Domain basis: Difference between revisions

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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.

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.

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>.

More is left to be said about canonicalizing interval bases, but it is beyond the scope of this "Interval subspaces as subspaces of other interval subspaces" subsection. We will address it here: [[User:Cmloegcmluin/Interval basis#Canonicalization]].

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 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.
-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.
+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>.
 More is left to be said about canonicalizing interval bases, but it is beyond the scope of this "Interval subspaces as subspaces of other interval subspaces" subsection. We will address it here: [[User:Cmloegcmluin/Interval basis#Canonicalization]].