Defactoring: Difference between revisions

Line 557:

* The case where the only equalities are between RREF and IRREF and between HNF and DCF is impossible, because if RREF=IRREF that suggests that all entries are multiples of their pivots, which is easy if the temperament is enfactored, but if HNF=DCF then it is not.

=== sum-and-difference defactoring ===

=== SAD (sum-and-difference) defactoring ===

When in development on an ideal defactoring method — the effort which culminated in column Hermite defactoring — Dave and Douglas invested a great deal of time in a defactoring method which still has some advantages over column Hermite defactoring but was ultimately rejected. This other method is called '''"sum-and-difference" defactoring''', or '''SAD defactor''' (it is sad partially because it didn't work out).

Line 640:

];</nowiki>

=== MADAM defactoring ===

=== MADAM (minors and divide-out-GCD, anti- minors) defactoring ===

Another technique which was experimented with took advantage of the fact that the list of minor determinants (or simply "minors") of a mapping is guaranteed to include any common factor as its entries' GCD. So, if one simply converted a mapping to its list of minors, removed the GCD (at which point you would have what in RTT is called a [[User:Cmloegcmluin/RTT_How-To#multimaps|canonical multimap]], or [[wedgie]]), and then performed an "anti-minors" operation to get back to a mapping form, any common factors should be removed.

Inspired by Gene Ward Smith's method for computing anti-minors as described [[Mathematical_theory_of_regular_temperaments#Wedgies|here]] and [[Basic_abstract_temperament_translation_code|here]], an anti-minors method was implemented in Wolfram Language. It was found that a defactoring algorithm based on '''M'''inors '''A'''nd '''D'''ivide-out-GCG, '''A'''nti-'''M'''inors, or '''MADAM defactoring''', does indeed work. However, it runs 10 to 20 times slower than Smith defactoring and column Hermite defactoring, and it is not compellingly easier to understand than either of them, so it is not considered to be of significant interest.

~~=== duality in LA and VEA ===~~

RTT could be said to be practiced in two major flavors: LA, or Linear Algebra, and [[User:Cmloegcmluin/VEA|VEA]], or Varianced Exterior Algebra. The former uses only vectors, covectors, and matrices. The latter uses multivectors and multicovectors instead of matrices, where a key example of a multivector is a "wedgie". Each RTT flavor has a notion of a dual.

LA's dual it is the null-space operation, which takes you back and forth between the two matrix representations of a temperament: its mapping and its comma-basis<ref>with the stipulation that the anti-null-space operation that gets you from the comma-basis back to the mapping requires an anti-transpose sandwich.</ref>. VEA's dual, on the other hand, is closely related to the Grassman/orthogonal complement in exterior algebra as well as the complement operation from MLA (multilinear algebra) which is sometimes referred to as the "Hodge dual" or "Hodge star", and it takes you back and forth between the two multi(co)vector representations of a temperament: the multimap and the multicomma.

One of the primary motivations for developing a canonical form for mappings (and comma-bases) was to achieve for the LA flavor of RTT a key thing it was missing that the VEA flavor had: a way to uniquely identify temperaments. In VEA, this was easy, because any enfactoring would be readily identifiable as a GCD>1 in the multimap or multicomma. In fact, the definition of the "wedgie" calls for removing such GCDs when computing it from a mapping.

Something that was observed while developing the canonical form for LA was that the dual operation for LA, null-space, eliminated any enfactoring. On the other hand, the dual operation for VEA preserves it. This might be another strike against VEA.

~~However, because the VEA wedgie calls for removing GCDs, we can assume that its dual would also build in the removal of any GCD. So it's not really an additional strike.~~

It could be construed as an extra point for LA, however, because it shows how its dual operation automatically eliminates the pathological aspect of enfactoring. This captures the truth of how an enfactored mapping has no effect on tempered commas, and vice versa.

== references ==

@@ Line 557: / Line 557: @@
 * The case where the only equalities are between RREF and IRREF and between HNF and DCF is impossible, because if RREF=IRREF that suggests that all entries are multiples of their pivots, which is easy if the temperament is enfactored, but if HNF=DCF then it is not.
-=== sum-and-difference defactoring ===
+=== SAD (sum-and-difference) defactoring ===
 When in development on an ideal defactoring method — the effort which culminated in column Hermite defactoring — Dave and Douglas invested a great deal of time in a defactoring method which still has some advantages over column Hermite defactoring but was ultimately rejected. This other method is called '''"sum-and-difference" defactoring''', or '''SAD defactor''' (it is sad partially because it didn't work out).
@@ Line 640: / Line 640: @@
 ];</nowiki>
-=== MADAM defactoring ===
+=== MADAM (minors and divide-out-GCD, anti- minors) defactoring ===
 Another technique which was experimented with took advantage of the fact that the list of minor determinants (or simply "minors") of a mapping is guaranteed to include any common factor as its entries' GCD. So, if one simply converted a mapping to its list of minors, removed the GCD (at which point you would have what in RTT is called a [[User:Cmloegcmluin/RTT_How-To#multimaps|canonical multimap]], or [[wedgie]]), and then performed an "anti-minors" operation to get back to a mapping form, any common factors should be removed.
 Inspired by Gene Ward Smith's method for computing anti-minors as described [[Mathematical_theory_of_regular_temperaments#Wedgies|here]] and [[Basic_abstract_temperament_translation_code|here]], an anti-minors method was implemented in Wolfram Language. It was found that a defactoring algorithm based on '''M'''inors '''A'''nd '''D'''ivide-out-GCG, '''A'''nti-'''M'''inors, or '''MADAM defactoring''', does indeed work. However, it runs 10 to 20 times slower than Smith defactoring and column Hermite defactoring, and it is not compellingly easier to understand than either of them, so it is not considered to be of significant interest.
-=== duality in LA and VEA ===
-RTT could be said to be practiced in two major flavors: LA, or Linear Algebra, and [[User:Cmloegcmluin/VEA|VEA]], or Varianced Exterior Algebra. The former uses only vectors, covectors, and matrices. The latter uses multivectors and multicovectors instead of matrices, where a key example of a multivector is a "wedgie". Each RTT flavor has a notion of a dual.
-LA's dual it is the null-space operation, which takes you back and forth between the two matrix representations of a temperament: its mapping and its comma-basis<ref>with the stipulation that the anti-null-space operation that gets you from the comma-basis back to the mapping requires an anti-transpose sandwich.</ref>. VEA's dual, on the other hand, is closely related to the Grassman/orthogonal complement in exterior algebra as well as the complement operation from MLA (multilinear algebra) which is sometimes referred to as the "Hodge dual" or "Hodge star", and it takes you back and forth between the two multi(co)vector representations of a temperament: the multimap and the multicomma.
-One of the primary motivations for developing a canonical form for mappings (and comma-bases) was to achieve for the LA flavor of RTT a key thing it was missing that the VEA flavor had: a way to uniquely identify temperaments. In VEA, this was easy, because any enfactoring would be readily identifiable as a GCD>1 in the multimap or multicomma. In fact, the definition of the "wedgie" calls for removing such GCDs when computing it from a mapping.
-Something that was observed while developing the canonical form for LA was that the dual operation for LA, null-space, eliminated any enfactoring. On the other hand, the dual operation for VEA preserves it. This might be another strike against VEA.
-However, because the VEA wedgie calls for removing GCDs, we can assume that its dual would also build in the removal of any GCD. So it's not really an additional strike.
-It could be construed as an extra point for LA, however, because it shows how its dual operation automatically eliminates the pathological aspect of enfactoring. This captures the truth of how an enfactored mapping has no effect on tempered commas, and vice versa.
 == references ==