Meet and join: Difference between revisions

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.

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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.

Note that in this definition, it doesn't matter what kind of group G is - it could represent musical intervals, or vals, or anything (it need not even be abelian). When working with temperaments, we have at least two relevant groups - the group of vanishing commas, and the group of supporting vals - both of which have a relevant lattice of subgroups. As we will see, we will get two different notions of meet and join based on which one we'd like to do.

Given some JI group G and dual group of vals G^, each temperament of G can be defined either as a subgroup of supporting vals within G^, or a subgroup of vanishing commas within G, also called a kernel. We will get two different notions of "meet" and "join" depending on if we are joining the kernels or the supporting vals. These are basically identical except the meaning is swapped; a join of vals is equal to a meet of kernels and so on.

This definition assumes both A and B have no torsion or contorsion, and that torsion/contorsion is removed after taking the meet/join; otherwise the definition is more complicated.

If A and B are two temperaments on different subgroups, then there are similarly two natural operations that we can use to combine them: informally, we can look for the "largest" temperament supported by both, and the "smallest" temperament that supports both, in a sense to be made precise below.

These definitions assume that both A and B have no torsion or contorsion and that such things are removed after either type of join or meet; if one desires to keep torsion and contorsion the definitions get much more complicated.

The operations of join and meet give the set of temperaments the structure of a partially ordered set, sometimes called a [[Wikipedia:Partially_ordered_set|poset]].

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.

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.  

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, everything is tempered out, and we may also call it [[Trivial temperament|'''OM'''_G]].  

The set of subgroup temperaments also, similarly, forms a lattice, from which we can derive the same basic poset operation above. Again, we have that if A ≤ B iff A & B = A, or equivalently, A ≤ B iff A | B = B. But now, the minimal subgroup temperament is the "trivial subgroup temperament" which has no intervals except for 1/1 and tempers out nothing; we may simply call this subgroup <'''1'''>. For all subgroup temperaments, we have that A & <'''1'''> = <'''1'''>. Similarly, the maximal temperament is Q/Q, meaning the subgroup temperament formed by taking infinite-limit JI and tempering the entire thing out; we may call this '''OM'''.

Similarly, if we had gone with the val-join instead of kernel-join, we would have gotten the same result, except the minimal and maximal temperaments would be flipped.

Originally, the "meet" and "join" of two temperaments was proposed [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19399.html on the tuning-math list] by Keenan Pepper, who used the terms "meet" and "join" to refer to what is above called the "val-meet" and "val-join," along with the caveat that it doesn't really matter which you call meet or join.