Mediant

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In the context of JI ratios, the mediant M of two ratios a/c and b/d in lowest terms is M=(a+b)/(c+d). It will always be between the two ratios (a/c < M < b/d, assuming a/c < b/d).

Examples

The following table shows the mediant m of some fraction pairs f1, f2.

f1 f2 m intermediate step(s)
3/2 5/4 4/3 (3+5)/(2+4) = 8/6
3/2 4/3 7/5 (3+4)/(2+3)
5/4 6/5 11/9 (5+6)/(4+5)
9/8 10/9 19/17 (9+10)/(8+9)
9/8 19/17 28/25 (9+19)/(8+17)
19/17 10/9 29/26 (19+10)/(17+9)

Generalization

The mediant operation can also be used to find generators and scales in edos representing temperaments. For example, the perfect fifth (3/2) in 12edo which supports meantone is 7 steps out of 12, and the fifth in 19edo, another meantone tuning, is 11 steps out of 19. Hence the perfect fifth in 31edo (which is a meantone tuning because 31 = 12 + 19; more precisely, the 5-limit val of 31edo is the sum of the 5-limit vals of 12edo and 19edo) is (7+11)\(12+19) = 18\31, which is in between the sizes of the 12edo fifth and the 19edo one.

See also