Mediant (operation): Difference between revisions

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== Generalization ==

The mediant operation can also be used to find generators and scales in [[edo]]s representing temperaments. For example, the perfect fifth ([[3/2]]) in 12edo which [[support]]s [[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]].

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 == Generalization ==
 The mediant operation can also be used to find generators and scales in [[edo]]s representing temperaments. For example, the perfect fifth ([[3/2]]) in 12edo which [[support]]s [[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) &minus; x](m+n) = (a+b)\(m+n) &minus; x(m+n)\(m+n) = [(a-xm)+(b-xn)]\(m+n).
+The relative error in edo steps is thus
+[(a+b)\(m+n) &minus; 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]].