From Schritte and Wechsel to Coxeter Groups

The PLR-moves of neo-Riemannian theory, when considered as reflections on the edges of an equilateral triangle, define the Coxeter group \(\widetilde{S}_3\). The elements are in a natural one-to-one correspondence with the triangles in the infinite Tonnetz. The left action of \(\widetilde{S}_3\) on the Tonnetz gives rise to interesting chord sequences. We compare the system of transformations in \(\widetilde{S}_3\) with the system of Schritte and Wechsel introduced by Hugo Riemann in 1880. Finally, we consider the point reflection group as it captures well the transition from Riemann’s infinite Tonnetz to the finite Tonnetz of neo-Riemannian theory.

The author would like to thank Benjamin Brück from Bielefeld, Germany, for helful comments regarding the Coxeter group. He is particularly grateful to Thomas Noll from Barcelona since his thoughtful advice, in particular regarding the action of interesting elements and subgroups of the Coxeter group on the Tonnetz, has led to substantial improvements of the paper (which about doubled in size).

Authors and Affiliations

1. Florida Atlantic University, Boca Raton, FL, 33431-0991, USA

   Markus Schmidmeier

Corresponding author

Correspondence to Markus Schmidmeier .

Editors and Affiliations

1. Georgia State University, Atlanta, GA, USA

   Mariana Montiel

2. Technical University of Madrid, Madrid, Spain

   Francisco Gomez-Martin

3. Technological University of the Mixteca, Oaxaca, Mexico

   Octavio A. Agustín-Aquino

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