Date of Award

Spring 5-2020

Level of Access

Open-Access Thesis

Degree Name

Master of Arts (MA)

Department

Mathematics

Advisor

Tyrone Crisp

Second Committee Member

Andrew Knightly

Third Committee Member

William Snyder

Abstract

In the branch of Western music theory called serialism, it is desirable to construct chord progressions that use each chord in a chosen set exactly once. We view this problem through the scope of the mathematical theory of Gray codes, the notion of ordering a finite set X so that adjacent elements are related by an element of some specified set R of involutions in the permutation group of X. Using some basic results from the theory of permutation groups we translate the problem of finding Gray codes into the problem of finding Hamiltonian paths and cycles in a Schreier coset graph of the permutation group generated by the involutions R. Having made this translation we can use known results about Hamiltonian paths in Schreier (and Cayley) graphs of groups to generate serialism-like chord progressions. We illustrate the method by examining two theorems from the literature on Hamiltonian paths, due to Conway, Sloane, and Wilks (Graphs Combin. 5 (1989), no. 4, 315–325), and to Eades and Hickey (J. Assoc. Comput. Mach. 31 (1984), no. 1, 19–29). We give proofs of these theorems that complement the published proofs by filling in some details and clarifying some potentially confusing points, and we then use the algorithms extracted from these proofs to produce chord progressions.

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