|
|
| Line 944: |
Line 944: |
|
| |
|
| The rest of the material in this article is not strictly necessary to understand canonical form and defactoring, but for posterity the work Dave and Douglas did to attain their insights has been summarized here in case it may be helpful to anyone else who might want to iterate on this later. | | The rest of the material in this article is not strictly necessary to understand canonical form and defactoring, but for posterity the work Dave and Douglas did to attain their insights has been summarized here in case it may be helpful to anyone else who might want to iterate on this later. |
|
| |
| == criteria ==
| |
|
| |
| 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><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>
| |
|
| |
| === exclusion of generator size target criteria; mingen form ===
| |
|
| |
| Initially, Dave and Douglas considered the proposal of a new standard RTT mapping form as an opportunity to include everything and the kitchen sink — in this case, to additionally massage generator sizes to fit a desirable rubric. It was ultimately decided to stay conceptually focused when drafting the proposal, leaving generator size out of the mix. Also, the particular generator size target they sought — named "mingen" — turned out to be a bit of a rabbit hole, especially above rank-2, both in terms of mathematical reality and engineering practicality. The results of those efforts are documented here: [[Generator size manipulation#mingen_form]]
| |
|
| |
|
| == relationship between various matrix echelon forms == | | == relationship between various matrix echelon forms == |