The geometry of melodic, harmonic, and metrical hierarchy

Abstract

Music is hierarchically structured in numerous ways, and all of these forms of organization share essential mathematical features. A geometrical construct called the Stasheff polytope or associahedron summarizes these similarities. The Stasheff polytope has a robust mathematical literature behind it demonstrating its wealth of mathematical structure. By recognizing hierarchies that arise in music, we can see how this rich structure is realized in multiple aspects of musical organization. In this paper I define hierarchic forms of melodic, harmonic, and metrical organization in music, drawing on some concepts from Schenkerian analysis, and show how each of them exhibits the geometry of the Stasheff polytope. Because the same mathematical construct is realized in multiple musical parameters, the Stasheff polytope not only describes relationships between hierarchies on a single parameter, but also defines patterns of agreement and conflict between simultaneous hierarchies on different parameters. I give musical examples of conflict between melodic and rhythmic organization, and show how melodic and harmonic organization combine in melody and counterpoint. © 2009 Springer Berlin Heidelberg.