Bayesian tonal-state geometry in Western art music: Uncertainty and displacement jointly determine tonal continuity

Abstract Historical change in Western art music is often described as a gradual expansion beyond the pitch and harmonic resources of the prevailing key. Existing studies have traced corresponding developments in pitch organization and key-transition structure, but have not directly quantified tonal continuity across successive local contexts. This study develops a Bayesian framework in which each temporal window is represented as a posterior distribution over 24 major and minor keys. Summed posterior concentration S Σ serves as the concentration counterpart to within-context uncertainty, whereas squared Euclidean distance D ² provides a Bayesian-surprise-like measure of between-context displacement. Their joint geometry determines successive-state overlap and, through a monotonic transformation, the leakage threshold λ: the fraction of inherited prior weight that must be replaced by a uniform, key-neutral component before the two receive equal posterior support. Thus, λ indexes the tonal contribution to continuity across successive contexts, with lower values indicating more fragile tonal continuity. The evidence-strength parameters governing posterior sharpness were bracketed from two parallel-mode modulations in Mozart by requiring each modulation to register as a distinct tonal event while retaining continuity with the preceding tonal context. The framework was applied to 26 subcorpora of the Distant Listening Corpus, composed between 1736 and 1925, at a 4-bar integration span. Mean λ declined with subcorpus median composition year, indicating that tonal continuity between successive contexts weakened in later repertoire. Episodes with λ < .5 were uncommon but theoretically informative, spanning a continuum from high posterior concentration with large displacement to lower concentration with smaller displacement. Under such conditions, emerging tonal interpretations may gain support with weaker constraint from the preceding tonal context, although listener-side attenuation may also depend on other musical and cognitive factors.