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2021 | OriginalPaper | Chapter

Modelling 4-Dimensional Tonal Pitch Spaces with Hopf Fibration

Authors : Hanlin Hu, David Gerhard

Published in: Perception, Representations, Image, Sound, Music

Publisher: Springer International Publishing

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Abstract

The question of how to arrange harmonically related pitches in space is a historical research topic of computational musicology. The primitive of note arrangement is linear in 1-D, in which ordered ascending pitches in one direction correspond to increasing frequencies. Euler represented harmonic relationships between notes with a mathematical lattice named Tonnetz, which extends the 1-D arrangement into 2-D space by reflecting consonances. Since then, mathematicians, musicians, and psychologists have studied this topic for hundreds of years. Recently, pitch-space modelling has expanded to mapping musical notes into higher-dimensional spaces. This paper aims to investigate existing tonal pitch space models, and to explore a new approach of building a pitch hyperspace by using the Hopf fibration.

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Title: Modelling 4-Dimensional Tonal Pitch Spaces with Hopf Fibration
Authors: Hanlin Hu
David Gerhard
Publisher: Springer International Publishing
Book: Perception, Representations, Image, Sound, Music
Print ISBN: 978-3-030-70209-0

Electronic ISBN: 978-3-030-70210-6

Copyright Year: 2021
DOI: https://doi.org/10.1007/978-3-030-70210-6_10

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