a theory of voice-leading sets for post-tonal music

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A Theory of Voice-leading Sets for Post-tonal Music by Justin Lundberg Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Supervised by Professor David Headlam Department of Music Theory Eastman School of Music University of Rochester Rochester, NY 2012

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Page 1: A Theory of Voice-leading Sets for Post-tonal Music

A Theory of Voice-leading Sets for Post-tonal Music

by

Justin Lundberg

Submitted in Partial Fulfillment

of the

Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor David Headlam

Department of Music Theory

Eastman School of Music

University of Rochester

Rochester, NY

2012

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ii

Curriculum Vitae

Justin Lundberg was born in Eau Claire, Wisconsin on May 4th

, 1982. He received a

Bachelor of Music in Music Theory from the University of Wisconsin-Eau Claire in

2004, where he was a national merit scholar. In 2006 he received a Master of Arts in

Music Theory from the Pennsylvania State University, where he was a Graham

fellowship recipient. His Master’s thesis, on sonata form in three late works by

Joseph Haydn, was supported by a summer residency at the Institute for the Arts and

Humanities. He began pursuing the Doctor of Philosophy in Music Theory at the

Eastman School of Music in 2006, and received a Raymond N. Ball dissertation

fellowship in 2010-2011.

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Acknowledgements

I wish to thank my friends and family for their support during the preparation of this

dissertation, as well as throughout my graduate studies. In addition, I am grateful to

my professors and classmates at Eastman who offered their encouragement, criticism,

comments, and suggestions during various stages of my work. This dissertation was

also supported in part by a Raymond A. Ball fellowship. I wish to thank the scholars

to whom the ideas presented in this dissertation are most indebted: John Roeder,

Joseph Straus, and Dmitri Tymoczko. Finally, I would like to thank my advisor,

Dave Headlam. I first began to explore the subject of post-tonal voice leading in

earnest in an independent study with him in the summer of 2007, and am grateful for

the enthusiasm, support, and patience he continues to provide.

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Abstract

This dissertation develops tools for the analysis and representation of voice leading in

post-tonal music. A voice leading is defined as an ordered series of mappings of one

pitch-class set onto another. Because voice leadings are represented by ordered sets,

they are independent of the harmonic structure of the pitch-class sets they span.

Further, each mapping in a voice-leading set represents an individual pitch-class

voice. The dissertation presents a theory of voice leading that first asserts that voice-

leading sets may be organized into set-classes under transposition, inversion, and/or

rotation, then, following a discussion of the groups formed by voice-leading sets,

demonstrates the construction of voice-leading spaces based on individual voice-

leading sets, as well as reduced spaces which represent voice-leading classes. A

number of distinct spaces are explored, with an emphasis on the complete spaces, that

is, spaces which include all voice-leading sets or classes in a given cardinality.

Selected analyses demonstrate that the diverse pitch and intervallic material of post-

tonal pieces may be unified by the consistent use of a small number of voice-leading

classes. The voice-leading spaces formed by these classes may then be used to model

the voice-leading transformations of specific pieces. The dissertation concludes by

suggesting a few avenues of future research.

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Table of Contents

Chapter 1 Introduction 1

Introduction to Voice-leading Sets 3

Background 9

Chapter 2 Voice-leading Set Theory 19

Contrapuntal Motion in Pitch-class Space 22

Voice-leading Set-classes 26

Voice-leading Group Theory 33

Group Representations 39

Chapter 3 Voice-leading Spaces 44

Dyadic Spaces 47

Trichordal Spaces 65

Chapter 4 Analysis 74

Arnold Schoenberg’s Op. 11 n. 1 79

Anton Webern’s Op. 5 n. 2 82

Anton Webern’s Op. 5 n. 3 89

Alban Berg’s Op. 5 n. 1 96

Alban Berg’s Op. 5 n. 2 101

Chapter 5 Summary, Conclusion, and Future Work 106

Bibliography 221

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List of Tables

Table 2-1 Subgroups of the group of dyadic vlsets 138

Table 2-2 Group table for the subgroup generated by <17> 139

Table 2-3 Subgroup <02> with inversion 141

Table 2-4 Examples of four-member subgroups of [C12 C12] C2 142

Table 2-5 Examples of two-member subgroups of [C12 C12] C2 143

Table 2-6 C12 C12 subgroups combined with inversional vlsets 143

Table 2-7 Group table for {<04>, <40>} + <61>i 146

Table 3-1 Dyadic vlset spaces 164

Table 3-2 Trichordal voice-leading spaces 177

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List of Figures

Figure 1-1 Mm. 11-13 of Arnold Schoenberg’s Op. 11 n. 2 119

Figure 1-2a Fuzzy transposition as in Straus 2003 119

Figure 1-2b Pitch voice leadings as in Roeder 1994 119

Figure 1-2c Vlsets from chord a to b and d to e 120

Figure 1-3 Values for the moving voice when retaining two common tones 121

Figure 1-4a The trichordal orbifold 122

Figure 1-4b One-quarter of the trichordal orbifold 123

Figure 2-1 Number of vlsets by cardinality 124

Figure 2-2 Vlsets spanning reordered pcsets 125

Figure 2-3 Parallel pitch-class voice leading 126

Figure 2-4 Oblique pitch-class voice leading 126

Figure 2-5 Contrary pitch-class voice leading 126

Figure 2-6 Similar pitch-class voice leading 127

Figure 2-7 Vlset transposition 127

Figure 2-8a Mm. 1-13 of Arnold Schoenberg’s Op. 11 n. 1 128

Figure 2-8b Vlclass [002] between ordered trichords 128

Figure 2-9 Ordered pcset tn-classes paired by ordered vlclasses 129

Figure 2-10 Ordered tn-class cycles 131

Figure 2-11 An ordered tn-class [04] tn-class cycle 132

Figure 2-12 Ordered pcset and tn-class cycles formed by vlset <14> 132

Figure 2-13 Transpositions of vlset <01>i and <0e>i 133

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Figure 2-14 Ordered tn-class paired by ordered vlclass [01]i and [10]i 134

Figure 2-15 Ordered pcset-classes paired by ordered vlclasses [04]i and [40i] 135

Figure 2-16 Tn-class pairs 136

Figure 2-17 Ordered tn-classes paired by ordered vlclass [025] 137

Figure 2-18 Subgroup <17> with three sets of pcsets as objects 140

Figure 2-19 Cyclic groups and rotational symmetry, C3: T0, T4, T8 148

Figure 2-20 Dihedral groups, D3: T0, T4, T8, In, In+4, In+8 148

Figure 2-21 T/I group representation 149

Figure 2-22 The dodecagonal torus 150

Figure 2-23 Interlocking dodecagons 151

Figure 2-24 Vlset <t27> on interlocking dodecagons 152

Figure 3-1 Straus’ Trichordal set-class space 153

Figure 3-2a The dyadic orbifold 154

Figure 3-2b Dyadic permutational equivalence 154

Figure 3-2c Dyadic Moebius strip 155

Figure 3-3 Torus knot of singleton dyads 156

Figure 3-4a Moebius strip dyadic orbifold 157

Figure 3-4b Dyadic set-class line 158

Figure 3-4c Dyadic tn-class circle 158

Figure 3-5a <05> space 158

Figure 3-5b <05> Moebius strip 159

Figure 3-5c Adjacent voice leadings along the <05> Moebius strip 160

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Figure 3-5d Adjacent and compound moves along the <05> Moebius strip 160

Figure 3-6a <05> tn-class circle 161

Figure 3-6b <05> set-class line 161

Figure 3-7a <2e>, {00} toroidal voice-leading space 162

Figure 3-7b <2e>, {00} toroidal voice-leading space 163

Figure 3-8 <02>, {00} toroidal voice-leading space 165

Figure 3-9 <02>, {11} toroidal voice-leading space 165

Figure 3-10 <02>, {01} toroidal voice-leading space 166

Figure 3-11 <02>, {10} toroidal voice-leading space 166

Figure 3-12 <02>, {11} Moebius strip 167

Figure 3-13 Vlset <02> spaces 168

Figure 3-14 <68>, {00} toroidal voice-leading space 168

Figure 3-15 <67>, {00} toroidal voice-leading space 169

Figure 3-16 <67> Moebius strip 170

Figure 3-17 <12>, {47} toroidal voice-leading space 171

Figure 3-18a <42>, {11} toroidal voice-leading space 171

Figure 3-18b <42>, {11} toroidal voice-leading space 172

Figure 3-19 <12>, {47} Moebius strip 173

Figure 3-20 <42>, {11} permutation space 174

Figure 3-21 <06>, {11} permutation space 174

Figure 3-22 <06>, {01} voice-leading space 174

Figure 3-23 <57>, {02} permutation space 174

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Figure 3-24 <26>, {11} toroidal voice-leading space 174

Figure 3-25 <001>, {000} hypercube 175

Figure 3-26 The trichordal orbifold 176

Figure 3-27a Vlclass [047] ordered tn-class space 179

Figure 3-27b Motion among adjacent ordered tn-classes in [047] ordered

tn-class space 179

Figure 3-28 Vlclass [047] ordered set-class space 180

Figure 3-29 Examples of complete ordered set-class spaces 181

Figure 3-30 Vlclass [002] ordered set-class spaces 184

Figure 4-1a <442>, {000} voice-leading space 185

Figure 4-1b <442>, {001} voice-leading space 186

Figure 4-1c Vlclass [002] whole-tone ordered set-class space 187

Figure 4-1d Vlclass [002] mixed whole-tone ordered set-class space 188

Figure 4-1e Ordered set-class interpretations from Figure 2-8 189

Figure 4-2a [026] in Webern’s op. 5 n. 2 190

Figure 4-2b Network of [026]s in Webern’s op. 5 n. 2 190

Figure 4-3 Melodic statements of vlclass [043] in Webern’s op. 5 n. 2 191

Figure 4-4 Network of melodic vlsets in Webern’s op. 5 n. 2 192

Figure 4-5 Vlclass [043] in the accompaniment in Webern’s op. 5 n. 2 193

Figure 4-6 Vlclass [043] transformation graph 195

Figure 4-7a Webern’s op. 5 n. 3, mm. 1-4 196

Figure 4-7b Vlset <t16> mappings 196

Figure 4-8 Variations of A 197

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Figure 4-9 B, mm. 9-10 198

Figure 4-10a [038] voice-leading space 199

Figure 4-10b Melody B’s complete voice-leading {2t1} – {0et} – {t59} 200

Figure 4-11a Reduction of opening [034]s in Violins and Viola, mm. 1-3 201

Figure 4-11b Reduction of Violin II, Viola, and Cello, mm. 15-17 201

Figure 4-12 Berg’s op. 5 n. 1 opening clarinet gesture 202

Figure 4-13a Berg’s op. 5 n. 1 clarinet melody 203

Figure 4-13b Vlclass [002] ordered set-class space 204

Figure 4-14 Vlclass [022] in m. 2 LH 205

Figure 4-15 Vlclass [022] in m. 2 RH 206

Figure 4-16 Opening clarinet gesture in the accompaniment, m. 9 206

Figure 4-17 Accompaniment, mm. 3-5 207

Figure 4-18a Vlclass [002] in mm. 5-6 208

Figure 4-18b Vlclass [002] in m. 7 208

Figure 4-19 Berg’s op. 5 n. 2 209

Figure 4-20 Berg’s op. 5 n. 2 melody 210

Figure 5-1 Pentachordal vlsets in Berg’s op. 5 n. 2 211

Figure 5-2 Normal order interpretation of Berg’s op. 5 n. 2 melody 213

Figure 5-3a Vlsets on the trichordal orbifold 214

Figure 5-3b Vlsets on the trichordal orbifold 215

Figure 5-3c Vlsets on the [013] ordered set-class space 216

Figure 5-4 Hyper-vlset graph 217

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Figure 5-5 Alban Berg’s Lyric Suite, VI, mm. 1-6 218

Figure 5-6 Lyric Suite, VI, mm. 39-46 219

Figure 5-7 4-3 suspension as diatonic vlset 220

Figure 5-8 Transposition of 4-3 diatonic vlset 220

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Chapter 1: Introduction

The study of voice leading has long been a prominent concern of music

theory. Although there are many differences between theories of voice leading in

tonal and post-tonal music, almost all emphasize either or both the harmonic

constraints and the minimal melodic distances available. In Schenkerian theory,

analytically-defined voices demonstrate the prolongation of tonic harmony at multiple

levels of structure. Neo-Riemannian theory focuses on harmonic successions which

have been described by Richard Cohn (and others) as parsimonious, or smooth voice

leadings; this voice-leading smoothness is inextricably linked with the harmonic

structure of tonal materials—triads and seventh chords.1

In post-tonal theory, studies of voice-leading typically focus on harmonic

similarity, as in the transformational voices formed by transposition and inversion, or

on total interval-class displacement.2 The more recent geometrical approach to voice

leading, as shown in the work of Clifton Callender, Ian Quinn, and Dmitri Tymoczko,

creates “voice-leading spaces,” by mapping pitch or pitch-class sets onto dimensional

coordinates with the semitone as a metric.3 This mapping produces minimal voice

1 “Maximally Smooth Cycles, Hexatonic Systems and the Analysis of Late-Romantic

Triadic Progressions” in Music Analysis, 15, n. 1 (1996): 9-40 and “Neo-Riemannian

Operations, Parsimonious Trichords and Their Tonnetz Representations” in Journal of

Music Theory 41, n. 1 (1997): 1-66. 2 See, for example, John Roeder’s “A Theory of Voice Leading for Atonal Music,”

(Ph.D. Diss., Yale University, 1984) and Joseph Straus’ “Uniformity, Balance, and

Smoothness in Atonal Voice Leading” in Music Theory Spectrum, 25, n. 2 (2003):

305-352. 3 Clifton Callender, Ian Quinn, and Dmitri Tymoczko “Generalized Voice-leading

Spaces” in Science, 320, (2008): 346-348, and Dmitri Tymoczko “The Geometry of

Musical Chords” in Science, 313 (2006): 72-74. Robert Morris “Voice-leading

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leading between adjacent chords. In this dissertation, however, instead of basing the

metric for voice leading on a fixed unit, voice-leading distances will be redefined as

contextual distances and relative measures. This transformational approach to voice

leading follows David Lewin’s more generalized notions of “interval” as some

distance between two points in a musical space.4

The approach to post-tonal music in this dissertation defines voice leadings as

ordered sets of individual pitch-class mappings from one pitch-class set to another.

Although the possible voice-leading interpretations between two chords are

constrained by their pitch-class content, no particular interpretation is privileged a

priori. Voice leadings are thus extracted from their harmonic contexts and examined

on their own terms. Individual voice-leading sets are used to generate alternative

voice-leading spaces, in which the unit distance or metric is the generating set rather

than semitonal offset. The analytical goal of this theory is to define pitch-class voices

in post-tonal pieces. In the analyses shown here, voice-leading sets are used to unify

the pitch and intervallic diversity that characterizes the surface of many post-tonal

pieces. These motivic voice-leadings are then represented by motion within some

contextually-generated voice-leading space. After a brief introduction to voice-

leading sets, this chapter continues with a survey of secondary literature. Chapter 2 is

Spaces” in Music Theory Spectrum, 20, no. 2 (1998): 175-208. The authors’ build on

earlier work by Robert Morris and John Roeder, see Morris’ “Voice-leading Spaces”

in Music Theory Spectrum, 20, no. 2 (1998): 175-208 and Roeder’s “A Geometric

Representation of Pitch-class Series” in Perspectives of New Music, 25, no. 1/2

(1987): 362-409. 4 See, for example, Generalized Musical Intervals and Transformations (New Haven:

Yale University Press, 1987, reprint New York: Oxford University Press, 2007): 16.

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devoted to the properties and group structure of voice-leading sets, and chapter 3

explores some of the voice-leading spaces these sets create. Chapter 4 demonstrates

the analytical applications of voice-leading sets and spaces, and chapter 5 suggests

some avenues for future research.

Introduction to Voice-leading Sets

In order to discuss voice leading, we must first describe the concept of a voice.

Traditionally, a voice is an actual voice-part, either a soprano, alto, tenor, or bass,

with a defined register and range. This concept is extended to instrumental music; a

voice is defined by the individual instrument or section. Just as in vocal music,

voices are distinguished by register and timbre.

There is a long history of analytical interpretations which differ from the

traditional definition of voices. Some of these analysts approach voices from a

harmonic perspective, demonstrating the parsimonious motions formed by harmonic

reductions. Others demonstrate linear connections formed over longer time-spans. In

Ernst Kurth’s discussion of polyphonic melody, he distinguishes between

Realstimmen, or actual voices, and Scheinstimmen, or apparent voices. Realstimmen

are formed by the actual sequence of notes on the score, and Scheinstimmen are

voices formed by registral connections. Kurth demonstrates, for instance, that in mm.

21-30 of the Allegro of Bach’s Violin Sonata n. 3, the solo violin portrays a G pedal

point as well as a lyrical melody in the upper register.5

5 Ernst Kurth: Selected Writings, ed. and translated by Lee Rothfarb, (New York:

Cambridge University Press, 2008), 81.

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The concept of polyphonic melody is central to Schenkerian analysis. The

voices defined by analysis demonstrate the overarching prolongation of the tonic

triad. Voices are defined by step-wise motion, and a number of techniques and

concepts, such as reaching-over, register transfer, coupling, and cover-tone, allow for

octave equivalence. If the tonic triad at the highest level is viewed as having three

voices—root, third, and fifth—all foreground and middleground material expresses

these voices through the transformations shown in the analysis—arpeggiation,

neighbor motion, linear progressions, etc. The end result of analysis is to demonstrate

the unique way in which the tightly bound voice leading of the Ursatz is unraveled to

form the musical surface. 6

Joseph Straus incorporates the concept of polyphonic melody into his

discussion of voice leading in post-tonal music, distinguishing between a voice and a

line. For Straus, a line is determined by register, while a voice is formed by mapping

one pitch-class onto another through transposition or inversion. Straus describes this

mapping as transformational voice leading, which projects an underlying “pitch-class

counterpoint.” 7

William Benjamin discusses this concept in the context of tonal

music, stating that a harmonic progression is made up of pc lines, while traditional

6 An important distinction between Schenkerian and post-tonal voice leading is the

accompanying harmonic prolongation through the levels of a Schenkerian analysis.

Straus has argued that post-tonal voice leading is associational, rather than

prolongational, by which he means that we make associations between non-adjacent

notes, rather than connect them by prolongation, in “The Problem of Prolongation in

Post-tonal Music” Journal of Music Theory, 31, n. 1 (1987): 13-15. 7 As Straus 2003 points out, the concept of “pitch-class counterpoint” is from William

Benjamin’s “Pitch-class Counterpoint in Tonal Music” in Topics in Music Theory,

Richmond Browne, editor (New York: Academic Press, 1981): 4.

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counterpoint is made up of pitch lines. A harmonic reduction, therefore,

demonstrates the pc counterpoint formed by harmonic motion underneath the musical

surface.8

In Straus’ distinction between voice and line, voices are analytical choices.

Because voices are formed by mappings, voice-leading analysis defines voices. The

idea of transformation, or a mathematical mapping, as voice leading originates in the

work of David Lewin. For example, in a discussion of the opening two chords of

Schoenberg’s Op. 19, n. 6, Lewin describes all of the transposition and inversion

operations that map at least two notes of the first chord onto two notes of the second

chord. The two chords are not members of the same set-class, so there is no

transposition or inversion that maps all three members onto three members of the

following chord. Lewin then shows how the potential of some of these near

transformations is realized later in the piece.9

In the analysis of post-tonal music, Straus follows up on Lewin’s earlier work

by proposing that voice leadings be described according to how transposition- or

inversion-like they are. Three criteria, uniformity, balance, and smoothness, evaluate

voice leadings. Transposition results in voice leading that is uniform, as all voices

move by the same interval, while inversion produces balanced voice leading, in that

all voices flip around the same abstract axis of symmetry. Uniformity and balance

8 Benjamin 1981: 1-32. For an extended analytical example, see Benjamin’s

“Debussy’s ‘pour les Sixtes’: an Analysis,” in Journal of Music Theory, 22, n. 2

(1978): 253-290. 9 “Transformational Techniques in Atonal and Other Music Theories” in Perspectives

of New Music, 21, n. 1/2 (1982-1983): 336-342; see especially examples 19 and 20 on

pp 339 and 340, respectively.

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both involve the concept of fuzzy transformations, that is, transformations which are

some measurable distance from the crisp transformations they resemble.10

Smoothness is simply semitonal distance—the sum of the interval-classes traversed

by the voices. Pitch-class mappings that are near-transpositions, such as pcsets {014}

and {015}, have a high degree of uniformity; in this case, {014} maps onto {015} by

T0 with one semitone offset. Balance is similar: pcset {016} is nearly inverted by It

to pcset {48t}. Straus uses an asterisk to show that the transformation is not “pure,”

and lists the number of offsets in parentheses; the two previous examples are labeled

*T0 (1) and *It (1), respectively.11

Figure 1-1 shows five chords, labeled a-e, from Arnold Schoenberg’s Drei

Klavierstuecke Op. 11 n. 2, mm. 11-13.12

Straus’ method for describing voice

leadings may be applied to this passage by considering the harmonic structures of the

chords, and the pitch-class transformation(s) that most closely approximate “crisp”

transposition or inversion. Chords a and b are not members of the same set-class, but

both contain a member of [036]. As Figure 1-2a shows, the C# diminished triad in

chord a is mapped onto the B diminished triad in chord b by Tt, while the singleton

10

Ian Quinn, “Fuzzy Transposition of Pitch Sets” Paper presented at the Society for

Music Theory Conference, Baton Rouge (1996), cited in Straus 2003. 11

Straus 2003, 314-320. 12

David Lewin’s “A Tutorial on Klumpenhouwer Networks, Using the Chorale in

Schoenberg’s Op. 11, n. 2” in Journal of Music Theory, 38, n. 1 (1994): 79-101

features this passage and the preceding two measures. Edward Jurkowski describes

the voice leading of this passage using what he terms I-DIFF Networks; these

partition the tetrachords into dyads, and are linked by the difference between the

interval-class values of the dyads. See “A Theory of Harmonic Structure and Voice-

leading for Atonal Music,” (Ph.D Diss., Eastman School of Music, 1998): 93-99.

Lewin 1994 also focuses on an ic 4/5 interpretation of the passage, partitioning the

harmonies in many different ways to do so; see pp. 86-93.

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F# moves to D# by T9; therefore, the voice leading between chords a and b may be

described by *Tt (1).

John Roeder’s work provides another way to view voice leading in post-tonal

music. Roeder uses vectors to describe individual motions among chord members.13

These vectors map registrally-corresponding chord members onto one another.

Unlike Straus, Roeder focuses on pitch voice leadings in his analyses, as he uses his

method to model voice leadings on the surface of post-tonal pieces.14

As Figure 1-2b

shows, the pitch voice leading between chords a and b in Figure 1-1, ordered from

lowest to highest, is <1,1,4,-3>; that is, the lowest chord members ascend by

semitone, the next by four, and the highest descends by three.

The present study combines the approaches of Straus and Roeder to develop a

theory of voice-leading sets.15

A voice-leading set (vlset) is an ordered set of

transpositions or inversions which operates upon a pitch-class set or multiset. Like

Lewin, Straus, and Roeder, voice leadings are understood as transformational

mappings from one set to another. As in Straus’ work, these mappings define an

underlying pitch-class counterpoint, that is, the individual pitch-class voices

interpreted from a succession of pitch-class sets. Unlike Straus, no emphasis is

placed upon either harmonic similarity or voice-leading smoothness. Vlsets employ

Roeder’s notation, but apply to pitch-class, rather than pitch, voice leadings.

13

Roeder 1984, 162. 14

“Voice Leading as Transformation,” in Musical Transformation and Musical

Intuition: Essays in Honor of David Lewin, ed. Raphael Atlas and Michael Cherlin

(Roxbury, MA: Ovenbird Press, 1994): 41-43. 15

I am grateful to Andrew Wilson for sharing an unpublished paper with me that

overlaps with some of the ideas presented in this chapter considerably.

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Vlsets express ordered sets of pitch-class transpositions or inversions as

vectors. For example, if chords a and b of Figure 1-1 are placed in normal order,

vlset <tt9t> maps ordered pcset <1467> onto <e235>. Vlset <tt9t> is a vector of

transpositions; each maps an individual member of ordered pcset <1467> onto the

corresponding member of <e235>.16

This series of mappings defines the pitch-class

voices between the two chords, as shown in Figure 1-2c. If the vlset from pcset

<1467> to <e235> is interpreted as a series of inversional mappings, the vlset

spanning them is <0690>i. The lowercase i designates that the vlset is inversional.

This alternative is also shown in Figure 1-2c. Vlsets may also span multisets. For

example, in Figure 1-2c, chord d is a hexachordal multiset, <569tt1>, and chord e is a

hexachord, <578t01>; using this ordering, chord d is mapped onto chord e by vlset

<01e020>, forming the pc voices shown in the figure.

This brief demonstration has shown that vlsets explicitly define the pc voices

formed by transformational mappings. In doing so, post-tonal voice leading may be

studied on its own terms, without reference to harmonic similarity or semitonal offset.

As will be shown in chapter 3, vlsets may be used to generate voice-leading spaces in

which they form the unit distance. This defines voice-leading parsimony

contextually, which may be more appropriate for the representation of individual

post-tonal works than a semitonal model.

16

In order to avoid confusion between pcsets and vlsets, in this study vlsets will

always be expressed in bold using angle brackets.

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Background

Post-tonal music has been interpreted in many ways which generally reduce to

two approaches: 1) forms of extended tonality, and 2) discussions of symmetry and

cycles, with properties stemming from the operations and transformations of set and

group theory.17

The first methodology is more or less compelling depending on the

piece, and the skill of the analyst.18

Post-tonal pieces may have a focal pitch or pitch-

class, but typically do not demonstrate the same relationships as tonal pieces.19

The second method above, the interpretation of post-tonal music in terms of

symmetries and group properties, has proven analytically fruitful. Many artifacts of

post-tonal study, from interval-class vectors and lists of M-related sets to

Klumpenhouwer networks, originate in group theory. This category includes serial

operations, which form a symmetrical Klein group among the four transformations of

17

See for example: Allen Forte “Pitch-class Set Genera and the Origin of Modern

Harmonic Species” in Journal of Music Theory, 32, no. 2 (1988): 187-270, Lewin

2007, and George Perle, Twelve-Tone Tonality 2nd

ed. (Berkeley: University of

California Press, 1996). 18

Felix Salzer applies Schenkerian methodology to post-tonal music in Structural

Hearing (New York: Dover, 1982). Heinrich Schenker himself provides an analysis

of a passage from Stravinsky’s Concerto for Piano and Winds in “Fortsetzung der

Urlinie-Betrachtungen,” in Das Meisterwerk in der Musik: Jahrbuch II (Munich: Drei

Masken Verlag, 1926); trans. by John Rothgeb as “Further Consideration of the

Urlinie: II,” in The Masterwork in Music: A Yearbook, vol. 2, ed. William Drabkin

(New York: Cambridge University Press, 1996), 1–22. Schenker’s work differs from

Salzer, in that his purpose is to show the ways in which Stravinsky’s piece is lacking;

that is, why it is poorly-composed. A number of more modern analysts have tackled

post-tonal music using Schenkerian techniques, including James Baker’s study of the

first of Schoenberg’s Six Little Piano Pieces, Op. 19 in Music Analysis, 9, n. 2 (1990):

177-200, and Olli Vaisala’s “Concepts of Harmony and Prolongation in Schoenberg’s

Op. 19/2” in Music Theory Spectrum, 21, n. 2 (1999): 230-259 and “Prolongation of

Harmonies Related to the Harmonic Series in Early Post-tonal Music” in Journal of

Music Theory, 46, n. 1/2 (2002): 207-283. 19

Joseph Straus critiques post-tonal prolongational analysis in Straus 1987: 1-21.

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P, I, R(P), R(I). Stemming originally from Milton Babbitt, then through his student

David Lewin, group theory concepts continue to dominate the post-tonal theoretical

and analytical landscape.

The group structure of triadic space, as well as the space of tonal chords in

general, has been explored extensively in the Neo-Riemannian literature of David

Lewin, Brian Hyer, and Richard Cohn, among others.20

Drawing upon ideas and

vocabulary from 19th

-century theorists, Lewin uses group theory to develop a system

of triadic transformations. While Lewin’s approach is mathematical and uses the

language, if not always the categories, of set theory, he includes familiar tonal labels

such as dominant and mediant. Lewin’s goal appears to be to encompass as much

music as possible, tonal, post-tonal, or otherwise, using transformations.

The starting point for Neo-Riemannian theory is the voice leading by the

smallest intervals of 1 and 2 between inversionally-related triads, which is often

found as such in late 18th

and 19th

century representations of tonal space. Using the

triadic transformations P (Parallel) (C-E-G, C-Eb-G), L (Leittonwechsel) (C-E-G, B-

E-G), and R (Relative) (C-E-G, C-E-A), Brian Hyer, constructs the space of triadic

transformations and demonstrates their group structure, drawing on the theories of

Riemann, Oettingen, and others.21

As Hyer shows, P, L, and R transform triads by

maintaining two common tones, and moving one by step. Of course, not all triads

may be linked this way, but the distance between any two triads may be viewed as

20

See, for example, Cohn 1996 and 1997, Hyer’s ”Reimag(in)ing Riemann” Journal

of Music Theory, 39, n. 1 (1995): 101-138, and Lewin 1987. 21

Hyer 1995.

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some combination of the three transformations. Richard Cohn also focuses on

parsimonious voice leading and demonstrates that triads are the only trichords in

which all three maximal-common-tone preserving transformations involve motion by

step (interval-class 1 or 2) in the moving voice. Figure 1-3 replicates Cohn’s Figure

4, showing the interval-class by which each voice of a trichord must move to produce

another member of the set-class that shares two common tones.22

The actual pitch-

class interval required is determined in part by the Tn form of the set-class in

question. For example, P transforms a C minor triad into a C major triad through

moving the third by interval 1, while it transforms a C major triad into a C minor triad

through the interval 11.

As shown in Figure 1-3, triads and parsimonious voice leading are

interdependent. While other trichords, such as [014], may be capable of

parsimonious voice leading in one particular transformation, [037] is the only one in

which all three transformations involve motion by step.23

As Cohn shows, this is

related to the triad’s internal intervallic structure; the triad’s largest interval, the fifth,

is the most nearly symmetrical division of the octave possible. Similarly, the third is

22

Cohn 1997, p. 6. Figure 1 differs slightly from Cohn’s Figure 4 in a few ways:

step-intervals are listed as ics instead of directed pitch-class intervals, and voice-

crossing is allowed. Interval-classes are used because the value is different when

moving from one Tn-class to another, {037} moves to {047} by ascending semitone,

but {047} moves to {037} by descending semitone, and the purpose of the table is to

show minimal motion; the non-distinct-invertibility of the tritone results in a voice-

leading possibility not included by Cohn’s Figure, that of transposition or inversion

with maximal common-tone retention where the moving voice crosses one of the

common tones. For example: {016}—{076}, where the “third” moves by tritone. 23

Joseph Straus generalizes PL and R and constructs tonnetze for all other trichordal

set-classes, then demonstrates their analytical usefulness in “Contextual-Inversion

Spaces,” in Journal of Music Theory 55, n. 1 (2011): 43-88.

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the most nearly symmetrical division of the fifth possible. These features allow for

very smooth voice leading when the fifth is inverted within the octave, and when the

third changes quality within the fifth. Thus, smooth voice leading and maximal but

asymmetrical spacing depend upon one another.24

Because parsimonious voice

leading and triads are inextricably linked, an emphasis on one supports the other.25

Therefore, it is not necessarily appropriate to emphasize stepwise voice leading

among non-triadic set-classes.

Julian Hook’s work expands upon that of the previous authors by studying

group structure among triadic transformations without focusing on voice leading.

Hook defines a “Uniform Triadic Transformation” as an ordered triple, <x,y,z>,

where y and z indicate the pitch-class interval by which a triad’s root changes, with

each number acting upon major or minor triads, respectively, while x indicates

whether the mode is preserved or changed. The Neo-Riemannian operation P

corresponds to <-,0,0,>; the root stays the same, while the mode is changed. Hook

then shows that other trichordal set-classes may be substituted for triads using the

transformations.26

Although not restricted to triads, Hook’s transformations may still

only be used between members of the same set-class, or between the members of two

24

Cohn’s work intersects in interesting ways with that of John Clough and Jack

Douthett in “Maximally Even Sets” Journal of Music Theory, 35, 1/2, (1991): 93-173.

See especially pp. 169-172, where the authors show that major and minor triads are

maximally even with respect to the diatonic scale—what they call second-order

maximally even sets. 25

Cohn 1997 describes the triad as “overdetermined” due to the incidental

relationship between its voice-leading properties and acoustic properties, 5. 26

“Uniform Triadic Transformations” Journal of Music Theory, 46, n. 1-2 (2002):

57-126.

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set-classes. The groups formed by voice-leading sets, as shown in chapter 2, may be

seen as expanding the use of group theory to encompass all pitch-class sets and

multisets in a given cardinality. In addition, the objects—pitch-class sets and

multisets—corresponding to the various subgroups formed by vlsets consist of

multiple set-classes.

Robert Morris’ work on voice-leading spaces is very influential on the

generation of voice-leading spaces proposed in chapter 3. Morris defines general

spaces with relative distances as “contour spaces” as well as modular and linear

spaces with defined spans, such as pitch and pitch-class spaces. More recently, his

work on voice-leading spaces defines properties of these spaces and the restrictions

and definitions of tonnetz spaces. He also shows Perle spaces as an outgrowth of

realignments of the cyclic spaces which constitute tonnetze.27

Morris’s definition of

voice-leading spaces can be interpreted as defining the transformations that relate

forms of the same set-class, as he shows in his “Riemann wreath” of trichords that

combine to create members of [013478].28

José Martins generates spaces built on scales from the medieval Enchiriadas

treatises, termed dasian and Guidonian spaces, respectively. He then shows that the

spaces may be useful in the analysis of post-tonal music, focusing chiefly on the

music of Bela Bartok. After showing that the spaces derive from interlocking

27

Morris 1987 and 1998. 28

Morris 1998, 198.

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interval-cycles, Martins generalizes their structure in order to produce a large number

of new spaces which may be used to analyze post-tonal music.29

The most recent explorations of voice leading incorporate geometrical models.

Figures 1-4a and b display the trichordal orbifold as described by Tymoczko, Quinn,

and Callender.30

This work intersects with Perle spaces, as well as an earlier article

by Walter O’Connell where the geometry of set-class space, here the all-interval-class

tetrachords, is examined in modern terms for the first time.32

Figure 1-4b provides a

close-up of one-quarter of the orbifold, corresponding to three of the triangular

sections shown in Figure 1-4a. As Figure 1-4b shows, each trichord is located in

closest proximity to chords that have two common tones and minimal offset in the

intervals involved. A chord’s location also corresponds to its evenness, that is, how

evenly it divides the octave. For instance, [048] divides the octave into three equal

parts, and is located at the center. Triads, [037], are nearly symmetrical, and

therefore orbit the central [048]s. Given this feature, traditionally defined

“parsimonious” voice leading may take place within one particular region of the

29

See Dasian, Guidonian, and Affinity Spaces in Twentieth-Century Music, (PhD

diss., The University of Chicago, 2006), “Stravinsky’s Discontinuities, Harmonic

Practice and the Guidonian space in the ‘Hymne’ for the Serenade in A” in Theory

and Practice, 31, (2006): 39-64, and “Affinity Spaces and Their Host Set Classes” in

Mathematics and Computation in Music 38, (2009): 496-508. 30

Clifton Callender, Ian Quinn, and Dmitri Tymoczko “Generalized Voice-leading

Spaces” in Science, 320, (2008): 346-348, and Dmitri Tymoczko “The Geometry of

Musical Chords” in Science, 313 (2006): 72-74. The images were generated using

the program Matlab a scientific imaging and calculating program. 32

Dave Headlam “Introduction” in Theory and Practice, 33, (2008): 1-47, and Walter

O’Connell “Tone Spaces” in Die Reihe, 8, (1968): 35-67.

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orbifold, such as among the triads at the center. This spatial display of chordal voice

leading and harmonic properties is a compelling feature of the orbifold.

Andrew Wilson also uses set theory to describe voice leadings in post-tonal

music. Wilson shows pitch voice-leadings as unordered sets, then demonstrates how

they may be grouped into equivalence classes using permutation, inversion, register,

transposition, and cardinality, describing the possible perceptual issues raised by each

equivalence.33

As mentioned above, the description of voice leading as an ordered set is also

indebted to John Roeder’s formalization of voice leadings as vectors. Roeder

explores some of the relationships between harmonic structure and voice leading, and

categorizes various types of voice leading. A T-class voice leading connects two

chords that have a relative transpositional level of zero, that is, their lowest pcs are

the same. A transpositional voice leading connects two members of a tn-class, and a

permutational voice leading connects two chords which contain the same pcs, but in a

different registral order.34

Roeder focuses on pitch voice leadings, rather than pitch-class, in his

analyses, in order to describe voice leadings as they occur on the surface of the

music.35

One of the challenges in analyzing pitch-class voice leadings is that there

are multiple possible interpretations of a passage. For example, any member of one

33

“Voice Leading as Set,” unpublished manuscript, (2011). Although we worked

independently, Wilson and I developed very similar ideas, extending even to terms

such as vlset and vlclass. 34

Roeder 1984, 162-165. 35

Roeder 1994, 41-43.

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pc set may map onto any member of another pc set. This difficulty is faced by those

using set theory to analyze pc sets as well, as they must decide how the music should

be segmented and partitioned.

Roeder’s ordered interval spaces are similar in structure to the orbifold spaces

discussed above. One important difference is that Roeder interprets the coordinates

of his spaces as ordered pc intervals. For example, <1,2> represents the

transposition-class [013]. Using transpositional equivalence, Roeder is able to

represent pc sets in a lower dimension, so that the two-dimensional space, of which

<1,2> is a member, represents trichordal transposition-classes.36

Klumpenhouwer-networks, or K-nets, as defined by Henry Klumpenhouwer

and described by David Lewin, are networks of nodes and arrows, in which nodes are

populated by pitch-classes and arrows represent transposition or inversion operations.

Networks are related to one another through the similarity of their graphs, known as

isography. The arrows of strongly isographic K-nets have the same T and I values.

Positively isographic K-nets have the same T arrows, and the same difference

between their I arrows. The value of this difference, known as <T>, is used to

describe the relationship between two K-nets. Negatively isographic K-nets have

inverse T arrows, and their I arrows add up to the same sum. This sum is known as

<I>.37

36

“A Geometric Representation of Pitch-Class Series” in Perspectives of New Music,

25, ns. 1/2 (1987): 362-409. 37

See, for example, “A Tutorial on Klumpenhouwer Networks, Using the Chorale in

Schoenberg’s Op. 11, n. 2” in Journal of Music Theory, 38, n. 1 (1994): 79-101.

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As Shaugn O’Donnell has shown, the <T> and <I> relationships among K-

nets may be seen as dual operations—two transpositions or two inversions,

respectively.38

The pcset a K-net represents may be divided into two subsets, which

are mapped onto the subsets of some other pcset. <T> and <I> values are equal to the

sum of the two operations. Therefore, to form isographic K-nets, two pcsets must be

divisible into the same two Tn-classes, or inversionally-related Tn-classes. Pcset

pairs that are subdivided into the same Tn-classes form strongly or positively

isographic K-nets, while those with inversionally-related Tn-class subsets form

negatively isographic K-nets. Therefore, using K-nets, one may describe

relationships among pcsets which are not members of the same set-class.

Dual operations provide relationships among the same pcsets as K-nets, but do

not form recursive network structures. Like K-nets, they are useful in post-tonal

repertoire for describing relationships among pcsets which do not belong to the same

set-class. One analytical advantage dual operations have over K-nets, is that they

explicitly describe pcset mappings. K-net <T> and <I> relationships, on the other

hand, describe operations upon network arrows. This is useful for building recursive

levels of networks, but does not describe the musical surface of a piece.39

38

Transformational Voice Leading in Atonal Music, (PhD diss., CUNY, 1997). In a

response to a critique of K-nets by Michael Buchler, Henry Klumpenhouwer outlines

the differences between dual transformations and K-nets, noting that the <T>/<I>

automorphisms of K-nets map the arrows of one network onto another, while dual

transformations map the contents of the nodes of one network onto another, among

other differences. 39

Headlam 2008 points out connections between Perle’s compositional materials and

K-nets; see pages 30-32.

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K-nets and dual operations, are analytically useful in post-tonal music because

they may be applied to passages which employ multiple set-classes, and demonstrate

relationships involving combinations of transformations. Voice-leading sets take this

idea a step further; all voice-leadings are interpreted as the combination of individual

pitch-class transformations. Dyadic vlsets are essentially identical to dual operations,

except that they apply to individual pitch-classes, rather than pitch-class sets.

Joseph Straus has also written and presented geometrical visualizations of

“set-class” voice leading, where we may move systematically through some sequence

of set-classes by a direction in the geometrical space presented.40

Straus’s ideas are

described more fully below. Finally, Michael Callahan demonstrates connections

between Perle spaces and Knet transformations by creating a lattice in which adjacent

trichords are related by strong-isography and/or minimal pc offset; graphically

demonstrating consistent relationships among pc sums and differences.41

40

“Voice Leading in Set-Class Space” in Journal of Music Theory 49, no. 1 (2005):

45-108. The space will be discussed in detail below. 41

“Mapping Sum-and-Difference Space: Parallels Between Perle and Lewin” in

Theory and Practice 33, (2008): 181-217.

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Chapter 2: Voice-leading Set Theory

As discussed in the previous chapter, the study of voice leading in post-tonal

music has focused mainly on harmonic similarity, using semitonal offset as a metric.

While this emphasis is appropriate for some musical contexts—especially triadic

contexts, as discussed in chapter 1—it is insufficient for post-tonal music. The

following discussion will demonstrate that voice-leading sets provide a way in which

to discuss post-tonal voice leading without relying on harmonic similarity or

semitonal offset; instead, voice-leading sets explicitly define pitch-class voices

produced by mapping one pcset onto another. Voice-leading sets will then be

compared to traditional pitch-class set transformations, such as transposition and

inversion, followed by a brief analytical example using Arnold Schoenberg’s Drei

Klavierstuecke, Op. 11, n. 1; the chapter will conclude with a discussion of the group

properties of voice-leading sets.

A voice-leading set (vlset) is an ordered set of transpositions or inversions

which operates upon a pitch-class set or multiset. Let us begin with pc dyads {01}

and {5t}. There are two possible voice leadings between them; either pc 0 maps onto

5 and 1 onto t, or 0 maps onto t and 1 onto 5. If the first pcset is ordered, <01>, and

the mappings are induced by transposition, they form vlsets <59> and <t4>. If the

mappings are induced by inversion, they form vlsets <5e>i and <t6>i.42

Because vlsets are ordered sets of transformations, they naturally order the

pcsets they operate upon. That is, vlsets do not operate consistently upon the various

42

Vlsets will always be shown in bold in angle brackets. Inversional vlsets include a

lower case i.

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permutations of a pcset, unless the vlset or pcset has some symmetrical property

which produces order equivalence. This is a significant difference between vlsets and

other proposed methods for studying voice leadings. A transposition or inversion

maps the set of pcs onto itself consistently, that is, by the same interval or sum. For

example, T1 maps every pc by T1. Vlsets do not operate uniformly on the set of pcs,

but on individual pcs. Therefore, they may be seen as operating upon multiple sets of

pcs, equal to the cardinality of the vlset. For instance, vlset <356> operates upon

three distinct sets of pcs simultaneously.43

Imagine that ordered pcset <000> is a slot

machine and each reel contains pcs 0-e; vlset <356> directs the first reel to spin three

places, the second five places, and the third six, producing pcset <356>. The analogy

breaks down with inversional vlsets, as these direct the reels to flip, rather than spin.

While the ordered structure of vlsets requires strict ordering of pcsets, it also allows

for multisets; each doubling in a multiset represents an independent set of twelve pcs.

The number of vlsets in each cardinality n is equal to (12^n)*2. For example,

the dyadic vlsets can have any of the twelve pcs in either position, and there are two

types of each vlset, transpositional and inversional; Figure 2-1 shows the number of

vlsets in each cardinality. The vlsets in cardinality one correspond to the twenty-four

transposition and inversion operations, with the exception that they apply to single

pcs as vlsets. Because vlsets include multisets, they can be any cardinality, although

it is difficult to imagine what usefulness a ten-thousand-member vlset might have.

43

The group structure of vlsets is addressed at the end of this chapter.

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Traditional transposition and inversion are preserved by vlsets as singleton

multisets; for instance, <000> is equivalent to T0, <44> is equivalent to T4, <3333>i

is equivalent to I3, etc. However, vlset notation does not necessarily demonstrate a

transpositional or inversional relationship between pcsets. Vlsets define individual

transpositional or inversional mappings. They are therefore independent of the

harmonic structure of the pcset mappings they produce. This is significantly different

than the transposition and inversion of pcsets, which preserves harmonic structure.

For example, pcsets {014} and {458} are both members of tn-class [014]. If placed

in normal order, <014> and <458>, the sets are spanned by vlset <444>, showing that

they are harmonically equivalent under transposition. In any other ordering, however,

the sets’ harmonic relationship is not shown by the vlset. Figure 2-2 shows the

possible mappings between pcsets {014} and {458}, including inversional vlsets.

Each of these mappings is a viable interpretation of the voice leading between the two

pcsets. Unlike transposition and inversion, the harmonic relationship between any two

sets is not addressed by vlsets, except for the way in which it limits the possible

mappings between them.44

Using vlsets, therefore, voice leading may be removed from its harmonic

context and examined more abstractly. The above examples begin with pcsets and

demonstrate how vlsets may map one onto another. As a transformation, with no a

44

Given two pcsets, no vlset spanning them is the “correct” interpretation. All

possible vlsets, such as those shown in Figure 2-3, are equally viable. An analyst,

however, may choose one particular voice-leading interpretation which conveys some

meaningful data about a piece. Analytical issues will be addressed further in chapter

four.

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priori pcsets, a particular vlset spans many pcset pairs. For example, in addition to

mapping pcset <1467> onto <e235>, vlset <tt9t> maps pcset <0245> onto <t013>, or

<0167> onto <te35>, or <0022> onto <tte0>, etc.45

Contrapuntal Motion in Pitch-class Space

Pitch voice leading categories such as parallel, oblique, contrary, and similar

motion may be adapted for pitch-class voice leading, and articulated with vlsets.

These categories are altered by their transformation from linear pitch space to

modular pitch-class space. Pc space counterpoint also introduces interpretive

challenges; the individual motions between pcsets may be interpreted in a number of

ways, producing multiple contrapuntal interpretations. In addition, due to octave

equivalence pitch-class counterpoint may or may not be reflected in pitch-space on

the musical surface.

Parallel motion in pitch space is formed by multiple voices moving in the

same direction by the same distance.46

In pc space, parallel motion corresponds to

transposition and inversion, which move each pc the same distance relative to some

axis. This axis is rotational for transposition and reflective for inversion.47

Parallel

motion occurs whenever a vlset’s values are the same. For example, vlsets <222>

45

The first example is shown in Figures 1-3 and 1-4. 46

This distance may be measured in diatonic pitch space as well, as in the parallel

thirds of a major scale. 47

Parallel motion corresponds to crisp transposition or inversion; see Joseph Straus’

“Uniformity, Balance, and Smoothness in Atonal Voice Leading” in Music Theory

Spectrum, 25, n. 2 (2003): 318.

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and <777>i create parallel motion, as they transform each pc voice by the same

amount; see Figure 2-3.48

Oblique motion in pitch-space is the movement of one voice while another

maintains the same pitch. Transpositional vlsets which include a zero, such as <05>,

show that one voice maintains the same pc, while another moves. Inversional vlsets

may also produce oblique motion, but are defined in part by the specific pcset they

operate upon. In order to map a pc onto itself, the index of inversion must be double

the pc value. See Figure 2-4 for some examples.

Contrary motion in pitch-space is defined as motion in opposite directions;

one voice ascends while the other descends. In modular pc space, this can be more

rigorously defined as motion by complementary intervals, that is, when one voice

moves by ordered pc interval X, the other voice moves by 12-X. This motion is

easily demonstrated by transpositional vlsets, such as <39>. Cardinalities greater

than two may produce sum-0 vlsets which are not made up of complementary pairs,

such as <et96>. Nevertheless, the total ordered pc distance invoked by the vlset is 0,

therefore the definition may be generalized to include these sets; a few examples are

shown on Figure 2-5.49

48

The lines on Figures 2-3 through 2-6 demonstrate the pc voices created by the

vlsets. Some of the examples demonstrate their contrapuntal motion in pitch space as

well as pitch-class. 49

Strongly isographic K-nets also involve the concept of contrary motion, as the T-

related nodes may be transposed by complementary transpositions from one to

another. Mike Callahan organizes these motions on a torus he terms sum-and-

difference space. One circle of this torus is formed by strongly isographic K-nets,

whose constituents, an ic 5 dyad and a singleton, move by T1 and Te, respectively;

the other circle is formed by positively isographic K-nets, where the ic 5 dyad is

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Inversional vlsets may also create pc contrary motion. However, this motion

results from the relationship between the sum of the inversional index numbers and

the total sum of the pcset. For an inversional vlset to produce contrary pc motion, it

must be double the total sum of the pcset it operates upon. For example, the total sum

of pcset {24} is 6, therefore any inversional vlset of sum 0 will produce contrary

motion. This is demonstrated by Figure 2-5. Although the Figure shows wedge

motion, each individual mapping from the first set onto a later set forms contrary

motion.

Just as contrary-motion producing inversional vlsets must be double the sum

of the pcs they operate upon, the total sum of transpositional vlsets which produce

contrary motion must be 0. Due to these sums, the vlsets preserve the sums of the

pcsets they operate upon. If Q is the sum of a pcset, inversional vlset sums are 2*Q,

and 2*Q – Q = Q; for transpositional vlsets, 0 + Q = Q. The relationship between the

total sum of two pcsets also demonstrates this relationship. Any pcsets of the same

sum will demonstrate contrary pc voice leading, no matter how the voice leading is

interpreted. For example, all of the transpositional vlsets shown in Figure 2-2,

between {014} and {458}, are sum 0, and all of the inversional vlsets are sum 10,

double the sum-5 pcsets.50

maintained, and the singleton moves by T1. “Mapping Sum-and-Difference Space:

Parallels Between Perle and Lewin,” in Theory and Practice, 33 (2008): 181-217.

The torus is shown on page 198. 50

On the orbifold spaces developed in Tymoczko 2006 and Callender, Quinn, and

Tymoczko 2008, pcsets of the same sum are grouped together; they are aligned on the

dyadic space, and in the same plane on the trichordal space. Therefore, any

movement along a same-sum line or within a sum plane may be characterized as

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Similar pitch motion is simply defined as two voices moving in the same

direction. It is distinguished from parallel motion in that the voices move by different

intervals; in horn fifths, for instance, one voice moves by step while the other moves

by arpeggio in the same direction. Pitch-class intervals make no distinction between

ascent and descent, of course, but do specify the total distance traversed in pitch-class

space. Therefore, similar pc motion may be defined by the difference between the

intervals of each voice. This is equivalent to Straus’ uniformity and balance,

discussed above.51

For example, vlset <12> moves one pc voice by T1, and the other

by T2, this voice leading is very similar, without being parallel or oblique. Similarity

among inversional vlsets may be defined in the same manner, as the difference of

inversional sums; <78>i would produce very similar motion.52

Pc voice-leading

similarity is a continuum, with a maximum dissimilarity of six in two voices, as in

<17>, and a midpoint of three, as in <t1>. For higher cardinalities, relative similarity

contrary. “Generalized Voice-leading Spaces” in Science, 320, (2008): 346-348, and

“The Geometry of Musical Chords” in Science, 313 (2006): 72-74. David Headlam

notes this organization of the orbifold and demonstrates that it corresponds with some

of the theories of George Perle, most notably his triadic arrays in his introduction to

Theory and Practice, 33 (2008): 32-39. This definition of contrary pc motion is also

related to Richard Cohn’s directed voice-leading sums, or DVLS, which are the total

ordered pitch-class sums produced by the voice-leading from one triad to another; see

“Square Dances With Cubes” in Journal of Music Theory, 42, n. 2 (1998): 283-296.

A DVLS value of 0 corresponds to the definition of contrary motion proposed here. 51

Straus 2003. The vlsets shown on Figure 2-7 correspond to fuzzy transposition or

inversion with very little offset. 52

Of course, it may also produce oblique motion, depending on the pcset.

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is given by the range of values; as shown in Figure 2-6, vlsets <1122> and <011>i,

with a range of one, produce very similar pc voice leading.53

Voice-leading Set-classes

The vlsets in any cardinality may be organized into voice-leading set-classes

(vlclasses) through various equivalences, such as transposition, and inversion. For

example, vlsets <01>, <23>, <45>, <43>, and <87> are members of vlclass [01]. The

set of vlclasses formed through transposition and inversion is isomorphic to the set of

set-classes of ordered pcsets, including multisets. Vlclasses preserve the intervals

among the pc voices produced by their vlset members. For example, Vlset <025>

transposes the first pc by T0, the second by T2, and the third by T5; the second pc

moves two steps further than the first and the third pc moves five steps further.

Another member of vlclass [025], such as <247>, increases each pc transposition by

two; the first pc moves by T2, the second moves two steps further, and the third pc

moves five steps further. Vlclasses thus preserve the harmonic structure of voice

leadings, rather than pcsets.

Vlset transformations are linked to pcset transformations when operating upon

ordered pcsets. For example: pcset <02> is mapped onto <03> by vlset <01>. The

same pcset, <02>, is mapped by <23> onto <25>; the T2 transposition of the vlset is

matched by a T2 transposition of the resulting pcset. Similarly, vlset <45> produces

53

Above I suggested that contrary motion might be broadly defined to include sum-

zero transpositional vlsets which are not made up of complementary transposition

pairs, as the total voice-leading is balanced among the pc voices. In my opinion, pc

voice-leading similarity should be defined by the relationships among all of the

individual voices, so the range, not the total sum should be used. This differs from

Straus, who measures the total offset in a fuzzy transformation.

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pcset <47> from <02>, a T4 transposition of <03>. When operating upon ordered

pcset <02>, the twelve transpositions of vlset <01> produce the twelve transpositions

of pcset tn-class [03], as Figure 2-7 shows. Therefore, vlclass [01], under

transposition, maps pcset tn-class [02] onto tn-class [03]. Inversions of vlset <01>,

such as <10>, reverse the direction of the voice leading. That is, they transform

ordered pcset <03> into some transposition of <02>. For example: ordered pcset

<03> is transformed by vlset <10> into pcset <13>.

As is evident from the above definition, the voice leading theory proposed

here uses an expanded form of the set theory currently in practice, but applies it to

voice leadings in addition to pc sets.54

A brief discussion of Arnold Schoenberg’s

Drei Klavierstuecke Op. 11 no. 1 will show some of the ways in which the ideas

presented thus far may be applied. The piece will be revisited in chapter 4. Figure 2-

8a shows the first eleven measures with ten trichords labeled a-i, and Figure 2-8b

demonstrates that voice-leading class [002] links many pairs of trichords; that is, the

vlsets are some transposition or inversion of one another. This preserves the

relationship among the intervals traversed by the individual moving voices. In the

case of vlclass [002], two voices move by the same interval, and one voice moves two

semitones further, or two semitones less. For example, the [04] dyad in chord a, {7e}

moves by Tt to the [04] dyad in chord b, {59}, and the G# in chord a moves by T8, a

difference of two semitones, to E in chord b. Similarly, two members of chord c,

{56}, move by T4 to {9t} in chord d, while the B in chord c moves by T2 to Db in

54

A focus on a painting’s negative space is a possible analogy.

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28

chord d. The voice-leading consistency provided by [002] ensures that two voices of

each trichord move by the same interval, and the third voice moves by an interval that

differs by two semitones. This consistency unifies the passage despite the harmonic

diversity presented by the trichords, which represent six different Tn-classes—[012],

[014], [015], [016], [024], and [026].

The voice leadings shown in Figures 2-8a and b are not necessarily intended

to be heard in real time. Rather, they are analytical interpretations of the paths

between various pcsets. In other words, I am not suggesting that a listener attempt to

“hear” the individual pcs of chord a mapping onto chord c and chord b at the same

time. Instead, vlclass [002] may be understood as a motive, which is varied through

transposition, inversion, and rotation. This motivic interpretation changes the way in

which I understand the piece, whether or not I am concentrating on hearing individual

voice leadings in a performance.55

The unique properties of vlclass [002] also affect

the way in which I understand the piece’s succession of trichords. Because [002] is a

multiset, each trichordal pair must share one interval, such as the [04] dyads in chords

a and b. Further, [002] preserves the whole-tone content of a pcset; either each pc is

mapped onto another within the same whole-tone scale, or each is mapped onto the

other whole-tone scale. These properties explain why, for example, chord e cannot be

mapped onto chord f by a member of vlclass [002]; the three members of chord e

55

A number of theorists, such as those involved in the Society for Music Perception

and Cognition, study music perception in many contexts, including transformation

theory. In this dissertation I concentrate on the presentation of a theory of voice

leading with analytical applications, and do not engage with issues of perception at

length; I hope to pursue these issues in the future.

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29

belong to the same whole-tone scale, while chord f contains two even whole-tone

members, and one odd.

Each vlset aligns the set of ordered pcsets with itself in specific pairs. As

mentioned above, any member of ordered vlclass [01] maps any member of ordered

pcset tn-class [02] onto some member of [03]. The remaining pairings are shown in

Figures 2-9a and b. As shown, each tn-class-pair in ordered vlclass [01] is reversed

by inversion to ordered vlclass [10], and vice versa. For instance, pcset <49> is

mapped onto <26> by vlset <t9>, and is mapped onto <17> by vlset <9t>. This

feature is similar to the relationship between a transposition operation and its

complement: T1({04}) = {15}, and Te({15}) = {04}, for example. The difference is

that the relationship among vlsets can be applied to pcset tn-classes, as there are

twelve transpositions and inversions of most vlsets. Like pcset transpositions, each

transpositional vlset has a complement when operating upon pcsets—the vlset which

is the specific sum-zero inversion: <34>(<16>) = {4t}, and <98>(<4t>) = {16}.

Figures 2-9c – l demonstrate the tn-class pairings produced by the remaining

dyadic ordered vlclasses. The vlclasses are aligned with their inversions to

demonstrate their reciprocal relationships. Among transpositions, T0 and T6 are

unique. T0 transposes every set onto itself, and T6 is its own inverse. These

properties are maintained to some extent by vlclasses [00] and [06]. While vlset

<00> has the same effect as T0—transposing any dyad onto itself—the remaining

members of vlclass [00] <11>, <22>, etc, preserve the tn-class of any pcset they

operate upon, as shown in Figure 2-9l. This is not surprising given that vlsets of

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30

vlclass [00] behave like traditional transpositions. In addition, vlset <66> is its own

inverse, just as T6 is. Figure 2-9k demonstrates that vlclass [06] also behaves in

some ways like pcset transposition by T6. Each tn-class is paired with only one

other, that is, members of vlclass [06] are their own pcset tn-class complements. This

is true regardless of any transposition of the vlset. For example: pcset <38> is

mapped by <39> onto <65>, and <65> is mapped by <4t> onto <t3>; vlclass [06]

maps ordered tn-class [05] onto [10], and [10] onto [05]. In addition, specific vlsets

<06> and <60> are their own complements.

Because they are made up of multiple transpositions, transpositional vl-classes

produce ordered tn-class interval cycles. Figures 2-10a – j display the cycles created

by each vlclass.56

Each vlclass creates an ordered tn-class cycle which is isomorphic

to a pcset interval cycle. Like pc interval cycles, vlclasses create cycles according to

the way their individual transpositions divide twelve. For example, [04] creates a

three-member tn-class cycle, because four (T4) divides twelve into three. 1, 5, 7, and

e are prime to twelve, therefore the ordered vlclasses with these transpositions, [01],

[05], [50], and [10], cycle through all twelve ordered pc tn-classes.

The tn-class cycles shown in the figures simply tabulate the results of the tn-

class pairings shown earlier. As in the tn-class pairs, vlclasses are aligned by

inversion, such as [01] and [10]; this demonstrates that their cycles are also inversions

of one another. In addition, vlclasses related by multiplication, such as [01] and [05],

create multiplication-related cycles. As their name suggests, the ordered tn-class

56

[00] and [06], discussed in the previous paragraph, create one- and two-member

cycles, respectively, and are left off of the figure.

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31

cycles are independent of the transposition of the vlset involved. For example, Figure

2-11 shows the ordered tn-class cycle created by three transpositions of vlclass [04]

applied to pcset <34>. As shown, the three transpositions create an ordered tn-class

cycle of ([01] – [05] – [30]).

Vlsets also create cycles of pcsets. Like the tn-class cycles just discussed,

these are directly linked to the individual transposition values in the particular vlset.

These pcset cycles operate independently of their ordered tn-class cycles. For

example, ordered vlclass [03] creates a four-member ordered tn-class cycle. Figure 2-

12 displays the pcset cycle and tn-class cycle created by vlset <14>. As the figure

shows, the tn-class cycle, ([01] – [04] – [50] – [20]) cycles through three times before

the pcset cycle is complete. The pcset cycle is twelve sets in length, because the first

member of the vlset, T1, cycles through all twelve pcs. The cycle can be seen as a

result of the combination of cycles formed by T1 and T4.57

Like transpositional vlsets, inversional vlsets align the set of ordered pcsets

with itself. For example, vlset <01>i maps pcset <34> onto <99>, and pcset <87>

onto <46>. Inversional vlset operations are involutions: vlset <01>i maps pcset <34>

onto <99>, and <99> onto <34>. Like transpositional vlsets, a transposition of an

inversional vlset results in a transposition of the pcset produced. Figures 2-13a and b

show the transpositions of vlset <01>i and <0e>i applied to pcset <04>. The pcsets

produced are members of the same tn-class, [30] and [50], respectively.

57

These cycles intersect with George Perle’s work in a number of ways, especially

his difference tables, which produce interval cycles through arrangements of multiple

cyclic sets. See Twelve-Tone Tonality, 2nd

edition, (Los Angeles: University of

California Press, 1996), 31-32.

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Transpositional and inversional vlsets are not commutative, just like their

counterparts—pc transposition and inversion. For example, <01>(<36>) = {37}, and

<01>i(<37>) = {96}, while <01>i(<36>) = {97}, and <01>(<97>) = {98}, therefore

<01>i(<01>(<36>)) ≠ <01>(<01>i(<36>)).

As mentioned above, invesionally-related transpositional vlsets produce

inversonally-related tn-class pairings. No such relationship exists for inversional

vlsets, because they are their own inverses. Instead, inverting an inversional vlset

produces a new pairing, as Figures 2-14a and b show. The two pairings replicate the

vlsets’ inversional relationship; that is, the paired pcsets in each collection are

inversions of one another. For example, ordered vlclass [01]i pairs ordered tn-classes

[04] and [30], while [10]i pairs [40] and [03]. For another example, see Figures 2-15a

and b, which demonstrate the tn-class pairings produced by ordered vlclasses [04]i

and [40]i. Figures 2-16a – h show the remaining tn-class pairings produced by

inversional vlclasses.

The properties of dyadic vlsets just discussed are also true for trichordal

vlsets. The main difference, of course, is the greatly expanded number of both pcsets

and vlsets. There are 1728 ordered pcsets, transpositional vlsets, and inversional

vlsets. The pcsets reduce to 144 ordered tn-classes, which, like the dyads, are paired

by the individual vlsets. Figure 2-17 demonstrates the tn-class pairings produced by

ordered vlclass [025]. As with dyadic vlsets, these pairs are reversed by inverting the

vlset, producing ordered vlclass [0t7]. For example, vlset <247> maps pcset <016>

onto <251>, and vlset <t85> maps pcset <251> onto <016>. Triadic inversional

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vlsets also share some of the properties with their dyadic counterparts. Each

individual inversional vlset is an involution: <278>i(<027>) = {251}, and

<278>i(<251>) = {027}.

As a result of the volume of trichordal vlsets, many distinct vlsets are similar.

For example, <046>, <460>, and <604> are rotations of one another—the same

vector of transpositions in the same order—but produce very different results when

applied to pcset <014>: {05t}, {474}, and {618}, respectively. Three more vlsets

share the same transpositions, but not the order: <064>, <640>, and <406>, which

produce {078}, {654}, and {417}, respectively. The same is true of ordered pcsets,

of course; <014>, <140>, <401>, <041>, <410>, and <104> all belong to different

ordered tn-classes. This permutational redundancy is eliminated by moving from

ordered to unordered pcsets.58

Depending on the analytical situation, it may be useful

to use either or both unordered or ordered vlsets, just as it is with pcsets. Analytical

issues will be addressed further in chapter four.

Voice-leading Group Theory

A group is a set of objects and transformations with a few special properties:

closure, associativity, identity, and inverse. A group must have closure, that is, the

result of every transformation must be a member of the group. If two transformations

are performed in succession, the result must be the same no matter which

transformation is performed first, for example: (a + b) + c = a + (b + c). There is an

58

This is the reduction used to move from 1728 ordered pcsets to 364 unordered

pcsets in the triadic orbifold.

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identity element in every group, which leaves the other elements unchanged. Finally,

each member of the group must have an inverse.

The most familiar groups used in music theory are formed by transposition

and inversion. In group theory, these transformations are equivalent to the groups

formed by the rotation and reflection of symmetrical objects. The twelve

transpositions operating upon pitch-classes satisfy the requirements listed above; the

transposition of any pc results in another pc, (T1 + T3) + T4 = T8 = T1 + (T3 + T4),

T0 is the identity, and each transposition has an inverse mod 12. T6 is

its own inverse. The transpositions form a cyclic group which is equivalent to the

rotations of a dodecagon, C12. The combination of transpositions and inversions

forms a twenty-four-member dihedral group, D12. Each inversion is its own inverse,

and T0 remains the identity.

The T/I groups contain a number of subgroups. For example, the rotations

and reflections of a triangle form a dihedral group (D3) with six members: T0, T4,

T8, In, In+4, and In+8. These subgroups have fewer elements than the larger group,

but must fulfill the same requirements—closure, associativity, identity, and inverse.

As mentioned above, the 144 dyadic transpositional vlsets behave much like

traditional pcset transpositions. The group formed by the transpositional vlsets is

simply the direct product of this group and itself, C12 C12, essentially squaring the

group of transpositions. This produces a simply transitive Abelian group of order

144, with <00> as the identity. Although it is the product of two cyclic groups, the

group is not cyclic, as it must be generated by more than one element: {<01>, <10>}.

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There are 90 subgroups of C12 C12, including the full group, and trivial

identity group. Table 2-1 lists the subgroups by order, and provides a generating set

for each. The subgroups generated by singleton multisets are isomorphic to the

familiar groups of PC transpositions found in transformation theory: C2 {<66>}, C3

{<44>}, C4 {<33>}, C6 {<22>}, and C12 {<11>}. In most cases, the other

subgroups share features with the transposition subgroups of the same order. For

example, the remaining subgroups of order 4 are generated by <03>, <30>, <36>,

<39>, <63>, and {<06>, <60>}. The latter subgroup is the direct product of 2

subgroups of order 2. The subgroups with more than 12 elements are less familiar, as

they are all formed by multiplying smaller subgroups.

Vlsets, and the groups they form, operate upon any pcsets of the same

cardinality. Therefore, the subgroups create networks of pcsets. Figure 2-18 shows

the subgroup generated by vlset <17> with three sets of dyadic pcsets as objects. As

the figure shows, the group of vlsets operates independently of the specific pcsets

used as objects; the first set of pcsets is made up of the even whole-tone singletons

and the odd whole-tone tritones, while the second is made up of the even whole-tone

[02] sets, and the odd whole-tone [08] sets. The third set of pcsets begins with <01>,

but is similarly constructed; it contains the even whole-tone [01] dyads, and the odd

whole-tone [50] dyads. The group table is shown in Table 2-2.

Each corresponding collection of ordered pcsets forms a Generalized Interval

System along with the <17> subgroup. The pcsets form the space, and the subgroup

is the set of intervals between each ordered pair of sets in the space. Each subgroup

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36

and corresponding set of ordered pcsets forms a GIS in this fashion, including the full

group of 144 transpositional vlsets.61

The whole-tone features of the sets in Figure 2-18 result from the properties of

vlset <17> and the group it generates. The cycles created by intervals 1 and 7

alternate odd and even intervals, and overlap on the evens. This partitions the vlset

operations into even singletons and odd tritones—in this case, the tritone is the

difference between the intervals. The even singletons always maintain the whole-

tone membership of the pcsets they operate on, while the odd tritones always change

the whole-tone membership. As mentioned above, vlclass [06] is its own tn-class

involution; therefore, the specific pcsets the <17> group operates on belong to only

two tn-classes.

There are 288 dyadic voice-leading sets; these correspond to the ordered

dyadic pcsets, including multisets, which may be organized into vlclasses. The full

set of 288 vlsets form a semidirect product group, [C12 C12] C2. Vlset <00> is

the identity, each transpositional vlset has an inverse, and each inversional vlset is its

own inverse. The C2 group acts upon C12 C12 through the inverse automorphism.

This could be shown using 0 and 1; all elements of the group would be of the form

{a, b, 0} and {c, d, 1}, where a, b, c, and d are members of 12. Group operations

combine differently depending on whether they have a 0 or 1; {a, b, 0} + {c, d, 0} =

{a + c, b + d, 0}, {a, b, 0} + {c, d, 1} = {-a + c, - b + d, 1}, {a, b, 1} + {c, d, 0} = {a +

c, b + d, 1}, and {a, b, 1} + {c, d, 1} = {-a + c, -b + d, 0}. This is the same way in

61

Lewin 2007.

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which transposition and inversion operations combine. The vlset group

representation simply substitutes the lowercase “i” for “1”: <12> + <48> = <5t>,

<12> + <48>i = <36>i, <12>i + <48> = <5t>i, and <12>i + <48>i = <36>. This may

be easily verified using any ordered pc dyad, for example: <12> (<04>) = {16},

<48>i (<16>) = {32}, and <36>i (<04>) = {32}.

There are many subgroups of [C12 C12] C2. The 90 subgroups of C12

C12 are subgroups of the full group as well, of course, and also combine with

inversional vlsets to form other subgroups. The set of inversional vlsets in any

subgroup is calculated by adding any individual vlset to the cycle of transpositional

vlsets formed by the subgroup’s generator. The generator of each subgroup

essentially acts as a pair of interval cycles. For example, the transpositional vlsets in

the subgroup generated by <02> all contain a member of the 0-cycle—which is

always 0, of course—followed by a member of the even 2-cycle, 0, 2, 4, 6, 8, and t.

A set of corresponding inversional vlsets is generated by adding any particular vlset

to this transposition cycle. For n = , and vlsets <x,y> and <a,b>i, the

transpositional vlsets = <x*n, y*n>, and the inversional vlsets = <x*n+a, y*n+b>i.

Therefore, the inversional vlsets for the <02> subgroup are <a+(n*0), b+(n*2)>i, or

<a, b+2n>i, as addition by 0 has no effect. This produces the full set of inversional

vlsets for this subgroup: <a, b+(2*0)>i, <a, b+(2*1)>i, <a, b+(2*2)>i, <a, b+(2*3)>i,

<a, b+(2*4)>i, <a, b+(2*5)>i, <a, b+(2*6)>i, <a, b+(2*7)>i, <a, b+(2*8)>i, <a,

b+(2*9)>i, <a, b+(2*t)>i, and <a, b+(2*e)>i. If <25>i is arbitrarily chosen to serve

as an example, the remaining vlsets are <27>i, <29>i, <2e>i, <21>i, and <23>i along

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with <02>, <04>, <06>, <08>, <0t>, and <00>. The group table is shown in Table 2-

3.

For another example, the three two-member subgroups, generated by

<06> <̧60>, and <66>, respectively, each combine with inversional vlset pairs to

form many subgroups. <06> combines with any <a, b+6n>i to form a four-member

subgroup with <00> as the identity. See Table 2-4 for some example group tables.

Because addition by six is an involution, each generator partitions the 144 inversional

vlsets into 72 pairs; <a, b>i and <a, b+6>i, <a, b>i and <a+6, b>i, and <a, b>i and

<a+6, b+6>i. These produce 216 four-member subgroups.

The procedure used to generate subgroups above may be extended for any of

the remaining subgroups of C12 C12. Because inversional vlsets are their own

inverses, all 144 combine with the trivial subgroup consisting solely of <00>, to

produce two-member subgroups; an example is shown in Table 2-5. Table 2-6 lists

the subgroups formed by combining inversional vlsets with the subgroups of C12

C12; the order is listed first, then the transpositional generator and inversional vlsets,

and finally the number of subgroups. In order to save space, inversional vlsets are

shown as multiples of n, where n = .

Some of the subgroups shown below cannot be generated by a single element.

For these subgroups, the method for calculating inversional vlsets shown above is

insufficient. The two transpositional generators form the subgroup through all

combinations of their cycles, and the inversional vlsets must match all of these

combinations. For example, the subgroup generated by {<04>, <40>} includes the

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39

two cycles formed by these vlsets, <04> - <08> - <00> and <40> - <80> - <00>, as

well as all of their combinations: <44>, <84>, <48>, <88>. Each of these vlsets must

have a corresponding inversional vlset. On Table 2-6 below, these subgroups are

shown as simply adding some inversion <a,b>i to the full spectrum of vlsets

generated by the transpositional vlset cycles, for example: {<04>, <40>} + <a,b>i

should be understood as showing that the values of each transpositional vlset

generated are added component-wise to some inversional vlset. Table 2-7 shows the

group table for the {<04>, <40>} + <61>i subgroup.

The groups formed by vlsets of larger cardinalities are similar to those

discussed above. The transpositional trichordal vlsets form the direct product group

C12 C12 C12, of order 1728, with <000> as the identity; the group is generated

by {<001>, <010>, <100>}. Like the dyadic direct product group, this group is

abelian and simply transitive; that is, it is commutative, and any pair of pcsets is

spanned by a unique member of the group. Similarly, the full group of transpositional

and inversional vlsets is the semidirect product [C12 C12 C12] C2 of order

2456. Larger vlset cardinalities result in larger groups, of course.

Group Representations

Groups formed by traditional transpositions are cyclic groups—C2, C3, C4,

C6, and C12. These groups correspond to the rotational symmetries of regular

polygons. The order of the group is equal to the number of sides; C2 is the rotations

of a line, C3 the triangle, C4 the square, C6 the hexagon, and C12 the dodecagon.

Figure 2-19 shows the symmetries of the triangle and the accompanying C3 group.

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40

T4 is a single clockwise rotation, moving A to B, B to C, and C to A, T8 is a single

counterclockwise rotation, moving A to C, C to B, and B to A, and T0 is no rotation.

While these rotations change the positions of the vertices, they preserve the structure

of the equilateral triangle. If a rotation between T4 and T8 were added, such as T6,

the rotation would result in an upside-down triangle. Because this operation does not

preserve the symmetry of the triangle, it is not a member of C3.

Dihedral groups are made up of transpositions and inversions. These groups

also correspond to the symmetries of regular polygons, but include reflections in

addition to rotations. Figure 2-20 shows the three axes of reflection included in D3—

the symmetries of the triangle. As shown, each inversion maps one vertex into itself,

and the other two onto one another. The inversions are named for the vertex they

preserve, therefore IA maps B and C onto one another, IB maps C and A onto one

another, and IC maps B and A onto one another. When operating upon pcs or pcsets,

these inversions follow the cyclic structure of the transpositions, In, In+4, and In+8,

for any n in . The D12 group works the same way, but corresponds to the

rotations and reflections of a dodecagon. Figure 2-21 shows a dodecagon with pcs at

the vertices. A transposition Tn rotates the figure n*30 degrees clockwise, and an

inversion In reflects the figure about some axis, where each pc a maps onto the pc n –

a. For instance, I0 reflects the figure about a vertical axis, and each pc maps onto its

mod 12 inverse 0 – 0, 1 – e, etc.

The direct product group C12 C12 essentially multiplies the rotational

symmetries of the dodecagon with itself. This produces a dodecagonal torus—a

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41

dodecagon that traces the path of a larger dodecagon; see Figure 2-22. Each member

of a transpositional dyadic vlset operates upon either the vertical or horizontal

dodecagon of the torus. For example, vlset <23> describes a vertical rotation of two

vertices and a concurrent rotation of three. This maps each pcset onto the location

determined by the vlset.

The semidirect product group [C12 C12 C12] C2 includes reflectional

symmetries as well as rotational symmetries. Like the rotational symmetries, each

inversional vlset may be understood as inverting the vertical and horizontal

dodecagons at the same time. For example, vlset <00>i creates an axis of symmetry

that runs straight through the torus through pcsets {00}, {06}, {66}, and {60}, all of

which map onto themselves. The torus is then flipped about this axis, mapping each

pcset onto its <00>i—related pair. Other axes of symmetry are not as simple, as they

do not form straight lines through the torus. For example, <11>i produces different

axes of symmetry at different parts of the torus. It may be helpful to decompose the

inversion operation in order to make the representation simpler. Just as T5I produces

the same result as I5, <11>i may be interpreted as <11><00>i, where the torus is

flipped about the <00>i axis, then rotated by the transposition <11>.

The toroidal representation of vlset groups is somewhat unwieldy, and

virtually impossible to conceive of for higher cardinalities; the transpositional triadic

vlset group, C12 C12 C12, for example, corresponds to a dodecagonal 3-torus, a

figure which is created by gluing each of the opposing faces of a cube to one another.

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This requires four dimensions.62

As an alternative to the higher-dimensional tori, the

vlset groups may be represented by the combination of dodecagons rotating and

flipping next to one another. This is what the groups essentially represent—multiple

C12 and/or D12 groups operating at the same time.

Figures 2-23a – c show three dodecagons aligned with one another.

Transpositional vlsets slide each dodecagon by the specified amount, while

inversional vlsets flip each one. Figure 2-23b demonstrates that vlset <216> maps

pcset <000> onto {216}. The vlset is applied to the dodecagons from outside in; the

outer dodecagon rotates by T2, the middle by T1, and the innermost by T6. In this

particular representation, the pcset is always determined from the vertical box which

the vertices of the dodecagons rotate through. Transpositions rotate the dodecagons

counterclockwise to move their vertices through the box in ascending order. Figure

2-23c shows that vlset <358>i maps pcset <000> onto {358}; like the transpositional

vlset, the inversional vlset begins with the outermost dodecagon and works inward.

As the figure shows, each dodecagon is flipped about a different axis, reversing the

order of the pcs. This group representation could be created for any vlset cardinality,

as each vlset member operates upon a different dodecagon.

Although the natural toroidal group representations are unwieldy, and even

unrepresentable in higher dimensions, one advantage they have over the interlocking

62

The toroidal objects discussed in this section are the same as the orbifolds in the

work of Tymozcko 2006, and Callender, Quinn, and Tymoczko 2008. In their work,

the authors use permutational equivalence to reduce the dyadic torus to a Moebius

strip, and the four-dimensional triadic torus to a torus in the shape of a triangular

prism with a half twist.

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dodecagons is that each pcset is represented by some point on the figure. The

interlocking dodecagons must be arranged so that their alignment produces a

particular pcset. Figure 2-24 demonstrates the mapping <238> - <t27> - {053}; as

shown, the dodecagons must be arranged so that pcset 238} is aligned, before the

rotations of the vlset may be applied. One advantage of the toroidal representations is

that they show the vlset groups operating upon one figure, much like the cyclic and

dihedral groups they are products of.

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Chapter 3: Voice-leading Spaces

Voice-leading sets provide a way to describe voice leadings on their own

terms, as they do not privilege any specific voice leading characteristic, such as

uniformity, balance, efficiency, contrary or similar motion, etc. This chapter

demonstrates some of the ways in which voice-leading spaces may be used to model

and represent these disparate voice-leading types. By generalizing the concept of

voice-leading distance, it becomes possible to create spaces for individual analytical

or compositional contexts.

The voice-leading spaces produced here may be used as representational,

analytical, or compositional resources. Pitch-class voices may be seen as “surfing”

the pcsets of a voice-leading space.63

The paths these voices travel will change if the

voice-leading space is changed, demonstrating the use of different metrics of voice-

leading distance. As an analytical tool, a voice-leading space may aid in the voice-

leading interpretation of musical surfaces, as will be discussed in the next chapter.

Finally, the spaces may be useful for composition in a variety of ways. For example,

if a composer wishes to maintain the same voice leading at various locations within a

piece, the space shows all of the possible pcset pairs that correspond to that voice

leading. Alternatively, a composer could maintain the same path through multiple

spaces, transforming the individual voice leadings, but preserving the relationships

among them; this creates a type of voice-leading contour transformation. Just as

63

David Lewin suggests that members of set-class [013] surf an [013] tonnetz in the

subject of Bach’s fugue in F# minor, WTC I in “Notes on the Opening of the F#

Minor Fugue from WTC I,” Journal of Music Theory, 42, n. 2 (1998): 235-239.

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contour theory reduces individual melodies to their relative intervals, a voice-leading

contour reduction would reduce a passage to the transformations of its voice-leadings.

For example, imagine a three-chord passage which consists of two vlsets: X and

T2(X). If a given vlset Y substitutes for X, the chords will be transformed, but the

transpositional relationship between the voice-leadings—Y and T2(Y)—will be

preserved. Some ideas for future work in voice-leading contour will be suggested in

chapter 5.

Edward Gollin uses chess to describe the various conceptions of distance in

transformational music theories. Gollin distinguishes between Kingspace, which

involves motion between adjacent squares, and Knightspace, which is produced by

the Knight’s L-shaped motion.65

Possible musical analogs of these spaces are the

chromatic scale and circle of fifths, where the King moves by semitone, and the

Knight by fifth.66

The following will demonstrate some of the other chessboards

created by the remaining players—the vlsets.

The voice-leading spaces discussed in chapter 1, Tymoczko, Quinn, and

Callender’s orbifolds, Straus’ set-class space, and Roeder’s ordered interval spaces,

focus on harmonic objects organized in space. In the orbifolds, as well as Straus’

spaces, "voice leading" itself is understood as a means to connect these objects, and

voice-leading distances are measured by their total semitonal content. Although his

65

“Representations of Space and Conceptions of Distance in Transformational Music

Theories” (Ph.D. Diss., Harvard University, 2000): xii-xv. 66

Gerald Balzano compares the chromatic scale and circle of fifths to the motions of

the Queen and Knight, respectively, in “The Group-theoretic Description of 12-fold

and Microtonal Pitch Systems” in Computer Music Journal, 4, 4 (1980): 66-84.

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presentation of ideas is significantly different, Straus’ space is geometrically identical

to the orbifold spaces presented by Tymoczko, Quinn, and Callender.67

As the latter

authors show, the set of ordered pcsets in any cardinality, and corresponding multi-

dimensional space, may be reduced through various equivalences, including

permutation (pc order), transposition, and inversion. The dyadic Moebius strip,

triadic toroidal prism, tetrachordal tetrahedral torus, and higher-dimensional

structures are all the result of permutational reduction.68

In Joseph Straus' trichordal set-class space, shown in Figure 3-1, each set-

class is connected to the set-classes it may move to by a fuzzy transposition or

inversion with a single semitonal offset.69

Alternatively, these operations may be

thought of as employing voice-leading class <01>, <001>, <0001>, etc. under

rotation, transposition, and inversion. Thus, Straus uses a type of parsimony to

generate his space, but generalizes that parsimony; each line segment represents

vlclass [001], whose constituents are organized through T, I, and rotational

equivalence. For example, the voice leadings <223>, <232>, and <9tt>, when

applied to ordered pcset <026>, produce {249}, {258}, and {904}, members of set-

classes [027], [036], and [037], respectively; these relationships are demonstrated by

67

“Generalized Voice-leading Spaces” in Science, 320 (2008): 346-348. As

discussed above, Roeder’s geometrical models are identical in structure, but

interpreted differently. Rather than modeling voice-leadings, Roeder interprets the

coordinates of his spaces as ordered intervals, uses them to generate pcsets, tn-classes,

and set-classes, and discusses the properties of these harmonic materials. 68

For a discussion of this reduction, as well as a comparison with the theories of

George Perle, see Dave Headlam’s introduction to Theory and Practice, 33 (2008):

32-39. 69

Straus 2005, 45-108.

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the scs’ locations in the space: [026] is connected to [027], [036], and [037] (among

others). Straus then examines this space, as well as others in higher dimensions, and

explores the properties of the set-classes that it demonstrates.

Dyadic Spaces

The orbifold spaces proposed by Tymoczko, Quinn, and Callender provide a

starting point for the creation of voice-leading spaces. Figure 3-2a shows the full,

unreduced dyadic orbifold. As shown, each pair is an ordered pcset, and they are

connected by semitonal voice leading—<01> along the x-axis, and <10> along the y-

axis. The sum 0 inversions of the vlsets, <0e> and <e0> respectively, describe the

motions along these axes in the opposite direction. This space wraps around

vertically and horizontally to form a torus, discussed in the previous chapter as the

symmetrical object rotated and flipped by the semidirect product group of dyadic

vlsets [C12 C12] C2.

The torus is reduced to a Moebius strip through permutational equivalence; in

other words, (x, y) = (y, x). The space may be folded over along the diagonal, which

is made up of the singleton multisets, as shown on Figure 3-2b. Figure 3-2c

rearranges the pcsets to demonstrate the way in which the edges twist together. As

shown, the pcsets on the upper right side, {66}, {57}, {48}, {39}, {2t}, {1e}, and

{00}, map onto the pcsets on the lower left side with a twist. The other two sides are

the singleton multisets, which trace a circular path once around the torus, and once

through the center. This line becomes the edge of the Moebius strip when reduced

through permutational equivalence. Imagine that the line of multisets is a rubber

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band. When the torus is reduced, this rubber band is twisted and realigned with itself,

so that tritone-related multisets are directly across from one another, as shown in

Figure 3-3. The line of multisets is a torus knot, a continuous line which wraps

around the outside of a torus. The Moebius strip lies in the plane formed between

each tritone-related pair as shown on Figure 3-4a.70

As Figure 3-4a shows, twelve lines of pcsets stretch across the Moebius strip.

The pcsets on these lines all add up to the same sum. Each line is a diagonal on the

toroidal space, corresponding to y = -x + n, where n = . For example, the line

between {00} and {66} is the diagonal y = -x + 0, and contains the zero-sum dyads

{1e}, {2t}, {39}, {48}, and {57}. The five other even-sum lines are transpositions of

this line, just as the six odd-sum lines are transpositions of one another. The

singletons form the edge of the strip, because they can only map onto two other

unordered pcsets by either rotation of the spaces’ generating voice leading. For

example, pcset {33} moves by <01>, <10>, <0e>, and <e0> to either {34} or {23}.

Non-singleton pcsets move to one of four unordered pcsets; for example, pcset {34}

may move to {35}, {44}, {33}, or {24}.71

70

Callender, Quinn, and Tymoczko 2008, and Tymoczko 2006 and 2011 discuss this

space, but only show the Moebius strip in its unfolded form. See “The Geometry of

Musical Chords,” in Science, 313 (2006): 72-74 and A Geometry of Music: Harmony

and Counterpoint in the Extended Common Practice (New York: Oxford University

Press, 2011). Rachel Hall states that the Moebius strip is twisted three times,

therefore the torus knot is a trefoil in “Geometrical Music Theory,” Science, 320

(2008): 328-329. 71

This space intersects with some of the theories of George Perle. Each of the same-

sum-dyad diagonals is equivalent to what Perle calls an inversionally complementary

cycle; Perle first shows these cycles as the combination of chromatic scales, as in the

orbifold, then demonstrates that they are reordered in the order-position pairs formed

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The dyadic Moebius strip may be further reduced to a set-class line by

inversion, as shown on Figure 3-4b. This is equivalent to Straus’ set-class space, as

well as the tn-class cycle shown in the previous chapter, Figure 2-10a. The original

torus may also be reduced by transpositional equivalence, which results in a circle.

As mentioned above, the orbifolds locate pcsets by maximal harmonic

similarity. Voice leadings may be traced from one pcset to another as pitch-class

intervals. These intervals are easily understood as transpositions, but are more

difficult to translate to inversion, as the indices of inversion are not consistent among

adjacent pcsets. As Straus shows, all adjacent pcsets may be inverted onto one

another by some index with minimal offset. This is logical, as adjacent pcsets only

differ by one semitone. In Straus’ fully reduced set-class space, each set-class is

connected to those which it transposes or inverts onto with minimal offset. In terms

of vlsets, the dyadic orbifold can be seen as the space of <01>, and vl-class [01]i. All

inversions with minimal offset are simply some transposition or inversion of this vl-

class, such as <23>i, <65>i, etc.

The dyadic orbifold provides a model by which voice-leading spaces may be

created for other vlsets. These spaces differ from the orbifolds in the way in which

voice-leading distance and spatial distance are aligned; most of the spaces do not

equate voice-leading distance and semitonal offset. Rather, voice-leading parsimony

is defined by the vlset which generates the space. In other words, the space’s “unit”

distance is the generating vlset.

by two inversionally-related twelve-tone rows; Twelve-Tone Tonality 2nd

ed.

(Berkeley: University of California Press, 1996) 7-11.

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The toroidal representation of dyadic pcsets, unrolled in Figure 3-2, can be

seen as generated by vlset <01> and pcset <00>. The x-axis is determined by vlset

<01>, the y-axis by <10>, and pcset <00> is placed at the origin. All other pcsets in

the space are generated by these elements, as all adjacent pcsets are related by single

semitone voice leading. In addition, adjacent pcsets map onto one another by some

member of vlclass [01]i. As Figure 3-2 shows, the transposition-class cycle shown in

chapter 2, Figure 2-10 forms a line along the x-axis. In fact, all lines parallel to the x-

axis contain some rotation of this cycle, and as well as all lines parallel to the y-axis.

Other vlsets may be substitued for <01> to generate their own voice-leading

spaces. To return to the chess analogy, if <01> produces Gollin’s Kingspace, <05>

could be used to produce Knightspace.72

For example, if vlset <05> is used with

pcset <00> at the origin, the resulting space locates pcsets by single-voice fifth voice

leading, rather than semitone. All adjacent pcset pairs are mapped onto one another

by some member of vlclass [05]i. This particular space, shown in Figure 3-5a, is an

M5 transformation of the original space. Just as the <01> space is reduced by

permutational equivalence to a Moebius strip, the <05> space may be folded along

the x=y diagonal and wound into a Moebius strip, as shown in Figure 3-5b. As on the

full space, adjacent pcsets are mapped onto one another by single-voice motion by

fifth, which corresponds to the rotations of vlsets <05> and <07>. Figures 3-6a and b

72

This is for conceptual comparison only—Gollin’s Kingspace is a two-dimensional

5x5 chessboard, but Knightspace is a four-dimensional object. The <01> and <05>

spaces defined here are similar in structure.

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show the tn-class circle and set-class line produced by vlset <05>. Like the toroidal

space and Moebius strip, the figures are simply M5 of the <01> spaces.73

Although the space in Figure 3-5 is generated by vlset <05>, this is by no

means the only voice leading the space represents. Moving west parallel to the x-axis

inverts the vlset—<07>. Moving south parallel to the y-axis similarly produces

<70>. <05> is simply the space’s unit distance. The voice leading between any two

pcsets on the space can be measured in terms of this vlset. For example, two unit

distances (<05> + <05>) along the x-axis produce the vlset <0t>. This is also true for

inversional voice leadings; the unit distance is a member of vlclass [05]i, and a

straight line of four unit distances is a member of vlclass [04]i. Where n = ,

moving along any y = x + n diagonal is equivalent to transposition by the total sum of

the vlset, and motion along any y = -x + n diagonal is equivalent to wedge motion by

the difference between the members of the vlset.

The unit distance on a voice-leading space demonstrates the contextualization

of voice-leading parsimony. Within the <05> space, single-voice motion by fifth is

parsimonious, and single-voice motion by semitone is not. As Figures 3-5a and b

show, semitonal offset requires five moves; only single-voice motion by tritone

requires more steps. This is summarized nicely by the tn-class circle and set-class

73

This equivalence is also shown by Walter O’Connell. Using the all-interval-class

tetrachords as a model, O’Connell creates two M-related six-dimensional interval

spaces, which he calls tone-lattices; he then demonstrates the relationships among a

number of geometrical figures—symmetrical pitch-sets—on the spaces, as well as an

all-interval twelve-tone row as a bent-wire figure. He then describes the intervallic

transformations that result from rotations of the figures within the lattices. See “Tone

Spaces,” Die Reihe, 8 (1968): 53-64. Originally published in German in 1962.

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lines shown in Figures 3-6a and b. The minimum distance from ordered tn-class [00]

to [01] or [10], for example, is five. Figure 3-6c shows a dyadic passage among

adjacent pcsets in the space shown in 3-5b; to produce a smooth motion on the space,

one voice moves by ic 5 while the other maintains its pc. Figure 3-6d shows another

passage which mixes smooth and compound moves in the space.

Toroidal voice-leading spaces may be produced by the combination of any

vlset <ab> and any pcset <cd>. The x-axis is determined by n*<ab>(<cd>), where n

= , and the y-axis by n*<ba>(<cd>). Adjacent pcset pairs may be mapped onto

one another by some member of vlclass [ab]i. Any two pcsets on the space are

related by some member of the subgroup generated by {<ab>, <ba>}, and combined

with their inversional equivalents.

Although any transpositional vlset may be used to generate a voice-leading

space, not all spaces include the full collection of pcsets.74

Complete spaces may

only be generated by vlsets whose sum and difference is prime to twelve—1, 5, 7, or

e. All others produce incomplete spaces. For example, the sum of vlset <2e> is

prime to twelve—1, but the difference is not—3 or 9. As a result the y = -x + n

diagonals of the space include some redundancy. As Figure 3-7a shows, these

diagonals are formed by <2e> - <e2> = <39>. The cycle produced by vlset <39> has

only four members, therefore the diagonals only contain four distinct pcsets each.

74

In a discussion of K-nets and Perle-Lansky cycles, David Lewin describes a

method of moving from one space to another as “hyperwarping,” then demonstrates

some ways in which a space may be transformed into another in “Thoughts on

Klumpenhouwer Networks and Perle-Lansky Cycles,” in Music Theory Spectrum, 24,

n. 2 (2002): 228-230.

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The interaction of the two cycles of the vlset along the x-axis also shows redundancy;

the fourth member of the 2-cycle is 8, and the fourth member of the e-cycle is 8. If

{00} is at the origin, the fifth pcset along the x-axis is {88}; this pcset is replicated

along the y = x diagonal.

The <2e>, <00> voice-leading space shown in Figure 3-7a is different than

the spaces discussed thus far. A 4 x 12 box has been placed on the figure, which

shows the 48 discrete pcsets of the space.75

When forming a torus, the 12 pcsets of

the y axis wrap around vertically. The x axis, however, requires more maneuvering.

As the figure shows, the fourth pcset of the x axis, <69>, proceeds to pcset <88>, not

<00>. <88> is the fourth pcset along the y axis, therefore, the cycle begun along the

x axis overlaps with the cycle along the y axis. Four pcsets later in the cycle involves

another shift up four pcsets along the y axis, to {44}. Another four pcsets later, the

cycle returns to <00> at the origin. As discussed above, the <01> torus is made up of

twelve vertical and twelve horizontal dodecagons; pcsets are located at their vertices,

which intersect. In the <2e>, <00> space, the dodecagons overlap. Figure 3-7b

shows the torus formed by the space. The four y axis cycles are shown as vertical

dodecagons, which must be rotated around the torus, so that the x axis cycles are

aligned. These cycles are dodecagons that have been folded twice, as their twelve

vertices are now grouped in the four vertical planes of the y axis cycles.

75

This box could be drawn in other ways as well, such as 12 x 4. As long as the box

has these dimensions, it will produce 48 unique pcsets no matter where it is located.

All of these are equivalent.

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Table 3-1 lists the dyadic spaces by vlset, demonstrating which are complete

by including the number of distinct ordered pcsets included in the toroidal space

generated by each vlset. Vlsets which are sum-zero inversions of one another are

aligned horizontally. These spaces are simply reflections of one another, provided

inversionally-related sets of pcsets are used as objects. Complete spaces are

unchanged no matter which pcset is placed at the origin, and their inversional

equivalents can always be mapped onto one another through flips and rotations.

Table 3-1 also demonstrates that vlsets related by multiplication generate spaces with

the same properties.

Many vlsets partition the pcsets into multiple collections, which require

multiple spaces to represent. For example, Figure 3-8 shows the toroidal space

generated by vlset <02> with pcset <00> at the origin. As the figure shows, the space

only includes even pcsets, that is, only those formed by the even whole-tone scale.

This results in a space which is six by six, rather than 12 by 12, one-quarter of the

size of the <01> space. Unlike the <2e>, <00> space shown above, the x and y

cycles of the <02> space do not overlap. As a result, the spaces fit onto a torus

formed by six vertical and six horizontal hexagons.

The small size of the <02>, <00> voice-leading space is a result of the

symmetries of the whole-tone scale. Although adjacent pcsets must be related by

either <02>, <20>, <0t>, or <t0>, any pcset pairs in the space are related by some

member of the subgroup generated by {<02> <20>} and combined with inversion.

This subgroup has 72 members—36 transpositional vlsets and their corresponding

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inversional vlsets—which results in four distinct voice-leading spaces of 36 pcsets

each. The four spaces are determined by the whole-tone content of their pcsets. Two

of the spaces contain pcsets made up of one whole-tone scale; in addition to the all-

even space shown in Figure 3-8, the all-odd space, shown in Figure 3-9, may be

generated with pcset <11> at the origin. The other two spaces contain all pcsets made

up of both whole-tone scales, one space is even-odd, and the other odd-even—<01>

and <10> are representative pcsets (see Figures 3-10 and 3-11). Due to the group

structure of the spaces, any pcset member of a space may be placed at the origin, and

the full space will result. For example, if pcset <76> is at the origin, a rotated version

of the full odd-even space is generated. As these spaces are toroidal, the rotations

show that they are equivalent.

The toroidal <02> spaces shown in Figures 3-8 and 3-9 may also be reduced

to Moebius strips. Like the spaces they are reduced from, they contain fewer pcsets

than the <01> version. Figure 3-12 demonstrates the Moebius strip formed by <02>

with <11> at the origin. As the figure shows, there are only six lines of pcsets on this

space. Like the <01> Moebius strip, the singletons form a trefoil knot, and each line

of pcsets between them adds to the same sum. Each blue line on the space represents

the rotations and sum-zero inversions of vlset <02>: <02>, <20>, <0t>, and <t0>.

Not all of the <02> spaces may be reduced to a Moebius strip. Figures 3-10

and 3-11 show the toroidal voice-leading spaces formed by <02> with pcset <01> and

<10>, respectively. As mentioned above, vlset <02> preserves the whole-tone

contents of any pcset it operates upon. Therefore, each pcset in the spaces is made up

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of one odd and one even whole-tone member. All pcsets in the <01> space are even-

odd, while all the pcsets in the <10> space are odd-even. Therefore, no pcsets may be

reduced by permutational equivalence. Instead, the two spaces are the permutational

equivalents of one another. Of course, either space may be reduced by transpositional

and/or inversional equivalence, but these form a circle (or hexagon), and line,

respectively, as shown in Figures 3-13a and b. Like the toroidal spaces, vlset <02>

produces multiple transpositional and inversional spaces, and pcset members of one

space cannot be linked to the other by any rotation or inversion of vlset <02>. In fact,

they cannot be linked by any transposition of <02> either; as mentioned in chapter 2,

vlclass [02] produces the transposition-class cycles shown in Figure 3-13a. Although

the toroidal spaces produced by vlsets <02> and <24>, for example, are different,

they both reduce to the transposition-class and set-class spaces shown below.

It is useful to describe transformations between voice-leading spaces, as a

given piece may “modulate” from one to another, or back and forth between a set of

spaces. As discussed above, inversion and multiplication are replicated throughout a

voice-leading space. This is only true for sum-zero inversion, and multiplication

without transposition. Other indices of inversion, and multiplication with

transposition change the sum of a vlset, therefore the changes to the voice-leading

space produced are more drastic.

As discussed in chapter 2, the transposition of a vlset—excluding T0—

produces a new vlset which maps the set of transposition-classes onto itself in the

same way. For example, <02> maps pcset <23> onto <25>, and <68> maps <23>

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onto <8e>; both vlsets map the members of tn-class [01] onto [03], and the difference

between the mappings is equal to the transposition of the vlset. That is, <8e> – <25>

= <66>, and <02> + <66> (or T6) = <68>. This is replicated in the voice-leading

spaces produced by any vlsets related by transposition. T6 is a unique

transformation, in that it preserves the sum of the dyadic set it operates upon, as 6 + 6

= 12, and preserves the difference between the members of the set by adding six to

both of them. This produces pairs of voice-leading spaces in each sum which are

related to one another by <66>.

As Figure 3-14 shows, the toroidal space generated by <68>, <00> is similar

in structure to that formed by <02>, <00> in Figure 3-8. Although some of the pcsets

have been relocated, the transposition-class structure is preserved. Motion along a

diagonal in either space is the same, because <02> + <20> = <68> + <86> = <22>.

Motion parallel to the x or y axis alternates transposition by <66> with <00>. The

cause of this feature is easily shown if vlset <68> is split into (<66> + <02>): <02> +

<02> = <04>; <66> + <02> + <66> + <02> = <04>. Any even number of vlset

<66>s sums to <00>. The result of these relationships is a space in which the even

diagonals of the <02> space are preserved, and the odd diagonals are rotated by <66>.

Figure 3-15 shows the space generated by <67>, <00>, for comparison with

Figure 3-2a, the <01>, <00> space. Like the <02> and <68> spaces, every other

diagonal is transposed by <66>. The <67>, <00> space may be reduced to a Moebius

strip as shown in Figure 3-16. As mentioned above, the pcset lines on the space are

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the diagonals y = -x + n. Therefore, the odd-sum lines are <66> transpositions of the

corresponding lines in the <01> Moebius strip, as shown in Figure 3-4a.

Nearly all vlsets paired by <66> generate similar spaces. Members of vlclass

[06] are exceptions, as <66> simply transforms them by rotation; this swaps the x and

y axes, producing a space where each pcset is a rotation of the corresponding set on

the other space. Some vlsets whose sums are 4, 8, and 0 also form exceptions,

because transposition by tritone drastically changes the pc cycles they form. For

example, vlset <04> produces a space which includes nine pcsets, while <6t>

includes eighteen. The other unequal sum-four pair is <22> and <88>, for the same

reason—the 2-cycle has six members, while the 8-cycle has 3. The sum-eight

unequal pairs are the inversions of these pairs: <08> and <26>, and <44> and <tt>.

The final exceptions are the sum-zero vlsets <00> and <66>, and <2t> and <48>.

Transpositions which change the sum of a vlset may change the size of the

voice-leading space that set produces. However, the spaces will still have the same

transposition-class structure. As mentioned above, the tn-class cycles of a vlclass

form the lines parallel to the x and y axes of the toroidal space. As shown in chapter

2, only ordered vlclasses [01], [05], [50], and [10] form complete tn-cycles. Because

the cycles are complete, any transposition of the vlclass combined with any pcset at

the origin will produce rotations of the vlclass’s cycle in lines parallel to the x and y

axes. This is true even for spaces which produce pcset redundancies, such as the

voice-leading space <12>, <47>, as shown in Figure 3-17. As the figure shows, the

vlclass [01] tn-class cycle along the x-axis begins with tn-class [03], and continues

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through the remainder of the cycle to [02]. Just as in the <01> space, each line

parallel to the x axis is a rotation of this cycle, and each line parallel to the y axis is a

retrograde rotation.

The <12>, <47> toroidal space in Figure 3-17 contains 48 unique pcsets,

which have been boxed. This space is similar to the <2e>, <00> space shown above,

in that the x and y cycles overlap. The two spaces’ tori can both be represented using

four dodecagons with a twist. In this case, the dodecagons are twisted downward, as

the cycle along the x axis leaps to the eighth pcset of the y axis, and then the fourth

before returning. Because x + y = <33>, y = x diagonals begin to repeat after four

pcsets.

As shown in chapter 2, the remaining vlclasses form multiple tn-class cycles.

As mentioned above, these vlclasses also form multiple voice-leading spaces. The tn-

cycles these spaces produce is dependent upon the pcset used to generate the space.

For example, vlclass [02] produces two tn-class cycles, one which contains the even

tn-classes, and one odd. As Figures 3-8 through 3-11 demonstrate, if an even tn-class

is placed at the origin, such as <00> or <11>, the space produces the even cycle.76

If

an odd tn-class is at the origin, such as <01> or <10>, the space produces the odd

cycle. These two tn-class cycles are produced by any transposition of vlclass [02],

and the voice-leading space may be more or less complete depending on the sum of

the vlset. As table 3-1 shows, the odd sums contain all of the odd set-classes, while

the even sums contain all even set-classes. Therefore, each odd sum contains two

76

Although {11} is an odd pcset in terms of whole-tone membership, it is a member

of the even tn-class [00].

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members of vlclasses (under transposition and inversion) [01], [03], and [05]; each

even sum contains two members of [00], [02], and [04], and one member of [06].

The two members of each vlclass in each sum are paired by <66>, as mentioned

above, and vlclass [06] maps onto itself under <66>.

As mentioned above, the sum-zero inversion of a vlset, <01> and <0e> for

example, simply retrogrades the x and y axes of a voice-leading space. In terms of

the torus, this is a reflection about the sum-zero axis. Inversion by other sums is

equivalent to this reflection, plus some rotation produced by transposition. As in the

discussion of transposition immediately above, this may change the size of the space,

but preserves the tn-class cycles it is made up of. The inversion, or reflection,

retrogrades the tn-cycles. For example, vlset <02> is transformed by <44>i into

<42>; Figure 3-18a shows the toroidal space <42>, <11>. A comparison of this

space with that of <02>, <11>, shown in Figure 3-9, demonstrates that the spaces

contain the same tn-class cycles in opposite order. Of course, the <42> space only

contains 12 pcsets, which are repeated throughout the 12 x 12 grid. The <02> space

has been reduced, but also would repeat itself in the full 12 x 12 space.

The <42>, <11> space is another in which the x and y cycles overlap. Due to

the small size of the space, the cycles form hexagons, rather than dodecagons. This is

not easily represented as a torus. As an alternative, the y hexagons may be aligned,

and the two x hexagonal paths traced between them, as shown in Figure 3-18b. If the

adjacent vertices of the hexagons were connected, this would form a hexagonal prism.

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As mentioned above, multiplication is another way to navigate among voice-

leading spaces. Multiplication operates very much like inversion. When it is not

combined with transposition, multiplication preserves the structure of a voice-leading

space, and can be seen as operating upon each individual pcset; this is true whether

the multiplication operation maps the chromatic scale to the circle of fourths or fifths.

When combined with transposition, however, multiplication may change the sum of

the vlset it operates upon, and thus change the size of the voice-leading space it

produces.

In the following discussion, TnM5 will denote the mapping of the chromatic

scale to the cycle of fourths, and vice versa, followed by Tn; TnM7 will denote the

mapping of the chromatic scale to the cycle of fifths, and vice versa, followed by Tn.

For example, T1M5 maps vlset <45> onto <92>, as 5 * <45> = <81>, and <81> +

<11> = <92>; TtM7 maps vlset <45> onto <29>, as 7 * <45> = <4e>, and <4e> +

<tt> = <29>. Using TnM5 and TnM7, any member of vlclass [01] or [10] may be

mapped onto any member of [05] or [50], and vice versa. Although multiplication

may be used for other vlclasses, it is equivalent to either transposition or inversion.

For example, T3M5 maps <48> onto <e7>, which is equivalent to <33>i.

It is unnecessary to show all of the dyadic voice-leading spaces here.

However, it is helpful to show some representatives of the different sizes of possible

spaces. Figure 3-19 shows the Moebius strip reduction of Figure 3-17. The four

singletons—<11>, <44>, <77>, and <tt>—do not reduce by permutational

equivalence, but the remaining 44 reduce to 22, forming a Moebius strip of 26 pcsets.

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The strip contains four lines of pcsets, which correspond to the y = -x diagonals of the

torus; each line contains all pcsets of the same sum. The sums follow the cycle given

by the vlset’s sum, 3. The generating pcset <47> sums to e, so the remaining sums

are 2, 5, and 8.

A small space—only 12 pcsets—is the <42>, <11> torus shown in Figure 3-

18. Like the previous space, this space may also form a Moebius strip, but only

contains 7 pcsets when reduced through permutational equivalence. This space has

only two lines of pcsets, which correspond to sums 2 and 8. Figure 3-20 shows the

reduced space. The two lines shown in the figure would be on opposite sides of the

Moebius strip; for ease of reading, they are shown as parallel.

Many of the small voice-leading spaces do not translate to a Moebius strip as

well as other, simpler shapes. Generally, spaces with less than twelve pcsets are

better represented some other way. For example, vlset <00> produces spaces with

single pcsets, which translates into a single point. Spaces with two pcsets, such as

those formed by vlset <66>, reduce either to a line, or a point, depending on the

pcsets involved. If a member of set-class [06] is a generating pcset, <66> will map

the pcset into itself under permutational equivalence. Toroidal spaces with three

pcsets may be represented by triangle, or a line if permutational equivalence is

involved. For example, the toroidal space <48>, <00> includes three pcsets, <00>,

<48>, and <84>, which reduces to <00> and <48> by permutation. The <48>, <01>

space—<01>, <49>, <85>—does not reduce further.

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Some three-member spaces are not best expressed by a triangle, but a line,

such as the space formed by <06>, <11>, which includes pcsets <11>, <17>, <71>,

and <77> (see Figure 3-21). The space could be further reduced through

permutational equivalence. Both <11> and <77> are connected to <17> by the

generating vlset, but require two statements of the vlset to map onto one another. Of

course, vlset <06> also generates spaces that are not reducible, such as <06>, <01>,

which is made up of pcset <01>, <07>, <61>, and <67>. Figure 3-22 shows the

space as a square. This is the toroidal space—it cannot be reduced by permutation.

Because there are only four members, the circles which connect the pcsets can be

simplified to lines.

The voice-leading spaces formed by singletons may be represented by the

familiar geometric figures of their traditional transpositional counterparts; <11>,

<00> forms a dodecagon, <22>, <00> a hexagon, etc. These spaces are generally not

reducible by permutation, except for the special case of set-class [06]. For example,

<33>, <28> has four members, <28>, <5e>, <82>, <e5>, which reduce to two under

permutation equivalence. This changes the space from a square to a line. Other

twelve-member toroidal spaces generally reduce to one or two lines of pcs, if they are

reducible. As in the other spaces, the lines are made up of the pcsets which belong to

the same sum. Sum-zero vlsets, such as <1e> and <57>, produce pcsets in a single

sum, and therefore may reduce from a circle to a line, as shown in Figure 3-23.

Two vlsets form toroidal spaces with 18 members, <26> and <6t>; as

inversions, these vlsets form spaces which are reflections of one another. Because the

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vlsets are members of vlclass [04], they produce four distinct tn-class cycles, as

discussed in chapter 2. Therefore, the vlsets produce four types of voice-leading

spaces. As a result of the 2-cycle in the vlset, each space preserves the whole-tone

membership of the generating pcset. This produces two versions of each basic space,

just as in the <02> spaces discussed above. Therefore, there are eight spaces of

eighteen pcsets each, adding up to the total of one hundred and forty-four.

Figure 3-24 shows the <26>, <11> toroidal voice-leading space. The boxed

portion of the space shows the 18 unique pcsets repeated throughout the figure. Like

the previous spaces, the x and y cycles overlap; in this case, they form six hexagons.

The six singletons do not reduce by permutational equivalence, but the remaining

twelve pcsets reduce to six. The space’s unusual structure results in a different

Moebius strip organization than shown by the other spaces. As Figure 3-24 shows,

both the y = x and y = -x diagonals contain only three distinct pcsets. The y = -x

diagonals are reduced to 2 through permutational equivalence. In the previous

spaces, all the pcsets of a particular sum are contained in a single y = -x diagonal.

These diagonals then become lines of pcsets across the Moebius strip. In the <26>,

<11> space, however, the pcsets of a particular sum are split into multiple diagonals.

For example, the diagonal containing pcset <77> contains the other sum-two dyads

<e3> and <3e>, but not <11>, <59>, or <95>. Although there are only three sums

represented by the pcsets of the space—2, 6, and t—they are divided into six distinct

diagonals. This is due to the fact that the voice leading represented by descending

along the diagonal is x – y, <26> - <62>, or <84>. <84> is a highly symmetrical

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vlset, which maps the singletons onto [04] dyads, and the [04] dyads onto themselves

(under permutation) or back onto the singletons: <84> + <11> = <95>, <84> + <95>

= <59>, and <84> + <59> = <11>. Therefore, the two pairs of the same sum cannot

be mapped onto one another by this voice leading. The voice leading x – y forms the

lines of the Moebius strip, so as a result, the reduced space contains six pairs of

equally spaced pcsets. The two pairs of the same sum are located across from one

another.

Trichordal Spaces

Trichordal voice-leading spaces share many properties with the dyadic spaces.

It is more difficult to represent them, however, given that the toroidal spaces are four-

dimensional hypercubes. Through various equivalences, such as rotation,

transposition, inversion, and permutation, many of these spaces may be reduced to

smaller structures. Figure 3-25 shows the hypercube formed by <001>, <000>.77

The x, y, and z axes are generated by the three rotations of the vlset—<001>, <010>,

and <100>. As discussed above, this space may be reduced through permutational

equivalence, from 1728 pcsets to 364. Permutational equivalence includes rotation;

for example, the permutations of a, b, and c in the set <abc> include all six orderings.

Rotational symmetry alone differentiates between orderings of the set, therefore

<abc> is equivalent to <bca> and <cab>, but distinct from <acb>, <cba>, and <bac>.

Three-member trichordal sets thus reduce 6:1 through permutation, and 3:1 through

rotation alone.

77

This hypercube is equivalent to the unreduced trichordal orbifold.

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The diagonal running through the origin contains the singleton multisets,

<000>, <111>, <222> etc., reflecting the sum of the voice leadings along each axis: x

+ y + z = <111>. This line, as in the dyadic Moebius strips, becomes a torus knot on

the edges of the triangular prism. The pcsets are sorted by their sums and placed in

twelve triangular planes. The least symmetrical pcsets, are placed at the edges, and

the most symmetrical sets at the center. The reduced trichordal orbifold is shown in

Figure 3-26.78

As in the dyadic spaces above, one may use alternative voice leadings to

construct spaces that are similar in structure to the orbifold. Like the dyadic spaces,

many of the trichordal vlsets produce incomplete spaces through pcset redundancy.

Table 3-2 lists the vlsets by sum and the size of the spaces they create. Vlsets are

grouped by T4, as well as inversional equivalence. Transposition by T4 and T8

produces the members of a vlclass who share the same sum. This creates networks of

T4-related vlsets whose spaces have the same properties. Like the dyadic spaces

paired by T6 above, these spaces share the same tn-class locations, and transform the

pcsets according to the relationship between the vlsets. T4 cycles the diagonal planes

of a space by T4, while T8 cycles them by T8. For example, diagonal planes 1, 4, 7,

and 10 in the <445> space are identical to the same planes in the <001> space

(assuming the same pcset is placed at the origin), diagonal planes 2, 5, 8, and 11 are

transformed by T4, and 3, 6, 9, and 12 are transformed by T8.

78

Callender et al. 2008.

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As discussed above, sum-zero inversions simply flip a voice-leading space.

Alternatively, the sum-zero inversion of a vlset describes motion within a space in the

opposite direction. For example, vlset <001> maps pcset {024} onto {025}, while

vlset <00e> maps {025} onto {024}. In addition, inversions I4 and I8 preserve a

vlset’s sum. These vlsets are equivalent to the sum-zero inversion of the T4 or T8

transformation of the vlset. Therefore, the vlsets may be grouped by the Dihedral

group D3, to form networks of six spaces that have similar properties. These

networks are listed in each line of Table 3-2. As the table shows, the only vlsets that

do not participate in such a network are those of vlclass [048], as they map onto

themselves under transposition or inversion by multiples of index number 4.

Equivalent permutations have been left off the table, such as <015> and

<051>. The rotations of a vlset are expressed in its space, so <015> represents both

<150> and <501>. The permutations are left off of the table because they produce

spaces in which the pcsets are also permutations of the other space; that is, pcset

{02t} in the first space is replaced with {0t2} in the second.

As table 3-2 shows, each sum contains ten sets of three vlsets related by

T4/T8. Sums 0, 3, 6, and 9 also contain the augmented vlsets—<048>, <159>,

<26t>, and <37e>—for a total of thirty-one. Like the dyadic vlsets, complete spaces,

that is, spaces with all 1728 ordered pcsets, are only formed by vlsets whose sums are

1, 5, 7 or e; they must also be a member of vlclass [001], [013], [025], [037], [016],

[034], or [005], under transposition and inversion. Sums 1, 5, 7, and e produce three

vlclasses which do not generate the full collection of pcsets due to the interaction of

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their cycles: [002], [004], and [026]. These vlclasses are all-even, meaning they

preserve a pcset’s whole-tone membership. [002] and [026] create two distinct voice-

leading spaces, much like dyadic vlclass [02] above. Vlclass [004] is even more

limiting, preserving the augmented triad content of a pcset. Let a, b, c, and d

represent the four augmented triads, where b = T1(a), c = T2(a), and d= T3(a). The

five distinct spaces of tn-classes may be represented by their augmented triad make-

up; 1) [aaa], such as [000], [004], and [048], 2) [aab], such as [045], [001], and [037],

3) [aac], such as [002], [026], and [024], 4) [aad], such as [003], [034], and [015], and

5) [abc], such as [012], [016], and [036].

Transposition increases the sum of a vlset by some multiple of three, as it adds

the same number to each member of the vlset. For example, a sum-one vlset such as

<02e> is mapped by T1 (or <111>) onto a sum-four vlset, <130>. Therefore, vlset

sums linked by three-cycles have the same vlclass content. As the table shows,

vlclasses of sum 1, 4, 7, and t are the same, as well as their inversions, sums 2, 5, 8,

and e. The remaining three-cycle—sums 0, 3, 6, and 9—are made up of vlclasses

[000], [003], [006], [012], [015], [024], [027], [036], and [048] under transposition

and inversion.

None of the vlclasses within sums 0, 3, 6, or 9 produce spaces with all 1728

pcsets, as their sums are factors of twelve. They do, however, differ in the number of

discrete tn-class spaces produced. Vlclass [000] is equivalent to traditional

transposition, and preserves the tn-class of any pcset it operates upon. [006]

preserves the tritone content of a pcset, and therefore links tn-classes whose

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constituents belong to the same tritones. For example, let a, b, c, d, e, and f stand for

tritones, where b = T1(a), c = T2(a), d = T3(a), e = T4(a), and f = T5(a). The non-

intersecting tn-class spaces created by [006] may be represented by 1) [aaa], 2) [aab],

3) [aac], 4) [aad], 5) [aae], 6) [aaf], 7) [abc], 8) [abd], 9) [abe], and 10) [ace]. [003]

and [036] both preserve the diminished-seventh membership of a pcset. Let a, b, and

c represent diminished-seventh chords, where b = T1(a), and c = T2(a). The distinct

tn-class spaces are given by 1) [aaa], 2) [aab], 3) [aac], and 4 [abc].

The remaining tn-class spaces are more unusual than those discussed above.

[024], like [002], preserves the whole-tone content of a space; however, due to the

interaction of its cycles, it generates six distinct tn-class spaces—three each of the

whole-tone tn-classes and mixed whole-tone tn-classes. Although it contains three

distinct cycles, [012] behaves very much like [003], in that the tn-class spaces it

produces are partitioned in the same way as the 3-cycles of sums listed in Table 3-2.

That is, [012] generates three separate tn-class spaces, one with the sum-0, 3, 6, and 9

tn-classes, one with the sum-1, 4, 7, and t tn-classes, and one with the sum-2, 5, 8,

and e tn-classes. [015] produces the same tn-class partitioning as [012], although the

tn-classes are not in the same locations in the spaces. [027] generates similar spaces

to [012], as they are related by multiplication, and each of the spaces it generates are

related by multiplication to [012]’s spaces. [048] is extremely restrictive, given the

highly symmetrical structure of the vlset. When combined with itself under rotation,

[048] either produces the identity, [000], or maps onto itself. Therefore, the tn-class

spaces created by [048] are all pairs of tn-classes.

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As shown above, the trichordal orbifold is reduced from a hypercube through

permutational equivalence. While any voice-leading set can produce a hypercube,

many of these cannot be reduced through permutational equivalence. Multiset vlsets,

such as <001>, have only three distinct permutations, <001>, <010>, and <100>,

which are rotations of one another. Along with their sum-zero inversions, <00e>,

<0e0>, and <e00>, these orderings transform a three-member pcset in at most six

ways, no matter the permutation of the vlset. For example, if the six vlsets are

applied to pcset {014}, they produce pcsets {015}, {024}, {114}, {013}, {004}, and

{e14}. If the pcset is re-ordered, {104}, the vlsets produce {105}, {114}, {204},

{103}, {1e4}, and {004}. If the rotations and inversions of a three-member vlset,

such as <025>, <250>, <502>, <0t7>, <t70>, and <70t> are applied to {014} and

{104}, the results are not equivalent under permutation. {014} is mapped onto

{039}, {264}, {516}, {0ee}, {t84}, and {712}, while {104} is mapped onto {129},

{354}, {606}, {1te}, {e74}, and {800}. As a result, the voice-leading hypercube

produced by vlset <025> cannot be reduced to a toroidal prism like the <001> space,

because the three-member pcsets could be mapped onto up to twelve other pcsets.

While three-member vlset spaces cannot be reduced by permutational

equivalence, they may still be reduced by rotation, transposition, and inversion. All

two and three-member pcsets in a space have three distinct rotations, such as {001},

{010}, and {100}, or {258}, {582}, and {825}. The singleton multisets, such as

{444} have one rotation. Therefore, under rotational equivalence alone, the 1728

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trichordal pcsets reduce to 584; the 12 singleton multisets do not reduce, and the

remaining 1716 sets reduce 3:1 to 572.

The complete spaces, as shown on Table 3-2, reduce from four-dimensional

hypercubes to cones with two points at one end by rotation and transpositional

equivalence. These are similar to the set-class cone shown by Callender, Quinn, and

Tymoczko, but differ because the authors’ spaces are reduced by permutational

equivalence. In their trichordal set-class space, [000] and [048] are at opposite ends

of the cone, as they represent the least and most even distribution of pcs within the

octave, respectively. The spaces reduced only by rotation and transposition form

cones with two points at one end, as [000] is opposite both [048] and [084]. Figure 3-

27a demonstrates the vlclass [047] ordered tn-class space. Figure 3-27b shows a

trichordal passage among adjacencies within the space, the specific vlsets and

mappings are shown below the score.

As Figure 3-27a shows, there are 50 ordered tn-classes under rotational

equivalence. These are made up of 12 multiset tn-classes, and 38 trichordal tn-

classes. The 19 unordered tn-classes each have two forms as ordered classes, such as

[013], [031], [023], and [032]. The 12 multiset tn-classes are the dyadic intervals and

singleton sets: [000], [001], [002]... [011]. Ordered voice-leading tn-classes [001],

[005], [055], [011], [013], [023], [031], [032], [014], [034], [041], [043], [016], [056],

[061], [065], [025], [035], [052], [053], [037], [047], [073], [074] produce complete

ordered tn-class spaces.

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Each of the complete ordered tn-spaces may be reduced further by ordered

inversion. Figure 3-28 shows the ordered set-class space for vlclass [047]. For the

symmetrical tn-classes listed above, each ordering is an inversion of the other. For

instance, ordered tn-classes [012] and [021] are inversions of one another under

rotational equivalence. They are not transpositions, because they may not be

reordered. Non-symmetrical tn-classes reduce from four forms to two; [025] and

[053] are inversionally-related pairs, as are [035] and [052]. Inversional equivalence

reduces the space to a more manageable 26 ordered set-classes. [000] and [006] are

already in their most reduced form, and the remaining 48 ordered tn-classes reduce by

half. As Figure 3-28 shows, when the spaces are reduced by inversion, the open end

of the cone is stitched together, and the two augmented triad points are merged.

Figures 3-29a-e show some representative ordered set-class spaces. As the Figures

show, these spaces are identical in structure, but their ordered set-classes are

redistributed to reflect their differences in voice leading. Highly symmetrical set-

classes, such as [000], [006], and [048] are located in the same positions in each

space.

Many ordered vlclasses produce multiple ordered tn-class and set-class

spaces. These spaces are disjunct due to the interaction of their generating vlclass’s

interval cycles. For example, all spaces produced by members of vlclass [002] reduce

by rotation, transposition and inversion to one of two ordered set-class spaces, shown

in Figures 3-30a and b, respectively. These spaces partition the trichordal tn-classes

by their whole-tone contents, Figure 3-30a includes ordered set-classes made up of

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members of one whole-tone scale, while the ordered set-classes in 3-32b are made up

of both whole-tone scales. Both spaces are three-dimensional; the two sides of Figure

3-30a fold onto one another, and Figure 3-30b wraps around itself to form an

irregular hexagonal prism, as shown.

The scope of the present study does not allow a full examination of the spaces

produced by the remaining trichordal vlclasses. As shown on Table 3-2, the

remaining vlclasses partition the pcsets, ordered tn-classes, and ordered set-classes

into multiple spaces. The previous examples have shown the construction of voice-

leading spaces for many distinct types of voice leading. These form alternatives to

the spaces based on harmonic similarity found elsewhere in contemporary theory.

The complete dyadic and trichordal spaces are especially useful, as they may be used

to model the voice leading between any pcsets or set-classes an analyst chooses.

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Chapter 4: Analysis

Post-tonal music is, in general, characterized by surface diversity. As the

following discussion will show, many analytical methods for post-tonal music

attempt to reduce the amount of information presented in each piece to a small

number of unifying elements, or principles governing behavior. Voice-leading sets fit

within this analytical tradition by demonstrating underlying unity through

transformational voice leading. After the opening of Arnold Schoenberg’s Drei

Klavierstucke Op. 11, n. 1 is briefly revisited, this chapter presents analyses of

movements 2 and 3 from Anton Webern’s Five Movements for String Quartet Op. 5

and movements 1 and 2 of Alban Berg’s Four Pieces for Clarinet and Piano, Op. 5.

Set theory is often used for the analysis of non-serial post-tonal music. Set

theory is especially useful for demonstrating harmonic relationships in various

musical contexts. In Forte’s formulation, an analyst shows how much of the

harmonic material of a piece is related to a central nexus set, most often through

inclusion relations.79

Set theory is also useful simply as a labeling system for the

harmonic materials of post-tonal music, and therefore may be used in conjunction

with other analytical systems, such as developing variation, basic cell analysis, and

transformation theory. In the following analyses, instead of a pitch-class set-class

nexus, the voice-leading set-class presented by prominent motivic material is treated

as a type of nexus set. Because any two pcsets have multiple vlset interpretations,

many diverse pcset pairs may provide voice leadings which belong to the same

79

These inclusion relations may be abstract. See The Structure of Atonal Music (New

Haven: Yale University Press, 1977).

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75

vlclass; the choice of which vlset “path” to follow is partly a statistical choice and

partly an interpretive choice. That is, it is based on the extent to which the path

accounts for an appreciable amount of the piece, how convincing the path is, and

whether it accounts for the most salient materials.

The voice leadings interpreted through my analyses are presented in a wide

variety of ways, with a wide range of salience. The goal of these analyses is not to

“hear” vlsets per se, although it is possible to derive a perceptual path based on the

intervals outlined. In the analytical context proposed here, voice leading is treated

motivically, through the “variation” (as in motivic variation) of one vlset, or a small

group of sets. The pitch information in the rest of the piece may then be thought of as

filtered through the vlclass, establishing relationships among a wide variety of

contexts within a piece.80

Salience does factor into the decision of which vlsets may

be primary in a piece, but this decision has multiple inputs.81

Developing variation and grundgestalt, or basic shape, are two important

concepts from Arnold Schoenberg’s analytical practice. Both concepts provide ways

in which to interpret a composition as a process of organic growth. In the most

80

This is not unlike Allen Forte’s octatonic filter employed in The Atonal Music Of

Anton Webern (New Haven: Yale University Press, 1998). 81

Common criticisms of pcset analysis are based on issues of segmentation, and the

analytical method proposed here does not offer a solution. My analysis admits a wide

variety of segmentation possibilities; as in pcset analysis, the analyst is responsible

for making meaningful choices. Other analysts could, of course, employ more

rigorous segmentation methods if they wish. See Ethan Haimo’s criticism of Allen

Forte in “Atonality, Analysis, and the Intentional Fallacy” in Music Theory Spectrum,

18, n. 2 (1996): 167-199. For an earlier discussion of segmentation, see Christopher

Hasty’s “Segmentation and Process in Post-tonal Music,” in Music Theory Spectrum,

3, (1981): 54-73.

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76

conventional interpretation, developing variation refers to a method of motivic

variation, while the basic shape is a musical figure presented near the outset of a

composition, whose “endless reshaping” forms the remainder of the piece.82

The size

and musical parameters of a basic shape may vary from piece to piece, while

developing variation typically refers to the transformations of a small collection of

pitches or intervals.83

In post-tonal music, Arnold Schoenberg uses his Op. 22 songs

to demonstrate ways in which a small motivic idea may be altered and transformed to

create new motives. There are many possibilities for variation of a motive, including

re-ordering, expansion, contraction, rearrangement, etc.84

This type of variation often

changes the harmonic content of a motive. For example, in the course of his talk,

Schoenberg shows two trichords from the second and third phrase of the piano

accompaniment in his song Seraphita, Op. 22 n. 1; ordered pcsets <032> and <478>.

As Schoenberg shows, the second trichord employs the same interval succession

(labeled in tonal terms) —minor 3rd

followed by minor 2nd

—as the first. In the

second trichord, however, the motive is varied by reversing the direction of the minor

2nd

.85

The two motives thereby belong to different set-classes, [013] and [014]

respectively, and cannot be related by transposition and inversion. Voice-leading

sets, however, provide a method for describing the systematic transformations of the

82

“Linear Counterpoint” in Style and Idea edited by Roy Carter with translations by

Leo Black (New York: St. Martins Press, 1975, reprint Los Angeles: University of

California Press, 1984). 83

See for example, Schoenberg’s analysis of Brahms’ F-major Cello Sonata, Op. 99

in “The Orchestral Variations, Op. 31: A Radio Talk,” The Score, 27 (1960): 28. 84

Arnold Schoenberg “Analysis of the Four Orchestral Songs Op. 22” translated by

Claudio Spies in Perspectives of New Music, 3, n.2 (1965): 1-21. 85

ibid., 5.

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77

motive. Vlset <446> maps pcset <032> onto <478>. The double T4s in the set show

that the first interval is unaltered, as both pcs are transposed by the same amount, and

the six shows the difference between them.

George Perle’s basic cell, related to developing variation, is essentially a pitch

or pitch-class set which provides unity through transposition, inversion, and

permutation. Perle focuses on small units and demonstrates their replication in

various harmonic and melodic guises. For example, in his discussion of

Schoenberg’s Op. 11 n. 1, Perle shows many ways in which the opening [014] can be

found in the opening three measures, as well as at important formal junctures

throughout the work. Perle’s analysis is not limited to one cell, but he discusses

multiple important cells in the piece.86

Although Perle does not employ the language

of set theory, it is easy to interpret his work as focusing on a piece’s primary sets; that

is, Perle demonstrates how the transformations of a piece’s most important pcsets

shape the work. The analyses shown below use Perle’s method, but apply it to voice

leading. That is, a small number of specific vlclasses form a piece’s basic cells.

As David Lewin has shown, transformation theory may be used in conjunction

with set theory, where pitch-class sets are mapped onto one another using various

transformations. Some of the transformations Lewin demonstrates are operations,

meaning they are one-to-one and onto, such as transposition, inversion, and

multiplication; others are more exotic, such as wedging-to-E, which moves each

86

Serial Composition and Atonality 6th

ed. (Berkeley: University of California Press,

2001) 9-15.

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78

pitch-class a semitone closer to E while preserving E and Bb.87

The latter is a many-

to-one transformation, and therefore not an operation, as it maps three different pitch-

classes, F, E, and D#, onto E. Lewin also describes a number of contextual

transformations, whose mappings change depending on the object they operate upon;

for example, the neo-Riemannian operation L, or Leitonwechsel maps major triads

onto minor triads a major third above, and minor triads onto major triads a major third

below, such as C major – E minor, and Eb minor – B major.88

Because they describe

individual pitch-class mappings, vlsets eliminate the distinction between interval-

preserving operations such as transposition and inversion, many-to-one or one-to-

many mappings, and contextual transformations, and they can model essentially any

pcset relationship.

The following analyses demonstrate ways in which voice-leading sets unify

pitch and interval diversity in post-tonal music. In each piece, much of the musical

surface can be viewed as expressing a small number of voice-leading classes, that is,

voice-leading transformations related by transposition or inversion. These vlclasses

are determined by the ordered voice-leading transformations presented by prominent

motivic material; in three of the pieces, this is the ordered voice leading between

trichordal members of the opening melodic gesture. The vlclass thus forms a type of

basic cell, or basic shape, which unifies seemingly disparate surface material. The

analyses presented here are, again, not intended to supplant those of previous authors,

87

Injection function, Generalized Musical Intervals and Transformations (New

Haven: Yale University Press, 1987, reprint New Haven: Yale University Press,

2007), Chapter 6. 88

Lewin 2007, 178.

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79

but may be used in conjunction with them to provide a richer interpretation of the

works.

In another view, the goal of the analytical method presented here is to

determine the underlying pitch-class voices, which make up the surface of post-tonal

pieces. This is not unlike the analysis of tonal music; in Schenkerian analysis, the

voices demonstrate ways in which the overarching tonic is prolonged using various

diminution techniques. As in tonal music, the interpreted voices in the following

analyses are understood as unfolding multiple pitch-class voices in tandem.

Compound melodies in tonal music are typically interpreted as projecting multiple

stepwise lines. In the following analyses, the voices of polyphonic melodies are

determined by their relationship to the main voice-leading motive.

Schoenberg Op. 11 n. 1

The [002] voice-leading interpretation presented in Figures 2-8a and b

highlight a recursive connection between melody and accompaniment in

Schoenberg’s piece. The voice leadings between the pairs of accompanimental

trichords, c-d (<442>) and g-h (<113>), reinforce the intervallic contrast presented by

the opening trichords of the antecedent and consequent phrases, ms 1-3 (<t99>) and

9-11 (<388)>) respectively. In ms. 1-2, the antecedent (trichord a) descends by pc int

3, followed by int 1; in ms. 9-10, the consequent (trichord e) descends by int 4,

followed by int 2. These interval pairs then form the basis for the voice leading

between the melodies’ accompanimental trichords: c moves by vlset <442> to d, and

g moves by vlset <113> to h. Thus, the intervals within the melodic trichords are

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replicated as voice leadings between accompanimental trichords; the technique of

intervals becoming T-levels is a standard recursive one in tonal music as well as in

post-tonal music.89

Many properties of vlclass [002] are highlighted in the passage, emphasizing

its prominence as a featured voice-leading set-class. As a closer examination of

Figure 2-8 will show, voice leading by any member of [002] preserves the whole-tone

content of a pcset. In particular, even members of [002] preserve whole-tone

membership among the pcs of a set, while odd members of [002] modulate all pcs

from one collection to the other. For example, pcset c contains two members of the

odd whole-tone collection, {5e}, and one even {6}; because vlset <442> is

exclusively even, pcset d has the same combination: two odd pcs, {91}, and one even

{t}. Likewise, pcset g is entirely even <468>, and moves by vlset <113> to odd pcset

h {57e}.

Given the whole-tone preserving property of vlclass [002], the paired pcsets in

Figure 2-8 must contain the same ratio of whole-tone members to one another.

Trichords e and f, which make up the second melodic statement in ms. 9-11, do not

share the same whole-tone ratios, and therefore cannot be spanned by a member of

vlclass [002]; e is a member of [026] and f is the chromatic trichord [012]. The

melody as a whole is made up of a whole-tone pentachord with one additional pc,

{689t02}, and is accompanied by the whole-tone trichords g {468} and h {57e}. The

consequent’s predominantly whole-tone sonorities, and change from [002]-based

89

This recursion could be shown in Klumpenhouwer-network analysis.

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81

voice leading contrast with the antecedent’s mix of whole-tone collections; this

contrast closes the opening section of the piece.90

Despite the whole-tone content retention aspects of vlset [002], each of the

seven members of vlclass [002] shown in Figure 2-8 create a unique voice-leading

space where adjacent pcsets are mapped onto one another by the generating vlset.

For example, Figures 4-1a and b show the spaces produced by vlset <442> with

pcsets {000} and {001} at the origin, respectively. The three-dimensional spaces

form six by six hypercubes, cubes whose opposing faces are adjacent to one another.

This diversity is built on top of a unity, however; as shown in chapter 3, all spaces

produced by members of vlclass [002] reduce by rotation, transposition, and inversion

to one of two ordered set-class spaces, shown in Figures 4-1c and d. These spaces

partition the trichordal tn-classes by their whole-tone contents, Figure 4-1c includes

ordered set-classes made up of members of one whole-tone scale, while the ordered

set-classes in 14-1d are made up of both whole-tone scales. Both spaces are three-

dimensional; the two sides of Figure 4-1c fold onto one another, and Figure 4-1d

wraps around itself to form an irregular hexagonal prism, as shown.

Each of the voice leadings shown in Figure 2-8 may be traced as a single line

segment along one of the vlclass [002] ordered set-class spaces shown in Figures 4-1c

and d; the ordered tn-class and ordered set-class interpretations of these chords are

shown in Figure 4-1e. It is important to note that these spaces are not reduced

90

George Perle argues that the melody’s shift to a predominantly whole-tone

collection is analogous to the variation of a melody in a tonal composition, where

additional statements are presented within new harmonic contexts in Twelve-tone

Tonality 2nd

ed. (Berkeley: University of California Press, 1996) 162-163.

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82

through permutational equivalence, therefore ordered tn-classes [014] and [041] are

not represented by the same point. When reduced by rotation and ordered inversion,

however, [041] is equivalent to [034]. As a result, the spaces are very flexible

analytical tools, as pcsets may be reordered and represented by different points on the

space. For example, ordered set-classes [014] and [015] are not adjacent on the

space, but [034] and [015] are. The first voice leading shown in Figure 2-8 interprets

chord a as ordered tn-class [041], which is an inversion of ordered set-class [034].

This same chord is reinterpreted as an ordered tn-class [014] in order to show its

voice-leading mapping onto chord c. Such multiple interpretations are part of the

richness of vlset analysis. 91

Webern Op. 5 n. 2

In his discussion of the second of Anton Webern’s Five Pieces for String

Quartet, David Lewin shows a network of relationships interpreted from the various

forms of the opening gesture, G-B-G-C#, pcset {G,B,C#}, tn[046], sc [026],

presented in the first three measures of the piece (as is typical for Lewin, he does not

label the tn or sc). As Lewin shows, these trichords, whether melodic, harmonic, or

some combination of the two, all feature pcs from the odd, or C#, whole-tone scale.

The transformations upon the network, therefore, preserve whole-tone membership:

91

In my opinion, it is the analyst’s responsibility to determine if the relationships

formed by multiple interpretations are meaningful and worthy of inclusion in an

analysis. All of the transformational mappings shown in Figure 4 contribute to my

understanding of the opening section of Schoenberg’s piece.

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83

T4, T6, T8, Tt, I0, I4, and I6, and are taken from the T0-t, I0-t dihedral subgroup.92

Figures 4-2a and b show the statements of [026] Lewin culls from the score, as well

as the network of transformations among them. Lewin identifies another important

note group, adding a C to G-B-G-C# (tn [0456] and set-class [0126]), formed by

filling in the whole-step of each [026]; the transformational network can also be

considered in the context of this set. Lewin’s article is intended to demonstrate the

construction of a network and its use in a portion of the piece, not to present an

analysis of the piece as a whole; therefore, he does not discuss the later portions of

the piece.93

The following analysis of Webern’s Op. 5 n. 2 will show that the piece is

governed by the transformations of vlclass [043]. These transformations create

networks of relationships among both pcsets and vlsets in Webern’s piece. Just as

Lewin focuses on the viola’s opening [026] and its intervals, the primacy of vlclass

[043] is determined by the ordered transformational voice leading between the viola’s

first and second trichords—vlset <910>, which is T9(<043>).94

The two trichords are

made up of pcs <7e1> and <401> in score order. Vlset <910> preserves this

92

Lewin uses the letter names I, J, and K as labels for “inversion about” D (I4), F#

(I0), and A (I6), respectively. “Transformational Techniques in Atonal and other

Music Theories” in Perspectives of New Music 21, no 1/2 (1982): 312-324. Another

study of this short piece comes from Bruce Archibald, who interprets a series of

symmetries spanning the piece, in various states of “completeness” which can be

thought of analogously as “unbalanced” and “balanced” – terms Lewin uses in his

study of inversional balance in Schoenberg’s music and also a term from

Schoenberg’s formal theories; see “Some Thoughts on Symmetry in Early Webern:

Op. 5 N. 2” in Perspectives of New Music, 10, n. 2 (1972): 159-163. 93

Lewin 1982, 312-325. The Figures appear on pgs 318 and 320, respectively. 94

Although the viola melody begins with a tetrachordal multiset (G-B-G-C#), the

second G is interpreted as the arpeggiation of a single pc voice in my analysis.

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84

ordering, with mappings 7 – 4, e – 0, and 1 – 1. This voice leading is then varied

through transposition and inversion, and used to span pcsets in melodic and

accompanimental gestures throughout the piece. While Lewin’s [026]/whole-tone

network provides a good model for the opening phrase, and is also relevant to the

third phrase, it is not applicable to the remainder of the piece. By focusing on

transformational voice leading, rather than harmonic similarity, the network of

relationships formed by members of vlclass [043] can be applied to the entire work.

The texture of Webern’s piece is largely melody and accompaniment, and it is

divided into four distinct phrases which are marked by changes in the

accompaniment: mm. 1-4, 4-7, 7-10, and 10-13. The first, second, and last phrases

feature chordal accompaniment, while the third consists of a slow Eb-F trill in the

second violin as a counterpoint to the melody. The second phrase is demarcated from

the first and third by semitonal sixteenth-note motives in mm. 4-5 and 6-7.95

Figure 4-3 shows the statements of vlclass [043] in the melody in the

remainder of the piece. As shown, the melody is transferred to the 2nd

violin in mm.

5-6, the 1st violin in mm. 7-12, and returns to the 2

nd violin in mm. 12-13. The pitch-

class voices are shown below the staff, with lines demonstrating the pc mappings.

95

Although Lewin does not demonstrate a transformational network for the last three

phrases of the piece, the third phrase, mm. 7-10, fits comfortably with Lewin’s

[026]—whole-tone interpretation of mm. 1-3. The second violin alternates Eb and F,

while the first violin supplies the remaining members of the odd whole-tone scale. If

G#, D, and E are seen as non-harmonic tones, the first violin presents three

overlapping [026]s: {9e3}, {391}, and {917}. It is possible to interpret many more

[026]s in conjunction with the second violin.

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Pcs are listed vertically in registral order for ease of reading, with the corresponding

vlset shown below.96

The pitch-class voices shown in Figure 4-3 correspond to the polyphonic lines

expressed by the melody. For example, each pc of the opening trichord, {7e1},

represents a voice, which moves by a different interval to the next pc; G moves by T9

to E, B moves T1 to C, and C# moves T0 C#, although descending an octave in pitch.

These pc voice motions correspond to the mappings induced by vlset <910>. There

are many possible vlset interpretations of any two pcsets, and there are many possible

pcset interpretations of any given musical passage. My analysis focuses on the

ordered voice leading produced by the viola’s opening gesture, and shows how that

voice leading is transformed throughout the piece.

Each of the subsequent voice leadings listed in Figure 4-3 is a member of

vlclass [043] under transposition or inversion. Therefore, the pc voices project

variations of the opening voice leading which unifies the melody. The second vlset

for example, <845>, is an I5 inversion of the opening voice leading. In other words,

my analysis differs from those of previous analysts by focusing on the harmonic

consistency expressed by pc voices, rather than pcsets.97

Figure 4-4 shows the

ordered harmonic relationships given by the vlsets highlighted in Figure 4-3.

96

This method of pitch-class voice-leading representation is indebted to Joseph

Straus, who employs it in multiple articles, including 2003. 97

As stated above, this is not intended to supplant pcset analysis, but rather to provide

an additional perspective. It is valuable to point out, for instance, that {589} is a

subset of the opening accompanimental tetrachord, and that the two pcsets in question,

{014} and {589} are inversions of one another. This is expressible as vlset <999>i,

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86

Two of the vlsets, <265>i and <401>i in mm. 5 and 8, are inversional

mappings. That is, the vlsets map the individual pcsets onto one another by

inversional sums. <265>i maps ordered pcset <t40> onto <425>, which is formed by

the last three pcs of m. 6. <401>i maps <89e> onto <832>.98

If transpositional and

inversional vlsets are thought of as two transformational modes, we may define an

operation, changemode, which produces a mode change from one vlset to another,

along with transposition or inversion. As shown on Figure 4-4, CI0 maps

transpositional vlset <t67> onto inversional vlset <265>i; the index numbers are

transformed by I0, and the mode is changed by C. Likewise, CT6 maps vlset <401>i

onto <t67>.99

In order to produce the voice-leading consistency shown in Figure 4-4, the

surface of the piece must be interpreted as projecting three pc voices, that is,

trichords. Some gestures, such as mm. 5-6 are easily interpreted as trichords. The

first two trichords in score order, <t40> and <625>, are inversions of the opening

trichords, {7e1} and {401}, by Ie and I6 respectively. As a result, the vlset spanning

the trichords in mm. 5-6, <t67>, is an I7 inversion of the opening <910>; I7 results

from the difference between inversional sums: 6 - e = 7.

but the purpose of this analysis is to demonstrate the variations of vlset <910> that

underlie the pitch material of Op. 5 n. 2. 98

As the pitch-class voices demonstrate, the common-tone G# is mapped onto itself.

This echoes the common C# in the opening voice-leading. 99

C forms a group with transposition and inversion of order 48 on the set of 48

transpositional and inversional members of vlclass [043]. CI operations are

involutions, and CT operations are not, excepting CT0 and CT6. T0 is the identity.

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87

Elsewhere, the melody is more difficult to parse into trichords. As Figure 4-3

shows, in both mm. 3 and 9, vlclass [043] maps a trichord from the melody onto a

melodic dyad plus a pitch-class “borrowed” from the 2nd

violin—A and F,

respectively. In addition, many successive trichords are overlapping. The first

example occurs in m. 3, where G and C# are each members of two trichords.

The trichordal interpretation of mm. 10-11 is shown by beams in Figure 4-3.

The melody begins and ends with chromatic [012] trichords, <543> AND <e0t>

which are T7-related; these bookend an imbricated -4/-1 interval cycle <G#ED#B>

(<G#ED#> AND <EDB>), which produces an [0158] tetrachord. As the beams

show, the repeated E and D# in m.11 are interpreted as members of pcset <345>, and

the G# forms pcset <8e0> with B and C, producing vlset <487>.100

B and C are

common tones with the next trichord, although the C is repeated on the downbeat of

m. 12.

Vlclass [043] is also expressed in the accompaniment of the piece. Figure 4-5

reduces the score to two staves, and demonstrates some of the pc voices formed by

vlclass [043] in the accompaniment, as well as between the melody and

accompaniment. The opening is accompanied by two chords sustained by the cello

and second violin—{2589} and {379}. As shown in the voice-leading mapping

below the staff of Figure 4-5, pc A is interpreted as belonging to the second chord,

and not the first, the two opening trichords are <258> and <379>; the A in the

opening is interpreted as an anticipation, a member of the following chord. The vlset

100

In terms of polyphonic melody, the G# is an anticipation of the next pcset, {8e0}

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88

spanning the two trichords is <7et>, a Tt transformation of the <910> voice leading it

accompanies. This voice leading follows the registral mapping on the surface of the

piece. As Figure 4-5 shows, the pc voices do not cross in pitch space. Two

interpretations of the symmetrical tetrachords in mm. 4-5 (<F#GBbB> and

<FF#BC>) are shown below the staff. Both demonstrate the voice-leading mapping

between ordered tn-classes [015] to [016].

Figure 4-5 also demonstrates vlclass [043] mappings between the melody and

accompaniment. The second and third accompanimental trichords (<GEbA> and

<GBC#>), are the same tn-class, [046], as the opening melody. These pcsets interact

with the melody in mm. 2 and 3. As shown, the vlset which maps pcset <012> onto

<7e1>, <7et>, is identical to the first accompanimental voice-leading mapping.

Two of the voice leadings produced between the melody and accompaniment

are identical to the opening vlset, <910>. In m. 6, <910> maps accompanimental

pcset {519} onto melodic pcset {256}. The same vlset maps {590} onto {e34} in m.

11. These repetitions of the opening vlset help to punctuate the piece’s phrases, as

they correspond to clear variations of the opening melody.

Figures 4-6 displays an alternative transformational graph of Webern’s Op. 5

n. 2. The nodes of this graph are the vlsets presented by the piece, and the arrows

demonstrate some of the relationships among them. Like Lewin’s network, the graph

follows the outline of the score. This graph of relationships demonstrates the

transformation of the basic voice leading throughout the piece, and the way in which

this transformation unifies the piece. While Lewin’s network focuses on a small

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89

portion of the piece, and the whole-tone D6 subgroup of the T/I group, the arrows of

my graph reflect the use of the full group formed by transposition, inversion, and

modechange.101

Webern Op. 5 n. 3

Webern’s Op. 5 n. 3 features one predominant voice leading—vlclass [038].

This voice leading is presented in various forms throughout the piece. As in the

analysis of Op. 5 n. 2 above, the primary voice leading is first presented in prominent

melodic material at the opening of the piece; vlclass [038] is produced by the ordered

pc voices in the violin 1 melody (A) in m. 4. As Figures 4-7a and b show, A may be

broken up into two trichords, <254> and <06t>—members of tn-classes [023] and

[046] respectively. As in the analysis above, the pc voices expressed by the motive

are interpreted in score order; D maps onto C, F - F#, and E - Bb. Vlset <t16> recurs

whenever A is repeated, provided both trichords are transposed by the same interval.

For example, the viola enters one eighth note after the first violin, transposed by T7;

this is also shown in Figures 4-7a and b.

The opening melody, A, is repeated in mm. 10-11 in the cello, followed by the

liquidation of A in all parts through m. 14, shown in Figure 4-8. In mm. 10-11 and

12-13, the cello and second violin repeat the T7 pairing from m. 4. As Figure 4-8

shows, this produces two statements of vlset <t16>. The viola and second violin

phrases that follow each present a new variation of A. In the viola, the first trichord

101

A smaller portion of the graph, with alternative node-connecting arrows, may be

used to demonstrate smaller subgroups; this may not model the piece as effectively,

however. Alternative voice-leading interpretations of the piece would also produce

alternative graphs, of course.

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90

is expanded by repeating the opening minor third, creating a 3, -1 pitch interval cycle;

this cycle produces two possible tn-class [023]s, the voice leading from the second,

{t01}, is shown on the Figure. In addition, the variation omits the final interval. The

first violin’s statement also omits the final interval, but does not add an additional

interval 3, shortening A to five pitches. Both the viola and first violin voice leadings

have common-tones which map onto themselves. This is reflected by the T0 in vlset

<470>, T6 of the opening <t16>.

The opening melody, and trichordal [023]/[046] juxtaposition it presents,

gives way to another melody (B)—first presented in mm. 9-10—which presents a

new trichordal pair: {t12} and {59t}, members of tn-classes [034] and [045]. Both A

and B are varied through the second half of the piece, and the movement closes with a

fortississimo statement of B in octaves in all instruments. Although B is made up of

different tn-classes than A, it projects the same vlset. As Figure 4-9 shows, pcset

{2t1} maps onto {t95} by vlset <8e4>; like vlset <t16>, <8e4> is a member of

vlclass [038]. Unlike A, this voice-leading mapping does not take place in score

order, as the second and third pcs of the first pcset are reversed in the second pcset.

This rotation on the surface is a variation of the opening voice leading.

One may choose to interpret melodies A and B in a variety of ways. They

may be antecedent and consequent, conflicting themes, or B may be viewed as a

variation of A, among many other interpretations. However the melodies are viewed,

the main interpretation in this analysis is that the voice leading presented in A is

subjected to variation in its realization in B. A and B are the two main melodic ideas

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presented by the piece. Throughout the work, they are mined for motivic material.

Vlclass [038] forms the common thread between these various gestures.

Figure 4-10a shows the ordered tn-class space formed by vlclass [047]. As

discussed in chapter 3, the ordered tn-class spaces are reduced through rotational

equivalence, therefore the space represents the various rotations of vlclass [047],

including [038].102

The voice-leading mappings projected by A and B are highlighted

on the space; A’s <t16> voice leading maps ordered tn-class [032] onto [046], while

B’s <8e4> maps [034] onto [045]. The voice leadings are understood as the motion

from one ordered tn-class to another along the line segment that links them. As

discussed in chapter 3, the tn-classes on the space are equivalent under transposition

and rotation, but not permutation, as vlsets such as <t16> do not operate consistently

upon the two distinct permutations of many pcsets. For example, as the space shows,

vlclass [038] maps [023] onto ordered tn-classes [016], [054], [042], [046], [007], and

[063], and maps [032] onto [006], [046], [054], [025], [034], and [072].

B contains an intermediate trichord, <t0e>, between <2t1> and <t59>. This

interpolation may be seen as a way of extending and elaborating the voice leading

between the two chords, <8e4>, which governs the phrase. Vlclass [038] cannot

provide the mapping between this intermediate trichord and the other two members of

B. Ordered tn-class [021] is at least four steps away from both [034] and [045] on the

[038] tn-class space shown in Figure 4-10a. One possible route from [034] to [021]

102

My analysis of Webern’s piece highlights the specific ordered voice-leading

presented by melody A, therefore I have chosen to describe the piece’s voice-leadings

using [038].

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begins with the vertical motion of three consecutive inverse voice leadings, <094>,

followed by a rotation <380>, forming a total voice leading of <3e0>, of vlclass

[014]. The specific transposition and rotation is given by <2t1> + <t19> = <0et>.

The voice leading from <0et> to <t59> is the combination of two <038>s and two

<940>s, for a total of <624>, of vlclass [024]; <t0e> + <79e> = <59t>; see Figure 4-

10b.

The intermediate voice leadings in B, <t19> and <79e>, are interpreted by the

[038] tn-class space in the way in which voice leadings on the traditional orbifold are

interpreted in terms of the semitone. Vlclass [038] forms the metric by which the

voice-leading distance is judged. Because [038] forms a complete tn-class space, it is

possible to represent any trichordal voice leading culled from the piece in terms of

[038] distance.

In addition to interpolating an additional trichord, B varies A’s voice leading

through the introduction of invariance, as all three trichords contain Bb. For the first

and last trichords, this is a combination of the transposition of the vlset and the tn-

class changes. Vlset <8e4> maps the first trichord, in score order, onto the last, and

the trichord begins with pc interval 8. In addition to the intermediary trichord, this

invariance helps to contrast B with A. The distinguishing features of B—the added

trichord and use of invariance with vlclass [038]—are immediately used in the

variation of A which begins in the next measure, discussed briefly above. As Figure

4-8 shows, in m. 11, the viola’s overlapping [023]s, {8et} and {t10}, share Bb, the

same pc shared by all three of melody b’s trichords. Vlclass [038] maps both pcset

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<8et> and <t10> onto <280>, by vlsets <692> and <470>, respectively. Note that the

transpositional difference between the two vlsets is Tt. This is the same difference

between A’s vlset <t16>, and B’s <8e4>. All three pcsets have some kind of

invariant relationship with the other two. {t10} and {082} overlap with pc 0, and

{8et} and {082} both contain pc 8. Therefore, both characteristics of melody B are

incorporated into this variation of melody A.

The accompaniment to melody B also introduces invariance in m. 9-10. As

Figure 4-9 shows, the viola presents trichords {e03} and {643}, pc 3 is shared. Vlset

<470> spans the trichords, a T6 transposition of the original vlset in m. 4, while the

vlset spans a new pair of tn-classes, [014] and [013].

The accompanimental material in the second violin in m. 9-11 demonstrates

that vlclass [038] may also be used to navigate between melodies B and A, as it

echoes the features of melody B, and combines them with melody A. As shown on

Figure 4-9, the second violin begins with pcset {034}, followed by {234}, T2 and T4

of the first two trichords of B. The trichords share two pcs, 3 and 4. The gesture

continues to a straightforward statement of melody A in m. 11, a Tt transposition of

the first violin in m. 4, shown in Figure 4-8. The first tn-class of melody b, pcset

{034}, is mapped onto the first pcset of melody A, {023}, by vlset <e83>, an inverted

form of [038].

Melody B’s tn-class juxtaposition, [034] and [045] is foreshadowed by the

opening three measures as well as the material following melody A. The piece begins

with the upper three parts sounding [034]s, shown in a reduction in Figure 4-11a.

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Excluding the [014] in m. 3, the three string parts all move by the same intervals—pc

int 8, followed by int 5. These intervals form three [045]s horizontally, and are

repeated in m. 10 in retrograde: B’s [045] is retrograde-inverted.

A more extensive variation of A occurs in the first violin in mm. 12-14. A is

presented in a new contour—the gesture is entirely descending, with the intervals in

their smallest forms, and the pcsets are re-ordered (see Figure 4-8). Vlset <149>

maps the first pcset, <032> onto the second, <17e>. The melody continues with an

ascending [026], pcset {359}, and ends with a [023], pcset {467}. Melody A’s

original [023]/[046] juxtaposition is thus presented in retrograde, with the second

trichord inverted. The resulting voice leading cannot be a transpositional member of

vlclass [038], and it is inversional vlset <74e>i which maps <395> onto <476>.

The accompaniment in mm. 13-15 continues to demonstrate a derivation of

material from melodies A and B. The second violin and cello both provide

statements of A, followed by 8/6 pc-interval cycles in the viola and cello. Each cycle

is four pcs long, creating [0268] tetrachords, which may be seen as overlapping [046]

and [026] trichords, matching the material in the first violin. These are followed by

ascending pitch interval 8s, {08} and {95}, in the second violin and viola, and a

descending [015]. The {5890} tetrachord may be viewed as overlapping [034] and

[014] trichords. Although there are no explicit tn-class [034]s, the two [0268]s in the

previous measures are separated by semitone, and may be combined to form both

[014]s and [034]s. This serves to revisit the juxtaposition presented in the opening, as

well as in melody B.

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The variation of A in mm. 12-14 is extended in the phrase beginning with the

pickup to m. 18. The melody begins with the tn-classes of melody A, pcsets {578}

and {046}, linked by vlset <149>. This is followed by a series of inversions with pc

invariance between [046]s and [026]s. The melody ends in m. 21 with pcset {9e0}.

Vlset <529>i maps pcset {8t2} at the end of m. 20 onto {9e0}.

As in the earlier sections of the piece, the melody in mm. 18-21 is

accompanied by gestures based on the voice leadings presented by melodies A and B.

The accompaniment begins in m. 15, three bars earlier, where the viola maps pcset

{89e} onto {489} by vlset <850>i, while the second violin and cello arpeggiate

(separately) [014]s and [034] ; a reduction is shown in Figure 4-11b. When the first

violin enters, the second violin and viola both present arpeggios of melody a’s tn-

class juxtaposition [023]/[046] with invariance. Both use the voice leading <470>.

As mentioned earlier, the piece closes with a statement of melody B in all four parts.

The preceding analysis of Webern Op. 5 n. 3 demonstrates that the piece is

unified by the use of vlclass [038] in both transpositional and inversional forms.

Presented by the opening and closing trichords in the piece’s main melodic

statements, A and B in m. 4 and mm 9-10, respectively, these voice leadings can also

be found throughout the piece in accompanimental gestures. Much like the variation

of pitch material, vlset <t16> forms a motive which is transposed, inverted,

reordered, fragmented, and condensed throughout the piece. The consistent use of

vlclass [038] can be visualized using the ordered tn-class voice-leading space it

creates.

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Berg Op. 5 n. 1

Alban Berg’s Four Pieces for Clarinet and Piano Op. 5, have been described

by earlier analysts as a miniature four-movement sonata.103

In keeping with this

interpretation, Douglas Jarman interprets the opening clarinet gesture, shown in

Figure 4-12, as the source of motivic material for all four pieces. Jarman first divides

the melody into an [012569] hexachord and [037] trichord, then extracts five

trichords from the [012569] hexachord: [012], [014], [015], [016], and [037]. He then

demonstrates how material in the remainder of the movement, as well as other

movements, may be derived from transformations and combinations of these

principal set-classes.104

Dave Headlam also shows a way in which the opening

clarinet gesture may serve as generative material for much of the remainder of Op. 5

n. 1. In lieu of Jarman’s trichords, Headlam points out that the gesture employs

primarily interval-classes 1, 4, and 5. Nearly all of the adjacent pcs are separated by

ic 4 or 5, and each of the first three notes are linked by pitch-interval 1 to some later

pitch, Ab5 – A5, Eb5 – E5, and G4 – F#4.105

Headlam identifies six additional

generative features presented by the gesture: 1) pcset {8349} and its set-class [0156];

2) pcset {7689} and its sc [0123]; 3) the wedge motion between the first and third

trichords, Ab5 – A5, Eb5 – D5, and G4 – F#4; 4) the ordered interval series it forms,

<-5, (-5+-8=-1), 2, 3, 4, 5, -7, -8, -9>, the final -9 interval occurs between the clarinet

F#4 and Piano A3; 5) an emphasis on D; and 6) sc [012569], the SCHoenBErG

103

Douglas Jarman, NG. 104

The Music of Alban Berg (Berkeley: University of California, 1979) 23-27. 105

The Music of Alban Berg (New Haven: Yale University Press, 1996) 183-185.

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hexachord. Headlam then discusses how these gestures are featured in the remainder

of the piece.106

My analysis also focuses on the opening clarinet gesture, and demonstrates

that the voice leading presented therein forms the primary motivic material for the

piece. As Figure 4-13a shows, the first and third trichords, <837> and <926>, are

linked by contour and register, and vlset <1ee> maps the first trichord onto the third

in score order. Vlclass [200], of which <1ee> is a member, forms the basic cell

which unifies the remaining material presented on the surface of the piece. Figure 4-

13b revisits the vlclass [002] ordered set-class space. As in the Webern analysis

above, [002] and [200] are equivalent under rotation, and their spaces are identical.

The arrows on the space represent two of the voice leadings presented by the opening

clarinet gesture: the <1ee> mapping between the first and third trichords, ordered set-

classes [045] and [047], and the <022> mapping between the second and third

trichords, <904> and <926>, ordered set-classes [037] and [047]. The remaining

voice leadings may be traced upon the space as well.

The first and third trichords of the piece are also linked to the second trichord

through vlclass [200]. The second trichord, <904>, maps onto the third, <926>, in

score order by vlset <022>. The first trichord cannot be mapped onto the second

using a transpositional vlset member of vlclass [200], but requires an inversional

vlset, <577>i, which inverts the first note of each trichord at I5, {8} – {9}, and the

common ic4 at I7, {37} – {40}.

106

ibid, 183-186.

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The opening trichords are echoed harmonically in the left hand of the piano in

m. 2, as shown in Figure 4-14. The <1ee> voice leading originally projected by the

clarinet’s [045] and [047] trichords is now featured vertically between their

inversions, [015] and [037]. Vlset <133> maps <67e> onto <7t2>. As the Figure

shows, the I2 mapping between vlsets <1ee> and <133> is equal to the difference

between the inversional sums of the respective trichords; {378} inverts by I2 onto

{67e}, and {269} by I4 onto {7t2}.

The registral and dynamic climax of Op. 5 n. 1 in mm. 7-8 presents a new set

of three trichords in the clarinet, pcsets {912}, {023}, and {e34}, or tn-classes [045],

[023], and [045]. The second and third trichords imitate the descending gestures of

the first and third trichords at the opening. The trichords are mapped onto one

another in succession by inversionally-related vlsets; <311> maps <912> onto <023>,

and <e11> maps <023> onto <e34>. See Figure 4-13.

Using vlsets, prominent motivic features that are not clearly related to one

another can be shown as expressing the same underlying voice leading. For example,

the right hand piano gesture in m. 2 can be divided into two trichordal pcsets, {9t8}

and {334}. As Figure 4-15 shows, vlset <577> maps <t98> onto <343>. This voice

leading requires both a re-ordering of the first pcset as well as the interpretation of a

doubled Eb. The doubling requirement is ameliorated by the echoing gesture by the

clarinet in m. 3. As Figure 4-13 shows, vlset <355> maps <t89> onto <112>.

The remainder of the clarinet melody expresses vlclass [200] in a variety of

ways. The voice leading is subject to motivic variation techniques such as

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transposition, inversion, and rotation. Some of these voice leadings are obvious,

while others require more subjective interpretation of the surface of the piece.

As Figure 4-13 shows, in m. 4 pcset <121> is mapped onto <7t9> by vlset

<688>, and in turn <t79> onto <435> by <688>. A rotated version of the voice

leading is unfolded in mm. 5-6, as the clarinet features two ic3 dyads followed by ic

1. This extends the <99e> voice leading between <519> and <2t8> to the space of a

full measure. Vlset <8tt> maps pcset <t76> onto <654> in the second half of m. 6;

F# is a common tone. The ascending gesture in m. 7 is made up of two inversionally

related [0148], inverted in register about Ab4, followed by an ascent of five

semitones. The imbricated trichords are closely related to the opening clarinet

gesture, although they are presented in a new contour; the first trichord is an

augmented triad which shares two pcs {37} with the opening [015], and it is followed

by two triads—Gb major and D minor. The D minor triad shares two pcs, {29}, with

the D major triad in the opening gesture. The juxtaposed triads in m. 7 present a

retrograde member of the vlclass; the Gb triad, pcset {803} is mapped onto the D

minor triad in score order by vlset <ee1>. The climactic gesture begun in the pickup

to m. 8 is discussed above. The third trichord, pcset <e34>, is mapped onto pcset

<t45> by vlset <e11>. Finally, the clarinet ends by unfolding a rotated version of the

opening vlset, <1e1>, with a series of ordered dyads <45> – <98> – <67>

representing pcsets <496> and <587>.

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The piano accompaniment following the fermata in m. 9 is based on the

opening clarinet gesture, as Headlam shows in his analysis.107

The opening

hexachord is transposed down four semitones in pitch, and the two trichords are

layered vertically. In the opening of the piece, this hexachord is followed by a D

major triad, which shares a common A with the minor triad of the hexachord. In m.

9, pcs G and E, which follow the F minor triad in the left hand, create a C major triad

with C as a common tone. This is a Tt pitch-class transposition of the opening

gesture’s D major triad. Therefore, the three trichords of the opening are mapped

onto the piano accompaniment in m. 9 by T8, T8, and Tt, a retrograded member of

vlclass [200].

The piano accompaniment in Berg’s Op. 5 n. 1 is contrapuntal in texture, and

there are many ways in which vlclass [200] may be embedded. Some of the more

salient examples are given by wedge motions, characteristic features of Berg’s atonal

music. These wedge motions are especially relevant given that they replicate the

wedge motion of the clarinet’s three pc voices at the outset of the piece. As shown in

Figure 4-17 in m. 3 pcset <t01> is mapped onto <912> by vlset <e11>, <912> is then

mapped onto <803> by <ee1>. <e11> maps pcset <382> in m. 4 across the barline

onto <293> on the downbeat of m. 5. There are many further examples in the piece.

In addition to its use in wedge motions, vlclass [200] unifies the contrapuntal

surface of the piano accompaniment in more abstract ways. As Figure 4-18a shows,

the last beat of m. 5 includes tn-classes [045] and [047], echoing the first and third

107

Headlam 1996, 132-133.

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trichords of the opening clarinet melody. These trichords are followed by a tn-class

[016] in the left hand, and [045] at the beginning of m. 6, followed by another wedge-

motion <1ee>. This sequence of trichords is produced by the mappings of vlclass

[200], as shown on the figure. Another example occurs in m. 7. As Figure 4-18b

shows, pcset <e04> is mapped onto <e26> by vlset <022>, <e26> onto <7t0> by

<886>, and <654> onto <898> by <244>. In the left hand, vlset <577> maps pcset

<834> onto <1te>.

As shown by the previous analysis, the [200] voice leading presented by the

clarinet at the outset of the piece is manipulated and varied in both melody and

accompaniment throughout the piece. Although the piece is highly contrapuntal in

texture, the underlying pitch-class counterpoint is derived from vlclass [200].

Berg Op. 5 n. 2

As mentioned above, Jarman and Headlam discuss multi-movement

connections in Berg’s Op. 5 pieces. As both point out, the opening clarinet

hexachord of n. 1, pcset <837904> sc [012569], is included in the clarinet line in mm.

1-4 of n. 2. The remaining pcs, D and F#, are given to the piano at the opening.108

Headlam also points out that the clarinet begins with pcset <310874>, member of

[013478], which is the Z-pair of [012569], as well as another [012569] hexachord,

pcset {108749}; this hexachord combines three ic5s—the clarinet’s E – A at the end

of m. 3 and the right hand Ab – Db and G – C.109

Another important harmonic

connection is the partitioning of pcset {01378}, a subset of [013478], into tn-classes

108

Jarman 1979, 24. Headlam 1996, 94-95. 109

Headlam 1996, 95-96.

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[037] and [045] in the right hand of the piano in mm. 2-4. This replicates the

trichords of the opening clarinet gesture of the first piece, and preserves maximal

common tones Ab, Eb, G, and C.

Alban Berg’s Four Pieces for Clarinet and Piano Op. 5 n. 2 is static in texture.

The left hand features dyads nearly exclusively, while the right hand is entirely made

up of trichords. Each hand is nearly entirely homophonic, forming a clear harmonic

accompaniment to the clarinet melody. The clarinet and two hands of the piano

create 3 voice-leading strata, each of which projects its own pc voices. These strata

are unified by the nearly exclusive use of whole-tone vlsets.

The left hand features only three dyadic set-classes, [04], [02], and [06] in

order of appearance. These dyads are all subsets of the whole-tone scale, therefore

the voice leading among them is drawn from the whole-tone scale. The voice-leading

sets featured in the left hand are members of voice-leading classes [02], [04], and

[06]. Figure 4-19 arranges all of the pcsets in the left hand into two horizontal pitch-

class voices. As shown, all of the voice leadings may be interpreted as some member

of vlclass [02], [04], or [06]. Furthermore, the four [04] vlsets, <7e> <̧e3>, and

<37>, form a network where each vlset is related to the next by T4. This mirrors the

augmented triad which closes the piece, as well as the [04] dyads featured throughout.

The piece begins with a repeated {26} dyad in the left hand. As shown in

Figure 4-19, this moves by T3 to {59} in m. 4, or vlset <33>. If the second dyad is

reordered, {95}, the mapping instead produces vlset <7e>, a member of vlclass [04].

Because dyads are inversionally symmetrical, the same inversional voice-leading

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classes map {26} onto {59}, <73>i and <ee>i, respectively. In the second half of m.

4, the left hand oscillates between {59} and {26}. <59> is mapped onto <26> by

vlsets <99> and <73>i, and <59> onto <62> by <15> and <ee>i.

In m. 5, a harmonic change occurs, as the [04] dyads give way to [02] dyads;

these are spelled as minor sevenths in pitch space. As Figure 4-19 shows, pcset <59>

is mapped onto <68> by vlset <1e>, a member of vlclass [02]. The minor sevenths

then ascend by half-step. If the voice leading in this passage is interpreted as a series

of voice crossings, each successive seventh is produced by vlset <e3>. Vlset <e3> is

another member of vlclass [04].

The final left hand dyad, {06}, is introduced following the ascending minor

sevenths in m. 5. The left hand then alternates between three pcsets, {06}, {15}, and

{9e}, which summarize the whole-tone set-classes featured in the piece, [02], [04],

and [06]. As Figure 4-19 shows, <9e> is mapped onto <06> by vlset <37>.

The right hand voice-leading stratum is made up entirely of trichords. In mm.

2-4, the right hand alternates between pcsets {037} and {801}. Although these

trichords are not whole-tone subsets, they may be linked by similar voice leading.

Vlset <155> maps pcset <037> onto <180>. This voice leading demonstrates the

common [04] dyad—{37} and {80}—between the two pcsets. Vlclass [044] is very

similar to the [04] voice leadings in the left hand. The I0 inversion of the opening

vlset, <e77>, maps pcset <180> back onto <037>.

As Figure 4-19 shows, additional trichords are introduced in m. 5. Two of

these, {913} and {t04}, are whole-tone subsets. All other trichords in the right hand

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are made up of members of both whole-tone scales. Vlclass [044] preserves the

whole-tone contents of a set, therefore it is impossible for any vlset members of [044]

to map {037} onto {913}, or {t04} onto {812}. Each pair of pcsets has an interval in

common, however, so it is possible to show their mappings using another multiset

vlclass, [033]. Vlset <366> maps <037> onto <391>, and <1tt> maps <04t> onto

<128>. The two whole-tone trichords, however, are spanned by a member of vlclass

[044]; <391> is mapped onto <t04> by vlset <733>. M. 5 returns to pcset {037} on

beat three. Vlset <ee1>, a member of vlclass [002], maps pcset <812> onto <703>.

An inversional vlset, <155>i, maps <370> onto <tt5>, and <t66> maps <tt5> onto

<84e>. This succession is repeated on beat two of m. 6, and m. 6 ends with pcsets

{48e} and {037}. These trichords may be mapped onto one another by members of

vlclass [002]; vlset <ee1> maps <48e> onto <370>, and vlset <e11> maps <037>

onto <e48>.

As the previous discussion has shown, the left and right hands of the piano

part can be interpreted as featuring predominantly whole-tone voice-leading. The

clarinet melody matches the piano part, as it utilizes whole-tone vlclasses [006],

[0066], and [044]. As discussed above, the clarinet line is interpreted as a

contrapuntal melody, which projects multiple pc voices.

Figure 4-20 shows the way in which the clarinet melody expresses the above

vlclasses. The first clarinet phrase, from the pickup to m. 2 to the downbeat of m. 3,

may be divided into two trichords, pcsets <013> and <487>. Vlset <771> maps the

first trichord onto the second; the first two pcs of the second trichord, Ab and G, are

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reversed in order. A T1 transposition of the first vlset, <882>, maps <847> onto

<409> in m. 4; the trichords share E at the end of m. 3. The clarinet melody in mm.

5-6 is shaped by three pitch interval-cycles: -4/-1, -2/-3, and -1/-1. Each cycle forms

a successively more compact tetrachod, and the first cycles overlap by one note, F#.

The first tetrachord, <3t6e>, is mapped onto the next tetrachord, <4e16>, by vlset

<1177>. T1 of this vlset, <2288>, maps <641e> onto the next tetrachord, <8697>,

which extends to the downbeat of m. 7. Another pcset interpretation of mm. 6-7,

which follows the phrase markings, shows that pcset <987> is mapped onto <698> by

vlset <911>. While the repetition of the descending trichord <987>, and its

replacement <986> suggests <00e> voice leading on the surface of the piece, the

vlclass [044] interpretation matches the whole-tone voice-leadings and harmonies of

the accompaniment. Another vlclass [006] vlset, <228>, maps <698> onto <8e4>,

which, in score order <4e8>, is mapped onto the final trichord, <652>, by vlset

<266>.

As the previous discussion has shown, Berg’s Op. 5 n. 2 is easily divided into

three voice-leading strata. While two of these layers are homophonic and one is

melodic, all three are unified through their use of whole-tone voice-leading classes.

This forms a recursive structure with the whole-tone pcsets used on the surface of the

piece, especially the augmented triad. The whole-tone voice-leading consistency

among pc voices and pcsets provides another multi-movement connection in Berg’s

Op. 5, as the first piece, discussed above, also features a whole-tone voice-leading

class, [200].

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Chapter 5: Summary, Conclusion, and Future Work

As discussed above, many studies of voice leading in post-tonal music focus

on harmonic similarity and semitonal offset. While this works well for some musical

contexts, it is not satisfactory for the intervallic diversity displayed by many post-

tonal pieces. Voice-leading sets provide a way in which to separate the study of voice

leading from harmony, and demonstrate that pc voices may unify disparate musical

surfaces.

The generalized definition of voice-leading parsimony proposed by this study

is best exemplified by the voice-leading spaces generated in chapter 3. The unit

distance on each space is defined by its generative vlset. For example, motion by one

unit of distance, therefore the “smoothest” voice leading, upon the vlset <23> space is

motion by the vlset itself. Less-smooth voice leadings are defined by combinations of

this generative vlset. Individual pieces may be modeled by one or more vl spaces;

representation on multiple spaces may be proposed to change voice-leading

interpretations and thereby change the metric for relative smoothness in different

formal sections. These spaces are useful tools for the type of analyses shown in

chapter 4, as adjacent pcsets or set-classes on the space reflect some version of the

underlying voice leading. The voice-leading spaces are also useful compositional

tools, as they provide a composer with a clear method for voice-leading manipulation

and variation.

Future Work

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The method for generating voice-leading spaces in chapter 3 should be

expanded to higher cardinalities. This is difficult, due to the number of dimensions

required, as well as the exponential expansion of the number of members of the

ordered pcset space. This expansion can be reduced through equivalence classes, as

shown by the ordered tn-class and set-class spaces above, but the representational

power of the voice-leading spaces may be diminished by further layers of abstraction,

such as multiplication-class equivalence.

The voice-leading spaces shown in chapter 3 may be used for the

representation of voice leadings interpreted from a variety of musical works. These

voice leadings are understood as the motion between pcsets or set-classes on the

space; this motion corresponds to the pc voices expressed by the voice-leading

mapping. Therefore, animation is required to adequately represent these voice

leadings. The next step towards improving both the spaces’ visual representation and

explanatory power is the development of user-friendly software which will allow

users to animate voice leadings within the wide variety of spaces introduced here.110

Such animation could prove particularly useful in the representation of larger

cardinalities and their corresponding higher-dimensional structures; additionally, the

program could be designed to display only a small portion of a larger space at a given

time.

110

I have produced a number of visualization tools using Matlab, and plan to expand

them to encompass the full range of voice-leading sets. For an example of

transformational animation, see Stephanie Lind and John Roeder’s “Transformational

Distance and Form in Berg’s Schlafend traegt man mich,” in Music Theory Online 15,

1 (2009), as well as other contributions to the volume.

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Segmentation Techniques

The problem of segmentation in the analysis of post-tonal music is

exacerbated by vlsets, as they exponentially increase the number of possible

relationships between musical events. The analyses shown in the previous chapter

address this issue by identifying some prominent voice-leading motive or motives,

and demonstrating the transformations of that motive throughout the piece. As is the

case for pcset theory and K-nets, vlsets are merely analytical tools; it is the analyst’s

responsibility to sort through the large quantity of data a piece presents in order to

construct an analysis from the most meaningful information.

One way in which to limit the great number of possible voice-leading

interpretations of a piece is to use consistent interpretive techniques. John Roeder’s

emphasis on pitch voice leadings provides a good example. Pitch space is ordered by

register, and therefore adjacent chord tones are well-defined using this ordering.

These chord-tone adjacencies are relative distances, however, as the “lowest” pitch

may be any frequency, so long as it is lower than the remaining pitches. Pitch-class

space is modular, eliminating the low-to-high ordering pitch space provides.

Harmonies in pitch-class space are ordered with respect to the chromatic scale,

however; just like pitch space, pitch-class space is ordered by relative frequency, that

is, the distance from one chord member to another. Just as the “lowest” pitch is

determined by its relationship to the remaining chord members, the members of a

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pitch-class set are ordered with respect to one another. Normal order thus forms a

natural “register” for pitch-class space.111

Describing voice leadings among normal orders, rather than ordered pcsets,

greatly reduces the volume of information a piece presents. For example, if restricted

to normal orders, all minor triads are mapped onto major triads by vlclass [010]; all

tn-class [0147]s are mapped onto tn-class [0246]s by vlclass [0101], etc. This

reduction of pc space may be essential for the voice-leading analysis of larger

cardinalities, as one unordered pcset may have hundreds of possible mappings onto

another. There are only two possible mappings between two pcsets in normal order;

either the pc voices are transpositions or inversions. For example, <014> maps onto

<379> by either <365> or <38e>i.

In an analytical example, Figure 5-1 reinterprets the voice leading of the

accompaniment to Berg’s Op. 5 n. 2, with its right-hand trichords and left-hand

dyads, as a series of pentachords. The pc voices are produced by mapping

corresponding normal-order members onto one another. The two possible vlsets for

each mapping are shown below the score. The varied vlsets from this mapping show

some patterning, and can be considered a “first pass” through the piece.

Spaces of Voice leadings

Each of the voice-leading spaces explored in this study organize pcsets or

their ordered equivalence classes in a space defined by a specific vlset or vlclass.

111

Pitch-class space may also be ordered by the circle of fifths, where pc numerals

stand for the number of fifths from pc C. This would simply map the set of pcsets

onto itself through multiplication.

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Because vlsets are isomorphic to the set of ordered pcsets, the objects on these spaces

could also be interpreted as organizing voice leadings, and demonstrating

relationships among them. This produces a metric for transformations between voice

leadings, and greatly expands the spaces’ analytical usefulness.

The generalization of the spaces from pcsets to vlsets is similar to the

generalization of voice leading proposed in this study. The analyses shown in chapter

4 privilege harmonic similarity among voice leadings, just as most contemporary

studies of voice leading privilege harmonic similarity among pcsets. The spaces may

therefore be used to model pieces which demonstrate many distinct voice leadings.

Figure 5-2 revisits mm. 1-4 of the clarinet melody from Alban Berg’s Op. 5 n.

1. The segmentation into trichords is the same as that shown in chapter 4, but the

pcsets are ordered by normal order; all are interpreted as transpositional vlsets. In the

figure, the pcsets are listed in normal order (ascending), showing the three pc voices

in pc register. Figures 5-3a and b show the unordered vlsets as locations on the

trichordal orbifold.112

The lines on the figure represent distances between vlsets. The

full orbifold is shown in Figure 5-3a for perspective. Figure 5-3c shows the vlsets on

the [013] ordered set-class space. The sets on this space are equivalent through

rotation, ordered transposition, and ordered inversion. As shown, the four ordered

vlclasses presented by the clarinet, [023], [001], [035], and [002], are located in one

112

This space is reduced by permutation, so the vlsets are unordered.

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region of the space, demonstrating that the vlsets in the passage are very similar,

when [013] is used as a metric.113

The arrows on the space shown in Figure 5-3c imply that some transformation

other than transposition or inversion may map one vlset onto another. The set of

vlsets themselves form a model for the construction of such transformations.

Following Lewin’s creation of recursive K-net transformations, a hyper-vlset is an

ordered series of transpositions or inversions which maps one vlset onto another.

Hyper-vlsets are listed with double angle brackets to differentiate them from vlsets,

such as <<013>>. Figure 5-4 shows a graph of the vlsets in question. As shown,

each hyper-vlset is equivalent to <<013>> under rotation, ordered transposition, and

inversion, except for the transformations between vlclasses [001] and [002]; this

transformation is measured in two <<013>>s.

Serial Music

The study of voice leading as an ordered set of transformations finds its

natural extension in serial music. The ordered intervals of a series of any length may

be expressed using vlset notation. Ordered pc transformations may also be used to

demonstrate how rows are mapped onto one another. For example, the sixth

movement of Alban Berg’s Lyric Suite uses two rows: Row 1, or 1PF, is

<56t41928730e> and Row 2, or 2PF, is <54120e6t9873>. These rows, and their It

inverted forms—which causes all four rows begin on F—are layered successively in

mm. 1-6, shown in Figure 5-5.

113

This is the trichordal space shown in Tymoczko 2006.

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Row 1 is mapped onto Row 2 by <0t3te2422574>.114

This mapping is

expressed vertically between the rows when the viola enters in m. 2. In m. 6, when

the inverted forms of the rows are given the same treatment in the violins, the same

vertical dyads are sounded. In mm. 4-5, the first violin plays the inverted form of

Row 2—2IF—above 1PF in the cello. If the mapping of 1PF onto 2IF is interpreted

as an ordered series of inversions, rather than transpositions, the ordered set is

<t070e8688536>i, an It inversion of the transpositional mapping. The remaining pair

is in m. 6 between the viola and second violin, 2PF and 1IF, respectively, which

produce an ordered inversional series, <t81890200352>i, a Tt transposition of the

original series, and an I8 inversion of the preceding pair. Inversion by index t is

important in other ways in the passage. Of course, row pairs {1PF 1IF} and {2PF

2IF} map onto themselves by It; in addition, in 1PF, order position-pairs 4-5, 6-7, and

8-9 are sum-t dyads, therefore they exchange places under It, and the first and last pcs

of the row, 5 and e, map onto themselves under It.115

The ordered mappings between the two rows of the movement form a third

row, which is subjected to transformation and manipulation just as the two pc rows

through which it is formed are. For example, the end of the movement returns to the

opening’s dyadic texture and reverses the layering used at the beginning; instruments

are removed one by one over the last five bars, until only the viola is left. The score

highlights invariant collections among horizontal and vertical row forms in this

114

Following Berg’s notes on the Lyric Suite given to the Kolisch Quartet and his

letter to Schoenberg in the derivation of material, George Perle derives the

hexachords of Row 2 from Row 1 through an ordered partition. Perle 2001, 11. 115

Various row relationships from this movement are given in Headlam 1996.

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section. In m. 40, the second violin and viola play rows 1PC# and 2IEb, respectively;

the dual transformations T8 and I8 from 1PF and 2IF transform the ordered mapping

from Row 1 onto Row 2 to <461652022e90>i, an I4 transformation of the original

transpositional row. As Figure 5-6 shows, the cello and 2nd

violin present the original

form of the transformational row, <0t3te2422574>, although they are offset by an

eighth note. The viola and violin 1, offset by two eighth notes, begin a tritone apart,

and therefore present T6 of the row, <649458t88e1t>. This row is It of the row

between the second violin and viola.

Due to the pitch-class/order-position exchange, many more mapping

relationships among rows are possible.116

For example, the order position mapping

from 1PF onto 2PF is expressed by <055tt3922266>; each interval in the series

demonstrates the op interval from one row to another. This mapping only produces a

form of Row 2 which begins on the same pitch. If this harmonic relationship between

the rows is altered, the op mapping must be used in conjunction with some other

transformation. For example, in m. 40, Row 1IA maps onto 2IA under the op

mapping, and then may be transposed by T6 to 2IEb.

Diatonic and Cyclic Voice leading

116

For example, see Walter O’Connell’s “Tone-spaces” in Die Reihe, 8 (1968): 35-67,

Andrew Mead “Some Implications of the Order-number/Pitch-class Isomorphism

Inherent in the Twelve-tone System” in Perspectives of New Music27, n. 1 (1989):

180-233, Larry Solomon “New Symmetric Transformations” in Perspectives of New

Music 2, n. 2 (1973): 257-264, and Michael Stanfield “Some Exchange Operations in

Twelve-tone Theory: Part One” in Perspectives of New Music 23, n. 1 (1984): 258-

277.

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The method for interpreting voice leadings proposed here may also be applied

to tonal music, whether using the full chromatic, or diatonic collection. Voice leading

in tonal music is typically the purview of Schenkerian analysis, as it provides a means

for showing the prolongation of the tonic triad at all levels of a piece. However,

voice-leading sets may be used to provide a unique contrapuntal perspective.

A four-voice chorale includes a limited number of diatonic sets, as mod-seven

space is considerably smaller than the full chromatic. In diatonic sets, root-doubled

triads are members of [0024], and seventh chords [0135]. In harmonic analysis, triads

and seventh chords may be presented in a number of different guises, with options for

doublings and incomplete chords. While these sonorities may be interpreted as

expressing the same basic harmony, they have distinct voice leading characteristics.

For example, the tripled root cadence results in multiset [0002], which is not capable

of the same types of voice leading as [0024].

A typical step towards harmonic analysis identifies non-harmonic tones, such

as appogiaturas, suspensions, passing tones, etc. All other notes fit into some

consonant sonority—either a triad or seventh chord. Voice-leading sets provide a

way in which to include non-tertian sonorities in the contrapuntal analysis of diatonic

voice leading. For example, Figure 5-7 shows that a 4-3 suspension projects two

diatonic set-classes: [0034] and [0024]. This motion can be expressed by vlset

<0060>, which shows that three pc voices are static, and one moves by diatonic

interval 6. Just as in the earlier post-tonal demonstrations, this vlset may be subjected

to variation through transposition and inversion. If the suspension figure recurs, and

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the vlset is transposed by T1 to <1101>, the suspended voice remains motionless,

while the three remaining chord members ascend by diatonic step; see Figure 5-8.

Figures 5-7 and 5-8 have very different harmonic implications, but present the same

underlying contrapuntal motion through the relationships between the voices. Just as

in their application to post-tonal music, diatonic vlsets are largely independent of the

harmonic objects they operate upon.

The process used to create voice-leading spaces shown in chapter 3 may be

used to create diatonic voice-leading spaces. One advantage of doing this is the

reduction in the size of the space involved; there are only 2,401 ordered diatonic

tetrachords, compared to 20,736 ordered chromatic tetrachords. These produce much

more manageable spaces.

Just as voice leading among diatonic sets may be described by vlsets, voice

leadings among cyclic collections such as the octonic, hexatonic, whole-tone scale,

Olivier Messiaen’s Modes of Limited Transposition, etc, may be represented by vlsets

and voice-leading spaces. The sets and voice leadings are simply defined modulo the

collection; for example, the hexatonic collection is made up of mod-6 sets and voice

leadings. Vlset <112> maps hexatonic set {023} onto {135}, which correspond to

pcsets {045} and {159}, respectively.

Contour Spaces

Voice-leading spaces may be created for any modulus. If the sets in these

spaces are interpreted as scale-degree sets, rather than pc sets, they may be populated

by any collection of pcs which are mapped onto the scale-degrees. For example, if a

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mod-7 vlset <013> is applied to a C major triad, diatonic set {024}, it maps the triad

onto {030}, a C-E multiset. If the diatonic sets are mapped onto some other

collection, say a chromatic septachord, the same mapping will produce different

musical results. Generalized contour spaces may be created, through which any

pitch-class collections may be mapped, just as pitch-classes are mapped onto an

ordered series in serial music.

Transformation Vectors

The analyses presented in chapter four demonstrate some of the ways in which

pitch-class voices may be defined in post-tonal music using vlsets—vectors of

transformational mappings. Vectors of transformations need not be limited to pitch-

class mappings, however. Any musical parameter, such as rhythm, meter,

articulation, texture, etc, may be described as an ordered series of mappings.117

Let the positive integers be mapped onto durations such that the integers

correspond to sixteenth-note durations, that is, 0 = no duration, 1 = one sixteenth-

note, 2 = one eighth-note, etc. Just as vlsets provide a method for mapping pcsets

onto one another regardless of harmonic similarity, vectors of transpositions and

inversions may be used to map various rhythmic patterns onto one another. As in the

analyses shown above, these transformations may be used to depict the variation of

rhythmic motives throughout a piece. The idea of duration voices is more difficult to

117

In GMIT, Lewin describes a number of GIS constructions whose objects are non-

pitch musical parameters, such as rhythm and timbre. Roeder 1994 demonstrates

some additional ways in which vectors of transformations may be employed.

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conceptualize, but nevertheless, the simultaneous transformations of individual

rhythms forms a type of duration counterpoint.

For example, returning to Anton Webern’s Op. 5 n. 2, the rhythm of the

opening viola melody is expressed by durationset {4624}, or quarter, dotted-quarter,

eighth, quarter. This durationset is varied in the melody in m. 5; beginning with Bb

and continuing to F# on the downbeat of m. 6, the 2nd

violin plays durationset

{2313}. This is very nearly half of the opening durationset. The first three values of

this set are exactly half of the opening three durations, while the final value is

extended by one sixteenth-note.

If the opening durationset is extended to include the eighth rest in m. 1, the

ordered set is {46240}, where 0 denotes no duration. Although the rest consists of

two sixteenth notes, there is no melody duration during the rest. Simply for clarity of

demonstration, let us assume mod-twelve.118

This pentachord is mapped onto the

{22244} gesture which immediately follows by <t8004>.

The rhythm of the melody in the final measure of the piece consists of three

eighths and a quarter, or {2224}; this interpretation ignores the eighth rest in the

middle of the measure. The opening gesture is mapped onto the closing gesture by

<t800>. This mapping represents the expanding and contracting of the rhythmic

values as they are transformed from the opening of the piece to the end.

Another tetrachordal rhythmic collection occurs in the melody in m. 6, in the

2nd

violin, {3133}. As an un-ordered set, it is an inversion of the durationset

118

The longest rhythm appears to be the tied chords in the accompaniment at the end,

which consists of 18 sixteenth-notes, so mod-nineteen may be a better choice.

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highlighted in the final measure of the piece <5555>i maps one set onto the other. It

is difficult to imagine how a duration may be inverted, but it is important to

remember that the mapping takes place between the numbers, which are

representatives of the durations. If the durations are arranged, like pitch-classes, in a

mod-12 circle, <5555>i forms the axis of inversion which the durations exchange

places around.

Any musical feature may be treated transformationally the way durations are

in the above example, provided it is parameterized, that is, mapped onto an ordered

series. This creates a feature vector; a space which maps musical objects onto

locations given by coordinates. The voice-leading spaces discussed above are feature

spaces.

As the above discussion shows, vlsets offer great promise for the analysis of

post-tonal and other musics. The analytical techniques outlined here and in chapter 4

reflect some possible methods for identifying pc voices in post-tonal pieces. These

voices may be represented as motion in some of the voice-leading spaces developed

above. Using vlsets, voice-leading parsimony may be redefined contextually for

individual musical contexts. The voice-leading spaces then show this relative

parsimony through the proximity of their objects.

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Figure 1-1: Arnold Schoenberg’s Drei Klavierstuecke Op. 11 n. 2, mm. 11-13

a b c d e

{1467} {e235} {4580} {569tt1} {578t01}

Figure 1-2a: fuzzy transposition as in Straus 2003

6 3

7 e

4 5

1 2

*Tt(1)

a b

Figure 1-2b: pitch voice leadings as in Roeder 1994

6 -3 3

7 +4 e

4 +1 5

1 +1 2

<1,1,4,-3>

a b

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Figure 1-2c: vlsets from chord a to b and d to e

9 0

5 5

6 3 6 3 1 t

7 e 7 e t 8

4 5 4 5 6 1

1 2 1 2 t 7

<tt9t> <0690>i <01e020>

a b a b d e

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Figure 1-3: values for the moving voice when retaining two common tones,

from Richard Cohn.

Set-class “Root” “Third” “Fifth”

[012] 3 0 9 {012}-{312}; {012}-{012}; {012}-{01e}

[013] 4 1 5 {013}-{413}; {013}-{023}; {013}-{01t}

[014] 5 2 5 {014}-{514}; {014}-{034}; {014}-{019}

[015] 6 3 3 {015}-{615}; {015}-{045}; {015}-{018}

[016] 5 2,4,6* 1 {016}-{716}; {016}-{056}; {016}-{0e6};

{016}-{076}; {016}-{017}

[024] 6 0 6 {024}-{624}; {024}-{024}; {024}-{02t}

[025] 5 1 4 {025}-{725}; {025}-{035}; {025}-{029}

[026] 4 2,4,6* 2 {026}-{826}; {026}-{046}; {026}-{0t6};

{026}-{086}; {026}-{028}

[027] 3 3 0 {027}-{927}; {027}-{057}; {027}-{027}

[036] 3 6 3 {036}-{936}; {036}-{096}; {036}-{039}

[037] 2 1 1 {037}-{t37}; {037}-{047}; {037}-{038}

[048] 0 0 0 {048}-{048}; {048}-{048}; {048}-{048}

*Because the tritone divides the octave evenly, four members of [016] and [026]

share the same tritone, resulting in more common-tone preserving transformations

when moving the other voice. These additional transformations are between

transpositionally-related sets. Symmetrical sets, such as [024], have only two

possible operations, as the interval between “root” and “fifth” can only form a

member of the set-class with one other pitch-class; [036] has three possible

transformations because of the symmetry of the tritone.

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Figure 1-4a: the trichordal orbifold.

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123

Figure 1-4b: one-quarter of the trichordal orbifold

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124

Figure 2-1: vlsets in each cardinality

1: 12 * 2 = 24

2: 12 * 12 * 2 = 288

3: 12 * 12 * 12 * 2 = 3,456

4: 12 * 12 * 12 * 12 * 2 = 41,472

5: 12 * 12 * 12 * 12 * 12 * 2 = 497,664

6: 12 * 12 * 12 * 12 * 12 * 12 * 2 = 5,971,968

7: 12 * 12 * 12 * 12 * 12 * 12 * 12 * 2 = 71,663,616

8: 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 2 = 859,963,392

9: 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 2

= 10,319,560,704

10: 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 2

= 123,834,728,448

11: 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 2

= 1,486,016,741,376

12: 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 2

= 1.78322009 * 1013

13: 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 2

= 2.13986411 * 1014

∞: (12^∞) * 2 = ∞

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125

Figure 2-2: vlsets spanning reordered pcsets

<014> - <444> - <458> <014> - <460>i - <458>

<014> - <471> - <485> <014> - <499>i - <485>

<014> - <570> - <584> <014> - <598>i - <584>

<014> - <534> - <548> <014> - <550>i - <548>

<014> - <831> - <845> <014> - <859>i - <845>

<014> - <840> - <854> <014> - <868>i - <854>

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Figure 2-3: parallel pitch-class voice leading

Figure 2-4: oblique pitch-class voice leading

Figure 2-5: contrary pitch-class voice leading

x = pcset <24> <39>i(x) <2t>i(x) <1e>i(x) <00>i(x) <e1>i(x)

<222> <777>i

<05> <10>i

<39> <1e> <et96>

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127

Figure 2-6: similar pitch-class voice leading

<21> <78>i

<1122> <011>i

Figure 2-7: vlset transposition, or ordered vlclass [01]

<02> - <01> - <03> <02> - <67> - <69>

<02> - <12> - <14> <02> - <78> - <7t>

<02> - <23> - <25> <02> - <89> - <8e>

<02> - <34> - <36> <02> - <9t> - <90>

<02> - <45> - <47> <02> - <te> - <t1>

<02> - <56> - <58> <02> - <e0> - <e2>

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Figure 2-8a: Drei Klavierstuecke Op. 11 no. 1, by Arnold Schoenberg, mm. 1-5

Figure 2-8b: Drei Klavierstuecke Op. 11 no. 1, by Arnold Schoenberg, mm. 6-13

e 9 e 1 e e 9 1

8 5 5 9 8 5 5 9

7 4 6 t 7 6 4 t

<tt8> <442> <tt0> <446> a b c d a c b d

8 5 0 8 6 8 8 0

4 e e 2 2 4 t e

6 7 3 9 0 6 9 3

<113> <99e> <442> <335> g h i j e g f i

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Figure 2-9: ordered pcset tn-classes paired by ordered vlclasses

a). [01] b). [10]

[00] – [01] [06] – [50] [00] – [10] [06] – [05]

[01] – [02] [50] – [40] [01] – [00] [50] – [06]

[02] – [03] [40] – [30] [02] – [01] [40] – [50]

[03] – [04] [30] – [20] [03] – [02] [30] – [40]

[04] – [05] [20] – [10] [04] – [03] [20] – [30]

[05] – [06] [10] – [00] [05] – [04] [10] – [20]

c). [02] d). [20]

[00] – [02] [06] – [40] [00] – [20] [06] – [04]

[01] – [03] [50] – [30] [01] – [10] [50] – [05]

[02] – [04] [40] – [20] [02] – [00] [40] – [06]

[03] – [05] [30] – [10] [03] – [01] [30] – [50]

[04] – [06] [20] – [00] [04] – [02] [20] – [40]

[05] – [50] [10] – [01] [05] – [03] [10] – [30]

e). [03] f). [30]

[00] – [03] [06] – [30] [00] – [30] [06] – [03]

[01] – [04] [50] – [20] [01] – [20] [50] – [04]

[02] – [05] [40] – [10] [02] – [10] [40] – [05]

[03] – [06] [30] – [00] [03] – [00] [30] – [06]

[04] – [50] [20] – [01] [04] – [01] [20] – [50]

[05] – [40] [10] – [02] [05] – [02] [10] – [20]

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g). [04] h). [40]

[00] – [04] [06] – [20] [00] – [40] [06] – [02]

[01] – [05] [50] – [10] [01] – [30] [50] – [03]

[02] – [06] [40] – [00] [02] – [20] [40] – [04]

[03] – [50] [30] – [01] [03] – [10] [30] – [05]

[04] – [40] [20] – [02] [04] – [00] [20] – [06]

[05] – [30] [10] – [03] [05] – [01] [10] – [50]

i). [05] j). [50]

[00] – [05] [06] – [10] [00] – [50] [06] – [01]

[01] – [06] [50] – [00] [01] – [40] [50] – [02]

[02] – [50] [40] – [01] [02] – [30] [40] – [03]

[03] – [40] [30] – [02] [03] – [20] [30] – [04]

[04] – [30] [20] – [03] [04] – [10] [20] – [05]

[05] – [20] [10] – [04] [05] – [00] [10] – [06]

k). [06] l). [00]

[00] – [06] [06] – [00] [00] – [00] [06] – [06]

[01] – [50] [50] – [01] [01] – [01] [50] – [50]

[02] – [40] [40] – [02] [02] – [02] [40] – [40]

[03] – [30] [30] – [03] [03] – [03] [30] – [30]

[04] – [20] [20] – [04] [04] – [04] [20] – [20]

[05] – [10] [10] – [05] [05] – [05] [10] – [10]

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Figure 2-10: ordered tn-class cycles

a). [01]

([00] – [01] – [02] – [03] – [04] – [05] – [06] – [50] – [40] – [30] – [20] – [10])

b). [10]

([00] – [10] – [20] – [30] – [40] – [50] – [06] – [05] – [04] – [03] – [02] – [01])

c). [02] d). [20]

([00] – [02] – [04] – [06] – [40] – [20]) ([00] – [20] – [40] – [06] – [04] – [02])

([01] – [03] – [05] – [50] – [30] – [10]) ([01] – [10] – [30] – [50] – [05] – [03])

e). [03] f). [30]

([00] – [03] – [06] – [30]) ([00] – [30] – [06] – [03])

([01] – [04] – [50] – [20]) ([01] – [20] – [50] – [04])

([02] – [05] – [40] – [10]) ([02] – [10] – [40] – [05])

g). [04] h). [40]

([00] – [04] – [40]) ([00] – [40] – [04])

([01] – [05] – [30]) ([01] – [30] – [05])

([02] – [06] – [20]) ([02] – [20] – [06])

([03] – [50] – [10]) ([03] – [10] – [50])

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i). [05]

([00] – [05] – [20] – [03] – [40] – [01] – [06] – [10] – [04] – [30] – [02] – [50])

j). [50]

([00] – [50] – [02] – [30] – [04] – [10] – [06] – [01] – [40] – [03] – [20] – [05])

Figure 2-11: an ordered tn-class [04] tn-class cycle

<34> – <15> – <49> – <37> – <74> – <t2> – <56>;

tn-classes: [01] – [05] – [30] – [01]

Figure 2-12: ordered pcset and tn-class cycles formed by vlset <14>

<12> – <14> – <26> – <14> – <3t> – <14> – <42> – <14> – <56> – <14> – <6t> –

<14> – <72> – <14> – <86> – <14> – <9t> – <14> – <t2> – <14> – <e6> – <14> –

<0t> – <14> – <12>

tn-classes:

[01] – [04] – [50] – [20] – [01] – [04] – [50] – [20] – [01] – [04] – [50] – [20] – [01]

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Figure 2-13: transpositions of vlset <01>i and <0e>i

a). <01>i b). <0e>i

<04> - <01>i - <09> <04> - <0e>i - <07>

<04> - <12>i - <1t> <04> - <10>i - <18>

<04> - <23>i - <2e> <04> - <21>i - <29>

<04> - <34>i - <30> <04> - <32>i - <3t>

<04> - <45>i - <41> <04> - <43>i - <4e>

<04> - <56>i - <52> <04> - <54>i - <50>

<04> - <67>i - <63> <04> - <65>i - <61>

<04> - <78>i - <74> <04> - <76>i - <72>

<04> - <89>i - <85> <04> - <87>i - <83>

<04> - <9t>i - <96> <04> - <98>i - <94>

<04> - <te>i - <t7> <04> - <t9>i - <t5>

<04> - <e0>i - <e8> <04> - <et>i - <e6>

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Figure 2-14: ordered tn-classes paired by ordered vlclasses [01]i and [10]i

a). [01]i b). [10]i

[00] – [01] [00] – [10]

[02] – [10] [01] – [20]

[03] – [20] [02] – [30]

[04] – [30] [03] – [40]

[05] – [40] [04] – [50]

[06] – [50] [05] – [06]

[00]

[06]

[03] [30]

[01]

[02]

[04]

[05]

[10]

[20]

[40]

[50]

[00]

[06]

[03] [30]

[01]

[02]

[04]

[05]

[10]

[20]

[40]

[50]

[01]i [10]i

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Figure 2-15: ordered pcset-classes paired by ordered vlclasses [04]i and [40]i

a). [04]i b). [40]i

[00] – [04] [00] – [40]

[01] – [03] [10] – [30]

[02] – [02] [20] – [20]

[05] – [10] [50] – [01]

[06] – [20] [06] – [02]

[50] – [30] [05] – [03]

[40] – [40] [04] – [04]

[00]

[06]

[03] [30]

[01]

[02]

[04]

[05]

[10]

[20]

[40]

[50]

[00]

[06]

[03] [30]

[01]

[02]

[04]

[05]

[10]

[20]

[40]

[50]

[04]i [40]i

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Figure 2-16: tn-class pairs

a). [02]i b). [20]i c). [03]i d). [30]i

[00] – [02] [00] – [20] [00] – [03] [00] – [30]

[01] – [01] [01] – [30] [01] – [02] [01] – [40]

[03] – [10] [02] – [40] [04] – [10] [02] – [50]

[04] – [20] [03] – [50] [05] – [20] [03] – [06]

[05] – [30] [04] – [06] [06] – [30] [04] – [05]

[06] – [40] [05] – [05] [50] – [40] [20] – [10]

[50] – [50] [10] – [10]

a). [05]i b). [50]i c). [06]i d). [00]i

[00] – [05] [00] – [50] [00] – [06] [00] – [00]

[01] – [04] [01] – [06] [01] – [05] [01] – [10]

[02] – [03] [02] – [05] [02] – [04] [02] – [20]

[06] – [10] [03] – [04] [03] – [03] [03] – [30]

[50] – [20] [40] – [10] [50] – [10] [04] – [40]

[40] – [30] [30] – [20] [40] – [20] [05] – [50]

[30] – [30] [06] – [06]

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Figure 2-17: ordered tn-classes paired by ordered vlclass [025]

[000] – [025] [010] – [035] [020] – [045] [030] – [055] [040] – [065] [050] – [075]

[001] – [026] [011] – [036] [021] – [046] [031] – [056] [041] – [066] [051] – [076]

[002] – [027] [012] – [037] [022] – [047] [032] – [057] [042] – [067] [052] – [077]

[003] – [028] [013] – [038] [023] – [048] [033] – [058] [043] – [068] [053] – [078]

[004] – [029] [014] – [039] [024] – [049] [034] – [059] [044] – [069] [054] – [079]

[005] – [02t] [015] – [03t] [025] – [04t] [035] – [05t] [045] – [06t] [055] – [07t]

[006] – [02e] [016] – [03e] [026] – [04e] [036] – [05e] [046] – [06e] [056] – [07e]

[007] – [020] [017] – [030] [027] – [040] [037] – [050] [047] – [060] [057] – [070]

[008] – [021] [018] – [031] [028] – [041] [038] – [051] [048] – [061] [058] – [071]

[009] – [022] [019] – [032] [029] – [042] [039] – [052] [049] – [062] [059] – [072]

[00t] – [023] [01t] – [033] [02t] – [043] [03t] – [053] [04t] – [063] [05t] – [073]

[00e] – [024] [01e] – [034] [02e] – [044] [03e] – [054] [04e] – [064] [05e] – [074]

[060] – [085] [070] – [095] [080] – [0t5] [090] – [0e5] [0t0] – [005] [0e0] – [015]

[061] – [086] [071] – [096] [081] – [0t6] [091] – [0e6] [0t1] – [006] [0e1] – [016]

[062] – [087] [072] – [097] [082] – [0t7] [092] – [0e7] [0t2] – [007] [0e2] – [017]

[063] – [088] [073] – [098] [083] – [0t8] [093] – [0e8] [0t3] – [008] [0e3] – [018]

[064] – [089] [074] – [099] [084] – [0t9] [094] – [0e9] [0t4] – [009] [0e4] – [019]

[065] – [08t] [075] – [09t] [085] – [0tt] [095] – [0et] [0t5] – [00t] [0e5] – [01t]

[066] – [08e] [076] – [09e] [086] – [0te] [096] – [0ee] [0t6] – [00e] [0e6] – [01e]

[067] – [080] [077] – [090] [087] – [0t0] [097] – [0e0] [0t7] – [000] [0e7] – [010]

[068] – [081] [078] – [091] [088] – [0t1] [098] – [0e1] [0t8] – [001] [0e8] – [011]

[069] – [082] [079] – [092] [089] – [0t2] [099] – [0e2] [0t9] – [002] [0e9] – [012]

[06t] – [083] [07t] – [093] [08t] – [0t3] [09t] – [0e3] [0tt] – [003] [0et] – [013]

[06e] – [084] [07e] – [094] [08e] – [0t4] [09e] – [0e4] [0te] – [004] [0ee] – [014]

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Table 2-1: subgroups of the group of dyadic vlsets

1 {<00>} 6 {<42>} 12 {<23>} 24 {<06>, <12>}

6 {<46>} 12 {<31>} 24 {<06>, <13>}

2 {<06>} 6 {<62>} 12 {<32>} 24 {<06>, <14>}

2 {<60>} 6 {<64>} 12 {<34>} 24 {<06>, <15>}

2 {<66>} 6 {<82>} 12 {<41>} 24 {<21>, <60>}

12 {<43>} 24 {<31>, <60>}

3 {<04>} 8 {<03>, <60>} 12 {<61>} 24 {<41>, <60>}

3 {<40>} 8 {<06>, <30>} 12 {<81>}

3 {<44>} 8 {<06>, <33>} 12 {<91>} 36 {<01>, <40>}

3 {<48>} 12 {<t1>} 36 {<02>, <20>}

9 {<04>, <40>} 12 {<02>, <60>} 36 {<04>, <10>}

4 {<03>} 12 {<06>, <20>} 36 {<04>, <11>}

4 {<30>} 12 {<01>} 12 {<06>, <22>} 36 {<04>, <13>}

4 {<33>} 12 {<10>} 12 {<06>, <24>} 36 {<04>, <21>}

4 {<36>} 12 {<11>} 36 {<12>, <20>}

4 {<39>} 12 {<12>} 16 {<03>, <30>}

4 {<63>} 12 {<13>} 48 {<01>, <30>}

4 {<06>, <60>} 12 {<14>} 18 {<02>, <40>} 48 {<03>, <10>}

12 {<15>} 18 {<04>, <20>} 48 {<03>, <11>}

6 {<02>} 12 {<16>} 18 {<04>, <22>} 48 {<03>, <12>}

6 {<20>} 12 {<17>}

6 {<22>} 12 {<18>} 24 {<01>, <60>} 72 {<01>, <20>}

6 {<24>} 12 {<19>} 24 {<02>, <30>} 72 {<02>, <10>}

6 {<26>} 12 {<1t>} 24 {<03>, <20>} 72 {<02>, <11>}

6 {<28>} 12 {<1e>} 24 {<06>, <10>}

6 {<2t>} 12 {<21>} 24 {<06>, <11>} 144 {<01>, <10>}

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Table 2-2: group table for subgroup generated by <17>

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Figure 2-18: subgroup <17> with three sets of pcsets as objects

If originating with pc {00}:

{00}---{17}---{22}---{39}---{44}---{5e}---{66}---{71}---{88}---{93}---{tt}---{e5}

If originating with pc {02}:

{02}---{19}---{24}---{3e}---{46}---{51}---{68}---{73}---{8t}---{95}---{t0}---{e7}

If originating with pc {01}:

{01}---{18}---{23}---{3t}---{45}---{50}---{67}---{72}---{89}---{94}---{te}---{e6}

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Table 2-3: subgroup <02> with inversion

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Table 2-4: examples of four-member subgroups of [C12 C12] C2

<00> <06> <00>i <06>i

<00> <00> <06> <00>i <06>i

<06> <06> <00> <06>i <00>i

<00>i <00>i <06>i <00> <06>

<06>i <06>i <00>i <06> <00>

<00> <06> <a,b>i <a,b+6>i

<00> <00> <06> <a,b>i <a,b+6>i

<06> <06> <00> <a,b+6>i <a,b>i

<a,b>i <a,b>i <a,b+6>i <00> <06>

<a,b+6>i <a,b+6>i <a,b>i <06> <00>

<00> <66> <29>i <83>i

<00> <00> <66> <29>i <83>i

<66> <66> <00> <83>i <29>i

<29>i <29>i <83>i <00> <66>

<83>i <83>i <29>i <66> <00>

<00> <06> <12>i <18>i

<00> <00> <06> <12>i <18>i

<06> <06> <00> <18>i <12>i

<12>i <12>i <18>i <00> <06>

<18>i <18>i <12>i <06> <00>

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Table 2-5: examples of two-member subgroups of [C12 C12] C2

<00> <20>i

<00> <00> <20>i

<20>i <20>i <00>

Table 2-6: C12 C12 subgroups combined with inversional vlsets

2 {<00>}, <a,b>i 144 24 {<23>}, <a+2n, b+3n>i 12

24 {<31>}, <a+3n, b+n>i 12

4 {<06>}, <a, b+6n>i 72 24 {<32>}, <a+3n, b+2n>i 12

4 {<60>}, <a+6n, b>i 72 24 {<34>}, <a+3n, b+4n>i 12

4 {<66>}, <a+6n, b+6n>i 72 24 {<41>}, <a+4n, b+n>i 12

24 {<43>}, <a+4n, b+3n>i 12

6 {<04>}, <a, b+4n>i 48 24 {<61>}, <a+6n, b+n>i 12

6 {<40>}, <a+4n, b>i 48 24 {<81>}, <a+8n, b+n>i 12

6 {<44>}, <a+4n, b+4n>i 48 24 {<91>}, <a+9n, b+n>i 12

6 {<48>}, <a+4n, b+8n>i 48 24 {<t1>}, <a+tn, b+n>i 12

24 {<02>, <60>} + <a,b>i 12

8 {<03>}, <a, b+3n>i 36 24 {<06>, <20>} + <a,b>i 12

8 {<30>}, <a+3n, b>i 36 24 {<06>, <22>} + <a,b>i 12

8 {<33>}, <a+3n, b+3n>i 36 24 {<06>, <24>} + <a,b>i 12

<00> <15>i

<00> <00> <15>i

<15>i <15>i <00>

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8 {<36>}, <a+3n, b+6n>i 36

8 {<39>}, <a+3n, b+9n>i 36 32 {<03>, <30>} + <a,b>i 9

8 {<63>}, <a+6n, b+3n>i 36

8 {<06>, <60>} + <a,b>i 36 36 {<02>, <40>} + <a,b>i 8

36 {<04>, <20>} + <a,b>i 8

12 {<02>}, <a, b+2n>i 24 36 {<04>, <22>} + <a,b>i 8

12 {<20>}, <a+2n, b>i 24

12 {<22>}, <a+2n, b+2n>i 24 48 {<01>, <60>} + <a,b>i 6

12 {<24>}, <a+2n, b+4n>i 24 48 {<02>, <30>} + <a,b>i 6

12 {<26>}, <a+2n, b+6n>i 24 48 {<03>, <20>} + <a,b>i 6

12 {<28>}, <a+2n, b+8n>i 24 48 {<06>, <10>} + <a,b>i 6

12 {<2t>}, <a+2n, b+tn>i 24 48 {<06>, <11>} + <a,b>i 6

12 {<42>}, <a+4n, b+2n>i 24 48 {<06>, <12>} + <a,b>i 6

12 {<46>}, <a+4n, b+6n>i 24 48 {<06>, <13>} + <a,b>i 6

12 {<62>}, <a+6n, b+2n>i 24 48 {<06>, <14>} + <a,b>i 6

12 {<64>}, <a+6n, b+4n>i 24 48 {<06>, <15>} + <a,b>i 6

12 {<82>}, <a+8n, b+2n>i 24 48 {<21>, <60>} + <a,b>i 6

48 {<31>, <60>} + <a,b>i 6

16 {<03>, <60>} + <a,b>i 18 48 {<41>, <60>} + <a,b>i 6

16 {<06>, <30>} + <a,b>i 18

16 {<06>, <33>} + <a,b>i 18 72 {<01>, <40>} + <a,b>i 4

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72 {<02>, <20>} + <a,b>i 4

18 {<04>, <40>} + <a,b>i 16 72 {<04>, <10>} + <a,b>i 4

72 {<04>, <11>} + <a,b>i 4

24 {<01>}, <a, b+n>i 12 72 {<04>, <13>} + <a,b>i 4

24 {<10>}, <a+n, b>i 12 72 {<04>, <21>} + <a,b>i 4

24 {<11>}, <a+n, b+n>i 12 72 {<12>, <20>} + <a,b>i 4

24 {<12>}, <a+n, b+2n>i 12

24 {<13>}, <a+n, b+3n>i 12 96 {<01>, <30>} + <a,b>i 3

24 {<14>}, <a+n, b+4n>i 12 96 {<03>, <10>} + <a,b>i 3

24 {<15>}, <a+n, b+5n>i 12 96 {<03>, <11>} + <a,b>i 3

24 {<16>}, <a+n, b+6n>i 12 96 {<03>, <12>} + <a,b>i 3

24 {<17>}, <a+n, b+7n>i 12

24 {<18>}, <a+n, b+8n>i 12 144 {<01>, <20>} + <a,b>i 2

24 {<19>}, <a+n, b+9n>i 12 144 {<02>, <10>} + <a,b>i 2

24 {<1t>}, <a+n, b+tn>i 12 144 {<02>, <11>} + <a,b>i 2

24 {<1e>}, <a+n, b+en>i 12

24 {<21>}, <a+2n, b+n>i 12 288 {<01>, <10>} + <a,b>i 1

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Table 2-7: group table for {<04>, <40>} + <61>i

<00> <04> <08> <48> <40> <44> <84> <88> <80>

<00> <00> <04> <08> <48> <40> <44> <84> <88> <80>

<04> <04> <08> <00> <40> <44> <48> <88> <80> <84>

<08> <08> <00> <04> <44> <48> <40> <80> <84> <88>

<48> <48> <40> <44> <84> <88> <80> <00> <04> <08>

<40> <40> <44> <48> <88> <80> <84> <04> <08> <00>

<44> <44> <48> <40> <80> <84> <88> <08> <00> <04>

<84> <84> <88> <80> <00> <04> <08> <48> <40> <44>

<88> <88> <80> <84> <04> <08> <00> <40> <44> <48>

<80> <80> <84> <88> <08> <00> <04> <44> <48> <40>

<61>i <61>i <65>i <69>i <t9>i <t1>i <t5>i <25>i <29>i <21>i

<65>i <65>i <69>i <61>i <t1>i <t5>i <t9>i <29>i <21>i <25>i

<69>i <69>i <61>i <65>i <t5>i <t9>i <t1>i <21>i <25>i <29>i

<t9>i <t9>i <t1>i <t5>i <25>i <29>i <21>i <61>i <65>i <69>i

<t1>i <t1>i <t5>i <t9>i <29>i <21>i <25>i <65>i <69>i <61>i

<t5>i <t5>i <t9>i <t1>i <21>i <25>i <29>i <69>i <61>i <65>i

<25>i <25>i <29>i <21>i <61>i <65>i <69>i <t9>i <t1>i <t5>i

<29>i <29>i <21>i <25>i <65>i <69>i <61>i <t1>i <t5>i <t9>i

<21>i <21>i <25>i <29>i <69>i <61>i <65>i <t5>i <t9>i <t1>i

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Table 2-7 continued

<61>i <65>i <69>i <t9>i <t1>i <t5>i <25>i <29>i <21>i

<61>i <65>i <69>i <t9>i <t1>i <t5>i <25>i <29>i <21>i

<69>i <61>i <65>i <t5>i <t9>i <t1>i <21>i <25>i <29>i

<65>i <69>i <61>i <t1>i <t5>i <t9>i <29>i <21>i <25>i

<25>i <29>i <21>i <61>i <65>i <69>i <t9>i <t1>i <t5>i

<21>i <25>i <29>i <69>i <61>i <65>i <t5>i <t9>i <t1>i

<29>i <21>i <25>i <65>i <69>i <61>i <t1>i <t5>i <t9>i

<t9>i <t1>i <t5>i <25>i <29>i <21>i <61>i <65>i <69>i

<t5>i <t9>i <t1>i <21>i <25>i <29>i <69>i <61>i <65>i

<t1>i <t5>i <t9>i <29>i <40> <25>i <65>i <69>i <61>i

<00> <04> <08> <48> <48> <44> <84> <88> <80>

<08> <00> <04> <44> <48> <40> <80> <84> <88>

<04> <08> <00> <40> <44> <48> <88> <80> <84>

<84> <88> <80> <00> <04> <08> <48> <40> <44>

<80> <84> <88> <08> <00> <04> <44> <48> <40>

<88> <80> <84> <04> <08> <00> <40> <44> <48>

<48> <40> <44> <84> <88> <80> <00> <04> <08>

<44> <48> <40> <80> <84> <88> <08> <00> <04>

<40> <44> <48> <88> <80> <84> <04> <08> <00>

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Figure 2-19: cyclic groups and rotational symmetry, C3: T0, T4, T8

Figure 2-20: dihedral groups, D3, T0, T4, T8, In, In+4, In+8

A

B C

T8 T4

A

B C

IB IC

IA

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Figure 2-21: T/I group representation

0 e 1

t 2

9 3

8 4

7 5 6

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Figure 2-22: the dodecagonal torus

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Figure 2-23: interlocking dodecagons

a). pcset {000}

b). + vlset <216> c). + vlset <358>i

0

e 1

t 2

9 3

8 4

7 5

6

0 e 1

t 2

9 3

8 4

7 5 6

0 e 1

t 2

9 3

8 4

7 5 6

2

1 3

0 4

e 5

t 6

9 7

8

1 0 2

e 3

t 4

9 5

8 6 7

6 5 7

4 8

3 9

2 t

1 e 0

T2

T1

T6

3

4 2

5 1

6 0

7 e

8 t

9

5 6 4

7 3

8 2

9 1

t 0 e

8 9 7

t 6

e 5

0 4

1 3 2

I8

I5

I3

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Figure 2-24: vlset <t27> on interlocking dodecagons;

{238} - <t27> - {053}

2

1 3

0 4

e 5

t 6

9 7

8

3 2 4

1 5

0 6

e 7

t 8 9

8 7 9

6 t

5 e

4 0

3 1 2

0

e 1

t 2

9 3

8 4

7 5

6

5 4 6

3 7

2 8

1 9

0 t e

3 2 4

1 5

0 6

e 7

t 8 9

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Figure 3-1: Straus’ trichordal set-class space

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Figure 3-2a: the dyadic orbifold

Figure 3-2b: dyadic permutational equivalence

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Figure 3-2c: dyadic Moebius strip

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Figure 3-3: torus knot of singleton dyads

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Figure 3-4a: Moebius strip dyadic orbifold

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Figure 3-4b: dyadic set-class line

[00] – [01] – [02] – [03] – [04] – [05] – [06]

Figure 3-4c: dyadic tn-class circle

Figure 3-5a: <05> space

[00]

[09] [03]

[06]

[0e] [01]

[07] [05]

[0t] [02]

[08] [04]

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Figure 3-5b: <05> Moebius strip

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Figure 3-5c: adjacent voice leadings along the <05> Moebius strip

0 0 5 5 0 e e e

9 4 4 e e 7 2 9

<07> <05> <07> <07> <07> <07> <07>

Figure 3-5d: adjacent and compound moves along the <05> Moebius strip

0 5 4 e 6 6 1 8

9 9 e 9 5 0 7 2

<05> <67> <05> <87> <07> <77> <77>

(7 moves) (5 moves) (2 moves) (2 moves)

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Figure 3-6a: <05> tn-class circle

Figure 3-6b: <05> set-class line

[00] – [05] – [02] – [03] – [04] – [01] – [06]

[00]

[09] [03]

[06]

[07] [05]

[0e] [01]

[02] [0t]

[04] [08]

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Figure 3-7a: <2e>, {00} toroidal voice-leading space

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Figure 3-7b: <2e>, {00} toroidal voice-leading space

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Table 3-1: dyadic vlset spaces

Vlset Sum Number of

Pcsets

Tn-class

cycles

Vlset Sum Number of

Pcsets

Tn-class

cycles

<01> 1 144 1 <0e> e 144 1

<2e> 1 48 3 <1t> e 48 3

<3t> 1 144 1 <29> e 144 1

<49> 1 144 1 <38> e 144 1

<58> 1 48 3 <47> e 48 3

<67> 1 144 1 <56> e 144 1

<05> 5 144 1 <07> 7 144 1

<14> 5 48 3 <8e> 7 48 3

<23> 5 144 1 <9t> 7 144 1

<6e> 5 144 1 <16> 7 144 1

<7t> 5 48 3 <25> 7 48 3

<89> 5 144 1 <34> 7 144 1

<02> 2 36 2 <0t> t 36 2

<11> 2 12 12 <ee> t 12 12

<3e> 2 36 4 <19> t 36 4

<4t> 2 12 6 <28> t 12 6

<59> 2 36 4 <37> t 36 4

<68> 2 36 2 <46> t 36 2

<77> 2 12 12 <55> t 12 12

<03> 3 16 3 <09> 9 16 3

<12> 3 48 1 <te> 9 48 1

<4e> 3 48 1 <18> 9 48 1

<5t> 3 48 1 <27> 9 48 1

<69> 3 16 3 <36> 9 16 3

<78> 3 48 1 <45> 9 48 1

<04> 4 9 4 <08> 8 9 4

<13> 4 36 2 <9e> 8 36 2

<22> 4 6 12 <tt> 8 6 12

<5e> 4 12 6 <17> 8 12 6

<6t> 4 18 4 <26> 8 18 4

<79> 4 36 2 <35> 8 36 2

<88> 4 3 12 <44> 8 3 12

<00> 0 1 12 <06> 6 4 6

<1e> 0 12 2 <15> 6 12 4

<2t> 0 6 4 <24> 6 12 2

<39> 0 4 6 <33> 6 4 12

<48> 0 3 4 <7e> 6 12 4

<57> 0 12 2 <8t> 6 12 2

<66> 0 2 12 <99> 6 4 12

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Figure 3-8: <02>, {00} toroidal voice-leading space

Figure 3-9: <02>, {11} toroidal voice-leading space

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Figure 3-10: <02>, {01} toroidal voice-leading space

Figure 3-11: <02>, {10} toroidal voice-leading space

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Figure 3-12: <02>, {11} Moebius strip

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Figure 3-13: vlset <02> spaces

a). transposition-class cycles

b). set-class lines

[00] – [02] – [04] – [06] [01] – [03] – [05]

Figure 3-14: <68>, {00} toroidal voice-leading space

[00]

[06]

[0t] [02]

[08] [04]

[01]

[07]

[0e] [03]

[09] [05]

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Figure 3-15: <67>, {00} toroidal voice-leading space

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Figure 3-16: <67> Moebius strip

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Figure 3-17: <12>, {47} toroidal voice-leading space

Figure 3-18a: <42>, {11} toroidal voice-leading space

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Figure 3-18b: <42>, {11} toroidal voice-leading space

11 77 17 71

9e 53

59 95

35 e9

e3 3e

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Figure 3-19: <12>, {47} Moebius strip

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Figure 3-20: <42>, {11} permutation space {e3} {9e} {77} {17} {59} {35} {11} Figure 3-21: <06>, {11} permutation space {11}—{17}—{77} Figure 3-22: <06>, {01} voice-leading space {61} {67} {01} {07} Figure 3-23: <57>, {02} permutation space {77}—{02}—{59}—{t4}—{3e}—{86}—{11} Figure 3-24: <26>, {11} toroidal voice-leading space

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Figure 3-25: <001>, {000} hypercube

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Figure 3-26: the trichordal orbifold

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Table 3-2: trichordal voice-leading spaces

Vlset Inversional Pair Sum Number of

Pcsets

Tn-class

spaces

<001>, <445>, <889> <00e>, <443>, <887> 1/e 1728 1

<02e>, <346>, <78t> <01t>, <245>, <689> 1/e 1728 1

<03t>, <247>, <68e> <029>, <146>, <58t> 1/e 1728 1

<049>, <148>, <058> <038>, <047>, <48e> 1/e 1728 1

<067>, <4te>, <238> <056>, <49t>, <128> 1/e 1728 1

<11e>, <355>, <799> <1ee>, <335>, <779> 1/e 432 2

<12t>, <256>, <69t> <2te>, <236>, <67t> 1/e 1728 1

<139>, <157>, <59e> <39e>, <137>, <57e> 1/e 432 2

<166>, <5tt>, <229> <66e>, <3tt>, <227> 1/e 1728 1

<337>, <77e>, <3ee> <599>, <119>, <155> 1/e 108 5

<005>, <449>, <188> <007>, <44e>, <388> 5/7 1728 1

<014>, <458>, <089> <08e>, <034>, <478> 5/7 1728 1

<023>, <467>, <8te> <09t>, <124>, <568> 5/7 1728 1

<06e>, <34t>, <278> <016>, <45t>, <289> 5/7 1728 1

<07t>, <24e>, <368> <025>, <469>, <18t> 5/7 1728 1

<113>, <557>, <99e> <9ee>, <133>, <577> 5/7 432 2

<122>, <566>, <9tt> <tte>, <223>, <667> 5/7 1728 1

<15e>, <359>, <179> <17e>, <35e>, <379> 5/7 432 2

<16t>, <25t>, <269> <26e>, <36t>, <27t> 5/7 1728 1

<33e>, <377>, <7ee> <199>, <115>, <559> 5/7 108 5

<002>, <446>, <88t> <00t>, <244>, <688> 2/t 216 2

<011>, <455>, <899> <0ee>, <334>, <778> 2/t 864 1

<03e>, <347>, <78e> <019>, <145>, <589> 2/t 864 1

<04t>, <248>, <068> <028>, <046>, <48t> 2/t 216 2

<059>, <149>, <158> <037>, <47e>, <38e> 2/t 864 1

<077>, <4ee>, <338> <055>, <499>, <118> 2/t 864 1

<12e>, <356>, <79t> <1te>, <235>, <679> 2/t 864 1

<13t>, <257>, <69e> <29e>, <136>, <57t> 2/t 864 1

<167>, <5te>, <239> <56e>, <39t>, <127> 2/t 864 1

<22t>, <266>, <6tt> <2tt>, <226>, <66t> 2/t 54 5

<004>, <448>, <088> <008>, <044>, <488> 4/8 27 5

<013>, <457>, <89e> <09e>, <134>, <578> 4/8 432 1

<022>, <466>, <8tt> <0tt>, <224>, <668> 4/8 108 2

<05e>, <349>, <178> <017>, <45e>, <389> 4/8 432 1

<06t>, <24t>, <268> <026>, <46t>, <28t> 4/8 108 2

<079>, <14e>, <358> <035>, <479>, <18e> 4/8 432 1

<112>, <556>, <99t> <tee>, <233>, <677> 4/8 432 1

<15t>, <259>, <169> <27e>, <36e>, <37t> 4/8 432 1

<23e>, <367>, <7te> <19t>, <125>, <569> 4/8 432 1

<277>, <6ee>, <33t> <55t>, <299>, <116> 4/8 432 1

<000>, <444>, <888> 0 1, 3

<01e>, <345>, <789> 0 144 3

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<02t>, <246>, <68t> 0 36 6

<039>, <147>, <58e> 0 16, 48 4

<048> 0 9

<057>, <49e>, <138> 0 144 3

<066>, <4tt>, <228> 0 8, 12 10

<11t>, <255>, <699> <2ee>, <336>, <77t> 0 16, 48 4

<129>, <156>, <59t> <237>, <67e>, <3te> 0 144 3

<003>, <447>, <88e> <009>, <144>, <588> 3/9 64, 192 4

<012>, <456>, <89t> <0te>, <234>, <678> 3/9 576 3

<04e>, <348>, <078> <018>, <045>, <489> 3/9 576 3

<05t>, <249>, <168> <027>, <46e>, <38t> 3/9 576 3

<069>, <14t>, <258> <036>, <47t>, <28e> 3/9 64, 192 4

<111>, <555>, <999> <eee>, <333>, <777> 3/9 4, 12

<13e>, <357>, <79e> <19e>, <135>, <579> 3/9 144 6

<159> <37e> 3/9 36

<177>, <5ee>, <339> <55e>, <399>, <117> 3/9 16, 48 10

<22e>, <366>, <7tt> <1tt>, <225>, <669> 3/9 64, 192 4

<23t>, <267>, <6te> <29t>, <126>, <56t> 3/9 576 3

<006>, <44t>, <288> 6 8, 24 10

<015>, <459>, <189> <07e>, <34e>, <378> 6 288 3

<024>, <468>, <08t> 6 72 6

<033>, <477>, <8ee> <099>, <114>, <558> 6 32, 96 4

<123>, <567>, <9te> 6 288 3

<16e>, <35t>, <279> 6 288 3

<17t>, <25e>, <369> 6 32, 96 4

<222>, <666>, <ttt> 6 2, 6

<26t> 6 18

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Figure 3-27a: vlclass [047] ordered tn-class space

Figure 3-27b: Motion among adjacent ordered tn-classes in [047] ordered tn-class

space

[000] [047] [012] [056] [043] [044] [007] [042]

7 4 1 1 8 4 5 e

7 1 0 0 9 0 0 9

7 9 e 7 5 4 5 7

<269> <730> <7e2> <158> <3e8> <158> <269>

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Figure 3-28: vlclass [047] ordered set-class space

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Figure 3-29: examples of complete ordered set-class spaces

a) vlclass [056]

b) vlclass [025]

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c) vlclass [014]

d) vlclass [013]

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e) vlclass [001]

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Figure 3-30: vlclass [002] ordered set-class spaces

a)

[000]

[002] [002]

[004] [024] [004]

[006] [026] [046] [006]

[048]

b)

[023] [025] [027]

[045] [047] [037]

[016] [036] [056]

([045]) [014] [034] [015] ([037])

([023]) [012] [013] [035] ([027])

[001] [003] [005]

([023] [025] [027])

[001]

[003]

[005]

[012] [014]

[027] [037]

[056]

[015]

[034]

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Figure 4-1a: <442>, {000} voice-leading space

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Figure 4-1b: <442>, {001} voice-leading space

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Figure 4-1c: vlclass [002] whole-tone ordered set-class space

[000]

[002] ([002])

[004] [024] ([004])

[006] [026] [046] ([006])

[048]

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Figure 4-1d: vlclass [002] mixed whole-tone ordered set-class space

[023] [025] [027]

[045] [047] [037]

[016] [036] [056]

([045]) [014] [034] [015] ([037])

([023]) [012] [013] [035] ([027])

[001] [003] [005]

([023] [025] [027])

[001]

[003]

[005]

[012] [014]

[027] [037]

[056]

[015]

[034]

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Figure 4-1e: ordered set-class interpretations from Figure 2-8

a <7 e 8> <t t 8> b <5 9 4> a <7 8 e> <t t 0> c <5 6 e>

[041] – [034] [015] [014] [016]

c <5 6 e> <4 4 2> d <9 t 1> b <5 9 4> <4 4 6> d <9 1 t>

[016] [014] [015] [041] – [034]

g <4 6 8> <1 1 3> h <5 7 e> e <0 2 6> <4 4 2> g <4 6 8>

[024] [026] [026] [024]

i <e 0 3> <9 9 e> j <8 9 2> f <8 9 t> <3 3 5> i <e 0 3>

[014] [016] [012] [014]

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Figures 4-2a and b: [026] in Webern’s Op. 5 n. 2 and a network of transformations linking them

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Figure 4-3: melodic statements of vlclass [043] in Webern’s Op. 5 n. 2

Vla

Vln II

Vln II

Vln I

Vln II

Vln I Vln II

191

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Figure 4-4: Network of melodic vlsets

I5 I6 I9

<910> <845> <t21> <e78>

Te

CI0 I6 CT6 I1

<t65> <265>i <401>i <t67> <376>

T1

I7 T1

<487> <3e0> <401>

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Figure 4-5: vlclass [043] in the accompaniment to Webern’s Op. 5 n. 2

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Figure 4-5 continued

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Figure 4-6: vlclass [043] transformation graph

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Figure 4-7a: Webern Op. 5 n. 3, mm. 1-4

Figure 4-7b: vlset <t16> mappings

5 0 0 7

2 t 9 5

4 6 e 1

<t16> <t16>

T7

{90e} {715}

A

{254} {06t}

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Figure 4-8: variations of A

8 3 e 4

5 1 8 0

7 9 t 6

<t16> <t16>

e 1 2

t t 8

8 0 0

<470>

3 t 4 e

0 8 1 9

2 4 3 5

<t16> <t16>

8 9 3 1 5 7

5 3 2 e 9 6

7 7 0 7 3 4

<470> <149> <74e>i

T7 T7

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Figure 4-9: B, mm. 9-10

B

2 t

1 5

t 9

<8e4>

0 6

e 4

3 3

<470>

{034} {234}

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Figure 4-10a: [038] voice-leading space, equivalent to [047] under rotation

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Figure 4-10b: melody B’s complete voice-leading {2t1} - {0et} - {t59}

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Figure 4-11a: reduction of opening [034]s in Violins and Viola, mm. 1-3

Figure 4-11b: reduction of Violin II, Viola, and Cello, mm. 15-17

{014} {901}

[014] [034]

e 9

9 8

8 4

<850>i

{7te}

[034]

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Figure 4-12: Berg Op. 5 n. 1, opening clarinet gesture, in C

{789034} {269}

[012569] [037]

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Figure 4-13a: Berg’s Op. 5 n. 1, Clarinet melody in C

203

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204

Figure 4-13b: vlclass [002] ordered set-class spaces

[000]

[002] [002]

[004] [024] [004]

[006] [026] [046] [006]

[048]

[023] [025] [027]

[045] [047] [037]

[016] [036] [056]

([045]) [014] [034] [015] ([037])

([023]) [012] [013] [035] ([027])

[001] [003] [005]

([023] [025] [027])

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205

Figure 4-14: vlclass [022] in m. 2 LH

{837} {67e} e 2

I2 7 t

{926} {7t2} 6 7

I4 <133>

<1ee> I2

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206

Figure 4-15: vlclass [022] in m. 2 RH

Figure 4-16: opening clarinet gesture in the accompaniment, m. 9

{e34} {590}

[012569]

Clar.

{047} {837} {4e3}

[037] {904} {590}

{926} {740}

<t88>

t 4

9 3

8 3

<577>

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207

Figure 4-17: accompaniment, mm. 3-5

0 2 3 2 3

t 1 0 8 9

1 9 8 3 2

<e11> <1ee> <200> <e11>

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208

Figure 4-18a: vlclass [002] in mm. 5-6

Figure 4-18b: vlclass [002] in m. 7

t 0 t 3 4

9 7 3 7 6

5 4 9 8 7

<977> <533> <0tt> <1ee>

e 6 0 6 9

4 2 t 5 8

0 e 7 4 8

<022> <688> <244>

{348} <577> {1te}

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Figure 4-19: Berg’s Op. 5 n. 2

209

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Figure 4-20: Berg’s Op. 5 n. 2 melody

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211

Figure 5-1: pentachordal vlsets in Berg’s Op. 5 n. 2

0 1 0 0 1 0

7 8 7 7 8 7

3 0 3 3 0 3

6 6 6 9 6 9

2 2 2 5 2 5

<0ee01> <0110e> <33435> <98798> <34534>

<03505>i <03505>i <37t37>i <36938>i<36938>i

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212

Figure 5-1 continued

0 3 t 1 7 t 4 7 t 4 7 4 6

7 9 4 8 3 5 e 3 5 e 3 e 2

3 1 0 2 0 t 8 0 t 8 0 8 t

9 6 7 8 9 6 5 9 6 5 6 5 t

5 8 9 t e 0 1 e 0 1 0 1 t

<etee9> <41132> <t9et9> <34312> <eet11> <45556> <65235>

<34323> <e11e1> <eet11> <t9et9> <87776>

<48159>i <037e2>i <039t3>i <e27e8>i <9e691>i <45e58>i <23617>i

<5t369>i <35915>i <9e691>i <039t3>i <45e58>i

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213

8 4 9 t 2 t 5

7 0 6 9 1 9 4

3 9 2 8 1 7 3

<658> <565> <631> <544> <688> <877>

Figure 5-2 vlsets among normal orders in Berg’s Op. 5 n. 1 melody

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Figure 5-3a vlsets on the trichordal orbifold

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Figure 5-3b: vlsets on the trichordal orbifold

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Figure 5-3c: vlsets on the [013] ordered set-class space

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217

Figure 5-4 hyper-vlset graph

<<e0t>> <<198>> <<t23>> <<10t>>i <<134>>i

<658> <565> <631> <544> <688> <877>

(<<301>>+<<t97>>) (<<235>>+<<8et>>)

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218

Figure 5-5: Alban Berg’s Lyric Suite, VI, mm. 1-6

Viola: Row 2PF 5 4 1 2 0 e 6 t 9 8 7 3

Cello: Row 1PF 5 6 t 4 1 9 2 8 7 3 0 e

T-row <0 t 3 t e 2 4 2 2 5 7 4>

Vln I: Row 2IF 5 6 9 8 t e 4 0 1 2 3 7 Vln II: Row 1IF 5 4 0 6 9 1 8 2 3 7 t e

Cello: Row 1PF 5 6 t 4 1 9 2 8 7 3 0 e Vla: Row 2PF 5 4 1 2 0 e 6 t 9 8 7 3

I-row <t 0 7 0 e 8 6 8 8 5 3 6>i I-row <t 8 1 8 9 0 2 0 0 3 5 2>i

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Figure 5-6: Lyric Suite, mm. 39-46

Vln II: Row 1PC# 1 2 6 0 9 5 t 4 3 e 8 7

Viola: Row 2IEb 3 4 7 6 8 9 2 t e 0 1 5

I-row <4 6 1 6 5 2 0 2 2 e 9 0>i

Vln I: Row 1IA 9 8 4 t 1 5 0 6 7 e 2 3

Viola: Row 2IEb 3 4 7 6 8 9 2 t e 0 1 5

T-row <6 4 9 4 5 8 t 8 8 e 1 t>

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Figure 5-7: 4-3 suspension as diatonic vlset

4 4

3 2

0 0

0 0

Diatonic vlset <0060>

Figure 5-8: transposition of 4-3 diatonic vlset

4 5

3 3

0 1

0 1

T1(<0060>) = <1101>

Page 233: A Theory of Voice-leading Sets for Post-tonal Music

221

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