Defactoring: Difference between revisions
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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"]]. | A regular temperament mapping is in '''defactored canonical form (DCF)''', or '''canonical form''' for short, 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 has 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 has 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<ref>This is because dividing rows is not a permitted elementary row operation when computing the HNF. See: https://math.stackexchange.com/a/685922</ref>. The | The critical flaw with HNF is its failure to defactor matrices<ref>This is because dividing rows is not a permitted elementary row operation when computing the HNF. See: https://math.stackexchange.com/a/685922</ref>. The canonical form 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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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 unenfactored 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. | 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 unenfactored 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. | ||
By the way, | By the way, canonical form is not only for mappings. Comma-bases may also be put into canonical form, as long as they are first antitransposed[20], and then antitransposed again at the end, or in other words, you sandwich the defactoring and HNF operations between antitransposes. | ||
=== conclusion === | === conclusion === | ||
So due to the high likelihood for confusion when conceptualizing enfactored temperaments, we believe that using HNF as the unique ID for temperaments, i.e. treating temperoids as true temperaments, is a dangerous and unhelpful road. Instead, | So due to the high likelihood for confusion when conceptualizing enfactored temperaments, we believe that using HNF as the unique ID for temperaments, i.e. treating temperoids as true temperaments, is a dangerous and unhelpful road. Instead, canonical form should be used, essentially HNF but with defactoring built in, so that practitioners of RTT can focus on working with true temperaments. | ||
== identifying enfactored mappings == | == identifying enfactored mappings == | ||
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==== DCF ==== | ==== DCF ==== | ||
'''Defactored Canonical Form''', or '''DCF''' is closely related to HNF, because the second step of finding the DCF is taking the HNF. So the DCF is always ''a'' HNF, and therefore it has all the same properties of being echelon, integer, and normalized, and in turn therefore it also provides a unique representation. However it is not necessary ''the'' same HNF of the original mapping, due to the first step being defactoring; it is the same as as HNF except when the original mapping is enfactored. | '''Defactored Canonical Form''', or '''DCF''' (listed here not because it predates itself of course, but for completeness) is closely related to HNF, because the second step of finding the DCF is taking the HNF. So the DCF is always ''a'' HNF, and therefore it has all the same properties of being echelon, integer, and normalized, and in turn therefore it also provides a unique representation. However it is not necessary ''the'' same HNF of the original mapping, due to the first step being defactoring; it is the same as as HNF except when the original mapping is enfactored. | ||
In the below example, <span><math>p_{ij}</math></span> represents any positive integer, and <span><math>a_{ij}</math></span> represents any nonnegative integer less than the <span><math>p</math></span> in its column. | In the below example, <span><math>p_{ij}</math></span> represents any positive integer, and <span><math>a_{ij}</math></span> represents any nonnegative integer less than the <span><math>p</math></span> in its column. | ||