Musical Offerings, Fall 2018

Musical Offerings ⦁ 2018 ⦁ Volume 9 ⦁ Number 2 69 At first glance, Figure 4 seems quite ambiguous. Taking a closer look, notice that the left column lists the twelve notes (from C to C), and Columns 3, 5, and 7 list the interval size in cents. For example, in Aron’s 1/4 comma temperament, C in 76 cents from C, D is 193 cents from C, and so on. Columns 2, 4, and 6 indicate how far that note is from Pythagorean intonation (in terms of a syntonic comma). Notice that in each system, the fifth (C to G) is tempered down by 1/4, 2/7, or 1/3 of a syntonic comma—hence the name of each temperament. Tempering each fifth results in deviations from Pythagorean intonation. Consequently, the error of all other notes can be found. Zarlino also worked extensively on various tuning systems. A contemporary of Salinas, Zarlino is mentioned alongside Marin Mersenne and Jean-Philippe Rameau as one of “the great music theorists. ” 24 Barbour makes the interesting point that these three “presented just intonation as the theoretical basis of the scale, but temperament as a practical necessity. ” 25 This common position demonstrates the dilemma that composers like Zarlino were facing. However, while a perfect answer remained elusive, the temperaments Zarlino used were still regarded as satisfactory. While equal temperament was used for fretted instruments, meantone temperament was used for keyboard instruments . 26 While meantone temperament is rarely used to tune keyboards today, Zarlino considered it “very pleasing for all purposes.” 27 Specifically, Zarlino created the 2/7 comma meantone temperament. This system has a few positives, but overall, it is “inferior to the 1/4 comma system.” 28 Essentially, the greater amount of tempering (2/7 > 1/4) causes intervals to be less pure. Why use it then? Based on the design of Zarlino’s system, the impurities are regular; major and minor thirds and sixths are all 1/7 comma off. 29 This small detail demonstrates why so many meantone temperaments arose; theorists sought to minimize slight discrepancies to get a better sound overall. These temperaments do allow for modest modulation, but unlike ET, they completely fail in far-off keys. Of these three, 1/4 comma temperament is the superior model. All the intervals are closer to true intonation, particularly the major third. A major third from C to E is 386 cents, which corresponds to a frequency ratio of 5/4. Note that 24 Barbour, Tuning and Temperament: A Historical Survey , 11. 25 Ibid. 26 Ibid. 27 Ibid., 27. 28 Ibid., 33. 29 Ibid.