A spectrum and a scale are 'related' when the spectrum's dissonance curve has minima at the scale's steps

Sethares’ key design principle: a spectrum S and a scale T are ‘related’ if, when two sounds with spectrum S are sounded at each interval in T, the resulting sensory dissonance is low (i.e., the dissonance curve of S has local minima at the scale steps of T). Related pairs are ‘natural’ for each other: the scale exploits the consonant intervals of the spectrum, and the spectrum reinforces the harmonic logic of the scale. This is a two-way relationship: given a spectrum, compute the related scale (find minima of the dissonance curve). Given a scale, find or synthesize a related spectrum (solve the inverse problem). The principle unifies Western, gamelan, Thai, and xenharmonic music under one framework: each is a different matched spectrum-scale pair.

Examples

Harmonic spectrum → related scale has minima near JI ratios → Western diatonic/chromatic scales. Gamelan metallophone spectrum (inharmonic) → related scale has minima near pelog or slendro intervals → Javanese gamelan scales. Ideal bar spectrum → related scale has minima near 7-tet steps → Thai pi phat tuning.

Assessment

You analyze an instrument and find its spectrum has prominent partials at 1.0, 1.87, 3.51, 5.67 (relative to fundamental). Describe the procedure to determine its related scale. What would you expect to find, qualitatively, if this instrument evolved in a musical tradition — what scale would that tradition likely use?