Defactoring: Difference between revisions

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== "normal" vs. "canonical" ==

A mapping in ''canonical'' form uniquely identifies a set of mappings that are equivalent to it. Historically, the xenharmonic community has most often used the word ''normal'' for this idea, and evidence of this can be found on many pages across this wiki<ref>I've started a discussion about how to handle the existing normal form material on the wiki here: https://en.xen.wiki/w/Talk:Normal_lists</ref>. And this is not wrong; normal forms are indeed often required to be unique. However, canonical forms are required to be unique even more often that normal forms are<ref>According to [https://en.wikipedia.org/wiki/Canonical_form the Wikipedia page for canonical form], 'the distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness.'</ref>, and so we prefer the term canonical to normal for this purpose.

A mapping in ''canonical'' form uniquely identifies a set of mappings that are equivalent to it (for some definition of equivalence). Historically, the xenharmonic community has most often used the word ''normal'' for this idea, and evidence of this can be found on many pages across this wiki<ref>I've started a discussion about how to handle the existing normal form material on the wiki here: https://en.xen.wiki/w/Talk:Normal_lists</ref>. And this is not wrong; normal forms are indeed often required to be unique. However, canonical forms are required to be unique even more often that normal forms are<ref>According to [https://en.wikipedia.org/wiki/Canonical_form the Wikipedia page for canonical form], 'the distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness.'</ref>, and so we prefer the term canonical to normal for this purpose.

Also, using "canonical" helps establish a clear distinction from previous efforts to establish unique representations of equivalent mappings; due to its lack of historical use in [[RTT]], it appears to be safe to simply use "canonical form" for short to refer to matrices in defactored canonical form.

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== motivation ==

A key goal of introducing this canonical form is to achieve for the linear-algebra-only school of RTT ~~practioners~~ a unique ID for temperaments, which previously was only available by using lists of minor determinants AKA wedge products of mapping rows.

A key goal of introducing this canonical form is to achieve for the linear-algebra-only school of RTT practitioners a unique ID for temperaments, which previously was only available by using lists of minor determinants AKA wedge products of mapping rows. For more information, see: [[Varianced Exterior Algebra#lack of importance to RTT]]

= terminology change proposal =

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 == "normal" vs. "canonical" ==
-A mapping in ''canonical'' form uniquely identifies a set of mappings that are equivalent to it. Historically, the xenharmonic community has most often used the word ''normal'' for this idea, and evidence of this can be found on many pages across this wiki<ref>I've started a discussion about how to handle the existing normal form material on the wiki here: https://en.xen.wiki/w/Talk:Normal_lists</ref>. And this is not wrong; normal forms are indeed often required to be unique. However, canonical forms are required to be unique even more often that normal forms are<ref>According to [https://en.wikipedia.org/wiki/Canonical_form the Wikipedia page for canonical form], 'the distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness.'</ref>, and so we prefer the term canonical to normal for this purpose.
+A mapping in ''canonical'' form uniquely identifies a set of mappings that are equivalent to it (for some definition of equivalence). Historically, the xenharmonic community has most often used the word ''normal'' for this idea, and evidence of this can be found on many pages across this wiki<ref>I've started a discussion about how to handle the existing normal form material on the wiki here: https://en.xen.wiki/w/Talk:Normal_lists</ref>. And this is not wrong; normal forms are indeed often required to be unique. However, canonical forms are required to be unique even more often that normal forms are<ref>According to [https://en.wikipedia.org/wiki/Canonical_form the Wikipedia page for canonical form], 'the distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness.'</ref>, and so we prefer the term canonical to normal for this purpose.
 Also, using "canonical" helps establish a clear distinction from previous efforts to establish unique representations of equivalent mappings; due to its lack of historical use in [[RTT]], it appears to be safe to simply use "canonical form" for short to refer to matrices in defactored canonical form.
@@ Line 27: / Line 27: @@
 == motivation ==
-A key goal of introducing this canonical form is to achieve for the linear-algebra-only school of RTT practioners a unique ID for temperaments, which previously was only available by using lists of minor determinants AKA wedge products of mapping rows.
+A key goal of introducing this canonical form is to achieve for the linear-algebra-only school of RTT practitioners a unique ID for temperaments, which previously was only available by using lists of minor determinants AKA wedge products of mapping rows. For more information, see: [[Varianced Exterior Algebra#lack of importance to RTT]]
 = terminology change proposal =