Mediant (operation)

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

• mediant of 3/2 and 5/4: (3+5)/(2+4) = 8/6 = 4/3
• mediant of 5/4 and 6/5: (5+6)/(4+5) = 11/9
• mediant of 9/8 and 10/9: (9+10)/(8+9) = 19/17
• mediant of 9/8 and 19/17: (9+19)/(8+17) = 28/25
• mediant of 19/17 and 10/9: (19+10)/(17+9) = 29/26

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

• The Noble Mediant: Complex ratios and metastable musical intervals, by Margo Schulter and David Keenan
• Wikipedia article on the mediant