Patent val/Properties - Revision history

Overthink: Add more efficient method; improve first section

2026-04-07T16:22:34Z

Add more efficient method; improve first section

← Older revision		Revision as of 16:22, 7 April 2026
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	For every prime ''q'' in the ''p''-limit, solve for ''n'':		For every prime ''q'' in the ''p''-limit, solve for ''n'':

	<math> \displaystyle		<center><math> \displaystyle
	\operatorname {round} (n \log_2 q) = v_{\pi (q)} </math>		\operatorname {round} (n \log_2 q) = v_{\pi (q)} </math></center>

	The solution is		Note that <math>\pi (n)</math> is the {{W\|prime counting function}}; that is, the number of prime numbers less than or equal to <math>n</math>. The solution is

	<math> \displaystyle		<center><math> \displaystyle
	\frac {v_{\pi (q)} - 1/2}{\log_2 (q)} < n < \frac {v_{\pi (q)} + 1/2}{\log_2 (q)} </math>		\frac {v_{\pi (q)} - 1/2}{\log_2 (q)} \le n < \frac {v_{\pi (q)} + 1/2}{\log_2 (q)} </math></center>

	Denote the solution sets as ''N''<sub>1</sub>, ''N''<sub>2</sub>, …, ''N''<sub>π(''p'')</sub>. Find their {{w\|intersection (set theory)\|intersection}} ''N'', that is,		Denote the solution sets as ''N''<sub>1</sub>, ''N''<sub>2</sub>, …, ''N''<sub>π(''p'')</sub>. Find their {{w\|intersection (set theory)\|intersection}} ''N'', that is,

	<math> \displaystyle		<center><math> \displaystyle
	N = \bigcap_{i = 1}^{\pi (p)} N_i </math>		N = \bigcap_{i = 1}^{\pi (p)} N_i </math></center>

	Then ''V'' is a GPV of every edo in ''N'' if ''N'' is non-empty ~~and not a singleton (i.e. a single point)~~; otherwise it is not a GPV.		Then ''V'' is a GPV of every edo in ''N'' if ''N'' is non-empty; otherwise it is not a GPV.

	== Cardinality ==		== Cardinality ==
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	Notice that the all-zero val is a GPV, so you can always enter it for the first one. It is guaranteed that you can iterate through all GPVs in this method. In practice it is recommended to start with the last entry and going backwards in step 2 for better performance, because the largest prime is most likely to increment.		Notice that the all-zero val is a GPV, so you can always enter it for the first one. It is guaranteed that you can iterate through all GPVs in this method. In practice it is recommended to start with the last entry and going backwards in step 2 for better performance, because the largest prime is most likely to increment.

			== A more efficient method ==
			We consider the values of <math>n</math> for which the mapping of a prime increments by a step. The just tuning of a prime <math>p</math> is <math>n\log_2(p)</math> steps of <math>n</math>-edo, and as <math>n</math> increases, the mapping of prime <math>p</math> will increase by a step when <math>n\log_2(p)</math> is a half-integer. In particular, if prime <math>p</math> is mapped to <math>m_p</math> steps of <math>n</math>-edo, the smallest value of <math>n</math> where prime <math>p</math> is instead mapped to <math>m_p + 1</math> steps occurs when

			<center><math>n\log_2 (p) = m_p + \dfrac {1}{2},</math></center>

			and solving for <math>n</math>, we get

			<center><math>n = \dfrac {m_p + \tfrac{1}{2}}{\log_2(p)}.</math></center>

			If we start from any GPV <math>⟨m_2 \: m_3 \: m_5 \: \dots ]</math>, we find the list

			<center><math>\left[ \dfrac {m_2 + \tfrac{1}{2}}{\log_2(2)} \:
			\dfrac {m_3 + \tfrac{1}{2}}{\log_2(3)} \:
			\dots \right],</math></center>

			and the next GPV occurs when <math>n</math> reaches the smallest number in that list, say the one corresponding to prime <math>p</math>. Then the next GPV increments the mapping of prime <math>p</math> by one step, while keeping the mappings of all other primes the same. We also add <math>\tfrac {1}{\log_2(p)}</math> to the entry of the list corresponding to prime <math>p</math>. We repeat the steps of finding the smallest entry in the list, incrementing the mapping of the corresponding prime by one step, and adding <math>\tfrac {1}{\log_2(p)}</math> to the corresponding entry in the list to repeatedly find the next GPV.

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Val]]		[[Category:Val]]

FloraC: Undo revision 167678 by ArrowHead294 (talk). There's no point in using a template argument separator where it's supposed to be an ordinary comma (punctuation), be it breaking or nonbreaking

2024-11-27T13:33:55Z

Undo revision 167678 by ArrowHead294 (talk). There's no point in using a template argument separator where it's supposed to be an ordinary comma (punctuation), be it breaking or nonbreaking

← Older revision		Revision as of 13:33, 27 November 2024
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	# For any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in ''Q'', there is ''not'' a point of ''n'' at which both ''v''<sub>''i''</sub> and ''v''<sub>''j''</sub> get an increment;		# For any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in ''Q'', there is ''not'' a point of ''n'' at which both ''v''<sub>''i''</sub> and ''v''<sub>''j''</sub> get an increment;

	where an increment of ''f''(''x'') at ''x''<sub>0</sub> is defined as lim {{nowrap\|''x'' ~~→~~ ''x''<sub>0</sub><sup>+</sup> ''f''(''x'')}} {{=}} {{nowrap\|lim ''x'' ~~→~~ ''x''<sub>0</sub><sup>−</sup> ''f''(''x'') + 1}}.		where an increment of ''f''(''x'') at ''x''<sub>0</sub> is defined as lim {{nowrap\|''x'' → ''x''<sub>0</sub><sup>+</sup> ''f''(''x'')}} {{=}} {{nowrap\|lim ''x'' → ''x''<sub>0</sub><sup>−</sup> ''f''(''x'') + 1}}.

	<nowiki/>#1 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 holds immediately following the definition of the round function, and the point is {{nowrap\|''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 {{nowrap\|''y'' {{=}} round(''x'')}} occurs only if {{nowrap\|2''x'' ∈ '''Z'''}}. Thus, for any distinct primes {{nowrap\|''q''<sub>''i''</sub>\|''q''<sub>''j''</sub> ∈ ''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 {{nowrap\|(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 #2, let us assume there exists such an ''n''. By the definition of the round function, an increment of {{nowrap\|''y'' {{=}} round(''x'')}} occurs only if {{nowrap\|2''x'' ∈ '''Z'''}}. Thus, for any distinct primes {{nowrap\|''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> ∈ ''Q''}}, {{nowrap\|2''n'' log<sub>2</sub>(''q''<sub>''i''</sub>) ∈ '''Z'''}}, and {{nowrap\|2''n'' log<sub>2</sub>(''q''<sub>''j''</sub>) ∈ '''Z'''}}. If that is the case, then their quotient {{nowrap\|(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.
	}}		}}

ArrowHead294: Look at the template and the associated module—«nowrap» with multiple arguments only makes the text in the arguments fixed; by default, the comma and spaces separating the text in the arguments are not non-breaking

2024-11-26T13:56:16Z

Look at the template and the associated module—«nowrap» with multiple arguments only makes the text in the arguments fixed; by default, the comma and spaces separating the text in the arguments are not non-breaking

← Older revision		Revision as of 13:56, 26 November 2024
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	# For any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in ''Q'', there is ''not'' a point of ''n'' at which both ''v''<sub>''i''</sub> and ''v''<sub>''j''</sub> get an increment;		# For any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in ''Q'', there is ''not'' a point of ''n'' at which both ''v''<sub>''i''</sub> and ''v''<sub>''j''</sub> get an increment;

	where an increment of ''f''(''x'') at ''x''<sub>0</sub> is defined as lim {{nowrap\|''x'' → ''x''<sub>0</sub><sup>+</sup> ''f''(''x'')}} {{=}} {{nowrap\|lim ''x'' → ''x''<sub>0</sub><sup>−</sup> ''f''(''x'') + 1}}.		where an increment of ''f''(''x'') at ''x''<sub>0</sub> is defined as lim {{nowrap\|''x'' → ''x''<sub>0</sub><sup>+</sup> ''f''(''x'')}} {{=}} {{nowrap\|lim ''x'' → ''x''<sub>0</sub><sup>−</sup> ''f''(''x'') + 1}}.

	<nowiki/>#1 holds immediately following the definition of the round function, and the point is ~~{{nowrap\|~~''n'' {{=}} (''v''<sub>''i''</sub> + 1/2)/log<sub>2</sub>(''q''<sub>''i''</sub>)}}.		<nowiki/>#1 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 #2, let us assume there exists such an ''n''. By the definition of the round function, an increment of {{nowrap\|''y'' {{=}} round(''x'')}} occurs only if {{nowrap\|2''x'' ∈ '''Z'''}}. Thus, for any distinct primes {{nowrap\|''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> ∈ ''Q''~~}}, {{nowrap~~\|2''n'' log<sub>2</sub>(''q''<sub>''i''</sub>) ∈ '''Z'''~~}},~~ and ~~{{nowrap\|~~2''n'' log<sub>2</sub>(''q''<sub>''j''</sub>) ∈ '''Z'''}}. If that is the case, then their quotient {{nowrap\|(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 {{nowrap\|''y'' {{=}} round(''x'')}} occurs only if {{nowrap\|2''x'' ∈ '''Z'''}}. Thus, for any distinct primes {{nowrap\|''q''<sub>''i''</sub>\|''q''<sub>''j''</sub> ∈ ''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 {{nowrap\|(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.
	}}		}}

FloraC: Don't abuse nowrap across commas (punctuation)

2024-11-26T08:17:47Z

Don't abuse nowrap across commas (punctuation)

← Older revision		Revision as of 08:17, 26 November 2024
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	\frac {v_{\pi (q)} - 1/2}{\log_2 (q)} < n < \frac {v_{\pi (q)} + 1/2}{\log_2 (q)} </math>		\frac {v_{\pi (q)} - 1/2}{\log_2 (q)} < n < \frac {v_{\pi (q)} + 1/2}{\log_2 (q)} </math>

	Denote the solution sets as ''N''<sub>1</sub>, ''N''<sub>2</sub>, …, ''N''<sub>~~π~~(''p'')</sub>. Find their {{w\|intersection (set theory)\|intersection}} ''N'', that is,		Denote the solution sets as ''N''<sub>1</sub>, ''N''<sub>2</sub>, …, ''N''<sub>π(''p'')</sub>. Find their {{w\|intersection (set theory)\|intersection}} ''N'', that is,

	<math> \displaystyle		<math> \displaystyle
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	# For any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in ''Q'', there is ''not'' a point of ''n'' at which both ''v''<sub>''i''</sub> and ''v''<sub>''j''</sub> get an increment;		# For any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in ''Q'', there is ''not'' a point of ''n'' at which both ''v''<sub>''i''</sub> and ''v''<sub>''j''</sub> get an increment;

	where an increment of ''f''(''x'') at ''x''<sub>0</sub> is defined as lim {{nowrap\|''x'' ~~→~~ ''x''<sub>0</sub><sup>+</sup> ''f''(''x'')}} {{=}} {{nowrap\|lim ''x'' ~~→~~ ''x''<sub>0</sub><sup>−</sup> ''f''(''x'') + 1}}.		where an increment of ''f''(''x'') at ''x''<sub>0</sub> is defined as lim {{nowrap\|''x'' → ''x''<sub>0</sub><sup>+</sup> ''f''(''x'')}} {{=}} {{nowrap\|lim ''x'' → ''x''<sub>0</sub><sup>−</sup> ''f''(''x'') + 1}}.

	<nowiki />#1 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 holds immediately following the definition of the round function, and the point is {{nowrap\|''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 {{nowrap\|''y'' {{=}} round(''x'')}} occurs only if {{nowrap\|2''x'' ∈ '''Z'''}}. Thus, for any distinct primes {{nowrap\|''q''<sub>''i''</sub>\|''q''<sub>''j''</sub> ∈ ''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 {{nowrap\|(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 #2, let us assume there exists such an ''n''. By the definition of the round function, an increment of {{nowrap\|''y'' {{=}} round(''x'')}} occurs only if {{nowrap\|2''x'' ∈ '''Z'''}}. Thus, for any distinct primes {{nowrap\|''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> ∈ ''Q''}}, {{nowrap\|2''n'' log<sub>2</sub>(''q''<sub>''i''</sub>) ∈ '''Z'''}}, and {{nowrap\|2''n'' log<sub>2</sub>(''q''<sub>''j''</sub>) ∈ '''Z'''}}. If that is the case, then their quotient {{nowrap\|(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.
	}}		}}

ArrowHead294: Formatting

2024-11-25T21:06:40Z

Formatting

← Older revision		Revision as of 21:06, 25 November 2024
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	{{Breadcrumb}}		{{Breadcrumb}}

	This page shows some properties of [[patent val]]s as well as [[generalized patent val]]s (GPVs), although currently it is all about the latter.		This page shows some properties of [[patent val]]s as well as [[generalized patent val]]s (GPVs), although currently it is all about the latter.

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	\frac {v_{\pi (q)} - 1/2}{\log_2 (q)} < n < \frac {v_{\pi (q)} + 1/2}{\log_2 (q)} </math>		\frac {v_{\pi (q)} - 1/2}{\log_2 (q)} < n < \frac {v_{\pi (q)} + 1/2}{\log_2 (q)} </math>

	Denote the solution sets as ''N''<sub>1</sub>, ''N''<sub>2</sub>, …, ''N''<sub>π (''p'')</sub>. Find their {{w\|intersection (set theory)\|intersection}} ''N'', that is,		Denote the solution sets as ''N''<sub>1</sub>, ''N''<sub>2</sub>, …, ''N''<sub>π(''p'')</sub>. Find their {{w\|intersection (set theory)\|intersection}} ''N'', that is,

	<math> \displaystyle		<math> \displaystyle
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	== Sorting property ==		== Sorting 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 {{nowrap\|''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 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 }}.
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	{{Proof		{{Proof
	\| title=Proof		\| title=Proof
	\| contents=		\| contents=By definition, the ''p''-limit GPV of ''n''-edo is {{nowrap\|''V''(''n'') {{=}} round(''n'' log<sub>2</sub>(''Q''))}}, where ''Q'' is the prime basis {{val\| 2 3 5 … ''p'' }}.

	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
	# ~~for~~ any prime ''q''<sub>''i''</sub> in ''Q'', there is a point of ''n'' at which ''v''<sub>''i''</sub> gets an increment; and		# For any prime ''q''<sub>''i''</sub> in ''Q'', there is a point of ''n'' at which ''v''<sub>''i''</sub> gets an increment; and
	# ~~for~~ any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in ''Q'', there is ''not'' a point of ''n'' at which both ''v''<sub>''i''</sub> and ''v''<sub>''j''</sub> get an increment;		# For any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in ''Q'', there is ''not'' a point of ''n'' at which both ''v''<sub>''i''</sub> and ''v''<sub>''j''</sub> get an increment;

	where an increment of ''f''(''x'') at ''x''<sub>0</sub> is defined as lim ''x''→''x''<sub>0</sub><sup>+</sup> ''f''(''x'') ~~=~~ lim ''x''→ ''x''<sub>0</sub><sup>-</sup> ''f''(''x'') + 1.		where an increment of ''f''(''x'') at ''x''<sub>0</sub> is defined as lim {{nowrap\|''x'' → ''x''<sub>0</sub><sup>+</sup> ''f''(''x'')}} {{=}} {{nowrap\|lim ''x'' → ''x''<sub>0</sub><sup>−</sup> ''f''(''x'') + 1}}.

	<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 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 {{nowrap\|''y'' {{=}} round(''x'')}} occurs only if {{nowrap\|2''x'' ∈ '''Z'''}}. Thus, for any distinct primes {{nowrap\|''q''<sub>''i''</sub>\|''q''<sub>''j''</sub> ∈ ''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 {{nowrap\|(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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	Given a finite prime limit, the above properties offer a way to iterate through all GPVs. To roll forwards:		Given a finite prime limit, the above properties offer a way to iterate through all GPVs. To roll forwards:

	# Enter a GPV. Set ''i'' = 1.		# Enter a GPV. Set {{nowrap\|''i'' {{=}} 1}}.
	# Copy the input and increase its ''i''-th entry by 1.		# Copy the input and increase its ''i''-th entry by 1.
	# Test if it is a GPV.		# Test if it is a GPV.
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	To roll backwards:		To roll backwards:

	# Enter a GPV. Set ''i'' = 1.		# Enter a GPV. Set {{nowrap\|''i'' {{=}} 1}}.
	# Copy the input and decrease its ''i''-th entry by 1.		# Copy the input and decrease its ''i''-th entry by 1.
	# Test if it is a GPV.		# Test if it is a GPV.

FloraC: Adopt template: Proof

2024-11-10T16:22:44Z

Adopt template: Proof

← Older revision		Revision as of 16:22, 10 November 2024
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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.

	{{~~Databox~~\|Proof\|		{{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
			\| contents=

			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]]
	~~[[Category:Pages with proofs]]~~

FloraC: Style and misc. cleanup

2024-11-10T16:13:51Z

Style and misc. cleanup

← Older revision		Revision as of 16:13, 10 November 2024
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			{{Breadcrumb}}

	This page shows some properties of [[patent val]]s as well as [[generalized patent val]]s (GPVs), although currently it is all about the latter.		This page shows some properties of [[patent val]]s as well as [[generalized patent val]]s (GPVs), although currently it is all about the latter.

	== To tell if a val is a GPV ==		== To tell if a val is a GPV ==
	Suppose we have a ''p''-limit val V, to tell if it is a GPV:		Suppose we have a ''p''-limit val ''V'', to tell if it is a GPV:

	For every prime ''q'' in the ''p''-limit, solve		For every prime ''q'' in the ''p''-limit, solve for ''n'':

	<math> \displaystyle		<math> \displaystyle
	\operatorname {round} (n \log_2 q) = v_{\pi (q)} </math>		\operatorname {round} (n \log_2 q) = v_{\pi (q)} </math>

	~~for ''n''.~~ The solution is		The solution is

	<math> \displaystyle		<math> \displaystyle
	\frac {v_{\pi (q)} - 1/2}{\log_2 (q)} < n < \frac {v_{\pi (q)} + 1/2}{\log_2 (q)} </math>		\frac {v_{\pi (q)} - 1/2}{\log_2 (q)} < n < \frac {v_{\pi (q)} + 1/2}{\log_2 (q)} </math>

	~~Let~~ N<sub>1</sub>, N<sub>2</sub>, …, N<sub>π (''p'')</sub> ~~denote the solution sets~~. Find		Denote the solution sets as ''N''<sub>1</sub>, ''N''<sub>2</sub>, …, ''N''<sub>π (''p'')</sub>. Find their {{w\|intersection (set theory)\|intersection}} ''N'', that is,

	<math> \displaystyle		<math> \displaystyle
	~~\mathrm {~~N} = \bigcap_{i = 1}^{\pi (p)} ~~\mathrm {N}_i~~ </math>		N = \bigcap_{i = 1}^{\pi (p)} N_i </math>

	Then V is a GPV of every edo in N if N is non-empty and not a singleton (i.e. a single point); otherwise it is not a GPV.		Then ''V'' is a GPV of every edo in ''N'' if ''N'' is non-empty and not a singleton (i.e. a single point); otherwise it is not a GPV.

	== Cardinality ==		== Cardinality ==
	Given a finite prime limit, the set of all GPVs are ~~[[Wikipedia: Countably infinite~~\|countably infinite]].		Given a finite prime limit, the set of all GPVs are {{w\|countably infinite}}.

	== Sorting property ==		== Sorting 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 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, 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 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.

	{{Databox\|Proof\|		{{Databox\|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'' }}.		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~~ sorting property ~~is equivalent~~ to		To prove the sorting property, we need to prove
	# 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 prime ''q''<sub>''i''</sub> in ''Q'', there is a point of ''n'' at which ''v''<sub>''i''</sub> gets an increment; 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~~;		# for any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in ''Q'', there is ''not'' a point of ''n'' at which both ''v''<sub>''i''</sub> and ''v''<sub>''j''</sub> get an increment;

	where an increment of ''f'' (''x'') at ''x''<sub>0</sub> is defined as lim ''x''→''x''<sub>0</sub><sup>+</sup> ''f'' (''x'') = lim ''x''→ ''x''<sub>0</sub><sup>-</sup> ''f'' (''x'') + 1.		where an increment of ''f''(''x'') at ''x''<sub>0</sub> is defined as lim ''x''→''x''<sub>0</sub><sup>+</sup> ''f''(''x'') = lim ''x''→ ''x''<sub>0</sub><sup>-</sup> ''f''(''x'') + 1.

	<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 ~~[[Wikipedia: Gelfond–Schneider theorem~~\|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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	# Enter a GPV. Set ''i'' = 1.		# Enter a GPV. Set ''i'' = 1.
	# Copy the input and ~~increment~~ its ''i''-th entry by 1.		# Copy the input and increase its ''i''-th entry by 1.
	# Test if it is a GPV.		# Test if it is a GPV.
	# If it is, return it and back to step 1; otherwise, ~~increment~~ ''i'' by 1 and back to step 2.		# If it is, return it and go back to step 1; otherwise, increase ''i'' by 1 and go back to step 2.

	To roll backwards:		To roll backwards:

	# Enter a GPV. Set ''i'' = 1.		# Enter a GPV. Set ''i'' = 1.
	# Copy the input and ~~decrement~~ its ''i''-th entry by 1.		# Copy the input and decrease its ''i''-th entry by 1.
	# Test if it is a GPV.		# Test if it is a GPV.
	# If it is, return it and back to step 1; otherwise, ~~increment~~ ''i'' by 1 and back to step 2.		# If it is, return it and go back to step 1; otherwise, increase ''i'' by 1 and go back to step 2.

	Notice that the all-zero val is a GPV, you can always enter it for the first one. It is guaranteed that you can iterate through all GPVs in this method. In practice it is recommended to start with the last entry and going backwards in step 2 for better performance, because the largest prime is most likely to increment.		Notice that the all-zero val is a GPV, so you can always enter it for the first one. It is guaranteed that you can iterate through all GPVs in this method. In practice it is recommended to start with the last entry and going backwards in step 2 for better performance, because the largest prime is most likely to increment.

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

Inthar at 22:31, 9 February 2024

2024-02-09T22:31:19Z

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	[[Category:Math]]		[[Category:Math]]
	[[Category:Val]]		[[Category:Val]]
	[[Category:~~Articles~~ with proofs]]		[[Category:Pages with proofs]]

BudjarnLambeth: Category:Articles with proofs

2024-01-29T01:17:10Z

Category:Articles with proofs

← Older revision		Revision as of 01:17, 29 January 2024
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	[[Category:Math]]		[[Category:Math]]
	[[Category:Val]]		[[Category:Val]]
			[[Category:Articles with proofs]]

FloraC: /* Application */

2022-06-09T13:26:19Z

Application

← Older revision		Revision as of 13:26, 9 June 2022
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	== Application ==		== Application ==
	Given a finite prime limit, the above properties offer a way to iterate through all GPVs.		Given a finite prime limit, the above properties offer a way to iterate through all GPVs. To roll forwards:

	# Enter a GPV. Set ''i'' = 1.		# Enter a GPV. Set ''i'' = 1.
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	# If it is, return it and back to step 1; otherwise, increment ''i'' by 1 and back to step 2.		# If it is, return it and back to step 1; otherwise, increment ''i'' by 1 and back to step 2.

	Notice that the all-zero val is a GPV, you can always enter it for the first one. It is guaranteed that you can iterate through all GPVs in this method. In practice it is ~~suggested starting~~ with the last entry and going backwards for better performance, because the largest prime is most likely to increment.		To roll backwards:

			# Enter a GPV. Set ''i'' = 1.
			# Copy the input and decrement its ''i''-th entry by 1.
			# Test if it is a GPV.
			# If it is, return it and back to step 1; otherwise, increment ''i'' by 1 and back to step 2.

			Notice that the all-zero val is a GPV, you can always enter it for the first one. It is guaranteed that you can iterate through all GPVs in this method. In practice it is recommended to start with the last entry and going backwards in step 2 for better performance, because the largest prime is most likely to increment.

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Val]]		[[Category:Val]]