Given a target scale, the inverse problem finds a spectrum whose dissonance curve has minima at those scale steps

The forward problem is: given a spectrum, find the related scale (compute dissonance curve minima). The inverse problem — harder and more interesting — is: given a target scale (set of intervals), find or synthesize a spectrum that makes those intervals most consonant. Sethares addresses this via ‘symbolic timbre selection’ using s-tables (spectra that satisfy algebraic compatibility constraints with the scale). For n-tet scales, the structure of the s-table must form a closed algebraic system. This is why equal temperaments are easier to solve than irregular scales. The inverse problem enables composition in any scale with musically rich timbres: specify your scale first, then synthesize compatible timbres for it.

Examples

For 10-tet: construct a spectrum with partials at the 10-tet scale steps’ frequency ratios — then sounds with this spectrum will have dissonance curve minima exactly at 10-tet intervals. Chapter 12 shows how to compute the spectrum from the scale via the s-table formalism.

Assessment

Why is the inverse problem (scale → spectrum) harder than the forward problem (spectrum → scale)? For a 12-tet scale, the harmonic series is a related spectrum — explain why using your knowledge of the forward problem. Would a 10-tet scale also have a related harmonic spectrum? Why or why not?