The Keyboard Tuning of Domenico Scarlatti

• John Sankey and William A. Sethares, "A Consonance-Based Approach to the Harpsichord Tuning of Domenico Scarlatti". Journal of the Acoustical Society of America 101(4):2332-2337 (1997). The paper discusses a quantitative method for the study of historical keyboard instrument tunings that is based on a measure of the perceived dissonance of the intervals in a tuning and their frequency of occurrence in the compositions of Domenico Scarlatti. We conclude that the total dissonance of a large volume of music is a useful tool for studies of keyboard instrument tuning in a historical musical context, although it is insufficient by itself. Its use provides significant evidence that Scarlatti used French tunings of his period during the composition of his sonatas. Use of total dissonance to optimize a 12-tone tuning for a historical body of music can produce musically valuable results, but must at present be tempered with musical judgment, in particular to prevent overspecialization of the intervals.
• William A.Sethares, "Local consonance and the relationship between timbre and scale". JASA 94:1218-1228 (1993). The principle of local consonance is based on an explicit parameterization of Plomp and Levelt's consonance curves. It explains the relationship between the spectrum of a sound (its timbre) and a tuning (or scale) in which the timbre will appear most consonant. Computational techniques are presented to find the most consonant scale for any timbre, including nonharmonic timbres.
• Pierre-Yves Asselin, "Musique et temperament" (Editions Costallat 1985). A sound presentation of historical European tunings based on tuning instructions of the time (in French).
• John Barnes, "Bach's keyboard temperament; Internal evidence from the Well-Tempered Clavier". Early Music 7:236-249 (1979). The avoidance of bad major thirds is a principal constraint in the use of a circular temperament. Barnes analyses these thirds, and concludes that Bach used a temperament similar to Werckmeister III, not equal tempering, even for the WTC.
• Carl Sloane, Continuo 16(6):15-16 (1992). The choices of tonics by Scarlatti indicates that he viewed the G#-Eb fifth as less consonant than other fifths, and their distribution that he may have retuned his instrument for a few keys.
• R.Plomp and W.J.M.Levelt, "Tonal Consonance and Critical Bandwidth". JASA 38:548-560 (1965). Why is consonance related to simple frequency ratio? Experiments support the hypothesis that consonance is related to beats of adjacent partials of complex tones, and that the transition from consonance to dissonance is related to a frequency-dependent critical bandwidth of the ear. In addition, for many musical instruments, the density of simultaneous partials alters as a function of frequency in the same way as does critical bandwidth.
• Brian McLaren, "Psychoacoustics and Tuning". to be published in Xenharmonikon. Preliminary version in the archives of the tuning list at listproc@eartha.mills.edu, files digest.508 through digest.535 Somewhat polemical and negative, but has many references to studies of the ear relevant to the tuning of musical instruments.

With regard to tunings actually used in the past for harpsichords, I must regretfully note that "Tuning In: Microtonality in Electronic Music" by Scott R. Wilkinson (1988) has a number of serious errors (Werckmeister III, for example).

For those with electronic tuners, the tuning used for my Scarlatti recordings is C=0, 85.6, 193.4, 291.4, 386.3, 498.0, 584.7, 696.8, 787.5, 888.7, 994.9, 1086.5 cents, my personal evaluation of a tuning described by a number of French documents of the period, I believe first by d'Alembert. For Bach, I use C=0, 90.2, 192.3, 294.1, 390.2, 498.1, 588.3, 696.2, 792.2, 888.3, 996.1, 1092.2 cents (Werckmeister III); for Wm.Byrd C=0, 76.2, 193.2, 310.2, 386.4, 503.4, 579.6, 696.6, 772.8, 889.8, 1006.8, 1083.0 cents (quarter-comma meantone).

John Sankey
Harpsichordist to the Internet