From Latent Cascade to Note Stream: How Scaling Exponents Survive Musical Quantization

Abstract

Generative systems built on multifractal processes make a quiet assumption: that the scaling structure of the latent process survives the discrete, nonlinear observation path that turns it into events. In the Fenophone — an instrument whose latent core is a Markov-switching multifractal (MSM) — that path is explicit and deterministic: band products are read into [0,1] lanes, mixed through a routing matrix into per-voice drives, thresholded into onsets, quantized onto a metric grid, and gated by harmony. We measure, across a control grid and seed panel, the detrended-fluctuation (DFA-1) exponents of the four latent lanes alongside those of the emitted onset and velocity streams, and the MFDFA width of both. We characterize the transformation empirically — routing decides which latent scaling the note stream inherits (only the routed mid-band lane's α transports; fixed-effects coefficient ≈ +0.32 against ≈ 0 for the unrouted fast band), while the channel's contrast decides how much of it registers (emitted α climbs ≈ 0.51 → 0.83 across the Intensity axis with every latent lane α constant, including exact low-contrast dead zones where adjacent dial positions emit bit-identical streams) — and explain the contrast axis with a level-crossing transfer model: the channel's quantizer is a monotone integer staircase, hence Hermite rank 1, so the latent exponent is preserved asymptotically at every contrast, and what contrast actually sets is the amplitude of the exponent-carrying component against the quantization-noise floor — the crossover scale a fixed measurement ladder reads as an effective α. A no-free-parameter mixture predictor built on that decomposition reproduces the measured climb cell by cell (median |Δα| = 0.009 on the model's theory rung). Three further maps complete the channel's characterization: the per-cell surrogate comparison shows the mid-Intensity width dip is a collapse into the finite-size floor (the dip cell beats neither its shuffle nor its IAAFT null), the latent-vs-emitted width map refutes the natural hypothesis that thresholding compresses multifractal width (emitted ≈ lane width; median per-run ratios 0.88–1.18), and the degeneracy map locates the usable region (the entire primary control plane is non-degenerate at 32 seeds, with all exact dead zones on one pack). The result is a practical design rule for multifractal instruments: a fitted α(Intensity, Spread) surface (R² 0.89–0.996) to invert for targets, which latent scaling survives musical quantization, and by what mechanism.