allen forte - context and continuity in an atonal work

Context and Continuity in an Atonal Work: A Set-Theoretic ApproachAuthor(s): Allen ForteSource: Perspectives of New Music, Vol. 1, No. 2 (Spring, 1963), pp. 72-82Published by: Perspectives of New MusicStable URL: http://www.jstor.org/stable/832105Accessed: 12/01/2009 10:39

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CONTEXT AND CONTINUITY IN AN ATONAL WORK

A SET-THEORETIC APPROACH

ALLEN FORTE

WHEN Schoenberg abandoned triadic tonality in 1908 and began to compose so-called atonal music he introduced into the historical flux certain problems that remain unsolved to this day. His own evi- dent concern as to whether composing was a logical or a sensory process may have obstructed any personal efforts to find solutions. Whatever the case, we know that he did not explain adequately the structural bases of his atonal works, but seemed to regard the discovery and development of the "method of composing with twelve tones" as a satisfactory denouement. It now appears that although we may have by-passed the problems of the atonal period quite success- fully we are still left with a large body of problematic music from that period-some of which is performed regularly, little of which is understood.

In an effort to attack the problem of atonal music more strategically the following discussion departs from the familiar terms context and continuity. One need only reflect for a moment upon the degree to which both context and continuity were determined by triadic tonality-indeed, were essential conditions-in order to realize that the abandonment of that musical system endangered what many regarded as fundamental properties of musical composition. Certain observers

responded to the threat by denying the "musicality" of atonal works. More recently others have attempted to show that the logic of such atonal works still resides in triadic tonality, so that one is to under- stand atonal configurations in terms of an implicit triadic norm. Both

responses have only further obscured the problems. Both have failed to recognize Schoenberg's truly revolutionary spirit, on the one hand, and his remarkable intuitive grasp of musical essentials on the other. But perhaps most misguided of all is the condescending observer whose stock-in-trade is that precious commodity "style," for Schoen- berg has indicated the fruitlessness of such an approach:

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In my music there has never been a concern for "style," but rather a constant concern for content [Inhalt] and its most precise representation [Wiedergabe]. Therefore my youthful works prepare for the understanding of my musical thought, and one would do well to familiarize himself with those works [written] before my means of expression became so concise-as in my latest works. (From a letter to Leo Kestenberg, 1939)

The above quotation will also serve to introduce the composition which is used here as an example of atonal music: Schoenberg's Sechs kleine Klavierstiicke, Op. 19. With the exception of the sixth piece this work is dated February 19, 1911, at which time the composer was thirty-six. The six famous miniatures form a single work. They belong together, for they are all compositional projections of the same relational system. The properties of that system are set forth below, together with several sample passages from Opus 19. As indicated in the title of the present article, mathematical set theory underlies the analytical approach. The set-theoretic formulation, in turn, reflects the general viewpoint that the analysis of a structural system begins with the determination of a set of elements and the combinational relations which they exhibit.

DESCRIPTION OF THE SYSTEM

We assume the usual partition of the available equal-tempered pitches, such that there is an equivalence relation based on the interval of the octave. The universal collection or set of our system, then, is the chromatic scale, and its elements are placed in one-to-one corre- spondence with zero and the positive integers from 1 to 11 as shown in Ex. 1.

x , ^ o - $t o " o ? #.. ? I .0 1 2 3 4 5 6 7 8 9 10 1

Ex. 1

This universal set contains a set of five subsets designated A whose members are selected in the following way. First, a subset of two elements x1 and x2 is defined as the "interval-of-reference." This subset, designated X in Ex. 2, consists of the pitches G, B, which in traditional parlance form the interval of a major third and in the numerical language used here the interval 4. The meaning of the term interval-of-reference becomes clearer when the selection of all the elements of set A [the collection of subsets A1 . . . A5] has been ex-

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PERSPECTIVES OF NEW MUSIC

plained. Each subset of set A contains the pitch elements of a single interval relation on X. Thus A1 contains all the half-step relations (1-relations), A2 contains all the whole-step relations (2-relations), . . Each element forms a pair (not ordered) either with G or with B or in a few cases with both. The basis of association will be evi- dent in a moment. First it is important to state two limitations upon this relational process. 1) As a result of the principle of octave equivalence the process does not continue beyond the 6-relation. 2) The

process observes what may be called the condition of proximity: Any pitch associated with xi must be either nearer to xl than to x2 or equidistant from both. The same condition holds for pitches associated with x2. As a result of this condition one otherwise possible subset is excluded, the subset [2, 4] which is based upon the 5-relation. In this way both xl and x2 are deprived of possible "dominant associates."

A =A1 . A5]

XJ, r X1 ^2

X = Interval-of-Reference

j$J-

3fiL u

L bJ

A1 = set of all 1-relatlons on X

A2 = set of all 2-relations on X

A3.= set of all 3-relations on X



[7,11]

0o, 6,8,10]

15,s,9]

2, 4]

[3]

[1,5]

Ex. 2

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A


Ex. 2 displays these subsets in musical and in corresponding numerical notation. The musical notation will facilitate study of the excerpts from the score, while the numerical notation enables us to see certain properties more readily. These structurally important properties are most efficiently discussed if we first extract another set B of subsets assumed to have special interest. This set is displayed in Ex. 3.

B =B1 . . . Bs

B1 = set of all elements associated [4, 6,8] ~- y' ^' *' 1 =?:~ ?~?;:,?~? ~~???-with G only

i Lf f ~ B 2 = set of all elements associated [0, 2,10] with B only

J J1 J J -==B3 = set of all elements associated [0,2,4,6,8,10] g - J~., -^-V " ~either with G or B

^O- |; i J J - B4 = set of all elements associated [1, 3, 5,9] with both G and B

* ' 'bltj B5 = set with only one number [3]

J| |-| J

>tW xj; A " -X = complement of X [0,1,2,3,4,5,6.8,9,10] 1 tJ} ~J' -tf J (all pitch-classes not contained in X)

Ex. 3

Set B may be of interest when we come to examine the composition, since it sorts out in various ways the relations on x1 and x2, where xl and x2 are regarded as discrete elements, whereas set A shows the classes of interval relations on X without emphasizing the individual roles of its members. It should be remarked here that in the composition the interaction of elements associated exclusively with pitch G or with pitch B, contrasted with the interaction of elements associated with both, is a significant aspect of continuity and of the controlled transformation of context. Both set A and set B are essential to an understanding of the relational system, and the explanation of important properties that follows will consider them individually and in combination.

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Equivalence Set A contains no two equivalent (identical) subsets. Set B likewise

contains no two equivalent subsets. This suggests that each subset has in some way a special structure with respect to X. From the analytic standpoint this is advantageous, for we can deal with specific and differentiated relations rather than with an undifferentiated collection of pitch elements.

Equivalence relations on A and B are as follows: 1) A =B 2) A4 = B5 (That is to say, the set with only one member [Eb]

is the same as the set of all 4-relations on X. As might be expected, this unique property is ex- ploited in the composition.)

3) A=X 4) B=X

Inclusion

Taking sets A and B separately we see that 1) A5 c A2 (i.e. A5 is included in A2) 2) B2 c B3 3) B1 CBs Inclusion involving both A and B yields 4) Al cB3 5) A2 C B4 6) A3 c B 7) A4 C B4 8) A5 cB4.

Intersection

The operation intersection perhaps is of more compositional interest since it suggests possibilities for connecting one subset to another or, conversely, detaching certain elements without endangering continuity. Intersection follows naturally from the list of subsets immediately above, for in all cases the set resulting from intersection is the same as the smaller subset of each of the pairs listed. That is, A2 n A5 = [1, 5], that is [Db, F], and A2 n B4 = [1, 5, 9], that is [C#, F, A],... One might raise a question at this point regarding intersection of more than two subsets of A and B, since presumably this occurrence would tend to make certain pitch elements more redundant than others. It happens that every element occurs in three different subsets, with the exception of the pitch-element A, which occurs only in A2 and B4.

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Every element occurs in one subset of A and two subsets of B, with the exception of the pitch-element F which occurs in A2, A5, and B4. Both pitches F and A are prominent in Opus 19.

Complementation The complement of X has already been displayed (Ex. 3) and it has

been remarked that X = A. Another kind of partition, one that is more important in the work, involves two whole-tone hexachords which arise within the system. This is explained in the next section.

Union

Unions are easily read from the numerical representation in Exam- ples 2 and 3 since a subset contains numbers drawn either from the set 0 and the even numbers, or it contains only odd numbers. Thus 1) A U A3 = B3 (The union of the set of all 1-relations and

the set of all 2-relations is the same as the set of all elements associated with either G or B.)

2) A2 UA4= B4 (The union of the set of all 2-relations and the set of all 4-relations is the same as the set of all elements associated with both G andB.)

3) B1UB2= B3, and 4) A1UA3 B1UB2 5) B4 U X B3

The last subset listed, B3, is so important in the composition that it will be designated Y. In terms of pitch elements Y is the whole-tone hexachord which together with B3 forms a partition of the universal set.

It should now be apparent that the relational system offers a multi- tude of compositional possibilities, suggesting ways of combining elements into larger units as well as ways of extracting smaller units from larger. In addition, it provides effective analogues to harmonic progression and voice-leading in the tonal system. For demonstrations of some of these attributes let us turn now to the compositional projections in Opus 19.

COMPOSITIONAL PROJECTIONS

As we approach the composition in terms of the relational system it is important to regard the system as a formalized statement or sum- mary of the structural basis of the composition. It is not an analysis of the work. In order to emphasize this distinction and at the same time

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relate the musical events to the abstract sets of the system the term compositional set is here introduced.

It is also important to realize that there are no "independent" elements in this composition, just as there are none in a tonal work. There are, however, elements which are less determinative than others, as may be inferred from the foregoing explanation of the system. Such elements stand forth in a variety of contexts and thus make clear a certain structural hierarchy, which, in turn, makes it possible to describe the structural meaning of a compositional set in terms of the elements which dominate it. For instance, at the beginning of the fifth piece (Ex. 4) we discover that the first phrase is controlled by F, A, and D#, while the lower (accompanimental) part of the phrase is controlled by Db and G. These strategically placed pitch elements are the union of X and B4, the special subset Y.

Etwas rasch (b)

-zart, abet voll p

Ex. 4

To bring these preliminary remarks to an end, a question: What, then, determines the structural meaning of a given note in this composition? Response: Its context as interpreted by the underlying relational system shown in Exx. 2 and 3.

The first excerpt to be discussed is the opening phrase of the first piece.

Leicht, zart ( ) 1

2 3 4

Ex. 5

For the sake of convenient reference the compositional sets are enclosed and numbered to correspond to the remarks which follow.

1. The first three pitches of this "melodic" set belong to the special subset Y. The last element, F#, is associated only with G (as in B1),

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so that over the span of the phrase the direction of the melody is from B toward G. The concluding melodic succession F-F# is one of three similar pairs prominent in the work. These semitone successions may be interpreted as unions of two elements, one drawn from each of the whole-tone hexachords B3 and Y.

2. This set provides a context for B of the melody, which we can take to be the controlling element here. It departs from A, a pitch element from subset Y. Of the remaining elements of the set the pitch C associates exclusively with the pitch B (as in B2), G belongs to X, and G# associates exclusively with G (as in B1). Taken together with B in the treble staff this compositional set exhibits a characteristic feature which pervades the work: equal distribution or balance of pitch elements with respect to the interval-of-reference. In this case each element of the interval-of-reference has its own exclusive associate, while the pitch A associates with both. Balanced sets of this kind, which occur at crucial structural points, are analogous to fundamental harmonies in the triadic system.

3. In terms of the relational system the fundamental pitch B is combined with two pitches from set B4 (the set of all elements associated with both G and B). The remaining element E is one of the three pitches associated only with G; in this context it serves to prepare the progression to the final compositional set, which centers on G. In this connection it is important to recognize that, as a continuity factor, the repetition of the melodic succession B-D# (in set 1) by the harmonic pair B-D# is secondary in significance to the transformation of context which takes place by virtue of the presence of the two addi- tional elements E and F.

4. The final simultaneity of the phrase consists of the fundamental pitch G together with one exclusive pitch associate and one pitch exclusively associated with B, the other fundamental pitch. Unlike the balanced sonority at the beginning of the phrase this sonority is weighted toward G.

In the second piece the interval-of-reference serves as a kind of ostinato against which various compositional sets are projected. The closing part of the piece is shown in Ex. 6.

1. Here again the elements of this initial compositional set are distributed equally with respect to the interval-of-reference. Only register and rhythmic accent differentiate. The emphasized pair C-Eb, in particular, is important here (and throughout the first part of the piece). The pitch C associates exclusively with B, while Eb is the only set with only one member and associates either with G or with

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Ex. 6

B depending upon context. Here it associates with both since both are present.

2. At this point the progression culminates in a harmony which is weighted in favor of the pitch B. Observe that although the pair C-Eb descends to the pair D-B in the upper part, it is retained (notated enharmonically) in the lower part of the harmony.

3. The restatement of the interval-of-reference at the beginning of this phrase represents the final "resolution" of the pair C-Eb. This entire set is a complete projection of Y.

4. The final set consists of an almost complete statement of B3, the complement of Y. Only Gj is lacking. The complementation is expressed instead by the juxtaposition of two trichords, a distribution consistent with the texture of thirds that characterizes this piece.

In the third piece we encounter a different texture as well as different techniques of projection. Ex. 7 summarizes the controlling compositional sets.

i I-opp v -

a b d

Ex. 7

Both in Ex. 7 and in Ex. 8 the different elements of the controlling sets are distinguished by notational means. The interval-of-reference is notated in half notes and the pitches associated with it are shown as 8th notes. The other controlling set F-A is given in quarter notes, while its pitch associates are shown as 16ths. The complement of the

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union of sets a and b is designated d. Let us now see how these sets interact in the composition. Example 8 is a sketch of the first four bars. Secondary harmonic notes (from subset d) have been omitted in order to show the controlling sets more clearly.

I b:r ^tj I J4 -^-J-, (-d ,i:- bg^

tB b^ ^ Ex. 8

Observe that the same elements of the subsets are consistently counter- pointed (vertical arrows), guaranteeing continuity over the longer span. Observe also that the interval-of-reference, which controls the first part of the passage, gradually yields to the interval F-A, in a manner reminiscent of tonal modulation. To extend the analogy, this might be called a natural modulation since F-A belongs to the set B4, the set of all elements associated with both G and B. The entire passage terminates in a whole-tone context consisting of the trichord F-A-Db played by the right hand and a disjunct tetrachord (partial statement of the complement) in the left hand. (The separateness of the two subsets is emphasized in this instance by an idiosyncratic performance instruction: "Throughout the first four measures the right hand is to play f, the left hand pp.") This whole-tone context prepares the first pitches of the following phrase (not shown), the interval-of-reference.

i 'jp - -4. y" -i pp23 r _^==> ' ^- '

7 7 1 _ p r 1U ^-1* _ ___, ^

Ex. 9

The closing passage of the third piece, shown in Ex. 9, begins with another balanced harmony, exploits the unique pitch set B5 (Eb) and ends with melodic emphasis placed upon the pairs Eb-G and F-A. The final harmony is weighted toward G, yet two of the four elements, D and Bb, associate only with B.

As a final illustration Ex. 10 shows the closing set of the last piece. Not only are the elements of the complete compositional set perfectly balanced with respect to the interval-of-reference but each of the

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Ex. 10

subsets is also distributed in that way. Thus the first subset, which contains the pitch B, contains as counterbalance an associate of G and an element of B4. The elements of the second subset are similarly balanced off against G. The third subset contains one element of B2 and one element of B1. Had this piece been composed some fifteen years later this final gesture would have been followed by the whole- tone dyads E-D and Eb-Db, the complement of the final set, so strongly suggested by the concluding pair.

It is hoped that the foregoing has indicated that with the assistance of new techniques atonal music can be studied effectively. To explore other aspects of the particular system unfolded here or to draw conclusions regarding the historical development of the art based upon the material presented would exceed both the physical bounds as well as the intent of the article.

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allen forte - context and continuity in an atonal work

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