Tropical Generalized Interval Systems

  • Giovanni AlbiniEmail author
  • Marco Paolo Bernardi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11502)


This paper aims to refine the formalization of David Lewin’s Generalized Interval System (GIS) by the means of tropical semirings. Such a new framework allows to broaden the GIS model introducing a new operation and consequently new musical and conceptual insights and applications, formalizing consistent relations between musical elements in an original unified structure. Some distinctive examples of extensions of well-known infinite GIS for lattices are then offered and the impossibility to build tropical GIS in the finite case is finally proven and discussed.


Tropical semiring Ordering Generalized Interval System 



We thank Claudio Bernardi (Università di Roma “La Sapienza”, Department of Mathematics) and the reviewers for their useful suggestions and remarks.


  1. 1.
    Adhikari, M.R., Adhikari, A.: Basic Modern Algebra with Applications. Springer, New Delhi (2014). Scholar
  2. 2.
    Ahsan, J., Mordeson, J.N., Shabir, M.: Fuzzy Semirings with Applications to Automata Theory. Springer, Heidelberg (2012). Scholar
  3. 3.
    Babbitt, M.: Twelve-tone invariants as compositional determinants. Music. Q. 46, 246–259 (1960)CrossRefGoogle Scholar
  4. 4.
    Bailhache, P.: Music translated into Mathematics: Leonhard Euler. In: Proceedings of Problems of Translation in the 18th Century, Nantes (1997)Google Scholar
  5. 5.
    Cuninghame-Green, R.: Minimax Algebra. Springer, Heidelberg (1979). Scholar
  6. 6.
    Euler: Tentamen novae theoriae musicae ex certissimis harmoniae principiis dilucide expositae (1737)Google Scholar
  7. 7.
    Forte, A.: The Structure of Atonal Music. Yale University Press, London (1973)Google Scholar
  8. 8.
    Gunawardena, J.: Idempotency. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  9. 9.
    Halmos, P., Givant, S.: Introduction to Boolean Algebras. Undergraduate Texts in Mathematics. Springer, New York (2009). Scholar
  10. 10.
    Hofmann-Engl, L.: Consonance/Dissonance - a historical perspective. In: Demorest, S.M., Morrison, S.J., Campbell, P.S. (eds.) Proceedings of the 11th International Conference on Music Perception and Cognition (ICMPC 2011), Seattle, Washington, USA (2011)Google Scholar
  11. 11.
    Krivulin, N.: Tropical optimization problems. In: Advances in Economics and Optimization: Collected Scientific Studies Dedicated to the Memory of L. V. Kantorovich, pp. 195–214. Nova Science Publishers, New York (2014)Google Scholar
  12. 12.
    Lewin, D.: A label free development for 12-pitch class systems. J. Music Theory 21, 29–48 (1977)CrossRefGoogle Scholar
  13. 13.
    Lewin, D.: On generalized intervals and transformations. J. Music Theory 24(2), 243–251 (1980)CrossRefGoogle Scholar
  14. 14.
    Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press, New Haven and London (1987)Google Scholar
  15. 15.
    Lewin, D.: Generalized interval systems for Babbitt’s lists, and for Schoenberg’s string trio. Music Theory Spectr. 17(1), 81–118 (1995)CrossRefGoogle Scholar
  16. 16.
    Popoff, A., Andreatta, M., Ehresmann, A.: A categorical generalization of Klumpenhouwer networks. In: Collins, T., Meredith, D., Volk, A. (eds.) MCM 2015. LNCS (LNAI), vol. 9110, pp. 303–314. Springer, Cham (2015). Scholar
  17. 17.
    Popoff, A., Agon, C., Andreatta, M., Ehresmann, A.: From K-nets to PK-nets: a categorical approach. Perspect. New Music 54(2), 5–63 (2016)CrossRefGoogle Scholar
  18. 18.
    Rahn, J.: Cool tools: polysemic and non-commutative nets, subchain decompositions and cross-projecting pre-orders, object-graphs, chain-hom-sets and chain-label-hom-sets, forgetful functors, free categories of a net, and ghosts. J. Math. Music 1(1), 7–22 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Vandiver, H.S.: Note on a simple type of algebra in which the cancellation law of addition does not hold. Bull. Am. Math. Soc. 40, 914–920 (1934)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Speyer, D., Sturmfels, B.: Tropical Mathematics. Math. Mag. 82, 3 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Conservatorio “Jacopo Tomadini”UdineItaly
  2. 2.Eesti Muusika- ja TeatriakadeemiaTallinnEstonia
  3. 3.ISSM “Franco Vittadini”PaviaItaly
  4. 4.Università degli Studi di PaviaPaviaItaly

Personalised recommendations