Defactoring: Difference between revisions
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A regular temperament mapping is in '''defactored canonical''' (DC) form when it is put into [https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form] (HNF) after being [[#defactoring|"defactored"]]. | A regular temperament mapping is in '''defactored canonical''' (DC) form when it is put into [https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form] (HNF) after being [[#defactoring|"defactored"]]. | ||
= defactored canonical form = | |||
== vs. normal form == | == vs. normal form == | ||
=== normal vs. canonical === | === '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. 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. 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. 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. | ||
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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. Instead we recommend that they be considered to represent mere "temperoids": temperament-like structures. | ||
== vs. | == defactored & enfactored vs. saturated and (con)torted == | ||
If you've studied RTT extensively, you've probably encountered the terms [[Saturation|"saturated" and "contorted"]] that are sometimes used to describe mappings. These two terms each have several flaws, and so this article presents alternative terms that are clearer and more descriptive: "defactored" and "enfactored", respectively. These new terms were coined by [[Dave Keenan]] in collaboration with [[Douglas Blumeyer]] in June of 2021. | If you've studied RTT extensively, you've probably encountered the terms [[Saturation|"saturated" and "contorted"]] that are sometimes used to describe mappings. These two terms each have several flaws, and so this article presents alternative terms that are clearer and more descriptive: "defactored" and "enfactored", respectively. These new terms were coined by [[Dave Keenan]] in collaboration with [[Douglas Blumeyer]] in June of 2021. | ||
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### show the examples we tried, like in the big defactoring table | ### show the examples we tried, like in the big defactoring table | ||
== canonical comma-bases == | |||
DC form is not only for mappings. Comma-bases — the duals of mappings — may also be put into DC form, as long as they are first antitransposed<ref>See a discussion of the antitranspose here: https://en.xen.wiki/w/User:Cmloegcmluin/Sandbox#null-space</ref>, and then antitransposed again at the end, or in other words, you sandwich the defactoring and HNF operations between antitransposes. | |||
DC form is arguably even more important for comma-bases than it is for mappings, because enfactored mappings at least have clear musical meaning, while enfactored comma-bases are little but a wellspring of confusion. In other words, {{map|24 38 56}} may not be a true temperament, but it still represents a temperoid and an EDO. However, {{ | |||
= related topics = | |||
== relationship between various matrix echelon forms == | == relationship between various matrix echelon forms == | ||
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# The RREF and IRREF are the same, but the HNF is different. Example: [⟨2 3 5] ⟨7 6 13]⟩ (I haven't found a realistic one yet) | # The RREF and IRREF are the same, but the HNF is different. Example: [⟨2 3 5] ⟨7 6 13]⟩ (I haven't found a realistic one yet) | ||
# The IRREF and HNF are the same, but the RREF is different. Example: hanson. | # The IRREF and HNF are the same, but the RREF is different. Example: hanson. | ||
== generator manipulation == | == generator manipulation == | ||