Mathematics and Soft Computing in Music

  • Teresa León
  • Vicente Liern
Chapter
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 273)

Abstract

Mathematics is the fundamental tool for dealing with the physical processes that explain music but it is also in the very essence of this art. Musical notes, the first elements which music works with, are defined for each tuning system as very specific frequencies; however, instrumentalists know that small changes in these values do not have serious consequences. In fact, sometimes consensus is only reached if the entire orchestra alters the theoretical pitches. The explanation for this contradiction is that musicians implicitly handle very complex mathematical processes involving some uncertainty in the concepts and this is better explained in terms of fuzzy logic. Modelling the notes as fuzzy sets and extending the concept of tuning systems lead us to a better understanding on how musicians work in real-life. The notes offered by a musician during a performance should be compatible with the theoretical ones but not necessarily equal. A numerical experiment, conducted with the help of a saxophonist, illustrates our approach and also points to the need for considering sequential uncertainty.

Keywords

Fuzzy Logic Fuzzy Number Soft Computing Trapezoidal Fuzzy Number Tuning System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Apel, W.: Harvard Dictionary of Music: Second Edition, Revised and Enlarged, 2nd edn. The Belknap Press of Harvard University Press, Cambridge, Massachussets (1994)Google Scholar
  2. 2.
    Arcos, J.L., De Mantaras, R.L.: An interactive case-based reasoning approach for generating expressive music. Applied Intelligence 14, 115–129 (2001)MATHCrossRefGoogle Scholar
  3. 3.
    Benson, D.: Music: a Mathematical Offering. Cambridge University Press, Cambridge (2006), http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf Google Scholar
  4. 4.
    Borup, H.: A History of String Intonation, http://www.hasseborup.com/ahistoryofintonationfinal1.pdf
  5. 5.
    Bosteels, K., Kerre, E.E.: A fuzzy framework for defining dynamic playlist generation heuristics. Fuzzy Sets and Systems 160, 3342–3358 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Christina, B.: A robust fuzzy logic-based step-gain control for adaptive filters in acoustic echo cancellation. IEEE Transactions on Speech and Audio Processing 9(2), 162–167 (2001)CrossRefGoogle Scholar
  7. 7.
    Cádiz, R.F.: Fuzzy logic in the arts: applications in audiovisual composition and sound synthesis. In: Proceedings of NAFIPS 2005 - 2005 Annual Meeting of the North American Fuzzy Information Processing Society, pp. 551–556. IEEE/IET Electronic Library, IEL (2005)Google Scholar
  8. 8.
    Civanlar, M.R., Joel Trussel, H.: Digital restoration using fuzzy sets. IEEE Transactions on Acoustics, Speech and Signal Processing ASSF-34(4), 919–936 (1986)CrossRefGoogle Scholar
  9. 9.
    Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980)MATHGoogle Scholar
  10. 10.
    Elsea, P.: Fuzzy logic and musical decisions (1995), http://arts.ucsc.edu/EMS/Music/research/FuzzyLogicTutor/FuzzyTut.html
  11. 11.
    Goldáraz Gaínza, J.J.: Afinación y temperamento en la música occidental. Alianza Editorial, Madrid (1992)Google Scholar
  12. 12.
    Hall, R.W., Josíc, K.: The mathematics of musical instruments. The American Mathematical Monthly 108, 347–357 (2001)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Haluska, J.: Equal temperament and Pythagorean tuning: a geometrical interpretation in the plane. Fuzzy Sets and Systems 114, 261–269 (2000)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Jan, H.: Uncertainty measures of well tempered systems. International Journal of General Systems 31, 73–96 (2002)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Haluska, J.: The Mathematical Theory of Tone Systems. Marcel Dekker, Inc., Bratislava (2005)Google Scholar
  16. 16.
    Kiranyaz, S., Qureshi, A.F., Gabbouj, M.: A Generic Audio Classification and Segmentation Approach for Multimedia Indexing and Retrieval. IEEE Transactions on Audio, Speech, and Language Processing 14(3), 1062–1081 (2006)CrossRefGoogle Scholar
  17. 17.
    Lattard, J.: Gammes et tempéraments musicaux. Masson Éditions, Paris (1988)Google Scholar
  18. 18.
    Lesaffre, M., Leman, M., Martens, J.-P.: A User-Oriented Approach to Music Information Retrieval. Dagstuhl Seminar Proceedings 06171 (2006), http://drops.dagstuhl.de/opus/volltexte/2006/650
  19. 19.
    Vicente, L.: Fuzzy tuning systems: the mathematics of the musicians. Fuzzy Sets and Systems 150, 35–52 (2005)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Piles Estellés, J.: Intervalos y gamas. Ediciones Piles, Valencia (1982)Google Scholar
  21. 21.
    Wold, E., Blum, T., Keislar, D., Wheaton, J.: Content-Based Classification, Search, and Retrieval of Audio. IEEE MultiMedia Archive 3(3), 27–36 (1996)CrossRefGoogle Scholar
  22. 22.
    Yilmaz, A.E., Telatar, Z.: Potential applications of fuzzy logic in music. In: Proceedings of the 18th IEEE International Conference on Fuzzy Systems, pp. 670–675. IEEE Press, Piscataway (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Teresa León
    • 1
  • Vicente Liern
    • 2
  1. 1.Department of Statistics and Operations ResearchUniversity of ValenciaValenciaSpain
  2. 2.Departmento de Matemática para la Economía y la EmpresaUniversity of ValenciaValenciaSpain

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