Meet and join: Difference between revisions

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== Mathematical preliminaries: join and meet of subgroups ==

In general, given some group G, the subgroups of G form an order-theoretic structure called a [https://en.wikipedia.org/wiki/Lattice_of_subgroups lattice of subgroups], where here "lattice" means [https://en.wikipedia.org/wiki/Lattice_(order) lattice in the order theory sense]; "~~trellis~~" in French, "Verband" in German. A lattice is a partially ordered set in which for two subgroups A and B of group G, we have A ≤ B iff A is itself a subgroup of B.

In general, given some group G, the subgroups of G form an order-theoretic structure called a [https://en.wikipedia.org/wiki/Lattice_of_subgroups lattice of subgroups], where here "lattice" means [https://en.wikipedia.org/wiki/Lattice_(order) lattice in the order theory sense]; "treillis" in French, "Verband" in German. A lattice is a partially ordered set in which for two subgroups A and B of group G, we have A ≤ B iff A is itself a subgroup of B.

Given two subgroups A and B, the '''join''' of A and B is the smallest subgroup of G containing both; this is sometimes also called the '''subgroup generated by A and B.''' The '''meet''' of A and B is the intersection of both.

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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.

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 chord]]s and [[comma pump]]s 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.

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 == Mathematical preliminaries: join and meet of subgroups ==
-In general, given some group G, the subgroups of G form an order-theoretic structure called a [https://en.wikipedia.org/wiki/Lattice_of_subgroups lattice of subgroups], where here "lattice" means [https://en.wikipedia.org/wiki/Lattice_(order) lattice in the order theory sense]; "trellis" in French, "Verband" in German. A lattice is a partially ordered set in which for two subgroups A and B of group G, we have A ≤ B iff A is itself a subgroup of B.
+In general, given some group G, the subgroups of G form an order-theoretic structure called a [https://en.wikipedia.org/wiki/Lattice_of_subgroups lattice of subgroups], where here "lattice" means [https://en.wikipedia.org/wiki/Lattice_(order) lattice in the order theory sense]; "treillis" in French, "Verband" in German. A lattice is a partially ordered set in which for two subgroups A and B of group G, we have A ≤ B iff A is itself a subgroup of B.
 Given two subgroups A and B, the '''join''' of A and B is the smallest subgroup of G containing both; this is sometimes also called the '''subgroup generated by A and B.''' The '''meet''' of A and B is the intersection of both.
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 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.
+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 chord]]s and [[comma pump]]s 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.