BLÄTTLER, Damian. 2013. a Voicing-centered Approach to Additive Harmony in Music in France, 1889-1940

A bstract

A Voicing-Centered Approach to Additive Harmony in Music in France, 1889-1940 Damian Blattler 2013

Music written in France during La Belle Epoque and the interwar period is remarkable in part for its development of additive harmony, i.e. its incorporation of novel chords into conventional tonal contexts. The conventional explanatory apparatus for this harmonic phenomenon, the extended-triad model of ninth, eleventh, and thirteenth chords, poorly describes those features that allow certain novel chords to participate in tonal progressions; this study develops a model of additive harmony that better accounts for the structure and behavior of the additive chords found in this repertoire. Chapter 1 examines the history of theories of additive harmony, tracing three distinct strategies for explaining simultaneities that are not triads or seventh chords: adapting basic chord types, identifying non-chord elements, and formulating new basic chord types.

Chapter 2

presents a model of additive-harmonic chord structure in which voicing plays a foundational role; chords are conceived of as pitch-space voicings constrained by the pragmatic considerations of tonal plausibility and chordability. Chapter 3 investigates the role voicing plays in the special additive-harmonic case of polychordal polytonality. The dissertation closes by discussing the connections between the small details at the musical surface that are the focus of this study and several larger music-theoretical issues.

A Voicing-Centered Approach to Additive Harmony Music in France, 1889-1940

A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy

by Damian Joseph BlSttler

Dissertation Director: Daniel Harrison

December 2013

UMI Number: 3578315

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Contents Examples, Tables, and Figures

i

Acknowledgements

viii

Introduction

1

Chapter 1 - Strategies in Additive Harmonic Theory 1.1 Additive Harmonic Thinking in Rameau’s Traite 1.2 The Modified-Basic-Types Strategy 1.3 The Non-Chord-Elements Strategy 1.4 The New-Basic-Types Strategy

13 17 26 51 68

Chapter 2 - A Vertical-Domain Model for Additive Harmony in Music in France, 1889-1940 2.1 Overview of the Model 2.2 Anchor Structures and Tonal Plausibility 2.3 Chordability 2.4 Additive Chords and Non-Chord Verticalities

75

Chapter 3 - The Special Case of Polychordal Polytonality 3.1 Theories of Polychordal Polytonality 3.2 Implications of Chordability Restrictions and Anchor Structures for Polychordal Polytonality

132 134 149

Conclusion

165

Appendix - C++ Program for Parsing All Verticalities in a 28Semitone Span

169

Bibliography

171

75 79 99 123

Illustrations Figures 1.1

Reduction of the final cadence of Ravel’s L ’enfant et les sortileges, R154-end.

1

1.2

Reproduction of example 27.1 from Roig-Francoli’s Harmony in Context.

5

1.3

The ill fit of extended-triad labels when applied to certain chords in the Parisian modernist repertoire.

7

1.4

Competing functional elements in the penultimate chord of L 'enfant et les sortileges.

8

1.5

Comparison of the impact of inversion upon an additive chord and upon a triad.

9

1.6

Over-applicability of the extended-triad model in the absence of guidelines for voicing.

11

1.1

Spectrum between competing ideals in additive harmonic theory.

15

1.2

Realization of the chord types described in Rameau’s Traite other than the perfect chord and the seventh chord.

18

1.3

Rameau’s example II.6; an irregular cadence.

20

1.4

Transcription of Rameau’s example II. 11 - explanation of a suspended fourth as a heteroclite eleventh chord.

21

1.5

Transcription and annotation of Rameau’s examples II.41 and 11.42; use of supposition to explain voice-leading novelties.

23

1.6

Rameau’s inability to explain ninths resolving over fallingfifth bass motion.

24

1.7

Rameau’s difficulty explaining suspensions in the bass voice.

25

1.8

Heinichen’s use of invertible ninth chords to explain novel dissonance treatment.

27

1.9

Set of extended chords from the Ist edition of Marpurg’s Handbuch.

28

1.10 Marpurg’s first-inversion thirteenth chord with fifth, seventh, and ninth omitted.

29

1.11

31

James M. Martin’s transcription of Sorge’s Plate VI, examples 4 and 5; presentation of the dominant ninth chords and their relationship to the leading-tone seventh chord.

1.12 Dehn’s ninth, eleventh, and thirteenth chords realized in C major.

33

1.13

34

Dehn’s prohibited voicings of the ninth chord.

1.14 The harmonic-series justification for the extended-triad model of additive harmony.

37

1.15 Lobe’s inversions of altered ii9 in A minor.

38

1.16 Transcription of Ziehn’s set of chromatic seventh chords.

39

1.17 Harmonic treatment of the whole-tone scale in Schoenberg’s Harmonielehre.

41

1.18 Schoenberg’s four- and five-note quartal chords, and their resolutions as substitute dominants.

42

1.19 Quartal chords in Schoenberg’s Kammersymphonie.

42

1.20 Ottman’s figure 10.18.

43

1.21

44

Piston’s example 22-12; various inversions of a G major-ninth chord.

1.22 Ulehla’s voicing guidelines for second-inversion major-ninth chords.

45

1.23 Ulehla’s analysis of the implications of chromatic alteration to the thirteenth above a dominant seventh.

46

1.24 Prout’s three fundamental chords.

48

1.25 Sixth-inversion thirteenth chord in Day.

48

1.26 Free treatment of the ninth in Prout’s Harmony.

49

1.27 Day’s explanation of a “ii®7” chord as a third-inversion dominant eleventh.

51

ii

1.28

Kimberger’s Figure XXXVI - non-essential dissonances.

52

1.29 Kimberger’s explanation for seventh chords that resolve upwards by step.

53

1.30 Catel’s chord formed by the suspension of the octave, the sixth, and the fourth.

55

1.31

55

Catel’s chord formed by raising the chordal fifth and lowering the chordal third.

1.32 Fetis’ concept of substitution.

56

1.33 Fetis’ presentation of Catel’s inversions of the so-called “fundamental” chord of the seventh.

57

1.34 Fetis’ analysis of Beethoven’s “improper” inversion of the leading-tone seventh.

58

1.35 Fetis’ derivation of the ii7 chord.

58

1.36 Ravel’s analysis of an unresolved appoggiatura in Les Grands Vents venus d 'Outre-mer.

61

1.37 Ravel’s analysis of an unresolved appoggiatura in “Oiseaux tristes” from Miroirs.

61

1.38 Ravel’s analysis of an unresolved appoggiatura in Vaises nobles et sentimentales, vii.

62

1.39 Lenormand’s derivation of an unresolved appoggiatura chord.

64

1.40 Casella’s set of Ravel ’s “genuine appoggiature chords.”

65

1.41

Ravel Sonatine/i, mm. 85-87; a final tonic chord with added ninth.

66

1.42 Kitson’s derivation of an unresolved appoggiatura chord.

67

1.43 Hindemith’s example 62 - non-triadic chords.

68

1.44 Preferring a new-basic-type label to tenuous modified-basic-types or non-chord-elements labels.

69

1.45 Persichetti’s example 4-4; the roots of quartal chords determined by melodic motion.

70

iii

1.46

Persichetti’s conditions for incorporating quartal chords with triads.

71

1.47

Hindemith’s chord groupings.

73

1.48

A Hindemith harmonic-fluctuation graph.

74

2.1

The pitch classes of the L 'enfant chord arranged as a cluster, as an integrated chord, and as disassociated strata.

78

2.2

The relationship between tonally plausible additive chord and common-practice chord modeled as a two-step process.

80

2.3

The set of common-practice chord types.

82

2.4

First-order anchor structures and the common-practice chord types they can stand in for.

84

2.5

Ravel, “Laideronette, imperatrice des pagodes” from Ma mere I ’Oye, mm. 16-24: |t|-anchored substitute supertonic.

86

2.6

Chords with distinct pitch-class content but a shared tonal plausibility.

87

2.7

Different tonal plausibilities arising from distinct voicings of a single pitch-class set.

89

2.8

Hull’s example 194; an example of the benefit of abandoning a stacked-thirds conception of additive chords.

89

2.9

Conflict between the chord identities projected by the bass motion and those projected by the verticalities themselves

91

2.10

Use of seventh chords in parallel motion in the Parisian modernist repertoire.

92

2.11

Ravel Jeux d ’eau/iii; |e|-anchored final tonic

2.12

Chromatically altered chordal third in Satie’s Gymnopedie No. 1, mm. 36-39.

93

2.13

“Omitted” chordal third in Debussy’s “La fille aux cheveux de lin,” mm. 18-19.

93

2.14

Second-order anchor structures.

95

iv

92

2.15

Examples of second-order anchor structures.

96

2.16

Interaction between anchor structures of different orders.

97

2.17

Second-order anchor structure cadence in Stravinsky, L 'histoire du soldat, p. 6.

98

2.18

Interaction of anchor structures of the same order.

98

2.19

First-order anchor structures not disturbed by the presence of |7 |.

99

2.20

Unlikely anchor-structure voicings.

100

2.21

Ravel, Miroirs, “Alborada del gracioso,” mm. 130-134.

101

2.22

Ravel, “Pavane de la Belle au bois dormant” from Ma mere I 'Oye; first-order anchor-structure chord supporting ro-interval 1.

102

2.23

Ravel Vaises nobles et sentimentales i, mm. 1-2.

104

2.24

Jazz voicings from Levine’s The Jazz Piano Book.

105

2.25

Ravel, Pavane pour une infante defunte; use of ro-interval 1 to destabilze a previously stable cadence.

107

2.26

Marked use of ro-interval 1 in Chabrier’s “Les Cigales.”

109

2.27

Scarcity of adornment options for second-order anchor structures.

113

2.28

Ravel, String Quartet, end of 1st movement.

114

2.29

Whole-step adjacencies in final tonics in the Parisian modernist repertoire.

118

2.30

Debussy, “Les collines d’Anacapri,” mm. 1-8; use of verticalized consecutive whole-step adjacencies as a marked event.

119

2.31

Debussy, “Feuilles mortes,” mm. 6-10; adjacent-whole-tone sonority treated as a passive musical object.

120

2.32

Reductive power of the model’s three voicing constraints.

123

2.33

Types of verticalities in the Parisian modernist repertoire.

124

v

2.34

Ravel, Vaises nobles et sentimentales, mm. 56-61; non-chordable verticalities interpreted as the coincidence of a chordal background and non-essential dissonance.

125

2.35

Debussy, “La Puerta del Vino,” mm. 5-12; viable additive chords interpreted as passing tones.

127

2.36

Reduction of Poulenc, “Rag Mazurka” from Les biches, R89; suspension of a major tenth above the bass.

128

2.37

Prokofiev, Sonata for Flute and Piano, mm. 1-4.

129

2.38

Debussy, “Feux d’artifices,” mm. 85-90.

131

3.1

Anchor-structure, stacked-thirds, and polychordal readings of an extended triad.

132

3.2

Milhaud’s table of bitonal combinations.

136

3.3

Febre-Longeray’s monotonal derivation of all bichordal combinations of two major triads.

138

3.4

Koechlin’s context-dependent polychord.

139

3.5

Persichetti’s hierarchy of bichordal combinations.

140

3.6

Examples of Persichetti’s four types of permissible trichords.

142

3.7

Persichetti’s demonstration of “grouping as a requisite of polyharmony.”

143

3.8

Ulehla’s analysis of the assimilation of the upper parts by a dominant seventh chord.

144

3.9

An exception to the assimilation effect of Figure 3.8 - a tonic triad above its dominant seventh.

144

3.10

Thompson and Mor’s Fig. 1 - excerpt from Theodore Dubois’s Circus.

148

3.11

Bichordal superimpositions that run afoul of Chapter 2’s chordability restrictions.

153

3.12

Analysis of bitonality in Milhaud’s “Botafogo” from Saudades do Brasil.

158

vi

3.13

Polytonal effect generated by an (M/M, 6) chord superimposition; mm. 34-46 of “Ipanema” from Milhaud’s Saudades do Brasil.

161

3.14

Analysis of non-bitonal polychordal construction in Ravel’s Sonata for Violin and Cello.

162

3.15

Analysis of Milhaud’s “Corcovado” from Saudades do Brasil, mm. 1-8.

164

3.1

All potential bichordal combinations involving major triads, minor triads, and dominant seventh chords.

151

3.2

The bichordal combinations of Figure 3.11 tabulated by ro-interval distance between chord roots.

157

Tables

vii

Acknowledgments

First and foremost, I would like to thank Daniel Harrison, the advisor to this dissertation, for all of his guidance, patience, and encouragement during the last several years; this work has benefitted enormously from his insight and mentorship. I also extend thanks to my committee members, Peter Kaminsky and Patrick McCreless, who have given generously of their time and knowledge to this project. Thanks go too to all of my graduate student colleagues, who have provided invaluable information, critique, and camaraderie through the various stages of this project. In particular I would like to thank the members of Professor Harrison’s lab group - Chris Brody, Jennifer Darrell, Megan Kaes Long, Elizabeth Medina-Gray, John Muniz, Jay Summach, and Christopher White - for their collegial consideration of my work and for the inspiration and motivation their work provided for me. Finally, I am eternally grateful to my family and to my wife Jackie for their faith in my musical pursuits and for their love and support.

Introduction

The music of composers working in France between 1889-1940 is shot through with harmonic moments like the one in Figure 1.1, the final cadence of Ravel’s 1925 opera L ’enfant et les sortileges. These are moments where the functional sense of a

F igure 1.1. Reduction of the final cad en ce of Ravel, L’enfant et les sortileges, R154-end.

The Child Ma - man!

Chorus

est

doux.

oboes

Orchestra

PP z

IP

strings

chord progression is clear, but the chords used in that progression are unconventional by common-practice standards. In the Ravel example, we hear the two bracketed chords as completing an authentic cadence; the chords follow a clear tonic-to-subdominant-todominant root progression of G-E-C-A-D that begins at R153. The first bracketed chord, however, is neither a triad nor a seventh chord (and instead involves a scalar subset B-CD-E, with the pitches of the chorus evaporating upon the strings’ entrance), while the final G-major triad is accompanied by a dissonant F it.

Jeremy Day-0’Connell has termed this incorporation of novel chords into tonal contexts additive harmony.1 It is an innovative feature of the music of this time and place; while there is a lot of variety and stylistic change in music in France between 1889-1940, there is also a common fascination with colorful harmonies and their potential tonal applications.2 This feature can be seen to stem in part from a latenineteenth-century desire to establish a compositional style distinct from the dense counterpoint of German expressive maximalism, one that involved, among other things, a renewed focus on pleasure, sensation, and the sensuous potential of chords.3 For the purposes of this study, the 1889 Exposition Universelle - an event which capped the rehabilitation of French national pride after the humiliation of the Franco-Prussian War4 serves as a symbolic start date; it is around this time that Debussy and other composers of his generation started to differentiate their style from German music and the “Germanized” French music of Franck and his contemporaries.5 The additive-harmonic innovations of this generation were taken up both by subsequent generations of French composers as well as by foreign-born composers who wrote for Parisian audiences (e.g. Stravinsky, Martinu, and Prokofiev at various points in their careers); additive harmonic practice developed in a relatively cohesive soundscape up until the German occupation of France during World War II (1940), after which the European compositional mainstream turned more comprehensively toward serial and atonal composition.

Since this

repertoire’s focus on the sensuous potential of novel chords participates in French 1 Jeremy Day-O’Connell, Pentatonicism from the Eighteenth Century to Debussy (Rochester, New York: University o f Rochester Press, 2007), 147. 2 Jim Samson, Music in Transition: a Study o f Tonal Expansion and Atonality, 1900-1920 (New York: W.W. Norton, 1977), 35. 3 Richard Taruskin, The Oxford History o f Western Music, vol. 4 (Oxford: Oxford Univeristy Press, 2005), 59-61. 4 Martin Cooper, French Music (London: Oxford University Press, 1951), 18-21 and 34-77. 5 Samson, 34.

2

modernism’s general privileging of pleasure and beauty over passion and the sublime,6 and in order to avoid confusing that music written by French composers during 18891940 with the overlapping-but-not-coterminous set of music written for French audiences at the same time (with the majority of notable composers of the period working in Paris), I shall refer to this repertoire as the Parisian modernist repertoire. Although additive harmony is commonplace in the Parisian modernist repertoire, it is poorly accounted for by modem music theory. This is due in part to a simple lack of attention. Most discussion of the development of tonal language in the late-nineteenth and early-twentieth centuries focuses on horizontal-domain procedures; the most common narrative is of how adherence to certain common-practice voice-leading principles (e.g. conjunct or parsimonious voice-leading, or a circumscribed set of teleological background stmctures) allowed for the incorporation into tonal contexts of new harmonic successions.7 In revealing these underlying tonal logics, however, this type of work often bypasses or reduces-out exactly that phenomenon - additive harmony’s vertical-domain surface details - which interests this study.

(The

6 Taruskin, 69-72. 7 The majority o f this work involves extended Schenkerian procedures, and most o f this literature focuses on the chromatic German repertoire o f the time period. A few studies have applied these techniques to the repertoire in France that is the focus o f this study: Matthew Brown, “Tonality and Form in Debussy’s Prelude a ‘L ’Apres-midi d ’un fa u n e,'" Music Theory Spectrum 15, no. 2 (Fall, 1993): 127-143; Sylvain Caron, “Tonal composition and new perspectives on Faure’s harmony,” Canadian University Music Review 22, no. 2 (2002): 48-76; Eddie Chong, “Extending Schenker’s ’Neue musikalische Theorien und P h a n t a s i e n Towards a Schenkerian Model for the Analysis o f Ravel’s Music” (Ph.D. diss., Eastman School o f Music, 2002); Charles Francis Navien, “The harmonic language o f L ’horizon chimerique by Gabriel Faure” (Ph.D. diss., University o f Connecticut, 1982); Adele Katz, Challenge to Musical Tradition (New York: Alfred A. Knopf, 1945); Edward R. Phillips, “ Smoke, Mirrors, and Prisms: Tonal Contradiction in Faure,” Music Analysis 12, no. 1 (March 1993): 3-24; Boyd Pomeroy, “Tales o f Two Tonics: Directional Tonality in Debussy’s Orchestral Music,” Music Theory Spectrum 26, no. 1 (Spring, 2004): 87-118; Felix Salzer, Structural Hearing: Tonal Coherence in Music (New York: C. Boni, 1952); Jim Samson, Music in Transition: A Study o f Tonal Expansion and Atonality, 1900-1920 (New York: Norton, 1977); Avo Somer, “Chromatic Third-Relations and Tonal Structures in the Songs o f Debussy,” Music Theory Spectrum 17, no. 2 (Autumn, 1995): 215-241”; and Avo Somer, “ Musical Syntax in the Sonatas o f Debussy: Phrase Structure and Formal Function,” Music Theory Spectrum 27, no. 1 (Spring, 2005): 67-96.

3

bypassing/reduction of additive chords often relies on the conventional extended-triad model of additive harmony, a model whose shortcomings will be discussed presently.) There is an analogous difference of focus in work that analyzes the Parisian modernist repertoire in terms of systematizable properties of pitch collections.8 While this work is eminently valuable in developing alternate lenses with which to view this repertoire and in detailing the sonic resources in play, it does not directly address activity at the musical surface - activity whose structures and tonal implications are essential components in this repertoire’s aesthetic effect.9 The explanation of additive harmony is therefore left to harmony textbooks (and generally then to back-of-the-book chapters in common-practice harmony texts). The majority of these textbooks account for additive harmony with the extended-triad model 8 The vanguard work in this area was done by Richard S. Parks (Richard S. Parks, “Pitch Organisation in Debussy: Unordered Sets in ‘Brouillards,’“ Music Theory Spectrum 2 [Spring, 1980]: 119-134; Richard S. Parks, “Tonal Analogues as Atonal Resources and Their Relation to Form in Debussy’s Chromatic Etude,” Journal o f Music Theory 29, vol. 1 [Spring, 1985]: 33-60; Richard S. Parks, The Music o f Claude Debussy [New Haven: Yale University Press, 1990]). Believing traditional tonal theory unable to adequately analyze Debussy’s works, Parks instead made analytic use o f an adapted form o f Allen Forte’s pitch-class-set genera; deep structure in Debussy’s music - the generator o f its surface detail - is conceived o f as recurring pitch collections and scales. Another study examining abstracted pitch-class collections in Debussy’s music is Mark McFarland, “Transpositional Combination and Aggregate Formation in Debussy,” Music Theory Spectrum 27 (2005), 187-220. Many o f the nexus set-classes found by Parks (such as 5-35, 6-35, and 8-28) are readily conceived o f as scales, and it is this thread o f pitch-collection analysis that has been most followed since Parks’ initial work. Studies that deal with scales in the French 1889-1940 repertoire include: Steven Baur, “Ravel’s ‘Russian’ Period: Octatonicism in His Early Works, 1893-1908,” Journal o f the American Musicological Society 52, no. 3 (Autumn, 1999), 531-592; Day-O’Connell; Allen Forte, “Debussy and the Octatonic,” Music Analysis 10, no. 1/2 (March-July 1991), 125-169; David Kopp, “Pentatonic Organization in Two Piano Pieces by Debussy,” Journal o f Music Theory 41, no. 2 (Autumn, 1997): 261-287; the chapter on “ Feux d’artifice” in David Lewin, Musical Form and Transformation (New Haven: Yale University Press, 1993): 97-159; Dmitri Tymoczko, “The Consecutive-Semitone Constraint on Scalar Structure: A Link between Impressionism and Jazz,” Integral 11 (1997): 135-179; and Dmitri Tymoczko, “Scalar Networks and Debussy,” Journal o f Music Theory 48, no. 2 (Autumn, 2004): 219-294. 9 The case for retaining a focus on tonal structures in analysis o f Debussy’s music in particular, made in reaction to the Parks-led atonal analysis project, is also argued by Avo Somer and Douglass M. Green. Somer writes “ Despite pervasive and often radical chromaticism that may well demand a fresh evaluation o f its tonal coherence, Debussy's musical language can be fully understood only in the light o f its allegiance to tonal centricity and its transformations o f classical harmonic practices” (Somer, “Chromatic ThirdRelations and Tonal Structure in the Songs o f Debussy,” p. 215); Green more simply states that “ How Debussy’s music is perceived is a knotty problem that is not helped by ignoring tonal associations in his works” (Douglass M. Green, “ Review o f The Music o f Claude Debussy by Richard S. Parks,” Music Theory Spectrum 14, no. 2 [Autumn, 1992]: 214-222, p. 217).

4

of ninth, eleventh, and thirteenth chords.10 While the details differ from presentation to presentation, the basic logic of the model is that the seventh chord can accommodate further dissonances arranged in a stack of thirds upwards from the chordal root. Figure 1.2, a reproduction of example 27.1 from Miguel A. Roig-Francoli’s Harmony in Context, is one example of this model; Roig-Francoli writes of these chords that “the ninth chord Figure 1.2. Reproduction of exam ple 27.1 from RoigFrancoli, Harmony in Context, extended tertian chords

generated by stacking thirds on top of the seventh chord.

(ffl *--------- & ----- -----........80 ----..... ........ ^ ;................. : ®------

DM:

ft

ft

ft

-t*

7 V

v?

v 'i

V 11

is used as an independent, nonlinear chord which results from adding one more third on top of a seventh chord . . . If we add one more third on top of the ninth chord, we will have an eleventh chord, and yet one more third will produce a thirteenth chord.”11 These extended triads can then substitute for the seventh chord, allowing for new dissonant

10 Post-WWII university-level textbooks which discuss additive harmony (however briefly) using the extended-triad model include Edward Aldwell and Carl Schachter, Harmony and Voice Leading, 3rd edition (Belmont, CA: Wadsworth Group, 2003); Thomas Benjamin, Michael Horvit, and Robert Nelson, Techniques and Materials o f Tonal Music: From the Common Practice Period to the Twentieth Century, 6lh edition (Belmont, CA: Thomson-Wadsworth, 2003); Leon Dallin, Techniques o f Twentieth Century Composition (Dubuque, IA: Wm. C. Brown Company, 1957); Allen Forte, Tonal Harmony in Concept and Practice, 3rd edition (New York: Holt, Rinehart and Winston, 1979); Robert Gauldin, Harmony Practice in Tonal Music, 2nd edition (New York: W.W. Norton and Company, 2004); Stefan Kostka, Materials and Techniques o f Twentieth Century Music 3rd (Columbus, OH: Prentice Hall College Division, 1989); Stefan Kostka and Dorothy Payne, Tonal Harmony, 4Ih edition (Boston: McGraw-Hill, 2000); Robert W. Ottman, Advanced Harmony, 5th edition (New Jersey: Prentice Hall, 2000); Vincent Persichetti, Twentieth-Century Harmony (N ew York: W.W. Norton and Company, 1961); Walter Piston, Harmony, 4th edition, revised and expanded by Mark DeVoto (New York: W.W. Norton and Company, 1978); Miguel A. Roig-Francoli, Harmony in Context (New York: McGraw Hill, 2003); Peter Spencer, The Practice o f Harmony, 5th edition (Englewood Cliffs, NJ: Prentice Hall, 2003); Greg A. Steinke, Bridge to Twentieth-Century Music: A Programmed Course, revised edition (Needham Heights, MA: Allyn and Bacon, 1999); and Ludmila Ulehla, Contemporary Harmony: Romanticism through the Twelve-Tone Row (New York: Free Press, 1966). 11 Roig-Francoli, 786-787.

5

chords to participate in tonal progressions. (The nuances of different versions of the extended-triad model will be explored in chapter 1.) While justification for the extended-triad model is occasionally provided either by locating chord extensions in the overtone series or by attributing generative significance to the stacking of thirds, the main features that recommend the model are its simplicity and its facile compatibility with common-practice theories of harmony. The fact that the model’s chord types can be arranged in sequence - triads are followed by seventh chords, seventh chords by ninth chords, and so on - has been used both as a pedagogical sequence and as a narrative about the development of chord types.12 By positing that taller chords are built upon seventh chords, the extended-triad model can lean on common-practice ideas about dissonance resolution, the origin of dissonance, and dissonance’s typical role in harmonic progression; the model can therefore explain novel verticalities without having to generate too many new principles of chord structure and/or behavior. For all its sequential neatness and adaptability, however, the extended triadic model struggles to adequately describe both the range of additive harmonies found in the Parisian modernist repertoire and the features of those harmonies that allow for their participation in tonal progressions. For one, there are many chords in this repertoire that can only awkwardly be accounted for by the extended-triad model, either as gapped taller chords or as tertian chords voiced in a non-tertian manner. Figure 1.3 provides two examples; (a) shows how the extended triad model has no better option than labeling the 12 A particularly clear-cut narrative claim is made by Alfredo Casella: “Assuredly the chord o f the major ninth, introduced by Weber, gave a totally different complexion to the entire musical language o f the 19th century. Nor is it less evident that the exploitation o f this chord reaches its culminating point in D eb u ssy .. . . The following harmonic concept [the augmented 11th chord] . . . it is only in Ravel that the new chord is finally used in a constant, conscious, and spontaneous manner.” Alfredo Casella, “Ravel’s Harmony,” The Musical Times 67, no. 996 (February 1, 1926): 125.

6

penultimate chord of L ’enfant et les sortileges as a D 13th chord with missing third, fifth, and eleventh, while (b) shows how the model might read the final tonic chord of Koechlin’s rondel “L’eau” as a C 13th chord with missing seventh and eleventh in which the ninth is voiced below the third.13 The ill fit of these labels calls into question the F igure 1.3. The ill fit of extended-triad labels w hen applied to certain chords in the Parisian modernist repertoire. a) reduction of the final two chords of L ’enfant et les sortileges and an extended-triad interpretation of the highlighted chord

------------------ =

a

h —1------ f n — *------

.....^ ......m.......... o

------------- *:--- >-----.' ....w .................................. ^

3 13th chord missing

**

3rd, 5th, and 11th?

b) reduction of Koechlin’s “L’e a u ” from his 9 Rondels, op. 14, mm. 42-44, and an extended-triad interpretation of the highlighted chord

8™...................................

a

J J J J SI

J

J

n J j j j

rr

^ rs -e-

1

if ’.................... ■□t r ~ r\ —1 — *-----1----------\—I— ------------------------- 1L*---------------------------1 ...' ........

r

^ J

t

r ~

H

C 13th chord missing 7th and 11th, with the 9th voiced below the 3rd?

centrality of thirds-stacking in a model of additive harmony. Already for triads and seventh chords the stacking of thirds is better understood as a convenient descriptor rather than as the generator of those chord structures. With taller chords, not only are

13 A common alternative explanation for the Koechlin chord would be to label it a “tonic chord with added sixth and ninth” ; this necessarily puts it in a separate category (“chord with added notes”) to the Ravel chord (“extended chords”). One o f the aims o f this dissertation is to be able to discuss these two chords as instances o f the same phenomenon - vertically enriched tonal progression.

7

generative claims more dubious - would-be eleventh and thirteenth chords appear more commonly with gaps than without - but many of the labels generated by the extendedtriad model, such as those in Figure 1.3, are inelegant at best. Another flaw of the extended-triad model is that it inadequately addresses the crucial impact voicing has on the identity of additive chords. While voicing is already significant for common-practice textures, the characteristic of additive chords that makes voicing particularly crucial is that many of them include pitch classes from two or more functional categories; Figure 1.4 shows that the penultimate chord of L ’enfant et les sortileges contains a dominant base in the lowest voice as well as powerful functional agents from both the subdominant and tonic functions.14

With these competing

F igure 1.4. Competing functional elem ents in the penultimate chord of L ’enfant et les sortileges. Tonic agent

Subdominant base & agent

XT Dominant base

functional elements present, the functional character of a chord progression is highly dependent on the vertical ordering of these pitch classes. This is why the progression in Figure 1.5a does more violence to the character of the original progression in Figure 1.1 than the progression in Figure 1.5c does to that in Figure 1.5b. In Figure 1.5a, the pitch classes of the chord are inverted to create a “root-position C major-ninth chord with

14 The language o f functional bases and agents is taken from Daniel Harrison, Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account o f its Precedents (Chicago and London: University o f Chicago Press, 1994).

8

Figure 1.5. Comparison of the impact of inversion upon an additive chord and upon a triad. a) inversion of the penultim ate chord of Figure 1.1

t r? b) replacem ent of the penultim ate chord in Figure 1.5a with a V chord

h

c) inversion of penultim ate chord of Figure i.5b

m

ff? missing fifth”; this replaces the authentic character of the original progression with a plagal one. In Figure I.5b the penultimate chord of L 'enfant et les sortileges is replaced with a dominant triad, which is then inverted in Figure 1.5c. Because the scale degrees in the new penultimate chord in both Figure I.5b and Figure I.5c are all dominantfunctional, Figure 1.5c changes the degree of finality of Figure 1.5b but not its basic functional character.15

Another example o f the functional importance o f the vertical ordering o f pitch classes is the “fourth inversion” o f the major-mode Iadd6 chord. Henry Martin has noted that placing the added note 6 in the bass prevents the chord from serving as a major-mode tonic; the resulting vi7 chord cannot be a secure

9

This brief discussion suggests that pitch-class content alone is an inadequate marker of additive chords’ identity and behavior, and suggests instead that voicing plays a crucial role (an idea that will be explored further in Chapter 2). The extended-triad model, however, struggles to account for this fact; since the model is designed to be easily compatible with readily invertible common-practice chord types, it accommodates voicing information only as a cumbersome add-on. The following passage is one small example of this type of after-the-fact list of voicing restrictions, taken from Ludmila Ulehla’s book Contemporary Harmony, one of the more thorough treatments of the extended-triad model: Due to the upward clim b o f thirds, the thirteenth rightfully dem ands the highest position. Occasionally, it will perm it the ninth to preside above, with the thirteenth directly beneath. The seventh o f the chord m ust not be placed next to the thirteenth . . . With inversions, the root may be placed in the upperm ost voice. . . N estled next to the seventh, [the thirteenth] becom es unclear and perm its the upper tones to predom inate.16

Although such rules are cumbersome (Ulehla is forced to provide equivalent rules for each type of extended triad and each of their chromatic alterations), forgoing any discussion of voicing/inversion means that the extended-triad model forfeits its ability to draw connections between chord structure and chord behavior. The consequence of allowing both chord-tone omissions and free invertibility is that any pitch combination can have any step class as a potential root. Figure 1.6 demonstrates: on the basis of pitchclass content alone, the L 'enfant chord naively could be labeled a C ninth chord with missing fifth in fourth inversion (chord a), an E thirteenth chord with missing third, ninth, tonic-function chord, as the elevated subdominant influence created by the presence o f scale degree 6 in the bass suppresses the influence o f the upper voice tonic elements. Henry Martin, “ From Classical Dissonance to Jazz Consonance: The Added Sixth Chord,” unpublished draft o f June 8,2005 (distributed in the course Theory and Analysis o f Contemporary Tonality, Yale University, Fall 2007): 40. 16 Ulehla, 100-101.

10

and eleventh in third inversion (chord b), a B eleventh chord missing fifth and seventh in first inversion (chord c), and so on. There is no reason inherent in the extended-triad

F igure 1.6. Over-applicability of the extended-triad model in the a b sen ce of guidelines for voicing. a)-c) potential alternate readings of the penultimate chord of L’enfant et les sortileges.

v

& (*)

(b)

(c)

d)-e) implausible chord labels available with the extendedtriad model i

|o

.

_fl

(d)

ilM

(e)

model as to why one reading is preferable to another. Furthermore, the fact that all diatonic step classes are present in a thirteenth chord means that, again allowing for invertibility and chord-tone omissions, any combination of tones could be considered some type of extended triad. The chromatic cluster in Figure I.6d could ostensibly be analyzed as a Bb raised-thirteenth chord missing third, fifth, ninth, and eleventh in sixth inversion (Figure I.6e), even though the verticality in (d) is unlikely to be found participating in conventional tonal progressions. Unable then to describe the full range of additive harmonies or to effectively discuss voicing’s crucial role in those chords’ structure and behavior, the extended-triad model is not a consistently applicable analytic tool for the Parisian modernist repertoire. This dissertation endeavors to develop a more effective model of additive harmony that, by making voicing a central concern, better accounts for the range of verticalities found

11

in this repertoire and for how and why they work. Chapter 1 examines the history of additive harmonic theory, tracing three distinct strategies for explaining simultaneities that are not triads or seventh chords: adapting basic chord types, identifying non­ harmonic tones, and formulating new basic chord types. Chapter 2 presents a model of additive-harmonic chord structure in which chords are conceived of as pitch-space voicings constrained by the considerations of tonal plausibility and chordability; this pragmatic figured-bass approach to additive harmony explains how a chord’s voicing relates to its most frequent behaviors and to its suitability for tonal use. Tertian structure is shown to be one of many potential adornments of an additive chord, and not an essential feature in its identity. Chapter 3 investigates the role voicing plays in the special additive-harmonic case of polychordal polytonality, wherein tertian structure is an essential feature but the desired effect is not the production of a single composite chord, but rather the separation of the music into distinct auditory streams. I close by drawing bigger-picture parallels between vertical-domain innovations in music in France and horizontal-domain innovations in Germanic music, and suggest that paying attention to the intricate surface textures of this repertoire adds a new dimension to our understanding of the development of tonal language in the late-nineteenth and early-twentieth centuries.

12

Chapter 1 Strategies in Additive Harmonic Theory

While this dissertation develops a model of additive harmony for the harmonic writing of the Parisian modernist repertoire, the fundamental question of additive harmonic theory - how can one explain those musical moments that do not immediately look like a tonal system's basic chord types? - is valid for all tonal theory. That question is a necessary byproduct of the reductive impulse in tonal theory. The central conceit of tonal theory is that music is organized and impelled by its chordal successions; that claim is more easily sustained when the tonal system involves a limited set of objects and behaviors. (The case that a certain chord plays a determinative role in musical succession is most convincing when that chord’s use consistently produces one or a few possible outcomes, rather than a wide range of potential events.) Hence the impulse to reduce the number of objects in a theorized tonal system: the fewer objects there are, the fewer outcomes/behaviors there are that need to be explained and the more tightly formulated and compelling theories of chord progression can be. Such a reduction of types enabled the construction of the first comprehensive tonal theory, Jean-Phillipe Rameau’s 1722 Traite de I ’harmonie; using a theory of chord inversion, Rameau pares the wide variety of figures found in earlier thoroughbass treatises down into only two chords - the triad and the seventh chord - which can appear on every scale degree. The tremendous power of this theoretical move has been hinted at above and is discussed in depth elsewhere, and the privileged central status of invertible triads and seventh chords has been foundational for the majority of tonal theory since

13

Rameau.1 Additive harmonic theory is a necessary byproduct of this reductive move. Musical practice throws up many verticalities that are not (or do not immediately appear to be) triads and seventh chords (or simple passing/neighbor figurations thereof); additive harmonic theory is required to explain how these moments fit into the tonal system. This chapter defines three foundational strategies for explaining non-standard verticalities - the modified-basic-types strategy, the non-chord-elements strategy, and the new-basic-types strategy - and uses them to examine the history of additive harmonic theory.

The

modified-basic-types

strategy

explains

novel

verticalities

as

modified/expanded versions of conventional chords (as in the extended-triad model of ninth, eleventh, and thirteenth chords discussed in the introduction). By positing that these modified/expanded chords and their progenitors behave similarly (e.g. that a C9 chord can substitute for the C7 chord from which it is dervied), this strategy allows for an expanded range of verticalities to participate in conventional progressions. The non-chord-elements strategy explains novel verticalities as the coincidence of conventional chords and non-chord-elements. This approach is more theoretically parsimonious than the modified-basic-types strategy; by labeling certain pitches in a verticality as being outside the tonal system, this strategy leaves the core objects of the system untouched. The new-basic-types strategy accounts for novel verticalities by defining new chord types that must then be worked into the tonal system; the strategy folds novel verticalities into the system as entities unto themselves (i.e. not as derivatives as triads

1 A fine discussion o f the historical origins and theoretical basis for Rameau’s inversion theory in the Traite, and o f how that theory allowed Rameau to be the first theorist to truly examine harmonic succession, is in Joel Lester, Compositional Theory in the Eighteenth Century (Cambridge, Massachusetts: Harvard University Press, 1992), 100-108 and 115-122.

14

and seventh chords). This “back to the drawing board” approach reexamines the basic primacy of triads and seventh chords established by Rameau; a measure of theoretical parsimony is sacrificed so that other chord types can be recognized as foundational. While this strategy is rarely used as basis for a complete system of harmony (i.e. not every vertical formation is allowed to be a chord unto itself), discussion of alternative chord types (e.g. quartal, quintal, or secundal chords) is found sprinkled through twentieth-century harmony texts. F igure 1.1. Spectrum betw een competing ideals in additive harmonic theory. Modified-basic-types strategy WWnmg i ng A appticabity

I |

I

I

^ |

New-basic-types strategy

Thsoretical parsimony

Non-chord-elements strategy

These three strategies can be understood as marking different positions on a continuum between two theoretical ideals (Figure l.l).2 On one side of the continuum

2 These strategies share ideas with but are not identical to (and indeed were developed independently from) the “four general strategies for modeling dissonance” discussed by Richard Cohn in Chapter 7 o f his book Audacious Euphony: Chromatic Harmony and the Triad’s Second Nature (New York: Oxford University Press), 139-168. Cohn defines four general strategies for modeling dissonance: suppression (reducing dissonant chords away from deeper levels o f structure), substitution (analyzing dissonant chords as modifications-by-substitution o f other dissonant chords, e.g. analyzing the diminished seventh chord as substituting i>6 for 5 in a dominant-seventh chord), reduction to a subset (analyzing dissonant tetrads as one o f their consonant triadic subsets, i.e. analyzing an F#0 7 chord as an A-minor triad), and combination to a superset (analyzing dissonant tetrads as the combination o f two triads). The main difference between my categories and Cohn’s is that his describe the various ways dissonant tetrads are folded into theories o f progression for nineteenth-century German music, while mine describe the ways o f accounting for any sort o f novel verticality in theory ranging from Rameau to the present. This difference in aim accounts for the ways in which the two sets o f strategies interact. Reduction-to-a-subset and substitution are both forms o f the modified-basic-types strategy o f additive harmonic theory, in that both try to explain a novel formation as a modified form o f a more common quantity. However, as will be explored in this chapter, the modified-basic-types strategy has more manifestations than the reduction-to-a-subset and substitution applications found in nineteenth-century

15

lies the ideal of wide-ranging applicability - the desire for a “non-reduced” theoretical system that can account for the entire variety of found verticalities. Opposite it lies the ideal of theoretical parsimony - the desire for an elegant and concisely formulated system.

The three strategies strike different balances between the two ideals.

The

modified-basic-types strategy is a centrist position; it stretches/loosens the definitions of some objects in the system so that the system can accommodate new sonorities without significantly expanding its basic set of objects. The non-chord-elements strategy is more “right-leaning” in that it leaves the basic objects and mechanics of the system undisturbed - novel verticalities are not admitted into the system as integrated objects. In examining these two strategies in detail below, we will see how the non-chord-elements strategy interfaces with the modified-basic-types strategy. One approach ends where the other begins - a pitch must be either a chord tone or a non-harmonic tone - and the position of the line that separates what is inside the system from what is outside the system is a key issue for additive harmonic theory. The new-basic-types strategy is more liberal than the modified-basic-types strategy in that it allows a greater range of found verticalities to be considered basic chords. Incorporating those new chord types into the tonal system, though, requires formulating additional rules of progression that can dilute the predictive power of the tonal model. (Figure 1.1. shows these strategies only in the abstract - the work of individual theorists may utilize one or more of these strategies. While some writers pursue one strategy and suppress or belittle others, other writers keep two German theory, and so I collapse those two categories into one broader one. Similarly, combination-to-asuperset is but one o f the novel types o f chord structures devised in order to explain the broader range o f non-standard verticalities found in the Parisian modernist repertoire. And because I am interested in the novel verticalities at the musical surface, I do not deal with the strategy o f suppression or other aspects o f theory that deal with deeper tonal structure beyond the explanations they give for why surface detail is suppressable. So while Cohn and I do share a similar approach to categorizing families o f theories, the differences between our two sets o f categories are significant enough that I will retain the use o f my own terms in this chapter.

16

approaches in productive equilibrium or foreground one while using others to paper over any cracks. Individual theories, then, can be mapped anywhere on the continuum or indeed across several points on the line, and not just at the nodes of Figure 1.1.) Reading the history of additive harmonic theory in terms of these three strategies clarifies what is at stake in the often confusing and/or trivial-seeming rules about the invertibility of ninth chords or debates about the viability of the unresolved appoggiatura as a type of non-harmonic tone. In play is an issue fundamental to theorizing musical systems - the challenge of developing a system that both a) has efficient and consistent internal mechanics and b) is applicable to a range of real musical situations. Viewing the history of additive harmonic theory in terms of these three strategies also allows connections to be drawn between theories that work with different repertoires. In the sections that follow, I investigate each of the strategies in turn and examine how their ideas adapt to evolving musical styles, and also explore moments of interaction between strategies. First, though, I will examine additive harmonic thinking in Rameau’s Traite in order to shed light on the interrelated genesis of the modified-basic-types and non-chordelements strategies.

1.1 - Additive Harmonic Thinking in Rameau’s Traite As the first theorist to develop a fully formed theory of tonality, Rameau is also the first to grapple with additive harmonic theory. While Rameau’s system is built on the primacy of the perfect chord (triad) and the seventh chord and the interactions between them, he does discuss the generation and behavior of three additional chord types, realized here in Figure 1.2: the ninth chord (chord al), the eleventh chord (bl) and the chord of the large sixth (c). All three of these additional chord types are derived from the

17

F igure 1.2. Realization of th e chord types described in R am eau’s Traite other than the perfect chord and the seventh chord. (Locations of R am eau’s written descriptions are given in the in-text footnotes.) Open note-heads com prise th e initial basic chord; black note h ead s are th e added/subposed notes. The fundam ental b a ss of any chord is the lowest open notehead.

&

if-

1 —1L

---

J L .... - i | gg___ .. .

6 in D or a chord with suspended 11th, 9th, and minor 6th that pulls towards f minor. 5 Mark Jerome Yeary, “Perception, Pitch, and Musical Chords,” Ph.D. dissertation, The University o f Chicago (2011), 54.

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that is readily perceived as an integrated entity) or from that of disassociated strata of pitches spread too far apart in pitch space. Yeary’s work also suggests that no every verticality is readily perceived as a chord - “a chord is a sonority [Yeary’s definition of a sonority is “multiple simultaneous notes”], but a sonority need not be heard as a chord” and that chords are certain specific sonorities that are perceived as Gestalts/integrated entities.6 As with anchor structures, chordability is a product of the real spacing of tones, not of pitch class content; Harrison writes that “‘chordness’ . . . is not a property of pitchclass sets but rather of pitch-space dispositions (voicings, realizations) of such sets.”7 Figure 2.1 briefly demonstrates the differences in these qualia by arranging the pitch classes of the L ’enfant chord from Figure 1.1 (B, C, D, E) first as a cluster (a), then as a chordable chord (b), and lastly as disassociated strata (c); the majority of additive harmonies in the Parisian modernist repertoire are voiced like (b). F igure 2.1. The pitch c la sse s of the L’enfant chord arranged a s a cluster in (a), a s an integrated chord in (b), and a s disassociated strata in (c). ‘

...... - g t ft

.......................it.............- S - .... -...... 11

(a)

....

=* (c)

The following sections demonstrate how the concepts of anchor-structureproduced tonal plausibility and chordability can be used to explain the behavior and structure of additive harmonies in the Parisian modernist repertoire. Before presenting the details of these concepts, though, I want to preemptively address the problem of “only 6 Ibid., 3. 7 Harrison, 8.

78

seeing what you want to see” and shoehoming the complexities of the repertoire into common-practice categories (one of the flaws of the extended triadic model). My model is not designed to be stringently proscriptive; the goal is not to definitively say which chords “work” and which do not, but rather to discuss common formations, usages, and hearings. Such is the range of verticalities produced by these composers’ fascination with color and sonority that accounting for every single chord in the repertoire is only an ideal towards which this project can strive, and not a fully attainable goal; Michael Russ has discussed how the success of models of early 20th-century harmony can only ever be partial, because “every one of the 344 possible note-combinations [from Simon Harris’ exhaustive classification of chord types] has an awkward tendency to turn up somewhere.”8 Not all verticalities used in tonal contexts will be plausibly tonally participatory or even chordable. These chords are not banished from the repertoire or labeled aberrations; they are simply less common, and require more contextual incorporation. My model captures how certain pitch voicings provide opportunities for tonal listening in this repertoire; the aim is to remain sensitive to the novel characteristics of this music while still highlighting its dependence on and allusions to common-practice structures.

2.2 - Anchor Structures and Tonal Plausibility In order to potentially participate in conventional tonal contexts - to be tonally plausible - a novel verticality must be relatable to a common-practice triad or seventh chord, so that it can assume the common-practice chord’s conventional tonal role. This

8 Michael Russ, “ Review o f Skryabin and Stravinsky, 1908-1914: Studies in Theory and Analysis by Anthony Pople and A Proposed Classification o f Chords in Early Twentieth-Century Music by Simon Harris,” Music & Letters 74, no. 2 (May, 1993): 325.

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link between the novel verticality and the common-practice chord is produced by an anchor structure - a voicing, involving the bass note, that is taken from the set of common-practice voicings (i.e. common-practice chords and their inversions).9 This anchor structure strongly evokes its common-practice progenitor, and allows the novel verticality built around/upon it to plausibly stand in for that progenitor. The pitches of the verticality that do not form an anchor structure are adorning tones; they produce new harmonic colors and voice-leading lines without disturbing the stability and tonal suggestiveness of the anchor structure. Figure 2.2 shows how the relationship between tonally plausible additive chords and common-practice chords can therefore be modeled F ig u re 2.2. The relationship betw een tonally plausible additive chord and com mon-practice chord modeled a s a tw o-step process. jComposition^ Process j ► Dominant chord stripped down to its anchor structure (the It) anchor structure D3-C4)

Common-practice cadence

(c)

(b) L *5?

^

Additive cadcacc; anchor tones are empty note beads, adorning tones flDed-in note heads

(8)

2 .........a .....'... :..- f " " ^

.J ' 4

*

—

—— —+— ■ —

—

..... • - ..............

Anchor structure related to a common-practice progenitor

f -

t * -----........ - 3 -----------------

-------------— _— V

................................ ----- & ---------------

Anchor structure extracted from the additive chord

Additive cadence

«----------------------------------Listening Process

as a two-stage process. Read from left to right, using the captions on top of the staff, the 9 Anchor structures always involve the bass note because the bass’ perceptual salience is essential to tonal listening habits; a change o f bass is almost always o f harmonic consequence. (Olli VaisSla, “Concepts o f Harmony and Prolongation in Schoenberg’s Op. 19/2,” Music Theory Spectrum 21, no. 2 [Autumn, 1999]: 234.)

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process suggests a model for generating tonally plausible additive chords. In order to generate a tonally plausible additive chord from the common-practice progression in Figure 2.2a, one first takes the common-practice chord and strips it down until only its most basic, emblematic voicing remains; here, the V7 chord in G major is stripped down to its D3-C4 anchor structure (Figure 2.2b). Next, that anchor structure is decorated with adorning tones - here in Figure 2.2c with an E4 and a G#4. (Anchor-structure tones are shown as open note-heads, adorning tones as filled-in note-heads. The arrived-at tonic chord in the additive cadence in Figure 2.2c is also an adorned anchor-structure chord; the set of possible anchor structures will be discussed in the section that follows.) Amidst the new pitches, it is the presence of the anchor structure - the actual voicing D3-C4, not just the pitch classes D and C - that “anchors” the verticality into the progression as a stand-in for its progenitor; the additive cadence of Figure 2.2 maintains the functional character of the initial common-practice cadence. When read from right-to-left using the captions beneath the staff, Figure 2.2 suggests a model for how anchor structures work for listeners: when confronted by the novel verticality in (c), the listener latches on to the anchor structure in (b) and so is able to relate the novel first chord from (c) to the common-place first chord in (a).10 Additive-harmonic versions are possible for every common-practice chord type. Figure 2.3 lists the complete set of common-practice chord types; it comprises all types of triads and sevenths that can be created from the diatonic scale, as well as the

10 This process is perhaps analogous to Justin London’s concept o f meter. In London’s theory, when a listener confronts a set o f heard durations, the listener’s mind entrains to them and filters them through a set o f coherent metrical patterns. With this model o f the anchor-structure listening process, the listener confronts heard new chords and filters them through a set o f coherent common-practice chordal frameworks. (Justin London, Hearing in Time: Psychological Aspects o f Musical Meter, [New York, Oxford University Press, 2004]: 9-16.)

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Figure 2.3. The set of common-practice chord types. -

th e th e th e the the the the the the

major triad and its inversions minor triad and its inversions diminished triad and its inversions dom inant-seventh chord and its inversions minor-seventh chord and its inversions half-diminished-seventh chord and its inversions major-seventh chord and its inversions fully dim inished-seventh chord and its inversions se t of augm ented-sixth chords

diminished seventh chord produced by the raised leading tone in the minor mode/flattened sixth degree in major and the augmented sixth chords produced by raising the fourth scale degree in the minor mode. The anchor structures that evoke these chord types divide into two categories - first-order anchor structures that evoke the rootposition versions of the common-practice chord types (save for the special cases of the diminished triad and the diminished seventh chord), and second-order anchor structures that evoke inversions of the common-practice chord types. The mechanisms of these two categories and the relationship between first-order and second-order anchor structures will now be discussed in turn.

First-Order Anchor Structures There are three first-order anchor structures: registrally-ordered interval |7 | above the bass, registrally-ordered interval |t| (ten) above the bass, and registrally-ordered interval |e| (eleven) above the bass."

(/?o-intervals that form anchor structures are

11 The term registrally-ordered interval is taken from Olli Vaisaia’s article “ Prolongation o f Harmonies Related to the Harmonic Series in Early Post-Tonal Music.” Shortened to “ro-interval,” a registrally ordered interval is an equivalence class in which an interval and its inversion are not considered equivalent, but a simple interval and a compound version o f that same interval are. /fo-intervals are measured in semitones, not scale steps. (Olli VSisSla, “ Prolongation o f Harmonies Related to the Harmonic Series in Early Post-Tonal Music,” Journal o f Music Theory 46, no. 1/2 [Spring-Autumn, 2002]: 207-283.) When

82

marked in bold font and placed in beams.)

These three anchor structures can support

tonally plausible additive-chord stand-ins for each of the root-position common-practice chords, save for the special cases of the diminished triad and the diminished seventh chord. Novel verticalities that have an ro-interval |7| anchor structure can assume the behavior of root-position consonant triads, novel verticalities that have an ro-interval |t| can assume the behavior of all augmented sixth chords and all seventh chords excepting the major seventh, and novel verticalities that have ro-interval |e| can assume the behavior of major-seventh chords (Figure 2.4). There are three reasons why these three anchor structures are particularly wellsuited to serve as the foundation for additive harmonies. First, as discussed in the work of Ernst Terhardt, Richard Pamcutt, and Ludger Hofmann-Engl, the three ro-intervals are all root-candidate-producing intervals, meaning that the upper pitch reinforces some of the overtones of the lower pitch;12 these three intervals, then, share a certain acoustic sturdiness. Yeary has pointed out that this harmonicity is “taken as an ecological cue to objecthood” because most real-world sound sources (including the human voice) produce complex tones, meaning that verticalities featuring these ro-intervals are more likely to perceived as integrated chords.13 Second, these ro-intervals, even in their narrowest realizations, are wide enough to fit a range of pitches between their boundary pitches without creating dissonant

used exclusively above the bass, ro-intervals are chromatic figured bass numbers. These are called “FB classes” (for figured-bass classes) and are formalized in an atonal context in Morris, 218-221. 12 Richard Pamcutt, “Revision o f Terhardt’s Psychoacoustical Model o f the Root(s) o f a Musical Chord,” Music Perception 6, no. 1 (Fall, 1988): 65-94; Ludger Hofmann-Engl, “Virtual pitch and pitch salience in contemporary composing,” http: \vw\v.c h am e leo n gro u p .org.uk research/virtual p itc h .p d f (accessed March 3, 2009). Root-producing intervals 13 Yeary, “Perception, Pitch, and Musical Chords,” 23.

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Figure 2.4. First-order anchor structures and the common-practice chord types they can stand in for. First-order anchor structures: th e s e t of root-position com mon-practice chords partitioned by largest ro-interval anchor structure | 7 | . . .

4

^ °

1

. . . supports stand-ins for root-position consonant triads

»

i

■

'

anchor structure | t | . . .

4 $

. . . supports stand-ins for augm ented-sixth chords and all root-position seventh chords sav e for m ajor-seventh chords

:- .=*B= f.-.-.-.-it.-.— P ====?t8 o===f ----ato--#= =1l==^=JL==f!§*==ll anchor structure | e | . . .

I" •:: • . . . supports stand-ins for root-position m ajor-seventh chords

4‘ » clusters. This feature - absent in the other root-candidate-producing ro-intervals, the major third and the major second - is a desirable one for composers looking to expand the palette of tonally participatory chords. It allows composers to substitute a wide

84

variety of colors for any single common-practice chord type without incurring harsh acoustic roughness (this issue will be discussed further in section 2.3). Third and most importantly, though, these ro-intervals have strongly defined roles in common-practice harmony. In common-practice contexts a fifth above the bass is always the root and fifth of a root-position chord; it is not susceptible to either enharmonic reinterpretation (doubly augmented fourths or doubly diminished sixths are extremely rare) or reinterpretation as different chord members of an alternate chord.14 The minor seventh is almost always the root and seventh of a seventh chord (dominant, minor, or half-diminished); it is the strength of that implication which allows for the common enharmonic reinterpretation of an augmented sixth chord as a dominant seventh and vice versa.15 The major seventh, like the perfect fifth, is insusceptible to enharmonic reinterpretation and is always the root and seventh of a major-seventh chord. It is this strength and clarity of these intervals’ common-practice connotations that allows them to serve as anchor structures. The intervals act like lightning rods for our tonal listening habits. When we hear a chord with a G#4-F#4 |t| anchor structure, such as that in mm. 21 of Ravel’s “Laideronette, imperatrice des pagodes” from his Ma mere

14 To take this approach to common practice harmony means that this chord (in C major) must be called a

subdominant major triad with added sixth (foundational fifth-above-the-bass plus added tones), and not a supertonic seventh chord in first inversion; I find this an acceptable reading, as the tendency o f 1 to resolve to 7 in the standard progression to V can be explained not as the essential dissonance o f the seventh chord resolving downwards, but instead as motivated by the need to avoid parallel fifths when the bass note moves up to 5. 15 Furthermore, the chromaticism and the modal and modulatory flexibility o f the Parisian modernist repertoire means that the functional distinction between these chords-with-a-minor-seventh is blurred; this interchangeability in practice further justifies them being placed into the same general category.

85

VOye (Figure 2.5), that anchor structure brings to bear our knowledge of commonpractice chords built with that same minor seventh.

We are then equipped with

expectations for the heard chord’s behavior. Given the Fit-major tonal context to this

F igure 2.5. Ravel, “Laideronette, imp6ratrice d es pagodes" from Ma mere I’Oye, mm. 16-24: |t|-anchored substitute supertonic. 16

# MM

• ■

>

ilniVrrf.i i£ £ | V"* ------------ =

r_r

&— V fc.-y-.^T I t r_r

20

point, we can hear supertonic-seventh potential in the chord, even in the absence of a clear stacking of thirds or conventional treatment of the dissonant Fit; the anchor structure has created a tonal plausibility.

In this instance Ravel chooses to actualize that

plausibility; the G#3-F|t4-anchored chord effectively stands in for a conventional supertonic seventh as it moves to the C#-Gtt open fifth. The absence of a clear common-practice connotation for the largest ro-interval of the diminished triad (ro-interval 6) and the diminished seventh chord (ro-interval 9) is the reason those two chords do not have first-order anchor-structure representatives; both rointervals are too functionally flexible. In common-practice contexts, ro-interval 9 above

86

the bass, in common-practice contexts, could comprise the chordal third and root of a first-inversion minor triad, the seventh and fifth of a third-inversion dominant seventh chord, or the fifth and root of a second-inversion major triad/cadential six-four chord. /?o-interval 6 is even more ambiguous, as it could be the third and seventh of either a first-inversion dominant seventh or a third-inversion dominant seventh, root and fifth of a diminished triad, any two pitches of a diminished seventh chord, or root and fifth of either a root-position or second-inversion half-diminished seventh chord.

Neither

interval, then, when featured above the bass of a novel verticality, can activate a clear set of expectations about chord identity and behavior.16 There are several benefits of conceiving of additive chords as the combination of a tonally plausible anchor structure and adorning tones. For one, the fact that a first-order anchor structure can support a wide range of colorations, without destabilizing the established tonal plausibility, explains how chords of divergent pitch content can function similarly. Figure 2.6 demonstrates: chords (a), (b), and (c), save for the shared D3-C4 |t|

F igure 2.6. C hords with distinct pitch-class content but a sh ared tonal plausibility. A

1%------ M

(c)

(b )

(3 )

----------b ty o M

-----

■ e-

4 |t|2 6 / 5

|t|1 38 / 5

-€»-

|t|5 2 / 5

anchor interval, have distinct pitch-class materials: (a) is a whole-tone sonority of interval 16 The same lack o f common-practice connotations is true o f the other two root-candidate-producing intervals, the major second and major third; the major second is not a common ro-interval above the bass in common-practice contexts, and major third is too functionally flexible - it can be root and third o f a major triad, third and fifth o f a minor triad in first inversion, the fifth and seventh o f a half-diminished seventh chord, etc.

87

vector 0404.02, (b) is an octatonic sonority with vector 223111, and (c) is a diatonic construction of vector 021120. Yet the voicing of the three verticalities is such that each can be easily understood as a D-rooted dominant of G (the common-practice progenitor of the anchor structure), and would be most likely used as such in the Parisian modernist repertoire. Figure 2.6 also displays a labeling system designed for this model. The rointervals above the bass of a chord are listed in their upwards order of appearance, forming what Alan Chapman has termed an AB (above bass) interval set, were these numbers arranged in order from smallest to largest, they would be one of Robert Morris’ FB (figured-bass) classes.17 While labeling the additive chords as AB sets rather than as FB classes is a slight hindrance when comparing two verticalities’ FB classes, it maintains ro-interval content within of the upper voices of the chord - something that will be important in section 2.3’s discussion of chordability.18 The pitches of the anchor structure are set off in bold font and beams; this makes visually prominent how anchor structures serve as equivalence classes, partitioning the vast set of voicings into analytically productive categories of tonal plausibility. The bass notes of the additive chords are labeled, where possible, by scale degree. Another benefit of the anchor-structure approach is that it can explain how different voicings of a single pitch-class set can suggest different plausible behaviors. Figure 2.7 illustrates. Chord (a) has an ro-interval |7 | anchor structure, and so behaves

17 Alan Chapman, “Some Intervalic Aspects o f Pitch-Class Set Relations,” Journal o f Music Theory 25, no. 2 (Autumn, 1981): 276-277; Morris, 218-221. 18 Preserving the ordering o f upper voices also brings the model in line with Arthur Samplaski’s finding that the ordering o f adjacent pitches plays a large role in listener judgments o f chord similarity. (Arthur G. Samplaski, “A Comparison o f Perceived Chord Similarity and Predictions o f Selected Twentieth-Century Chord-Classification Schemes, Using Multidimensional Scaling and Clustering Techniques,” Ph.D. dissertation [Indiana University, 1999]: 217.)

88

F igure 2.7. Different tonal plausibilities arising from distinct voicings of a single pitch-class set.

-X suggested by chord types & pitch-class content

r ------ 1.........dzu. ] J f ............. %...............^ ^ --------- u------------ b 9------ -------^-©--------------------iy ;--------- ^--------------1--------

ii-V-I in G suggested by bass motion

horizontal contexts actually strengthens the premise that motivates this model - that the majority of novel sonorities used in the Parisian modernist are designed with reference to common-practice chord types, so that those novel sonorities can be incorporated into tonal contexts. Another benefit of the first-order anchor-structure concept is that it places sevenths on a structural par with the perfect fifth, a move which acknolwedges the newly stable status of seventh chords in Parisian modernist repertoire. The seventh is no longer exclusively an essential dissonance in this repertoire and, freed of its requirement to resolve downwards, instead serves as a foundational chord member. The repertoire is full of passages such as that in Figure 2.10a from Ravel’s Sonatine, or in Figure 2.10b from Samuel Rousseau’s La cloche du Rhin, in which seventh chords move in parallel motion and therefore showcase the stability of the seventh chord as an entity.20 Seventh chords are also on occasion found as final tonics, the most stable chord of all; one example is

20 The Rousseau example is reproduction o f Figure 3 from Lenormand, 15

91

F igure 2.10. U se of seventh chords in parallel motion in the Parisian modernist repertoire. a) Ravel Sonatineli, mm. 6-13

— -rm

Sm

m

b) Figure 3 from C hapter 2 of Lenormand A Study of Modem Harmony, p. 15; SAMVEL ROUSSEAU. C alm ato.

I

La cloche du Rhin. ______

^ 1 - j ^ - lr V - -

(“ )

r

Page 136. -

——•«=====— - _ :

:»-U

(W m JxUM-i-

(Cboodeos, Pubr.)

IJ 4 t T T W

F ig u re 2.11. Ravel Jeux d ’Eau/iii; |e|-an ch o red final tonic

I

found in the last movement of Ravel’s Jeux d ’eau, which ends on an |e|-anchored majorseventh chord (Figure 2.11). 21

21 The elevation o f |t|-anchored chords to integrated chord status (i.e. not chords containing essential dissonances) is also tied to the greater variety o f scalar resources in the repertoire; no perfect fifths are

92

Another consequence of the anchor-structure model is that it frees thirds from the responsibility of generating chordal structure. The core component of an additive chord is its acoustically robust anchor-structure voicing and its attendant common-practice implications, not the presence of tertian construction; thirds are merely adorning tones, more akin to a ninth or a sixth above the bass than to the perfect fifth. The anchorstructure model’s conception of the third as coloration rather keystone is borne out on the Parisian modernist repertoire in the frequency with which the third is either chromatically altered (as in the cadence in Figure 2.12, from Satie’s Gymnopedie No. 1) or absent altogether (as in the cadence in Figure 2.13, from Debussy’s “La fille aux cheveux de lin,” mm. 18-19; the dominant of Eb contains no leading tone).

F ig u re 2.12. Chrom atically altered chordal third in S a tie ’s Gymnopedie No. 1, mm. 36-39.

1

36

F igure 2.13. “Omitted” chordal third in D ebussy’s “La fille aux cheveux d e lin,” mm. 18-19.

18

available in the whole-tone collection, but |t|-anchored chords are the most stable and tonally connotative chords available in whole-tone passages.

93

Second-Order Anchor Structures and Anchor-Structure Interaction

Second-order anchor structures are voicings that evoke those common-practice chord types not represented by the three first-order anchor structures: inversions of triads and seventh chords, the diminished triad, and the diminished seventh chord. Whereas first-order anchor structures only involve two pitches, second-order anchor structures require three pitches. This is because the individual ro-intervals above the bass of these common-practice chord types are neither as acoustically robust nor as strongly connotative of a specific chord type as are the ro-intervals involved in first-order anchor structures. A third pitch is therefore necessary to strongly reference a common-practice chord type, meaning that the common-practice progenitors of second-order anchor structures cannot really be stripped down beyond the omission of either the chordal third or chordal fifth from a seventh chord.

The complete set of second-order anchor

structures is shown in Figure 2.14; the common-practice chord types referenced are listed below each structure.

The upper two pitches of second-order anchor structures can

appear in either order (e.g. the |38| anchor structure can be voiced as an |83 |) save for those marked with an asterisk; those two structures, if the upper parts are flipped, run afoul of the chordability constraints discussed in section 2.3.

Those two structures

marked with double asterisks are “impossible” structures in that they run afoul of chordability constraints in any voicing; they are included here for the sake of completeness. Second-order anchor structures, in part because they are less stable and immediately evocative than first-order anchor structures, appear far less frequently in the repertoire than do first-order anchor structures. Indeed, as will be discussed at the end of

94

Figure 2.14. Second-order anchor structures.

o-----m

i f ------

|38|

first-inversion m ajor triad

49

|58|

« -----

second-inverson major triad

second-inversion m inor triad

first-inversion m inor triad

|36|

rb?

|39|

|69|

Diminished triads/seventh chords and their inversions

|88| first-inversion dominant 7th

$

|79| first-inversion m inor 7th/ half-diminshed 7th

I

|3S|

26

------

third-inversion dominant 7th

second-inversion dominant 7th/ minor 7th

|48| g 1

1*81

~

second-inversion half-diminished 7th

third-inversion minor 7th/ half-diminished 7th

TT

|S7|* first-inversion m ajor 7th

|S4|*

H8|** third-inversion m ajor 7th

second-inversion major 7th

|29|

-----

third-inversion dominant 7th/ m inor 7th

|28| third-inversion half-diminished 7th

1181** third-inversion m ajor 7th

the section 2.3, there are few ways of adorning second-order anchor structures that do not either (a) suggest first-order anchor structures or (b) run afoul of chordability restrictions. For this reason then, second-order anchor-structure chords often appear near first-order anchor-structure chords in situations where the common-practice progenitors of both the first- and second-order anchor structures have the same root (e.g. a |68|-anchored chord evoking a first-inversion dominant seventh chord will appear near a |t|-anchored chord

95

evoking a root-position version of that same dominant seventh). A pair of examples from the literature is shown in Figure 2.15. Figure 2.15a is the opening phrase of the Rigaudon F igure 2.15. Exam ples of second-order anchor structures, a) Ravel, Le tombeau de Couperin, “Rigaudon” mm. 1-8 Assez vif

r r

n

r

n

f U J ii( iJ- * #

p f r r'f

* _■ 3 ^-... ... t ........................

—

► 1

p w ff A ^ f--- j|—£ij cr » E >» >► —

•

|59|2

257|t|

b) D ebussy, F&tes galantes, “En sourdine" mm. 39-43 Reveusement lent

en se perdant

39 *

J J «brJ" J B

pp

f—

— r

JB

r r

F|6 |3 9 |8 |m i

|S3|97 /A2

|t|427 /A5

|7|49

n

from Ravel’s Le tombeau de Couperin', it features a |59|-anchor structure in the last chord, which gives the chord a tonic-with-added-sixth effect even in the unconventional cadence echo pattern. (That G is tonic has been established in part by the pentatonic |t|anchored dominant in m. 7.) Figure 2.15b, from Debussy’s “En sourdine” from his Fetes galantes, features two second-order anchor-structure chords. The |6|39|8| / tt l, chord of

96

m. 39, standing in for an Ftt-rooted dominant seventh, is used to create “dominant-totonic” drive to the bass’ scale degree 2 in m. 40.22 That 2 is used to support the |53|97 chord that sets up its own more-stable version, the |t|427 /

5

dominant that cadences into

m. 42. It is possible for several anchor structures to exist over a bass note, and therefore for a single verticality to possess several tonal plausibilities.

Which plausibility is

actualized depends of course on context, but there are two rules-of-thumb for anchor structure interaction. The first is that tonal plausibilities created by first-order anchor structures, by dint of their relative acoustic strength and the clarity of their commonpractice connotations, generally take precedence over those created by second-order anchor structures.

Figure 2.16 demonstrates this point using the chord from

“Laideronette, imperatrice des pagodes”; chord (a) in Figure 2.16 could serve as a |25 |

F igure 2.16. Interaction betw een anchor structures of different orders. (a)

(b)

|25|

257|t|

second-order anchor structure . . .

. . . potentially "overriden" by first-order anchor structure

anchor structure for a verticality that behaves as a minor seventh chord in third inversion. Adding the rest of the pitches in (b), though, creates a |t| first-order anchor structure that is more acoustically robust and functionally suggestive; the resultant tonal plausibility -

22 This analysis incorporates the F#5 into the anchor structure because o f its registral and rhetorical prominence; an alternate reading for the chord in m. 39 would be a |63|98 chord, which fulfills the same functional role as a |6|39|8|.

97

that the chord could serve as a Git-rooted seventh chord - is more prominent, and is the plausibility actualized in the Ravel passage. An actualization of a |25| second-order anchor structure is shown in Figure 2.17, from Stravinsky’s L ’histoire du soldat; the

9|52| chord stands in for a leading-tone seventh chord of C at the cadence.

F igure 2.17. Second-order anchor structure cad en ce in Stravinsky, L’histoire du soldat, p. 6. \k %

■j:

s— i n , »r | 7 p U - ... — If*

.. h..7.... >....."TI

» t ! £ _ f - -i---- imh =

L*

h-------1------

17 49

9|52|

The other rule-of-thumb is that, when two anchor structures of the same order are present, the one voiced lower in the verticality generally takes precedence. Figure 2.18 demonstrates. In chord (a), the second-order |68| anchor structure (formed by the E3,

F igure 2.18. Interaction of anchor structures of th e sam e order; the anchor structure closer to the b a ss generally tak es precedence. unlikely hearing/usage

....^ 8 ------ ^ “ -O* -4► - bo |

likely hearing/usage

t = \ E & jB — */& ---------- o : -----—

--

(a)

(b)

(c)

|368|92 / 7 (in F)

2|6|93|8| 7 (in F)

|269|3S / 4 (in B)

Bb3, and C4) is low in the texture, and the chord plausibly serves as a dominant of F. In

98

chord (b), that |68 | structure is spread out in the texture, and so that usage is less likely; the available |26 | second-order anchor structure is now lower in the texture, as shown in chord (c), making the chord more plausible as a dominant of B rather than of F. There is no conflict between anchor structures of the same order that reference the same commonpractice progenitor; the available |26 | and |29 | second-order anchor structures in chord (c) co-exist peacefully. Also, there is no conflict between anchor interval |7| and other first-order anchor intervals, since the fifth is a component of the common-practice seventh chords that the |t|- and |e|-anchored chords are standing in for (Figure 2.19).

F igure 2.19. First-order anchor structures not disturbed by th e p resen ce of |7|.

ln im . I n in A

J f S

A

|«|29

I lS :

7|*|29

2.3 - Chordability The anchor-structure apparatus explains how certain skeletal voicings allow additive chords to stand in for common-practice chord types in tonal contexts. However, not all verticalities with anchor structures are equally likely to be found in the Parisian modernist repertoire’s tonal passages - the verticalities of Figure 2.20, for instance, are unlikely to find tonal use in the repertoire despite having first-order anchor structures above their bass notes. We see then, that there are additional elements of chord structure that shape the additive harmonic surfaces of this repertoire - constraints on what adornments of anchor structures are possible.

99

Figure 2.20. Unlikely anchor-structure voicings. tl£X

TT

2468| t| 5

123456|7|

There are three such constraints that generally hold true in this repertoire. Together, they preserve the qualia of chordability, the capacity for a verticality to be perceived of as an integrated whole, and not merely as a simultaneous collection of pitches that might be more readily understood as a tone cluster (e.g. Fig. 2.20a), as the coincidence of diassociated pitch strata (e.g. Figure 2.20b), or as some other type of non­ integrated verticality.23 (Non-chord verticalities will be discussed in section 2.4.) The chordability of novel sonorities is vital in maintaining additive harmonic passages’ connection to common-practice contexts. Conventional tonal progression is built with chords (i.e. auditory objects wherein multiple tones are perceived as an integral Gestalt), and so those sonorities that insert themselves into and participate in those structures of chord succession must also be chords; the majority of novel harmonies in the Parisian modernist repertoire are chordable verticalities. The three constraints on voicing in this repertoire that preserve chordability are (1) there are no ro-interval Is in a chordable voicing, save for above the bass note of first-order anchor structures; (2) there are no consecutive whole-step adjacencies; and (3) there is no more than an octave between 23 Simon Harris has formulated a similar definition o f chord, and notes its necessary malleability: “ [a chord is] a collection o f notes that seems to form a unit o f simultaneous sounds - with all the difficulties and possible disagreements inherent in a such a subjective definition.” Simon Harris, A Proposed Classification o f Chords in Early Twentieth-Century Music (New York: Garland, 1990): 87.

100

upper pitches (there may be more than an octave between the bass note and the secondlowest pitch). These constraints will now be discussed in turn.

(1) The RO-Interval 1 Constraint The only ro-interval whose simple presence undermines the qualia of chordability is ro-interval 1. The harshness of ro-1 is a flashpoint in any sonority and, rather than fostering the cohesion between tones that defines chordability, introduces acoustic roughness that legislates against hearing a sonorous composite chord. For example, the biting effect of mm. 130-134 of Ravel’s “Alborada del gracioso” from Miroirs (Figure 2.21), with two adjacent instances of ro-interval 1 (Ejt3, F#3, G3), is far removed from the chordal

blend of Figure

2.19;

the

verticality

in “Alborada

del

gracioso”

F ig u re 2.21. Ravel, Miroirs, “Alborada del gracioso” mm. 130-134.

HP

Gardez la Ped. jusqu ■ a

♦

is not a chord that participates in chord progression, but a tone cluster which reboots the harmonic palette between two of the prelude’s themes. The only common occurrence of ro-interval 1 in chordable sonorities is as a minor 9th above the bass of chord with a first-order anchor structure (e.g. Figure 2.22, from Ravel’s Ma mere I ’oye). The root-sensation-producing acoustic robustness of those sonorities can accommodate the harsh dissonance of ro-1, provided it is not present as a

101

direct half-step adjacency to the bass note (which would cloud the acoustic salience of the

bass). F igure 2.22. Ravel, “P av an e d e la Belle au bois dorm ant” from M a mere /’Oye; first-order anchor-structure chord supporting ro-interval 1.

H 12A 12

t r i - i j~|Tf i [ in | { 1|t|3 / 5

11*|3 / 5

1|t|3 / 5

1|7|3 / 5

This stylistic practice of avoiding ro-interval 1 when building chords has roots in physiology and cognition. For one, human physiology is such that the half-step is always acoustically rough: the critical band (the frequency range within which two signals compete for the same receptor cells on the basilar membrane), although it varies in size depending on position in the spectrum, is never smaller than a major second.24 That acoustic roughness contributes to the half-step consistently being ranked as the most dissonant equal-tempered interval in listener-opinion studies; in an index produced by David Huron that collates the results obtained by asking listeners to rate the relative stability of intervals, the minor second ranks as three times more unstable than the next closest interval.25 (The same study revealed that listeners do not give the equivalent

24 Keith Mashinter, “Calculating Sensory Dissonance: Some Discrepancies Arising from Models o f Kameoka & Kuriyagawa, and Hutchinson & Knopoff,” Empirical Musicology Review 1, No. 2 (2006): 66 and McGowan, 117. 25 David Huron, “Interval-class content in equally-tempered pitch-class sets: Common scales exhibit optimum tonal consonance.” Music Perception 11, No. 3 (1994): 290-293.

102

stability ratings to inversionally related intervals, a finding that buttresses this model’s "yft focus upon ro-intervals rather than interval classes. ) The acoustic roughness of ro-interval 1 means it features prominently in listener engagement with chords; Arthur Samplaski’s 2004 dissertation, which used listener experiments to evaluate atonal models of chord similarity, determined that the presence/absence of a ro-interval 1 was the single most important element in listener judgments about chord similarity.27 Given that perceptual salience, ro-interval 1 forms a significant obstacle to chordal stability. In a 2006 study, Norman Cook and Takashi Fujisawa asked subjects to rate the stability of all possible trichords, with each trichord presented in a variety of voicings. The two elements found to be responsible for low stability ratings in the experimental results were a) the presence of a half-step and b) two intervals of equal size; the presence of a half-step, however, was the far more significant marker.

Further studies by Keith Mashinter and Joos Vos corroborate Cook and

Fujisawa’s findings;

this line of research supports the present theory that the presence

of ro-interval 1 in a verticality undermines chordability. While it would not suffice to cherry-pick a few examples of additive chords that do not feature ro-interval, the relevance of the ro-interval-1 constraint for the Parisian modernist repertoire can be corroborated in a few other ways. One is to note that the constraint obtains even in those chords highlighted by contemporary theorists as being particularly dissonant. Figure 2.23 shows the opening bars of the first of Ravel’s Vaises

26 Huron, “ Interval-class content in equally-tempered pitch-class sets,” 293. 27 Samplaski, 217. 28 Norman Cook and Takashi X. Fujisawa, “The Psychphysics o f Harmony Perception: Harmony is a Three-Tone Phenomenon,” Empirical Musicology Review 1, No. 2 (2006): 112-113. 29 M ash in ter, 6 5 -8 4 and Joos Vos, “Commentary on ‘Calculating Sensory Dissonance’: Some Discrepancies Arising from Models o f Kameoka & Kuriyagawa, and Hutchinson & K nopoff by Keith Mashinter,” Empirical Musicology Review 1, no. 3 (2006): 180-181.

103

nobles et sentimentales, a passage discussed Roland-Manuel, who writes the passage that “all discords are left bare; their harsh gaiety will never be surpassed.”30 Yet while the F igure 2.23. Ravel Vaises nobles et sentimentales i, mm. 1-2

4 = nhw ...

r »—

i f . . —0----------- L -p— p-

:::

j

passage is highly dissonant, ro-interval 1 is restricted to appearances above the bass note with a first-order anchor structure present; given that ro-interval 1 is the most dissonant interval, this suggests that incorporating ro-interval 1 into the upper portions of chords was a line rarely crossed even in composers’ most sonically adventurous moments. Another piece of evidence in support of the ro-interval 1 constraint is the fact that prescriptions against minor ninths appear in jazz theory.

There is well-documented

cross-pollination between jazz and the Parisian modernist repertoire, and much of jazz theory concerns the practical (rather than speculative or analytical) codification of additive chords.31 Often the prescription against ro-interval 1 is not explicitly stated in

30 Roland-Manuel, Maurice Ravel (London: Dennis Dobson Limited, 1947): 116. 31 Links between jazz and the Parisian modernist repertoire include the popularity o f the ragtime craze in Europe from the 1890s onwards (leading to pieces such as Debussy’s “Golliwog’s Cakewalk” or the ragtime for the Wedgewood Mug in Ravel’s L 'enfant et les sortileges), the European tours o f Will Marion Cook’s Southern Syncopated Orchestra and the Paul Whiteman Orchestra, Ravel’s 1927 trip to America, and the fascination o f jazz composers such as Bix Beiderbecke and Hoagy Carmichel with recordings o f French “ Impressionist” music. For detailed discussion, see Mervyn Cooke, “Jazz among the classics, and the case o f Duke Ellington” in The Cambridge Companion to Jazz, edited by Mervyn Cooke & David Horn (Cambridge: Cambridge University Press, 2002): 153-173; David A. Franklin, “A Preliminary Study o f the Acceptance o f Jazz by French Music Critics in the 1920s and Early 1930s,” Annual Review o f Jazz Studies 4 (1988): 1-8; Frank Murphy, “ Bix Beiderbecke: Composer for the Pianoforte,” Jazzforschung/Jazz

104

jazz theory, but can be inferred from lists of allowed and disallowed chords.

For

instance, in Henry Martin’s discussion of notes than can be integrated into tonic triads, two of the scale degrees that he names as unable to become tonic chord tones are

and b

4, i

a

6 - exactly those tones that are at ro-interval 1 above the chordal third and fifth.

^

7

can be a potential chord tone (which emphasizes the distinction between ro-interval 11 and ro-interval 1), as can b 2 because, if it is not placed directly adjacent to the tonic chord’s root in the bass, to ro-interval 1 formed by b 2 would lie above the bass note of a chord with a first-order anchor structure. Implicit avoidance of half-steps also guides Mark Levine’s discussion of “upper structure chords” in The Jazz Piano Book. An upper-structure chord is formed by taking the tritone of a dominant seventh chord and then superimposing a triad atop it; the chords are labeled by the relationship of the upper triad to the (absent) root of the tritone’s dominant seventh (an upper-structure II chord is shown in Figure 2.24a; the absent root is F ig u re 2.24. Jazz voicings from Levine’s The Jazz Piano Book; (a) is an upper-structure II chord, (b) is a Phrygian chord. r r ... ...... .. .............. i f --------- W ----------~....... ¥ bn

(a)

shown as a filled-in note head).

-

(b)

Levine lists nine possible upper-structure chords;

although Levine never explicitly states so, all the upper triads left off of that list

( b ll,

ii,

Research 21 (1989): 71-81; and Tymoczko, “The Consecutive-Semitone Constraint on Scalar Structure: A Link between Impressionism and Jazz,” 135-179. 32 Martin, “From Classical Dissonance to Jazz Consonance: The Added Sixth Chord,” 43.

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iii, III, vii, VII, bvii, bVII) are exactly those that create ro-interval 1 above either pitch of the foundational tritone, whereas those upper triads that create ro-interval 11 are permitted.33 Those voicings in Levine’s book that do include ro-interval 1, such as the Phrygian chord shown as Figure 2.24b, generally do so in root-position chords (i.e. chords with first-order anchor structures). The case against the minor ninth in jazz theory is explicitly made by Adam Ricci, who suggests that a minor ninth is undesirable because it presents the upper of its two pitches as the root and thereby confuses the rootedness of a tertian verticality; he then demonstrates that the only tertian chords common to jazz harmonic practice which feature the minor ninth do so above their root.34 James John McGowan finds that the destabilizing effect of the minor ninth explains the avoidance in jazz practice of bl3ths over major triads and b7ths in the add6 “dialect” of consonance (i.e. contexts where a tonic triad with added sixth is the baseline level of consonance).35 Both McGowan and Levine also note the frequency of half-step dissonances in the music of Thelonious Monk; that Monk’s distinct and dissonant music is marked as a special case emphasizes the avoidance of ro-interval 1 in most jazz harmony.36 Given the close relationship between jazz harmony and the additive harmonies of the Parisian modernist repertoire, the importance of the minor ninth the former suggests ro-interval 1 might be of similar importance in the latter.

33 Mark Levine, The Jazz Piano Book (Petaluma, CA: Sher Music, 1995): 116-119. 34 Adam Ricci, “W hat’s Wrong with the Minor Ninth?”, written copy o f a talk delivered at the annual Society o f Music Theory Conference in Madison WI, 2003 (distributed in the course Dissertation Reading and Research, Yale University, Fall 2010). 35 McGowan, 107. 36 Levine, 147 and McGowan, 107.

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One final way of corroborating the importance of the ro-interval 1 constraint is to note that when ro-interval 1 does appear, its use is often a marked event deployed to specific expressive ends.37 A small-scale example is found in Ravel’s Pavane pour une infante defunte. In mm. 17-19 (Figure 2.25a) the piece, having started in G major and next moved through B minor, cadences authentically in D. The melody from mm. 13-19 F igure 2.25. Ravel, Pavane pour une infante dPfunte\ u se of ro-interval 1 to destabilze a previously stable cadence. a): mm. 17-19

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is then repeated in mm. 20-26 - with the upper stave’s notes remaining the same and the bass line shifting slightly to more strongly emphasize D major (there is A-to-D motion in m. 23) - and sets up to repeat mm. 17-19’s cadence in mm. 24-26. Measure 25 and the

37 The concept o f markedness generating expressive meaning comes from Robert Hatten, Musical Meaning in Beethoven: Markedness, Correlation, and Interpretation, (Bloomington, IN: Indiana University Press, 2004).

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first two beats of measure 26 are identical to mm. 18-19 save for the insertion of C#4 into the chords marked with arrows (Figure 2.25b); the larger arrows indicate the places where the C#4 creates ro-interval 1 against D4. The unsettling effect of these new rointerval 1 dissonances destabilizes the cadence in m. 26; with that moment of repose having been compromised by the ro-interval 1 dissonances, the cadential melody is then repeated in order to head off in another tonal direction, arriving at a half cadence in G in m. 27 that sets up the return of the main theme in m. 28. A larger-scale marked use of ro-interval 1 is found in Chabrier’s song “Les Cigales.” Here, the presence/absence of ro-interval 1 is central in the song’s dramatic progression. The settings of the poem’s verses features heavy use of ro-interval 1 between E5 and F5 in mm. 1-5 and 7, between F#5 and G5 in mm. 13-16, and over the bass voice in mm. 16-17 (Figure 2.24a, pp. 109-110).

These unusual and harshly

dissonant chords can be read as mimicking the “rauques ululees” of the titular insects that so aggravate the poet.38 When the poet’s position on the insects softens in the refrain, and rather than characterizing the insect’s sounds as “raucous hooting” he/she says that “Les cigales, ces bestioles / ont plus d'ame que les violes / les cigales, les cigalons / chantent mieux que les violons!”, ro-interval 1 evaporates (the first refrain is shown in Figure 2.24b, pp. 111-112). There are no ro-interval Is in the refrain; by the end of the refrain (mm. 32-35), the A-Ctt-E-F chord with ro-interval-1 chord of mm. 1-4 has softened into an A-Ctt-E-Fn chord with ro-interval-2. The use of ro-interval 1 is marked, then, in that it represents not only the sound of the insects, but the poet’s frame of mind when the poet is able to let go of his/her annoyance and view the insects with affection, 38 That these chords were unusual for this repertoire can be seen in the fact Chabrier’s publisher requested that the song’s dissonances to be removed so as to make the song more palatable and marketable to amateur musicians. (Foreword to the Dover Edition, p. ix)

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F igure 2.26. Marked u se of ro-interval 1 in C habrier’s “Les Cigales. »

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the harsh ro-interval-1 chords of the verse are replaced with more pleasurably consonant sonorities. The ro-interval 1 constraint, when combined with the rules for anchor-structure interaction, helps explain why second-order anchor structures are limited in use. The |35| anchor structure in Figure 2.27 can only support one additive-harmonic adornment, A. Of the other 8 potential adorning pitches, three would produce first-order anchor F igure 2.27. Scarcity of adornm ent options for secondorder anchor structures.

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-

pitches that could produce com peting first-order plausibilities: 0 , F, Ft pitches that would produce ro-interval 1: G#, B, C# pitches that would merely com plete a com m on-practice chord: E, Eb

structures above the bass note that could override the second-order tonal plausibility, three more are “off-limits” because they would create harsh ro-interval Is, and the remaining two are relatively uninteresting from an additive harmonic standpoint because they would complete common-practice chords. This demonstrates how there are limited ways of presenting novel inverted chords in additive harmonic environments, and helps explain the popularity of root-position-heavy textures in the Parisian modernist repertoire (something that is true of jazz harmonic practice as well). Figure 2.28, from the ending of the first movement of Ravel’s string quartet, is an example of this type of texture; the unequivocal grounding of the cello part provides stability for the upper-part adornments.

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Figure 2.28. Ravel, String Quartet, end of 1st movement.

The relevance of the ro-interval 1 constraint for describing the types of additive chords commonly found in Parisian modernist repertoire has been supported by research into psychoacoustics, by the importance of similar constraints in jazz theory, and by the marked nature of ro-interval l ’s use in the repertoire.

The fact that the acoustic

roughness of ro-interval 1 plays a central role in this model of additive harmony, though, does not mean that chordability is a simple issue of composite consonance and dissonance. For one, it is impossible to formulate an absolute scale of consonance-todissonance for multi-note sonorities.39 It is impossible to separate aesthetic from psychophysical judgments in listener responses about a chord’s consonance level; one example of this effect is that listeners rank the dyad C-B as more “dissonant” than the

39 David Huron, “Consonance and Dissonance - The Main Theories,” http:, w w w .in u s ic -c o a .o h io sta te .e d u -M u sic82 9B m ain.theo ries.htm l (accessed February 23, 2009).

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more frequently heard and “prettier” yet inherently acoustically rougher C-E-G-B tetrad.40

The level of listener experience also plays a role in judgments about

consonance/dissonance; David Huron has shown that trained musicians rate perfect fourths, fifths, and octaves as the most consonant intervals, while non-musicians rate thirds and sixths as more consonant41 Lastly, while an absolute scale of the consonance of simple dyads is attainable, chordal consonance/dissonance is not a simple summation of the dissonance of a chord’s composite dyads 42 The research of both Cook and Fujisawa and Andrzej Rakowski has demonstrated that listener judgments about the stability of multi-note sonorities are not correlated with those sonorities’ summed intervallic dissonance; this again suggests that a chordability is a distinct qualia, and not a simple aggregate of constituent parts.43 So while acoustic factors do play a role in this model by buttressing the prescription against ro-interval 1, they are not the end-all of a system of additive harmony. The relationship between acoustic factors and musical practice is elegantly encapsulated by Olli Vaisala: “While it is clear that functional norms cannot be deduced from perceptual properties . . . neither are perceptions irrelevant to functional norms.”44

40 Mashinter, 66. 41 David Huron, “Consonance and Dissonance - The Effect o f Culture,” http: w w vv.m usic-cou.ohiostate.edu M usic 8 2 9 B c ulture.htm l (accessed February 23, 2009). 42 David Huron, “ Interval-class content in equally-tempered pitch-class sets,” 293. 43 Cook and Fujisawa write: “Most significantly, the fact that some chords sound stable, final and resolved, while others sound unstable, tense and unresolved cannot be explained solely on the basis o f the summation o f interval dissonance among tones and their upper partials.” Cook and Fujisawa, 117; and Andrzej Rakowski, “Chordal harmoniousness is determined by two distinct factors: interval dissonance and chordal tension,” International Musicological Society, Report o f the Eleventh Congress (Copenhagen, 1972): 597603. 44 Vaisala, “Concepts o f Harmony and Prolongation in Schoenberg’s Op. 19/2,” 234.

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The Adjacent-Whole-Tone Constraint Another general constraint on voicing that preserves the qualia of chordability is the adjacent-whole-tone constraint. This states that, while whole-step adjacencies abound in additive harmonic textures, chordable verticalities rarely have consecutive adjacent whole steps.

This constraint maintains a distinction between chord and scale;

consecutive whole steps do not present a “chord-able” qualia, and are instead most likely heard as a tone cluster or as the verticalization of a scale segment. Joel Lester, Joseph Straus, and Oily Vaisala, have all discussed how maintaining a distinction between stepwise motion and arpeggiation, and therefore between voiceleading and harmony, is necessary for prolongation.45 It is an essential condition for prolongation because, in its absence, “it is virtually impossible to determine the voiceleading function of the melodic motions”;46 one can no longer distinguish between motion within a prevailing harmony and motion away from that harmony. In common-practice music, the line between harmonic interval and voice-leading intervals is between three half-steps and two half-steps, meaning that the triad enjoys privileged status as “the maximal subset of the diatonic collection consisting entirely of non-adjacent elements.”47 Vaisala has suggested, however, that the definition of step and leap can be flexible and contextually defined; in an analysis of Schoenberg’s Kleines

45 Joel Lester, “A Theory o f Atonal Prolongations as Used in an Analysis o f the Serenade, Op. 24 by Arnold Schoenberg”, Ph.D. dissertation, Princeton University (1970); Joseph Straus “The Problem o f Prolongation in Post-Tonal Music,” Journal o f Music Theory 31, no. 1 (Spring, 1987): 1-21; Vaisala, “Concepts o f Harmony and Prolongation in Schoenberg’s Op. 19/2,” 230-259. 46 Straus, “The Problem o f Prolongation in Post-Tonal Music,” 3. 47 Straus, 5.

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Klavierstuck op. 19/2, he allows context to shift the line, making ro-interval 2 a harmonic interval and leaving ro-interval 1 as the only voice-leading interval.48 The line necessarily shifts in the Parisian modernist repertoire as well, because whole-tone adjacencies are commonplace and so must be viewed as valid chordal intervals (i.e. not just as the byproducts of octave replication or inversion). One telltale usage that supports this new status for the major-second adjacency is the use of major chords with added major sixth as the final tonic of a piece, as in the last two movements of Debussy’s La Mer (Figure 2.29a and 2.29b) or in “Les Cigales” (Figure 2.29c). These chords are not in inversion - the heard tonic and root pitches here are the bass notes (an impression further reinforced in Figure 2.29b by the falling-fifth approach in the bass). But nor is the major-second adjacency a dissonance that requires resolution - these chords are the contextually stable final tonics of their respective pieces. A distinction between chord and scale still persists in the Parisian modernist repertoire - in contrasting this repertoire to the saturated chromaticism of the Germanic school (Wagner, Strauss, and early Schoenberg), Dmitri Tymoczko writes that “composers like Rimsky-Korsakov, Debussy, and Ravel preserved a more conventional understanding of the relation between chord and scale, but with a significantly expanded musical vocabulary.”49 The point of division has simply been shifted to allow for the verticalization of single scale-step adjacencies. There is still one pitch interval reserved exclusively for step motions (pitch-interval 1; the ro-interval 1 constraint explains why this pitch-interval does not feature in chord construction). And while the verticalization

48 “The borderline between the small, ‘stepwise,’ and the larger intervals is, naturally, flexible and may vary according to the context.” (Vaisala, “Concepts o f Harmony and Prolongation in Schoenberg’s Op. 19/2,” 236.) 49 Tymoczko, “Scale Networks in Debussy,” 220.

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F igure 2.29. W hole-step adjacencies in final tonics in the Parisian m odernist repertoire. a) reduction of D ebussy, La merit, final th ree bars.

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118

of one scale-step adjacency is now permitted (pitch-interval 2), the verticalization of consecutive scale steps is not.50 Adjacent whole tones are infrequent because verticalized scale steps produce a qualia that is too scale-like to be readily used in additive harmonic contexts; maintaining a distinction between chord and scale is vital not only for prolongation but, more pertinently for this study, in maintaining connections between additive harmonies and common-practice chord types. As with ro-interval 1, the use of consecutive vertical whole tones is often a marked event. One example is in the opening of Debussy’s “Les collines d’Anacapri” from the first book of Preludes (Figure 2.30). The piece opens by sustaining the verticality B3-Ctt4-E4-F#4-G#4-B4, which contains the consecutive whole steps E4-F#4-Gtt4. This sonority does not serve as additive-harmonic stand-in for a

F igure 2.30. D ebussy, “Les collines d’Anacapri,” mm. 1-8; u se of verticalized consecutive w hole-step adjacencies a s a marked event. Vif (d' = 184) Tres mndere

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common-practice chord, though, but rather as an explicitly scalar gesture - David Kopp

50 This constraint need only legislate against consecutive whole steps, since the combination o f a whole step and a h alf step or o f two h alf steps is covered by the ro-interval 1 constraint; furthermore, consecutive semitones are not found in any o f the common scales o f the Parisian modernist repertoire. (Tymoczko, “The Consecutive-Semitone Constraint on Scalar Structure,” 142.)

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has detailed how these measures lay out the pentatonic collection that controls the first and last sections of the tripartite piece.51 In another of the Debussy preludes, “Feuilles mortes,” when adjacent whole tones coalesce in the right-hand in m. 6 (Figure 2.31), they do not form an additive chord that participates in chord progression.

Instead, the sonority is treated as a passive

object/oddity, first inverted about C#/G so as to preserve its adjacaent-whole-tone set (FttB-O-D# in m. 7

B-C#-Dtt-Gtt in m. 8), and then planed through a shifting scalar space

in mm. 8-10. This marked treatment of an adjacent-whole-tone verticality again speaks to its general novelty in the repertoire. F igure 2.31. Debussy, "Feuilles m ortes,” mm. 6-10; adjacent-w hole-tone sonority treated a s a passive musical object. verticalization o f adjacent scale steps

I------------------------------------------------------------------------ 1

This model’s accommodation of ro-interval-2 adjacencies in chordable verticalities means that it differs from the definition of chordability presented by Daniel Harrison in his manuscript “On the Chordability of Pitch-Class Sets.” This divergence in the definition of a chordable voicing is attributable to differences in the two models’ aims. Harrison is investigating which pitch-class sets can be voiced in a “chordable”

51 Kopp, “Pentatonic Organization in Two Piano Pieces by Debussy,” 268

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manner (i.e. producing the qualia of “chordness”), and defines three types: T-chordability with either pitch-interval 3 or 4 between voiced pitches, A-chordability allowing pitchintervals 3, 4, or 5, and B-chordability allowing A-chordable voicings that also have at least one pitch-interval 6 between adjacent tones.52 He lists which pitch-class sets are chordable and how, and also presents observations about potential symmetries in chord realizations. Harrison’s definition, then, is designed not to parse musical surfaces but to examine the properties of abstract pitch-class sets. Taken from its intended environment, it is only an imperfect fit for actual musical surfaces, as the use of chords with pitchinterval-2 adjacencies demonstrates. My definition is expressly designed to deal directly with music surfaces, and so better performs that task.53

The Proximate-Upper-Voices Constraint The last constraint that preserves the qualia of chordability is that the upper voices of a chordable verticality generally are voiced within an octave of one another. This condition also obtains in common-practice chordal writing, and has a partial basis in acoustics. Work by Reinier Plomp and Willem Levelt, later revisited and reworked by first Akio Kameoka and Mamora Kuriyagawa and then David Huron and Peter Sellmer, has shown that conventional four-part voicing generally produces partials “whose

52 Daniel Harrison, “On the Chordability o f Pitch-Class Sets,” unpublished draft o f August 21, 2009 (obtained in personal correspondence): 1-18. 53 One sees that there are Harrison-chordable voicings that are not particularly tonally plausible (e.g. the Bchordable 0-3-9-1’), and there are also common chords that are either not Harrison-chordable (e.g. 0-4-B2 ’), or most commonly voiced in non-Harrison-chordable ways (e.g. 0-7-9-4’). One might argue that in the last case, the chordability o f the chord’s interval vector renders any voicing sufficiently mellow; this would run up against experimental data such as that in Samplaski’s dissertation that shows voicing is o f greater perceptual import in judgm ents about chord similarity than is pitch-class content. An intuitive version o f this response would be to show that not all instantiations o f a set-class are created equal: o f the B-chordable versions o f 4-15z, 0-4-A-3 is a common chord (C7#9), while 0-5-B-3 is not.

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distribution would be homogenous with regard to critical bands”;54 this means that chords are commonly voiced with a relatively large gap between the bass note and the next highest tone and smaller gaps between upper voices. There are also cognitive reasons as to why this constraint promotes chordability - Yeary has discussed how sonorities with tones separated by large distances in frequency may encourage the segregation of a single sonority into multiple steams/perceptual objects.55 This constraint, then, serves as another connection to the common-practice chord types that inform the construction of additive harmonies in this repertoire. This condition is the most flexible of those listed here; as in common-practice writing, upper voices will temporarily move further than an octave from each other before returning soon after. Taken together, the three chordability constraints have a powerful reductive effect on the number of potential additive chords.

Figure 2.32 displays the results of an

investigation of all possible verticalities spanning two octaves plus a major third (a 28semitone-span, a semi-arbitrary ergonomic constraint devised to represent the largest possible close-position piano chord playable with average-sized hands).

In order to

process the vast number of possible verticalities (there are 228 = 268,435,456 possible verticalities with in a 28-semitone span), I wrote a computer program that produces and then filters all possible 29-bit-long bit-strings (the bit strings in my program are 29 bits long to cover that 28-semitone span; the program is attached as Appendix 1). The bit string represents chromatic space; a 1 is a pitch in the chromatic slot, 0 the absence of a pitch; the first bit of the string is always a 1, representing the bass note of a verticality. The program first parses out all verticalities with fewer than four notes, in order to focus 54 David Huron and Peter Sellner, “Critical bands and the spelling o f vertical sonorities,” Music Perception 10. No. 2(1992): 130. 55 Yeary, 103.

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on additive chords. Of the 134,217,700 remaining verticalities, only a small fraction six hundredths of a percent - fulfill all three constraints; this demonstrates that voicing information need not be an add-on to theories of chord structure, but can instead be a point of departure. Of further note is that a large majority of “chordable” voicings possess first-order anchor structures; this demonstrates how first-order anchor intervals can support a range of colorations without running afoul of chordability restrictions, and attests to the variety of voicings that can be harnessed to tonal progressions when using those skeletal elements. F igure 2.32. Reductive power of the model’s three voicing constraints. For a verticality spanning two octaves and a major third, there are 268,435,456 possible pitch combinations. -

Verticalities of m ore than 3 pitches . . . : 134,217,700 . . . and no ro-interval 1s : 153,996 . . . and no adjacent whole ste p s . . . : 83,178 . . . and no g ap s > octave betw een upper voices: 80,136 . . . and have a first-order anchor structure: 71,632

2.4 - Additive Chords and Non-Chord Verticalities The anchor structure model holds that, in the Parisian modernist repertoire, those novel verticalities that participate in tonal chord progression - i.e. additive chords - are (1) tonally plausible and (2) chordable. This does not mean, though, that every novel verticality in this repertoire is an additive chord. There are two broad categories of non­ chord verticalities in the repertoire (and two categories of chords; see Figure 2.33): those non-chord verticalities that ornament chordal structure (i.e. verticalities created by ornamenting a chord progression with non-chord-elements) and those non-chord

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verticalities that exist apart from those chordal structures (e.g. clusters, polychords, and the like). This section discusses these two categories in turn.

F ig u re 2.33. Types of verticalities in the Parisian m odernist repertoire.

C hords - com m on-practice chords - additive chords (chordable verticalities built around anchor structures) Non-Chord Verticalities - non-chord verticalities that ornam ent chordal structure - non-chord verticalities that exist ap arat from chordal structures o clusters o verticalized scales o disassociated sim ultaneous strata o polychords

Non-Chord Verticalities that Ornament Chordal Structure The increased dissonance and relative density of additive harmonic textures means that non-chord tones are more context-dependent in the Parisian modernist repertoire than in common-practice contexts.56 In common-practice contexts, isolating non-chord ornamentation is relatively a simple process - any tone that does not belong to a triad or a seventh chord is a non-chord tone. In additive harmonic textures, though, many more notes can be part of a chord; indeed, one of the strengths of the anchorstructure model is how it allows for multiple adornments of a single basic tonal function.

56 This phenomenon is akin to how the sharp dividing line between chord intervals and voice-leading intervals is also necessarily more context-dependent in additive-harmonic textures than in common-practice contexts, as was highlighted in the discussion o f the adjacent-whole-tone chordability constraint in section 2.3

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This does not mean, however, that non-chord ornamentation of chord structures is no longer viable. There are several factors that can contribute to the perception of non­ chord ornamentation in this repertoire. One is the use of non-chordable verticalities; since such verticalities resist integrated hearings as chords, they suggest alternate interpretations, and, in the midst of a chordal passage, interpreting a brief non-chord verticality as the coincidence of non-chord-elements (i.e. nonessential dissonance) with a chordal background is a likely hearing. Figure 2.34, mm. 56-61 from the first of Ravel’s Vaises nobles et sentimentales, is one passage where such a hearing is likely. The F igure 2.34. Ravel, Vaises nobles et sentimentales, mm. 56-61; non-chordable verticalities interpreted a s the coincidence of a chordal background and non-essential dissonance.

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highlighted chords have a non-chordable ro-interval 1 between the outer pitches of their right-hand notes. These chords are not outside of the chord progression as a cluster or polychord might be - they participate in the passage’s chromatic circle-of-fifths sequence of |t|-anchored chords - and the consistent voice-leading of the upper part means that these chords are interpreted as the coincidence of the circle-of-fifths sequence with the upper voice’s melodic process (i.e. the chromatic ascent from A4 to G5). Another element that can contribute to hearing non-chord tones in additive harmonic textures is gestural reference to common-practice ornamentation. Commonpractice non-chord tones are defined by certain types of characteristic melodic motion; 125

gestures that emulate that motion in additive harmonic textures, even in the absence of a sharp dividing line between consonant chord intervals and dissonant ornamenting intervals, suggest the non-chord ornamentation of a chordal background structure. Dallin goes as far as to claim that this melodic mimicry is the only way non-chord tones can be perceived in dissonant textures: “No clear distinction is possible when every conceivable dissonance is permissible within the harmony.

Notes are perceived as being

nonharmonic in modem music only when their resemblance to one of the familiar categories [or nonharmonic tones] is obvious.”57 These gestures must be present at foreground levels of musical structure - the density of additive harmonic textures means one could potentially find a stepwise resolution at some level of background structure for almost any surface pitch, but this study aims to capture the novelty and vitality of this repertoire’s musical surfaces. The last element that helps project non-chord ornamentation in additive harmonic textures is contextual stability conditions, which allow for dissonant additive chords to serve as stable sites of resolution. In his paper “Analyzing Tonal Embellishment in PostTonal Music” Peter Silberman defines a set of contextual stability conditions for posttonal music that can be directly retrofitted for the Parisian modernist repertoire. Silberman defines “tonal embellishment” as the stepwise resolution of a pitch/pitch-class from a less stable context into to a more stable one,58 and gives four conditions that project a chord as being contextually stable: a chord A is more stable than chord B if chord A is either (1) more consonant/less dissonant than chord B; (2) more frequently 57 Dallin, 146. 58 Peter Silberman, “Analyzing Tonal Embellishment in Post-Tonal Music,” written copy o f a talk delivered at the annual West Coast Conference o f Music Theory and Analysis in Salt Lake City, 2003 (distributed in the course Contemporary Tonality, Yale University, Fall 2007), 10. Silberman allows for intervals larger than ic2 to be considered steps, should a passage feature consistent use a gapped collection such as the pentatonic scale.

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found at the beginning and/or end of formal units; (3) more salient in a domain other than consonance/dissonance and formal position (e.g. non-pitch characteristics, or through parallelism with other stable events); or (4) suggestive of a common-practice chord that would be stable in a common-practice context (the “tonal mimicry” condition).59 These conditions are all valid in the Parisian modernist repertoire, and the “tonal mimicry” condition interfaces nicely with the anchor-structure model, suggesting that first-order anchor-structure chords, because they allude to stable, root-position common-practice chords, can serve as stable sites for the resolution of non-chord ornamentation. A few examples from the Parisian modernist repertoire will now demonstrate how chordability, gestural reference to common-practice voice-leading, and contextual stability conditions can interact to produce non-chord ornamentation in an additive harmonic context. In Figure 2.35, mm. 5-12 from Debussy’s prelude “La Puerta del Vino,” all the verticalities produced by the melody’s ascent are chordable (the ro-interval 1 created by Dt)4 over Db 2 is above the bass of a first-order anchor-structure chord). The

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between the B and E chord tones of the 74|t|-cum-74|t|3 chord. The Bn and En (dissonant against the Db and F respectively) are understood as contextually stable chord tones because they are salient in terms of duration (Silberman’s condition 3) and in terms of formal position in the phrase (Silberman’s condition 2).

The O and Dti are then

understood as passing tones, even though the 74|e| and |7|41 verticalities they create over the left hand would be just as viable as additive harmonies as are the 74|t| and 74|t|3 harmonies they pass between. In Figure 2.37, a reduction of the opening bars of the “Rag mazurka” from Poulenc’s ballet Les biches, the additive harmonic context turns a major tenth into a dissonant suspension. The marked chord is a non-chordable verticality, with an rointerval 1 between B3 and C4/C5. This dissonance means that the C has the potential to F igure 2.36. Reduction of Poulenc, “Rag M azurka” from Les biches, R89; suspension of a major tenth above the

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be heard as a non-chord ornament, even though it would otherwise be consonant against the Ab and Eb in the lower parts. Hearing the C as a non-chord tone is then reinforced by the melodic gesture it participates in - it is prepared as a consonance and then resolved

128

downwards by step, and so one hears it as novel major-tenth suspension to the 736|e| chord that comes together on the second beat of the bar. Gestural reference to common-practice voice-leading, used in the previous example to present a pitch as a non-chord ornamentation, can also be used to present a dissonant pitch as a stable chord tone. In Figure 2.38, the opening of Prokofiev’s Flute Sonata, the downward leap of a perfect fourth in the melody presents the A5 as a stable chord tone over the B\>2 - virtually no common-practice dissonance would be arrived at via an accented downward perfect fourth. Without that downward leap, the A5 might be

F igure 2.37. Prokofiev, S onata for Flute and Piano, mm. 1-4; the melodic g esture p resen ts the dissonant A5 a s a chord tone.

more readily interpreted as a long suspension to the G5 of a g minor-seventh chord in first inversion. The passage as written, though, more strongly suggests Bi> as a root, and Bi>ends up being a significant key for the piece - the movement modulates to Bi> (with an attendant change of key signature) at R6+4 in order to present a reprise of the second theme.

Non-Chord Verticalities that Stand Apart from Chordal Structures

129

The preceding discussion has shown how certain contextual conditions allow for the perception of non-chord ornamentation in additive harmonic textures, which means that certain verticalities (both chordable and non-chordable) in this repertoire can be are interpreted as non-chord ornamentation of a passage’s chordal structure. There are other non-chord verticalities, though, that do not directly interact with chord progression and indeed stand apart from chordal structures. These non-chord verticalities, unlike some of those verticalities produced by contextual non-chord ornamentation, are generally nonchordable (often emphatically so) in order to distinguish themselves from a piece’s chordal passages. Most of these types (listed at the bottom of Figure 2.34) have been mentioned already: clusters, verticalized scales/scale segments, and simultaneities created by disassociated strata are all non-chord verticalities that perform roles away from structures of chord progression. (Polychords are a special type of non-chord verticality, generated by superimposing one chordal structure atop another, and will be discussed in Chapter 3.) When these types of sonorities appear within pieces that primarily involve chordal structures, these non-chord, non-chordable verticalities serve as a form of punctuation, i.e. as a harmonic palate-cleanser between passages of chord progression (see Figures 2.21 and 2.31). Another example of this kind of usage if found at the end of Debussy’s “Feux d’artifice” (Figure 2.38); the huge tone cluster in m. 87 (produced by combining a white-key glissando and a black-key glissando) completely obliterates the harmonic energy that has been built up from the thematic reprise in m. 79 and then worked into a froth by the parallel dominant minor-ninth chords (with 6-5 accented passing tones in the uppermost part) in mm. 85-86. This is not a triumphant chordal resolution of dramatic build-up, as one might find in a German Romantic composition;

130

F igure 2.38. D ebussy, “Feux d’artifices," mm. 85-90; u se of a non-chordable verticality a s a harmonic reboot.

T res reten u

this is a harsh explosion (the end of the fireworks show) that erases all energy from before; the piece reverts back to the F-G-A, Bi>-Ai>-Gb repeating scalar pattern (now with added D3) that opened the prelude.60 Non-chord verticalities that stand apart from structures of chord-progression can participate in logics other than those that govern chord progression; these sorts of structures lie outside the realm of additive harmony, and belong more to the strand of pitch-collection, scalar, and transformational analysis mentioned in the introduction’s footnote 8.

60 F o r a tran sfo rm atio n al an aly sis o f “ Feux d ’artifice” th a t ex p lain s the relatio n sh ip o f m m . 8 7 ’s “clim actic b o o m ” to a h o st o f o th er stru ctu res in th e piece, see L ew in, “A Transformational Basis for Form and Prolongation in Debussy’s ‘Feux d ’artifice,” ’, 97-159.

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Chapter 3 The Special Case of Polychordal Polytonality

Chapter 2 has argued that triadic construction is not an essential feature of additive harmonic structure and function in the Parisian modernist repertoire, and that any triadic chords that are used, such as the chord in Figure 3.1a, are better conceived of as the combination of a voiced anchor structure with a constrained range of adorning tones (as in Figure 3.1b) than as a stack of thirds generated upwards from a chordal root (as in Figure 3.1c). In the special case of polychordal polytonality, though, not only is

F igure 3.1. Anchor-structure, stacked-thirds, and polychordal readings of an extended triad.

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156

T able 3.2. The bichordal com binations of Figure 3.11 tabulated by ro-interval distance betw een chord roots. ro -in terv al b etw ee n c h o rd ro o ts

# of ap p ea ran ces of th e given ro-interval root distance in Figure 3.11

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Unsurprisingly, having the root of the upper chord at a distance of ro-interval 1 from the root of the lower chord produces the most instances of ro-interval 1. Other root-distances that appear frequently and that generate many instances of ro-interval 1 are ro-intervals 4, 6, 8, and 9. The appearance of 6, in conjunction with the significance of root-distance 1, supports Harrison’s claim that placing tonal materials in T1 or T6 relationships is conducive to bitonal effects; the relatively high frequency of root-distances along either the 0-4-8 or 0-3-6-9 cycles of thirds suggests that juxtapositions of key areas further apart in circle-of-fifths distance produce chord superimpositions better suited to polychordal polytonality than do more tonally proximate chord pairings. The significance of Figure 3.1 l ’s particular set of chordal superimpositions in creating polytonal effects can now be illustrated in a few musical examples. Figure 3.12a is the opening of the piece “Botafogo” from Milhaud’s set of piano pieces Saudades do Brasil. It is the first example of true “dual priority” that Kaminsky presents in his article;

157

for Kaminsky, the passage produces a dual-priority hearing because “the voice leading of the treble is largely autonomous (i.e., not drawn into the orbit of F minor), and hence the elements relating to Ff-minor retain a measure of association with that key.”36 His analysis is reproduced as Figure 3.12b; the treble line is annotated both with Roman numerals indicating the implied independent harmonies in Ft minor and with lines between the staves indicating “possible (albeit weak) points of assimilation of the treble notes by the bass „37 F igure 3.12. Analysis of bitonality in Milhaud’s “Botafogo" from Saudades do Brasil. a) opening of “Botafogo” D oucem ent 84 -

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Kaminsky’s analysis is convincing: independence of the voice-leading structures plays an enormous role in this passage’s establishment of dual-priority streaming. (Another element that contributes to the bitonal effect is the simple fact that the streams start playing at different times; Yeary has noted how, throughout the history of Western polyphony, staggering the onset of events has helped ensure that the events are perceive

159

io

as separate musical objects. ) The specific chordal superimpositions used in the passage, though, also play a vital role.

The chord superimpositions in the five explicitly

polychordal measures in Figure 3.12a (mm. 7-11) have been labeled using the labels from Figure 3.11. The superimpositions used both occur at the root distance of ro-interval 1, and produce multiple instances of ro-interval 1 in the texture (three in the superimposition of F# minor over F minor triads, and two in the superimposition of E# diminished, implying 0 7 , over C7 missing its third); these harsh dissonances are essential in establishing a sense of dual priority. Figure 3.12c illustrates by removing those dissonances; the entire treble staff has been transposed upwards one step in measures 3-12, into g minor. The ro-interval 1-producuing chord superimpositions of (m/m, 1) and (7/7, 2) have been replaced by the chord superimpositions (m/m, 2) and (7/7, 2), neither of which appears in Figure 3.11. Consequently, the bitonal effect of the passage is severely attenuated. This is despite the voice-leading of the treble part still remaining relatively independent with respect to the bass.

Figure 3.12d displays a

reduction equivalent to that in Figure 3.12b. While the upper part does now land on chord tones during some of the V7 measures, the upper-part voice leading is still highly unconventional (e.g. moving to

4

over the tonic, or moving from

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to

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in a minor­

mode authentic cadence) if interpreted in F minor. However, in the absence of the nonchordable element ro-interval 1, a potential integrated additive-harmonic hearing, in which the upper pitches are understood as colorations of the F-minor and C7 chords, is more available than in Milhaud’s original.

38 Mark Jerome Yeary, “ Perception, Pitch, and Musical Chords,” Ph.D. dissertation, The University o f Chicago (2011), 12.

160

“Botafogo,” then, showcases the ability of ro-interval 1 chord superimpositions to foster polytonal effects. “Ipanema,” from the same suite, showcases the dual-prioritygenerating potential of chord superimposition by ro-interval 6. In Figure 3.13, when the El; appears over a Gb triad, it is first heard as a bluesy flattened seventh appropriate for the sultry atmosphere created by habanera rhythm of the left hand.

When that “Fb” is

revealed in m. 38 to be the third of a right-hand C-major triad, ro-interval Is are created between Gb4 in the treble and Gb2 in the bass, and then between Dt|4 in the treble and Db3 in the bass in mm. 40 and 42, that help promote a split-stream hearing of the passage. When those troublesome notes vacate the texture in mm. 43-44, an integrated, flattenedseventh hearing of the Eb is able to reassert itself and the section closes with a perfect authentic cadence in Gb.

F igure 3.13. Polytonal effect generated by an (M/M, 6) chord superimposition; mm. 34-46 of “Ipanem a” from Milhaud’s Saudades do Brasil. 34

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As is the case in integrated chord voicings, ro-interval 1 behaves differently in polychordal contexts than does ro-interval 11. Figure 3.14a, measures 142-148 from the second movement of Ravel’s Sonata for Violin and Cello, is polychordal in construction; the violin outlines a B-E-A-D-G circle-of-fifths progression at ro-interval 11 above the cf-bb-eb-ab progression outlined by the cello. The passage does not split into separate auditory streams, though; this is because the (M/m, 11) superimposition used in the passage is not one of Figure 3.11 ’s set of ro-interval superimpositions that assist dual­ priority hearings by resisting integrated chordable ones. (That dual priority does not

F ig u re 3.14. Analysis of non-bitonal polychordal construction in Ravel’s Sonata for Violin and Cello. a) Sonata for Violin and Cello/ii, mm. 132-138 -f-

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occur here is not my opinion alone - Mark DeVoto writes of the passage that its “notation in two simultaneous different keys . . . results in a sense of polychordal harmony in which one or the other key momentarily predominates, rather than two

162

perceptible keys at the same time.”39)

Figure 3.14b replaces the (M/m, 11)

superimposition of Figure 3.14a with a (m/M, 1) superimposition drawn from Figure 3.11. This switch flips the harmonic content of the parts - the violin keeps the same gestures but now cycles through the chords c-f-bb-eb-ab, while the cello’s chords are now B-E-A-D-G - and produces a much stronger bitonal impression than the original passage; this further emphasizes that the types of chord superimpositions used in polychordal writing, and specifically the presence or absence of ro-interval 1 in those chord pairings, have a influence in whether those passages produce true dual-priority effects.40 The preceding discussion is not meant to suggest that a polychordal passage can only be polytonal if it uses chord superimpositions from Figure 11; there are other contextual ways for a piece to establish distinct auditory streams (e.g. by staggering object entrances, separating objects in register, or presenting different layers in distinctly different timbres). Nor does the discussion suggest that the presence of ro-interval 1 immediately renders a polychordal passage polytonal; shades of gray are possible in dual­ priority hearing, and a passage’s separation into distinct auditory streams can flicker in

39 Mark DeVoto, “Harmony in the Chamber Music,” in The Cambridge Companion to Ravel, ed. Deborah Mawer (Cambridge, UK: Cambridge University Press, 2000): 110. 40 Another detail that emphasizes the importance o f chord superimpositions’ voicing and ro-interval 1 in particular is the fact that in Febre-Longeray’s integrated-chord origins for all bitonal triad combinations (Figure 3.3), ro-interval 1 appears only in the chordable, integrated way outlined by chapter 2s voicing model, i.e. as a minor ninth above the bass o f first-order anchor-structure chords (or in some cases, such as in the juxtaposition o f C and Db major, as a minor ninth above the bass o f a dominant-function first-order anchor-structure chord that appears above a tonic pedal). While DeVoto does not read true bitonal effect in the passage from the Sonata fo r Violin & Cello, he does write that “subtle bitonal harmony is a later development in Ravel’s music, notably in [his final opera] L ’enfant et les sortileges ” (Ibid., 110). With that in mind, then, it is instructive to note that the two passages Kaminsky analyzes as producing true double-priority hearings, the duet o f the Squirrel and the Frog and the duet o f the Chinese Teacup and the Wedgewood Teapot, do both involve chord superimpositions that produce ro-interval 1; in the Squirrel-Frog duet, the A# o f the Eb-clarinet’s Ad-minor triad forms ro-interval 1 with the At| in the Dmaj7 chord in the orchestra, while in the Teacup-Teapot duet, placing the Teacup’s F-major pentatonic music on top o f the Teapot’s Ab-minor music the produces clashes between the pitches A/Ab, D/Db, and G/Gb.

163

and out. In “Corcovado,” another of the Saudades, the presence of ro-interval 1 between C# and C in does not produce a clear polytonal hearing (Figure 3.15). All of the other right-hand tones - even though they outline D major when played by themselves - are diatonic to and easily integrable into the left hand’s G-major chords (see the anchorstructure annotations on Figure 3.15), and the Cjt can be read as participating in a D-Ctt-D neighbor-note voice-leading that is not uncommon in the key of G.

So despite the

F ig u re 3.15. Analysis of Milhaud’s “C orcovado” from Saudades do Brasil, mm. 1-8.

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passage’s clear polychordal construction and use of a (7/7, 7) chord superimposition from Figure 3.11 ’s list, these ro-interval Is do not precipitate a split into distinct auditory streams, but instead just serve as intriguing bits of dissonance in an otherwise integrated passage. But while the voiced interval of polychordal superimposition and is not the beall and end-all of chordal polytonality, the above discussion has shown that it is one of several contextual factors that needs to be properly calibrated in order to generate polytonal effects; superimposition that produces intervallic clashes not found in chordable sonorities is a key feature of most passages that produce true split-stream polytonal effects.

164

Conclusion

Voicing and additive harmony are topics on the margins of music-theoretical discourse; conventional models of harmony privilege chord inversion, reduction, and deeper levels of tonal structure rather than surface details. On the face of it, then, a project that aims to devise a voicing-centered model of additive harmony for the relatively under-examined Parisian modernist repertoire barely seems worth the effort. This project, though, has hopefully not only incorporated voicing into a model of additive harmony that better explains the structure and function of additive harmony in this repertoire, but also revealed how the seemingly small, surface-detail topics of voicing and additive harmony relate to several larger music-theoretical issues. Chapter 1 discussed how the minutiae of additive-harmonic theory can reveal fundamental truths about the aims of a theoretical model. Is the model intended to describe musical practice as closely as possible? formulation of tonal mechanisms?

Is its goal to present an elegant

Is it designed to train composers/performers/

improvisers? The answers to these foundational questions influence the lengths to which writers are willing to go to explain a particularly novel dissonance formation, and that in turn influences technical decisions about the chromatic malleability of an added thirteenth or the contextual root of a quartal chord. It is interesting to note that those theoretical works that deal most heavily with voicing (e.g. Heinichen, Ulehla, Persichetti, or Levine) are those texts most expressly aimed at practical musicians; voicing, then, has generally been considered something useful for producing music, but not essential for analyzing it. The anchor structure model rehabilitates voicing as an analytic concern by recognizing how certain voicings are predictive of certain chord behaviors.

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The anchor-structure model itself incorporates strands of thought from each of the three strategies discussed in chapter 1. The model is rooted in the modified-basic-types strategy; its claim is that those novel chords that participate in tonal contexts can do so because they are relatable to their anchor structures’ common-practice progenitors. But only the pitches of the anchor structure itself are necessarily derived from that commonpractice progenitor; anchor structures can support multiple potential arrangements of adorning tones. Conceiving of additive chords in this way, i.e. as the coincidence of a structurally essential anchor structure with various coloristic adorning tones, is an application of the non-chord-elements strategy. It allows a host of differently shaded verticalities to participate in tonal contexts without having to attribute functional significance to any of those shadings.

(To read functional significance into every

potential additive-harmonic shading would dilute the intelligibility of the underlying tonal mechanisms on which additive harmony depends.) Lastly, the model involves elements of the new-basic-types strategy in that, by establishing that tertian construction is not essential for tonally plausible additive chords, the model dislodges extended triads from their privileged position in systems of additive harmony. This project has also demonstrated how musical practices can relate to and be informed by our psychoacoustic apparatus.

It has assembled a set of research that

suggests that our psychoacoustic apparatus is highly sensitive to voicing and other details at the musical surface. Many of the mechanisms of conventional theory, though, rely on concepts (such as interval class) that operate at a level of remove from the musical surface. The anchor structure model bypasses much of that abstraction, and makes use of our fundamental cognitive processes - e.g. our ability to recognize acoustically salient

166

intervals, or our sensitivity to the roughness of ro-interval 1 - to describe how we interact with additive harmonic surfaces. Put in crude terms, the anchor structure model codifies which tonally participatory vertical things immediately sound like other tonally participatory vertical things. A good portion of this may be common sense, but it is common sense that is often either left unsaid or obscured by underdeveloped, back-ofthe-book models of additive harmony. A similar directness of approach informs chapter 2’s approach to non-chord ornamentation; chordability, gestural reference, and Silberman’s stability conditions combine to describe how certain additive-harmonic structures can sound like the non-harmonic tones listeners are familiar with from common-practice contexts. The anchor structure model also with interfaces Robert Gjerdingen’s methodological premise that one of the aims of music theory is to describe how musical structures relate to a shared set of cultural assumptions between audiences and producers;1 chord structure in the anchor structure model is conceived of as a set of stylistic constraints that ensured that additive chords resembled the common-practice structures with which French audiences would have been familiar. (These constraints also prove useful in dealing with the sheer number of verticalities in play when dealing with pitch-space voicing.) This study’s focus on simple foreground structures rather than on deeper levels of musical structure is another echo of Gjerdingen’s work. Given this connection, an intriguing avenue for further research might be to see to what extent Parisian modernist harmonic practice involved patterns of the type found in galant composition. Are there additive-harmonic harmonic-contrapuntal patterns, equivalent to Gjerdingen’s schemata, that recur in this repertoire? (One would perhaps be the perfect 1Gjerdingen, 16-19.

167

authentic cadence with in 5 the upper voice of both chords that becomes a cliche in the chansons of Chabrier, Duparc, and Faure.) Do any galant patterns translate into additive harmonic contexts? And how might certain schemata be translated into novel scalar contexts? To close, I will suggest that the anchor structure model adds a new dimension to our understanding of the development of tonal language in the late nineteenth and early twentieth centuries. The most common narrative of that era is of how adherence to certain common-practice voice-leading principles allowed for the incorporation into tonal contexts of new harmonic successions. My model allows for the development of additive harmony to be read in parallel fashion: it was adherence to certain vertical-domain common-practice principles - skeletal voicings derived from common-practice chords and generalized principles of chordable pitch combination - that allowed for the incorporation into tonal contexts of new chords.

168

Appendix - C++ Program for Parsing All Verticalities in a 28-Semitone Span

Thanks go to Francis Song for his help in building this program.

linclude u s i n g n a m e s p a c e std; bool

BitSet(const

unsigned

int& x,

const

int& n)

{ return x &

(l