Delta-N ratio - Revision history

FloraC: Style and link improvements

2024-12-05T08:28:56Z

Style and link improvements

← Older revision		Revision as of 08:28, 5 December 2024
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			{{DISPLAYTITLE:Delta-''N'' ratio}}
	{{Beginner\|Abc, high quality commas, and epimericity}}		{{Beginner\|Abc, high quality commas, and epimericity}}
	{{Wikipedia\|Superpartient ratio}}		{{Wikipedia\|Superpartient ratio}}

	The '''delta''' of a [[ratio]] is simply the difference between its numerator and its denominator. (Delta is also known as degree of epimoricity.) A ratio with a delta of N is called a '''delta-N ratio'''.		The '''delta''' of a [[ratio]] is simply the difference between its numerator and its denominator. Delta is also known as '''degree of epimoricity'''. A ratio with a delta of ''N'' is called a '''delta-''N'' ratio'''.

	{\| class="wikitable" style="text-align:center;"		{\| class="wikitable" style="text-align:center;"
	\|+ Examples		\|+ Examples
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	\|}		\|}

	Thus [[superparticular]] ratios are delta-1 ratios, and '''superpartient ratios''' are all ratios ''except'' delta-1 ratios. The delta-N terminology was coined by [[Kite Giedraitis]].		Thus [[superparticular]] ratios are delta-1 ratios, and '''superpartient ratios''' are all ratios ''except'' delta-1 ratios. The delta-''N'' terminology was coined by [[Kite Giedraitis]].

	More particularly, a superpartient ratio takes the form:		More particularly, a superpartient ratio takes the form:
	:<math>\dfrac{n + d}{n} = 1 + \dfrac{d}{n}</math>,		:<math>\dfrac{n + d}{n} = 1 + \dfrac{d}{n}</math>,
	where <math>n</math> and <math>d</math> are [[Wikipedia:Positive integer\|positive integer]]s, <math>d > 1</math> and <math>d</math> is ~~[[Wikipedia:Coprime~~\|coprime]] to <math>n</math>.		where <math>n</math> and <math>d</math> are [[Wikipedia:Positive integer\|positive integer]]s, <math>d > 1</math> and <math>d</math> is {{w\|coprime}} to <math>n</math>.

	== Etymology ==		== Etymology ==
	In ancient Greece, they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".		In [[ancient Greek music\|ancient Greece]], they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".

	== Definitions ==		== Definitions ==
	In ancient Greece, fractions like 3/1 and 5/1 were not considered to be epimeric ratios because of their additional restriction that [[~~Harmonic~~\|multiples of the fundamental]] cannot be epimeric. Epimeric ratios were considered to be inferior to epimoric ratios.		In ancient Greece, fractions like 3/1 and 5/1 were not considered to be epimeric ratios because of their additional restriction that [[harmonic\|multiples of the fundamental]] cannot be epimeric. Epimeric ratios were considered to be inferior to epimoric ratios.

	== Superpartient subcategories ==		== Superpartient subcategories ==
	Superpartient ratios can be grouped into subcategories based on the exact difference between the numerator and the denominator. This is known as the '''degree of epimoricity''' (not to be confused with ''epimericity''~~—~~see link below), or '''delta''' (proposed by [[Kite Giedraitis]]). This is particularly useful when considering ratios that are [[comma]]s.		Superpartient ratios can be grouped into subcategories based on the exact difference between the numerator and the denominator. This is known as the ''degree of epimoricity'' (not to be confused with ''epimericity'' – see link below), or ''delta'' (proposed by Kite Giedraitis). This is particularly useful when considering ratios that are [[comma]]s.

	These subcategories are named as superbipartient, supertripartient, superquadripartient, etc., or in ~~[[Delta-N\|~~delta-N terminology]] as delta-2, delta-3, delta-4, etc. Superparticular or epimoric ratios can likewise be named delta-1.		These subcategories are named as superbipartient, supertripartient, superquadripartient, etc., or in delta-''N'' terminology as delta-2, delta-3, delta-4, etc. Superparticular or epimoric ratios can likewise be named delta-1.

	=== Examples ===		=== Examples ===
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	:''The general formula for this factorization is <math>\prod\limits_{i = 1}^K \frac {K A + i N} {K A + (i - 1) N} = \frac {A + N} A</math>. Here you can see more clearly that actual delta of factors will be <math>N / \operatorname{gcd}(K, N)</math>.''		:''The general formula for this factorization is <math>\prod\limits_{i = 1}^K \frac {K A + i N} {K A + (i - 1) N} = \frac {A + N} A</math>. Here you can see more clearly that actual delta of factors will be <math>N / \operatorname{gcd}(K, N)</math>.''

	~~[[Wikipedia:Størmer's theorem~~\|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.		{{W\|Størmer's theorem}} can be extended to show that for each [[prime limit]] ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.

	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Ratio]]		[[Category:Ratio]]

Arseniiv: /* Properties */ yet another minor clarification

2024-06-15T12:15:52Z

Properties: yet another minor clarification

← Older revision		Revision as of 12:15, 15 June 2024
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	* 4/1 = (12/9) (9/6) (6/3) = '''(4/3) (3/2) (2/1)''' ''— now we can’t get delta-3 because there are 3 factors.''		* 4/1 = (12/9) (9/6) (6/3) = '''(4/3) (3/2) (2/1)''' ''— now we can’t get delta-3 because there are 3 factors.''
	* 4/1 = (16/13) (13/10) (10/7) (7/4).		* 4/1 = (16/13) (13/10) (10/7) (7/4).
	Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors.		Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors (including their order).

	:''The general formula for this factorization is <math>\prod\limits_{i = 1}^K \frac {K A + i N} {K A + (i - 1) N} = \frac {A + N} A</math>. Here you can see more clearly that actual delta of factors will be <math>N / \operatorname{gcd}(K, N)</math>.''		:''The general formula for this factorization is <math>\prod\limits_{i = 1}^K \frac {K A + i N} {K A + (i - 1) N} = \frac {A + N} A</math>. Here you can see more clearly that actual delta of factors will be <math>N / \operatorname{gcd}(K, N)</math>.''

Arseniiv: changed \frac to \dfrac at the start of the article to make the fractions possibly more readable, the formula being basically a display-style there

2024-06-15T12:13:17Z

changed \frac to \dfrac at the start of the article to make the fractions possibly more readable, the formula being basically a display-style there

← Older revision		Revision as of 12:13, 15 June 2024
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	More particularly, a superpartient ratio takes the form:		More particularly, a superpartient ratio takes the form:
	:<math>\~~frac~~{n + d}{n} = 1 + \~~frac~~{d}{n}</math>,		:<math>\dfrac{n + d}{n} = 1 + \dfrac{d}{n}</math>,
	where <math>n</math> and <math>d</math> are [[Wikipedia:Positive integer\|positive integer]]s, <math>d > 1</math> and <math>d</math> is [[Wikipedia:Coprime\|coprime]] to <math>n</math>.		where <math>n</math> and <math>d</math> are [[Wikipedia:Positive integer\|positive integer]]s, <math>d > 1</math> and <math>d</math> is [[Wikipedia:Coprime\|coprime]] to <math>n</math>.

Arseniiv: /* Properties */ another clarifying addition

2024-06-15T12:09:12Z

Properties: another clarifying addition

← Older revision		Revision as of 12:09, 15 June 2024
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	Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors.		Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors.

	:''The general formula for this factorization is'' <math>\prod\limits_{i = 1}^K \frac {K A + i N} {K A + (i - 1) N} = \frac {A + N} A</math>.		:''The general formula for this factorization is <math>\prod\limits_{i = 1}^K \frac {K A + i N} {K A + (i - 1) N} = \frac {A + N} A</math>. Here you can see more clearly that actual delta of factors will be <math>N / \operatorname{gcd}(K, N)</math>.''

	[[Wikipedia:Størmer's theorem\|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.		[[Wikipedia:Størmer's theorem\|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.

Arseniiv: /* Properties */ small tweak #2

2024-06-15T12:05:08Z

Properties: small tweak #2

← Older revision		Revision as of 12:05, 15 June 2024
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	Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors.		Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors.

	:The general formula for this factorization is <math>\~~prod_~~{i = 1}^K \frac {K A + i N} {K A + (i - 1) N} = \frac {A + N} A</math>.		:''The general formula for this factorization is'' <math>\prod\limits_{i = 1}^K \frac {K A + i N} {K A + (i - 1) N} = \frac {A + N} A</math>.

	[[Wikipedia:Størmer's theorem\|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.		[[Wikipedia:Størmer's theorem\|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.

Arseniiv: /* Properties */ check if an inline-style formula is less distracting from other properties

2024-06-15T12:04:13Z

Properties: check if an inline-style formula is less distracting from other properties

← Older revision		Revision as of 12:04, 15 June 2024
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	Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors.		Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors.

	:The general formula for this factorization is <math>~~\displaystyle~~\prod_{i = 1}^K \frac {K A + i N} {K A + (i - 1) N} = \frac {A + N} A</math>.		:The general formula for this factorization is <math>\prod_{i = 1}^K \frac {K A + i N} {K A + (i - 1) N} = \frac {A + N} A</math>.

	[[Wikipedia:Størmer's theorem\|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.		[[Wikipedia:Størmer's theorem\|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.

Arseniiv: /* Properties */ an explicit product formula

2024-06-15T12:03:18Z

Properties: an explicit product formula

← Older revision		Revision as of 12:03, 15 June 2024
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	* 4/1 = (16/13) (13/10) (10/7) (7/4).		* 4/1 = (16/13) (13/10) (10/7) (7/4).
	Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors.		Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors.

			:The general formula for this factorization is <math>\displaystyle\prod_{i = 1}^K \frac {K A + i N} {K A + (i - 1) N} = \frac {A + N} A</math>.

	[[Wikipedia:Størmer's theorem\|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.		[[Wikipedia:Størmer's theorem\|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.

Arseniiv: factorization of delta-N ratios

2024-06-15T11:55:03Z

factorization of delta-N ratios

← Older revision		Revision as of 11:55, 15 June 2024
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	<math>\displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P</math>		<math>\displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P</math>

			Likewise, you can also express any delta-''N'' ratio as a product of any number of delta-''M'' ratios with ''M'' being a divisor of ''N''. For example,
			* 5/3 = (10/8) (8/6) = '''(5/4) (4/3)''' ''— can’t get delta-2 because we have an even number of factors.''
			* 5/3 = (15/13) (13/11) (11/9).
			* 5/3 = (20/18) (18/16) (16/14) (14/12) = '''(10/9) (9/8) (8/7) (7/6)''' ''— again only delta-1.''
			* 5/3 = (25/23) (23/21) (21/19) (19/17) (17/15).
			* 4/1 = (8/5) (5/2).
			* 4/1 = (12/9) (9/6) (6/3) = '''(4/3) (3/2) (2/1)''' ''— now we can’t get delta-3 because there are 3 factors.''
			* 4/1 = (16/13) (13/10) (10/7) (7/4).
			Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors.

	[[Wikipedia:Størmer's theorem\|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.		[[Wikipedia:Størmer's theorem\|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.

Fredg999: Categories

2024-06-10T14:58:19Z

ArrowHead294 at 19:09, 8 May 2024

2024-05-08T19:09:04Z

← Older revision		Revision as of 19:09, 8 May 2024
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	{{Beginner\| Abc, high quality commas, and epimericity }}		{{Beginner\|Abc, high quality commas, and epimericity}}
	{{Wikipedia\| Superpartient ratio }}		{{Wikipedia\|Superpartient ratio}}

	The '''delta''' of a [[ratio]] is simply the difference between its numerator and its denominator. (Delta is also known as degree of epimoricity.) A ratio with a delta of N is called a '''delta-N ratio'''.		The '''delta''' of a [[ratio]] is simply the difference between its numerator and its denominator. (Delta is also known as degree of epimoricity.) A ratio with a delta of N is called a '''delta-N ratio'''.
	{\| class="wikitable" style="text-align:center;"		{\| class="wikitable" style="text-align:center;"
	\|+		\|+ Examples
	~~examples~~		! Delta-1 ratios
	!~~delta~~-1 ratios		\| 2/1
	\|2/1		\| 3/2
	\|3/2		\| 4/3
	\|4/3		\| 5/4
	\|5/4		\| 6/5
	\|6/5		\| 7/6
	\|7/6		\| etc.
	\|etc.
	\|-		\|-
	!~~delta~~-2 ratios		! Delta-2 ratios
	\|3/1		\| 3/1
	\|5/3		\| 5/3
	\|7/5		\| 7/5
	\|9/7		\| 9/7
	\|11/9		\| 11/9
	\|13/11		\| 13/11
	\|etc.		\| etc.
	\|-		\|-
	!~~delta~~-3 ratios		! Delta-3 ratios
	\|4/1		\| 4/1
	\|5/2		\| 5/2
	\|7/4		\| 7/4
	\|8/5		\| 8/5
	\|10/7		\| 10/7
	\|11/8		\| 11/8
	\|etc.		\| etc.
	\|-		\|-
	!~~delta~~-4 ratios		! Delta-4 ratios
	\|5/1		\| 5/1
	\|7/3		\| 7/3
	\|9/5		\| 9/5
	\|11/7		\| 11/7
	\|13/9		\| 13/9
	\|15/11		\| 15/11
	\|etc.		\| etc.
	\|}		\|}

	Thus [[superparticular]] ratios are delta-1 ratios, and '''superpartient ratios''' are all ratios ''except'' delta-1 ratios. The delta-N terminology was coined by [[Kite Giedraitis]].		Thus [[superparticular]] ratios are delta-1 ratios, and '''superpartient ratios''' are all ratios ''except'' delta-1 ratios. The delta-N terminology was coined by [[Kite Giedraitis]].

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	== Superpartient subcategories ==		== Superpartient subcategories ==
	Superpartient ratios can be grouped into subcategories based on the exact difference between the numerator and the denominator. This is known as the '''degree of epimoricity''' (not to be confused with ''epimericity'' – see link below), or '''delta''' (proposed by [[Kite Giedraitis]]). This is particularly useful when considering ratios that are [[comma]]s.		Superpartient ratios can be grouped into subcategories based on the exact difference between the numerator and the denominator. This is known as the '''degree of epimoricity''' (not to be confused with ''epimericity''—see link below), or '''delta''' (proposed by [[Kite Giedraitis]]). This is particularly useful when considering ratios that are [[comma]]s.

	These subcategories are named as superbipartient, supertripartient, superquadripartient, etc., or in [[Delta-N\|delta-N terminology]] as delta-2, delta-3, delta-4, etc. Superparticular or epimoric ratios can likewise be named delta-1.		These subcategories are named as superbipartient, supertripartient, superquadripartient, etc., or in [[Delta-N\|delta-N terminology]] as delta-2, delta-3, delta-4, etc. Superparticular or epimoric ratios can likewise be named delta-1.

← Older revision		Revision as of 14:58, 10 June 2024
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	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Ratio]]		[[Category:Ratio]]
	~~[[Category:Greek music]]~~