Wedgie/Archived version: Difference between revisions

Line 25:

# 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 ''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>

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'' 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''''j'', ''j'' = 1, ..., ''r'', in the temperament's lattice.

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

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 # 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(q_{k_1}, \ldots, q_{k_r}) = \det[V_i(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> 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>''j''</sub></sub>, ''j'' = 1, ..., ''r'', in the temperament's lattice.
 == How the period and generator falls out of a rank-2 wedgie ==