Non-12-tet equal temperaments (7-tet, 10-tet, 19-tet, etc.) are viable musical systems matched to specific timbres

The octave can be divided into any number of equal steps, not just 12. Common alternatives include: 7-tet (used in Thai classical music, matched to ideal bar spectra), 10-tet (featured in Sethares’ compositions), 19-tet (approximates just thirds better than 12-tet), 22-tet, 31-tet, and many others. Each n-tet system has a different set of available intervals, different approximations to just ratios, and will pair best with specific timbres whose dissonance curve has minima at that system’s scale steps. A spectrum can be designed (or found in acoustic instruments) that makes any n-tet system sound consonant. The choice of tuning is not aesthetically arbitrary — it must match the timbre for sensory consonance to arise.

Examples

Thai pi phat orchestra uses instruments with bar-like spectra (ideal bar partials at 1:2.76:5.40:8.93) — these spectra have dissonance curve minima near 7-tet steps, which is why Thai music naturally uses ~7-tet. Carlos’ piece ‘Beauty in the Beast’ uses 8-tet and stretched timbres matched to that scale.

Assessment

Given a spectrum with partials at [1, 2.4, 3.8, 5.0] (relative frequencies), describe the procedure you would use to find which equal temperament it is related to. Why would 12-tet be a poor choice for this timbre?