Pathology of enfactoring: Difference between revisions

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In this article~~, we will use lattices~~ to ~~visualize~~ [[~~enfactoring|enfactored~~]] ~~temperaments~~, to demonstrate the musical implications of mappings ~~with~~ common factors, and the lack of musical implications of comma bases ~~with~~ common factors.

In this article that relates to [[regular temperaments]], we will use lattices to demonstrate the musical implications of [[mappings]] that contain common factors, and the lack of musical implications of [[comma basis|comma bases]] that contain common factors.

== Defactored case ==

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[[File:Unenfactored mapping.png|365px|thumb|right|A 3-limit tempered lattice, superimposed on the JI lattice]]

First, let's look at a defactored mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.

First, let's look at a [[defactoring|defactored]] mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.

This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket|{{map|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector|-3 2}}, AKA 9/8. And so the comma basis for this temperament is [{{vector|-3 2}}].

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== Enfactored mappings vs. enfactored comma bases ==

One may pose the question: ~~what~~ is ~~the~~ relationship between an enfactored mapping and an enfactored comma basis? ~~Can you have~~ one ~~but not~~ the other~~? Must you? Or must you not~~? Or ~~does the question even make sense? Certainly at least some have suggested these cases~~ are ~~meaningfully~~ independent~~<ref>such as the page [[Color_notation/Temperament_Names|color notation]], which reads "it's possible that there is both torsion and contorsion"</ref>.~~

One may pose the question: is there any relationship between an enfactored mapping and an enfactored comma basis? Does one imply the other. Does one imply the absence of the other? Or are they completely independent?

~~The conclusion we arrive at here~~ is ~~that because~~ enfactored ~~comma bases don't make any sense~~, ~~or at least don't represent any legitimately new musical information of any kind that their defactored version doesn't already represent~~, ~~it is not generally useful~~ to ~~think of enfactored mappings~~ and ~~enfactored comma bases~~ as independent ~~phenomena~~. ~~It only makes sense to speak~~ of ~~enfactored temperaments~~. ~~Of course~~, ~~one will often use~~ the ~~term "~~enfactored ~~mapping" because~~ enfactored mappings ~~are the kind~~ which ~~do have some musical purpose~~, ~~and often~~ the ~~enfactored mapping will be being used to represent~~ the ~~enfactored temperament — or~~ temperoid, that is.

We note that there is no such thing as an enfactored temperament, only enfactored matrices. And these essentially result from an arithmetic oversight, namely failing to defactor, and as such they are completely independent<ref>as observed on the page [[Color_notation/Temperament_Names|color notation]], which reads "it's possible that there is both torsion and contorsion"</ref>. When we use matrix math for RTT we defactor as a matter of course. For example, defactoring is built in to the NullSpace matrix operation in the Wolfram Language.

However, if you have an artistic or other reason to generate scales using generators derived directly from enfactored mappings, and therefore to treat enfactored mappings as representing different musical objects, which we call temperoids, not temperaments, then you might choose to encode the same common factor into the comma basis representation of the temperoid with the understanding that this is a mere bookkeeping exercise and has no mathematical basis.

=Footnotes=

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-In this article, we will use lattices to visualize [[enfactoring|enfactored]] temperaments, to demonstrate the musical implications of mappings with common factors, and the lack of musical implications of comma bases with common factors.
+In this article that relates to [[regular temperaments]], we will use lattices to demonstrate the musical implications of [[mappings]] that contain common factors, and the lack of musical implications of [[comma basis|comma bases]] that contain common factors.
 == Defactored case ==
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 [[File:Unenfactored mapping.png|365px|thumb|right|A 3-limit tempered lattice, superimposed on the JI lattice]]
-First, let's look at a defactored mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.
+First, let's look at a [[defactoring|defactored]] mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.
 This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket|{{map|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector|-3 2}}, AKA 9/8. And so the comma basis for this temperament is [{{vector|-3 2}}].
@@ Line 73: / Line 73: @@
 == Enfactored mappings vs. enfactored comma bases ==
-One may pose the question: what is the relationship between an enfactored mapping and an enfactored comma basis? Can you have one but not the other? Must you? Or must you not? Or does the question even make sense? Certainly at least some have suggested these cases are meaningfully independent<ref>such as the page [[Color_notation/Temperament_Names|color notation]], which reads "it's possible that there is both torsion and contorsion"</ref>.
+One may pose the question: is there any relationship between an enfactored mapping and an enfactored comma basis? Does one imply the other. Does one imply the absence of the other? Or are they completely independent?
-The conclusion we arrive at here is that because enfactored comma bases don't make any sense, or at least don't represent any legitimately new musical information of any kind that their defactored version doesn't already represent, it is not generally useful to think of enfactored mappings and enfactored comma bases as independent phenomena. It only makes sense to speak of enfactored temperaments. Of course, one will often use the term "enfactored mapping" because enfactored mappings are the kind which do have some musical purpose, and often the enfactored mapping will be being used to represent the enfactored temperament — or temperoid, that is.
+We note that there is no such thing as an enfactored temperament, only enfactored matrices. And these essentially result from an arithmetic oversight, namely failing to defactor, and as such they are completely independent<ref>as observed on the page [[Color_notation/Temperament_Names|color notation]], which reads "it's possible that there is both torsion and contorsion"</ref>. When we use matrix math for RTT we defactor as a matter of course. For example, defactoring is built in to the NullSpace matrix operation in the Wolfram Language.
+However, if you have an artistic or other reason to generate scales using generators derived directly from enfactored mappings, and therefore to treat enfactored mappings as representing different musical objects, which we call temperoids, not temperaments, then you might choose to encode the same common factor into the comma basis representation of the temperoid with the understanding that this is a mere bookkeeping exercise and has no mathematical basis.
 =Footnotes=