Fourier (Common-Tone) Phase Spaces are in Tune with Variational Autoencoders’ Latent Space

Expanding upon the potential of generative machine learning to create atemporal latent space representations of musical-theoretical and cognitive interest, we delve into their explainability by formulating and testing hypotheses on their alignment with DFT phase spaces from \(\{0,1\}^{12}\) pitch classes and \(\{0,1\}^{128}\) pitch distributions – capturing common-tone tonal functional harmony and parsimonious voice-leading principles, respectively. We use 371 J.S. Bach chorales as a benchmark to train a Variational Autoencoder on a representative piano roll encoding. The Spearman rank correlation between the latent space and the two before-mentioned DFT phase spaces exhibits a robust rank association of approximately \(.65 \pm .05\) for pitch classes and \(.61 \pm .05\) for pitch distributions, denoting an effective preservation of harmonic functional clusters per region and parsimonious voice-leading. Furthermore, our analysis prompts essential inquiries about the stylistic characteristics inferred from the rank deviations to the DFT phase space and the balance between the two DFT phase spaces.