Meet and join: Difference between revisions

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== ~~Definition~~ ==

== Mathematical Preliminaries ==

~~Meet~~ and join are defined in terms of the [https://en.wikipedia.org/wiki/Lattice_of_subgroups lattice of subgroups] ~~of G, consisting of groups of [[Smonzos and Svals|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 [https://en.wikipedia.org/wiki/Lattice_(order) lattice in the order theory sense]; "trellis" in French, "Verband" in German~~. Either of these subgroup lattices serves to define the temperaments of G~~.

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.

~~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^.

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.

~~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.

== 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.

=== Poset Properties ===

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.

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-== Definition ==
+== Mathematical Preliminaries ==
-Meet and join are defined in terms of the [https://en.wikipedia.org/wiki/Lattice_of_subgroups lattice of subgroups] of G, consisting of groups of [[Smonzos and Svals|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 [https://en.wikipedia.org/wiki/Lattice_(order) lattice in the order theory sense]; "trellis" in French, "Verband" in German. Either of these subgroup lattices serves to define the temperaments of G.
+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.
-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^.
+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.
-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.
+== 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.
+=== Poset Properties ===
 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.