Domain basis - Revision history

Dummy index: /* Column Hermite normal form */ correct link

2024-12-27T13:56:06Z

Column Hermite normal form: correct link

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	== Column Hermite normal form ==		== Column Hermite normal form ==
	To be exact, we want to use the ''column-style'' Hermite normal form, sometimes called column Hermite normal form for short. All this means is that we put the HNF call in an antitranspose sandwich, [[Normal_lists#~~Defactored_Hermite_form_2~~\|as described here]]).		To be exact, we want to use the ''column-style'' Hermite normal form, sometimes called column Hermite normal form for short. All this means is that we put the HNF call in an antitranspose sandwich, [[Normal_lists#Antitransposed defactored Hermite form\|as described here]]).

	== Vs. canonical form for temperaments themselves: do not defactor ==		== Vs. canonical form for temperaments themselves: do not defactor ==

Lériendil: chromatic pairs link

2024-12-20T05:13:38Z

chromatic pairs link

← Older revision		Revision as of 05:13, 20 December 2024
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	So, for instance, a temperament in the 2.3.5 domain cannot map the intervals 7/6 or 11/8, because there is no way to represent either of those intervals using only the primes 2, 3, and 5. It could, however, temper 6/5, 5/4, 10/9, or 9/8, etc., because those intervals ''can'' be represented using only those three primes.		So, for instance, a temperament in the 2.3.5 domain cannot map the intervals 7/6 or 11/8, because there is no way to represent either of those intervals using only the primes 2, 3, and 5. It could, however, temper 6/5, 5/4, 10/9, or 9/8, etc., because those intervals ''can'' be represented using only those three primes.

	A basis may be used to label the columns of a mapping, with one basis element per column. Here's [[~~Chromatic pairs~~#Slendric\|slendric]], a temperament of the 2.3.7 domain:		A basis may be used to label the columns of a mapping, with one basis element per column. Here's [[Gamelismic clan#Slendric\|slendric]], a temperament of the 2.3.7 domain:

	<math>		<math>

ArrowHead294 at 21:43, 23 May 2024

2024-05-23T21:43:34Z

← Older revision		Revision as of 21:43, 23 May 2024
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	</math>		</math>

	Note for comparison that a comma basis is also a type of basis. In the same way that a domain basis is a minimal representation of all the ''intervals'' in the temperament, a comma basis is a minimal representation of all the ''commas'' in the temperament{{mdash}}to be precise, the subspace of all commas that are made to vanish.		Note for comparison that a comma basis is also a type of basis. In the same way that a domain basis is a minimal representation of all the ''intervals'' in the temperament, a comma basis is a minimal representation of all the ''commas'' in the temperament—to be precise, the subspace of all commas that are made to vanish.

	In the case of a comma basis, both the basis vectors and all of the spanned vectors are commas. But in the case of a domain basis, neither of these things is true. The basis vectors constitute an identity matrix, which is why they're our "mother of all bases"; at the point one hits basis identity matrix bedrock like this, the only place to go is defining what the entries of these vectors actually stand for, which in our case is prime bases of exponents.		In the case of a comma basis, both the basis vectors and all of the spanned vectors are commas. But in the case of a domain basis, neither of these things is true. The basis vectors constitute an identity matrix, which is why they're our "mother of all bases"; at the point one hits basis identity matrix bedrock like this, the only place to go is defining what the entries of these vectors actually stand for, which in our case is prime bases of exponents.
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	== Basis matrix conversion ==		== Basis matrix conversion ==
	The reduction method we will use as part of canonicalization is the [[Hermite normal form]]. If you are previously familiar with it, you may be surprised to see it here, because you may realize that it is defined for matrices, not lists of numbers. So far, when we've looked at subspaces{{mdash}}or at least looked at the bases that represent them{{mdash}}we've simply notated them as lists of numbers, such as 2.3.7. And in most contexts this number list notation is sufficient. However, in order to merge domains, we need to temporarily convert their bases them into matrix form, in order to use the Hermite normal form.		The reduction method we will use as part of canonicalization is the [[Hermite normal form]]. If you are previously familiar with it, you may be surprised to see it here, because you may realize that it is defined for matrices, not lists of numbers. So far, when we've looked at subspaces—or at least looked at the bases that represent them—we've simply notated them as lists of numbers, such as 2.3.7. And in most contexts this number list notation is sufficient. However, in order to merge domains, we need to temporarily convert their bases them into matrix form, in order to use the Hermite normal form.

	Well, let's get to the matrix-ifying!		Well, let's get to the matrix-ifying!
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	We can do this by factorizing the basis elements in just the same way we factor intervals into [[prime-count vector]]s, such as 5/4 factorizing to {{vector\|-2 0 1}}. This is also the same way we represent comma intervals within the other key RTT basis: the comma basis.		We can do this by factorizing the basis elements in just the same way we factor intervals into [[prime-count vector]]s, such as 5/4 factorizing to {{vector\|-2 0 1}}. This is also the same way we represent comma intervals within the other key RTT basis: the comma basis.

	But here, we're going one step deeper down! Now we're breaking down our basis elements{{mdash}}the building blocks of our intervals{{mdash}}into ''their own building blocks''. And these, finally, are just actual prime numbers.		But here, we're going one step deeper down! Now we're breaking down our basis elements—the building blocks of our intervals—into ''their own building blocks''. And these, finally, are just actual prime numbers.

	Then, each resulting vector becomes a column of our desired matrix.		Then, each resulting vector becomes a column of our desired matrix.

ArrowHead294 at 13:44, 23 May 2024

2024-05-23T13:44:39Z

Show changes

Cmloegcmluin: /* Terminology: domain basis vs. subgroup */ deleting per Flora's critique

2023-05-04T00:39:17Z

Terminology: domain basis vs. subgroup: deleting per Flora's critique

← Older revision		Revision as of 00:39, 4 May 2023
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	The behavior of these are not yet well-described. For instance, is it best to represent them as matrices? If a basis element is <math>π/ɸ</math>, does the vinculum allow us to treat the two irrational numbers as separate basis elements (where the basis elements are expanded to include not only prime numbers)? Perhaps, but as far as this author understands, this hasn't been pinned down yet. And so, for example, irrational numbers are not supported yet in the RTT library in Wolfram Language.		The behavior of these are not yet well-described. For instance, is it best to represent them as matrices? If a basis element is <math>π/ɸ</math>, does the vinculum allow us to treat the two irrational numbers as separate basis elements (where the basis elements are expanded to include not only prime numbers)? Perhaps, but as far as this author understands, this hasn't been pinned down yet. And so, for example, irrational numbers are not supported yet in the RTT library in Wolfram Language.

	~~= Terminology: domain basis vs. subgroup =~~

	In some other RTT writings on the wiki, the term "subgroup", is used in various senses. Sometimes it seems to be used to refer to nonstandard domain bases. Other times it seems to be used as a synonym for doman. Sometimes it seems to be used for both at once. In any case, this article breaks from that terminology, considering it to be confusing and unhelpful, for a number of reasons.

	~~== Mixing terminology from different mathematical fields ==~~

	The term "subgroup basis" mixes mathematical terminology from different mathematical fields: "[[Wikipedia:Subgroup\|subgroup]]" comes from [[Wikipedia:Group_theory\|group theory]], while "[[Wikipedia:Linear_basis\|basis]]" comes from [[Wikipedia:Linear_algebra\|linear algebra]]. The equivalent term for "subgroup" in linear algebra is "[[Wikipedia:Linear_subspace\|subspace]]", and the equivalent term for "basis" in group theory is "[[Wikipedia:Minimal_generating_set\|minimal generating set]]". So the consistent terminology would be to either call something a "subspace basis" (such as a "nullspace basis") or to call it a "subgroup minimal generating set". So "subgroup basis" is already problematic as an inconsistent mix of different fields' terminology.

	~~== Simple vs. advanced math: linear algebra vs. group theory ==~~

	Regarding the choice between these two internally consistent versions of this term, then, this article prefers "subspace basis". This is because group theory is a relatively obscure and advanced field of mathematics, and this article prefers to leverage terminology from the more well-known and basic field of linear algebra whenever possible.

	"Subgroup" and "subspace" are indeed analogous, but due to differences between group theory and linear algebra, they are not completely synonymous. Essentially, group theory takes some of the convenient assumptions which we rely on when doing linear algebra and sets them aside. Doing so can be powerful, and some argue that RTT cannot be sufficiently described using only linear algebra. This article, however, prioritizes pedagogy of the basics over any potential considerations arising from such advanced RTT problems.

	~~== Generic math terms vs. specialized application terms ==~~

	~~For the previous two reasons — consistency, and simplicity — choosing "subspace basis" over "subgroup basis" would be preferable. But this article thinks it can do even better.~~

	Setting aside the specialized use it has taken on in these RTT writings, a subgroup (or subspace) in the general mathematical sense is just a generic mathematical structure, like a matrix or vector. This article prefers to use specialized terminology for objects in our RTT application, so that we can clearly discuss them independently from the mathematical structures that represent them. Just like how we call certain objects represented by matrices "mappings" and certain objects represented by vectors "intervals", this article prefers using a specialized term for this RTT object — one that cannot be confused with a generic mathematical structure.

	A common need when dealing with domains is determining whether they are subspaces of other domains (as discussed in [[Cross-domain temperament merging#Interval subspaces as subspaces of other domains]]). If the name for the specialized RTT object was simply "subspace" instead of "domain", then each use of the word "subspace" could be unclear whether it was referring to the specialized RTT object or to the generic mathematical structure. Communicating about such things would become terribly confusing (as it is at present, in existing writings that use the term "subgroup" in both senses).

	~~== A fresh start re: not excluding standard prime-limit domains ==~~

	"Subgroup" in many typical RTT usages is apparently intended to exclude the standard prime-limit subgroups. This makes it more difficult than necessary to communicate about the standard prime-limit basis, which are still very much subgroups — of the entire space of primes, for one example. So we think this is unnecessary complexity with no clear benefit.

	= Domain basis operations =		= Domain basis operations =

Cmloegcmluin: /* Basis matrix conversion */ key term

2023-01-30T17:06:01Z

Basis matrix conversion: key term

← Older revision		Revision as of 17:06, 30 January 2023
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	Then, each resulting vector becomes a column of our desired matrix.		Then, each resulting vector becomes a column of our desired matrix.

	So, for example, 2.9/7.5 in the form of a matrix <math>B</math> ("basis matrix") looks like this. For convenience, we've labeled each column with the basis element, and each row with the prime:		So, for example, 2.9/7.5 in the form of a matrix <math>B</math> (a '''domain basis change matrix''', or "basis matrix" for short) looks like this. For convenience, we've labeled each column with the basis element, and each row with the prime:

Cmloegcmluin: /* Basis matrix conversion */

2023-01-30T17:03:58Z

Basis matrix conversion

← Older revision		Revision as of 17:03, 30 January 2023
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	Then, each resulting vector becomes a column of our desired matrix.		Then, each resulting vector becomes a column of our desired matrix.

	So, for example, 2.9/7.5 in the form of a matrix <math>F</math> ("basis matrix") looks like this. For convenience, we've labeled each column with the basis element, and each row with the prime:		So, for example, 2.9/7.5 in the form of a matrix <math>B</math> ("basis matrix") looks like this. For convenience, we've labeled each column with the basis element, and each row with the prime:


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	To make a popular culture reference, you may be starting to get an [[Wikipedia:Inception\|Inception]] vibe here: we're breaking primes into deeper primes (perhaps we could call this "intervalception"?). Indeed, this might all seem dizzyingly abstract, but fortunately, we don't need to go any deeper than this. And we assure you that this matrix representation of the domain basis (again, called the "basis matrix") will be quite helpful for comparing different domain bases.		To make a popular culture reference, you may be starting to get an [[Wikipedia:Inception\|Inception]] vibe here: we're breaking primes into deeper primes (perhaps we could call this "intervalception"?). Indeed, this might all seem dizzyingly abstract, but fortunately, we don't need to go any deeper than this. And we assure you that this matrix representation of the domain basis (again, called the "basis matrix") will be quite helpful for comparing different domain bases.

	== Column Hermite normal form ==		== Column Hermite normal form ==

Cmloegcmluin: /* Basis elements and nonstandard domain bases */ fix link

2023-01-24T15:41:34Z

Basis elements and nonstandard domain bases: fix link

← Older revision		Revision as of 15:41, 24 January 2023
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	= Basis elements and nonstandard domain bases =		= Basis elements and nonstandard domain bases =

	Bases are expressed as lists of '''[[basis ~~elements~~]]''', the building blocks of the corresponding space, and domain bases are no different. They are notated by separating these basis elements with periods. The most common bases consist of prime numbers, which are the building blocks of the [[Wikipedia:Rational_number\|rational numbers]] we use for [[just intonation\|justly intoned]] interval ratios. Examples of such bases are 2.3.5, or 2.3.7.11.		Bases are expressed as lists of '''[[basis element]]s''', the building blocks of the corresponding space, and domain bases are no different. They are notated by separating these basis elements with periods. The most common bases consist of prime numbers, which are the building blocks of the [[Wikipedia:Rational_number\|rational numbers]] we use for [[just intonation\|justly intoned]] interval ratios. Examples of such bases are 2.3.5, or 2.3.7.11.

	When left unspecified, the basis is the <math>d</math> [[prime limit]], or in other words, the sequence of the first <math>d</math> primes where <math>d</math> is the [[dimensionality]]. So, for example, given the [[mapping\|regular temperament mapping]] {{rket\|{{map\|1 0 -4 -13 -25}} {{map\|0 1 4 10 18}}}}, we should assume that its domain basis is 2.3.5.7.11. This is called a '''standard basis''', and any other basis would therefore be a '''nonstandard basis'''.		When left unspecified, the basis is the <math>d</math> [[prime limit]], or in other words, the sequence of the first <math>d</math> primes where <math>d</math> is the [[dimensionality]]. So, for example, given the [[mapping\|regular temperament mapping]] {{rket\|{{map\|1 0 -4 -13 -25}} {{map\|0 1 4 10 18}}}}, we should assume that its domain basis is 2.3.5.7.11. This is called a '''standard basis''', and any other basis would therefore be a '''nonstandard basis'''.

FloraC: Resolve name conflict

2023-01-08T03:04:55Z

Resolve name conflict

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			:''"Domain" redirects here. For the temperament, see [[Domain (temperament)]].''

	A '''domain basis''' is a [[basis]] for a ''domain'', or space of inputs to a function, such as a [[regular temperament]] [[mapping]]. Since the basis for the domain is more fundamental than other types of bases, such as [[comma basis\|comma bases]] or [[unchanged-interval basis\|unchanged-interval bases]], we can typically shorten this to simply "basis", and we will be largely adopting that shorthand for the remainder of the article.		A '''domain basis''' is a [[basis]] for a ''domain'', or space of inputs to a function, such as a [[regular temperament]] [[mapping]]. Since the basis for the domain is more fundamental than other types of bases, such as [[comma basis\|comma bases]] or [[unchanged-interval basis\|unchanged-interval bases]], we can typically shorten this to simply "basis", and we will be largely adopting that shorthand for the remainder of the article.

Cmloegcmluin: temper out → vanish

2022-12-27T18:17:49Z

temper out → vanish

← Older revision		Revision as of 18:17, 27 December 2022
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	= Bases for temperaments =		= Bases for temperaments =

	In the context of a regular temperament, a domain basis serves as a minimal representation of all the intervals this temperament can map (some of which it completely [[~~temper out\|vanishes~~]]). The full set of these mappable intervals is called the domain; it, in turn, is a ''sub''space with respect to a theoretically ''full'' domain which would include all conceivable intervals able to be built from the infinitude of greater and greater primes.		In the context of a regular temperament, a domain basis serves as a minimal representation of all the intervals this temperament can map (some of which it completely makes to [[vanish]]). The full set of these mappable intervals is called the domain; it, in turn, is a ''sub''space with respect to a theoretically ''full'' domain which would include all conceivable intervals able to be built from the infinitude of greater and greater primes.

	So, for instance, a temperament in the 2.3.5 domain cannot map the intervals 7/6 or 11/8, because there is no way to represent either of those intervals using only the primes 2, 3, and 5. It could, however, temper 6/5, 5/4, 10/9, or 9/8, etc., because those intervals ''can'' be represented using only those three primes.		So, for instance, a temperament in the 2.3.5 domain cannot map the intervals 7/6 or 11/8, because there is no way to represent either of those intervals using only the primes 2, 3, and 5. It could, however, temper 6/5, 5/4, 10/9, or 9/8, etc., because those intervals ''can'' be represented using only those three primes.
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	Note for comparison that a comma basis is also a type of basis. In the same way that a domain basis is a minimal representation of all the ''intervals'' in the temperament, a comma basis is a minimal representation of all the ''commas'' in the temperament — to be precise, the subspace of all commas that are ~~tempered out~~.		Note for comparison that a comma basis is also a type of basis. In the same way that a domain basis is a minimal representation of all the ''intervals'' in the temperament, a comma basis is a minimal representation of all the ''commas'' in the temperament — to be precise, the subspace of all commas that are made to vanish.

	In the case of a comma basis, both the basis vectors and all of the spanned vectors are commas. But in the case of a domain basis, neither of these things is true. The basis vectors constitute an identity matrix, which is why they're our "mother of all bases"; at the point one hits basis identity matrix bedrock like this, the only place to go is defining what the entries of these vectors actually stand for, which in our case is prime bases of exponents.		In the case of a comma basis, both the basis vectors and all of the spanned vectors are commas. But in the case of a domain basis, neither of these things is true. The basis vectors constitute an identity matrix, which is why they're our "mother of all bases"; at the point one hits basis identity matrix bedrock like this, the only place to go is defining what the entries of these vectors actually stand for, which in our case is prime bases of exponents.
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	== Vs. canonical form for temperaments themselves: do not defactor ==		== Vs. canonical form for temperaments themselves: do not defactor ==

	There's an important difference between the canonical form of comma bases and mappings and the canonical form of domain bases (in their basis matrix form). With the former, it's typical to fully [[defactoring\|defactor]] them as well as put them into [[normal form]], because [[The_pathology_of_enfactoring\|enfactored representations of temperaments are pathological]]. Enfactored basis matrices, however, are ''not'' pathological; they represent meaningfully distinct domains.<ref>Here's a key difference between an enfactored comma basis and an enfactored basis matrix, by example. 2-enfactored meantone is [{{vector\|-8 8 -2}}], representing a [[temperoid]] where somehow 6561/6400 = (81/80)² is ~~tempered out~~ but 81/80 is not, a situation which is musically absurd, and this is the crux of why comma basis enfactoring is pathological. We can achieve a similar but non-pathological situation with a nonstandard domain basis. In the 2.3.25 domain basis, the comma basis [{{vector\|-8 8 1}}] represents the temperament where 6561/6400 is ~~tempered out~~. But this isn't absurd, because the temperament doesn't explicitly say that 81/80 is ''not'' ~~tempered out~~. In this temperament, 81/80 doesn't even exist! The page [[Sane and insane temperaments]] contains some more discussion of ideas in this vicinity.</ref>		There's an important difference between the canonical form of comma bases and mappings and the canonical form of domain bases (in their basis matrix form). With the former, it's typical to fully [[defactoring\|defactor]] them as well as put them into [[normal form]], because [[The_pathology_of_enfactoring\|enfactored representations of temperaments are pathological]]. Enfactored basis matrices, however, are ''not'' pathological; they represent meaningfully distinct domains.<ref>Here's a key difference between an enfactored comma basis and an enfactored basis matrix, by example. 2-enfactored meantone is [{{vector\|-8 8 -2}}], representing a [[temperoid]] where somehow 6561/6400 = (81/80)² is made to vanish but 81/80 is not, a situation which is musically absurd, and this is the crux of why comma basis enfactoring is pathological. We can achieve a similar but non-pathological situation with a nonstandard domain basis. In the 2.3.25 domain basis, the comma basis [{{vector\|-8 8 1}}] represents the temperament where 6561/6400 is made to vanish. But this isn't absurd, because the temperament doesn't explicitly say that 81/80 is ''not'' made to vanish. In this temperament, 81/80 doesn't even exist! The page [[Sane and insane temperaments]] contains some more discussion of ideas in this vicinity.</ref>

	For example, if we were to defactor the basis matrix for the 2.9.5 domain basis, we'd get 2.3.5. But 2.9.5 is a perfectly reasonable domain basis that we don't wish to conflate with 2.3.5<ref>Even 4.9.25 is an acceptable domain basis. It's not a special situation where there's a common factor in the powers on each basis element, which in this case is 2.</ref>.		For example, if we were to defactor the basis matrix for the 2.9.5 domain basis, we'd get 2.3.5. But 2.9.5 is a perfectly reasonable domain basis that we don't wish to conflate with 2.3.5<ref>Even 4.9.25 is an acceptable domain basis. It's not a special situation where there's a common factor in the powers on each basis element, which in this case is 2.</ref>.