Patent val/Properties: Difference between revisions

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== Adjacent GPVs property ==

Given a finite prime limit, the set of all GPVs can be ordered in this way such that all but one entry in the GPV vk and its next GPV vk + 1 are the same, and for the different entry, the latter increments the former by 1.

Given a finite prime limit, the set of all GPVs can be ordered in this way such that all but one entry in the GPV v''k'' and its next GPV v''k'' + 1 are the same, and for the different entry, the latter increments the former by 1.

This property states that, for example, if it is known that {{val| 12 19 28 }} is a GPV, then the next GPV is one of {{val| 13 19 28 }}, {{val| 12 20 28 }}, or {{val| 12 19 29 }}.

This also holds for any rationally independent subgroups, such as 2.3.7 and 2.9.7. It does not hold, however, for rationally dependent subgroups, such as 2.3.9.7, where at certain points of edo number ''N'', both the ~~mapping~~ for 3 and 9 increment.

This also holds for any rationally independent subgroups, such as 2.3.7 and 2.9.7. It does not hold, however, for rationally dependent subgroups, such as 2.3.9.7, where at certain points of edo number ''N'', both the mappings for 3 and 9 increment.

=== Proof ===

By definition, the ''p''-limit GPV of ''N''-edo is v (''N'') = round (''N'' log2 (q)), where q is the prime basis {{val| 2, 3, 5, …, ''p'' }}.

The adjacent GPVs property is equivalent to

# for any prime ''q''''i'' in q, there is a point of ''N'' to cause ''v''''i'' to increment to ''v''''i'' + 1; and

# for any distinct primes ''q''''i'', ''q''''j'' in q, there is ''not'' a point of ''N'' to cause both ''v''''i'' and ''v''''j'' to increment to ''v''''i'' + 1 and ''v''''j'' + 1, respectively.

<nowiki>#1</nowiki> holds because the point is ''N'' = (''v''''i'' + 1/2)/log2 (''q''''i'').

To prove <nowiki>#2</nowiki>, let us assume there exists such an ''N''. By the property of the round function, an increment of ''y'' = round (''x'') occurs iff 2''x'' ∈ '''Z'''. Thus, for any distinct primes ''q''''i'', ''q''''j'' in q, 2''N'' log2 (''q''''i'') ∈ '''Z''', and 2''N'' log2 (''q''''j'') ∈ '''Z'''. If that is the case, then their quotient (2''N'' log2 (''q''''i''))/(2''N'' log2 (''q''''j'')) = log''q''''j'' (''q''''i'') ∈ '''Q''', which contradicts [[Wikipedia: Gelfond–Schneider theorem|Gelfond–Schneider theorem]]. Therefore, the hypothesis is false, and such an ''N'' does not exist.

== Application ==

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 == Adjacent GPVs property ==
-Given a finite prime limit, the set of all GPVs can be ordered in this way such that all but one entry in the GPV v<sub>k</sub> and its next GPV v<sub>k + 1</sub> are the same, and for the different entry, the latter increments the former by 1.
+Given a finite prime limit, the set of all GPVs can be ordered in this way such that all but one entry in the GPV v<sub>''k''</sub> and its next GPV v<sub>''k'' + 1</sub> are the same, and for the different entry, the latter increments the former by 1.
 This property states that, for example, if it is known that {{val| 12 19 28 }} is a GPV, then the next GPV is one of {{val| 13 19 28 }}, {{val| 12 20 28 }}, or {{val| 12 19 29 }}.
-This also holds for any rationally independent subgroups, such as 2.3.7 and 2.9.7. It does not hold, however, for rationally dependent subgroups, such as 2.3.9.7, where at certain points of edo number ''N'', both the mapping for 3 and 9 increment.
+This also holds for any rationally independent subgroups, such as 2.3.7 and 2.9.7. It does not hold, however, for rationally dependent subgroups, such as 2.3.9.7, where at certain points of edo number ''N'', both the mappings for 3 and 9 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'' }}.
+The adjacent GPVs property is equivalent to
+# for any prime ''q''<sub>''i''</sub> in q, there is a point of ''N'' to cause ''v''<sub>''i''</sub> to increment to ''v''<sub>''i''</sub> + 1; and
+# for any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in q, there is ''not'' a point of ''N'' to cause both ''v''<sub>''i''</sub> and ''v''<sub>''j''</sub> to increment to ''v''<sub>''i''</sub> + 1 and ''v''<sub>''j''</sub> + 1, respectively.
+<nowiki>#1</nowiki> holds because 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 property of the round function, an increment of ''y'' = round (''x'') occurs iff 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 [[Wikipedia: Gelfond–Schneider theorem|Gelfond–Schneider theorem]]. Therefore, the hypothesis is false, and such an ''N'' does not exist.
 == Application ==