Domain basis: Difference between revisions
Cmloegcmluin (talk | contribs) better to assume, even at this advanced level of RTT, that formal primes are rational, and so their basis elements are simply primes; no need to abstract to the point of generic "basis elements" |
Cmloegcmluin (talk | contribs) "formal primes matrix" is a clearer name for this object than "interval basis" (which can be reserved for the simple numeric list) and less likely to be confused with "comma basis"; in other places, "interval subspace" remains a good choice |
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The canonical form of an interval basis requires a few steps to achieve: | The canonical form of an interval basis requires a few steps to achieve: | ||
# Find the matrix representation of the interval basis. | # Find the matrix representation of the interval basis, which we can call a formal primes matrix. | ||
# Put | # Put the formal primes matrix into column Hermite normal form. This step has the effect of sorting the formal primes so that those with higher primes in their factorizations come later, e.g. so that 7 comes after 9 even though 9 is a bigger number, because 9 factorizes into 3's. | ||
# Eliminate any columns that are all zeros. | # Eliminate any columns that are all zeros. | ||
# Convert the matrix back into a list of numbers (separated by periods). | # Convert the formal primes matrix back into a list of numbers (separated by periods). | ||
# Take the [https://forum.sagittal.org/viewtopic.php?p=1296#undirected-value undirected value] of each number; that is, if it is less than 1, replace it with its reciprocal (which will be greater than 1). So this would flip e.g. the "subratio" 3/5 into its "superratio" 5/3, or little phi φ (~0.618) into big phi Φ (~1.618). | # Take the [https://forum.sagittal.org/viewtopic.php?p=1296#undirected-value undirected value] of each number; that is, if it is less than 1, replace it with its reciprocal (which will be greater than 1). So this would flip e.g. the "subratio" 3/5 into its "superratio" 5/3, or little phi φ (~0.618) into big phi Φ (~1.618). | ||
== | == Formal primes 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 — 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 interval subspaces, we need to temporarily convert 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 interval subspaces, we need to temporarily convert them into matrix form, in order to use the Hermite normal form. | ||
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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 interval basis will be quite helpful for comparing different interval 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 interval basis (again, called the "formal primes matrix") will be quite helpful for comparing different interval bases. | ||
== Column Hermite normal form == | == Column Hermite normal form == | ||
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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 interval bases. 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 | 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> | ||
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>. | 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>. | ||
== Example == | == Example == | ||