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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Hu, H., Gerhard, D. (2021). Modelling 4-Dimensional Tonal Pitch Spaces with Hopf Fibration. In: Kronland-Martinet, R., Ystad, S., Aramaki, M. (eds) Perception, Representations, Image, Sound, Music. CMMR 2019. Lecture Notes in Computer Science(), vol 12631. Springer, Cham. https://doi.org/10.1007/978-3-030-70210-6_10
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