Making use of music theory, group theory, and category theory

From Musical Actions of Dihedral Groups

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 chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.

Summary:
This paper connects the twelve musical tones to elements in the dihedral group of order 24 (the symmetries of a regular dodecagon). The translation from pitch classes to integers modulo 12 allows for the modeling of musical works using abstract algebra. The first action on major and minor chords described in the paper is based on the musical techniques of transposition and inversion. A transposition moves a sequence of pitches up or down and an inversion reflects a melody about a fixed axis. The other action arises from the P, L, and R operations of the 19th-century music theorist Hugo Riemann. It is through these operations that the dihedral group of order 24 acts on the set of major and minor triads. The paper also describes how the P, L, and R operations have beautiful geometric presentations in terms of graphs. In particular the authors describe a connection between the PLR-group and chord progressions in Beethoven’s 9th Symphony, which leads to a proof that the PLR-group is dihedral. Another musical example is Pachelbel’s Canon in D. In summary, the paper gives a very pretty explanation of what we commonly hear in tonal music in terms of elementary group theory.