Meet and join: Difference between revisions

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=== Same Subgroup ===

Given that the set of temperaments of some subgroup G forms an order-theoretic lattice, we can look at the resulting poset and derive some interesting related ideas.

In the notation below, A ⊓ B refers to the "kernel-meet" of A and B, and A ⊔ B refers to the "kernel-join." (It does not really matter which convention we choose, as we get the same basic result either way.)

There is a partial order on the temperaments of G, given by A ≤ B iff A ⊓ B = A, or equivalently, A ≤ B iff A ⊔ B = B. Since A ⊓ G = G, G is the minimal temperament - it is JI and tempers out no commas. Similarly, if we denote G/G as the "rank-0" temperament of G in which ''everything'' is tempered out, we have that A ⊔ G/G = G/G, thus G/G is the maximal temperament.

A ≤ B may be expressed by "A is supported by B", in the sense that every comma tempered out by A is also tempered out by B, thus all of the [[essentially tempered chords]] and [[comma pumps]] from temperament A are also playable in B.

If we had gone with the other convention for meet and join, we would have gotten the same result, except this would be flipped: G would now be the maximal temperament and G/G the minimal.

In the temperament defined by G, nothing is tempered out, and we may also call it '''JI'''; and in the temperament defined by G/G, nothing is tempered out, and we may also call it [[Trivial temperament|'''OM'''<sub>G</sub>]].

@@ Line 38: / Line 38: @@
 === Same Subgroup ===
 Given that the set of temperaments of some subgroup G forms an order-theoretic lattice, we can look at the resulting poset and derive some interesting related ideas.
+In the notation below, A ⊓ B refers to the "kernel-meet" of A and B, and A ⊔ B refers to the "kernel-join." (It does not really matter which convention we choose, as we get the same basic result either way.)
 There is a partial order on the temperaments of G, given by A ≤ B iff A ⊓ B = A, or equivalently, A ≤ B iff A ⊔ B = B. Since A ⊓ G = G, G is the minimal temperament - it is JI and tempers out no commas. Similarly, if we denote G/G as the "rank-0" temperament of G in which ''everything'' is tempered out, we have that A ⊔ G/G = G/G, thus G/G is the maximal temperament.
 A ≤ B may be expressed by "A is supported by B", in the sense that every comma tempered out by A is also tempered out by B, thus all of the [[essentially tempered chords]] and [[comma pumps]] from temperament A are also playable in B.
+If we had gone with the other convention for meet and join, we would have gotten the same result, except this would be flipped: G would now be the maximal temperament and G/G the minimal.
 In the temperament defined by G, nothing is tempered out, and we may also call it '''JI'''; and in the temperament defined by G/G, nothing is tempered out, and we may also call it [[Trivial temperament|'''OM'''<sub>G</sub>]].