Mediant (operation): Difference between revisions
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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). | 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). | ||
The mediant operation can also be used to find generators and scales in [[edo]]s representing temperaments. For example, the | 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 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. | ||
[http://en.wikipedia.org/wiki/Mediant_(mathematics) Wikipedia article on the mediant] | [http://en.wikipedia.org/wiki/Mediant_(mathematics) Wikipedia article on the mediant] | ||
Revision as of 09:03, 19 June 2020
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).
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.
Wikipedia article on the mediant
also see:
The Noble Mediant: Complex ratios and metastable musical intervals, by Margo Schulter and David Keenan