Superpartient ratio - Revision history

Fredg999: Categories

2024-06-10T14:58:42Z

FloraC: -redundant categories

2023-08-29T09:22:52Z

-redundant categories

← Older revision		Revision as of 09:22, 29 August 2023
Line 1:		Line 1:
	#REDIRECT [[Delta-N ratio]]		#REDIRECT [[Delta-N ratio]]

	~~[[Category:Ratio]]~~
	[[Category:Terms]]		[[Category:Terms]]
	~~[[Category:Greek]]~~

Fredg999: Fix double redirect

2023-08-22T05:44:43Z

Fix double redirect

← Older revision		Revision as of 05:44, 22 August 2023
Line 1:		Line 1:
	#~~redirect~~ [[Delta-N]]		#REDIRECT [[Delta-N ratio]]

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

Inthar: Redirected page to Delta-N

2023-05-20T22:21:43Z

Redirected page to Delta-N

@@ Line 1: / Line 1: @@
-{{Wikipedia|Superpartient ratio}}
+#redirect [[Delta-N]]
 [[Category:Ratio]]
 [[Category:Terms]]
 [[Category:Greek]]

Inthar at 22:16, 20 May 2023

2023-05-20T22:16:48Z

@@ Line 11: / Line 11: @@
 == 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.
 == Superpartient subcategories ==

TallKite: /* Superpartient subcategories */ added a link to the delta-N page

2023-03-01T09:34:23Z

Superpartient subcategories: added a link to the delta-N page

← Older revision		Revision as of 09:34, 1 March 2023
Line 15:		Line 15:
	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 delta-2, delta-3, delta-4~~, etc., or as superbipartient, supertripartient, superquadripartient~~, 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.

	=== Examples ===		=== Examples ===

TallKite: cleanup/rewrite. Is there a typo in the last sentence? n refers to both epimericity and epimoricity, yet they are rarely (never?) identical.

2023-03-01T07:18:23Z

cleanup/rewrite. Is there a typo in the last sentence? n refers to both epimericity and epimoricity, yet they are rarely (never?) identical.

← Older revision		Revision as of 07:18, 1 March 2023
Line 8:		Line 8:
	== Etymology ==		== Etymology ==
	In ancient Greece, they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".		In ancient Greece, they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".

	[[Kite Giedraitis]] has proposed the term delta-1 (where [[delta]] means difference, here the difference between the numerator and the denominator) as a replacement for superparticular, delta-2 for ratios of the form <math>\frac{n+2}{n}</math>, likewise delta-3, delta-4, etc.

	== 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 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 delta-2, delta-3, delta-4, etc., or as superbipartient, supertripartient, superquadripartient, etc. Superparticular or epimoric ratios can likewise be named delta-1.

			=== Examples ===
			* Delta-2 (superbipartient) ratios: [[3/1]], [[5/3]], [[7/5]], [[9/7]], [[11/9]], [[13/11]], etc.
			* Delta-3 (supertripartient) ratios: [[4/1]], [[5/2]], [[7/4]], [[8/5]], [[10/7]], [[11/8]], etc.
			* Delta-4 (superquadripartient) ratios: [[5/1]], [[7/3]], [[9/5]], [[11/7]], [[13/9]], [[15/11]], etc.

	== Properties ==		== Properties ==
Line 19:		Line 27:
	<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>

	When considering ratios, and particularly when they are ratios for [[comma]]s, it can be useful to introduce the notion of the '''degree of epimoricity''' (not to be confused with ''epimericity'' – see link below), or '''delta''' (proposed by [[Kite Giedraitis]]). In terms of ''p''/''q'' reduced to lowest terms it is ''p'' - ''q''. An epimoric ratio has degree 1, the 7-limit comma 245/243 degree 2, the 5-limit comma 128/125 degree 3, and so forth. [[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''.

	~~== Examples ==~~
	* Delta-2 (superbipartient) ratios: [[3/1]], [[5/3]], [[7/5]], [[9/7]], [[11/9]], [[13/11]], etc.
	* Delta-3 (supertripartient) ratios: [[4/1]], [[5/2]], [[7/4]], [[8/5]], [[10/7]], [[11/8]], etc.
	* Delta-4 (superquadripartient) ratios: [[5/1]], [[7/3]], [[9/5]], [[11/7]], [[13/9]], [[15/11]], etc.

	== See also ==		== See also ==

Fredg999: Fix typo

2023-02-26T07:17:59Z

Fix typo

← Older revision		Revision as of 07:17, 26 February 2023
Line 19:		Line 19:
	<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>

	When considering ratios, and particularly when they are ratios for [[comma]]s, it can be useful to introduce the notion of the '''degree of epimoricity''' (not to be confused with ''epimericity'' – see link below), or '''delta'''(proposed by [[Kite Giedraitis]]). In terms of ''p''/''q'' reduced to lowest terms it is ''p'' - ''q''. An epimoric ratio has degree 1, the 7-limit comma 245/243 degree 2, the 5-limit comma 128/125 degree 3, and so forth. [[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''.		When considering ratios, and particularly when they are ratios for [[comma]]s, it can be useful to introduce the notion of the '''degree of epimoricity''' (not to be confused with ''epimericity'' – see link below), or '''delta''' (proposed by [[Kite Giedraitis]]). In terms of ''p''/''q'' reduced to lowest terms it is ''p'' - ''q''. An epimoric ratio has degree 1, the 7-limit comma 245/243 degree 2, the 5-limit comma 128/125 degree 3, and so forth. [[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''.

	== Examples ==		== Examples ==

Fredg999: Move examples from Delta, reformat as lists instead of table (table may cause a false sense of correspondence between elements in the same column)

2023-02-26T07:14:51Z

Move examples from Delta, reformat as lists instead of table (table may cause a false sense of correspondence between elements in the same column)

← Older revision		Revision as of 07:14, 26 February 2023
Line 19:		Line 19:
	<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>

	When considering ratios, and particularly when they are ratios for [[comma]]s, it can be useful to introduce the notion of the '''degree of epimoricity''' (not to be confused with ''epimericity'' – see link below). In terms of ''p''/''q'' reduced to lowest terms it is ''p'' - ''q''. An epimoric ratio has degree 1, the 7-limit comma 245/243 degree 2, the 5-limit comma 128/125 degree 3, and so forth. [[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''.		When considering ratios, and particularly when they are ratios for [[comma]]s, it can be useful to introduce the notion of the '''degree of epimoricity''' (not to be confused with ''epimericity'' – see link below), or '''delta'''(proposed by [[Kite Giedraitis]]). In terms of ''p''/''q'' reduced to lowest terms it is ''p'' - ''q''. An epimoric ratio has degree 1, the 7-limit comma 245/243 degree 2, the 5-limit comma 128/125 degree 3, and so forth. [[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''.

			== Examples ==
			* Delta-2 (superbipartient) ratios: [[3/1]], [[5/3]], [[7/5]], [[9/7]], [[11/9]], [[13/11]], etc.
			* Delta-3 (supertripartient) ratios: [[4/1]], [[5/2]], [[7/4]], [[8/5]], [[10/7]], [[11/8]], etc.
			* Delta-4 (superquadripartient) ratios: [[5/1]], [[7/3]], [[9/5]], [[11/7]], [[13/9]], [[15/11]], etc.

	== See also ==		== See also ==

Fredg999: Added delta etymology

2023-02-26T06:54:26Z

Added delta etymology

← Older revision		Revision as of 06:54, 26 February 2023
Line 8:		Line 8:
	== Etymology ==		== Etymology ==
	In ancient Greece, they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".		In ancient Greece, they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".

			[[Kite Giedraitis]] has proposed the term delta-1 (where [[delta]] means difference, here the difference between the numerator and the denominator) as a replacement for superparticular, delta-2 for ratios of the form <math>\frac{n+2}{n}</math>, likewise delta-3, delta-4, etc.

	== Definitions ==		== Definitions ==

← Older revision		Revision as of 14:58, 10 June 2024
Line 2:		Line 2:

	[[Category:Terms]]		[[Category:Terms]]
			[[Category:Ancient Greek music]]