Defactoring: Difference between revisions

Line 1:

'''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-basis|comma-bases]], the duals of mappings, where it instead checks its columns for enfactoring.

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 [[defactoring#~~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.

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 [[defactoring#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 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.

Line 15:

= Motivation =

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]], and for more information on the canonical form developed, see [[defactored Hermite form]].

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]], and for more information on the canonical form developed, see [[defactored Hermite form]].

= Terminology change proposal =

Line 319:

After we know how to do these two things individually, we'll learn how to tweak them and assemble them together in order to perform a complete column Hermite defactoring.

Fortunately, both of these two processes can be done using a technique you may already be familiar with if you've learned how to calculate the null-space of a mapping by hand (as demonstrated [[Douglas_Blumeyer%27s_RTT_How-To#~~null~~-space|here]]):

Fortunately, both of these two processes can be done using a technique you may already be familiar with if you've learned how to calculate the null-space of a mapping by hand (as demonstrated [[Douglas_Blumeyer%27s_RTT_How-To#Null-space|here]]):

# augmenting your matrix with an identity matrix

# performing elementary row or column operations until a desired state is achieved<ref>For convenience, here is a summary of the three different techniques and their targets:<br>

Line 863:

= Canonical comma-bases =

Canonical form is not only for mappings; comma-bases may also be put into canonical form. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed<ref>See a discussion of the antitranspose here: [[Douglas_Blumeyer%27s_RTT_How-To#~~null~~-space]]</ref>once at the beginning, and then antitransposed again at the end.

Canonical form is not only for mappings; comma-bases may also be put into canonical form. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed<ref>See a discussion of the antitranspose here: [[Douglas_Blumeyer%27s_RTT_How-To#Null-space]]</ref>once at the beginning, and then antitransposed again at the end.

For example, suppose we have the comma-basis for septimal meantone:

@@ Line 1: / Line 1: @@
 '''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-basis|comma-bases]], the duals of mappings, where it instead checks its columns for enfactoring.
-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 [[defactoring#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.
+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 [[defactoring#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 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.
@@ Line 15: / Line 15: @@
 = Motivation =
-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]], and for more information on the canonical form developed, see [[defactored Hermite form]].
+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]], and for more information on the canonical form developed, see [[defactored Hermite form]].
 = Terminology change proposal =
@@ Line 319: / Line 319: @@
 After we know how to do these two things individually, we'll learn how to tweak them and assemble them together in order to perform a complete column Hermite defactoring.
-Fortunately, both of these two processes can be done using a technique you may already be familiar with if you've learned how to calculate the null-space of a mapping by hand (as demonstrated [[Douglas_Blumeyer%27s_RTT_How-To#null-space|here]]):
+Fortunately, both of these two processes can be done using a technique you may already be familiar with if you've learned how to calculate the null-space of a mapping by hand (as demonstrated [[Douglas_Blumeyer%27s_RTT_How-To#Null-space|here]]):
 # augmenting your matrix with an identity matrix
 # performing elementary row or column operations until a desired state is achieved<ref>For convenience, here is a summary of the three different techniques and their targets:<br>
@@ Line 863: / Line 863: @@
 = Canonical comma-bases =
-Canonical form is not only for mappings; comma-bases may also be put into canonical form. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed<ref>See a discussion of the antitranspose here: [[Douglas_Blumeyer%27s_RTT_How-To#null-space]]</ref>once at the beginning, and then antitransposed again at the end.
+Canonical form is not only for mappings; comma-bases may also be put into canonical form. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed<ref>See a discussion of the antitranspose here: [[Douglas_Blumeyer%27s_RTT_How-To#Null-space]]</ref>once at the beginning, and then antitransposed again at the end.
 For example, suppose we have the comma-basis for septimal meantone: