The Markov-Switching Multifractal as a Generative Model of Musical Intensity: An Instrument, Measurements, and a Research Programme

Abstract

Musical intensity — the joint trajectory of onset density, dynamics, and harmonic tension — shares a distinctive phenomenology with financial volatility: bursts and lulls, clustered at every time scale, with no single characteristic period. We propose taking this correspondence literally. We adopt the Markov-Switching Multifractal (MSM) of Calvet and Fisher — a stationary, causal Markov construction of multifractal volatility — not as a metaphor or a texture source but as the generative core of a musical instrument: we discard the return process entirely and sonify the volatility state itself, read through a fixed band map into four control lanes (dynamics, density, timbre energy, harmonic tension) that drive an authored musical scaffold. We describe a complete reference implementation (the Fenophone), whose event layer is integer/fixed-point and bit-identical across platforms, whose player controls are exactly the MSM parameters (m_hi, f_slow, S, k), and whose "conducting" gesture is clamping individual latent states. We report measured properties of the emitted note streams: with the authored bar template removed as a seasonal trend, per-scaffold median DFA exponents α(onset) of 0.695–0.770 and α(velocity) of 0.676–0.702, every seed exceeding its shuffled-surrogate null; MFDFA widths h(−2) − h(2) of 0.054–0.126, exceeding matched IAAFT surrogate nulls on every seed for three of the four measured packs (the fourth, the least intermittent by design, is consistent with linear long memory at this length and reported as such); and a monotone dose–response of α on the time-scale-span control, with a median gap ≥ 0.09 between span extremes. All measurements are CI-gated, surrogate-compared, estimator-anchored on known-Hurst ground truth, and reproducible from pinned seeds. We then lay out a six-experiment research programme — surrogate discrimination, a preference surface over the (α, width) plane, tension-lane validation, boundary perception against ground-truth latents, counterfactual intervention pairs, and corpus inversion — each with predictions and explicit falsification conditions. The corpus half of the inversion experiment now has first results (2026-07-12; companion analysis): 1,260 MAESTRO piano performances measure median α(onset) 0.717 and width 0.118 under the same protocol — inside the instrument's published band — with fitted parameters identifying composer above chance, while drum-kit onset counts are near-white with the cascade structure appearing on the velocity channel instead. The five perceptual experiments (E1–E5) remain unrun, and the corpus inversion (E6) is complete only on its statistical half; this paper is a preregistration-grade programme, not a report of finished perceptual results.