Profinite hierarchical patterns and prime-indexed multiscale invariants in symbolic music

We introduce a framework grounded in profinite completions and p-adic towers for analysing symbolic temporal sequences at prime-power indexed scales. For each prime p and level n, a kNN similarity graph on length-p^n patterns yields topological invariants—connected components (beta_0) and a giant-component fraction—that together quantify hierarchical coherence. A 26-piece MIDI corpus of J.S. Bach's Cello Suites serves as a benchmark: the beat-synchronised axis under p=2 and p=3 produces systematic prime separation, while seconds-axis and control primes p=5,7 do not, consistent with the metrical hypothesis. Profinite-map coherence audits, phase-sensitivity checks, and random-permutation null models corroborate the signal. An open reproducibility bundle accompanies this paper.SECIHTI (Mexico), grant CF-2019/217367