Musical Actions of Dihedral Groups
Alissa S. Crans, Thomas M. Fiore, Ramon Satyendra
The American mathematical monthly
INTRODUCTION. Can you hear an action of a group? Or a centralizer? If knowledge of group structures can influence how we see a crystal, perhaps it can influence how we hear music as well. In this article we 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. 1 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
... 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 The first musical action of the dihedral group of order 24 we consider arises via the familiar compositional techniques of transposition and inversion. A transposition moves a sequence of pitches up or down. When singers decide to sing a song in a higher register, for example, they do this by transposing the melody. An inversion, on the other hand, reflects a melody about a fixed axis, just as the face of a clock can be reflected about the 0-6 axis. Often, musical inversion turns upward melodic motions into downward melodic motions. 2 One can hear both transpositions and inversions in many fugues, such as Bernstein's "Cool" fugue from West Side Story or in Bach's Art of Fugue. We will mathematically see that these musical transpositions and inversions are the symmetries of the regular 12-gon. The second action of the dihedral group of order 24 that we explore has only come to the attention of music theorists in the past two decades. Its origins lie in the P, L, and R operations of the 19th-century music theorist Hugo Riemann. We quickly define these operations for musical readers now, and we will give a more detailed mathematical definition in Section 5. The parallel operation P maps a major triad 3 to its parallel minor and vice versa. The leading tone exchange operation L takes a major triad to the minor triad obtained by lowering only the root note by a semitone. The operation L raises the fifth note of a minor triad by a semitone. The relative operation R maps a major triad to its relative minor, and vice versa. For example, P(C-major) = c-minor, L(C-major) = e-minor, R(C-major) = a-minor. It is through these three operations P, L, and R that the dihedral group of order 24 acts on the set of major and minor triads. The P, L, and R operations have two beautiful geometric presentations in terms of graphs that we will explain in Section 5. Musical readers will quickly see that 1 The composer Milton Babbitt was one of the first to use group theory to analyze music. See  . 2 A precise, general definition of inversion will be given later. 3 A triad is a three-note chord, i.e., a set of three distinct pitch classes. Major and minor triads, also called consonant triads, are characterized by their interval content and will be described in Section 4.