Defactoring: Difference between revisions
Cmloegcmluin (talk | contribs) →temperament states: include DCF |
Cmloegcmluin (talk | contribs) DCF, not DC form (like HNF) |
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A regular temperament mapping is in '''defactored canonical''' ( | A regular temperament mapping is in '''defactored canonical form''' (DCF) when it is put into [https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form] (HNF) after being [[#defactoring|"defactored"]]. | ||
== vs. normal form == | == vs. normal form == | ||
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More importantly, and perhaps partially a result of this weak understanding of the difference between the conventions for normal and canonical forms, the xenharmonic community ha mistakenly used HNF as if it provides a unique representation of equivalent mappings. To be more specific, HNF does provide a unique representation of ''matrices'', i.e. from a perspective of pure mathematics, and so you will certainly find throughout mathematical literature that HNF is described as providing a unique representation, and this is correct. However, when applied to the RTT domain, i.e. to ''mappings'', the HNF sometimes fails to identify equivalent mappings as such. | More importantly, and perhaps partially a result of this weak understanding of the difference between the conventions for normal and canonical forms, the xenharmonic community ha mistakenly used HNF as if it provides a unique representation of equivalent mappings. To be more specific, HNF does provide a unique representation of ''matrices'', i.e. from a perspective of pure mathematics, and so you will certainly find throughout mathematical literature that HNF is described as providing a unique representation, and this is correct. However, when applied to the RTT domain, i.e. to ''mappings'', the HNF sometimes fails to identify equivalent mappings as such. | ||
The critical flaw with HNF is its failure to defactor matrices. The | The critical flaw with HNF is its failure to defactor matrices. The DCF that will be described here, on the other hand, ''does'' defactor matrices, and therefore it delivers a truly canonical result. | ||
=== defactoring === | === defactoring === | ||
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### talk about the criteria, like has to be integer, full-rank | ### talk about the criteria, like has to be integer, full-rank | ||
In addition to being canonical and defactored, | In addition to being canonical and defactored, DCF has other important properties: | ||
* It is integer, i.e. contains only integer terms. | * It is integer, i.e. contains only integer terms. | ||
* reduced | * reduced | ||
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== canonical comma-bases == | == canonical comma-bases == | ||
DCF is not only for mappings. Comma-bases — the duals of mappings — may also be put into DCF, 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. | |||
DCF 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, {{ | |||
== relationship between various matrix echelon forms == | == relationship between various matrix echelon forms == | ||
There are several well-known echelon forms for matrices that predate | There are several well-known echelon forms for matrices that predate DCF. Let's review them and their properties. | ||
=== list of forms === | === list of forms === | ||