Yale University Department of Music

Space and Symmetry in BartókAuthor(s): Jonathan W. BernardReviewed work(s):Source: Journal of Music Theory, Vol. 30, No. 2 (Autumn, 1986), pp. 185-201Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/843574 .Accessed: 11/03/2012 05:10

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SPACE AND SYMMETRY IN BARTOK

Jonathan W. Bernard

The phenomenon of symmetry in the music of B61a Bart6k has received considerable attention in the critical literature of the past thirty years. In the process, a fascinating variety of information has come to light. Symmetry has been depicted as an influence upon structure on many levels, ranging from the smallest to the largest of contexts. The widespread interest in sym- metry might well be expected, not only because of its prominence in the audible surface of much of Bart6k's work-a fact that cannot help but pro- voke speculation about its deeper implications - but also because of the po- tential power of symmetry to control musical structure. More surprising, perhaps, is that only rarely has symmetry been addressed on its own terms, as an independently functioning system of organization. Rather, theorists have tended to incorporate it into more general frameworks, often designed to assert the primacy of some other dimension of the music.'

Of the analytical works that fall into this category, by far the most exten- sive and probably the most significant is Ern6 Lendvai's B&la Bart6k: An Analysis of His Music, published in English translation in 1971? Many of Lendvai's various models of structure serve to identify symmetrical patterns of durations, chordal functions, and intervals. Symmetry even plays a role in the delineation of form. Lendvai, however, does not attach any special significance to symmetry per se, or to any other single domain of structure,

185

for that matter. As a result, the formal nature of his theory remains some- what diffuse.

One notable exception to the usual treatment of symmetry as a secondary issue is George Perle's "Symmetrical Formations in the String Quartets of B61a Bart6k." Here, the author considers the possibility that Bart6k's use of symmetry represents a compositional breakthrough, raising symmetry to a level of importance far above its (alleged) insignificance in the music of certain of his predecessors. Perle's conclusions, however, are distinctly pes- simistic:

Impressive as these procedures are, it must be observed that Bart6k's sym- metrical formations are only an incidental aspect of his total compositional means. Even in those few works where they perform a significant structural role they do not ultimately define the context, which is determined instead by a curious amalgam of various elements. . . . Can symmetrical formations generate a total musical structure, as triadic relations have done tradition- ally? The implications of Bart6k's work in this, as in other aspects, remain problematical?

But the reader might well disagree. Perle's statement seems strangely at odds with the evidence that he cites-evidence which, though hardly con- clusive, certainly invites further investigation. One is reluctant, after en- countering so many striking instances of symmetrical construction, to dismiss them as mere curiosities.

Why, exactly, should symmetry be a "problematical" issue? Perhaps the problem resides in the definition of the phenomenon itself. One point that has not been stressed in the relevant literature is that there is a difference between instances of symmetry in which the relationships are consistently represented in literal, registral terms, and instances in which registral repre- sentation is only partially or not at all present. Consider Example 1. Here two chords with identical pitch content are shown side by side. What distin- guishes one from the other is, of course, the registral placement of the pitches; in the first chord the disposition is symmetrical (since the same interval of eleven semitones separates Db and C as does C and B), while in the second it is not. Recognition of this difference justifies the provisional conclusion that, of the two types, registrally represented symmetry is more significant structurally. Symmetry is, after all, first and foremost a spatial phenomenon.

We have long been accustomed to the concepts of inversional and octave equivalence in tonal music, and for the most part we have retained these concepts in analysis of the post-tonal repertoire. Nevertheless, in a sym- metrical context there is no reason why the metaphorical space of the modulo-12 pitch-class system should be equated with real space. The con- vergence or divergence of instrumental parts, for example, or the transposi- tion or inversion of a chord or motive, is most clearly perceived when

186

Example 1

tlP

re

r~P

Example 2. Illustrations from Bl61a Bart6k, "The Problem of the New Music"

mm. 1 5 9 13 17 27 27

, •O

Example 3. Music for Strings, Percussion, and Celesta, I, mm. 1-27, head-

notes of successive fugal entrances

187

absolute intervallic dimensions are preserved. Examples of this sort of literal symmetry in Bart6k's compositions are, of course, numerous. Their very existence, in fact, suggests that relationships based entirely upon ab- solute intervallic dimensions might well operate, in ensemble, as a general method of symmetrical procedure throughout entire works.

Certainly, there is no shortage of evidence tending to support this hypothesis. Representative samples are presented below, by way of defining types of symmetry that will be applied in the longer analyses to follow.

Some of this evidence is provided by Bart6k himself. In his "The Prob- lem of the New Music" (1920), Bart6k departs briefly from his usual reticence about matters theoretical and presents several examples of atonal chords (Ex. 2). Of his four chords, three are symmetrical about an axis of either one or two pitches and exhibit what we will call henceforth mirror symmetry. Commenting on these sonorities, Bart6k notes that registral arrangement is crucial to their effect."

One well-known instance of symmetry in Bart6k's music is the pattern of fugal entrances in the first movement of Music for Strings, Percussion, and Celesta (1936) (Ex. 3). The order of these entrances produces two diverging chains of perfect fifths, radiating in alternating order from the initial A. Even more significant is that all of these fifths are exactly seven semitones in size-that is, no inverted or compound fifths are substituted. The overall design, then, is one of mirror symmetry about A.

In the opening of the second movement of the Second Piano Concerto (1931), the string parts are disposed in two groups moving for the most part in contrary motion (Ex. 4a). The symmetry involved in this motion seems at first only approximate, but if all of the notes in mm. 1-5 are arranged in a scale, preserving the registral positions in which they actually appear, then a mirror-symmetrical pattern emerges, as shown in Example 4b. Arrangement of the contents of mm. 6-8 in similar fashion reveals another symmetry, different from that of mm. 1-5. Here, mirror symmetry (as re- vealed by the presence of a midsegment) is combined with parallel sym- metry between the two outer segments, so called because both exhibit the same order of intervals from bottom to top (Ex. 4c).

The title of Mikrokosmos No. 141, "Subject and Reflection" (1933),5 is of course suggestive from the standpoint of symmetry. The piece consists of a series of short sections, each of which is symmetrical about a single pitch or a pair of pitches one or more octaves apart. Example 5a shows the beginning of the first section. On a larger scale, the axes of symmetry them- selves, if taken as a series, form a symmetrical pattern of their own. The axes, in order, are Bb, B, D, Eb, FS, G, and Bb (Ex. 5b). This series is equivalent to Lendvai's 1:3 scale6 and is of the parallel-symmetrical type. Besides this pattern, in which the pitches appear in various different octave locations, other features control registral structure. First, the opening posi- tions of the Bb's encompass exactly one octave less space, both above and

188

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Example 4a. Concerto No. 2 for Piano and Orchestra, II, mm. 1-8

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Example 4b. mm. 1-5, summary of pitch content

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Example 4c. mm. 6-8, summary of pitch content 189

below, than the ensemble of Bb's in the final section (mm. 63-82). Over the duration of the work, then, an expansion has taken place (Ex. 5c). Second, the 1:3 scale of axes, traced through the entire work, can be found continu- ously in only one octave, beginning on the Bb below middle C. The end- points of this octave scale are the same pair of Bb's that opened the piece. The doubling of notes in the scale (excluding the final Bb, which is treated somewhat differently from the other axes) fall into a symmetrical pattern as well, shown in Example 5d. This pattern may be read as follows: Bb and B doubled above, then D doubled both above and below, then Eb and F$ doubled below, then G doubled above and below.

Closely related to parallel and mirror symmetry respectively are replica- tion and inversion. The only difference is that replication and inversion are better suited to describing order of events, in which a given configuration may be said to give rise to another. An example of replication and inversion at work in the same passage occurs at the beginning of the third movement of the Second Piano Concerto (Ex. 6a). In m. 3, the D below middle C is the lowest note in the ensemble chord of octaves and unisons, and it is im- mediately followed by C and Eb in the timpani-pitches which, repeated, turn into an ostinato. The group C-D-Eb is labeled (A) in example 6b. The piano re-enters in m. 7 with Eb and Gb, eventually (m. 10) filling in this interval with F. At this point, the pitches Eb-F-Gb, labeled (B), may be re- garded as a replication of (A). Two measures later, the piano takes up the timpani Eb and C, but fills the interval with Db instead of D, forming a group (C). With respect to (B), then, (C) represents an inversion. Finally, in m. 14, the piano reaches the A below these pitches, completing an expan- sion from the original C-Eb.

The foregoing are, of course, only examples, and we should be wary of drawing unwarranted conclusions from them. No matter how compelling an example may be, the possibility always remains that it is atypical: a special instance of a procedure that cannot be generalized. To go beyond the isolated example, we need to consider more closely the concept of sym- metry itself, and the unexamined assumptions involved in its customary definition.

Others besides Perle have noted the potential poverty of musical rela- tionships inherent in a symmetrically governed "total musical complex." Stravinsky, for example, is said to have remarked that "to be perfectly sym- metrical is to be perfectly dead."' What these two statements have in common is an implicit definition of symmetry as a state in which all com- ponents of the musical texture are subsumed under a single relation, a single axis of symmetry.

However, other definitions of symmetry are feasible. An expansion of the single-axis conception would change the situation radically. Let us postulate instead a multivalent symmetry, involving a matrix of possibilities to be exploited successively or in free combination. Such a matrix, if it

190

Allegro J = 136-144

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Example 5a. Mikrokosmos No. 141, "Subject and Reflection," mm. 1-7

c. )-d.

•• b. m. 1 15 23 30 40 47 63

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l 3

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Example 5b,c,d. "Subject and Reflection," schematic representations of symmetrical relationships

sostenuto

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t TimpAL - - o I

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Example 6a. Concerto No. 2 for Piano and Orchestra, III, mm. 3-15 A

. . t, -, .•. •. -- ;•. t,. _ , •. -

•, •-. _•,•...... '

. .

Example 6b. mm. 3-15, schematic reduction 191

exists, will imply a compositional emphasis upon the procedures of sym- metrical progression themselves rather than upon the axes per se. That is, more significance will be attributed to the fact that there are axes (hence symmetries) than to the pitches that actually constitute these axes. This sys- tem therefore will not engage the pitch (or pitch-class) identity of the axes, either at particular moments or over time.

Multivalent symmetry has other implications as well, already suggested by some of the examples presented above. Such a symmetrical system may work to create unity by operating, not exclusively on one level, but rather in many different ways at once, ranging from control of single chords or motives to control of large sections of music. Indeed, it should be possible, if symmetry is a significant force, to discover a hierarchy of relationships in which smaller symmetries contribute to larger ones, which in turn con- tribute to even larger ones, and so on, across ever longer spans of time.

Discussion of any such hierarchy must be preceded by an enumeration of criteria for segmentation, or the division of a musical texture into its component parts. Usually there are several ways of segmenting a musical passage; some of these segmentations may be better than others. On the other hand, as a function of the matrix of possibilities mentioned earlier, several different equally valid segmentations may well co-exist in the same passage. Among the relevant criteria for segmentation in a spatial context are, first of all, lowest and highest notes, or lower and upper boundaries. These notes delimit the portion of the musical space in use over some length of time, whether for the duration of a single chord or for the duration of several measures. These two possibilities are illustrated in Examples 7a and 7b respectively; the circled pitches are the lower and upper boundaries in each case. Second, timbre is also an important means of segmentation. In the first of the longer excerpts below, the opening sonority may be divided into brass and percussion groups, with important consequences for the anal- ysis. Of course, since these two groups occur at the same time, they may also be considered in terms of their aggregate effect: one possibility does not exclude the other. Third, rhythmic characteristics may serve to dif- ferentiate musical elements, especially for temporally simultaneous events. Besides the explicit sort of contrast apparent, for example, between sus- tained and rapid notes, there are other, more intricate possibilities. At the opening of Sonata for Two Pianos and Percussion (see Ex. 11), the second piano enters in canon to the first. Even though registral overlap occurs be- tween the two, their parts may be treated, on one level, as independent regions of activity.,

Finally, other factors such as phrase markings, dynamics, changes in texture, or separation by rests are by no means rendered meaningless by the criteria enumerated above. In fact, a segmentation often stands or falls de- pending upon the degree of support provided by these factors.

192

a) (b

8va bassa 8va ......... J

Example 7. Sonata for Two Pianos and Percussion, I, mm. 14-15, piano II

mm 1-6 7-9 10-13 13-18 19-21 22-29

timp. A ~

hns. bns.

. hns. brass - brass -

"pno. ..brass

8va bassa ......................................

Example 8. Concerto No. I for Piano and Orchestra, I, mm. 1-29, sche- matic reduction

12

10 1

12

10

8V(1&LU~

Example 9

a2 A A sustained

a i 2 1sustained

sustained

Example 10. Concerto No. for Piano and Orchestra, I, mm. 13-19, horns 193

The following two analyses serve to illustrate, in different ways, how symmetrical constructs interact in larger and more intricate contexts. In the First Piano Concerto (1926), first movement, mm. 1-29 (Ex. 8), the first six measures are devoted to a sonority to be called (X), for the sake of con- venient reference. (X) consists of the pitch B doubled in the piano and tim- pani, two octaves apart, and A doubled in the horns and trombones one octave apart. The size of the total space in use here, expressed in semitones, is [24].Y In mm. 7-9 a brass chord, to be called (Y), takes the place of (X). The highest note in this sustained chord is A, in the trumpet; thus the in- terval between the highest note in the brass of mm. 1-6 is exactly [24] lower than the highest note of (Y). This relationship is marked in Example 8 by brackets with single cross-marks.

Symmetrical construction is evident as well in the internal structure of (Y) and its placement with respect to (X): from bottom to top, it encom- passes a span of [23]. This interval corresponds to two others of equal size: one is the vertical distance from the top pitch of the horns in (X) to the top pitch of the horns in (Y), the interval A-Ga; the other is the vertical distance from the lower limit of (X) to the lower limit of (Y). Note the brackets with double cross-marks in Example 8. All of these correspondences are in- stances of replication.

The juxtaposition of (X) and (Y) is interesting in other ways as well. Let us consider the piano and brass notes of (X) as internal detail of the vertical span between the lowest notes of (X) and (Y). Example 9 shows that the intervallic structure formed by the juxtaposition of (X) and (Y) is replicated in (Y) itself.

The passage continues in mm. 10-13 with a second occurrence of (X), followed by further development of (Y) in mm. 13-18. Here, to the col- lection of sustained notes in mm. 7-9 is added a doubling of the high A one octave below, and the octave Ga's in the horns become the first notes of a passage in octaves (Ex. 10). Again, symmetrical considerations are vital- in two ways. First, the overall framework in these measures, articulated both by the sustained notes and the upper and lower boundaries of the horns' octave passage, is DS-A-D%-A: space is uniformly divided into tritones. Second, the internal structure of the horns' music is symmetrical. The notes included in this passage form a series of intervals [1] [2] [1] [2] in each octave, an instance of parallel symmetry.

After a third appearance of (X) in mm. 19-21, (Y) returns, also for the third time, but in a slightly different guise. With respect to mm. 13-18, everything in mm. 22-29 is an octave lower, and the bassoons have assumed the motivic role of the horns, piano. The implications for spatial structure are considerable; Example 8 shows that contraction, the opposite of expan- sion, takes place at this point with respect to the previously delineated outer boundaries of the entire passage. This contradiction might seem at first to be only approximate, since the upper distance of contraction is a semitone

194

larger than the lower. However, if instead of hearing the interval between the upper A and the lower A we focus upon the succession of attack points, then the last G$ of the upper horn part in m. 18 is followed by the upper A in mm. 22-29. Heard this way, the two distances of contraction are exactly the same. The contraction itself may be regarded as serving the function of closing the introductory section and preparing for the transition to the Allegro which follows directly thereafter.

In the Sonata for Two Pianos and Percussion (1937), first movement, mm. 1-8 (Ex. 1la, score; Ex. lib, reduction), symmetry operates not so much by replication of vertical spans as by expansion from and contraction to single pitches that, by other criteria, emerge as prominent. This happens in the first three measures, where the initial solo F$ in the timpani even- tually becomes the midpoint of the space delineated by the lowest (Dc) and the highest (A) notes in the first piano, a space which then stands as an ex- pansion from the F).

Cumulatively speaking, up to the downbeat of m. 4 the interval DS-A has been filled chromatically in three octaves (the lowest three of the piano's range), and the timpani part has supplied the pitch B between the upper two of these three filled intervals. This B prefigures the next development, in m. 4: the pitch succession Bb-A-B in three adjacent octaves, enlarging each of the chromatic segments described above by two semitones. The midpoint of this new formation is the middle Bb, also the midpoint of the first vertical in m. 4. The upper two terminal B's, however, become more significant to subsequent events in the remainder of m. 4 and all of m. 5. Here, overlap- ping spatial areas are outlined independently by each of the two pianos. Piano I's stands as a symmetrical expansion from the previously mentioned B, and Piano II's as a symmetrical expansion from the B one octave higher. Note that Piano II enters in canon to I, at the tritone, with an imitative period of two eighths. Piano I's material, however, is in quadruple octaves, while II's is only in triple. One likely reason for this discrepancy is that Piano II's range, if further expanded, would not stand in symmetry to the pitch B.

At this point, we might pause in our progress through the excerpt to ex- amine the melodic material of mm. 2-5 more closely. In Example 12, measures 2-3 are viewed as an interlocking collection of three trichords. These trichords, besides filling chromatically the space D$-A, exhibit other interesting features. Taking the circled groups of Example 12a and vertical- izing them (Ex. 12b), we find that group 3 is a replication of group 1, while group 2 is of a different type. This in itself, of course, is a symmetrical ar- rangement. Example 13 shows that mm. 4-5 are an extension of mm. 2-3; again, when the groups of Example 13a are verticalized (in Ex. 13b), it be- comes clear that the fourth trichord, the new one, is an inversion of the second trichord. Thus mm. 4-5, in a sense, complete mm. 2-3, and one symmetrical design gives way to another.

195

Assai lento 5

= ca. 70

Piano I

Piano II

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I* 0

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-••I

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with the heavy end ?f

a drum

stick' ?n the

dome"

?I $: **senzacorda: snares off.

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Example 1 la. Sonata for Two Pianos and Percussion, I, mm. 1-8 196

P II

t p P.'P.I"

'P.P I

. P[17]

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Example 1 lb. mm. 1-8, schematic reduction.

197

The sudden forte and fortissimo dynamics of m. 6 herald, among other events, an abrupt shift in registral location, to an area bounded by D's three octaves apart. The lower boundary of the sonority in Piano I is symmetric- ally placed with respect to the boundaries of mm. 4-5 (that is, Pianos I and II taken together)-and, hence, to those of the entire piece up to this point. Furthermore, the triple-octave span of D's, explicitly subdivided into octave segments, is a replication of the final position of Piano I in m. 5 (C$'s).

Also in m. 6, Piano II presents a double run of rapid notes. This sudden burst of activity has several features worth noting. First, it is symmetrical in itself: an ascent is made from the B below middle C to the A above it and back again; and at each point in the run, two notes a semitone apart sound together, except at the apex, where three are played simultaneously. Second, the beginning and ending B is the very pitch that stood in sym- metry to the space delineated by Piano II in m. 5. Previous material, then, continues to affect structure here. The connection makes sense musically, since Piano II's material, being the comes of the canon in mm. 4-5, ends after Piano I and has therefore been heard more recently. Third, the apex trichord G-G$-A of the run in m. 6 stands exactly halfway between the outer boundaries of the measure (D's). The prominence of this trichord is guar- anteed further by its placement at the point of maximum loudness in the crescendo-diminuendo pattern played by Piano II.

The space filled by the run in m. 6 continues to exert influence upon structure in mm. 6-7. The melody in Piano II, at the dynamic of piano, extends from the D above middle C down to the E below it, replicating the space B-A of the run of rapid notes. The composite of these two spans of [10] is E-A, [17], which is exactly the same as the distance of contraction from the triple-octave D's to the apex trichord G-G$-A. The melody in Piano II has, as well, a clearly symmetrical layout of its own (see Ex. 14): verticalization of its contents shows that its two component trichords are inversionally related and, further, are of the same type as numbers 1 and 3 in Example 12.

With the diminuendo to the downbeat of m. 8 comes the end of a series of spatial contractions, as the upper boundary is successively lowered, from the initial high D to the D an octave lower, then finally to the D an octave below that. At the same time the original low D, while still sounding, has faded more quickly than the other notes in the texture, owing to the fact that after m. 6 it is not struck again. The D immediately above middle C, then, is the center of a contraction carried out from m. 7 to m. 8, from the boundary D's one octave below and above.

To claim that these symmetrical procedures are universally applicable to

Bart6k's music would certainly be premature. The reader will have noticed

198

(1) (2) (3)

Example 12

(4)

Example 13

Example 14

199

that the repertoire from which the foregoing examples were drawn is cir- cumscribed in two ways. First, all but one of the pieces chosen feature the piano prominently in the instrumentation. It is possible that the special demands of writing for the piano, with its unbroken range of more than seven octaves and its uniform rate of timbral transformation throughout that range, may entail special controls upon delineation of registral space-or, at least, may have suggested to Bart6k the possibility of instituting such controls. Thus, the string quartet literature, heavily favored by other investi- gators of symmetry, may be precisely the wrong place to look for extensive use of registral symmetry, given the lack of timbral uniformity, the smaller total range, and the limitations imposed by size of the ensemble together with the (presumed) need to promote distinctiveness and individuality of parts.

Second, many of the excerpts quoted (and both of the longer ones) are taken from works or movements with slow tempos. This is not an accidental circumstance, but rather a reflection of the origins of this study in the per- ceived character of Bart6k's "slow music" and the impression that it conveys of steady and relentless motion from one point to the next. For pragmatic reasons, too, such music is ideal for illustration of the ideas in this paper; because the rate of pitch change with respect to time is slow, large move- ments or sections may be more easily treated.

Given these limits, however, one must conclude nonetheless that sym- metrical processes do have a thoroughgoing influence upon compositional design in the passages cited, and probably in much of Bart6k's other music as well. No claim is made here that these are the only aspects of his music worth discussing-only that symmetry has greater power than might have been suspected to explain not only the presence of many specific elements of Bart6k's musical textures (that is, single notes and small groups of notes) but also the disposition of entire passages and their relative locations in musical space. The implications seem well worth exploring in future analytical work.

200

NOTES

1. Among the more important studies dealing with symmetry, besides those listed in notes 2 and 3 below, are: Milton Babbitt, "The String Quartets of Bart6k," The Musi- cal Quarterly 35 (1949): 377-385; Colin Mason, "An Essay in Analysis: Tonality, Symmetry and Latent Serialism in Bart6k's Fourth Quartet," The Music Review 18 (1957): 189-201; Leo Treitler, "Harmonic Procedure in the Fourth Quartet of Bla Bart6k," Journal of Music Theory 3 (1959): 292-298; Allen Forte, "Bart6k's 'Serial' Composition," The Musical Quarterly 46 (1960): 233-245; Arnold Whittall, "Bar- t6k's Second String Quartet," The Music Review 32 (1971): 265-270; George Perle, Twelve-Tone Tonality (Berkeley: University of California Press, 1977), esp. pp. 10-12;. Elliott Antokoletz, "Principles of Pitch Organization in Bart6k's Fourth String Quar- tet," In Theory Only 3/6 (1977): 3-22; Antokoletz, The Music of Bdla Bartdk: A Study of Tonality and Progression in Twentieth-Century Music (Berkeley: University of Cali- fornia Press, 1984).

2. Ern6 Lendvai, B&la Bartdk: An Analysis of His Music (London: Kahn and Averill, 1971).

3. George Perle, "Symmetrical Formations in the String Quartets of Bl61a Bart6k," The Music Review 16 (1955): 300-312.

4. Bl61a Bart6k, "The Problem of the New Music," reprinted in B&la Bartdk Essays, ed. Benjamin Suchoff (London: Faber and Faber, 1976), pp. 455-459.

5. The date of composition is supplied in John Vinton, "Toward a Chronology of the Mikrokosmos," Studia Musicologica 8 (1966): 41-69.

6. Lendvai, B&la Bart6k, p. 51. 7. Igor Stravinsky and Robert Craft, Conversations with Igor Stravinsky (New York:

Knopf, 1959), p. 20. 8. Numbers in brackets express the size of intervals in semitones. 9. Bart6k's use of canon here can be regarded as yet another aspect of symmetrical con-

struction (replication).

201