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Musical Actions of Dihedral Groups

by Alissa S. Crans (Loyola Marymount University), Thomas M. Fiore (University of Michigan-Dearborn), and Ramon Satyendra (University of Michigan)

Award: Hasse

Year of Award: 2011

Publication Information: The American Mathematical Monthly, vol. 116, no. 6, June 2009, pp. 479-495.

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.

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About the Authors: (From the Prizes and Awards booklet, MathFest 2009)

Alissa S. Crans is assistant professor of mathematics at Loyola Marymount University. She earned a B.S. from the University of Redlands and Ph.D. from UC Riverside. Alissa's research is in the field of higher-dimensional algebra; her current work, funded by an NSA Young Investigators Grant, involves categorifying algebraic structures called quandles with the goal of defining new knot and knotted surface invariants. She is also interested in the connections between mathematics and music. Alissa is extremely active in helping students increase their appreciation and enthusiasm for mathematics through co-organizing the Pacific Coast Undergraduate Mathematics Conference and her mentoring of young women in the Summer Mathematics Program at Carleton College and the EDGE Program. Alissa was an invited speaker at her local MAA Spring Section Meeting and the keynote speaker at the University of Oklahoma Math Day and the UCSD Undergraduate Math Day.

Thomas M. Fiore received a B.S. in Mathematics and a B.Phil. in German at the University of Pittsburgh. He completed his Ph.D. in Mathematics at the University of Michigan in 2005 under the direction of Igor Kriz. He was an NSF Postdoctoral Fellow and L.E. Dickson Instructor at the University of Chicago, and a Visiting Professor at the Autonomous University of Barcelona. He then joined the faculty at the University of Michigan-Dearborn in 2009. His research interests include higher category theory, algebraic topology, and mathematical music theory.

Ramon Satyendra received his doctorate in the History and Theory of Music from the University of Chicago. He is Associate Professor of Music Theory at the University of Michigan. His interests include music and mathematics, Liszt, jazz, South Asian music, and model composition in classical styles of the 18th and 19th centuries. He performs as a classical pianist and jazz organist.

Subject classification(s): Algebra and Number Theory | Abstract Algebra | Groups
Publication Date: 
Tuesday, August 23, 2011

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