Does the Markov-Switching Multifractal describe real music? A fitting pipeline and a statistical Turing test

Abstract

The Markov-Switching Multifractal (MSM) is a parsimonious volatility model — a product of k independent two-state switching multipliers — that the Fenophone instrument repurposes as a generative music engine, producing onset streams with measured DFA exponents alpha ~ 0.70-0.78 and multifractal widths ~ 0.05-0.13. We ask whether the MSM also describes real music. We build a likelihood-based fitting pipeline that treats the MSM as a 2^k-state hidden Markov model, recover its parameters from synthetic data within ~10% at 10^4 ticks, and pose the closing question as a statistical Turing test: resynthesise from the fitted parameters and ask whether a classifier can separate real from surrogate on held-out summary statistics. Fitting 1,264 MAESTRO piano performances and 198 Groove drum takes (beat-time grid, bar covariate), we find a clean dissociation: real piano onset streams are strongly persistent and multifractal (median DFA alpha = 0.717, width 0.118 — inside the instrument's own published band), the fitted cascade is genuinely non-flat (m_hi_eff = 1.62), and resynthesis retains most of the persistence (alpha = 0.691) — while drum-kit onset counts are near-white (alpha = 0.475) with the cascade structure appearing instead on the velocity channel (vel_m_hi ~ 1.36-1.52 in both corpora). The fitted model is still not sufficient at classifier resolution (accuracy 0.758 piano / 0.932 drums vs a ~0.50 permutation null), so MSM-Cox describes real musical intensity at second order without yet being a complete generator of it. A competitive arm (§5.1) places both the engine and the resynthesis on Zhang et al.'s (2025) audio-envelope battery against 381 Billboard hits and three neural generators, domain-matched: the asymmetry we predicted inverts — the engine holds spectrum-shape dispersion where the neural systems lose it (alpha_peak 1.25x the human IQR against their 0.70-0.86x — under the IQR ratio only; the comparison reverses under the std and span ratios in the same file, §5.1) while our arms fall outside the human band on the DFA alpha family that all three of them match. A subsequent envelope-domain IAAFT null (§5.1, E8) — the first placed under the width family in that domain — leaves 5 of 12 shipped packs' Delta-alpha indistinguishable from a linear surrogate.