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 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.

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]~~

== See also ==

~~also see~~:

* [http://dkeenan.com/Music/NobleMediant.txt The Noble Mediant: Complex ratios and metastable musical intervals], by [[Margo Schulter]] and [[David Keenan]]

* [http://en.wikipedia.org/wiki/Mediant_(mathematics) Wikipedia article on the mediant]

[~~http~~:~~//dkeenan.com/Music/NobleMediant.txt The Noble Mediant~~: ~~Complex ratios and metastable musical intervals~~]~~, by~~ [[~~Margo_Schulter|Margo Schulter~~]] ~~and~~ [[~~David_Keenan|David Keenan~~]]

[[Category:Theory]]

[[Category:Term]]

[[Category:Interval ratio]]

[[Category:Scales]]

[[Category:Practical help]]

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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 &lt; M &lt; b/d, assuming a/c &lt; 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 &lt; M &lt; b/d, assuming a/c &lt; b/d).
-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.
+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]
+== See also ==
-also see:
+* [http://dkeenan.com/Music/NobleMediant.txt The Noble Mediant: Complex ratios and metastable musical intervals], by [[Margo Schulter]] and [[David Keenan]]
+* [http://en.wikipedia.org/wiki/Mediant_(mathematics) Wikipedia article on the mediant]
-[http://dkeenan.com/Music/NobleMediant.txt The Noble Mediant: Complex ratios and metastable musical intervals], by [[Margo_Schulter|Margo Schulter]] and [[David_Keenan|David Keenan]]
+[[Category:Theory]]
+[[Category:Term]]
+[[Category:Interval ratio]]
+[[Category:Scales]]
+[[Category:Practical help]]