Defactoring: Difference between revisions

Line 17:

== defactoring ==

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). 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 [[~~User:Cmloegcmluin/Defactored_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.

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). 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.

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.

Line 366:

\end{array} \end{array} \right]</math>

Now we begin applying elementary row operations until the part on the left is in Hermite Normal Form. If you need to review the definition of HNF and its constraints, you can find that in detail later in this article [[~~User:Cmloegcmluin/Defactored_canonical_form~~#HNF|here]]. The quick and dirty is:

Now we begin applying elementary row operations until the part on the left is in Hermite Normal Form. If you need to review the definition of HNF and its constraints, you can find that in detail later in this article [[canonical_form#HNF|here]]. The quick and dirty is:

# all pivots > 0

Line 1,025:

=== HNF ===

'''[https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form]''', or '''HNF''': this one's constraints begin with echelon form and integer, therefore every HNF is also IREF. But HNF is not exactly reduced (as discussed above; [[~~User:Cmloegcmluin/Defactored_canonical_form~~#RREF|link here]]); instead, it is ''normalized'', which — similarly to reduced — is a two-part constraint. Where reduced requires that all pivots be exactly equal to 1, normalized requires only that all pivots be positive (positive integers, of course, due to the other integer constraint). And where reduced requires that all entries in pivot columns besides the pivots are exactly equal to 0, normalized requires only that all entries in pivot columns below the pivots are exactly equal to 0, while entries in pivot columns above the pivots only have to be strictly less than the pivot in the respective column (while still being non-negative).<ref>The exact criteria for HNF are not always consistently agreed upon, however.</ref><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. 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><ref>We are using "row-style" Hermite Normal Form here, not "column-style"; the latter would involve simply flipping everything 90 degrees so that the echelon requirement was that pivots be strictly ''below'' the pivots in the previous ''column'', and that pivot ''rows'' are considered for the normalization constraint rather than pivot ''columns''.</ref> In other words, elements above the pivot have to be reduced modulo the pivot. The normalization HNF uses is cool because this constraint, while strictly less strict than the reduced constraint used by RREF, is still strict enough to ensure uniqueness, but loose enough to ensure the integer constraint can be simultaneously satisfied, where RREF cannot ensure that.

'''[https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form]''', or '''HNF''': this one's constraints begin with echelon form and integer, therefore every HNF is also IREF. But HNF is not exactly reduced (as discussed above; [[canonical_form#RREF|link here]]); instead, it is ''normalized'', which — similarly to reduced — is a two-part constraint. Where reduced requires that all pivots be exactly equal to 1, normalized requires only that all pivots be positive (positive integers, of course, due to the other integer constraint). And where reduced requires that all entries in pivot columns besides the pivots are exactly equal to 0, normalized requires only that all entries in pivot columns below the pivots are exactly equal to 0, while entries in pivot columns above the pivots only have to be strictly less than the pivot in the respective column (while still being non-negative).<ref>The exact criteria for HNF are not always consistently agreed upon, however.</ref><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. 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><ref>We are using "row-style" Hermite Normal Form here, not "column-style"; the latter would involve simply flipping everything 90 degrees so that the echelon requirement was that pivots be strictly ''below'' the pivots in the previous ''column'', and that pivot ''rows'' are considered for the normalization constraint rather than pivot ''columns''.</ref> In other words, elements above the pivot have to be reduced modulo the pivot. The normalization HNF uses is cool because this constraint, while strictly less strict than the reduced constraint used by RREF, is still strict enough to ensure uniqueness, but loose enough to ensure the integer constraint can be simultaneously satisfied, where RREF cannot ensure that.

So HNF has a lot in common with IRREF, which is the IREF you find by converting the RREF, but it is not always the same as IRREF.

Line 1,138:

=== problem 2. edge case: well-hidden common factors ===

SAD defactoring was found to fail in the case of [[~~User:Cmloegcmluin/Defactored_canonical_form~~#well-hidden_enfactoring|well-hidden common factor]]s, i.e.

SAD defactoring was found to fail in the case of [[canonical_form#well-hidden_enfactoring|well-hidden common factor]]s, i.e.

ones where there are no common factors to be found in sums or differences of the rows, but are to be found in sums or differences of ''multiples of'' these rows. In order to solve this problem, SAD defactoring would need to be reworked to check every possible combination of integer multiples of each row, potentially up to the largest absolute value of integer in the given mapping, which would cause it to become intractably slow to compute.

@@ Line 17: / Line 17: @@
 == defactoring ==
-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). 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 [[User:Cmloegcmluin/Defactored_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.
+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). 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.
 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.
@@ Line 366: / Line 366: @@
 \end{array} \end{array} \right]</math>
-Now we begin applying elementary row operations until the part on the left is in Hermite Normal Form. If you need to review the definition of HNF and its constraints, you can find that in detail later in this article [[User:Cmloegcmluin/Defactored_canonical_form#HNF|here]]. The quick and dirty is:
+Now we begin applying elementary row operations until the part on the left is in Hermite Normal Form. If you need to review the definition of HNF and its constraints, you can find that in detail later in this article [[canonical_form#HNF|here]]. The quick and dirty is:
 # all pivots > 0
@@ Line 1,025: / Line 1,025: @@
 === HNF ===
-'''[https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form]''', or '''HNF''': this one's constraints begin with echelon form and integer, therefore every HNF is also IREF. But HNF is not exactly reduced (as discussed above; [[User:Cmloegcmluin/Defactored_canonical_form#RREF|link here]]); instead, it is ''normalized'', which — similarly to reduced — is a two-part constraint. Where reduced requires that all pivots be exactly equal to 1, normalized requires only that all pivots be positive (positive integers, of course, due to the other integer constraint). And where reduced requires that all entries in pivot columns besides the pivots are exactly equal to 0, normalized requires only that all entries in pivot columns below the pivots are exactly equal to 0, while entries in pivot columns above the pivots only have to be strictly less than the pivot in the respective column (while still being non-negative).<ref>The exact criteria for HNF are not always consistently agreed upon, however.</ref><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. 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><ref>We are using "row-style" Hermite Normal Form here, not "column-style"; the latter would involve simply flipping everything 90 degrees so that the echelon requirement was that pivots be strictly ''below'' the pivots in the previous ''column'', and that pivot ''rows'' are considered for the normalization constraint rather than pivot ''columns''.</ref> In other words, elements above the pivot have to be reduced modulo the pivot. The normalization HNF uses is cool because this constraint, while strictly less strict than the reduced constraint used by RREF, is still strict enough to ensure uniqueness, but loose enough to ensure the integer constraint can be simultaneously satisfied, where RREF cannot ensure that.
+'''[https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form]''', or '''HNF''': this one's constraints begin with echelon form and integer, therefore every HNF is also IREF. But HNF is not exactly reduced (as discussed above; [[canonical_form#RREF|link here]]); instead, it is ''normalized'', which — similarly to reduced — is a two-part constraint. Where reduced requires that all pivots be exactly equal to 1, normalized requires only that all pivots be positive (positive integers, of course, due to the other integer constraint). And where reduced requires that all entries in pivot columns besides the pivots are exactly equal to 0, normalized requires only that all entries in pivot columns below the pivots are exactly equal to 0, while entries in pivot columns above the pivots only have to be strictly less than the pivot in the respective column (while still being non-negative).<ref>The exact criteria for HNF are not always consistently agreed upon, however.</ref><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. 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><ref>We are using "row-style" Hermite Normal Form here, not "column-style"; the latter would involve simply flipping everything 90 degrees so that the echelon requirement was that pivots be strictly ''below'' the pivots in the previous ''column'', and that pivot ''rows'' are considered for the normalization constraint rather than pivot ''columns''.</ref> In other words, elements above the pivot have to be reduced modulo the pivot. The normalization HNF uses is cool because this constraint, while strictly less strict than the reduced constraint used by RREF, is still strict enough to ensure uniqueness, but loose enough to ensure the integer constraint can be simultaneously satisfied, where RREF cannot ensure that.
 So HNF has a lot in common with IRREF, which is the IREF you find by converting the RREF, but it is not always the same as IRREF.
@@ Line 1,138: / Line 1,138: @@
 === problem 2. edge case: well-hidden common factors ===
-SAD defactoring was found to fail in the case of [[User:Cmloegcmluin/Defactored_canonical_form#well-hidden_enfactoring|well-hidden common factor]]s, i.e.
+SAD defactoring was found to fail in the case of [[canonical_form#well-hidden_enfactoring|well-hidden common factor]]s, i.e.
 ones where there are no common factors to be found in sums or differences of the rows, but are to be found in sums or differences of ''multiples of'' these rows. In order to solve this problem, SAD defactoring would need to be reworked to check every possible combination of integer multiples of each row, potentially up to the largest absolute value of integer in the given mapping, which would cause it to become intractably slow to compute.