Dissertation Index

Author: Gollin, Edward H.

Title: Representations of Space and Conceptions of Distance in Transformational Music Theories

Institution: Harvard University

Begun: September 1997

Completed: June 2000


The dissertation examines metaphors of space and distance in transformational music theories, formalizing the notion of a music-transformational space. A music-transformational space is a particular arrangement of musical elements whose structure is determined by a family of normative relations or transformations acting on those elements. Such spaces may take the form of maps or networks of musical elements or they may be expressed through the symbolic language used to represent transformations: normative relations are expressed by unitary symbols whereas more complex relations—those relating distant elements in a space—are expressed as combinations of the space’s normative relations. A music- transformational space, for example, underlies the familiar symbology of twelve-tone operations, in which unitary operations such as R and I are considered normative while the relation RI is understood to be composite or derivative. The dissertation explores how music-transformationl spaces and the transformational pathways therein reflect the distinct ways of experiencing a given transformation in particular musical contexts. The dissertation also uses music-transformational spaces as tools for historical interpretation, examining how explicit graphic and symbolic accounts of musical relationships in compositional and theoretical treatises of the eighteenth through twentieth centuries reflect the often implicit theoretical priorities of their creators. Particularly central are the writings of Hugo Riemann for whom the notion of space was fundamental to ideas of harmonic relatedness and progression. The work concludes by illustrating applications of spatial models in the analysis of musical works in both harmonic and non-harmonic structural domains.

Keywords: Riemann, neo-Riemannian, transformation, group theory, graph theory, combinatorial, Heinichen, Schubert, Prokofiev


0. Introduction
1. Some Examples of Spaces and Problems of Distance
1.1. Some Serial Spaces
1.2. Intervals and Tone Spaces
1.3. Some Hexatonic Spaces
2. The Mathematics of Spaces and Pathways
2.1. Words, Group Presentations and Graphs
2.2. Pathways, Equivalence and Transformation Classes
2.3. Conceptions of Distance: Representative Words and Characteristic Path Length
3. Some Eighteenth Century Spaces
3.1. The Mediant Circle: Werckmeister’s Paradoxal Discourse
3.2. Heinichen’s Musicalische Circul
3.3. David Kellner’s Musicalische Circul
3.4. Vial’s Arbre Genealogique de l’Harmonie
4. Tone Spaces and Pathways in the Nineteenth Century: The Theories of Hugo Riemann Part I
4.1. The Development of the Table of Relations: Oettingen and the Acoustical Perspective
4.2. Riemann’s Changing Perspective: Toward the Table as Map
4.3. Tonbestimmung and the Combinatorial Conception of Interval
4.4. A Historical Curiosity: Euler’s Speculum Musicum
4.5. Ottokar Hostinsky’s Tonschema
5. Triadic and Key Space in the Nineteenth Century: the Theories of Hugo Riemann Part II
5.1. Riemann’s System of Harmonieschritte
5.2. Space, Enharmonicism and Aesthetics in Riemann’s Harmonic Theories
6. Music-Transformational Spaces in Neo-Riemannian Theories
6.1. Parsimonious Generators, Cohn Functions and Tonnetz Duals
6.2. The Harmonic Spaces of Brian Hyer and David Kopp
7. Some Analytical Applications of Music-Transformational Spaces
7.1. Text and Space in Three Schubert Lieder
7.2. Automorphisms and Some Non-Traditional Harmonic Spaces
7.3. Distance, Pathway and Analysis in Some Permutation Spaces
Appendix A. Basic Definitions and Concepts: Functions, Operations, Transformations and Groups


Department of Music
Harvard University
Cambridge, MA 02138

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