Enumeration and construction in music theory |
Abstract: In this paper we describe in a more or less complete way how to apply methods from algebraic combinatorics to the classification of different objects in music theory. Among these objects there are intervals, chords, tone-rows, motives, mosaics etc. The methods we are using can be described in a very general way so that they can be applied for the classification of objects in different sciences. For instance for the isomer enumeration in chemistry, for spin analysis in physics, for the classification of isometry classes of linear codes, in general for investigating isomorphism classes of objects. Here we present an application of these methods to music theory.
The main aim of this paper is to show how the number of essentially different (i. e. not similar) objects can be computed and how to construct a (complete) system of representatives of them. Similarity is defined by certain symmetry operations which are motivated by music theory.
In other words, we try to enumerate or construct a list of objects such that objects of this list are pairwise not similar and each possible object is similar to (exactly) one object of this list.
The objects we are interested in belong to (or are constructed in) the n -scale Z_{n} consisting of exactly n pitch-classes, which generalizes the concept of 12 tones in one octave to n tones in one octave. After describing the symmetry operations as permutations we can apply different methods from combinatorics under group action (e. g. Pólya's Theory of counting, Read's method of orderly generation etc.) for the classification of these objects.
Enumeration and construction in music theory |