Domain basis: Difference between revisions

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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 interval bases (via formal prime matrices). With the former, it's typical to fully [[defactoring|defactor]] them as well as [[normal form~~|normalize~~]] ~~them~~, because [[The_pathology_of_enfactoring|enfactored representations of temperaments are pathological]]. Enfactored formal prime matrices, however, are ''not'' pathological; they represent meaningfully distinct interval bases.<ref>Here's a key difference between an enfactored comma basis and an enfactored formal prime matrix, by example. 2-enfactored meantone is {{bra|{{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 interval basis. In the 2.3.25 interval basis, the comma basis {{bra|{{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 interval bases (via formal prime matrices). 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 formal prime matrices, however, are ''not'' pathological; they represent meaningfully distinct interval bases.<ref>Here's a key difference between an enfactored comma basis and an enfactored formal prime matrix, by example. 2-enfactored meantone is {{bra|{{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 interval basis. In the 2.3.25 interval basis, the comma basis {{bra|{{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>

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

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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 interval bases (via formal prime matrices). With the former, it's typical to fully [[defactoring|defactor]] them as well as [[normal form|normalize]] them, because [[The_pathology_of_enfactoring|enfactored representations of temperaments are pathological]]. Enfactored formal prime matrices, however, are ''not'' pathological; they represent meaningfully distinct interval bases.<ref>Here's a key difference between an enfactored comma basis and an enfactored formal prime matrix, by example. 2-enfactored meantone is {{bra|{{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 interval basis. In the 2.3.25 interval basis, the comma basis {{bra|{{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 interval bases (via formal prime matrices). 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 formal prime matrices, however, are ''not'' pathological; they represent meaningfully distinct interval bases.<ref>Here's a key difference between an enfactored comma basis and an enfactored formal prime matrix, by example. 2-enfactored meantone is {{bra|{{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 interval basis. In the 2.3.25 interval basis, the comma basis {{bra|{{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>
 For example, if we were to defactor the formal prime matrix for the 2.9.5 interval basis, we'd get 2.3.5. But 2.9.5 is a perfectly reasonable interval basis that we don't wish to conflate with 2.3.5<ref>Even 4.9.25 is an acceptable interval basis. It's not a special situation where there's a common factor in the powers on each formal prime, which in this case is 2.</ref>.