Patent val/Properties: Difference between revisions
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This also holds for any rationally independent subgroups, such as 2.3.7 and 2.9.7, and even ''some'' rationally dependent subgroups, such as 2.3.9.7. It does not hold, however, for other rationally dependent subgroups, such as 2.3.27.7, where at certain points of edo number ''n'', both the mappings for 3 and 27 get an increment. | This also holds for any rationally independent subgroups, such as 2.3.7 and 2.9.7, and even ''some'' rationally dependent subgroups, such as 2.3.9.7. It does not hold, however, for other rationally dependent subgroups, such as 2.3.27.7, where at certain points of edo number ''n'', both the mappings for 3 and 27 get an increment. | ||
{{ | {{Proof | ||
By definition, the ''p''-limit GPV of ''n''-edo is ''V'' (''n'') = round (''n'' log<sub>2</sub> (''Q'')), where ''Q'' is the prime basis {{val| 2 3 5 … ''p'' }}. | | title=Proof | ||
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By definition, the ''p''-limit GPV of ''n''-edo is ''V'' (''n'') = round (''n'' log<sub>2</sub>(''Q'')), where ''Q'' is the prime basis {{val| 2 3 5 … ''p'' }}. | |||
To prove the sorting property, we need to prove | To prove the sorting property, we need to prove | ||
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<nowiki>#1</nowiki> holds immediately following the definition of the round function, and the point is ''n'' = (''v''<sub>''i''</sub> + 1/2)/log<sub>2</sub> (''q''<sub>''i''</sub>). | <nowiki>#1</nowiki> holds immediately following the definition of the round function, and the point is ''n'' = (''v''<sub>''i''</sub> + 1/2)/log<sub>2</sub> (''q''<sub>''i''</sub>). | ||
To prove <nowiki>#2</nowiki>, let us assume there exists such an ''n''. By the definition of the round function, an increment of ''y'' = round (''x'') occurs only if 2''x'' ∈ '''Z'''. Thus, for any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in ''Q'', 2''n'' log<sub>2</sub> (''q''<sub>''i''</sub>) ∈ '''Z''', and 2''n'' log<sub>2</sub> (''q''<sub>''j''</sub>) ∈ '''Z'''. If that is the case, then their quotient (2''n'' log<sub>2</sub> (''q''<sub>''i''</sub>))/(2''n'' log<sub>2</sub> (''q''<sub>''j''</sub>)) = log<sub>''q''<sub>''j''</sub></sub> (''q''<sub>''i''</sub>) ∈ '''Q''', which contradicts {{w|Gelfond–Schneider theorem}}. Therefore, the hypothesis is false, and such an ''n'' does not exist. | To prove <nowiki>#2</nowiki>, let us assume there exists such an ''n''. By the definition of the round function, an increment of ''y'' = round (''x'') occurs only if 2''x'' ∈ '''Z'''. Thus, for any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in ''Q'', 2''n'' log<sub>2</sub>(''q''<sub>''i''</sub>) ∈ '''Z''', and 2''n'' log<sub>2</sub>(''q''<sub>''j''</sub>) ∈ '''Z'''. If that is the case, then their quotient (2''n'' log<sub>2</sub>(''q''<sub>''i''</sub>))/(2''n'' log<sub>2</sub>(''q''<sub>''j''</sub>)) = log<sub>''q''<sub>''j''</sub></sub>(''q''<sub>''i''</sub>) ∈ '''Q''', which contradicts {{w|Gelfond–Schneider theorem}}. Therefore, the hypothesis is false, and such an ''n'' does not exist. | ||
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[[Category:Math]] | [[Category:Math]] | ||
[[Category:Val]] | [[Category:Val]] | ||