From Matrix to Map

Edward Gollin

in The Oxford Handbook of Neo-Riemannian Music Theories

Published in print December 2011 | ISBN: 9780195321333
Published online September 2012 | | DOI:

Series: Oxford Handbooks

From Matrix to Map

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This article examines the changing meaning of the Tonnetz in Riemann's writings over the course of his career. It examines how Riemann came to reconcile the literal-acoustical and the spatial-metaphorical views of the Tonnetz—how the Tonnetz as a literal matrix to represent and calculate relative frequencies of the tones in just intonation evolved in his later writings into traversable landscape of tones. This article begins by examining the table from Arthur von Oettingen's Harmoniesystem in dualer Entwickelung, which provided not only the graphic model for the table of relations in Riemann's dissertation but also shaped its underlying acoustical conception. The article then examines how two developments in the following decades were significant for Riemann's conceptual reformation of the Tonnetz. The transition from a literal to a metaphorical understanding of the Tonnetz did not simply mirror Riemann's shift from an acoustical to a psychological view of the foundation of harmony, but rather made possible the transition, providing Riemann a mechanism to mediate between the phenomenal world of musical practice and the unbounded noumenal realm of musical meanings.

Keywords: Tonnetz; intonation; Arthur von Oettingen; conceptual reformation; Entwickelung; tones; literal-acoustical views; spatial-metaphorical views

Article.  8604 words. 

Subjects: Music ; Music Theory and Analysis

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