Dual list: Difference between revisions

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This page gives a more mathematical take on this topic. For a basic introduction, see [[comma-basis]].

=Definition=

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The interest of the dual list stems from the fact that if A is a list of vals defining a temperament, then Dulist(A) is a list of monzos defining the same temperament, and conversely--Dulist applied to a list of monzos defining a temperament gives a list of vals defining the same temperament.

~~=Equivalence to null-space=~~

If one is previously familiar with linear algebra concepts, the list of vals defining a temperament can be thought of as a mapping, and its dual list (the list of monzos, or comma-basis) can be thought of as a basis for its null-space.

~~Null-space operations are readily available in many math libraries.~~

To reverse the null-space operation, that is, to find a mapping from a basis for the null-space, you can also use the null-space operation; the relationship between the two matrices works both ways. However, most math libraries' null-space operation is designed to work for mappings, and so if you want correct results, you must transform the basis for the null-space into a mapping-like form, perform the null-space operation, and then undo the initial transformation. This initial transformation you must do and undo is called the anti-transpose, which is just like the typical transpose of a matrix, except instead of reflecting the matrix's values across the main diagonal (starting from either the top-left or bottom-right corner), you reflect them across the anti-diagonal (starting from either the top-right or bottom-left corner).

You can remember this because most mappings and comma bases have zeroes in the bottom-left corner, and you want to keep them there; some kind of transpose is necessary to convert the constituent comma vectors columns of the comma basis into rows as if they were constituent generator mapping rows of a mapping, but a normal transpose of the comma basis would flip its zeroes into the top-right corner instead.

=Examples=

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+This page gives a more mathematical take on this topic. For a basic introduction, see [[comma-basis]].
 =Definition=
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 The interest of the dual list stems from the fact that if A is a list of vals defining a temperament, then Dulist(A) is a list of monzos defining the same temperament, and conversely--Dulist applied to a list of monzos defining a temperament gives a list of vals defining the same temperament.
-=Equivalence to null-space=
-If one is previously familiar with linear algebra concepts, the list of vals defining a temperament can be thought of as a mapping, and its dual list (the list of monzos, or comma-basis) can be thought of as a basis for its null-space.
-Null-space operations are readily available in many math libraries.
-To reverse the null-space operation, that is, to find a mapping from a basis for the null-space, you can also use the null-space operation; the relationship between the two matrices works both ways. However, most math libraries' null-space operation is designed to work for mappings, and so if you want correct results, you must transform the basis for the null-space into a mapping-like form, perform the null-space operation, and then undo the initial transformation. This initial transformation you must do and undo is called the anti-transpose, which is just like the typical transpose of a matrix, except instead of reflecting the matrix's values across the main diagonal (starting from either the top-left or bottom-right corner), you reflect them across the anti-diagonal (starting from either the top-right or bottom-left corner).
-You can remember this because most mappings and comma bases have zeroes in the bottom-left corner, and you want to keep them there; some kind of transpose is necessary to convert the constituent comma vectors columns of the comma basis into rows as if they were constituent generator mapping rows of a mapping, but a normal transpose of the comma basis would flip its zeroes into the top-right corner instead.
 =Examples=