Some combinatorial aspects of the musical chords

Abstract

Group theory has been used in the study of various properties of musical scales. The cyclic group C12 provides a mathematical model for the tempered intervals (see for example Budden [1] Chapter 23, where the diminished seventh chords are shown to correspond to an appropriate subgroup of C12 and its cosets).

Musical Actions of Dihedral Groups

“…The result is a graph on the torus. Earlier, Waller studied this graph on the torus in [24], and observed that its automorphism group is the dihedral group of order 24. Waller's torus is pictured in Figure 7.…”

“…The result is a graph on the torus. Earlier, Waller studied this graph on the torus in [24], and observed that its automorphism group is the deleted, and a different region of the Tonnetz is displayed. Special thanks go to Richard Cohn for giving us permission to use this modified version of the figure.…”

“…Waller's torus from[24] has been reproduced in Figure7by kind permission of the U.K. Mathematical Association and the Mathematical Gazette.…”

Parsimonious graphs for the most common trichords and tetrachords

“…In that paper, the authors also provide several remarkable parsimonious graphs, particularly the Chicken-Wire Torus (the dual of the Tonnetz ) and Cube Dance for nearly and most even trichords, respectively, and the Towers Torus and Power Towers for nearly and most even tetrachords, respectively. Twenty years before, however, Waller (1978) published a torus equivalent to the Chicken-Wire, but which clearly shows its full hexagonal tessellation, as well as all P L, P R, and -although a bit harder to visualize -LR cycles. These and other P LR compound operations were later studied extensively by Cohn (1996Cohn ( , 1997Cohn ( , 1998Cohn ( , and, particularly, 2012.…”