Complex Multiplication, Structure, and Process: Harmony and Form in Boulez's Structures II

Abstract

This article clarifies the process behind the operation known as complex multiplication and demonstrates that it is less abstract and more closely linked to conventional serial procedures than was previously thought. The operation results from one of many transpositional schemes that Boulez regularly incorporated in his precompositional tables. An understanding of this scheme and the way it derives from the partial ordering in register of the multiplication factors reveals isomorphic relationships that are crucial to the structural properties of the music. Through an analysis of Chapitre I of Boulez's Structures II, this article brings to light the nature of the harmonic relationships that shape the musical language, shows how these relate to the precompositional tables, and describes how they are essential to his sophisticated concept of form.

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