The Circle of Fifths as Latent Geometry in Bach's Well-Tempered Clavier

Najla Sadek, Joseph Bakarji

Keywords: Deep Learning, Music Information Retrieval, Autoencoders, Latent Space Analysis, Circle of Fifths, Bach

TL;DR: A dimensionality reduction using deep autoencoder and PCA reveals the circle of fifths in the two dominant modes of Bach's Well-Tempered Clavier

Abstract: Can unsupervised deep learning methods encode fundamental music-theoretic features? We answer this question by training an autoencoder on J.S. Bach’s Well-Tempered Clavier and analyzing its latent space via principal component analysis. Sequences in the first two principal components are clustered hierarchically into pieces and keys that spontaneously arrange in a circle-of-fifths geometry. Quantitatively, relative major-minor key pairs (sharing pitch collections) lie more than three times closer than non-relative pairs, and circle-of-fifths distance correlates strongly with learned distances. This structure emerges entirely from reconstruction loss, with no harmonic labels or supervision. Our results suggest that the circle of fifths is an intrinsic property of tonal relationships, demonstrating that unsupervised representation learning can recover harmonic principles that open the door for interpretable data-driven exploration of latent spaces across diverse musical traditions.

Submission Number: 30