Prokofiev’s Symphony no. 2, Yuri Kholopov, and the Theory of Twelve-Tone Chords

Christopher Segall

KEYWORDS: Sergei Prokofiev, twelve-tone technique, twelve-tone chord, aggregate chord, Russian music theory, Yuri Kholopov

ABSTRACT: A small collection of works, including Prokofiev’s Symphony no. 2 (1924), include chords with all twelve pitch classes. Yuri Kholopov, the foremost late-Soviet theorist, considered twelve-tone chords a branch of twelve-tone technique. Taking Prokofiev and Kholopov as a starting point, and building on prior scholarship by Martina Homma, I assemble a history and theory of twelve-tone chords. The central theoretical problem is that of differentiation: as all twelve-tone chords contain the same twelve pitch classes, there is essentially only one twelve-tone chord. Yet twelve-tone chords can be categorized on the basis of their deployment in pitch space. Twelve-tone chords tend to exhibit three common features: they avoid doublings, they have a range of about 3 to 5.5 octaves, and their vertical interval structure follows some sort of pattern. This article contextualizes twelve-tone chords within the broader early-twentieth-century experimentation with aggregate-based composition.

Received May 2017
Volume 24, Number 2, July 2018
Copyright © 2018 Society for Music Theory


[1] This paper bases a history and theory of twelve-tone chords around an unlikely starting point: Sergei Prokofiev, whose Symphony no. 2 (1924) features a chord with all twelve pitch classes.(1) Scholars have developed sophisticated models for various aspects of twelve-tone technique—such as tone-row structure, invariance, hexachordal combinatoriality, and rotation—but with isolated exceptions have not to date examined the phenomenon of twelve-tone verticals. Prokofiev, of course, is not widely associated with twelve-tone technique, and twelve-tone composition has not been a major area of Prokofiev studies. In contextualizing Prokofiev’s twelve-tone chord, this article will investigate the history of twelve-tone chords, the theorists who have described them, and the composers in whose works they have appeared. I draw on the work of Yuri Kholopov (1932–2003)—longtime professor at the Moscow Conservatory, and the foremost Soviet and post-Soviet theorist of his era—whose writings connect Prokofiev to twelve-tone technique in two ways.(2) First, Kholopov viewed Prokofiev’s “twelve-step” chromatic system, with independent harmonic functions on all degrees of the chromatic scale, as the true innovation of twentieth-century aggregate-based composition.(3) Second, Kholopov argued more generally in several writings for a broad conception of twelve-tone technique that included not only serial dodecaphony (his term for Schoenberg’s practice) but also a variety of other techniques, including twelve-tone chords (dodekakkordï), the topic of this article.(4) Kholopov’s (1983a) chronology of Schoenbergian precursors and parallels includes the twelve-tone chord from Prokofiev’s symphony. In this article, I survey the issues involved in analyzing twelve-tone chords and assemble a theory of twelve-tone chords. I return to Kholopov’s more detailed analysis of Prokofiev’s twelve-tone chord, in order to decipher its contradictions and propose an alternative. Finally, I argue that twelve-tone technique is a construction, rather than a fixed concept, that can be used in various ways by various authors.

[2] In considering twelve-tone chords as an independent topic, my paper engages not only with Kholopov, but also with other composers and theorists who have written on twelve-tone chords. Several authors of compositional treatises—namely Alois Hába (1927), Vincent Persichetti (1961), Ctirad Kohoutek (1976), and Eugen Suchoň (1979)—outlined possible forms and uses of twelve-tone chords. Among theorists, Martina Homma (1999, 2001) has most thoroughly investigated the issue, having traced the history of twelve-tone chords and their theoretical explication. Homma has advanced two arguments that will be taken up in the course of this article: first that twelve-tone chords constitute an independent and continuous compositional practice, and second that twelve-tone chords are a vertical analogue to the horizontal technique of twelve-tone rows. In general prior writers have addressed the problem of differentiating twelve-tone chords from one another, given that they all contain the same notes, as well as a variety of additional issues (explored below). As they demonstrate, there is more to consider regarding twelve-tone chords than simply how many notes they contain.

[3] The theoretical formulation proposed here draws on a catalogue of twelve-tone chords found in the repertoire. In describing twelve-tone chords as a compositional practice, several writers have referred to specific examples in the repertoire. This study collates all such examples for the purpose of drawing larger conclusions. Although there are exceptions, twelve-tone chords tend to feature the following properties: they avoid doubled pitch classes, they have a range of about 3 to 5.5 octaves, and their internal intervallic structure follows some sort of pattern. These features support the concept of a twelve-tone chord as an intentional compositional object, deliberately structured in such a way as to reinforce its identity as a twelve-tone chord. For this reason, chords containing only 10 or 11 distinct pitch classes are excluded from the study. Twelve-tone chords are special cases of highly chromatic sonorities, as through their purposeful exhaustion of the aggregate they forge a connection to twelve-tone technique more broadly, reinforcing the historically emergent conceit of composing with all twelve pitch classes.

[4] Kholopov (2009) offers Prokofiev’s Symphony no. 2 as a case study for analyzing twelve-tone chords, and I address both his analysis and an alternative analysis in this essay. Whereas early sections of the article will focus on ways in which twelve-tone chords can be differentiated from each other, the Prokofiev symphony offers a testing ground to consider how a twelve-tone chord can be analyzed in the context of an individual work, particularly a work that does not otherwise consist of twelve-tone chords. A twelve-tone chord appears in the symphony’s second movement. Kholopov’s attempts to relate the intervallic content of Prokofiev’s twelve-tone chord to other motives in the same movement leads to conflicting interpretations of the same chord. As a twelve-tone chord contains every pitch class, so can it be parsed to contain a variety of motives and intervals.

[5] Finally, I’ll propose a perspective of twelve-tone technique as a construction, with a shifting, negotiable definition. What “counts” as twelve-tone technique can change over time and from author to author, and the issue is worth considering from a historiographic standpoint. Specifically, a gambit such as Kholopov’s that broadens the conception of twelve-tone technique beyond serial dodecaphony allows connections to be drawn between the Schoenbergian approach and other aggregate-based practices of the early twentieth century. This perspective is thus historical as well as theoretical.


[6] Yuri Kholopov, with the title to an article, poses the question: “Who Invented Twelve-Tone Technique?” (1983a).(5) The answer, he assures us at the outset, is Arnold Schoenberg, if by “twelve-tone technique” we mean serial dodecaphony, the technique of composing with transformationally manipulated twelve-tone rows. But he nonetheless outlines a chronology of precursor techniques to Schoenberg’s compositional method, most created independently of Schoenberg and therefore not casting a direct influence on the later method. Composers in the early twentieth century experimented with a number of ideas that seemed to coalesce in the music of Schoenberg and his followers. Kholopov notes early instances of several techniques, including the following: “microserial” technique (a three-note cell in Webern’s String Quartet [1905], a four-note cell in Stravinsky’s Firebird), isolated melodic statements of the aggregate (twelve-tone rows in early Berg and Webern), referential non-twelve-tone vertical sonorities used also for horizontal content (Scriabin’s Mystic Chord, Roslavets’s synthetic chords), and twelve-tone fields in which all twelve pitch classes freely circulate (Nikolai Obukhov, Yefim Golïshev). He also includes the well-known tropes of Josef Matthias Hauer. Through Kholopov’s article, Schoenberg’s technique appears to emerge from a swirling cloud of ideas in the air at the time. Far from presenting Schoenberg as a visionary genius who invented an important method on his own—although Schoenberg, it must be said, is given due credit for the depth of his contribution—Kholopov historicizes Schoenberg, accounting for the way in which he solved problems that composers had been grappling with for decades.

[7] In this chronology that leads up to (and slightly past) Schoenberg’s first serial works of 1921–24, we read the following entry: “In the Finale of the Symphony no. 2 (1924) Prokofiev wrote a contrapuntal layer of sonoristic twelve-tone chords” (Kholopov 1983a, 56).(6) (The chords, as will be shown below in Example 1, consist of a single twelve-tone chord transposed via parallel planing.) This is striking, not only because Prokofiev is not typically associated with twelve-tone technique (nor, for that matter, with the sonorism of the later Polish avant-garde), but also because Kholopov himself used Prokofiev as a foil for twelve-tone practices in earlier writings. Kholopov had argued that Prokofiev extended tonal harmony to allow for independent harmonic functions on all twelve degrees of the chromatic scale—what Kholopov called dvenadtsatistupennost’, or the “twelve-step” or “tonal chromatic” system. Thus Prokofiev, with his twentieth-century tonality, had been considered the true innovator of aggregate-based composition.(7) Now Prokofiev is marshaled into service not as an alternative to twelve-tone composition, but rather as a progenitor.

Example 1. Prokofiev, Symphony no. 2, II, m. 134 (strings only)

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[8] In subsequent writings, Kholopov develops his chronology of twelve-tone techniques into twelve categories of “twelve-toneness,” one of which is twelve-tone chords.(8) Later textbooks and analysis manuals introduce each twelve-tone technique separately.(9) The adumbration of twelve-tone chords in Garmonicheskiy analiz [Harmonic Analysis] (2009) is particularly notable in the present context, for it uses Prokofiev’s Symphony no. 2 as its primary example. In this chapter, as in every chapter, Kholopov presents a brief theoretical overview of the topic at hand (that for twelve-tone chords is very short, just a few paragraphs) and then applies his comments to the musical excerpt (again, in the present case, very briefly), before inviting the student-reader to complete the analysis.(10) Example 1 shows the Prokofiev excerpt under discussion, from the second variation (in A major) of the second movement.(11) In the commentary, Kholopov remarks how the variation’s main melodic motive (see the top violin line) is harmonized in parallel with twelve-tone chords, played by twelve-part divisi strings, with each part playing a different transposition of the melody.(12) On every sixteenth note, the divisi strings project the aggregate, each pitch class appearing once among the twelve parts. The right side of Example 1 shows the pitches of the twelve-tone chord as it sounds on the second sixteenth note of the measure; this chord is transposed to form the remaining chords.

[9] Of what does Kholopov’s brief theoretical overview consist? He writes that there are many ways one could organize a twelve-tone chord, bottom to top. One could arrange it with common subsets (e.g., superimposed diatonic or whole-tone subsets, diminished seventh chords, or augmented triads, or as a verticalized interval cycle) or with diverse subsets (e.g., different triad types, sonoristic sound blocks, combinations of the above subsets). The emphasis on subsets suggests that there tends to be—or perhaps, is supposed to be—some logical principle underlying the construction of twelve-tone chords. For some construction types, a specific work or composer is identified (although vaguely, as with an overtone series chord attributed only to Karamanov), but mostly Kholopov lists compositional possibilities, whether realized in known works or not. The possibility of writing a twelve-tone chord not based on a logical principle is not explicitly excluded, but neither is it clearly stated as an option. Prokofiev’s chord is parsed as involving several quartal subsets, although this interpretation will be addressed further below.

[10] In focusing on possibilities (in interval and subset construction) rather than actualities (in chords drawn from the repertoire, though a couple are cited), Kholopov is drawn into the orbit of several earlier compositional or practical theorists, in whose treatises the possibilities of writing twelve-tone chords of various sorts were laid out. Kholopov does not cite any of these authors; they each formulated theories of twelve-tone chords independently of the others. Alois Hába (1927, 89–92) extensively lists twelve-tone chords based on 3rds (of various qualities), 4ths, 5ths, 6ths, and 7ths.(13) The compositional opportunities are not dwelt upon, although in the introduction to the same volume, Hába does point out a (quintal) twelve-tone chord from one of his own early orchestral compositions dated 1913 (so, according to the New Grove, probably Mládí), in a bid to be credited as the first composer ever to think of such a thing (but, as shown below, he wasn’t).

[11] Other treatises go beyond cataloguing the possible interval types. Vincent Persichetti (1961, 87–90) provides a thoughtful exposition on twelve-tone chords, explaining for composers-in-training their potential use (harmonic punctuation, quiet and sustained tension, contrasts with unison or two-part writing) and construction (equal-interval, heterogeneous, mirror-symmetrical, polychordal). His musical examples are self-composed excerpts, and he appends a list of several examples from the repertoire, which are considered in the discussion below.

[12] In more recent scholarship, Martina Homma (1999, 2001) has addressed the problem of twelve-tone chords most directly and comprehensively. Her important work undertakes a recovery of early theoretical descriptions of twelve-tone chords, beginning with the accordo dodecafonico of Domenico Alaleona, an Italian theorist who first formulated the conception in an article published in 1911. Homma identifies several compositions that incorporate early twelve-tone chords, including those by composers in possible contact with Alaleona, leading to an argument that twelve-tone chords form their own tradition in compositional history. Much of Homma’s theoretical discussion involves differentiating twelve-tone chords on the basis of their interval content.


[13] A twelve-tone chord is, to provide a working definition, a simultaneity that contains all twelve pitch classes. Various authors have discussed the issues involved in treating twelve-tone chords theoretically.(14) These extend to concerns regarding their internal structure, perceptibility, function, and their relationship to twelve-tone technique more broadly (including to serial dodecaphony). The present section summarizes the issues involved in analyzing twelve-tone chords; the following section discusses twelve-tone chords from the repertoire.

[14] Differentiation. Every twelve-tone chord contains the same twelve pitch classes. If a twelve-tone chord is defined by its pitch-class content, then there can be only one twelve-tone chord; it cannot be transposed.(15) But, conversely, as a variety of twelve-tone chords exists in the repertoire, then they must be differentiable according to their internal interval structure. Homma (2001, 102) summarizes the issue best, explaining that somewhere between the “only one” perspective and the “infinitely many” perspective lies the theory of twelve-tone chords, which must account for how the same set of pitch classes can be organized in perceptibly different ways.

[15] Interval structure. To the extent that twelve-tone chords can be compared on the basis of internal characteristics, their intervallic profile (or subset content), ordered vertically from bottom to top, offers a basis. Although there is technically no need for any twelve-tone chord to follow a predetermined pattern in its intervallic construction, it turns out that many such chords in the literature do so, exactly as predicted by Kholopov’s short synopsis. Some chords are based on a single interval size (e.g., 3rds, 4ths, 5ths). Others are based on patterns in the intervals (e.g., all-interval chords that contain every interval from 1 to 11 semitones, or symmetrical patterns of pitches based around an axis). Of course it is entirely possible for a twelve-tone chord to emerge as a compositional accident, if a large number of polyphonic layers happen to sound each of the twelve pitch classes, all at the same moment. But perhaps the reason for such commonly encountered intervallic patterns is that composers’ decisions to write twelve-tone chords are deliberate, and thus when they do not appear by chance, they may very well be organized in some logical manner—logical, that is, from the standpoint of the theory able to describe them (and not necessarily from the standpoint of the scoreless listener).

[16] Doubling. In Alaleona’s description of twelve-tone chords, doubling was expressly forbidden (Homma 2001).(16) And in most twelve-tone chords in the repertoire, doublings do not exist. The reason seems similar to that proposed above, namely that writing a twelve-tone chord is a deliberate act, and likely composers treat the issue as one of arranging each of the twelve pitch classes vertically into an intervallic pattern. Avoiding doublings draws attention to the fact that the chord is a twelve-tone chord. To admit doublings would be tantamount to allowing repetitions in a twelve-tone row—that is, repetitions not in the presentation of the row, but within the row itself. The prohibition on doublings helps determine the intentionality of a twelve-tone chord. In those cases where exactly twelve tones are used, the chord may seem an intentional twelve-tone chord, whereas in cases where more than twelve notes sound simultaneously, the question is raised of whether the twelve-tone chord is merely incidental.

[17] Perceptibility. The fact that a given chord contains twelve pitch classes, as opposed to ten or eleven, may not be particularly perceptible to listeners, and its patterned intervallic structure may be likewise aurally obscure. For this reason, Mikhail Tarakanov (1968, 108) and Kholopov (1979, 107; 2009, 14) characterize Prokofiev’s twelve-tone chord, with its sixteenth-note planing, as producing a swirling sound mass, relating the technique to avant-garde sonorism. Although the specific intervals within a twelve-tone chord may not be perceived by ear, their arrangement and distribution affects the sound character of the overall chord.(17)

Example 2. Progressions from twelve-tone chords to tertian sonorities

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[18] Function. The earliest descriptions of twelve-tone chords discuss the possibilities of hearing them as conveying harmonic function. Example 2 shows two progressions from the theoretical literature in which twelve-tone chords “resolve” to tertian sonorities. As shown in the first progression, Alaleona organized twelve-tone chords as dominant sevenths with a full spectrum of extensions, and then resolved the upper notes as appoggiaturas (Homma 2001, 101–2). The twelve-tone chord was thus considered not a stable entity, but rather an embellishing one. In an analysis of the second progression (from an unpublished composition by Jean Huré), which begins with a quartal twelve-tone chord and arrives eventually (beyond the excerpt shown) on a first-inversion D major added-sixth harmony, René Lenormand (1940, 127) writes that the progression “defies analysis.” By contrast, Taruskin (2010, 225–26, 283–84), rather than describing twelve-tone chords in terms of harmonic function, relates them to a significatory function, by which they are understood to contain anything and everything, their identities remaining stable as their parts move around. Thus for Scriabin (in the unrealized Acte préalable) and Ives (in the unrealized “Universe” Symphony), the twelve-tone chord was to signify the all-encompassing One.

[19] Relationship to twelve-tone practice. Kholopov attempts to subsume twelve-tone chords under a larger category of twelve-tone practices. The deliberate or systematic use of the aggregate of twelve pitch classes was a constant concern among early twentieth-century composers, and both twelve-tone chords and twelve-tone rows, he seems to argue, are reflective of that. Martina Homma advances a more directed argument toward associating twelve-tone chords with twelve-tone rows, viewing the former as the verticalization of the latter. This argument will be addressed in further detail below.

Example 3. Britten, The Turn of the Screw, act I, mm. 39–47 (reduction)

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[20] We might point out that any twelve-tone row can produce a twelve-tone chord if its pitches are sustained, so that the culminating sonority is the full chromatic. We may want to consider whether such resultant chords are twelve-tone chords in the sense under discussion here, or whether the blurred-together notes of a twelve-tone row should be considered a distinct category. There is an irony here, however. When a twelve-tone chord is presented as a chord (that is, a block chord), it may not be perceptually clear that it contains twelve tones. When, on the other hand, a twelve-tone chord is presented as a series of twelve tones—that is, sustained tones introduced one at a time, in a perceptually salient manner—it may not be clear that the resultant conglomeration is meant to be understood as a chord. Britten’s The Turn of the Screw (1954) features such a sustained-series chord (Example 3). The row’s presentation highlights its ic5 pairs—every discrete dyad is a perfect 4th or 5th, and every discrete tetrachord forms a four-note quartal harmony (i.e., prime form [0257]). But its identity as a twelve-tone chord may be less clear.

Example 4. Positional twelve-tone chords

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[21] Positional twelve-tone chords. Beginning in the 1960s, several composers—such as Lutosławski and Carter—began to use twelve-tone chords as “background” structures to be arpeggiated in the following sense. Each pitch class was assigned to a fixed register, and every instance of that pitch class would sound in that register, but there was no requirement that all twelve pitches be sounded simultaneously in the manner of a chord. Although twelve-tone chords rarely appear in a given positional span, the general practice has been related to the earlier twelve-tone chords under discussion. Homma (1999) refers to this background-structure technique as “pitch-positioning,” as it fixes the registral position of each note. I adopt her terminology and refer to the resultant structures as “positional” twelve-tone chords. Example 4 shows positional twelve-tone chords from the works of various composers, although given the extent to which they employed them—Stucky (1981, 116–17) catalogues dozens of Lutosławski’s twelve-tone chords based on the interval classes they emphasize—this example should be taken as a representative sample only.(18) Carter has a predilection for all-interval twelve-tone chords spanning five-and-a-half octaves from bottom to top.(19) Magnus Lindberg also uses positional twelve-tone chords, having stated a preference for both all-interval chords and overtone chords modeled on the natural harmonic series; the example demonstrates the latter.(20) Alfred Schnittke’s Variations on a Chord uses an all-interval positional chord as the titular “chord.” (See Example 4; here and in similar examples later in the paper, accidentals affect only the pitches to which they apply, and natural signs are omitted for ease of reading.)

[22] Several authors have associated positional chords with twelve-tone chords more generally. Homma in particular has argued that pitch positioning forms a crucial link between twelve-tone chords and twelve-tone rows—the former is the vertical analogue of the latter. Because positional twelve-tone chords control the pitch content of extended passages of music, but do not necessarily sound as simultaneities, they have a different function, sound, and conceptual basis than the other twelve-tone chords considered in this study. The question as to whether a positional twelve-tone background structure is truly a chord may be debatable.

[23] Problem cases. Some twelve-tone chords identified as such in the literature raise particular problems. Above, with reference to Britten’s Turn of the Screw, I discussed the issue of considering a blurred-together twelve-tone row as a twelve-tone chord. Along similar lines, Suchoň (1979, 162–63) identifies a twelve-tone chord in Boulez’s Sonatina for Flute and Piano that not only contains doubled pitch classes—a red flag in twelve-tone chord analysis—but (more to the point) results from four measures of music pedaled together. The “chord” is never articulated as such (when Chopin’s pedaling indications invite a blurring together of harmonies, do we refer to polychords?), and perhaps Kholopov’s twelve-tone field would be a better characterization of the passage.(21)

Example 5. Polychords with whole-tone roots

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[24] Persichetti (1961, 90) identifies a twelve-tone chord in Milhaud’s Piano Sonata no. 1 that warrants close analytical scrutiny (Example 5). The pianist pedals together three polychords on successive beats. At first, eleven of twelve possible pitch classes are engaged—only eleven, even as the pianist articulates 22 different notes! Finally the pianist changes the first polychord and the twelfth pitch class enters the fray. But the logic seems not to be that of the twelve-tone chord, despite the aggregate completion. Rather the pianist appears to complete the whole-tone scale, which provides the root of every constituent triad within the polychords. The aggregate is a by-product of the fifth of a major triad belonging to a different whole-tone collection than its root, and therefore when the roots spell out one complete whole-tone collection, the fifths spell out the other one. One could imagine a similar technique with different constituent harmonies failing to produce an aggregate—for instance, the sixteen-note (but six-pitch-class) chord in Ernest Fanelli’s unpublished Tableaux symphoniques (Lenormand 1940, 130–31), of which the constituent chord type is the dominant seventh
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