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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Cohn, R., Dempster, D.: Hierarchical unity, plural unities: Towards a reconciliation. In: Bergeson, K., Bohlman, P.V. (eds.) Disciplining Music: Musicology and its Canons. University of Chicago Press, Chicago (1992)
Komar, A.: Theory of Suspensions: A Study of Metrical and Pitch Relations in Tonal Music. Princeton University Press, Princeton (1971)
Lee, C.W.: The associahedron and triangulations of the n-gon. European Journal of Combinatorics 10/6, 551–560 (1989)
Larson, S.: The problem of prolongation in ‘tonal’ music: Terminology, perception, and expressive meaning. Journal of Music Theory 41(1), 101–136 (1997)
Lerdahl, F.: Issues in prolongational theory. Journal of Music Theory 41(1), 141–155 (1997)
Lerdahl, F., Jackendoff, R.: A Generative Theory of Tonal Music. MIT Press, Cambridge (1983)
Loday, J.-L.: Realization of the Stasheff polytope. Archiv der Mathematik 83, 267–278 (2004)
Loday, J.-L.: Inversion of integral series enumerating binary plane trees. Séminaire Lotharingien de Combinatoire 53 Article B53d, 1–16 (2005)
Loday, J.-L.: Parking functions and triangulation of the associahedron. Contemporary Mathematics 431, 327–340 (2007)
Loday, J.-L., Ronco, M.O.: Order structure on the algebra of permutations and of planar binary trees. Journal of Algebraic Combinatorics 15, 253–270 (2002)
Marsden, A.: Generative structural representation of tonal music. Journal of New Music Research 34(4), 409–428 (2005)
Marsden, A.: Empirical study of Schenkerian analysis by computer. In: The joint meeting of the Society for Music Theory and the American Musicological Society, Nashville (2008)
Proctor, G., Lee Riggins, H.: Levels and the reordering of chapters in Schenker’s Free Composition. Music Theory Spectrum 10, 102–126 (1988)
Rahn, J.: Logic, set theory, music theory. College Music Symposium 19(1), 114–127 (1979)
Schachter, C.: Rhythm and Linear Analysis: A Preliminary Study. In: Mitchell, W., Salzer, F. (eds.) Music Forum 4 (1976); reprinted in Straus, J. (ed.): Unfoldings: Essays in Schenkerian Theory and Analysis, pp. 17–53. Oxford University Press, New York (1999)
Schachter, C.: Rhythm and Linear Analysis: Durational Reduction. In: Mitchell, W., Salzer, F. (eds.) Music Forum 5 (1980); reprinted in Straus, J. (ed.): Unfoldings: Essays in Schenkerian Theory and Analysis, pp. 54–79. Oxford University Press, New York (1999)
Schachter, C.: Rhythm and Linear Analysis: Aspects of Meter. In: Mitchell, W., Salzer, F. (eds.) Music Forum 6 (1987); reprinted in Straus, J. (ed.): Unfoldings: Essays in Schenkerian Theory and Analysis, pp. 79–117. Oxford University Press, New York (1999)
Schenker, H.: Free Composition (2 vols.), translated by Ernst Oster. Longman, New York (1979)
Schenker, H.: Counterpoint (2 vols.), translated by John Rothgeb and Jürgen Thym. Schirmer Books, New York (1987)
Schenker, H.: The Masterwork in Music: A Yearbook (3 vols.), edited by William Drabkin. Cambridge University Press, Cambridge (1997)
Smoliar, S.: A Computer Aid for Schenkerian Analysis. Computer Music Journal 4(2), 41–59 (1980)
Stasheff, J.D.: Homotopy associativity of H-spaces I. Transactions of the American Mathematical Society 108, 275–292 (1963)
Yeston, M.: The Stratification of Musical Rhythm. Yale University Press, New Haven (1976)
Yust, J.: Formal Models of Prolongation. PhD Diss. University of Washington (2006)
Yust, J.: The Step-Class Automorphism Group in Tonal Analysis. In: Proceedings of the First International Conference on Mathematics and Computation in Music. Springer, Berlin (forthcoming 2007)
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Yust, J. (2009). The Geometry of Melodic, Harmonic, and Metrical Hierarchy. In: Chew, E., Childs, A., Chuan, CH. (eds) Mathematics and Computation in Music. MCM 2009. Communications in Computer and Information Science, vol 38. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02394-1_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-02394-1_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02393-4
Online ISBN: 978-3-642-02394-1
eBook Packages: Computer ScienceComputer Science (R0)