Wedgie/Archived version: Difference between revisions

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

The following is a procedure for finding a period and a generator for a rank-2 regular temperament on the 2.''q''1.(…).q''n'' [[JI subgroup]]. We also give a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the [[equave]] is the octave, but non-octave JI equaves can be substituted for the octave if needed, by substituting the appropriate JI ratio for 2/1.

The following is a procedure for finding a period and a generator for a rank-2 regular temperament on the 2.''q''1.(…).q''n'' [[JI subgroup]]<ref>Note that ''q''1 is not 2, but the next formal prime in the group</ref>. We also give a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the [[equave]] is the octave, but non-octave JI equaves can be substituted for the octave if needed, by substituting the appropriate JI ratio for 2/1.

The following assumes that:

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=== The procedure ===

Consider the rank-2 temperament a&b, where a and b are two [[val]]s. Then the entries of the wedgie W corresponding to a&b are W('''2''', '''q'''1), …, W('''2''', '''q'''''n''), and W('''q'''''i'', '''q'''''j'') for ''i'' < ''j'', and the entry W('''p''', '''q''') is given by a('''p''')b('''q''') − a('''q''')b('''p''').

Consider the rank-2 temperament a&b, where a and b are two [[val]]s. Then the entries of the wedgie W corresponding to a&b are W('''2''', '''q'''1)<ref>The W(p,g) function at its most basic level returns the value at the (p,g) index of the wedgie W. For example, for W = {{multimap|1 4 4}}, W('''2''','''3''') = 1, W('''2''','''5''') = 4, and W('''3''','''5''') = 4.

Note that the multicovector form we typically view wedgies in is compressed from its full tensor form, because the tensor form is antisymmetric and therefore has a bunch of useless zero entries along its diagonal and also an entire half of it is redundant with the other half (its negation). For example, a rank-2 5-limit wedgie W has three entries with indices W('''2''','''3'''), W('''2''','''5'''), and W('''3''','''5'''). But the other six permutations of two of these indices exist too. Those with duplicates all equal 0, i.e. W('''2''','''2''') = 0, W('''3''','''3''') = 0, and W('''5''','''5''') = 0. Those with indices that are reversals of the ones shown in the multicovector form have values that are negations of those shown in the multicovector, e.g. W('''3''','''2''') = -W('''2''','''3'''), W('''5''','''2''') = -W('''2''','''5'''), and W('''5''','''3''') = -W('''3''','''5'''). For more information about the relationship between the compressed multicovector form of wedgies and their full tensor form, see: [[Varianced Exterior Algebra#as compressed antisymmetric tensors]]

As for arbitrary values of '''p''' and '''g''' in W('''p''','''g''') — such as non-integers like 3/2 or 10/9 — the value of W('''p''','''g''') can be understood as the volume the parallelogram spanned by '''p''' and '''g''', or in other words, that 1/|W('''2''','''g''')| is the unit fraction of the tempered lattice capable of being generated by '''p''' and '''g''', as is discussed in greater detail here: [[Varianced Exterior Algebra#tempered lattice fractions generated by prime combinations]]</ref>, …, W('''2''', '''q'''''n''), and W('''q'''''i'', '''q'''''j'') for ''i'' < ''j'', and the entry W('''p''', '''q''') is given by a('''p''')b('''q''') − a('''q''')b('''p''').

To find the '''period''': Let ''d'' = gcd(W('''2''', '''q'''1), …, W('''2''', '''q'''''n'')). Then your period is 1\''d''.

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 == How the period and generator falls out of a rank-2 wedgie ==
-The following is a procedure for finding a period and a generator for a rank-2 regular temperament on the 2.''q''<sub>1</sub>.(…).q<sub>''n''</sub> [[JI subgroup]]. We also give a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the [[equave]] is the octave, but non-octave JI equaves can be substituted for the octave if needed, by substituting the appropriate JI ratio for 2/1.
+The following is a procedure for finding a period and a generator for a rank-2 regular temperament on the 2.''q''<sub>1</sub>.(…).q<sub>''n''</sub> [[JI subgroup]]<ref>Note that ''q''<sub>1</sub> is not 2, but the next formal prime in the group</ref>. We also give a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the [[equave]] is the octave, but non-octave JI equaves can be substituted for the octave if needed, by substituting the appropriate JI ratio for 2/1.
 The following assumes that:
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 === The procedure ===
-Consider the rank-2 temperament a&b, where a and b are two [[val]]s. Then the entries of the wedgie W corresponding to a&b are W('''2''', '''q'''<sub>1</sub>), …, W('''2''', '''q'''<sub>''n''</sub>), and W('''q'''<sub>''i''</sub>, '''q'''<sub>''j''</sub>) for ''i'' < ''j'', and the entry W('''p''', '''q''') is given by a('''p''')b('''q''') &minus; a('''q''')b('''p''').
+Consider the rank-2 temperament a&b, where a and b are two [[val]]s. Then the entries of the wedgie W corresponding to a&b are W('''2''', '''q'''<sub>1</sub>)<ref>The W(p,g) function at its most basic level returns the value at the (p,g) index of the wedgie W. For example, for W = {{multimap|1 4 4}}, W('''2''','''3''') = 1, W('''2''','''5''') = 4, and W('''3''','''5''') = 4.<br><br>
+Note that the multicovector form we typically view wedgies in is compressed from its full tensor form, because the tensor form is antisymmetric and therefore has a bunch of useless zero entries along its diagonal and also an entire half of it is redundant with the other half (its negation). For example, a rank-2 5-limit wedgie W has three entries with indices W('''2''','''3'''), W('''2''','''5'''), and W('''3''','''5'''). But the other six permutations of two of these indices exist too. Those with duplicates all equal 0, i.e. W('''2''','''2''') = 0, W('''3''','''3''') = 0, and W('''5''','''5''') = 0. Those with indices that are reversals of the ones shown in the multicovector form have values that are negations of those shown in the multicovector, e.g. W('''3''','''2''') = -W('''2''','''3'''), W('''5''','''2''') = -W('''2''','''5'''), and W('''5''','''3''') = -W('''3''','''5'''). For more information about the relationship between the compressed multicovector form of wedgies and their full tensor form, see: [[Varianced Exterior Algebra#as compressed antisymmetric tensors]]<br><br>
+As for arbitrary values of '''p''' and '''g''' in W('''p''','''g''') — such as non-integers like 3/2 or 10/9 — the value of W('''p''','''g''') can be understood as the volume the parallelogram spanned by '''p''' and '''g''', or in other words, that 1/|W('''2''','''g''')| is the unit fraction of the tempered lattice capable of being generated by '''p''' and '''g''', as is discussed in greater detail here: [[Varianced Exterior Algebra#tempered lattice fractions generated by prime combinations]]</ref>, …, W('''2''', '''q'''<sub>''n''</sub>), and W('''q'''<sub>''i''</sub>, '''q'''<sub>''j''</sub>) for ''i'' < ''j'', and the entry W('''p''', '''q''') is given by a('''p''')b('''q''') &minus; a('''q''')b('''p''').
 To find the '''period''': Let ''d'' = gcd(W('''2''', '''q'''<sub>1</sub>), …, W('''2''', '''q'''<sub>''n''</sub>)). Then your period is 1\''d''.