Advertisement

Insiders’ Choice: Studying Pitch Class Sets Through Their Discrete Fourier Transformations

  • Thomas NollEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11502)

Abstract

This contribution responds to a growing interest in the application of Discrete Fourier Transform (DFT) to the study of pitch class sets and pitch class profiles. Theoretical fundaments, references to previous work and explorations of various directions of study have been eloquently assembled by Emmanuel Amiot. Recent pioneering work in the application to music analysis and the reinterpretation of theoretical knowledge has been accomplished by Jason Yust. The intention of this paper is to show ways to make Yust’s strategies and methods more easily accessible and reproducible for a broader readership, especially students. This includes the introduction of concepts as well as interactive experiments with the help of computation and visualization tools.

The theoretical starting point is the interpretation of pitch class sets in terms of their characteristic functions, i.e. as pitch class profiles with values 0 and 1. Apart from the magnitudes of the respective partials, the study of their phases is particularly illuminating. The paper shows how the contents of this approach can be made accessible in a four steps proceedure.

Keywords

Discrete Fourier analysis Pitch class sets Pitch class profiles Two-phase-plots 

Notes

Acknowledgement

I thank my students of the course Teoria musical dels segles XX i XXI at ESMUC in Barcelona for their interest, commitment and feedback during the development of this material.

References

  1. 1.
    Amiot, E.: David Lewin and maximally even sets. J. Math. Music 1(3), 157–172 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Amiot, E.: The Torii of phases. In: Yust, J., Wild, J., Burgoyne, J.A. (eds.) MCM 2013. LNCS (LNAI), vol. 7937, pp. 1–18. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39357-0_1CrossRefzbMATHGoogle Scholar
  3. 3.
    Amiot, E.: Music Through Fourier Space: Discrete Fourier Transform in Music Theory. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-45581-5CrossRefzbMATHGoogle Scholar
  4. 4.
    Lewin, D.: Intervallic relations between two collections of notes. J. Music Theory 3(2), 298–301 (1959)CrossRefGoogle Scholar
  5. 5.
    Lewin, D.: Special cases of the interval function between pitch-class sets X and Y. J. Music Theory 42(2), 1–29 (2001)CrossRefGoogle Scholar
  6. 6.
    Noll, T., Amiot, E., Andreatta, M.: Fourier oracles for computer-aided improvisation. In: 2006 Proceedings of the ICMC: Computer Music Conference. Tulane University, New Orleans (2006)Google Scholar
  7. 7.
    Noll, T., Carle, M.: Fourier scratching: SOUNDING CODE. In: SuperCollider Conference, Berlin (2010)Google Scholar
  8. 8.
    Quinn, I.: General equal-tempered harmony. Perspect. New Music 44(2), 114–158, 45(1), 4–63 (2006/2007)Google Scholar
  9. 9.
    Yust, J.: Schubert’s harmonic language and fourier phase space. J. Music Theory 59(1), 121–181 (2015)CrossRefGoogle Scholar
  10. 10.
    Yust, J.: Distorted continuity: chromatic harmony, uniform sequences, and quantized voice leadings. Music Theory Spectr. 37(1), 120–143 (2015)CrossRefGoogle Scholar
  11. 11.
    Yust, J.: Applications of DFT to the theory of twentieth-century harmony. In: Collins, T., Meredith, D., Volk, A. (eds.) MCM 2015. LNCS (LNAI), vol. 9110, pp. 207–218. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-20603-5_22CrossRefGoogle Scholar
  12. 12.
    Yust, J.: Harmonic qualities in Debussy’s ‘Les sons et les parfums tournent dans l’air du soir’. J. Math. Music 11(2–3), 155–173 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Yust, J.: Probing questions about keys: tonal distributions through the DFT. In: Agustín-Aquino, O.A., Lluis-Puebla, E., Montiel, M. (eds.) MCM 2017. LNCS (LNAI), vol. 10527, pp. 167–179. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-71827-9_13CrossRefGoogle Scholar
  14. 14.
    Yust, J.: Geometric generalizations of the Tonnetz and their relation to fourier phases spaces. In: Montiel, M., Peck, R. (eds.) Mathematical Music Theory: Algebraic, Geometric, Combinatorial, Topological and Applied Approaches to Understanding Musical Phenomena. World Scientific (2018)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departament de Teoria, Composició i DireccióEscola Superior de Música de CatalunyaBarcelonaSpain

Personalised recommendations