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Geometry, Iterated Quantization and Filtered Voice-Leading Spaces

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Mathematics and Computation in Music (MCM 2015)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9110))

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Abstract

A recent special issue of the Journal of Mathematics and Music on mathematical theories of voice leading focused on the intersections of geometrical voice-leading spaces (GVLS), filtered point-symmetry (FiPS) and iterated quantization, and signature transformations. In this paper I put forth a theoretical model that unifies all of these approaches. Beginning with the basic configuration of FiPS, allowing the n points of a filter or beacon to vary arbitrarily yields the continuous chord space of n voices (\(T^n/S_n\)). Each point in the filter space induces a quantization or Voronoi diagram on the beacon space. The complete space of filter and beacon is a singular fiber bundle, combining the power and generalization of GVLS with the central FiPS insight of iterated filtering by harmonic context. Additionally, any of the sixteen types of generalized voice-leading spaces described by Callender, Quinn, and Tymoczko can be used as filters/beacons to model different contexts.

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Notes

  1. 1.

    This paper does not touch upon the potential intersections of Fourier-based approaches to harmony/chord quality and voice leading discussed in [1, 4, 11, 12, 16].

  2. 2.

    In the usual depiction of such configurations, the output of one ring passes through the nearest point of the next ring in a counter-clockwise direction. Allowing outputs to pass through the nearest point in either direction makes the connection with geometrical voice-leading and, in particular, Voronoi diagrams clearer.

  3. 3.

    However, because the filter and beacon are strictly even divisions of the octave, the region shown in Fig. 1 serves as a fundamental region for the entire configuration space, which can be formed by identifying the vertical and horizontal boundaries to form a torus.

  4. 4.

    There are cases involving singularities in either the beacon or the filter space where these properties break down, but the details are not essential for present purposes.

  5. 5.

    I would like to thank my colleagues Eriko Hironaka and Paolo Aluffi for a number of helpful mathematical suggestions.

  6. 6.

    Indeed, depending on how the filters associated with these paths are ordered, there will be n! different configuration spaces.

  7. 7.

    OpenMusic is a visual programming environment for computer assisted composition designed and developed by the Musical Representations Team (Gerard Assayag, head) at IRCAM. Visit http://forumnet.ircam.fr/product/openmusic/.

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Author information

Authors and Affiliations

  1. College of Music, Florida State University, Tallahassee, FL, 32306, USA

    Clifton Callender

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  1. Clifton Callender
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Correspondence to Clifton Callender .

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Editors and Affiliations

  1. De Montfort University, Leicester, United Kingdom

    Tom Collins

  2. Aalborg University, Aalborg, Denmark

    David Meredith

  3. Utrecht University, Utrecht, The Netherlands

    Anja Volk

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Callender, C. (2015). Geometry, Iterated Quantization and Filtered Voice-Leading Spaces. In: Collins, T., Meredith, D., Volk, A. (eds) Mathematics and Computation in Music. MCM 2015. Lecture Notes in Computer Science(), vol 9110. Springer, Cham. https://doi.org/10.1007/978-3-319-20603-5_27

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  • DOI: https://doi.org/10.1007/978-3-319-20603-5_27

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20602-8

  • Online ISBN: 978-3-319-20603-5

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