From Contour Similarity to Motivic Topologies

This paper is devoted to a qualitative and quantitative study of topological spaces built on premotif collections of musical scores. These motivic topologies are related to similarity concepts in the American music set theory. Through shapes, imitations and gestalts, and similarity relations between any two shapes, we obtain a motivic hierarchy of a score. This model of motivic analysis is completed by proposing different tools, such as weighted shapes and motivic evolution trees (MET), for visualizing these non-intuitive topologies. The concept MET, which is related to a systematic variation of the similarity parameter, is of a highly cognitive flavor. In the light of our approach, the still debated question concerning the length of the main theme in Bach's Kunst der Fuge is addressed. As a result, we can state that the extended 12-tone theme is essentially equivalent to the shorter 8-tone version when viewed from its motivic substance.