Mcdonald

I would like to thank Jessica Berson, Rick Cohn, Sumanth Gopinath, Lynne Rogers, DavidRothenberg, H. Colin Slim, Pieter van den Toorn, and the anonymous reviewers for this Journalfor their comments on earlier versions of this essay. Parts of the essay are derived from papers pre-sented at the Seventy-Third Annual Meeting of the American Musicological Society in QuebecCity, November 2007, and the Thirtieth Annual Meeting of the Society for Music Theory inBaltimore, November 2007. Archival research was supported by a British Academy SmallResearch Grant. Materials from the facsimile of the sketchbook for The Rite of Spring are repro-duced by the kind permission of Boosey & Hawkes.

1. The original French reads: “[M]a liberté sera d’autant plus grande et plus profonde que jelimiterai plus étroitement mon champ d’action et que je m’entourerai de plus d’obstacles . . . etl’arbitraire de la contrainte n’est là que pour obtenir la rigueur de l’exécution” (86).

2. Pasler, “Music and Spectacle in Petrushka and The Rite of Spring,” 67.3. Rehearsal numbers in Stravinsky’s score are indicated by R followed by the number. For

example, “R13” refers to the bar designated by the rehearsal number (i.e., the first bar of the ac-cent pattern here), “R13+1” to the following bar, and so on. The rehearsal numbers are continu-ous throughout the score.

Journal of the American Musicological Society, Vol. 63, Number 3, pp. 499–552 ISSN 0003-0139, electronic ISSN1547-3848. © 2010 by the American Musicological Society. All rights reserved. Please direct all requests for permissionto photocopy or reproduce article content through the University of California Press’s Rights and Permissions website,www.ucpressjournals.com/reprintInfo.asp. DOI: 10.1525/jams.2010.63.3.499.

Jeux de Nombres : Automated Rhythm in The Rite of Spring

MATTHEW McDONALD

My freedom will be so much the greater and more meaningful the more nar-rowly I limit my field of action and the more I surround myself with obstacles . . . and the arbitrariness of the constraint serves only to obtain precision of execution.

—I. Stravinsky, Poetics of Music in the Form of Six Lessons, 871

The idea of games points to the real subject matter of the ballet—the creativeact itself.2

Igor Stravinsky’s The Rite of Spring long ago secured its status as a corner-stone of musical modernity. Among the many musical features to whichthis enduring distinction can be traced, perhaps most important are the

aggressively dissonant chord and irregular accent pattern that constitute theopening bars of the ballet proper, the “Augurs of Spring,” an icon of whatscandalized audiences at the work’s premiere in 1913. Accordingly, both theharmony and the rhythm of these bars have attracted various analyses. In re-gard to the rhythm, some analysts have focused on the interaction betweenthe irregularly spaced accents and the regular downbeats of the duple meter, asestablished in the bars that precede R13.3 Pierre Boulez, for instance, divided

4. That is, the metric positions of accents in the third cell are reversed in the fourth: seeBoulez, “Stravinsky Remains,” 68–70.

5. See, for instance, Messiaen, Traité de rythme 2:100–101. Messiaen undertook an extensiverhythmic analysis of the entire score for The Rite: see Traité de rythme 2:93–147. Pieter van denToorn presented analyses that alternately emphasize and downplay the meter of “Augurs,” com-menting on their relative merits and advocating recognition of each perspective: see van denToorn, Stravinsky and the Rite of Spring, 68–71.

6. This and all subsequent examples condense the score into either one, two, or three stavesand frequently simplify the texture in the process. In all cases where I analyze the intervallic con-tent of a particular harmony or collection of pitches, however, I have not omitted any pitchesfound in the full score. Unless otherwise noted, all score excerpts are based on the 1921 edition ofthe score (actually published in 1922), the earliest printed edition of the full orchestral score. Myanalyses, however, are based upon a full consideration of the sketchbook, the autograph full andshort scores (both housed in Basel at the Paul Sacher Foundation), and the four-hand score (pub-lished in 1913 and now available in a Dover edition), all crucial sources for addressing the compo-sitional process considered in this essay. I refer to these earlier versions of the score in all cases inwhich discrepancies between the 1921 edition and the earlier versions are relevant to my analyses.Stravinsky’s post-1921 revisions do not come into play: it is difficult to establish what any of theserevisions might have to tell us about Stravinsky’s original compositional thought-processes, andthey rarely differ from earlier versions in ways that would affect my analyses. For the most thor-ough consideration of the various sources for The Rite, see Cyr, “Le sacre du printemps: Petite his-toire d’une grande partition,” 89–148.

7. Here and elsewhere, I refer to specific pitches using the scientific nomenclature that desig-nates middle C as C4; in this system, C5 denotes the pitch one octave above middle C, and B �3the pitch one whole step below middle C.

8. A compelling precedent for this way of conceptualizing the relationship between intervalsand durations can be found in the theoretical work of the Russian composer and theorist SergeyTaneyev, whose music was influential to the young Stravinsky. Taneyev devised a single numericalsystem for the measurement of both intervals and durations. In analyzing a contrapuntal work, forinstance, Taneyev would designate the relationship between a pair of imitative voices with two

the first eight bars into four two-bar “cells,” noting a retrograde relationshipbetween the rhythms of the third and fourth cells.4 Others have ignored themeter, focusing exclusively on how the accents delineate an irregular groupingof eighth notes, an approach encouraged by the rigidity of the eighth-notepulse that defines this passage.5

The latter perspective is illustrated in Example 1a, which reproduces theopening bars of the “Augurs of Spring.”6 Example 1b reproduces the begin-ning of the ostinato that commences immediately after the famous accent pat-tern at R14. The overall pitch content is a close variant of the repeated chordthat begins the “Augurs of Spring”: rather than an E-flat dominant seventh superimposed over an F-flat-major triad, here the E-flat dominant is superim-posed over a C-major triad. (I will refer to these two collections of pitches asthe “Augurs” chord and its C-major variant.) The resulting pitch collection isoctatonic, unlike the chord with which “Augurs” begins. The intervals formedby this collection of pitches (with the addition of C57 and omission of B �3,both of which will be addressed below), measured in semitones from top tobottom, produce the same numerical series as the durations produced by theaccent pattern.8

500 Journal of the American Musicological Society

Jeux de Nombres : Automated Rhythm in The Rite of Spring 501

This correspondence, striking as it is, has not been noted in the vast litera-ture on The Rite, nor has such a correspondence been noted in much music atall composed before World War II; the earliest examples are generally found inserial music of the late 1940s.9 It is unlikely to be coincidental, judged simply

numbers, one representing what he called “vertical” distance in pitch space, the other represent-ing “horizontal” distance measured in beats. See Carpenter, “Contributions of Taneev, Catoire,Conus, Garbuzov, Mazel, and Tiulin.”

9. One exception is Richard Cohn’s recent study of Béla Bartók’s Sonata for Two Pianos andPercussion (1937), in which Cohn explored what he called “pitch-time analogies,” albeit using amore complex theoretical apparatus than is necessary here. In the same essay, Cohn also cited inpassing a four-note ostinato found at the beginning of Maurice Ravel’s “Une barque sur l’océan”from Miroirs (1904–5), which features a correspondence between intervals and durations ex-tremely similar to that found in “Augurs.” This example would need to be bolstered by additionalevidence to be significant here, but given the relationship between Stravinsky and Ravel, and in

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in terms of mathematical probability. Given that the accent pattern as well asthe “Augurs” chord and its octatonic, C-major variant were likely among Stra -vinsky’s very first musical ideas for The Rite, as recorded on the first page of hissketchbook and as Stravinsky himself confirmed,10 and given that Stravinskylater singled out the accent pattern as “the foundation of the whole [work],”11

this connection between pitch and rhythm was perhaps even a fundamental as-pect of his conception of the music for the ballet. In discussions of the open-ing rhythm of “Augurs”—as in discussions of Stravinsky’s compositionalpractice generally—the implicit assumption is almost always that Stravinsky de-liberately crafted every musical detail to achieve a specific effect. But my analy-sis suggests that, at an initial compositional stage, Stravinsky may have used amechanical procedure to derive the accent pattern, in nearly its ultimate form,from an independently generated succession of intervals. In what follows, Iwill present extensive evidence suggesting that this type of strategy was essen-tial to Stravinsky’s composition of The Rite, a conclusion that has importantconsequences for how we think about the ballet and its composer.

I proceed, however, aware that such an endeavor is likely to be met with someresistance. Nearly a half century ago, Robert Craft offered discouraging wordsfor anyone proposing revelations about Stravinsky’s score: “There is no longerany novelty in The Rite of Spring. . . . Its musical mysteries are now profaneknowledge (the desacralization complete), I have no new theory to pro-pound, and in any case the music is neither neglected nor in need of re-evaluation.”12 But this statement was so wildly and predictably off-the-markthat we might wonder whether or not Craft actually believed it. More likely itwas a smokescreen, perhaps a reflection of Stravinsky’s own worries aboutwhat secrets his sketches (long held “under lock and key,” as Craft explained)might reveal. “To anyone interested in musical embryology, these facsimile

particular the fact that Ravel accompanied Stravinsky when the latter purchased his sketchbook forThe Rite in Varese, Italy, the example from Miroirs is nonetheless suggestive in the present con-text. See Cohn, “Pitch-Time Analogies and Transformations in Bartók’s Sonata for Two Pianosand Percussion.” Justin London has provided an overview of twentieth-century composers andtheorists interested in creating isomorphisms between pitch and time; his earliest example is from1944 (Oliver Messiaen’s written commentary on his compositional methods). See “Some Non-Isomorphisms between Pitch and Time.”

10. See Craft, “Commentary to the Sketches,” 4. Some sketches for the Introduction to Part I of the ballet may predate Stravinsky’s sketches for the “Augurs”; see Taruskin, Stravinskyand the Russian Traditions 1:890–91. Because Stravinsky undertook so much of the composi-tional process at the piano rather than on paper, however, it is difficult to establish with absolutecertainty the chronology by which the score evolved; see ibid. 1:894.

11. See the documentary film Portrait of Stravinsky (1966), cited by Chua, “Rioting withStravinsky,” 100n28.

12. Craft, “Rite of Spring,” xv. This essay was adapted from a 1966 lecture.

13. Ibid., xv.14. Forte, “Letter to the Editor in Reply to Richard Taruskin,” 326.15. Forte, The Harmonic Organization of “The Rite of Spring.”16. Richard Taruskin, “Myth of the Twentieth Century,” 7.17. See Chua, “Rioting with Stravinsky”; and Code, “Synthesis of Rhythms.” The Rite was

also recently featured prominently in a popular non-academic book, Lehrer’s Proust Was aNeuroscientist.

18. See V. Stravinsky and Craft, Stravinsky in Pictures and Documents (hereinafter referred toas Stravinsky in Pictures and Documents), 524–26, which reproduces Edward B. Hill’s Englishtranslation of 1916. Taruskin provided his own translation in Russian Traditions 1:877–78. Theoriginal essay in French, “Ce que j’ai voulu exprimer dans Le sacre du printemps,” was published inthe magazine Montjoie! (29 May 1913) and is reprinted in Lesure, Le sacre du printemps: Dossierde presse, 13–15. Stravinsky denied authorship of the essay, but it seems for the most part to haverepresented his ideas accurately. See Taruskin, Russian Traditions 2:995–1006, for a thoroughdiscussion of the matter.

pages are a major document,”13 Craft wrote, but a different message wasclearly discernible: look away, musicologists, there’s nothing to see here.

In more recent years, still other scholars have attempted to stamp outanalyses of The Rite, if not in such an absolute fashion. Surely Allen Forte wasspeaking tongue-in-cheek when, with regard to some well-studied bars from“The Ritual of the Rival Tribes,” he announced: “subsequent to [my] analysisthere will be no further analyses of this passage, as decreed by the joint ad hocAnglo-American Committee on Definitive Analyses, the members of whichwere appointed by the present author.”14 Or was he? Forte had authored, af-ter all, a monograph titled The Harmonic Organization of the Rite of Spring(emphasis mine).15 Richard Taruskin called upon Forte to renounce the“viewpoint” of this particular book as “incorrect,” and elsewhere Taruskin hasattempted to demonstrate how “much of [the] literature [on The Rite] is soeasily and instructively falsified.”16 One might be forgiven for concluding thatno one wants anyone else to say much about The Rite at all.

But, of course, no one has succeeded in enforcing a moratorium. Countlessessays, chapters, and books have been devoted to The Rite over the past fewdecades, including two substantial analytical studies since 2007.17 The litera-ture continually expands, despite many of the contributors themselves seem-ing uncomfortable with the expansion. This dichotomy can perhaps be tracedto Stravinsky himself, beginning with the Montjoie! program attributed to thecomposer, “What I Wished to Express in The Consecration of Spring.”18 Here,Stravinsky described the ballet’s introduction as the “Birth of Spring”:

In the Prelude, before the curtain rises, I have confided to my orchestra thegreat fear which weighs on every sensitive soul confronted with potentialities,the “thing in one’s self,” which may increase and develop infinitely.

And later:

In short, I have tried to express in this Prelude the fear of nature before the arising of beauty, a sacred terror at the midday sun, a sort of pagan cry. The



musical material itself swells, enlarges, expands. Each instrument is like a budwhich grows on the bark of an aged tree; it becomes part of an imposingwhole.19

This “imposing whole” can easily be read as a metaphor for Stravinsky’s entirescore for The Rite: we are meant to be terrified of this “primitive” musical lan-guage. Indeed, this fear might help to explain the audience’s disquiet at thepremiere. But a more striking metaphor is suggested by the first part of thisquotation. The “great fear” Stravinsky described bears a remarkable resem-blance to his characterization of his own creative angst many years later in ThePoetics of Music:

As for myself, I experience a sort of terror when, at the moment of setting towork and finding myself before the infinitude of possibilities that present them-selves, I have the feeling that everything is permissible to me. If everything ispermissible to me, the best and the worst; if nothing offers me any resistance,then any effort is inconceivable, and I cannot use anything as a basis, and con-sequently every undertaking becomes futile.20

Taking a cue from this later statement, I would propose that the fear of “po-tentialities” Stravinsky referred to in the Montjoie! program may have beenfundamentally his own, an anxiety that he had allowed his musical ideas to de-velop without proper limitations, and thereby to take on a life of their own.

This anxiety would not have been eased by the critical reception of TheRite. From the beginning, Stravinsky’s score for The Rite has often been char-acterized as rough around the edges, lacking in systematization—qualities thatare easily linked to the primitivist impulse behind the ballet.21 Not surprisingly,in the decades after the premiere, Stravinsky increasingly cultivated an oppos-ing image for the score, insisting that it be divorced from the original balletand regarded instead as a concert piece, and gradually revising the score sothat certain relationships would appear more logical and systematic.22

This revisionism helps to account for the apparent desire among analysts tostunt the development of analytical research devoted to Stravinsky’s score.

19. Stravinsky in Pictures and Documents, 525.20. I. Stravinsky, Poetics of Music, 85.21. As described in an early review: “. . . this is the most dissonant music ever written. I

would say . . . that never has the system and cult of the wrong note been practised with such zealand persistence as in this score; that from the first bar to the last whatever note one expects isnever the one that comes, but the note to one side, the note which ought not to come . . . themusic is rough and violent. . . . These are not pretty, elegant harmonies, combined with patientsubtlety. They are hard, loud, dense, freely invented by a cruel and fertile musical nature.” PierreLalo, in Le Temps, 5 August 1913, quoted in Lesure, Dossier de presse, 33–34; translation fromHill, Stravinsky: The Rite of Spring, 93.

22. On Stravinsky’s evolving attitude toward The Rite, as reflected in his writings, interviews,and his revisions of the score, see Taruskin, “Myth of the Twentieth Century”; and Fink, “ ‘Rigoroso (� = 126).’ ” A revised version of Taruskin’s essay is featured as the first part of thechapter “Stravinsky and the Subhuman” in his book Defining Russia Musically.


Following in the composer’s footsteps, some analysts have appeared deter-mined to counter the idea that any details of a work held in such high esteemmight not have been carefully planned out by the composer—that the musicmight somehow be “arbitrary,” as Theodor Adorno famously described it, ajudgment that could be seen to expose the entire analytical endeavor as nomore than a fool’s errand (a “futile undertaking,” to borrow Stravinsky’swords). In some cases, these analysts have attempted to deliver a swift and per-manent blow to the notion of arbitrariness, identifying coherent and audiblestructures that explain Stravinsky’s compositional procedures in comprehen-sive ways—Forte’s set theory, for example, or Boulez’s notion of durationalsymmetry.23 Such studies might have been more abundant in an earlier, morepositivist era of music analysis, but more recent examples are easily found aswell.24

According to Taruskin, this disciplinary tendency can be attributed to anunderlying fear of what he saw as the true nature of The Rite: “a rational complexity,” Taruskin argued, “is far less disturbing than a mystifying simplic-ity.”25 Simplicity is a quality rarely attributed to The Rite, but it is crucial inTaruskin’s characterization: The Rite “is not a complex score,” he stated, andthe music is distinguished by a “radical simplification of texture.”26 But thereis, I would argue, a more intuitive explanation for the persistent desire to ex-plain Stravinsky’s Rite with overarching theoretical systems or analytical narra-tives: a rational complexity is preferable to an irrational one. The apparentreckless abandon of Stravinsky’s score—a musical “riot,” as Daniel Chua hasrecently characterized it27—has proven difficult to reconcile with the largerreputation of Stravinsky’s oeuvre, promoted by the composer himself, as aparagon of logic and objectivity.

One source of the problem, perhaps, is the faulty assumption that thehighly controlled compositional methods we might associate with Stravinskynecessarily result in orderly musical surfaces. Whereas Stravinsky clearlyreigned in the “infinitude” of potential compositional strategies to a muchlesser extent in The Rite than in later works, the singular unruliness of thescore should not be taken simplistically as evidence of a lack of systematic com-positional planning. The analysis above and those that follow shed light on therelative presence of order and disorder in Stravinsky’s score, suggesting that

23. See Forte, Harmonic Organization; van den Toorn, Stravinsky and the Rite of Spring(esp. chaps. 5–7); and Boulez, “Stravinsky Remains.”

24. David Code, for instance, in his 2007 study of the “Augurs of Spring,” concluded:“What seems like a dance of utter visceral and hormonal immediacy is found to be tightly boundby the fetters of compositional control”; “Synthesis of Rhythms,” 162. Similarly, Code identifiedthe “Augurs” as “the perfect exemplar of the structural investments Stravinsky attributed to thewhole ballet some years later, when he described it as an ‘architectonic’ work”; ibid., 113.

25. Taruskin, “Myth of the Twentieth Century,” 19.26. Ibid.27. Chua, “Rioting with Stravinsky.”


one of Stravinsky’s central compositional strategies was systematic and yet stillyielded somewhat unpredictable musical results; I propose that he employedan “automated” procedure to generate musical ideas, in which entities fromone realm of music (pitch) are translated into another realm (rhythm) wheretheir meaning is altogether different and any desirable musical result is largelyfortuitous.28 That is to say, Stravinsky sacrificed part of his role as an organizerof musical sound in favor of a more detached engagement with music as a sys-tem of abstract, quantifiable relationships.

Analyses

“Augurs of Spring” (continued) and Introduction to Part I

The procedure for converting pitch intervals into durations is evident, as wehave seen, at the very outset of the “Augurs of Spring.” Given its primacywithin the compositional history of The Rite—and arguably the canon oftwentieth-century music as well—the opening of the “Augurs” merits furtherattention here. First, it is instructive to consider why the accent pattern wouldrelate most closely to the C-major variant rather than to the initial “Augurs”chord itself. As already noted, the C-major variant (unlike the initial chord) isoctatonic, and thus characteristic of Stravinsky’s harmonic language in TheRite.29 It is also an important chord beyond its initial appearances in“Augurs”: it is featured prominently, for instance, from R16 to R18 and inthe first several bars of the “Ritual of Abduction.” But most importantly, it isthe C-major variant, and not the repeated chord with which “Augurs” begins,that dominates Stravinsky’s initial sketches, appearing in three musical frag-ments at the top of the first page of the sketchbook, possibly the first musicalideas Stravinsky committed to paper (they appear on the upper third of page 3, which features five fragments sketched in black ink). Thus the C-majorvariant may have had conceptual priority over its more famous sibling in theearly compositional stages of The Rite.

The famous accent pattern, just as we know it today, is laid out on the mid-dle of the first page of sketches. Surprisingly, on the next two pages (4–5),Stra vinsky wrote out the accent pattern two more times, but with an addi-tional accent on the downbeat of the fourth bar. On the second system of

28. I should state up front that I have found no compelling evidence of a similar composi-tional approach in any of Stravinsky’s other music, although my investigation of this possibility hasnot been thorough. My sense is that the approach was closely linked to Stravinsky’s specific ideasabout The Rite (an idea I will develop in my conclusion below) and therefore may have beenunique to this work. But it remains an open question whether or not he employed this approachelsewhere.

29. On octatonicism in The Rite, see Berger, “Problems of Pitch Organization in Stravinsky”;van den Toorn, Music of Igor Stravinsky, 100–137; and idem, Stravinsky and the Rite of Spring,chaps. 5–7.


page 4, the accent has apparently been erased from both staves and is barelylegible, but it is easily seen on page 5 (see the rhythmic sketch just above thehorizontal blue lines near the top of the page). Example 2a reproduces thesecond of these sketches, in Stravinsky’s bare rhythmic notation; I havemarked the offending accent with an asterisk. This blasphemous pattern couldbe “derived” from the pitch collection shown in Example 1b with an addedC4 inserted between B �3 and D �4. This new C, together with the surround-ing E �, D �, and B �, would form an instance of the “minor” tetrachord (tone–semitone–tone) that is crucial to the harmonic and melodic language of TheRite, a subset of both the diatonic and octatonic collections and a melodic ba-sis of many of The Rite’s folk-music sources.30 The E �–D �–C–B � tetrachord infact features prominently among the musical fragments at the top of the firstpage of the sketchbook. In three of these, it results from the interaction of thefamiliar D �–B �–E �–B � ostinato and an arpeggiated C-major triad; one of thesefragments is reproduced in Example 2b. In the score, the tetrachord is fea-tured prominently seven bars before the “Augurs” begins, where a premoni-tion of the ostinato is superimposed over a trilled C. All of this evidencestrengthens the notion that the C-major variant may have been linked to theaccent pattern early on.

Note that in the final bar of the sketch reproduced in Example 2b, anarpeggio adds an additional C above the D �–B �–E �–B� ostinato, and an equiv-alent upper C can be found in the fragment occupying the upper right-handcorner of the first page of sketches. These upper Cs correspond to the one Ihave added provisionally in the intervallic analysis shown in Example 1b. Thisalternative voicing of the C-major triad (preserved at R16+2 of the printedscore) may have influenced Stravinsky’s derivation of the accent pattern, as Ihave suggested by including the upper C in Example 1b. This C is also thevery first pitch of The Rite, the opening note held by the bassoon. Just as thiselemental C gives rise to the slowly thickening texture of the introduction (the“Birth of Spring”), it can be understood as spawning the intervallic seriesshown in Example 1b, which in turn gives birth to the famous accent pattern.This detail suggests that there may have been a link between Stravinsky’s com-positional approaches and his programmatic ideas for The Rite, an idea I willreturn to at the end of this essay.

As noted above, the intervallic analysis shown in Example 1b passes overthe B � of the English horn’s ostinato at R14 (B �3), a note which has no corre-sponding accent. This sort of imperfection will turn up in several of the analy-ses below as well, and in general I will not speculate about potential causes. In

30. Taruskin has rightly made much of Stravinsky’s “discovery” that two minor tetrachords,separated by a semitone, comprise a complete octatonic collection, and thus that his folk-musicsources could easily be adapted to an octatonic environment. See Russian Traditions 1:937–41.Van den Toorn emphasized repeatedly the importance of the minor “(0235)” tetrachord in TheRite (as a subset of the octatonic collection) in Stravinsky and the Rite of Spring.


this case (as in similar cases below), the correlation between intervals and du-rations is still extremely strong. We could push further and observe that the B �would generate an accent on the third eighth note of R13+3, resulting inthree consecutive groups of three eighth notes, a regularity that would be atodds with the irregularity Stravinsky obviously desired. One could imagineother plausible rationales as well. But here and elsewhere, my assumption issimply that generating durational patterns from intervallic ones was a compo-sitional starting point for Stravinsky, that he modified the results of the initialgeneration to whatever extent necessary in order to achieve the musical resultshe desired, and that he saw no need to deny himself flexibility when employ-ing these generations.

Finally, before proceeding further, my assumption that Stravinsky deriveddurational patterns from intervallic ones (as opposed to the reverse process)requires explanation. In the case of my initial example, it seems extremely im-probable that Stravinsky conceived of the accent pattern first and then derivedthe succession of intervals from this pattern. Stravinsky described composingthe “Augurs” chord at the piano, and there is no particular reason to doubtthis story. If this chord and rhythm were among Stravinsky’s first musical ideasfor The Rite (as all evidence suggests), then this first example bears special im-portance in informing our understanding of all the examples presented below.Furthermore, the C-major variant of the “Augurs” chord is both octatonicand derived from tertian harmonies, and is in this way characteristic of TheRite’s harmony in general, whereas the rhythmic pattern has no obviousprecedent or corollary: it is the rhythm, not the chord, that seems to require aspecial explanation. Likewise, the collections of pitches analyzed below are in

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Example 2 “Augurs of Spring”: (a) alternate accent pattern from the sketchbook, p. 5; (b) frag-ment from the top of the first page of sketches


general easily accounted for vis-à-vis octatonicism, diatonicism, and extendedtertian harmony, whereas the corresponding rhythmic patterns tend to resistany comparably straightforward explanations. I would not close off the possi-bility that in some instances Stravinsky may have worked “in both directions”when relating pitch formations and rhythmic patterns, but the evidence pointstoward intervallic patterns having generally been the basis for durational ones.

One possible critique of my analysis of “Augurs” is that the correspondencebetween intervals and durations is best perceived intellectually, not aurally.Taruskin has identified (and vehemently objected to) a tendency among ana-lysts of The Rite to find logical, but inaudible, structures underlying the com-plex musical surfaces.31 He cited as evidence Elliott Antokoletz’s analysis ofthe opening bassoon melody, which showed how the notated rhythms of thefirst three bars create a quasi-symmetrical pattern.32 Taruskin rejected thisanalysis as a “fiction,” arguing that Antokoletz’s groupings have little to dowith how the music actually sounds, but rather are “entirely an artifact of thebar-placement, a notational convenience”: in other words, the phrasing of thebassoon melody, as indicated by fermatas and slurs, is incompatible withAntokoletz’s groupings.33

Taruskin’s complaint relies on the assumption (often implicit in analyticalwriting) that analysis should relate to how music is heard: “What is revealed[in Antokoletz’s analysis] is a germ-free vivarium, entirely ‘closed off ’ . . .from the world in which the music was composed and in which it is experi-enced.”34 But this statement suggests the basis of its own critique: we can gainaccess to “the world in which the music was composed” partly through its notation—a trace of the compositional process—some aspects of which maybear little or no relation to musical experience. In the case of the opening bas-soon melody, Stravinsky’s notation is in many ways counterintuitive. Onecould imagine different ways of notating these bars to achieve a similar soundwhile reinforcing visually the phrasing of the melody—for instance, followingJuszkiewicz’s transcription of the folk tune from which Stravinsky derived themelody, by placing the second and third Cs, which instigate repetitions of thebasic melodic idea, on metrical strong beats. The notation Stravinsky chosebegs the question of its meaning or significance. Consider the first bar: boththe beaming of notes and the 4/4 meter suggest the rhythmic grouping1+4+3+3, as shown in Example 3a. It is unlikely that anyone would hear thesegroupings without benefit of the score, but they are logical in terms of the no-tation. Example 3 compares this grouping with an intervallic analysis of the

31. “Method is inferred from ‘structure’ and then attributed to the composer, whose work isthus rationalized and rendered abstract.” See Taruskin, “Myth of the Twentieth Century,” 10.

32. See Antokoletz, Twentieth-Century Music, 95. Boulez arrived at essentially the sameanalysis decades earlier; see “Stravinsky Remains,” 60–62.

33. Taruskin, “Myth of the Twentieth Century,” 11.34. Ibid., 11–12.


pitch collection outlined by the opening melody, as anchored by the horn’s C�on the downbeat of the second bar (see Ex. 3b): once again, the intervalsmeasured in semitones correspond to the rhythmic groupings.35

A significant difference between this analysis of the opening bassoonmelody and my analysis of “Augurs” is the nature of the isomorphism be-tween rhythm and pitch: in the “Augurs” example, it is based on a durationalpattern, whereas with the bassoon melody, it is based on rhythmic groupings.The rhythmic nature of the two passages, in fact, could hardly be more differ-ent. One features a metronomic beat, its rhythmic interest based solely on apattern of irregular emphases; this circumstance is much more prevalent inThe Rite as a whole. The other sounds improvisational, with little or no audi-ble metrical grounding; this quasi-rubato effect virtually disappears from thework when the dancing begins.

The similarities between the two analyses, however, are crucial and charac-terize the body of analyses below as well:

1. In each analysis, an intervallic pattern derived from an important chordor pitch collection corresponds to a durational or rhythmic pattern. Allbut one of the analyses concerns the beginning of a titled section of TheRite, where much of the best known and most provocative music of theballet resides. In fact, all but one of the fourteen titled sections feature thissort of correspondence in their opening bars.36 Due to limitations of space,I have excluded some of the more complex of these examples,37 but theubiquity of the procedure at the beginnings of movements is an impor-tant element informing my thesis. Furthermore, the presence of this

35. The main intervallic analysis passes over the bassoon’s A, which would generate a 2+2subdivision of the sixteenth notes in bar 1, as indicated on the example. This subdivision is notsuggested by Stravinsky’s notation, although perhaps it is implied by the time signature as a simpledivision of the quarter-note beat.

36. In two cases, the correspondence is found in an early incarnation of a section’s openingbars but not in the ultimate printed version, as a result of Stravinsky either repositioning themovement title (“Procession of the Sage”) or inserting new opening material (Introduction toPart II).

37. These include the “Ritual of Abduction” (bars 1–5), the “Mystic Circle of the YoungGirls” (bars 1–7), and the “Ritual Action of the Ancestors” (bars 1–14).

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sort of correspondence at the openings of movements suggests that thisparticular means of deriving rhythmic material was a compositionalstarting point for Stravinsky, a way of setting ideas in motion for eachnew section.

2. In each of the examples, intervals are measured from top to bottom (neverbottom to top), and measurements originate from the uppermost pitchof a chord or collection of pitches (or, in one exception, a semitone be-low the uppermost pitch).38

3. In all cases, the unit of durational measurements is the largest possible notevalue that allows all measurements to be made in whole numbers. For in-stance, in Example 1a, the eighth note is the largest such note value: ifdurations between accents were measured in quarter notes, this wouldyield the numerical succession 4.5, 1, 3, 1.5, etc., and no correspon-dence with semitonal measurements would be possible. In each of thepassages in which the meter changes from bar to bar, my durationalgroupings follow the notated meter.

4. Finally, as I will discuss further below, most of the chords or pitch collec-tions under consideration share important intervallic characteristics thathave already been noted in the literature, particularly the intervallicstructures 5+6 and 6+5 (that is, the major seventh partitioned into a per-fect fourth and tritone).

All of these consistencies suggest a coherent set of compositional strategies, al-though I should emphasize again that Stravinsky’s approach clearly allowedfor flexibility, experimentation, and play. Although this procedure for derivingrhythms seems to have been mechanical, there can be no doubt that Stra vin -sky, in every case, evaluated and shaped the derived rhythms so as to arrive at amusical result that he found satisfactory.

“Sacrificial Dance”

The opening of the “Sacrificial Dance,” like the opening of “Augurs,” is char-acterized by an explosive, irregular accent pattern featuring an iconic chordthat becomes a sonic emblem of the movement (it is, in fact, a virtual transpo-sition of the “Augurs” chord). Unlike the opening of “Augurs,” however,

38. The notion of Stravinsky measuring intervals always by counting downwards harmonizesnicely with Chua’s characterization of the primary “Augurs” chord: “the chord is ‘upside down,’requiring some kind of topsy-turvy tonal theory”; “standing tonal theory on its head in some waywould make more sense of Stravinsky’s E � key-signature that prefaces the ‘Augurs of Spring’; the‘root’ is on the top and the chord hangs upside down from the E� surface.” See Chua, “Riotingwith Stravinsky,” 69. David Lewin conceived of the opening bassoon melody of The Rite in a sim-ilar way, citing B as the “root” of the arpeggiated E-minor triad in bar 1; see “Formal Theory ofGeneralized Tonal Functions,” 41–43. Van den Toorn’s method of notating and referring to theoctatonic scale in The Rite similarly assumes a fundamental orientation downwards; see Stravinskyand the Rite of Spring, 144–45.


here Stravinsky reinforced the pattern with changing meter, as opposed toshifting accents within a stable meter.39 My durational groupings, shown inExample 4a, begin after the call to attention of the first bar, which is set off bya fermata. (Durations are measured in sixteenth notes.) The repeated materialat R143 was not present in Stravinsky’s original sketch (see the top systems ofpages 84–85 of the sketchbook); therefore my durational analysis passes overthese four bars. The durational analysis corresponds to the notation of the me-ter from R142 to R144, which reinforces the groupings a listener is likely to hear: the meter is made audible by various features, most prominently thepattern of low, accented Ds followed by unaccented, offbeat harmonies.40 AtR144, a new accented, marcato thematic idea enters, as though on a down-beat, and thus at odds with the meter as notated (see my durational analysis);when this idea recurs at R145, its entrance is adjusted to occur on a notateddownbeat. In his original sketch on page 85 of the sketchbook, Stravinsky didnot include the low D found at R144 of the printed score, thereby minimiz-ing the conflict between the notated downbeat of this bar and the entrance ofthe new motive. Example 4b analyzes the intervallic construction of the pri-mary chord of this passage (see the first two bars of Ex. 4); the intervallicanalysis generates precisely the same numerical series as the durational analysis.

Introduction to Part II and “Glorification of the Chosen One”

Embedded in each of the pitch collections analyzed thus far is the 5+6 or 6+5trichordal skeleton fundamental to harmony in The Rite (that is, the perfectfourth + tritone or vice versa); Taruskin has referred to this trichord as “theRite chord” or the “Grundgestalt” of The Rite of Spring, and van den Toornhas likewise called it The Rite’s “basic sonority.41 In the “Augurs” chord andits C-major variant, the trichord is formed by the pitches E �–B �–E � (“5+6,”measured from top to bottom); in the related chord of the “Sacrificial

39. The following analysis incorporates the metrical subdivisions of the 1921 printed edition,in which the meter of the second and third bars (2/16+3/16) subdivides what had been a single5/16 bar in earlier versions. This slightly revised metrical scheme simply reinforces the groupingsalready asserted strongly by the low Ds.

40. When the Ds are absent, other features reinforce the notated meter. For instance, acrescendo leads to an accent on the downbeat of the fifth bar of the movement, punctuated by thebrass, thereby articulating the downbeat in the absence of a low D. The downbeat of the fourthbar is a prime example of what Grosvenor Cooper and Leonard Meyer called an “accented rest,”where one has been conditioned to hear an emphasis despite the absence of sound. The passagefrom the development of the “Eroica” Symphony that constitutes their primary example of thephenomenon might in fact be regarded as a progenitor of the beginning of the “SacrificialDance.” See Cooper and Meyer, Rhythmic Structure of Music, 137–40.

41. Taruskin, Russian Traditions 1:939–40; van den Toorn, Stravinsky and the Rite ofSpring, 145–46. Van den Toorn’s notion of a “basic sonority” is somewhat more complex, en-compassing other features, particularly the trichord’s interaction with the octatonic collection (ofwhich it is a subset). In van den Toorn’s view, the “basic sonority” and related harmonic featuresconstitute a fairly consistent harmonic language in The Rite.


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Dance,” by the pitches D–A–E � (although here the latter pitch sounds an octave lower); and in the Introduction to Part I, by the pitches C–G–C�. Atsome points in the score, the coordination of 5+6 or 6+5 intervallic and dura-tional patterns is quite obvious. Example 5a concerns the beginning of theIntroduction to Part II of the ballet, not as we know it today, but as Stravinskyoriginally conceived it: in the full and short versions of the autograph score,R79–R86 of the printed score were not initially present.42 Thus the originalIntroduction began with the juxtaposition of two contrasting ideas at whatwould become R86: a legato duet in the trumpets followed by an accompani-mental pattern in the low strings, the latter comprised of two alternating stac-cato trichords over a four-note pizzicato bassline (these two ideas are heardtogether a few bars later). The changing meter of the trumpet duet and the in-tervallic content of the string accompaniment both feature 5+6 and 6+5groupings, as shown in the example. More specifically, the symmetry of themetrical scheme (5/4–6/4–5/4) also has a close analogue in the realm ofpitch. Example 5b outlines symmetrical aspects of the relationship betweenthe second staccato trichord and the pizzicato bassline: the embedded 5+6+5pattern is highlighted by the shared inner tritone B �–F � and the E �s occupyingthe registral extremes. (Note that the upper trichord is presented linearly inthe example and the tritones superimposed for analytic clarity.)

A related passage occurs in the “Glorification of the Chosen One.”R112+3 introduces a musical idea (reminiscent of the “Danse infernale” fromThe Firebird) comprised of off-beat trichords over a short ostinato pattern; itis first heard for only a bar but soon returns for several bars at R114. This syn-copated idea alternates between five-beat and six-beat versions, notated in 5/4and 6/4, as shown in Example 6a (see R114–R115 for the full passage).Throughout this passage, each of the upper trichords is based on either the5+6 or 6+5 intervallic pattern, as shown in Example 6b; the 5+6 and 6+5 tri-chords are also heard in alternation (see Ex. 6a). Thus, quite similarly to whatwe have just seen in the Introduction to Part II, the alternation between 5+6and 6+5 trichords can be related to an alternation between 5/4 and 6/4 barsthat generates various 5+6 and 6+5 durational pairings.

“Procession of the Sage,” “Adoration of the Earth,” and “Dance of the Earth”

More subtle instances of 5+6 and 6+5 relationships can be found in the lastthree movements of Part I, linking these movements together as a larger unit.

42. Stravinsky completed the autograph full score for Part I of the ballet on 29 February1912, and then began to write out Part II, beginning with what would become R86. About ayear later, on four pages dated 29 March 1913, he wrote out the missing passage (R79–R86) andinserted it in the autograph; see Cyr, “Petite histoire,” 104. The last page of the autograph isdated 8 March 1913, and thus it appears that Stravinsky returned to the Introduction to Part II,adding the opening material after having completed (or believing that he had completed) thedraft in the autograph.


In the early stages of composition, Stravinsky’s working title for what wouldbecome the “Procession of the Sage” was “Idut-vedut” (“They Are Coming,They Are Bringing Him” in Taruskin’s translation43), a title that he used atseveral points in the sketchbook and preserved in the autograph scores. In theprinted score, however, the “Procession of the Sage” begins at R67, whereasin the autograph scores, “Idut-vedut” begins at the equivalent of R64 in theprinted score; there is no title at R67 in the autographs.44 The opening of“Idut-vedut” is shown in Example 7a. This passage features two seemingly in-dependent musical layers, an upper melody in thirds (woodwinds and stringsin the full score) and a lower tuba melody centered firmly on G�. Example 7a

43. Taruskin, Russian Traditions 1:873.44. The sketchbook suggests some early equivocation between these two potential starting

points for the movement: here, Stravinsky places titles at both points on different pages (see pp. 12, 14, and 16).

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ends at the point where the melody in thirds drops out, resulting in a suddenthinning of the texture that demarcates the beginning of a new musical sec-tion. Each layer features an initial musical idea that is repeated several times, either in its entirety or in abridged forms. The melody in thirds consists of atwo-bar idea that is sometimes repeated in full and other times in one-bar frag-ments. The tuba melody features a five-bar idea that is subsequently shortenedto four- and two-bar versions; all three versions end with the identical two-bartag. My durational analysis in Example 7a outlines these various statementsand repetitions, treating the whole note as the basic durational unit (thelargest possible note value that allows durations to be measured in wholenumbers). Example 7b analyzes the complete pitch content of this passage,omitting only the lower neighbors to the tuba’s G�s, which might be brack-eted as ornamental given their relatively short duration compared to the sus-tained G�s and their weak metrical placement. In the full printed score, thelower voice of the melody in thirds reaches down to a B � (enharmonicallyequivalent to the tuba’s A�); however, this note was a B � in Stravinsky’s four-hand arrangement and thus constituted part of the overall pitch content of the passage at an earlier stage (this B � is in fact retained in the full score in the four bars before R62, an earlier occurrence of the same melody in thirds).Including this B � in Examples 6a and 6b, the intervallic pattern generated bythe overall pitch content of this passage replicates the durational measure-ments, as can be seen by comparing Examples 6a and 6b. The intervallic seriesshown above the upper staff in Example 7b is very similar to the series gener-ated by the upper portion of the C-major variant of the “Augurs” chord, as

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shown in Example 7c (I have included parenthetically the C4 found on thefirst page of sketches, as discussed above); both are elaborations of the familiar5+6 pattern.

More important to the intermovement connections under considerationhere is the 5+4+2 pattern shown between the staves on Example 7b (also asubdivision of 5+6), common to the intervallic measurements and the dura-tional analysis of the tuba melody. In the extremely short section that follows,the “Adoration of the Earth,” this 5+4+2 durational pattern occurs in reversein the only moving part, as shown in my analysis of the contrabassoon melodyin Example 8, which treats the quarter note as the basic durational unit. TheD �–D�–D � motion of the contrabassoon recalls the tuba’s G�–G�–G� motionin the “Procession of the Sage” (see Ex. 7a), thereby reinforcing the connec-tion between these two melodic figures.

This 2+4+5 durational pattern recurs a few bars later at the beginning ofthe “Dance of the Earth,” now coordinated with a 2+4+5 intervallic patternin a more elaborate correspondence, as shown in Example 9. The durationalgroupings in Example 9a (measured in eighth notes) follow the irregular itera-tions of the sforzando chords, as articulated by the vast majority of the instru-ments: a C-major triad with added F� followed by its “neighboring” chord.The excerpt analyzed in the example spans from R72 to R73, including thefirst three bars featuring the “C-major” chord followed by three bars featuringits “D-major” neighbor. In some cases, a glissando directs its energy into adownbeat rest; I have treated these silences as “accented rests,” in analogy tomy analysis of the fourth bar of the “Sacrificial Dance.” The first of these rests(the downbeat of R72+2) was actually a sforzando chord in Stravinsky’s origi-nal sketch of the movement,45 supporting my treatment of these rests as comparable to the sforzando chords (Stravinsky similarly equivocated betweenlow Ds and accented rests in the “Sacrificial Dance,” as is evident from com-parison of his multiple versions of the movement). The intervallic measure-ments shown in Example 9b, identical to the durational measurements ofExample 9a, are based on the overall pitch content of this passage, includingboth the initial repeated sforzando chord and the whole-tone ostinato under-neath it. Compromising this analysis to some degree, the intervallic measure-ments shown in Example 9b (unlike all others presented in this essay) do notbegin with the uppermost pitch, G, but a semitone below (F�); furthermore,the intervallic analysis omits the G� of the bassline ostinato (reflecting its possi-ble status as a passing tone from F� to A�), as well as some upper and lowerdoublings of the primary chord. Although I am hesitant to offer possible ex-planations for these discrepancies, they might indicate that Stravinsky wasworking with a condensed, preliminary version of the sforzando chord and ac-companying ostinato (as opposed to the fully realized orchestral version)when he arrived at the rhythm of this passage.

45. See the penultimate system on p. 35 of the sketchbook, Stravinsky’s first sketch of themovement.


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“Glorification of the Chosen One”

I now turn to the simplest and the most complex of my examples, both ofwhich feature several more instances of 5+6 and 6+5 intervallic and durationalpatterns. Both examples are drawn from the beginning of “Glorification of theChosen One,” extended to include the two bars that immediately precede it(the “Glorification” proper begins at R103). During the first of these twobars, much of the orchestra participates in a one-bar glissando that leads intoperhaps the most obviously “primitive” moment in Stravinsky’s score, an11/4 bar consisting of eleven repetitions of a single dissonant chord. In mostof the participating instruments, this glissando is written out as a quintupletfollowed by a sextuplet, which Stravinsky emphasized in his original sketchwith a disproportionately large “5” and “6” above the uppermost line (seepage 61 of the sketchbook). This rhythmic treatment partitions the chromaticglissando into 5+6 semitones. Vertically, each note of the glissando is harmo-nized with a 5+6 trichord. Example 10a illustrates by reproducing the fluteparts: the linear 5+6 intervallic partition is shown above the uppermost staff,and the harmonic one is shown on the left side of the example. In the follow-ing 11/4 bar, the repeated chord consists of two superimposed, inversionallyrelated versions of the 5+6 and 6+5 trichord, as shown in Example 10b; theexample reproduces Stravinsky’s notation of the chord in the four-hand score,which emphasizes its derivation from the 5+6 and 6+5 trichords.46 Note, ofcourse, that each trichord spans eleven semitones from top to bottom, and thechord is repeated eleven times. Thus these two bars feature a pervasive coordi-nation of units of eleven and 5+6 subdivisions in rhythmic and intervallicspace.

A closer examination of the intervallic patterns embedded within the chordfrom the 11/4 bar suggests a close relationship between these patterns andthe wildly irregular metrical scheme of “Glorification.” In addition to the der -ivation of the chord from 6+5 and 5+6 trichords, the same pitches, measuredfrom top to bottom, generate the intervallic pattern shown on the far right ofExample 10b, 6+3+2+3+6. The bottom portion of Example 10b outlines sev-eral other possible groupings of these intervals: 9+5+6, 14+6, and the inver-sions of these. Example 10c represents the first 16 bars of the “Glorification,”charting the duration of each bar in eighth notes (5, 9, 5, etc.). The secondbar of the printed score is omitted from the example: this bar is an exact repeatof the first and was not part of Stravinsky’s original sketch of the passage (seethe drafts on pages 61 and 67 of the sketchbook). Underneath these durationsare possible groupings that correspond to the intervallic groupings outlined

46. This chord is extremely similar to a chord featured prominently in what may have beenone of Stravinsky’s very first sketches for The Rite. In a sketch that according to Craft predated thesketchbook, Stravinsky composed a repeating octatonic chord based on symmetrically related 6+5and 5+6 trichords, one in each hand. See the first bar of the second system of the sketch, as repro-duced in Stravinsky in Pictures and Documents, 597.


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Note: The published score begins with a repeated 5/8 bar; this diagram corresponds to the origi-nal version of the sketchbook, in which the 5/8 bar is not repeated.


above (5+9, 14+6, 6+5+9, etc.). Analyzed in this way, virtually every bar ofthis section can be accounted for in relation to the intervallic structure of the11/4 chord. The bottom of the example outlines a five-part symmetrical formbased on these durations: the section begins and ends with the durational pat-tern 5+9+5, and the pattern 14+6+14 occupies the center, flanked by two in-stances of the pattern 3+4.47 Likewise, the chord from the 11/4 bar has afive-part symmetrical intervallic structure (6+3+2+3+6). In sum, the elemental11/4 bar seems to have served as a progenitor of the following section, withthe harmonic construction of the chord from this bar providing a model forthe overall metrical structure of the first sixteen bars of the “Glorification.” Itis noteworthy that this chord returns prominently three times during thesesixteen bars and in fact dominates the center of the symmetrical scheme (seethe two 14-beat bars at R106+1 and R107+1, bars 7 and 9 in Ex. 10c). Moregenerally, this example suggests that Stravinsky may have used intervallic mea-surements to derive rhythmic and metric patterns on a large scale, a possibilityI will revisit below in my investigation of Stravinsky’s sketches, which containevidence of Stravinsky having tracked durational patterns (in some cases hun-dreds of beats at once) in large sections of The Rite.

“Spring Rounds” and “Evocation of the Ancestors”

My final two examples involve largely diatonic environments where subdivi-sions of the octave are more prominent than the characteristic 5+6 and 6+5subdivisions of the major seventh we have seen thus far. Example 11a repro-duces the first three bars of “Spring Rounds” as they appear in Stravinsky’ssketchbook (see page 9). This early sketch of the opening clarinet melody dif-fers from the printed version in two details: it contains a grace note in the firstbar48 and shorter slurs in the first two bars (in the printed version, the notes ofeach bar are contained under a single slur). The groupings indicated by theslurs of the sketchbook are maintained in Stravinsky’s rhythmic notation of theflutes in the printed version, which sustain a trilled E � during the first two bars,notated as shown in Example 11c. My durational groupings in Example 11aare suggested by the placement of Stravinsky’s original slurs and grace notes,

47. It is worth noting, additionally, that the first, third, and fifth parts of the symmetricalscheme are each symmetrical in themselves (5+9+5; 14+6+14; 5+9+5), and that all three parts aresymmetrical in their musical content as well (in each case, the first and third bars contain identicalmaterial). Finally, the 5+9+5 bars at the beginning of the passage are identical in musical contentto those at the end, as is true for the paired 3+4 bars, further enhancing the overall symmetry. Thepervasive presence of quasi-symmetrical durational structures in The Rite was a major conclusionof Boulez’s “Stravinsky Remains”; although in my view Boulez overestimated the significance ofsuch structures, there can be no doubt that symmetry (and quasi symmetry) played a role inStravinsky’s compositional decisions.

48. This omission in the printed score is curious, given that the transposed return of themelody at R56+2 contains grace notes in exactly the same positions as the sketchbook does.


the notated meter, and by iterations of the pitch center, E �, the primary pitchcenter of the entire “Spring Rounds” (each E � initiates a durational group inmy analysis). The intervallic measurements of the complete pitch collectionoutlined by this melody correspond precisely to the durational measurements,as can be seen by comparing parts (a) and (b) of the example.

The opening fanfare of the “Evocation of the Ancestors” (beginning atR121+3) is more ambiguous in terms of its internal phrasing.49 The dura-tional analysis shown in Example 12a simply follows the meter of the printededition. These durational groupings correspond to intervallic measurementsderived from the primary chord of this passage, as shown in Example 12b.Note that these measurements are based on a cross section of the chord (simi-lar to my analysis of the primary chord of “Dance of the Earth”) but preservethe uppermost pitch-class, E. Note also the provisional suggestion of a corre-spondence between the single quarter note that ends the fanfare and the com-pound semitone formed by the low D� pedal against which the fanfare isheard. In the intervallic analysis of Example 12b, this semitone complicateswhat would otherwise be a straightforward subdivision of the octave,4+3+2+3. The motivic organization of the fanfare conveys an even more basic

49. This ambiguity is reflected by Stravinsky’s apparent difficulty in deciding how to bar thefanfare: Stravinsky’s original sketch, the autograph and four-hand scores, and the 1921 editioneach bar the fanfare somewhat differently (although Stravinsky’s barring of a related passage on p. 74 of the sketchbook suggests that he had already conceived of the 1921 barring of the fanfareat an early stage). Van den Toorn has gone to great lengths to demonstrate how the revised bar-ring of the 1929 score (which maintained the barring of the 1921 score) conveys a degree of sub-tlety missing in the earlier versions of the fanfare: see Stravinsky and the Rite of Spring, 49–51,86–94. For Taruskin’s take on these different barrings, see Russian Traditions 1:959–61.

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7+5 subdivision, the reverse of the 5+7 subdivision featured in the “SpringRounds”; we might regard this pair as a domesticated cousin of the 5+6 and6+5 “Grundgestalt.”

To summarize the upshot of the preceding analyses: each is a hypothesissuggesting a relationship between intervallic and durational patterns that mayhave provided Stravinsky with a compositional starting point, a means of “lim-iting the field of action” in order to unlock creative impulses, to glossStravinsky’s famous statement that is cited at the beginning of this article. Inno example would I claim to have presented Stravinsky’s exact conception of a musical relationship; instead, each example is an educated guess at what hisconception was, based on the evidence of the sketches, the autographs, andthe earliest printed editions of the score. My argument is located not so much

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in the specific details of any one analytical example, but in the totality of theseexamples (including those I have been unable to present here due to limita-tions of space). Taken as a whole, they provide powerful evidence that Stra vin -sky used intervallic series to generate rhythmic ideas. And thus an unfamiliarpicture of Stravinsky emerges, in which the composer is an almost passive ob-server at the conception of some of the most famous rhythmic innovations ofthe twentieth century.

Sketches

On Sunday, 17 November 1912, Igor Stravinsky, with an unbearable tooth -ache, finished the music for The Rite of Spring. We know this, of course, be-cause Stravinsky declared it triumphantly (if a bit prematurely) in blue and redpencil on pages 96–97 of his sketchbook. The size of Stravinsky’s handwritingsuggests that he was understandably excited to write these words, and indeed,just as these pages record a decisive moment in the composition of The Rite,they mark the climax of the sketchbook itself, a work of art in its own right.Surrounding Stravinsky’s announcement, and amongst his sketches for theend of the “Sacrificial Dance,” we find an explosion of other marginal jottings,including two unusual symbols in blue (one at the top of each page) that func-tion as continuity indicators50 (symbols that indicate connections betweennon-consecutive sketches), and numerous unexplained calculations. Acrescendo of activity marks the preceding pages as well, where Stravinskydrafted earlier portions of the “Sacrificial Dance”: quasi-hieroglyphic symbolsand numerical notations turn up frequently in these sketches. Many of thesemarkings are cryptic, and most have received no more than casual interpreta-tions, sometimes being dismissed altogether as trivial.

These sorts of notations are not limited to the sketchbook: the autographfull and short scores for The Rite and the four-hand reduction on whichStravinsky recorded choreographic instructions for Nijinsky are marked bysimilar scribbles, although they are less pervasive. One finds, for instance, over-sized metrical indications hovering prominently over groups of bars, additiontables recording the precise size of instrumental forces at a given point in thescore, and many other unlabeled calculations. These sorts of markings aresprinkled through numerous other manuscripts of Stravinsky’s as well, fromvarious stages of the composer’s career. An overwhelming impression afterviewing these documents is that of a composer preoccupied with numbers andarithmetic. This is precisely how Stravinsky described himself late in life, in a1966 interview with Robert Craft:

All composers eventually become obsessed with numbers, the rapport ex-pressed between them being so much greater than most expressions of rapport

50. I am grateful to David Grayson for suggesting this term.


in reality . . . certain musical relationships are more clearly expressed as num-bers. It may well be that my love of combining twos, threes, fours, and sixes iscompulsive and that a composer who works this way is behaving in music likethe man who has to lock his door three times, or step on all of the cracks in thesidewalk. But if this is true, and musical composition involves nervous disor-ders, I would not want to be cured.51

Perusal of the sketches and manuscripts for The Rite suggests that Stravinsky’snumerical compulsive disorder was already at an advanced stage by 1913; yetit remains far from obvious to what extent the numerical notations in thesedocuments might provide insights into his compositional method. However,as I will argue, careful study of the numerical and symbolic notations in thesketchbook reveals what seems to be an elaborate set of durational calculationsthat operate on a scale much larger than those considered thus far, in somecases involving tallies of hundreds of beats at a time. Interpretation of thesenotations helps to support the idea outlined above that Stravinsky conceivedof durational patterns in The Rite in terms of abstract numerical relationships,while suggesting further that these calculations held important symbolic sig-nificance for Stravinsky: that is to say, Stravinsky’s numerically based composi-tional strategies seem to be linked to private meanings.

The sketches for the “Sacrificial Dance” (pages 84–102 of the sketchbook) areparticularly rich with marginalia and thus provide a convenient site for explor-ing the relevance of such annotations.52 I will begin by examining the sketcheson pages 90–95, where Stravinsky drafted continuously what would becomeR149–R162, music characterized by a staccato-chord vamp overlaid with spo-radic descending chromatic figures marked marcato in the brass. Pages 90–91are distinguished by the presence of numerous quasi-hieroglyphic characterswith no corollaries elsewhere in the sketchbook. Craft identified the cluster ofdoodles in the lower right margin of page 90 as Stravinsky’s attempt to createan “impresa” out of his initials,53 and on closer inspection, we can see thatStravinsky was also experimenting with a rendering of his first initial and entirelast name, perhaps imagining some ancient alphabet (see Figure 1 for one of

51. Craft, Stravinsky: Chronicle of a Friendship, 445. Elsewhere, Stravinsky explained to Craft:“. . . the way composers think—the way I think—is, it seems to me, not very different from math-ematical thinking. I was aware of the similarity of these two modes while I was still a student; and,incidentally, mathematics was the subject that most interested me in school”; see I. Stravinsky andCraft, Conversations with Igor Stravinsky, 17.

52. A short melodic fragment can also be found in the lower left corner of page 64. Cyr iden-tified some possible fragments on pages 66, 81, and 83; see “Petite histoire,” 103. Given the myr-iad challenges of deciphering Stravinsky’s sketches, this section of the my essay often requires thecareful consideration of minutiae; here and elsewhere, the finer points of these discussions areconfined to the footnotes. Some readers will undoubtedly find it helpful to have the sketchbookfacsimile at hand while reading this discussion.

53. Craft, “Commentary to the Sketches,” 24.


these doodles). The three characters enclosed in a box appear to constitute afurther abstraction of Stravinsky’s name. This boxed signature may have beena “seal” inspired by Stravinsky’s recent fascination with Japanese art and poetry, an interest that led to his composition of the Three Japanese Lyrics atthe same time he was working on The Rite. The larger red symbols on pages90–91—perhaps related to the cluster of doodles on page 90—appear to de-lineate a group of beats. The music sketched between the first symbol and thelast corresponds to R149–R151 of the printed score (where Stravinsky subdi-vided all of the 4/8 and 5/8 bars from the original sketch), a distinct musicalsection. This passage spans 29 eighth-note beats, and the five red symbols inthe sketchbook are arranged in a near-symmetrical pattern over these bars (the first two of these, occupying the top of page 90, are shown in Figure 2).54

The importance of this durational unit is confirmed on the following pages,where Stravinsky marked the next two units of 29 beats with circled “29s”(see the top of page 92 and the upper right margin of page 93)—the only in-stances of circled numbers in the sketchbook.55 The remainder of the passageis unmarked by “29s,” but it spans 58 beats—that is, exactly two more groupsof 29.56

In sum, the complete section sketched on pages 90–95 can be partitionedinto exactly five groups of 29 beats, following Stravinsky’s delineation of thefirst three of these groups.57 The division of the music into groups of 29 beats

54. If these red characters represent a durational analysis, then a potentially significant prece-dent can be found in Georgy Konyus’s theory of “metrotechtonicism,” which began to circulatein the first decade of the twentieth century and was known to such figures as Sergey Taneyev andAnton Arensky as early as 1902. Konyus believed that musical works of the eighteenth and nine-teenth centuries were essentially symmetrical in their temporal construction: his analyses dividedworks into durational segments, measured in beats, which formed symmetrical patterns at variouslevels of structure. See in particular his analysis of a passage from Chopin’s F-minor Ballade, re-produced in Carpenter, “Contributions of Taneev,” 296–97. This analysis is extremely similar,conceptually, to my analysis of the “Glorification of the Chosen One” as represented in Example10, and the details of its durational analysis are virtually identical to those of Stravinsky’s “analysis”on pp. 90–91 of the sketchbook. The latter correlation is likely coincidental, but it pointsnonetheless to an intersection between idiosyncratic Russian theoretical ideas and Stravinsky’smusical thought.

55. The “29” on p. 93 is not positioned precisely (i.e., after the fourth beat of the secondbar), perhaps due to the lack of space at the top of the page.

56. The first of these is completed after the first bar (3/8) of the second system on p. 94. Thesecond is completed by the 6/8 bar (later eliminated) at the end of the first system on p. 95, atwhich point Stravinsky abandoned the sketch, writing “etc.” over every staff and adding a dra-matic symbol in blue, apparently as a continuity indicator. On the following page, this symbol re-curs in the upper left corner, but here it is unassociated with any musical sketch (the sketches onthis page all pertain to the ostinato section from R174 to R186 in the printed score).

57. This potential for 29-beat groups is maintained in the printed score. Here, Stravinskyomitted the 6/8 bar (i.e., 6 beats) from the end of the section sketched on pp. 90–95, but headded a new passage (R162–R165) which spans 35 beats. In other words, he lengthened the section by a total of 29 beats, and thus R149–165 of the printed score spans precisely 6 groups of29 beats (174 beats total).


was almost certainly related to Stravinsky’s choreographic conception of thispassage. As recorded in Stravinsky’s copy of the four-hand piano score, thecomposer often conceived of the choreography in terms of large groups ofbeats, which tended not to correspond to logical musical sections: “the princi-

Figure 1 Stravinsky’s rendering of his name (“I. Stravinsky”) on page 90 of the sketchbook

Figure 2 Two red symbols from the top of page 90 of the sketchbook


pal choreographic accents and phrase units . . . were seldom coterminous withthe accents and phrases of the music,” as he explained in his commentary onthe four-hand score.58 Stravinsky in fact partitioned a group of 29 beats earlierin the ballet, beginning one bar before R138 (“The women continue therhythm, 29/4,” as he wrote in his later commentary).59 R149 seems to havemarked a meaningful choreographic division in Stravinsky’s conception, themoment at which the Chosen One first moves from her fixed position on thestage, and thus Stravinsky may have intended to have the commencement ofhis groups of 29 mark this event.60 These groups of 29 beats are not recordedin the four-hand score, however, nor do Stravinsky’s annotations in that scoreinclude any unusual symbols such as those found in the sketchbook. Further -more, the sketchbook does not seem to have been a place where Stravinskygenerally recorded choreographic ideas. We might suspect, then, that thequasi-hieroglyphs in the sketchbook have meaning beyond the realm ofchoreography, and perhaps held personal and symbolic significance forStravinsky.

Pages 96–97 correspond to R174–R186 of the printed score, the sectiondominated by an ostinato pattern in the low strings, low brass, and percussion.Unlike the circled “29s” on pages 92–93, however, the calculations in themargins of pages 96–97 are not visibly tied to any musical considerations.These calculations are grouped in two clusters, one in the left margin of page 96 and one in the left margin of page 97. For most of the calculations onpage 97, Stravinsky seems to have multiplied a number first by 3 and then by 5—that is, by 15 (see Fig. 3). This operation yields three products from top to bottom: 225, 180, and 210, derived from the numbers 15, 12, and 14,respectively. (Stravinsky seems to have ordered these products, writing a “1”next to “210,” a “2” next to “225,” and a “3” next to “180.”) Were we tomultiply the number 10 by 15, this would yield the product 150, and thisnumber features prominently in the cluster on page 96, occurring three times(see Figure 4). Here we also see the numbers 225 and possibly 135 (the latteris not entirely legible)—both multiples of 15—the latter written over, or over-written by, the number 133. A likely scenario is that 133 was a corrective to135, given that the former is featured in the calculation 133+150 = 283 whichseems to bring together the free-floating “150” and “133” at its left. Perhapsrelated to this calculation is the aborted addition of 150 and 130 at the bot-tom of this cluster. Finally, one other calculation is situated at the top of thecluster on page 96: 40+13 = 53+1 = 54.

Craft found all of these numbers mystifying: “whether the arcane calcula-tions to the left are concerned with overdue rent at the Châtelard, railroadtimetables, musical minutage, or numbers of instruments, I have no idea.”61

58. See Stravinsky, “Stravinsky-Nijinsky Choreography,” 35.59. Ibid., 42.60. Ibid.61. Craft, “Commentary to the Sketches,” 25.


There are examples of Stravinsky both calculating monetary sums and jottingdown train schedules in sketches for other pieces, but the fact that he seems tohave jotted down these numbers at such a critical stage of his composition ofThe Rite, soon after tracking the groups of 29 on pages 90–95, raises the pos-sibility that the numbers on pages 96–97 might indeed concern minutage—specifically, durations measured in beats.

Table 1 shows the duration in beats of seven segments comprising thecomplete “Sacrificial Dance,” both as drafted in the sketchbook and as it ap-pears in the printed score. Most of these segments are derived from the defini-tive drafts of the sketchbook—those that conform most closely to theautograph scores and four-hand reduction.62 The only exceptions are: (1) the

62. Note that some fragmentary sketches on pp. 84–85 and 87 are superseded by thesketches referred to in Table 1, and that Stravinsky fleshed out the orchestration for much of thesketch on p. 97 on pp. 98–101.

Figure 3 Left margin of the sketchbook, page 97

Figure 4 Left margin of the sketchbook, page 96


transposition of R142–R149 down a semitone at R167–R174, whichStravinsky indicated in the sketchbook but did not write out (see the refer-ences to “C� major” at the bottoms of pages 87 and 95); and (2) R162–R165and R186–end, which Stravinsky did not sketch.63 Note that all seven seg-ments correspond to logical musical sections in the printed score. In fact,Table 1 represents the most obvious segmentation of the movement, save forthe division it shows at R165 (which is best viewed as secondary).

Before we examine these beat counts and comparing them with the num-bers on pages 96–97, some preliminary issues must be addressed. To begin, itis not clear at what stage Stravinsky jotted down the numbers on pages 96–97.Presumably he did not write them before reaching page 96 of the sketchbook;that is, presumably the earliest he wrote these numbers was immediately aftersketching the music on page 95. But he may also, of course, have written thesenumbers while in the midst of sketching the music on pages 96–97, or atsome later point. If these numbers do represent durations of the “SacrificialDance” measured in beats, they may represent Stravinsky’s computations ofbeats in segments he had already drafted, or they may represent targets formusic Stravinsky had not yet drafted or planned to revise. And if the latter,Stravinsky may or may not have later realized these targets. Therefore, in whatfollows, I will consider the durations in beats of segments of the “SacrificialDance” not only as drafted in the sketchbook (and here there are revisions toconsider as well—erasures, overwritings, and the like), but also as they wouldultimately appear in the printed score. The numbers on pages 96–97 may referto one or the other, or to both.

My identification of beats in the “Sacrificial Dance” adheres to the principleestablished above, which seems to have been fundamental to Stravinsky’scomposition of The Rite: the “beat” in a given segment is the largest possible

63. It appears that Stravinsky intended to sketch a draft of R162–165 next to the blue symboloccupying the upper left corner of page 96 but dispensed with the sketch for some reason.

Table 1 Total Beats of “Sacrificial Dance” Segments in the Sketchbook and the 1921 Edition

Rehearsal Page nos. Value of Value of beat No. of beats No. of beats nos. (score) (sketchbook) beat (score) (sketchbook) (score) (sketchbook)

R142–R149 pp. 86–87 � � 112 111R149–R162 pp. 90–95 � � 139 145R162–R165 (no sketch) � — 35 —R165–R167 p. 95 � � 21 21 (orig. 23?)R167–R174 (no sketch) � — 112 —R174–R180 p. 97 � � 77 72R180–R181 p. 89 � � 23 � 148 22 � 128 (orig. 98)R181–R186 pp. 96–97 � � 48 34R186–end pp. 87–89 � � 265 132


note value that allows all durational measurements to be made in whole num-bers. In the sketchbook, this principle designates the eighth note as the beat inmost of the segments outlined in Table 1 (see the fourth column). Stravinsky’smetric notations in the sketchbook support these beat designations precisely:that is, the lower numbers of his time signatures always correspond to the notevalues shown in the fourth column of Table 1. In the autographs and subse-quent versions of the score (up until his 1943 revision of the “SacrificialDance”), Stravinsky renotated several of the segments outlined in Table 1such that all rhythmic values are cut in half. In these sections, a sixteenth noteis equivalent to what had been an eighth-note beat in the sketches, and thus Itreat the sixteenth note as the beat in these sections (see the third column ofTable 1).

It should be noted that this interpretation of the beat is not entirely consis-tent with the regular pulse that underlies the movement in the printed editionsof the full orchestral score. The tempo indications of the printed score, includ-ing metric modulations at R174, R180, R181, and R186, indicate that a basic pulse of 126 beats per minute—associated with different note valuesduring different sections—should remain constant throughout the entire“Sacrificial Dance” (fermatas notwithstanding). The beats identified abovecorrespond to this pulse in some sections but not in others, and thus the“beat” as I have defined it is not reliable as a unit upon which to measure ab-solute durations across sections of the “Sacrificial Dance” in the printed score.

Yet as Robert Fink has argued, the immutable beat of the printed editions(“modernist geometry,” in Fink’s words64) is an outgrowth of Stravinsky’slater efforts to divorce his score for The Rite from the original ballet, not therealization of any initial inclination toward a systematic treatment of the beat.On the contrary, it is unlikely that Stravinsky had a system of beat relationshipsor even many specific tempo markings in mind when composing the work. AsFink explained:

It would be comforting to let the composer convince us that he had an idealontological tempo in mind for every dance before the first rehearsal of the Rite,but it is more likely . . . that loose tempo ranges were honed collaborativelyduring the dance and orchestral preparations.65

In other words, as Stravinsky was composing and revising the “SacrificialDance,” there was likely no possibility of his using beats to measure absolutedurations or proportions on a large scale. And there is no particular reason tothink that Stravinsky wished to keep track of absolute durations in the“Sacrificial Dance” anyway. As I will argue below, Stravinsky’s beat tallies inthe “Sacrificial Dance” seem to have related to his conception of the choreog-raphy (in which dancers were required to count large groups of beats) and,

64. Fink, “Rite of Spring,” 314.65. Ibid., 325.


in some instances, to have possessed abstract or symbolic significance.Further more, if the beat is construed as I have suggested, then a crucial char-acteristic of the “Sacrificial Dance” emerges: in the printed score, every bar ofthis movement (save for the final three) consists of 2, 3, 4, or 5 beats (all well-represented durational units in the analyses above); the situation is virtually thesame in the sketchbook as well, where Stravinsky often keeps track of thesemetrical types by writing the numbers 2, 3, 4, and 5 over the appropriate bars(see, for instance, the top of page 88).66 Each segment of the “SacrificialDance” outlined in Table 1 is characterized by the interaction of all four metri-cal types. Thus, although one might be tempted, for example, to treat the beatat R174 of the printed score as the eighth note rather than the quarter (in or-der to preserve a consistent pulse), to construe the duration of the first threebars at R174 as 10+6+10 rather than 5+3+5 would seem to violate Stra -vinsky’s basic conception of meter in the “Sacrificial Dance, and indeed in TheRite as a whole.

We are now ready to take a closer look at Table 1. The beat counts shownin the table reflect several discrepancies between how the various segments appear in the sketchbook and in the printed score:

1. The downbeat D of R142 is not present in Stravinsky’s sketch on page86; this accounts for the one-beat difference between R142–R149 ofthe printed score and the equivalent passage in the sketchbook.67

2. Stravinsky originally included an extra 6/8 bar at the end of what wouldbecome R149–R162 in the printed score; this accounts for the six-beatdifference between the two versions (139 versus 145 beats).

3. Also as discussed above, Stravinsky seems originally to have intended tosketch some version of R162–R165 of the printed score on the top ofpage 96, but ultimately did not sketch this segment.

4. On the middle system of page 95 of the sketchbook, Stravinsky scrib-bled out a 2/8 bar; thus it appears that he may have originally conceivedof the passage from R165 to R167 as spanning 23 beats rather than 21,as indicated in the table.

5. Following the sketchbook, Table 1 subdivides R174–R186 into threesegments that correspond to the initial ostinato passage, the interpola-tion of music from the opening of the movement, and the return of the

66. Taruskin referred to Stravinsky’s play with the various permutations of these metricaltypes in the “Sacrificial Dance” as “mosaic”-like; Russian Traditions 1:962–63. One might char-acterize in the same way Stravinsky’s play with permutations of 3, 4, and 5 in intervallic space. Forinstance, the three variants of the “Augurs” chord at R14 are distinguished by the intervallic parti-tioning of the lower E octave in the bassoons and cellos: 5+3+4 (the primary “E-major” chord),4+5+3 (the C-major variant), and 5+4+3 (an E-minor variant).

67. This omission is curious, given the presence of a downbeat D in an earlier sketch on p. 84.It would seem that the sketch on p. 86 was meant to proceed from the descending chromatic sep-tuplet on the second system of p. 83, which ends with a low D (compare with the clarinet line thatinitiates the Sacrificial Dance in the printed score).


ostinato figure; these segments are delineated by metric modulations inthe printed score. In the sketchbook, the first ostinato segment beginson page 97 and ends with the penultimate bar in green near the end ofthe penultimate system (the equivalent of R180 in the printed score).68

The interpolation begins with the following bar in green and continueswith the first four bars on page 89. (Note that this interpolation differssignificantly from the published version, although it is nearly identical induration.) Finally, Stravinsky’s sketch of the second ostinato segmentbegins with the final system on page 97 (including the initial bar ingreen), continues at the bottom of page 96 (as indicated by a circled red“X”), and ends with the penultimate bar on that page (that is, just be-fore the incomplete 5/8 bar, which he sketched in full to the immediateright of the blue “X” on page 88, the beginning of a new segment). Theseven 2/4 bars that follow the interpolation in the printed score (R181–R182+2) are absent from Stravinsky’s sketch; this omission accounts forthe fourteen-beat discrepancy between the two versions of the secondostinato segment (48 versus 34 beats).

6. Stravinsky seems to have originally sketched the ostinato section onpages 96–97 without an interpolation. The sketch in green at the end ofthe penultimate system on page 97 overwrites several bars sketched inblack, which seem to proceed directly to the final system. It appears thatthese original bars were essentially the same as the five bars that immedi-ately precede the sketches in green, which Stravinsky enclosed in brack-ets, indicating a repetition of these bars. I assume that Stravinskyoriginally wrote out the repetition at the end of the system (remnants ofthe final bar are still clearly visible, which he would resketch in green as the first bar of the final system), but later wrote over the repetitionwith the new bars in green, and was thus forced to indicate the repeti-tion with brackets around the relevant five bars. The original version ofthe ostinato section would total 98 beats, as indicated in the table.69

7. The printed score reflects substantial revisions and expansions of theconcluding section of The Rite that Stravinsky drafted on pages 87–89.The meter of the second bar on page 89, which features the familiar

68. Note that the second bar on p. 97 is marked “bis.” In a revision on the following page,Stravinsky substituted 5/4 and 3/4 bars for the “bis” bar, a revision he maintained in the printedscore. My beat calculations for the sketches for this passage do not incorporate this revision, as itmore than likely was made after Stravinsky jotted down the calculations on pp. 96–97 (althoughthe exact chronology is not clear, as discussed above).

69. The sketch spans 56 beats from the first bar of p. 97 up to the sketch in green ink on thepenultimate system of the page; under the sketch in green, Stravinsky presumably wrote out thefive-bar repetition (10 beats long) that he later indicated with brackets. The original sketch inblack from the bottom system on p. 97 through the bottom of p. 96 (up to the incomplete 5/8bar) spans 32 beats. Thus, the total duration of the original sketch on pp. 96–97 is 98 beats(56+10+32).


woodwind tremolo and ascending glissando figure from the conclusionof The Rite, is ambiguous. Two strong clues, however, suggest that theintended meter here is 3/8. First, 3/8 is the meter of the previous threebars (see the second system on page 88, the last two bars of which are resketched with full orchestration on page 89). Second, in the four-hand piano score (where the rhythmic values are halved), the music inquestion spans three sixteenth-note beats, here notated as the last threebeats of a 5/16 bar (see the third-to-last bar of the four-hand score).70

As shown in Table 2, a simple proportional pattern can be derived from thebeat totals shown in Table 1. In two instances, contiguous segments combineto form a larger group of 56 beats: R162–R167 of the printed score and thedraft of R180–R186 in the sketchbook. The number 56 is exactly half of 112,the duration of R142–R149 and R167–R174 in the printed score. The draftof R174–R180 in the sketchbook spans 72 beats, almost exactly half the beattotal of the draft of R149–R162 (145 beats). Although Stravinsky did notsketch R162–R165 or R167–R174, if we assume that he planned these seg-ments in their published forms (35 and 112 beats, respectively; these numbersare enclosed in parentheses in Table 2), and if we redistribute a single beatfrom the second sketchbook segment (the draft of R149–R162) to the first(the draft of R142–R149), then the proportional pattern outlined in Table 2can be discerned, a pattern that accounts for the complete original sketch ofthe “Sacrificial Dance” save for the final segment (what would become R186–end). The presence of these proportional relationships in Stravinsky’s sketchessupports two of my larger claims: (1) that the segmentation of the “SacrificialDance” in the sketchbook (as represented in Table 1) is meaningful; and (2) that the beat totals outlined in Table 1 are meaningful as well.71

These two claims underlie Table 3, which suggests numerous connectionsbetween the durations charted in Table 1 and Stravinsky’s calculations onpages 96–97 of the sketchbook:

1. Stravinsky’s original sketch of R149–R162 (145 beats, partitioned29�5 as discussed above) and the 35-beat segment from R162 to

70. Some other aspects of this sketch may require clarification. The final two bars of thepiece, as Stravinsky indicated in blue on p. 89, are sketched on p. 87, after which he wrote “Endof the Sacre”; each of these bars is in 3/4 (the printed score similarly ends with two 3/4 bars, al-though the music of these bars is much different). My calculation of 132 beats assumes that thebars labeled by letters in the bottom right corner of p. 88 correspond with those sketched to theright of the blue “X” on the same page. Note also that the arrow leading from the bottom rightcorner of the page to the upper left points toward the second bar of the first system; the first bar isredundant with the bar labeled “E” in the lower right corner.

71. Code has uncovered numerous simple proportional relationships on many levels in hisexhaustive study of the “Augurs of Spring” movement; see “Synthesis of Rhythms.” JonathanKramer has asserted the importance of proportional relationships in Stravinsky’s music generally;see in particular “Discontinuity and Proportion in the Music of Stravinsky.”


R165 (which Stravinsky did not sketch but may have planned) togetherspan 180 beats; 180 appears as the product of 12�15 in the cluster ofcalculations on page 97.

2. The printed version of R149–R162 spans 139 beats; this segment andthe following two (R162–R167) span 195 beats in the printed score, amultiple of 15 (13�15 = 195).

3. In the printed score, R165–R174 spans 133 beats. Stravinsky’s originalsketch of R165–167 was two beats longer than in the published version,and thus R165–R174 may have spanned 135 beats (a multiple of 15) inhis original conception (recall that R167–R174 is indicated in thesketchbook but not sketched). The duration of this section, then, mayaccount for the “135” that seems to have been overwritten by “133”on page 96 of the sketchbook; furthermore, this revision may corre-spond to Stravinsky’s deletion of the 2/8 bar on the bottom of page 95.

4. Although in the printed score, the duration of R165–R174 is not amultiple of 15, R165–R180 spans 210 beats; 210 appears as the prod-uct of 14�15 in the cluster of calculations on page 97.

5. Assuming that Stravinsky planned R167–R174 to span 112 beats (assuggested above), the passage from R167 to R186 would also havespanned 210 beats in Stravinsky’s original sketch in black on pages 96–97.

6. Stravinsky’s revised sketch of R167–R186, incorporating the sketches ingreen ink and the fragment at the top of page 89, spans 240 beats(again, assuming that R167–R174 spanned 112 beats in his originalconception); 240 is the product of 16�15, and 16 appears as a free-floating number (perhaps awaiting multiplication by 15) near the bot-tom of the cluster of calculations on page 97.

7. In the printed score, R167–end spans 525 beats; 525 is the product of35�15, and 35 appears underneath 16 on page 97 (also possibly await-ing multiplication by 15).

A few aspects of the beat totals charted in Table 3 merit special attention.These totals commence from three different starting points: R149, R165, and

Table 2 Proportional Pattern Derived from Sketchbook Segments

Rehearsal nos. Total beats

R142–R149 111 (+ 1) = 56 � 2R149–R162 145 (� 1) = 72 � 2R162–R167 (35) + 21 = 56

R167–R174 (112) = 56 � 2R174–R180 72 = 72R180–R186 22 + 34 = 56

Note: Rehearsal numbers are derived from the corresponding passages in the printed score.


R167. Note that the first of these (R149) marks the beginning of the onlysegment in the sketchbook that contains concrete evidence of Stravinskycounting large groups of beats (the segment partitioned into groups of 29 beats on pages 90–95).72 As mentioned above, in Stravinsky’s conceptionof the ballet’s choreography, the Chosen One begins to move laterally only atR149, which perhaps provides a clue as to why Stravinsky’s beat measure-ments appear to begin at this point. Also note that all but the first two beat totals in Table 3 include the 112-beat segment from R167 to R174, suggest-ing that this segment, although only implied in the sketchbook, was especiallycrucial to Stravinsky’s beat calculations. Stravinsky associated this segmentwith the I. S. “impresa” at the bottom of page 87, which I discuss further below.

Table 3 provides a potential explanation for every number and calculationon pages 96–97, except for the product 225 on page 97, the three prominent“150s” on page 96, and the calculation “40+13 = 53+1 = 54” on the samepage. As for the latter calculations, in the annotated four-hand score,Stravinsky delineated a 54-beat segment with a slash one beat after the equiva-lent of R162 and another slash labeled “54” one beat before the equivalent ofR167. As Stravinsky elaborated in his 1967 commentary on the four-handscore, the “true rhythmic phrase” (at odds with the notated meter) consists of“two measures of 7/8 . . . followed by one of 6/8.”73 This phrasing suggestsa 20+20+14 subdivision of the 54-beat segment (two 7+7+6 units followedby an incomplete 7+7 unit), which may help to explain Stravinsky’s calculationon page 96; Stravinsky’s addition of first 13, then 1, to 40 might suggest addi-tionally his equivocation between ending with a 7+6 or 7+7 unit.

If the “133” on page 96 of the sketchbook is indeed related to the dura -tion of R165–R174 (as suggested in Table 3), we might assume that the cal-culation “133+150 = 283” on the same page refers to durations as well. As

72. Although R142–R149 does not factor into any of the correspondences outlined in Table 3, it may be worth noting that in the printed score, R142–R143 and R143–R144 bothspan precisely fifteen beats.

73. Stravinsky, “Stravinsky-Nijinsky Choreography,” 43.

Table 3 Correspondences among Durations Charted in Table 1 and Marginal Calculationsfrom Stravinsky Sketchbook, pp. 96–97

Rehearsal nos. Source Total beats

R149–R165 sketchbook + score 145+35 = 180 (12 � 15)R149–R167 score 139+35+21 = 195 (13 � 15)R165–R174 sketchbook + score 23+112 = 135 ( 9 � 15)R165–R174 score 21+112 = 133R165–R180 score 21+112+77 = 210 (14 � 15)R167–R186 score + sketchbook 112+98 = 210 (14 � 15)R167–R186 score + sketchbook 112+128 = 240 (16 � 15)R167–end score 112+148+265 = 525 (35 � 15)


reckoned in Table 1, the duration of the ostinato section in the printed score(R174–R186) is 148 beats; thus the duration of R165–186 would be133+148 = 281—tantalizingly close to Stravinsky’s calculation. Looking againat page 96 of the sketchbook, we might consider whether the last bar ofStravinsky’s sketch, the incomplete 5/8 bar (equivalent to the first two beatsat R186 in the printed score), should be included in the total duration of theostinato segment rather than the following segment as sketched on page 88(recall that the incomplete 5/8 bar is sketched in full on page 88, where it be-gins a new segment). That is to say, the precise ending of the ostinato segmentmight best be designated by the arrival of G (completing the upper melody)on the downbeat of what would become R186. Following this logic in theprinted score would add one eighth-note beat to the duration of the segmentfrom R174 to R186, bringing the total to 149—or perhaps two sixteenth-note beats (given that the beat shifts to the sixteenth note at R186), bringingthe total to 150.74 The addition of two sixteenth-note beats to the ostinatosegment would necessitate an accompanying subtraction of two beats fromthe following segment; in the sketchbook, this would bring the total durationof the final segment of the “Sacrificial Dance” to 130 beats (as opposed to132, as indicated in Table 1), providing a potential explanation for the appar-ently aborted calculation “150+130” at the bottom of the cluster of numberson page 96.75

Each of the correspondences between beat counts and Stravinsky’s numericaljottings that I have shown is inherently speculative, and none of them makes a convincing argument about Stravinsky’s compositional practice on its own.As with the musical analyses above, it is the totality of these correspondencesthat is most compelling. And what, exactly, do these correspondences tell us?Clearly, Stravinsky was counting and keeping track of large groups of beats:Stravinsky’s annotations to the four-hand score established this fact long ago,and my findings add an additional layer of evidence. But what is even moresignificant in the context of the compositional strategies outlined earlier in thisessay is that Stravinsky seems to have shaped the durations of large sections tomeet abstract, numerical specifications. He seems to have targeted, for in-stance, multiples of fifteen and perhaps made musical adjustments in order to

74. Stravinsky used quarter notes in most voices in his sketch of the incomplete 5/8 bar; theeighth-note G in the upper melody follows Stravinsky’s erroneous use of eighth notes two andfour bars earlier. Thus Stravinsky seemed to have planned for the duration of the final G to span aquarter note, the equivalent of an eighth note in the printed score, although in the printed scorehe adjusted the rhythm to a sixteenth note followed by a sixteenth-note rest. Nonetheless,R181–R186 conditions one to hear the duration of the concluding G at R186 as a full eighthnote: this G is always a quarter note it its previous incarnations (see, for example, R185 andR185+2).

75. This interpretation would assume that Stravinsky had conceived of the ultimate durationof R174–R186 upon jotting down these numbers.


meet these targets; and these targets seem to have been important both whenStravinsky was drafting these segments in the sketchbook and later when hewas working on the complete version of the score. Some of these numericalspecifications seem to have been tied to Stravinsky’s choreographic ideas: thesum of 54 scribbled on page 96; the groups of 29 beats sketched on pages90–95; and, by analogy, perhaps the multiples of 15 targeted on page 97 aswell. If Stravinsky was in fact composing large sections of music with specificpredetermined durations in mind, then it is a small step to imagine him doingthe same thing on a smaller scale—that is, creating rhythmic or metrical pat-terns based on durations derived from intervallic structures. Indeed, Stravin -sky seems to have been engaged with counting beats and creating durationalpatterns on various musical levels, from surface rhythms to large-scale formalsegments.

This preoccupation with durations and counting seems to have intensifiedas Stravinsky approached the finish line, just as his compositional efforts weregrowing more intense as well. The spectacular tableau of pages 96–97—thevast musical sketch crammed into page 97, the triumphant declaration onpage 96, the proliferation of idiosyncratic symbols, and the calculations, all ofthese in black, blue, red, and (for the first time in these pages) green—pointsto a composer brimming over with enthusiasm and creative energy, the accu-mulation of which can be witnessed on the previous pages as well. Amongst all of this graphic activity, particularly intriguing is the blue monogram “I. S.”positioned as a continuity indicator at the top of the first bar on page 97 (seeFig. 5). The inspiration for the monogram may have come from Stra vin sky’sassociation with the art scene in Paris: numerous Parisian artists, fascinated byJapanese art and design, signed their works in this manner (recall the discussion above of Stravinsky’s “seal” on page 90). The sign appears onceearlier on page 87, where Stravinsky reminded himself to proceed from thesection sketched on pages 86–87 (what would become R167–R174) to the ostinato section sketched on pages 96–97. In his reminder, Stravinsky des-ignated the ostinato section by sketching the six-note motive upon which theostinato is based.

Example 13 reproduces this motive as Stravinsky sketched it on page 87,next to the blue monogram. The first three notes (E �, E �, B �) form the famil-iar 5+6 trichord, the very same pitch classes, in fact, that constitute the sharedskeleton of the “Augurs” chord and its C-major variant. As an encapsulationof his first and most crucial harmonic idea for The Rite, Stravinsky may haveregarded this motive as a musical signature, or as a “monogram” referring to alarger set of compositional materials and strategies. And thus the prominenceof the blue “I. S.” monogram on page 97, perched at the top of the page, mayindicate that Stravinsky, as he committed to paper what he believed to be hisfinal sketches for The Rite, regarded the six-note motive that saturates thispage as a personal stamp. The presence of Stravinsky’s monogram in closeproximity to the various multiplications by 15 in the left margin, at the end of


a movement permeated by various segments spanning multiples of 15 beats,might even make us wonder whether Stravinsky imagined a symbolic connec-tion between the letters “I S” and the numerals “1 5,” and whether the com-poser’s preoccupation with the number 15 as he was completing The Rite mayhave been a way of inscribing himself symbolically into the music at this criticalcompositional stage.76 Should this connection seem far-fetched, consider thatwhile Stravinsky was composing The Rite, fellow Parisian Lili Boulanger isknown to have fixated upon what she conceived as a symbolic connection be-tween her initials “L. B.” and the number thirteen, frequently rendering her

76. The significance of the number fifteen may extend to the realm of intervals as well: the C-major variant of the “Augurs” chord can be understood in terms of overlapping spans of fifteensemitones, anchored by the critical E�–B�–E� trichord:

Figure 5 Stravinsky’s monogram from the top of page 97 of the sketchbook

Ý Ł− Ł− Ł− Ł Ł¦ Ł Ł Ł¦

15

15

15

Ý Ł− Ł¦ Ł− Ł Ł− Ł�¹3

Example 13 Ostinato figure from the “Sacrificial Dance” (as represented in the sketchbook, p. 87)


monogram in a manner resembling Stravinsky’s impresa (see Fig. 6), and insome cases encoding the number thirteen within virtually every quantifiableparameter of her compositions, particularly in Clairières dans le ciel, a cycle ofthirteen songs that she began composing in 1913.77

Conclusions

My ideas about Stravinsky’s composition of The Rite rely on the accumulationof evidence from the score and sketches. But however compelling the weightof this evidence may be, to my knowledge there is no “smoking gun” thatprovides an explicit, unequivocal indication that Stravinsky was using interval-lic series to generate rhythms. This absence is hardly surprising. Stravinsky isnotorious for obscuring aspects of his personal history and compositionalmethod, and this tendency is pronounced in the case of The Rite, whose 1913premiere Stravinsky followed with various denials. These included his dis-avowal of the Montjoie! program, and, later, his downplaying of the use of folkmaterial and dismissal of the choreography and program as impertinent to themeaning of the work; each of these refutations was extreme and has been exposed as deceptive.78 “Whence this rage to cover tracks?” Taruskin has won-dered.79 In the present case, the most obvious answer would be that Stra vin -sky preferred no one suspect that the novel musical landscape of The Riteowed anything to mechanical compositional procedures that could be seen tobypass the composer’s own musical imagination.

77. See Dopp, “Numerology and Cryptography in the Music of Lili Boulanger,” an essay de-voted to the ways in which the number thirteen informed this and other works of Boulanger. I amgrateful to Carlo Caballero for bringing this source to my attention.

78. Of the folk and octatonic elements in The Rite, Taruskin wrote: “The deeper they went—the more they thus, as it were, receded from view—the more pervasive and determinant their in-fluence became”; Russian Traditions 1:949.

79. Ibid. 1:948.

Figure 6 Lili Boulanger’s monogram from the title page of Clairières dans le ciel


Whatever his motivations, Stravinsky seems to have taken pains to lay thegroundwork for the myth of The Rite’s “virgin birth” (as Taruskin has dubbedit80) while he was composing the score. As Stephen Walsh has noted in refer-ence to the sketches for the “Sacrificial Dance”:

As usual with Stravinsky, the music appears on the page—albeit in draft form—like an object plucked out of the air, exactly as the composer seems to havewanted to suggest when he described himself sententiously as “the vesselthrough which Le Sacre passed.” But these notations were actually the endproduct of a laborious process of sonic experimentation, carried out always at the keyboard, and involving much trial and error, much sounding and resounding. . . .81

Stravinsky’s habit of composing and experimenting at the piano may in factprovide an important clue about how he arrived at the idea to relate intervalsand durations in the manner described above. Stravinsky, as is well known,composed The Rite at a small piano in Clarens, and he may well have hap-pened upon the specific idea to measure intervals in semitones at the key-board, where the association of eighty-eight keys with eighty-eight pitchespromotes and facilitates intervallic measurement and numerical conceptions ofmusical space. This way of measuring intervals was by no means new in theearly twentieth century: European music theorists such as Heinrich Vincentand Anatole Loquin had begun to conceive of intervals in terms of their semi-tonal spans as many as forty to fifty years before Stravinsky composed TheRite.82

Much later in life, Stravinsky articulated the importance of the piano in hisapproach to composition, describing it as particularly conducive to the mi-crolevel analysis of musical materials: “The instrument itself is the centre of mylife and the fulcrum of all my musical discoveries. Each note that I write istried on it, and every relationship of notes is taken apart and heard on it againand again.”83 Here, the “relationship of notes” might refer specifically to in-tervallic relationships. Indeed, Stravinsky’s comments about intervals at vari-ous stages in his life point toward the conclusion that he conceived of theinterval, not the individual pitch, as the most basic unit of musical material.He described his compositional process, for instance, as “composing with in-tervals”84 and “relating intervals rhythmically,”85 and he spoke of remember-ing compositional ideas by “repeating to myself their intervals and rhythm.”86

And consider the following anecdote that Stravinsky related to Craft, certainly

80. Taruskin, “Myth of the Twentieth Century,” 7.81. Walsh, Stravinsky: A Creative Spring, 187.82. See Nolan, “Music Theory and Mathematics,” 286–87.83. I. Stravinsky and Craft, Expositions and Developments, 51–52.84. See Babbitt, “Remarks on the Recent Stravinsky,”167; and idem,Words About Music, 20.85. I. Stravinsky and Craft, Conversations with Igor Stravinsky, 11.86. Ibid., 13.


one of the more bizarre quotations ever attributed to Stravinsky or any othermajor composer:

After working late one night I retired to bed still troubled by an interval. Idreamed about this interval. It had become an elastic substance stretching ex-actly between the two notes I had composed, but underneath these notes at ei-ther end was an egg, a large testicular egg. The eggs were gelatinous to thetouch (I touched them) and warm, and they were protected by nests. I wokeup knowing that my interval was right. (For those who want more of thedream, it was pink—I often dream in color. . . .)87

I hesitate to comment on this story at all, let alone attach any significance to it.Yet, an egg serves as an apt metaphor for the interval and its generative role inStravinsky’s conception of his music.88 If, as Stravinsky’s dream suggests, theinterval was significant to him not only as a basic unit of music but as a sym-bolic object as well, then perhaps Stravinsky’s treatment of intervals in TheRite might also serve a symbolic function. That is to say, Stravinsky’s numeri-cal procedures may have been not merely a means of generating musical mate-rial but an expressive device as well.

Regardless of Stravinsky’s revisionist commentary on the matter, the ex-pressive underpinnings of his score for The Rite are inseparable from theshared artistic vision of Stravinsky and his principal collaborators, NicholasRoerich and Vaslav Nijinsky.89 As Jann Pasler has documented, this vision wasfounded in part on the desire for “correspondences” among the music, set de-sign, costumes, and choreography. Of particular interest here is what might bedescribed (echoing Viktor Shklovsky90) as the “stoniness” of these elements asconceived by the principal artists, a feature of the ballet that emerges forcefullyin Pasler’s essay: Roerich’s set design featured large boulders; Nijinksy

87. Ibid., 14. Here, it is worth remembering Taruskin’s warning that Stravinsky’s commentsin the series of “conversation” books he coauthored with Craft often tell us more about Stravinskyat the time the conversations were held than about Stravinsky at the times he was reminiscingabout. See Taruskin, Russian Traditions 1:11–12. Taruskin concluded, however, that “at least thefirst four books . . . quite faithfully reflect the composer’s thinking” (1:12n42).

88. The biological metaphor for Stravinsky’s composition of The Rite seems particularly aptwhen considered in tandem with contemporaneous developments in genetic research. In 1913,Alfred Sturtevant, a student of Thomas Hunt Morgan, revealed for the first time a “genetic map”:the linear arrangement of genes on a chromosome, with fixed distances between contiguousgenes. The intervallic structure of Stravinsky’s chords, whose precise arrangement of pitches pre-figure durational patterns—the rhythmic “life” of The Rite—offer a striking analogue to these ge-netic maps. See Sturtevant, “The Linear Arrangement of Six Sex-Linked Factors in Drosophila.”

89. Taruskin has written extensively on the contributions of Stravinsky, Roerich, and othersto the ideas behind the ballet; see Russian Traditions 1:849–91. For an account that focuses onNijinsky’s role, see Garafola, Diaghilev’s Ballets Russes, 63–73.

90. Shklovsky’s notion of “defamiliarization” in fact provides a useful means of conceptualiz-ing various aspects of The Rite, particularly the ways in which Stravinsky used traditional tonal ele-ments (triads and seventh chords, diatonic melodies) to create his novel musical sounds. SeeShklovsky, “Art as Technique.”


described The Rite as “the life of the stones and the trees” and “a thing of con-crete masses,” and cited “the dead weight of the body” as a crucial feature ofhis choreography; and Stravinsky characterized his rhythms as “lapidary” anddescribed The Rite as a “stone sculpture.”91 Pasler drew many specific associa-tions among these various elements of the ballet production. For example, in acomparison of the set design and the musical score, she wrote:

In order that his art form corroborate the visual effect of the ballet, Stravinskyattempted to create musical equivalents for the instantaneity and simultaneityprojected by the set design. For example, thirty-two percussive repetitions ofone chord (a combination of the dominant seventh on E � major over an F �major chord in the bass) accompany the first visual moment, the rising of thecurtain in part 1. This passage affects the public with the same immediacy andstasis as the backdrop being revealed for the first time. The sound forms a blockwith the massive power of the boulders in the set design.

Taking this idea further, correspondences within the musical realm (that is, be-tween intervals and durations) can be understood to operate toward the sameaesthetic end—stoniness—as correspondences among the different artisticmediums that constituted the original ballet. Stravinsky’s derivation of theopening rhythm of the “Augurs,” for instance, may be viewed as an attempt tocreate rhythmic life out of a static object—namely, the C-major variant of the“Augurs” chord from which the opening accent pattern seems to have beenderived, a harmonic fossil whose intervallic imprint is linearized in the firsteight bars of the movement. Indeed, the derivation of durational patternsfrom intervallic ones was a viable compositional strategy in large part becauseStravinsky’s intervallic structures were fixed, lacking the flexible voicings oftonal harmonies. As Taruskin has observed, “To an extent previously unthink-able in ‘cultured’ music, chord and motif [in The Rite] were hypostatized,turned into stone, timbrally and registrally so fixed that even transposition—let alone transformation or transition—were inconceivable.”92

The notion of The Rite’s rhythms emerging from sources in Nature is a re-curring feature of the Montjoie! program, the most detailed account of theballet’s scenario. Rhythm is a primary element of ritual in the ballet, and atseveral points in the program, this element is characterized as a direct expres-sion of Nature. Early in the ballet, an “old woman . . . who knows the secretsof nature . . . teaches her sons Prediction,” and they go on to “mark in theirsteps the rhythm of Spring”; later in Part I, a “Sage gives a benediction to theEarth, . . . becoming one with the soil,” and this benediction “is as a signal foran eruption of rhythm,” leading the surrounding adolescents to “[pour] forthin numbers, like the new energies of nature.”93 In each case, a wise elder

91. Pasler, “Music and Spectacle,” 69–70, 71, 74.92. Taruskin, Russian Traditions 1:957.93. Stravinsky in Pictures and Documents, 525.


serves as a medium through which the secrets of Nature can be passed on toothers, and these secrets take on the form of rhythm and dance. These twoscenes refer specifically to the “Augurs of Spring” and the “Dance of theEarth,” both of which begin with a repeated chord whose intervallic structureseems to generate the opening rhythm of the passage. In Stravinsky’s imagina-tion, then, these intervallic series may have represented “secrets of Nature”that engendered the rhythm of each dance. Or, to borrow Stravinsky’smetaphor, the intervallic “eggs” of each chord give birth to the rhythms. Or,to suggest another metaphor, the dissonant chord that begins each section—primitive and elemental—is the “augur” of the rhythm expressed via repeti-tions of that chord. We could interpret the openings of several of themovements in much the same way, with the 5+6 intervallic construction link-ing most of these together as variations on a single “secret of Nature.” Inmany of the movements, the disclosure of the secret is preceded by a pause ormoment of hesitancy, as though the force of Nature is being gathered, before“erupting” into rhythm or “pouring forth in numbers,” to cite the program(see R12, R71, R102, and R139, which precede the “Augurs,” “Dance ofthe Earth,” “Glorification,” and “Sacrificial Dance,” respectively).

Thus, in The Rite, numbers would seem to be a crucial feature underlyingStravinsky’s conceptions of nature, ritual, and musical relationships, as well as a means of signifying “primitive” knowledge: the numerical codes of Naturepass through human conduits and emerge as pure rhythm.94 For ancientGreek philosophers and many others since, asserting a numerical basis forNature has been a means of establishing a masculine order behind the inher-ently feminine, thereby reducing Nature to the knowable and transcendingher.95 We might detect a related impulse behind Stravinsky’s numerical proce-dures. “They are not entirely formed beings; their sex is single and double likethat of the tree,” Stravinsky said of the Adolescents in the “Augurs of Spring,”but it would be a mistake to assign any gender-neutrality to The Rite. The hi-erarchy of the ballet’s scenario is obvious, most evident in the treatment of theChosen One, sacrificed at the hands of the Ancestors (“rapacious monsters,”as they are described in the Montjoie! program96) and driven to death by the

94. In this way, the basic compositional procedure under consideration here is a prime exam-ple of what Robert Morgan has termed the “secret languages” of modernism, alternatives to theshared language of tonality developed by fin-de-siècle composers; see Morgan, “Secret Languages:The Roots of Musical Modernism.” Morgan’s comments about Schoenberg’s music are surpris-ingly relevant here: “[Schoenberg] attempted to transform musical language from an essentially‘public’ vehicle, susceptible to comprehension by ordinary people (but thereby also limited tomore or less ordinary statement), to an essentially ‘private’ one capable of speaking the unspeak-able. Music became an incantation, a language of ritual that, just because of its inscrutability, revealed secrets hidden from normal understanding” (458).

95. See Wertheim, Pythagoras’ Trousers: God, Physics, and the Gender Wars (esp. 27–30), anacclaimed study of the historical roots of gender bias in the field of physics.

96. Stravinsky in Pictures and Documents, 525.


rhythms of the “Sacrificial Dance.” This subject matter may threaten to impli-cate The Rite’s creators as complicit in a misogynist program,97 but such an analysis would be much too simplistic: surely one can depict misogynywithout participating in it. Extended to a metaphorical level, however, thecontaining and controlling of the feminine depicted in The Rite is less easilyseparated from the views and objectives of the artists who portrayed it. Themale-female opposition was fundamental to modernist conceptions of “primi-tive” cultures in relation to “modern” ones,98 just as Romanticism was oftenviewed as a feminine counterpart to Modernism itself; thus, the treatment ofthe “Chosen One” can be read as a symptom of larger desires to harness theprimitive (an impulse hardly unique to The Rite) and to stamp out Roman -ticism as well.99 Stravinsky’s numerical procedures are tied up with all of theseideas: they lie behind the rhythms of the “Sacrificial Dance” to which theyoung virgin falls victim, the representation of Nature and her “secrets,” theconception of primitive ritual, and the triumph of Modernist objectivity overRomantic subjectivity.

For some critics, the putative objective detachment that pervades The Ritehas implicated Stravinsky and his collaborators precisely because this detach-ment can be seen to distance the creators from these themes of control, con-cealing the will to dominate by suppressing any suggestion of human agency.Lynn Garafola, for instance, focusing on Nijinsky’s choreography, has writtenof how “fate—powerful, atavistic, aleatory—[is] the ruler of a godless uni-verse” in The Rite: “implacable and omnipresent, [fate] has been wrested froman absent god . . . above all, it holds power over the individual.”100 Garafola’scritique assimilates the basic position of Adorno as outlined in Philosophy ofNew Music, a work whose influence is strongly felt in Taruskin’s writing aswell. Adorno’s arguments are familiar enough to require only a brief summaryhere. Recall that for Adorno, Stravinsky’s score, particularly its rhythms, wasfundamentally “arbitrary” in nature, a quality he attributed to a perniciousagenda: the simulation of objectivity, a manufactured illusion that ultimatelyimplied the lack of an organizing psychology behind the music. He drovehome this idea with a litany of negative assessments of Stravinsky’s composi-tional method and of musical effects in The Rite. For example, with regard to

97. Garafola, in her polemical commentary on The Rite from a choreographic perspective, in-terpreted the “Chosen One” as a symptom of “twentieth-century male sexual anxiety,” theSymbolist femme fatale submitted to male domination. See Diaghilev’s Ballets Russes, 72.

98. As Marianna Torgovnick has written, “Sooner or later those familiar tropes for primitivesbecome the tropes conventionally used for women.” See Gone Primitive, 17.

99. On modernist notions of the feminine, see Nicholls, Modernisms, particularly the intro-duction and chaps. 1–2, which provide a useful framework for conceiving of The Rite within thelarger context of European modernism. As Nicholls wrote, “the ‘elimination’ of the feminine”emblematic of much modernist literature was “the very mark of that triumph of form over ‘bodily’ content on which one major strand of modernism . . . depend[ed]”; Modernisms, 4.

100. Garafola, Diaghilev’s Ballets Russes, 68, 71.


Stravinsky’s practice of repeating a melodic idea numerous times in close suc-cession with a variable pattern of accents, Adorno wrote that such passages“give the effect of resulting from a throw of the dice” and that “the melodiccells stand under a spell.”101 In this way, Adorno characterized rhythm in TheRite as dehumanized and dehumanizing, and this formed the basis of a highlycharged ideological critique of Stravinsky’s music—a critique that, in itsgravest moments, linked the music to fascism.

Pieter van den Toorn, in his 1987 monograph on The Rite and especially ina more recent essay, has objected strenuously to Adorno’s claims and soughtto discount them based on their lack of grounding in detailed musical analy-sis.102 In several analyses, van den Toorn has attempted to show how shiftingmelodic accents and groupings in The Rite should be heard in relation to anunderlying expectation of regularity. As he put it, disruptions in rhythmic andmetric regularity are “dependent on preestablished or readily inferrable impli-cations of pulse, periodicity, and fixed metric identity.”103 That is to say, therhythm plays with our expectations, engaging us on a psychological level.What Adorno heard as arbitrary, van den Toorn heard as purposeful.

A decision about who is correct here is neither desirable nor possible. But ifwe accept the notion that Stravinsky generated rhythmic and metric patternsthrough a mechanical process that severely limited the composer’s ability toshape the psychological experience of listeners in specific ways, then Adorno’sargument has the advantage of bringing to the fore an important element ofStravinsky’s compositional process. Indeed, an understanding of rhythm inThe Rite informed by the mechanical procedure I have described coexists un-easily with van den Toorn’s conception. Yet it is strikingly compatible withAdorno’s, and even suggests a concrete explanation for the musical qualities(primarily rhythmic) that he identified. Adorno cited, for instance, “a rulingcontradiction between the moderation of the horizontal dimension and theaudacity of the vertical dimensions,” and he argued that Stravinsky employeda “trick” in which “complexes of time are presented as if they werespatial”104—strangely appropriate phrases in the context of the present analy-ses. Or we might say, borrowing Adorno’s words, that Stravinsky’s rhythmsare “under the spell” of intervallic structures, derived from a “game ofchance” in which pitch measurements dictate rhythmic patterns.

And yes, following Adorno, this manner of composing might accurately bedescribed as “dehumanized.” But this assessment need not be pejorative.When it came to generating rhythmic ideas for The Rite, to “dehumanize” thecompositional process seems to have been precisely Stravinsky’s point. To at-tempt to convey specific musical ideas in such a way that they can be grasped

101. Adorno, Philosophy of New Music, 114.102. Van den Toorn, “Stravinsky, Adorno, and the Art of Displacement.”103. Van den Toorn, Stravinsky and the Rite of Spring, 113.104. Adorno, Philosophy of New Music, 114, 143.


by listeners is necessarily to interact with culturally determined musical codes;but in Stravinsky’s idealized vision of The Rite, he would bypass these codes inan effort to engage listeners on a more fundamental, “primitive” level. Indeed,the primitivist ideal provides one of the strongest explanations for Stravinsky’snumber games in The Rite. Garafola, Joan Acocella, and Jonnie Greene pro-vided a succinct definition of this ideal:

Of these ideas [contained in The Rite], the most important is primitivism, thebelief that society does not elevate or improve the human soul but, on the contrary, corrupts it and that it is those things that are least socialized, least civilized—children, peasants, “savages,” raw emotion, plain speech—that areclosest to the truth.105

And how does a composer as cultured as Stravinsky bypass culture? By at-tempting to exclude his own musical imagination, steeped as it was in the clas-sical tradition, from the compositional process at certain critical stages.Stravinsky’s generation of rhythmic ideas might therefore be understood as asort of “automatic writing” (a practice obliquely referenced in the title of thisessay) designed to transfer control from the composer to his elemental har-monies, those mysterious objects found by Stravinsky’s own hands in hisarcheological expeditions at the keyboard in Clarens. When composing hisrhythms, Stravinsky may have imagined his role as that of a transcriber ratherthan a composer, a recorder of patterns dictated by some entity outside ofhimself. Or, as the composer himself would famously declare: “I heard and Iwrote what I heard. I am the vessel through which Le Sacre passed.”106

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Abstract

The novel rhythms of The Rite of Spring, according to Theodor Adorno, seemto have arisen from “the throw of the dice.” Many scholars have argued other-wise, characterizing Stravinsky’s rhythms as logically patterned and deliber-ately crafted. This essay, however, suggests that Stravinsky did in fact employ afundamentally mechanical, “automated” procedure for generating his irregu-lar rhythms and metrical schemes. Specifically, Stravinsky seems to have con-verted intervallic series, derived from prominent chords or melodies, intodurational ones, equating intervals measured in semitones with durationsmeasured in beats—a relationship that is best perceived intellectually, not aurally.

This claim is supported by a wealth of evidence drawn from the sketches,autographs, and earliest printed editions of the score, and it applies to theopening of virtually every titled section of The Rite, where much of its best-known music resides. Close analysis of these sources, considered in conjunc-tion with the copious symbolic and numerical notations in the margins ofStravinsky’s sketchbook, Stravinsky’s programmatic commentary on The Riteand his idiosyncratic ideas about intervals, and the critical commentary ofAdorno and others, sheds new light on Stravinsky’s conception of The Riteand its symbolic meanings.

Keywords: Igor Stravinsky, The Rite of Spring, rhythm, sketch study, modernism.