From Schritte and Wechsel to Coxeter Groups

Abstract

The PLR-moves of neo-Riemannian theory, when considered as reflections on the edges of an equilateral triangle, define the Coxeter group ${\tilde{S}}_3$. The elements are in a natural one-to-one correspondence with the triangles in the infinite Tonnetz. The left action of ${\tilde{S}}_3$ on the Tonnetz gives rise to interesting chord sequences. We compare the system of transformations in ${\tilde{S}}_3$ 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.

14 pages for the Mathematics and Computation in Music Conference in June 2019 in Madrid, the revised version extends the music theoretic discussion