Wedgie/Archived version: Difference between revisions
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Divisibility by ''d'' and the fact that '''e'''<sub>1</sub> and '''e'''<sub>2</sub> represent JI ratios in the 2.''q''<sub>1</sub>.[...].''q''<sub>''n''</sub> subgroup imply that ''λ''<sub>1</sub> and ''λ''<sub>2</sub> are both divisible by ''d'', and hence 2/1 is a ''d''th power in ''M' '' (the temperament space). Since gcd(W('''2''', '''q'''<sub>1</sub>), ..., W('''2''', '''q'''<sub>''n''</sub>)) = ''d'', we can always find a linear combination ''g'' = ''c''<sub>1</sub>'''q'''<sub>1</sub> + ... + ''c''<sub>''n''</sub>'''q'''<sub>''n''</sub> such that W('''2''', '''g''') = ''c''<sub>1</sub>W('''2''', '''q'''<sub>1</sub>) + ... + ''c''<sub>''n''</sub>W('''2''', '''q'''<sub>''n''</sub>) = ''d'' using the extended Euclidean algorithm. Then since W('''2''', '''g''') = W(''d'''''p''', '''g''') = ''d''W('''p''', '''g''') = ''d'', we have W('''p''', '''g''') = 1. Ta-da! | Divisibility by ''d'' and the fact that '''e'''<sub>1</sub> and '''e'''<sub>2</sub> represent JI ratios in the 2.''q''<sub>1</sub>.[...].''q''<sub>''n''</sub> subgroup imply that ''λ''<sub>1</sub> and ''λ''<sub>2</sub> are both divisible by ''d'', and hence 2/1 is a ''d''th power in ''M' '' (the temperament space). Since gcd(W('''2''', '''q'''<sub>1</sub>), ..., W('''2''', '''q'''<sub>''n''</sub>)) = ''d'', we can always find a linear combination ''g'' = ''c''<sub>1</sub>'''q'''<sub>1</sub> + ... + ''c''<sub>''n''</sub>'''q'''<sub>''n''</sub> such that W('''2''', '''g''') = ''c''<sub>1</sub>W('''2''', '''q'''<sub>1</sub>) + ... + ''c''<sub>''n''</sub>W('''2''', '''q'''<sub>''n''</sub>) = ''d'' using the extended Euclidean algorithm. Then since W('''2''', '''g''') = W(''d'''''p''', '''g''') = ''d''W('''p''', '''g''') = ''d'', we have W('''p''', '''g''') = 1. Ta-da! | ||
== | == Gene Ward Smith's introduction == | ||
An alternating [[Wikipedia: Multilinear map|multilinear map]] which is a multilinear function taking a certain number n of [[monzos]] as arguments and returning an integer as a value we may call an '''n-map'''. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory. | An alternating [[Wikipedia: Multilinear map|multilinear map]] which is a multilinear function taking a certain number n of [[monzos]] as arguments and returning an integer as a value we may call an '''n-map'''. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory. | ||