Vol. 8, No. 2, 2015

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Counting set classes with Burnside's lemma

Joshua Case, Lori Koban and Jordan LeGrand

Vol. 8 (2015), No. 2, 337–344
Abstract

Mathematical tools from combinatorics and abstract algebra have been used to study a variety of musical structures. One question asked by mathematicians and musicians is: how many d-note set classes exist in a c-note chromatic universe? In the music theory literature, this question is answered with the use of Pólya’s enumeration theorem. We solve the problem using simpler techniques, including only Burnside’s lemma and basic results from combinatorics and abstract algebra. We use interval arrays that are associated with pitch class sets as a tool for counting.

Keywords
set classes, pitch class sets, Burnside's lemma, group actions
Mathematical Subject Classification 2010
Primary: 00A65, 05E18
Milestones
Received: 14 August 2013
Revised: 24 October 2013
Accepted: 23 December 2013
Published: 3 March 2015

Communicated by Kenneth S. Berenhaut
Authors
Joshua Case
Mathematics Department
University of Maine at Farmington
228 South Street
Farmington, ME 04938
United States
Lori Koban
Mathematics Department
University of Maine at Farmington
228 South Street
Farmington, ME 04938
United States
Jordan LeGrand
Mathematics Department
University of Maine at Farmington
228 South Street
Farmington, ME 04938
United States