Normal forms: Difference between revisions

Line 36:

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 {{rket| {{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. <br>

And for a comma basis the form this would take is rotated 90 degrees: a row of all zeros below all other nonzero entries, e.g. [{{vector| 4 -4 1 0 }}].<br>

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 ~~antinullspace~~ of [{{vector| 4 -4 1 }}], or in other words its mapping, as we know well is {{rket| {{map| 1 0 -4 }} {{map| 0 1 4 }} }}. But that is not a basis for the ~~antinullspace~~ of [{{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> }} }}.</ref>. We now have a <math>(r,d)</math>-shaped matrix, with <math>r</math> rows where <math>r</math> is the ''rank''.

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 nullspace of [{{vector| 4 -4 1 }}], or in other words its mapping, as we know well is {{rket| {{map| 1 0 -4 }} {{map| 0 1 4 }} }}. But that is not a basis for the nullspace of [{{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> }} }}.</ref>. We now have a <math>(r,d)</math>-shaped matrix, with <math>r</math> rows where <math>r</math> is the ''rank''.

# Then, put this result into HNF.

@@ Line 36: / Line 36: @@
 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 {{rket| {{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. <br>
 And for a comma basis the form this would take is rotated 90 degrees: a row of all zeros below all other nonzero entries, e.g. [{{vector| 4 -4 1 0 }}].<br>
-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 antinullspace of [{{vector| 4 -4 1 }}], or in other words its mapping, as we know well is {{rket| {{map| 1 0 -4 }} {{map| 0 1 4 }} }}. But that is not a basis for the antinullspace of [{{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> }} }}.</ref>. We now have a <math>(r,d)</math>-shaped matrix, with <math>r</math> rows where <math>r</math> is the ''rank''.
+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 nullspace of [{{vector| 4 -4 1 }}], or in other words its mapping, as we know well is {{rket| {{map| 1 0 -4 }} {{map| 0 1 4 }} }}. But that is not a basis for the nullspace of [{{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> }} }}.</ref>. We now have a <math>(r,d)</math>-shaped matrix, with <math>r</math> rows where <math>r</math> is the ''rank''.
 # Then, put this result into HNF.