Polytonal Closure in the Music of Darius Milhaud and Howard Swanson 

 

 

by 

Connor Fast 

 

 

A thesis accepted and approved in partial fulfillment of the 

requirements for the degree of 

Master of Arts 

in Music Theory 

 

Thesis Committee: 

Dr. Stephen Rodgers, Chair 

Dr. Drew Nobile, Member 

Dr. Abigail Fine, Member 

 

University of Oregon 

Spring 2025 

 

  



2 

 

 

 

 

 

 

 

 

 

 

 

 

 

© 2025 Connor Fast 

This work is openly licensed via CC BY 4.0. 

 

 

 

 

 

 

 

 

 

 

 

 

 



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THESIS ABSTRACT 

Connor Fast 

 

Master of Arts in Music Theory 

 

Title: Polytonal Closure in the Music of Darius Milhaud and Howard Swanson 

Recent years have seen a renewed interest in cadence and closure in post-Classical-era 

music, especially Romantic and neo-tonal repertoire (Caplin 2018 and 2024, Rodgers and 

Osborne 2020, Eng 2019). One subset of neo-tonal music that poses special challenges regarding 

closure is polytonal music. Polytonality relies on the establishment of separate key centers that 

can work both independently and in conjunction with one another, which raises many questions 

concerning cadence and closure—principally, what do we consider a cadence in polytonal music, 

and is closure possible in only one key or can it happen in multiple keys at once? 

My thesis addresses these questions by examining the closural strategies utilized by 

French composer Darius Milhaud and African-American composer Howard Swanson. Drawing 

upon studies of polytonality by Peter Kaminsky (2004) and studies of neo-tonal closure by Clare 

Eng (2019), I outline three types of closure that appear in Milhaud’s and Swanson’s polytonal 

works: (1) monoclosure, where closure is reached in one of the established keys, but the other 

key(s) does not achieve any closure, either stopping mid phrase or continuing uninterrupted; (3) 

polyclosure, where simultaneous closure occurs in two (or more) separately established keys; and 

(3) converging closure, where multiple established keys converge at the point of cadence to 

achieve closure in one unified key.  

After developing this analytical approach, I apply it to large-scale works by Milhaud and 

Swanson. First, drawing on the Sonata Theory of James Hepokoski and Warren Darcy (2006), as 



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well as writings about twentieth-century sonata form by Joseph Straus (1990) and Damien 

Blättler (2024), I analyze the first movement of Darius Milhaud’s Sonata for Flute, Oboe, 

Clarinet, and Piano, Op. 47-I; this analysis includes a meta-narrative analysis founded on 

theories of agency theory by Robert Hatten (2018) and Edward Klorman (2016). Then, I turn to 

Swanson’s Fantasy for Soprano Saxophone and String Orchestra–a work that has never been 

published or performed, which I found among Howard Swanson’s papers in the Amistad 

Research Center at Tulane University. After developing a model of analyzing Swanson’s unique 

polytonal idiom in general terms, building upon Marsha Reisser (1989) and Lee Cronbach 

(1981), I use my theory of polytonal closure to show how multiple layers of formal function 

emerge out of this 350-measure single-movement work. 

 



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ACKNOWLEDGMENTS 

 I am deeply indebted to my committee chair and advisor Dr. Steve Rodgers for his 

dedication in overseeing this project. I could not have done this without him; his care and 

guidance were invaluable in navigating this process. I am also extremely grateful to the other 

members of my committee, Dr. Drew Nobile and Dr. Abigail Fine, for their insights and patience 

through the development of this thesis. Additionally, this project could not have progressed 

without the research support of the School of Music and Dance (SOMD) at the University of 

Oregon.  

 A very special thank you to the Amistad Research Center at Tulane University. The 

invaluable work of the archivists/librarians there made my research into Howard Swanson 

possible; they provided every manuscript I have from Howard Swanson. Amistad is the largest 

archive of Black history in America, and they are experiencing constant attacks from hegemonic 

powers on their funding, legacy, and institution; learn more at saveblackhistory.org.   

 Lastly, I’d like to express my gratitude to my friends, family, and loved ones for keeping 

me loved the whole time. My family and friends displayed a continued enthusiasm in my work, 

whether they have musical training or not, and listened to my stream of consciousness with care 

and attention.  

    

  



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DEDICATION 

Dedicated to my fiancée, Mireya Leilani, without whom I could not have written this. 

This thesis would not exist today if she had not encouraged me to continue my education, and 

she continues to support me through every step of the way. 

Through the arduous process of graduate school, you have supported me and kept me 

sane. Through the process of writing this thesis, you stayed with me during the late nights and 

difficult times. You planned a wedding while I couldn’t get my head out of the books. We 

educate ourselves for a better future, and I can’t wait to live and celebrate that future with you. 

I’ll see you there, my Love.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 



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Table of Contents 
Chapter 1: Introduction ............................................................................................................. 12 

Methodology ..................................................................................................................... 16 

On Key Priority ..................................................................................................... 17 

On Closure ............................................................................................................ 19 

Chapter 2: Three Methods of Polytonal Closure: Analytical Vignettes ................................. 23 

Monoclosure ..................................................................................................................... 23 

Dirge, from Suite for Cello and Piano, Howard Swanson .................................... 23 

Polyclosure ........................................................................................................................ 29 

Le Boeuf Sur le Toit, Darius Milhaud................................................................... 30 

Joyeux, from Sonata for Flute, Oboe, Clarinet, and Piano, Darius Milhaud ........ 32 

Converging Closure .......................................................................................................... 37 

Botafogo, from Saudades de Brasil, Darius Milhaud ........................................... 37 

Pantomime, from Suite for Cello and Piano, Howard Swanson ........................... 40 

Dirge, from Suite for Cello and Piano, Howard Swanson .................................... 45 

Chapter 3: Case Study: Sonata Form ....................................................................................... 49 

Introduction ....................................................................................................................... 49 

Polytonality, Sonata Theory, and Disability .......................................................... 49 

Contextual Conceptions of Neoclassicism, Polytonality, and Sonata Form ......... 52 

Exposition ......................................................................................................................... 57 

The Primary Theme .............................................................................................. 58 

The Transition ....................................................................................................... 64 

The Secondary Theme .......................................................................................... 65 

Development ..................................................................................................................... 72 

Recapitulation ................................................................................................................... 75 

Conclusion ........................................................................................................................ 79 

Chapter 4: Case Study: Simple Forms ...................................................................................... 81 

Introduction ....................................................................................................................... 81 

Noret as Rosetta Stone ...................................................................................................... 83 

Closure in Fantasy for Soprano Saxophone and String Orchestra .................................. 97 

Section A ............................................................................................................. 101 

Global Closure ........................................................................................ 103 



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Intermediate Closure ............................................................................... 107 

Section B ..............................................................................................................114 

Global Closure .........................................................................................115 

Intermediate Closure ............................................................................... 122 

Section C ............................................................................................................. 128 

Global Closure ........................................................................................ 131 

Intermediate Closure ............................................................................... 135 

Chapter 5: Conclusion .............................................................................................................. 138 

References .................................................................................................................................. 142 

 



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List of Figures 

Figure 1: Milhaud’s Le Boeuf sur le Toit, closure in Gb major and C major. ............................... 13 

Figure 2: Dirge, beginning of the B section. ................................................................................. 24 

Figure 3: Dirge, closure in the middle of the B section. ............................................................... 25 

Figure 4: Various cello monocloses within Dirge against the Db-major strand’s inner 

counterpoint. ................................................................................................................................. 29 

Figure 5: Milhaud’s Le Boeuf, closure in Gb major and C major, condensed and annotated. ....... 30 

Figure 6: Joyeux, mm. 70–78. ....................................................................................................... 33 

Figure 7: Milhaud’s Botafogo, mm. 1–14. .................................................................................... 38 

Figure 8: Recomposition of Botafogo, mm. 1–14......................................................................... 39 

Figure 9: Swanson’s Pantomime, B section. ................................................................................. 41 

Figure 10: Polytonal map of Pantomime, B section. .................................................................... 41 

Figure 11: Converging closure in Pantomime, beginning of B section. ....................................... 42 

Figure 12: Converging closure in Pantomime, end of B section. ................................................. 44 

Figure 13: Converging closure in Dirge. ...................................................................................... 45 

Figure 14: Two possible interpretations of closure in Dirge’s B section. ..................................... 48 

Figure 15: Op. 47-I, mm. 1–19. .................................................................................................... 59 

Figure 16: Op. 47-I, mm. 20–30, MC and S-zone start. ............................................................... 67 

Figure 17: Op. 47-I, mm. 38–42. .................................................................................................. 70 

Figure 18: Op. 47-I, mm. 43–52. .................................................................................................. 71 

Figure 19: Op. 47-I, mm. 65–68. .................................................................................................. 74 

Figure 20: Op. 47-I, mm. 76–90. .................................................................................................. 77 

Figure 21: Op.47-I, map of C strand’s assimilation into Ab strand. .............................................. 78 



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Figure 22: Swanson’s Noret (Undated). ........................................................................................ 84 

Figure 23: Diatonic analysis of Noret. .......................................................................................... 85 

Figure 24: Swanson’s Short Compositions, pg. 1. ........................................................................ 88 

Figure 25: Short Composition’s palindrome in E & B. ................................................................. 89 

Figure 26: Noret palindrome. ........................................................................................................ 90 

Figure 27: Noret mm. 1–3, tonal strands annotated. ..................................................................... 93 

Figure 28: Nocturne 1, unstratified strands................................................................................... 95 

Figure 29: Nocturne 1, stratified. .................................................................................................. 96 

Figure 30: Noret, stratified. ........................................................................................................... 97 

Figure 31: Fantasy form diagram. .............................................................................................. 100 

Figure 32: Fantasy, opening. ...................................................................................................... 101 

Figure 33: Fantasy, A section, Moderato.................................................................................... 102 

Figure 34: Fantasy, Section A, global close. .............................................................................. 104 

Figure 35: Fantasy, Andante 1 closure, mm. 16–17. .................................................................. 108 

Figure 36: Fantasy, Moderato, mm. 44–46. ............................................................................... 109 

Figure 37: Fantasy, Moderato, harmonic closure. .......................................................................110 

Figure 38: Fantasy, Andante 2, mm. 67–69................................................................................. 111 

Figure 39: Fantasy, Andante 2, mm. 68–69.................................................................................113 

Figure 40: Fantasy, formal diagram of the A Section/first movement. .......................................114 

Figure 41: Fantasy, Larghetto, mm. 193–202. ............................................................................116 

Figure 42: Fantasy, Larghetto, rhythmic churn, mm. 145–152. ..................................................118 

Figure 43: Fantasy, Larghetto, mm. 198–203. ........................................................................... 121 

Figure 44: Fantasy, Largo, mm. 135–137. ................................................................................. 123 



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Figure 45: Fantasy, Largo, mm. 130–134 .................................................................................. 126 

Figure 46: Fantasy, formal diagram of the B section/second movement. .................................. 128 

Figure 47: Fantasy, formal diagram C section/third movement. ................................................ 128 

Figure 48: Fantasy, Agadio, mm. 201–204. ............................................................................... 129 

Figure 49: Fantasy, Andante (3), mm. 231–234. ........................................................................ 130 

Figure 50: Fantasy, Adagio Expressivo, mm. 255–257. ............................................................. 130 

Figure 51: Fantasy, Andante Espressivo, mm. 222–224. ........................................................... 131 

Figure 52: Fantasy, global closure in Largo, mm. 351–355. ...................................................... 132 

Figure 53: Fantasy, Largo (2), mm. 345–347. ............................................................................ 134 

Figure 54: Fantasy, Adagio, mm. 228–230. ............................................................................... 136 

Figure 55: Fantasy, Andante Espressivo, mm. 328–331. ........................................................... 137 

  



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Chapter 1: Introduction 

Figure 1 shows the ending of a phrase within Darius Milhaud’s Le Boeuf sur le Toit (simply Le 

Boeuf from here), where two keys are cadencing simultaneously. As the strings and woodwinds, 

in the key of Gb major, approach rehearsal D, they move through a cadential progression 

(V6/4=5/3–I), landing on a root-position tonic at the downbeat of rehearsal D. All the while, the 

French horns and trumpets play the same line as the woodwinds a tritone away, in the key of C 

major. They move through the same cadential progression and land on the tonic at rehearsal D. 

Taken in isolation, each key center—grouped according to instrumentation (Gb major: 

strings/woodwinds; C major: brass)—communicates a sense of ending through the same 

mechanism of closure, a textbook perfect authentic cadence (PAC). However, within the whole 

texture, these key centers are at odds; in fact, they are as far apart as two key centers can be, 

situated at opposite ends of the circle of fifths. How do we experience closure in a passage like 

this, which closes in different keys at the same time? How does it communicate a sense of 

ending? And how do the closural techniques in this passage relate to closural techniques used in 

other polytonal works that also close in multiple keys at once? 

My thesis explores these questions, focusing on polytonal works by French composer 

Darius Milhaud (1892–1974) and African-American composer Howard Swanson (1907–1978). I 

choose these two composers for their diverse applications of polytonal closure, as they both 

employ similar methods but do so to different expressive effects. Through these composers 

specifically, I construct a theory of polytonal closure that describes mechanisms of closure, the 

expressive effect/utilization of closure, and the form building function of polytonal closure. 



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Figure 1: Milhaud’s Le Boeuf sur le Toit, closure in Gb major and C major. 

 

Darius Milhaud was a principal purveyor of polytonality in the early-to-mid twentieth-

century. His contribution to polytonality spurred the rise of neoclassicism and worked its way 

through his connection to the composer collective, Les Six. With Milhaud at the helm, 

polytonality became representative of neoclassicism and was integral to the development of the 

post-war neoclassical movement, with every composer of Les Six experimenting with it (Médicis 



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2005, 575). Milhaud’s philosophy on polytonality upheld neoclassical ideals, as he claimed it 

was the natural progression from diatonicism (Milhaud 1923, 7).1  

 Howard Swanson was a Black American composer who saw great success in his day but 

fell into relative obscurity following his death. Swanson received a master’s degree in 

composition from the Cleveland Institute of Music and spent a year studying with Nadia 

Boulanger. On the surface, his musical aesthetic is defined by strong neo-classical influences, 

structural forms, contrapuntal textures, and polytonality. However, deeper insight into his music 

has revealed a strong connection to Afro-American idioms and Black culture (Reisser 1989, 15; 

1982, 6). I have chosen the music of Howard Swanson as one of my primary focal points 

because of his distinct use of polytonality, which—through its interaction with African-American 

idioms—is unlike that of any other composer. 

 Milhaud, insofar as my project is concerned, is therefore a stand-in for the normative use 

of polytonality and polytonal closure in neo-classical music. Swanson, by contrast, represents a 

more idiosyncratic use of polytonality and polytonal closure. Together, they show the wide 

diversity of polytonal closure in the twentieth century. 

In researching polytonal closure, I expand upon and integrate two different realms of 

scholarly inquiry that have tended to remain separate but, when brought together, can deeply 

enrich our understanding of the mechanics of closure in passages like Milhaud’s Le Boeuf 

(Figure 1). The first scholarly realm is research on “key priority” in polytonal music, which 

explores the various factors that allow us to hear one key (among several superimposed keys) as 

predominant—but which has not generally considered closure to be one of those factors. The 

 
1 This parallels Schoenberg who considered atonality to be the natural progression from chromaticism, Milhaud 

explicitly draws this parallel in his article “Atonality and Polytonality,” 1923.  



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second area of research is work in the subfield of “new Formenlehre,” which has in recent years 

developed new frameworks for understanding closure in music after the Classical era but has not 

yet grappled with the many different closural strategies used in polytonal music (Caplin 2018). 

I engage with each of these bodies of scholarship to build the foundation my framework 

emerges from, and outline three types of polytonal closure: (1) monoclosure, where closure is 

reached in one of the established keys but not in the other key(s); (2) polyclosure, where 

simultaneous closure occurs in two (or more) separate keys; and (3) converging closure, where 

two or more established keys converge at the point of cadence to achieve closure in one unified 

key center. As I will show, Milhaud and Swanson utilize these techniques in a variety of different 

ways, with different expressive effects. Looking at their music with these categories in mind 

reveals the wide range of strategies that these composers use to create moments of closure, and 

also provides a possible starting point to explore similar strategies in the music of other 

composers who make use of polytonality. 

The second chapter of this thesis explores specific examples of polytonal closure, moving 

through my three categories one by one, to define the scope that polytonal closure covers. I 

analyze specific moments of closure, continuing to draw connections between theories of 

polytonality and priority, and theories of Formenlehre and closure. In doing so, I showcase the 

breadth of expressive effects that Milhaud and Swanson created through their use of polytonal 

closure.  

The third chapter is a case-study analysis of a polytonal sonata. I undertake the analysis 

of sonata form because of Sonata Theory’s interaction with, and requirement of, closure. I 

analyze Milhaud’s Sonata for Flute, Oboe, Clarinet, and Piano through my framework of closure, 

all the while considering how the work engages with sonata-form conventions. In my analysis of 



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Milhaud’s sonata, I examine the use of polytonal closure through the lens of virtual agency and 

multiple agency (Hatten 2018; Klorman 2016). I also incorporate perspectives from disability 

studies to produce a nuanced analysis that avoids a therapeutic conflict–resolution narrative 

(Straus 2018).  

The fourth chapter examines how polytonal closure functions within Howard Swanson’s 

music, and by extension, how polytonal closure works to project form. This chapter starts with 

an initial exploration of Swanson’s unique idiom and approach to tonality and polytonality 

through an analysis of his unpublished piece, Noret. Then, I dive into the levels of closure within 

Swanson’s Fantasy for Soprano Saxophone and String Orchestra to show how Swanson uses 

polytonal closure as a form-generating device. 

 

Methodology 

This thesis combines two disparate areas of music theory scholarship to construct a framework of 

polytonal closure: key priority and closure. Key priority is a loose framework of analysis 

developed for describing the relative strength of competing key centers in polytonal music. Prior 

to a theory of key priority, music theorists lacked a framework for analyzing the strength and 

perceptibility of competing tonal centers. Methods for analyzing how different keys were set 

against each other did and do exist, and date back to Milhaud’s own conception of polytonality.2 

This framework of key perceptibility is combined with modern approaches and conceptions of 

closure in post Classical music.  

 
2 See Milhaud’s “Atonality and Polytonality,” 1923. 



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On Key Priority 

Polytonality refers to the presence of multiple key centers perceived simultaneously; however, 

these key centers are not necessarily understood as having equal strength. First explored by 

David Huron (1989), later built upon by William Forde Thompson and Shulamit Mor (1992), and 

finally described in detail by Peter Kaminsky (2004), the concept of key priority provides a 

hierarchical framework for ordering keys in polytonal music according to their relative 

importance. In other words, it helps us to discern which keys are audible and dominant within a 

particular polytonal passage. Realistically, the relative dominance that one key holds over 

another can change from moment to moment, depending on contextual factors like dynamics, 

voice leading, and instrumentation. Since these factors change over the course of a piece, so too 

can key priority, making this a fluid analytical framework.  

Kaminsky argues that in a superimposed moment, multiple keys constantly vie for 

priority, and priority can change quickly. Kaminsky often uses the word “superimposed,” rather 

than “polytonal,” to describe such passages because, in his mind, “polytonal” implies that 

multiple key centers are being perceived simultaneously, whereas “superimposition” simply 

refers to the compositional act of putting two (or more) keys in contrast with each other, without 

making a claim as to whether one key dominates, or two keys are perceived at the same time. 

Works of superimposition may be polytonal, but they may also lend themselves to monotonal 

readings. In other words, all polytonal works come from superimposition, but not all 

superimposition results in polytonality. The act of superimposition is relatively straightforward: 

contrasting key centers are set against each other as superimposed chords, scales and scale 

fragments, and/or key signatures. Within any moment of superimposition, many factors 



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contribute to priority. The one most cited by Kaminsky is voice leading, but others include the 

choice of keys being superimposed, the instruments associated with the keys, and the ranges of 

the instruments. Through all these factors (and more), a moment of superimposition may be 

heard in multiple keys at once, i.e., polytonally. If those keys are heard as more or less equal in 

importance, then the passage may be referred to as exhibiting dual priority (Kaminsky 2004, 

241). If those keys are heard hierarchically (i.e., with one key predominating), then one key 

would exhibit primary priority and the other would exhibit secondary priority (Kaminsky 2004, 

240). Kaminsky does not carve out space for a tertiary priority, since his levels of 

superimposition include bass voice and upper voice(s), without room for a third layer. 

If music exists within a polytonal space, the concept of priority is always in effect. In 

Thompson and Mor’s research on the matter, they note that “listeners draw on long-form 

knowledge of tonal organization when listening to polytonal music. Internal representations of 

key structure are established rapidly by the immediate context” (1991, 70). When listening to a 

passage of polytonal music, a listener is constantly re-analyzing and re-understanding their 

perception of key importance, hierarchy, and priority. Kaminsky makes a similar point, writing 

that “the support for such a weighting process by a listener will hinge on contextual factors, 

including the manner of presentation of the conflicting parts, and the extent of their respective 

pitch organizations, rhythm, register and contour” (2004, 240). These are the factors that must be 

considered when determining the hierarchy of keys. However, one factor that Kaminsky does not 

consider is the effect of closure and cadence on priority. Nor do other scholars of polytonality 

treat closure as a central feature of their analyses.3  

 
3 In general, scholars writing about polytonality do not address cadence/closure directly, if they address it 

at all. Where closure is mentioned, it is not thematized as a main topic but mentioned briefly in the context of 

 



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On Closure 

Recent strides in the so-called “new Formenlehre” have sparked new conversations about 

closure in post-Classical era music. For example, in his article “Beyond the Classical Cadence” 

(2018), William Caplin explores the “fate” of the Classical cadence in early Romantic music, 

identifying seven characteristics of Romantic style that affect cadence and closure. Aspects of 

post-Classical closure are further explored by Hyland (2009), Smith (2016 & 2020), Rodgers and 

Osborne (2020), Rodgers (2022), and Caplin (2024). As studies of closure stretch into the music 

of the twentieth century, the sheer diversity in the modes of closure becomes an analytical 

challenge. Some twentieth-century composers still write music that is in dialogue with Classical 

closural conventions, of course, and Caplin’s ideas about Classical and Romantic closure thus 

remain relevant to the analysis of this music. However, for those neo-tonal composers who don’t 

rely on Classical conventions, the study of closure in their music requires a wider outlook and 

approach.   

Clare Eng, in her 2019 article “The Problem of Closure in Neo-Tonal Music,” offers just 

such an approach. Her work bears heavily on my own as I address strategies of closure in 

repertoire that could be considered neo-tonal. Eng addresses shared systems of closure in the 

music of Benjamin Britten and Béla Bartók, exploring how both composers use similar systems 

in different ways. The “problem” that Eng identifies is that neo-tonal closure engages with so 

many conventions of closure that it can be difficult to define closural strategies that apply to a 

 
individual analyses, in order to show how closure is achieved in a particular passage. Kaminsky (2004), 

for example, contains only two mentions of cadence/closure. In both instances, cadence/closure is not treated as a 

factor that contributes to key priority but rather as a byproduct of key priority (e.g., two superimposed keys have 

already established dual-priority; therefore, they can successfully pull off a cadence in two keys). Other scholars 

mention how particular polytonal passages close—with rhythmic resolution (Reisser 1989) and dissonance 
vectors (MacFarland 2009), for instance—but do not generalize about strategies of polytonal closure as a whole.  



20 

 

broad range of neo-tonal repertoire.4 Different composers, in short, use different strategies, and 

even where the same strategies appear in two different works, the effects of those strategies may 

be radically different. She writes:  

I propose that closure is a relational construct that engages with conventions on multiple 

levels. Conventions that influence the quality of an ending come not only from a stylistic 

corpus or piece, but also from a composer’s normative practice. […] Because similar 

endings engage in different conventions, and because conventions can interact in 

different ways to influence the quality of an ending, neo-tonal endings with similar 

features should not ipso facto be assumed to have the same closural quality (2019, 285). 

Building upon Eng’s discussion, I would argue that the polytonal conventions of closure that I 

have defined above (monoclosure, polyclosure, and converging closure) apply principally to the 

patterns of closure within Milhaud’s and Swanson’s music (i.e., they are first and foremost 

composer conventions). And even where similar conventions appear across the music of these 

two composers, those conventions can’t be assumed to have the same effect (or, to borrow from 

Eng, the “same closural quality”). It is the way these conventions are used in the context of 

individual works that matters most. I also do not claim that all polytonal music uses the three 

strategies I have outlined (i.e., I do not argue that they are corpus conventions). Still, I hope that 

these categories might be useful to other music analysts exploring how other polytonal 

composers use closure, should they wish to establish broader corpus conventions. Furthermore, I 

make no claims as to the hierarchy or strength of any closural pattern over another (i.e., 

polyclosure isn’t necessarily stronger closure than monoclosure). In other words, I do not rank 

these types of closure according to syntactical strength, as Caplin does with Classical cadences 

(see especially Caplin 2004). 

 
4 Eng outlines three layers of convention in neo-tonal music: (1) opus conventions, conventions that exists within 

few or single pieces; (2) composer conventions, conventions utilized by a specific composer across their 

compositions; and (3) corpus convention, genre-level conventions that are ubiquitous within a particular genre. 



21 

 

That being said, I do borrow Agawu and Caplin’s terminology of rhetorical strength. 

Which is defined by Agawu as “the set of devices that emphasize the close…” (1991, 67). This 

use of “rhetorical”—in describing closure—is expanded by Caplin to mean, “[the cadence that] 

has a unique compositional realization entailing the entire range of musical parameters, including 

rhythm, meter, texture, intensity, and instrumentation” (2004, 107). The conventions below may 

not have a set hierarchy of syntactical strength, but we can nonetheless gauge their rhetorical 

strength on a case-by-case basis as they are applied by Milhaud and Swanson. I do so by 

examining the same elements relevant to key priority, such as rhythm, dynamics, melodic 

contour, register, texture, and timbre. I consider rhetorical strength in two layers: first, the 

rhetorical strength of individual key strands and their ability to communicate closure in isolation, 

and second, the rhetorical strength of the whole polytonal close (whether mono, poly, or 

converging). In other words, I look at the rhetorical strength of closure in the individual key 

strands, and in the total composite of key strands.   

Finally, I define closure much more broadly than many scholars working in the subfield 

of “new Formenlehre,” for the precise reason that composers of polytonal music enact musical 

closure in so many diverse ways. Sometimes these composers do indeed evoke Classical-style 

cadences (PACs, IACs, and HCs), but just as often they use non-Classical harmonic progressions 

to create a sense of ending, or they use parameters other than harmony, such as rhythm, register, 

melodic contour, timbre, and so on. Therefore, I use the term “closure” quite loosely, more in 

line with Eng’s conception of closure. I define Closure as a feeling of rest, or a feeling that 

something has ended. Adopting a loose definition such as this enables me to focus on the 

closural mechanisms of each individual work, and, in turn, weigh the strength of various 

moments of closure intuitively, considering the musical parameters most relevant to any 



22 

 

particular situation (rather than importing Classically-derived definitions based largely on 

harmonic progression). This approach parallels the fluidity of the key priority framework, as both 

call for case-by-case analysis of present and relevant factors, ultimately allowing the music to 

dictate whether we are hearing an example of monoclosure, polyclosure, or converging closure. 

  



23 

 

Chapter 2: Three Methods of Polytonal Closure: Analytical 

Vignettes 

 

Monoclosure 

Monoclosure is when one key within a polytonal work reaches a moment of closure, but the 

other key does not (or, in cases where more than two keys are present, when only one key 

achieves closure). In the music of Milhaud and Swanson, this method of closure most often 

appears as local closure, at the end of a phrase, and occasionally as intermediate closure, at the 

end of a section, to borrow terminology from Kofi Agawu (1987, 6). This single key center can 

come to a close through various methods. In the case of Milhaud, a full cadence often appears in 

a superimposed key while the other voices (in other keys) continue their themes. In the music of 

Howard Swanson, a melodic line often appears in a single voice and ends contrapuntally while 

the other voices continue. As we’ll see, Swanson uses this technique extensively with different 

expressive effects, both rhetorically strong and weak.  

 

Dirge, from Suite for Cello and Piano, Howard Swanson 

The third movement of Howard Swanson’s Suite for Cello and Piano, titled Dirge, is built upon 

this idea of monoclosure and uses it consistently, particularly in the B section. In this movement 

there are four layers of superimposition as analyzed by Marsha Reisser (1989). The cello is in F 

minor, the piano’s treble clef is in Db major, and the piano’s bass clef is in two keys in different 

registers, A minor in the upper register and E in the lower (the lower register projects neither 

major nor minor because it presents only open fifths and octaves). Figure 2 depicts the beginning 

of the B section where each key strand is initially expressed. The accompanying rhythm is 



24 

 

consistent throughout the section: the piano (in 6/4 time) hits every beat in a repetitive pattern 

that descends in register as we move through each polytonal strand in the piano. Beat 1 is a chord 

in the key of Db major (with some chromatic motion), beat 2 is an A-minor chord, and beat 3 is 

an open fifth over E with the octave doubled. Beats 4–6 repeat this same pattern descending 

through the register. The consistency and unyielding energy of this cascading accompaniment 

pattern provides a backdrop for the monocloses to come. 

 

Figure 2: Dirge, beginning of the B section. 

 

One of the more prevalent moments of monoclosure in Dirge occurs in the middle of the 

B section, depicted in Figure 3. The F-minor cello plays an improvisatory-type melody over the 

cascading polychords of the piano. For this phrase to be an example of monoclosure, one of the 

strands will have to exhibit some type of closure that the other strands must not, and that closing 

strand must have enough priority to be understood as its own individual key strand. In other 

words, whatever strand closes does not have to have primary priority (it can have secondary 

priority), it just cannot be assimilated into the other strand(s).   



25 

 

 

Figure 3: Dirge, closure in the middle of the B section. 

 

The cello plays a single phrase conveying a sense of closure in F minor. The arch of this 

phrase builds tension and anticipation, which is resolved in the final bar. Without harmonic 

support, the closural strength of this example is achieved chiefly by the contour of the line. The 

phrase opens on scale degree 5 before abruptly jumping up a fourth to scale degree 1, then 

immediately jumps up again, this time an octave. Within the first two beats, a substantial amount 

of energy is built up. The phrase then descends to 1, the octave below, touching on every note of 

an F minor-pentatonic collection in a single sextuplet gesture: F–Eb–C–Bb–Ab–F. That quick 

descent includes the fastest rhythmic motion in the movement thus far, and the cello’s next 

measure (after the trill) increases the surface rhythm even more. But finally, that tension is 

released in the final bar of the phrase, with a slow 3–1 gesture. Notice that this final F completes 

a longer-range descent through an F minor-pentatonic scale (marked in the pink overlay). The 

contour, combined with the relaxing of motion, creates a strong sense of rest and closure at that 

last F. 



26 

 

Beneath this melodic line that comes to a point of rest are dissonant polychords that do 

not reach a resolution. The key centers of E and A minor are static—they simply play single 

chords and fifths—and while the Db strand does exhibit some harmonic motion, it too does not 

reach a tonal resolution (or any other type of resolution) in the final bar of the example. Nor is 

there rhythmic closure, as the cascading pattern keeps consistently moving forward. Finally, 

there is no closure contrapuntally. Throughout this section, the middle voice of the Db-major 

strand descends by step before leaping up and restarting its descent (most often as Fb–Eb–D–Db). 

This pattern is initially depicted at the beginning of the movement in Figure 2 above (D–Db–Eb–

D–Db). As it coincides with the F-minor cello’s closure, the Db-major strand is restarting its 

contrapuntal descent, not ending it. Therefore, against the descending melody of the F-minor 

cello, the cascading piano makes no indication that it’s reaching any sort of rest or conclusion.   

I have shown that only one of the four strands of this excerpt achieves closure, but what 

about the relative priority of those strands? When we listen to music like this, we are constantly 

listening to strands vying for priority and recalculating their hierarchy (Thompson and Mor, 

1992). Because the E and A-minor strands are so static, they have the least priority.5 Therefore, 

the F-minor cello strand and the Db-major piano strand, which do feature some motion, are the 

two strands vying for the most priority. Db major and F minor are closely related keys, so if one 

strand is going to break away from the other, it is going to require a considerable effort.  

It is the F-minor cello that does the work to break away from Db major. The interaction 

between the Db-major piano strand and the F-minor cello is deeply engaging, as the cello is 

constantly distancing itself from Db-major harmony. This is the most telling at the harmonic 

 
5 While Kaminsky doesn’t outline a tertiary priority, that is what I would label the E and A-minor strands here. As 

key strands/centers that may be audible but aren’t really pulling focus.  



27 

 

changes in the Db-major strand. In the first measure of Figure 3, the cello’s C is part of the 

piano’s Vb5, but it quickly moves away from that association by moving to F, resisting 

assimilation. Later, on the fourth beat of that same measure, the cello again finds itself a part of 

the Db strand as it plays an Ab against a Db tonic chord, but it again resists the harmony by trilling 

up to Bb, further thwarting harmonic assimilation. Finally, at the last bar, as the cello finds itself 

as the root of the Ab chord, it calmly moves away to its own tonic. In doing so, the cello 

consistently resists Db major with contrapuntal motion more indicative of F-minor.  

However, the cello doesn’t just distance itself through harmonic or contrapuntal means; 

its timbre and effect are also completely separated from the piano. As discussed above, the cello 

line is restless in how it builds tension early and quickly, holds onto it, then finally releases the 

energy at the very end of the phrase. Compared to that, the Db-major piano strand does not 

exhibit the same energy. The Db-major strand’s only motion is chromatic or small leaps in the 

middle voice, and its rhythm stays mostly constant. Timbrally, there are no dynamic or 

articulative indications that set it apart from the static harmonies of A and E. Through rejected 

harmonies and individualized tensions, the F-minor cello affirms primary priority, and the Db-

major piano claims secondary. In short, the F-minor cello puts in the work to separate itself from 

the other strands, claiming primary priority, where the Db-major piano does not exhibit the same 

work. 

A singular key strand that distances itself from others and communicates a sense of 

rest/closure is an example of monoclosure. In this example, it isn’t just any distance; it is a 

substantially defying distance. And it isn’t just a sense of rest created through contour; it is a 

sense of rest built through growing and resolved tension. Swanson uses monoclosure to great 



28 

 

rhetorical and expressive effects in this example. Despite playing in each other’s range and in 

closely related keys, there is a feeling of great distance between the two strands. Even as the 

cello resolves, the distance remains, and the strong sense of rest in the cello can predominate. 

The predominance of this monoclosure furthers the cello’s primary priority which is utilized 

continually throughout the B section. As the cello keeps coming to moments of monoclosure, it 

reinforms to the ear that it’s something to listen to. The monoclosure informs the status of 

primary priority as much as it relies on it. 

Dirge contains numerous similar examples of monoclosure within its B section, which 

comes to define the expressive effect of the movement. In each instance of monoclosure, the 

piano continues its progression of polychords through its repetitive comping rhythm, while the F-

minor cello continually defies Db major and reaches similar moments of closure on scale degrees 

1, 3, and 5. A few are recreated below in Figure 4, where the contrapuntal line in Db is notated 

beneath. In the greater context of this piece, this monoclosure exhibits the idea of the F-minor 

cello fiercely rejecting Db major and coming to rest against such an unyielding dissonance. It 

could be speculated that this pattern relates to the name of this movement, Dirge: a movement 

depicting an improvisatory lament set against a dissonant marching backdrop refusing to relent. 

Each monoclosural moment in the cello is a brief moment of local closure, but the rhetorical 

strength of that closure can only get so far against the dissonance and momentum of the piano. 

This feud will eventually come to a head at the end of the B-section—more on that in the section 

on converging closure.   



29 

 

 

Figure 4: Various cello monocloses within Dirge against the Db-major strand’s inner 

counterpoint. 

 

Polyclosure 

Polyclosure is the presence of two (or more) tonal centers closing simultaneously. This type of 

closure has a wide-ranging ambiguity that is often dependent on the relationship between the 

superimposed keys; keys that are distant (not closely related, lacking common tones) from each 

other may project relatively weak polyclosure, because dissonance between the keys undercuts 

true closure in one or both keys. On the other hand, rhetorically stronger polyclosure is typically 

achieved when the superimposed keys have a more consonant relationship with each other. 

 



30 

 

Le Boeuf Sur le Toit, Darius Milhaud 

The introductory example to this thesis provided an instance of rhetorically weak polyclosure 

(condensed and annotated below in Figure 5). To reiterate, in this excerpt from Milhaud’s Le 

Boeuf two strands are superimposed: a Gb-major strand played by the strings and woodwinds and 

a C-major strand played by the brass. The measure before rehearsal D, both strands progress 

through a cadential progression, landing on their individual tonics on beat 1 of D. 

 

Figure 5: Milhaud’s Le Boeuf, closure in Gb major and C major, condensed and annotated. 

 

The Gb-major strand assumes primary priority over C major through a number of factors. 

Registrally, Gb major covers the widest range, as it is expressed as low as the double bass and as 

high as the flute, whereas the trumpets and French horns of the C-major strand stay within the 

same register. Moreover, Gb major is prevalent across a variety of timbres, including the whole 

range of strings, the double reeds, the clarinet, and the flute. By contrast, C major is expressed 



31 

 

only in the high brass, trumpet and French horn. Finally, C major is simply outnumbered by Gb 

major: more voices sound in the Gb-major strand, reinforcing it as primary priority.    

However, the C-major strand is able to resist assimilation into Gb major and keep 

individual secondary priority. The tritone-related keys of C major and Gb major are, of course, far 

apart intervallically—far enough that it is difficult to rationalize and hear C major as an 

extension of Gb major within this context. There are no contextual clues that would indicate C as 

a distant altered extension to Gb, so they remain separate key centers. Also contributing to this 

sense of separation is the timbral difference between the Gb-major strand and the C-major strand. 

As mentioned above, C major is expressed only in the high brass, but those instruments exist 

within a timbral space that isn’t encroached upon by the rest of the orchestra. The brass may be 

outnumbered, but they’re still very clearly audible within the texture of the orchestra.6 

This hierarchical priority continues through the phrase until the dual cadence occurs at 

rehearsal D. As I noted earlier, both keys proceed through the same harmonic progression 

concurrently, each leading to a perfect authentic cadence. Formally, this is also the end of a 

section, as the very next melody heard is the refrain of the rondo form (Corrêa do Lago 2002, 

20). So, this fits firmly within my definition of closure: something has ended, which is further 

confirmed by a new beginning. This is characteristic polyclosure: two keys, tonally separated 

from each other, achieving individual closure simultaneously, and in the composite, express an 

overall closure.   

 
6 Rudolph Johnson (2011) categorizes the trumpet and horn as part of the “Power” class, as opposed to the 

“standard” class that the strings, flute, and bassoon fall under. They sound within their own distinct timbral spaces. 



32 

 

What, however, is the effect of that overall closure? Though each key closes emphatically 

(with the most syntactically strong cadence, a PAC, utilizing a syntactically strong cadence per 

Caplin 2004, in dialogue with Classical closural conventions); the combined effect is rather 

rhetorically weak, in large part because these two emphatic cadences seem to cancel each other 

out. The Gb-major strand may have had primary priority on the approach to the moment of 

closure—because of register, timbral diversity, etc.—but the sheer strength of closure in C major 

puts the strands ever so briefly on equal footing. In Caplinian terms (relevant because the 

cadence plays on classical convention), the syntactical strength of the C-major cadence is the 

same as that of the Gb-major cadence. In turn, the two cadences are competing for priority at the 

moment of polyclosure, thus muddying and weaking the composite closure.   

 

Joyeux, from Sonata for Flute, Oboe, Clarinet, and Piano, Darius Milhaud 

In contrast with Le Boeuf, the second movement of Milhaud’s Sonata for Flute, Oboe, Clarinet, 

and Piano, titled Joyeux, ends with an example of polyclosure that is rhetorically strong. By the 

end of the movement, shown in Figure 6, the oboe has dropped out and the piano plays in F# 

major. Superimposed against it, the flute is centered around A (there are no Cs or C#s in the flute 

melody to indicate whether it is in A major or A minor). These key centers are certainly closer to 

each other than C and Gb and can achieve closure that is overall rhetorically stronger.  

Both key centers are established independently from each other, stratifying the strands. 

The piano’s key center of F# major is expressed through repeated root position F#-major chords, 

it’s monotonous, but steadily focused. The flute’s key center of A is established melodically. The 

flute only plays 1, 4, and 5 of the key of A (until the penultimate measure). The absence of other 



33 

 

scale degrees in the flute part precludes the passage from a modal interpretation– major, minor, 

or otherwise– but does still point towards a tonal center. In application, the scarce number of 

scale degrees present in use does little else other than point towards a tonal center.  

 

Figure 6: Joyeux, mm. 70–78. 

 

While the flute and piano are firmly in their key centers, the clarinet hovers between the 

two tonal centers, fluctuating between F# minor and A major. The ambiguity in the clarinet part 

effectively contributes harmonic information to both the piano’s F# strand and the flute’s A 

strand. In the first few measures, the clarinet is playing an arpeggiated F#m7 chord (F#–A–C#–E). 

This could be interpreted as firmly residing in F#, albeit polymodally to the F#-major piano, but it 



34 

 

is better interpreted as alternating between F#-minor and A-major triads. The clarinet can 

therefore be fully understood as participating in two separate tonal contexts simultaneously, F# 

and A.  

More on the tonal ambiguity of the clarinet, motion in the clarinet part alternates with the 

motion in the piano and flute. When the flute and piano are both moving, such as the second and 

fourth measures in Figure 6, the clarinet idles on a common tone to both A major and F# major; 

when the piano and flute stop moving (either resting or idle), the clarinet restarts its motion with 

the F#m7. The effect is that the flute and piano are directly pitted against each other, in conflict, 

and the clarinet is offering a compromise. With the strands established, they must work towards 

their independent modes of closure. 

Motion and rhythm play a chief role in the F#-major strand’s closure, as its motion calms 

over the course of the phrase, before coming to rest. The piano plays F#-major chords, alternating 

between measures of quicker motion and measures of relative calm. The first five measures in 

Figure 6 alternate between half notes and syncopated rhythms. The sixth measure slows the 

syncopations down to quarter notes. More motion is expected in m. 8, but instead, the piano 

continues its half notes. This slowing of motion is accompanied by a long decrescendo from p to 

ppp, giving an all-around effect of winding down. Moreover, the half notes in the piano part 

create their own pattern of closure. Each second half note in mm. 1, 3, and 5 raises in octaves 

above A#3 and C#4. Once the motion has slowed in m. 7, the half note pattern plays through 

again, echoing the previous measures and projecting closure of the F#-major strand.  

The flute utilizes the same slowing of motion as it moves towards closure. It alternates 

between motion and rest, and every iteration of motion slowed. The motion is first in triplets, 



35 

 

then 8th notes, then quarter notes, and finally, closure is achieved in half notes. Along with the 

slowing of motion is a long dynamic descent that was also present in the piano. The A flute and 

F#-major piano parallel each other in their modes of closure, they both close in the same way. 

These parallels contribute towards the rhetorical strength of closure.  

The clarinet’s contribution to closure mostly goes towards the flute. When the clarinet is 

considered contributing to the A-strand, there is a more complete harmonic understanding. With 

the addition of the clarinets C# and F#, the A strand can fully be understood as A-major. The final 

pitch of the clarinet, C#, is easily interpreted as 3 of A major. Such an interpretation would give 

the flute harmonic closure as well. Moreso, the flute introduces its first new pitches in the 

penultimate bar, F# and G#. The introduction of these pitches, paired with the A/C# dyad in the 

final bar, is a typical diatonic closing gesture: 5 –6 –7–1. These rhythmic, dynamic, harmonic 

modes allow for both strands to reach a healthy amount of closure.  

Regarding priority, the flute is fighting an uphill battle in establishing its own tonal center 

without being assimilated into the piano. As mentioned above, the lowest voice is more naturally 

going to overtake the higher voices, and the burden of separation falls on the higher voice to 

distance itself from the lower (Kaminsky 2004).  

There are a variety of ways that the A-major tonal center distances itself from the F#-

major strand. Until the very end, the flute is playing only A’s, D’s and E’s, which are all distant 

overtones from F# (19th, 13th, and 7th respectively). While D and E are closer harmonics than A, 

they are deeply out of tune, almost a quarter tone out of tune in equal temperament. Therefore, 

the notes of the flute are less likely to be heard and understood as distant extensions over the F# 

chords in the bass. To understand these notes as harmonic extensions above F# major, the As, Ds, 



36 

 

and Es need to be interpreted as b3 (split third), b7, and b6. Such an interpretation would be a leap 

and imply F# minor more than anything. Moreso, the clashing of A in the flute against A# in the 

piano provides a healthy amount of dissonance to permeate between the two parts and keeps 

them separated.  

Furthermore, the A-strand flute actively resists F# minor assimilation. Kaminsky cites 

voice leading as one of the most important factors in resisting the assimilation of the lower voice 

(2004, 240). The flute here, with its limited pitch catalog, doesn’t commit to F#-minor voice 

leading. I would argue that it’s simpler for the ear to understand its melody as all 1, 4 and 5 in A, 

rather than a melody consisting only of b3, b6, and b7. If the flute melody had any D’s resolve 

down to C#, its voice leading may imply a more F# minor centered tonality, as D–C# would be 

interpreted as b6–5 in F#, rather than a 4–3 motion in A. Note that the clarinet does have this D–

C# motion, and is interpreted as centralizing both A and F#. By never having this motion, or any 

like it, the flute resists assimilating into F#-minor, effectively circumnavigating any draw towards 

an F# center, major or minor.7 

 As the voices oscillate between motion and rest, so too does the priority change. When 

the flute and piano are both in motion, priority is split into dual priority. When both are resting or 

idling, the clarinet begins its motion and ambiguously contributes to both tonal strands. This 

effectively makes the priority oscillate between a measure of dual priority and a measure of 

ambiguity, where both strands are vying.  

 
7 This seems to be one of the drawbacks of superimposition of close key centers. The Le Beouf above example 

couldn’t even accidentally voice lead like its counter, because the key centers are so far. The closeness of F# minor 

and A mean the composer must make heed to not accidentally imply even a relative/parallel relationship.  



37 

 

 The movement ends without either winning out. The final chord of the movement is a 

split third chord, containing both A# and A. Both the F#-major strand and A-strand are still 

present, yet despite such dissonance, the movement ends restfully. The two strands are able to 

remain separate from each other, but through their parallel modes of closure, can wind down to a 

rhetorically strong sense of rest.  

 

Converging Closure 

Converging closure is an instance where two (or more) keys have split priority but join together 

in one unified key at or before a moment of closure. As a result, converging closure interacts 

differently with key priority than both monoclosure and polyclosure. Both monoclosure and 

polyclosure involve the feeling of closure in (still) separated key strands that are continually 

vying for priority up to and through the moment(s) of closure. By contrast, in converging closure 

key strands converge into each other before, or at, the moment of closure, effectively ending the 

striving for priority. In many cases, because the closure stemming from converging closure can 

be heard and analyzed monotonally, it is often rhetorically stronger closure.  

 

Botafogo, from Saudades de Brasil, Darius Milhaud 

An example of this occurs in the first phrase of Milhaud’s “Botafogo,” the second piece in his 

Saudades de Brasil. In this excerpt, shown in Figure 7, the left hand of the piano is in the key of 

F minor, playing an ostinato pattern alternating between i and V. After two introductory 

measures, the right hand enters in F# minor, playing an ascending melody that quickly becomes 



38 

 

descending block chords. Kaminsky analyzes these descending block chords as alternating i and 

V7 in F# minor, though the dominant chords have no root (Kaminsky 2004, 242).  

 

Figure 7: Milhaud’s Botafogo, mm. 1–14. 

 

In Kaminsky’s own analysis of this excerpt, he awards it the coveted status of dual-

priority (coveted because it’s harder to achieve), where the F#-minor strand and F-minor strand 

have equal tonal focus for most of the phrase. Unlike previous examples, both strands in 

“Botafogo” are played by the piano, so timbre is far less of a contributing factor in the separation 

of key centers, and dual-priority is established by other elements of the superimposition—such as 

the distance between the key centers, the textural difference of the two lines, and the registral 

difference.   

If we agree with Kaminsky that these two keys have dual priority, then for converging 

closure to occur, they will have to unify at or before a moment of closure. These keys converge 



39 

 

together at m. 12, when the right hand of the piano breaks its pattern and drops down to a single 

note, C#. Within the F#-minor strand, this C# is first understood as 5, and underneath it, the F-

minor ostinato has reached the V in its pattern. Without any other harmonic information in the 

right hand to keep up dual prioritization, that 5 in F# minor is quickly reinterpreted as b9 in the 

dominant beneath it (i.e., Db). Making the composite of that measure a Vb9 (Cb9) in F minor. That 

b9 then immediately resolves down to a C over the ostinato’s tonic chord, as would be expected. 

In short, the F#-minor strand forfeits its priority and is assimilated into F minor by filling in a 

tone in F minor’s dominant. This forfeit of priority works in conjunction with the Vb9 to create a 

rhetorically strong cadence.   

Converging closure in polytonal music is often rhetorically strong, because the moment 

of closure can typically be understood within a monotonal context (like Vb9 to i in the key of F 

minor). To iterate that point, Figure 8 provides a recomposition of the above excerpt to feature a 

different type of polytonal closure. In this case, rather than the F#-minor voice dropping down to 

a single voice and relinquishing priority, it instead keeps up its pattern of descending tonic and 

dominant chords.   

 

Figure 8: Recomposition of Botafogo, mm. 1–14. 



40 

 

 

  This recomposition could be analyzed as either polyclosure or monoclosure. As a method 

of polyclosure, the F#-minor strand ends its phrase on a i chord, coming from a V7. 

Simultaneously, the F-minor ostinato plays its V7 to i below. However, this is not particularly 

strong closure, it is rhetorically weak. As seen with polyclosure examples above, the dual priority 

of this excerpt undercuts feelings of closure in both keys. A monoclosural reading of this 

recomposition would include the F#-minor strand closing as described above, but the movement 

of the F-minor ostinato doesn’t allow it a feeling of closure, as it’s a continuing ostinato. 

Regardless of which analysis, the closure of the recomposition is rhetorically weaker than the 

original. Closure is present, particularly in the F#-minor strand, but the feeling of rest is much 

less than the original composition. Therefore, it is reasonable to conclude that the process of 

converging closure leads the ear towards a more concise rhetorically strong close.  

 

Pantomime, from Suite for Cello and Piano, Howard Swanson 

Another case of converging closure is found in the B section of the second movement of 

Swanson’s Cello Suite, Pantomime. In this section, the piano’s tonal centers are fluid, and each 

passing center implements extensive use of blue notes and modal mixture (Reisser, 1998). The 

cello maintains a key center of C, also with extensive modal mixture, expressing both C major 

and minor. Reisser analyzes the key areas of the piano as A minor, G major, C major/minor, and 

A minor (1998); see Figure 9. These centers are not necessarily projected with full harmonic 

progressions, rather, they are more centralizing the harmonic areas.  

 



41 

 

 

Figure 9: Swanson’s Pantomime, B section. 

Figure 10 is a map of this polytonal section. The top strand represents the cello in C. The 

bottom strand represents the piano’s tonal space as it travels through its various tonal centers, 

moving toward and away from the tonal center of the cello. The most intriguing parts of the 

example occur in m. 3 and m. 12, where the piano converges with the cello. Relative distance 

between the tonal strands denotes the relative distance of the tonal centers. 

 

Figure 10: Polytonal map of Pantomime, B section. 

 



42 

 

In the first few measures of this section, recreated in Figure 11, the cello and piano are 

harmonically separate from one another. While the cello may use modal mixture and blue notes, 

its key center is definitively C. By contrast, the piano uses so much chromatic alteration that its 

key center is more nebulous, though it has been analyzed as closer to A minor (Reisser, 1989). 

Simply through that juxtaposition, the cello’s clearer center gives it primary priority. Now for 

this example to be considered converging closure, these two strands (C-cello/A-minor-ish piano) 

must assimilate into each other and reach closure together.  

 

Figure 11: Converging closure in Pantomime, beginning of B section. 

Tackling these requirements individually, let us first analyze their level of convergence. 

The cello moves through its improvisatory melody before outlining a G-major chord in m. 3, 

reaching a climax on B, highlighted in pink. At the same time, the piano floats its way into a G9 

chord, with the third omitted, highlighted in orange. Like puzzle pieces, the climactic B fits right 

into the G9 of the piano, and the two successfully converge to create something like a half 

cadence in C. 

However, this pseudo half cadence doesn’t project a strong sense of closure. On the very 

next beat, the strands separate into C and A minor again, and the very next measure restarts the 

same melody and chord progression. The half cadence that the cello and piano formed together is 



43 

 

rejected, preventing this from being a true moment of converging closure. As there may have 

been convergence, but there was no closure. This brief rejected converging closure is depicted in 

the map as a peak in the piano strand at m. 3.  

Following this missed converging closure, the piano (separated from the cello) begins to 

float around to a few more key centers. It first creates more distance from the C cello by moving 

to G major (m. 5). It then converges around a C tonal center (m. 8). While both parts project C 

simultaneously, they still avoid fully converging. When the piano is in C major, the cello is in C 

minor, and then they switch. Throughout these measures, neither strand achieves full 

convergence or closure. This section feels almost like a cat and mouse game where the piano is 

trying to converge with the cello. In this game, it’s almost as if the piano is trying to predict 

where the cello is going to be. This back and forth is congruent with the programmatic title, 

“pantomime,” which would evoke some sort of imagery of imitation.   

Finally, after multiple failed attempts at converging the cello and piano return to the 

material that got them the closest to closure the first time, that initial near convergence. Figure 12 

depicts the final bars of the B section of Pantomime; you’ll notice the material in the last two 

bars of B is identical to the first two bars of B (Figure 11) effectively bookending the section. 

The cello and piano strive for convergence again, just as they did before, but can they achieve 

closure this time? The only thing that differentiates these ending measures from the opening 

measures is, (1) a dynamic drop, and (2) a fermata on the composite G9 chord. This time around, 

the long pause on the converged half cadence achieves closure. By book-ending the section, 

returning material can signpost an ending. The short pause on the created HC, that then moves on 

to A-section material indicates the end of the section, as a type of intermediate closure. Thus, the 

whole cadence can be considered converging closure.  



44 

 

 

Figure 12: Converging closure in Pantomime, end of B section. 

 The expressive effect of this section hinges on these moments of convergence and 

closure. It opens with a split priority that quickly converges together but misses its moment of 

closure. It rebounds to more distant keys before trying to converge again. This time the 

convergence doesn’t line up as the strands aren’t modally on the same page. Finally, they return 

to the material that secured their near miss at the beginning, and, on their second attempt, 

converge and pause on the convergence, dwelling on it before moving on. This example of 

converging closure is one I will refer to in Chapter 4 as it encapsulates a distinct mode of 

converging closure. A mode of convergence where, rather than an assimilation of one tonal 

strand into another (as in “Botafogo”), disparate tonal strands effectively float into each other 

and create a composite chordal structure–something that Lee Cronbach would call “vertical 

dynamism” (1981). The composite chord is then both a moment of convergence and a 

structurally harmonic chord. As I will show, Swanson uses this mode of convergence extensively 

to great expressive, closural, and formal effects. 

 



45 

 

Dirge, from Suite for Cello and Piano, Howard Swanson 

For my final example of converging closure, I return now to the B section of Dirge. Earlier, I 

explored the incessant use of monoclosure throughout this middle area as local closes. As the 

section comes to its end, converging closure is used to resolve more strands than just the F-minor 

cello’s and provides an intermediate close that is rhetorically stronger.  

The F-minor cello reaches the final phrase of its improvisatory melody against the 

polytonal piano part (Db major, A minor, E); see Figure 13. At the end of the cello’s phrase, it 

comes to a sustained Ab, scale degree 3. Held scale degree 3 was a common enough feature of 

monocloses for this section as the cello had previously come to monoclosure on Ab; see Figure 4 

above. The difference being, this final pitch in the phrase (Ab) is held far longer than any of the 

monocloses heard so far. This longer drawn-out end contributes to a sense of rhythmic closure. 

While the cello holds this note, the piano continues the same cascading pattern it has been using 

throughout the entire section, including the contrapuntal descent within the Db-major strand.  

 

Figure 13: Converging closure in Dirge. 

 



46 

 

This moment of closure is different from the monocloses already heard, as the Db-major 

piano strand also reaches a moment of closure coinciding with the cello. Harmonically, the Db-

major piano strand moves from a dominant on beat 1 of the penultimate piano measure, to a tonic 

on beat 1 of the final measure (of the section), giving harmonic closure. Meanwhile, the inner 

voice follows its prescribed pattern, descending chromatically down from Fb to Db, providing a 

sense of contrapuntal closure; as compared to Figure 3, this time the contrapuntal closure is 

concurrent with the cello’s closure. Finally, the continuous cascading accompaniment pattern 

stops at this contrapuntal arrival on a Db-major chord, providing rhythmic closure that coincides 

with the cello’s closure. This ending then exhibits (at least) two strands coming to close 

simultaneously. 

If the closing of these two strands happened independently in separated keys, this would 

be an example of polyclosure. For this to be considered converging closure, the cello and piano 

would have to assimilate into each other in some way, which they do. The previous examples 

from this movement depicted the F-minor cello constantly distancing itself from the piano, but in 

contrast, the cello is no longer actively resisting assimilation by Db major. The cello reaches and 

holds a note which is very prominent in the Db-major strand below it and doesn’t shy away as it 

did before. Not only is it on the common tone of Ab, it is in full unison with the piano. Moreover, 

where the cello was initially separated from the piano by tension, motion, and dynamics, it now 

parallels the piano by being calmer and quieter. Through this calm and quiet unison, the Db-major 

piano can assimilate the cello part into its key. In effect, as the Ab is held in the cello it is initially 

heard as scale degree 3 in F minor, but as the piano line continues under it, that held Ab is 

reinterpreted as 5 of Db major. With reinterpretation, the cello and piano successfully converge 



47 

 

on Db-major and reach a moment of converging closure, ending the B section of Dirge with a 

rhetorically powerful joint sense of rest.   

This instance of converging closure is unique in that it itself functions as a component of 

another type of closure. While the F-minor cello and Db-major piano come to a moment of 

converged closure in Db major, the other piano strands in A minor and E do not converge or 

assimilate into Db major. In short, the composite closure of this entire section ends with 3 

independent strands, where just one of the closing strands is Db major.8 Thus, the whole of this 

closure is either polyclosure or monoclosure. The distinction between polyclosure and 

monoclosure in this case depends on whether these ancillary strands (A minor and E) achieve 

closure. The only thing that changes about those strands in that moment is that the cascading 

accompanying rhythm stops and the strands hold their chords. In turn, this provides the same 

sense of rhythmic closure awarded the Db-major piano strand above. Whether or not this single 

mode of closure is enough to say these strands reach a state of rest is ambiguous and leads to two 

possible analyses: (1) if the rhythmic closure of the A-minor and E strands is enough to say they 

have reached sense of rest, the whole composite would be polyclosure, as three independent 

(unassimilated) strands would be reaching a sense of closure and rest simultaneously; on the 

other hand; (2) if the rhythmic closure in those strands is not enough to create a true sense of rest 

(if, in other words, they seem to stop but not to close), the whole composite would be 

monoclosure, as only one of the composites strands is reaching closure among the group of 3. 

 
8 I’ve affectionately (unofficially) dubbed this compounded converging closure. An instance where a converging 

closure sits within the composite of another type of closure. Within this corpus there is not enough examples of this 

(that I have found) to constitute adding it to my framework outright. But it has the potential to describe polytonal 

closure when 3 or more strands are in play.   



48 

 

Figure 14: Two possible interpretations of closure in Dirge’s B sectiondiagrams these two 

possible analyses.    

 

Figure 14: Two possible interpretations of closure in Dirge’s B section. 

 

 All the modes of polytonal closure will be used to great expressive effects in the 

following case studies in Chapters 3 and 4. The intention of this chapter was to analyze each of 

these closural methods in isolation to uncover the distinct mechanisms of closure at play. While 

the later chapters—which still examine the internal closural mechanisms to an extent—are more 

concerned with the rhetorical, narrative, and form building properties of polytonal closure.  

  



49 

 

Chapter 3: Case Study: Sonata Form 

Introduction 

Polytonality, Sonata Theory, and Disability 

With a developed framework in place to analyze and describe modes of polytonal closure, I now 

have the tools to describe and analyze polytonality in longer musical forms that rely on cadence 

and closure, such as sonata form. Milhaud wrote numerous compositions that utilize 

superimposition and are in dialogue with elements of sonata form. For this case study I will look 

primarily at the first movement of his Sonata for Flute, Oboe, Clarinet, and Piano, Op. 47 (1918).   

I follow the formal framework set forth by Hepokoski and Darcy in their 2006 book, 

Elements of Sonata Theory. However, as this is a twentieth-century sonata, I take certain liberties 

in analyzing the form as outlined in Joseph Straus’s book, Remaking the Past (1990). As Straus 

explains, “The interesting twentieth-century sonatas are those that struggle most profoundly with 

the tradition” (1990, 132). It is Milhaud’s reconciliation of superimposition and sonata form—the 

profound struggle—that makes this piece the perfect case study for understanding how polytonal 

closure functions in a narrative/formal context. In analyzing the narrative, I also refer to the work 

of Damien Blättler and his corpus study of twentieth-century Parisian sonatas (2024), in which 

he analyzes the first movement of Milhaud’s Op. 47. While I construct a different narrative 

around the function of polytonality in this piece, his analysis of irony in the French neoclassicists 

has influenced my analysis. Synthesizing Blättler’s analysis of French irony with Straus’s ideas 

on twentieth-century composers profoundly struggling with sonata form, I construct a narrative 

reading within Milhaud’s sonata that engages with a meta-understanding of sonata form and its 

seeming incongruence with polytonality. 



50 

 

When superimposition is utilized in a sonata form, regardless of whether such a 

superimposition is heard polytonally or not, questions related to the larger form arise from 

smaller technical questions—What does Essential Expositional Closure look like in a polytonal 

sonata? What would the key structure of a polytonal sonata be?—to broader questions—What is 

the role of closure within a sonata form? Can polytonal closure fill that role? Can a sonata even 

be polytonal? This case study intends to address those questions and more, in order to further 

develop an understanding of closure and cadence within polytonal music as it relates to larger 

forms. Over the course of this case study, I will identify the conventions of sonata form utilized 

within Milhaud’s sonata and show how polytonal closure contributes to defining the form. I will 

also approach the analysis of this sonata through a lens of virtual/multiple agency and disability 

studies to construct a narrative analysis of the sonata through the formal/closural conventions 

analyzed. 

The first movement of Milhaud’s Sonata for Flute, Oboe, Clarinet, and Piano (Op. 47-I 

from here on) is my first case study because of its interactions between polytonal closure and 

conventions of eighteenth-century sonata form. Within this movement, Milhaud uses all three 

modes of polytonal closure with various rhetorical strengths. As I will show, the rhetorical 

strength of internal polytonal closure maps to conventional structures of sonata forms. In other 

words, rhetorically strong polytonal closure correlates with where strong harmonic closure would 

be expected in a conventional sonata. Conventional sonata forms tend to rely on conventional 

harmonic structures, wherein harmonic closure plays a pinnacle role in the mapping of the form 

(Hepokoski and Darcy 2006, 174). Without traditional harmonic closure, it could be difficult to 

interpret or understand a polytonal composition as a sonata, and a conventional sonata-form 



51 

 

analysis would have to come to terms with a lack of closure as a very substantial deformation.9 I 

choose to analyze Op. 47-I through the framework of Sonata Theory since the movement 

contains the narrative functions of a traditional sonata form and a near-standard sonata layout, 

per Hepokoski and Darcy (2006). At the same time, I substitute my framework of polytonal 

closure for the more conventional methods of harmonic closure typically seen in a sonata form. 

Analyzing the piece through the dual framework of Sonata Theory and polytonal closure allows 

us to fully understand its dynamics of closure. 

So, I argue, does viewing the sonata through the lenses of virtual agency and multiple 

agency. Robert Hatten describes “agents” working in music as fictitious characters within a story 

(2018). Agents are often musical gestures but can include specific instruments and melodies. 

These agents develop their own identities through the course of a piece of music with their own 

roles and motivations, like protagonist and antagonist (Hatten 2018). Edward Klorman expands 

on a different theory of agency, one where the agential properties of gesture and melody are 

personified in the performer playing those parts (2016). For example, in the case of a striking 

musical gesture played in the viola part of a string quartet, Hatten would be concerned with the 

narrative role of the gesture, and Klorman would be concerned with the narrative role of the 

viola. In my analysis, I give agency to both instruments and gestures in such a way that Seth 

Monahan would call work-personas; in that these agents have a longer formed understanding and 

perception of the narrative unfolding and are active in it (2013). By applying the theories of 

virtual agency/multiple agency to polytonal works, I expand on their theories to include the 

unique function of key areas within an agency narrative. Key areas in Op. 47-I are not so much 

agents themselves but are like masks that musical agents don and doff in the progression of the 

 
9 Blatter concedes to this deformation. He doesn’t acknowledge any closure in his sonata analysis. 



52 

 

story. Essentially, key areas in a polytonal work are like props that various work persona agents 

use to build out narrative.  

Finally, I bring a disability studies focus to my analysis. Much research into polytonal 

music has approached this repertoire from the perspective of overcoming a challenge or solving a 

problem—i.e., the so-called “therapeutic model” of disability (Straus 2018, 155–84).10 Op. 47-I 

itself has previously been analyzed as overcoming polytonality; Blättler argues that “the sense of 

resolution at the recapitulation is here produced by the taming of polytonality” (2024, 122). I 

analyze this sonata differently, viewing polytonality not as something to be overcome but instead 

as an essential feature of the piece that doesn’t need to be solved or “tamed.” In the analytical 

vignettes from Chapter 3, I intentionally avoided applying a conflict narrative to the role of 

polytonality or polytonal closure, for the precise reason that polytonality does not always pose a 

conflict in these works. In many of my examples from Chapter 3, including the second 

movement of this same piece (Joyeux), polyclosure is utilized to genuine and congenial effect, 

without any sense of conflict. The same is true of Op. 47-I. 

 

Contextual Conceptions of Neoclassicism, Polytonality, and Sonata Form 

Before I begin the analysis, I must first situate Milhaud in relation to conventions of 

neoclassicism, polytonality,11 and sonata form. To accurately analyze Milhaud’s op. 47-I, it is 

important to contextualize its composition date (1918) within the broader scope of conventions 

occurring at the time. This piece was written during a so-called “revolt against impressionism” in 

 
10 A prime example of what Straus calls a disabling discourse in polytonal studies could be found in Jennifer 

Beavers 2016 article “Integrating Incompatibilities” wherein pitches that are perceived as “wrong” are “normalized” 

through polytonality.   
11 “Polytonality” in Milhaud’s time would have been a catch-all term for all applications of superimposition.  



53 

 

post-war France and thus finds itself at an aesthetic turning point, toward neoclassicism. It uses 

extensive polytonality, a deeply controversial practice tied closely to Les Six and French musical 

nationalist ideology—though Les Six wouldn’t be formed for two more years (Rašín 1957, 165; 

De Médicis 2005, 574). And it uses sonata form, a deeply Classical form with centuries of 

history that no composer of the twentieth century could be ignorant of (Straus 1990, 132). 

 Neoclassicism is difficult to define and often means different things to different people. 

A common definition of neoclassicism is that it is a return to eighteenth-century aesthetics and 

ideals, but it was mostly used by critics, not composers (Messing 1991, 481). A conception of 

neoclassicism as a purely aesthetic movement would be inaccurate. As Messing argues, the 

Neoclassical movement is a resistance to the predominant aesthetics that preceded it (1991, 489). 

It is not an aesthetic in and of itself, it is a backlash. Marianne Wheeldon writes that 

“neoclassicism was a relational term, deriving much of its meaning from the way it was 

positioned against a series of imagined antagonists in composers’ and critics’ discourse” (2017, 

436). The composers that neoclassicists were resisting were primarily Debussy and Schoenberg.  

Milhaud’s op. 47-I was then written during the rising pressure of this backlash, so what 

was Milhaud’s role in resisting the pressures of Debussy and Schoenberg? Before Schoenberg 

popularized atonality and serialism, Milhaud and the rest of Les Six were reacting to the 

Debussian aesthetic (Wheeldon 2017, 435). This reaction was emphasized through more reliance 

on counterpoint than harmony, and a paradigm shift of timbral importance. According to critics 

and observers at the time, such as Louis Laloy (1920), new composers (referring to Les Six and 

Stravinsky) preferred the “dryness of intersecting melodies.” Even more, their contrapuntal 

technique evolved; Laloy goes on to say that “this counterpoint is extremely simple and basically 

consists of pedals, which cling stubbornly to a note despite the modulation of the other parts” 



54 

 

(1920, 2). He is talking about superimposition, and the splitting of tonal strands. This aesthetic 

preference is highly prevalent in op. 47-I, as we shall see.  

Timbrally, the neoclassical reaction to Debussy includes a change in instrument 

emphasis. Another contemporary critic, Émile Vuillermoz (1923), described it as a de-emphasis 

of the violin and a reranking of timbral hierarchy. Wind instruments and percussion were given 

more dominant roles in the orchestral music of the time. Timbral stratification became a priority 

in reaction to string orchestras' timbral homogeneity. This too is evident in Milhaud’s op. 47-I. 

Revived recognition of contrapuntal importance and new desire for timbral variety leave room 

for new compositional techniques that make strong use of both. In essence, the aesthetic 

trajectory of early twentieth-century France was the perfect space for polytonality to emerge.   

 Milhaud’s attachment to neoclassicism ties directly into how he understood and 

developed his polytonal technique. Polytonality was in Milhaud’s mind the natural progression 

of tonal music, making it intrinsically an extension of the tonal tradition, not a breaking from it. 

In his article “Polytonality and Atonality” (1923), he states in explicit terms that the atonality of 

Schoenberg was the natural progression and next step of chromaticism–a belief that Schoenberg 

shared to an extent. Continuing on, he concludes that since atonality was the natural progression 

of chromaticism, then polytonality must of course be the natural progression of diatonicism. He 

rationalizes polytonality as being the next step in a tonal tradition. When viewed within the 

reactions against Debussy, polytonality is a technique that relies heavily on counterpoint and 

timbre for the tonal separation of strands. Refer to Chapter 2, Figures 5 and 6 (Le Boeuf and Op. 

47-II respectively), where timbre and counterpoint are explicitly called out for establishing 

independent tonal strands. In Figure 5, the secondary tonal strand of Gb major is stratified 

through timbral separation by keeping that tonal strand solely in the high brass–trumpet and 



55 

 

French horn. In Figure 6, it is the contrapuntal movement of the flute (in A-major) that resists the 

draw of F#-major in the piano. 

 Milhaud did not lay out this approach and conceptual ideas of polytonality until years 

after writing Op. 47 and being scrutinized for it. The use of polytonality was highly politicized, 

per de Médicis (2005). And despite being widely accepted by Les Six, many other composers in 

France, and indeed many French critics, saw polytonality as synonymous with atonality. Many 

articles and reviews equating polytonality to German atonal ideals were published at the time. 

This would lead to stronger reactions from polytonalists to establish polytonality as a purely 

French idea (De Médicis, 2005). In the midst of this backlash Milhaud published his articles 

(referenced above) explicitly separating the polytonal aesthetic from Schoenberg’s atonal 

aesthetics. The explicit separation from Schoenberg and atonality encourages a purely polytonal 

understanding of Op. 47-I. There are areas of Op. 47-I where so much superimposition is 

occurring that it may be heard and conceived of as atonal. Milhaud himself warned listeners that 

this was possible in his 1923 article, wherein too much or too distant superimposition of keys 

would destroy a perception of tonic and create something atonal. Such is the case with Milhaud’s 

fourth string quartet, which consisted of three to four simultaneous superimpositions and would 

ultimately be dedicated to Schoenberg. 

An area of neoclassical ideals I have yet to touch on is the prevalence and importance of 

Classical influence. Music that is Neoclassical is in dialogue with older aesthetics. In essence, it 

is synthesizing contemporary ideas of counterpoint and harmony with older models of the 

baroque and Classical eras (Hyde 1996, 202). This drive for modernizing older forms comes 



56 

 

from an anachronistic desire to imitate mastery.12 In her article, “Neoclassic and Anachronistic 

Impulses in Twentieth-Century Music,” Martha Hyde outlines four different imitative practices 

that neoclassicists used to modernize older forms (1996). In brief, those four practices are: (1) 

reverential imitation, where a composer sticks to older forms/practices in a textbook to an exact 

manner; (2) eclectic imitation, a loose, more allusionary reference to older forms and aesthetics; 

(3) heuristic imitation, highlighting and accentuating older anachronisms (this separates itself 

from reverential imitation by allowing itself to stray from models, where reverential imitation is 

a dogmatic adherence); and (4) dialectical imitation, where the form is used simply as a backdrop 

for a different musical idea to be expressed (often a conflict narrative or dialogue narrative) 

(Hyde 1996). I understand Milhaud’s Op. 47-I as lying somewhere between dialectical and 

heuristic imitation. In such that, the sonata form of the piece maintains the parts of a traditional 

sonata form but produces a different rhetorical model of such parts (dialectically imitative), but 

that different rhetorical model essentializes and even parodies the sonata form model 

(heuristically imitative). As I’ll show, Op. 47-I synthesizes the sonata form model with 

polytonality, and in doing so, spins a rhetorical metanarrative of sonata form.  

Permission for such a distant analysis for a sonata form comes from Joseph Straus 

(1990). Straus claims, “it is no longer possible in this [20th] century, however, to write a sonata 

form that arises organically, spontaneously, and seamlessly from the musical relationships. [...] 

Twentieth century composers inevitably approach the sonata self-consciously and often, as we 

have seen, with malice aforethought” (1990, 132). It is then reasonable to assume that Milhaud 

would synthesize sonata form with polytonality and create an alternative metanarrative. 

 
12 “Mastery” implicitly referring to composers that Justin London refers to as “BHMB”–Bach, Hayden, Mozart, 

Beethoven– in his 2022 article, “A Bevy of Biases.” 



57 

 

Milhaud then finds himself at the center of dueling aesthetics. In his reaction against 

Debussy, he developed polytonal technique. But in developing a polytonal technique, local 

criticism compared him, and the rest of Les Six, to German aesthetics (De Médicis, 2005). 

Milhaud’s Op. 47-I finds itself right in the middle of this swing. Meanwhile alternative 

approaches to neoclassicism, having to do not with aesthetics, but with form, are developing in 

French, German, and Russian composers alike. As such, Milhaud makes his mark by composing 

a polytonal sonata, one that plays on formal convention through parody and metanarrative.  

 

Exposition 

The exposition of Sonata for Flute, Oboe, Clarinet, and Piano unfolds following many formal 

conventions. It includes a primary theme zone (P), a transition (TR), a secondary theme zone (S), 

with expected Medial Caesura (MC), and Essential Expositional Closure (EEC).13 As all these 

formal areas play out sequentially, the initial characters/agents and their corresponding motives 

are established within the sonata’s rhetorical narrative chronologically.  

The exposition of Milhaud’s Op. 47 ranges from mm. 1–52. It opens with conventionally 

tonal language but quickly moves into the polytonal soundscape. From there, the deformations of 

the traditional sonata form include the utilization of polytonal closure as well as the atypical key 

structure of the primary and secondary themes. When these deformations are accounted for under 

 
13 The medial caesura is a mid-exposition break between the transition and the start of a secondary theme, where the 

sonata slows to a point of rest before starting the second part of the form; this often coincides with a half cadence in 

the key of the dominant but is not required. Essential Expositional Closure is the first PAC in an a sonata form 

exposition occurring at the end of the secondary theme; it is often the first most significant moment of closure 

(Hepokoski and Darcy 2006, 176). 



58 

 

the lens of virtual/multiple agency, a narrative revolving around the polytonal key areas and their 

closure begins to take shape.  

 

The Primary Theme 

The movement opens with some tonal ambiguity (mm. 1–19 depicted in Figure 15). The piano 

opens with a comping figure, alternating Eb/Ab dyads and F/C dyads. At the outset, it’s not 

immediately clear whether the piece is in Ab major or F minor. The piano continues its static 

ambiguous harmony while the P theme enters in m. 2, played by the oboe. The oboe’s melody is 

also harmonically ambiguous, it could either be analyzed as 4–7–5–b7–1 in F minor, or as 2–5–

3–5–6 in Ab major. However, the clarinet enters in m. 3 to clear up the ambiguity with a clearly 

centralized Ab. Therefore, despite an opening ambiguity, I analyze the primary theme in Ab 

major.14 However, the conventionality of the P theme is muddied by the entrance of the flute.  

The flute enters in m. 6, outlining a C-major triad, and continues into a superimposing C-

major melody against the Ab-major primary theme, developing a polytonal soundscape. The C-

major countermelody hovers mostly around E and G (3 and 5) providing an abundance of 

dissonance against the Ab theme, which emphasizes Eb and F (5 and 6). Moreover, the flute 

sounds in a radically different timbral space than the oboe, clarinet, and piano, i.e., it is timbrally 

distinct and separated (Johnson 2011, 6). Therefore, there is enough disparity and dissonance 

between the two tonal centers to keep the flute from being assimilated into the Ab-major strand, 

making this opening primary theme affirmably polytonal. By strength in numbers and registral 

dominance, the pre-established Ab-major strand has primary priority, and C-major strand has 

 
14 Blättler (2024) analyzes the primary theme similarly. 



59 

 

secondary. These two superimposed keys continue their thematic material all the way to a 

moment of monoclosure in m. 10.  

 

Figure 15: Op. 47-I, mm. 1–19. 



60 

 

 

Figure 15 cont. 



61 

 

 The Ab-major melody in the oboe and clarinet reach monoclosure against the C-major 

flute, in m. 10. The monoclosure of the Ab-major oboe is achieved through contrapuntal and 

melodic means. It is the P theme’s opening motive restated, but resolving to Ab instead of F. The 

clarinet’s final utterance is a diminutive echo of the oboes final line, closing in the same fashion. 

This monoclose retains some rhetorical strength, as the Ab-major theme doesn’t surrender 

primary priority, even when the oboe drops out, and in fact the restatement of the oboe’s closing 

line pulls more focus towards the clarinet. In this instance, the monoclosure is effectually 

contributing to the maintaining of priority. After the clarinet monocloses out, effectively ceasing 

the Ab-strand and returning to ambiguous piano harmony, the C-major flute is readily handed 

primary priority. But rather than continuing to a cadence in C major, the flute immediately shifts 

to resolve in Ab, m. 11. The transition starts in the following bar, m. 12. 

 Two questions related to sonata form deformation must be asked: (1) How does a 

polytonal primary theme work? And (2) is the monoclosure achieved in the clarinet and oboe 

enough cadence attainment to consider this sonata tonally closed (or tonally determined, to 

borrow Hepokoski and Darcy verbiage)?  

 The primary theme could be functioning polytonally in one of two ways, and the answer 

won’t be clear until the recapitulation. It is first possible, and likely, that the primary theme is 

just the Ab-major strand/theme, and C-major strand is ancillary to P. In this case, the flutes C-

major theme is serving as polytonal flavor and accompaniment. However,