Intervals

Note: Modern books on tuning denote ascending intervals by ratios between 1 and 2, corresponding to frequency ratios. Here, however, as we are focusing on the monochord, we follow the ancient convention, and denote descending intervals by length ratios between 1 and 2.

The most important intervals, with special symbols and associated length ratios, are:

• pythagorean comma, K = 81/80 = 3⁴/(2⁴*5)
• pythagorean limma, or Greek hemitone, H = 256/243 = 2⁸/3⁵
• just limma, or semitone, S = HK = 16/15
• pythagorean major tone, T = 9/8
• thin major tone, t = T/K = 10/9
• just, consonant, perfect, or true major third, Tt = tT = 5/4
• pythagorean, or major third, TT = 81/64
• perfect major fourth, 4/3 = TTH = T(T/K)KH = TtS = tTS
• perfect major fifth, 3/2 = TTHT = TtST