Musical Offerings, Fall 2018

Musical Offerings ⦁ 2018 ⦁ Volume 9 ⦁ Number 2 67 However, while Salinas conceived the idea theoretically, he did not advocate for it outside the making of the viol. Instead, Salinas supported just intonation which meant that “all intervals are derived from the pure fifth and the pure major third. ” 19 This contrasted with the Pythagorean system which based small intervals solely off the division of the fifth. Practically, this meant that the Pythagorean system was more complex and less acoustically viable. Salinas’s work was instrumental in the development of the music of his time. His tuning system, which used the pure fifth and pure major third, led to the acceptance of the triad as the “basic building block of late Renaissance music. ” 20 This step had consequences for many years to come. Although Salinas’s just intonation was aurally pleasing, it failed to fully and perfectly tune a keyboard instrument. As mentioned previously in this article, any type of modulation will destroy the sound of just intonation. Harmonies may sound consonant in one key, but after a half- step modulation, all kinds of problems arise. Salinas was aware of this and he knew that he had to compromise somehow when tuning keyboard instruments. According to Arthur Daniels, “Salinas recommends three systems of meantone temperament for keyboard instruments, the first of which was his own invention: the 1/3 comma, 2/7 comma, and the 1/4 comma temperament systems. ” 21 The second meantone temperament was invented by Gioseffo Zarlino and the third by Pietro Aron. 22 Overall, these three temperaments are constructed using the same process, yet with slightly different specifics. To understand the details of meantone temperament, one must delve slightly into mathematics. The definition of meantone is more complex than equal temperament. First, some terms must be clarified. In this situation, “comma” refers to a syntonic comma, or the difference between a Pythagorean third (81/64) and a just third (5/4). Written as a ratio, the syntonic comma is 81/80 (81/64 x 80/81 = 5/4). In terms of cent values, the syntonic comma is equal to 21.5 cents (3986 x log [81/80] = 21.5). Meantone temperaments all slightly flatten the fifth by some amount—the three discussed here flatten the fifth by 1/4 comma, 2/7 comma, and 1/3 comma respectively. In 1/4 comma temperament, this is 19 Daniels, “Microtonality and Mean-Tone Temperament,” 5. 20 Ibid., 6. 21 Ibid. 22 Mimi S. Waitzman, “Mean-Tone Temperament in Theory and Practice,” In Theory Only: Journal of the Michigan Music Theory Society 5, no. 4 (May 1981): 7, 12 , https://quod.lib.umich.edu/g/genpub/0641601 .