Mediant (operation): Difference between revisions

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~~In the context of [[just intonation]] [[ratio]]s, 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''' ''M'' of two ratios ''a''/''c'' and ''b''/''d'' in lowest terms is ''M'' = (''a'' + ''b'')/(''c'' + ''d''). The result is always between the two ratios (''a''/''c'' < ''M'' < ''b''/''d'', assuming ''a''/''c'' < ''b''/''d'').

The mediant operation can be applied to [[just intonation]] ratios, to [[step]]s of an [[edo]], etc.

== Examples ==

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== ~~Generalization~~ ==

== Applications ==

The mediant operation can also be used to find generators and scales in [[edo]]s representing [[temperament]]s. For example, the [[3/2|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.

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* [[Mediant hull]]: mediants applied infinitely to generators of several tunings to comprise a range of tunings

~~[[Category:Terms]]~~

[[Category:Interval]]

[[Category:Elementary math]]

[[Category:Method]]

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 {{Wikipedia|Mediant (mathematics)}}
-In the context of [[just intonation]] [[ratio]]s, 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''' ''M'' of two ratios ''a''/''c'' and ''b''/''d'' in lowest terms is ''M'' = (''a'' + ''b'')/(''c'' + ''d''). The result is always between the two ratios (''a''/''c'' &lt; ''M'' &lt; ''b''/''d'', assuming ''a''/''c'' &lt; ''b''/''d'').
+The mediant operation can be applied to [[just intonation]] ratios, to [[step]]s of an [[edo]], etc.
 == Examples ==
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 |}
-== Generalization ==
+== Applications ==
 The mediant operation can also be used to find generators and scales in [[edo]]s representing [[temperament]]s. For example, the [[3/2|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.
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 * [[Mediant hull]]: mediants applied infinitely to generators of several tunings to comprise a range of tunings
-[[Category:Terms]]
 [[Category:Interval]]
 [[Category:Elementary math]]
 [[Category:Method]]