Pentacircle chords - Revision history

FloraC: It's important to single ou the palindromic triad cuz it's the most fundamental form of these chords. The rest can be viewed as variations on it

2024-10-11T13:55:25Z

It's important to single ou the palindromic triad cuz it's the most fundamental form of these chords. The rest can be viewed as variations on it

← Older revision		Revision as of 13:55, 11 October 2024
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	Pentacircle chords are of [[Dyadic chord/Pattern of essentially tempered chords\|pattern 1b]] in the 2.9.7.11 [[subgroup]] [[11-odd-limit]], meaning that there are 3 [[triad]]s, 6 [[tetrad]]s and 2 [[pentad]]s, for a total of 11 distinct chord structures.		Pentacircle chords are of [[Dyadic chord/Pattern of essentially tempered chords\|pattern 1b]] in the 2.9.7.11 [[subgroup]] [[11-odd-limit]], meaning that there are 3 [[triad]]s, 6 [[tetrad]]s and 2 [[pentad]]s, for a total of 11 distinct chord structures.

	The pentacircle triads ~~are~~		The three pentacircle triads include a palindrome since it identifies [[14/11]] by a stack of two [[9/8]]'s:
	* 1–9/~~7–16~~/9 with steps 9/7, 11/8, 9/8;		* 1–9/8–14/11 with steps 9/8, 9/8, 11/7.
	* 1–9/~~7–16~~/11 with steps 9/7, 9/8, 11/8;
	* ~~1–11~~/~~7–16~~/9 with steps 11/7, 9/8, 9/8.		And an inversely related pair:
			* 1–9/8–16/11 with steps 9/8, 9/7, 11/8;
			* 1–9/8–14/9 with steps 9/8, 11/8, 9/7;

	The tetrads include the palindromic		The tetrads include the palindromic

FloraC: Improve readability; recategorize

2024-10-11T13:43:17Z

Improve readability; recategorize

← Older revision		Revision as of 13:43, 11 October 2024
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	The pentacircle triads are		The pentacircle triads are
	* ~~1-9~~/~~7-16~~/9 with steps 9/7-11/8-9/8;		* 1–9/7–16/9 with steps 9/7, 11/8, 9/8;
	* ~~1-9~~/~~7-16~~/11 with steps 9/7-9/8-11/8;		* 1–9/7–16/11 with steps 9/7, 9/8, 11/8;
	* ~~1-11~~/~~7-16~~/9 with steps 11/7-9/8-9/8.		* 1–11/7–16/9 with steps 11/7, 9/8, 9/8.

	The tetrads include the palindromic		The tetrads include the palindromic
	* ~~1-9~~/~~8-14~~/~~9-7~~/4 with steps 9/8-11/8-9/8-8/7;		* 1–9/8–14/9–7/4 with steps 9/8, 11/8, 9/8, 8/7;
	* ~~1-9~~/~~8-11~~/~~8-14~~/9 with steps 9/8-11/9-9/8-9/7.		* 1–9/8–11/8–14/9 with steps 9/8, 11/9, 9/8, 9/7.

	And the inversely related pairs		And the inversely related pairs
	* ~~1-11~~/~~8-11~~/~~7-16~~/9 with steps 11/8-8/7-9/8-9/8, and its inverse		* 1–11/8–11/7–16/9 with steps 11/8, 8/7, 9/8, 9/8, and its inverse
	* ~~1-11~~/~~8-14~~/~~9-7~~/4 with steps 11/8-9/8-9/8-8/7;		* 1–11/8–14/9–7/4 with steps 11/8, 9/8, 9/8, 8/7;
	* ~~1-11~~/~~9-11~~/~~7-16~~/9 with steps 11/9-9/7-9/8-9/8, and its inverse		* 1–11/9–11/7–16/9 with steps 11/9, 9/7, 9/8, 9/8, and its inverse
	* ~~1-11~~/~~9-11~~/~~8-14~~/9 with steps 11/9-9/8-9/8-9/7.		* 1–11/9–11/8–14/9 with steps 11/9, 9/8, 9/8, 9/7.

	Finally, there are two pentacircle pentads, inversely related:		Finally, there are two pentacircle pentads, inversely related:
	* ~~1-11~~/~~9-11~~/~~8-11~~/~~7-16~~/9 with steps 11/9-9/8-8/7-9/8-9/8, and its inverse		* 1–11/9–11/8–11/7–16/9 with steps 11/9, 9/8, 8/7, 9/8, 9/8, and its inverse
	* ~~1-11~~/~~9-11~~/~~8-14~~/~~9-16~~/9 with steps 11/9-9/8-9/8-8/7-9/8.		* 1–11/9–11/8–14/9–16/9 with steps 11/9, 9/8, 9/8, 8/7, 9/8.

	[[~~EDO\|~~Equal ~~divisions of the octave~~]] with pentacircle chords include {{EDOs\| 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, and 208 }}, with 208edo giving the [[optimal patent val]].		[[Equal temperament]]s with pentacircle chords include {{EDOs\| 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, and 208 }}, with 208edo giving the [[optimal patent val]].

	[[Category:11-odd-limit]]		[[Category:11-odd-limit chords]]
	[[Category:Essentially tempered chords]]		[[Category:Essentially tempered chords]]
	[[Category:Triads]]		[[Category:Triads]]

FloraC: +pattern of chords

2023-07-27T05:19:39Z

+pattern of chords

← Older revision		Revision as of 05:19, 27 July 2023
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	A '''pentacircle chord''' is an [[essentially tempered dyadic chord]] in the 2.9.7.11 [[subgroup]] in the [[11-odd-limit]], [[tempering out]] the pentacircle comma, [[896/891]].		A '''pentacircle chord''' is an [[essentially tempered dyadic chord]] in the 2.9.7.11 [[subgroup]] in the [[11-odd-limit]], [[tempering out]] the pentacircle comma, [[896/891]].

	~~The pentacircle~~ [[triad]]s are ~~three in number:~~		Pentacircle chords are of [[Dyadic chord/Pattern of essentially tempered chords\|pattern 1b]] in the 2.9.7.11 [[subgroup]] [[11-odd-limit]], meaning that there are 3 [[triad]]s, 6 [[tetrad]]s and 2 [[pentad]]s, for a total of 11 distinct chord structures.

			The pentacircle triads are
	* 1-9/7-16/9 with steps 9/7-11/8-9/8;		* 1-9/7-16/9 with steps 9/7-11/8-9/8;
	* 1-9/7-16/11 with steps 9/7-9/8-11/8;		* 1-9/7-16/11 with steps 9/7-9/8-11/8;
	* 1-11/7-16/9 with steps 11/7-9/8-9/8.		* 1-11/7-16/9 with steps 11/7-9/8-9/8.

	~~There are six pentacircle [[tetrad]]s, including~~ the palindromic		The tetrads include the palindromic
	* 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7;		* 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7;
	* 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7.		* 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7.
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	* 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7.		* 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7.

	Finally, there are two pentacircle ~~[[pentad]]s~~, inversely related:		Finally, there are two pentacircle pentads, inversely related:
	* 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8, and its inverse		* 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8, and its inverse
	* 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8.		* 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8.

	~~The count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.~~

	[[EDO\|Equal divisions of the octave]] with pentacircle chords include {{EDOs\| 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, and 208 }}, with 208edo giving the [[optimal patent val]].		[[EDO\|Equal divisions of the octave]] with pentacircle chords include {{EDOs\| 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, and 208 }}, with 208edo giving the [[optimal patent val]].

FloraC: Recategorize

2023-05-23T11:44:05Z

Recategorize

← Older revision		Revision as of 11:44, 23 May 2023
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	[[Category:11-odd-limit]]		[[Category:11-odd-limit]]
	[[Category:Essentially tempered chords]]		[[Category:Essentially tempered chords]]
	[[Category:~~Triad~~]]		[[Category:Triads]]
	[[Category:~~Tetrad~~]]		[[Category:Tetrads]]
	[[Category:~~Pentad~~]]		[[Category:Pentads]]
	[[Category:Pentacircle]]		[[Category:Pentacircle]]

FloraC: Cleanup

2023-04-28T15:20:27Z

Cleanup

← Older revision		Revision as of 15:20, 28 April 2023
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	A '''pentacircle chord''' is an [[essentially tempered dyadic chord]] in the 2.9.7.11 [[~~Just intonation subgroup\|~~subgroup]] in the [[11-odd-limit]], [[tempering out]] the [[pentacircle comma]], [[896/891]].		A '''pentacircle chord''' is an [[essentially tempered dyadic chord]] in the 2.9.7.11 [[subgroup]] in the [[11-odd-limit]], [[tempering out]] the pentacircle comma, [[896/891]].

	The pentacircle [[~~Triad\|triads~~]] are three in number:		The pentacircle [[triad]]s are three in number:
			* 1-9/7-16/9 with steps 9/7-11/8-9/8;
	* 1-9/7-16/9 with steps 9/7-11/8-9/8		* 1-9/7-16/11 with steps 9/7-9/8-11/8;
	* 1-9/7-16/11 with steps 9/7-9/8-11/8
	* 1-11/7-16/9 with steps 11/7-9/8-9/8.		* 1-11/7-16/9 with steps 11/7-9/8-9/8.

	There are six pentacircle [[~~Tetrad\|tetrads~~]]:		There are six pentacircle [[tetrad]]s, including the palindromic
			* 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7;
	* ~~the [[palindromic]]~~ 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7		* 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7.
	* ~~the palindromic~~ 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7
	* the [[inverse pair]] 1-11/8-11/7-16/9 with steps 11/8-8/7-9/8-9/8 and 1-11/8-14/9-7/4 with steps 11/8-9/8-9/8-8/7
	* the inverse pair 1-11/9-11/7-16/9 with steps 11/9-9/7-9/8-9/8 and 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7

	~~Finally~~, ~~there are two pentacircle [[Pentad\|pentads]]~~, ~~inversely related:~~		And the inversely related pairs
			* 1-11/8-11/7-16/9 with steps 11/8-8/7-9/8-9/8, and its inverse
			* 1-11/8-14/9-7/4 with steps 11/8-9/8-9/8-8/7;
			* 1-11/9-11/7-16/9 with steps 11/9-9/7-9/8-9/8, and its inverse
			* 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7.

	* 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8		Finally, there are two pentacircle [[pentad]]s, inversely related:
	* 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8		* 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8, and its inverse
			* 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8.

	The count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.		The count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.

	[[EDO\|Equal divisions of the octave]] with pentacircle chords include ~~[[22edo~~\|22]], ~~[[27edo\|~~27]], ~~[[41edo\|~~41]], ~~[[46edo\|~~46]], ~~[[58edo\|~~58]], ~~[[68edo\|~~68]], ~~[[80edo\|~~80]], ~~[[87edo\|~~87]], ~~[[121edo\|~~121]], ~~[[145edo\|~~145]], ~~[[167edo\|~~167]], ~~[[208edo\|~~208]], [[~~266edo\|266e]] and [[433edo\|433bce~~]].		[[EDO\|Equal divisions of the octave]] with pentacircle chords include {{EDOs\| 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, and 208 }}, with 208edo giving the [[optimal patent val]].

	[[Category:11-odd-limit]]		[[Category:11-odd-limit]]
	[[Category:Essentially tempered chords]]		[[Category:Essentially tempered chords]]
			[[Category:Triad]]
			[[Category:Tetrad]]
			[[Category:Pentad]]
	[[Category:Pentacircle]]		[[Category:Pentacircle]]
	~~[[Category:Tetrad]]~~

Fredg999 category edits: Removing from Category:Chords using Cat-a-lot

2023-04-28T03:51:55Z

Removing from Category:Chords using Cat-a-lot

← Older revision		Revision as of 03:51, 28 April 2023
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	[[Category:11-odd-limit]]		[[Category:11-odd-limit]]
	~~[[Category:Chords]]~~
	[[Category:Essentially tempered chords]]		[[Category:Essentially tempered chords]]
	[[Category:Pentacircle]]		[[Category:Pentacircle]]
	[[Category:Tetrad]]		[[Category:Tetrad]]

BudjarnLambeth: Added links (as requested by the to:do notice)

2023-04-20T07:43:48Z

Added links (as requested by the to:do notice)

@@ Line 1: / Line 1: @@
-A '''pentacircle chord''' is an [[essentially tempered dyadic chord]] in the 2.9.7.11 subgroup in the 11-odd-limit, tempering out the [[pentacircle comma]], 896/891. The pentacircle triads are three in number, 1-9/7-16/9 with steps 9/7-11/8-9/8; 1-9/7-16/11 with steps 9/7-9/8-11/8; and 1-11/7-16/9 with steps 11/7-9/8-9/8. There are six pentacircle tetrads: the palindromic 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7; the palindromic 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7; the inverse pair 1-11/8-11/7-16/9 with steps 11/8-8/7-9/8-9/8 and 1-11/8-14/9-7/4 with steps 11/8-9/8-9/8-8/7; and the inverse pair 1-11/9-11/7-16/9 with steps 11/9-9/7-9/8-9/8 and 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7. Finally, there are two pentacircle pentads, inversely related: 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8 and 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8.
+A '''pentacircle chord''' is an [[essentially tempered dyadic chord]] in the 2.9.7.11 [[Just intonation subgroup|subgroup]] in the [[11-odd-limit]], [[tempering out]] the [[pentacircle comma]], [[896/891]].
 The count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.
-Equal divisions with pentacircle chords include 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, 208, 266e and 433bce.
+[[EDO|Equal divisions of the octave]] with pentacircle chords include [[22edo|22]], [[27edo|27]], [[41edo|41]], [[46edo|46]], [[58edo|58]], [[68edo|68]], [[80edo|80]], [[87edo|87]], [[121edo|121]], [[145edo|145]], [[167edo|167]], [[208edo|208]], [[266edo|266e]] and [[433edo|433bce]].
 [[Category:11-odd-limit]]
@@ Line 10: / Line 28: @@
 [[Category:Pentacircle]]
 [[Category:Tetrad]]

Xenwolf: recat

2021-05-20T08:09:47Z

recat

← Older revision		Revision as of 08:09, 20 May 2021
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	Equal divisions with pentacircle chords include 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, 208, 266e and 433bce.		Equal divisions with pentacircle chords include 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, 208, 266e and 433bce.

	[[Category:11-limit]]		[[Category:11-odd-limit]]
	[[Category:Chords]]		[[Category:Chords]]
	[[Category:Essentially tempered chords]]		[[Category:Essentially tempered chords]]

Xenwolf: +cat

2021-05-20T07:12:06Z

+cat

← Older revision		Revision as of 07:12, 20 May 2021
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	A '''pentacircle chord''' is an [[~~Dyadic chord#Essentially tempered dyadic chords\|~~essentially tempered dyadic chord]] in the 2.9.7.11 subgroup in the 11-odd-limit, tempering out the [[pentacircle comma]], 896/891. The pentacircle triads are three in number, 1-9/7-16/9 with steps 9/7-11/8-9/8; 1-9/7-16/11 with steps 9/7-9/8-11/8; and 1-11/7-16/9 with steps 11/7-9/8-9/8. There are six pentacircle tetrads: the palindromic 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7; the palindromic 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7; the inverse pair 1-11/8-11/7-16/9 with steps 11/8-8/7-9/8-9/8 and 1-11/8-14/9-7/4 with steps 11/8-9/8-9/8-8/7; and the inverse pair 1-11/9-11/7-16/9 with steps 11/9-9/7-9/8-9/8 and 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7. Finally, there are two pentacircle pentads, inversely related: 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8 and 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8.		A '''pentacircle chord''' is an [[essentially tempered dyadic chord]] in the 2.9.7.11 subgroup in the 11-odd-limit, tempering out the [[pentacircle comma]], 896/891. The pentacircle triads are three in number, 1-9/7-16/9 with steps 9/7-11/8-9/8; 1-9/7-16/11 with steps 9/7-9/8-11/8; and 1-11/7-16/9 with steps 11/7-9/8-9/8. There are six pentacircle tetrads: the palindromic 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7; the palindromic 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7; the inverse pair 1-11/8-11/7-16/9 with steps 11/8-8/7-9/8-9/8 and 1-11/8-14/9-7/4 with steps 11/8-9/8-9/8-8/7; and the inverse pair 1-11/9-11/7-16/9 with steps 11/9-9/7-9/8-9/8 and 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7. Finally, there are two pentacircle pentads, inversely related: 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8 and 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8.

	The count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.		The count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.

	Equal divisions with pentacircle chords include 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, 208, 266e and 433bce.		Equal divisions with pentacircle chords include 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, 208, 266e and 433bce.

	[[Category:11-limit]]		[[Category:11-limit]]
	[[Category:Chords]]		[[Category:Chords]]
			[[Category:Essentially tempered chords]]
	[[Category:Pentacircle]]		[[Category:Pentacircle]]
	[[Category:Tetrad]]		[[Category:Tetrad]]
	[[Category:Todo:add links]]		[[Category:Todo:add links]]

Xenwolf: bold lemma

2020-08-27T07:24:05Z

bold lemma

← Older revision		Revision as of 07:24, 27 August 2020
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	A ''pentacircle chord'' is an [[Dyadic chord#Essentially tempered dyadic chords\|essentially tempered dyadic chord]] in the 2.9.7.11 subgroup in the 11-odd-limit, tempering out the [[pentacircle comma]], 896/891. The pentacircle triads are three in number, 1-9/7-16/9 with steps 9/7-11/8-9/8; 1-9/7-16/11 with steps 9/7-9/8-11/8; and 1-11/7-16/9 with steps 11/7-9/8-9/8. There are six pentacircle tetrads: the palindromic 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7; the palindromic 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7; the inverse pair 1-11/8-11/7-16/9 with steps 11/8-8/7-9/8-9/8 and 1-11/8-14/9-7/4 with steps 11/8-9/8-9/8-8/7; and the inverse pair 1-11/9-11/7-16/9 with steps 11/9-9/7-9/8-9/8 and 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7. Finally, there are two pentacircle pentads, inversely related: 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8 and 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8.		A '''pentacircle chord''' is an [[Dyadic chord#Essentially tempered dyadic chords\|essentially tempered dyadic chord]] in the 2.9.7.11 subgroup in the 11-odd-limit, tempering out the [[pentacircle comma]], 896/891. The pentacircle triads are three in number, 1-9/7-16/9 with steps 9/7-11/8-9/8; 1-9/7-16/11 with steps 9/7-9/8-11/8; and 1-11/7-16/9 with steps 11/7-9/8-9/8. There are six pentacircle tetrads: the palindromic 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7; the palindromic 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7; the inverse pair 1-11/8-11/7-16/9 with steps 11/8-8/7-9/8-9/8 and 1-11/8-14/9-7/4 with steps 11/8-9/8-9/8-8/7; and the inverse pair 1-11/9-11/7-16/9 with steps 11/9-9/7-9/8-9/8 and 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7. Finally, there are two pentacircle pentads, inversely related: 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8 and 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8.

	The count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.		The count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.