Generalizations of the Tonnetz
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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.