Musical Actions of Dihedral Groups

Authors

Alissa S. Crans, Loyola Marymount UniversityFollow
Thomas M. Fiore, University of Chicago
Ramon Satyendra, University of Michigan-Ann Arbor

Document Type

Article - post-print

Publication Date

2008

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.

Comments

This is the post-print version of the article.

Original Publication Citation

Crans, A.; Fiore, T.; and Satyendra, R. “Musical Actions of Dihedral Groups.” The American Mathematical Monthly, Vol. 116 (2009), No. 6: 479 – 495.

Digital Commons @ LMU & LLS Citation

Crans, Alissa S.; Fiore, Thomas M.; and Satyendra, Ramon, "Musical Actions of Dihedral Groups" (2008). Mathematics, Statistics and Data Science Faculty Works. 59.
https://digitalcommons.lmu.edu/math_fac/59