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Using Fundamental Groups and Groupoids of Chord Spaces to Model Voice Leading

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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

We model voice leading using tools from algebraic topology, principally the fundamental group, the orbifold fundamental group, and related groupoids. Doing so is a natural extension of modeling voice leading by continuous paths in chord spaces. The resulting algebraic precision in the representation of voice leadings and their concatenations allows for new distinctions between voice crossing cases, and enhanced connections with other approaches to voice leading.

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Notes

  1. 1.

    It may seem that by invoking topological invariants, we lose important geometrical information that allows for computation of the sizes of voice leadings. However, in the cases we consider, each path class can be represented by a unique geodesic, through which the geometrical information can be recovered.

  2. 2.

    Admittedly, if subsequent notes in a voice are staccato or separated by a rest, modeling the voice as a continuous function of time breaks down; however, if one allows, as suggested in a footnote in [3], that the first note is retained in the listener’s memory for some time beyond that of actual sound production, at least a perceptual sense of continuity can be retained. This idea was also suggested to the author by Richard Cohn in a recent conversation.

  3. 3.

    Topologists see this immediately by noticing that \(S^1\) is a deformation retract of the Möbius band.

  4. 4.

    i.e., where not all ordered pairs can be combined using an operation that would otherwise be a group operation.

  5. 5.

    This reverses the usual order for functional composition, for compatibility with path composition.

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Acknowledgments

I am especially grateful to George Dragomir for explaining orbifold paths and related matters. I am also grateful for communications with Hans Boden, Richard Cohn, Michael Davis, Guerino Mazzola, and Dmitri Tymoczko. I thank the anonymous reviewers for their many helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematical and Computer Sciences, Elizabethtown College, 1 Alpha Drive, Elizabethtown, PA, 17022, USA

    James R. Hughes

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  1. James R. Hughes
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Correspondence to James R. Hughes .

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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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Hughes, J.R. (2015). Using Fundamental Groups and Groupoids of Chord Spaces to Model Voice Leading. 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_28

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  • DOI: https://doi.org/10.1007/978-3-319-20603-5_28

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