Classification of motives: a mathematical approach |
Abstract: In this paper we show in a more or less complete way how to apply methods from algebraic combinatorics to the classification of motives. These methods 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 concept of motives we are using in this paper was introduced by G. Mazzola. It is a mathematical precise definition which is useful for investigating both tonal and rhythmical aspects of music. Then we present some mathematical definitions for operations like transposing, inversion, temporal shift, retrograde inversion etc. They can be applied to a motive such that from a given motive we can construct many other motives. They all will be called similar or equivalent motives.
The main aim of this paper is to show how the number of essentially different (i. e. not similar) motives can be computed and how to construct a (complete) system of representatives of motives. In other words, we try to get a list of motives such that motives of this list are pairwise not similar and each possible motive is similar to (exactly) one motive of this list.
Classification of motives: a mathematical approach |