Defactoring: Difference between revisions

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Following 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]], the anti-minors method was developed. Unfortunately it required defactoring to work. So it couldn't be used as part of a defactoring solution itself.

=== duality in LA and VEA ===

RTT could be said to be practiced in two major flavors: LA, or Linear Algebra, and 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 ==

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 Following 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]], the anti-minors method was developed. Unfortunately it required defactoring to work. So it couldn't be used as part of a defactoring solution itself.
+=== duality in LA and VEA ===
+RTT could be said to be practiced in two major flavors: LA, or Linear Algebra, and 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 ==