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| A regular temperament mapping is in '''defactored canonical form (DCF)''', or '''canonical form''' for short, when it is put into [https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form] (HNF) after being [[#defactoring|"defactored"]].
| | '''Defactoring''' is a operation on the mapping for a regular temperament which ensures it represents the same information but without any enfactoring, or in other words, redundancies due to a common factor found in its rows. It is also defined for comma-bases, the duals of mappings, where it instead checks its columns for enfactoring. |
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| = vs. normal form =
| | Being enfactored is a bad thing. Enfactored matrices — those in the RTT domain, at least — are sick, in a way<ref>According to [[saturation]], "...if [an RTT matrix] isn't saturated the supposed temperament it defines may be regarded as pathological..." </ref>; it's no accident that "enfactored" sounds sort of like "infected". We'll discuss this pathology in detail in [[canonical_form#the_pathology_of_enfactoredness|a later section of this article]]. Fortunately, the remedy is simple: all one has to do is "defactor" it — identify and divide out the common factor — to produce a healthy mapping. |
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| == "normal" vs. "canonical" ==
| | Due to complications associated with enfactored matrices which we'll get into later in this article, we discourage treating them as representations of true temperaments.<ref>As Graham Breed writes [http://x31eq.com/temper/method.html here], "Whether temperaments with contorsion should even be thought of as temperaments is a matter of debate."</ref> Instead we recommend that they be considered to represent mere "temperoids": temperament-like structures. |
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| A mapping in ''canonical'' form uniquely identifies a set of mappings that are equivalent to it (for some definition of equivalence). Historically, the xenharmonic community has most often used the word ''normal'' for this idea, and evidence of this can be found on many pages across this wiki<ref>I've started a discussion about how to handle the existing normal form material on the wiki here: https://en.xen.wiki/w/Talk:Normal_lists</ref>. And this is not wrong; normal forms are indeed often required to be unique. However, canonical forms are required to be unique even more often that normal forms are<ref>According to [https://en.wikipedia.org/wiki/Canonical_form the Wikipedia page for canonical form], 'the distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness.'</ref>, and so we prefer the term canonical to normal for this purpose.
| | = more specific definition = |
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| Also, using "canonical" helps establish a clear distinction from previous efforts to establish unique representations of equivalent mappings; due to its lack of historical use in [[RTT]], it appears to be safe to simply use "canonical form" for short to refer to matrices in defactored canonical form.
| | A mapping is enfactored if linear combinations of its rows can produce another row whose elements have a common factor (other than 1).<ref>The counts of rows that are being linearly combined in this way to check for enfactoring may not share a common factor (again, other than 1), or else enfactoring would be introduced.</ref> |
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| == vs. HNF == | | For example, {{vector|{{map|3 0 -1}} {{map|0 3 5}}}} is enfactored, because {{map|3 0 -1}} - {{map|0 3 5}} = {{map|3 -3 -6}}, which has a common factor of 3. |
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| More importantly, and perhaps partially a result of this weak understanding of the difference between the conventions for normal and canonical forms, the xenharmonic community has mistakenly used HNF as if it provides a unique representation of equivalent mappings. To be more specific, HNF does provide a unique representation of ''matrices'', i.e. from a perspective of pure mathematics, and so you will certainly find throughout mathematical literature that HNF is described as providing a unique representation, and this is correct. However, when applied to the RTT domain, i.e. to ''mappings'', the HNF sometimes fails to identify equivalent mappings as such.<ref>There is also a rarely mentioned Hermite Canonical Form, or HCF, described here: http://home.iitk.ac.in/~rksr/html/03CANONICALFACTORIZATIONS.htm, which sort of combines the HNF's normalization constraint and the RREF's reduced constraint (all pivots equal 1, all other entries in pivot columns are 0, both above and below the pivot), but we didn't find it useful because due to its constraint that all pivots be 1, it does not preserve periods that are genuinely unit fractions of an octave (at first glance, when a pivot is not equal to 1, it might trigger you to think that the mapping is enfactored. But temperaments can legitimately have generators that divide primes evenly, such as 5-limit Blackwood, {{vector|{{map|5 8 0}} {{map|0 0 1}}}}, which divides the octave into 5 parts. So any form that enforces pivots all be 1's, such as HCF and RREF, would fail this criteria.) It also doesn't qualify as an echelon form, which becomes apparent only when you use it on rank-deficient matrices, because it doesn't require the rows of all zeros to be at the bottom; instead it (bizarrely, though maybe it's related to how the SNF requires all pivots exactly along the main diagonal) requires the rows to be sorted so that all the pivots fall on the main diagonal.</ref>
| | This definition includes mappings whose rows themselves include a common factor, such as {{vector|{{map|24 38 56}}}}, which already has a clearly visible common factor of 2. |
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| The critical flaw with HNF is its failure to defactor matrices<ref>This is because dividing rows is not a permitted elementary row operation when computing the HNF. See: https://math.stackexchange.com/a/685922</ref>. The canonical form that will be described here, on the other hand, ''does'' defactor matrices, and therefore it delivers a truly canonical result.
| | = motivation = |
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| == defactoring ==
| | A major use case for defactoring is to enable a [[canonical form]] for temperament mappings, or in other words, to achieve for the linear-algebra-only school of RTT practitioners a unique ID for temperaments. Previously this was only available by using lists of minor determinants AKA wedge products of mapping rows, which by virtue of reducing the information down to a single list of numbers, could be checked for enfactoring by simply checking the single row's GCD. For more information on this historical situation, see: [[Varianced Exterior Algebra#lack of importance to RTT]] |
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| Defactoring a matrix means to perform an operation on it which ensures that it is not "enfactored". And a matrix is considered to be "enfactored" if linear combinations of its rows can produce another row whose elements have a common factor (other than 1)<ref>(and of course the counts of rows that are being linearly combined do not share a common factor)</ref>. This definition includes matrices whose rows already include a common factor, such as {{map|24 38 56}} which has a common factor of 2. Being enfactored is a bad thing. Enfactored matrices — those in the RTT domain, at least — are sick, in a way<ref>According to [[saturation]], "...if [an RTT matrix] isn't saturated the supposed temperament it defines may be regarded as pathological..." </ref>; it's no accident that "enfactored" sounds sort of like "infected". We'll discuss this pathology in detail in [[canonical_form#the_pathology_of_enfactoredness|a later section of this article]]. Fortunately, the remedy is simple: all one has to do is "defactor" it — identify and divide out the common factor — to produce a healthy mapping.
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| Due to complications associated with enfactored mappings which we'll get into later in this article, we discourage treating them as representations of true temperaments.<ref>As Graham Breed writes [http://x31eq.com/temper/method.html here], "Whether temperaments with contorsion should even be thought of as temperaments is a matter of debate."</ref> Instead we recommend that they be considered to represent mere "temperoids": temperament-like structures.
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| == vs. IRREF ==
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| Elsewhere, [[Normal_lists|Integer Reduced Row Echelon Form]], or IRREF, has been proposed as a normal form for mappings. It has a similar problem as HNF does, however, in that it does not always defactor matrices. Worse, even, sometimes IRREF introduces enfactoring where before there was none! For example, consider this mapping for 5-limit porcupine, {{vector|{{map|7 11 16}} {{map|22 35 51}}}}. This mapping is not enfactored, but its IRREF is {{vector|{{map|3 0 -1}} {{map|0 3 5}}}}, which is 3-enfactored. More on this later.
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| == motivation ==
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| A key goal of introducing this canonical form is to achieve for the linear-algebra-only school of RTT practitioners a unique ID for temperaments, which previously was only available by using lists of minor determinants AKA wedge products of mapping rows. For more information, see: [[Varianced Exterior Algebra#lack of importance to RTT]]
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| = terminology change proposal = | | = terminology change proposal = |