Meet and join: Difference between revisions

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Meet and join are a pair of binary operations which combine two [[abstract regular temperament]]s on a JI group G into another temperament on G. The operations are commutative and associative. More concretely, for any of the standard ways of representing an abstract regular temperament (normal val lists, normal comma lists, wedgies, Frobenius projection maps, and reduced row echelon form) we can regard them as taking any pair of such defined on G and producing another also defined on G.

Notably, the notion of meet and join can also be extended to an arbitrary pair of subgroup temperaments, even if on different subgroups.

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In mathematical order theory, meet and join are denoted by ∨ and ∧. We avoid doing that for two reasons; the first is to avoid confusion with the interior and wedge products of multivals. The second is that meet and join are operations on abstract temperaments; ordering by increasing size of the group of commas and decreasing size of the group of vals is regarded and notated as the same.

== Subgroup Temperament Meet and Join ==

If A and B are two subgroup on different temperaments, then there are similarly two natural operations that we can use to combine them: informally, we can look for the most complex temperament supported by both, or the simplest temperament that supports both.

The first is found by taking the intersection of the two subgroups and the intersection of the two temperament kernels, independently, producing another subgroup temperament. This is the ''join'' of the two subgroup temperaments, which reduces to the prior definition of the join if the two subgroups are equal. The join is the most complex temperament supported by both A and B, in the sense that any other temperament supported by both A and B is also supported by the join. Every comma tempered out by 'both' A and B is also tempered out in the join, and vice versa.

The second is found by extending the two subgroups to the simplest subgroup which includes both, and then repeating with the two kernels. This is the ''meet'' of the two subgroup temperaments. The meet is the simplest temperament that supports both A and B, in the sense that if any other temperament also supports both A and B, it supports the meet. Every comma tempered out by 'either' A or B is also tempered out in the meet, and vice versa.

== Examples ==

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 Meet and join are a pair of binary operations which combine two [[abstract regular temperament]]s on a JI group G into another temperament on G. The operations are commutative and associative. More concretely, for any of the standard ways of representing an abstract regular temperament (normal val lists, normal comma lists, wedgies, Frobenius projection maps, and reduced row echelon form) we can regard them as taking any pair of such defined on G and producing another also defined on G.
+Notably, the notion of meet and join can also be extended to an arbitrary pair of subgroup temperaments, even if on different subgroups.
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@@ Line 11: / Line 13: @@
 In mathematical order theory, meet and join are denoted by ∨ and ∧. We avoid doing that for two reasons; the first is to avoid confusion with the interior and wedge products of multivals. The second is that meet and join are operations on abstract temperaments; ordering by increasing size of the group of commas and decreasing size of the group of vals is regarded and notated as the same.
+== Subgroup Temperament Meet and Join ==
+If A and B are two subgroup on different temperaments, then there are similarly two natural operations that we can use to combine them: informally, we can look for the most complex temperament supported by both, or the simplest temperament that supports both.
+The first is found by taking the intersection of the two subgroups and the intersection of the two temperament kernels, independently, producing another subgroup temperament. This is the ''join'' of the two subgroup temperaments, which reduces to the prior definition of the join if the two subgroups are equal. The join is the most complex temperament supported by both A and B, in the sense that any other temperament supported by both A and B is also supported by the join. Every comma tempered out by 'both' A and B is also tempered out in the join, and vice versa.
+The second is found by extending the two subgroups to the simplest subgroup which includes both, and then repeating with the two kernels. This is the ''meet'' of the two subgroup temperaments. The meet is the simplest temperament that supports both A and B, in the sense that if any other temperament also supports both A and B, it supports the meet. Every comma tempered out by 'either' A or B is also tempered out in the meet, and vice versa.
 == Examples ==