Meet and join

Meet and join are a pair of binary operations which combine two abstract regular temperaments 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.

Definition

Meet and join are defined in terms of the lattice of subgroups of G, consisting of groups of smonzos defining the commas of the temperaments of G, or equivalently and dually, the lattice of subgroups of the dual group G^ of svals, where here "lattice" means lattice in the order theory sense; "trellis" in French, "Verband" in German. Either of these subgroup lattices serves to define the temperaments of G.

Given two temperaments A and B defined in terms of normal comma lists, the join A ⊔ B is the reduction to a normal comma list of the concatenation of A and B, which is to say, the Hermite reduction of the list of commas of A with the commas of B. The join, in terms of commas, tempers out those commas both in A and B, as well as all linear combinations thereof, and in terms of vals it is defined by the intersection of the supporting vals of A and those of B, within the dual group of vals G^.

The meet A ⊓ B is defined by taking the intersection of the commas tempered by A with the commas tempered by B. Or, if A and B are defined by vals, then the meet A ⊓ B is defined by taking the normal val list for A and that of B, concatenating them, and reducing the result to a normal interval list. Since temperaments expressed as normal val lists can be converted to temperaments expressed as normal interval lists and back again via the dual list function, this defines both join and meet as operations on normal val lists.

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.

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 <1>.

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 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.

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

A note on terminology

Originally, the "meet" and "join" of two temperaments was proposed on the tuning-math list by Keenan Pepper, who used the opposite convention from the above: the "meet" was the meet of vals (and hence the join of kernels) and vice versa. If the two temperaments are on the same subgroup, one can think about "joining" or "meeting" either their kernels, or the subgroups of supporting vals (the "join" in one convention is the "meet" in the other and so forth), so it makes little difference which convention is chosen (as noted in the tuning-math post above). However, when looking at temperaments on different subgroups, the other convention makes things much more natural, since then the "join" of two temperaments is simply the "join" of kernels and subgroups independently. We have chosen that convention here, although it should be noted that in older emails and posts, the "join" and "meet" were often chosen using the other convention.

Examples

Suppose we take G to be the 11-limit group. Then we have the following:

Meantone ⊔ Meanpop = [<31 49 72 87 107|] = 31, where "31" is the shorthand notation for the 31edo patent val (or, equivalently, its subgroup of vanishing commas).

Meantone ⊓ Meanpop = [<1 0 -4 -13 0|, <0 1 4 10 0|, <0 0 0 0 1|] = <81/80, 126/125>, where <S> for a set of commas S denotes the temperament of the group G tempering out the given commas.

Meantone ⊔ Marvel = 31, Meantone ⊓ Marvel = <225/224>

Meantone ⊔ Porcupine = G/G = <1>, Meantone ⊓ Porcupine = <176/175>

In the 7-limit, that becomes Meantone ⊔ Porcupine = <1>; Meantone ⊓ Porcupine = <JI>, hence, we may consider 7-limit meantone and porcupine to be totally unrelated.

Miracle ⊔ Magic = 41, Miracle ⊓ Magic = Marvel.

In the 7-limit, again Miracle ⊔ Magic = 41, Miracle ⊓ Magic = Marvel

Miracle ⊔ Mothra = 31, Miracle ⊓ Mothra = Portent. In the 7-limit, Miracle ⊓ Mothra = Gamelan.

Meantone ⊔ Magic = <1>, Meantone ⊓ Magic = <225/224>

Note that in terms of wedgies, Meantone ∧ Magic = <<<<0 1 2 -2 -5||||, which represents Meantone ⊓ Magic. This is an instance of the general proposition that if A ⊔ B = <1>, then A ⊓ B is represented by A ∧ B.

See also

• &: the ampersand operator