Volume 13, Number 4, December 2007
Copyright � 2007 Society for Music Theory
Exploring Complementation in Bartók’s Third Quartet
KEYWORDS: complementation, post-tonal theory, set theory, Bartók, quartet
ABSTRACT: Despite, on the one hand, a healthy if circumscribed tradition of Bartók music studies and, on the other, a continuing analytical concern with notions of complementation and contrast, I argue that these two domains have not been brought together to full effect, and that doing so can significantly enhance a reading of Bartók’s Third Quartet of 1927. In this setting, complementation applies not only to sets and pitches, but also to modes and, more broadly interpreted, to parameters such as texture, rhythm and form. Complementation across several parameters, or at different levels of significance, may operate as a multi-faceted complex which serves to heighten, or on occasion diffuse, tensions in the music.
Received May 2007
 Bartók scholarship relevant to the six quartets enjoys a long, illustrious history in Hungary, the Unites States and United Kingdom.(1) Its span extends from writings published soon after Bartók’s death (see Abraham 1945 and Seiber 1945), through to those which point up the composer’s continuing status in the twenty-first century, especially the fine volume edited by Antokoletz, Fischer, and Suchoff (2000), supplemented by that of Amanda Bayley (2001).(2) Nonetheless, as noted in the comprehensive article by Malcolm Gillies (2001, 808): “The history of Bartók analysis has been one of slow changes in trend: from early postwar concerns with style analysis, mainly in the pitch domain, through to the more structural concerns of the 1950s to 70s ... During the 1980s and 90s, despite an apparently ever-growing divergence of methods, the tendency has again been to concentrate on more ‘micro’ levels of compositions.”
 Equally, in furthering approaches to post-tonal music, whilst still questing after an elusive system of the stature and universality of Schenkerian voice-leading within tonal music, analysts have increasingly recognized complementation as a useful concept. Properly embraced, complementation can facilitate a breakaway from restrictive notions of large-scale unity and a corresponding move toward a dynamic, coherent diversity, by elucidating contrasts and discontinuities rather than overstating weak similarities.(3) However, despite on the one hand an ongoing healthy interest in Bartók studies and on the other a growing awareness of complementation, I argue that the concept’s full potential and applicability has not been realized in reading Bartók’s Third Quartet of 1927, especially its Prima parte. Although I might be accused of furthering research at “more ‘micro’ levels,” it is precisely through this attention to detail that Bartók’s special qualities are revealed, in particular the operation of complementary complexes within and beyond the domain of pitch.
I. Charting complementation and Bartók Studies
 Complementation involves a process whereby the whole is composed of distinct, yet in some sense matching, parts which co-exist within a dynamic balance.(4) The term is derived from the Latin complere, to fill up, thus to achieve a state of completeness. Each element implies the existence of, and indeed completes the other.(5) Ultimately, there is a contradiction because the elements may be, at the same time, both opposites and yet intricately connected.(6) Since an element may in fact be articulated as powerfully by its absence as by its presence, we may also connect complementation with the post-structuralist thought of Derrida, fulfilling specifically one of the two conditions of his concept of “différance.”(7) “Différance” encapsulates a pun since, in its French inflection, the word suggests both difference and deferral, and it is the notion of constant, ultimately infinite, deferral which is germane here. This might further suggest an affinity with the psychoanalytical critical concept of deriving meaning through omission or suppression. In this fulfilling of expectation, there is a parallel, unconnected, evolution in the notion of “implication-realization” propounded by Leonard B. Meyer (1978, 1989). And in just the same way, implied complementations may themselves be fully or only partially realized, resulting in incompletion, instability, and scope for further extension.
 The history of complementation in music theory commenced, effectively, with its deployment by Allen Forte (1973, 73–74), specifically in a pc-set-oriented context. In the 1980s, complementation was discussed more widely in a useful volume by Jonathan Dunsby and Arnold Whittall (1988), together with coverage by James Baker (1986), Richard Parks (1989) and Marianne Kielian-Gilbert (1982–83, 1991), focusing mainly on music by Stravinsky, Scriabin and Debussy.(8) Moving out from strict pc-set complementation, the concept evolved within the pitch domain to embrace octatonic complementation, that between harmonic units, and motivic and structural complementations. Joseph Straus (1982) in particular advocated “complementary axes.” An undervaluing of complementation was evidenced, however, by its limited inclusion in Bent and Drabkin (1987, 62, 105, 106) with references confined to set theory; similarly, in the otherwise admirable text by Lester (1989), the concept was also undersold.(9) More positively, in the later 1990s, David Lefkowitz (1997) considered the crucial role of hexachordal complementation, while more recent research on pc-set complementation, founded on the work of Robert D. Morris, includes that of Michael Buchler (2001) and Paul Nauert (2003), which considers through a process of modeling the effect of complementation on pc-sets within a progression.
 Outside of pitch relations, instrumental complementation may result from solo versus group, with textural oppositions of polyphony and “heterophony”; distinctions between textural polyphony and “heterophony” were explored quite early on by Dunsby (1989). Similarly, stylistic complementation may arise from the confrontation of old and new, which, in the language of Whittall (1987, 1989) may “converge precariously” to form a certain “symbiosis.”(10)
 In Bartók studies, the cultural and stylistic synthesis of East and West had long been a concern of scholars such as Ernö Lendvai (1971).(11) But more analytically-focused research on inversional pitch symmetry, as a distinct complementary subset of mirrored, rather than opposed elements, has also proved very fruitful. The notion was initiated in the early post-war period by George Perle (1955, 1977), then followed up by Wallace Berry (1980) in his extensive article on the Third Quartet itself.(12) The substantial contribution of Antokoletz (1984) with his “Intervallic cells,” emerged from work on the Fourth Quartet (1977),(13) and was supplemented by more specific research on symmetry, combined with other phenomena, by Jonathan Bernard (1986), Richard Cohn (1988), and Christopher Mark (1992). Detailed contributions by Gillies (1989, 1993) to understanding Bartók’s tonal structuring should also be properly credited. Research on these and broader issues was then provided in the crucial, enduring book on analyzing Bartók’s music, including the Third Quartet, by Paul Wilson (1992). Around the same time, our pitch understanding of the Third Quartet was enhanced by further research of Antokoletz (1993).
 Beyond the strict considerations of a music-theory-led history of complementation and particular applications to Bartók, a survey of this type would not enjoy any measure of completeness without acknowledging the profound influence of Olivier Messiaen on this area of musical thought. Messiaen’s example is relevant both in a general sense—as a theorizing of musical practice, and, specifically, in terms of his enthusiasm for palindromic manipulations.
 In addition to drawing together the types already discussed, we may add those of “modal” and “bimodal complementation.” “Modal complementation” may be defined as that which involves a division, or partitioning, within a given collection,(14)
and “bimodal complementation” as that which may exist between similar collections at different transpositions, e.g. between pentatonic collections on
 We need too to consider the relationships involved in strongly contrasting timbre and register, and the potential for formal and rhythmic complementation, where a clear sense of “opposition” or, alternatively, symmetry may be perceived. In fact, in discussing “Symmetrical Interval Sets,” Berry (1980, 325) mentions “opposition and textural-rhythmic separation of two combinatorially associated pentachords,” although he does not really develop the idea. Specifically on rhythm, in his own volume on Hungarian Folk Music published just four years after the Third Quartet, Bartók (1931, 29, 30) noted occurrences of reversed patterns, perceiving two schemata as among the most common for openings of folk melodies: (form no.7) “short–long, long–short” and its converse (form no.8) “long-short, short-long.” Thus the notion of applying this to his music is hardly inappropriate, and similar reversed patterns do occur in the Quartet.
 Presented simultaneously or successively, these bare ingredients can produce a whole variety of effects and cross-relationships, whether it be a simple juxtaposition, or collage (devoid of tension), a precarious polarity (imbued with tension), or more rarely a “symbiosis,” even synthesis. These states are to some extent comparable to “stages” outlined by Kielian-Gilbert (1991, 451–52), although she is more concerned with establishing a neoclassical aesthetic to relate old and new. Complementation across several parameters may operate as a multi-faceted complex, and at different levels of significance—an idea apparently not discussed elsewhere. Thus the effect of a complementation may be heightened, or, conversely in different circumstances, contradicted, confused or diffused. Across a section, movement or work, particular complementations may cause and maintain instability, although they may also represent a powerful means of achieving a new “unity” or completeness.
 From this point on, the reader is referred to the miniature score of the Third Quartet published by Universal Edition in 1929.
II. Founding principles: the Prima parte (mm. 1–6)
 Complementation has a role to play throughout the Quartet, so that discussion may usefully commence with the opening measures, as shown in Example 1. Although their pitch structure has been well documented, especially in Berry (1980), Wilson (1992, 89) and Antokoletz (1993, 260), they still merit further consideration and offer a strikingly lucid illustration of complementation.
 In fact mm. 1–6, concluded by Bartók’s articulative modulus, neatly encapsulate the activity of the whole Quartet. Cello, viola and violin II present and sustain a semitonal germinal cell:
 In addition to these discrete unordered pitch segments, there is, in the motivic patterning, a bimodal complementation of two distinct collections which operate across the Quartet,(16)
that is, total chromaticism and pentatonicism, generally indicative of Forte’s pc-set Genus 5 (chroma) and Genus 11 (dia). The point is illustrated by looking again at ordered presentations within the opening measures. Segment A is entirely chromatic, albeit with octave displacements and semitonal pairings (Genus 5, Supragenus II). Meanwhile the violin I melody that comprises segment A' (violin I) tends to avoid adjacent semitones, outlining rather the major second/minor third intervals characteristic of pentatonic syntax (a more tentative Genus 11, certainly Supragenus IV). Particularly striking are the violin pitches marked with a tenuto stress:
Example 1. Bartók, String Quartet No. 3, Prima parte, mm. 1�6
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Example 2. Pc-set and pitch complements (Prima parte, mm. 1–6)
Example 3. Bimodal complementation and symmetry
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 This bimodality to some extent articulates a stylistic complementation between Western European modernism, probably inspired by Berg’s Lyric Suite which Bartók had heard in July 1927, and Bartók’s Eastern European heritage of folksong. Bartók (1976, 323) acknowledges this differentiation in his lecture on “The Relation of Folk Song to the Development of the Art Music of our Time,” commenting that “genuine folk music of Eastern Europe is almost completely diatonic and in some parts, such as Hungary, even pentatonic.” And yet there is “Curiously enough, at the same time, an apparently opposite tendency ... towards the emancipation of the twelve sounds comprised within our octave.”
 Having launched the analysis, I want to step back briefly to introduce some more theoretical “models”: Figures 1–4. These models aim to show how aspects of Bartók’s practice, particularly the fundamental bimodal complementation of chromaticism and pentatonicism, may be underpinned by theory. (They will also point up how the apparently opposed worlds of chromaticism and pentatonicism enjoy a certain mediation, crucial to the notion of complementation itself.) The applicability of Figure 1 to mm. 1–6 is considered here, while the extent of instantiation within Bartók’s score of the remainder of Figure 1 and Figures 2–4 is clarified below, especially in section III. An initial “Chromatic/pentatonic matrix” (Figure 1) offers a possible representation of the “chromatic-non-chromatic dialectic” identified by Berry (1980, 359).(17)
 We can map the opening segments A (set 4-1) and A' (set 8-1) onto the left-hand vertical, chromatic axis, where
Figure 1. Chromatic/pentatonic matrix
 The potential “interaction,” or intersection, between chromatic and pentatonic collections is illustrated in a slightly different way in Figure 2, still bearing in mind the initial chromatic collection (4-1/8-1). Pentatonic constructs on
 Figure 3 maps the tritonal relations which can feature as a result of interaction between pentatonic constructs (on
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Figure 3. Tritonal relations from interaction of pentatonic
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Figure 4. Octatonic perspective
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 A final model (Figure 4) shows a theoretical octatonic perspective where the collection
 Of course the opening of the Quartet thrives not solely because of various polarized elements. As in the theoretical models, there are also subtle aspects of modal synthesis in the presentation of the complementary pc-sets, associating the apparently quite different domains of chromaticism and pentatonicism. On the one hand, the most sustained pitches of the violin I phrase (refer again to Example 1), i.e. the initial re-sounded pitch (
Example 4. Aspects of modal synthesis (Prima parte, mm. 1–6)
(click to enlarge)
 Where the interaction of Genus 5 and Genus 11 is concerned, it is worth pointing out that these genera, as the main sources of reference across the work, have a notably high Difference Quotient (Difquo) among the 66 possible pairs of genera: 0.79227. Thus the intuitive perception of these as broadly “complementary,” or at least close to being maximally dissimilar, is borne out by Forte’s calculations. Even so, the genera combination is not quite as differentiated as it might be, this distinction being reserved for Genus 4 and Genus 11 with the highest Difquo of 0.87058. Again, we have to accept a measure of relativity: this cannot be a true, absolute complementation.
 This opening phrase is already demonstrably complementary in three ways: firstly, pitches and sets; secondly, modality; and thirdly, symmetrical motivic patterning. But there are also further extensions of complementation in other parameters. Cello, viola and violin II present segment A as an uninflected four-part sonority: essentially devoid of rhythmic identity, the four pitches simply articulate the quarter note pulse. In marked contrast, segment A' is presented by the first violin as a lyricized solo line, with a distinct rhythmic identity, articulated by a leaning into the first note of each clearly marked, anacrusic phrase. Timbre too is used in complementary (or at least oppositional) fashion: the lower strings are designated
con sord, while violin I is to be unmuted. Additionally, segment A is presented with a timbral heterogeneity, each pitch separately articulated by a new instrumental timbre (
III. Scope elsewhere in the Prima parte
 Having given some idea of localized complementation, I wish now to show its scope and extent by summarizing other instances from the
Prima parte, concentrating primarily on complementary types, complexes and effects not examined above. The imitative section across mm. 6–20 uses complementary pitch processes that are increasingly concerned with, in the terminology of Whittall (1995, 34), “black-note/white-note” collections of pc-set 5-35:
 Across mm. 9–12 (just before Rehearsal Mark 1), we may contrast the F, G, D, C pitches of violin I with those of violin II and cello:
 A new instance of set and pitch complementation occurs at the Quasi a tempo tranquillo (Rehearsal Mark 2, m. 21). Here Violin I presents a chromatic segment: 5-1 (
 Pentatonic complementations return at the Più andante (Rehearsal Mark 4; mm. 35–42): viola on
 In terms of a complementary complex, the aspects just described which differentiate lower and upper strings are supported by a third parameter—a looser timbral contrast (con sord versus senza sord, the latter exaggerated in its harshness by means of sul ponticello). A fourth parameter is that of broadly opposed rhythmic identities. Viola and cello supply a regular background of eighth notes, against which are counter-pointed the violins’ intricate patterns of quintuplet, triplet sixteenth notes and dotted groupings. A further, fifth, dimension appears across mm. 37–38 and 40–41, with the simultaneous statement and reversion (i.e. mirror-imaging) of two sets of rhythmic patterns between the violins, interspersed each time by rests; see Example 6. Pattern 1—a repeated thirty-second note plus double-dotted eighth note—is particularly apt in view of Bartók’s observations about this “short–long, long–short” schema in his Hungarian Folk Music. In an act of calculated overloading is added a disruptive sixth parameter: that of dynamic scaling. In m. 40 (Rehearsal Mark 5), the mf marking of the repeated, rather straightforward lower-string figures, threatens to undermine the more intricate upper lines, marked merely mp, thus offering a useful illustration of emphasis through suppression. The effect is to cue change.
 Bold textural complementation is focused upon in the ensuing Tempo I, in modo ordinario (mm. 43–46); it operates across a horizontal plane, with upper strings imitated by lower strings within a biting, entirely chromatic pitch context. There is a sense of progression from the Più andante since all instruments share the same figurations and syntax, presented at f dynamic, without mutes or sul ponticello effects. Overlapping 4-1 units are presented in a block-type (partial) complementary complex: each unit involving symmetry through contrary motion, and so confirming the potential inter-relationship suggested above. All twelve pitches are embraced across the first four units (mm. 43�44) in a reworking of the opening, as represented in Figure 5. The pitches are given in a uniformly ascending format in order to clarify the collective pitch saturation.
Example 5. Chromatic contradiction and symmetry
Example 6. Rhythmic reversion (Prima parte, mm. 37–38)
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Figure 5. Schema to show reordered 4-1 segments
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 Textural complementation is followed through in the poco a poco allargando portion (see Rehearsal Marks 7–8). Functioning now primarily on a vertical plane, linear/legato fragments, at
piano through to forte dynamic, are contrasted by chordal/detached fragments, with reiterated down-bows, at
ff dynamic, their opposition articulated by eighth-note rests. This procedure combines with the typical pitch opposition of “white”-note modality in violins, contrasted by “black”-note in lower strings (occasionally a chord reverses the natural-over-flat layering with flats presented on top, e.g. m. 55), which maintains the earlier horizontal plane. Quartal and quintal constructs are evident within a pentatonic syntax, as is the continuing larger-scale preoccupation with semitonal polarity, as occurs with G versus
 This negating of one dimension by the other actually serves to diffuse the complementation and achieve an increased modal fusion through mm. 57–59, necessary so as to shift the focus onto a so far less explored complementation—that of pitch symmetry, especially in violins: initially A, G, F, E, F, G, A. Two further examples of symmetrical pitch-patterning, achieved through melodic (and chordal) reversion, are worthy of mention in the latter portion of the
Prima parte. Mm. 79–80 yield a brief pattern: F, G,
IV The Quartet’s large-scale design
 Having discussed particular examples of localized complementation pertaining to pitch (i.e. chromatic and modal collections), rhythm and timbre, operating both singly and compositely, it now makes sense to consider possible, admittedly more speculative, large-scale formal complementation. The overall formal design of the Quartet has been discussed often enough in the analytical literature, but as far as I am aware no one has yet suggested that the concept of complementation might have something useful to offer here. Interestingly, though, there may be a precedent in the work of Colin Mason (1957, 189), who argued of the Fourth Quartet that “the general tonal scheme [of the middle movement in D, with prominent
 I would suggest that two complementary levels operate across the work: the first concerns the parallel relationships between the Prima parte (section A) and its shortened Ricapitulazione (A'), and between the substantial Seconda parte (B) and its corresponding recapitulation (B') embraced within the brief Coda. The second, more fundamental, level involves the possible relationship between sections A and A' (viewed collectively as alpha), and sections B and B' (viewed collectively as alpha'). It is this idea of a hierarchical complex, owing much to complementation, which I believe may offer a useful means of viewing the form of the Third Quartet.
 In terms of the first formal level, we need to be convinced that the two recapitulations are sufficiently different from their respective expositions to render them somehow “opposite.” The recapitulations of the First and Second Parts are in no sense literal; rather they balance their expositions in somewhat condensed format across far shorter temporal spans. The recapitulation of the Seconda parte has in any case to fulfill a dual function, re-invoking ideas from the central Allegro and, in its role as Coda, concluding the whole work.
 It is as though the essential material for each part of the Quartet existed in some paradigmatic or Grundgestalt state, conceivably as a pre-compositional notion.(19) Bartók seems to be offering the listener and performer alternative views of this material. In terms of the relationship between the First and Second Parts and their respective recapitulations, as Whittall (1995, 34) acknowledged, the third and fourth sections do offer a “remarkably free rethinking of the issues.”
 If the explanation of complementation is to be deemed plausible, there needs also to be an element of correspondence and inevitability in terms of exposition and subsequent recapitulation. Both expository parts are left initially incomplete, and there is a consequent sense of implication, or expectation, which is at least partially realized in their recapitulations. Although the recapitulations are quite strikingly different in their presentation from the expositions, there is of course a common core between the instances which we have considered in the Prima parte and their alternative readings in the Ricapitulazione della prima parte.
 We may broadly compare mm. 1–8 of the Ricapitulazione with the opening of the Quartet through to Rehearsal Mark 1. As Wilson (1992, 109) observes, the “goal tones” here, i.e.
 Mm. 17–18 of the recapitulation reintroduce pitch-exclusive sets of 4-1 reminiscent of Part One (both its opening and mm. 43–44), here inversionally related in their pitch ordering and producing a composite 8-1 (viola/cello: A,
 The possibility of the larger-scale complementary relationship between the First and Second Parts is of course more speculative, but may still make a useful contribution to understanding this work. Even at the most basic level, the contrasts between the two parts are extreme enough to render them opposites. As Whittall (1995, 34) emphasized: “Part One is slow, Part Two is fast; Part One is lyrical and builds to a melodic statement of gravely simple beauty; Part Two is hectic, dramatic (and distinctly more folk-like).” Walsh (1982, 40) developed the point further, referring to a “drastic contrast between two autonomous kinds of music,” and “a more serious duality of the kind associated previously above all with late Beethoven.”
 Yet beyond the extreme contrasts and oppositions, in order to satisfy a complementary perspective, the two parts must be intricately related. This crucial criterion is fulfilled by indisputable motivic correspondence, all the more so since, despite this, in the words of Walsh (1982, 44), the character of the motives in the Second Part is “altered beyond recognition.” The point is illustrated by examining the opening measures of the
Allegro, apparently unrelated to the First Part, but which yet exhibit a similar polarity between chromatic (Genus 5) and diatonic (Genus 12, Supragenus IV) syntax: pc-set 5-1 played
quasi glissando on viola (B, C,
 The Seconda parte reveals a further correspondence with the Prima parte in that those germinal opening pitches of the First (
 The occurrence of the Seconda parte, rather as a sonata form “second subject,” before the
Ricapitulazione della prima parte undoubtedly influences the presentation of the latter material, notably more spacious, yet succinct, as a reaction against the large-scale frenetic activity of the
Seconda parte. Equally, elements of the Seconda parte have been subsumed in the
Ricapitulazione della prima parte: most strikingly the focal
 It should also be acknowledged that once the listener has experienced the completion of the Prima parte, by virtue of the occurrence of its Ricapitulazione, there is an increased sense of inevitability or complementary pull operative where the recapitulation (Coda) of the Seconda parte is concerned. It is as though the work is presented as a jigsaw, with certain key pieces missing at first, but, gradually, we can perceive the outline shape of what is missing and are therefore expecting its subsequent appearance.
 Finally, it is significant that the Coda (B') is so called: although at one level it functions as the recapitulation of the
Seconda parte, it also relates to the Prima parte (and its Ricapitulazione), re-establishing the pitch centre on
 It might be argued that I am in danger of seeing complementation at every turn, and so I emphasize that it involves a special relationship, achieved only by fulfilling specific criteria of opposition, inter-relation and wholeness; that it operates at very different structural levels, and across a continuum from looseness to rigor, from ambiguity to clarity of application. Bartók’s Third Quartet provides an exceptional example of such complementation functioning as a structural determinant on both small and large scales, where the rich and diverse embeddings may result in juxtaposition, polarity or synthesis. Although in his pitch study of this piece, Berry (1980, 379) was convinced that “the pentatonic, diatonic, or chromatic content of these sets—and of their ‘combinatorial’ or overlapping associations—is a vital style-defining factor in Bartók’s work,” the Quartet is atypical in the extent and sheer intensity of its complementary obsession.
 Finally, while I am not suggesting that the concept of complementation alone offers some neat answer to the many questions raised by the work, I conclude that an understanding of the musical processes of the Third Quartet is significantly enhanced through its use.
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1. In the course of developing this article, I have benefited from feedback offered by various scholars; in particular, I should like to thank the anonymous
MTO readers and Tim Koozin. I am also very grateful to Ronald Woodley for his accomplished setting of the music examples and figures.
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2. Especially relevant in Antokoletz, Fischer, and Suchoff 2000 is the contribution of Malcolm Gillies, “Analyzing Bartók’s Works of 1918–1922: Motives, Tone Patches and Tonal Mosaics,” 43–60. For a review of Bayley 2001, including comparison with Antokoletz, Fischer, and Suchoff 2000, see Novak 2001, in this journal. For other pertinent literature, see Antokoletz 2005 and Bayley 2000.
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4. A definition offered by Kielian-Gilbert 1982–83, 210, seems today unnecessarily restrictive: “Complementation refers to the process of pairing entities on either side of a center of symmetry.”
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5. As rightly acknowledged early on by Berger 1963, 25, the term “polarity ... cannot accurately be applied to one thing without its opposite.” I maintain that the same is true of complementation.
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14. Partitioning of collections is a useful notion, as employed by van den Toorn 1983, 48ff., and most probably originated by Milton Babbitt. Note too the early interest of Babbitt 1949 in Bartók’s quartets.
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16. A further sense in which the opening embodies characteristics of the Quartet as a whole lies in the clear articulation of a tritone B–
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18. “Black/white-key” pentatonic, octatonic and whole-tone collections are also discussed by Antokletz 1993, 261–64, who considers black/white “polarization” as part of a broader concern with “transformational processes” (259). Antokoletz explores Bartók’s “extension in range,” as the diatonic expansion of chromatic material and its converse, “chromatic compression,” referring to interval-cycles (1/11 and 5/7) rather than set theory.
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