Bach autoencoder latent space recovers the circle of fifths

Sadek & Bakarji train an autoencoder on J.S. Bach's Well-Tempered Clavier (24 preludes and fugues per book, all 24 major/minor keys) with reconstruction loss only, using no harmonic labels [sadek-bakarji-2026-circle-of-fifths] Projected onto the first two principal components, pieces cluster hierarchically by piece and by key, and the key clusters arrange into the classical circle-of-fifths ordering [sadek-bakarji-2026-circle-of-fifths] Relative major-minor key pairs (sharing a pitch collection) sit more than three times closer in latent space than non-relative pairs [sadek-bakarji-2026-circle-of-fifths] Classical circle-of-fifths distance between keys correlates strongly with learned latent distance, so the structure is a byproduct of compression rather than a supervised target [sadek-bakarji-2026-circle-of-fifths] Full text was inaccessible (OpenReview challenge, unparseable PDF mirrors), so claims rest on the peer-reviewed PMLR abstract and no causal-validation claim is made [sadek-bakarji-2026-circle-of-fifths]