Domain basis: Difference between revisions

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==== Vs. canonicalization in temperament merging: do not defactor ====

There's an important difference between the canonicalization step as it's done with temperament merging versus with interval basis merging. When merging temperaments, it's typical to fully [[defactor]] them as well as [[normal form|normalize]] them, because [[The_pathology_of_enfactoring|enfactored representations of temperaments are pathological]]. Enfactored interval bases, however, are ''not'' pathological; they represent meaningfully distinct interval bases.

There's an important difference between the canonicalization step as it's done with temperament merging versus with interval basis merging. When merging temperaments, it's typical to fully [[defactor]] them as well as [[normal form|normalize]] them, because [[The_pathology_of_enfactoring|enfactored representations of temperaments are pathological]]. Enfactored interval bases, however, are ''not'' pathological; they represent meaningfully distinct interval bases.<ref>Here's a key difference between an enfactored comma basis and an enfactored interval basis, 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>

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For example, if we were to defactor 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. canonicalization in temperament merging: do not defactor ====
-There's an important difference between the canonicalization step as it's done with temperament merging versus with interval basis merging. When merging temperaments, it's typical to fully [[defactor]] them as well as [[normal form|normalize]] them, because [[The_pathology_of_enfactoring|enfactored representations of temperaments are pathological]]. Enfactored interval bases, however, are ''not'' pathological; they represent meaningfully distinct interval bases.
+There's an important difference between the canonicalization step as it's done with temperament merging versus with interval basis merging. When merging temperaments, it's typical to fully [[defactor]] them as well as [[normal form|normalize]] them, because [[The_pathology_of_enfactoring|enfactored representations of temperaments are pathological]]. Enfactored interval bases, however, are ''not'' pathological; they represent meaningfully distinct interval bases.<ref>Here's a key difference between an enfactored comma basis and an enfactored interval basis, 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>
+</ref>
 For example, if we were to defactor 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>.