  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.
