Wedgie/Archived version: Difference between revisions

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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>

<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''.)q

where bolded variables and numbers represent the ordinary numbers written in [[monzo]] form.

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

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

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For example, a 5-limit wedgie is of the form

<math>\langle \langle W(2, 3) \ W(2,5) \ W(3, 5)]],</math>

<math>\langle \langle \mathrm{W}(\mathbf{2}, \mathbf{3}) \ \mathrm{W}(\mathbf{2},\mathbf{5}) \ \mathrm{W}(\mathbf{3}, \mathbf{5})]],</math>

and a 7-limit wedgie is of the form

<math>\langle \langle W(2, 3) \ W(2,5) \ W(2, 7) \ W(3, 5) \ W(3, 7) \ W(5, 7)]].</math>

<math>\langle \langle \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''1, …, ''V''''r'' in a rank-''n'' [[JI subgroup]] ''q''1.[…].''q''''n'', then the wedgie for the rank-''r'' temperament ''V''1&…&''V''''r'' is defined by:

More generally, if one takes ''r'' independent [[vals]] V1, …, V''r'' in a rank-''n'' [[JI subgroup]] ''q''1.[…].''q''''n'', then the wedgie for the rank-''r'' temperament V1&…&V''r'' is defined by:

# Take the [[Wikipedia: Wedge product|wedge product]] of the vals, producing an ''r''-'''multival'''.

# Divide out the greatest common divisior of the entries.

# If the first non-zero entry of the result of step 2 is negative, every entry of the multival is multiplied by −1, changing the sign of the first non-zero entry to be positive.

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

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

<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>

<math>\mathrm{W}(\mathbf{q}_{k_1}, \ldots, \mathbf{q}_{k_r}) = \det[\mathrm{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''''1'', ..., '''q'''''k''''r'' in the temperament's lattice.

where <math>[\mathrm{V}_i(\mathbf{q}_{k_j})]_{i,j}</math> denotes the ''r''×''r'' matrix whose (''i'', ''j'') entry is <math>\mathrm{V}_i(\mathbf{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 ==

@@ Line 5: / Line 5: @@
 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>\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>
+<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''.)q
+where bolded variables and numbers represent the ordinary numbers written in [[monzo]] form.
+(Note that by the alternating property, W('''q'''<sub>''i''</sub>, '''q'''<sub>''i''</sub>) = 0 for all ''i''.)
 For the ''p''<sub>''n''</sub>-prime limit, the entries of ''W'' are conventionally listed in the order
@@ Line 15: / Line 17: @@
 For example, a 5-limit wedgie is of the form
-<math>\langle \langle W(2, 3) \ W(2,5) \ W(3, 5)]],</math>
+<math>\langle \langle \mathrm{W}(\mathbf{2}, \mathbf{3}) \ \mathrm{W}(\mathbf{2},\mathbf{5}) \ \mathrm{W}(\mathbf{3}, \mathbf{5})]],</math>
 and a 7-limit wedgie is of the form
-<math>\langle \langle W(2, 3) \ W(2,5) \ W(2, 7) \ W(3, 5) \ W(3, 7) \ W(5, 7)]].</math>
+<math>\langle \langle \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:
 # Take the [[Wikipedia: Wedge product|wedge product]] of the vals, producing an ''r''-'''multival'''.
 # Divide out the greatest common divisior of the entries.
 # If the first non-zero entry of the result of step 2 is negative, every entry of the multival is multiplied by &minus;1, changing the sign of the first non-zero entry to be positive.
-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):
+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(\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>
+<math>\mathrm{W}(\mathbf{q}_{k_1}, \ldots, \mathbf{q}_{k_r}) = \det[\mathrm{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>''1''</sub></sub>, ..., '''q'''<sub>''k''<sub>''r''</sub></sub> in the temperament's lattice.
+where <math>[\mathrm{V}_i(\mathbf{q}_{k_j})]_{i,j}</math> denotes the ''r''×''r'' matrix whose (''i'', ''j'') entry is <math>\mathrm{V}_i(\mathbf{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 ==