Defactoring: Difference between revisions
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== vs. normal form == | == vs. normal form == | ||
=== | === "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. IRREF === | |||
Elsewhere, [[Normal_lists|Integer Reduced Row Echelon Form]], or IRREF, has been proposed as a normal form for mappings. It has a similar problem as HNF does, however, in that it does not always defactor matrices. Worse, even, sometimes IRREF introduces enfactoring where before there was none! For example, consider this mapping for 5-limit porcupine, {{vector|{{map|7 11 16}} {{map|22 35 51}}}}. This mapping is not enfactored, but its IRREF is {{vector|{{map|3 0 -1}} {{map|0 3 5}}}}, which is 3-enfactored. More on this later. | |||
== defactored & enfactored vs. saturated and (con)torted == | == defactored & enfactored vs. saturated and (con)torted == | ||