Learning the helix topology of musical pitch

TL;DR

This paper presents a method to uncover the helical structure of musical pitch from unlabeled audio by using manifold learning, revealing the octave-related helix in musical notes and speech.

Contribution

It introduces a novel approach combining correlation analysis and Isomap to discover the helix topology of pitch directly from audio data.

Findings

The manifold resembles a helix, completing a full turn at each octave.

A circular pitch shape is observed in speech but not in urban noise.

Design choices like instrument type and neighbor count affect visualization.

Abstract

To explain the consonance of octaves, music psychologists represent pitch as a helix where azimuth and axial coordinate correspond to pitch class and pitch height respectively. This article addresses the problem of discovering this helical structure from unlabeled audio data. We measure Pearson correlations in the constant-Q transform (CQT) domain to build a K-nearest neighbor graph between frequency subbands. Then, we run the Isomap manifold learning algorithm to represent this graph in a three-dimensional space in which straight lines approximate graph geodesics. Experiments on isolated musical notes demonstrate that the resulting manifold resembles a helix which makes a full turn at every octave. A circular shape is also found in English speech, but not in urban noise. We discuss the impact of various design choices on the visualization: instrumentarium, loudness mapping function, and number of neighbors K.