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2015 | OriginalPaper | Chapter

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

Author : James R. Hughes

Published in: Mathematics and Computation in Music

Publisher: Springer International Publishing

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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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previous chapter Geometry, Iterated Quantization and Filtered Voice-Leading Spaces

next chapter All-Interval Structures

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.

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.

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

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

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

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Title: Using Fundamental Groups and Groupoids of Chord Spaces to Model Voice Leading
Author: James R. Hughes
Publisher: Springer International Publishing
Book: Mathematics and Computation in Music
Print ISBN: 978-3-319-20602-8

Electronic ISBN: 978-3-319-20603-5

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-3-319-20603-5_28

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