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(2015) Mathematics and computation in music, Dordrecht, Springer.
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 class="EmphasisTypeItalic ">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.
Publication details
DOI: 10.1007/978-3-319-20603-5_27
Full citation:
Callender, C. (2015)., Geometry, iterated quantization and filtered voice-leading spaces, in T. Collins, D. Meredith & A. Volk (eds.), Mathematics and computation in music, Dordrecht, Springer, pp. 257-266.
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