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A Mathematical Model of Tonal Function (I): Voice Leadings

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Mathematics and Computation in Music (MCM 2022)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13267))

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Abstract

This paper is the first in a series of two papers on mathematical models of tonal function. In this first paper, we present a mathematical model of tonal function whose scope is limited to classic music from the common practice period. After a formalization of some harmonic elements (pitch classes, chords, arrangements, voicings, voice leadings), the model of tonal function is described. Our model is based on voice leadings and the tonal function is defined in terms of them. A combinatorial optimization algorithm is used to determine the tonal function. In this work, only chords with the same number of voices are considered. The general case was left for the second paper of the series; in the second paper the model of tonal function is generalized.

Universidad Politécnica de Madrid.

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Notes

  1. 1.

    \(\varDelta \) here denotes the major seventh chord, not the nabla distance. The symbol is the same, but the context determines its meaning without confusion.

References

  1. Agustín-Aquino, O.A., Mazzola, G.: Modulation in tetradic harmony and its role in jazz. J. Math. Music 14(1), 58–65 (2020). https://doi.org/10.1080/17459737.2019.1655673

    Article  Google Scholar 

  2. Benson, D.: Music: A Mathematical Offering. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  3. Bry, H., et al.: Harmonic function. http://openmusictheory.com/harmonicFunctions.html, Access Dec 2021

  4. Caplin, W.: Tonal function and metrical accent: a historical perspective. Music Theory Spect. 5, 1–14 (1983)

    Article  Google Scholar 

  5. Carpenter, P.: Grundgestalt as tonal function. Music Theory Spect. 5, 15–38 (1983)

    Article  Google Scholar 

  6. Cohn, R.: Neo-riemannian operations, parsimonious trichords, and their tonnetz representations. J. Music Theory 41(1), 1–66 (1997)

    Article  Google Scholar 

  7. Cohn, R.: An introduction to neo-riemannian theory: a survey and historical perspective context and harmonic function in tonal music. J. Music Theory 42(2), 167–180 (1998)

    Article  Google Scholar 

  8. Cohn, R.: Square dances with cubes. J. Music Theory 42(2), 283–296 (1998). http://www.jstor.org/stable/843879

  9. Dudeque, N.: Schoenberg on tonal function (1997). http://www.rem.ufpr.br/_REM/REMv2.1/vol2.1/Schoenberg/Schoenberg_on_Tonal.html

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

    Google Scholar 

  11. Gollin, E., Rehding, A.: Oxford Handbook of Neo-Riemannian Music Theories. Oxford University Press, New York (2011)

    Book  Google Scholar 

  12. Harrison, D.G.: Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of Its Precedents. University of Chicago Press, Chicago (1994)

    Google Scholar 

  13. Hyer, B.: Reimag(in)ing riemann. J. Music Theory 39(1), 101–138 (1995)

    Article  Google Scholar 

  14. Hyer, B.: Tonality. Grove Music (2001). https://doi.org/10.1093/gmo/9781561592630.article.28102

  15. Kopp, D.: On the function of function. Music Theory Online 1(3), 1–8 (1955)

    Google Scholar 

  16. Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Res. Logist. Q. 2, 83–97 (1955)

    Article  Google Scholar 

  17. Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press, New Haven (1987)

    Google Scholar 

  18. Piston, W.: Harmony. Gollancz, London (1950)

    Google Scholar 

  19. Pozo, I., Gómez, F.: Formalization of voice-leadings and the nabla algorithm. In: Proceedings of Mathematics and Computation in Music 2019, Madrid, Spain (2019)

    Google Scholar 

  20. Pozo, I., Gómez, F.: A mathematical model of tonal function (I): voice leadings. In: Proceedings of Mathematics and Computation in Music 2019, Atlanta, USA (2022)

    Google Scholar 

  21. Quinn, I.: Harmonic function without primary triads. In: Paper Delivered at the Annual Meeting of the Society for Music Theory in Boston (2005)

    Google Scholar 

  22. Riemann, H.: Harmony simplified: or, The theory of the tonal functions of chords. Augener Ltd., London (1896). Translation of Vereinfachte Harmonielehre (1893). Republished by Alpha Editions, September 2019, London

    Google Scholar 

  23. SMCM: Proceedings of mathematics and computation in music (2022). http://mcm2022.org/

  24. Wason, R.E.: Viennese Harmonic Theory from Albrecthsberger to Schenker and Schoenberg, Ann Arbor, London (1985)

    Google Scholar 

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Acknowledgements

We thank the reviewers for the extremely lucid and valuable comments they provided us with.

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Authors and Affiliations

  1. Universidad Politécnica de Madrid, Madrid, Spain

    Isaac del Pozo & Francisco Gómez-Martín

Authors
  1. Isaac del Pozo
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  2. Francisco Gómez-Martín
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Corresponding author

Correspondence to Francisco Gómez-Martín .

Editor information

Editors and Affiliations

  1. Georgia State University, Atlanta, GA, USA

    Mariana Montiel

  2. Technological University of the Mixteca, Huajuapan de León, Mexico

    Octavio A. Agustín-Aquino

  3. Universidad Politécnica de Madrid, Madrid, Spain

    Francisco Gómez

  4. Life University, Marietta, GA, USA

    Jeremy Kastine

  5. National Autonomous University of Mexico, Mexico City, Distrito Federal, Mexico

    Emilio Lluis-Puebla

  6. Georgia State University, Atlanta, GA, USA

    Brent Milam

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Pozo, I.d., Gómez-Martín, F. (2022). A Mathematical Model of Tonal Function (I): Voice Leadings. In: Montiel, M., Agustín-Aquino, O.A., Gómez, F., Kastine, J., Lluis-Puebla, E., Milam, B. (eds) Mathematics and Computation in Music. MCM 2022. Lecture Notes in Computer Science(), vol 13267. Springer, Cham. https://doi.org/10.1007/978-3-031-07015-0_18

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  • DOI: https://doi.org/10.1007/978-3-031-07015-0_18

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