Defactoring: Difference between revisions

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In addition to being canonical and defactored, DCF has other important properties, which probably go without saying in the context of RTT mappings, but here they are just in case:

* '''integer''': contains only integer terms.

* '''full-rank'''<ref>Interesting tidbit regarding full-rank matrices and unimodular matrices: for square matrices, full-rank implies unimodularity, and vice-versa.</ref>: removes rank deficiencies, or in other words, rows that are all zeros (upon any linear combination of rows)

* '''full-rank'''<ref>Interesting tidbit regarding full-rank matrices and unimodular matrices: for square matrices, full-rank implies unimodularity, and vice-versa.</ref><ref>Another interesting tidbit here is that canonical form does not remove what one might call dimensionality deficiencies. In the case of a mapping, this would take the form of an extra column of all zeros to the right of any non-zero entries, or in other words, an unmapped prime higher than other mapped prime. For example you could have {{vector|{{map|1 0 -4 0}} {{map|0 1 4 0}}}} which is just 5-limit meantone but represented in the 7-limit even though prime 7 is not used. And for a comma-basis the form dimensionality deficiency would take is rotated 90 degrees: a row of all zeros below all other nonzero entries, e.g. {{map|{{vector|4 -4 1 0}}}}. The reason these additional zeros should be preserved and these temperaments be treated as different from their untrimmed counterparts is made clear when we consider the difference in the anti-null-spaces of comma-bases; the presence of extra dimensions implies the presence of extra generators that are unbound to the other generators. And if we decide this way re: comma-bases, we can't hardly make an inconsistent call re: mappings.</ref>: removes rank deficiencies, or in other words, rows that are all zeros (upon any linear combination of rows)

* '''preserves genuine unit-fraction-of-an-prime periods''': at first glance, when a pivot is not equal to 1, it might trigger you to think that the mapping is enfactored. But temperaments can legitimately have generators that divide primes evenly, such as 5-limit Blackwood, {{vector|{{map|5 8 0}} {{map|0 0 1}}}}, which divides the octave into 5 parts.<ref>Any form that enforces pivots all be 1's would fail this criteria.</ref>

@@ Line 877: / Line 877: @@
 In addition to being canonical and defactored, DCF has other important properties, which probably go without saying in the context of RTT mappings, but here they are just in case:
 * '''integer''': contains only integer terms.
-* '''full-rank'''<ref>Interesting tidbit regarding full-rank matrices and unimodular matrices: for square matrices, full-rank implies unimodularity, and vice-versa.</ref>: removes rank deficiencies, or in other words, rows that are all zeros (upon any linear combination of rows)
+* '''full-rank'''<ref>Interesting tidbit regarding full-rank matrices and unimodular matrices: for square matrices, full-rank implies unimodularity, and vice-versa.</ref><ref>Another interesting tidbit here is that canonical form does not remove what one might call dimensionality deficiencies. In the case of a mapping, this would take the form of an extra column of all zeros to the right of any non-zero entries, or in other words, an unmapped prime higher than other mapped prime. For example you could have {{vector|{{map|1 0 -4 0}} {{map|0 1 4 0}}}} which is just 5-limit meantone but represented in the 7-limit even though prime 7 is not used. And for a comma-basis the form dimensionality deficiency would take is rotated 90 degrees: a row of all zeros below all other nonzero entries, e.g. {{map|{{vector|4 -4 1 0}}}}. The reason these additional zeros should be preserved and these temperaments be treated as different from their untrimmed counterparts is made clear when we consider the difference in the anti-null-spaces of comma-bases; the presence of extra dimensions implies the presence of extra generators that are unbound to the other generators. And if we decide this way re: comma-bases, we can't hardly make an inconsistent call re: mappings.</ref>: removes rank deficiencies, or in other words, rows that are all zeros (upon any linear combination of rows)
 * '''preserves genuine unit-fraction-of-an-prime periods''': at first glance, when a pivot is not equal to 1, it might trigger you to think that the mapping is enfactored. But temperaments can legitimately have generators that divide primes evenly, such as 5-limit Blackwood, {{vector|{{map|5 8 0}} {{map|0 0 1}}}}, which divides the octave into 5 parts.<ref>Any form that enforces pivots all be 1's would fail this criteria.</ref>