Meet and join - Revision history

Fredg999: Markup

2025-07-28T18:59:04Z

Markup

← Older revision		Revision as of 18:59, 28 July 2025
Line 48:		Line 48:
	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 chord]]s 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 chord]]s and [[comma pump]]s from temperament A are also playable in B.

	If we had gone with the other convention for meet and join, we would have gotten the same result, except this would be flipped: G would now be the maximal temperament and G/G the minimal.		If we had gone with the other convention for meet and join, we would have gotten the same result, except this would be flipped: G would now be the maximal temperament and G/G the minimal.

Fredg999: Markup

2025-07-28T18:56:56Z

Markup

← Older revision		Revision as of 18:56, 28 July 2025
Line 48:		Line 48:
	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.		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 chord]]s and [[comma pumps]] from temperament A are also playable in B.

	If we had gone with the other convention for meet and join, we would have gotten the same result, except this would be flipped: G would now be the maximal temperament and G/G the minimal.		If we had gone with the other convention for meet and join, we would have gotten the same result, except this would be flipped: G would now be the maximal temperament and G/G the minimal.

Fredg999: /* Mathematical preliminaries: join and meet of subgroups */ Fix typo in French term

2025-07-21T20:26:54Z

Mathematical preliminaries: join and meet of subgroups: Fix typo in French term

← Older revision		Revision as of 20:26, 21 July 2025
Line 10:		Line 10:

	== Mathematical preliminaries: join and meet of subgroups ==		== Mathematical preliminaries: join and meet of subgroups ==
	In general, given some group G, the subgroups of G form an order-theoretic structure called a [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. A lattice is a partially ordered set in which for two subgroups A and B of group G, we have A ≤ B iff A is itself a subgroup of B.		In general, given some group G, the subgroups of G form an order-theoretic structure called a [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]; "treillis" in French, "Verband" in German. A lattice is a partially ordered set in which for two subgroups A and B of group G, we have A ≤ B iff A is itself a subgroup of B.

	Given two subgroups A and B, the '''join''' of A and B is the smallest subgroup of G containing both; this is sometimes also called the '''subgroup generated by A and B.''' The '''meet''' of A and B is the intersection of both.		Given two subgroups A and B, the '''join''' of A and B is the smallest subgroup of G containing both; this is sometimes also called the '''subgroup generated by A and B.''' The '''meet''' of A and B is the intersection of both.

Hkm: fix title case

2025-04-11T00:48:43Z

fix title case

← Older revision		Revision as of 00:48, 11 April 2025
Line 1:		Line 1:
			{{Expert\|Temperament merging}}

	Meet and join are a pair of binary operations which combine two [[abstract regular temperament]]s 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.		Meet and join are a pair of binary operations which combine two [[abstract regular temperament]]s 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.

Line 7:		Line 9:
	__TOC__		__TOC__

	== Mathematical ~~Preliminaries~~: ~~Join~~ and ~~Meet~~ of ~~Subgroups~~ ==		== Mathematical preliminaries: join and meet of subgroups ==
	In general, given some group G, the subgroups of G form an order-theoretic structure called a [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. A lattice is a partially ordered set in which for two subgroups A and B of group G, we have A ≤ B iff A is itself a subgroup of B.		In general, given some group G, the subgroups of G form an order-theoretic structure called a [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. A lattice is a partially ordered set in which for two subgroups A and B of group G, we have A ≤ B iff A is itself a subgroup of B.

Line 14:		Line 16:
	Note that in this definition, it doesn't matter what kind of group G is - it could represent musical intervals, or vals, or anything (it need not even be abelian). When working with temperaments, we have at least two relevant groups - the group of vanishing commas, and the group of supporting vals - both of which have a relevant lattice of subgroups. As we will see, we will get two different notions of meet and join based on which one we'd like to do.		Note that in this definition, it doesn't matter what kind of group G is - it could represent musical intervals, or vals, or anything (it need not even be abelian). When working with temperaments, we have at least two relevant groups - the group of vanishing commas, and the group of supporting vals - both of which have a relevant lattice of subgroups. As we will see, we will get two different notions of meet and join based on which one we'd like to do.

	== Intra-~~Subgroup Temperament Meet~~ and ~~Join~~ ==		== Intra-subgroup temperament meet and join ==

	Given some JI group G and dual group of vals G^, each temperament of G can be defined either as a subgroup of supporting vals within G^, or a subgroup of vanishing commas within G, also called a kernel. We will get two different notions of "meet" and "join" depending on if we are joining the kernels or the supporting vals. These are basically identical except the meaning is swapped; a join of vals is equal to a meet of kernels and so on.		Given some JI group G and dual group of vals G^, each temperament of G can be defined either as a subgroup of supporting vals within G^, or a subgroup of vanishing commas within G, also called a kernel. We will get two different notions of "meet" and "join" depending on if we are joining the kernels or the supporting vals. These are basically identical except the meaning is swapped; a join of vals is equal to a meet of kernels and so on.
Line 26:		Line 28:
	This definition assumes both A and B have no torsion or contorsion, and that torsion/contorsion is removed after taking the meet/join; otherwise the definition is more complicated.		This definition assumes both A and B have no torsion or contorsion, and that torsion/contorsion is removed after taking the meet/join; otherwise the definition is more complicated.

	== Inter-~~Subgroup Temperament Meet~~ and ~~Join~~ ==		== Inter-subgroup temperament meet and join ==

	If A and B are two temperaments on different subgroups, then there are similarly two natural operations that we can use to combine them: informally, we can look for the "largest" temperament supported by both, and the "smallest" temperament that supports both, in a sense to be made precise below.		If A and B are two temperaments on different subgroups, then there are similarly two natural operations that we can use to combine them: informally, we can look for the "largest" temperament supported by both, and the "smallest" temperament that supports both, in a sense to be made precise below.
Line 36:		Line 38:
	These definitions assume that both A and B have no torsion or contorsion and that such things are removed after either type of join or meet; if one desires to keep torsion and contorsion the definitions get much more complicated.		These definitions assume that both A and B have no torsion or contorsion and that such things are removed after either type of join or meet; if one desires to keep torsion and contorsion the definitions get much more complicated.

	== Poset ~~Properties~~ ==		== Poset properties ==
	The operations of join and meet give the set of temperaments the structure of a partially ordered set, sometimes called a [[Wikipedia:Partially_ordered_set\|poset]].		The operations of join and meet give the set of temperaments the structure of a partially ordered set, sometimes called a [[Wikipedia:Partially_ordered_set\|poset]].

	=== Same ~~Subgroup~~ ===		=== Same subgroup ===
	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.

Line 52:		Line 54:
	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, everything 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, everything is tempered out, and we may also call it [[Trivial temperament\|'''OM'''<sub>G</sub>]].

	=== 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 & 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.		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.

	== A ~~Note~~ on ~~Terminology~~ ==		== A note on terminology ==
	Originally, the "meet" and "join" of two temperaments was proposed [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19399.html on the tuning-math list] by Keenan Pepper, who used the terms "meet" and "join" to refer to what is above called the "val-meet" and "val-join," along with the caveat that it doesn't really matter which you call meet or join.		Originally, the "meet" and "join" of two temperaments was proposed [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19399.html on the tuning-math list] by Keenan Pepper, who used the terms "meet" and "join" to refer to what is above called the "val-meet" and "val-join," along with the caveat that it doesn't really matter which you call meet or join.

Sintel: -legacy

2025-03-29T18:09:17Z

-legacy

← Older revision		Revision as of 18:09, 29 March 2025
Line 1:		Line 1:
	~~{{Legacy}}~~
	Meet and join are a pair of binary operations which combine two [[abstract regular temperament]]s 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.		Meet and join are a pair of binary operations which combine two [[abstract regular temperament]]s 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.

Lériendil at 16:55, 17 February 2025

2025-02-17T16:55:20Z

← Older revision		Revision as of 16:55, 17 February 2025
Line 1:		Line 1:
	{{Legacy~~\|Meet_and_join~~}}		{{Legacy}}
	Meet and join are a pair of binary operations which combine two [[abstract regular temperament]]s 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.		Meet and join are a pair of binary operations which combine two [[abstract regular temperament]]s 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.

Lériendil: deploying new cat

2025-02-17T16:10:39Z

deploying new cat

← Older revision		Revision as of 16:10, 17 February 2025
Line 1:		Line 1:
			{{Legacy\|Meet_and_join}}
	Meet and join are a pair of binary operations which combine two [[abstract regular temperament]]s 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.		Meet and join are a pair of binary operations which combine two [[abstract regular temperament]]s 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.

Lériendil: Changed protection level for "Meet and join" ([Edit=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)))

2025-02-17T16:01:13Z

Changed protection level for "Meet and join" ([Edit=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:01, 17 February 2025
(No difference)

Mike Battaglia at 21:10, 23 December 2021

2021-12-23T21:10:14Z

← Older revision		Revision as of 21:10, 23 December 2021
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.		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, 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 && 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.		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 ===		=== 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.		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 ==		== 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'''<sub>G</sub>, Meantone && Porcupine = <176/175>		Meantone \| Porcupine = G/G = '''OM'''<sub>G</sub>, Meantone & Porcupine = <176/175>

	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 = <225/224>		Meantone \| Magic = '''OM'''<sub>G</sub>, 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'''<sub>G</sub>, 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'''<sub>G</sub>, then A & B is represented by A ∧ B.

	== See also ==		== See also ==

Mike Battaglia: change kernel-meet and kernel-join symbols

2021-12-21T21:19:00Z

change kernel-meet and kernel-join symbols

← Older revision		Revision as of 21:19, 21 December 2021
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.		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.		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 ===		=== 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.		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 ==		== 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'''<sub>G</sub>, Meantone ⊓ Porcupine = <176/175>		Meantone \|\| Porcupine = G/G = '''OM'''<sub>G</sub>, Meantone && Porcupine = <176/175>

	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 = <225/224>		Meantone \|\| Magic = '''OM'''<sub>G</sub>, 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'''<sub>G</sub>, 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'''<sub>G</sub>, then A && B is represented by A ∧ B.

	== See also ==		== See also ==

← Older revision		Revision as of 21:10, 23 December 2021
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.		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, 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 && 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.		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 ===		=== 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.		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 ==		== 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'''<sub>G</sub>, Meantone && Porcupine = <176/175>		Meantone \| Porcupine = G/G = '''OM'''<sub>G</sub>, Meantone & Porcupine = <176/175>

	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 = <225/224>		Meantone \| Magic = '''OM'''<sub>G</sub>, 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'''<sub>G</sub>, 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'''<sub>G</sub>, then A & B is represented by A ∧ B.

	== See also ==		== See also ==