A Topological Approach of Musical Relationships

Abstract : The notion of space is often summoned in music theory, both for the analysis of existing pieces and for the composition of new ones. The use of space as a metaphor is instrumental in conveying musical insights but it is also a very effective heuristic for developing new computer tools to assist musicians in their creative processes. This chapter offers some applications of spatial representations of musical notions based on elementary concepts in algebraic topology: the representation by cell spaces of the melodic and harmonic content for the classification or the generation of musical phrases. The basic idea is to represent classes of simple musical objects (e.g., pitches, chords or intervals) as elementary spatial domains and to represent their relationships (e.g., their co-occurrence or their succession) as neighborhood relationships. The topological notions of incidence, path, boundary, obstruction, etc., are then used to unravel the musical structure. For instance, a musical sequence can be represented as a path in a cell space that represents the chord structure; the form of that path reveals some information concerning the musical strategies used by the composer.