Meet and join: Difference between revisions

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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.

In the temperament defined by G, nothing is tempered out, and we may also call it JI; and in the temperament defined by G/G, nothing is tempered out, and we may also call it [[Trivial temperament|OMG]].

In the temperament defined by G, nothing is tempered out, and we may also call it '''JI'''; and in the temperament defined by G/G, nothing is tempered out, and we may also call it [[Trivial temperament|'''OM'''G]].

=== 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 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'''.

== A Note on Historical Terminology ==

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Meantone ⊔ Marvel = 31, Meantone ⊓ Marvel = <225/224>

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

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

In the 7-limit, that becomes Meantone ⊔ Porcupine = OMG; Meantone ⊓ Porcupine = ~~OM<sub~~, 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.

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Miracle ⊔ Mothra = 31, Miracle ⊓ Mothra = Portent. In the 7-limit, Miracle ⊓ Mothra = Gamelan.

Meantone ⊔ Magic = OMG, 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 = OMG, 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 33: / Line 33: @@
 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.
-In the temperament defined by G, nothing is tempered out, and we may also call it JI; and in the temperament defined by G/G, nothing is tempered out, and we may also call it [[Trivial temperament|OM<sub>G</sub>]].
+In the temperament defined by G, nothing is tempered out, and we may also call it '''JI'''; and in the temperament defined by G/G, nothing is tempered out, and we may also call it [[Trivial temperament|'''OM'''<sub>G</sub>]].
 === 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 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&gt;.
+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'''.
 == A Note on Historical Terminology ==
@@ Line 52: / Line 52: @@
 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 = OM<sub, 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.
@@ Line 62: / Line 62: @@
 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 ==