Wedgie/Archived version: Difference between revisions

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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'''1)}}<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.]]

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.]]

As for arbitrary values of '''p''' and '''g''' in {{nowrap|W('''p''', '''g''')}}—such as non-integers—the value of {{nowrap|W('''p''', '''g''')}} can be understood as the volume the parallelogram spanned by '''p''' and '''g''', or in other words, that {{frac|1|{{!}}{{nowrap|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: [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT#Multicomma entries: tempered lattice fractions generated by prime combinations]]</ref>, …, {{nowrap|W('''2''', '''q'''''n'')}}, and {{nowrap|W('''q'''''i'', '''q'''''j'')}} for {{nowrap|''i'' < ''j''}}, and the entry {{nowrap|W('''p''', '''q''')}} is given by {{nowrap|a('''p''')b('''q''') − a('''q''')b('''p''')}}.

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 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.]]
+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.]]
 As for arbitrary values of '''p''' and '''g''' in {{nowrap|W('''p''', '''g''')}}&mdash;such as non-integers&mdash;the value of {{nowrap|W('''p''', '''g''')}} can be understood as the volume the parallelogram spanned by '''p''' and '''g''', or in other words, that {{frac|1|{{!}}{{nowrap|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: [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT#Multicomma entries: tempered lattice fractions generated by prime combinations]]</ref>, …, {{nowrap|W('''2''', '''q'''<sub>''n''</sub>)}}, and {{nowrap|W('''q'''<sub>''i''</sub>, '''q'''<sub>''j''</sub>)}} for {{nowrap|''i'' &lt; ''j''}}, and the entry {{nowrap|W('''p''', '''q''')}} is given by {{nowrap|a('''p''')b('''q''') &minus; a('''q''')b('''p''')}}.