Musical Offerings, Fall 2018

64 True ⦁ Intonation and Modulation just intonation represent examples of the former; equal temperament and mean-tone temperament represent examples of the latter. Throughout the course of music history, hundreds of tuning and temperament systems have been suggested. Several important ones will be summarized here. Pythagorean intonation is a tuning system in which the perfect fifth ratio (3:2) is used to generate the relationship between other notes of a musical scale. In Pythagorean intonation, the major third is represented by a frequency ratio of 81/64 (noticeably different than the 5/4 ratio of the pure major third), and notes such as A and G are not necessarily the same frequency . 8 Just intonation involves simpler ratios for each interval. Figure 2 : Frequency ratios for just intonation . 9 Note C D E F G A B C’ Frequency 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 Interval 9/8 10/9 16/15 9/8 10/9 9/8 16/15 N/A The top row of numbers represents the frequency ratio between that note and C. The bottom row of numbers represents the frequency difference between that note and the note to its right. For example, to get from F to G, one must multiply by 9/8: therefore, 4/3 × 9/8 = 3/2. To get from F to A, multiply by 9/8 and 10/9. 4/3 × 9/8 × 10/9 = 5/3. This definition looks very nice on paper and sounds adequate for simpler music. However, to put it bluntly, “the compromise breaks down when one wants to play in another key. ” 10 See the chart below to understand some of the differences between Pythagorean and just intonation. The noticeable difference between them comes in the definition of the fifth and the resulting wolf fifth. 8 Vicente Liern, “On the Construction, Comparison, and Exchangeability of Tuning Systems,” Journal of Mathematics & Music 9, no. 3 (November 2015): 201, doi : 10.1080/17459737.2015.1031468 . 9 Fauvel, Flood, and Wilson, Music and Mathematics , 21. 10 Ibid.