Generalizations of the Tonnetz
Abstract
In this paper, we attempt to explore mathematical structures of tonal music from
the 18th and 19th centuries. We review the known mathematical/musical structures
and, noting lacking features of this geometrical construct as a useful model of tonal
harmony, we propose generalizations that may be better suited to tonal music. We
bring several mathematical devices to bear on the Tonnetz in new ways, including
Cayley graphs and .Coxeter hyperbolic representations. We conclude with three-dimensional
geometric models that represent the four-note seventh chords which are
ubiquitous in 18th and 19th century music.