Wedgie/Archived version: Difference between revisions

Line 45:

=== 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 {{nowrap|W('''2''', '''q'''<sub>1</sub>)}}<ref>The {{nowrap|W(p, g)}} function at its most basic level returns the value at the ({{nowrap|p, g}}) index of the wedgie W. For example, for {{nowrap|W {{=}} {{multimap|1 4 4}}}}, {{nowrap|W('''2''','''3''') {{=}} 1}}, {{nowrap|W('''2''','''5''') {{=}} 4}}, and {{nowrap|W('''3''','''5''') {{=}} 4.

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 {{nowrap|W('''2''', '''q'''<sub>1</sub>)}}<ref>The {{nowrap|W(p, g)}} function at its most basic level returns the value at the ({{nowrap|p, g}}) index of the wedgie W. For example, for {{nowrap|W {{=}} {{multimap|1 4 4}}}}, {{nowrap|W('''2''','''3''') {{=}} 1}}, {{nowrap|W('''2''','''5''') {{=}} 4}}, and {{nowrap|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 many 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 {{nowrap|W('''2''', '''3''')}}, {{nowrap|W('''2''', '''5''')}}, and {{nowrap|W('''3''', '''5''')}}. But the other six permutations of two of these indices exist too. Those with duplicates all equal 0, i.e. {{nowrap|W('''2''', '''2''') {{=}} 0}}, {{nowrap|W('''3''', '''3''') {{=}} 0}}, and {{nowrap|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. {{nowrap|W('''3''', '''2''') {{=}} −W('''2''', '''3''')}}, {{nowrap|W('''5''', '''2''') = −W('''2''', '''5''')}}, and {{nowrap|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: [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT#As compressed antisymmetric tensors|Dave Keenan & Douglas Blumeyer's guide to exterior algebra for regular temperament theory.]]

@@ Line 45: / Line 45: @@
 === 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 {{nowrap|W('''2''', '''q'''<sub>1</sub>)}}<ref>The {{nowrap|W(p, g)}} function at its most basic level returns the value at the ({{nowrap|p, g}}) index of the wedgie W. For example, for {{nowrap|W {{=}} {{multimap|1 4 4}}}}, {{nowrap|W('''2''','''3''') {{=}} 1}}, {{nowrap|W('''2''','''5''') {{=}} 4}}, and {{nowrap|W('''3''','''5''') {{=}} 4.
+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 {{nowrap|W('''2''', '''q'''<sub>1</sub>)}}<ref>The {{nowrap|W(p, g)}} function at its most basic level returns the value at the ({{nowrap|p, g}}) index of the wedgie W. For example, for {{nowrap|W {{=}} {{multimap|1 4 4}}}}, {{nowrap|W('''2''','''3''') {{=}} 1}}, {{nowrap|W('''2''','''5''') {{=}} 4}}, and {{nowrap|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 many 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 {{nowrap|W('''2''', '''3''')}}, {{nowrap|W('''2''', '''5''')}}, and {{nowrap|W('''3''', '''5''')}}. But the other six permutations of two of these indices exist too. Those with duplicates all equal 0, i.e. {{nowrap|W('''2''', '''2''') {{=}} 0}}, {{nowrap|W('''3''', '''3''') {{=}} 0}}, and {{nowrap|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. {{nowrap|W('''3''', '''2''') {{=}} &minus;W('''2''', '''3''')}}, {{nowrap|W('''5''', '''2''') = &minus;W('''2''', '''5''')}}, and {{nowrap|W('''5''', '''3''') {{=}} &minus;W('''3''', '''5''')}}. For more information about the relationship between the compressed multicovector form of wedgies and their full tensor form, see: [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT#As compressed antisymmetric tensors|Dave Keenan & Douglas Blumeyer's guide to exterior algebra for regular temperament theory.]]