Defactoring: Difference between revisions
Cmloegcmluin (talk | contribs) |
Cmloegcmluin (talk | contribs) |
||
| Line 863: | Line 863: | ||
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: | 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. | * '''integer''': contains only integer terms. | ||
* '''full-rank''': removes rank deficiencies, or in other words, rows that are all zeros | * '''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) | ||
* '''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> | * '''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> | ||