# From Schritte and Wechsel to Coxeter Groups

## TL;DR

This paper explores the mathematical structure of neo-Riemannian transformations using Coxeter groups, linking historical musical concepts with modern algebraic frameworks to analyze chord progressions.

## Contribution

It introduces a Coxeter group perspective on neo-Riemannian transformations and compares it with Riemann's original Schritte and Wechsel system.

## Key findings

- Coxeter group $ ilde{S}_3$ models neo-Riemannian PLR-moves.
- The group action corresponds to transformations of the Tonnetz.
- The point reflection group connects infinite and finite Tonnetz models.

## Abstract

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.