Musical Offerings, Fall 2018

66 True ⦁ Intonation and Modulation mentioned by Ross Duffin, “Nothing can change the fact that the major third of ET is a long way from acoustical purity. ” 13 L. S. Lloyd says “Musicians know that equal temperament is an acoustical compromise, tolerated by many ears on the piano, and designed to satisfy as completely as possible three incompatible requirements—true intonation, complete freedom of modulation and convenience in practical use in keyed instruments—and that it sacrifices the first of these to the second and third. ” 14 ET provides complete freedom of modulation as well as practicality and convenience for keyed instruments. However, it does sacrifice true intonation. Consequently, while ET standardizes the distances between pitches, it took time for musicians to accept the aural impurities that ET embraces. One very early figure to speak on this issue was Aristoxenus, a Greek philosopher who lived about one hundred years after the time of Pythagoras. 15 He asked a very important question—one that is central to the issue of resolving the Pythagorean comma. Which should have priority—aural purity or mathematical perfection? Aristoxenus argued that aural perception should have authority over mathematical ratios. While this idea is a noble conjecture, it does not necessarily present a practical solution. Claudius Ptolemy, a second century mathematician, theorist, and author of the influential book Harmonics , believed differently. He disagreed with Aristoxenus, and instead thought that “tuning is best for which ear and ratio are in agreement. ” 16 A compromise must be reached between rigid mathematical definition and sensory aural perception. In 1577, Francisco Salinas first mentioned ET as a viable tuning system for certain instruments with fixed pitches. 17 Salinas was an early Spanish theorist whose writings influenced much of Renaissance and Baroque music. He realized that equal temperament was important during the construction of fretted string instruments, particularly the viol . 18 13 Ross W. Duffin, How Equal Temperament Ruined Harmony (and Why You Should Care) 1 st ed . (New York: W. W. Norton, 2007), 29. 14 L. S. Lloyd, Intervals, Scales and Temperaments (London: Macdonald & Co. Ltd., 1963), 66. 15 Staufer, “The Unifying Strands,” 33. 16 Barbour, Tuning and Temperament: A Historical Survey , 2. 17 Arthur Daniels, “Microtonality and Mean-Tone Temperament in the Harmonic System of Francisco Salinas,” Journal of Music Theory 9, no. 1 (1965): 5, doi : 10.2307/843148 . 18 Barbour, Tuning and Temperament: A Historical Survey , 6.