Wedgie/Archived version: Difference between revisions

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In geometric terms, given JI ratios ''u'' and ''v'', a rank-2 temperament's wedgie ''W'' is defined by the property that the number ''W''(''u'', ''v'') is (proportional to) the signed area of the parallelogram spanned by (tempered versions of) ''u'' and ''v''. This is the determinant of the tempered versions of ''u'' and ''v''. The musical interpretation of the parallelogram spanned by ''u'' and ''v'' is: If you want to consider intervals that are multiples of ''u'' apart the same note (for example, if you want an octave-equivalent scale), ''W''(''u'', ''v'') tells you how many generators of your rank-2 temperament it would take to get to ''v''. The reason that wedgies work as unique identifiers of temperaments is that the value ''W''(''u'', ''v'') only depends on what the temperament does to ''u'' and ''v'', and this dependence (in a sense) matches up exactly with what commas are tempered out by the temperament.

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''1.[…].''q''''n'' (where the ''q''''i'' need not be prime). Then the entries of the wedgie ''W'' corresponding to the rank-2 temperament ''a''&''b'' of the JI subgroup ''q''1.[…].''q''''n'' 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''1.[…].''q''''n'' (where the ''q''''i'' need not be prime). Then the entries of the wedgie W corresponding to the rank-2 temperament a&b of the JI subgroup ''q''1.[…].''q''''n'' are (ignoring sign and normalization):

<math>\mathrm{W}(\mathbf{q}_i, \mathbf{q}_j) = a(\mathbf{q}_i)b(\mathbf{q}_j) - a(\mathbf{q}_j)b(\mathbf{q}_i) \text{ for } i < j.</math>

(Note that by the alternating property, ''W''(''q''''i'', ''q''''i'') = 0 for all ''i''.)

(Note that by the alternating property, W('''q'''''i'', '''q'''''i'') = 0 for all ''i''.)q

For the ''p''''n''-prime limit, the entries of ''W'' are conventionally listed in the order

<math>\langle\langle W(2, 3) \ \ldots \ W(2, ~~p_n~~) \ W(3, 5) \ldots \ W(3, ~~p_n~~) \ldots W(p_{n-2}, p_{n-1}) \ W(p_{n-2}, ~~p_n~~)\ W(p_{n-1}, ~~p_n~~)]].</math>

<math>\langle\langle \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

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The result is the wedgie of the rank-''r'' temperament ''V''1&…&''V''''r'', whose entries are (ignoring steps 2 and 3):

<math>W(q_{k_1}, \ldots, q_{k_r}) = \det[V_i(q_{k_j})]_{i,j}, \ \text{for} \ 1 < k_j < n, </math>

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

where <math>[V_i(q_{k_j})]_{i,j}</math> denotes the ''r''×''r'' matrix whose (''i'', ''j'') entry is <math>V_i(q_{k_j})</math>. These are ''r''-dimensional quantities, the volumes of the ''r''-dimensional parallelograms spanned by ''q''''k''''j'', ''j'' ~~= 1, ...,~~ ''r'', in the temperament's lattice.

where <math>[V_i(q_{k_j})]_{i,j}</math> denotes the ''r''×''r'' matrix whose (''i'', ''j'') entry is <math>V_i(q_{k_j})</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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 In geometric terms, given JI ratios ''u'' and ''v'', a rank-2 temperament's wedgie ''W'' is defined by the property that the number ''W''(''u'', ''v'') is (proportional to) the signed area of the parallelogram spanned by (tempered versions of) ''u'' and ''v''. This is the determinant of the tempered versions of ''u'' and ''v''. The musical interpretation of the parallelogram spanned by ''u'' and ''v'' is: If you want to consider intervals that are multiples of ''u'' apart the same note (for example, if you want an octave-equivalent scale), ''W''(''u'', ''v'') tells you how many generators of your rank-2 temperament it would take to get to ''v''. The reason that wedgies work as unique identifiers of temperaments is that the value ''W''(''u'', ''v'') only depends on what the temperament does to ''u'' and ''v'', and this dependence (in a sense) matches up exactly with what commas are tempered out by the temperament.
-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'') = &minus;''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''') = &minus;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>W(q_i, q_j) = a(q_i)b(q_j) - a(q_j)b(q_i) \text{ for } i < j.</math>
+<math>\mathrm{W}(\mathbf{q}_i, \mathbf{q}_j) = a(\mathbf{q}_i)b(\mathbf{q}_j) - a(\mathbf{q}_j)b(\mathbf{q}_i) \text{ for } i < j.</math>
-(Note that by the alternating property, ''W''(''q''<sub>''i''</sub>, ''q''<sub>''i''</sub>) = 0 for all ''i''.)
+(Note that by the alternating property, W('''q'''<sub>''i''</sub>, '''q'''<sub>''i''</sub>) = 0 for all ''i''.)q
 For the ''p''<sub>''n''</sub>-prime limit, the entries of ''W'' are conventionally listed in the order
-<math>\langle\langle W(2, 3) \ \ldots \ W(2, p_n) \ W(3, 5) \ldots \ W(3, p_n) \ldots W(p_{n-2}, p_{n-1}) \ W(p_{n-2}, p_n)\ W(p_{n-1}, p_n)]].</math>
+<math>\langle\langle \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
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 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>W(q_{k_1}, \ldots, q_{k_r}) = \det[V_i(q_{k_j})]_{i,j}, \ \text{for} \ 1 < k_j < n, </math>
+<math>W(\mathbf{q}_{k_1}, \ldots, \mathbf{q}_{k_r}) = \det[V_i(\mathbf{q}_{k_j})]_{i,j}, \ \text{for} \ 1 < k_j < n, </math>
-where <math>[V_i(q_{k_j})]_{i,j}</math> denotes the ''r''×''r'' matrix whose (''i'', ''j'') entry is <math>V_i(q_{k_j})</math>. These are ''r''-dimensional quantities, the volumes of the ''r''-dimensional parallelograms spanned by ''q''<sub>''k''<sub>''j''</sub></sub>, ''j'' = 1, ..., ''r'', in the temperament's lattice.
+where <math>[V_i(q_{k_j})]_{i,j}</math> denotes the ''r''×''r'' matrix whose (''i'', ''j'') entry is <math>V_i(q_{k_j})</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 ==