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.

There are actually two different notions of meet and join, depending on if one is "joining" the kernels of the two temperaments, or the subgroups of supporting vals. The two satisfy duality relationships in that the "join" in one convention is the meet in the other. Both conventions are presented below, as well as the larger picture of how they interrelate.

Both notions of meet and join can also be extended to an arbitrary pair of subgroup temperaments, even if on different subgroups, also explained below.

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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.
+There are actually two different notions of meet and join, depending on if one is "joining" the kernels of the two temperaments, or the subgroups of supporting vals. The two satisfy duality relationships in that the "join" in one convention is the meet in the other. Both conventions are presented below, as well as the larger picture of how they interrelate.
+Both notions of meet and join can also be extended to an arbitrary pair of subgroup temperaments, even if on different subgroups, also explained below.
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