Wedgie/Archived version: Difference between revisions
ArrowHead294 (talk | contribs) mNo edit summary |
ArrowHead294 (talk | contribs) mNo edit summary |
||
| Line 8: | Line 8: | ||
A wedgie is written as a list of entries that give the values of the wedgie on the [[basis]] elements of the [[JI subgroup]] that the temperament is on. By the alternating property [i.e. W('''u''', '''v''') = −W('''v''', '''u''')] and bilinearity [W is linear in each argument separately], specifying the values on basis elements of the JI subgroup is enough to define W as an alternating bilinear form on all of the JI subgroup. The simplest example is rank-2 wedgies: Let a and b be (non-[[contorted]]) vals on a [[JI subgroup]] ''q''<sub>1</sub>.[…].''q''<sub>''n''</sub> (where the ''q''<sub>''i''</sub> need not be prime). Then the entries of the wedgie W corresponding to the rank-2 temperament a&b of the JI subgroup ''q''<sub>1</sub>.[…].''q''<sub>''n''</sub> are (ignoring sign and normalization): | A wedgie is written as a list of entries that give the values of the wedgie on the [[basis]] elements of the [[JI subgroup]] that the temperament is on. By the alternating property [i.e. W('''u''', '''v''') = −W('''v''', '''u''')] and bilinearity [W is linear in each argument separately], specifying the values on basis elements of the JI subgroup is enough to define W as an alternating bilinear form on all of the JI subgroup. The simplest example is rank-2 wedgies: Let a and b be (non-[[contorted]]) vals on a [[JI subgroup]] ''q''<sub>1</sub>.[…].''q''<sub>''n''</sub> (where the ''q''<sub>''i''</sub> need not be prime). Then the entries of the wedgie W corresponding to the rank-2 temperament a&b of the JI subgroup ''q''<sub>1</sub>.[…].''q''<sub>''n''</sub> are (ignoring sign and normalization): | ||
<math>\mathrm{W}(\mathbf{q}_i, \mathbf{q}_j) = \mathrm{a}(\mathbf{q}_i)\mathrm{b}(\mathbf{q}_j) - \mathrm{a}(\mathbf{q}_j)\mathrm{b}(\mathbf{q}_i) \text{ for } i < j,</math> | <math>\mathrm{W}\left(\mathbf{q}_i, \mathbf{q}_j) = \mathrm{a}(\mathbf{q}_i)\mathrm{b}(\mathbf{q}_j) - \mathrm{a}(\mathbf{q}_j)\mathrm{b}(\mathbf{q}_i) \text{ for } i < j,</math> | ||
where bolded variables and numbers represent the ordinary numbers written in [[monzo]] form. | where bolded variables and numbers represent the ordinary numbers written in [[monzo]] form. | ||
| Line 16: | Line 16: | ||
For the ''p''<sub>''n''</sub>-prime limit, the entries of W are conventionally listed in the order | For the ''p''<sub>''n''</sub>-prime limit, the entries of W are conventionally listed in the order | ||
<math>\ | <math>\wedgie{\mathrm{W}(\mathbf{2}, \mathbf{3}) \ \ldots \ \mathrm{W}(\mathbf{2}, \mathbf{p}_n) & \mathrm{W}(\mathbf{3}, \mathbf{5}) \ \ldots \ \mathrm{W}(\mathbf{3}, \mathbf{p}_n) \ldots \mathrm{W}(\mathbf{p}_{n-2}, \mathbf{p}_{n-1}) & \mathrm{W}(\mathbf{p}_{n-2}, \mathbf{p}_n) & \mathrm{W}(\mathbf{p}_{n-1}, \mathbf{p}_n)}.</math> | ||
For example, a 5-limit wedgie is of the form | For example, a 5-limit wedgie is of the form | ||
| Line 24: | Line 24: | ||
and a 7-limit wedgie is of the form | and a 7-limit wedgie is of the form | ||
<math>\ | <math>\wedgie{\mathrm{W}(\mathbf{2}, \mathbf{3}) & \mathrm{W}(\mathbf{2},\mathbf{5}) & \mathrm{W}(\mathbf{2}, \mathbf{7}) & \mathrm{W}(\mathbf{3}, \mathbf{5}) & \mathrm{W}(\mathbf{3}, \mathbf{7}) & \mathrm{W}(\mathbf{5}, \mathbf{7})}.</math> | ||
More generally, if one takes ''r'' independent [[vals]] V<sub>1</sub>, …, V<sub>''r''</sub> in a rank-''n'' [[JI subgroup]] ''q''<sub>1</sub>.[…].''q''<sub>''n''</sub>, then the wedgie for the rank-''r'' temperament V<sub>1</sub>&…&V<sub>''r''</sub> is defined by: | More generally, if one takes ''r'' independent [[vals]] V<sub>1</sub>, …, V<sub>''r''</sub> in a rank-''n'' [[JI subgroup]] ''q''<sub>1</sub>.[…].''q''<sub>''n''</sub>, then the wedgie for the rank-''r'' temperament V<sub>1</sub>&…&V<sub>''r''</sub> is defined by: | ||
| Line 40: | Line 40: | ||
The following assumes that: | 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 | * you know what a "period" and a "generator" of a rank-2 temperament are, and | ||
* you know what | * you know what [[monzo]]s and [[val]]s are and how to work with them. | ||
=== The procedure === | === The procedure === | ||