Mediant hull: Difference between revisions

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The '''mediant hull''' or '''Farey hull''' of a set of [[equal-step tuning]]s for a given scale structure is the tuning range defined as the set of all tunings that can be obtained by ~~traversing~~ a finite (resulting in another equal-step tuning) or infinite (resulting in a tuning with irrational step ratio) Farey sum of the starting values. We write the mediant hull of tunings by putting double bars <code>||</code> between the tunings. For example, <code>5L 2s(7||12)</code>, the mediant hull of 7edo [[5L 2s|diatonic]] and 12edo diatonic, is [[TAMNAMS|soft-of-basic]] diatonic. (Note: Double bars are used for formatting on Discord, so you should use <code>``</code> or escape the double bars.)

The '''mediant hull''' or '''Farey hull''' of a set of [[equal-step tuning]]s for a given scale structure is the tuning range defined as the set of all tunings that can be obtained by taking a finite (resulting in another equal-step tuning) or infinite (resulting in a tuning with irrational step ratio) Farey sum of the starting values. We write the mediant hull of tunings by putting double bars <code>||</code> between the tunings. For example, <code>5L 2s(7||12)</code>, the mediant hull of 7edo [[5L 2s|diatonic]] and 12edo diatonic, is [[TAMNAMS|soft-of-basic]] diatonic. (Note: Double bars are used for formatting on Discord, so you should use <code>``</code> or escape the double bars.)

Formally, given a 2-step scale structure ''S'' with equave ''E'' and a set of equal-step tunings ''T1, ..., Tr'', given by generators ''m1\n1edE, ..., mr\nredE'' for S, the ''mediant hull'' or ''Farey hull'' of ''T1, ..., Tr'' is

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This works similarly for rank-3 and higher scales: for example, <code>diasem(7||26||31)</code> specifies the mediant hull of the 7edo (L:m:s = 1:1:0), [[26edo]] (L:m:s = 4:2:1) and [[31edo]] (L:m:s = 5:2:1) tunings for [[diasem]].

== See also ==

* [[Mediant]]

[[Category:Terms]]

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-The '''mediant hull''' or '''Farey hull''' of a set of [[equal-step tuning]]s for a given scale structure is the tuning range defined as the set of all tunings that can be obtained by traversing a finite (resulting in another equal-step tuning) or infinite (resulting in a tuning with irrational step ratio) Farey sum of the starting values. We write the mediant hull of tunings by putting double bars <code>||</code> between the tunings. For example, <code>5L 2s(7||12)</code>, the mediant hull of 7edo [[5L 2s|diatonic]] and 12edo diatonic, is [[TAMNAMS|soft-of-basic]] diatonic. (Note: Double bars are used for formatting on Discord, so you should use <code>``</code> or escape the double bars.)
+The '''mediant hull''' or '''Farey hull''' of a set of [[equal-step tuning]]s for a given scale structure is the tuning range defined as the set of all tunings that can be obtained by taking a finite (resulting in another equal-step tuning) or infinite (resulting in a tuning with irrational step ratio) Farey sum of the starting values. We write the mediant hull of tunings by putting double bars <code>||</code> between the tunings. For example, <code>5L 2s(7||12)</code>, the mediant hull of 7edo [[5L 2s|diatonic]] and 12edo diatonic, is [[TAMNAMS|soft-of-basic]] diatonic. (Note: Double bars are used for formatting on Discord, so you should use <code>``</code> or escape the double bars.)
 Formally, given a 2-step scale structure ''S'' with equave ''E'' and a set of equal-step tunings ''T<sub>1</sub>, ..., T<sub>r</sub>'', given by generators ''m<sub>1</sub>\n<sub>1</sub>edE, ..., m<sub>r</sub>\n<sub>r</sub>edE'' for S, the ''mediant hull'' or ''Farey hull'' of ''T<sub>1</sub>, ..., T<sub>r</sub>'' is
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 This works similarly for rank-3 and higher scales: for example, <code>diasem(7||26||31)</code> specifies the mediant hull of the 7edo (L:m:s = 1:1:0), [[26edo]] (L:m:s = 4:2:1) and [[31edo]] (L:m:s = 5:2:1) tunings for [[diasem]].
+== See also ==
+* [[Mediant]]
 [[Category:Terms]]