Wedgie/Archived version: Difference between revisions

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The result is the wedgie of the rank-''r'' temperament V1&…&V''r'', whose entries are (ignoring steps 2 and 3):

<math>\mathrm{W}\left(\mathbf{q}_{k_1}, \ldots, \mathbf{q}_{k_r}\right) = \det[\mathrm{V}_i\left(\mathbf{q}_{k_j}\right)]_{i,j}, \ \text{for} \ 1 < k_j < n, </math>

<math>\mathrm{W}\left(\mathbf{q}_{k_1}, \ldots, \mathbf{q}_{k_r}\right) = \det\left[\mathrm{V}_i\left(\mathbf{q}_{k_j}\right)\right]_{i,j}, \ \text{for} \ 1 < k_j < n, </math>

where <math>[\mathrm{V}_i\left(\mathbf{q}_{k_j}\right)]_{i,j}</math> denotes the ''r''×''r'' matrix whose (''i'', ''j'') entry is <math>\mathrm{V}_i\left(\mathbf{q}_{k_j}\right)</math>. These are ''r''-dimensional quantities, the volumes of the ''r''-dimensional parallelograms spanned by '''q'''''k''''1'', ..., '''q'''''k''''r'' in the temperament's lattice.

where <math>\left[\mathrm{V}_i\left(\mathbf{q}_{k_j}\right)\right]_{i,j}</math> denotes the ''r''×''r'' matrix whose (''i'', ''j'') entry is <math>\mathrm{V}_i\left(\mathbf{q}_{k_j}\right)</math>. These are ''r''-dimensional quantities, the volumes of the ''r''-dimensional parallelograms spanned by '''q'''''k''''1'', ..., '''q'''''k''''r'' in the temperament's lattice.

== How the period and generator falls out of a rank-2 wedgie ==

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The following assumes that:

* ~~you~~ can think of JI ratios as vectors living in the ''n''-dimensional lattice of the "JI subgroup",

* You can think of JI ratios as vectors living in the ''n''-dimensional lattice of the "JI subgroup",

* ~~you~~ know what a "period" and a "generator" of a rank-2 temperament are, and

* You know what a "period" and a "generator" of a rank-2 temperament are, and

* ~~you~~ know what [[monzo]]s and [[val]]s are and how to work with them.

* You know what [[monzo]]s and [[val]]s are and how to work with them.

=== The procedure ===

@@ Line 32: / Line 32: @@
 The result is the wedgie of the rank-''r'' temperament V<sub>1</sub>&…&V<sub>''r''</sub>, whose entries are (ignoring steps 2 and 3):
-<math>\mathrm{W}\left(\mathbf{q}_{k_1}, \ldots, \mathbf{q}_{k_r}\right) = \det[\mathrm{V}_i\left(\mathbf{q}_{k_j}\right)]_{i,j}, \ \text{for} \ 1 < k_j < n, </math>
+<math>\mathrm{W}\left(\mathbf{q}_{k_1}, \ldots, \mathbf{q}_{k_r}\right) = \det\left[\mathrm{V}_i\left(\mathbf{q}_{k_j}\right)\right]_{i,j}, \ \text{for} \ 1 < k_j < n, </math>
-where <math>[\mathrm{V}_i\left(\mathbf{q}_{k_j}\right)]_{i,j}</math> denotes the ''r''×''r'' matrix whose (''i'', ''j'') entry is <math>\mathrm{V}_i\left(\mathbf{q}_{k_j}\right)</math>. These are ''r''-dimensional quantities, the volumes of the ''r''-dimensional parallelograms spanned by '''q'''<sub>''k''<sub>''1''</sub></sub>, ..., '''q'''<sub>''k''<sub>''r''</sub></sub> in the temperament's lattice.
+where <math>\left[\mathrm{V}_i\left(\mathbf{q}_{k_j}\right)\right]_{i,j}</math> denotes the ''r''×''r'' matrix whose (''i'', ''j'') entry is <math>\mathrm{V}_i\left(\mathbf{q}_{k_j}\right)</math>. These are ''r''-dimensional quantities, the volumes of the ''r''-dimensional parallelograms spanned by '''q'''<sub>''k''<sub>''1''</sub></sub>, ..., '''q'''<sub>''k''<sub>''r''</sub></sub> in the temperament's lattice.
 == How the period and generator falls out of a rank-2 wedgie ==
@@ Line 40: / Line 40: @@
 The following assumes that:
-* you can think of JI ratios as vectors living in the ''n''-dimensional lattice of the "JI subgroup",
+* You can think of JI ratios as vectors living in the ''n''-dimensional lattice of the "JI subgroup",
-* you know what a "period" and a "generator" of a rank-2 temperament are, and
+* You know what a "period" and a "generator" of a rank-2 temperament are, and
-* you know what [[monzo]]s and [[val]]s are and how to work with them.
+* You know what [[monzo]]s and [[val]]s are and how to work with them.
 === The procedure ===