Such a spatial map would be an improvement over a map showing SCs of only one cardinality, as a great deal of—if not most—music regularly adds and drops voices as it progresses in time. In order to keep the model simple, and to ground it in common practice found in a wide range of musical styles, in the next chapter I will present a 3D-cum-2D spatial map of all optimal offset connections between all trichordal SC types and all tetrachordal SC types. Formalizing functions from sets of lower cardinality to those of greater cardinality can be problematic. Consider the following (standard) definitions: Let f : A → Β. f is injective or is an injection if whenever a1 ≠ a2, then f(a1) ≠ f(a2). f is surjective or is a surjection if for all b ∈ B there is some a ∈ A such that f(a) = b, i.e., the image of f is all of B. [. . .] f is bijective or is a bijection if it is both injective and surjective. (Dummit and

28

Chapter Two

Foote 2005, 2)

If we map a set A of cardinality n to a set B of cardinality n – 1, and if the image of B is equal to the codomain, we have a surjective mapping. This happens frequently in music, as, for instance, when all the pitches of a tetrachord voice-lead to all the pitches of a trichord. However, the reverse situation—where all the pitches of a trichord lead to all the pitches of a tetrachord—cannot be justified mathematically. There is no such thing as a “one-to-many” function. We could envision an injective function from a trichord A to a tetrachord B, but only three of the four elements of the codomain would be the image of A under f. It would seem to be important to find a way to mathematically formalize transformations that involve sets of differing cardinalities—particularly the problematic mappings from a set A of cardinality n to a set B of cardinality n + 1—and at the same time give emphasis to those that result in parsimonious voice leadings. The algebraic apparatus that elevates these intuitive (and frequently audible) voice-leading constructs to the level of formal transformations will be developed in chapter 3. Straus’s formulation of the concepts of “near”-transposition and -inversion constitutes an important step forward in theories of transformational voice leading. Following the precedents of O’Donnell and Lewin, Straus proposes a new way of tracing the voice leading between chords of different SC types, and does so in a way that is more sensitive to smoothness, a quality that is frequently at odds with “traditional” chord transposition- and inversion-based pitch-class counterpoint. Additionally, in both Cope’s and Straus’s recent studies, voice-leading transformations do not rely upon registral ordering of voices as Klumpenhouwer’s and Roeder’s transformations are compelled to. Even with these improvements, there still remains a problematic issue that Straus’s transformational apparatus is not completely successful in addressing, namely, its inability to describe voice-leading specificity. Whether the transformation is a fuzzy transposition or a fuzzy inversion, and even if a transformation is chosen so as to optimize smoothness, Straus’s analytical labels (e.g., *T3 (2) and *I1 (2)) tell us the transposition or inversion index that most of the voices map by17 and the amount of total displacement, but they do not specify individual voice leadings (mappings of individual lines). In examples in which all but one of the voices map by an operation different from the convergence point of the label (e.g., a *T7 (6) operation in which individual voices map by T5, T7, and T11) there remains very little aural sense of a “warped” transposition relation between the two pc sets. Even from a purely analytical standpoint, one cannot trace voiceleading paths between two sets merely by reading Straus’s analytical labels

Some Transformational Models of Voice-Leading Space

29

without first performing a series of mental gymnastics. As previously discussed, as individual voices increasingly trend toward motion that reinforces their individuality (i.e., via similar, oblique, or contrary motion), the parallelmotion Tn and TnI set-class operations increasingly appear to be a less-thaneffective analytical tool for describing the voice leading of the separate contrapuntal lines. The concept of near-transposition and near-inversion offers a degree of flexibility to such operations but ultimately seems to be an only partially successful modification of a system that is not particularly well suited to describing and quantifying the motion of individual voices. It is my aim here to introduce a system in which individual set members (pcs)—and not the pc sets themselves—are privileged, undergoing transformations which produce differentiated, dynamic voice-leading paths. Many composers view voice-leading smoothness as a desirable quality, and the proposed system is centered around (but not strictly limited to) parsimonious transformations. This system is described and formalized in the following chapters. For a preview of the single-voice-transformation (SVT) system, let us conclude by examining a 1994 study by John Roeder. The article, “Voice Leading as Transformation,” is a refinement of ideas from Roeder’s dissertation, and proposes a transformational approach to voice-leading that shares some similarities with Klumpenhouwer’s model, but dispenses with its permutational aspects. Individual pcs are gathered as an ordered n-tuple in angle brackets, arranged in register from low to high. Roeder calls this an “element series.” We have already seen that Roeder defines a “voice” as a succession of pitches in the same registral order position (41). This definition of the term voice does not allow for voice crossing. “Voice leading” is defined as another set of ordered intervals which represents the pitch-class successions between corresponding elements in two element series (i.e., corresponding to the transposition level between each element pi of the element series P = ¢p1,p2,...,pn² and each element qi of the element series Q = ¢q1,q2,...,qn²). This list of ordered transpositions is called an “element-series interval,” and is arranged in angle brackets, i.e., ¢3,2,11,2,3². More formally, it is described as int(P,Q) = V, and V = ¢v1,v2,...,vn². Roeder defines transposition operations on element series as the group action of V on P. His Theorem 2 states: Given an element series P = ¢p1,p2,...,pn², and an element-series interval V = ¢v1,v2,...,vn². Then an element series that is TV(P) is Q = ¢q1,q2,...,qn² in which each element is the transposition of the corresponding element of P by the corresponding entry of V; symbolically, qi = Tvi(pi).

A partial proof shows:

30

Chapter Two

int(P,Q)= int(¢p1,p2,...,pn²,¢Tv1(p1),Tv2(p2),...,Tvn(pn)²) = ¢int(p1,Tv1(p1)),int(p2,Tv2(p2)),...,int(pn,Tvn(pn))² = ¢v1,v2,...,vn² (Roeder 1994, 46–7, italics added and spacing adjusted to conform to conventions used in this study).

Inversion operations on element series are formalized similarly. Here then, Roeder has developed a way in which to account for separate voiceleading paths between corresponding members of two different chords which do not have to belong to the same set-class type. This approach marks a radical departure from Klumpenhouwer’s, Straus’s, O’Donnell’s, and Lewin’s studies, which rely upon set transposition and inversion to determine the voice-leading paths. Roeder’s system emphasizes the independence of the voices and their potential to create new harmonies unrelated to previous harmonies through their composite motion, as opposed to the monolithic transposition or inversion levels to which the entire chord is subject (with possibly one non-conforming voice in the case of Lewin’s “if-only transposition/inversion” or Straus’s “near transposition/inversion”) in “traditional” set-transformational approaches to voice leading. One of the most attractive by-products of Roeder’s system is the way in which it clearly shows that two separate progressions involving chords belonging to different SC types can be and frequently are composed of identical element-series interval transformations. My SVT system, detailed in chapter 3, owes a great deal to Roeder’s element-series transformational model. There are several differences between the two approaches, the principal one being that the SVT system develops SVT-related chord-succession classes, which generalize pairs of SVT-related chords into equivalence classes—not necessarily involving chords from the same set class—to show voice-leading relationships that are not immediately apparent to the analyst. SVT-related chord-succession classes correlate directly to the geometric regularity of certain network transformation graphs of SC types, as will be seen in chapter 4. While Roeder’s system relies on registral ordering without voice crossing in order to function, mine imposes no such restrictions. There is also no accounting for “splits” or “fuses”—operations that increase or decrease the cardinality of a chord—in Roeder’s theory, as there is in the SVT system. Unlike Roeder’s model, the SVT system does not include inversion operations. The SVT system also focuses largely (though not entirely) on parsimonious voice-leading transformations, or SSTs, and thus has close ties to much of the recent research that has been done in the area of neo-Riemannian theory. To conclude this section, it seems apropos to quote David Clampitt’s keen observation on the suitability or unsuitability of a transformational system to adequately model what it claims to explicate—in this case, voice-

Some Transformational Models of Voice-Leading Space

31

leading transformations between representatives of different set classes: “The point of invoking any GIS is that its consistency serves as a warrant that something coherent can be heard and/or said in the intervalic or transformational terms that the GIS presumes, assuming that the GIS is wellgrounded in musical reality” (1998, 327). One needs look only as far as the hypothetical T7I transformation of a representative of SC [0258] mapping the Tristan chord to the dominant-seventh chord in the Tristan Prelude (Figure 2-4b) to recognize the cogency of this statement. On an abstract theoretical level the two chords are indeed related by inversion, but does one actually hear such a relation when listening to the Prelude? And does a transformational system of Tn/TnI operations on pc sets produce voiceleading paths that adequately reproduce perceivable lines grounded in the listener’s musical “reality”?

CHAPTER 3 THE SINGLE-VOICE-TRANSFORMATION MODEL Qualities Privileged in Voice-Leading Models Before proceeding with the construction of a formal model for transformational voice leading, we should first describe some of the qualities of the voices that we hope to model. Let us return to Figure 2-4, which provides a good example of what these qualities are we are looking for. From repeated listenings to these three measures, we are likely to hear the voices “progressing” as the stem directions in Figure 2-4a imply: the initial melodic line in the ’cello turns out to be an inner voice, doubled by English horn on the pitch D4 and then slipping down a semitone to D94 in measure 3. The treble line (oboe) rises chromatically from G4 to B4. The bass line descends chromatically from F3 to E3 in the second bassoon (doubled by clarinet on the first chord), and the remaining inner voice, B3, leaps down a minor third to G3 (once again bassoon, doubled by clarinet on the first chord). This explanation of the voice-leading in this passage is the “standard” one that most musicians would recognize. Here our “voices” are identical to the “lines” identified in the previous paragraph. What exactly are the factors that so strongly reinforce this interpretation? First, this particular reading is backed by an audio/visual correlation between the music that is heard and the note-by-note lines read in the score. On a more general level, what exactly are the factors that allow us to perceive (whether by reading the score or by listening to a performance of the score) four completely differentiated, individually coherent musical lines? It would seem that each line is a perceptual construct given a distinct identity by a combination of timbre and pitch register. We may add a third factor to this explanation—of lesser importance to this particular reading but possibly of greater importance to other readings—of voice-leading smoothness. It can easily be seen that this voice leading is the smoothest between the two principal chords in pitch space, but it is not the smoothest in pitch-class space. Turning now to the voice leading in the pitch-class-counterpoint of Figure 2-4b (arising as the result of a T7I transformation on the Tristan chord of measure 2) we see a different voice-leading map. As these voices are generated exclusively by the mapping of elements between two sets of the same set-class type in pitch-class space and not by perceptual factors such as

The Single-Voice-Transformation Model

33

timbre and register, any correlation of the resultant individual voices to the lines of the “standard” reading would be coincidental. We see that two of these voices are in fact identical between the two readings, those which (for convenience’s sake) could be called the “soprano” and “tenor” lines. Conversely, we also note that both the “alto” and “bass” transformational voice leadings differ markedly from those of the standard “perceptual” reading. It is exactly this difference—as pointed out in chapter 2—that calls attention to one of the principal deficiencies of the set-transformational model of voice leading, namely, its frequent lack of congruence with a listener’s music-perceptual experience of linear motion. This may be purely speculative, but it seems highly unlikely that a composer would intend that a listener perceive voices being spun out by a continuous mapping though pitch-class space as a result of transpositions or inversions of the pitch-class sets to which they belong. This is especially evident when one considers the nearimpossibility that the ear would be able to trace these paths. The ear has no way of analyzing exactly what the Tn or TnI operation is between any two chords, except in retrospect, and very few listeners have the musical skills to do this “on the fly,” especially when listening to music in which chords change at a high rate of speed.1 (The task is much simplified if the operations are limited to transpositions by small intervals (by directed ic ±1 or ±2) in pitch space.) Whether by cultural conditioning or by being “hard-wired” that way, it seems reasonable to hypothesize that the brain would have an easier time processing musical relationships in pitch space than in the more abstract pitch-class space, and thus would be more likely to group successive notes in similar registers to form a musical line than successive notes in different registers, even if there exists a theoretical logic that would suggest the latter. It would only be through the addition of a second (or maybe even a third) unifying factor that a non-registrally contiguous pitch-class counterpoint would be perceived. The most obvious candidates for such a unifying factor would be timbre, dynamics, articulation, or even rhythm; and here the music of Webern comes to mind, where musical lines are frequently constructed between pitches widely separated in pitch space, but the result of a logical compositional process, linked together by one of these non-registral factors. It must be admitted that here I may be venturing onto shaky ground with this claim of compositional intent (or non-intent, as the case may be). It is entirely conceivable that certain composers may have viewed their voice leading as a result of pc-set transformations, and there seems to be a greater possibility of this in the years since Klumpenhouwer’s, Roeder’s, and Straus’s studies first saw the light of day. Even if a composer were to take such an approach, it is entirely possible she would seek to “mask” the voices derived from the pitch-class counterpoint by creating faux-lines, grouping

34

Chapter Three

resultant pitches by timbre and/or register. Such a case would have the added attractiveness of possessing the complexity of two apparent voice-leading complexes rather than just one. In any case, it seems it may now be beneficial to attempt to explain voice leading by using an approach different from pc-set mapping, one that has much in common with a perceptual hearing of musical lines, but that at the same time retains the powerful tools of transformational theory.

Fig. 3-1: Voice leading which privileges pitch-class smoothness

Figure 3-1 shows voice-leading connections between the Tristan chord and the dominant seventh that follows in a reading which privileges pitchclass smoothness over all other factors. There are some striking similarities to and differences from the pc-set transformational mapping shown in Figure 2-4b. In each figure, two voices follow the “standard” perceptual voiceleading paths in pitch space while the other two voices diverge from them markedly. In Figure 2-4b the soprano and tenor parts lead as their register and instrumentation would suggest we should expect, while the alto and bass parts each map to a note a considerable distance from its origin in pitch space but by a much smaller interval in pitch-class space. In Figure 3-1 it is the alto and bass parts that lead as expected, while the soprano and tenor parts each traverse a fairly large interval in pitch space but absolutely none in pitch-class space. The motion of the individual lines in this latter example is particularly interesting inasmuch as not only do they demonstrate the smoothest voice-leading connection in pitch-class space (with a total displacement of only two semitones) but, in terms of pitch space, they also proceed in a manner with some similarities to the most significant prolongational voice-leading model. In this respect the parsimonyprivileging voice leading in Figure 3-1 is closer to a non-transformational Schenkerian interpretation, with a “faux voice exchange” occurring between

The Single-Voice-Transformation Model

35

soprano and tenor parts. As previously mentioned, my aim in the construction of a model that privileges voice-leading parsimony is not to attempt to produce a formalized transformational system for Schenkerian theory. Although in many cases the voice-leading paths in the parsimony-privileging model and a Schenkerian reading will be identical, they also frequently are not, and the aims of the two theories are quite different. At its core the Schenkerian model is—despite a number of attempted modifications over the years—a theory of voice leading in functional tonal music, while my transformational model can be applied equally profitably to tonal, modal, or atonal music. Its focus on voice-leading smoothness above any other parameter sets it apart from previous work in transformational theories of voice leading, thus relating it more closely to so-called neo-Riemannian theory than to the transformational models discussed in chapter 2. The effectiveness of the different approaches to analyzing a passage’s voice-leading corresponds largely to whichever aspect or quality one wishes to emphasize in one’s analysis. In the standard music-perceptual reading (not a formalized model), pitch register and timbre are privileged and thus define individual lines; in a parsimonious reading, smoothness of connections in pitch-class space are privileged; in the pc-set transformational reading, it is the pc-set transformations (the TTOs) themselves which are privileged. It is for this reason that a parsimonious transformational approach could be considered a more suitable model for voice-leading analysis than the set-class transformational approach, because smoothness is a quality that is often of considerable importance in a wide variety of compositional styles and is an inherent, readily perceivable component of musical lines, in the same way as are register and timbre. Tracing voice-leading paths as a by-product of TTOs seems to be a much less accurate description of the way we experience music, and in practical analysis appears to be limited to atonal music, functioning optimally only in very short passages comprised of chords all of the same set class. It is with the assumption that a transformational model—based on the perceivable quality of smoothness—can be a good way of describing voice leading that we proceed to the formal construction of such a model. But first we must examine some recent publications which anticipate certain aspects of the SVT model.

Callender 2004 and Tymoczko 2006 We have already seen discussed (in chapter 2) two studies which hint at the yet-to-be-formalized SVT model, namely Cope 2002 and especially Roeder 1994. Both authors use ordered lists of pcs to represent chord members and apply individual transpositions to each order position to arrive at a

36

Chapter Three

cumulative voice leading between two chords. Roeder’s method is a theoretical model rather than a compositional algorithm, and it has the added advantage of being a formalized GIS. Recent work on geometrical models of harmonic and voice-leading space by Clifton Callender and Dmitri Tymoczko also anticipate aspects of the SVT model. Callender’s 2004 article, “Continuous Transformations,” examines gradual transformations over time, using as its starting point the fifteen-minute pitch glissandi in three voices of Kaija Saariaho’s Vers le blanc and then devising a generalized compositional space in which continuous transformations are mapped as trajectories through that space. After examining continuous transformations in the pitch domain, Callender uses the same principle to map trajectories in the domain of tempo, to measure distances between justintonation and equal-tempered set classes, to plot distances between the different hexachordal transposition-classes, and finally to effect continuous transformations between beat-class sets. Its most relevant sections for the SVT model are parts 2 and 3 in which Callender constructs the pitch and pitch-class spaces in which the transformations operate. While Callender’s focus is on gradual transformations over time, the compositional space he constructs serves equally well for discrete transformations. Like Roeder (1994), Callender considers verticalities as n-tuples (he calls them “ordered sets”), subscripted to identify each order position, i.e., A = ¢a1,...,an². The class of n-tuples equivalent to A under transposition is denoted /A/T and the class of n-tuples equivalent to A under transpositional, inversional, permutational, and mod-12 equivalence is designated /A/. Much of section 3 of his paper is concerned with the construction of a geometrical representation of a compositional space that shows these equivalences. Figure 3-2 is a reproduction of Callender’s Figure 10 showing trichordal (multi)set-classes /A/ in the “fundamental region” of the geometrical representation. In this figure, the border forming the X axis represents voice 1’s equivalence with voice 2 (permutational equivalence between a1 and a2). The diagonal boundary shows equivalence between the differences (voice 3 – voice 2) and (voice 2 – voice 1), which in musical terms represents inversional equivalency. The vertical boundary represents the line at which (voice 1 + voice 2 + 12) has twice the value of voice 3, and is also an inversional boundary. The fundamental region of the equivalence-class space is denoted ∏3 for trichordal set classes. Callender is interested in finding the minimal distance between set-class types, but he takes the somewhat unusual step of using a Euclidean metric rather than the (much more common to neo-Riemannian theory) “city-block” or “taxicab” metric to measure distances. The former is much more conducive to transformations involving all real numbers rather than just integers, and is better suited for continuous rather than discrete transformations.

The Single-Voice-Transformation Model

37

[048]

[036]

v 3 - v 2= v 2 - v 1 [024]

[012]

[000]

[001]

[013]

[002]

[025]

[014]

[003]

[026]

[015]

[004]

[037]

[027]

v 1 + v 2 + 12= 2v 3

[016]

[005]

[006]

v 1 = v2

Fig. 3-2: Fundamental region of transpositionally, inversionally, permutationally, and mod-12-equivalent voice-leading space with trichordal SC types plotted (from Callender 2004)

Both n-tuples and their set classes enjoy a special relationship if one can be transformed into another by changing the value of a single member; in musical terms, a single voice moves a distance h, and the n-tuples or set classes are said to be Δh-related.2 Distances in ∏3 are automatically the minimal distance between any and all members of respective set classes by Callender’s formula ρ(/Α/,/Β/) = min(ρ(A′ ,B′ )), for all A′ ∈ /A/ and B′ ∈ /B/ (Callender 2004, 11). These concepts—in particular the minimal distance formula—will become quite important as the theory of SST-succession classes is developed later in this chapter. We turn now to a recent article that was something of a sensation when it first appeared in the summer of 2006, being discussed not only in the usual music-theoretic circles but also receiving coverage in the mainstream media.3 Dmitri Tymoczko’s “The Geometry of Musical Chords” was the first music theory publication to appear in the journal Science in its then126-year history. Tymoczko represents chords (sets of pcs) as coordinates in n-dimensional geometrical spaces called orbifolds. The global-quotient orbifold Tn/Sn contains all unordered n-note chords of pitch classes, where Tn represents the quotient space (n-torus) (R/12Z)n and Sn is the symmetric

38

Chapter Three

Fig. 3-3: The T3/S3 orbifold (from Tymoczko 2006). At the center of the orbifold are the four Weitzmann regions, linked by augmented triads into the Cube Dance (see Douthett and Steinbach 1998, 253–4; Cohn 1998; and Cohn 2000).

group of n (Tymoczko 2006, 73). Minimal voice leading (in which a single chord voice moves by a single semitone) is shown by lines connecting plotted chords in the orbifold. Figure 3-3 is a reproduction of Tymoczko’s Figure S5, from the supporting online material for his paper. Tymoczko’s orbifolds are elegant geometric models which show parsimonious voice leading between any two sets of the same cardinality. Much of the focus of the latter part of the article is on maximally even chords and their proximity to the center of the orbifold to which they belong, and on symmetrical or nearly symmetrical chords and their distribution throughout the orbifolds. The “meatier” part of the paper, containing mathematical formalization of the principles briefly summarized in the main article, is found in the online supporting material.4 Let us examine how Tymoczko models voice leading between pitch classes using both ordered sets (n-tuples) and unordered sets. Tymoczko starts by ordering chords into n-tuples, which he calls “multisets of pitch classes.” This is somewhat inaccurate. A multiset differs from a set in that each member has a multiplicity, which is a number indicating how many times it is a member in the multiset.5 Thus a multiset—like an n-

The Single-Voice-Transformation Model

39

tuple—can contain duplicated elements. Multisets are special kinds of sets, which by definition are unordered. It is clear, however, that Tymoczko intends his “multisets” to function as n-tuples. His voice leadings associate pcs arranged by order position. Thus the chord A leads to the chord B through individual transformations affecting corresponding order positions: ¢a1,a2,...,an² → ¢b1,b2,...,bn².6 This is precisely what Roeder and Callender do, although we note here that the transformation that causes the mapping is undefined here, and indeed, is not defined throughout the entire paper. Converting the ordered n-tuples into unordered sets leads to the construction of an orbifold Tn/Sn by imposing additional equivalencies on the voiceleading space, in much the same way as Callender (2004) derives a fundamental region (Tymoczko calls these “fundamental domains”) of Tn-, TnI-, and permutationally equivalent pc-set types. Also of interest in the supporting materials is a dynamic programming algorithm which finds the minimal voice leading between two chords, using n-tuples to duplicate pc elements as necessary to find the closest match. Tymoczko’s algorithm is limited to pc sets and does not generalize to SC types, as Callender’s minimal-distance formula does. As an example of the algorithm, the minimal voice leading between the sets {4,7,0,11} and {4,8,11,3} can be expressed as a function between n-tuples ¢4,7,11,0,4² → ¢4,8,11,11,3² for a total displacement of 3 semitones. The derivation of the additional “voice” is never explained. Although Tymoczko does not directly address it, he is touching upon the problem of so-called “splits” and “fuses,” or how to account for a discrepancy in the number of voices constituting successive harmonies. Where does the “extra” voice come from? What happens to it after it has served its purpose and seemingly disappears? A later section of this chapter attempts to address this problem, presenting three possible formal models for increasing or decreasing the number of elements between chords.

Direct-Product Transformations All distances in the single-voice-transformation model are measured using the city-block (“taxicab”) metric rather than the Euclidean metric. In city-block geometry, the distance between any two n-tuples in n-dimensional Euclidean space with fixed Cartesian coordinate system is the sum of the lengths of the projections of the line segment between the points onto the coordinate axes.7 For example, the total distance between the point P1 with coordinates ¢x1,y1,z1² and the point P2 with coordinates ¢x2,y2,z2² is |x1 – x2| + |y1 – y2| + |z1 – z2|. In musical terms, we can describe the total voiceleading distance as the sum of the absolute values of the difference (measured in semitones) between corresponding order positions of two chords expressed as n-tuples. Thus the total voice-leading distance between two

40

Chapter Three

chords A (expressed as ¢a1,a2,...,an²) and B (expressed as ¢b1,b2,...,bn²) in mod-12 pitch-class space is expressed as: |b1 – a1| + |b2 – a2| + ... + |bn – an|. The city-block metric is more commonly found in theoretical models that measure voice-leading distance than is the Euclidean metric. A notable exception to this is Callender 2004. In order to construct a transformational model for voice-leading that privileges smoothness, we must first provide definitions that describe precisely what the term “parsimony” means.8 DEFINITION: A relation between two consecutive pitch classes in an ETS 12 system is said to be parsimonious if the unordered pitch-class interval between the two pcs equals 0, 1, or 2. A transposition operation that maps one pc to a parsimoniously related pc (in the same voice) is called a parsimonious voice-leading transformation. Thus T0, T1, T2, T10, and T11 are parsimonious functions (always in an equaltempered 12-tone universe) while T4, T7, and the like are not. It is easy to see that T12 = T2, and T112 = T10. Because Tn is a group, every transposition has a unique inverse. Thus we see the parsimonious transposition T1 has the inverse function T1–1 = T11. We can use T1 (or T11) as a generator of the cyclic quotient group Z/12Z of all transpositions, both parsimonious and nonparsimonious. T7 and its inverse T5 also generate this group, but neither T7 nor T5 are parsimonious functions, although their squared functions are (T72 = T2 and T52 = T10).9 We will now want to consider two concepts. The first is rather simple. As our voice-leading model will privilege half- or whole-step smoothness, it seems fitting that we should privilege T1 over T7 as the basic function in our system. This is not to say that T7 will not play a rôle in the model. For one thing, motion by ic 7 (or its inverse, ic 5) is very common (especially in the bass line) and could be argued to be the most fundamental “harmonic” motion in traditional tonal music; thus it would be useful to have a generating function handy for those occasions—especially common in “functional” triadic music but also frequent in non-tonal music—where it would be impractical to describe the voice-leading interval of T7 as (T1(T1(T1(T1(T1(T1(T1))))))) (= T17) or as (T1–1(T1–1(T1–1(T1–1 (T1–1))))) (= T1–5). We should also acknowledge that in many instances composers deliberately wish to avoid smooth voice leading and instead rely heavily on larger ics for intervallic motion.10 In cases like these, which we will see occur less frequently than the parsimonious functions, it can be helpful to employ a larger generating function such as T7 or even a non-generating

The Single-Voice-Transformation Model

41

function such as T3. The compounding of functions will allow us to account for the remaining intervals of transposition. For example, (T1–1(T7–1)) gives us the interval of transposition T4, and T3 may also be derived in a number of different ways: T13, (T1(T72)), (T1–2(T7–1)), and T7–3. Each interval of transposition can in fact be arrived at by an infinite number of combinations of individual generating transpositions. To adhere to the spirit of minimal voice leading, however, we will interpret composite transpositions as the most efficient composition of functions we can find among the infinite number possible. Up to this point we have been discussing a transposition-based model for voice-leading paths in a single voice. The second concept, more difficult to formalize, is the modeling of simultaneous voice-leading paths, independently occurring in two-, three-, four-, or more-part counterpoint. We have seen that in set-transformational voice-leading models, transposition functions are applied to complete sets; distinct individual transposition levels are not applied to individual pcs. Applying transposition operations to individual pcs constituting a chord can result in an space-consuming, over-determined analysis, especially if the chord is of a large cardinality, but doing so describes the voice-leading motion much more accurately than does the settransformational approach. A model which preserves the accuracy of the individual-pc-transpositional approach yet reduces some of its analytical clutter is one based on direct-product relations.11 DEFINITION: The direct product group G +H of the groups ¢G, 1² and ¢H, 2² is the set of 2-tuples ¢gi,hi² where gi ∈ G and hi ∈ H, with the binary operation of the direct product defined componentwise: ¢g1,h1² ¢g2,h2² = ¢g1 1 g2, h1 2 h2² (Dummit and Foote 2005, 18; 154). As we will be dealing with the same group of mod-12 pc transpositions for each voice, our direct product group will be G + G for a two-voice “chord,” where G = Tn, the group of musical transpositions. We can create direct product groups for three or more voices: G1 + G2 + ... + Gm, whose elements are an ordered m-tuple of the form ¢g1,g2,...,gm². The reason for using direct product groups is to produce an ordered series of transpositions which can operate on individual chord elements (pcs). DEFINITION: The SVT Group (designated Gm) is a direct product group G1 + G2 + ... + Gm acting on a set (chord) C of order m. Each gi ∈ G where G = Tn, the Z/12Z group of musical transpositions, and C is sorted into voices expressed as an m-tuple ¢c1,c2,...,cm².12

42

Chapter Three

Gm can be expressed more generally as Tnm, the direct product group Tn1 + Tn2 + ... Tnm, whose elements are an n-tuple of the form ¢Ti1,Tj2,Tk3,...,Tzn². The action of Gm on C is: Gm(Cm) = ¢g1,g2,...,gm² & ¢c1,c2,...,cm² = ¢g1(c1),g2(c2),...,gm(cm)² where each gi is some transposition operation Ti. DEFINITION: A single-voice transformation (SVT) is any element of Gm such that all but one of the entries in the direct product vector are the identity: ¢e1,e2,...,Tn,...,em². Since e = T0, SVT(Cm) = ¢T0(c1),T0(c2),...,Tn(ci),...,T0(cm)². The SVT maps individual intervals of transposition to individual ordered chord members. As can be seen above, the chord C is expressed as an ordered m-tuple ¢c1,c2,...,cm², where each ci is a pc in C. There are a few different ways in which this order could be determined.13 Our “ordering” could be nothing more than a random inventory of the pitch classes that make up the chord. However, in the analysis of a contrapuntal/harmonic passage it may be propitious to designate order positions based on the most immediately conspicuous linear organization of the passage. For example, if the passage is for saxophone and piano, we may wish to designate the initial saxophone note as voice 1, the highest piano pitch as voice 2, and so forth. If we choose an approach similar to that used by Klumpenhouwer or Roeder, the chord element ordering would be completely determined by pitch register. As the SVT system makes use of pitch-class space rather than pitch space, pitch register need not be the principal factor in determining our chord-member ordering, although it may be helpful in certain analytical situations. No matter what criteria we draw upon to determine the m-tuple ordering, we must realize that the demands of parsimony in the SVT system may at times overrule the apparent voice leading of the individual lines as displayed in the score. After we decide upon the initial m-tuple ordering of the initial chord of our analysis, and after we have traced each voice by the most parsimonious path possible to the end of the passage, it is quite possible that the voices will end up in a different registral configuration than that which we started from. Throughout this study, pitch classes will be indicated by integer notation where the arbitrarily chosen pitch class C = 0.14 m-tuples of pcs (chords) acted upon by SVTs will be notated with individual elements arranged in

The Single-Voice-Transformation Model

43

angle brackets, thus an m-tuple of m elements Cm = ¢c1,c2,...,cm². The Tn/TnI type of the ordered m-tuple may occasionally be indicated by a parenthetical subscript, simply for convenience’s sake. Thus a chord comprised of the pitch-class elements A, D, C, and F would be converted to integer notation and ordered within angle brackets, followed by a subscript indicating the set-class type to which it belongs: ¢9,3,1,6²[0258] In this example, the ordering commences with the pc A, but we could have arranged the pcs in any order. This particular ordering could possibly have been chosen to reproduce the registral ordering of pitch classes at the beginning of a piece. Let us return to our formula for single-voice transformations: SVT(Cm) = ¢T0(c1),T0(c2),...,Tn(ci),...,T0(cm)². DEFINITION: A single-semitone transformation (SST) is an SVT in which the sole non-identity operation in the action of the direct product group, Tn, is restricted to T1. Specific SSTs are indexed with a subscript i that identifies which order position of the pitch-class m-tuple is acted on by T1. The remaining order positions are transposed by T0. The subscript i varies from 1 to m: SSTi(Cm) = ¢T0(c1),...,T1(ci),...,T0(cm)². There are exactly m distinct SSTis in the SVT group Gm. As an example of an SST, SST2(¢9,3,1,6²[0258]) = ¢9,4,1,6²[0358]. Each SSTi operation has a unique inverse operation, SSTi–1 which transposes the ith order position by T11 and the remaining order positions by T0, e.g., SST2–1(¢9,4,1,6²[0358]) = ¢9,3,1,6²[0258]. We may now adapt our definition of a parsimonious relation between two pcs to a situation involving two chords of the same cardinality. DEFINITION: A relation between two chords (expressed as m-tuples) of the same cardinality m in an ETS 12 system is said to be parsimonious if the unordered pc interval between corresponding m-tuple order positions equals 0 (for any number of order positions), 1, or 2 (to a maximum of m/2 order positions). An SST or composition of SSTs that maps one m-tuple to a parsimoniously related m-tuple is called a parsimonious voice-leading transformation.

44

Chapter Three

DEFINITION: A relation between two chords of cardinality m (expressed as m-tuples) in an ETS 12 system is said to be strongly parsimonious if m – 1 pcs are the same with the remaining pc integer differing by ±1 or ±2. An SST or composition of SSTs that maps one m-tuple to a strongly parsimoniously related m-tuple is called a strongly parsimonious voiceleading transformation. By the second definition, all simple SSTs result in strongly parsimonious voice-leading transformations. As our definition of a strongly parsimonious relation between two chords allows for a difference of ±2 between one of the chords’ corresponding m-tuple order positions, a single voice may undergo the same SST twice and still be considered to be a strongly parsimonious relation. The double application of an SST may be notated as follows: SST2(SST2(¢9,3,1,6²)[0258]) = SST2(¢9,4,1,6²[0358]) = ¢9,5,1,6²[0148], SST2–1(SST2–1(¢9,5,1,6²)[0148]) = SST2–1(¢9,4,1,6²[0358]) = ¢9,3,1,6²[0258]. To save space, we can use exponent notation to show the composition of these functions: SST2(SST2(¢9,3,1,6²)) = SVT22(¢9,3,1,6²)) = ¢9,5,1,6²[0148], SST2–1(SST2–1(¢9,5,1,6²)) = (SVT2–1)2(¢9,5,1,6²)) = ¢9,3,1,6²[0258]. Notice that in the above example, compound SSTs that non-trivially transpose the same m-tuple order position are probably more properly called SVTs, as the affected voice is no longer transposed by a semitone, but by a whole step. Later in this chapter, we will see another type of SVT, the singlefifth transformation or SFT. Because the perfect fifth, like the semitone, is a generator of the ETS 12 pitch-class system, the composition of two SFTis results in the same transformation as the composition of two SSTis (as long as the non-trivial transposition affects the same m-tuple order position i in each). This leads us to an important aspect of the single-voice transformation model: SSTs may be combined through the composition of functions, not only to transpose a single voice by two semitones (to remain strongly parsimonious), but a composite transformation may also non-trivially transpose more than one voice (to a maximum of m/2 voices to remain parsimonious). For instance: SST4–1(SST2(¢9,3,1,6²)[0258]) = SST4–1(¢9,4,1,6²[0358]) = ¢9,4,1,5²[0148]. This composite operation could have been written SST2(SST4–1(¢9,3,1,6²))

The Single-Voice-Transformation Model

45

= SST2(¢9,3,1,5²) = ¢9,4,1,5²[0148], as the SVT group G4 (= G1 +G2 +G3 + G4) is commutative. Composite functions may be abbreviated as SST(i)(j), etc., where i = an order position and j = a different order position of the ordered m-tuple: SST(2)(4–1)(¢9,3,1,6²[0258]) = ¢9,4,1,5²[0148]. N.B.: (4–1) in the above example is an abbreviation for SST4–1. Here is an example involving different m-tuples but the same SSTs: SST(2)(4–1)(¢0,2,5,8²) = ¢0,3,5,7²[0247]. The transformation SST(2)(4–1), is only one of the multiple voiceleading “paths” that exist between m-tuple representatives of SC types [0258] and [0247]. Another path that is readily apparent is SST(3–1)(4–1), which when applied to ¢0,2,5,8² produces ¢0,2,4,7². Another, less readily apparent path is through SST2–1(SST1(SST1(SST1(¢0,2,5,8²)))), equivalent to SST(2–1)(13)(¢0,2,5,8²) = ¢3,1,5,8², ordered members of the set {1,3,5,8}, which itself is a representative of [0247]. This last voice-leading path prompts the question: at what point does the composition of SST functions begin to become excessive? As defined above, the number of voices that move by step or by half-step in a chord should not exceed half the cardinality of the chord in order for the voice leading between the two chords to be considered parsimonious. In the case of SST(2–1)(13)(¢0,2,5,8²), we can see that only two of the four voices undergo non-identity transformations, thus satisfying half of the definition of a parsimonious relation. However, the chord’s first order position receives three consecutive semitonal transpositions, disqualifying this composite operation from being a parsimonious voice-leading transformation. The answer to the question posed at the beginning of this paragraph then seems to be driven by yet another set of questions: what exactly does the analyst seek to show by the use of the SST model? Is the music being examined a good match for this particular voice-leading model? Are compounded SST functions audible, and if so, to how many consecutive iterations transposing a single voice nontrivially? An intuitive response to these questions would be that the SST model—based on its privileged interval—would work better with music that contains predominantly smooth voice leading. This consideration, along with a general assumption that—all other factors being equal—the audibility of a voice-leading path between pcs is inversely proportional to the size of the interval of displacement, would suggest that the closer the music in question adheres to exclusively parsimonious voice leading, the greater the effi-

46

Chapter Three

cacy of the SST model in describing individual voice-leading paths. Thus in the course of our analyses, we will see that most of the music can be effectively modeled by one or more SSTs to within parsimony, but that occasionally SST operations must be compounded three or more times in a single voice to explain “leaps” by larger intervals—voice leading which by our earlier assumption is less audible. Chords that contain more than six pitch classes are much less common in the literature than chords with fewer pitch classes; we will see that most of the works analyzed in chapter 6 are driven by trichordal, tetrachordal, and pentachordal harmonies, with the occasional hexachord or heptachord making an appearance. Thus in an analysis in which the maximum size of a chord Cm is limited to 7 pitch-class members, we may still allow for up to m/ 2, or 3 parsimonious voice-leading paths, and our nomenclature remains relatively uncluttered: SVT(i–1)2(j–1)2(k–1)2(¢c1,c2,...,c7²) = ¢g1(c1),g2(c2),...,g7(c7)² is about as unwieldy as our notational system can get and still remain parsimonious, with a total of six compounded SVT operations. Similarly to the compounding of SST operations within a single order position in excess of our self-imposed parsimonious boundaries, we may choose to allow nontrivial voice leading between more than m/2 members of set m—for example, in 4 of the 7 voices of our hypothetical heptachord C7. Stepping outside of these usual limits for a brief period allows the occasional atypical chord or voice-leading interval to be accommodated into our model of mostly parsimonious relations. Because of the historical importance of voice-leading motion by the interval of a fifth (T7 or its inverse, T5), we will also want to consider allowing for the single fifth transposition, or SFT, which is similar to the SST in all respects except for the interval of transposition of the indexed m-tuple member. DEFINITION: A single-fifth transposition (SFT) is an SVT that transposes the ith order position of the pitch-class m-tuple it is acting on by T7 and the remaining order positions by T0. The subscript i varies from 1 to m: SFTi(Cm) = ¢T0(c1),...,T7(ci),...,T0(cm)². This transformation and its inverse, SFTi–1, will occasionally be found in the analyses in the final chapter. The SFT can be an effective model for a single, simple, relatively common voice-leading transformation that would otherwise require multiply iterated SSTs extending beyond the established

The Single-Voice-Transformation Model

47

parameters of a parsimonious function.

Splits and Fuses One of the greatest problems inherent in any transformational theory of voice leading is that of linking sets of differing cardinalities. That we should want a system which allows us to do so is incontestable: most music—tonal as well as atonal—does not maintain a set number of voices across the entire span of a piece. Instead, voices may drop out, new voices may enter, and identical pitch classes may be represented by more than one voice simultaneously (doubling). The problem in moving from chords of one cardinality to chords of another cardinality—in set-transposition- and -inversion-based approaches such as those of Klumpenhouwer and Straus—consists of the familiar hurdle of trying to map chords belonging to one set-class type to chords belonging to a different set-class type. O’Donnell is able to work partially around this problem by simultaneously applying two distinct transformations to a set (1997, 49–50). Figure 3-4 shows how these “split transformations” can occasionally appear to convert sets of cardinality m to sets of cardinality m – 1. The figure displays what O’Donnell calls “dual transposition,” where some voices undergo a transposition transformation of Tn and others a transposition transformation of Tn + x. The composite transformation is called Tn/Tn + x. Following O’Donnell’s notational conventions, the first component Tn (the lowest in pitch space) is indicated by solid arrows in the analysis, and the second component Tn + x (the next highest in pitch space) is indicated by dashed arrows.

Fig. 3-4: A split transformation T10/T10 + 3 (after O’Donnell 1997)

O’Donnell’s work-around affords the analyst/composer a greater degree

48

Chapter Three

of flexibility than does straight set transposition, yet still it only offers two separate voice-leading transposition levels, each of which contains two or more voices moving in strictly parallel motion. It will be seen that split transformations give rise to two additional issues which we will now explore in more detail. We notice in Figure 3-4 that the “tenor” and “alto” voices both map to the same pitch, C4 (or if we are thinking in terms of pitch class, to 0), yet it is clear from the example that the number of voices does not decrease from 4 to 3. According to common practice, the pitch C4 or pitch class 0 would be said to be “doubled” by two separate voices. Another possibility would be to rewrite the second chord to show only three independent voices and call this convergence of voices a “fuse,”15 but what exactly does this mean in a mathematical sense? The most significant mathematical ramification of a voice-leading fuse as a result of O’Donnell’s split transformation is that it is non-retrogradable. More specifically, a fuse is not a function, as it does not have a uniquely determined inverse. The reverse of a fuse (a “split” transformation in the sense of Callender (1998)) could not be a mapping at all, as one chord element in the domain would have to “map” to two different elements in the codomain. Let us consider this problem by imagining the reverse of Figure 3-4. If we wanted to move from the trichord consisting of the pcs {3,0,9} to the tetrachord consisting of the pcs {5,11,2,8}, one of the trichord members would have to map by two different transpositions to two different members of the tetrachordal set. True voice-leading splits—where the number of voices increases—cannot be the result of applying any function to the chord. We have already seen, in chapter 2, how Joseph Straus’s introduction of “near-transposition” and “near-inversion” allows for multiple independent voice-leading paths to a greater degree than did previous transformational theories. Straus’s *Tn and *In operations also allow for voice splits and fuses, although Straus does not attempt to offer mathematical explanations for voice-leading between sets of differing cardinalities.16 There are at least four distinct possibilities for modeling split and fuse voice-leading functions. I am grateful to John Rahn for suggesting the first three of these,17 which we will examine in turn. For each model we must clearly distinguish between the number of voices and the number of pitch classes in play at any given moment. In all models each chord is an m-tuple with each order position representing a voice. The m-tuple pitch-class content is represented by variables a, b, c (and sometimes d) in the first chord and variables x, y, z (and sometimes w) in the second chord. We will be examining the interaction of three- and four-voice chords, as this very commonly occurs in a great deal of music, in both tonal and atonal styles. The first model, illustrated in Figure 3-5, maintains a constant, minimal

The Single-Voice-Transformation Model

a)

49

b) z

d

y

c

x

b

w

a

c

z

b

y

a

x ¢a,b,c² o¢w,x,{y,z}²

¢a,b,{c,d}² o¢x,y,z²

Fig. 3-5: Constant min-m-voice transformational model: a) for splits; b) for fuses

number of voices while the number of pcs change in the course of a transformation. From the figure we see that although the number of pcs in the chords vary between 3 and 4, the number of voices remains constant and equals that of the chord containing the smallest number of distinct pcs, hence the term “min-m” to describe this particular model of splits and fuses. This model is attractive in that it is perhaps the closest to previous descriptions of the terms “split” and “fuse.”18 One can very clearly see the derivation of the split voice and the provenance of the fused voice in this model. But at the same time the min-m model is possibly the most problematic of the four we will examine. The reason for this is simply that it contains the least precise information about specific voice leading between pcs. In Figure 3-5a we can clearly see that the pc set {y,z} is the content of the third voice in the second chord, being led to from pc c in the first chord. The problem arises if we wish to continue with a progression to another four-pc chord. Voices w and x may each map to single pcs in the third chord, but what do we do with the third voice, set {y,z}? We certainly can map this voice to another two-pc set {g,h}, but what we cannot do is show specific voice leading from individual pcs y and z to individual pcs g and h. In other words, we cannot convert m voices into m + 1 true voices using the min-m model. Figure 3-6 shows a second possibility. In this model, corresponding order positions (voices) map to each other. If there is an additional order position in one chord but not the other, there is no mapping to or from the pc in the additional order position. Thus the minimal number of pcs will participate in the mapping—3 pcs in the case of Figure 3-6—with an additional pc not involved in the mapping. The advantage to the variable m-voice model is that, unlike the min-m model and the max-m model (below), the number of

50

Chapter Three

a)

b) z

d

y

c

x

b

w

a

c

z

b

y x

a ¢a,b,c² o¢w,x,y,z²

¢a,b,c,d² o¢x,y,z²

Fig. 3-6: Variable m-voice transformational model: a) for splits; b) for fuses

voices (order positions) actually does increase or decrease by one with a split or fuse, respectively. This is a positive development if one wishes to continue to a third chord and maintain the same number of voices as found in the second chord. Despite this advance over the min-m-voice model, there remains one major problem with the variable m-voice model, namely, accounting for the derivation of the voice that does not participate in the transformation. In Figure 3-6a, the pc z is a creatio ex nihilo, and pc d disappears just as inexplicably during the fuse in Figure 3-6b. .

a)

b) c

z

d

z

c

y

c

z

b

x

b

y

a

w

a

x

¢a,b,c,c² o¢w,x,y,z²

¢a,b,c,d² o¢x,y,z,z²

Fig. 3-7: Constant max-m-voice transformational model: a) for splits; b) for fuses

The Single-Voice-Transformation Model

51

The third model for splits and fuses is shown in Figure 3-7. Here the number of voices (m-tuple order positions) remains constant and equals that of the chord containing the largest number of distinct pcs, hence the term “max-m” to describe this particular model. It should be pointed out that the voice leadings c → z and c → y in Figure 3-7a are not in error; both are legitimate functions with inverses as each c represents a distinct voice (order position) even though they share pitch-class content. To avoid confusion, one could label the first chord in the figure as ¢a,b,c,d² and then stipulate that, in terms of pitch-class content, d = c, to emphasize that the voice leadings c → y and d → z occur in separate voices. The constant max-m-voice model is perhaps the best suited for musical textures which maintain a near-constant number of voices, with occasional pc duplication (“doubling”). Trouble arises, however, if a passage continues with max-m – 1 voices for some time after a fuse; in such a scenario it makes little sense to continue modeling the voice leading with max-m voices, as the independence of two of the voices is completely destroyed. Additionally, if one wishes to model voice leading to a chord of one-greater cardinality than a max-m chord, one would have to switch to either the constant min-m-voice or the variable m-voice model to account for the additional pc. An ad-hoc mixture of the three models like this, while certainly possible, leaves much to be desired as it does not contribute to a consistent, unified explanation of split and fuse voice leading. Let us turn now to the fourth model for splits and fuses, which is the one that will be adopted in this study. This model combines aspects of the variable m-voice model and the other two models in that the number of voices varies by ±1 with each split or fuse, but the derivation of the previously unaccounted-for voice is now clearly shown. The following shows how this is possible. DEFINITION: Let SPLITi be a transformation on m-tuples which increases the size of an m-tuple by creating a duplicate of the content of the ith order position in the (i + 1)th order position, shifting all subsequent order positions by +1 and creating an (m + 1)-tuple. The composite operation of a SPLITi transformation, followed by an SST or SFT that non-trivially transposes one of the (m + 1)-tuple order positions with duplicate content, is called a split. It should be noted that while SPLITi itself creates the actual split voices, I am defining the (lowercase) term “split” as the composition of functions SVT(SPLITi(Cm)). The reason for this is to bring the definition more closely in line with previous definitions of the term (for example, as used in Callender 1998). An example follows:

52

Chapter Three

SST3(SPLIT2(¢0,3,7²)) = SST3(¢0,3,3,7²) = ¢0,3,4,7². The function SPLITi is derived from the standard identity function e but creates an additional order position in the m-tuple. In the example above, it should be noted that the transformation SST3 affects what initially appears to be the pc in the second order position of ¢0,3,7² (3), not the third order position (7). This is because the SPLITi function, by its very nature, does not commute with an SVT. In other words, if we apply the SST3 transformation to ¢0,3,7² before the SPLIT2 function we would obtain SPLIT2(SST3(¢0,3,7²)) = SPLIT2(¢0,3,8²) = ¢0,3,3,8²[037] which is an m-tuple belonging to set-class type [037], but it is clear that ¢0,3,3,8² is a different m-tuple than the original m-tuple ¢0,3,7², and we have not succeeded in increasing its cardinality (of non-replicated elements) as would have happened had the SPLIT2 function and the SST3 function been reversed. A fuse transformation is simply the inverse of a split, and may be defined likewise: DEFINITION: Let SPLITi–1 be a transformation on m-tuples which decreases the size of an m-tuple by eliminating duplicate pitch-class content in the (i + 1)th order position, shifting all subsequent order positions by –1 and creating an (m – 1)-tuple. The composite operation of a SPLITi–1 transformation and a preceding SST or SFT that non-trivially transposes one of the m-tuple order positions to produce duplicate content with an adjacent position, is called a fuse. The following example shows the effect of the composite fuse on the mtuple ¢0,3,4,7²: SPLIT3–1(SST3–1(¢0,3,4,7²)) = SPLIT3–1(¢0,3,3,7²) = ¢0,3,7². The index i in SPLITi–1 may at first seem somewhat superfluous, as we could have defined a function SPLIT–1 (without subscript) that eliminates any m-tuple order position(s) containing the same content as another order position. There are, however, a couple of reasons why we may want to continue to use a subscripted index number for the fuse function. The first and most readily apparent reason is to provide an inverse function for SPLITi that affects the same m-tuple order position, even if it does not affect the same index number. In the above example we could just as easily have used the function SPLIT2–1 instead of SPLIT3–1 to achieve the same resultant m-

The Single-Voice-Transformation Model

53

tuple, ¢0,3,7². However, SPLIT3–1—like SPLIT2—creates or removes content in the third order position of an m-tuple. SPLIT3–1, therefore, is the inverse of SPLIT2, despite the counterintuitive nature of this relationship.

Fig. 3-8: A fuse function SPLIT5–1(SST(3–1)(5–1)(¢0,3,4,7,8²)) = ¢0,3,3,7²

The second reason is to provide for exactness when an m-tuple contains two or more instances of duplicated elements. Consider the m-tuple ¢0,3,4,7,8². We may wish to allow for a function that maps ¢0,3,4,7,8² to ¢0,3,3,7² (see Figure 3-8). Applying a hypothetical “generic” SPLIT–1 (affecting all order positions equally) after the SST—SST(3–1)(5–1) (¢0,3,4,7,8²) = ¢0,3,3,7,7²—would produce ¢0,3,7². To achieve our intended m-tuple ¢0,3,3,7² we would then have to apply a SPLIT2 transformation, thus our composite function would be SPLIT2(SPLIT–1(SST(3–1)(5–1)(¢0,3,4,7,8²))) = SPLIT2(SPLIT–1(¢0,3,3,7,7²)) = SPLIT2(¢0,3,7²) = ¢0,3,3,7² Although this is certainly a possibility, it makes little sense to change the cardinality of the set twice, applying a split function to the order position we just finished fusing. By specifying an index number for SPLIT–1 we can streamline this formula, removing one of the functions and avoiding the “overcompensation” that comes with a generic SPLIT–1 transformation.19 Examining our SPLIT5–1 function on the m-tuple that results after the SST transformations in Figure 3-8: SPLIT5–1(¢0,3,3,7,7²) = ¢0,3,3,7²

54

Chapter Three

we notice the fifth order position (and one of the pitch-class elements 7) is eliminated while order positions 2 and 3 (and pitch-class elements 3 and 3) remain unchanged.

Fig. 3-9: Continuation of voice leading in four voices after a fuse

What would be the advantage, in this example, of maintaining the duplicate pc elements 3 when we could have reduced the m-tuple to ¢0,3,7²? This would have been a simple and intuitive step when one considers the equivalence relationship shared by octave-related tones in pitch-class space, in this case the E(3 and E(4 in Figure 3-8. The answer, it seems to me, should be based on the immediate musical context of the passage in question. How many active, independent voices continue in the subsequent music after the fuse occurs? If the answer, in the case of Figure 3-8, is three (with “alto” and “tenor” moving in parallel octaves, one functioning merely as a “doubling” of the other) then a reading which reduces the multiset ¢0,3,3,7,7² to ¢0,3,7² through a “global” fuse operation SPLIT–1 might be more appropriate than an indexed fuse operation SPLITi–1. Figure 3-9 shows the voice leading of Figure 3-8 with a hypothetical continuation in four voices. From this example it is clear that pc 3 of the second chord is best interpreted as a duplicated element occurring in two order positions in the 4-tuple ¢0,3,3,7², rather than as a 3-tuple ¢0,3,7², as the third chord confirms that each instance of pc 3 in the second chord functions as part of two independent voices in a four-voice texture. In the case of Figure 3-9, it thus seems more appropriate to use the indexed fuse function SPLIT5–1 to obtain ¢0,3,3,7² rather than a generic fuse function SPLIT–1, obtaining ¢0,3,7². With the preceding discussion in mind, the analyst must proceed carefully whenever encountering a pitch-class duplication: is the duplication the

The Single-Voice-Transformation Model

55

result of a split or fuse, or is it merely a momentary pitch-class doubling in a texture in which the number of voices remains constant? It is only by examining the subsequent music that an answer to this question may be obtained.

SST-Succession Classes For reasons which will become apparent in subsequent chapters, it will prove helpful to define classes of SST-related chord successions, in much the same way as equivalence classes are imposed on pitch-class sets to form Tn and Tn/TnI types. In brief, SST-succession classes provide a meaningful way of describing minimal “distances” between set-class types by generalizing a pair of SST-related m-tuples to a pair of classes of SST-related m-tuples (or Tn/TnI types). This is akin to the idea of using Tn/TnI equivalence on transformation-related chord progressions to produce classes of chord progressions, as recently discussed by Dmitri Tymoczko and others.20 SSTsuccession classes subsume all transposed, inverted, or permuted forms of two pair-ordered m-tuples that are SST-related in any way. We recall from an earlier section of this chapter that Callender (2004) defines Δh as a relation between two m-tuples or between the set classes to which they belong, and which is obtained if the first can be transformed into the second by changing the value of a single member by the amount (distance) h. When we obtain this relation between two m-tuples, Δh is equal to one or more SVTs which non-trivially transpose a single voice. /A/T and /B/T are classes of m-tuples equivalent to m-tuples A and B respectively under transpositional equivalence, and /A/ and /B/ are classes of m-tuples equivalent to m-tuples A and B respectively under transpositional, inversional, permutational, and mod-12 octave equivalence.21 Callender’s Assumption 2a reads: If a member of one T-class can be transformed into a member of another Tclass by moving a single voice, [a] distance between the two T-classes should be the distance this voice moves. That is, if /A/T Δh /B/T, then ρ(/A/T, /B/T) = h.22

We can generalize this assumption to Tn/TnI types, substituting /A/ and /B/ for /A/T and /B/T. We will also make some minor changes to Callender’s notation, substituting ¢A² and ¢B² for A and B to represent m-tuples, (A) and (B) for /A/T and /B/T to represent Tn classes of m-tuples, and [A] and [B] for /A/ and /B/ to represent Tn/TnI classes of (unordered) sets, as well as using angle brackets instead of parentheses to denote ordered pairs. Thus, if [A] Δh [B], then ρ¢[A],[B]² = h. Further investigation shows this formula is problematic, for when we talk about distance between set classes (which is

56

Chapter Three

another way to describe Straus’s “offset”; see p. 24 in chapter 2), we have to take into account ALL of the Δh relations between ALL the m-tuple members of [A] and [B] (two set-class types). The distance h varies for each ¢A′ ² ∈ [A] and each ¢B′ ² ∈ [B]. To keep things simple, let’s just consider all voice-leading relations between 2-tuple members of dyadic SCs [A] = [02] and [B] = [03], arbitrarily indexed as ¢A1²,¢A2²,¢...²,¢A24 ²and ¢B1²,¢B2²,¢...²,¢B24² (Table 3-1): Table 3-1: List of all Tn/TnI representatives of SCs [02] and [03] ¢A1² ¢A2² ¢A3² ¢A4² ¢A5² ¢A6² ¢A7² ¢A8² ¢A9² ¢A10² ¢A11² ¢A12² ¢A13² ¢A14² ¢A15² ¢A16² ¢A17² ¢A18² ¢A19² ¢A20² ¢A21² ¢A22² ¢A23² ¢A24²

¢0,2² ¢1,3² ¢2,4² ¢3,5² ¢4,6² ¢5,7² ¢6,8² ¢7,9² ¢8,10² ¢9,11² ¢10,0² ¢11,1² ¢0,10² ¢11,9² ¢10,8² ¢9,7² ¢8,6² ¢7,5² ¢6,4² ¢5,3² ¢4,2² ¢3,1² ¢2,0² ¢1,11²

¢B1² ¢B2² ¢B3² ¢B4² ¢B5² ¢B6² ¢B7² ¢B8² ¢B9² ¢B10² ¢B11² ¢B12² ¢B13² ¢B14² ¢B15² ¢B16² ¢B17² ¢B18² ¢B19² ¢B20² ¢B21² ¢B22² ¢B23² ¢B24²

¢0,3² ¢1,4² ¢2,5² ¢3,6² ¢4,7² ¢5,8² ¢6,9² ¢7,10² ¢8,11² ¢9,0² ¢10,1² ¢11,2² ¢0,9² ¢11,8² ¢10,7² ¢9,6² ¢8,5² ¢7,4² ¢6,3² ¢5,2² ¢4,1² ¢3,0² ¢2,11² ¢1,10²

There are a total of 576 voice leadings between all 2-tuple members of [A] and [B]. There are a total of 96 Δh relations between all 2-tuple members of [A] and [B].23 Here we clearly see the problem of different voice-leading distances between the various dyadic m-tuples. For example, for two 2tuples not related by Δh, ¢A1² and ¢B6², the total voice-leading distance sums to 11. For two Δh-related 2-tuples ¢A1² and ¢B13², the total voice-leading distance sums to 5: ρ¢¢A1²,¢B13²² = 5. But for the Δh-related 2-tuples ¢A1² and

The Single-Voice-Transformation Model

57

¢B1², ρ¢¢A1²,¢B1²² = 1, which equals the shortest cumulative distance possible between any two 2-tuple members of [A] and [B]. ρ¢¢A1²,¢B1²² is not the only instance of a Δh relation with a total distance of 1. There are actually 48 Δh relations where h = 1 between members of [A] and [B]. How can we identify the shortest possible “distance,” or offset, between set classes [A] and [B] when the distance between certain m-tuple representatives ¢A′ ²∈ [A] and ¢B′ ²∈ [B] equals one SST (where h = 1) but other representatives have a greater h value or may not even be Δh-related? Callender considers the “distance” ρ between two set classes [A] and [B] to be the minimal distance between members of the respective set classes: ρ¢[A],[B]² = min(ρ¢¢A′ ²,¢B′ ²²) for all ¢A′ ² ∈ [A] and ¢B′ ² ∈ [B]. Notice that this definition does not stipulate that the m-tuples or set classes must be separated by a distance of 1, let alone be Δh-related to each other. Here is the same definition, re-written to show the individual elements in the m-tuple: ρ¢[Am],[Bm]² = min(ρ¢¢a′1,a′2,...,a′m²,¢b′1,b′2,...,b′m²²) In the case of SCs [02] and [03], ρ¢[02],[03]² = min(ρ¢¢a′1,a′2²,¢b′1,b′2²²) = 1. This obtains between 48 members ¢A′ ² ∈ [A] and ¢B′ ² ∈ [B]. We can see from Table 3-1 that the following are a few of the 48 min(ρ) voice leadings involving m-tuple representatives of [02] and [03]: ρ¢¢1,3²,¢0,3²² = 1 ρ¢¢0,2²,¢0,3²² = 1 ρ¢¢0,2²,¢11,2²² = 1 ρ¢¢9,7²,¢9,6²² = 1 ρ¢¢2,0²,¢3,0²² = 1 If we examine all 48 of these, we see that min(ρ) voice leadings result in only four scenarios: A) the second order position of ¢A′ ² increases by 1; B) the first order position of ¢A′ ² decreases by 1; C) the first order position of ¢A′ ² increases by 1, or D) the second order position of ¢A′ ² decreases by 1. (Scenarios 3 and 4 result from operations on inverted members of the SC type.) Can min(ρ) ever equal anything other than 1? Not between SCs [02] and [03], but it is frequently the case between other set classes, as we can easily determine when we consider ρ¢[01], [03]², where min(ρ) = 2. This measurement may result from motion in a single voice, such as ρ¢¢0,1²,¢0,3²² = 2, where the 2-tuples are Δh-related, or as a result of motion in different voices,

58

Chapter Three

as can be seen where ρ¢¢0,1²,¢11,2²² = 2, where the 2-tuples are not Δhrelated. Let us, for the moment, consider ρ to be the result of one or more SVTs between transposed or inverted versions of the 2-tuples ¢0,2² and ¢0,3²: SST1–1(¢1,3²) = ¢0,3²; ρ = 1 SST2(¢0,2²) = ¢0,3²; ρ = 1 SST1–1(¢0,2²) = ¢11,2²; ρ = 1 SST2–1(¢2,0²) = ¢2,11²; ρ = 1 SST1(¢2,0²) = ¢3,0²; ρ = 1 and now between 2-tuples belonging to the SCs [A] = [01] and [B] = [03]: SVT22(¢0,1²) = ¢0,3²; ρ = 2 SST(1–1)(2)(¢0,1²) = ¢11,2²; ρ = 2 All SVTs between 2-tuples belonging to set classes [02] and [03], and again, between set classes [01] and [03], exhibit strongly parsimonious voice leading, but we note that ρ¢[01], [03]² = 2 and thus requires a composite SVT operation. We will now commence showing how the SST-succession classes are derived. We begin with a pair-succession of m-tuples, which is simply an ordered pair of m-tuples ¢A² and ¢B² and which may or may not be SSTrelated: ¢¢A²,¢B²²

example: ¢¢0,3,7²,¢0,6,8²².

If we choose m-tuples related by some SSTi, we append the subscript SSTi to the pair succession, creating a pair-succession of SSTi-related mtuples: ¢¢A²,¢B²²SSTi

example: ¢¢0,4,7²,¢11,4,7²²SST1–1.

This is simply a re-notation of the SST nomenclature we have been using up to this point. The example would have been expressed SST1–1(¢0,4,7²[037]) = ¢11,4,7²[037] using such a notation. We next create a pair-succession of m-tuple equivalence classes [A] and [B] which include the m-tuple elements ¢A² and ¢B², respectively. The equivalence classes each contain all transposed, inverted, and permuted versions of the m-tuple elements ¢A² and ¢B². More precisely, [A] is the Tn/TnI type of the permutation class Sm of ¢A². This class-type contains all possible

The Single-Voice-Transformation Model

59

orderings of all members of the Tn/TnI type of the unordered set of pitch classes in ¢A² ({A}): ¢[A],[B]² example: ¢[037],[026]² (c.f. the first pair-succession example). In our final step, we can limit our equivalence classes ¢[A],[B]² to only those pairs of m-tuple members which are in some SSTi relation to each other: DEFINITION: A pair-succession of SSTi-related m-tuple equivalence classes (called an SST-succession class for short) is an ordered pair of set classes [A] and [B] whose elements are all Tn/TnI and permuted forms of the m-tuples ¢A² ∈ [A] and ¢B′ ² ∈ [B] that are SSTi-related (¢¢A²,¢B′ ²²SSTi) and where ¢A² names the set class [A]: ¢[A],[B]²SSTi

example: ¢[037],[037]²SST3

Comparing this last example with the second pair-succession example, ¢¢0,4,7²,¢11,4,7²²SST1–1, is instructive. Both ¢0,4,7² and ¢11,4,7² are m-tuple members of SC [037]. The SSTi motion involves a half-step descent in the first order position. However, the naming m-tuple ¢0,3,7² ∈ [A], of which ¢0,4,7² is an inversional form, participates in an SST3 transformation with its m-tuple partner, ¢0,3,8² ∈ [B]. The change in “direction,” from a negative SST to a positive one, and the change in m-tuple order position from 1 to 3, occurs as a result of moving from an inversional form of the naming m-tuple ¢0,3,7², to the non-inverted naming m-tuple itself. As this m-tuple represents SC [A] = [037], the SST-succession class adopts its associated subscript, SST3. It is important to remember that an SST-succession class contains only those pairs of m-tuples that are themselves SST-related. The subscript SSTi acts as a filter to ensure that only the pairs of m-tuple members belonging to the equivalence classes [A] and [B] that maintain the SSTi-type relation between m-tuples ¢A² and ¢B′ ² will be included in this pair-succession.24 Continuing to use the example of ¢[037],[037]²SST3, which is equivalent to a neo-Riemannian L operation, we see that this particular SST-succession class contains the following elements (among many others): a) ¢¢0,3,7²,¢0,3,8²²SST3 c) ¢¢11,4,7²,¢0,4,7²²SST1 e) ¢¢8,0,3²,¢7,0,3²²SST1–1

b) ¢¢1,4,8²,¢1,4,9²²SST3 d) ¢¢0,3,8²,¢0,3,7²²SST3–1 etc.,

of which b) involves transposition of both m-tuples in the representative SST-related pair-succession ¢¢0,3,7²,¢0,3,8²²SST3, c) involves permutation

60

Chapter Three

and inversion of ¢A² and ¢B′ ², respectively, d) involves permutation and inversion of both ¢A² and ¢B′ ², and e) involves inversion and permutation of ¢A² and ¢B′ ², respectively. Only the transposed pair-succession b) retains the same SSTi-relation as the representative pair-succession a), in this case, SST3. This demonstrates the general principle that only Tn-related pairsuccessions will share the same internal SSTi relation with the representative pair-succession ¢¢A²,¢B′ ²²SSTi. The notation ¢[A],[B]²SSTi in our definition of an SST-succession class does not mean that the representative m-tuples ¢A² ∈ [A] and ¢B² ∈ [B] are privileged in any way; we are merely identifying the set types to which the various SSTi-related m-tuples belong. The apparent privileging of a specific SSTi in forming the SST-succession class ¢[A],[B]²SSTi is likewise illusory. Theoretically, we could have used the SSTi operation associated with any of the SST-related pair-successions a)–e) on p. 59 (as well as SST2 and SST2–1 for other permuted forms of ¢A² and ¢B′ ²) for the subscript identifying the SST-succession class, but ultimately identification of the class will be given by the SSTi which relates the m-tuple ¢A² to an m-tuple ¢B′ ², where ¢A² provides the name for its own SC type [A]. The notation ¢[A],[B]²SSTi should be adequate for identifying specific SST-succession classes throughout this study. However, in cases in which space is limited—such as with analytical overlay in music examples, in tables, and in graphs—it may be helpful to identify the SST-succession class with an abbreviated form of the notation. We may also wish to study or model music in which SST-succession classes occur—well, in succession. In all of these cases the notation will take the form of set classes interspersed with a particular SSTi subscript in square brackets, thus [A] [SSTi] [B] [SSTj] [C] [SSTk] ... which is understood to signify interlocking SST-succession classes, more properly notated as ¢[A],[B]²SSTi ¢[B],[C]²SSTj ¢[C],[D]²SSTk ... Whenever the set classes are unambiguously understood, for example, in the middle of an extensive discussion of the neo-Riemannian P operation, we may temporarily dispense with the SC labels (which we know will be [037]) and refer simply to [SST2], which is a shorthand notation for the SSTsuccession class ¢[037],[037]²SST2. If we wish to consider two or more distinct SST-succession classes that share the same subscripted SSTi relation we

The Single-Voice-Transformation Model

61

may refer to the subscript alone as [SSTi]. The reasons we may want to refer to the subscript alone will become clear later in this study. By Callender’s definition and the SST-succession-class definition, the distance ρ between the set classes in an SST-succession class will always equal 1: ρ¢[A],[B]² = min(ρ¢¢A′ ²,¢B′ ²² = 1 if ¢[A],[B]²SSTi because SSTs, by their very nature, involve min(ρ)-related m-tuples which are also in the Δh relation where h = 1. It should be briefly reinforced that SST-succession classes do not directly show voice-leading between SC types (an impossibility); they only show that min(ρ) voice-leading distances between certain members of the SC types they identify are equal to 1. Using an earlier example, we can easily see that SST3(¢0,3,7²) → ¢0,3,8² and that SST1–1(¢0,4,7²) → ¢11,4,7²; it is less readily apparent that both successions, expressed as SST-related pairsuccessions, are members of the SST-succession class ¢[037],[037]²SST3. The notation ¢[037],[037]²SST3 does not show us any specifics of the voiceleading possibilities between m-tuple members of SC [037] other than the somewhat vague stipulation that the order position corresponding to the third order position of ¢0,3,7² in a transposed, inverted, and/or permuted form of that m-tuple is affected. We must examine individual SST-related pairsuccessions belonging to the SST-succession class ¢[037],[037]²SST3 in order to see that, for instance, SST3(¢0,3,7²) → ¢0,3,8² and SST1–1(¢0,4,7²) → ¢11,4,7² are both operations which change the mode from a minor triad to a major triad (and vice versa) and that both correspond to the neo-Riemannian operation L. Thus the “dualistic” nature of the neo-Riemannian operation L is preserved in the SST-succession class subscript [SST3]. [SST3] is not the only SST-succession class of SST transformations which convert m-tuple members of [037] into other m-tuple members of [037]. Take for example ¢[037],[037]²SST2, which is a generalization of the min(ρ) voice leading that obtains by increasing by 1 the value of the second order position of the m-tuple that names the set class, i.e., SST2(¢0,3,7²) → ¢0,4,7², or by decreasing by 1 the value of the second order position of an inverted version of the m-tuple, i.e., SST2–1(¢0,4,7²) → ¢0,3,7². SSTsuccession class ¢[037],[037]²SST2 is thus equivalent to the neo-Riemannian operation P. Additional investigations will show that the composite SSTsuccession class ¢[037],[037]²(SVT1–1)2 is equivalent to the strongly parsimonious neo-Riemannian operation R. Speaking generally, we observe that more than one SST-succession class type may describe the same kind of voice-leading action between m-tuple pair-successions when the first m-tuple belongs to a symmetrical set class.

62

Chapter Three

As an example of this, consider ρ¢[036],[025]². We notice that ρ¢¢0,3,6²,¢0,2,5²²—the representative 3-tuples—equals 2, which is not min(ρ). However, we can list some 3-tuples belonging to SCs [036] and [025] which are min(ρ)-(SST-)related: ρ¢¢0,3,6²,¢0,3,5²² = 1 ρ¢¢0,3,6²,¢1,3,6²² = 1 and so forth. The SST-succession classes in this case would be, respectively, ¢[036],[025]²SST3–1 and ¢[036],[025]²SST1. However, the SST operations SST1 and SST3–1 have inversionally equivalent results when applied to ¢0,3,6² ∈ [036]: they each reduce the size of one outermost interval in the m-tuple from ic 3 to ic 2, resulting in an m-tuple ¢B′ ² ∈ [025]. Because we cannot tell whether the SST—and by extension, the SST-succession class—involves the SST3–1 or the SST1 relation, we may use either [SST1] or [SST3–1] as the SST-succession-class subscript. The development of SST-succession classes provides us with a way of noting optimal offset between the various pc set classes. This subscript SSTi assumes importance in the construction of geometrically regular optimal offset network graphs in two or more dimensions showing strongly parsimonious voice-leading possibilities between set classes of the same cardinality. This will be seen in chapter 4. Another possibility—which unfortunately lies outside the scope of the present study—would be the construction of SFTsuccession classes and the attendant spatial graphs generalizing min(ρ) voice-leading connections between the different set classes. In such a system, “parsimony” would necessarily be measured in units of ic 5 rather than ic 1.

Split-Succession and Fuse-Succession Classes Just as SST-succession classes could be generalized from SST-related mtuple pair-successions, we can imagine split-succession and fuse-succession classes as ways of relating set-class pair-successions containing representatives that themselves are related by a split or fuse transformation. This allows us to fashion network graphs showing parsimonious voice-leading connections between members of set classes of different cardinalities. As we saw earlier in this chapter, a split is actually the composition of two functions, SPLITi followed by an SVT. Likewise, a split-succession class is comprised of two pair-ordered SCs whose m-tuple members are related by some operation SPLITi followed by an operation SVTj.

The Single-Voice-Transformation Model

63

DEFINITION: A pair-succession of split-related m-tuple equivalence classes (called a split-succession class for short) is an ordered pair of equivalence classes [A] and [B] whose elements are all Tn/TnI and permuted forms of the m-tuples ¢A² ∈ [A] and ¢B′ ² ∈ [B] that are splitrelated (SVTj((SPLITi)(¢A²))) → ¢B′ ²) and where ¢A² names the equivalence class [A]: ¢[A],[B]²split

example: ¢[037],[0347]²split

We can more precisely identify the split-succession class type by dividing the subscript split into its constituent parts: ¢[A],[B]²SVTj(SPLITi), but this may not always be desirable, especially if space limitations (such as in graphs or musical examples) are a factor. As with SST-succession classes, split-succession classes may take the form of a shorthand notation, especially if one wishes to present a series of interlocking split-succession classes, arranged so as to minimize the space consumed by analytical clutter: [A] [split] [B] [split] [C] [split] [D] ..., or, showing more detail: [A] [SVTj(SPLITi)] [B] [SVTl(SPLITk)] [C] [SVTn(SPLITm)] [D] ... This is more properly notated as ¢[A],[B]²SVTj(SPLITi) ¢[B],[C]²SVTl(SPLITk) ¢[C],[D]²SVTn(SPLITm) ... As an example of a split-succession class, consider two separate split transformations on m-tuples, SST3(SPLIT2(¢0,3,7²)) = SST3(¢0,3,3,7²) = ¢0,3,4,7² and SST2–1(SPLIT2(¢5,9,0²)) = SST2–1(¢5,9,9,0²) = ¢5,8,9,0². In this example, both transformations employ the same SPLITi operation, SPLIT2, but each is followed by a different SSTj operation. However, both composite operations can be generalized to the same split-succession type, ¢[037],[0347]²SST3(SPLIT2). We note that this split-succession class’s subscript is identified by the split operation involving [A]’s m-tuple representative ¢A². As with the SST-succession classes, the split-succession class subscript SPLITi does not privilege a specific operation SPLITi on m-tuples, but rather merely adopts its name to identify the class of all SPLITi-related pair-

64

Chapter Three

successions between transposed, inverted, or permuted forms of ¢A² and ¢B′ ². The sub-subscript i references the m-tuple order position of ¢A² ∈ [A] which undergoes the content duplication. The subsequent subscript SVTj usually takes the form of an SSTj on the jth order position of the m-tuple ¢A², here generalized to all equivalence-class members ¢A² of [A] that preserve the SSTj relation with some m-tuple member ¢B′ ² ∈ [B]. i may or may not be equal to j. Fuse-succession classes can be derived in a similar manner. Like a split, a fuse is the composition of two functions, SPLITj–1 preceded by an SVT. A fuse-succession class is comprised of two pair-ordered SCs whose m-tuple members are related by some operation SPLITj–1 preceded by an operation SVTi. DEFINITION: A pair-succession of fuse-related m-tuple equivalence classes (called a fuse-succession class for short) is an ordered pair of equivalence classes [A] and [B] whose elements are all Tn/TnI and permuted forms of the m-tuples ¢A² ∈ [A] and ¢B′ ² ∈ [B] that are fuserelated (SPLITj–1((SVTi)(¢A²))) → ¢B′ ²) and where ¢A² names the equivalence class [A]: ¢[A],[B]²fuse

example: ¢[0347],[037]²fuse

We can more precisely identify the fuse-succession class type by separating the subscript fuse into its constituent parts: ¢[A],[B]²SPLITj–1(SVTi), but this may not always be desirable, especially if space limitations (such as in graphs or musical examples) are a factor. As with the split-succession classes, we may use a shorthand notation for fuse-succession classes, particularly if one wishes to present a series of interlocking fuse-succession classes, arranged so as to minimize the space consumed by analytical clutter: [A] [fuse] [B] [fuse] [C] [fuse] [D] ..., or, showing more detail: [A] [SPLITj–1(SVTi)] [B] [SPLITl–1(SVTk)] [C] [SPLITn–1(SVTm)] [D] ... This is more properly notated as ¢[A],[B]²SPLITj–1(SVTi) ¢[B],[C]²SPLITl–1(SVTk) ¢[C],[D]²SPLITn–1(SVTm) ...

The Single-Voice-Transformation Model

65

As an example of a fuse-succession class, let us look at the reverse of the split-succession class scenario described above. Consider SPLIT3–1(SST3–1(¢0,3,4,7²)) = SPLIT3–1(¢0,3,3,7²) = ¢0,3,7² and SPLIT3–1(SST2(¢5,8,9,0²)) = SPLIT3–1(¢5,9,9,0²) = ¢5,9,0². In this example, both transformations employ the same SPLITj–1 operation, SPLIT3–1, but the preceding SSTi differs in each case. Nevertheless, both composite operations can be generalized to the same fuse-succession type, ¢[0347],[037]²SPLIT3–1(SST3–1). We note that this fuse-succession class’s subscript is identified by the fuse operation involving [A]’s representative m-tuple ¢A². The fuse-succession class subscript SVTi usually takes the form of an SSTi relation between m-tuples ¢A² and ¢B′ ², a relation that holds between all pair-ordered equivalence-class members ¢A² of [A] that preserve the SSTi relation with some m-tuple member ¢B′ ² ∈ [B]. Again, i may or may not be equal to j in the associated subscript SPLITj–1.

CHAPTER 4 GRAPHICAL REPRESENTATIONS OF PARSIMONIOUS VOICE-LEADING SPACES Graphs of Voice-Leading Transformations on PCs and PC Sets of the Same Type In recent years a number of diagrams or graphs have appeared that chart voice-leading connections in pitch-class space between chords that differ by a single semitone. These graphs of voice-leading space are consciously modeled upon a much older construct, Hugo Riemann’s Tonnetz, which elegantly portrays strongly parsimonious voice leading between major and minor triads. Figure 4-1 is a version of the Tonnetz that appears in Riemann’s “Ideen zu einer ‘Lehre von den Tonvorstellungen’” dating from 1914–15.1 The diagram can be read as follows: each diamond represents the four pitches that constitute the notes of a major and a minor triad with the same root (which is labeled in the center of the diamond). Although Riemann does not explicitly label his Tonnetz in this way, it may help to imagine a series of horizontal lines that bisect each diamond. These additional lines help delineate distinct major and minor triads (major triads pointing upward, minor triads pointing downward). Specific pitches are located at the vertices. The exact pitch content of the vertices can be inferred from Riemann’s diagram: in the case of the diamond labeled “c,” pitches are, starting at the left-most vertex and moving clockwise: C, E, G, and E(.2 Figure 4-2 displays a reconfigured Tonnetz, modified to show 1) the horizontal lines that represent fifth relations, 2) pitch classes at the vertices of the triangles comprising major and minor triads, and 3) equal temperament. The arrows labeled L, P, and R show the voice leadings that would obtain if their corresponding neo-Riemannian transformations were effected on a C-major triad, flipping a triangle about one of its sides and obtaining a new triad of minor mode. Both the L (Leittonwechsel) [“leading-tone exchange”] and P (Parallel) operations hold two common tones and exchange the third for a pitch a semitone away, while the R (Relative) operation holds two common tones and exchanges the third for a pitch two semitones away. The Tonnetz can thus be thought of as a kind of diagram of voice leading between triads of the same set-class type. In this case the set class is [037], but Tonnetze can also be constructed in two and three dimensions to show voice leading between chords of other set-class types.3 These Tonnetze do not always maintain the property of smooth voice leading.

Graphical Representations of Parsimonious Voice-Leading Space

67

Fig. 4-1: The Riemann Tonnetz (from “Ideen zu einer ‘Lehre von den Tonvorstellungen’” (1914–15), reproduced in Hyer 1995)

Fig. 4-2: A “traditional” Tonnetz (from Gollin 1998)

As useful as Tonnetze can be in displaying a limited number of voiceleading transformations in a specific compositional space, they are by their very nature somewhat limited. Most importantly, they only show voice leading between chords of the same SC type, thus making them as limited an

68

Chapter Four

Fig. 4-3: Voice leading as determined by set inversion operations L, P, R, and R′ (from Morris 1998)

analytical/compositional tool as are Tn/TnI-based set transformations in describing voice leading.4 In fact, if we are to interpret the voice leading of L, P, and R transformations in this way (as being determined by set inversion),5 the individual pc mappings between sets do not preserve common tones and only one pc maps by parsimonious motion. This disjunct between inversional set transformations and the apparent parsimony of the voice leading suggested by the Tonnetz is familiar to us from chapters 2 and 3, and is exemplified by Figure 4-3, from Morris 1998. In this figure we see that common tones are not held as “pivot” pitches but must map to each other in order to effect a “flip” of the triangle in an inversion. Morris writes, “Note how the common pcs between triads are not presented by the parts of [Figure 4-3]; in other words, the transformational voice-leading frequently ‘contradicts’ the proximate voice-leading.”6 Taken as a whole, then, a contextual inversion operation such as L, P, or R on a triadic set does indeed result in another triadic set that preserves two common tones with the previous set, with the remaining pc in the second triad being parsimoniously related to its corresponding pc in the first set. It is when one tries to trace individual voice-leading paths by looking at the pcs inside the sets that it is revealed that such apparent parsimony is for the most part illusory. On a more general level, Tonnetze can be thought of as types of “ball-

Graphical Representations of Parsimonious Voice-Leading Space

69

and-stick” graphs in which the “balls” or nodes represent pitch classes and the “sticks” or connective lines represent a consistent, geometrically replicated intervallic relation between pitch classes. This type of pc-space representation is akin to but not quite the same as Lewin’s “node-and-arrow” network graphs of GISs.7 Both are transformational, but with the distinction that the Tonnetz lines are not exactly the same as Lewin’s transformational network arrows. The latter are unidirectional and are associated with a specific operation; the former are not really transformations but are rather an specific unordered interval between pitch classes. If we wish to find something more closely related to a Lewin arrow in the Tonnetz of Figure 4-2 we should look no further than the L, P, and R arrows. Unlike Lewin network arrows, these bi-directional transformational arrows do not point toward the nodes of the graph but rather to the pc triads that form the triangular spaces in the Tonnetz.8 The “nodes” of the Tonnetz are exclusively pitch classes, but the nodes of Lewin networks can be anything from individual pcs to pitchclass sets to lynes of pcs (Lewin networks’ node content usually consists of triads in neo-Riemannian theory). This observation leads us to the next type of graph, which on its surface resembles the Tonnetz but is actually more closely related to Lewin transformational network graphs. If we replace the pitch classes of each node in a Tonnetz with a pitch-class set, and the connective lines of the Tonnetz with specific voice-leading transformations (not necessarily parsimonious) we obtain a graph which shows a periodic arrangement of one or more voiceleading transformations between chords of the same SC type. This graph is a geometric dual of the Tonnetz, “constructed by mapping each [triangular] face in the Tonnetz to a vertex and joining vertices with an edge if their corresponding faces in the Tonnetz share an edge.”9 It should be noted that the structure of the Riemannian Tonnetz will not be isomorphic with that of our single-SC voice-leading graph. The structure of the latter will be dependent on the SC employed and the types of voice-leading transformations connecting the pc sets. Figure 4-4 shows a graph which is the geometric dual of the standard Riemannian Tonnetz. In the upper, two-dimensional lattice, major triads correspond to nodes with upper-case designations and minor triads to those with lower-case designations. All voice leading is by semitone (L and P relations) except for the horizontal dashed lines, which represents the R transformation with voice leading by whole step. The lower part of Figure 44 shows the toroidal nature of the two-dimensional lattice. This can be obtained by joining the chords at the top of the lattice to their pitch-class equivalents near the bottom of the lattice to make a cylinder, then joining the open ends of the tube to complete the torus (being sure to make a third-of-arevolution twist at one end to ensure that corresponding chords line up). A similar torus can be fashioned from the Tonnetz of Figure 4-2.

70

Chapter Four

Fig. 4-4: A geometric dual graph of the Tonnetz (“chicken-wire torus,” from Douthett and Steinbach 1998)

In moving from pitch classes in Figure 4-2 to pitch-class sets in Figure 44 as the nodes of the graph, we also lose a little bit of information. Specifically, it is no longer immediately apparent which member of the triad moves

Graphical Representations of Parsimonious Voice-Leading Space

71

parsimoniously while the other two pitches preserve common tones. Admittedly, this is not too difficult to figure out in Figure 4-4, but it does require a modicum of thought. We will see that this lack of specificity in showing the exact voice leading between two chords becomes more of a problem as we begin to discuss graphs that show parsimonious voice leading between chords which are members of set classes with larger cardinalities and especially between chords belonging to different set-class types.

Fig. 4-5: Graph showing parsimonious voice leading between members of SC [0258] where two voices move by semitone (“pipeline” torus from Douthett and Steinbach 1998)

As an example of this, consider Figure 4-5, a three-dimensional graph that Douthett and Steinbach call the “pipeline.” The pipeline shows all voice leading between members of set class [0258]—dominant and halfdiminished seventh chords—where two chord members are retained as common tones and two move by semitone. This type of relationship between two chords is thus parsimonious according to our definition in chapter 3 and also by Douthett and Steinbach’s formulation, where it is given the label Pm,n = P2,0, in which P stands for “parsimonious,” m stands for the number of voices that move by half step, and n stands for the number of voices that move by whole step.10 In the pipeline graph, solid lines connect P2,0-related seventh chords. Eight additional voice-leading connections between pairs of circular crosssections are not shown, but the diagram at the bottom of the figure shows

72

Chapter Four

where these occur (in the panels between cross-sections). The pipeline is really a torus, as the ends of the pipe may be joined if one end is twisted a quarter turn. We can still determine which pitch classes are retained as common tones and which move by half step between any two nodes on the pipeline, but this determination now takes approximately twice as long to calculate as did the graph of [037]-type triads in Figure 4-4 due to the increased cardinality of the pc sets in the network.

Graphs of Voice-Leading Transformations between PC Sets of Different Types Let us now turn to graphs that display parsimonious voice leading between chords pertaining to a limited number of set classes of the same cardinality. Such graphs are useful inasmuch as they show smooth voiceleading connections between chords which commonly appear juxtaposed in various combinations in certain styles of music, most notably in lateRomantic/early-Modernist pieces by composers such as Liszt, Wagner, Franck, Debussy, and Scriabin.

Fig. 4-6: “Cube dance” showing parsimonious voice leading between major, minor, and augmented triads (from Douthett and Steinbach 1998)

Graphical Representations of Parsimonious Voice-Leading Space

73

The graph in Figure 4-6 shows strongly parsimonious voice leading between all major, minor, and augmented triads and is called the “cube dance” by Douthett and Steinbach (1998). Although the “distance” between augmented triads and neighboring major/minor triads appears to be greater than distances between hexatonic major/minor-chord cycles (represented by bold lines in the graph), this is merely a result of the geometry of the graph; all connective lines show voice leading by one semitone in voice of each triad. A related graph (not shown here) is the so-called “Weitzmann’s waltz,” which again displays voice-leading transformations between major, minor, and augmented triads, but instead is structured so as to show P2,0 relations between the triads (i.e., two of the three voices in each triad move by half step and thus are neither parsimonious nor strongly parsimonious, at least according to the criteria established in the previous chapter). Several graphs showing strongly parsimonious voice leading between commonly used seventh chords have been constructed by Douthett and Steinbach (1998, 246–56). Three of these are shown in Figure 4-7. The first of these, Figure 4-7a, is one of four EnneaCycles in which major-minor/halfdiminished and minor seventh chords (of SC types [0258] and [0358]) belonging to the same enneatonic set are strongly parsimoniously related. All adjacent pairs of chords in this graph differ by a single semitone (P1,0), except for the voice leading between half-diminished and major-minor seventh chords, in which one voice moves (parsimoniously) by two semitones (P0,1). There are four distinct Ennea-Cycles, the other three being transpositions of Figure 4-7a by T3, T6, and T9, and which account for all 36 distinct major-minor, half-diminished, and minor seventh triads. A related construct, not shown here, are the three transpositionally equivalent OctaTowers, which show major-minor, half-diminished, and minor seventh chords related solely by P1,0. Strongly parsimonious voice-leading bridges exist between the four EnneaCycles. These are shown in what Douthett and Steinbach call the “towers torus,” of which Figure 4-7b is a two-dimensional representation. The towers torus shows four distinct P1,0 voice-leading operations and one P0,1 operation (the horizontal dashed lines in the graph). Like the chickenwire torus, pitch classes that are common to chords marking the corners of a face are listed in the center of that face. As with some of the other graphs we have examined (the pipeline, the chicken-wire torus, and the EnneaCycles, for instance), we can still determine exactly which chord members are held as common tones along the tower torus’s face edges and which pitch class changes, but again, this identification is not explicitly stated and requires a little thought. Figure 4-7c introduces members of a third set-class type, [0369], to the graphs of strongly parsimonious relationships between members of set

74

Chapter Four

a)

b)

c)

Fig. 4-7: Three representations of strongly parsimonious seventh-chord voice-leading space: a) an EnneaCycle; b) the towers torus; and c) the power towers (from Douthett and Steinbach 1998)

Graphical Representations of Parsimonious Voice-Leading Space

75

classes [0258] and [0358] that we have already examined in Figures 4-7a and b. This representation is called the “power towers” graph by Douthett and Steinbach, and shows all P1,0 voice-leading connections between members of the three set classes. In the power towers graph, fully diminished seventh chords act as parsimonious bridges between the three individual OctaTowers graphs. What are not shown here, however, are the P0,1 voiceleading connections we have previously seen in the EnneaCycles and towers torus graphs. Perhaps this is just as well, as this would have resulted in a bewildering clutter of connective lines, whereas the single-move P0,1 transformation very clearly subsumes a pair of consecutive P1,0 transformations. A subtle problem inherent in the EnneaCycles graph is its failure to distinguish clearly between two different “sizes” of voice-leading connections, in this case between P1,0 and P0,1 parsimonious relations. This proves not to be a problem with the chicken-wire and tower tori, as they make use of different styles of connecting lines to differentiate between single-semitone displacement and two-semitone displacement (the latter represented by dashed horizontal lines in both graphs). In the cube dance and power towers graphs this issue does not even arise, as all chords are related by the same strongly parsimonious relation, P1,0. In order to simplify visual presentation, the graphical representations of voice-leading space which we will examine in the remainder of this chapter will all make use of connective lines showing P1,0 relations exclusively. In other words, all voice-leading transformations between objects in these graphs are strongly parsimonious and can be explained by a single SST. P0,1 relations can be obtained by tracing connective lines (transformations) through two nodes or objects, provided that the voice leading occurs in the same voice. Should the two-node move involve more than one voice, the beginning and ending objects are related by P2,0. The reader may have noticed that I have now moved away from using the term “chord” and have substituted the more general, somewhat vague term “object.” The reason for this will soon become readily apparent. In our discussion of graphical representations of parsimonious voice leading, we began by examining the Tonnetz, which demonstrates voice-leading transformations on triads by equating contextual inversion operations with “flips” about the edges of triangles whose vertices are occupied by specific pitch classes. We than progressed to a discussion of geometric duals of the Tonnetz and other related structures (such as the pipeline) where each Tonnetz face maps to a vertex and connective lines represent parsimonious transformations between representatives of the same set class. The principal difference, then, between the Riemannian Tonnetz and the chicken-wire torus is that vertices (or nodes) in the latter represent pitch-class sets instead of singular pitch classes, and that the connective lines in the latter represent a voice-

76

Chapter Four

leading transformation rather than an interval between pitch classes as in the former. We then modified the resulting graphs by allowing for the use of chords belonging to more than one set-class type while retaining the use of lines as symbolic of parsimonious voice-leading transformations.

Graphs of Optimal Offset between SC Types of the Same Cardinality The next series of graphs completes our move toward displaying parsimonious connections between larger categories of objects, from single pitch classes as objects in the Tonnetz, to pitch-class sets as objects in the other graphs examined up to this point, and now to set classes of a specific cardinality themselves as objects. Instead of showing voice-leading parsimony between all the set-class representatives of a limited number of set classes, these graphs show all strongly parsimonious min(ρ) voice-leading distances between all set classes of the same cardinality. In other words, connective lines in the previous graphs represented SSTs; in the following graphs the lines represent distinct SSTi relations associated with SST-succession classes. Figure 4-8 shows a two-dimensional graph of optimal offset between trichordal set classes as constructed by Joseph Straus. As we have seen in chapter 2, “optimal offset” is Straus’s term for the minimum voice-leading displacement between any two set classes. The optimal offset (Callender’s Δh) between set classes directly connected by a line in this graph is equal to 1; offset increases through simple addition as one moves through x number of set-class nodes to a particular set-class destination. Straus’s graph is as notable for what it excludes as for what it includes. For example, as with previous graphs, nowhere does it explicitly show which “voice” undergoes the non-trivial transformation that leads to the parsimoniously related set class, although this is impossible to determine now that we are dealing with generalized SST-succession classes and not individual SSTs. We can, however, begin to note which lines in the graph correspond to the SSTi relation associated with specific SST-succession classes ¢[A],[B]²SSTi—abbreviated according to the conventions in chapter 3 as [A] [SSTi] [B] or even simply [SSTi] when it is clear which SCs are involved in the SST-succession class—when we recall that ¢A² names the set class [A] in the SSTi-related pair-succession ¢¢A²,¢B′ ²²SSTi. At first glance this appears easy to deduce, especially if one follows either horizontal or vertical lines on the map. [014] [SST3] [015], as SST3(¢0,1,4²) → ¢0,1,5². Upon second and subsequent glances, however, the specific [SSTi] paths are not nearly so simply accounted for. A couple of examples will adequately demonstrate this. Set class [027] appears to have a

Graphical Representations of Parsimonious Voice-Leading Space

77

3-1 012

3-2 013

3-3 014

3-6 024

3-4 015

3-7 025

3-5 016

3-8 026

3-10 036

3-9 027

3-11 037

3-12 048

Fig. 4-8: Graph of voice-leading space showing optimal offset between trichordal SCs (from Straus 2003)

total of 3 single-[SST] connections with other set classes: one each to [016], [026], and [037]. However, if we simply look at the graph this way, we are momentarily forgetting that if we consider [027] as the equivalence class of the 3-tuple ¢0,2,7², the element comprising each order position may map by either T1 or by T1–1 as part of an SST operation for a total of six voiceleading possibilities. Let us see what the results are of systematically applying T1 and T1–1 to each order position of the 3-tuple in turn: SST1(¢0,2,7²) = ¢1,2,7²[016]

SST1–1(¢0,2,7²) = ¢11,2,7²[037]

78

Chapter Four

SST2(¢0,2,7²) = ¢0,3,7²[037] SST3(¢0,2,7²) = ¢0,2,8²[026]

SST2–1(¢0,2,7²) = ¢0,1,7²[016] SST3–1(¢0,2,7²) = ¢0,2,6²[026]

Before performing these calculations it was already fairly easy to see, just by looking at Figure 4-8, that ¢0,2,7² maps to ¢0,3,7² via a single SST operation that transposes the pitch class in the second order position by T1. But it was not nearly as immediately clear that an SST that transposed the pc in the second order position by T1–1 would result in an ordered 3-tuple belonging to set class [016]. Likewise, by looking at the vertical line extending from [027] to [026] in Figure 4-8, one readily sees that the voice-leading function SST3–1 on ¢0,2,7² transposes the pc in the third order position by T1–1, but it is not immediately apparent that an application of SST3 to the 3tuple results in another representative of the same set class, [026]. For an extreme case of this phenomenon, look no further than SC [048], which appears to have a single min(ρ) [SSTi] connection to [037], when in reality any SSTi or its inverse SSTi–1 applied to ¢0,4,8² (transposing any order position by T1 or by T1–1) results in a 3-tuple representative of SC [037]. When accounting for optimal offset between set class types, we can speak of the total number of distinct voice-leading paths between any two set classes.11 These paths are all the distinct [SSTi] subscripts associated with the SST-succession classes comprising the same pair-succession classes [A] and [B]. For example, in the case of [027] there exist two distinct voiceleading paths to SC [016], two to SC [026], and two to SC [037]. With SC [048], six paths can be traced through distinct [SSTi]s that result in representatives of SC [037]. With SC [026] = [A] as part of an SST-succession class, on the other hand, there are no duplicate paths. There will always be a total of 2m paths leading from a set class of cardinality m if all possible [SSTi]s are accounted for, some of which may lead to the same set class and some of which may lead to a set class of apparent lower cardinality (more properly a multiset class). Appendix A lists the number of distinct SCs [B] which can form an SST-related equivalence-class pair-succession with a given SC [A] for all trichordal and tetrachordal set classes, allowing for splits and fuses. The appendix also contains a list of the distinct [SSTi] paths associated with the SST-succession classes. Figure 4-9 shows how information about specific [SSTi] paths may be incorporated into Straus’s graphical representation. Some of the more obvious features of Figure 4-8 are now confirmed in this graph: motion “downward” or in a “southerly” direction represents, as we previously intuited, [SST3] relations. Likewise, horizontal motions to the right (or “east”) are instances of [SST2] relations between equivalence-class pair-successions. The key of Figure 4-9 shows us the remaining relations associated with the

Graphical Representations of Parsimonious Voice-Leading Space

79

Key:

3-1 012

3-2 013

[SST1] =

and red

[SST1–1] =

and dotted red

[SST2] =

and yellow

[SST2–1] =

and dotted yellow

[SST3] =

and green

[SST3–1] =

and dotted green

3-3 014

3-6 024

3-4 015

3-7 025

3-5 016

3-8 026

3-10 036

3-9 027

3-11 037

3-12 048

Fig. 4-9: Graph of specific [SSTi]s between trichordal equivalence-class pair-successions (¢[A],[B]²SSTi). A color version of this Figure is shown in the Centrefold. solid black lines on Straus’s map: directed motion to the “north” manifests [SST3–1], motion to the “west” manifests [SST2–1], motion to the “northwest” manifests [SST1], and motion to the “southeast” manifests [SST1–1]. In addition to these “normal,” geometrically regular [SSTi]s, there exist instances in which one or more additional [SSTi]s duplicate these paths. These duplicate paths are shown by color-coded arrows and rings in Figure 4-9; the figure’s key identifies these. The trichordal SST-succession-class graph, marked with additional [SSTi] paths (Figure 4-9) shows us an interesting property associated with

80

Chapter Four

SST-succession classes. This property involves the retrogradability of SSTsuccession classes. In other words, what is the result of swapping order positions in an SST-related equivalence-class pair-succession? This could almost be thought of as an “inverse” operation, were it not for the fact that we are dealing with objects (SST-succession classes) and not functions (SSTs) between m-tuples. In any case, the ordered-pair position swap has the effect of reversing the voice-leading direction between the m-tuples which compose the equivalence classes. Curiously, the SSTi relation between the equivalence-class pairs sometimes takes the form of a true inverse SSTi–1 relation when the equivalence-class pair order positions are swapped, but at other times the inverse SSTi–1 relation does not obtain between [B] and [A]. In this second scenario, what one assumes would be the inverse relation SSTi–1 instead obtains between [B] and some other SC [C]. I.e., ¢[A],[B]²SSTi, when reversed, sometimes takes the form ¢[B],[A]²SSTi–1, but at other times ¢[B],[C]²SSTi–1. From Figure 4-9 we see that reversed SSTsuccession classes assume the form ¢[B],[A]²SSTi–1 (with the inverse SSTi–1 relation) when they are aligned in parallel as part of the regular, geometric lattice in the trichordal graph (indicated by solid black lines), but if one seeks an “inverse” SSTi–1 relation to a duplicate path (indicated by colored arrows) ¢[A],[B]²SSTi, the alternate “inverse” SST-succession-class relation ¢[B],[C]²SSTi–1, obtains instead. As an example of this property, let us consider the SST-succession-class ¢[024],[025]²SST3. As this voice-leading path is part of the regular geometry of the graph, if we “undo” the voice leading by swapping the SSTsuccession-class order positions, the SSTi-relation subscript becomes the inverse of its previous form: ¢[025],[024]²SST3–1. However, the graph also shows us the “additional” SST-succession-class ¢[024],[025]²SST1–1 which gives us another way of relating members of SCs [024] and [025] parsimoniously. But the inverse SSTi-relation subscript takes us on a voice-leading path away from [025] to a completely different equivalence class if we place the second pair-succession position in the first ordered-pair slot: ¢[025],[014]²SST1. Let us examine some of the more interesting properties of SSTsuccession-class optimal-offset paths on this graph of trichordal voiceleading space. Three of the twelve set classes [A] do not appear to have duplicate [SSTi]-related voice-leading paths to [B], namely, [014], [015], and [026]. This is misleading, as we recall that all trichords have the potential to be transformed by 3 +2, or 6 possible SSTis. Only [026], at the center of the map, enjoys [SSTi] relations with six distinct SCs [B] and has no duplicate paths. The two “missing” [SSTi] relations between [014] and [015] and some SC [B]—[SST1] and [SST2–1]—are duplicate fuse-like paths, each relating [014] and [015] to the multiset classes [003] and [004], or [004] and [005]

Graphical Representations of Parsimonious Voice-Leading Space

81

respectively. In three of the SCs—[013], [025], and [037]—m-tuple members are transformed to another member of the same set class under [SST2] (¢[A],[A]²SST2), and m-tuple members of [037] are also transformed to another member of [037] under [SST3].12 This latter property, unique to [037] among trichordal set classes, is indicative of a Cohn-function set. This designation, as described by Lewin, is based upon the premise that pc sets can be represented by functions. For example, the major triad can be represented by the mod-3 function (435), where the integers represent interval sizes between members of the triad. According to Lewin, “A mod N function is Cohn flipped when an exchange of different (NB) values at adjacent arguments gives rise to a rotated retrograde of the original function.” In the case of the major triad (435), if 5 is exchanged with 3 the result is (453), a rotated retrograde of the original. Lewin goes on to say, “A Cohn function is a function with arguments mod N that can be Cohn-flipped in two different ways—i.e. at two different adjacent-argument locations.”13 The major triad (435) may also be flipped by exchanging 4 and 3, thus becoming (345), another rotated retrograde of the original. The practical upshot of this is that {037} may be inverted two different ways: by preserving the interval 7 while exchanging that interval’s vertices, and by preserving the interval 3 while exchanging that interval’s vertices. We need look no further than Figure 4-3, Morris’s flips of [037] triangles about the Tonnetz, to see Cohn functions in operation. Here we notice that the first operation, L, inverts the C-major triad about interval 3 (between pcs 4 and 7) but also that the vertices must be exchanged in order for the triangle to achieve the proper orientation, substituting pc 11 for 0 and inverting to an E-minor triad. Next comes an R operation, where the interval that is preserved is 4 (between pcs 0 and 4); the vertices are exchanged, and the E-minor triad inverts to a G-major triad. The third operation, P, does not require an exchange of vertices, but the set class [037] is nonetheless a Cohn function, as only two of the three flips are required to be Cohn flips. Looking more closely at Figure 4-9, it is fairly easy to recognize that ¢[037],[037]²SST2 corresponds to a set-classpreserving neo-Riemannian P operation on members of [037], and that ¢[037],[037]²SST3 corresponds to the set-class-preserving L operation. The R operation, as it requires a whole-step transposition in one voice (corresponding to Douthett and Steinbach’s P0,1 designation), is not directly shown on this map but can be traced by following two [SST1–1]-relation segments, moving southeastward once to [048], and then returning along the same path (the dotted red arrow in Figure 4-9). Set classes—like [037]—that are Cohn functions are also the only set classes capable of participating in what Richard Cohn calls maximally smooth cycles (Cohn 1996, 15–6). Looking at Figure 4-9, it is also rather striking that SC [037] appears to

82

Chapter Four

be the center of “action” in terms of optimal offset. No other trichordal set class [B] enjoys a greater number of [SSTi] relations with some SC [A] than does [037], where it serves in the [B] order position of 13 distinct SSTsuccession classes. Additionally, [037] as [A] in an SST-succession class enjoys more distinct [SSTi] relations with some SC [B] than any other trichord, if split-succession classes are allowed for (see Appendix A). Not only has [037] been the trichord of primary importance in the history of Western music for a variety of well-documented reasons (its Tn form (047)’s derivation from simple harmonic ratios in the overtone series being one of the most obvious), its significance in studies of parsimonious voice leading is second to none. Whether mapping to themselves as Cohn functions through L, P, and R neo-Riemannian operations, or mapping from or being mapped to as part of SST-related m-tuple pair-successions, members of SC [037] will play a significant rôle not only in the voice leading of late Romantic composers such as Liszt and Wagner (as is to be expected), but also in that of many late twentieth-century composers who make use of triadic sonorities divorced from a system of functional tonality. For more on the voice-leading potential of the elements of this very special set class, see Appendix B, which tabulates all possible single [SSTi] relations associated with all possible trichordal SST-succession classes ¢[A],[B]²SSTi. Figure 4-10 is Straus’s graph of optimal offset between tetrachordal set classes.14 At first glance it appears to be similar to the graph of trichordal optimal offset, but there are some important differences. Specific [SSTi] relations between pair-ordered SCs are not shown in this figure, as they are in Figure 4-9, partly because there are so many duplicate paths that would need to be recognized, but mostly because the geometry of this graph is not consistent, as it is in the trichordal graph. We can observe this if we attempt to associate motion in specific directions with specific [SSTi] relations. It is clear that motion to the south is by [SST4] and motion to the north is of course its inverse, [SST4–1]. Likewise, motion to the east is by [SST3] and motion to the west is by [SST3–1]. We then note that motion to the southwest is by [SST2] and to the northeast is by [SST2–1]. Problems begin to arise, however, when we try to account for [SST1] and its inverse. For one thing, there are two distinct southeast-to-northwest-trending linear systems, one oriented slightly more toward the north-northwest and south-southeast than the other. Upon closer examination, it appears both systems work the same way in that southeast-to-northwest motion between certain set classes—for instance, between [0235] and [0124] and then again between [0347] and [0236]—is indeed by the [SST1] relation. But not all southeast-to-northwest voice leading is relatable by [SST1]. In the larger, “true” southeast-tonorthwest system, [SST1] relations alternate with [SST2] relations, while in the other system, ¢[0148],[0347]²SST4. Thus identically oriented line

Key:

3-1 012

3-2 013

[SST1] =

and red

[SST1–1] =

and dotted red

[SST2] =

and yellow

[SST2–1] =

and dotted yellow

[SST3] =

and green

[SST3–1] =

and dotted green

3-3 014

3-6 024

3-4 015

3-7 025

3-5 016

3-8 026

3-10 036

3-9 027

3-11 037

3-12 048

Fig. 4-9: Graph of specific [SSTi]s between trichordal equivalence-class pair-successions (¢[A],[B]²SSTi)

Key: [SST1] and [SST1–1] = red [SST2] and [SST2–1] = yellow [SST3] and [SST3–1] = green [SST4] and [SST4–1] = blue

Fig. 4-12: Three-dimensional graph of specific [SSTi]s between tetrachordal equivalence-class pair-successions (¢[A],[B]²SSTi)

Key: [SST1] and [SST1–1] = red [SST2] and [SST2–1] = yellow [SST3] and [SST3–1] = green [SST4] and [SST4–1] = blue Solid arrows = [SSTi] Dashed arrows = [SSTi–1]

Fig. 4-13: Three-dimensional graph of specific [SSTi]s between tetrachordal equivalence-class pair-successions (¢[A],[B]²SSTi), including additional parallel [SSTi] paths

Fig. 4-14: Cohn’s tetrahedral graph of tetrachordal voice-leading space (from Cohn 2003).

Layer 1

Layer 2 013 012 024

0235

0124

0134 0123 Layer 4

Layer 3 014

015

025

026

0126

0156 036

0145

0125 0236

0246

0237

0257 0347 0247 0146

0135 Layer 5

0136

016 027

0167

0127

037

0137

0268 0148

0158 0157

0258

0248

0137

0147

Fig. 4-16: Sectional layers comprising the trichordal split-succession-class graph

Fig. 4-17: Top and side views of superimposed layers of the incipient trichordal-tetrachordal split-succession-class graph

Fig. 4-18: Graph of trichordal SST-succession classes and trichordal splitsuccession classes

Fig. 4-19: Graph of trichordal split-succession classes

Key: [SST1] and [SST1–1] = red

[SST3] and [SST3–1] = green

[SST2] and [SST2–1] = yellow

[SST4] and [SST4–1] = blue

Figure 4-20a: Three-dimensional graph of trichordal SST-succession classes (¢[A],[B]²SSTi) and trichordal/tetrachordal split-succession classes (¢[A],[B]²SSTj(SPLITi)), identified by [SSTi] or [SSTj] relation only

Key: [SST1] and [SST1–1] = red

[SST3] and [SST3–1] = green

[SST2] and [SST2–1] = yellow

[SST4] and [SST4–1] = blue

Fig. 4-20b: Graph of trichordal SST-succession classes and trichordal/ tetrachordal split-succession classes, identified by [SSTi] or [SSTj] relation only (rear view)

Graphical Representations of Parsimonious Voice-Leading Space

83

4-1 0123

4-3 0134

4-2 0124

4-10 0235

4-11 0135

4-4 0125

4-7 0145

4-21 0246

4-12 0236

4-5 0126

4-15 0146

4-13 0136

4-8 0156

4-22 0247

4-14 0237

4-23 0257

4-17 0347 4-6 0127

4-18 0147

4-29 0137

4-9 0167

4-16 0157

4-24 0248

4-25 0268

4-27 0258 4-26 0358 4-19 0148

4-20 0158

4-28 0369

Fig. 4-10: Graph of voice-leading space showing optimal offset between tetrachordal SCs (based on Straus 2003)

segments may signify different [SSTi] relations, unlike the line segments in the trichordal SST-succession-class graph. The principal reason for this inconsistency is explained when one considers that certain set classes [A] would be [SST1]-related to multiset classes [B] (these could possibly be re-interpreted as fuses), and Straus did not account for multiset classes in his system at the time the graph was originally published (2003). For example, consider ¢0,1,4,6² ∈ [0146] acted on by SST1: SST1(¢0,1,4,6²) = ¢1,1,4,6²[0035].

84

Chapter Four

Fig. 4-11: Three-dimensional graph of voice-leading space showing optimal offset between tetrachordal SCs (from Straus 2005)

A SPLIT1–1 function would then need to be applied to reduce the 4-tuple to a 3-tuple, and the corresponding multiset-class type would change from [0035] to the set-class type [025]. In order to display the SST-succession-class relations unambiguously in a geometric optimal offset graph, the orientation of connective line segments must be consistently aligned with distinct [SSTi] relations. As this is not possible with tetrachordal SST-succession classes in a two-dimensional graph, let us examine a three-dimensional representation. The extension of Figure 4-10 from two to three dimensions seems only natural, given that we have increased the cardinality of the pair-succession equivalence classes by one. Straus’s revised tetrachordal optimal offset graph (Figure 4-11) was first presented at the annual SMT meeting in Boston in November 2005, and it solves a number of the problems found in the two-dimensional graph. First, and most importantly, it precisely matches the orientation of connective lines with specific [SSTi] relations associated with SST-succession classes. These are not marked on Straus’s graph but are easily recognized: “diagonal” descending motion is by [SST1] and ascending by [SST1–1], ascending vertical motion is by [SST2] and descending by [SST2–1], horizontal motion from the “front” of the graph toward the rear (if SCs 4-1 to 4-6 are considered to be the front row) is by [SST3] and from the rear to the front by [SST3–1], and horizontal motion at 90 degrees to [SST3] connections is by [SST4] and [SST4–1]. The second problem that is (partially) corrected in the 3D graph is that of SC [0148], which in the 2D map required a curved line segment to

Graphical Representations of Parsimonious Voice-Leading Space

85

reproduce the [SST1] and [SST1–1] offset represented by straight lines in the rest of the map. The reason this problem is only “partially” corrected is due to SC [0148]’s duplication in the 3D graph. In order to maintain the proper orientation of the line segments in the graph, [0148] must appear in two separate locations, once as the Tn type (0148) and again as the Tn type (0348). We will see that this is a recurring problem associated with SC [0148] in any representation of tetrachordal SST-succession-class space, as it tends to “warp” the space around it, necessitating either a curvature of lines (as in Figure 4-10), duplication of the set class in the graph (as in Figure 4-11), or the introduction of lines whose orientation does not match those of other, regular SST-succession-class relations [SSTi] in the graph. We can see this last point in action in Figure 4-12, a three-dimensional graph of tetrachordal SST-succession-class space similar in most ways to Straus’s 3D graph.15 The principal differences are the following. First—and most superficially—the orientation of the two graphs is somewhat different. In Straus’s graph, the plane of set classes of the type [01xy] is on the “bottom” of the lattice; in Figure 4-12 it forms the far vertical plane. Additionally in Straus’s graph, equivalence-class pair-successions related by [SST4] and [SST3] form orthogonal planes, whereas in Figure 4-12 the orthogonal planes are comprised of equivalence-class pair-successions related by [SST4] and [SST1]. Another way in which the graphs differ is by the labeling of specific [SSTi] relations in Figure 4-12, where they are color coded for easy identification. The third difference between the two graphs—and perhaps the most significant—is that Figure 4-12 avoids duplication of SC [0148]. As we have previously seen, this particular set class is highly problematic in graphic representations of tetrachordal SST-succession-class space. In Figure 4-10 the layout of the 2D map did not allow for a straight line connecting [0148] to [0237], so a curved line was necessarily introduced; in Figure 4-11 the geometry of the 3D graph would have been compromised if [0148] had not been represented twice. In Figure 4-12 the consequences of a single representation of [0148] in the 3D graph can be seen. It is clear the geometry of the lattice is distorted in three of the SST-succession-class relations ¢[A],[B]²SSTi where [0148] = [A]: the [SST] lines leading to [0358], [0237], and [0347] do not run parallel to other lines of the same [SSTi]-relation type. In addition, different [SSTi]s are required when tracing optimal offset to and from [0148] and these three SCs. As an example of this, the line segment connecting SCs [0358] and [0148] represents ¢[0358],[0148]²SST2, but ¢[0148],[0358]²SST1–1 and is not the expected inverse ¢[0148],[0358]²SST2–1. For this reason two of the three non-orthogonal lines originating from [0148] are colored grey.16 No matter what type of two- or three-dimensional

86

Chapter Four

Key: [SST1] and [SST1–1] = red [SST2] and [SST2–1] = yellow [SST3] and [SST3–1] = green [SST4] and [SST4–1] = blue

Fig. 4-12: Three-dimensional graph of specific [SSTi]s between tetrachordal equivalence-class pair-successions (¢[A],[B]²SSTi). A color version of this Figure is shown in the Centrefold. representation of tetrachordal SST-succession-class space we choose, there will necessarily be some degree of spatial distortion associated with [0148], the closest tetrachordal approximation to the trichordal set class that equally divides the octave, [048]. The only way to avoid the graphic anomalies

Graphical Representations of Parsimonious Voice-Leading Space

87

associated with [0148]—both the “warping” of SST-succession-class space it induces or its “ghost” duplicate in Figure 4-11—is to modify Straus’s 3D graph by “bending” it so that both instances of [0148] are conjoined.17 This, however, creates an entirely new set of problems such as SST-succession classes with the same [SSTi] relation not being represented by parallel lines, and it does not result in a toroidal lattice similar to the one we can obtain from bending the Tonnetz. Figure 4-13 shows other specific [SSTi] relations associated with SSTsuccession classes, in addition to those already delineated in Figure 4-12 (parallel paths). The nomenclature is similar to that found in the key to Figure 4-9. Some of the more interesting properties of these multiple SSTsuccession-class paths are reminiscent of those found in the earlier figure. For example, the location of a specific [SSTi] occurring as an additional path is often replicated geometrically in the graph, e.g., the yellow [SST2] arrows relating pair-successions ¢[0167],[0157]², ¢[0156],[0146]², and so forth. As is to be expected, set classes whose members relate parsimoniously to members of only one other tetrachordal set class—[0167] and [0369]—have multiple paths emanating from them. Similarly to the trichordal graph, a number of [SSTi] connections are not shown, as they lead to multiset classes that can potentially be reduced to trichordal set classes. Two interrelated aspects of Figure 4-13 deserve further discussion. It should be noted that all symmetrical set classes have multiple paths associated with them, and in general, symmetrical set classes have a larger number of multiple paths emanating from them than do non-symmetrical set classes. At one end is [0123], with one additional subscript, [SST1–1] as a second path to [0124], duplicating the [SST4]-relation path inherent in the geometry of the graph. SC [0123] also contains multiple paths to multiset classes [0012], [0023], [0114], [0113], and [0034], which are not shown in Figure 413. At the other extreme is [0369], with seven additional [SSTi]-paths to [0258], in addition to the “autochthonous” [SST1] relation. This observation is closely related to the second one I wish to make here, namely, that certain set classes are at the “receiving end” ([B] in each SST-succession class ¢[A],[B]²SSTi) of a large number of multiple [SSTi] paths, most notably [0247], [0157], and especially [0258], which is [B] in an amazing 23 separate SST-succession classes. Once again, it seems significant to note the important rôle played by [0258] in the history of Western music, both as the chief tetrachordal sonority as well as acoustically, with its close pitch-class approximation to the first ten partials of the overtone series. Appendix B tabulates all the single [SSTi] relations associated with all possible tetrachordal SST-succession classes ¢[A],[B]²SSTi. Here it is interesting to note that unlike certain trichordal m-tuples, no tetrachordal m-tuple maps to another m-tuple of the same set-class type under SSTs. The

88

Chapter Four

Key: [SST1] and [SST1–1] = red [SST2] and [SST2–1] = yellow [SST3] and [SST3–1] = green [SST4] and [SST4–1] = blue Solid arrows = [SSTi] Dashed arrows = [SSTi–1]

Fig. 4-13: Three-dimensional graph of specific [SSTi]s between tetrachordal equivalence-class pair-successions (¢[A],[B]²SSTi), including additional parallel [SSTi] paths. A color version of this Figure is shown in the Centrefold. symmetry of certain set classes [A] (indicated in bold in Appendix B) is also manifested by a palindromic list of [SSTi] relations, with [SST1] and

Graphical Representations of Parsimonious Voice-Leading Space

89

[SST3–1] conjoining equivalent pair-ordered SCs in SST-succession classes, as well as the relations [SST1–1] and [SST3], and so forth for all symmetrical trichordal set classes [A] in ¢[A],[B]²SSTi. This is merely one example of the phenomenon, which is also discussed at the end of chapter 3. Those who would like to obtain a closer view of the tetrachordal SSTsuccession-class graph are advised to visit http://www.c-s-p.org/extras/14438-1860-7/home.html. This webpage contains a link to the home page of the developer of the CAD software, 3D Canvas, which was used to develop the graph as well as other graphs discussed in this study. As this software is freeware, one can easily and legally download it and install it (on Windows XP or Windows 2000 operating system), then download and open my 3D graph of tetrachordal SST-succession-class space (tetrachords.3DC). With the 3D Canvas software, one can rotate the graph, zoom in or out, and even record animated movies of various camera paths through the graph, one of which is provided as an .avi file on the website (flyby.avi). Constructing graphs of optimal-offset space for set classes with cardinality greater than 4 is much more problematic than for trichordal and tetrachordal set classes. One notable attempt at this is found in Straus’s 2005 paper, in which he constructs 3D lattices for pentachordal and for hexachordal SCs that resemble Figure 4-11, but with more complex “rules” for reading each graph due to the increased cardinality of the set classes. Ultimately the best hope for a clean presentation of optimal offset between pentachordal set classes would be a graph in four dimensions, and for hexachordal SCs, a graph in five dimensions, reflecting the general principle that a graph of m-chordal SST-succession-class space adequately models that space in m – 1 dimensions. As graphs in higher dimensions cannot be produced in a two-dimensional medium such as the printed page, and in general are outside the realm of our three- or four-dimensional everyday existence, I will not attempt to graph SST-succession-class spaces for set classes of cardinalities higher than four. Before beginning our discussion of graphs of parsimonious SSTsuccession-class space involving set classes belonging to different cardinalities, let us examine one final graph showing optimal offset between tetrachordal set classes devised by Richard Cohn (2003, Figure 4-14). Cohn is able to achieve a perfect tetrahedron in his graph but only through duplication of a number of set class nodes—[0137], [0147], [0157], [0167], [0258], and [0268]—which are colored violet and lie along or adjacent to the C–D axis of the tetrahedron. Cohn himself points out the problem of duplicate set classes, noting that although they produce an elegant symmetry in his graph, if the buff-colored set classes are to be considered equivalent to their violetcolored doppelgängers, one needs to visualize a creasing of the BCD face of the tetrahedron so that corresponding set classes “fold” over to each other

90

Chapter Four

Fig. 4-14: Cohn’s tetrahedral graph of tetrachordal voice-leading space (from Cohn 2003). A color version of this Figure is shown in the Centrefold.

and are fused. The BCD face then recedes into the interior of the structure, the remaining faces become distorted, and the graph loses its classical symmetry (Cohn 2003, 13). This should not pose a conceptual problem, as we can imagine similar bendings of parsimonious voice-leading space with graphs such as the chicken-wire torus and the pipeline. We must be vigilant about reminding ourselves when viewing Figure 4-14 that the tetrahedron is not in its “final” state, and that the “duplicate” set classes are not really duplicates (if we accept equivalency between corresponding violet and buff set classes).18 It is also interesting to note that set class [0247] is at the center of Cohn’s tetrahedral graph. Cohn calls this set class the “queen bee,” as it has the property of being the only set class that enjoys parsimonious voice-leading

Graphical Representations of Parsimonious Voice-Leading Space

91

relations with the maximum possible number of distinct set classes: 8 (see also my Appendix A). In comparison, SC [0247] also appears close to the center of Figure 4-12, but the actual geometric center of Figure 4-12 is occupied by [0258], which we have already seen possesses a number of unique properties. Finally, the “sticks” of Cohn’s ball-and-stick graph do not indicate specific [SSTi] relations associated with SST-succession classes ¢[A],[B]²SSTi. Motion along the A-to-D axis is by [SST4], and motion along the B-to-D and C-to-D axes is by [SST3–1], but—as we have seen with Straus’s 2D graph in Figure 4-10—parallel SST-succession-class paths do not consistently correspond to single [SSTi] relations. As an example of this, consider the straight line connecting SCs [0126], [0237], [0148], and [0158] in Figure 4-14 (on the ABD face). We may note that the SST-succession-class subscript leading from [0126] to [0237] would be [SST1–1], and that the subscript from [0237] to [0148] would also be [SST1–1]. So far, so good. However, if we continue along the same trajectory from [0148] to [0158], the SST-succession-class subscript is now [SST3]. We might dismiss this inconsistency with [0148]’s tendency to throw our graphs out of kilter, so we trace another unidirectional path that does not involve [0148]: [0137] through [0248] and [0258] to [0268] (on the BCD face of Figure 4-14). ¢[0137],[0248]²SST1–1, but the remaining subscripts show an [SST3]-relation between pair-ordered SCs. If we are looking for the consistent alignment of SST-succession-class subscripts in the geometry of an optimal offset graph, graphs such as Figures 411 and 4-12 are better suited to the purpose than is Cohn’s tetrahedral graph.

Graphs of Optimal Offset between SC Types of Differing Cardinality Figure 4-15 is a three-dimensional graph of parsimonious voice-leading space that is Straus’s modification of his tetrachordal graph (Figure 4-11) by the introduction of trichordal set classes. In order to do this without disrupting the geometry of the graph, the trichordal set classes first had to be conceived of as multiset classes. Although Straus does not describe them as such, the trichordal set classes must be considered to be multiset classes to allow for split-succession and fuse-succession classes when explaining parsimonious voice leading, respectively, to and from tetrachordal set classes. Once this adjustment is made, the trichordal SCs can easily and elegantly be accommodated in the structure of the graph, preserving the optimal offset connections associated with specific SST-succession-class [SSTi] subscripts that are aligned in four distinct orientations. The only drawback to Figure 4-15 is an amplification of the minor problem with Figure 4-11: whereas the earlier figure contained a duplicate set

92

Chapter Four

Fig. 4-15: Graph of parsimonious tri- and tetrachordal voice-leading space (from Straus 2005)

class, [0148], the trichordal–tetrachordal graph contains sixteen duplicates: [0148] and tetrachordal multiset-class variants of all trichordal SCs except [012], [024], and [048]. Indeed, multiset-class variants of SCs [014], [015], [016], [025], [026], and [027] cause each to be represented three times in the graph. If we are looking for a graph that accommodates both trichordal and tetrachordal set- or multiset-classes, in which the SST-succession class subscripts are consistently organized according to a regular geometric grid, we need look no further than Figure 4-15. If we wish to eliminate the duplicate set classes, however, radical changes must be made to the graph, disrupting the regular geometry of the SST-succession-class subscripts. To construct such a graph, let us begin—as Straus does—with the trichordal set classes 3-1 to 3-5 ([012] to [016]). Each one of these SCs will serve as the point of departure for the remaining trichordal and tetrachordal set classes involved in SST-, split-, and fuse-succession classes, and collectively they will form the “axis” or backbone of the graph. Figure 4-16 shows how this is accomplished. Each trichordal set class, 3-1 to 3-5, is placed at the top or “north” end of a disk that will later serve as a sectional layer in the 3D graph. Tetrachordal set classes of the form [01xy] are placed around the rim of each layer, where x begins with a value of 2 and increases by 1 as one moves clockwise around the disk; the value of y is always one greater than the last digit of the trichordal set class anchoring its disk. In the first layer,

Graphical Representations of Parsimonious Voice-Leading Space

Layer 1

93

Layer 2 013 012 024

0235

0124

0134 0123 Layer 4

Layer 3

015

014

026

025

0126

0156 036

0145

0125

0237

0257

0236

0246

0347 0247 0146

0135 Layer 5

0136

016 027

0167

0127

037

0137

0268 0148

0158 0157

0258

0248

0137

0147

Fig. 4-16: Sectional layers comprising the trichordal split-succession-class graph. A color version of this Figure is shown in the Centrefold.

only one tetrachordal set class can satisfy the requirements for x and y where the vertebral trichord is [012], and it is placed halfway around the disk in order to divide the disk’s circumference evenly. In the second layer, two

94

Chapter Four

tetrachordal set classes meet our requirements and are spaced evenly so the disk’s circumference is divided into thirds. The process continues for the remaining layers, each successive disk containing 1 additional tetrachordal SC type than the previous layer. Let us now add to the mix trichordal set classes 3-6 through 3-9 ([024] to [027]). Like SCs 3-1 to 3-5, each of these will serve as the point of departure for a series of tetrachordal set classes of the form [02xy], where x begins with 3 and increases by 1 as one moves clockwise along a new, inner disk (indicated by green in each layer of Figure 4-16), and where the value of y is one greater than the last digit of the trichordal set class in its disk.19 This inner system of disks first appears with layer 2. These new trichords form a secondary, inner spine along the “northern” edge of the inner disks. A third system of inner disks begins to appear with layer 4; these disks are shaded blue in Figure 4-16 and are centered on trichordal SCs 3-10 and 3-11 ([036] and [037]), with associated tetrachordal SCs of the form [03xy], where x begins with 4 and increases by one as one moves clockwise around the disk, and the value of y is one greater than the last digit of the trichordal set class in its disk. [036] and [037] are aligned to form a tertiary spine, again oriented toward the north on the perimeter of the blue disks. Although we have still not accounted for [048], we can place it at the center of the disk in a new layer (6) as the unique representative of this layer—but this time in a disk of diameter 0, collapsed to a single point (or set class). The attentive reader will notice that two tetrachordal set classes have not yet made an appearance in any of the five layers of Figure 4-16, both of which are much closer in terms of internal spacing to the maximally even tetrachordal set class, [0369], than to the extremely compact, maximally uneven SC [0123].20 These set classes are [0358] and the maximally even [0369] itself; they are excluded from the graph we are currently constructing only because they are not involved in any split/fuse-succession-class relations involving trichordal set classes and thus need not be represented. In addition to these two SCs, from Appendix A we can see that several of the tetrachordal set classes closer to the maximally even end of the spectrum and appearing in Figure 4-16 do not lead to trichordal set classes through fuseclass relations. These set classes are retained, at least for the moment, as placeholders to preserve the structure of the incipient trichordal-tetrachordal graph. Let us now place the layers in a vertical stack with layer 5 at the bottom and with an object representing the dyadic set class [01]—or [02]—at the top of the stack.21 Figure 4-17 shows views of the resulting graph from the top and from the side, without set-class labels for the moment to avoid clutter. As each successive layer contains a greater number of set classes than the previous layer, the disk sizes increase toward the bottom of the graph,

Graphical Representations of Parsimonious Voice-Leading Space

95

Fig. 4-17: Top and side views of superimposed layers of the incipient trichordal-tetrachordal split-succession-class graph. A color version of this Figure is shown in the Centrefold. creating a conical structure resembling a Christmas tree. Let us now add some tinsel to the tree, tracing SST-succession-class connections between pair-ordered SCs and adding SC labels to the graph (see Figure 4-18). As we are interested in optimal offset between set classes of differing cardinality, any multiset classes that result from parsimonious voice leadings can be reduced to trichordal SCs if each SST-succession-class is re-cast as a fuse-succession-class; e.g., ¢[0123],[0012]²SST1 and ¢[0012],[012]²SPLIT1–1, or in shorthand notation: [0123] [SST1] [0012] [SPLIT1–1] [012]. One of the first things we notice about this graph is the presence of tetrachordal set classes with no connective fuse-succession-class lines. These are the set classes which do not participate in split/fusesuccession-class relations with trichordal set classes and are retained—as previously mentioned—for the time being as placeholders to preserve the structure of the graph. These set classes are, however, SST-related to other tetrachordal set classes as ordered pairs in SST-succession classes, but these are not shown in order to keep an already complex graph from becoming unnecessarily cluttered. (These missing tetrachordal SST-succession-class connections can be seen in Figure 4-12.) As one example of the complexity we have avoided, set class [0258] alone would require seven distinct SSTsuccession-class subscript connections ([SSTi]) leading from it were we to

96

Chapter Four

Fig. 4-18: Graph of trichordal SST-succession classes and trichordal splitsuccession classes. A color version of this Figure is shown in the Centrefold. connect it to other SST-related tetrachordal set classes. Another aspect of the graph that is not quite as readily apparent from Figure 4-18 is the larger number of split/fuse-succession-class connections on the left side of the outer [01x] trichordal spine than on the right. This is more easily visualized by rotating the graph in the original .3DC file, available at http://www.c-s-p.org/extras/1-4438-1860-7/home.html. The reason for this is simply that most of the tetrachordal set classes (as listed in Appendix A)

Graphical Representations of Parsimonious Voice-Leading Space

97

that do not relate to trichordal set classes through fuse-succession-class voice leading are arranged on the “right” side of the trichordal spine. These set classes all lie closer to the maximally even end of the ic spacing spectrum and do not contain ic 1 in their interval vectors.

Fig. 4-19: Graph of trichordal split-succession classes. A color version of this Figure is shown in the Centrefold.

One of the most noticeable features of the graph in Figure 4-18 is the prominent spine of trichordal SCs extending toward the center of the cone from trichordal SCs of the form [01x] on the outer surface through [02x] trichordal types along the first inner cone to the two [03x] trichordal types forming the spine of the second inner cone. A quick glance at Figure 4-8 confirms that Straus’s graph is a 2D slice (with slightly modified orthography) of the trichordal section of the Christmas-tree graph of Figure 4-18.

98

Chapter Four

Figure 4-19—which eliminates SST-succession-class connections between trichordal SCs, any tetrachordal SCs that do not participate in split/fusesuccession-classes, and the colored disks—is a simplification of Figure 418’s graph, focusing exclusively on trichordal split-succession classes. Although the main purpose of the trichordal-tetrachordal split/fusesuccession-class graph (Figure 4-18) is not to show “regular” SSTsuccession classes (indeed, it cannot show a geometrically regular lattice of consistently oriented SST-succession classes), we can nevertheless note where they occur. Figure 4-20a shows the same optimal offset connections as does Figure 4-18, but without the tetrachordal set classes that are not [B] components of trichordal split-succession classes, and without the colored disks, for clarity’s sake. The coloring of the SST-succession class and split/ fuse-succession-class “tinsel” ([SSTi] or [SSTj] subscripts) remains the same as in previous examples, with the addition of striped beams to represent duplicate paths that can be explained by two or more [SSTi]s or [SSTj]s associated with a particular pair-ordering of equivalence classes. This would be a good point to remind ourselves of three important considerations when tracing optimal offset paths through Figure 4-20a: first, this particular graph shows split-succession classes (with trichordal SCs as [A] in each pair-succession) only, and not fuse-succession classes (with tetrachordal SCs as [A] in each pair-succession). A separate graph would be required for the latter, which would preserve the geometry of the trichordal split-succession-class graph, but the [SSTi] subscripts in a tetrachordal fusesuccession-class graph would have different values from the ones in the trichordal split-succession-class graph, as fuse-succession classes are not simply retrogressions of or “inverses” of split-succession classes. As an example of this, consider a pair-ordering of SST-related set-class types. Near the top of the graph we see a red and blue striped line connecting set classes [012] and [0123]. This identifies the split-succession class ¢[012],[0123]²split, with the equivalence-class pair identified by the red and blue colorings as being either [SST1–1]- or [SST4]-related (after some kind of unidentified SPLITi operation). We can easily ascertain that the parallel paths are in fact more precisely identified as ¢[012],[0123]²SST1–1(SPLIT1) and ¢[012],[0123]²SST4(SPLIT3). The reversal of these split-succession classes would be an apparent “inverse” pair of fuse-succession-class paths: ¢[0123],[012]²SPLIT1–1(SST1) and ¢[0123],[012]²SPLIT3–1(SST4–1). In these fuse-succession classes, the retrogrades are literal inverses of the two splitsuccession classes’ voice leading paths. Unfortunately, as we have previously seen with pair-succession order-position swapping (in conjunction with Figure 4-9, p. 79) this is not always the case. Consider the splitsuccession classes ¢[027],[0157]²SST4(SPLIT3) and ¢[027],[0157]²SST3–1 (SPLIT3) near the bottom of Figure 4-20a. Because the split-succession-class

Graphical Representations of Parsimonious Voice-Leading Space

99

Key: [SST1] and [SST1–1] = red

[SST3] and [SST3–1] = green

[SST2] and [SST2–1] = yellow

[SST4] and [SST4–1] = blue

Figure 4-20a: Three-dimensional graph of trichordal SST-succession classes (¢[A],[B]²SSTi) and trichordal/tetrachordal split-succession classes (¢[A],[B]²SSTj(SPLITi)), identified by [SSTi] or [SSTj] relation only. A color version of this Figure is shown in the Centrefold.

subscript components are determined by the SST relation between the representative m-tuple ¢A²—which names the first SC ([A]) in each ordered pair of

100

Chapter Four

equivalence classes—and some m-tuple ¢B′ ², the “inverse” fuse-succession classes may have entirely different [SSTi] subscripts, as these are determined by the representative m+1-tuple ¢B² which names the SC [B] which occupies the first order position in the retrograde fuse-related pair-succession. The reverse fuse-succession class in the case of the current example would be ¢[0157],[027]²SPLIT1–1(SST2–1). This illustrates the principle that split- and fuse-succession classes generally cannot be reversed by fuse- and splitsuccession classes, respectively, with which they share “inverse”-related subscripts.22 This is sometimes the result of the change in m-tuple (and corresponding set-class) cardinality and at other times the result of the lack of an retrograde ¢[B],[A]²SSTi–1 for a duplicate (non-geometric) SST-succession class path ¢[A],[B]²SSTi . The second consideration we have just now seen in action and is familiar to us from the discussion of fuse- and split-succession classes in chapter 3, namely, that the order of the components of the split-succession-class and fuse-succession-class relational subscripts are reversed when the voice leading is reversed. In generic terms, ¢[A],[B]²SSTj(SPLITi) is a split-succession class but ¢[A],[B]²SPLITj–1(SSTi) is a fuse-succession class. The third observation is that the SPLITi subscript components of the split-succession classes in Figure 4-20a are not identified anywhere on the graph. This is to reduce clutter, but they can fairly easily be extrapolated by examining the SSTj subscripts. One immediately notices the green coloring of the [SST3] relations forming the “spine” of the graph between trichordal SCs of the form [01x], and along the subsidiary spines linking trichordal SC pair-successions of the form [02x], and again [03x]. Some other readily apparent [SSTi]-relation patterns include the red [SST3] relations between SC pair-successions of the form [01x] and [02y] and the yellow [SST2] relations between SC pairsuccessions of the form [01x] and [02x]. Less obvious from the point of view of Figure 4-20a is the geometry of the other [SSTi] relations. While their arrangement is not nearly as regular as that of the [SSTi] subscripts in Figures 4-15 and 4-12, a view from another angle is illuminating. Figure 4-20b considers the same graph from a viewpoint located immediately behind the outer spine of trichordal SCs. From this vantage point, we see that [SST3] relations branch out from either side of the trichordal spines in a perfectly symmetrical pattern, while [SST1] relations are confined entirely to the left side of the spine and centered along the spine itself. [SST4] relations, on the other hand, appear on both sides of the spine in what appears at first glance to be a less regular pattern. On closer examination, however, it can be seen that [SST4] voice leadings occur on the right side of the graph in mirrorimage positions to the red [SST1] relations on the left side. The only exceptions to this symmetry between split-succession-classes related by [SST4]

Graphical Representations of Parsimonious Voice-Leading Space

101

Key: [SST1] and [SST1–1] = red

[SST3] and [SST3–1] = green

[SST2] and [SST2–1] = yellow

[SST4] and [SST4–1] = blue

Fig. 4-20b: Graph of trichordal SST-succession classes and trichordal/ tetrachordal split-succession classes, identified by [SSTi] or [SSTj] relation only (rear view). A color version of this Figure is shown in the Centrefold.

and by [SST1] occur as a result of the presence of two [SST4] relations on the left side of the graph, as part of the split-succession class ¢[025],[0136]²split (where the subscript component [SST4] indicates a

102

Chapter Four

duplicate path to the “normal” geometric [SST1] relation), and as part of the split-succession class ¢[048],[0148]²split, which involves an [SST1] relation that is paralleled by all the remaining [SST]s (as is to be expected between the two equivalence classes which compose this pair-succession). To view the graph from different angles, the interested reader is again encouraged to visit http://www.c-s-p.org/extras/1-4438-1860-7/home.html, where my 3D graph of trichordal-to-tetrachordal split-succession classes (tri-tetra.3DC) is located. With the 3D Canvas software, one can rotate the graph, zoom in or out, and even record animated movies of various camera paths through the graph, one of which is provided as an .avi file on the website (tri-tetra.AVI). While Straus’s graph (Figure 4-15) ultimately is better for showing consistent geometric [SSTi] orientation than the graph in Figure 4-20, by its having to resort to duplicate and even triplicate representation of certain set classes it does not as clearly model the rôle that specific set classes play in the trichordal/tetrachordal voice-leading system. As the best illustration of the strength of Figure 4-20’s graph, it is well known that SC [037] is of particular interest to a number of composers and theorists for its central place in both the tonal and post-tonal literature. An entire branch of transformational theory, neo-Riemannian theory, stems from the investigation of transposition and contextual inversion operations on the major/minor triad.23 More recently, neo-Riemannian theory has expanded to include studies of chords belonging to certain tetrachordal set classes that often appear in conjunction with major and minor triads in both tonal and in advanced “non-functional” harmonic idioms based on tonal materials but bordering on atonality. These tetrachordal set classes are for the most part those comprised of commonly used seventh chords which embed members of the trichordal set class [037], namely the dominant/half-diminished seventh [0258], the minor seventh [0358], and the major seventh [0158] set classes. Besides its inclusional relation to these seventh-chord set classes, [037] also enjoys more optimal offset connections with tetrachordal set classes (as part of a split-succession-class) than does any other trichordal set class (see Appendix A). The central place [037] has in the entire system of trichordal and tetrachordal SST- and splitsuccession classes is reflected by its position in Figure 4-20a, something which is not at all obvious from the geometrically regular graph of Figure 415. The tetrachordal SCs with which it enjoys split-succession-class relations—[0137], [0147], [0148], [0158], [0237], and [0347]—(with the exception of [0158] and possibly even [0148]) are more commonly encountered in atonal music than in tonal music, allowing for interesting voice-leading possibilities, especially in music that is neither completely tonal nor completely atonal. We will examine some of the implications of the system of SSTs and SST-succession classes on neo-Riemannian theory in the next chapter, and will conclude by taking a closer look at some of the post-tonal/“post-atonal”

Graphical Representations of Parsimonious Voice-Leading Space

103

music dating from the last twenty-five years of the twentieth century that mixes elements of tonal and atonal idioms—and that has received scant attention in the analytical literature—in the final chapter of this study.

CHAPTER 5 THE SST MODEL AND NEO-RIEMANNIAN THEORY This chapter assumes the reader has a basic familiarity with neoRiemannian theory, in particular with the seminal publications which instigated theoretical investigations in this area.1 The earliest and most fundamental studies in neo-Riemannian theory deal with transposition and inversion operations on the major and minor triad, SC [037]. Transposition operations on a triad (Klang) are called Schritten (anglicized as “schritts”), and inversion operations are called Wechseln (sometimes anglicized as “wechsels”), both terms originating with Riemann in his Skizze einer neue Methode der Harmonielehre (1880). Although most of these early studies investigate the use of both schritts and wechsels as transformations that could explain advanced harmonic progressions in late-Romantic tonal music, by the late 1990s many theorists had turned their attention away from schritts entirely, focusing instead on the wechsels, or contextual-inversion operations (cios) R, P, and L. Henry Klumpenhouwer was the first to question the expediency of using schritt operations such as MED, DOM, SDOM, and SMED in analysis, as they are so closely linked to the (functional) tonal system (1994, ¶¶ 5–7). This focus on cios as the most fundamental transformations in this emerging branch of theory led indirectly to two major developments in neo-Riemannian research: first, and most importantly, the jettisoning of the schritt operations effectively sundered the remaining operations from their historic tonal underpinnings, allowing for their application to a wider range of musical periods and styles than was possible with the schritts and wechsels together. As examples of this, only a few years later, in the “neo-Riemannian issue” of the Journal of Music Theory (Volume 42.2) we see, through the use of cios alone, analyses of decidedly “non-functional” music by Debussy, Scriabin, Stravinsky, and even Stockhausen. The second major development follows the first: as theorists begin to analyze music which is not necessarily predicated on major and minor triads and their associated cios R, P, and L, they must take into account different harmonic structures (from “tonal” seventh chords to octatonic collections to unmistakably non-tonal pc sets) and at the same time formulate new cios and other transformations which better model events in pieces belonging to this new repertoire. This last point is, perhaps, not quite what Richard Cohn was articulating when he wrote, “In assuming the a priori status of consonant triads, neo-

The SST Model and Neo-Riemannian Theory

105

Riemannian theory leaves unaddressed the reasons that late nineteenthcentury composers continued to favour triads as harmonic objects” (1996, 12). Cohn then goes on to propose that the internal structure of the [037] triad lends itself well to a group-theoretic approach in which parsimonious voice leading is obtained through Lewinian transformations. His exegesis of the voice-leading potential inherent in the major/minor triad provides fertile ground for this and a number of subsequent studies (Cohn 1996, 1997, 1998, 1999, and 2000), in response to his question “Why should triads have any status at all, except as optimal acoustic objects?” For the moment, let us reinterpret his question and answer it not to validate the important status of set class [037], but instead to take it at face value: why indeed limit our studies to a single set-class type when we can formulate a group-theoretic approach that can be applied to all set-class types of all cardinalities? The answer to the question is readily apparent: Lewin’s transformational approach, with a focus on harmonic transformations and voice-leading connections, reveals a great wealth of significant relations within and between set classes other than and including [037]. In this chapter we will examine some of these new developments in neoRiemannian research, with an emphasis on theorists’ attention to voice leading in a field in which the focus has historically been centered on harmonic transformations. It will be seen that most of these theorists leave unanswered questions about voice leading as a result of harmonic transformations, and when they do discuss voice leading it usually involves some variant of the set-class transformational approach discussed in chapter 2. As there is already an extensive body of literature exploring the triadic cios R, P, and L, we will pay special attention to neo-Riemannian operations on chords belonging to set classes other than [037]—especially tetrachordal set classes—and to operations which transform chords of differing cardinalities. We will conclude the chapter with some observations on how the theory of SSTs relates to neo-Riemannian theory, showing how the SST model shifts the emphasis from harmonic transformations to voice-leading transformations, and also how it encompasses and generalizes the disparate neoRiemannian labels that have proliferated in the last two decades.

A Brief Overview of Tetrachordal Neo-Riemannian Studies A passage from Julian (Jay) Hook’s dissertation is of interest on at least two counts: [O]nly a small minority of UTTs [(transformations)] behave in ways that have anything to do with the theories of Riemann. There are, I believe, many pitfalls in waving a person’s name on too broad a banner . . . In the present

106

Chapter Five

situation I suggest that UTTs be considered part of “harmonic transformation theory” rather than “neo-Riemannian theory.” (2002, 10)

The most noticeable thing about this passage is Hook’s aversion to the moniker “neo-Riemannian theory.” He is not alone in rejecting the by-nowstandard rubric: David Kopp also chooses to replace it, in his case with the term “chromatic transformation system” (2002, 165), although for different reasons than Hook. Nearly lost in the discussion of the term “neo-Riemannian” in Hook’s passage above is his choice of the adjective “harmonic” to describe the transformation operations that constitute his theory. As is the norm with the theories of set transformation which we investigated in chapter 2, the emphasis in Hook’s theory is on the transformation of one harmony to another, rather than on tracing the voice-leading paths that arise as a result of that particular transformation. We will see that this is, for the most part, the case with nearly all neo-Riemannian studies. Granted, the distinction that I make here between harmony and voice leading is largely an artificial construct with a long history in music theory. Both are aspects of the same process; they are two ways of looking at the same thing (a musical transformation). My focus on voice leading through using the SST model by no means necessitates disregarding the harmonic dimension in toto, in much the same way as Schenker’s emphasis of the linear dimension over the vertical dimension in his mature theory does not slight the role harmonic function plays in his analyses. As the term “neo-Riemannian” enjoys such currency, I will continue to use it in this study, but only as a description of the theoretical works examined in this chapter, instead opting to use the term “single-voice transformational model” to reference my own approach, which focuses on the voice leading of individual chord members rather than on operations which transform one Klang into another harmonic object. The logical place to begin is with the article that inaugurated the renewed interest in Riemann’s functions, David Lewin’s “A Formal Theory of Generalized Tonal Functions” (1982). Lewin defines a “Riemann system,” which is an ordered triple of the form ¢T,d,m², where T is a pc and d and m are intervals corresponding to the “dominant” and “mediant,” respectively. The canonical listing of the diatonic set of a Riemann system is an ordered series ¢T – d,T – d + m,T,T + m,T + d,T + d + m,T + 2d². The usual diatonic set (of SC [013568T] (Forte label 7-35)) may be generated by any of the Riemann systems ¢T,2,7², ¢T,5,8², or ¢T,5,9² or their mod-12 complements; thus an “Fmajor” Riemann system would be represented as ¢F,7,4². Its canonical listing would be

The SST Model and Neo-Riemannian Theory

107

B( d F a C e G. At the core of this Riemann system is the “tonic” triad, F a C. Lewin demonstrates how to obtain the “relative” triad (and by extension, Riemannian system): a TM-inversion (TMINV) exchanges the tonic and mediant notes of the F-major system with the “tonic” and “mediant” notes of the dual Aminor system.2 The tonic note F of the F-major system is reinterpreted as the mediant f of the dual A-minor system, and the mediant note a of the F-major system is reinterpreted as the tonic note A of the dual A-minor system. The function TMINV(¢F,C,a²) produces the new Riemann system ¢A,D,f². On a triadic level, TMINV corresponds to Riemann’s Terzwechsel, an operation that later becomes widely known as the R operation.3 Lewin formalizes two other operations that later become significant for neo-Riemannian theory: the MD-inversion (MDINV) exchanges the mediant and dominant notes of major and dual minor Riemann systems and corresponds to Riemann’s Leittonwechsel (later standardized as L)—i.e., inverting a major or minor triad about its minor third interval—and the TDinversion (TDINV) exchanges the tonic and dominant notes of major and dual minor Riemann systems and corresponds to Riemann’s Quintwechsel (later standardized as P)—i.e., effecting an inversion of the major or minor triad about its perfect fifth interval. This, then, is the starting point for what later developed into neoRiemannian theory. Although Lewin does not describe how seventh chord transformations operate in Riemann systems, we can extrapolate them here. Before we can proceed, we must first define the interval s, which corresponds to the interval of a seventh from the tonic. In the F-major canonical listing B( d F a C e G we can consider the notes F a C e to form a seventh chord. Let us see what happens when we apply TDINV to the seventh chord. F maps to C and vice versa, the note a is retained as the mediant of the seventh chord, and our new seventh s is the note d. We now see why we could not call the interval s “l” for “leading tone”: we cannot make our Riemann system into an ordered 4tuple ¢T,d,m,l² because the interval of a seventh is not consistently the same size. In the F-major Riemann system, s lies the distance of 11 semitones from T (if we consider the canonical listing to be a segment of a larger series in ETS 12 pitch-class space) but in the dual C-minor system, s lies the distance of 10 semitones from T (again, in ETS 12 pitch-class space). In the former system, the seventh would be considered a leading tone; in the latter, a subtonic. Thus we use s instead of l as a generic interval size (just like the mediant, m), and the “inversion” is contextual and not absolute.

108

Chapter Five

We can now acknowledge some of the limitations of Lewin’s Riemann systems. First, the transformational apparatus is still quite limited; we are only able to transform Riemann systems into other Riemann systems that are isomorphic with each other, through inversion operations like TDINV and through transposition operations such as SHIFT(N) (Lewin 1982, 48).4 Lewin does not account for seventh-chord transformations or transformations of Riemann systems other than ordered triples in his theory, and his incipient transformational theory sets the standard to which the approaches evaluated in chapter 2 closely adhere—for better or for worse—namely, the monolithic transformation of entire sets (chords) rather than discrete transformations acting on individual members of said sets. This gives rise to the problems noted in the earlier chapter involving voiceleading mapping and musical lines. Lewin himself was careful to point out that his new theory avoids conclusions on “voice leading and counterpoint in relation to tonal functionality . . . though one could of course work out protocols for voice-leading and counterpoint in connection with specific individual Riemann Systems other than the tonal ones” (1982, 57). Lewin’s principal investigations into what later became known as neoRiemannian theory are found in his GMIT, particularly in his chapter 8, where Riemann systems are now re-defined in the context of his new GIS concept. As these investigations are quite well known, it should suffice here to consider only those elements of his theory that have direct bearing on the present study. Here Lewin reconsiders the familiar Riemann functions DOM, SUBD, and MED, along with the contextual-inversion operations REL, PAR, and LT, “new” labels for the “old” functions on major/minor triads: TMINV, TDINV, and MDINV, respectively. These Riemannian transformations act upon triadic Klänge only, and—as in the earlier study—there appears to be no consideration of specific voice-leading paths resulting from these transformations. In addition, Lewin does not address the problem of mappings between sets of different cardinality. Some of the more important developments in the later chapters of GMIT include the introduction of definitions for a number of new transformations, including SLIDE (which occupies an important place in a surprising amount of late-twentieth-century music, as we will see in the subsequent chapter), as well as Lewin’s presentation of node-and-arrow network graphs, which later became a staple of neo-Riemannian analyses.5 In his later exploration of Cohn functions (1996), Lewin begins with the familiar case of [037] (the definition of a Cohn function was given on pp. 81–82 of chapter 4), which can be Cohn-flipped in two distinct inversions corresponding to Lewin’s PAR and LT operations; he then proceeds to devise a large number of additional examples of Cohn functions. In the last two parts of his paper, Lewin proposes a function he calls DOUTH1, after Jack

The SST Model and Neo-Riemannian Theory

109

Douthett, in which a relation between two pc sets can be obtained if one member pc of the first set is discarded in favor of a new pc a semitone away from the original pc. The remaining pcs are held in common between the first and the second pc sets. It is at once evident that there are some intriguing parallels between Lewin’s DOUTH1 function and my SST function. In fact, the preceding description of the DOUTH1 function quite accurately describes the effect of an SST operation, but without the additional specificity of 1) “voice-leading direction” and 2) identification of the order position that undergoes the semitone displacement, both of which are provided by the SST. Another function Lewin introduces in the same paper is closely related: DOUTH2 works the same way as DOUTH1 except that two pitches in the original pc set are replaced by two pitches a semitone away from the original pitches in the new pc set. As with DOUTH1, there is no apparatus for describing the specifics of the voice leading (which set member undergoes the change, and in which “direction” the semitone intervals proceed from the original set members), and again we note similarities to the SST model, specifically the compounding of two SST operations in the form SST(i)(j). Upon a suggestion from Douthett, Lewin shows how a dominant-seventh/halfdiminished-seventh chord {0,2,5,8} is in a DOUTH2 relation with three of its own transpositions and six of its own inversions (1996, 206–7), but he does not list, or even mention, the numerous other sets that do not belong to set class [0258] which can result from the DOUTH2 operation. Lewin’s introduction of the DOUTH2 function proved to be significant for later theorists: both DOUTH1 or DOUTH2 became integral parts of later neo-Riemannian studies (but under new names), and—perhaps even more important—for possibly the first time in this new area of inquiry, parsimonious voice-leading transformations were mapped between pc sets other than major/minor triads. This was a major conceptual breakthrough, and Lewin’s Example 19 (demonstrating DOUTH2 relations between the Tristan chord and its own transpositions and inversions) was to be particularly influential on studies extending the voice-leading parsimony of triadic neo-Riemannian operations to seventh chords and beyond. In concluding this review of Lewin’s considerable contributions to neoRiemannian theory, we should consider a variation on the DOUTH2 transformation in which the two replacement pitch classes lie the distance of a perfect fourth or a perfect fifth from the “discarded” pitch classes, instead of the distance of a semitone. Lewin describes sets so related as in a DOUTHCF2 relationship with each other, where “CF” stands for “circle of fifths.” Again, there are obvious parallels with my SFT function; specifically, DOUTHCF2 corresponds to SFT(i)(j). As with the DOUTH1 and DOUTH2 functions, specific information about the set members undergoing

110

Chapter Five

the transformation and their directed motion is not provided by DOUTHCF2. The Fall 1998 issue of JMT, a special issue devoted to neo-Riemannian theory, was remarkable for the appearance of a number of articles in which neo-Riemannian concepts were extended to set-theoretic objects other than major and minor triads—in particular, to seventh chords. Writings by Adrian Childs, Edward Gollin, Stephen Soderberg, Clifton Callender, and Jack Douthett with Peter Steinbach all offer important contributions to the thenrelatively-new branch of theory, with most studies taking Lewin’s DOUTH1 or DOUTH2 functions as their starting points. Callender’s article (1998) examines parsimonious voice-leading in Scriabin, in particular between representatives of the “mystic chord” hexachord [013579] and closely related set classes: the whole-tone hexachord [02468T], the “acoustic” heptachord [013468T], the octatonic set [0134679T], and others. In order to bridge the cardinality gap between these different collections, Callender proposes—and he may be the first to formulate—“split relations.”6 Callender’s splits differ from my splits in that his pitch class undergoing the split does not map to itself in the new chord. Instead it “maps” to the two pitch classes each a semitone away from the original pc, through contrary-motion voice leading.7 Thus in the case of a trichord X splitting to a tetrachord Y, Callender’s S(x) relation retains two common pitch classes while obtaining two new pitch classes in tetrachord Y. The original pitch class x (undergoing the split) is “obliterated” and is not a member of set Y. In the SVT model, the operation SPLITi on Xi, followed by some SST, results in the retention of three common tones and Y contains only one new pitch class. Callender’s split is useful in bridging the cardinality gap between common triads and seventh chords in that splitting the root of a major triad produces a half-diminished seventh, and splitting the fifth of a minor triad (its dualistic root) produces a dominant seventh (229). In addition to the split relation, Callender makes use of Lewin’s DOUTH1 operation, re-labeling it P1 (ostensibly “P” for “parsimonious,” not “parallel”) and using it to describe proximate voice leading between any two sets of the same cardinality (when compounded with S(x) it allows for smooth voice leading between sets of differing cardinality). The P1 function may be reiterated, producing P2, the equivalent of Lewin’s DOUTH2 relation, but with the caveat that the same “voice” may not map to a new pitch class twice in succession.8 The P1 and P2 functions, in effect re-labeled DOUTH1 and DOUTH2 relations, inherit the same principal drawback from Lewin: the half-step motion they describe is non-directed and they do not specify which voice is involved in the operation, although the split operation S(x) is a little more helpful in that it precisely identifies the pitch class that undergoes the split (x).

The SST Model and Neo-Riemannian Theory

111

Adrian Childs’s paper investigates transformational parsimonious voice leading between seventh chords, but only chords that are members of SC [0258]—dominant and half-diminished sevenths. Unlike many of his contemporaries (including Lewin), Childs is interested in the specifics of voice leading, and introduces a system that precisely identifies which set member moves and somewhat less precisely identifies the direction in which it moves. Childs’s nomenclature for parsimonious voice-leading transformations between members of SC [0258] takes the form Pn, where n equals the number of voices that move by half step as the other voices remain constant, and thus corresponds to Callender’s Pn relation. As P1 relations do not obtain between two distinct [0258] n-tuple representatives (see Appendix B and p. 87 of chapter 4), Childs only examines P2 relations, which correspond to Lewin’s DOUTH2 relation. To specify the voices that move and the type of motion they participate in, Childs uses the notation Sx(y) and Cx(y) for his transformations, where S stands for “similar motion”9 and C stands for “contrary motion.” The subscripted integers x and y correspond respectively to the ic between the pitches being held constant and the ic between the firstchord pitches that move. Childs’s S operations can be construed as a special form of O’Donnell’s dual transformation in which one of the transformations is equal to T0. The use of these subscripts is a significant step toward more precise identification of the voice-leading specifics of parsimonious transformations, and allows Childs the ability to construct a 3D graph of P2 relations between all dominant and half-diminished seventh chords within a single octatonic collection. Childs’s network graph is reproduced in Figure 5-1. In the graph, dominant seventh chords are designated by a pitch class letter name followed by a plus sign and half-diminished seventh chords are designated by a pitch class letter name followed by a minus sign. In addition to the similarmotion transformations that form the edges of the cube, contrary-motion transformations form a second network crisscrossing the faces of the cube. Each of Childs’s transformations has a SST-succession-class counterpart ¢[A],[B]²SST(i)(j): among the S transforms, S2(3) = the SST-succession-class subscript [SST(3)(4)], S4(3) = [SST(2)(3)], and S5(6) = [SST(2)(4)]; the C transformations, which preserve mode, include C6(5) = [SST(1–1)(3)], C3(2) = [SST(1)(2–1)], and C3(4) = [SST(1–1)(4)]. There are other P2 transformations on members of [0258] which are not shown in Figure 5-1 but can be seen in Figure 4-5, Douthett and Steinbach’s pipeline torus. Childs’s cube comprises the left-most segment of the pipeline, the octatonic collection {0,2,3,5,6,8,9,E}, including the crossing voiceleading connections on the four “panels” of a pipeline segment, not shown on the pipeline itself but demonstrated in the diagram at the bottom of the figure. P2 transformations that bridge the gap between different octatonic

112

Chapter Five

Fig. 5-1: Childs’s network of P2 transformations between set members of [0258] within an octatonic collection (from Childs 1998)

collections are (moving from cross-section 2 to cross-section 3 in Figure 45) S6(5) = [SST(1–1)(3–1)], S3(4) = [SST(1–1)(4–1)], and S3(2) = [SST(1–1)(2–1)]. It is readily apparent that Childs’s nomenclature has the attractive quality of singling out the precise pitch classes that participate in the parsimonious voice-leading transformations. In the case of the S transformations, it also has the advantage of assigning a single label to transformations with parallel voice leading, regardless of the direction of motion, similar to the SSTsuccession-class notation. Conversely, however, the nomenclature suffers slightly from its failure to provide the voice-leading direction of chord members participating in contrary or parallel voice leading. This can be determined easily enough in the case of dominant and half-diminished seventh chords, but if one is dealing with representatives of other, less familiar set classes, or labeling functions between members of different set-class types, it can require additional calculation time. All in all, Child’s nomenclature is probably no easier or more difficult to decipher than my SST-successionclass notation; both require some mental juggling with the chords’ structure,

The SST Model and Neo-Riemannian Theory

113

and both require some time to get used to their conventions. With practice, the SST-succession-class nomenclature can become as familiar to the analyst as the identification of Roman numerals and figured bass to the novice student of tonal harmony.

?

?

Fig. 5-2: Partial ambiguity associated with voice leading operation C3(2) when initial chord contains more than one ic 3

There are a few other consequences of Childs’s S and C transformations that are not readily apparent upon first appraisal of his system. We have already seen how the cubic network of P2 voice leading between dominant and half-diminished seventh chords shown in Figure 5-1 cannot accommodate relations outside of a single octatonic collection; these are more easily displayed in the pipeline torus graph. Another—perhaps unanticipated—potential problem with Childs’s nomenclature is exemplified in Figure 5-2, which is a representation of the voice-leading transformation C3(2) from F7. Unless one is carefully plotting the voice leading of each pair of parts, it is easy to make a mistake and map the F7 chord to an A(7 chord by overlooking the specification that the parts that participate in the contrary motion must lie the initial distance of ic 2 from each other. There are, in fact, two instances of ic 3 in the F7 chord, shown in brackets in Figure 5-2. Without considering the (2) part of the transformation, an erroneous C3(4) function can result. The greatest limitation to Childs’s transformational system is selfimposed: he limits himself to DOUTH2 transformations and to a single set class. In setting such strict limitations, however, he is able to examine this system in greater detail than would have been permitted by an attempt to generalize his system to a greater constellation of set-class types and transformational operations. We have already seen some of Douthett and Steinbach’s graphs in chapter 4 that show parsimonious relations in networks of dominant and half-diminished seventh chords (the pipeline torus), to

114

Chapter Five

which they add minor seventh chords (of SC [0358], in the EnneaCycles and tower torus), diminished seventh chords (of SC [0369], power towers) and even go so far as to theoretically allow for major seventh (SC [0158]) and French sixth chords (SC [0268]). We will now briefly consider additional aspects of their investigations into voice-leading parsimony involving seventh chords. Lewin (1996) credits Douthett with being the first to propose what he calls the DOUTH1 and DOUTH2 operations. We have seen these renamed Pn and Pn by Callender and Childs, respectively, where n = 1 or 2. Douthett (with Steinbach) revisits these operations and proposes yet another notational convention, which appears to be gaining currency as the standard way of referring to this type of operation. The new label takes the form Pm,n and refers to an operation between sets X and Y in which m pcs move by semitone from X to Y, n pcs move by whole step from X to Y, and the remaining pitch classes are held as tones common to both X and Y. The introduction of the element n allows whole-step voice leading to be considered parsimonious, and thus admits operations like the neo-Riemannian R, which is not a DOUTH1 or DOUTH2 operation. Thus R would be labeled a P0,1 operation under the new nomenclature, L and P and all other DOUTH1 functions would be P1,0-type operations, and Childs’s Sx(y), Cx(y), and all other DOUTH2 functions would be P2,0-type operations. Douthett and Steinbach’s functions are like Lewin’s rather than like Childs’s in that they do not show which set members move or in which direction they move. In SVT terms, P0,1 would correspond to some SVTi2, P1,0 would correspond to some SSTi, and P2,0 would correspond to some SST(i)(j). The SVT model allows for multiple simultaneous voice-leading transformations not discussed by Callender, Childs, or Douthett and Steinbach. For example, the generic SST(i)(j2)(k–1) would be labeled P2,1 by Douthett and Steinbach, but of course this notation does not show the direction of the voice-leading motion as the SVT model clearly does. The towers torus (Figure 4-11b) is a lattice displaying P1,0 and P0,1 relations among the set of all dominant, half-diminished, and minor seventh chords (this set is called S1 by the authors). In an interesting development, Douthett and Steinbach attempt to link specific seventh-chord transformations to the “canonical” neo-Riemannian triadic transformations L, P, and R. This is most easily seen by considering their R* operation between dominant and half-diminished seventh chords, which—like the triadic analogue R—is a P0,1-type operation. In SST-succession-class terms, R* corresponds to ¢[0258],[0258]²SVT(1–1)2. Slightly more difficult to conceptualize are the analogies between the triadic P and the tetrachordal P* operations, and also the triadic L and the tetrachordal L* operations. All are of the P1,0 type, but there are now two P* and two L* operations. P*1 connects half-diminished

The SST Model and Neo-Riemannian Theory

115

sevenths to minor sevenths with the same root; in SST-succession-class notation it is simply ¢[0258],[0358]²SST4. P*2 relates minor sevenths to dominant sevenths with the same root; because this succession is a retrograde inversion of P*1, the SST-succession-class notation for P*2 is ¢[0358],[0258]²SST4, utilizing the same relation [SST4] as the SSTsuccession class ¢[0258],[0358]²SST4. As with the triadic P operation, the concept of a chord’s “root” is indispensable to Douthett and Steinbach’s formulation, and all three of these P-type operations effect a change of chord “quality” in the functional tonal sense. The tonal concepts of “root” and “quality” are of course unnecessary in the SVT system. Like P*, there are two L-like operations: L*1 relates half-diminished and minor seventh chords, and L*2 relates minor seventh and dominant seventh chords. These are distinguished from P*1 and P*2 by the fact that the P* operations relate chords with a common root but the L* operations do not, in exactly the same way P and L differ as triadic operations. In SST-successionclass terms, L*1 and L*2 both correspond to ¢[0258],[0358]²SST2 and to ¢[0358],[0258]²SST2–1. Douthett and Steinbach go so far as to question the analytical value of these seventh-chord P1,0 and P0,1 operations, noting that most of the music that has been profitably analyzed using a seventh-chord transformational system relies upon P2,0-related voice leading. As examples of the latter, they cite analyses of music by Wagner, Chopin, Scriabin, and Stravinsky, and note that “as of yet a search for analytical examples of P1,0- and P0,1-related seventh chords has not been made” (257). In chapter 6 of this study I will present instances in which tetrachordal- or pentachordal-textured music can be profitably analyzed by noting P1,0- and P0,1-type relations. The adaptation of neo-Riemannian triadic operations to non-triadic—and specifically to tetrachordal—textures is a theme that we will see playing out in subsequent studies and a topic to which I will return later in this chapter. Like Childs’s, Gollin’s study focuses on what Douthett and Steinbach would call P2,0-type transformations on the dominant/half-diminished [0258] seventh chord. New in Gollin’s article is the construction of a 3D Tonnetz for this set class. A portion of this Tonnetz is shown in Figure 5-3. In the figure, upward-pointing tetrahedrons form dominant seventh chords, and downward-pointing tetrahedrons form half-diminished seventh chords. A central tetrachord, {C,E,G,B(}, is surrounded by what Gollin calls “edge flips”; these are inversion operations which map around two invariant pcs. In a kind of “neo-Hauptmannian” approach, Gollin labels the root of each seventh chord “i”; this pc corresponds to the Hauptmannian “I,” or Einheit. In the case of the “minor dual” of the dominant seventh chord, the seventh (dualistic root) of the half-diminished seventh chord is labeled “i.” The third of the chord is labeled “ii,” the fifth is labeled “iii,” and the seventh “iv.”10

116

Chapter Five

Figure 5-3: A portion of a Tonnetz for SC [0258] showing edge flips about central tetrachord {C, E, G, B(} (from Gollin 1998)

An edge flip about the chord members i and ii, for example, is labeled Iiii , where “I” stands for “inversion” and is not a Roman numeral. When the Iiii operation is applied to the central tetrachord of the figure, {C,E,G,B(}, an F half-diminished seventh chord results. Gollin’s approach differs from the other JMT articles we have considered up to this point in that instead of conceiving transformations between [0258]-type chords as resulting in semitone displacement, transformations are pc-set inversions which may or may not result in parsimonious voiceleading relations. Some of the ramifications of this re-conceptualization of the operative transformation should be explored here. First, there has been a sea change in emphasis from parsimonious voice leading to a more traditional pc set inversion operation. Nowhere in his study does Gollin discuss the implications of inversion operations on pc voice leading; we can infer that an inversion transformation on a pc set would map individual pcs in a kind of pitch-class counterpoint that closely resembles some of the approaches examined in chapter 2 (Klumpenhouwer, Straus, O’Donnell et al.). For example, in the case of the Iiii operation on {C,E,G,B(}, pcs C and E map to themselves, G maps to A, and B( maps to F, although the small staff at the bottom of Figure 5-3 clearly shows that the most efficient voice leading between the non-common-tone pcs would be a pair of semitone shifts

The SST Model and Neo-Riemannian Theory

117

from B( and A and from G to F. This would be a P2,0 operation after Douthett and Steinbach; in SVT terms it would be SST(3–1)(4–1) (¢0,4,7,10²[0258]) = ¢0,4,6,9²[0258], which belongs to SST-succession class ¢[0258],[0258]²SST(2)(3). Second, although Gollin goes on to create a generalized tetrachordal 3D Tonnetz, he admits that it would be difficult to adapt such a graph to set classes other than [0258]: “The family of common-tone preserving, contextual inversions of one set class is in general a different family of contextual inversions from any other” (203). Even if we were to find a set-class type that would work within the generalized Tonnetz, the lowercase-Romannumeral labels i through iv would have to be completely reassigned on a basis other than measurement from a tertian “root.” Thus there would be no generalization of contextual inversion functions to all tetrachordal set classes. This is in contrast to the more versatile SVT model in which the same SST or SFT function can be applied to any m-tuple, and their associated SVT-succession classes can encompass any set classes of any cardinality. Building on the foundation built by Lewin, Childs, and Douthett and Steinbach, Richard Bass explores voice leading to and from half-diminished seventh chords by taking a curious approach that fuses ostensibly “atonal” transformational theory with tonal theory (Bass 2001). In the first part of his paper, Bass examines a number of “optimally smooth” voice-leading progressions to and from the half-diminished seventh chord—here not inversionally equivalent to the dominant seventh chord. Then he shifts gears, pulling from the transformational toolbox a P2,0 operation (adapted from Childs’s C transform) which he calls an “ic 4-2 transform.” This operation, when applied to a half-diminished seventh chord, converts the pcs that form a dyad of ic 4 to pcs forming a dyad of ic 2 through contrary motion while the other two pcs remain invariant. Related to the ic 4-2 transform is the “ic 5-5 transform,” whose voice-leading specifics should be self-explanatory. Bass’s article is rather interesting as a study of a specific chord structure within a tonal harmonic context (in spite of being limited to only one type of chord), but ultimately it does not advance transformational theory to any great degree. Granted, the larger goal of his study is not necessarily an advancement of transformational theory but rather to show how halfdiminished seventh-chord progressions function within and between octatonic collections, how they participate in Cohn-like cycles, and how concepts of voice-leading parsimony aid a tonal analysis of late-romantic music. As the admittance of seventh-chord structures into triadic neo-Riemannian theory “presents a special problem” because of “the inherent inapplicability of Riemannian transformations like ‘parallel,’ ‘relative,’ and ‘Leittonwechsel’ to connections among seventh chords of any type,”11 Bass does not

118

Chapter Five

pursue a solution to this problem, presumably because the issue lies outside his area of focus. Attempts at a solution will be discussed presently in connection with studies by Amy Shimbo, Julian Hook, and Scott Baker, and I too will address the topic briefly at the end of the chapter.

a)

S2

S3

S4

S 3?

S3?

S3?

b)

Fig. 5-4: a) Shimbo’s S2, S3, and S4 transformations; b) hypothetical S3 transformations to other sonorities

A conference paper presented by Amy Shimbo (2001) advances a number of new split functions (after Callender 1998) linking major or minor triads with dominant/half-diminished seventh chords. Like Callender’s splits, Shimbo’s split functions do not preserve the splitting pc in the chords to which they “map,” and like Callender she uses the operational label Sx for these functions without providing a mathematical explanation for the splits themselves. However, Shimbo’s Sx function is not the same as Callender’s S(x) function: whereas Callender’s subscript x refers to the pitch class that undergoes the split, Shimbo’s x refers to the total displacement, in semitones, of the resultant pitch classes from the original pc undergoing the split. Thus, Shimbo’s S2 split corresponds exactly with Callender’s S(x) split, and she introduces the additional splits S4, in which the two new pitch classes each lie a whole step from the splitting pitch class, and S3, in which one part of the split tone moves by semitone and the other by a whole tone

The SST Model and Neo-Riemannian Theory

119

in contrary motion. These split functions are demonstrated on a C-major triad in Figure 5-4a (minor equivalents are shown in Shimbo’s paper but not in this figure). It is important to note how Shimbo determines which voice undergoes the split. Taking her cue from Hauptmann and from Gollin’s neoHauptmannian labels i, ii, iii, and iv, Shimbo sets the root of a triad equal to I, the Hauptmannian Einheit, the fifth equal to III, the Verbindung,12 and the third equal to II. She likewise labels the members of dominant or halfdiminished seventh chords as Gollin does, from i to iv, measuring from the root out to the seventh. (In the case of minor triads and half-diminished seventh chords, these are measured from their “dual roots.”) Shimbo then derives “rules” for the split transformations that ensure that only members of set classes [037] and [0258] are involved in the split. In the case of S2, I splits into i and iv; with S3, III (the fifth) splits into iii and iv; and with S4, I splits into i and ii (Figure 5-4a). Translated to SST-succession-class terminology, these would loosely correspond to ¢[037],[0258]²SST(3–1)(4)(SPLIT3), ¢[037],[0258]²SST(1–1)2(2)(SPLIT1), and ¢[037],[0258]²SVT(3–1)2(42)(SPLIT3), respectively. What would be the result of one of these new split transformations if Shimbo’s rules were relaxed? If the split transformation is allowed to lead to a set class other than [0258] some interesting possibilities can be obtained. The first measure of Figure 5-4b shows an S3 split that involves the root of the triad (I) instead of the fifth. The result is a minor seventh chord. Had the voices derived from the split pitch class swapped intervals, the result would have been a C fully diminished seventh chord (not shown). The second and third measures of Figure 5-4b show two possible results of a S3 split on the third of the chord. Shimbo also proposes neo-Riemannian operations on dominant/halfdiminished seventh chords which are tetrachordal analogues to the triadic L, P, and R transformations. Her L¡ function transforms a [0258]-type seventh chord to another of the same type related to it by P0,1 (e.g., a C dominant seventh chord to an E half-diminished seventh chord); the P¡ function toggles chord quality from dominant to half-diminished while maintaining the same root, and R¡ transforms a [0258]-type seventh chord to one related to it by P2,0 (e.g., a C dominant seventh chord to an A half-diminished seventh chord). The most appealing property of these “neo-neo-Riemannian” seventh-chord transformations is that they share the same root relations as ∅ L¡ their triadic counterparts. The seventh-chord operation C7 → e 7 shows a L clear relation to the triadic operation C+ → E–. The importance of this correlation between triadic and seventh-chord operations, however, is missing in non-tonal contexts. Shimbo’s ¡ functions are dependent on a definition of a chord “root” and thus are not generalizable to set classes that do not form traditional tertian sonorities. Furthermore, some of the “essential”

120

Chapter Five

identifying characteristics of the neo-Riemannian L, P, and R transformations are not carried over into the new seventh-chord operations. For example, the most identifiable characteristic of the triadic Leittonwechsel operation is the semitonal motion of the pitch class forming part of ic 4 to an interval of ic 5 and vice versa; or, to continue to use our C+/E– example, the chord member that leads between pcs C and B. In the L¡ operation this identifying characteristic is not to be found, and in its place is a whole-step displacement in the opposite direction, more reminiscent of the triadic R operation than L. An important approach to neo-Riemannian theory through the mathematical formalization of some of its concepts is found in Julian (Jay) Hook’s 2002 dissertation. Hook’s uniform triadic transformations (UTTs) are transposition and inversion operations on major or minor triads with clearly defined roots in a tonal system. The UTTs are composed of the “traditional” neo-Riemannian functions L, P, and R, as well as many others (288, to be exact). Hook’s UTTs map triads to triads (and by extension, to certain seventh chords) and are thus set-class transformations along the lines of those in writings we have examined earlier, such as those by Klumpenhouwer, Straus, O’Donnell, and Gollin, among others. The result of this reliance on set-class mapping is a theory which does not concern itself with aspects of voice-leading parsimony, in contrast to a good number of neo-Riemannian studies.13 A UTT is an ordered triple: a sign (σ) and two integers mod 12. The sign indicates whether the mode of the triad undergoing the transformation is preserved or “reversed” (toggled between major and minor). Thus the sign “+” indicates mode is preserved and “–” indicates mode is reversed. The integers specify the interval of transposition of the root of the triad: the first integer specifies the interval of transposition for major triads, and the second for minor triads. Thus the UTT ¢–,4,8² indicates an operation that changes the chord quality from major to minor and vice versa and transposes a major triad by T4 or a minor triad by T8. This particular UTT corresponds to the neo-Riemannian Leittonwechsel. The neo-Riemannian R operation is represented by the UTT ¢–,9,3² and P corresponds to ¢–,0,0². In the sixth chapter of his dissertation, Hook investigates seventh-chord transformations as well as transformations that map triads to seventh chords and vice versa. Like several other authors, Hook limits his discussion of seventh chords to the dominant/half-diminished seventh chord [0258]. The sign σ of the UTT ordered triple is now reinterpreted to act as a toggle between a major-minor seventh chord and a half-diminished seventh chord. Thus the ∅ Leittonwechsel UTT, ¢–,4,8² transforms a C7 chord to e 7, and Hook suggests that the L label could even be applied to this seventh-chord transformation, although he also notes “one might justifiably feel that the action of

The SST Model and Neo-Riemannian Theory

121

these transformations on seventh chords is not similar enough to warrant using the same labels here” (101). It should be noted that in adapting the triadic UTTs to [0258] transformations, the transposition levels remain the same as their triadic counterparts, being based on the chord’s root. Thus Hook’s tetrachordal “L,” “P,” and “R,” transformations correspond exactly to Shimbo’s L¡, P¡, and R¡ operations. Hook’s answer to the problem of splits and fuses is the cross-type transformation, which maps objects of one type to objects of another type through an inclusion transformation, usually coupled with another type of transformation. When applied to members of set classes [037] and [0258], the inclusion transformation maps a major triad to the unique major-minor seventh chord that contains it; a minor triad maps to the unique half-diminished seventh chord that contains it. This transformation is labeled ⊂, thus a C-minor triad operated on by ⊂ maps to an A half-diminished seventh chord. It should be noted that the cross-type transformation does not account for voice leading or the derivation of the (new) fourth pitch when mapping from a triad to a seventh chord. Additionally, this type of transformation is only defined for relations between members of set classes [037] and [0258]. The reverse of ⊂, a fuse-type operation, is labeled ⊃ and maps [0258]-type seventh chords to [037]-type triads. Hook’s UTT system offers an elegant mathematical model of triadic transformations; his cross-type transformations effortlessly integrate major and minor triads with common dominant and half-diminished seventh chords, and his investigations bring the full potential of the theory of mathematical groups to neo-Riemannian theory to a greater extent than any other study since Lewin’s GMIT. Hook’s UTTs provide a reasonable model of triadic progressions in tonal music (with the addition of the occasional dominant or half-diminished seventh chord) with one exception: relatively straightforward tonal progressions like I–IV–V–I appear somewhat “clumsy and heavy-handed” (172) when analyzed using this model. (The same could be said of most neo-Riemannian approaches in general.) This deficiency can be adequately compensated for in the SVT model by accounting for voice leading by the interval of a fifth through single fifth transformations (SFTs); the analyses presented in chapter 6 demonstrate the utility of this theoretical tool in a variety of musical styles. The “neo-Riemannian” publications we have considered up to this point have their origins in the analysis of a very limited palette of musical styles (mostly late classical to late romantic) and indeed have even purposely limited the types of harmonic objects studied (in most cases tertian representatives of set classes [037] and [0258]). In order for the theoretic systems developed by these authors to function, each harmonic object must contain a chord root, and therefore each system relates either closely or somewhat

122

Chapter Five

obliquely to the common-practice tonal tradition. The one exception to this general observation is Childs (1998); by divorcing his S and C functions from a tonal context, he is able to focus on the ic content of chords and sets, relating set classes through parsimonious connections in tonal as well as atonal music. Although he considers only set class [0258], his model has the potential to be applied to most other (non-symmetric) set classes as well. Another attempt at opening up neo-Riemannian theory to the analysis of atonal music is Baker 2003. Baker’s system does not go so far as to abandon functional tonality and the central rôle it plays in neo-Riemannian theory, but by logical extensions to now standard transformational constructs (such as the LPR family of operations) it theoretically allows for the analysis of nontonal repertoires by this approach. In practice, these extensions only admit triadic and seventh-chord “common-practice sonorities” into the neoRiemannian realm, but still this is a major advance over the limited number of set-class types previous systems had limited themselves to. Table 5-1: Baker’s tables of semitonal transformations

Algebraic transformation

Function

“Neo-Riemannian” function

Triad to Triad XM – Xm

1α1

P

XM – (X+4)m

1α2

L

XM – (X+1)°

1α3

(–L)

Xm – X°

1α4

(–L)

XM – (X+0, 4, 8)+

1α5

(R*)

Xm – (X+3, 7, 11)+

1α6

(R*)

Xdom7 – Xm7

1β1

P*2

Xdom7 – (X+9) m7

1β2

L*2 (R1)

Xdom7 – (X+1, 4, 7, 10)°7

1β3

(–L1)

Xdom7 – XM7

1β4

(R*1)

Seventh Chord to Seventh Chord

The SST Model and Neo-Riemannian Theory

Algebraic transformation

Function

123

“Neo-Riemannian” function

Xm7 – X∅7

1β5

P*1

Xm7 – (X+9)∅7

1β6

L*1 (R2)

X∅7 – (X+0, 3, 6, 9)°7

1β7

(R*2)

X∅7 – (X+11)M7

1β8

(–L2)

XM – XM7

1χ1

-

Xm – (X+8)M7

1χ2

-

Triad to Seventh Chord

Baker’s functions are displayed in Table 5-1. Each function represents displacement by the interval of a semitone in an individual chord voice. Functions operate on major or minor triads and dominant, minor, or halfdiminished seventh chords. As an example of Baker’s transformational shorthand, the function 1α5 stands for an operation that transposes a major triad X by ic 0, 4, or 8 and then converts it to an augmented triad. Although the voice which moves by semitone after transposition is not directly identified, it can easily be located. Traditional neo-Riemannian labels—where they exist—are listed for triadic functions; non-parenthetical seventh-chord functions are labeled after Douthett and Steinbach 1998. The last section of the table lists two possible split transformations. These splits differ from those proposed by Callender and Shimbo in that the “splitting” pc is retained and a new pc a semitone away is also obtained. In this way Baker’s split functions are closer to mine, even if their mathematical bases differ. Split functions that hold a minor-third interval invariant are called “L splits” (SL), and those that hold a major third invariant are “R splits” (SR).14 At first glance it could be easy to mistake Baker’s six triadic functions and eight seventh-chord functions for a systematic exploration of all possible semitonal voice leadings stemming from a single chord type. Had he chosen to do so, there would have been a total of twenty-four functions for the four triad types (if one counts all voice leadings, but only nine if one allows for transpositional and inversional equivalents) and forty functions for the five seventh-chord types (if one counts all voice leadings, but only fourteen if one allows for transpositional and inversional equivalents). Instead, the numbering of Baker’s functions does not follow any particular logical

124

Chapter Five

sequence, making his labels even more arbitrarily assigned than Forte’s setclass labels.15 Some functions other than the usual neo-Riemannian L and P and Douthett and Steinbach’s L*1, L*2, P*1, and P*2 are assigned additional labels by Baker. A function preceded by a minus sign resembles a Leittonwechsel or L-like seventh-chord function in every way except for the directed motion of the changing chord element, which instead moves in the opposite direction by semitone. An R function followed by an asterisk is called a “fuzzy R” transformation by Baker; it differs from an R or R-like seventh-chord function in that the voice that moves does so only by a semitone, instead of the expected whole step. These new labels are enclosed within parentheses in Table 5-1. Baker breaks with the rationale behind Shimbo’s, Hook’s, and Douthett and Steinbach’s extension of neo-Riemannian labels to seventh chords: instead of basing the new labels on chordal root relations or set-inclusional principles, they instead are based on the less-formal quality of aural similarity to their triadic analogues. For this reason, Baker re-labels Douthett and Steinbach’s L*2 function, calling it R1 because one is likely to hear the dominant-seventh and minor-seventh chords at a minor-third “relative” interval to each other, much as their triadic counterparts are commonly identified in tonal harmony. Baker’s paper appears significant in that it admits a number of new (common-practice) set classes into a neo-Riemannian transformational system, but this development simultaneously presents us with at least two new problems. The first—and less significant—issue deals with the increasing proliferation of new transformational labels, a process that is not uniquely Baker’s fault, but has continued nearly incessantly with each new publication that has followed Lewin’s seminal studies. In the next section of this chapter I will tabulate the various competing neo-Riemannian labeling systems and show their congruence with the decidedly non-neo-Riemannian single-voice transformational model. The second issue is a thorny one: how does one extend neo-Riemannian concepts to non-tonal music or to music from the last quarter of the twentieth century, much of which is either modal (where verticalities are usually less significant than the rhythmic structure, as is the case with most minimalism), neo-tonal (where tertian sonorities prevail but in a non-functional context, e.g., in the music of Paul Lansky and Daniel Lentz), or a juxtaposition of atonal and neo-tonal elements (George Rochberg, John Corigliano, Aaron Jay Kernis and others)? The reader is likely to discern where I am headed with this question. The SVT model retains many of the aspects of recent neoRiemannian researches, particularly an emphasis on semitonal voice leading, simple operations in which one voice moves and the remainder sustain com-

The SST Model and Neo-Riemannian Theory

125

mon tones, and split/fuse operations which allow mapping between chords of differing cardinalities. Unlike neo-Riemannian operations, however, single-voice transformations do not rely on a chord “root,” nor do they need to refer to tonal relations (e.g., MED, D, or R operations) in order to function. SVTs and SST-succession classes can be profitably used to analyze atonal as well as tonal passages, and even otherwise bewildering recent music which appears to contain elements of both organizational schemes. In the next section of this chapter I will explore how SVTs can be of particular use in pointing out neo-Riemannian-like operations in (neo-)tonal/atonal “hybrid” music.

Relating SST-Succession Operations

Classes

to

Neo-Riemannian

A Comparison of Neo-Riemannian Labeling Systems Let us turn our attention to the first of the problematic issues raised in our discussion of Baker’s paper: the increasing proliferation of labels for triadic and seventh-chord neo-Riemannian functions. Not all of these labels necessarily emphasize parsimonious voice leading and common-tone retention; one need look no further than Gollin’s [0258]-type inversional operation labels for evidence of this. Some of these labels—particularly those used by Brian Hyer (1995) for triads—have gained such currency that they now might properly be called “traditional” neo-Riemannian functional labels. The following tables (5-2, 5-3, and 5-4) list a number of relations between common triads and seventh chords that are described first in terms of one or more SST-succession classes, and then with various neo-Riemannian labels that have been attached to them over the years. Table 5-2 lists simple functions that involve motion of only one semitone (or an SST-succession class with subscript [SSTi]), Table 5-3 lists composite functions involving semitonal motion in two or more voices, and Table 5-4 lists three types of split functions connecting triads to seventh chords. Let us briefly mention some of the more interesting aspects of each of these tables. By the internal natures of the different labeling systems, frequently there will not be an exact one-to-one correspondence between two different authors’ functions. As is often the case, one system may achieve with one function a result that can be achieved two different ways in another system. Many of these instances arise out of a mismatch between the dualistic and non-dualistic (or transpositionally/inversionally equivalent and transpositionally equivalent only) pitch-class spaces associated with each theoretical system. This can be seen in the tables in cells in which two functions are

[037]

[036]

2

3

–1

[0369]

[0358]

[0258]

[037]

[0258]

[0258]

4

[0258]

3

all

[0258]

[0258]

1 2–1

–1

[0158]

[0358]

4

2

2–1

[0369]

[0358]

1–1

1–1, 2 [0258]

–1 [0158] 1, 2

all

[037]

[037]

1–1

[037]

3

[037]

[048]

1–1, 3

[036]

[048]

[B ]

[SSTi ]

[A ]

MDINV

TDINV LT

PAR

Lewin Lewin 1982 1987

P L

W4

P1*, P2*

L1*, L2*

P1*, P2*

L1*, L2*

L

P R

P

Hyer Douthett & Kopp 1995 Steinbach 2002 "Traditional" 1998

W0

Klumpenhouwer 1994



Hook 2002

1β 1, 1β 5

1β 2 (R1), 1β 6 (R2)

1β 4 (R*1), 1β 8 (–L2) 1β 1, 1β 5

1β 3 (–L1), 1β 7 (R*2) 1β 2 (R1), 1β 6 (R2)

1α 3, 1α 4

1α 2

1α 5, 1α 6 1α 1

Baker 2003

Table 5-2: Comparison of labels for single [SSTi]-related tertian SST-succession classes

126 Chapter Five

[0258]

[037]

[A ]

W9 R

[037]

[037]

(2)(3 )

(2 )(3 )

2

2 –1

[0258]

[0258]

[0258]

[0258]

[0258]

(1–1)(4–1)

(2)(3)

(2)(4)

(3)(4)

–1

(1 )(3 )

–1

[0258]

[0258]

–1

–1

(1 )(2 )

[0258]

(1–1)(3)

(1–1)(4)

[0258]

(1 )(2)

–1

(1 )

2

[0258]

[037]

2

[037]

(1 ) (3 )

-1

(1–1)(2)(3)

2 –1

[037]

W8

W7

S 3, S 9

S 9, S 3

LRP

PR

RP

PLP

LPR

RLP

W5

[037]

(1–1)(2–1)

(1–1)(3–1)

W1

LP

PL

S 4, S 8

S 8, S 4

Hyer 1995 "Traditional"

[037]

SLIDE

REL

Klumpenhouwer 1994

–1 (1 )(3)

TMINV

Lewin Lewin 1982 1987

[037]

[037]

[B ]

(1 )(2)

–1

(1 )

2 –1

[SSTi ]s

H

S

N

R

Cohn 1998

R*

R

Douthett & Steinbach 1998

R´

P´

L´

R

Morris 1998

Gollin 1998

F–1







S2(3)

S5(6)

P



(P)

(R)



m–1, m

(L)



m, m –1









Hook 2002

PM

SLIDE

F

M–1, M

M, M–1

r

Kopp 2002

ic4-2

ic5-5

Bass 2001

S4(3) R

L



Shimbo 2001

S3(4)

S6(5)

S3(2)

C3(4)

C6(5)

C3(2)

Childs 1998

Table 5-3: Comparison of labels for composite tertian SST-succession classes

The SST Model and Neo-Riemannian Theory 127

128

Chapter Five

separated by commas. For example, in Table 5-2, my SST-succession class ¢[037],[048]²SST1–1 corresponds to either of Baker’s functions 1α5 or 1α6 on members of [037]. In other words, the SST-succession-class subsumes both of Baker’s functions because Baker’s approach relies on Tn set-class equivalence, while the SST-succession-class approach assumes pairsucession equivalence under Tn/TnI and permutation. Even in cases in which there are no neo-Riemannian equivalents for specific SST-succession classes, there may be two SST-succession classes which achieve the same result through different voice-leading paths ([SSTi] subscripts). The two paths are set apart by commas, as in the separate [SST1–1] and [SST2] subscripts relating the same equivalence-class pair-succession ¢[0158],[0258]². (I have done this purely as a space-conserving measure in the tables.)

Table 5-4: Comparison of labels for tertian split-succession classes [A ] [037]

[SSTj ]/ ([SPLITi ])

[B ]

(4(SPLIT3))

[0158]

Callender Shimbo 1998

2001

((1)(2 ) (SPLIT1)) [0358] ((3 )(4)(SPLIT3))

[0258]

2003 1F1, 1F2 (SL)

2 –1

–1

Baker

SR S(x )

S2

A few elements of the tables merit further explanation. The subscripted S and W labels under the “Klumpenhouwer 1994” column correspond to Riemannian schritt and wechsel relations, with the subscripted integer corresponding to transpositional level. This notation is not actually Klumpenhouwer’s, who instead assigns each relation one of Riemann’s names plus an abbreviation (thus S9 is called a Kleinterzschritt and is assigned the label K). The notation used in Tables 5-2 and 5-3 follows the conventions used in Hook 2002 and other sources, but the identification of the different schritts and wechsels originated with Riemann and was carried to its logical conclusion by Klumpenhouwer. Of some interest in Table 5-2 are the subscripts [SST1] and [SST2–1] relating [0158] to [037] in a fuse-succession class. When we remember that each SSTi relation really exists between m-tuple members of SCs [A] and [B] in an SST-succession class, it becomes apparent that the SSTsuccession classes in reality involve the multiset class [0377][037]: ¢[0158],[0377]²SST1 and ¢[0158],[0377]²SST2–1. To obtain a true fuse each

The SST Model and Neo-Riemannian Theory

129

equivalence class [B] must subsequently appear in the [A] order position of a SPLIT–1-succession class: ¢[0377],[037]²SPLIT3–1. This of course can be notated as a composite fuse-succession class: ¢[0158],[037]²SPLIT3–1(SST1) or ¢[0158],[037]²SPLIT3–1(SST2–1). Also of note is the apparent doubling of Douthett and Steinbach’s L* and P* functions and some of Baker’s 1βn functions involving SCs [0258] and [0358]. This is a result of the difference between my transpositionally, inversionally, and permutationally equivalent SST-succession classes and the other authors’ transpositionally (only) equivalent systems. The L* operations are further affected by the structure of the [0358] set class, resulting in the need for two separate SSTsuccession classes whose subscripts are inversely related.16 It can also be seen from Table 5-2 that a large number of parsimonious relations can be explained by the SST model above and beyond even the expanded neo-Riemannian functions of Douthett/Steinbach and Baker. The table does not even take into account the ability of SSTs to model parsimonious voice leadings to and from non-tertian set-class members or tertian chords of cardinality 5 or greater. In contrast, Table 5-3 purposely limits itself to SST-succession classes in which [A] = [037] or [0258] because so many extended neo-Riemannian functions have been devised that map various members of each of these set-class types to other chords of the same SC type. The SVT model is capable of accounting for many other composite functions between common tertian sonorities, as we have seen in Table 5-2, but these are excluded here to keep the table from becoming prohibitively large. The tabulation of these additional composite SST-succession classes is left as an exercise for the interested reader. Perhaps the quirkiest aspect of Table 5-3 is the presence of the R function here. This function can be explained by all other labeling systems as a unitary mapping, but in the SST model it is a composite move requiring consecutive SSTs in the same order position. Nevertheless, it is still considered a strongly parsimonious voice leading according to our definition in chapter 3. A few of the Cohn labels require some explanation here. The N function stands for Nebenverwandt17 and H stands for “hexatonic pole,” the triad of opposite mode that lies precisely halfway around a hexatonic cycle from the triad of origin.18 S is short for SLIDE, but the designation is problematic not only because it can be confused with the S of the schritt/wechsel group but with the “subdominant” function S used by Lewin and Klumpenhouwer.19 As is well known, the traditional LPR family of operations is noncommutative; thus applying P, then L to a triad obtains a very different result than if one had applied L, then P to it. Another factor that must be considered when scanning Table 5-3 is whether each theorist takes a dualistic or a nondualistic approach to transformations. David Kopp’s mediant functions merit special mention here in the context of dualism versus non-dualism. In his

130

Chapter Five

“Dualistic” function

[SST(1–1)(2)] PL

3 2 1

Non-dualistic function

[SST(1–1)(2)] PL

1 2 3

M S8 ¢+,8,4²

M–1 S4 ¢+,8,4²

Fig. 5-5: Comparison of Kopp’s non-dualistic M function to the “dualistic” SST-succession class ¢[037],[037]²SST(1–1)(2)

book Chromatic Transformations in Nineteenth-Century Music (2002), Kopp quite convincingly argues that third relations be accorded a more prominent rôle in the tonal music of the nineteenth century than they previously had been given, and that in some of the music of late-Romantic composers in particular, progressions by major and minor thirds rival and may even surpass fifth progressions in importance as an organizational paradigm. Kopp notes the use of four distinct mode-preserving third progressions in tonal music corresponding to triadic transpositions by ics 3, 4, 8, and 9 and which he labels m–1, M–1, M, and m respectively. Unlike L, P, and R, these mediant functions commute with themselves. Also unlike L, P, and R—as generally understood in neo-Riemannian theory—Kopp’s mediant functions are non-dualistic. This is readily apparent when one compares Kopp’s M and its inverse with the Tn/TnI- and permutationally equivalent (in a sense, “dualistic”) ¢[037],[037]²SST(1–1)(2) SST-succession class as shown in Figure 5-5. The neo-Riemannian S8, Kopp’s M, and Hook’s ¢+,8,4² transformations are all shown to be equivalent in the first bar of Figure 5-5, where a C-major triad maps to an A(-major triad. The premise of each of these transformations is identification of the triadic root and transposition by ic 8 (thus the bold emphasis given to the “8” in Hook’s UTT). The mod-12 inverse of this operation is shown in the second bar of the figure, where a C-minor triad maps to an E-minor triad. Non-dualistic transformational labels include the neo-Riemannian S4, Kopp’s M–1, and Hook’s ¢+,8,4² (with emphasis given to the “4” to reflect the ic by which minor triads are transposed). In contrast to these labels, the [SST(1–1)(2)] and PL relation labels allow the same designation to be used for Tn/TnI-equivalent (dualistic) mappings. Integers 1, 2, and 3 in the figure identify the dualistic root, third, and fifth in both measures for the neo-Riemannian PL interpretation. Unlike the traditional neo-

The SST Model and Neo-Riemannian Theory

131

Riemannian label PL, which is essentially a pair of consecutively applied contextual set-inversion operations, the SST-succession-class mapping is determined exclusively by independent voice-leading paths (constituent SSTs between 3-tuple members of the equivalence classes). In light of the considerable emphasis given to voice-leading parsimony in the neo-Riemannian literature, it seems rather remarkable that the SST model is the only transformational system yet devised that is exclusively reliant on voice leading rather than on more traditional set-class transposition and inversion operations.20 Even Richard Cohn, who was the first to delve into the neo-Riemannian condition of voice-leading proximity in any depth, still views triadic functions principally as set-class cios with interesting voice-leading consequences, rather than the reverse. Table 5-4 shows three different types of split-succession-class relations. Although many more are possible through various combinations of other [SSTj] and [SPLITi] subscripts, these are the ones most commonly mentioned in the literature. The first in the table can be represented by a simple, single split-succession-class relation, and the remaining two require splitsuccession-class subscripts containing two [SST]s. The split function from a major/minor triad to a half-diminished/dominant seventh chord is the focus of Callender’s S(x) operation, where x equals the pitch-class integer which undergoes the split, where the split voices move in contrary motion by semitone, and in which the original pc x is annihilated. Shimbo calls this function S2 and gives somewhat complex rules for determining which triadic member undergoes the split, based on the chord’s dualistic root. The split-successionclass notation is more straightforward: in a 3-tuple transposition, inversion, or permutation of ¢0,3,7² the element corresponding to order position 3 of ¢0,3,7² is duplicated, then the duplicate members of the ensuing 4-tuple move in opposite directions by semitone. Both of Baker’s 1χn functions are subsumed in ¢[037],[0158]²SST4(SPLIT3), which converts a major or minor triad to a major seventh chord. Baker also labels functions of this type “SL” because the motion of one of the splitting voices resembles that of a triadic Leittonwechsel transformation. Likewise, his SR function closely resembles a triadic R transformation in its mapping of a major or minor triad to a minor seventh chord.21

“Neo-Riemannian” SST-Succession Inclusional principle

Classes

and

the

It seems to me that Baker is on to something fairly significant with his neo-Riemannian-like labeling of split transformations involving common seventh chords. This concept is also touched upon by Douthett and Steinbach (1998) and Shimbo (2001) with their L, P, and R-like seventh-

132

Chapter Five

chord transformations. After listening to a certain amount of principally tertian-organized music (both functional and non-functional), one begins to hear progressions that audibly resemble neo-Riemannian triadic transformations but that occur between chords of greater cardinality than three. Much of the time these chords are easily recognizable seventh chords, familiar from the tonal literature, but occasionally one hears a parsimonious progression between atonal tetrachords, pentachords, or hexachords that gives the convincing impression of, say, a SLIDE operation or a Leittonwechsel in a non-tonal context. When one analyzes the harmonies that created such an impression, a certain common denominator can be found. Table 5-5: P-, L-, and R-like tetrachordal SST-succession classes

[A]

P-like [SSTi]

[B]

0137

3

0147

Douthett and Steinbach

R-like

L-like [SSTi]

Result

[SVTi2]

Result

0147

4

0237

(1–1)2

0258

3–1

0137

1–1

0258

42

0347

0148

3

0158

4

0347

(1–1)2

0148

0158

3–1

0148

2–1

037

42

0237

0237

3

0247

4

0137

(1–1)2

0158

0247

3–1

0237

1–1

0358

42

0247

0258

4

0358

1

0147

(3–1)2

0137

0347

2

037

1–1

0148

(1–1)2

0147

3–1

037

4

0148

42

0147

1–1

0258

1

0247

22

037

4

0258

4–1

0247

(3–1)2

037

0358

P1*, P2*

P1*, P2*

This common denominator, quite reasonably, involves the embedding of a [037] set-class representative within the larger-cardinality chord. A neoRiemannian operation that affects the embedded [037]-type sonority will give a familiar transformational color to the entire harmony. Table 5-5 shows

The SST Model and Neo-Riemannian Theory

133

P-, L-, and R-like SST-succession classes involving tetrachords that include the [037] subset, along with some of the additional transformational labels proposed by Douthett and Steinbach, where they exist. Table 5-6: Additional neo-Riemannian-like tetrachordal SST-succession classes

[A]

SLIDElike [SST(i)(j)]

[B]

H-like [SST(i)(j)(k)]

[B]

N-like [SST(i)(j)]

[B]

0137

(1–1)(4–1)

0247

(1–1)(3)(4)

0358

(1–1)(2)

0148

0147

(1)(4)

037

(1–1)(3–1)(4)

0247

(3)(4)

0158

0148

(2–1)(4–1)

037

(2–1)(3)(4)

037

(2–1)(3–1)

037

0158

(2)(4)

0358

(2–1)(3–1)(4)

037

(3)(4)

0147

0237

(1–1)(4–1)

0347

(1–1)(3)(4)

0258

(1–1)(3–1)

037

0247

(1)(4)

0137

(1–1)(3–1)(4)

0147

(3)(4)

0258

0258

(1–1)(3–1)

0258

(1)(3–1)(4)

0237

(3–1)(4–1)

0247

0347

(1)(4)

0237

(1–1)(2)(4)

037

(1–1)(2–1)

0358

(1–1)(4–1)

0237

(1–1)(3–1)(4)

037

(3)(4)

0358

(1–1)(3–1)

0158

(1–1)(2)(4–1)

0137

(1)(3–1)

0237*

(2)(4)

0158

(1)(3–1)(4)

0137

(3)–1(4–1)

0347*

0358

I will make very brief mention of a couple of interesting aspects of the succession-classes displayed in Table 5-5. One has to do with the somewhat surprising fact that only four of the set-class types involved in these SSTsuccession classes commonly occur as recognizably triadic chord types: [037], [0158], [0258], and [0358]. The others do occur in tonal music but are more commonly encountered in atonal pieces. Some of the SST-succession classes, then, function almost as stylistic “bridges” between triadic and nontriadic sonorities, and can be of use in the analysis of a large cross-section of

134

Chapter Five

more recent music which is not strictly beholden to one harmonic system or the other. The second observation I would like to make simply has to do with the fact that set classes [0347] and [0358] each contain two representatives of SC [037] embedded within them; thus there are two separate voiceleading paths to distinct but type-equivalent chords. Table 5-6 takes the idea of neo-Riemannian-like tetrachordal relations one step further, showing SST-succession-classes that involve the nine tetrachordal SC types which embed [037]-type trichords related by SLIDE, N, and H transformations. Of interest is the fact that the SLIDE- and N-like transformations are what Douthett and Steinbach would call P2,0-type relations, but the H-like transformations would be P3,0-type relations where only one common tone is retained and three voices move by semitone. This is stretching the definition of voice-leading parsimony by just about anyone’s definition. Although no single voice moves more than a semitone, none of the studies we have examined model smooth voice leading past the P2,0 limit. Recalling our discussion of voice-leading parsimony in chapter 3, we recall that under the (admittedly somewhat arbitrary) rules found there, parsimonious voice leading between two chords occurs as a result of one or more applications of SSTs to a chord of cardinality m, with a maximum limit of m/2 voices undergoing SSTs, and with no more than two SSTs per voice. Thus the H-like tetrachordal SSTs are not strongly parsimonious transformations but the SLIDE-like and N-like tetrachordal SSTs are.22 Only one of these additional functions has previously been described in the literature, and that is the SLIDE-like ¢[0258],[0258]²SST(1–1)(3–1), which corresponds to Childs’s S6(5) operation.

N on embedded major triad Db E b F b Ab 1

2

3

4

4

3

2

1

C Eb F Ab CEFG

There are two [0237] ways [SST(i)(j)] of relating the sym[0358] [SST(1)(3–1)] metrical set class [0358] = [A] to some [B], only one of which is [0237] N-like.

N on embedded minor triad C Eb E G 1

2

3

4

4

3

2

1

[0347]

C Eb F Ab

[0358]

Db E F A b

[0347]

[SST(3–1)(4–1)]

Fig. 5-6: N-like SST-succession classes where [A] = [0358]

Of interest in Table 5-5 are the last two tetrachordal set classes listed, [0347] and [0358]. As these are symmetrical set classes their representatives participate in additional neo-Riemannian-like parsimonious functions. Additional functions are also found associated with the same set classes in Table 5-

The SST Model and Neo-Riemannian Theory

135

6, with a very special case involving N-like SST-succession classes where SC [0358] = [A], asterisked in the table. As [0358] contains two embedded type[037] trichords, we can apply an N operation in each in turn. The left side of Figure 5-6 shows the result of applying an N operation to the embedded major triad in a [0358]-type chord (subsumed in the SST-succession class ¢[0358],[0237]²SST(1)(3–1)), and the right side shows the result of applying an N operation to the embedded minor triad in a [0358]-type chord (subsumed in the SST-succession class ¢[0358],[0347]²SST(3–1)(4–1)). Somewhat surprisingly, two different set-class types form the SC [B] in each SST-succession class containing the embedded N operations—requiring two different [SST(i)(j)] subscripts—setting apart [0358] as unique among the tetrachordal set classes constituting the extended neo-Riemannian collection. Not only do two distinct set-class types result from the N operation on embedded [037]type trichords, but because of the symmetrical nature of the set class [0358], only half the constituent SST(i)(j)s belonging to an N-like SST-succession class will actually produce an N transformation on the embedded [037]-type trichord. As an example of this, consider the left side of figure 5-6, where SST(1)(3–1) on ¢0,3,5,8² produces ¢1,3,4,8²[0237] and where the embedded A(major triad in the former is related to the embedded D(-minor triad in the latter by N.23 However, another n-tuple pair-succession from the same SSTsuccession class (subscripted [SST(1)(3–1)]) on an inverted member of [0358], SST(1–1)(3)(¢8,5,3,0²) yields ¢7,5,4,0²[0237], where the embedded A(-major triad in ¢8,5,3,0² is not related to the embedded C-major triad in ¢7,5,4,0² by N. (In the figure, the small arabic numerals 1–4 above and below the pitchclass letter names show the two different ways the 4-tuple may be ordered.) Although chords belonging to the same set-class type result from the same SST(i)(j) on both transposed and inverted forms of the 4-tuple ¢0,3,5,8², SSTs belonging to the SST-succession class ¢[0358],[0237]²SST(1)(3–1) produce Nlike transformations only on the transposed forms of the 4-tuple. This phenomenon also holds for the SST-succession-class ¢[0358],[0347]²SST(3–1)(4–1) (on the right side of Figure 5-6), where an N relation is obtained between the embedded minor triad in 4-tuples of SC type [0358] = [A] and the embedded major triad in the 4-tuples of SC type [0347] = [B], but again only for transposed forms of the [0358]-type 4-tuple. To conclude this chapter, let us briefly consider aspects of the inclusional principle in respect to pentachords that contain our “extended neoRiemannian” tetrachordal collection. Although an in-depth study of neoRiemannian-like operations on pentachords is beyond the scope of this treatise, it may be helpful at least to identify exactly which pentachordal set classes include the extended neo-Riemannian tetrachordal set classes. These are shown in Table 5-7. Embedded trichords and tetrachords within these pentachordal set classes may undergo parsimonious transformations familiar

136

Chapter Five

Table 5-7: Inclusional relations between pentachordal and tetrachordal “extended neo-Riemannian” set classes [037] ⊂

[0137] ⊂

[01237], [01248], [01347], [01357], [01367], [01568], [02358], [02368]

[0147] ⊂

[01247], [01258], [01347], [01367], [01369], [01457], [01469], [01478]

[0148] ⊂

[01248], [01348], [01458], [01468], [01478], [02458], [03458]

[0158] ⊂

[01258], [01358], [01458], [01568]

[0237] ⊂

[01237], [01348], [01368], [01457], [01568], [02347], [02357]

[0247] ⊂

[01247], [01357], [01358], [01468], [02347], [02357], [02469], [02479]

[0258] ⊂

[01258], [01368], [01369], [01469], [02358], [02368], [02458], [02469]

[0347] ⊂

[01347], [01458], [01469], [02347]

[0358] ⊂

[01358], [01469], [02358], [02479], [03458]

from neo-Riemannian theory ranging from the commonplace (P, L, R) to the more exotic (P2*, –L2, etc.), all of which can be simply and consistently identified in terms of the SST vocabulary. Pentachordal set classes which can be the result of a split function are identified in bold in the table. Once again, of particular interest here is SC [0258], the quintessential tonal seventh chord, which itself includes the quintessential major/minor-triad set class [037]. The [0258] set class ties three other extended neo-Riemannian tetrachordal set classes for the maximum number of pentachordal set classes in which it can be embedded, and it is unique among its peers in terms of maximizing the number of possible pentachordal set class types to which its representatives can be transformed through split functions (seven). The importance of the triadic inclusional principle to progressions involving tetrachords and pentachords will become more evident in the next chapter, which contains a number of analyses of music ranging from a wellknown prelude by Chopin to a highly chromatic etude by Scriabin, and to music by John Adams that employs non-functional tertian harmonies. The utility of the SVT model will be further demonstrated by its applicability to

The SST Model and Neo-Riemannian Theory

137

atonal pieces such as Webern’s Second Cantata and Paul Lansky’s Modal Fantasy.

CHAPTER 6 ANALYSES In general terms, the effectiveness of the SVT system in modeling music is directly proportional to the degree to which a passage adheres to strongly parsimonious voice leading. Parsimonious, but not strongly parsimonious voice leading is also successfully modeled by the SVT system. Voice leading in diatonic music is somewhat less easily accounted for by SVTs, and disjunct voice leading is poorly modeled by the system. There are exceptions to this generalization, however. Music in which there is a predominance of voice-leading motion by ic 5 can comfortably be accounted for in the SVT model through the use of SFTs. The analyst may conceivably go one step further and modify the model by setting SVTs equal to whatever the principal voice-leading interval is in a particular piece. For example, if we imagine a piece in which most voice leading could be accounted for by steps of ic 3, we could use “single third transformations,” STTs, to model the voices’ motion, and the motion would in a sense be parsimonious, within a “minorthird space.” Because ic 3 is not a generator of 12, there would be three distinct minor-third sub-spaces of 12-tone equal-tempered space, and “nonparsimonious” transformations such as SSTs(!) would be necessary to bridge the gaps between the minor third cycles. One could even envision a composer aiming for maximally unsmooth voice leading by setting his or her principal voice-leading interval equal to ic 6. The result would be musically unsatisfying, of course, without occasional non-tritone voice leadings, as a voice would be stuck in a rut, continually fluctuating between two ic-6-related pitches. One could make a case that the SVT model could profitably analyze such a music through the coupling of SFT and SST operations. The analyses presented in this chapter assume that “voices” are distinguished from “lines,” as the former arise from transformations and the latter are usually defined by pitch register and by timbre.1 In the case of the SVT model, the transformations that spin out the voices privilege parsimony above all other factors. Thus, for the most part, when analyzing the progression from one chord to another, preference will be given to the transformation(s) that can account for the voice leading between the chords by using the fewest possible SVTs. We may not always want to do this, however. Momentary departures from the SVT model can occur whenever evidence strongly suggests that voice leading between two chords should be driven by factors other than parsimonious relationships. When there are two equally

Analyses

139

parsimonious ways to account for the voice leading between two chords, or even if there is a less parsimonious way that is strongly suggested by this evidence—which most frequently takes the form of a recurring pattern or process within the music—such a factor may temporarily override the “law of the shortest way” and suggest a less direct voice-leading path. These exceptions are rare, and examples of them will be pointed out in the individual analyses in this chapter.2 In each of the following analyses, chord progressions are interpreted as a series of interlocking SST-succession classes; in some of the analyses, the SVTs themselves are traced between chords. Although one runs the risk of falling into the trap of simply labeling chords and the transformations between them, resulting in an uninspired analysis, the keen ear/eye will be able to find interesting patterns, relations, and exceptional moments through the SVT- and SST-succession-class analyses that might not otherwise be noticeable using other systems of analysis. Through the exigencies of the parsimony-privileging approach, simple SVTs can be interesting when their result is voice leading that contradicts the musical lines in a score. SVTs in pitch-class space transcend register and timbre in pitch space and suggest voice-leading connections that would not normally be apparent simply by tracing “parts.” Another issue that will recur with some frequency in these analyses is the problem of splits and fuses. After an apparent change in cardinality, does the music continue with more or with fewer voices, or does it continue with the same number due to a momentary pitch-class doubling? It will be seen that it is important to distinguish between true splits or fuses and Rahn’s constant max-m voice leadings where the chord cardinality remains constant. Finally, in both tonal and post-atonal music, tertian sonorities are very common, and this allows for the possibility of neo-Riemannian or neoRiemannian-like voice leading. The types of chords that participate in these kind of transformations include, first and foremost, members of SCs [037] and [0258], other tertian chords from functional tonal harmony (SCs [036], [048], [0158], [0358], and [0369]), and several SCs that include [037], as listed in Tables 5-5 and 5-7. These neo-Riemannian-like progressions are noted where they occur, and I will point out their correspondence to transformational labels given by other theorists when such a relationship exists. To show the utility of the SVT model, analyses are presented of music from vastly different historic and stylistic periods. We will begin with analyses of chromatic tonal music (dating from 1838 and 1903), proceed to atonal examples (dating from 1943 and 1970), and conclude the chapter by examining some of the post-atonal minimalist literature from the last quarter of the twentieth century.

140

Chapter Six

Tonal Analyses Chopin, Prelude in E Minor, Op. 28 No. 4 Figure 6-1 presents a chart of SVT voice leading for the entire Prelude in E minor, Op. 28, no. 4 of Chopin (the score appears in Figure 6-3).3 To facilitate rapid reading of the chart, SVTs appear between m-tuples. Let us see if the SVT reading adds to what we already know about this well-known piece. The SVT analysis confirms a couple of things we notice from looking at the score. Most of the SSTs have “negative” values (SSTi–1), as the general tendency of each line is to move down by semitone or (occasionally) by whole step in each of the two halves of the piece. Also apparent from both the analysis and from score-reading is that most of the transformations are parsimonious (by our definition in chapter 3). About half of the parsimonious transformations are strongly parsimonious in that only one voice is transposed non-trivially by ic 1 or 2. The SVT reading does, however, reveal a number of things that are not so readily apparent to analyst. For example, in measures 1–8 the first transformation in each measure consists of a single SST, and non-parsimonious transformations (composed of 3 or more combined SSTs) only occur over the barline, moving into a subsequent measure. Measure 9 is interesting in that it marks the point where this pattern changes drastically. The two transformations in this measure are composite SSTs, each containing four constituent transformations, one of which involves a change in cardinality. Measure 9 also marks the point where the first triad occurs since measure 1, and where the melody makes its first significant deviation from the dotted-half/ quarter figure in measures 1–8. Here we find the first melodic ascent of any distance greater than a step-wise upper neighbor figure. The phrase reaches its harmonic goal in the first half of measure 10, where we notice all of the lines have also reached their nadirs.4 Measures 9–10 also mark the spot where the first discrepancy between the ordering of the musical lines (by register) and the ordering of the voices (voice leading as a result of SVTs) occurs. The transformation over the barline into measure 10 produces the 4tuple ¢B,E,F,A²; voices (order positions) 3 and 4 now correspond to musical lines 4 and 3, respectively (measuring from the bass).5 The break in the SVT analysis between measures 11 and 13 highlights one of the potential shortcomings of the SVT system: simple diatonic progressions are not easily expressed as SVTs. On the one hand, the progression ¢B,D,F,A² to ¢B,E,G,B² retains only one common tone and requires a total of four combined SSTs (plus a fuse operation if one wishes to reduce the cardinality of the chord). In SVT terms, this would correspond to SST(2)(3)(42) without the fuse.6 On the other hand, the tonal progression V7–i6 much more

Analyses

141

m.1: ¢G,B,E,B² SST4 ¢G,B,E,C² SST(1–1)(2–1)2(4–1) m.2: ¢F#,A,E,B² SST3–1 ¢F#,A,Eb,B² SST4 ¢F#,A,Eb,C² SST(1–1)(4–1) m.3: ¢F,A,Eb,B² SST3–1 ¢F,A,D,B² SST(2–1)(4) ¢F,G#,C,D² SST(1–1)(4–1) m.4: ¢E,G#,D,B² SST2–1 ¢E,G,D,B² SST(3–1)(4–1) ¢E,G,C#,Bb² SST(3–1)(4–1) m.5: ¢E,G,C,A² SST2–1 ¢E,F#,C,A² SVT42 ¢E,F#,C,B² SVT(4–1)2 m.6: ¢E,F#,C,A² SST1–1 ¢D#,F#,C,A² SVT42 ¢D#,F#,C,B² SST(1–1)(4–1)2 m.7: ¢D,F#,C,A² SST2–1 m.8: ¢D,F,C,A² SST3–1 ¢D,F,B,A² SST4–1 ¢D,F,B,G#² SST(1–1)2(2–1) m.9: ¢C,E,B,G#² (SPLIT4–1(SST(3–1)2(4))) ¢C,E,A² (SST(1–1)(32))(SPLIT2)) m.10: ¢B,E,F#,A² SST2–1 ¢B,D#,F#,A² SST(1)(2) ¢C,E,F#,A² SST(1–1)(2–1) m.11: ¢B,D#,F#,A² // (V7–i6) m.13: ¢G,B,E,B² SST4 ¢G,B,E,C² SST(1–1)(2–1)2(4–1) m.14: ¢F#,A,E,B² SST(1–1)(3–1) ¢F,A,Eb,B² SST4 ¢F,A,Eb,C² SST(2–1)(4–1) m.15: ¢F,Ab,Eb,B² SST3–1 ¢F,Ab,D,B² SST(1–1)(2–1) ¢E,G#,D,B² SST4 ¢E,G#,D,C² SST(2–1) (4–1)

m.16: ¢E,G,D,B² SST(3–1)(4–1) ¢E,G,C#,A#² SST(2–1)(3–1)(4–1) ¢E,F#,C,A² (SST(1–1)(3–1) (SPLIT3)) m.17: ¢D#,F#,B,C,A² (SPLIT3–1(SST(4–1)(5–1)2)) ¢D#,F#,B,G² (SPLIT4–1(SST(1)(2))) ¢E,G,B² SST(22)(3) m.18: ¢E,A,C ² (SVT22(SPLIT1)) ¢E,F#,A,C² SST(32)(4–1) m.20: ¢E,F#,B,B² SST1–1 ¢D#,F#,B,B² SVT(4–1)2 ¢D#,F#,B,A² SST(1)(2)(3)(4) (= T1) m.21: ¢E,G,C,Bb² SST(2)(4–1) ¢E,F#,C,A² SST(2–1)2(3–1) m.22: ¢E,E,B,A² SST4–1 ¢E,E,B,G#² SST4–1 ¢E,E,B,G² SVT22 ¢E,F#,B,G² SST(2)(3–1) SFT4–1 m.23: ¢E,G,Bb,C² (SPLIT4–1(SST(2)(3)(4–1))) m.24: ¢E,F#,B² SST1–1 ¢D#,F#,B² SST(1)(2) m.25: ¢E,G,B²

Fig. 6-1: SST analysis of Chopin, Prelude in E minor, Op. 28 no. 4

succinctly expresses the relationship between these two chords; although it does not directly specify the voice leading, this can easily be inferred by the conventions of traditional harmony. After a repeat of the music of the first measure and a half, we notice some interesting changes in the four-voice descent in measures 13–16. The rate of the descent seems to have accelerated, with SSTs occurring four times

142

Chapter Six

a measure (whereas they occurred only two or three times per measure in bars 1–9) and increasingly as composite SSTs. These result in new harmonies, not heard in the first half of the piece. In measure 16 Chopin re-voices the A diminished-seventh chord in mid-measure; since no SST occurs here, a discrepancy is introduced between the ordering of n-tuple voices and the registral ordering of the musical lines. Measure 17 corresponds to the climax of the prelude, where Chopin takes the piece to its extremities in the dimensions of intensity (the forte dynamic marking), ambitus (B1–C5, not exceeded in the bass until the final chord), tempo (the stretto marking), melodic activity, and harmonic cardinality (the dominant ninth chord). In the SST analysis, a series of fuse operations is necessary to reduce the cardinality of the harmonies from five in m. 17 to three in mm. 19–20. The deceptive progression leading into m. 21 is best modeled by parallel SSTs in all voices. This would correspond to a T1 operation in standard pc-set theory and necessitates an implied B( at the beginning of the measure. (The B( in fact does appear halfway through the measure.) This parallel voice leading, coupled with the unusual SST which sends the G to the F at the end of the measure, violates the “rules” of tonal voice leading and shows quite clearly that SVT voice leading is distinct from the conventions of traditional tonal counterpoint. Obviously this is not your grandmother’s voice leading. One final remark should suffice to conclude our discussion of the SVT analysis of the prelude. The “diminished third” chord in m. 23 could be regarded as one of the most dramatic moments of the piece, with the pregnant pause that follows it, preceding the final cadence. An SVT analysis which seeks to retain common tones where possible would keep the E and G invariant, and the B would clearly lead by semitone to the B(, but how does one account for the voice leading F–C? This could be accomplished by the combination of an SFT and an SST, but this tritone progression does not seem very intuitive. Another possible solution—the one adopted in Figure 61—is to retain E as a common tone, lead the B to the B(, send the F up a half step to G, and apply an SFT to the G, mapping it to C. This seems to be a much more plausible alternative, as the piece’s only SFT is “concealed” entirely within the inner voices. Let us now generalize the SSTs and related pair-succession m-tuples to SST-succession classes to see if this abstraction gives us any additional insights into the voice leading of the piece. Figure 6-2 is a chart similar to that of Figure 6-1 but showing interlocking SST-succession classes. A close examination of the chart allows us to see certain relationships the SST chart did not show. For example, we noticed in Figure 6-1 that the first transformation in each measure in the first eight measures consists of a single SST; in Figure 6-2 we see, perhaps somewhat surprisingly, that in five of these measures—including the first four bars—all of the SST-related m-tuple pair-

Analyses

143

m.1: [0377] [SST4] [0158] [SST(1–1)2(2–1)(4–1)] or [SST(1)(22)(3)] m.2: [0257] [SST4] or [SST1–1] [0258] [SST1–1] [0369] [SST(1–1)(3–1)] or [SST(2)(4)], etc. m.3: [0268] [SST4] or [SST1–1], etc. [0258] [SST(1–1)(2)] [0258] [SST(1–1)(3–1)] m.4: [0258] [STT4] [0358] [SST(1–1)(2–1)] or [SST(3)(4)] [0369] [SST(1–1)(4–1)] or [SST(1)(4)], etc. m.5: [0358] [SST3] or [SST2–1] [0258] [SVT32] [0157] [SVT(1–1)2] m.6: [0258] [SST1–1] [0369] [SVT42] or [SVT(3–1)2], etc. [0147] [SST(1–1)2(3–1)] m.7: [0258] [SST4] m.8: [0358] [SST3] or [SST2–1] [0258] [SST1–1] [0369] [SST(1)(22)] or [SST(1–1)2(2–1)], etc. m.9: [0148] ([SPLIT2–1]([SST(22)(3–1)])) [037] ([SST(2–1)(42)]([SPLIT3])) m.10: [0257] [SST4] or [SST1–1] [0258] [SST(1–1)(4–1)] [0258] [SST(1–1)(4–1)] m.11: [0258] // (V7–i6) m.13: [0377] [SST4] [0158] [SST(1–1)2(2–1)(4–1)] or [SST(1)(22)(3)] m.14: [0257] [SST(1–1)(2–1)] or [SST(3)(4)] [0268] [SST4] or [SST1–1], etc. [0258] [SST(3)(4)] m.15: [0258] [SST1–1] [0369] [SST1–1] or [SST3], etc. [0258] [SST3–1] [0248] [SST(1–1) (4–1)] or [SST(3)(4)] m.16: [0358] [SST(1–1)(2–1)] or [SST(3)(4)] [0369] [SST(2–1)(3–1)(4–1)], etc. [0258] ([SST(1–1)(4–1)]([SPLIT4])) m.17: [01369] ([SPLIT1–1]([SST(1)(32)])) [0148] ([SPLIT1–1]([SST(2–1)(3–1)])) [037] [SST(22)(3)] m.18: [037] ([SVT42]([SPLIT3])) [0258] [SST(32)(4–1)] m.20: [0277] [SST2] or [SST1–1] [0377] [SVT42] [0258] T1 m.21: [0258] [SST(2)(3)] [0258] [SST(22)(4–1)] m.22: [0277] [SST2] or [SST1–1] [0377] [SST2] [0037] [SVT22] [0237] ([SST(2)(4)]([SFT3–1])) m.23: [0258] ([SPLIT1–1]([SST(1)(2–1)(3)])) m.24: [027] [SST2] or [SST1–1] [037] [SST(1–1)(2–1)] m.25: [037]

Fig. 6-2: SST-succession-class analysis of Chopin, Prelude in E minor, Op. 28 no. 4

successions (representing three distinct SSTis), belong to SST-succession classes in which the subscript is [SST4].7 Although there are a very large number of possible combinations of [SSTi] subscripts, their number is limited in the prelude, and certain combinations occur with more frequency than others. [SST4] and [SST1–1] in particular occur many times, and [SST(3)(4)]

144

Chapter Six

[0258] [SST1–1] [0369] –L1 (Baker)

[037] ([SST4]([SPLIT3])) [0158] SL (Baker)

[SST(1–1)(3–1)] [0258] [SST4] [0358] P2* (D&S) S6(5) (Childs)

[0358] [SST2–1] [0258] L1*(D&S)

[0258] [SST1–1] [0369] R*2 (Baker)

[SST4] [0358] [SST2–1] [0258] [SST1–1] [0369] R*2 (Baker) P2* (D&S) L1* (D&S)

[0258]

[0258] [SST(1–1)(2)] [0258] C3(2) (Childs)

[0258] [SST(1–1)(4–1)] [0258] S3(4) (Childs)

[0377] [SST4] [0158] SL (Baker)

[0258]

[0258] [SST(1–1)(4–1)] S3(4) (Childs)

[0258][SST(3)(4)][0258][SST1–1] [0369] R*2 (Baker) P¡(Shimbo)

[0258][SST(1–1)(4–1)] [0258] S3(4) (Childs)

[0258] [SST(2)(3)] [0258] S4(3) (Childs)

[0377][SST2][0037] P [037] [SST(1–1)(2–1)] [037] N

Fig. 6-3: Chopin, Prelude in E minor, Op. 28 no. 4 with extended neoRiemannian labels

becomes increasingly important in the second half of the piece. It can also be seen that the number of split- and fuse-succession classes increases at the two climaxes of the piece (measures 9 and 16–8) where most of the changes in chord cardinality are clustered.

Analyses

145

Another aspect of the piece the SST-succession-class analysis reveals are recurrent successions that would not normally be spotted in a tonal analysis. The first chord in m. 3, a [0268]-type tetrachord,8 leads to a [0258]-type tetrachord through the descent of the “alto” part by a semitone. The same progression occurs in the second half of m. 14, but here the “soprano” part moves up by a half step. In m. 3 the second chord is a half-diminished seventh chord and in m. 14 the second chord is a dominant seventh chord. When we look at the SST transformations (in Figure 6-1) we see that they are listed as SST3–1 and SST4, respectively. But in Figure 6-2 it is seen that both mtuple pair-successions and associated SSTs belong to the same SSTsuccession class, namely ¢[0268],[0258]²SST4 (or ¢[0268],[0258]²SST1–1). It is clear from this example that the normalizing power of the SST-succession classes, by reducing to a single type all the minimal voice leadings between all transposed, inverted, and permuted versions of the m-tuple pairsuccessions, allows the analyst to draw meaningful connections between chord progressions that may play drastically different rôles in tonal harmonic analysis. As a further example of this principle, consider the SST-succession class ¢[037],[0258]²SVT42(SPLIT3) in m. 18 and the SST-succession class ¢[0377],[0258]²SVT42 in m. 20.9 In the first case, a minor triad (iv) is transformed to a half-diminished seventh chord (ii∅7, or iv with added 6th). In the second case a major triad (V) is transformed to a dominant seventh chord (V7). Both progressions are accounted for by the same SST-succession-class type, and furthermore, this type shows the derivational relationship between the subdominant added-sixth chord and the dominant seventh chord. Perhaps the most interesting feature of this prelude, as illuminated by the SST-succession-class analysis, is the sheer variety of parsimonious relationships obtained by voice leading to, from, and between members of [0258] and other, mostly tertian chords belonging to the greater “neo-Riemannian” collection. In mm. 10 through 12 an oscillation occurs between the dominant seventh chord ([0258]) and the aforementioned subdominant added-sixth chord (also [0258]). In the SST analysis this oscillation can be expressed by the transformations ¢11,3,6,9² SST(1)(2) ¢0,4,6,9² SST(1–1)(2–1) ¢11,3,6,9², where the second composite SST “undoes” the work of the first by virtue of being its mathematical inverse. In the SST-succession-class analysis, however, the same passage reads [0258] [SST(1–1)(4–1)] [0258] [SST(1–1)(4–1)] [0258]. The composite SST-succession class ¢[0258],[0258]²SST(1–1)(4–1), then, is its own inverse, always toggling between members of a single set class, in exactly the same way as the neo-Riemannian operations L, P, and R function. We can see from Table 5-3 that ¢[0258],[0258]²SST(1–1)(4–1) is a P2,0-type SST-succession class10 whose [SST(i)(j)] subscript has previously been labeled S3(4) by Childs, Iiiivi by Gollin, and ¢–,7,5² by Hook. From Table

146

Chapter Six

5-3 we see that there are ten distinct P2,0-type [SST(i)(j)] subscripts (paths) relating succession-pairs of the SC type [0258]; in a brief 25-bar prelude, Chopin explores fully half of these and even one that requires a conglomeration of more than two SSTs. These are: mm.3–4: [0258] [SST(1–1)(2)] [0258] [SST(1–1)(3–1)] [0258] mm.10–2: [0258] [SST(1–1)(4–1)] [0258] (four times) mm.14–5: [0258] [SST(3)(4)] [0258] mm.20–1: [0258] T1 [0258] [SST(2)(3)] [0258]. For each of these P2,0 relations there is an extended- or “neo-neoRiemannian” label of some type (as found in Tables 5-2 through 5-4). Figure 6-3 presents the score of the prelude with each of these listed where they occur, along with their equivalent abbreviated SST-succession-class nomenclature. A few of these merit brief discussion here. From Figure 6-3 we can see one of the most attractive features of SSTsuccession-class analysis: all the disparate extended neo-Riemannian labels devised by Cohn, Douthett and Steinbach, Morris, Gollin, Childs, Shimbo, Bass, Kopp, and Baker are easily accounted for by simple SST-successionclass nomenclature. In this sense SST-succession-class analysis resembles Hook’s UTTs in its versatility, but it goes one step further, accounting for all parsimonious relations beyond those obtained between members of SC [037] and a circumscribed number of tonal seventh chords, as Hook’s system is limited to. Additionally, analysis by SST-succession class has the advantage of maintaining the same [SSTi]-relation subscript for transpositionally, inversionally, or permutationally related set-class successions that would require two separate labels in other extended neo-Riemannian systems, such as those of Douthett and Steinbach, Kopp, and Baker. As an example of this, consider the SST-succession class ¢[0258],[0369]²SST1–1, which occurs in four places in the Chopin prelude (mm. 2, 6, 8, and 15). In its first incarnation, the constituent SST connects a major-minor seventh chord with a diminished seventh chord. This would receive the label –L1 in Baker’s system. The other three occurrences consist of a half-diminished seventh chord leading to a diminished seventh chord. These all receive the label R*2 in Baker’s system, but the SST-succession class remains the same, as the m-tuple pairsuccession in m. 2 merely involves inverted forms of the SST-related mtuples occurring later in the piece. In addition to the extended neo-Riemannian labels associated with certain SST-succession classes as found in Figure 6-3, some of these SSTrelated tetrachordal pair-successions contain “embedded” neo-Riemannian triadic operations such as those listed in Tables 5-5 and 5-6. The [SST4] sub-

Analyses

147

script in m. 4 and again, between mm. 7 and 8 signifies a “P-like” relation, and the subscript [SST(1–1)(3–1)] between mm. 3 and 4 identifies a “SLIDElike” succession. The final progression of the piece, with associated subscript [SST(1–1)(2–1)], is equivalent to a triadic neo-Riemannian N operation.

Scriabin, Etude in C-Sharp Minor, Op. 42 No. 5 Sixty-five years after Chopin’s prelude saw the light of day, Scriabin’s eight etudes, Op. 42, were published in 1903. The fifth of these, in C-sharp minor (“Affannato”), is one of Scriabin’s most-performed pieces. Stylistically it represents a middle point between his Chopinesque early pieces and his mature, near-atonal works such as the Op. 74 Preludes. The form of the etude is most akin to sonata form without a development section (Figure 64). The principal material of the piece (A), an eight-bar section from measures 1 to 8, repeats immediately after its first appearance but is extended, presents a new layer of inner-voice figuration, and modulates to the minor dominant. The principal interest in the A section is chromatic harmonic progression, as the melody for the most part is quite rudimentary, subservient to the active harmonies. The second theme (B) is much more active melodically but adheres more closely to conventional tonal harmonic progressions. The key of B major is briefly tonicized twice in measures 21–6; a four-bar extension leads back to C-sharp minor by measure 31.

mm.

1

9

material

A

key area

c#

21

31

39

47

A (extended) B

A

B

Coda

c# (modulates)g#

c#

c#

c#

Fig. 6-4: Formal plan of Scriabin, Etude in C-sharp minor, Op. 42 no. 5

At this point the A material recapitulates, rhythmically varied and at a much louder dynamic level (see Figure 6-5). Harmonically, it is almost identical to measures 1–8. Following this the B material returns, remaining in the tonic key but tonicizing E major twice. (The first part of this is shown in the second half of Figure 6-5.) This is followed by an eleven-measure coda. The chord progression in measures 1–4 (and repeated in measures 31–4) is very difficult to make sense of using tonal harmonic analysis. The

148

Chapter Six

A

Fig. 6-5: Scriabin, Etude in C-sharp minor, Op. 42 no. 5, measures 30–42

progression is, essentially: AM7 – A7 – F7 – AM7 – Am/M7 – B7 – G. The first chord in measure 5 (and 35)—¢A,D,E,B²—is particularly problematic as it is a whole-tone tetrachord and inexplicable as a tonal seventh chord.11 The only chord in this sequence that makes any sense in the key of C-sharp minor is the dominant at the half cadence concluding the first phrase of the A section (measures 4 and 34). Let us see if applying an SST analysis to this passage (measures 31–5) is any more informative than a tonal harmonic analysis. Figure 6-6 revisits these measures using SST labels similar to those used in Figure 6-1. (Again, limited space prevents us from annotating this directly on the score.) It

Analyses

149

B

Fig. 6-5 (continued)

would appear the SST analysis tells us little that we didn’t already know about the music. Unlike the harmonic analysis, the voice leading is described in detail in Figure 6-6, but no interesting patterns have yet emerged, other than the move from strongly parsimonious through parsimonious to nonparsimonious voice leading (by our previous definitions) in measures 34 and 35. The increased complexity of the voice leading and chord progressions is reinforced by the use of fuse and split operations in conjunction with multiply iterated SST functions in these measures. In retrospect, it would seem that SST analysis is just about as ill-suited for saying anything meaningful about this passage as the tonal harmonic analysis. But let’s not abandon all hope yet. Can an SST-succession-class analysis inform our reading of this passage? The lower part of Figure 6-6 shows such a reading. The merits of such an analysis are at first not as obvious as they were in the Chopin example. We notice that nearly all of the chords occurring in this excerpt are instances of set classes central to tonal harmonic practice: [037], [0258], and [0158]. Also appearing is a whole-tone tetrachord, [0268], and the graph-warping tetrachord [0148], which appears twice (just as it did in the Chopin prelude) in the span of four measures.

150

Chapter Six

SST Analysis m.30, last beat: ¢G#,E,C#,A² SST1–1 m.31: ¢G,E,C#,A² SST(1–1)(4) ¢F#,E,C#,A#² SST(12)(4–1) ¢G#,E,C#,A² SST3–1 m.32: repeats m. 31 m.33: ¢G#,E,C,A² SST(1–1)2(2–1)(3–1) ¢F#,D#,B,A² (SPLIT4–1(SST(12)(3)(4–1))) m.34: ¢G#,D#,B#² ((SST(1)(32)(4–1)(SPLIT2)) m.35: ¢A,D#,E#,B² SST2–1 ¢A,D,E#,B² SST-Succession-Class Analysis m.30, last beat: [0158] [SST1–1] or [SST2] m.31: [0258] [SST(1–1)(2)] (= Childs’s C3(2)) [0258] [SST(1–1)2(4–1)] [0158] [SST3–1] or [SST4] (= P-like) m.33: [0148] [SST(1–1)2(3–1)(4–1)] [0258] ([SPLIT2–1]([SST(1–1)(2)(3–1)2])) m.34: [037] ([SST(1–1)2(3)(4–1)]([SPLIT3]) m.35: [0268] [SST3–1] or [SST4] or [SST1–1] or [SST2] [0258] [SST(2–1)3(3–1)] m.36: [0148]([SPLIT2–1]([SST(2–1)(3–1)])) (= N-like) [037] ([SST4]([SPLIT3])) (= Baker’s SL) m.37: [0158]

Fig. 6-6: SST and SST-succession-class analyses of Scriabin, Etude in Csharp minor, Op. 42 no. 5, measures 31–6

Although relatively few of the pair-successions here are parsimoniously related (in the sense that no more than half the voices undergo stepwise voice leading), all voice leadings but one are completely smooth, with all voices moving by half step, whole step, or by common tone. Upon further reflection, we notice that both instances of [0148]-type tetrachords are associated with neo-Riemannian-like transformations. The [0148] chord in measure 33 comes on the heels of a major seventh chord via an [SST3–1] SST-succession-class relation (¢[0158],[0148]²SST3–1), which Table 5-5 informs us is P-like. However, we do not need to refer to this table in order to hear the P-like succession; this is immediately apparent when we listen to the passage, just as it was on its first occurrence in measures 2–3. The table merely identifies for us, in SST-succession-class terms, which transformations will have this audible effect. The [0148]-type tetrachord in measure 36 leads to an F-sharp minor triad via an N-like transformation (see Table 5-6). Interestingly, this transformation is also quite easily heard, as an N transformation leading from a major to a minor triad sounds like the tonal

Analyses

151

progression V – i.12 The voicing of the [0148] tetrachord strongly emphasizes its C-sharp-major triadic subset—the pc A is completely isolated from the other (mid-range) pcs by its consignment to the lowest octave of the piano—and by this voicing the N-like effect is enhanced. This very same progression is later repeated—in a completely different voicing and melodic and structural context—at the beginning of the B section in measure 39 (see Figure 6-5). One interesting aspect of this passage shown only by the SST-successionclass analysis is the gradual accretion of recurrent [SSTi] relations as the number of voices involved in parsimonious motion increases. The first pairsuccession in measure 31 belongs to either the SST-succession-class with subscript [SST1–1] or to the class with the subscript [SST2]. The next pairsuccession is related by a composition of both these SST-succession-class subscripts ([SST(1–1)(2)]). The third pair-succession maintains the subscript [SST1–1], doubles it ([SVT(1–1)2)], and adds to it the new subscript [SST4–1]. After the P-like succession, which introduces the new subscript [SST3–1], the complex SST-succession class subscript [SST(1–1)2(3–1)(4–1)] appears, which is the union of the two preceding subscripts. This process of gradual accretion of SST-succession-class relations is not apparent from the SST analysis alone.

Atonal Analyses Although not all of the atonal literature may profitably be analyzed through use of the SVT model, whenever smooth voice leading between verticalities is the norm, it is likely that SST and SST-succession-class analysis may illuminate some previously unheralded aspect of such music. The SST model can even be applied to twelve-tone music, provided that the internal structure of the row and the composer’s selection and arrangement of the row forms emphasize common-tone retention (invariants) and/or parsimonious voice leading. This suggests that some of the twelve-tone music that might prove to be promising for fruitful SST analyses could include works by Webern and Babbitt, among others. In this section, we will briefly consider what the SST model can tell us about twelve-tone music that would not be apparent through other analytical approaches.

Webern, Cantata No. 2, Op. 31: I Figure 6-7 shows the row forms used by Webern in the first movement of his Cantata No. 2, Op. 31. In the first movement, Webern uses only P0, I0, R0, and RI0 forms of the row. The derivation of P0 follows Joe Brumbeloe’s

152

Chapter Six

(1986) practice, with hexachords labeled a and b, and with the hexachords of the I0 form of the row labeled c and d, even though the first chord heard at the beginning of the piece receives Brumbeloe’s designation c.13 Webern achieves a high degree of invariance between corresponding hexachords of P0 and I0. In fact, five of the six pcs are held invariant while only pcs 1 and E are exchanged in each half of the row when moving from P0 to I0 and vice versa. We can think of the motion between hexachords a and c, and again, between hexachords b and d, as transformations involving parsimonious voice leading.

a P0

0

3

E

b T

2

9

1

5

c I0

0

9

1

4

8

7

6

4

5

6

d 2 T 3

E

7

8

Fig. 6-7: Webern, Cantata No. 2, Op. 31, I: row forms

Figure 6-8 shows a reduction of the first five measures of the movement. Hexachords a–d are labeled in the figure. The voice-leading transformation between hexachords c and a (encircled in the figure) can be read as an SST, namely, SVT(5–1)2 between 6-tuples ¢9,0,2,10,1,3² and ¢9,0,2,10,11,3², measured from the lowest pitch to the highest in the first hexachord. The second hexachord does not literally follow the ordering shown here in the second 6tuple; this is a good example of a voice leading that is projected and not manifest—or literally occurring between pitches in the same register—to use Lewin’s terms. Likewise, the voice-leading between hexachords d and b (enclosed in rectangles in the figure) corresponds to SVT12 between 6-tuples ¢11,7,8,4,5,6² and ¢1,7,8,4,5,6², if one constructs the first 6-tuple in the order of appearance of each pc. We note here that hexachords d and b take the form of a melody while hexachords c and a are chordal. To help delineate the subtle harmonic differences between hexachords c and a, Webern gives them distinct timbres: c is sounded by the winds and a appears in the strings. Similarly, hexachord d is played by the flute and hexachord b begins with a solo violin and concludes with staccato punctuation by the celesta and harp.

Analyses

153

SVT12

c d SVT(5–1)2

a b

Fig. 6-8: Webern, Cantata No. 2, Op. 31, I: measures 1–5 (reduction)

The first movement of the cantata differs from the subsequent five movements in that it is the only non-canonic movement of the whole. It also projects an extremely simple conception of harmony. Nearly the entire movement consists of nothing more than hexachords a–d appearing as vertical chords supporting horizontal versions of the same hexachords in the solo bass voice. Figure 6-9 is a reduction of measures 11 and 12. Here we see, in the instruments, a move from hexachord a back to hexachord c. The exact 6tuple ordering differs from the hexachords’ first appearance in measures 1 and 2. This time the 6-tuple ordering (projected in the case of the second 6tuple) and SST operation is SVT22(¢9,11,2,10,0,3²) → ¢9,1,2,10,0,3². The bass voice, coming from an F in measure 10, sounds hexachord d, after having intoned the pcs of hexachord b in measures 9–10. Once again Webern is careful to make a distinction between the very similar hexachords a and c through orchestration: a is heard in the strings, as it was in measure 2, while c is performed by the winds, as it was in measure 1. Although the specific SSTs between hexachords a and c differ in Figures 6-8 and 6-9, the SST-succession-classes are identical. In both cases the

154

Chapter Six

(F#) d c

a

SVT22

Fig. 6-9: Webern, Cantata No. 2, Op. 31, I: measures 11–2 (reduction)

individual SST operations SVT(5–1)2 and SVT22 are subsumed in the SSTsuccession class ¢[012356],[012356]²SVT32. This class is invariant upon swapping of equivalence-class order positions, much as neo-Riemannian L or P operations (which, when associated with two chords, could be reinterpreted as SST-succession classes) are their own inverses. The relation between hexachords d and b can also be generalized to an SST-succession class, namely ¢[012347],[012347]²SVT62. Finally, we should notice that each transformation between a hexachord and its I0 form is a strongly parsimonious transformation, by the definition given in chapter 3.

Lansky, Modal Fantasy, “Prelude” Figure 6-10 reproduces the first four measures of the first movement, “Prelude,” of Paul Lansky’s Modal Fantasy for piano, composed in 1970.14 Figure 6-11 presents an SST analysis of this passage, along the lines of Figure 6-1. The excerpt consists entirely of rapidly changing pentachords, formed through the moment-to-moment alignment of five independent voices which move mostly by half or whole step. Thus there are no fuse or split operations

Analyses

155

Fig. 6-10: Paul Lansky, Modal Fantasy, “Prelude,” mm. 1–4

in these measures, although pitch-class doubling does occur at the beginning of m. 3. Each voice makes a gradual ascent in pitch space, with the initial phrase of the prelude concluding with the original pentachord transposed by T12 in pitch space, or T0 in pitch-class space. The SST analysis shows a clear progression from exclusive single-SST voice leading in the first measure, to more complex composite transformations toward the end of m. 2 and extending into m. 3, and finally, to mostly two-fold transformations until the end of the excerpt. The mostly positive SST values reflect the musical lines’ gradual ascent in pitch throughout the phrase. SFTs are occasionally employed, with the first occurring over the barline leading into m. 2 and the second over the barline leading into m. 3. This second SFT is part of a much larger, complex transformation which could be explained as the decidedly non-parsimonious composition (SFT2–1)(SST(1–1)(3–1)(42)(5)). However, there is a simpler way to account for this operation: it is in essence an “near-transposition” (after Straus 1997) where all voices but one map by T10 to the next 5-tuple. The nonconforming pc, B, would then map by SST3–1 or SVT32 to B( or D(, but this “near-transposition” option is not shown in the SST analysis because it

156

Chapter Six

m.1: ¢C,G,B,Eb,Ab² SST4 ¢C,G,B,E,Ab² SST1 ¢Db,G,B,E,Ab² SST5 ¢Db,G,B,E,A² SST4 ¢Db,G,B,F,A² SST3–1 ¢Db,G,Bb,F,A² SST1 ¢D,G,Bb,F,A² SST2–1 ¢D,F#,Bb,F,A² (SST4)(SFT4) m.2: ¢D,F#,Bb,Db,A² SST3 ¢D,F#,B,Db,A² SST5 ¢D,F#,B,Db,Bb² SST1 ¢D#,F#,B,Db,Bb² SST4 ¢D#,F#,B,D,Bb² SST2 ¢D#,G,B,D,Bb² SVT(42)(52) ¢D#,G,B,E,C² SST2 ¢D#,Ab,B,E,C² (SFT2–1)(SST(1–1)(3–1)(42)(5)) m.3: ¢D,Db,Bb,Gb,Db² SST(1)(3) ¢Eb,Db,Cb,Gb,Db² SST(12)(3–1)2(5) ¢F,Db,A,Gb,D² SST3 ¢F,Db,Bb,Gb,D² SST4 ¢F,Db,Bb,G,D² SST5 ¢F,Db,Bb,G,Eb² SST3 ¢F,Db,B,G,Eb² SST1–1 ¢E,Db,B,G,Eb² SVT52 ¢E,Db,B,G,F² SVT42 ¢E,Db,B,A,F² SVT(1–1)2 ¢D,Db,B,A,F² SST5 ¢D,Db,B,A,F#² SST(1)(52) m.4: ¢Eb,Db,B,A,Ab² SST43 ¢Eb,Db,B,C,Ab² (SST2–1)(SFT2–1) ¢Eb,F,B,C,Ab² SVT22 ¢Eb,G,B,C,Ab²

Fig. 6-11: SVT analysis of Lansky, Modal Fantasy, “Prelude,” mm. 1–4

would result in a permutation of the 5-tuple order positions. No matter which way we approach the voice leading over the barline from m. 2 to m. 3 we will run into problems if we are trying to privilege parsimony. The composite SST shown in Figure 6-11, while adhering as closely to a parsimonious progression as is possible, nevertheless results in the first discrepancy between the ordering of the lines in the score (measured from the bass) and the voices in the 5-tuples. By the time we reach the end of the excerpt, order positions 1 and 4 have been exchanged. Let us see what insights an SVT-succession-class analysis of these measures may yield. Figure 6-12 presents such an analysis, modeled after the SVT-succession-class analysis of the Chopin prelude in Figure 6-2. As in Figure 6-2, it is interesting to note that from among the large number of possible SST-succession classes and their combinations, a very limited number are actually employed in the music. The subscript [SST5] in particular figures prominently, occurring no fewer than 10 times out of a total of 31 SC pair-successions. Similarly—and most likely more consciously on his part—Lansky has carefully limited the number of set-class types appearing in the opening phrase to only 11 of the possible 38 set classes, of which the final 2 are introduced only in m. 4. SC [01458] figures prominently in these four measures, not only beginning and ending the phrase but also appearing 11 times throughout, including its presence as the final chord of both the first and second measures. These two locations prove significant in that the SSTsuccession classes immediately following them are increasingly complex

Analyses

157

m.1: [01458] [SST5] [01458] [SST1–1] [01469] [SST2] [02469] [SST5] or [SST5–1] [02468] [SSTi] (any) [02458] [SST5] [01358] [SST3] [01458] ([SST4–1]([SFT4–1])) m.2: [01458] [SST3–1] [01358] [SST4–1] [01348] [SST1–1] [01358] [SST3] [01458] [SST5] [01458] [SVT(4–1)2(2–1)2] [01458] [SST5] [01458] (T10([SST4–1])) m.3: [01448] OR [01458] ([SST(1–1)2(2)(3–1)(4)]([SFT5])) m.3: [01148] [SST(1–1)(4–1)] [02247] [SST(1–1)2(3)(42)] [01458] [SST5] [01458] [SST1–1] [01469] [SST2] [02469] [SST5] or [SST5–1] [02468] [SSTi] (any) [02458] [SVT32] [02368] [SVT(1–1)2] [01468] [SVT22] [02458] [SST5] [01358] [SST(1–1)(5–1)2] m.4: [01357] [SST23] [02347] ([SST2]([SFT2])) [01469] [SVT42] [01458]

Fig. 6-12: SST-succession-class analysis of Lansky, Modal Fantasy, “Prelude,” mm. 1–4

types previously unencountered in the piece, corresponding to the first composite SST-succession classes (including an [SFTi] subscript) in mm. 1–2 and the “near-transposition” in mm. 2–3. There is also a string of 4 consecutive [01458]-type pentachords in m. 2. The SST-succession class analysis shows two possible readings of the problematic juncture between measures 2 and 3. The first shows the neartransposition by T10 which results in the multiset class [01448]. The second reading shows a more elaborate composition of subscripts resulting in the multiset class [01148]. Although the first reading seems to be the more intuitive of the two, it remains problematic in that it produces a pc doubling which is at odds with the one in the score. The second reading is thus closer to the surface voice leading as shown in the score. Of particular interest in the SST-succession class analysis, and not at all obvious by other means of analysis (including the SVT reading of Figure 611), is the reiteration of the opening succession midway through m. 3. This progression, identified in Figure 6-12 as [01458] [SST5] [01458] [SST1–1] [01469] [SST2] [02469] [SST5] or [SST5–1] [02468] [SSTi] (any) [02458], is reproduced exactly beginning with the third chord of m. 3 and is, in effect, a T6 transposition of the opening six chords. Although it is probably not profitable to examine neo-Riemannian-like transformations in this style of music, it should be pointed out that all but one of the pentachordal set classes Lansky uses contain [037] as a subset (see Table 5-7), and thus have the potential of admitting parsimonious neoRiemannian-like [SSTi] relations similar to those listed in Tables 5-5 and 56. The one pentachordal set class that is not part of the extended neo-

158

Chapter Six

Riemannian collection is the whole-tone hexachord, [02468], which occurs twice and can be combined with SC [02458] in a pair-succession relatable by any single [SSTi].

Post-Atonal Analyses Although the SST model can be profitably used to trace parsimonious voice-leading paths in both tonal and atonal music, its analytical potential is perhaps best realized when applied to the post-atonal repertoire. This is a very generic term meant to encompass all Western art music composed since the 1960s which consistently employs triadic harmonies, if not tonality that is “functional” in the common-practice-period sense. A notable sub-category of post-atonal music is most minimalist music, with its emphasis on diatonic scales/collections and tertian chord successions. Musical minimalism was perhaps the most significant school of compositional style post-dating the serial and aleatoric music of the 1950s and 1960s. Its considerable influence continued from its formative years (the mid-1960s through the 1970s) to the end of the century and beyond. Many now speak of a “post-minimalist” style that was heavily influenced by the earlier movement, and which is represented by composers such as William Duckworth, Michael Torke, Robert Moran, Janice Giteck, and Paul Dresher. The music of certain minimalist composers displays a conspicuous sensitivity to smooth voice leading, whether manifest or projected.15 Indeed, many of these composers’ chord progressions can be accounted for by various and sundry neo-Riemannian and neo-neo-Riemannian transformations ranging from the commonplace (R, P, and L) to the exotic (“SLIDE-like,” etc.). Minimalist composers whose music invites SVT analysis include Philip Glass, John Adams, Michael Nyman, Louis Andriessen, and Alvin Curran. Minimalist composers whose music does not make much use of parsimonious voice leading include La Monte Young, Terry Riley, and Steve Reich. As minimalism figures so prominently in music of the late twentieth century, and because it has received relatively little analytical attention in comparison to the serial music of the generation that preceded it, the remainder of this study will examine the application of the SVT system to some fairly recent minimalist music. By doing so, I hope to show the system’s capacity to model music belonging to this style. Although I will not analyze any of it here, a great deal of Philip Glass’s music dating from about 1980 to the present makes frequent use of neoRiemannian transformations and parsimonious voice leading. Glass’s second opera, Satyagraha, marks a significant shift from his earlier minimalist style in its adoption of repeated four-bar phrases as basic compositional units, as opposed to phrase lengths that gradually expanded or contracted through his

Analyses

159

“additive construction” process that figured so prominently in earlier works. Likewise, Satyagraha marks a shift in Glass’s harmonic language, from long-held chords over which additive construction processes work themselves out, to more rapidly changing harmonies (usually one chord per measure) that frequently are connected by smooth voice leading. Despite the stylistic change, Glass’s harmonic materials did not change. His harmonic choices are quite conservative, almost always being the familiar tonal triads or seventh chords. Occasionally non-tertian sonorities are formed as temporary contrapuntal constructs, the result of parsimonious voice leading between tertian chords. Triadic neo-Riemannian progressions are very common in Glass’s music, especially the L, P, and R operations, but N and H are also used frequently, as is Lewin’s SLIDE, a particular favorite of both Glass and Adams. Several of these neo-Riemannian operations can be found in works such as Akhnaten,16 Songs from Liquid Days (“Changing Opinion,” “Lightning,” “Liquid Days Part I,” and “Forgetting”), and music composed for the first two Candyman horror films. Like Glass’s music, John Adams’s early minimalist works favored diatonic collections (in much of his music these take the form of the modal scales corresponding to the seven church modes). Verticalities in Adams’s music frequently are constructed as pitch classes selected from a diatonic field, which may or may not take the form of tertian sonorities; Adams is noticeably less reliant on these than is Glass. (In this way his harmonic practice is closer to that of Steve Reich.)17 Adams’s rhythmic and formal practice is also much more complex than is Glass’s. Several passages from Adams’s early music would make good candidates for traditional neo-Riemannian (triadic) analyses. I will make brief mention of some of these now, although I will concentrate on some of Adams’s music richer in parsimonious relations than these pieces when I examine selections from Nixon in China later in this chapter. In Adams’s piano piece China Gates, the first of his minimalist works (1976), the entire composition is based on the strongly parsimonious interaction of four modes. “Gates,” or sudden shifts in harmony, occur as Adams takes advantage of the six common tones retained and the single semitonal displacement between pairs of diatonic collections (i.e., between G Aeolian and F Locrian). Adhering more closely to the neo-Riemannian tradition is Grand Pianola Music, which contains several L operations in the first movement; R relations feature prominently in the central slow movement, and the ending is largely built on a repeated SLIDE operation. Triadic neoRiemannian transformations can be found throughout Nixon in China, particularly in the ensemble piece “Flesh Rebels” from Act II (in which the H transformation stands out noticeably) and especially in the “Cheers” chorus at the end of Act I, where L, P, and R transformations appear in pairs (mm.

160

Chapter Six

662–829). Finally, mention must be made of the soprano aria “I Must Have Been Hysterical” from The Death of Klinghoffer, which has been analyzed with Lewinian network graphs by Mark Johnson (1999). The following section presents analyses of a number of sections from Adams’s first opera, Nixon in China, written between 1985 and 1987. This work, perhaps Adams’s best known composition (now rivaled by the 9/11 memorial piece On the Transmigration of Souls from 2002) in a sense represents the culmination of the minimalist compositional techniques he had been developing in his music during the preceding decade. An emphasis on voice-leading parsimony between tertian sonorities had been apparent in Adams’s scores from as early as the string septet Shaker Loops of 1978, and Nixon was one of his last pieces to use this relatively straightforward (some would say “accessible”) harmonic style, although it would occasionally resurface to a lesser extent in later pieces. I will begin by presenting short excerpts from Nixon that illustrate Adams’s use of extended neo-Riemannian progressions and the utility of SST analysis in describing the voice leading of these passages. I will conclude my analyses of the opera with two larger excerpts.

Adams, Nixon in China Figure 6-13 reproduces a short passage from Chou En-lai’s eloquent Act I aria, “Ladies and Gentlemen, Comrades and Friends.” Harmonically, measures 391–404 consist of a simple oscillation between two sonorities: a D major ninth chord and a D minor ninth chord. The smoothest possible way of connecting these two chords is by altering each chord’s third and seventh, moving them by parallel semitones. By the norms of functional tonality this would be forbidden, as the voices would have to move by parallel perfect fifths, but in Adams’s post-atonal style this voice leading is perfectly acceptable. We note that in the score (at least in the piano reduction) this voice leading is made manifest, or in other words, the semitonal motion is not displaced by one or more octaves and appears to occur within the same two musical lines (those occupying the two lowest registral positions). What does an SST-succession-class analysis of this passage reveal? Somewhat surprisingly, it turns out that minor ninth and major ninth chords belong to the same SC type, [01358]. Not only that, but the exact same [SSTi] relation holds between the two harmonies regardless of pair ordering. This is not all that surprising, however, as any SST-succession class that involves a pair-succession of the same set class will be involved in a cycle; on the n-tuple level, such a cycle will eventually return to the original chord. In the case of [01358], there are only two chords in the cycle, thus the sub-

Analyses

161

[01358] DM9

[SST(1–1)(4–1)] “SLIDE-like”

[SST(1–1)(4–1)] “SLIDE-like”

[01358] Dm9

[SST(1–1)(4–1)] “SLIDE-like”

[SST(1–1)(4–1)] “SLIDE-like”

[01358] DM9

[01358] DM9

[01358] Dm9

© 1987 Hendon Music, Inc., a Boosey & Hawkes company. Copyright for all countries. All rights reserved. Reprinted by permission.

Fig. 6-13: Adams, Nixon in China, Act I, Scene 3, mm. 390–404 with analytical overlay

scripted relation [SST(1–1)(4–1)] is its own inverse. From Table 5-7 we see that [01358] contains embedded SC [037], via the tetrachordal SCs [0158], [0247], and [0358]. We immediately note that [0158] corresponds to the major seventh chord and [0358] to the minor seventh chord. These two SCs relate to each other in an SST-succession class by a SLIDE-like [SSTi] subscript (Table 5-6) and thus the pentachordal SCs should be related likewise. But we did not really need to consult the tables to find this out; the

162

Chapter Six

embedded SLIDE operation grabs our attention as we listen to this passage. We find here that the sound of a SLIDE is so distinctive that it cannot be masked by increasing the chord cardinality to five. To this point I have not discussed how the voice part fits into (or refuses to fit into) the SST-succession-class analysis. Most of Chou’s pitches are chord tones, with a number of passing tones occurring in mm. 392–3. However, even after the first change of harmony at m. 394, Chou seemingly refuses to acknowledge the new pc, C9, which replaces the C of the previous measures. A similar thing occurs at m. 400, where once again Chou stubbornly persists with the C instead of the C9. This may be explained as a little bit of text painting on Adams’s part; Chou’s speech describes the “frontier we occupy” and “holding in perpetuity the ground our people won today.” At these points in the music, the D minor ninth chord’s territory is being uneasily shared with a persistent squatter, the pc C.

a)

b)

c)

Fig. 6-14: Adams, P2,0 cycles: a) Nixon in China, Act I, Scene 2, mm. 180–3; b) Nixon in China, Act II, Scene 2, mm. 271–2; c) Fearful Symmetries, mm. 297–8 (keyboard 1 part only)

Figure 6-13 illustrates one of Adams’s favorite parsimonious voice-

Analyses

163

leading progressions, the SLIDE. Adams is also very fond of what Douthett and Steinbach would call P2,0 progressions between major-minor seventh chords (SC [0258]). Some of these are shown in Figure 6-14. The first of these (Figure 6-14a), from Act I of Nixon, contains a succession of majorminor seventh chords with roots descending by minor thirds (mm. 180–1). The series is repeated, a half step higher, in mm. 182–3. Had Adams continued each of these progressions by an additional seventh chord, he would have obtained complete cycles of P2,0-related chords. Each of these would be represented in graphical space by one circuit around the circumference of Douthett and Steinbach’s pipeline torus (Figure 4-5). In SST-successionclass terms, each of these P2,0 progressions corresponds to the subscript [SST(1–1)(2)], which is also equivalent to Childs’s C3(2) transformation. Figure 6-14b, from Act II of Nixon, is the retrograde of the second half of Figure 6-14a. This progression makes a complete P2,0 cycle of major-minor seventh chords ascending by minor thirds, and concludes with a jarring [02469]-type chord which is the result of another move along the cycle combined with a SPLIT transformation. Each move between seventh chords here corresponds to the subscript [SST(1–1)(4)], which is equivalent to Childs’s C3(4) and Bass’s ic 4-2 transformations. Figure 6-14c, from the 1988 piece Fearful Symmetries, is another example of an [SST(1–1)(4)] cycle.18 Let us now turn to two lengthier analyses of music from Nixon in China. The first of these (Figure 6-15) is from the middle section of Nixon’s Act I aria “News Has a Kind of Mystery.” Because most of Adams’s early pieces—and a great deal of minimalist music in general—have a slow to very slow rate of harmonic motion, we can place our SST-succession-class analysis under the systems in the piano-vocal score, something that could not be accomplished in the analyses of the Chopin and Lansky pieces (due to spatial constraints), with their rapid harmonic changes. As we scan through the excerpt, we note a gradual increase in harmonic complexity. At first, each harmony is a major or a minor triad. At m. 520 the first tetrachord is heard (a major-minor seventh chord). Shortly thereafter, pentachords begin to appear, intermixed with tetrachords. Simple trichords do not reappear until somewhat later, with a return to the opening text and musical material in this ABA'-form aria (not shown in Figure 6-15). We also note that although these chords of higher cardinality move us away from simple triads, the tetrachords and even the pentachords remain subsets of diatonic collections. The SST-succession classes likewise become more complex as the excerpt progresses, involving single neo-Riemannian relations at the beginning and later making use of composite [SSTi] relations that press the SSTsuccession-class model to its useful limits by m. 551. Concurrent with the first move to a tetrachordal sonority in m. 521 comes the first of several whole-step [SSTi] relations and split-succession classes. By the end of the

164

Chapter Six

[037]

[SST3] [037] L

[SST3] [037] L

[SST3] [037] L

[SST2] [037] P

[SST3] [037] L

Fig. 6-15: Adams, Nixon in China, Act I, Scene 1, mm. 487–558, pianovocal score with analytical overlay

excerpt, the number of voices has been stabilized at four (through the only fuse-succession class in the figure), and [SSTi] subscripts are again limited to only those which affect a single voice. Neo-Riemannian and extended neo-Riemannian progressions abound in this passage. The first 33 measures comprise nothing but [037]-type trichords related by single L and P operations and one SLIDE. Toward the end of the excerpt Adams presents one instance each of what I am calling L-, P-, or R-like transformations. These involve tetrachordal set classes that contain [037] as a subset, as listed in Table 5-5. The second of these, the P-like transformation between mm. 552 and 553, is quite easy to hear: in the vocabulary of seventh chords, the third of a G( major seventh chord is lowered a half step to form an F minor-augmented seventh. One clearly hears the change of quality in the triadic component of these chords. This figure, as well as Figure 6-13, shows another advantage of SST-succession classes over the neo-Riemannian systems and graphs discussed in chapters 4 and 5: no other system would even recognize the “L-like-ness” or “SLIDE-likeness” of some of the tetrachordal or pentachordal voice leadings found in these examples, even though these relations are quite readily identified by

Analyses

165

[SST3] [037] L

[SST(1–1)(3–1)] [037] SLIDE

[SST3] [037] L

[SST2] [037] P

Fig. 6-15 (continued)

the ear. Perhaps the most interesting aspect of this passage involves the

166

Chapter Six

[037]

[037]

[SVT(3–1)2(42)]([SPLIT3]) [0258]

[037]

[0258]

[SVT(1 –1)2]([SPLIT1])

[SST3]([SPLIT2])

[0358]

[02358]

[SFT3]

[SST1] “L-like”

[02469]

[0247]

[SST(3–1)3] [0158]

Fig. 6-15 (continued)

transformations occurring between mm. 520 and 542. The first of these SSTsuccession classes with composite [SSTi] subscripts, ¢[037],[0258]²

Analyses

[SVT32]([SPLIT2]) [02469]

[0148]

167

[SPLIT2–1]([SST(1–1)(3–1)2(5–1)2]) [0158]

[SST3–1] “P-like”

[SVT(1–1)2] “R-like” [0258] © 1987 Hendon Music, Inc., a Boosey & Hawkes company. Copyright for all countries. All rights reserved. Reprinted by permission.

Fig. 6-15 (continued)

SVT(3–1)2(42)(SPLIT3),

is marked in the score at m. 520. The voice leading here, while parsimonious, hints at the limits of the SVT model: here five separate SVT relations must be combined in order to account for the SSTsuccession class. Were it not for the lack of functional harmony, necessarily containing a well defined tonic, it would be much simpler to describe this progression as ii–V7 in the key of E( major. This potential shortcoming of the SVT system is even more pronounced in m. 550, with the complex SSTsuccession class ¢[02469],[0158]²SPLIT2–1(SST(1–1)(3–1)2(5–1)2) appears. Here, six separate SVT relations must be combined to explain the voice leading between the two chords in terms of semitonal motion, where the progression instead could have been explained as a V9–I7 progression in the key of G( major. While the SVT system can model common diatonic progressions such as these, one has to wonder whether such a system should be used when a much more elegant solution already exists in an alternative analytical system. Keeping this problem in mind, let us return to mm. 520–1. The next ten measures are simply an oscillation between the same F-minor and B( majorminor seventh chords. I purposely have not shown the SST-succession class

168

Chapter Six

involving this pair of chords. Instead, we may note an interesting development in subsequent measures: the chord in m. 532 corresponds to the chord in m. 527 with an added note, and the chord in m. 537 retains four of m. 532’s tones while substituting a fifth; likewise, the chord in m. 534 corresponds to the chord in m. 529 with an added note, and the chord in m. 542 retains three of m. 534’s tones while substituting a fourth. The first “family” of related chords is identified in Figure 6-15 by encircled SCs, while the second family is identified with boxed SCs. If, instead of analyzing SSTsuccession classes involving pair-successions between these two families of chords, we instead note SST-succession-class relations within each family, we obtain much simpler [SSTi] relations. As this passage at its most basic level is simply an oscillation between two (slightly varying) chords, it makes sense to analyze the parsimonious voice-leading variations in each of the two chords separately. The SST-succession classes involving chordal variants in each family are shown at the center of the arrows drawn in Figure 615. The first succession-pair in each family is split-related, and the second is relatable by a single [SVTi]—an [SSTi] in one case and an [SFTi] in the other. In each family, each succession-class pair is relatable by the same subscript index number: in the encircled family [SVTi]s where i = 3 relate pairordered equivalence classes, and in the boxed family [SVTi]s where i = 1 relate pair-ordered equivalence classes. What would be Adams’s reason to move from major and minor triads related by L and P operations early in the aria to chords of greater cardinality related by more complex voice-leading transformations? Again, I would suggest a possible explanation lies in the composer’s sensitivity to the libretto. Nixon’s character is portrayed musically throughout most of the opera with triadic harmonies, a strong pulse, simple rhythms, and syllabic text setting. Typical of Nixon’s music is his Act I, scene 3 speech, “Mr. Premier, Distinguished Guests,” and his “News” aria, the middle section of which is reproduced here as Figure 6-15. The musical style suggests a somewhat stiff/square persona, but the text projects Nixon’s confident, almost ebullient optimism during his historic February 1972 trip. That is, until doubts begin to creep into his mind, as reflected by the text beginning with m. 517: “We live in an unsettled time. Who are our enemies? Who are our friends?” The music immediately turns to the minor, and it is at this point that the denser tetrachords and pentachords begin to appear, perhaps symbolizing Nixon’s consternation. The phrase that follows, “The Eastern hemisphere beckoned to us, and we have flown east of the sun, west of the moon,” is set to a more Romantic-sounding music, abounding with lush seventh and ninth chords as Nixon grows more pensive (almost waxing poetic) and less troubled. We will see, in the next example, that this last style returns in the highly introspective third Act of the opera.

Analyses mm. 748

752

757

759

169 762

764

768

[SST(2)(4)] [0258] [SST1–1] [0369] [0258] [SST(1–1)(3–1)] [0258] [SST(3)(4)] [0258] [SVT(1–1)2] [0258] [SST(1–1)(4–1)] [0258] S6(5) (Childs) C#‡7 P¡(Shimbo) Db7 L¡(Shimbo) F ‡7 S3(4) (Childs) Bb7 R¡(Shimbo) G‡7 R*2 (Baker) G°7 Eb7 775 782 794 796 800 804 807

[0258] E‡7

[SSTi]

[SST2–1] [0158] [SST4] [0148] [SST(1–1)3][0158] [SST(1–1)(22)] [0258] [SST(1–1)(2)] [0258] [SST(1–1)(2)] [0258] –L2 (Baker) EbM7

GM7

B‡7 C3(2) (Childs) D‡7 C3(2) (Childs) F‡7

Fig. 6-16: Adams, Nixon in China, Act III, mm. 748–809, reduction

Figure 6-16 is a reduction of the harmonies found in mm. 748–809 of Act III of Nixon. In the figure, open noteheads represent the primary chords (usually these can be identified as various types of seventh chords) and filled noteheads represent other “strongly presented” pcs.19 With the exception of the [0148]-type chord in m. 794, all of the harmonies in these measures conform to one of the “standard” types of seventh chords. Furthermore, the first six chords and the last three are all of SC type [0258]. Because the common seventh chord types in general—and seventh chords of SC type [0258] in particular—have been the subject of a good deal of recent neo-Riemannian studies, it is not at all surprising that nearly all of the parsimonious progressions between seventh chords in Figure 6-16 can receive an extended-neoRiemannian label. These transformational labels, from Childs, Shimbo, and Baker, appear below the SST-succession-class analysis. Once again, we see the utility of this type of analysis in its ability to succinctly summarize the voice-leading motion between sonorities while at the same time transcending the hodgepodge of competing labeling systems that have proliferated over the years. Additionally, SST-succession-class analysis can easily show parsimonious relations between any given set classes; all the extended-neoRiemannian labeling systems we have examined describe parsimonious relations only between type-[037] trichords or between tetrachords of types [0258], [0158], [0358], and [0369]. Nearly all the [SSTi] relations between pair-ordered SCs in Figure 6-16 are what Douthett and Steinbach would call P1,0 and P2,0 relations, with either one or two voices moving by semitone. It is interesting to note that in the chain of six [0258]-type tetrachords at the beginning of the figure,

170

Chapter Six

© 1987 Hendon Music, Inc., a Boosey & Hawkes company. Copyright for all countries. All rights reserved. Reprinted by permission.

Figure 6-17: Adams, Nixon in China, Act III, mm. 791–6 (piano–vocal score)

Adams is able to obtain chords of the same SC type through five distinct voice-leading operations involving the motion of two semitones. Addition-

Analyses

171

ally, a sixth distinct voice leading between type-[0258] tetrachords appears at the end of the excerpt. Two of these SSTis connect Tristan chords (mm. 759 and 807); these occur during instrumental interludes and probably do not have any special meaning other than perhaps to help evoke a lateRomantic sensibility. Act III of Nixon differs fundamentally from the previous acts in that it is less action-based and much more introspective. During the act, the six principal characters spend most of their time sitting on beds, talking to—but mostly talking “through”—each other.20 Through much of the act, it seems as if all the characters were delivering their own personal soliloquies simultaneously, only occasionally listening to and responding to each other. Much of what each character chooses to reflect upon is their past, with Nixon recalling his World War II days, Pat Nixon the time she spent waiting for Dick to return from the war, and Mao, Madame Mao, and Chou remembering the Revolution. Figure 6-17 presents a short excerpt from the piano–vocal score. Nixon’s and Pat’s lines conform very closely to the notes of the prevailing harmonies (with the exception of Nixon’s F in m. 794, perhaps a bit of word painting on “cry”). But Mao’s and Madame Mao’s parts are notable for the prominent, high-tessitura G(s appearing in cross-relation with the Gs in the accompaniment. One is tempted to interpret this as Adams’s way of emphasizing the nonconformism of the revolutionary couple in contrast to the conservatism of the Nixons. This passage then has clear parallels with the extra-harmonic Cs in Chou’s aria, as seen in Figure 6-13.

Conclusion The SVT and SST-succession-class models are logical extensions of systems proposed by Roeder (1994), Callender (2004) and Tymoczko (2006). The development of the SVT model was instigated by the perceived need to fill a notable gap in previous transformational theories of voice leading. Where the earlier studies relied principally on the transposition and inversion of sets to account for voice leading, the SVT model privileges voiceleading parsimony above all other factors, and thus can profitably be applied to a wide variety of musical styles in which linear motion by step or half step prevails. While most of the SVTs examined in this thesis are of the SST type, the potential exists for describing voice-leading transformations where parsimony is defined in terms of intervals other than ic 1. We have seen how the single-fifth transformation occasionally makes an appearance in SVT analyses and succinctly explains a voice-leading transformation that would otherwise require several composite SSTs. The SFT may be considered to be the parsimonious transformation in music in which most of the voice leading is by ic 5; in such a case all other voice leadings (including SSTs) would be

172

Chapter Six

considered non-parsimonious transformations. Additionally, the SVT model provides what is very likely the first mathematical formalization of splits and fuses. This problem—which has plagued transformational theory for years—receives a detailed examination in chapter 3. The subsequent generalization of SSTs, splits, and fuses (and the associated m-tuple pair-successions they relate) to SST-, split-, and fusesuccession classes links the SVT model to previous studies that plot optimal offset between set class types (Straus 2003 and 2005, Cohn 2003). The establishment of SST-succession classes takes this development one step further, showing for the first time that SST-succession classes relatable by the same subscript [SSTi] appear in geometrically regular positions in the optimal offset graphs constructed by Straus (2005). Another strength of the SVT and SST-succession-class models is their analytical applicability to a wide variety of musical styles, both tonal and atonal, and even to much recently composed music which—to a greater or lesser degree—rejects musical modernism by a partial return to the harmonic structures of the common-practice period. We have seen how much of the recent work in neo-Riemannian theory has attempted to adapt and apply Riemannian transformations to chords other than major or minor triads. This has given rise to a confusing proliferation of extended-neo-Riemannian labeling systems, as can be seen in Tables 5-2 through 5-4. The SSTsuccession classes subsume all the disparate extended-neo-Riemannian labels and account for voice-leading transformations between non-tertian sonorities that the other systems cannot model. Finally, the SST-succession-class model even has the intriguing possibility of serving as a compositional model. A composer may choose to limit voice-leading transformations to those found in a few select SST-succession classes and still be able to generate a large number of distinct harmonies, both tertian and non-tertian. The addition of a single split-succession class and fuse-succession class to the few pre-selected SST-succession classes would offer to the composer the possibility of an even greater number of sonorities and the flexibility to change chord cardinality as desired.

APPENDIX A SUMMARY OF [SSTI]-RELATIONS BETWEEN TRICHORDAL AND TETRACHORDAL EQUIVALENCE-CLASS PAIR-SUCCESSIONS Number of Distinct [SSTi]-Relatable Set Classes [B] in a Pair-Succession ¢[A],[B]²²SSTi SC [A]

To Trichords

012 (3-1 ) 013 (3-2) 014 (3-3) 015 (3-4) 016 (3-5) 024 (3-6) 025 (3-7) 026 (3-8) 027 (3-9) 036 (3-10) 037 (3-11) 048 (3-12)

1 3 4 4 3 3 5 6 3 3 4 1

To Dyads (Fuse) Subtotal To Tetrachords (Split) Total 2 2 2 2 2 0 0 0 0 0 0 0

3 5 6 6 5 3 5 6 3 3 4 1

1 3 4 4 4 2 5 5 3 3 6 1

4 8 10 10 9 5 10 11 6 6 10 2

(Note maximum number of possible split relations obtainable from [037].) SC [A] To Tetrachords To Trichords (Fuse) Subtotal To Pentachords (Split) Total 0123 (4-1) 0124 (4-2) 0125 (4-4) 0126 (4-5) 0127 (4-6) 0134 (4-3) 0135 (4-11) 0136 (4-13) 0137 (4-29) 0145 (4-7) 0146 (4-15) 0147 (4-18) 0148 (4-19) 0156 (4-8)

1 4 4 4 2 2 6 6 5 2 6 6 6 2

2 3 3 3 2 2 2 2 2 2 2 2 2 2

3 7 7 7 4 4 8 8 7 4 8 8 8 4

1 3 4 4 2 2 4 5 5 2 5 6 5 2

4 10 11 11 6 6 12 13 12 6 13 14 13 6

174

Appendix A

SC [A] To Tetrachords To Trichords (Fuse) Subtotal To Pentachords (Split) Total 0157 (4-16) 0158 (4-20) 0167 (4-9) 0235 (4-10) 0236 (4-12) 0237 (4-14) 0246 (4-21) 0247 (4-22) 0248 (4-24) 0257 (4-23) 0258 (4-27) 0268 (4-25) 0347 (4-17) 0358 (4-26) 0369 (4-28)

6 3 1 3 6 5 4 8 4 4 7 2 3 3 1

2 1 1 1 2 2 0 0 0 0 0 0 1 0 0

8 4 2 4 8 7 4 8 4 4 7 2 4 3 1

5 3 1 2 5 5 3 6 3 3 7 2 3 4 1

13 7 3 6 13 12 7 14 7 7 14 4 7 7 2

(Note maximum number of split relations obtainable from [0258].)

Equivalence-Class Pair-Successions Relatable through Some Single [SSTi] (Including Split- and Fuse-Succession Classes) SC [A] SC [B] 012 013 014 015 016 024 025 026 027 036 037

01 (2 paths), 02 (2 paths), 013 (2 paths), 0123 (2 paths) 02, 03, 012, (013), 014, 024, 0123 (2 paths), 0124, 0134 03, 04, 013, 015, 024, 025, 0124, 0125, 0134, 0145 04, 05, 014, 016, 025, 026, 0125, 0126, 0145, 0156 05, 06, 015 (2 paths), 026, 027, 0126, 0127, 0156, 0167 013 (2 paths), 014 (2 paths), 025 (2 paths), 0124 (4 paths), 0135 (2 paths) 014, 015, 024, (025), 026, 036, 0125 (2 paths), 0135, 0136, 0146, 0235 015, 016, 025, 027, 036, 037, 0126 (2 paths), 0137, 0146, 0157, 0236 016 (2 paths), 026 (2 paths), 037 (2 paths), 0127 (2 paths), 0157 (2 paths), 0237 (2 paths) 025 (2 paths), 026 (2 paths), 037 (2 paths), 0136 (2 paths), 0147 (2 paths), 0236 (2 paths) 026, 027, 036, (037–2 paths (Cohn function)), 048, 0137, 0147, 0148, 0158, 0237, 0347

Summary of [SSTi]-Relations Between Equivalence-Class Pair-Successions 175

SC [A] SC [B] 048

037 (6 paths), 0148 (6 paths)

SC [A] SC [B] 0123 0124 0125 0126 0127 0134 0135 0136 0137 0145 0146 0147 0148 0156 0157 0158 0167 0235 0236

012 (2 paths), 013 (4 paths), 0124 (2 paths), 01234 (2 paths) 013, 014, 024 (2 paths), 0123, 0125, 0134, 0235, 01234 (2 paths), 01235, 01245 014, 015, 025 (2 paths), 0124, 0126, 0135, 0236, 01235, 01236, 01245, 01256 015, 016, 026 (2 paths), 0125, 0127, 0136, 0237, 01236, 01237, 01256, 01267 016 (2 paths), 027 (2 paths), 0126 (2 paths), 0137 (2 paths), 01237 (2 paths), 01267 (2 paths) 013 (2 paths), 014 (2 paths), 0124 (2 paths), 0135 (2 paths), 01234 (2 paths), 01245 (2 paths) 024, 025, 0125, 0134, 0136, 0145, 0235, 0246, 01235 (2 paths), 01245 (2 paths), 01246, 01356 025, 036, 0126, 0135, 0137, 0146, 0236, 0247, 01236 (2 paths), 01247, 01346, 01356, 01367 026, 037, 0127, 0136, 0147, 0237 (2 paths), 0248, 01237 (2 paths), 01248, 01347, 01367, 01568 014 (2 paths), 015 (2 paths), 0135 (2 paths), 0146 (2 paths), 01245 (2 paths), 01256 (2 paths) 025, 026, 0136, 0145, 0147, 0156, 0246, 0257, 01246, 01256 (2 paths), 01257, 01346, 01367 036, 037, 0137, 0146, 0148, 0157, 0247, 0258, 01247, 01258, 01347, 01367, 01457, 01478 037, 048, 0147, 0158, 0237, 0248, 0347, 0358, 01248, 01348, 01458 (2 paths), 01478, 03458 015 (2 paths), 016 (2 paths), 0146 (2 paths), 0157 (2 paths), 01256 (2 paths), 01267 (2 paths) 026, 027, 0147, 0156, 0158, 0167, 0257, 0268, 01257, 01267 (2 paths), 01268, 01457, 01568 037 (2 paths), 0148 (2 paths), 0157 (2 paths), 0258 (2 paths), 01258 (2 paths), 01458 (2 paths), 01568 (2 paths) 016 (4 paths), 0157 (4 paths), 01267 (4 paths) 025 (2 paths), 0124 (2 paths), 0135 (2 paths), 0236 (2 paths), 01235 (4 paths), 01346 (2 paths) 026, 036, 0125, 0136, 0235, 0237, 0246, 0347, 01236 (2 paths), 01346, 01347, 01457, 02346

176

Appendix A

SC [A] SC [B] 0237 0246 0247 0248 0257 0258 0268 0347 0358 0369

027, 037, 0126, 0137 (2 paths), 0148, 0236, 0247, 01237 (2 paths), 01348, 01457, 01568, 02347 0135 (2 paths), 0146 (2 paths), 0236 (2 paths), 0247 (2 paths), 01246 (4 paths), 01357 (2 paths), 02346 (2 paths) 0136, 0147, 0237, 0246, 0248, 0257, 0347, 0358, 01247 (2 paths), 01357, 01358, 01468, 02347 (2 paths), 02357 0137 (2 paths), 0148 (2 paths), 0247 (2 paths), 0258 (2 paths), 01248 (4 paths), 01468 (2 paths), 02458 (2 paths) 0146 (2 paths), 0157 (2 paths), 0247 (2 paths), 0258 (2 paths), 01257 (4 paths), 01368 (2 paths), 02357 (2 paths) 0147, 0158, 0248, 0257, 0268, 0358 (2 paths), 0369, 01258 (2 paths), 01368, 01369, 01469, 02358, 02368, 02458 0157 (4 paths), 0258 (4 paths), 01268 (4 paths), 02368 (4 paths) 037 (2 paths), 0148 (2 paths), 0236 (2 paths), 0247 (2 paths), 01347 (2 paths), 01458 (2 paths), 02347 (2 paths) 0148 (2 paths), 0247 (2 paths), 0258 (4 paths), 01358 (2 paths), 01469 (2 paths), 02358 (2 paths), 03458 (2 paths) 0258 (8 paths), 01369 (8 paths)

APPENDIX B CHART OF [SSTI] SUBSCRIPTS RELATING TRICHORDAL AND TETRACHORDAL EQUIVALENCE-CLASS PAIR-SUCCESSIONS Equivalence Classes [B] Relatable to [A] by Single [SSTi]s as Part of an SST-Succession Class Symmetrical set classes are indicated in bold. [SSTi]s which relate an equivalence class [A] to an equivalence class [B] of the same SC type (where [A] = [B]) are enclosed in parentheses. Some equivalence classes [B] are multisetclasses, which may subsequently be reduced to classes of cardinality m – 1 as [A] components of SPLITj–1-succession classes. SST1–1 SC [B]

SST1 SC [A] SC [B] 012 013 014 015 016 024 025 026 027 036 037 048

001 002 003 004 005 013 014 015 016 025 026 037

013 024 025 026 027 025 036 037 037 037 048 037

SST2 SC [B]

SST2–1 SC [B]

002 (013) 024 025 026 014 (025) 036 037 026 027 037

002 003 004 005 006 014 015 016 016 026 (037) 037

SST3 SC [B] 013 014 015 016 015 025 026 027 026 037 (037) 037

SST3–1 SC [B] 001 012 013 014 015 013 024 025 026 025 036 037

SST1 SC [A] SC [B]

SST1–1 SST2 SC [B] SC [B]

SST2–1 SC [B]

SST3 SC [B]

SST3–1 SST4 SC [B] SC [B]

SST4–1 SC [B]

0123 0124 0125 0126 0127 0134

0124 0235 0236 0237 0137 0135

0133 0024 0025 0026 0027 0144

0133 0134 0135 0136 0137 0144

0113 0114 0115 0116 0016 0124

0012 0123 0124 0125 0126 0133

0012 0013 0014 0015 0016 0013

0113 0224 0225 0226 0227 0124

0124 0125 0126 0127 0126 0135

178

Appendix B

SST1 SC [A] SC [B]

SST1–1 SST2 SC [B] SC [B]

SST2–1 SC [B]

SST3 SC [B]

SST3–1 SST4 SC [B] SC [B]

SST4–1 SC [B]

0135 0136 0137 0145 0146 0147 0148 0156 0157 0158 0167 0235 0236 0237 0246 0247 0248 0257 0258 0268 0347 0358 0369

0246 0247 0248 0146 0257 0258 0358 0157 0268 0258 0157 0236 0347 0148 0247 0358 0258 0258 0369 0258 0148 0258 0258

0255 0036 0037 0155 0266 0377 0048 0166 0277 0377 0016 0135 0136 0137 0146 0147 0148 0157 0158 0157 0247 0258 0258

0145 0146 0147 0155 0156 0157 0158 0166 0167 0157 0116 0135 0246 0247 0146 0257 0258 0157 0268 0157 0247 0258 0258

0125 0126 0127 0135 0136 0137 0237 0146 0147 0148 0157 0225 0226 0227 0236 0237 0137 0247 0248 0258 0337 0148 0258

0134 0135 0136 0144 0145 0146 0147 0155 0156 0157 0166 0124 0235 0236 0135 0246 0247 0146 0257 0157 0148 0247 0258

0024 0025 0026 0144 0255 0036 0037 0155 0266 0377 0166 0124 0125 0126 0135 0136 0137 0146 0147 0157 0236 0247 0258

0235 0236 0237 0135 0246 0247 0248 0146 0257 0258 0157 0225 0336 0337 0236 0347 0148 0247 0358 0258 0347 0148 0258

0136 0137 0237 0146 0147 0148 0347 0157 0158 0148 0157 0236 0237 0137 0247 0248 0247 0258 0358 0258 0236 0258 0258

Equivalence Classes [B] Relatable to [A] by Single [SSTi]s as Part of an SST-Succession Class or Fuse-Succession Class This table is nearly identical to the one above, but allows for fusesuccession-class relations in lieu of multiset classes [B] of the same cardinality as [A]. Symmetrical set classes are indicated in bold. [SSTi]s which relate an equivalence class [A] to an equivalence class [B] of the same SC type (where [A] = [B]) are enclosed in parentheses. [SPLITj–1] subscripts are not shown (where they occur). SST1 SC [A] SC [B]

SST1–1 SC [B]

SST2 SC [B]

SST2–1 SC [B]

SST3 SC [B]

SST3–1 SC [B]

012 013 014 015

013 024 025 026

02 (013) 024 025

02 03 04 05

013 014 015 016

01 012 013 014

01 02 03 04

Chart of [SSTi] Subscripts Relating Equivalence-Class Pair-Successions

179

SST1 SC [A] SC [B]

SST1–1 SC [B]

SST2 SC [B]

SST2–1 SC [B]

SST3 SC [B]

SST3–1 SC [B]

016 024 025 026 027 036 037 048

027 025 036 037 037 037 048 037

026 014 (025) 036 037 026 027 037

06 014 015 016 016 026 (037) 037

015 025 026 027 026 037 (037) 037

015 013 024 025 026 025 036 037

SST1 SC [A] SC [B]

SST1–1 SST2 SC [B] SC [B]

SST2–1 SC [B]

SST3 SC [B]

SST3–1 SST4 SC [B] SC [B]

SST4–1 SC [B]

0123 0124 0125 0126 0127 0134 0135 0136 0137 0145 0146 0147 0148 0156 0157 0158 0167 0235 0236 0237 0246 0247 0248 0257 0258 0268 0347 0358 0369

0124 0235 0236 0237 0137 0135 0246 0247 0248 0146 0257 0258 0358 0157 0268 0258 0157 0236 0347 0148 0247 0358 0258 0258 0369 0258 0148 0258 0258

013 024 025 026 027 014 025 036 037 015 026 037 048 016 027 037 016 0135 0136 0137 0146 0147 0148 0157 0158 0157 0247 0258 0258

013 0134 0135 0136 0137 014 0145 0146 0147 015 0156 0157 0158 016 0167 0157 016 0135 0246 0247 0146 0257 0258 0157 0268 0157 0247 0258 0258

013 014 015 016 016 0124 0125 0126 0127 0135 0136 0137 0237 0146 0147 0148 0157 025 026 027 0236 0237 0137 0247 0248 0258 037 0148 0258

012 0123 0124 0125 0126 013 0134 0135 0136 014 0145 0146 0147 015 0156 0157 016 0124 0235 0236 0135 0246 0247 0146 0257 0157 0148 0247 0258

05 013 014 015 016 025 026 037

012 013 014 015 016 013 024 025 026 014 025 036 037 015 026 037 016 0124 0125 0126 0135 0136 0137 0146 0147 0157 0236 0247 0258

013 024 025 026 027 0124 0235 0236 0237 0135 0246 0247 0248 0146 0257 0258 0157 025 036 037 0236 0347 0148 0247 0358 0258 037 0148 0258

0124 0125 0126 0127 0126 0135 0136 0137 0237 0146 0147 0148 0347 0157 0158 0148 0157 0236 0237 0137 0247 0248 0247 0258 0358 0258 0236 0258 0258

NOTES

Chapter 1 1. There have been, however, a few attempts at formalizing aspects of Schenkerian theory. See for instance Michael Kassler, “A Trinity of Essays: Toward a Theory that Is the Twelve-Note Class System; Toward Development of a Constructive Tonality Theory Based on Writings by Heinrich Schenker; Toward a Simple Program Language for Musical Information Retrieval” (Ph.D. diss., Princeton University, 1967) and Proving Musical Theorems I: The Middleground of Heinrich Schenker’s Theory of Tonality (Sydney: University of Sydney, Basser Department of Computer Science, 1975); Allan Keiler, “The Syntax of Prolongation,” In Theory Only 3/5 (1979), pp. 13–27; John Rahn, “Logic, Set Theory, Music Theory,” College Music Symposium 19/1 (1979), pp. 114–24; Stephen Smoliar, “A Computer Aid for Schenkerian Analysis,” Computer Music Journal 4/2 (1980), pp. 41–59; James Snell, “Computerized Hierarchical Generation of Tonal Compositions,” in Informatique et musique, ed. H. Charnass (Ivry: Elmeratto/CNRS, 1984); and Jason Yust, “Formal Models of Prolongation” (Ph.D. diss., University of Washington, 2006). A well known approach that is related to Keiler’s but not purely Schenkerian is found in Fred Lerdahl and Ray Jackendoff’s A Generative Theory of Tonal Music (Cambridge, Mass.: MIT Press, 1983). 2. Chapter 3 presents more formal definitions for “parsimonious” and “strongly parsimonious” relations between chords. 3. O’Donnell also attempts to address this problem; see my chapter 2 for his solution. 4. To borrow (somewhat loosely) a term from William E. Benjamin. See his “PitchClass Counterpoint in Tonal Music,” in Music Theory: Special Topics, ed. Richmond Browne (Orlando: Academic Press, 1981), 1–32. 5. See n. 12 of chapter 3 for my reason for using the variable m instead of the more common n. 6. Benjamin Boretz explores aspects of the relationship between these two set classes in a classic analysis of the Tristan Prelude in his “Meta-Variations, Part IV” (1972, 162 ff). 7. Jack Douthett and Peter Steinbach were perhaps the first to include the minor seventh chord in a network of parsimonious voice-leading transformations, as it occurs in their “octatowers” lattice that also includes dominant and half-diminished seventh chords (1998, 245). 8. See Lewin 1987, pp. 195–6 and Fiore and Satyendra 2005, p. 16, n. 40.

Chapter 2 1. This line of investigation is extended by Lewin in his 1990 article “Klumpenhouwer Networks and Some Isographies that Involve Them,” Music The-

Notes

181

ory Spectrum 12/1, pp. 83–120; and two subsequent articles: “A Tutorial on Klumpenhouwer Networks, Using the Chorale in Schoenberg’s Opus 11, No. 2,” Journal of Music Theory 38/1 (1994), pp. 79–101; and “Thoughts on Klumpenhouwer Networks and Perle-Lansky Cycles,” Music Theory Spectrum 24/2 (2002), pp. 196–230. This last article is found in a special issue of Spectrum devoted to K-nets which also includes articles by Philip Lambert, Philip Stoecker, and David Headlam. 2. This idea is later developed in Straus 2003; see pp. 24–9 in this chapter. 3. See for example p. 68, where O’Donnell admits his analyses favoring K-net recursive structures sometimes override voice-leading logic, distancing themselves from the apparent lines at the musical surface. 4. Not to be confused with the so-called “split transformations” or simply “splits” that allow for voice leading from sets of lower cardinality to sets of one-greater cardinality. See Callender 1998 and the extensive discussion of these types of situations in the second half of chapter 3. See Figure 3-2 in particular for an illustration of a way in which O’Donnell’s split transformation may actually produce the opposite result, a “fuse” or voice-leading operation which results in a reduction of chordal cardinality by one. 5. This voice-leading quality is called simply “uniformity” by Straus (2003, 311–4). See also pp. 24–5 in this chapter. 6. See also chapter 1, pp. 8–9 and Lewin 1987, p. 1. 7. This and some of the following observations about Lewin’s article originate with John Rahn. The mathematical terms that follow are defined on p. 27. 8. Lewin defines the terms “maximally uniform surjective voice leading,” “maximally uniform injective voice leading,” and “maximally uniform bijective voice leading,” none of which can properly be spoken of when thinking in terms of relations rather than functions (1998, 57–62). 9. Discussed on pp. 29–30 of this chapter. 10. The term “lyne,” familiar from atonal theory, shares some similarities to but also some differences with both the terms “voice” and “line.” Andrew Mead writes, “The term ‘lyne’ refers specifically to strings of pitch classes which have not yet been compositionally interpreted in the surface of the music” (“Recent Developments in the Music of Milton Babbitt,” The Musical Quarterly 70/3 (Summer 1984), p. 312). By this definition, a lyne can thus be interpreted as a precompositional construct. Mead notes in a footnote that “‘Lyne’ was first coined by Michael Kassler, and is cited in Milton Babbitt, ‘Responses: A First Approximation,’ PNM 14:2/15:1.” A line, or “part,” traces a more-or-less readily discerned melodic path in sounding music. Both lynes and voices (“voice” as defined by Straus) are constructs somewhat abstracted from the musical score; like a voice, a lyne is not necessarily dependent on registral ordering. However, a lyne is a precompositional theoretic ordering which may or may not take the form of a melody. This subtle distinction between “voice” and “lyne” is not made by most theorists and composers today, and to add to the confusion, Straus’s specialized use of the term “voice” is not widely accepted—most would interpret the generic term “voice” to mean “part,” e.g., soprano, bassoon, etc. Thus for many, “voice” would equal what Straus would call a “line,” and “lyne” would correspond roughly to Straus’s “voice.”

182

Notes

11. The most famous of the analyses belonging to the second category appears in Boretz 1972. 12. There is actually a slight change in timbre: the ’cello plays the D simultaneously with the English horn but drops out before the D9enters. 13. It is listed, implicitly, in the head sub-list, but this intermediate analytical stage is bypassed, with only the origin and target set-class types annotated. 14. It is, however, standard (and simple) Lisp programming language. 15. Straus’s inversional index In corresponds to the more familiar inversional index TnI. For example, the first transformation of Figure 2-7, which Straus labels as *I1 (2), would be labeled *T1I in “traditional” notation, and as *IC C by Lewin (after his practice in GMIT) and his followers. 16. Notice that *Tn and *In operations are not given in this example, but these can be cross-referenced in Figures 2-6 and 2-7. 17. Or more precisely, the convergence point of all the individual transpositions/ inversions occurring simultaneously during a near-transposition/near-inversion. See Straus 2003, 316.

Chapter 3 1. This is of course more easily done by reading/analyzing the score than by listening, but this too is a slow process and in general cannot be done in “real time” (or more accurately, in musical time, reading/analyzing/tracing the voice leading at the tempo indicated in the score). 2. This is a generalization of Lewin’s DOUTH1 relation, also known variously as P1, P1, or P1,0. See chapter 4 for details. 3. Gareth Cook, “Music’s inner map revealed, with some help from geometry,” The Boston Globe, July 31, 2006 (http://www.boston.com/news/science/articles/2006/07/ 31/musics_inner_map_revealed_with_some_help_from_geometry/); Alan Boyle, “The geometry of music,” MSNBC, July 7, 2006 (http://cosmiclog.msnbc.msn.com/ archive/2006/07/07/950.aspx); Roger Highfield, “In search of the last chord and winter colds,” The Daily Telegraph (London), July 18, 2006 (http://www.telegraph. co.uk/connected/main.jhtml?xml=/connected/2006/07/18/edcolds18.xml). 4. On the Science website at http://www.sciencemag.org/cgi/content/full/313/5783/72/ DC1. 5. See www.wikipedia.org, s.v. “Multiset.” 6. I have replaced Tymoczko’s parentheses with angle brackets to conform with the conventions used throughout this study. 7. See www.wikipedia.org, s.v. “Taxicab geometry.” 8. Parsimonious: “exhibiting economy of action, effort, or process” (Oxford English Dictionary, http://dictionary.oed.com). 9. They are parsimonious functions in fourth- or fifth-space, i.e., compositional space in which “stepwise” motion is by five or seven semitones and constitutes the norm for that particular space, just as stepwise motion is by one or two semitones in second-space, which is our usual way of contrasting “steps” and “leaps.” For a discussion of fourth/fifth-space in a piece by Benjamin Boretz, see John Rahn, “UN(-)

Notes

183

Ravelled, or, The Hidden Dragon” (Perspectives of New Music 43/2–44/1 (2005–2006): 39–69). 10. Webern’s fondness for ic 6 and Romantic composers like Schubert’s and Liszt’s use of third relations (ics 3 and 4) are only two examples that immediately come to mind. 11. I am very much indebted to John Rahn for suggesting direct-product groups as a possible way of formalizing single-voice transformations. Important precedents include Callender 2004, Tymoczko 2006 (both discussed earlier in this chapter), and especially Roeder 1994 (discussed near the end of chapter 2). 12. Note that here and throughout the study, I use the variable m to identify chord order positions (voices) and the variable n to identify transposition levels. The result of this is the frequent use of the term “m-tuple” to describe an ordering of chord voices. 13. John Rahn has pointed out that it is always possible to index a set in this way, given the Axiom of Choice (Zorn’s Lemma). 14. While the pc C is not necessarily privileged in any hierarchical way in the pitchclass organization of the pieces analyzed in this study, it has become common practice in post-tonal set theory to designate C = 0, solely for convenience’s sake, and thus I will continue to make use of this familiar convention. 15. After Callender 1998, 224. 16. See Straus 2003, 314; 317 n. 32. 17. Personal communication, September 2006. 18. See the discussion of Callender 1998, Shimbo 2001, Hook 2002, and Baker 2003 in chapter 5 for more on splits and fuses. 19. I.e., in this case, going from an m-tuple of cardinality 5 to one of cardinality 3, before making a final adjustment to obtain one of cardinality 4. 20. Tymoczko calls these “chord progression set classes.” See the discussion with the subject header “LPR as objects, not transformations” as posted on the SMT-talk listserv September 20–November 3, 2006, archived on the internet at the Society for Music Theory’s website, http://mbe224.music.utexas.edu/pipermail/smt-talk/2006 September/thread.html. 21. Note that classes /A/T and /B/T preserve order positions of their constituent mtuples, while classes /A/ and /B/ do not, because of permutational equivalence. 22. Callender 2004, 6, with the last sentence slightly re-written (in accordance with Callender’s footnote 15) for accuracy. In Callender’s assumption, ρ signifies “distance.” As will be seen below, there can be more than one distinct “distance” between two T-classes or Tn/TnI classes. 23. All distances are measured here using the city-block metric, unlike Callender, who uses the Euclidean metric. See pp. 39–40. 24. ¢B′ ² may in some cases correspond to the naming m-tuple ¢B² ∈ [B], but just as frequently it does not.

Chapter 4 1. Reproduced in HTML format at http://www.8ung.at/fzmw/1999/1999_1.htm. This is one of the last of many versions of the Tonnetz that Riemann crafted in the years

184

Notes

1882–1916 and is quite different from the earlier versions. The idea of the Tonnetz actually pre-dates Riemann and appears to have first been proposed by Leonhard Euler in 1739 (see Cohn 1997, 7 and 62–3, n. 11). Arthur von Öttingen modified Euler’s Tonnetz in his Harmoniesystem in dualer Entwicklung of 1866, a study which most certainly was known to Riemann. See Hyer 1995, 101 and 136 n. 1. 2. It should be noted that Riemann’s Tonnetz is a map of pitch space, not pitch-class space, as it is constructed to show pitch/chord relations in a just-intonation system. The grid of diamonds theoretically covers an infinite plane, expanding outward from the center in just perfect fifths along the horizontal dimension, just major thirds along the southwest-to-northeast axis, and just minor thirds along the northwest-tosoutheast axis. 3. Examples of different Tonnetze are found in Gollin 1998 (SC [0258] in 3 dimensions), and especially in Morris (1998), who constructs 2-dimensional Tonnetze for SCs [014] and [016] and describes the process of creating them for other compositional spaces. 4. Perhaps the most interesting use of the [037] Tonnetz is as a “road map” through cycles of L, P, and R transformations, both individually and in combination. See the publications of Richard Cohn listed in the references for an exhaustive study of these cycles. 5. L, P, and R are all inversion operations. 6. Morris 1998, 187. The R′ transformation is the obverse of R, inverting the triad (and thus flipping the triangle) about the pc that would normally move in an R operation. 7. Although graphs like Figure 4-2 can easily be converted into a Lewin transformational network by adding transformational labels such as T7, T4, and T3 between pc nodes and converting the connective lines to unidirectional arrows. 8. As the L, P, and R transformations are all their own inverses their arrows are bidirectional. 9. Douthett and Steinbach 1998, p. 247. 10. Douthett and Steinbach’s nomenclature derives from the Pn relation, first proposed in an unpublished 1996 paper by Douthett and later developed in Lewin 1996 and Childs 1998 (see Childs 1998, p. 192 n. 5 and Douthett and Steinbach 1998, pp. 243–4 and 261, n. 4). Douthett and Steinbach point out that Pm,n-related chords need not belong to the same set class, although in this case they do. See chapter 5 for more on this topic. 11. Here and elsewhere, the term “voice leading” to describe optimal offset between set classes should be taken with a grain of salt. SST-succession classes are not voiceleading transformations, as they are equivalence classes of SST-related m-tuple pairsuccessions. Although “voice leading” more accurately describes the effect of SSTs on m-tuples and “optimal offset” is a better description for each [SSTi] subscript appended to SST-succession-classes, I will occasionally use the former term in place of the latter, as there will always be some SSTi(¢A²) → ¢B′ ² in each SST-succession class ¢[A],[B]²SST where ¢A² names the equivalence class [A]. i 12. This property is identified by colored rings within the set-class circles in Figure 49. 13. Lewin 1996, 183.

Notes

185

14. I have made one correction to the graph, adding the line segment connecting SCs [0237] and [0148]. Apparently this link was inadvertently left out; in any case it would not have been possible to draw a straight line between the two set classes based on the graph’s layout. Optimal offset is achieved by the [SST1–1] relation when moving to the southeast along this line, and by the [SST1] relation when moving to the northwest. 15. Straus and I arrived at the 3D graphs of tetrachordal voice leading independently. His graph first appeared in November 2005 at the annual SMT meeting, and mine dates from August 2004, but appears to the public for the first time here. 16. The third, between SCs [0148] and [0347], retains its blue color as the SSTsuccession class is related by [SST4] in each direction. 17. A 3D graph of “bent” (non-orthogonal) triadic Tn-class (not Tn/TnI-class) space can be seen in Dmitri Tymoczko’s ChordGeometries software, available for download at http://music.princeton.edu/~dmitri/ChordGeometries.html. 18. Cohn discusses some of the implications of regarding corresponding set classes as non-equivalent, using some of John Roeder’s investigations as a point of departure (Cohn 2003, 13–14). 19. The only exception to this last point is SC [0137] in layer 5, which is Tn-class (0238) by our formula but is [0137] under Tn/TnI equivalency. We will also see a similar situation with the tetrachordal SCs of the inner (blue) disk of layer 5, where [0148] would have been (0348) were it not for Tn/TnI equivalency. Additionally, SC [0158] takes the place of [0358] in this disk, as members of the latter do not participate in any fuse/split functions involving trichords; thus [0358] needs not be represented in our next graph, which shows split/fuse-succession-class optimal offset relations only. 20. See Straus 2005 for a discussion of the maximally compact–maximally even continuum of set classes and motion along the spectrum according to what Straus calls the “Law of Atonal Harmony.” The same phenomenon is discussed in terms of orbifold geometry in Tymoczko 2006. 21. In the trichordal split-succession-class graph (Figure 4-18), as with the graph of tetrachordal SST-succession-class relations in Figure 4-12, tetrachordal SCs are represented by cubes, trichordal SCs are represented by spheres, and the [01] (or [02]) dyadic SC is represented by a cone for easy identification of set-class cardinality. 22. Although a fuse does indeed undo a split at the m-tuple level; see p. 52 of chapter 3. 23. See Figures 4-1 through 4-4 for graphical Tonnetz representations of these operations on set members of [037].

Chapter 5 1. The first to apply a transformational approach to Riemann’s ideas was David Lewin in “A Formal Theory of Generalized Tonal Functions” (1982). Lewin subsequently expanded upon concepts adumbrated in this early article in his GMIT, chapter 8 (1987). Other important early studies include Lewin 1992, Klumpenhouwer 1994, Hyer 1995, and a series of articles by Richard Cohn (1996, 1997, 1998, 1999, and

186

Notes

2000). 2. Here Lewin is making use of Riemann’s dualistic theoretical system, which he himself inherited from earlier German theorists Mortiz Hauptmann and Arthur von Öttingen. For a brief overview of Riemann’s dualism, see Kopp 2002, Ch. 4, especially pp. 61–9; see also Kopp’s take on Lewin’s Riemann systems, pp. 142–51. For an in-depth study of dualism and harmonic function, see Harrison 1994. 3. TMINV is an R operation only if it transforms a major Riemann system to a dual minor Riemann system and vice versa. 4. To be fair, Lewin does explore other operations such as RET and CONJ in his 1982 paper, and adds many more transformations in Lewin 1987 and later publications. 5. Lewin even makes use of direct-product transformations (pp. 37–45), where spaces S1 and S2 of two GISs correspond to different musical parameters, e.g., pitch classes in the case of the former and time-points in the case of the latter (with corresponding differences between IVLS1 and IVLS2 and also between int1 and int2). The principal difference between Lewin’s direct-product transformations and SVTs in the present study is that all transformations in the latter operate in the same (pitch-class) space S and employ the same group of transposition operations Gm. John Rahn makes a number of categorical distinctions between various types of Lewin transformational networks in the forthcoming article “Approaching Musical Actions” (Perspectives of New Music 45/2 (Summer 2007)). 6. Callender appears to be the first to use the term “split” in a transformational context. 7. This “mapping” of course is not a function as the splitting pitch class x in chord X of cardinality n “maps” to two separate pcs y1 and y2 in chord Y of cardinality n + 1. Nowhere does Callender provide a mathematical explanation of his split transformations. 8. This is an arbitrary limitation imposed by Callender to define “conjunct voice leading” and to avoid problems with set classes such as [013579] which contain a high proportion of whole steps. In SVT notation, a two-step move in a single voice would correspond to SVTi2. 9. More accurately, parallel motion, but Childs wisely avoids an over-proliferation of the sign “P,” which he already uses in the form Pn for “parsimony,” and also to avoid confusion with the neo-Riemannian triadic P (parallel) function. 10. This represents something of a departure from Hauptmann’s triadic Einheit–Zweiheit–Verbindung adaptation of Hegel’s dialectic, where the root equals I, the fifth equals II, and the third equals III. Had he followed Hauptmann’s lead, the pcs in a dominant seventh triad would be labeled (going from root to seventh by third) i–iii–ii–iv. 11. Bass 2001, 41 and 46, n. 18 12. However, this is not done according to Hauptmann’s practice. See note 10. 13. E.g., Cohn 1996, 1997; Callender 1998, Childs 1998, and Douthett and Steinbach 1998 to name only a few. Hook discusses the lack of correlation between voice leading and the algebraic structure of the transformation group on his p. 101. 14. For reasons that remain unexplained, Baker allows whole-step motion in the splitting voice in an SR transformation, whereas all his other transformations are limited to semitonal voice leading.

Notes

187

15. The leading number 1 in all functions refers to the single semitonal displacement. Baker does not discuss transformations that result in whole-step displacement (except for his SR function; see previous note). 16. See also the discussion on pp. 134–5. 17. After Carl Friedrich Weitzmann, Der übermässige Dreiklang (Berlin: T. Trautweinschen, 1853). See Cohn 2000 for more on this relation. 18. See Cohn 1996, p. 19 for more on hexatonic systems and hexatonic poles. 19. Hook 2002, p. 40. 20. Precedents for a voice-leading transformational approach were set by Childs (1998) and Bass (2001), but as can be seen from Table 5-3, these are limited in scope, applicable to a maximum of only nine transformations between representatives of SC [0258]. 21. Baker does not establish 1χn function labels for this operation as motion is by whole step in one of the split voices rather than by semitone. (Perhaps he should have called these 2χ1 and 2χ2?) 22. In fact triadic H operations retain no common tones even though all voice leading is by semitone. See Cohn 1996, p. 19. 23. The triadic N relation between embedded trichords is highlighted by bold pc letter names in the figure.

Chapter 6 1. See the discussion of voice vs. line in chapter 2, pp. 19–22. 2. For those seeking a more formal way of determining the most parsimonious voice leading between two chords, Dmitri Tymoczko’s dynamic programming algorithm (2006, supporting online material, pp. 13–4 and Figure S12, p. 23) may be of interest. This algorithm is not particularly well suited for the SVT theory, however, as it does not admit SFTs as transformations. 3. The SVT and SST-succession-class analyses are presented here as charts as there is not sufficient space for analytical markings on the score itself. The abbreviated SSTsuccession-class nomenclature ([A] [SSTi] [B] [SSTj] ...) discussed in chapter 3 (p. 60)—which stands for a series of interlocking SST-succession classes—will be adopted here. 4. The phrase is extended through repetition to measure 12. 5. The derivation of pc F as the result of a split from pc E is tenable if one notices the prominent pitch E4 one quarter note before m. 10. It is very easy to hear the E4 (and not the A4) leading to the F4 on the downbeat while E3 is maintained as a common tone in the left hand. 6. Notice here that the pc A leads to B, not to G as would be the norm in tonal voice leading. This provides us with a second pc B in the 4-tuple that opens measure 13, obviating a fuse operation, as Chopin continues with 4-pc chords thereafter. 7. As is also the case with the first SST-succession class in measure 10. 8. Possibly interpreted as a ii∅7 in A minor with a 4-3 suspension. 9. These are essentially the same progression once the multiset-class [0377] has been obtained.

188

Notes

10. After Douthett and Steinbach 1998, pp. 243–4 and 261, n. 4. See the discussion of their Pm,n relation on pp. 71–2 of my chapter 4. 11. It is most akin to an “inverted” French augmented sixth chord, which one would expect to lead to a triad with a root on A, but this does not happen. In any case, such a triad would be difficult to reconcile with the key of C minor. 12. An N transformation of this type is also clearly heard at the end of the Chopin prelude. 13. Brumbeloe’s designations, regardless of their accuracy, will be adopted here chiefly for the sake of continuity with his article. 14. The term “modal” in the title refers to the “12-tone modal system” developed by George Perle and Lansky in the years 1969–73 and does not imply “mode” in the traditional sense(s). 15. See pp. 22–3 of chapter 2 for discussion of these (David Lewin’s) terms. 16. See, for instance, the climactic passage between rehearsal numbers 35.5 and 38. 17. For a summary of Adams’s harmonic practice in works up to and including Nixon in China, see Timothy A. Johnson, “Harmonic Vocabulary in the Music of John Adams: A Hierarchical Approach,” Journal of Music Theory 37/1 (Spring 1993), pp. 117–56. 18. P2,0 transformations figure prominently throughout Fearful Symmetries. 19. See T. Johnson (1993) for a hierarchical systematization of Adams’s harmonic practice. 20. Henry Kissinger leaves midway through the act after asking Chou En-lai for directions to the toilet.

REFERENCES Baker, Scott. 2003. “Toward a Unified System for the Analysis of Parsimonious Voice Leading.” Paper presented at the annual meeting of the Rocky Mountain Society for Music Theory, Tucson, Ariz. Bass, Richard. 2001. “Half-Diminished Functions and Transformations in Late Romantic Music.” Music Theory Spectrum 23/1: 41–60. Benjamin, William E. 1981. “Pitch-Class Counterpoint in Tonal Music.” In Music Theory: Special Topics, ed. Richmond Browne, 1–32. Orlando: Academic Press. Boretz, Benjamin. 1972. “Meta-Variations, Part IV: Analytic Fallout (I).” Perspectives of New Music 11/1: 146–223. Boyle, Alan. 2006. “The geometry of music.” MSNBC. July 7. (http:// cosmiclog.msnbc. msn.com/archive/2006/07/07/950.aspx) Brumbeloe, Joe. 1986. “Pitch Structures in Webern’s Second Cantata, Op. 31.” Indiana Theory Review 7/2: 30–47. Callahan, Michael R. 2006. “Measuring K-Net Distance: Parallels Between Perle and Lewin, and a Generalized Representation of Sum-andDifference Space.” Paper presented at the annual meeting of the Society for Music Theory, Los Angeles, Calif. Callender, Clifton. 1998. “Voice-leading Parsimony in the Music of Alexander Scriabin.” Journal of Music Theory 42/2: 219–33. ———. 2004. “Continuous Transformations.” Music Theory Online 10/3. Childs, Adrian. 1998. “Moving Beyond Neo-Riemannian Triads: Exploring a Transformational Model for Seventh Chords.” Journal of Music Theory 42/2: 181–93. Clampitt, David. 1998. “Alternative Interpretations of Some Measures from Parsifal.” Journal of Music Theory 42/2: 321–34. Clough, John. 1998. “A Rudimentary Geometric Model for Contextual Transposition and Inversion.” Journal of Music Theory 42/2: 297–306. Cohn, Richard. 1996. “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions.” Music Analysis 15/

190

References

1: 9–40. ———. 1997. “Neo-Riemannian Operations, Parsimonious Trichords, and Their Tonnetz Representations.” Journal of Music Theory 41/1: 1–66. ———. 1998. “Square Dances with Cubes.” Journal of Music Theory 42/2: 283–96. ———. 1999. “As Wonderful as Star Clusters: Instruments for Gazing at Tonality in Schubert.” 19th-Century Music 22/3: 213–32. ———. 2000. “Weitzmann’s Regions, My Cycles, and Douthett’s Dancing Cubes.” Music Theory Spectrum 22/1: 89–103. ———. 2003. “A Tetrahedral Graph of Tetrachordal Voice-Leading Space.” Music Theory Online 9/4. Cook, Gareth. 2006. “Music’s inner map revealed, with some help from geometry.” The Boston Globe. July 31. (http://www.boston.com/news/ science/articles/2006/07/31/musics_inner_map_revealed_with_some_ help_from_geometry/) Cope, David. 2002. “Computer Analysis and Composition Using Atonal Voice-Leading Techniques.” Perspectives of New Music 40/1: 126–46. Douthett, Jack and Peter Steinbach. 1998. “Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition.” Journal of Music Theory 42/2: 241–63. Dummit, David S. and Richard M. Foote. 2005. Abstract Algebra. 2d ed. New York: Wiley. Fiore, Thomas M. and Ramon Satyendra. 2005. “Generalized Contextual Groups.” Music Theory Online 11/3. Forte, Allen. 1988. “New Modes of Linear Analysis.” Paper presented at the Oxford University Conference on Music Analysis. Gollin, Edward. 1998. “Some Aspects of Three-Dimensional Tonnetze.” Journal of Music Theory 42/2: 195–206. Harrison, Daniel. 1994. Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of its Precedents. Chicago: University of Chicago Press. Hermann, Richard. 2005. “Parsimonious Equivalence-Classes for VoiceLeading Between Maximally Even and Near Maximally Even SetClasses.” Paper presented at the Colloquium on Transformational Theory

References

191

and Neo-Riemannian Analysis, IRCAM/Univ. Sorbonne, Paris. Highfield, Roger. 2006. “In search of the last chord and winter colds.” The Daily Telegraph (London). July 18. (http://www.telegraph.co.uk/ connected/main.jhtml?xml=/connected/2006/07/18/edcolds18.xml) Hook, Julian. 2002. “Uniform Triadic Transformations.” Ph.D. diss., Indiana University. Hyer, Brian. 1995. “Reimag(in)ing Riemann.” Journal of Music Theory 39/ 1: 101–38. Johnson, Mark. 1999. “Harmonic Design in the ‘Scherzo’ of John Adams’s The Death of Klinghoffer.” Paper presented at the Pacific Northwest Music Graduate Students Conference, State University of British Columbia, Vancouver. Johnson, Timothy A. 1993. “Harmonic Vocabulary in the Music of John Adams: A Hierarchical Approach.” Journal of Music Theory 37/1: 117–56. Jurkowski, Edward. 1998. “A Theory of Harmonic Structure and Voice Leading for Atonal Music.” Ph.D. diss., University of Rochester, Eastman School of Music. Kassler, Michael. 1967. “A Trinity of Essays: Toward a Theory that Is the Twelve-Note Class System; Toward Development of a Constructive Tonality Theory Based on Writings by Heinrich Schenker; Toward a Simple Program Language for Musical Information Retrieval.” Ph.D. diss., Princeton University. ———. 1975. Proving Musical Theorems I: The Middleground of Heinrich Schenker’s Theory of Tonality. Sydney: University of Sydney, Basser Department of Computer Science. Keiler, Allan. 1979. “Syntax of Prolongation.” In Theory Only 3/5: 13–27. Klumpenhouwer, Henry. 1991. “A Generalized Model of Voice-Leading for Atonal Music.” Ph.D. diss., Harvard University. ———. 1994. “Some Remarks on the Use of Riemann Transformations.” Music Theory Online 0/9. Kochavi, Jonathan. 1998. “Some Structural Features of ContextuallyDefined Inversion Operators.” Journal of Music Theory 42/2: 307–20. Kopp, David. 2002. Chromatic Transformations in Nineteenth-Century Music. Cambridge Studies in Music Theory and Analysis 17. Cambridge:

192

References

Cambridge University. Lerdahl, Fred and Ray Jackendoff. 1983. A Generative Theory of Tonal Music. Cambridge, Mass.: MIT Press. Lewin, David. 1982. “A Formal Theory of Generalized Tonal Functions.” Journal of Music Theory 26/1: 32–60. ———. 1982–3. “Transformational Techniques in Atonal and Other Music Theories.” Perspectives of New Music 21: 312–71. ———. 1987. Generalized Musical Intervals and Transformations. New Haven: Yale University Press. ———. 1990. “Klumpenhouwer Networks and Some Isographies that Involve Them.” Music Theory Spectrum 12/1: 83–120. ———. 1992. “Some Notes on Analyzing Wagner: The Ring and Parsifal.” 19th-Century Music 16/1: 49–58. ———. 1994. “A Tutorial on Klumpenhouwer Networks, Using the Chorale in Schoenberg’s Opus 11, No. 2.” Journal of Music Theory 38/1: 79–101. ———. 1996. “Cohn Functions.” Journal of Music Theory 40/2:181–216. ———. 1998. “Some Ideas about Voice-Leading Between Pcsets.” Journal of Music Theory 42/1: 15–72. ———. 2002. “Thoughts on Klumpenhouwer Networks and Perle-Lansky Cycles,” Music Theory Spectrum 24/2: 196–230. Mead, Andrew. 1984. “Recent Developments in the Music of Milton Babbitt.” The Musical Quarterly 70/3: 310–31. Morris, Robert. 1998. “Voice-Leading Spaces.” Music Theory Spectrum 20/ 2: 175–208. O’Donnell, Shaugn J. 1997. “Transformational Voice Leading in Atonal Music.” Ph.D. diss., City University of New York. Öttingen, Arthur von. 1866. Harmoniesystem in dualer Entwicklung. Dorpat: Gläser. Rahn, John. 1979. “Logic, Set Theory, Music Theory.” College Music Symposium 19/1: 114–24. ———. 1980. Basic Atonal Theory. New York: Macmillan. ———. 2005–6. “UN(-) Ravelled, or, The Hidden Dragon.” Perspectives of

References

193

New Music 43/2–44/1: 39–69. ———. 2007. “Approaching Musical Actions.” Perspectives of New Music 45/2 (Summer, forthcoming). Riemann, Hugo. 1880. Skizze einer neuen Methode der Harmonielehre. Leipzig: Breitkopf & Härtel. ———. 1914–15. “Ideen zu einer ‘Lehre von den Tonvorstellungen’.” Jahrbuch der Bibliothek Peters 21–2: 1–26. Roeder, John. 1984. “A Theory of Voice-Leading for Atonal Music.” Ph.D. diss., Yale University. ———. 1994. “Voice Leading as Transformation.” In Musical Transformation and Musical Intuition: Essays in Honor of David Lewin, ed. Raphael Atlas and Michael Cherlin, 41–58. Dedham, Mass.: Ovenbird Press. Schoenberg, Arnold. 1978. Theory of Harmony, trans. Roy E. Carter. Berkeley: University of California Press. Shimbo, Amy. 2001. “Some Transformations Between Triads and Seventh Chords.” Paper presented at the Symposium on Neo-Riemannian Theory, State University of New York at Buffalo. Smoliar, Stephen. 1980. “A Computer Aid for Schenkerian Analysis.” Computer Music Journal 4/2: 41–59. Snell, James. 1984. “Computerized Hierarchical Generation of Tonal Compositions.” In Informatique et musique, ed. H. Charnass. Ivry: Elmeratto/ CNRS. Soderberg, Stephen. 1998. “The T-Hex Constellation.” Journal of Music Theory 42/2: 207–18. Straus, Joseph N. 1997. “Voice Leading in Atonal Music.” In Music Theory in Concept and Practice, ed. David W. Beach, James M. Baker, and Jonathan W. Bernard, 237–74. Rochester: University of Rochester Press. ———. 2003. “Uniformity, Balance, and Smoothness in Atonal Voice Leading.” Music Theory Spectrum 25/2: 305–52. ———. 2005. “Atonal Pitch Space.” Paper presented at the annual meeting of the Society for Music Theory, Boston, Mass. Tymoczko, Dmitri. 2004. “Semitonal Voice-Leading Among Tertian Sonorities.” Paper presented at the annual meeting of the Society for Music Theory, Seattle, Wash.

194

References

———. 2005. “Voice Leadings as Generalized Key Signatures.” Music Theory Online 11/4. ———. 2006. “The Geometry of Musical Chords.” Science 313/5783: 72–4. Supporting online material found at http://www.sciencemag.org/cgi/ content/full/313/5783/72/DC1. Weitzmann, Carl Friedrich. 1853. Der übermässige Dreiklang. Berlin: T. Trautweinschen. Yust, Jason. 2006. “Formal Models of Prolongation.” Ph.D. diss., University of Washington.

INDEX

A Adams, John 8, 136, 158–9, 168, 188 China Gates 159 The Death of Klinghoffer 160 Fearful Symmetries 162–3, 188 Grand Pianola Music 159 Nixon in China 159–71, 188 On the Transmigration of Souls 160 Shaker Loops 160 Andriessen, Louis 158 B Babbitt, Milton 17, 151, 181 Baker, Scott 118, 122–5, 128–9, 131, 146, 169, 183, 186–7 balance, voice-leading 24–7 Bass, Richard 117, 146, 163, 186–7 Beethoven, Ludwig van 22 Benjamin, William E. 180 Boretz, Benjamin 180, 182 Boyle, Alan 182 Bruckner, Anton 2 Brumbeloe, Joe 151–2, 188 C Callahan, Michael 3, 13–4 Callender, Clifton 4–6, 39–40, 48, 51, 55, 57, 61, 76, 110–1, 114, 118, 123, 131, 171, 181, 183, 186 “Continuous Transformations” 36–7 chicken-wire torus 70, 73, 75, 90 Childs, Adrian 6, 110–4, 117, 122, 134, 145–6, 163, 169, 184, 186–7 Chopin, Frédéric 7, 115, 136, 147, 149, 156, 163, 187–8 Prelude in e minor, Op. 28 No. 4 140–7 Clampitt, David 30 Cohn function 81–2, 108, 174 Cohn, Richard 1, 5–6, 14, 38, 81, 89–91, 104–5, 117, 129, 131, 146, 172,

196

Index

184–7 Cook, Gareth 182 Cope, David 3, 28, 35 “Computer Analysis and Composition Using Atonal Voice-Leading Techniques” 23–4 Corigliano, John 124 cube dance 38, 72–3, 75 Curran, Alvin 158 D Debussy, Claude 72, 104 direct-product group 4, 41–3, 183, 186 DOUTH1 108–10, 114, 182 DOUTH2 109–11, 113–4 DOUTHCF2 109–10 Douthett, Jack 6, 38, 70–5, 81, 109–11, 113–5, 117, 123–4, 129, 131, 133–4, 146, 163, 169, 180, 184, 186, 188 Dresher, Paul 158 dualism 107, 110, 115, 119, 125, 129–31, 186 Duckworth, William 158 duplicate paths 5, 45, 78, 79–80, 87–8, 91, 98, 100, 102 E EnneaCycles 73–5, 114 Euclidean metric 36, 39–40, 183 Euler, Leonhard 184 F Fiore, Thomas M. 180 Forte, Allen 2, 16, 124 Franck, César 72 fuse 5, 30, 39, 47–55, 64, 78, 83, 121, 125, 128, 139–40, 142, 149, 154, 172–4, 181, 183, 185, 187 fuse-succession class 5, 62, 64–5, 91–2, 94–8, 100, 128–9, 144, 164, 172, 174, 177–8, 185 “fuzzy” transposition and inversion 3, 24–6

Index

197

G geometric dual 69–70, 75 Giteck, Janice 158 Glass, Philip 158–9 Akhnaten 159 Candyman 159 Satyagraha 158–9 Songs from Liquid Days 159 Gollin, Edward 6–7, 67, 110, 115–7, 119–20, 125, 145–6, 184 Gubaidulina, Sofia 22 H Harrison, Daniel 186 Hauptmann, Moritz 115, 119, 186 Headlam, David 181 Hegel, Georg Wilhelm Friedrich 186 hexatonic pole (H) 129 Highfield, Roger 182 Hook, Julian 6, 8, 105–6, 118, 120–1, 124, 128, 130, 145–6, 183, 186–7 Hyer, Brian 1, 125, 184–5 I inclusion relation 121, 131, 135–6 J Jackendoff, Ray 180 Johnson, Mark 160 Johnson, Timothy A. 188 Jurkowski, Edward 3 “A Theory of Harmonic Structure and Voice Leading in Atonal Music” 14 K Kassler, Michael 180–1 Keiler, Allan 180 Kernis, Aaron Jay 124 Klumpenhouwer network 1, 12–4, 16, 180–1 Klumpenhouwer, Henry 1, 3, 11, 14–5, 17, 19, 28–30, 33, 42, 47, 104, 116,

198

Index

120, 128–9, 185 “A Generalized Model of Voice-Leading for Atonal Music” 12–3 Kochavi, Jonathan 7 Kopp, David 106, 129–30, 146, 186 Chromatic Transformations in Nineteenth-Century Music 130 L Lambert, Philip 181 Lansky, Paul 7, 124, 163, 188 Modal Fantasy 137, 154–8 Lentz, Daniel 124 Lerdahl, Fred 180 Lewin, David 1–3, 6, 11, 14, 16, 28, 30, 69, 81, 105–11, 114, 117, 124, 129, 152, 159, 180–2, 184–6, 188 “A Formal Theory of Generalized Tonal Functions” 106 “Some Ideas about Voice-Leading Between Pcsets” 17–9, 22 Generalized Musical Intervals and Transformations 1, 8–9, 108, 121 line, musical 19–23, 31–2, 42, 108, 138–9, 160, 181, 187 Liszt, Franz 72, 82, 183 lyne 181 M manifest voice leading 22, 152, 158, 160 maximally smooth cycle 81 Mead, Andrew 181 minimalism 8, 139, 158–60, 163 Moran, Robert 158 Morris, Robert 3, 5, 68, 81, 146, 184 N near-inversion 3, 15–6, 24, 29–30, 48, 182 near-transposition 3, 15–6, 24, 29–30, 48, 155, 157, 182 Nebenverwandt (N) 129 neo-Riemannian theory 1–2, 6, 26, 30, 35–6, 59–61, 66, 69, 81–2, 102, 104–10, 114–5, 117, 119–25, 128–36, 139, 144–7, 150, 154, 157–60, 163–4, 169, 172, 186 network graph 5, 9, 13, 27, 62, 66, 69–70, 72–9, 82–102, 108, 111–4, 117, 164, 172, 184–5 Nyman, Michael 158

Index

199

O O’Donnell, Shaugn 3, 14, 28, 30, 47–8, 111, 116, 120, 180–1 “Transformational Voice Leading in Atonal Music” 15–7 OctaTowers 73, 75, 180 optimal offset 27, 62, 76–8, 80, 82–5, 89, 91, 95, 98, 102, 172, 184–5 Öttingen, Arthur von 184, 186 P parsimony, voice-leading 40, 42–5, 58, 62, 66, 68, 71–6, 82, 90–1, 95, 109–12, 114, 116–7, 125, 129, 131, 134, 138, 145, 149, 151–2, 158–60, 167–8, 171, 180, 187 Perle, George 13, 188 pipeline torus 71–3, 75, 90, 111, 113, 163 power towers 74–5, 114 projected voice leading 23, 152, 158 R Rahn, John 8, 48, 139, 180–3, 186 Reich, Steve 158–9 Riemann, Hugo 6, 105–7, 128, 183–6 “Ideen zu einer ‘Lehre von den Tonvorstellungen’” 66–7 Skizze einer neue Methode der Harmonielehre 104 Riley, Terry 158 Rochberg, George 124 Roeder, John 1, 3, 15, 19, 28, 33, 35, 36, 39, 42, 171, 183, 185 “Voice Leading as Transformation” 29–30 S Saariaho, Kaija Vers le blanc 36 Satyendra, Ramon 180 Schenker, Heinrich 1, 11, 106, 180 Schenkerian analysis 2, 11, 15, 35, 180 Schillinger, Joseph 11 Schoenberg, Arnold 2, 22 Schubert, Franz 183 Scriabin, Alexander 7, 17, 72, 104, 110, 115, 136

200

Index

Etude in c-sharp minor, Op. 42 No. 5 147–50 Preludes, Op. 74 147 SFT-succession class 62 Shimbo, Amy 7, 118–9, 121, 123–4, 131, 146, 169, 183 single-fifth transformation (SFT) 2, 44, 46, 109, 121, 138, 155, 171, 187 single-semitone transformation (SST) 2, 43–6, 109, 148–51, 154, 171 single-voice transformation (SVT) 2, 42–4, 46, 106, 124–5, 171–2, 183, 187 SLIDE 108, 129, 132, 134, 147, 158–9, 161–4 Smoliar, Stephen 180 smoothness, voice-leading 26–9, 32, 34–5 Snell, James 180 Soderberg, Stephen 110 split 5, 30, 39, 47–52, 55, 62–4, 78, 110, 118–9, 121, 123, 125, 131, 136, 139, 149, 154, 163, 172–4, 181, 183, 185–7 split-succession class 5, 62–4, 82, 91–102, 131, 144, 163, 172, 174, 185 SST-succession class 4, 55, 58–62, 76, 78–80, 82–9, 91–2, 95–6, 98–102, 111–5, 117, 125–35, 139, 142–3, 145–6, 149–51, 153–7, 160, 162–4, 166–9, 171–2, 177–8, 184–5, 187 Steinbach, Peter 6, 38, 70–5, 81, 110–1, 113–5, 117, 123–4, 129, 131, 133–4, 146, 163, 169, 180, 184, 186, 188 Stockhausen, Karlheinz 104 Stoecker, Philip 181 Straus, Joseph 1, 3, 5, 9, 11, 14–7, 19–20, 23–8, 30, 33, 47–8, 56, 76–9, 82–5, 87, 89, 91–2, 97, 102, 116, 120, 155, 172, 181–3, 185 Stravinsky, Igor 22, 115 SVT group 41, 43, 45 T “taxicab” metric 36, 39–40, 182–3 Tonnetz 5, 9, 66–9, 75–6, 81, 87, 115–7, 183–5 Torke, Michael 158 towers torus 73–5, 114 Tymoczko, Dmitri 4, 36, 55, 171, 182–3, 185, 187 “The Geometry of Musical Chords” 37–8 U uniform triadic transformation (UTT) 12–1, 130, 146 uniformity, voice-leading 24–7, 181

Index

V voice, musical 19–22, 29, 31–2, 42, 108, 138, 181, 183, 187 W Wagner, Richard 72, 82, 115 Tristan und Isolde, Prelude 21–2, 31 Webern, Anton 7, 33, 151, 183 Cantata No. 2, Op. 31 137, 151–4 Five Movements for String Quartet 19–20 Weitzmann regions 38, 73 Weitzmann, Carl Friedrich 187 Y Young, La Monte 158 Yust, Jason 180 Z Zorn’s Lemma 183

201