Defactoring: Difference between revisions

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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). 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". 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. Instead we recommend that they be considered to represent mere "temperoids": temperament-like structures.

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.

=== vs. IRREF ===

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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). 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". 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. Instead we recommend that they be considered to represent mere "temperoids": temperament-like structures.
+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.
 === vs. IRREF ===