Sensory dissonance analysis can help reconstruct the tuning a historical composer likely used

By computing the total sensory dissonance of a corpus of pieces under different historical tunings and applying gradient descent to minimize it, researchers can infer which tuning a composer likely heard. For Scarlatti’s keyboard sonatas, Sethares compares 12-tet, various meantone temperaments, well temperaments (Kirnberger, Vallotti, d’Alembert), and derived optimal tunings (TDA1, TDA2). The gradient algorithm starts from a known historical tuning and iteratively adjusts pitch values to reduce total dissonance. Results show that Vallotti and related well temperaments score better than 12-tet or extreme meantone, consistent with historical evidence about 18th-century tuning practice. The procedure also reveals which passages are most tuning-sensitive.

Examples

Applying the gradient algorithm to Scarlatti’s K380 starting from d’Alembert’s tuning produces TDA1, which improves dissonance scores while preserving musical shape. Over-optimization (TDA2) makes the most common intervals pure but produces unacceptably dissonant rare intervals — showing there is an optimal trade-off.

Assessment

Why would minimizing sensory dissonance over an entire corpus of pieces be a better reconstruction approach than choosing the tuning that makes a single piece sound best? What additional musical criteria (beyond dissonance) does the chapter suggest should also be considered in such reconstruction?