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clarify different types of join and meet
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== Mathematical Preliminaries ==
== Mathematical Preliminaries: Join and Meet of Subgroups ==
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 [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.
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.


Thus, given some JI group G, we can look at the subgroups of [[Smonzos and Svals|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.
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.


== Intra-Subgroup Temperament Meet and Join ==
== 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.
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.


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.
Given two temperaments A and B of group G, the '''kernel-join''' or '''val-meet''' of A and B is formed by taking the join of the kernels of A and B, or, equivalently, the intersection of the supporting vals of both. The resulting temperament, 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. It can be computed by starting with the normal comma lists of A and B, and reducing to a normal comma list 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.
 
Similarly, the '''kernel-meet''' or '''val-join''' of A and B is formed by taking the meet of the kernels of A and B, and thus equivalently, the join of the supporting vals of both. The resulting temperament, in terms of commas, tempers out only those commas tempered in both A ''and'' B, and in terms of vals, is supported by any linear combination of vals supporting either A ''or'' B. Dually to the above, we can compute this by taking the normal val lists for A and B, concatenating, and Hermite-reducing to another normal val list.
 
The kernel-join/val-meet of two temperaments A and B is the "simplest" temperament that supports both A and B, and kernel-meet/val-join is the "most complex" temperament that is supported by both A and B.
 
This definition assumes both A and B have no torsion or contorsion.


== Inter-Subgroup Temperament Meet and Join ==
== Inter-Subgroup Temperament Meet and Join ==
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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.
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.
We can take the intersection of the two temperaments' subgroups, and the intersection of the two temperaments' kernels, independently, to form the '''kernel-meet'''/'''val-join''' of both temperaments, which reduces to the prior definition of the meet if the two subgroups are equal. The kernel-meet/val-join 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 kernel-meet. Every comma tempered out by ''both'' A and B is also tempered out in the kernel-meet, and vice versa.
 
Similarly, we can take the join of both subgroups and kernels to get the '''kernel-join'''/'''val-meet'''. The kernel-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 kernel-join. Every comma tempered out by ''either'' A or B is also tempered out in the kernel-join, 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.
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.


== Poset Properties ==
== Poset Properties ==
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