![]() ![]() Loday, J.-L., Ronco, M.O.: Order structure on the algebra of permutations and of planar binary trees. Loday, J.-L.: Parking functions and triangulation of the associahedron. Séminaire Lotharingien de Combinatoire 53 Article B53d, 1–16 (2005) Loday, J.-L.: Inversion of integral series enumerating binary plane trees. Loday, J.-L.: Realization of the Stasheff polytope. Lerdahl, F., Jackendoff, R.: A Generative Theory of Tonal Music. Lerdahl, F.: Issues in prolongational theory. Larson, S.: The problem of prolongation in ‘tonal’ music: Terminology, perception, and expressive meaning. Lee, C.W.: The associahedron and triangulations of the n-gon. Princeton University Press, Princeton (1971) Komar, A.: Theory of Suspensions: A Study of Metrical and Pitch Relations in Tonal Music. University of Chicago Press, Chicago (1992) (eds.) Disciplining Music: Musicology and its Canons. This process is experimental and the keywords may be updated as the learning algorithm improves.Ĭohn, R., Dempster, D.: Hierarchical unity, plural unities: Towards a reconciliation. These keywords were added by machine and not by the authors. ![]() I give musical examples of conflict between melodic and rhythmic organization, and show how melodic and harmonic organization combine in melody and counterpoint. 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. 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. By recognizing hierarchies that arise in music, we can see how this rich structure is realized in multiple aspects of musical organization. The Stasheff polytope has a robust mathematical literature behind it demonstrating its wealth of mathematical structure. A geometrical construct called the Stasheff polytope or associahedron summarizes these similarities. Music is hierarchically structured in numerous ways, and all of these forms of organization share essential mathematical features. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |