Mediant (operation)
In the context of just intonation 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.
Given a target interval x (written logarithmically in octaves), the relative error of the mediant of two edo approximations a\m and b\n to x is the sum of the respective relative errors of a\m and b\n. Since x is exactly equal to xm\m in m-edo and xn\n in n-edo, the error of the approximation (a+b)\(m+n) is
[(a+b)\(m+n) − x](m+n) = (a+b)\(m+n) − x(m+n)\(m+n) = [(a-xm)+(b-xn)]\(m+n).
The relative error in edo steps is thus
[(a+b)\(m+n) − x](m+n) = (a-xm) + (b-xn),
which is the sum of the relative errors in m- and n-edo.
Edos admitting a 5L 2s diatonic MOS subscale can be generated by taking mediants of 4\7 (the fifth is too flat and 5L 2s equalizes (L = s) into 7edo) and 3\5 (the fifth is too sharp and 5L 2s collapses (s = 0) into 5edo), the first generation being the 12edo diatonic generator 7\12, the second generation being 10\17 and 11\19 fifths, and so on: see 5L 2s#Scale tree.
See also
- Merciful intonation
- Mediant hull — mediants applied infinitely to generators of several tunings to comprise a range of tunings