A b str a c t

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

ofYale University

in Candidacy for the Degree of Doctor of Philosophy

byDamian 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 131.1 Additive Harmonic Thinking in Rameaus Traite 171.2 The Modified-Basic-Types Strategy 261.3 The Non-Chord-Elements Strategy 511.4 The New-Basic-Types Strategy 68

Chapter 2 - A Vertical-Domain Model for Additive Harmony in 75Music in France, 1889-1940

2.1 Overview of the Model 752.2 Anchor Structures and Tonal Plausibility 792.3 Chordability 992.4 Additive Chords and Non-Chord Verticalities 123

Chapter 3 - The Special Case of Polychordal Polytonality 1323.1 Theories of Polychordal Polytonality 1343.2 Implications of Chordability Restrictions and 149

Anchor Structures for Polychordal Polytonality

Conclusion 165

Appendix - C++ Program for Parsing All Verticalities in a 28- 169Semitone Span

Bibliography 171

Illustrations

Figures

1.1 Reduction of the final cadence of Ravels L enfant et les sortileges, R154-end.

1.2 Reproduction of example 27.1 from Roig-Francolis Harmony in Context.

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

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

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

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

1.1 Spectrum between competing ideals in additive harmonic theory.

1.2 Realization of the chord types described in Rameaus Traite other than the perfect chord and the seventh chord.

1.3 Rameaus example II.6; an irregular cadence.

1.4 Transcription of Rameaus example II. 11 - explanation of a suspended fourth as a heteroclite eleventh chord.

1.5 Transcription and annotation of Rameaus examples II.41 and11.42; use of supposition to explain voice-leading novelties.

1.6 Rameaus inability to explain ninths resolving over falling- fifth bass motion.

1.7 Rameaus difficulty explaining suspensions in the bass voice.

1.8 Heinichens use of invertible ninth chords to explain novel dissonance treatment.

1.9 Set of extended chords from the Ist edition of Marpurgs Handbuch.

1

5

7

8

9

11

15

18

20

21

23

24

25

27

28

1.10 Marpurgs first-inversion thirteenth chord with fifth, seventh, 29and ninth omitted.

1.11 James M. Martins transcription of Sorges Plate VI, examples 314 and 5; presentation of the dominant ninth chords and their relationship to the leading-tone seventh chord.

1.12 Dehns ninth, eleventh, and thirteenth chords realized in C major. 33

1.13 Dehns prohibited voicings of the ninth chord. 34

1.14 The harmonic-series justification for the extended-triad model 3 7of additive harmony.

1.15 Lobes inversions of altered ii9 in A minor. 38

1.16 Transcription of Ziehns set of chromatic seventh chords. 39

1.17 Harmonic treatment of the whole-tone scale in Schoenbergs 41Harmonielehre.

1.18 Schoenbergs four- and five-note quartal chords, and their 42resolutions as substitute dominants.

1.19 Quartal chords in Schoenbergs Kammersymphonie. 42

1.20 Ottmans figure 10.18. 43

1.21 Pistons example 22-12; various inversions of a G major-ninth 44chord.

1.22 Ulehlas voicing guidelines for second-inversion major-ninth 45chords.

1.23 Ulehlas analysis of the implications of chromatic alteration to 46the thirteenth above a dominant seventh.

1.24 Prouts three fundamental chords. 48

1.25 Sixth-inversion thirteenth chord in Day. 48

1.26 Free treatment of the ninth in Prouts Harmony. 49

1.27 Days explanation of a ii7 chord as a third-inversion 51dominant eleventh.

ii

1.28 Kimbergers Figure XXXVI - non-essential dissonances. 52

1.29 Kimbergers explanation for seventh chords that resolve 53upwards by step.

1.30 Catels chord formed by the suspension of the octave, the sixth, 55and the fourth.

1.31 Catels chord formed by raising the chordal fifth and lowering 55the chordal third.

1.32 Fetis concept of substitution. 56

1.33 Fetis presentation of Catels inversions of the so-called 57fundamental chord of the seventh.

1.34 Fetis analysis of Beethovens improper inversion of the 58leading-tone seventh.

1.35 Fetis derivation of the ii7 chord. 58

1.36 Ravels analysis of an unresolved appoggiatura in Les Grands 61Vents venus d 'Outre-mer.

1.37 Ravels analysis of an unresolved appoggiatura in Oiseaux 61tristes from Miroirs.

1.38 Ravels analysis of an unresolved appoggiatura in Vaises nobles 62et sentimentales, vii.

1.39 Lenormands derivation of an unresolved appoggiatura chord. 64

1.40 Casellas 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 Kitsons derivation of an unresolved appoggiatura chord. 67

1.43 Hindemiths example 62 - non-triadic chords. 68

1.44 Preferring a new-basic-type label to tenuous modified-basic-types 69or non-chord-elements labels.

1.45 Persichettis example 4-4; the roots of quartal chords determined 70by melodic motion.

iii

1.46 Persichettis conditions for incorporating quartal chords with triads.

1.47 Hindemiths chord groupings.

1.48 A Hindemith harmonic-fluctuation graph.

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

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

2.3 The set of common-practice chord types.

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

2.5 Ravel, Laideronette, imperatrice des pagodes from Ma mere I Oye, mm. 16-24: |t|-anchored substitute supertonic.

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

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

2.8 Hulls example 194; an example of the benefit of abandoning a stacked-thirds conception of additive chords.

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

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

2.11 Ravel Jeux d eau/iii; |e|-anchored final tonic 92

2.12 Chromatically altered chordal third in Saties Gymnopedie No. 1, mm. 36-39.

2.13 Omitted chordal third in Debussys La fille aux cheveux de lin, mm. 18-19.

2.14 Second-order anchor structures.

iv

71

73

74

78

80

82

84

86

87

89

89

91

92

93

93

95

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 98du soldat, p. 6.

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; 102first-order anchor-structure chord supporting ro-interval 1.

2.23 Ravel Vaises nobles et sentimentales i, mm. 1-2. 104

2.24 Jazz voicings from Levines The Jazz Piano Book. 105

2.25 Ravel, Pavane pour une infante defunte; use of ro-interval 1 to 107destabilze a previously stable cadence.

2.26 Marked use of ro-interval 1 in Chabriers 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 118repertoire.

2.30 Debussy, Les collines dAnacapri, mm. 1-8; use of 119verticalized consecutive whole-step adjacencies as a marked event.

2.31 Debussy, Feuilles mortes, mm. 6-10; adjacent-whole-tone 120sonority treated as a passive musical object.

2.32 Reductive power of the models 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 125 verticalities interpreted as the coincidence of a chordal backgroundand non-essential dissonance.

2.35 Debussy, La Puerta del Vino, mm. 5-12; viable additive chords 127interpreted as passing tones.

2.36 Reduction of Poulenc, Rag Mazurka from Les biches, R89; 128suspension of a major tenth above the bass.

2.37 Prokofiev, Sonata for Flute and Piano, mm. 1-4. 129

2.38 Debussy, Feux dartifices, mm. 85-90. 131

3.1 Anchor-structure, stacked-thirds, and polychordal readings of an 132extended triad.

3.2 Milhauds table of bitonal combinations. 136

3.3 Febre-Longerays monotonal derivation of all bichordal 138combinations of two major triads.

3.4 Koechlins context-dependent polychord. 139

3.5 Persichettis hierarchy of bichordal combinations. 140

3.6 Examples of Persichettis four types of permissible trichords. 142

3.7 Persichettis demonstration of grouping as a requisite of 143polyharmony.

3.8 Ulehlas analysis of the assimilation of the upper parts by a 144dominant seventh chord.

3.9 An exception to the assimilation effect of Figure 3.8 - a tonic 144triad above its dominant seventh.

3.10 Thompson and Mors Fig. 1 - excerpt from Theodore Duboiss 148Circus.

3.11 Bichordal superimpositions that run afoul of Chapter 2s 153chordability restrictions.

3.12 Analysis of bitonality in Milhauds Botafogo from Saudades 158do Brasil.

vi

3.13 Polytonal effect generated by an (M/M, 6) chord superimposition; 161mm. 34-46 of Ipanema from Milhauds Saudades do Brasil.

3.14 Analysis of non-bitonal polychordal construction in Ravels 162Sonata for Violin and Cello.

3.15 Analysis of Milhauds Corcovado from Saudades do Brasil, 164mm. 1-8.

Tables

3.1 All potential bichordal combinations involving major triads, minor 151triads, and dominant seventh chords.

3.2 The bichordal combinations of Figure 3.11 tabulated by ro-interval 157distance between chord roots.

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 Harrisons 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 Ravels 1925

opera L enfant et les sortileges. These are moments where the functional sense of a

Figure 1.1. Reduction of the final cadence of Ravel,Lenfant et les sortileges, R154-end.

The Child

Ma - man!

Chorus

doux.est

oboes IPOrchestra PP z

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-to-

dominant 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-C-

D-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-0Connell 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 late-

nineteenth-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

repertoires focus on the sensuous potential of novel chords participates in French

1 Jeremy Day-OConnell, 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

modernisms 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 1889-

1940 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

harmonys 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 Debussys Prelude a L Apres-midi d un faune,'" Music Theory Spectrum 15, no. 2 (Fall, 1993): 127-143; Sylvain Caron, Tonal composition and new perspectives on Faures harmony, Canadian University Music Review 22, no. 2 (2002): 48-76; Eddie Chong, Extending Schenkers Neue musikalische Theorien und P h a n ta s ie n Towards a Schenkerian Model for the Analysis of Ravels 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 Debussys 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

repertoires 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 Debussys 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 Debussys works, Parks instead made analytic use o f an adapted form o f Allen Fortes pitch-class-set genera; deep structure in Debussys 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 Debussys 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, Ravels Russian Period: Octatonicism in His Early Works, 1893-1908, Journal o f the American Musicological Society 52, no. 3 (Autumn, 1999), 531-592; Day-OConnell; 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 dartifice 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 Debussys 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 Third- Relations and Tonal Structure in the Songs o f Debussy, p. 215); Green more simply states that How Debussys 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-Francolis Harmony in Context,

is one example of this model; Roig-Francoli writes of these chords that the ninth chord

Figure 1.2. Reproduction of example 27.1 from Roig- Francoli, Harmony in Context, extended tertian chords generated by stacking thirds on top of the seventh chord.

(ffl *--------- & ----- ........8 ..... ........ ^ ;................. :------f t

------0 -----

f t f t -t*

7DM: 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, I A: 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 (New 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

models 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

dissonances 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 ebussy.. . . 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, Ravels 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

Koechlins rondel Leau 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

Figure 1.3. The ill fit of extended-triad labels when applied to certain chords in the Parisian modernist repertoire.

a) reduction of the final two chords of Lenfant et les sortileges and an extended-triad interpretation of the highlighted chord

------------------ = a h 1------ f n *------.....^ ......m..........

------------- *:--- >------

o

.' ....w .................................. ^ ** 3 13th chord missing3rd, 5th, and 11th?

b) reduction of Koechlins Leau from his 9 Rondels, op. 14, mm. 42-44, and an extended-triad interpretation of the highlighted chord

8...................................

a J J J J S I J J n J j j j ^ 1

rs-e-t H

rr t r r ~ if ....................r\ 1 *-----1-------

r ~----\I ------------------------- 1L*---------------------------1

.^..' J........C 13th chord missing7th 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 extended-

triad 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

Figure 1.4. Competing functional elem ents in the penultimate chord of Lenfant et les sortileges.

Tonic agent

Subdominant base & agent

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).

X T

8

Figure 1.5. Comparison of the impact of inversion upon an additive chord and upon a triad.

a) inversion of the penultimate chord of Figure 1.1

tr?

b) replacement of the penultimate chord in Figure 1.5a with a V chord

h

c) inversion of penultimate chord of Figure i.5b

fm

f ?

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 dominant-

functional, 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

Ulehlas book Contemporary Harmony, one of the more thorough treatments of the

extended-triad model:

Due to the upward climb o f thirds, the thirteenth rightfully demands the highest position. Occasionally, it will permit the ninth to preside above, with the thirteenth directly beneath. The seventh o f the chord must not be placed next to the thirteenth . . . With inversions, the root may be placed in the uppermost voice.. . Nestled next to the seventh, [the thirteenth] becomes unclear and permits the upper tones to predominate.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 pitch-

class 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

Figure 1.6. Over-applicability of the extended-triad model in the absence of guidelines for voicing.

a)-c) potential alternate readings of the penultimate chord of Lenfant et les sortileges.

v &(*) (b) (c)

d)-e) implausible chord labels available with the extended- triad model

_fl i . |o ilM

(d) (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 voicings 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 chords 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 chords 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 Rameaus 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 Rameaus 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.

Figure 1.1. Spectrum between competing ideals in additive harmonic theory.

Modified-basic-typesstrategy

WWnmgi ng A I I I ^ Thsoreticalappticabity | | parsimony

New-basic-types Non-chord-elementsstrategy 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 Triads 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 Cohns 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-a- superset is but one o f the novel types of 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 of 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 Rameaus Traite in

order to shed light on the interrelated genesis of the modified-basic-types and non-chord-

elements strategies.

1.1 - Additive Harmonic Thinking in Rameaus 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 Rameaus 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

Figure 1.2. Realization of the chord types described in Ram eaus Traite other than the perfect chord and the seventh chord. (Locations of Ram eaus written descriptions are given in the in-text footnotes.)

Open note-heads comprise the initial basic chord; black note heads are the added/subposed notes. The fundamental bass of any chord is the lowest open note- head.

& 1 --- 1 8 *i f - 1L(al )

JL .... - i | -

_J--- -----m------(bl )

- m 1

fcjl --- ----i(c)

gg___ .. . - -... ... .....W~ -..... i

chords are generated by subposed bass tones, the fundamental bass of the chord remains

the fundamental of the initial seventh chord. This is the interpretation shown in Figure

1.2 for chords al and bl.4 In these two different derivations of the same chords, we are

already exposed to a tension in additive harmonic theorizing between, on the one hand,

generating modified chord types by using a structural principle consistent with that used

to generate the systems basic chord types, and on the other, having a chords identity be

defined by how it behaves in context.

Both the ninth and eleventh chords can be altered to involve the raised leading

tone of the minor mode, producing the chord of the augmented fifth (Figure 1.2, chord

a2) and chord of the augmented seventh (b2) respectively. Both chords can also appear

in stripped-down forms; Rameau writes that the ninth chord always appears without

seventh above the tonic note (a3, in C major), while the eleventh can appear without the

third above the fundamental, the fifth above the fundamental, or both (this last scenario is

shown as b3). Rameau calls this last form abnormal (heteroclite) since it is not divided

[by fifths and thirds] as are the other chords.5

The chord of the large sixth (chord c in Figure 1.2) is also produced from the

primary chord, specifically via the addition of a sixth above the triads fundamental.

While Rameau recognizes that this chord might be interpreted as an inversion of the

seventh chord built on the minor perfect chord, he stresses that the context of an irregular

cadence (Figure 1.3) demonstrates that the chord of the large sixth is distinct from a

4 Ibid., 88-89.5 Ibid., 108-109, 240, and 91.

19

minor seventh chord.6 Were D the fundamental of the first chord in Figure 1.3, not only

Figure 1.3. R am eau's example II.6; an irregular cadence.

Major thirdAdded sixth

Fifth

Fourth note Fundamental bass Tonic note

There is a dissonance of a second between these two parts.

Irregular cadence in the major mode

would the fundamental bass progression be weakened (the fundamental bass would not

be progressing by descending fifth, descending third, or ascending fifth), but the C5 in

the alto voice would need to resolve as a dissonance (since the dissonant sevenths of

seventh chords must resolve down by step). Instead, reading the chord in Figure 1.3 as a

chord of the large sixth, with a fundamental of F, produces a stronger progression in the

fundamental bass and means that the D in the upper voice is now the added dissonance,

explaining its upward motion out of the C-D second.

Rameau devotes individual chapters to the ninth, eleventh, and large-sixth chords,

and these adapted chord types do important work in relating non-standard verticalities to

the central triads-and-sevenths machinery of his theory. Ninth and eleventh chords are

used to explain various forms of dissonance treatment, including the behavior of

6 The chord formed by adding a sixth to the perfect chord is called the chord o f the large sixth. Although this chord may be derived naturally from the seventh chord, here it should be regarded as an original. Ibid., 75.

20

suspended fourths at cadences. In Figure 1.4, the resolution of D4 to C#4 over a sounding

static bass is explained as the downward resolution of the seventh in a progression of the

fundamental bass from E-A. This means that the bass-voice A of the second chord is a

subposed bass note, and that the chord is an A-E-D heteroclite form of the eleventh

chord.7

Figure 1.4. Transcription of R am eaus example 11.11 - explanation of a suspended fourth as a heteroclite eleventh chord.

The five upper parts are sounding voices; the fundamental bass is R am eau's analytic gloss.

p ........ = 1

k ........................................ = 1

.................................................. J

$ ..... .................. ..... ....

# = = = = ^

u

f t O -

Tenor*TWt-----------------------------O-----------------------

.......................( K ...............................

O O -----------------------1

""U.................................................................

r - & -------------------------------------------------- -

BassA

Fundamentalbass

4 7 i

7 7s

* The tenor represents the fundamental bass in chords by supposition. A This is the bass o f chords by supposition.

There is an additional wrinkle to Rameaus presentation of ninth and eleventh

chords, beyond his generating ninth and eleventh chords by subposition rather than

7 Ibid., 90.

21

upwards third-stacking, that differentiates it from modern-day extended-triad models.

Rameaus ninth and eleventh chords are generated from the primary chords using the

constituent intervals of those chords, and so their structure represents an application of

the modified-basic-types strategy for dealing with novel verticalities. Rameaus

treatment of these chords in context, however, involves the non-chord-elements strategy.

The non-chord elements are the subposed bass notes themselves. Rameau frequently

describes these bass notes as supernumerary, and they do not participate in the action

of the fundamental bass nor can they be moved from the lowest voice; in Rameaus

system, ninth and eleventh chords can not be inverted as primary chords can. The

subposed bass tones are coloristic rather than functional, and are therefore outside of the

system of chord progression. By way of contrast, the added sixth plays an integral role in

the chord of the large sixth - the dissonance it creates with the chordal fifth ensures that

the final chord of Figure 1.3 is understood as an arrived-at tonic, and not as the dominant

of the first chord with the fundamental of F.8 This means that, in analytical application,

Rameaus conception of ninth, eleventh, and large-sixth chords is the reverse of that

found in most modem textbooks, in which elevenths and especially ninths are freely

moveable chord members while the added sixth is a voicing-dependent ornamental tone.

The main benefit of conceiving of ninth and eleventh chords as the

combination of fundamental seventh chords with non-chord subposed basses is that it

simplifies the explanation of dissonance: if the ninth and the eleventh above the subposed

bass are both actually sevenths above the fundamental bass, then a) the rules for their

treatment are clear and b) the seventh is still the source of all dissonance, a fact central to

8 Ibid., 75.

22

Rameaus mechanistic conception of the tonal system.9 This allows Rameau to elegantly

explain unusual voice-leading structures like that in Figure 1.5a. The apparent

transgression of the F2-E3 seventh resolving to a D2-D3 octave in similar motion can

instead interpreted as a fifth above the true fundamental bass of A2 that correctly

resolving to the octave; the F2 is then interpreted as a subposed bass note (Figure 1.5b).

Locating the fundamental bass at A2 also means that the ninth in (a) is the true seventh,

which explains why it resolves downward by step to a third.10

Figure 1.5. Transcription and annotation of Ram eaus exam ples 11.41 and II.42; use of supposition to explain voice-leading novelties.

(a) (b)

3 9 3 3 7 3

Apparent transgression - F-E seventh resolving Rameaus explanation: the F in the progression on the leftto the D-D octave in similar motion is a subposed bass note; the real fundamental o f the middle

chord is A

But because the lowest tone of a ninth chord cannot participate in progressions of

the fundamental bass, Rameau is at pains to explain the dominant-to-tonic falling-fifth

bass-motion progression from the second to the third chord of Figure 1.6a. If ninths

above the bass are interpreted as sevenths above the fundamental bass, they must resolve

as sevenths, either to an octave above a first-inversion chord (Figure 1.6b) or to a third

above a root position chord (Figure 1.6c); the second of these acceptable resolutions

appears in the progression from the third to the fourth chord of Figure 1.6a. Motion

9 Ibid., 112. Although Rameau talks o f both major and minor dissonances, the minor dissonance o f theseventh is prior, in that its presence above a perfect major chord is what turns the third o f that triad into themajor dissonance.10 Ibid., 132-133.

23

Figure 1.6. R am eau's inability to explain ninths resolving over falling-fifth bass motion.

a) Ram eaus example III.87; improper resolution of the ninth chord above F3.

The ninth resolved by the fifth

Potential proper resolutions for the ninth chord above F3 (derived from Ram eaus example III.86).

b) resolves the seventh above the fundamental b ass to an octave

c) resolves that seventh to a third.

(b)

Bassocontinuo

Fundamentalbass

l=i(C)

from the ninth to a fifth above the bass, as between the second and third chords 1.6a, is an

improper resolution. Rameau seems to acknowledge that this is an instance where his

tightly constructed theoretical system fails to account for a relatively common

progression, writing We might further wish to resolve the ninth by the fifth, with the

bass ascending a fourth. The harmony arising from this, however, is improper and so we

leave this matter to the discretion of men of good taste.11

11 Ibid., 295-296 .

24

Because ninths and elevenths are not full chord tones and therefore cannot

participate in inversion, Rameau also cannot use them to explain dissonance treatment in

the bass voice; Rameau cannot draw a connection between the suspended A3-G3 bass

motion of Figure 1.7a and the suspended middle voice of Figure 1.4. The motion of A3-

Figure 1.7. Ram eaus difficulty explaining suspensions in the bass voice.

a) Ram eaus example IV.34

Same

or

b) what Ram eaus interpretation would be were the bass voice of (a) moved into an inner voice; E would then be the subposed bass voice

I ................7 4

Fundamental < 6) : 1-----

c) Ram eaus interpretation of (a); the A3 is a subposed bass note

X T65

FundamentalBass m

G3 in Figure 1.7a cannot be a resolution of the seventh of a chord with a fundamental

bass of B to the third of chord with a fundamental bass of E (the interpretation presented

25

in Figure 1.7b) because that would make the E4 of the suspension chord in Figure 1.7a a

subposed bass note that was not appearing in the lower voice. Instead, Rameau is forced

to label the A3 as a subposed bass note (his interpretation is presented as Figure 1.7c), an

interpretation that provides no explanation for that notes downward-resolving tendency.

Rameaus Traite, then, uses both the modified-basic-types and non-chord-

elements strategies to explain novel formations. The use of both stategies demonstrates

the care Rameau takes in considering non-standard verticalities within the context of his

tonal system, and each of the two strategies is further explored by Rameaus successors.

1.2 - The Modified-Basic-Tvpes Strategy

In German theory of the eighteenth century, Rameaus ideas combine with what

David A. Sheldon has described as a German compulsion to categorize and explain

striking departures from the theoretical norm to produce analytically more flexible

models of extended triads.12 One way of enabling extended triads to account for a

broader range of novel formations is to allow the bass notes of extended triads to

participate in inversion. Figure 1.8, a realization of a written description from Johann

David Heinichens 1728 Der General-Bass in der Composition, demonstrates this type of

usage.13 Heinichen explains the 7/4/2 and 7/5/2 chords in (b) as inverted forms of the

9/5/3 and 9/6/3 chords in (a). The tendency for D to resolve to C, present between D5

and C5 in the foundational forms, is preserved in inversion; this explains why D3

resolves to C3 in the lowest voice of the inverted chord forms, when sevenths above the

bass (found between D3 and C5 in the inverted forms) would generally be resolved by

12 David A. Sheldon, The Ninth Chord in German Theory, Journal o f Music Theory 26, no. 1 (Spring 1982), 61.13 Johann David Heinichen, Der General-Bass in der Composition (Dresden: the author, 1728), 208-209, quoted in Sheldon, 64.

26

Figure 1.8. Heinichen's use of invertible ninth chords to explain novel dissonance treatment.

a) foundational ninth chords; the C3-D5 ninth is resolved by the upper voice moving down by step

b) inversion of the chords in (a); D still resolves downward by step to C

8.....-.-......-1 m= :::.=.=. -------------------------------1.....r =\ t r ^

c) Heinichens alternate explanation of the 7/4/2 chord as being generated by an anticipated resolution in the upper parts

anticipated resolution normal

$

t

the upper voice moving downwards by step. Heinichens explanation requires a

conception of ninth chords as entities unto themselves - objects well-defined enough to

retain their identity in inversion.14 Heinichens further discussion of the 7/4/2 chord also

demonstrates the cheek-by-jowl relationship of the modified-basic-types and non-chord-

14 It is worth noting here that, unlike in modem conceptions o f chord-vs-non-harmonic tone, verticalities that resolve over the same bass note were commonly considered distinct chords in much eighteenth-century theory. (Lester, 209-210.) Examples include Rameaus chords o f the augmented fifth and chords o f the augmented seventh.

27

i

elements strategies. Figure 1.8c displays Heinichens description of how the same 7/4/2

chord can be understood as arising from an anticipated resolution in the upper parts.15

While the musical structures of Figures 1.8b and 1.8c are similar, their tales of origin are

different; in 1.8b, the melodic motion occurs because of the energies inherent in a ninth

chord, while in 1.8c the 7/4/2 chord coincidentally appears as the byproduct of a melodic

process.

Another way of expanding the analytic reach of extended triads is to read a thirds-

stacking process into the structure of triads and seventh chords, and then to extend that

process to produce not only the ninth chords and gapped eleventh chords found in the

Traite, but full eleventh and thirteenth chords as well. This variety of the extended-triad

model is developed in the 1755 first edition of Friedrich Wilhelm Marpurgs Handbuch

bey dem Generalbasse und der Composition. Figure 1.9 reproduces James Martins

Figure 1.9. Set of extended chords from the 18' edition of Marpurgs Handbuch. Fundamentals of the foundational seventh chords are underlined.

ninth chorda eleventh chorda th irteen th ohordedb &

db

rd

r td d

t&

at

ga g %g#a

b

b & / g

db

s 0 Q c e0

a ae o

a*1c ga

o a g

transcription of Marpurgs complete set of dissonant chords.16 The chords are created by

15 Heinichen, 187-189, reproduced in Sheldon, 66. A similar temporal explanation o f the 7/4/2 chord is found in the 1755 first edition o f Friedrich Wilhelm Marpurgs Handbuch bey dem Generalbasse und der Composition, with the sole difference being that the explanatory mechanism is delay o f the bass rather than anticipation o f the upper parts. (David Sheldon, M arpurgs Thoroughbass and Composition Handbook: A Narrative Translation and Critical Study [Stuyvesant, NY: Pendragon Press, 1989]: 20.)16 James M. Martin, The Compendium Harmonicum (1760) o f Georg Andreas Sorge: A Translation and Critical Commentary (Ph.D. dissertation, The Catholic University o f America, 1981): 45

28

consecutively stacking subposed thirds beneath the fundamental of a seventh chord.17

Unlike Rameaus ninth and eleventh chords, these extended triads are true chords; their

subposed pitches are full chord tones in that they can participate in inversions.

In the first edition of the Handbuch, Marpurg does not restrict how these extended

triads operate; he states that the chords can appear in any inversion, and that various

combinations of chord members can be omitted in reduced-voice textures.18 This

combination of maximally extended stacks of thirds, unrestricted invertibility, and

potential omissions lends Marpurgs extended-triad model tremendous analytic

flexibility. Marpurg can provide a chordal explanation for almost any dissonant

formation; for instance, the resolution of the 9/6/4 chords double suspension in Figure

1.10 can be explained as the downward resolution of the eleventh and thirteenth of a first-

inversion thirteenth chord on D with fifth, seventh, and ninth omitted.19

Figure 1.10. Marpurgs first-inversion thirteenth chord with fifth, seventh, and ninth omitted.

On the bottom staff, fundamental bass tones are shown with open note heads, and omitted chord tones with filled- in note heads.

Fundamental bass : omitted chord tones

Sheldon, M arpurgs Thoroughbass and Composition Handbook, 1.18 J. Martin, 46-47.19 Ibid., 15.

29

One of the drawbacks to such a liberal conception of extended chords, however, is

that many of the available pitch combinations a) rarely appear in practice and b) are

harshly dissonant. Marpurgs contemporaries Georg Andreas Sorge and Johann Philipp

Kimberger aired these criticisms in print, and in response Marpurg altered his extended-

triad model in the 1762 second edition of the Handbuch; most of the complete forms of

eleventh and thirteenth chords are omitted, while the number of potential inversions of

the taller chords is restricted.20 And instead of subposing continuous chains of thirds

beneath foundational seventh chords, Marpurgs 1762 edition reverts to Rameaus

gapped method of subposition: the eleventh chord is now generated by subposing a fifth

beneath a seventh chord (i.e. with no intervening third), and the thirteenth chord by

subposing a seventh beneath a seventh chord.21 These modifications both address

practical considerations (i.e. the rarity of full eleventh chords in practice) and highlight

the connection between Marpurgs theories and those of Rameau.22 The modifications to

the 1762 edition of the Handbuch did not stop criticism of Marpurgs work - he held

another full-blown public debate with Kimberger in the 1770s - but they do reveal

awareness on Marpurgs part of the potential pitfalls of overly liberal models of extended

triads.23

Another way to modify the extedend-triad model to account for certain types of

unusual dissonance treatment is to extend the stacking of thirds upwards rather than

downwards from the foundational seventh chord. This is first done in Sorges

20 Ibid., 10. For detailed discussion o f Sorge and Kimberger's criticisms o f Marpurg, see Lester, 196-197 and 247-249.21 Sheldon, The Ninth Chord in German Theory, 69.22 Marpurg valued this latter feature because positioning himself as the true heir to Rameau's ideas - a stance Joel Lester has shown generally to be false and based on poor transmission o f Rameaus work in Germany - was a central component in his defense against his critics. (Lester, 150.)23 For discussion o f the second Marpurg-Kimberger debates, see Lester, 250-256.

30

Compendium Harmonicum of 1760, where the dominant ninth chord is presented as the

upward extension of a dominant seventh.24 This allows Sorge to connect the behavior of

the leading-tone seventh chord and the dominant seventh chord, with the latter being seen

as the generator of the former. In Figure 1.11, example 4 shows how Sorge considers

leading-tone seventh chords to be rootless inversions of root-position dominant ninth

chords.25 The connection between the leading-tone seventh chords and the dominant

ninth chord is then further emphasized in example 5, which demonstrates how the

seventh of the leading-tone seventh chord, in any inversion and in both the minor and

major modes, often resolves to the root of the generating dominant chord.

Figure 1.11. Jam es M. Martins transcription of Sorges Plate VI, exam ples 4 and 5; presentation of the dominant ninth chords and their relationship to the leading-tone seventh chord.

Generating extended triads in the upward direction gives the ninth chord a firmer

acoustic foundation than does generating them by subposition: Sorge can extract the

24 Sheldon, "The Ninth Chord in German Theory, 73-75.25 J. Martin, 204 and 109.

dominant ninth from the harmonic series by tempering the fourth, fifth, sixth, seventh,

and ninth overtones of the fundamental,26 whereas Marpurgs subposition model is

dependent on a tenuous conception of the undertone series and combination tones

imported from Rameau.27 This solid acoustic foundation means that Sorges dominant

ninth chord is analytically more flexible; because it an independent vertical entity - the

ninth is free and unsuspended (i.e. what Kimberger will label an essential

dissonance) - the ninth chord is not limited to any specific preparatory context.28

The dominant ninth chord is the only extended triad Sorge admits into his theory

- all other dissonances are explained as produced by inverted sevenths, suspended

dissonances, or some combination of the two. When combined with Johann Philipp

Kimbergers clear formulation of the distinction between essential and non-essential

dissonances, Sorges model of additive harmony - with the upward-generated ninth as

the sole true extended chord, and all other dissonances explained as non-chord elements -

becomes the baseline for much nineteenth-century theory, including the entire

Conservatoire tradition running from Charles-Simon Catel in 1802 to Theodore Dubois

texts in the early twentieth century29

One theorist in the nineteenth century who does discuss extended triads beyond

the ninth chord is Siegfried Dehn. In his 1840 Harmonielehre, Dehn, like Sorge, limits

his discussion of extended triads to chords built around the dominant, but expands the

26 J. Martin, 37-38 and 44.27 Heinrich Christoph Kochs 1782 Versuch einer Anleitung zur Composition attempts to reconcile the subposition model with the overtone series. As in Marpurgs late theory, Koch produces ninth, eleventh, and thirteenth chords by subposing thirds, fifths, and sevenths beneath seventh chords. In these constructions, however, Koch acknowledges that the lowest note these chords is now the acoustic generator o f the seventh chords fundamental; the bass note is called the eigentlich fundamental, while the upper note is mitklingend. (Sheldon, The Ninth Chord in German Theory, 79.)28 J. Martin, 204.29 Mildred Freeman Rieder, Traite d harmonie by Theodore Dubois in the Context o f ^ -C e n tu ry French Harmonic Theory and Pedagogy, (M.M. thesis, The University o f Western Ontario, 1995): 21-24.

32

system to include eleventh and thirteenth chords. Ninth chords are generated not upward

from the root of the dominant triad, but by placing a third both above and below the

diminished triad of a key (Figure 1.12a; the added pitches are shown with filled-in note-

heads); eleventh and thirteenth chords are generated by placing the tonic note beneath the

dominant seventh and ninth chords respectively (Figure 1.12b and c).30 The focus on the

dominant allows Dehn to explain why the thirds are absent in the eleventh and thirteenth

chords - they are omitted because it is the tone of resolution of the upper-voice dominant

chords sevenths.31

Figure 1.12. Dehns ninth, eleventh, and thirteenth chordsrealized in C major.

(a) (b) (c)

Dehn limits his application of these extended triads more than either Marpurg or

Sorge - for Dehn these chords cannot appear in inversion nor can they sustain omissions

of their roots, uppermost tones, or leading tones - and justifies his restrictions on

practical grounds. The extended triads cannot be inverted because their identity is

dependent on a specific voicing. The chords do not belong to the family of invertible

Stammakkorden because the closest possible arrangement of their component pitch

classes does not form a stack of thirds.32 The extended triads therefore require a specific

30 Siegfried Dehn, Theoretisch-praktische Harmonielehre mil angefugten Generalbassbeispielen (Berlin: Verlag von Wilhelm Thome, 1840): 119-123.31 Ibid., 122.32 Ibid., 119-120: Zu den Stammakkorden kann er aber nicht gezShlt werden, weil seine Tone, wenn sie nach der natfirlichen Reihenfolge er diatonischen Tonleiter von C dur oder C moll geordnet wtirden, nicht terzenweise ilber einander zu liegen kommen, sondern in folgender Gestalt [figure showing G-A-B-D-F in major, G-Ab-B-D-F in minor].

33

voicing to produce a properly tertian chord structure, and so inversions of the ninth,

eleventh, and thirteenth chords that obscure that tertian structure are prohibited; the same

principle rules out voicings such as those in Figure 1.13, in which the would-be chordal

ninth is adjacent to the chord root.33 Omission of chord tones is dismissed for the simple

Figure 1.13. Dehns prohibited voicings of the ninth chord, with ninths placed directly above the chord roots.

reason that these extended triads cease to be themselves when tones of the ninth,

eleventh, or thirteenth are omitted, and because Dehn furnishes other labels for the chords

resulting from those omissions (e.g. vii07 for a dominant major ninth chord missing its

root).34

While it does elevate extended chords to the status of key-delineating

Hauptakkorde (I, V, or chords containing the 7 - 4 tritone), tethering extended chords so

Although not Stammakkorde, ninth, eleventh, and thirteenth chords are still true chords in Dehns system (and not the coincidence o f Stammakkorde and non-harmonic tones, as ninth and eleventh chords are theorized in Rameaus Traite)-, all three extended triads belong to the privileged set o f key-defining Hauptakkorde: I, vii, vii 7, vii7, V7, V9, V7/ I and V9/ 1. (Ibid., 121-122).33 Ibid., 223.34 Ibid., 120-121. Dehn is quite severe his denouncement o f the practice labeling chords based on absent notes, calling into question whether or not theory that involves such practices is worthy o f the term system: Wie schon weiter oben bemerkt wurde, konnen nur solche Intervalle eines Akkords weggelassen werde, die ihn nicht wesentlich von andem Akkorden unterscheiden. Nun unterscheidet sich der Nonenakkord eben durch seine None von den Vierklangen, die nach ihrem Sussersten Intervall Septimenakkorde genannt werden; wenn man daher von einem Nonenakkord den Basston fortlasst, so bleibt er kein Nonenakkord mehr, sondem die iibrigbleibenden T8ne bilden ihrer terzenweisen Lage entweder einen kleinen oder einen verminderten Septimenakkord; ISsst man hingegen die Oberstimme des Nonenakkords fort, so bleibt ein Dominantenakkord, dessen Umkehrungen man eben so wenig, wie die Umkehrungen der erst genannten Septimenakkorder, fur Umkehrungen eines Nonenakkords mit weggelassener Ober- order Unterstimme ansehen kann. Solche Annahme k6nnte aber auch leicht dahin verleiten, z. B. den Dreiklang h d f, oder den Dreiklang g h d, oder endlich den Dreiklang d f a, von Nonenakkord g h d f a abzuleiten, indem man von diesem entweder die Tftne g und a, oder f und a, oder endlich die T8ne g und h fortlSsst. - Von einer systematischen Construction der Akkorde kann bei einem solchen Verfahren gar nicht die Rede sein, oder man mOsste die Grundbedeutung des ursprlinglich griechischen Wortes Systema ganz ausser Acht lSssen.

34

closely to the diminished triad limits their explanatory power. If extended chords are

neither invertible nor transposable to other scale degrees, the model can only account for

a handful of progressions: V9 resolves to I and cadences with multiple suspended upper

voices.35 Dehns model, then, leans toward the opposite pole of Figure 1.1 than does

Marpurgs 1755 model; whereas Marpurgs chords are endlessly adaptable but often

theoretically dubious, Dehns chords are more ontologically secure but less analytically

adaptable.

Marpurg, Sorge, and Dehn have presented three different ways of and

justifications for generating extended triads: Marpurg generates them by subposing the

various intervals of a dominant seventh chords below a fundamental-bass carrying

dominant seventh, Sorge stacks thirds upwards from a chordal root by drawing pitches

from the overtone series, and Dehn places thirds on either side of a diminished triad with

the practical aim of categorizing the extended triads as Hauptakkorde. Even though we

have not yet looked at theorists who apply modified-basic-types thinking to the Parisian

modernist repertoire, in the analytic implications of these three ways of generating

extended triads, we see the range of issues that any application of modified-basic-types

strategy must contend with: What justifies/motivates the modification of the basic chord

types? How much modification can the basic chord types withstand (either in terms of

extension or chromatic alteration) before their identity is compromised? How flexible are

the resulting modified basic types (i.e. are they freely invertible, and on what scale

degrees/in what contexts can they occur)? These parameters will structure the following

35 Dehn specifically notes that the ninth chord is also not suitable for deceptive cadential progressions, as the necessary resolution o f the ninth ((die Auslosung der Terzdecime der Tonart) would produce a seventh chord (Nebenseptimenakkord) above the submediant. (Ibid., 223.)

35

discussion of late-nineteenth-, twentieth-, and twenty-first-century applications of the

modified-basic-types strategy.

Issues o f Generation

After Dehn, justification for extended-triad manifestations of the modified-basic-

types strategy is sought either in pragmatic appeal to a stacked-thirds approach or in the

overtone series. A pithy justification of the stacked-thirds approach is made by Robert

Ottman: The principle of chord construction by adding thirds can be continued past the

triad and the seventh chord to include chords of the ninth, the eleventh, and the thirteenth.

. . No further additions are possible, since another third duplicates the root two octaves

higher;36 similarly unadorned statements are found in the work of Emile Durand,

Theodore Dubois, and Leon Dallin.37 While this justification is parsimonioius and

pedagogically useful, it does not necessarily illuminate the true nature and behavior of

extended triads, a fact alluded to by Stefan Kostka and Dorothy Payne in their textbook

Tonal Harmony: Just as superimposed 3rds produce triads and seventh chords,

continuation of that process yields ninth, eleventh, and thirteenth chords (which is not to

say that this is the manner in which these sonorities evolved historically). The

introduction to this dissertation has also suggested that considering stacked thirds as

36 Robert W. Ottman, Advanced Harmony: Theory and Practice, 5th ed. (New Jersey: Prentice Hall, 2000): 292. Vincent Persichetti demonstrates how the stack o f thirds can be extended higher if the would-be- repeat tone is chromatically altered; he does admit, though, that these chords are so unwieldy as to be of limited use, mostly in parallel harmony or as harmonic punctuation. Vincent Persichetti, Twentieth- Century Harmony: Creative Aspects and Practice (New York: W.W. Norton and Co., 1961): 85-87.37 Emile Durand, Traite complet d harmonie theorique et pratique (Paris: Leduc, 1881 ):71; Theodore Dubois, Traite d harmonie theorique et pratique (Paris: Heugel, 1921): 23;; Leon Dallin, Techniques o f Twentieth Century Composition (Dubuque, I A: Wm. C. Brown Company, 1957): 60. Dallin adds a little more justification to the stacked-thirds model o f additive harmony by claiming it produces chords that are most similar to common-practice chords: Contemporary chords built in thirds have the closest relation to conventional harmonies since they continue the process by which triads and seventh chords are constructed, superimposition o f thirds. (Dallin, 60).38 Stefan Kostka and Dorothy Payne, Tonal Harmony, 4th ed. (Boston: McGraw-Hill, 2000): 436.

36

foundational for additive harmonies often fails to produce either elegant or pragmatic

labels for many additive chords in the Parisian modernist repertoire.39

A acoustic basis for extended triads is argued because, past the triad, the new

pitches that appear in the overtone series arrive in a stack-of-thirds order - the 6th partial

forms the (out-of-tune) interval of a compound minor 7th above the fundamental, the 8th

partial forms a compound major 9th, the 10th partial forms a compound augmented 11th,

and the 12th partial forms a compound major 13th (Figure 1.14). This theory has been put

forward by A. Eaglefield Hull, Ludmila Ulehla, and Rene Lenormand among others.40

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

Issues o f Modification

Creating extended triads out of basic chords is not the only way to modify basic

chord types in order to account for novel surface formations; a different application of the

modified-basic-types strategy is to chromatically alter chord tones in order to create new

verticalities whose identity and behavior is still related to a basic chord. This approach

39 Ludmila Ulehla locates the dissolution o f the triadic model at the 11th chord, where the inclusion o f the natural 11th above the root can necessitate the omission o f the chordal third (a situation Ulehla insists is explained by the desire not to have a suspensions projected tone o f resolution already present in another voice, not the presence o f a minor 9th between chordal third and 11th): The natural eleventh becomes a chord tone in Modem styles in which it frequently replaces the third, and in so doing, abandons the former Classical concept o f triadic sound. (Ludmila Ulehla, Contemporary Harmony [New York: The Free Press, 1966]: 90.)40 Ren6 Lenormand, A Study o f Modern Harmony, trans. Herbert Antcliffe (London: Joseph Williams, Ltd., 1915): 7;; A. Eaglefield Hull, Modern Harmony: Its Explication and Application (London: Augner, Ltd., 1915): 94; and Ulehla, 59. Dallin a