Reexamining PC-Set Multiplication, Complex Multiplication, and Transpositional Combination to Determine Their Formal and Functional Equivalence

Abstract

REEXAMINING PC-SET MULTIPLICATION, COMPLEX MULTIPLICATION, AND TRANSPOSITIONAL COMBINATION TO DETERMINE THEIR FORMAL AND FUNCTIONAL EQUIVALENCE CIRO SCOTTO 1. WHAT IS MULTIPLICATION? ULTIPLICATION IS A COMPOSITIONAL TECHNIQUE invented by Pierre Boulez that generates collections of pc-sets. As we see, the multiplicative process takes a pair of source pc-sets as inputs or arguments and produces additional pc-sets as the output or product of the operation.1 But Boulez’s own presentation of multiplication in his book, Boulez on Music Today, does not completely define or explain M PC-Set Multiplication, Complex Multiplication,Transpositional Combination 135 multiplication. Example 1, which reproduces Example 33 from Boulez’s book, contains a pair of input pitch-sets (or p-sets) and five psets that are the product of the operation.2 p-sets a = {7, 0, T} and e = {6, 9} are the source or input p-sets, and from them multiplication generates five output p-sets labeled (a, e, 1), (a, e, 2), (a, e, 3), (e, a, 1), and (e, a, 2).3 Although the example contains source and product p-sets, it does not explicitly define the operation that generates the product p-sets from the source sets. Boulez provides an abstract group multiplication table in an earlier example that represents the formal structure of multiplication, but the table does not reveal the operation or its structure. Nevertheless, certain relationships are discernible from the input and product p-sets. For example, the product p-sets (a, e, 1) through (e, a, 2) are all pitch transpositions of each other. In fact, product set (a, e, 1) almost contains a pitch transposition down a semitone of input p-set a (see Example 2 and compare with Example 1). Although replacing the pitch4 A4 in product set (a, e, 1) with A5 would support the pitch transpositional relationship between source set a and product set (a, e, 1), the A4 could actually have another generative source. Input p-set e contains an interval of three semitones. Transposing each pitch of the T-1 pitch transposition of input p-set a up three semitones produces the remaining pitches found in product set (a, e, 1). In other words, combining T-1(a) with it own T+3 transposition would generate product set (a, e, 1). However, product set (a, e, 1) as a pitch transposition of input set a, and subsequent pitch transposition by the interval contained in p-set e would be a multiset because it contains both A4 and A5. Since product set (a, e, 1) is not a multiset, Boulez’s multiplication operation is probably a pc operation, and will henceforth be referred to as pc-multiplication. (a, e, 1) (a, e, 1) (a, e, 1) (e, a, 1) (e, a, 1) EXAMPLE 1: PITCH-CLASS SETS GENERATED BY SOURCE SETS a AND e PC-MULTIPLICATION (BOULEZ, BOULEZ ON MUSIC TODAY, EXAMPLE 33, 80) & & œ œ œ b œ œ # œ œ œ œœ # œ œœ œ œ # # œ œœ œ œ # # # œ œ œ œœ b b b œ œ œ œœ b a e 136 Perspectives of New Music So, how does one pair of input pitch-class sets (henceforth pc-sets) generate multiple output pc-sets with a specific pc content from a given pair of input pc-sets with specific pc contents? Interestingly, the lowest pitches of (a, e, 1), (a, e, 2), and (a, e, 3) (F#4, C#4, and D#4) form a pc-set that is a member of the same SC (3-7[025]) as input pc-set a. Similarly, the lowest pitches of (e, a, 1) and (e, a, 2) (G4 and E4) are members of the same IC as input pc-set e. In fact, the operation T1I relates both “root” pc-sets to their respective input pc-sets. Nevertheless , even if the operation T1I applied to input pc-sets a and e generates the lowest note of product sets (a, e, 1) through (e, a, 2) and therefore the transpositional levels of (a, e, 1) through (e, a, 2), what mechanism of the operation determined the transformation T1I? Moreover, why not apply T1I only to input pc-set a, or only apply it to input pc-set e? In spite of the generative and transformational relationships linking input and product pc-sets...