Domain basis: Difference between revisions

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= 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.

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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.

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== 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>.

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 = 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.
@@ Line 102: / Line 102: @@
-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.
@@ Line 233: / Line 233: @@
 == 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>.