Dissertation Index

Author: Brown, Stephen C.

Title: Dual Interval Space in Twentieth-Century Music

Institution: Yale University

Begun: September 1994

Completed: May 1999


This study proposes a model of pitch-class space called "dual interval space." A dual interval space (or "DIS") is a two-dimensional array of pitch classes, in which each dimension corresponds to a unique (non-zero) interval class. Given some pitch-class collection, the members of that collection can be visualized as residing in various locations of a DIS. These locations can then be translated within the space or flipped about some axis. The flipping operations in particular offer new ways to relate set-classes, even set-classes of different cardinalities. Aside from its theoretical interest, the concept of dual interval space is suggestive for the analysis of certain twentieth-century pieces. Moreover, it can apply to music displaying a wide range of styles and techniques, including pitch-centric, freely atonal, and serial music.

Chapter 1 introduces the concept and develops the idea of operations in a DIS, exploring their effects on pitch-classes and pitch-class sets. It closes by relating some ideas from graph theory to the concept of dual interval space. Chapter 2 discusses precedents for dual interval space and traces connections with more recent developments in music theory. Chapter 3 shows how the concept can provide the basis for the detailed study of a single body of music, specifically, that of late Shostakovich. Finally, Chapter 4 demonstrates the broader applicability of the concept, using it to analyze pieces by Ruggles, Schoenberg, and Webern.

Keywords: atonal theory, post-tonal theory, graph theory, transpositional combination, Klumpenhouwer networks, Shostakovich, Ruggles, Schoenberg, Webern


Chapter 1 Theory and Methodology
Chapter 2 Other Music Theories: Precedents and Intersections
Chapter 3 Ic1/Ic5 Space in the Music of Shostakovich
Chapter 4 Dual Interval Space in Other Music


Stephen C. Brown
15 North Cedar Street
Oberlin, OH 44074
440-775-0249 (home)
440-775-8239 (office)

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