Defactoring: Difference between revisions

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And there's our canonical comma-basis.

~~= preserving trailing zeroes =~~

~~Note that canonicalizing a mapping does not remove trailing dimensions with only zeros.~~

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 {{ket|{{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. {{bra|{{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 duals. For a comma-basis, the extra dimension implies the presence of extra generators that are unbound to the other generators. For example, a basis for the anti-null-space of {{bra|{{vector|4 -4 1}}}}, or in other words its mapping, as we know well is {{ket|{{map|1 0 -4}} {{map|0 1 4}}}}. But that is not a basis for the anti-null-space of {{bra|{{vector|4 -4 1 <math>\color{red}0</math>}}}}; the mapping for that comma-basis would have to be {{ket|{{map|1 0 -4 <math>\color{red}0</math>}} {{map|0 1 4 <math>\color{red}0</math>}} {{map|<math>\color{red}0</math> <math>\color{red}0</math> <math>\color{red}0</math> <math>\color{red}1</math>}}}}.

= references =

@@ Line 934: / Line 934: @@
 And there's our canonical comma-basis.
-= preserving trailing zeroes =
-Note that canonicalizing a mapping does not remove trailing dimensions with only zeros.
-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 {{ket|{{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. {{bra|{{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 duals. For a comma-basis, the extra dimension implies the presence of extra generators that are unbound to the other generators. For example, a basis for the anti-null-space of {{bra|{{vector|4 -4 1}}}}, or in other words its mapping, as we know well is {{ket|{{map|1 0 -4}} {{map|0 1 4}}}}. But that is not a basis for the anti-null-space of {{bra|{{vector|4 -4 1 <span><math>\color{red}0</math></span>}}}}; the mapping for that comma-basis would have to be {{ket|{{map|1 0 -4 <span><math>\color{red}0</math></span>}} {{map|0 1 4 <span><math>\color{red}0</math></span>}} {{map|<span><math>\color{red}0</math></span> <span><math>\color{red}0</math></span> <span><math>\color{red}0</math></span> <span><math>\color{red}1</math></span>}}}}.
 = references =