Dissertation Index

Author: Hook, Julian L.

Title: Uniform Triadic Transformations

Institution: Indiana University

Begun: February 2000

Completed: May 2002


A simple algebraic framework is proposed for studying triadic transformations. Included are the neo-Riemannian transformations P, L, and R, and other transformations recently studied by Lewin, Cohn, Hyer, and others. Hyer’s group of 144 transformations is extended to a group of 288, in which composition of transformations may be defined in a simpler and more unified fashion. Each of these "uniform triadic transformations" (UTTs) is represented in componentwise fashion by an ordered triple consisting of a sign (indicating whether the transformation preserves or reverses mode) and two integers mod 12 (indicating the intervals through which the roots of major and minor triads are transposed). This formalism simultaneously generalizes and simplifies much recent work in triadic transformations, and provides some clarification of the relationship between those transformations that behave in characteristically "Riemannian" ways (such as P, L, and R) and others that do not (such as ! th! e "dominant" and "mediant" transformations)--a relationship that some have found confusing or disturbing. The algebraic structure of the UTT group is studied in some detail, with emphasis on the Riemannian subgroup and other simply transitive subgroups and on the many intersections with other recent studies in transformational theory. The interaction of UTTs with inversion operators is examined. The methods presented are applicable not only to consonant triads but in many other settings as well, including set classes other than triads, equal-tempered systems other than that with 12 notes, certain diatonic structures, and some serial relationships. A variety of examples demonstrate some of the analytical potential of the theory.

Keywords: algebra, group theory, transformations, neo-Riemannian, Lewin, Cohn, Hyer


1. Triads and UTTs
2. Riemannian UTTs
3. Simply Transitive Groups of UTTs
4. The Structure of the UTT Group
5. UTTs and Inversion Operators
6. Beyond Triads
7. Analytical Applications I: Twelve-Tone Examples
8. Analytical Applications II: Triadic Examples
Appendix: Glossary of Algebraic Terms


Julian Hook
School of Music
Penn State University
University Park, PA 16802
(814) 863-5392

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