Pitch-class set multiplication in Boulez's Le marteau sans maitre
Since its 1955 premiere, Pierre Boulez's Le Marteau sans mai tre has engendered a great deal of discourse, most notably in analyses by Lev Koblyakov, related to both its compositional processes and its serialist aesthetics. The exact nature of the process of pitch-class set multiplication that generates the pitch classes of the first cycle of Le Marteau has never been fully explored, with the result that the process appears to have an ad hoc quality in which the transpositional type of a multiplicative result is fixed but its pitch-class content is not. Coinciding with this appearance is the question of why a composer of Boulez's skill would care to use the process at all. The theory presented here not only refutes the appearance of capriciousness, but also demonstrates the attractiveness that the operation would have for a composer--an extension of traditional serialism that permits the generation of a large number of different, yet interrelated, unordered pitch-class sets.This study formalizes pitch-class set multiplication in three configurations. The first, called "simple multiplication," involves the construction of one operand set's intervallic structure on each pitch class of another operand set; this operation is examined not only as it pertains to Boulez, but also as a compositional/analytical tool that parallels, in pitch-class-specific terms, Richard Cohn's theory of transpositional combination. The second, "compound multiplication," illustrates schemata for the transposition of a simple multiplicative product. The third, "complex multiplication," is the elegant operation that generates pitch-class sets in the first cycle of Le Marteau; like arithmetical multiplication, this process is commutative. The operation, the twelve-tone row to which it is applied, and the resulting pitch-class sets are then examined with respect to their implications for a process-based listening strategy.
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