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On Linear Algebraic Representation of Time-span and Prolongational Trees

  • Satoshi TojoEmail author
  • Alan Marsden
  • Keiji Hirata
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11265)

Abstract

In constructive music theory, such as Schenkerian analysis and the Generative Theory of Tonal Music (GTTM), the hierarchical importance of pitch events is conveniently represented by a tree structure. Although a tree is easy to recognize and has high visibility, such an intuitive representation can hardly be treated in mathematical formalization. Especially in GTTM, the conjunction height of two branches is often arbitrary, contrary to the notion of hierarchy. Since a tree is a kind of graph, and a graph is often represented by a matrix, we show the linear algebraic representation of trees, specifying conjunction heights. Thereafter, we explain the ‘reachability’ between pitch events (corresponding to information about reduction) by the multiplication of matrices. In addition we discuss multiplication with vectors representing a sequence of harmonic functions, and suggest the notion of stability. Finally, we discuss operations between matrices to model compositional processes with simple algebraic operations.

Keywords

Time-span tree Prolongational tree Generative Theory of Tonal Music Matrix Linear algebra 

Notes

Acknowledgements

This work is supported by JSPS Kaken 16H01744 and BR160304.

References

  1. 1.
    Hamanaka, M., Hirata, K., Tojo, S.: Implementing methods for analysing music based on Lerdahl and Jackendoff’s Generative Theory of Tonal Music. In: Meredith, D. (ed.) Computational Music Analysis, pp. 221–249. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-25931-4_9CrossRefzbMATHGoogle Scholar
  2. 2.
    Hirata, K., Tojo, S., Hamanaka, M.: Cognitive similarity grounded by tree distance from the analysis of K.265/300e. In: Aramaki, M., Derrien, O., Kronland-Martinet, R., Ystad, S. (eds.) CMMR 2013. LNCS, vol. 8905, pp. 589–605. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-12976-1_36CrossRefGoogle Scholar
  3. 3.
    Hirata, K., Tojo, S., Hamanaka, M.: An algebraic approach to time-span reduction. In: Meredith, D. (ed.) Computational Music Analysis, pp. 251–270. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-25931-4_10CrossRefzbMATHGoogle Scholar
  4. 4.
    Kostka, S., Payne, D.: Tonal Harmony. Knopf, New York (2003)Google Scholar
  5. 5.
    Lehrdahl, F., Jackendoff, R.: A Generative Theory of Tonal Music. The MIT Press, Cambridge (1983)Google Scholar
  6. 6.
    Lerdhal, F.: Tonal Pitch Space. Oxford University Press, New York (2001)Google Scholar
  7. 7.
    Marsden, A., Hirata, K., Tojo, S.: Towards computable procedures for deriving tree structures in music: context dependency in GTTM and Schenkerian theory. In: Proceedings of the Sound and Music Computing Conference, pp. 360–367 (2013)Google Scholar
  8. 8.
    Matsubara, M., Kodama, and Tojo, S.: Revisiting cadential retention in GTTM. In: Proceedings of KSE/IEEE 2016 (2016)Google Scholar
  9. 9.
    Riemann, H.: Vereinfachte Harmonielehre. Augener, London (1893)Google Scholar
  10. 10.
    Sakamoto, S., Arn, S., Matsubara, M., Tojo, S.: Harmonic analysis based on Tonal Pitch Space. In: Proceedings of KSE/IEEE 2016 (2016)Google Scholar
  11. 11.
    Tojo, S., Hirata, K.: Structural similarity based on time-span tree. In: Aramaki, M., Barthet, M., Kronland-Martinet, R., Ystad, S. (eds.) CMMR 2012. LNCS, vol. 7900, pp. 400–421. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-41248-6_23CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Japan Advanced Institute of Science and TechnologyNomiJapan
  2. 2.Lancaster UniversityBailrigg, LancasterUK
  3. 3.Future University HakodateHakodate, HokkaidoJapan

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