Author: Alegant, Brian
Title: The Seventy-Seven Partitions of the Aggregate: Analytical and Theoretical Implications
Institution: Eastman School of Music
Begun: November 1989
Completed: October 1992
The total chromatic, or aggregate, is widely recognized as a fundamental building block of many different kinds of twelve-tone compositions, and the procedure of aggregate completion has been given much attention in the music-theoretic literature. Only recently, however, have theorists considered the following questions: how many distinct ways can the aggregate be partitioned into pitch-class sets? what properties do these partitions possess? and how do they establish coherence on both the surface and the deeper levels of a given work? The aim of this dissertation is to examine the theoretical and analytical implications of the seventy-seven partition classes of the aggregate: that is, the seventy-seven unique ways to divide the aggregate into disjoint pitch-class sets.
Part I provides the theoretical framework. Chapter 1 summarizes previous studies of twelve-tone partitioning and discusses issues and problems relating to a partitional approach to twelve-tone music. Chapter 2 establishes the groundwork for a general theory of partitions by enumerating and classifying the number of mosaics for each partition class of the aggregate, and by examining relations and similarities among and between them. Chapter 3 reviews techniques of combinatorics relevant to the study of partitions and discusses inherent difficulties in enumerating mosaics in the twelve-tone system.
Part II presents analyses of three different types of twelve-tone compositions to illustrate the analytical scope of the theory. Chapter 4 looks at compositions using even-partitioned arrays characteristic of Milton Babbitt's first-period works; Chapter 5, compositions organized by hexachordal I-combinatorial regions characteristic of Arnold Schoenberg's ouevre; and Chapter 6, *all-partition array: works of Robert Morris and Babbitt.
The analyses demonstrate the ability of a partitional approach to highlight the identification and transformation of pitch-class sets of rows and aggregates, to model the associations and correspondences between aggregates, and to provide insight into the coherence and underlying strategies of twelve-tone compositions.
Keywords: Hierarchy, Twelve-Tone, Partitions, Mosaics, Schoenberg, Babbitt, Music Theory, Analysis, Combinatorics
PART I: Theoretical Implications
Chapter 1. Partitions: An Historical Review
Chapter 2. A General Theory of Twelve-Tone Partitions
Chapter 3. Combinatorics and Mosaics
PART II: Analytical Implications
Chapter 4. Analytical Applications I: Symmetrical Array Compositions
Chapter 5. Analytical Applications II: Schoenberg and Partitions
Chapter 6. Analytical Applications III: All-Partition Compositions
Summary and Conclusion
Appendix One: Y/Z-related mosaics for the Three Partitions:
Appendix Two: Array of Babbitt's The Widow's Lament in Springtime
Appendix Three: Pc Array for Milton Babbitt's Sextets
Appendix Four: Listing of mosaics for the Even Partitions
Appendix Five: Mosaics for the remaining three-partitions:
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