Normal forms: Difference between revisions
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This is the "canonical form" for a temperament that was developed by [[Dave Keenan]] and [[Douglas Blumeyer]], formed from [[defactoring]] the matrix (a.k.a. removing [[contorsion]]) prior to putting it into Hermite form. | This is the "canonical form" for a temperament that was developed by [[Dave Keenan]] and [[Douglas Blumeyer]], formed from [[defactoring]] the matrix (a.k.a. removing [[contorsion]]) prior to putting it into Hermite form. | ||
We may write a list of vals (mapping) as a {{nowrap|(''k'', ''d'')}}-shaped matrix (read "k by d", i.e. with ''k'' rows and ''d'' columns), where the rows of the matrix are the vals (maps), and ''d'' is the dimensionality of the system<ref group="note">Calling the {{w|prime-counting function}}, written π(''x''), on the prime limit will give us this number. For examples, {{nowrap| π(2) {{=}} 1 }}, {{nowrap| π(3) {{=}} 2, {{nowrap| π(5) {{=}} 3 }}, {{nowrap| π(7) {{=}} 4 }}, {{nowrap| π(11) {{=}} 5 }}, etc. </ref>. To get the '''defactored Hermite form''', we do the following: | We may write a list of vals (mapping) as a {{nowrap|(''k'', ''d'')}}-shaped matrix (read "k by d", i.e. with ''k'' rows and ''d'' columns), where the rows of the matrix are the vals (maps), and ''d'' is the dimensionality of the system<ref group="note">Calling the {{w|prime-counting function}}, written π(''x''), on the prime limit will give us this number. For examples, {{nowrap| π(2) {{=}} 1 }}, {{nowrap| π(3) {{=}} 2 }}, {{nowrap| π(5) {{=}} 3 }}, {{nowrap| π(7) {{=}} 4 }}, {{nowrap| π(11) {{=}} 5 }}, etc. </ref>. To get the '''defactored Hermite form''', we do the following: | ||
# First, defactor it (a.k.a. make sure it is [[saturated]]).<ref group="note">Historically, this step was not explicitly recognized as necessary for normal forms. The vast majority of normal forms catalogued on the wiki are not contorted/enfactored in the first place, but specifically defining this canonical form to include this requirement is an important step toward ensuring that, which will prevent redundant temperaments from being catalogued. In various domains, normal forms are often required to be unique, however, canonical forms are required to be unique even more often that normal forms are; according to [[Wikipedia: 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.' This is the rationale behind defining ''canonical'' as opposed to merely ''normal''. To be more specific, The 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. And the critical flaw with HNF is its failure to defactor matrices – meaning that a contorted mapping matrix has a different Hermite normal form than a non-contorted one with the same kernel – and this is because dividing rows is not a permitted elementary row operation when computing the HNF. See: [https://math.stackexchange.com/a/685922]. The canonical form as described here ''does'' defactor matrices, and therefore it delivers a truly canonical result.<br> | # First, defactor it (a.k.a. make sure it is [[saturated]]).<ref group="note">Historically, this step was not explicitly recognized as necessary for normal forms. The vast majority of normal forms catalogued on the wiki are not contorted/enfactored in the first place, but specifically defining this canonical form to include this requirement is an important step toward ensuring that, which will prevent redundant temperaments from being catalogued. In various domains, normal forms are often required to be unique, however, canonical forms are required to be unique even more often that normal forms are; according to [[Wikipedia: 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.' This is the rationale behind defining ''canonical'' as opposed to merely ''normal''. To be more specific, The 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. And the critical flaw with HNF is its failure to defactor matrices – meaning that a contorted mapping matrix has a different Hermite normal form than a non-contorted one with the same kernel – and this is because dividing rows is not a permitted elementary row operation when computing the HNF. See: [https://math.stackexchange.com/a/685922]. The canonical form as described here ''does'' defactor matrices, and therefore it delivers a truly canonical result.<br> | ||
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== Footnotes == | == Footnotes == | ||
<references group="note" /> | <references group="note" /> | ||
== References == | |||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
[[Category:Math]] | [[Category:Math]] | ||
[[Category:Mapping]] | [[Category:Mapping]] | ||