Moment-Constrained Variable-Order Spectral Representations: Exact Attribute Control and a MUSHRA-Style Listening Test

When sinusoidal or resonator models change their number of components, conventional selection shifts the coarse spectral distribution abruptly. This note treats a frame’s components as a positive measure over log frequency and builds order-K representations that reproduce its total power, centroid, and spread to numerical precision: free Gaussian quadrature support, and a prescribed-support allocation that snaps the Gaussian nodes to the source’s partials and re-weights them non-negatively. Under a forced K = 16→8→4→8→16 schedule on synthetic and recorded sources, both methods hold the declared attributes to numerical precision in the representation (within 1e-11 cents) and within 0.01 cents (synthetic) and 6.4 cents (recorded) in phase-continuous renderings, and keep an auditory excitation pattern 2.8–4.9 dB from the full model, against 5.0–32.0 dB for four selection and quantization baselines. A MUSHRA-style listening test with a pre-specified analysis plan (after ITU-R BS.1534, with documented deviations; 12 screened listeners on headphones or earbuds; six excerpts) did not support the perceptual hypotheses, with or without post-screening: the prescribed-support method was rated farther from the full model than keeping the K strongest components (median paired difference −13 points) and not closer than masking-ranked selection or the Gaussian rule. Neither the excitation-pattern distance nor ISO 532-1 specific loudness predicted the ratings in this test. The demonstrated property of the method is therefore numerically exact control of declared coarse attributes through order changes, not perceptual closeness. It is implemented as the ORDER stage of the Ontogen effect (version 1.2) in Harmonic DAW. Supplementary archive: complete Python reference code (MIT), per-frame and summary results, renderings of the synthetic sources, and the listening-test plan, anonymous responses and analyses. The recordings used for the recorded sources are not included.