Séminaire Lotharingien de Combinatoire, B26a (1991), 14 pp.
[Formerly: Publ. I.R.M.A. Strasbourg, 1992, 476/S-26, p. 29-42.]

Harald Fripertinger

Enumeration in Musical Theory

Abstract. Being a mathematician and a musician (I play the flute) I found it very interesting to deal with Pólya's counting theory in my Master's thesis. When reading about Pólya's theory I came across an article, called "Enumeration in Music Theory" by D. L. Reiner [Amer. Math. Monthly 92 (1985), 51-54]. I took up his ideas and tried to enumerate some other "musical objects".

At first I would like to generalize certain aspects of 12-tone music to n-tone music, where n is a positive integer. Then I will explain how to interpret intervals, chords, tone-rows, all-interval-rows, rhythms, motifs and tropes in n-tone music. Transposing, inversion and retrogradation are defined to be permutations on the sets of "musical objects". These permutations generate permutation groups, and these groups induce equivalence relations on the sets of "musical objects". The aim of this article is to determine the number of equivalence classes (I will call them patterns) of "musical objects". Pólya's enumeration theory is the right tool to solve this problem.

In the first chapter I will present a short survey of parts of Pólya's counting theory. In the second chapter I will investigate several "musical objects".

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