Classification of motives: a mathematical approach

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 (cf. [16, 17]). Here we present an application of these methods to music theory. First it should be mentioned that in British English the word “motive” is often spelled as “motif”. Sometimes terms like “figure” or “subject” are used as synonyms for it. Moreover there is no unique definition of a motive. The meaning of this word was changing through the centuries so actually there exist many different definitions. In general a motive is the smallest part of a composition which makes some musical sense. In the second half of the eighteenth century such meaningful units of a composition were mentioned for the first time in music theory. The first definitions of motives were given in the first half of the nineteenth century. (Cf. [19, 18].) In H. Riemann’s definition [28] a motive includes rhythm, melody, harmony, dynamics and tone colour. ([3]) The concept of motives we are using in this paper was introduced by G. Mazzola in [20]. It is a mathematical precise definition which is useful for investigating both tonal and rhythmical aspects of music. (See section 6.) At the end of section 4 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 exact definition when two motives are called similar will be given in section 6. In order to describe the musical meaning of the operations mentioned above we demonstrate their application in the next three examples. We didn’t give any formal definitions so far. In some later sections when the corresponding operations are defined in our mathematical model the reader should check that these definitions coincide with the usual meaning in music theory. First three short pieces of music — we call them motives — are shown, which intuitively could be regarded as motives or as collections of motives, together with similar motives which were produced by applying certain of these