Meet and join: Difference between revisions
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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. | 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. | ||
In the notation below, A | In the notation below, A & B refers to the "kernel-meet" of A and B, aka the val-join, and A | B refers to the "kernel-join" or val-meet. It does not really matter which convention we choose, as we get the same basic result either way. Historically, using & to join vals has been the predominant convention, with some differences regarding the treatment of contorsion, although phrases like e.g. "2.3.5.7 81/80 & 126/125" have also seen occasional use. | ||
There is a partial order on the temperaments of G, given by A ≤ B iff A | 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. | 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. | ||
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=== Different Subgroups === | === 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 | 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'''. | ||
Similarly, if we had gone with the val-join instead of kernel-join, we would have gotten the same result, except the minimal and maximal temperaments would be flipped. | Similarly, if we had gone with the val-join instead of kernel-join, we would have gotten the same result, except the minimal and maximal temperaments would be flipped. | ||
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== Examples == | == Examples == | ||
Suppose we take G to be the 11-limit group. Then we have the following, where again | Suppose we take G to be the 11-limit group. Then we have the following, where again | and & are referring to "kernel-join" and "kernel-meet" in this particular usage. | ||
Meantone | 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 | 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 | Meantone | Marvel = 31, Meantone & Marvel = <225/224> | ||
Meantone | Meantone | Porcupine = G/G = '''OM'''<sub>G</sub>, Meantone & Porcupine = <176/175> | ||
In the 7-limit, that becomes Meantone | In the 7-limit, that becomes Meantone | Porcupine = '''OM'''<sub>G</sub>; Meantone & Porcupine = '''JI''', hence, we may consider 7-limit meantone and porcupine to be totally unrelated. | ||
Miracle | Miracle | Magic = 41, Miracle & Magic = Marvel. | ||
In the 7-limit, again Miracle | In the 7-limit, again Miracle | Magic = 41, Miracle & Magic = Marvel | ||
Miracle | Miracle | Mothra = 31, Miracle & Mothra = Portent. In the 7-limit, Miracle & Mothra = Gamelan. | ||
Meantone | Meantone | Magic = '''OM'''<sub>G</sub>, Meantone & Magic = <225/224> | ||
Note that in terms of wedgies, Meantone ∧ Magic = <<<<0 1 2 -2 -5 | 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 = '''OM'''<sub>G</sub>, then A & B is represented by A ∧ B. | ||
== See also == | == See also == | ||