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.
Mathematical Preliminaries
In general, the meet and join are defined for any two subgroups of some group. The meet of two subgroups is their intersection, and the join of two subgroups is the smallest subgroup generated by both. The terms "meet" and "join" come from order theory; the subgroups of a group form a lattice, called the lattice of subgroups, where here "lattice" means lattice in the order theory sense; "trellis" in French, "Verband" in German.
Thus, given some JI group G, we can look at the subgroups of smonzos, each of which can be thought of as a kernel for a temperament. These kernels define the commas of the temperaments of G and form a lattice in the aforementioned order-theoretic sense. Or, equivalently and dually, we could also look at the lattice of subgroups of the dual group G^ of svals, for which the subgroups can be thought of as corresponding to the supporting vals of some temperament and thus also define the temperaments of G. Either is sufficient and both form a lattice of subgroups.
Intra-Subgroup Temperament Meet and Join
Given two temperaments A and B, then, the join A ⊔ B is formed by simply "join"ing their kernels in the aforementioned sense. If A and B are defined in terms of normal comma lists, the join 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. If A and B are instead defined in terms of vals, the join is formed by taking the intersection of the supporting vals of A and B, which can also be expressed as a normal val list. The join of A and B, in terms of commas, tempers out those commas either in A or B, as well as any linear combination thereof. In terms of vals, it is supported by only those vals that support both A and B.
Similarly, the meet A ⊓ B is defined by taking the intersection of the kernels of A and B. The meet of A and B, in terms of vals, tempers out only those commas tempered in both A and B, and in terms of vals, is supported by linear combination of vals supporting either A or B. If A and B are defined by vals, 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, we can also us this to compute the normal comma list for the meet.
Inter-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.
Poset Properties
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.
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 OMG.
Different Subgroups
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.
A Note on Historical 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.
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.
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 = OMG, Meantone ⊓ Porcupine = <176/175>
In the 7-limit, that becomes Meantone ⊔ Porcupine = OMG; 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 = OMG, 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 = OMG, then A ⊓ B is represented by A ∧ B.