Meet and Join

Introduction

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.

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 val lists, the join A⋏B is the reduction to a normal val list of the concatination of A and B, which is to say, the Hermite reduction of the list of vals of A with the vals of B (with the obvious extension to svals if G is not a full p-limit group.) The join in terms of vals is the subgroup of G^ generated by A and B, and in terms of intervals is defined by intersection of the commas of A and B. The meet A⋎B is defined by the intersection of the group of vals generated by A with the group of vals generated by B. If A and B are defined by normal interval lists, then the meet A⋎B is defined by concatinating A and B, 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 and A≤B iff A⋎B = B. Since A⋎G = G, G is the maximal temperament, and since A⋏G^ = G^, G^ is the minimal temperament. In the temperament defined by G, everything is tempered out, and we may also call it <JI>; and in the temperament defined by G^, nothing is tempered out, and we may also call it <1>. A≤B may be expressed by "A is supported by B".

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.

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 = <JI>, Meantone⋏Porcupine = <176/175>

In the 7-limit, that become Meantone⋎Porcupine = <JI>, Meantone⋏Porcupine = <1>; 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 = <JI>, 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 = <JI>, then A⋏B is represented by A∧B.