Modelling 4-Dimensional Tonal Pitch Spaces with Hopf Fibration

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.