Skip to main content
SpringerLink
Log in

Lattice-based and topological representations of binary relations with an application to music

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

Formal concept analysis associates a lattice of formal concepts to a binary relation. The structure of the relation can then be described in terms of lattice theory. On the other hand Q -analysis associates a simplicial complex to a binary relation and studies its properties using topological methods. This paper investigates which mathematical invariants studied in one approach can be captured in the other. Our main result is that all homotopy invariant properties of the simplicial complex can be recovered from the structure of the concept lattice. This not only clarifies the relationships between two frameworks widely used in symbolic data analysis but also offers an effective new method to establish homotopy equivalence in the context of Q -analysis. As a musical application, we will investigate Olivier Messiaen’s modes of limited transposition. We will use our theoretical result to show that the simplicial complex associated to a maximal mode with m transpositions is homotopy equivalent to the (m−2)–dimensional sphere.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Wille, R.: Restructuring lattice theory: An approach based on the hierarchy of concepts. In: Rival, I. (ed.) Ordered sets: proceedings of the NATO Advanced Study Institute held at Banff, Canada, August 28 to September 12, 1981. D. Reidel Pub. Co. (1982)

  2. Barbut, M., Monjardet, B.: Ordre et Classification: Algèbre et Combinatoire. Hachette (1970)

  3. Dowker, C.H.: Homology groups of relations. Ann. Math 2nd Series 56(1), 84–95 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  4. Atkin, R.H.: From cohomology in physics to q–connectivity in social science. Int. J. Man Mach. Stud. 4(2), 139–167 (1972)

    Article  MathSciNet  Google Scholar 

  5. Casti, J.L.: Connectivity, Complexity, and Catastrophe in Large-Scale Systems. Wiley, New York (1979)

    MATH  Google Scholar 

  6. Freeman, L.C.: Q-analysis and the structure of friendship networks. Int. J. Man Mach. Stud. 12(4), 367–378 (1980)

    Article  MathSciNet  Google Scholar 

  7. Johnson, J.: Transport Planning and Control, Chapter The dynamics of Large Complex Road Systems. Oxford University Press, pp. 165–186 (1991)

  8. Duckstein, L., Nobe, S.A.: q-analysis for modeling and decision making. Eur. J. Oper. Res. 103(3), 411–425 (1997)

    Article  MATH  Google Scholar 

  9. Barcelo, H., Kramer, X., Laubenbacher, R., Weaver, C.: Foundations of a connectivity theory for simplicial complexes. Adv. Appl. Math. 26(2), 97–128 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kaburlasos, V.G.: Special issue on information engineering applications based on lattices. Inf. Sci. 181(10), 1771–1773 (2011)

    Article  MathSciNet  Google Scholar 

  11. Catanzaro, M.J.: Generalized Tonnetze. J. Math. Music 5(2), 117–139 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ganter, B., Wille, R.: Formal Concept Analysis. Mathematical Foundations. Springer–Verlag, Berlin and Heidelberg (1999)

    Book  MATH  Google Scholar 

  13. Atkin, R.H.: Q-analysis. A hard language for the soft sciences. Futures, 492–499 (1978)

  14. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  15. Munkres, J.R.: Elements of Algebraic Topology. The Benjamin/Cummings Publication Company, Menlo Park (1984)

    MATH  Google Scholar 

  16. Bigo, L., Giavitto, J.-L., Spicher, A.: Building topological spaces for musical objects. In: Mathematics and Computation in Music, volume 6726 of LNCS. Springer, Paris (2011)

  17. Rehding, A.: Hugo Riemann and the Birth of Modern Musical Thought. Number 11 in New Perspectives in Music History and Criticism. Cambridge University Press (2003)

  18. Lewin, D.: Generalized musical intervals and transformations. Yale University Press (2007 reedition by Oxford University Press) (1987)

  19. Halsey, G.D., Hewitt, E.: Eine gruppentheoretische Methode in der Musiktheorie. Jahresbericht der Deutschen Mathematiker-Vereinigung 80, 151–207 (1978)

    MathSciNet  Google Scholar 

  20. Collins, N.: Enumeration of chord sequences. In: Sound and Music Computing. Aalborg University Copenhangen, Denmark (2012). SMC

    Google Scholar 

  21. Reiner, D.L.: Enumeration in music theory. Am. Math. Mon., 51–54 (1985)

  22. Fripertinger, H., Voitsberg, G.: Enumeration in musical theory. Institut für Elektronische Musik (IEM) (1992)

  23. Fripertinger, H.: Enumeration of mosaics. Discret. Math. 199(1), 49–60 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Fripertinger, H.: Enumeration and construction in music theory. In: Proceedings of the Diderot Forum on Mathematics and Music (Vienna), pp. 170–203 (1999)

  25. Mazzola, G., Muzzulini, D., Hofmann, G.R.: Geometrie der Töne: Elemente der Mathematischen Musiktheorie. Birkhäuser (1990)

  26. Mazzola, G., et al.: The topos of music. Birkhäuser, Basel (2002)

    Book  MATH  Google Scholar 

  27. Tymoczko, D.: The geometry of musical chords. Science 313(5783), 72–74 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. Mazzola, G.: Gruppen und Kategorien in der Musik: Entwurf einer mathematischen Musiktheorie, volume 10 of Reasearch and Exposition in Mathematics. Heldermann (1985)

  29. Bigo, L., Andreatta, M., Giavitto, J.-L., Michel, O., Spicher, A.: Computation and visualization of musical structures in chord-based simplicial complexes. In: Yust, J., Wild, J., Burgoyne, J.A. (eds.) Mathematics and Computation in Music, volume 7937 of Lecture Notes in Computer Science, pp 38–51. Springer, Berlin Heidelberg (2013)

  30. Nestke, A.: Paradigmatic motivic analysis. In: Perspectives in Mathematical and Computational Music Theory, Osnabrück Series on Music and Computation, pp. 343–365 (2004)

  31. Wille, R.: Musik und Mathematik: Salzburger Musikgespräch 1984 unter Vorsitz von Herbert von Karajan, chapter Musiktheorie und Mathematik, pp. 4–31. Springer (1985)

  32. Noll, T., Brand, M.: Morphology of chords. Perspect. Math. Comput. Music Theory 1, 366 (2004)

    Google Scholar 

  33. Schlemmer, T., Andreatta, M.: Using formal concept analysis to represent chroma systems. In: Mathematics and Computation in Music, pp. 189–200. Springer (2013)

  34. Forte, A.: The Structure of Atonal Music. Yale University Press (1973)

  35. Lewin, D.: Forte’s interval vector, my interval function, and Regener’s common-note function. J. Music Theory, 194–237 (1977)

  36. Bresson, J., Agon, C., Assayag, G.: Openmusic: Visual programming environment for music composition, analysis and research. In: Proceedings of the 19th ACM International Conference on Multimedia, pp. 743–746. ACM (2011)

  37. Read, R.C.: Combinatorial problems in the theory of music. Discret. Math. 167, 543–551 (1997)

    Article  MathSciNet  Google Scholar 

  38. Broué, M.: Les tonalités musicales vues par un mathématicien. Le temps des savoirs (Revue de l’Institut Universitaire de France), pp. 37–78. Odile Jacob (2001)

  39. Schlemmer, T., Schmidt, S.E.: A formal concept analysis of harmonic forms and interval structures. Ann. Math. Artif. Intell. 59(2), 241–256 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  40. Borchmann, D., Ganter, B.: Concept lattice orbifolds — first steps. In: Ferré, S., Rudolph, S. (eds.) Formal Concept Analysis: 7th International Conference, ICFCA 2009 Darmstadt, Germany, May 21–24, 2009 Proceedings. Springer Verlag, Berlin and Heidelberg (2009)

  41. Fripertinger, H.: Remarks on rhythmical canons. In: Fripertinger, H., Reich, L. (eds.) Proceedings of the Colloquium on Mathematical Music Theory, volume 347 of Grazer Math. Ber. pp. 73–90. Graz, Austria (2004)

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, University of Leeds, Leeds, UK

    Anton Freund

  2. CNRS - UMR 9912, IRCAM & Sorbonne Universités, UPMC (Université Paris 6), Paris, 06, France

    Moreno Andreatta & Jean-Louis Giavitto

  3. INRIA Rocquencourt MuTAnt team, Paris, France

    Jean-Louis Giavitto

Authors
  1. Anton Freund
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Moreno Andreatta
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jean-Louis Giavitto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anton Freund.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Freund, A., Andreatta, M. & Giavitto, JL. Lattice-based and topological representations of binary relations with an application to music. Ann Math Artif Intell 73, 311–334 (2015). https://doi.org/10.1007/s10472-014-9445-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-014-9445-3

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation