Society for Music Theory
Pitch-Class Set Multiplication in Theory and Practice Author(s): Stephen Heinemann Source: Music Theory Spectrum, Vol. 20, No. 1 (Spring, 1998), pp. 72-96 Published by: on behalf of the Society for Music Theory Stable URL: http://www.jstor.org/stable/746157 . Accessed: 03/09/2014 00:53
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This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions Pitch-Class Set Multiplicationin Theory and Practice
Stephen Heinemann
Since its 1955 premiere, Le Marteausans maitre has been sults are unpredictable;compositional and analytical choices regarded as one of the most important compositions of the are capricious; and important properties arise from arith- post-war era. Certainly it is the best-known work of Pierre metical, as opposed to pitch-class, operations. And under- Boulez. Although many attempts have been made to analyze lying such assumptionsis a more fundamentalquestion: What Le Marteau'sserial organization, it is Lev Koblyakov who has compelled Boulez to use the operation at all? most effectively clarified the results, even if not the process, In this essay I will examine and explain Boulez's multi- of Boulez's technique.1 Koblyakov demonstrates that an op- plicative operations and thereby correct each of these mis- eration invented by Boulez called "multiplication"generates perceptions. I will present three different but related oper- the pitch-class sets that form the basis of the cycle of "L'Ar- ations called simple, compound, and complex multiplication. tisanat furieux" (movements 1, 3 and 7) in Le Marteau. The Simple and compound multiplication are treated cursorily composer himself, in his theoretical writings, has also de- (but neither differentiated as such nor usefully formalized) in lineated certain features of multiplication.Yet neither Boulez Boulez's writings.2I will demonstrate, however, that complex nor Koblyakov has described and examined the pitch-class multiplication, the mechanics of which were not divulged by sets that result from multiplicative operations. Boulez's re- Boulez, was employed to generate the pitch-class sets that luctance to elaborate on his technique and the apparent in- constitute the first cycle of Le Marteau.3I will also attempt ability of Koblyakov to decipher it can easily lead to the to clarify some aspects of Koblyakov's published analyses, as following misperceptions: the operation is arbitrary;the re- 2The most importantof these-at least as far as Le Marteauis concerned -is the chapter entitled "Musical Technique" that forms the bulk of Pierre This article expands on a paper presented at the annual meeting of the Boulez, Boulez on Music Today, trans. Susan Bradshawand RichardRodney Society for Music Theory, Kansas City, 1992, and is primarilydrawn from Bennett (Cambridge:Harvard University Press, 1971), 35-143. The title and Stephen Heinemann, "Pitch-ClassSet Multiplicationin Boulez's Le Marteau intent of the chapter recall the treatise of Boulez's principal teacher Olivier sans maitre" (D.M.A. diss., University of Washington, 1993). Messiaen, Techniquede mon langage musical (Paris: Alphonse Leduc, 1944). 1Lev Koblyakov, "P. Boulez 'Le marteau sans maitre,' analysis of pitch 3This operation's application will also be demonstrated with respect to structure,"Zeitschriftfiur Musiktheorie 8/1 (1977): 24-39. These early findings Boulez's earlier, subsequentlywithdrawn, Oubli signal lapide. Koblyakovlists were refined in Lev Koblyakov, "The World of Harmony of Pierre Boulez: nine works following Le Marteau in which multiplication is employed: the Analysis of Le marteau sans maitre" (Ph.D. diss., Hebrew University of Third Sonata, StructuresII, Don, Tombeau, Eclat, Eclat/Multiples,Figures- Jerusalem, 1981), and Pierre Boulez: a Worldof Harmony (New York: Har- Doubles-Prismes, Domaines, and cummings ist der Dichter (Koblyakov, wood Academic Publishers, 1990). Pierre Boulez, 32).
This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions Pitch-Class Set Multiplication in Theory and Practice 73 well as posit a process-based listening strategy for sections of In Cohn's formalization, the transpositional combination the first movement of Le Marteau, using techniques derived of two Tn-type sets can be calculated on an additive matrix from analyses of music by Slonimsky, Lutoslawski, and of the type shown in Example 1: each element of one operand Stravinsky. set is added to every element of the other, resulting in a set Boulez's multiplicationhas been cited by Richard Cohn as of integers analogous to pitch classes from which Tn- and an early and important formulation of transpositional com- Tn/TnI-typesets are derived. The operation is signified by * bination.4 This is accurate to the extent that both operations between operands. Undermining a tenet of much post-tonal can be understood informally as the construction of one set theory, the greater abstraction of the Tn/TnI-typeset is of upon each element of another. But while Cohn's theories limited value here. Example 1 shows that the transpositional concern multiple combinations of abstractly formulated Tn- combination of [0126] and [015], both of which are inver- and Tn/TnI-typesets, the multiplicative operations presented sionally asymmetrical,can be obtained in four different ways, here involve paired pitch-class sets drawn directly from spe- and can even yield sets of different cardinalities. cific musical representations. Regardless of such differences, the theory of transpositionalcombination provides an invalu- SIMPLEMULTIPLICATION OF PITCH-CLASSSETS able introduction to many of the workings and possibilities of multiplicationas practicedby Boulez and, in different sche- Even the is too abstractfor of mata, by other composers and theorists.5 Tn-type purposes pitch-class set multiplication; a theoretical tool is required at the next greater level of specificity. Since the Tn-type is derived from 4RichardL. Cohn, Combination in "Transpositional Twentieth-Century the normal form of an unordered set, this next level will Music" diss., Eastman School of Music, 48-50; "Inversional pc (Ph.D. 1987), be derived from a of set called an ini- Symmetryand TranspositionalCombination in Bart6k," Music TheorySpec- type partially-ordered trum 10 (1988): 23. tially ordered pitch-class set, or io set. This is a set in which 5Cohnhas observed that differencesin terminologycan camouflageclosely one pitch class (selected according to contextual criteria) is the resemblance of his combina- related techniques, citing "transpositional ordered as the first; the remaining pcs are unordered with tion" and Boulez's "multiplication" to, among others, Howard Hanson's respect to each other, but succeed the first. The first pc of "projection" and Jonathan Bernard's "parallel symmetry." See Cohn, "In- an io set is referred to as the initial class and is des- versional Symmetry,"23. To these can be added Nicolas Slonimsky's"inter-, pitch infra-, and ultrapolation" and Anatol Vieru's "composition." See Nicolas ignated by the letter r. The normal form of an io set is derived Slonimsky, Thesaurusof Scales and Melodic Patterns (New York: Schirmer by listing pc integers in ascending order and rotating this Books/Macmillan, 1987; originally published by Charles Scribner's Sons, order to begin with the initial pitch class; the resultant io set and Anatol "Modalism-A 'Third " 1947), ii; Vieru, World,' Perspectivesof is notated with a combination of ordered and unordered New Music 24/1 (1985): 65. George Perle appropriatesCohn's term but asserts set notations.6 For set and r = having "formulated the concept" in 1962 (correspondence, Music Theory example, given pc {1479} 4, Spectrum17/1 [1995]: 138), a claim contradictedby the earlier work of Slon- imsky, Boulez, and Hanson; conversely, Slonimskyidentifies Domenico Ala- accurate (if less concise), but to the originator goes the nomenclatorial leona, Alois Haba, and Joseph Schillingeras his theoreticalprecursors. Trans- prerogative. positional combinationserves ideally as the umbrellaunder which this variety 6Conventions of notation are taken principally from John Rahn, Basic of technical approachescan be understood. My own experience with Boulez's Atonal Theory(New York: Longman, 1980). The conventions most important technique suggests that the term "multiple transposition" would be more to this study include: C = 0;
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Example 1. Four realizations of [0126] * [015]
6 6 7 e 6 6 t e 6 6 7 e 6 6 t e 2 237 2 267 5 5 6 t 5 5 9 t 1 1 2 6 1 1 5 6 4 459 4 489 0 0 1 5 0045 0 0 045 * 0 4 5 0 1 5 * 0 1 5 0 4 5 * 0 1 5 >g 045
-- * (0126) * (015) -> (0126) * (045) (0456) * (015) -> (0456) (045) -e -- {e0123567} -> (te01245671 -> {45679te01 -> {45689te0) (01234678) -> (012346789) -> (012356789) -> (01245678) -> 8-5 [01234678] 9-5 [012346789] 9-5 [012346789] 8-5 [01234678] the io set is <4{791}>. The generalized form of such a set fewer initial orderings than permutations.7 Finally, these are draws from the Tn-type concept and is called the ordered hardly new concepts. Just as the "generic" dominant-seventh pitch-class intervallic structure, or ois. The ois lists the ordered chord is an example of a Tn-type set in tonal practice, so also pitch-class intervals from the initial pitch class to every el- is the specification of root position or inversion within such ement of an io set, including itself; for normal form, integers a chord an example of an ois. When, for instance, a second- are underlined and listed in ascending order. For example, inversion dominant-seventh chord is constructed on the bass the ois of <4{791}> is 0359. The value of the ois formulation (that is, initial) pc E, an initially ordered pc set results-the each of its six is apparent when calculations of multiplicative operations V4 chord in D major, shown in Example inn are made: unlike the transpositional combination of abstract permutations beginning with E, the first of which corresponds Tn-type sets, they will be shown to permit the generation of to io set normal form.8 pitch-class sets. At the same time, the initially ordered set derived from set A while eliminates the redundant calculations that result from dealing 7Thenumber of io sets that can be pc equals JA|, the number of ordered sets that can be derived from pc set A equals with ordered sets, since set than a dyad will have any larger IAl-factorial. The number of unique oiss that can be derived from any T,/ TnI-typeset is double the set's cardinality,divided by its degree of symmetry. 8The ordered pitch-class intervallic structurehas been formalized differ- pc set; (xyz) = Tn-type set; [xyz] = Tn/TnI-typeset, or set class; i
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Example 2. io set and ois in tonal theory Example 3. ois/pc constructions J a. ois < 5172} > =029 "J 029 0 6 = {368} Oil XJ J 'd 9 3 2 8 0 6 0 6 :Itoi r r f r b. ois < 7{25} > =07t D: V4 V43, V43 V43 V43 V43 07t 6 = {146} t 4 71 0 6 Unlike a Tn-typeset, any ois can be constructed on a pitch ? 6 class to form a single pc set. For example, while the note C could be a for one of three members of startingpoint (047)-- c.ois<5{72}> =029 forming major triads in which C is the root (C major), third 029 9 = { 69e } 96 (Ab major), or fifth (F major)-the construction of 047 on 2 e ? = 0 C results in C major. The construction of an ois on a fe !* S 9 only (0 9 given pitch class is signifiedby a circled multiplicationsymbol, 0 (a sign traditionally used for operations resembling arith- metical and is calculated in notation multiplication9), integer tiplication (AE,1; AE,2; AE,3; EA,1; and EA,2).10 The cal- the ois to the as shown by adding integers pitch-class integer, culation of these products is shown in staff and integer matrix in each matrix in 3. The construction of an ois on Example notations in Examples 4b through4f. In Examples 4b, 4c, and a class forms the basis of set pitch pitch-class multiplication. 4d, the oiss of each of the three io sets derived from mul- The of two sets consists simple multiplication pitch-class tiplicand A are constructed on each pc of multiplier E. In of the construction of the ois of one set, the mul- operand Examples 4e and 4f, the oiss of each of the two io sets derived on each of another set, the multiplier. tiplicand, pc operand from multiplicandE are constructed on each pc of multiplier The set the union of all from these comprising pcs resulting A. The different orderings of pcs on the staff are addressed constructions is called the The is product. operation again below; for now, products will be defined as unordered sets. 0. signified by While all of this seems straightforwardenough, there is 4a shows one of Boulez's illustrations of Example simple discomfiting inelegance lurking here. Perhaps the most ob- He two sets, labeled A and E, multiplication. gives operand viously inelegant feature is noncommutativity:A ? E $ E and the five that can result from their mul- products simple ? A. Further, Examples 4b, 4c, and 4d, collectively con- show and with identical Annual Meeting of the Society for Music Theory, Tallahassee, 1994). As with sidered, multiplicands multipliers figured bass, neither the AB set nor FB class includes the interval from r to pitch-class contents resulting in different products, as do 4e itself (part of the Tn-type modeling) necessary to the present formalization. and 4f. An io set's pc content is therefore less significant to 9Cohn differently defines A * B when A and B are pc sets. The use of Cohn's * will be retained for operations involving Tn-type sets. '?Boulez, Boulez on Music Today, 79.
This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions 76 Music Theory Spectrum multiplicationthan the ois it spawns;in fact, any of the twelve Example 4. Simple multiplicationof pitch-classsets transpositionsof the io sets in any of these examples will leave - each product unchanged, since every such transposition will a. From Boulez on Music Today: Trichord,dyad five products yield the same ois-a property called multiplicand redun- dancy. Based on the foregoing, the formula for the simple mul- A E tiplication of pitch-class sets is: where A and B are pc sets and B = {b,c,...m}, A ) B = (ois(A) ) b) U (ois(A) 0 c) U . . . (ois(A) () m). Of particular interest is the formula for fi.111 Ii . Xm I1iA t the simple multiplication of pitch classes: where a E A, b E AE AE,1 AE,2 AE,3 EA,1 EA,2 B, and r is the initial pc of A: a 0 b = i
1Ibid.
This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions Pitch-Class Set Multiplicationin Theory and Practice 77 pitch-spatialrepresentation of the products in Example 4a so The sequences forming the bulk of Slonimsky's Thesaurus that they reflect the property; perhaps this transpositional of Scales and Melodic Patternsconstitute perhaps the earliest equivalence led him to experiment furtherwith multiplication substantialand systematic applicationof simple multiplicative and ultimately to realize its potential. principles. Most of this speculative work (containing "a great number of melodically plausible patterns that are new"'3) SIMPLE MULTIPLICATION OF LINES involves the construction of line oiss on pcs within multiplier ordered sets that divide the octave into equal parts (and are There is an invaluable aspect to the superficiallynegative thus both transpositionally and inversionally symmetrical). property of multiplicand redundancy. The abstraction of an Example 5 shows two patterns from the Thesaurusthat will ois from a specific multiplicandpc set becomes possible, form- be discussed in terms of simple multiplication. ing one of the most important features of simple multipli- The theory of simple multiplication of pc sets is easily cation theory; many passages from a variety of composers can extended to construct patterns such as Slonimsky's; all that be described with methods derived from the foregoing. Un- is required is a modificationof the ois to account for linearity. like Boulez, for whom complex multiplication is essentially A combination of Rahn's "line equivalence class" notation'4 a precompositional technique generating unordered pitch- and ois notation will represent ordered pc intervals-again, class sets that will be given compositional order in a manner from an initial pitch class, but now defined as the first- that refers obliquely (at most) to their source, the composers occurring pc rather than the "bass"15-separated by dashes considered here have employed simple multiplication at the and underlined. These line oiss are multiplied by ordered pc musical surface in varyingdegrees of immediacy and technical sets by constructingthe line ois on each pc of the multiplier sophistication. Each case involves the construction of a mul- pc set, resulting in lines of pcs. In Example 5 the production tiplicand linear ordered pitch-class intervallic structure (line of the ascending line of Slonimsky's Pattern 196 is shown as ois) on pcs within a multiplier ordered set. The construction 0-5-9 0 <048> = 0-5-9-4-9-1-8-1-5, and the ascending of a line ois on a pitch class results in a line segment that can line of Pattern 395 as 0-5 ( <0369> = 0-5-3-8-6-e-9-2. be elaborated in several different ways. Line ois construction (An appropriatelyordered reading of the product in an ad- on a multiplier consisting of a repeated pitch class, resulting ditive matrix will display these lines. Further refinements of in repetition of the segment, produces an isomelos; if the line pitch and ordering would account for the octave repetition ois is constructed on a scalewise or arpeggiated multiplier, a and line retrograde that characterize Slonimsky's patterns, pattern is produced.12If consistent with respect to rhythmic and realization, an isomelos becomes more pitch-spatial spe- 13Slonimsky, Thesaurusof Scales and Melodic Patterns, i. cifically an ostinato and a pattern becomes a sequence. 14Rahn,Basic Atonal Theory, 139. 15Morris(in "Equivalenceand Similarityin Pitch") again provides a sim- ilar formalization,but one based on the lowest pitch within a segment rather '2This nonstandard usage of "isomelos" follows Vincent Persichetti, than the first-occurring.A pitch-class-basedtheory such as that presented here Twentieth-CenturyHarmony (New York: Norton, 1961), 217. The definition will trip over the "lowest pitch" requirement, especially when describing of "pattern"parallels Slonimsky's but subsumes his definition of "scale" as passages such as the first operation in the Stravinskyexample cited below, an interpolated progression (see note 16). Both terms are here extended to in which an initial pc is realized in pitch space three times as the lowest pitch include pitch class. and twice as the highest.
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Example 5. Slonimsky's Patterns Nos. 196 and 395 (Thesaurus of Scales ? 1947, 1974)
196 R L#II f f fHbFr#ir |I
0-5-9 ? < 048 > 395
0-5 ? < 0369 >
but the essential qualities of his system are sufficiently de- on pitch class C, every pattern contains (and, moreover, be- scribed through the present schema.) Ostinatos and se- gins and ends with) that pitch class, thus forming a C para- quences very frequently conclude with an incomplete state- tonality that is reinforced by the traditionally tonal associ- ment of the multiplicand line segment; this is indicated (in ations of the sequential gesture. later examples) in integer notation by a parenthetical final Unlike the union of io set ois/pc constructions, line ois/pc element in the multiplier, in staff notation by a dashed beam. constructions do not omit repetitions; Pattern 196, for in- As these patterns are lines, their characteristic use is me- stance, repeats pitch classes 9, 1, and 5. When repetitions lodic; in this context, their appearance becomes somewhat result from otherwise nonrepetitive operands, their appear- problematic. Multiplier replication is an obvious feature, ance can be predicted in the form of pairings. A single oc- particularly since the multipliers most extensively employed currence of a pitch class cannot be paired with itself. A by Slonimsky, the interval-cyclic referential collections twice-occurring pc, x1 and x2, forms one pairing, x1/x2. A <06>, <048>, <0369>, <02468t>, and certain permuta- thrice-occurring pc, x1, x2, and X3, forms three pairings, x1/x2, tions thereof, are so familiar.16 Since each multiplier is based x1/x3, and x2/x3. The number of pairings of n occurrences = (n2-2) + 2, so that four occurrences result in six pairings, five occurrences in ten pairings, and so forth. With one ex- division of the octave is for who con- 16Equal inescapable Slonimsky, ception, an interval class that appears in both operands will ceptualizes patterns as forming when pitches are temporallyinserted between result in one pairing in the product: 0-1 ( <01> = 0-1-1-2, pitches of a "progression"(his term for the multiplier), not constructedon 0-2 ( <02> = and so on until the ic a pitch or pc. In pitch space, these insertions are either between progression 0-2-2-4, exception, = pitches (called "interpolation"), below the lower progression pitch ("infra- 6, which results in two pairings: 0-6 0 <06> 0-6-6-0. polation"), or above the higher ("ultrapolation");the intervallicconsistency By extension, pairings in line multiplication can be predicted Pattern 196 of such insertions depends upon the cyclic progressions. exem- via the interval vectors of the operands: the sum of the ar- of two notes" and Pattern 395 "ultrapolationof one plifies "ultrapolation ithmetical multiplicative products of the quantities of like in- note." (Many of his patterns involve combinations of these techniques and are described by lengthy hyphenates [such as "infra-inter-ultrapolation"], appealing, no doubt, to Slonimsky's propensity toward what he called "ses- the use of asymmetrical multipliers, a technique employed frequently by quipedalian macropolysyllabification.")Examples 6c-e below demonstrate Boulez and well-explored by Cohn.
This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions Pitch-ClassSet Multiplicationin Theory and Practice 79 terval classes (with the product of ic 6 doubled) equals the As described above, the arithmetical multiplicative product number of pairings. Examples of the workings of the pairings of ic 6 must be doubled. Example 6d shows a tetrachord theorem follow. containing a tritone multiplied by itself (or "squared"). 0-1- In Slonimsky'sPattern 196, the interval vector of the mul- 5-7 0 <0157> = 0-1-5-7-1-2-6-8-5-6-t-0-7-8-0-2 tiplicand 0-5-9 is <0,0,1,1,1,0> and the interval vector of (nine pairings: 0/0/0, 1/1, 2/2, 5/5, 6/6, 7/7, 8/8): the multiplier <048> is <0,0,0,3,0,0>: <1, 1, 0, 1, 2, 1> <0, 0, 1, 1, 1, 0> x <1, 1, 0, 1, 2, 1> x <0, 0, 0, 3, 0, 0> 1+1+0+1+4+1x2 = 1+1+0+1+4+2 = 9 0+0+0+3+0+0 = 3 Just as other common-tone theorems17have their limita- Interval class 4 appears once in the multiplicand-i(-9) - tions, this one will predict neither the types of pairings (e.g., and thrice in the multiplier-i(0,4), i(4,8), and i(8,0)- whether three pairings are realized as three twice-occurring resulting in three pairings, 1/1, 5/5, and 9/9. pcs or as one thrice-occurringpc) nor, except as noted below, Example 6 shows five operations that further demonstrate the cardinalityof the unordered pc set comprising the union the pairings theorem as well as staff-notational conventions. of resulting line pcs. Example 6e inverts the multiplier of the Multiplicandpcs are joined by beams closer to the noteheads, previous example, which does not change that operand's in- while the more distant beams connect multiplier pcs. In Ex- terval vector: 0-1-5-7 0 <0267> = 0-1-5-7-2-3-7-9- ample 6a, 0-4-8 0 <159> = 1-5-9-5-9-1-9-1-5 (each 6-7-e-1-7-8-0-2. Nine pairings still result (0/0, 1/1, 2/2, of the three pcs occurs three times-1/1/1, 5/5/5, 9/9/9-for 7/7/7/7), but the cardinalityof the resultant unordered set is a total of nine pairings): greater by one. <0, 0, 0, 3, 0, 0> The theorem does predict, however, that two operand sets x <0, 0, 0, 3, 0, 0> A and B with no interval classes in common will produce a 0+0+0+9+0+0 = 9 set with a cardinality equal to IAIx IBI,since no pcs are was demonstratedin 6b shows with no interval classes held in duplicated. (This Example 6b, a product Example operands of 4 x 3 = when one common. 0-1-3-6 ? <048> = 0-1-3-6-4-5-7-t-8-9-e-2 cardinality 12.) Further, pairing is predicted, it must be realized as one so (no pairings): twice-occurringpc, that IA ? BI = (IAIx IBI) - 1; when two pairings are 1, 1> <1, 1, 2, 0, predicted, they must be realized as two twice-occurringpcs, x 0> <0, 0, 0, 3, 0, so that IA 0 BI = (IAI x IBI)- 2. Conversely, when IAI x 0+0+0+0+0+0 = 0 IBI > 12 (as is the case when one tetrachord is multiplied by Example 6c shows operands with several interval classes held another), the operands must have at least one interval class in common. 0-1-2-5 0 <014> = 0-1-2-5-1-2-3-6-4- in common.18 5-6-9 (four pairings: 1/1, 2/2, 5/5, 6/6): 0> '7For examples, see Rahn, Basic Atonal Theory, 97-123. <2, 1, 1, 1, 1, an of the of a tetrachord x '8Thisprovides explanation impossibility lacking <1, 0, 1, 1, 0, 0> both ic 3 and ic 6, since [0369], the sole tetrachord limited to two interval 2+0+1+1+0+0 = 4 classes, contains only those; the multiplication of [0369] by a hypothetical
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Example 6. Other line multiplications
b. c.
0-4-8 ? < 159 > 0-1-3-6 ? <048> 0-1-2-5 ? < 014 >
d. , e. _fe1X_ I,_ bh ?-_^ >.. r 'r^f-^^^-TT II . ^-^ * ^ ^ F bH t r rI 0 --r I < r1I57, r I I I I 0-1-5-7 ? < 0157 > 0-1-5-7O-1-5-7 ?M) < 0267 >
Another problematic area in line multiplication is that of shown in Example 7: the upper line is a brief oboe passage the sequence itself: at what point does a pattern become from Witold Lutoslawski's Concerto for Orchestra (1954), overly predictable? Tonal practice would suggest that Pattern and the lower line illustrates the two multiplicative operations 196 be broken off after the seventh note, the start of the third 0-2-5-3 ?
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Pitch-Class Set Multiplicationin Theory and Practice 81
Example 7. Lutoslawski,Concerto for Orchestra,mm. 471-478, oboes; analysisof line multiplication
471 ^r ^t^errrr r yIF IIF- r il-gT^TrF - I "i^LT-11 _ , _ ^ :*r ir e 1
0-2-5-3 ? < tel >
O-e-t? < 174 >
of these employs the line ois 0-3-e-2 (established in the first pitches in this first operation do not change octaves, but the movement in various realizations20)multiplied by a repeated successive statements of the multiplicand are varied rhyth- pc C which, after an opening C5, alternates between C5 and mically. The second operation, more obviously based in pitch C6-a simple yet effective disguise of an isomelos. The other space but disguised by ingenious rhythmic displacement, ro- tates the multiplicandof the first to the equivalent 0-8-e-9,21 20Accordingto Stravinsky,"The subject of the fugue was developed from and multiplies it by a descending-wholetone tetrachord, the sequence of thirds already introduced as an ostinato in the first move- <31e9>. ment," a described as "the root idea of the whole figure previously sym- Another analysis of is shown in Ex- phony." Igor Stravinskyand Robert Craft, "A Quintet of Dialogues," Per- Stravinsky's subject spectives of New Music 1/1 (1962): 16. Statements of this ostinato in the first ample 9. In contrast to the previous analysis's note-by-note movement occur most prominentlyat rehearsalnumbers 4 (oboes and English horn), 7 (oboes and English horn, with cellos and basses in augmentedrhythm 21Equivalentoiss are those which, when constructedon a pc, produce sets a tritone away, completing an octatonic collection), and 12 + 3 (like 7, but of the same T,-type. They are calculated by subtractingeach ois integer from harp replaces double reeds which move to 0-3-2-5, the other octatonic all ois integers (see Heinemann, "Pitch-ClassSet Multiplicationin Boulez," tetrachordalsegment in a rotated similarshape). Given its "root idea" status, 29-30); for example, all the oiss equivalent to 023e, in their normal forms, the motivic line ois 0-3-e-2 is curiously absent from the third movement of are: the Symphony of Psalms (the first-composedof the three), but the composer 023e - 0 = 023e; regarded the ois as having been "derived from the trumpet-harpmotive at 023e - 2 = 019t; the beginningof the allegro"of the thirdmovement (ibid.), an ois of 0-3-1-5, 023e - 3 = 089e; equivalent to 0-4-e-2; see note 21. The significance of both motives' set 023e - e = 0134. classes [0134] and [0135] to other Stravinskyworks is explored in Joseph N. In any such listing, one of the equivalent oiss (or more than one, in the case Straus, "A Principle of Voice Leading in the Music of Stravinsky," Music of transpositionallysymmetrical sets) will match the integer content of the Theory Spectrum4 (1982): 106-24 passim. common T,-type-here (0134).
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Example 8. Stravinsky, Symphony of Psalms, II, mm. 1-5; analysis of line multiplication
[Ob.] -
mf
0-8-e-9 ? <31e9>
1L
0-3-e-2 ? < 0000(0) >
Example 9. Stravinsky, Symphony of Psalms, II, mm. 1-5; analysis of compound-melodic line multiplication
0-1-t-e
0-3-2 < 000o0> ? 0-e-t-9 ? <3< 3e >
This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions Pitch-Class Set Multiplicationin Theory and Practice 83 approach, this one is influenced by pitch register and regards Example 10. Compound multiplicationin Boulez on Music the subject as a compound melody in which four separate line Today multiplicationsare found.22These operations are shown with the analytical beams and stems above the upper staff and is preserved between the staves. In both the upper and lower below the lower, again with the line oiss beamed closer to and 1 i L - the multiplierfurther from the noteheads; the original rhythm voices, the first and second operations overlap. Especially evocative here is the upper voice's second multiplication, -& I 1n 2 T I - ? HI L l where the operation O-1-t-e 0
The second type of operation, compound multiplication, b 'x X 4 L.l L,l . 1 i will be discussed only briefly because it appears to represent t~~t I-'L " b an intermediate stage in Boulez's thinking and is not par- ticularlyimportant to Le Marteau.This operation differs from simple multiplicationin that the simple multiplicativeproduct and these are is transposed according to some schema. The example that though multiple variable, complexes deduced from one another in the most functional in that Boulez of this is shown here as 10, which he way possible, they obey a gives Example coherent structure.23 describes as follows: logical, There are several different ways in which this can If the ensembleof all the complexes[pc sets in staff 1] is multiplied figure be but, in initial by a given complex[the boxed pc set in staff 1], this will resultin interpreted, any case, ordering, conceptually with one a seriesof complexesof mobiledensity [i.e., cardinality],of which, problematic just operand, is now required of both. in addition,certain constituents will be irregularlyreducible; al- The initial pitch classes of two of the sets are represented here as invented note-heads-circles with dots inside. Transposi- tion is determined 22Thiscompound-melodic subject has also been presented as an example by intervals between these initial pitch of octave displacement, a "modernization"of a more traditionally narrow- classes, as shown at the right of the sketch. The advantage ranged melody. See, for example, Harold Owen, Modal and Tonal Coun- of this operation is that multiplier replication is no longer an terpoint (New York: Schirmer, 1992), 346. Such an interpretation could be supported by the closely spaced treatment of the subject in the second flute, 23Boulez, Boulez on Music Today, 39-40. Sets inside the dashed box in mm. 16-17, but this is easily accounted for: the lower melody (as analyzed Example 10 are implied but not literally given by Boulez in the original, and here) dips below the range of the flute and is accordinglyraised an octave. the staff enumerationshave been added here for reference purposes. Example Owen observes that "much of the characterwould be lost" in narrowingthe 10 is discussedin greaterdetail in Heinemann, "Pitch-ClassSet Multiplication contour. in Boulez," 50-56.
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84 Music Theory Spectrum
issue; the multiplier, shown in this example as a boxed set Example 11. The Marteau row; derivation of V-sets in staff is not a subset of the 1, necessarily product. However, Domain Partitioning other pitfalls of simple multiplication remain: the operation P VDA VVDB VDC VDD VDE is noncommutative; variety of products is evident from the 1 2/4/2/1/3 " " lo ? 4 different staves, and there is no rationale for preferring a product in one staff to a product in another; and initial or- dering, problematic with only one operand, is now required VCA VCB VCC VCD VCE of both. of these the sketch in Regardless problems, Example 2 4/2/1/3/2 r I , s0.- * | |. | 10 was probably vital to Koblyakov's detective work: staff 1 consists of sets partitioned from the row that is the basis for the first cycle of Le Marteau, while staff 2 contains mul- VBA VBB VBC VBD VBE
that in the itself, as 0 tiplicative products appear composition 3 2/1/3/2/4" 0 I, "'" I I ~ I will be exemplified presently.
COMPLEX MULTIPLICATION A VAA VAB VAC VAD VAE 4 l I^ l" I V1/3/2/4/2 1| | I I ", As a prelude to discussing complex multiplication, it is necessary to delineate some organizational principles in the first cycle of Le Marteau sans maitre as distilled and modified A VEA VEB VEC VED VEE from The is based on the twelve- Koblyakov's writings. cycle 5 3/2/4/2/1 * b _~? I-tI1 I ?O"1 0 j I0 \\ I tone row 3521te908476 that is partitioned into five subsets with cardinalities of, in order, 2/4/2/1/3. This cardinality series is rotated while the row itself remains constant, and each such rotation splits the row into different subsets. The process of cardinality series; Y represents the first through fifth position partitioning by means of the rotating cardinality series is il- of the V-set within the cardinality series. V-sets are employed lustrated in Example 11. The partitioned subsets are called to generate pitch-class domains, precompositionally deter- V-sets and are represented generally by the three-letter des- mined groups of pc sets from which is drawn the pitch-class ignation VXY: the V simply indicates that the set is a V-set;24 material of the composition. The five rotations of the car- X represents the first through fifth position of the monad in dinality series each produce V-sets that in turn generate five partitioning, and thus varies depending on the rotation of the domains: V-sets from the original series generate Domain 1, V-sets from the next rotation generate Domain 2, and so 24The"V" designation follows Koblyakov, "Analysis of pitch structure." forth. Within the VXY designation, X and/or Y can be made He the of V-sets to a in PierreBoulez: changes designation bracketingsystem specific by replacing either or both with letters A through E; a World of Harmony, but this change appears to be based on the incorrect for instance, as seen in VXA is the first V-set notion that transpositionsof V-sets supersede V-set function in Le Marteau. Example 11, = See ensuing discussion and Heinemann, "Pitch-Class Set Multiplication in in any domain (since Y A, the first position in the car- Boulez," 63. dinality series), VCY is any V-set in Domain 2 (since X =
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Example 12. Domain matrix (adapted from Boulez on Music does not describe. His transposed V-sets do occur; they can Today) be seen in each domain in the row that has the same A through E identificationas the X in each VXY-for instance, VXA VXB VXC VXD VXE in Domain 1, field D is a transpositionof the Domain 1 VDY sets. However, these "transposed V-sets" are actually them- AA AB AC AD AE selves products of complex multiplication. The delineation of Example 10 and its compound multi- BA BB BC BD BE plicative method introduced a new consideration: the pos- that a within a can have CA CB CC CD CE sibility pc multiplicand importance beyond its ois-producing function. In Example 10, pc 8, the DA DB DC DD DE initial pc of the multiplicand, was paired with each element of {235} to determine transposition levels for each set in lines EA EB EC ED EE 2 through 4. An extension of this consideration might include the use of a transposition-determiningpc which remains con- stant and a part of the operation even if it is not an element C, the position of the monad in the 4/2/1/3/2 rotation), and of eitheroperand. Not coincidentally, this is exactly the "miss- VCA refers specifically to the first V-set in Domain 2. The ing link" in the theory of multiplication, the crucial compo- generalized structure of a domain is shown in Example 12, nent that disposes of every problematic area of simple and wherein V-sets are shown at the top of the matrix. The sets compound multiplication. This component is called the below these, each represented by a pair of letters, are called transposition-determining constant-a pitch class, chosen ac- domain sets; these sets are in fact products formed by the cording to some schema, that participates in a multiplicative multiplication of V-sets. Their two-letter designations are operation. The transposition-determining constant, desig- taken from the Y of each of the two VXY sets that generate nated k, may or may not be an element of the multiplicand, them: for example, domain set DC is produced by the mul- multiplier, or product. tiplicative operation VXD 0 VXC. The five actual pitch-class Complex multiplicationcombines the techniques of simple domains themselves are shown in Example 13. The realiza- and compound multiplicationinto an operation that is elegant tion of these sets in the musical surface can be traced through in its economy and constancy. This operation takes a product the domain matrix in a fairly systematic way that has been of simple multiplication, here called "simple AB," and trans- thoroughly explained by Koblyakov. The italicized sets-sets poses it via a method that seems likely to have evolved from AA through AE in the top row of the domain matrix-are compound multiplication. The formula for complex multi- not used in the composition, for reasons explored below. plication follows. I contend that all of the domain sets in these pitch-class WhereA and B are sets, r is the initial classof A, domains are generated by the V-sets by means of complex pitch-class pitch and k is the transposition-determiningconstant: multiplication.This position differs significantlyfrom that of Koblyakov, who believes that domain sets are generated from A 0 B = ois(A) ( B = simple AB; Tn(simple AB) = AB, where transpositionsof the V-sets according to some operation he n = i
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Example 13. Pitch-class domains for the first cycle of Le Marteau sans maitre
(C =0)
A B C D E A B C D E VD 135} 1tel2} 1901 18} {467} VC 1235} {te} 191 {0481 167} A {135} {89te012} {79t0} {68} {24567} A {01234568} {9teO12} {89t0} {345789e0O} {56789t} B 189te0121} 3456789te } 235689} 11245} 19te01234} 2 B (9te012} {678} {56} {014589} {234} C {79t0 {235689) {147} {03} {8tel2} C {89tO} 56} 14) 137e} 112} D {68} 11245} 103} le} {79t} D {345789e01 1014589} {37e} {26t} {014589} E {24567} 19te01234} {8te12} {79t} {356789} E {56789t}1 234} {12} 1014589}) teO}
A B C D E A B C D E
VB 135} {2}1 tel} {90} 14678} VA 13} 1125} {te} 18904} 1671 A {135} {02} {89tel} {79t0} {245678} A [3} {125} {te] {8904} [67} 3 B 102} {e} 178t} {69) {1345) 4 B 125) 1e01347) {89t01}) 678te23) {4,5,6,8,91 C {89tl} (78t} 1345679} {23568} {9te01234) c {te} {89t01 1567) {34578e0} 11,2,3} D {79t0} {69} 123568} {1471 {8te0123} D {89041 {678te231 {34578e0) {123569t} {e013478) E {245678} {1345} 19te01234} {8te0123} {356789te} E {67} {45689) 1123) {e013478) {9te)
A B C D E
VE {235) {tl} {89e0) 147) {6) A {234568} {tel24} {89te0123} {4578t} {679} 5 B {tel24} 1690} {4578te} {036} {25} c 189te0123} {4578te} {23456789t } te1245} 101341 D 14578t} {036} {te1245) 1690) {8e) E {679) {25} 10134) {8e) {t}
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The operation is thus a two-step process: first, the simple be transposed by the calculated interval prior to simple mul- multiplication of the multiplicand ois by the multiplier, as tiplication in order to arrive at the same product. described previously; and second, the transposition of The transposition-determiningconstant can be chosen by the simple multiplicative product by the ordered pitch-class any means the composer deems appropriate. In Le Marteau, interval from the transposition-determiningconstant to the a single constant governs each domain, chosen according to initial pitch class of the multiplicand. this criterion: the transposition-determiningconstant is the Example 14 illustrates complex multiplication and, for rightmost of the first three pcs of the row appearing in set sake of comparison, employs the same operands as in VXA, the firstV-set partitionedfrom the row in each domain. Example 4. Any pitch class could have been chosen as Thus (as can be inferred from Example 11) in Domains 1 and the transposition-determiningconstant; F is used here. The 3, k equals F; in Domains 2 and 5, k equals D; in Domain transposition-determiningconstant is shown first, followed by 4, k equals Eb. the ordered pitch-classinterval from k to r; next is r, followed There is a significant parallel between simple multiplica- by the simple multiplication. Finally, the simple multiplica- tion and complex multiplication. In the former operation, tive product is transposed by the calculated interval to be- there is multiplier replication-the property that the multi- come the complex multiplicativeproduct. An examination of plier is always a subset of the product. In complex multi- Example 14 will demonstrate that, given two operand pc sets plication, there is the propertyof k-setco-operand replication: and a transposition-determiningconstant, only one product if one operand is a "k-set"-that is, has k as an element-the is possible. This operation is commutative. Unlike in simple other operand will be a subset of the product. (Because com- multiplication, the pitch-class content of the multiplicand is plex multiplication is commutative, either the multiplicand now as vital as its ois, since the initial pc is a part of the or the multiplier may be a k-set-hence the use of the term transposition determination. Initial ordering of the multipli- "co-operand"here.) This property provides a reasonable ex- cand remains vital to the operation, but exactly which initial planation for Boulez's jettisoning of field A in every domain. ordering is chosen is irrelevant, thus rendering moot the ne- Using the V-sets along with the domain sets establishes a cessity for selection criteria. As in compound multiplication, literal connection with the original row and is thus concep- except as noted below, multiplier replication is not an issue; tually sound. Since k is an element of each set VXA, these in fact, in Example 14, the operands and the product do not V-sets are necessarily subsets of the corresponding field A intersect at all. In each operation shown, the transposition domain sets; eliminating the domain sets avoids unequal changes for the simple multiplicative product to result in the weighting of the V-set pitch classes. sole complex multiplicative product, with the transposition- Simple line multiplicationproduces inherently ordered pc determining constant acting as catalyst-not an element of sets. In contrast, the five complex multiplicative operations the operands, but allowing the operation to take place. shown in Example 14 show four different orderings of the In Example 14, the integer matrices to the right of the staff same product pcs. If the operand spacings are changed, still notations use transpositionsof the multiplier. These matrices other orderings are possible. Each product pc can thus be demonstrate the complex multiplication corollary: A ? B = shown in any order relative to every other product pc. Each ois(A) 0 Tn(B) = AB where n = i
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Example 14. Complex multiplication: where k = 5, {7t0} ) {69} = {8e124} k r r 14a. < 7{tO}> ? {69) = {8e124) |I i(k,r) i(k,r)1 5 14 ik, H = n u- -, 8 e =^R2---^2-og0- 0 8e ?w 2---I_
14b. < 0{7t1 > ? {69) = (8e1241
t 2 f |e
- T5 11 2 14 14c.
14d. < 6(91 >? (7t0 = (8e124} 3 e24 -- = bS 1_ u u-- --= 1 Lg $ iW-- 4, as Tl #11 0 8e I 1 ( 8e I
14e. <9(6} > {7t0} = (8e124)
-- --= 9 8e 1 ? o u bo u= bT4 0 e24 L------4 - - 4I e24
This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions Pitch-Class Set Multiplicationin Theory and Practice 89 preference. Complex multiplicative products are inherently than Le Marteau(and subsequently withdrawn) entitled Ou- unordered sets by nature of their generation, and this may bli signal lapide.27These pitch-class domains are shown in have been one of the operation's most appealing character- integer notation in Example 16. An examination of these istics to Boulez. It is even possible that his compositional domains reveals many parallels with the theory already pre- move toward what he termed "local indiscipline . .. a free- sented, including but not limited to the following. First, the dom to choose, to decide and to reject"25was a direct result generating row is the inversion of the Marteaurow. Second, of this operation. the cardinality sequence partitioning this row is the retro- The assumption has been implicit, but is, in fact, Boulez's grade of that of Le Marteau. Third, the A field is again re- own process accurately portrayed here? He alone could say placed by V-sets. Fourth, the transposition-determiningcon- with complete certainty, but it should be noted that, given stant is now the leftmost pc of set VXA-in other words, k the method for determining V-sets, the complex multiplica- equals El, throughout. Fifth, while the procedure for select- tion operation described herein, and the criterion for deriv- ing k has been altered slightly, the complex multiplication ing the transposition-determiningconstant, the domain sets operation itself is exactly the same as that described for Le shown in Example 13 are the only possible products; the cor- Marteau, under which only these domain sets will result; the respondence is complete. It has also been demonstrated that correspondence is again complete. the process can be carriedout using tools available to Boulez, By focusing on individual pitch classes within each oper- which would have included staff notation but presumably and, we can prove the assertions that complex multiplication not integer notation. Similarly, while the process is complex, is commutative and generates a single product. Recall that, it is not especially complicated-that is part of its elegance. in simple multiplication, a ? b = (a - r) + b. In complex Unfortunately, Boulez's sketches for Le Marteau have been multiplication, this product is transposed by the ordered lost-we must look elsewhere for further confirmation. pitch-class interval from k to r. Therefore, in complex mul- Example 15 shows a sketch from the Paul Sacher Foun- tiplication, dation's Pierre Boulez Collection; a reproductionof this page a 0 b = ab) appears in the tenth-anniversary Festschrift of the Arnold Ti
This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions ______
- I_
!L4,1 : i-% t 1 .''^N^--*
. . _
* . -UU ^-i*^*^j,~ ', I _I~~~~~~~I --.-,.,.: ti -. ?.1._ ' 'M " '1 1 I l r 4 ';gttA -: 4 mL- 4Z7t. t?) n t4tl't-~~~~~~~4'i I J i ap!dvl jlnuSs ilqno o10 surmuop ssup-qoi!d jo uoilwou s,zalnog 'I ajlduexg uwnJlads Ajoeql o!snlA 06 This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions Pitch-Class Set Multiplicationin Theory and Practice 91 Example 16. Pitch-class domains for Oubli signal lapide (C = 0) A B C D E A B C D E VB 1134} 15 {(78} 169t2} leO) VA 13} 114) {57891 16t} {e02} A {e12345} {356} {56789} {6789te0234} {9teOl} A [3} {14} {5789] {6t} {e02] 1 B 13561 171 19t} {8e041 121 B ( 14}) e25} {356789t}* 1478e) 19t013} C 156789} {9t} l{eO} {te12367} 1345) C {5789} 1356789t} 179te0123 ) 8te0234 {12345678} D 16789te0234} 8e04} {te12367} 10134589 123567te D 16t} 1478e} 18te0234} 1159} 1235679} E {9te01 } 121 (3451} 23567te 1 {789) E {e02} 19t0131 112345678} 1235679) 1789te } * shownas 1456789t1in sketch A B C D E A B C D E VE 113} (4578} {69} {te2} 10} VD 11345} 178} [69t} {21 {e0} A {el3} {2345678} {4679} [89te02} {t} A {e1234567} {56789t} 146789te0} {0234} [9teO12} 3 B 2345678} 156789te01} 178te12} {e0123467} 11245} B 156789t} le0ll {tel23} (67) 1345} C 146791 {78te12} {903} 112458} {36} C 146789te0}) te123) 19013451} 589} {23567} D 189te02} {e0123467} (12458} 15679tl} 178e} D {0234} (67} {589} 11} Ite} E {tO} 11245} {361 178e} 19} E {9te012} 13451 {23567} {tel 1789) A B C D E VC 1131 1458} 17} {69} {te02} A {el3} {234568} [57} {4679} 189te02} B {234568} 15679tl }t 890} 178te2 e0123457} C {57} 1890} {e} {tl} 12346} D {4679} 178te2 {tl} {903} {1234568} { 89te02 } e01234571 2346 } 12345681 56789te1 } t shownas 135679te0}in sketch This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions 92 Music Theory Spectrum simply add the operand integers and subtract k.) The only Example 17. Le Marteau:Interval class 3 in Domain 5 V-sets variable in the operation, the initial pc of the multiplicand, _ has canceled itself out, leaving just the constants-the op- A,L~4?-t J;;I erand pitch classes and the transposition-determiningcon- stant. VEA VEB VEC VED VEE COMPLEX MULTIPLICATION AND THE MUSICAL SURFACE or [036], or are segments of an octatonic scale; as such, they An examination of the domain sets and their realization are easily perceptible. in the first movement of Le Marteauwill reveal the outlines Well-documented octatonicism in Stravinsky provides an of a process-based listening strategy. Two sections of the introduction to an analytical method that can be applied to movement use Domain 5, generated by the V-sets shown in certain passages in Boulez despite the obvious dissimilarities Example 17. Occurrences of interval class 3 are prominent in musical surfaces. Example 20 shows another approach to here, as their beaming demonstrates. Since multiplication the Symphony of Psalms fugue subject, one that emphasizes preserves the unordered intervals of each operand, we can the interval-class 3 content and octatonic subset attribute expect interval class 3 to be as prominent in the domain sets, of the motivic set class [0134]. Here, the segmentation of sets and so it is; see the complete representation of Domain 5 in is indicated by barlines, and rhythms are preserved graphi- Example 18, where interval-class 3 relations are bracketed. cally by relating pitches to the evenly-spaced eighth-notes The pitch classes that are isolated in this regard are shown below each system, where measure numbers will also be as open noteheads. (The V-sets and field C in Domain 5 also found. The two mutually interactive listening strategies are appear in Example 10 above as, respectively, staves 1 and 2.) illustrated here: interval-class 3 is shown with beams as in Furthermore, all except two of these domain sets are also Example 17, and the octatonic analysis is shown above each subsets of the octatonic collection-or, to be more specific, system by a thick line indicating the significant appearance of one or two of the three octatonic pitch-class collections. of a collection. The compound operation T(t+ +t{e023} = The roman numerals in Example 18 refer to these collections T6{e023} = {5689} results in the completion of Collection II. as labeled by Pieter van den Toorn, shown in Example 19.28 Example 21 is a two-staff reduction of the first movement Moreover, these subsets are either of the Tn/TnItypes [03] of Le Marteausans maitre, mm. 11-20. The notational con- ventions of Example 20 obtain here. Only the interval-class 3 within domain sets are but others are 28Pietervan den Toorn, The Music of Igor Stravinsky(New Haven: Yale relationships given, University Press, 1983), and Stravinskyand The Rite of Spring: The Begin- easily found-for instance, Ft5 in the fourth set is repeated nings of a MusicalLanguage (Berkeley: University of CaliforniaPress, 1987). in the fifth, where it is associated with A3. The octatonic Since van den Toorn does not describe how he chose this taxonomy, it is analysisis more fluid than in the Stravinsky,because a change interesting to note that Collections I, II, and III can be constructed respec- of sets does not indicate a of collection. on and 3 with an ois that the necessarily change tively pcs 1, 2, duplicates Tn-type: Collection III can be as an Collection I: 0134679t ) 1 = {124578te} safely ignored important orga- Collection II: 0134679t 0 2 = {235689e0} nizing factor. The realization of set CC in m. 15 is particularly Collection III: 0134679t 0 3 = {34679t01} evocative: this chromaticnonachord is the only set employed This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions Pitch-Class Set Multiplicationin Theory and Practice 93 Example 18. Le Marteau: Interval class 3 in Domain 5, staff notation, with analytical markings A B C D E ,II I,III II I,III II,III VE - NI~~~~~~~~~~~~~~~~~~~~1 B ~~~II 7~~~~~~~~~ ~~~~~~I III C h II~~1,111 I # 1,111 1,11 D TIII 1,11 III 1,11 1,111 E *NI #.Tj. l'o Example 19. Octatonic pitch-class collections (van den Toorn's technique reminiscent of twelve-counting-is shown in Ex- taxonomy) ample 22. Examples 23 and 24 provide the same kinds of analysis of mm. 53-60; these measures are even more than those of 21. It Collection I straightforward analytically Example " ?0 fl- ?0 0- 0 c0 0 $00to should be noted, however, that the other domains do not parse as easily as Domain 5 and that such an analytical ap- Collection II proach is not without its obstacles. The aural "processing" in terms of interval-class 3 and octatonic structuring is com- " plicated by tonal and neotonal associations in the Stravinsky Collection III 41 z bo k?? , bo ?r ff 43p?it"tt? and by the sheer rapidity of change in the Boulez. Never- theless, these analyses are firmly rooted in the pitch-class content of each passage and constitute a natural outgrowth here which is not an octatonic subset, yet it is composed in of the processes of their composition. such a way that it continues the previously-established Col- Boulez chose to use multiplication in other works of this lection I for its first four pitches; when the foreign Elb is era; perhaps now this choice is understandable. Complex introduced, the remaining five pitches are all contained in multiplication is an elegant, logical extension of the serialist Collection II. The sets appearing here which are shown as tradition that provides the composer with a great deal of Collection III subsets are composed in a similar manner. flexibility at the local level. The resulting pitches are inter- Koblyakov's matrix tracing of this passage--an analytical related, but row-counting as a listening strategy is completely This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions 94 Music Theory Spectrum Example 20. Stravinsky, Symphony of Psalms, II, mm. 1-5; octatonic/ic-3 analysis I II III 1 -t t f - I -f __I-a . 1 T L,r r _r rylrII--. L;-I--- , I I -1 Tt Tt Tt e023) -* 19t01 - (178te} {- 15689} = 60 1 Ar r r r r r r r r r r r r r r r r r r I 2 3 4 5 inappropriate. The technique is individualistic, and Boulez progression and refinement of the vocabularies of musical was obviously no proselytizer for it. His work thus invites us expression.30 The reach has often exceeded the grasp but is to take it on its own terms-yet we can meaningfully bring all the more valuable for that; through this process of con- to it our own experience as listeners. As such, it represents tinual redefinition, the grasp has grown, not able to collect the best aims of the most significant music of our recent past. everything but better able to deal with that which has been Composer-percussionist William Kraft, who performed in collected. the American premiere of Le Marteau sans maitre (11 March 1957 in Los Angeles, Boulez conducting), wrote of the ex- ABSTRACT set in man- perience: This article examines pitch-class multiplication its various ifestations, formalized as simple, compound, and complex multi- and the The newness and originality of Boulez's style, consequent plication. Simple multiplicative techniques are related to their or- me to ask him challenge it placed on the average listener, prompted igins in sequence and ostinato, and different technical approaches His answer if he was at all concerned about public acceptance. was, by Slonimsky, Lutoslawski, and Stravinsky are exemplified. Com- think it will take?" to which "Yes." I went on, "How long do you plex multiplicationis posited as an elegant, uniquely Boulezian tech- he replied, "Eighty years."29 nique for generating unordered pitch-classsets. An analysis of sets within Le Marteausans maitre demonstrates Given the current state of "public acceptance," even this process-basedlistening strategies. estimate might seem overly optimistic. But we have, as of this writing, just passed its midpoint. Composers who challenge our cognitive capabilities and composers who stay within the limits of these have led an often putative capabilities uneasy 30Theseissues are confrontedin detail with respect to Le Marteauby Fred coexistence throughout the history of Western art music; Lerdahl, "Cognitive Constraintson Compositional Systems," in Generative perhaps the irreconcilability itself has incubated both the Processesin Music: ThePsychology of Performance,Improvisation, and Com- position, ed. John Sloboda (Oxford:Oxford UniversityPress, 1988), 231-259, and by Ciro G. Scotto, "Can Non-Tonal Systems Support Music as Richly as the Tonal System?" (D.M.A. diss., University of Washington, 1995), es- 29FromPierrot to Marteau, 55-57. pecially in Chapter 1. This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions Pitch-Class Set Multiplication in Theory and Practice 95 Example 21. Le Marteau sans maitre: first movement, mm. 11-20 Accidentalsapply only to notes they immediatelyprecede. I II III __ _ m :& VAL I |~~~~~~~~~~~~~~0# _*jb L 0_t Tff l^-c^~ry^^^I^1- aJ I KJyI -I 9: 1t t I^cj r, ^ j^ DE CE + BE VEEDD ED DC CC BC DB CB VEB DA EA VEC BB J=2o8g r r r rfr r r r r rAr r r r r rlr r r r r r r r rrrL r r r r r r r r r rr r r r r r 13 14 15 16 17 18 19 11 12 20 ----- poco rit. Example 22. Koblyakov's matrix tracing of mm. 11-20 (Pierre Boulez: a World of Harmony); original uses lower-case letters VEA VEB VEC VED VEE T /T T BA BB BC BD BE T / T/ T CA CB CC CD CE DA DB DC DD DE EA EB EC ED EE This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions 96 Music Theory Spectrum Example 23. Le Marteau sans maitre: first movement, mm. 53-60 Accidentalsapply only to notes they immediatelyprecede. I II III '* _ b;- >~1O' ~[ vL II^l[ I _I ? , CI 1tj ' VEE BE BD CD DD ED BC VEC EB DB CB e BB VEA BA =28 r r r r r r r r r r r r r r r r r r r r r r r g r r r r riz 53 54 55 56 57 58 59 60 pr-esser ------*D5 shown as E5 in score Example 24. Koblyakov's matrix tracing of mm. 53-60 (Pierre Boulez: a World of Harmony); original uses lower-case letters VEA VEB IVEC VED VEE , \ T I BA BB BC BD <- BE T / CA CB CC CD CE T \ DA DB DC \DD DE \ EA EB EC ED EE This content downloaded from 129.120.93.218 on Wed, 3 Sep 2014 00:53:19 AM All use subject to JSTOR Terms and Conditions