Musical Actions of Dihedral Groups

Abstract

Abstract. The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor triads. We illustrate both geometrically and algebraically how these two actions are dual. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.

Music in the pedagogy of mathematics

“…See Hook (2002), Crans, Fiore, and Satyendra (2009), and Fiore and Satyendra (2005). • Rhythmic canons, which include areas such as Galois theory, tiling, and Fourier analysis on finite groups.…”

From Schritte and Wechsel to Coxeter Groups

“…For example, going from ( * ) in Figure 5 to (1) is a reflection, to (2) a rotation, to (3) a glide reflection, and to (6) It turns out that the Coxeter group S 3 has a normal subgroup T of index 6 given by translations. The factor group S 3 / T is isomorphic to the symmetric group S 3 .…”

“…• The group P is naturally isomorphic to the opposite group of R. • There is a normal subgroup K in P with factor the dihedral group D 12 . Recall that D 12 is the group of 24 elements, generated by the PLRmoves on the finite Tonnetz, see [6,Chapter 5]. The point reflection group P is the subgroup of the group of Euclidean plane isometries generated by the 180 • rotations π 1 , π 2 , π 3 , where each π i is the point reflection about the midpoint of the edge of (*) on the s i -axis, see Figure 10.…”

Increasingly Enumerable Submonoids of R : Music Theory as a Unifying Theme

“…Apart from [7], to which the present article is a natural continuation, many other references can be cited. For instance, [3] gives a theoretical approach to music theory from the perspective of group theory; in [1] there is a connection between rhythmic canons and Galois theory; and in [11] there is a musical interpretation of the dihedral group of order 24. From a combinatorial point of view, [8] presents a connection between musical scale theory and word theory, while [20,22] analyze music intervals in terms of mathematical transformations.…”