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.

In the notation below, A ⊓ B refers to the "kernel-meet" of A and B, and A ⊔ B refers to the "kernel-join." (It does not really matter which convention we choose, as we get the same basic result either way.)

In the notation below, A && B refers to the "kernel-meet" of A and B, and A || B refers to the "kernel-join." (It does not really matter which convention we choose, as we get the same basic result either way.)

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.

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.

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=== 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'''.

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.

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== Examples ==

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.

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 ⊔ 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 = [<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 && 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 || Marvel = 31, Meantone && Marvel = <225/224>

Meantone ⊔ Porcupine = G/G = '''OM'''G, Meantone ⊓ Porcupine = <176/175>

Meantone || Porcupine = G/G = '''OM'''G, Meantone && Porcupine = <176/175>

In the 7-limit, that becomes Meantone ⊔ Porcupine = '''OM'''G; Meantone ⊓ Porcupine = '''JI''', hence, we may consider 7-limit meantone and porcupine to be totally unrelated.

In the 7-limit, that becomes Meantone || Porcupine = '''OM'''G; Meantone && Porcupine = '''JI''', hence, we may consider 7-limit meantone and porcupine to be totally unrelated.

Miracle ⊔ Magic = 41, Miracle ⊓ Magic = Marvel.

Miracle || Magic = 41, Miracle && Magic = Marvel.

In the 7-limit, again 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.

Miracle || Mothra = 31, Miracle && Mothra = Portent. In the 7-limit, Miracle && Mothra = Gamelan.

Meantone ⊔ Magic = '''OM'''G, Meantone ⊓ Magic = <225/224>

Meantone || Magic = '''OM'''G, 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 = '''OM'''G, then A ⊓ B is represented by A ∧ B.

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'''G, then A && B is represented by A ∧ B.

== See also ==

@@ Line 42: / Line 42: @@
 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 ⊓ B refers to the "kernel-meet" of A and B, and A ⊔ B refers to the "kernel-join." (It does not really matter which convention we choose, as we get the same basic result either way.)
+In the notation below, A && B refers to the "kernel-meet" of A and B, and A || B refers to the "kernel-join." (It does not really matter which convention we choose, as we get the same basic result either way.)
-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.
+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.
@@ Line 53: / Line 53: @@
 === 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 &lt;'''1'''&gt;. For all subgroup temperaments, we have that A ⊓ &lt;'''1'''&gt; = &lt;'''1'''&gt;. 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'''.
+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 &lt;'''1'''&gt;. For all subgroup temperaments, we have that A && &lt;'''1'''&gt; = &lt;'''1'''&gt;. 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.
@@ Line 67: / Line 67: @@
 == Examples ==
-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.
+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 ⊔ Meanpop = [&lt;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 = [&lt;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 = [&lt;1 0 -4 -13 0|, &lt;0 1 4 10 0|, &lt;0 0 0 0 1|] = &lt;81/80, 126/125&gt;, where &lt;S&gt; for a set of commas S denotes the temperament of the group G tempering out the given commas.
+Meantone && Meanpop = [&lt;1 0 -4 -13 0|, &lt;0 1 4 10 0|, &lt;0 0 0 0 1|] = &lt;81/80, 126/125&gt;, where &lt;S&gt; for a set of commas S denotes the temperament of the group G tempering out the given commas.
-Meantone ⊔ Marvel = 31, Meantone ⊓ Marvel = &lt;225/224&gt;
+Meantone || Marvel = 31, Meantone && Marvel = &lt;225/224&gt;
-Meantone ⊔ Porcupine = G/G = '''OM'''<sub>G</sub>, Meantone ⊓ Porcupine = &lt;176/175&gt;
+Meantone || Porcupine = G/G = '''OM'''<sub>G</sub>, Meantone && Porcupine = &lt;176/175&gt;
-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.
+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 ⊔ Magic = 41, Miracle ⊓ Magic = Marvel.
+Miracle || Magic = 41, Miracle && Magic = Marvel.
-In the 7-limit, again 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.
+Miracle || Mothra = 31, Miracle && Mothra = Portent. In the 7-limit, Miracle && Mothra = Gamelan.
-Meantone ⊔ Magic = '''OM'''<sub>G</sub>, Meantone ⊓ Magic = &lt;225/224&gt;
+Meantone || Magic = '''OM'''<sub>G</sub>, Meantone && Magic = &lt;225/224&gt;
-Note that in terms of wedgies, Meantone ∧ Magic = &lt;&lt;&lt;&lt;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.
+Note that in terms of wedgies, Meantone ∧ Magic = &lt;&lt;&lt;&lt;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 ==