Musical Actions of Dihedral Groups

The dihedral group of order 24 is the group of symmetries of a regular 12-gon, that is, of a 12-gon with all sides of the same length and all angles of the same measure. Algebraically, the dihedral group of order 24 is the group generated by two elements, s and t, subject to the three relations ssup12=1, tssup2=1, tst=ssup-1. Here, Crans et al explore how music may be interpreted in terms of the group structure of the dihedral group of order 24 and its centralizer by explaining two musical actions. They argue that the first musical action of the dihedral group of order 24 arises via the familiar compositional techniques of transposition and inversion; while the second action has only come to the attention of music theorists in the past two decades, where its origins lie in the P, L, and R operations of the 19th-century music theorist Hugo Riemann. Moreover, Crans et al illustrate these group actions and their duality in musical examples by Pachelbel, Wagner, and Ives.