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Computation and Visualization of Musical Structures in Chord-Based Simplicial Complexes

  • Louis Bigo
  • Moreno Andreatta
  • Jean-Louis Giavitto
  • Olivier Michel
  • Antoine Spicher
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7937)

Abstract

We represent chord collections by simplicial complexes. A temporal organization of the chords corresponds to a path in the complex. A set of n-note chords equivalent up to transposition and inversion is represented by a complex related by its 1-skeleton to a generalized Tonnetz. Complexes are computed with MGS, a spatial computing language, and analyzed and visualized in Hexachord, a computer-aided music analysis environment. We introduce the notion of compliance, a measure of the ability of a chord-based simplicial complex to represent a musical object compactly. Some examples illustrate the use of this notion to characterize musical pieces and styles.

Keywords

MGS simplicial complexes generalized Tonnetze compliance Hexachord chord spaces 

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References

  1. 1.
    Chew, E.: The Spiral Array: An Algorithm for Determining Key Boundaries. In: Anagnostopoulou, C., Ferrand, M., Smaill, A. (eds.) ICMAI 2002. LNCS (LNAI), vol. 2445, pp. 18–31. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Cohn, R.: Neo-Riemannian Operations, Parsimonious Trichords, and their “Tonnetz” Representations. Journal of Music Theory 41(1), 1–66 (1997)CrossRefGoogle Scholar
  3. 3.
    Callender, C., Quinn, I., Tymoczko, D.: Generalized Voice-Leading Spaces. Science 320(5874), 346 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Tymoczko, D.: The Geometry of Musical Chords. Science 313(5783), 72 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Giavitto, J.L., Michel, O.: MGS: a Rule-Based Programming Language for Complex Objects and Collections. In: van den Brand, M., Verma, R. (eds.) Electronic Notes in Theoretical Computer Science, vol. 59. Elsevier, Amsterdam (2001)Google Scholar
  6. 6.
    Giavitto, J.L.: Topological Collections, Transformations and their Application to the Modeling and the Simulation of Dynamical Systems. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 208–233. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Munkres, J.: Elements of Algebraic Topology. Addison-Wesley (1984)Google Scholar
  8. 8.
    Spicher, A., Michel, O., Giavitto, J.-L.: Declarative Mesh Subdivision Using Topological Rewriting in mgs. In: Ehrig, H., Rensink, A., Rozenberg, G., Schürr, A. (eds.) ICGT 2010. LNCS, vol. 6372, pp. 298–313. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Giavitto, J.L., Spicher, A.: Simulation Of Self-Assembly Processes Using Abstract Reduction Systems. In: Krasnogor, N., Gustafson, S., Pelta, D.A., Verdegay, J.L. (eds.) Systems Self-Assembly: Multidisciplinary Snapshots, pp. 199–223. Elsevier, Amsterdam (2008)CrossRefGoogle Scholar
  10. 10.
    Bigo, L., Giavitto, J., Spicher, A.: Building topological spaces for musical objects. In: Agon, C., Andreatta, M., Assayag, G., Amiot, E., Bresson, J., Mandereau, J. (eds.) MCM 2011. LNCS, vol. 6726, pp. 13–28. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  11. 11.
    Mazzola, G., et al.: The Topos of Music: Geometric Logic of Concepts. In: Theory, and Performance. Birkhäuser (2002)Google Scholar
  12. 12.
    Morris, R.: Composition with Pitch Classes: a Theory of Compositional Design. Yale University Press, New Haven (1987)Google Scholar
  13. 13.
    Estrada, J.: La teoría d1, MúSIIC-Win y algunas aplicaciones al análisis musical: Seis piezas para piano, de Arnold Schoenberg. In: Lluis-Puebla, E., Agustín-Aquinas, O. (eds.) Memoirs of the Fourth International Seminar on Mathematical Music Theory, Huatulco (2011)Google Scholar
  14. 14.
    Andreatta, M., Agon, C.: Implementing Algebraic Methods in openmusic. In: Proceedings of the International Computer Music Conference, Singapore (2003)Google Scholar
  15. 15.
    Catanzaro, M.: Generalized Tonnetze. Journal of Mathematics and Music 5(2), 117–139 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Gollin, E.: Some Aspects of Three-Dimensional “Tonnetze”. Journal of Music Theory 42(2), 195–206 (1998)CrossRefGoogle Scholar
  17. 17.
    Albini, G., Antonini, S.: Hamiltonian Cycles in the Topological Dual of the Tonnetz. In: Chew, E., Childs, A., Chuan, C.-H. (eds.) MCM 2009. CCIS, vol. 38, pp. 1–10. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Louis Bigo
    • 1
    • 2
  • Moreno Andreatta
    • 2
  • Jean-Louis Giavitto
    • 2
  • Olivier Michel
    • 1
  • Antoine Spicher
    • 1
  1. 1.LACL/Université Paris-Est CreteilFrance
  2. 2.UMR CNRS STMS 9912/IRCAMFrance

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