Palingenetic chords - Revision history

FloraC: +pattern of chords

2023-07-26T15:44:11Z

+pattern of chords

Xenllium: Fixed typo

2023-07-23T22:16:24Z

Fixed typo

← Older revision		Revision as of 22:16, 23 July 2023
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	They can be extended to the following tetrads, with two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are		They can be extended to the following tetrads, with two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are
	* 1-18/17-10/9-20/17 chord with steps 18/17-21/10-18/17-17/10;		* 1-18/17-10/9-20/17 chord with steps 18/17-21/20-18/17-17/10;
	* 1-21/20-10/9-7/6 chord with steps 21/20-18/17-21/10-12/7.		* 1-21/20-10/9-7/6 chord with steps 21/20-18/17-21/20-12/7.

	The inversely related pairs of chords are		The inversely related pairs of chords are

FloraC: Readability

2023-07-21T06:37:07Z

Readability

← Older revision		Revision as of 06:37, 21 July 2023
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	There is an inversely related pair which is even more squeezed:		There is an inversely related pair which is even more squeezed:
	* 1-18/17-10/9 with steps 18/17-21/20-9/5 and its inverse 1-21/20-10/9 with steps 21/20-18/17-9/5.		* 1-18/17-10/9 with steps 18/17-21/20-9/5 and its inverse
			* 1-21/20-10/9 with steps 21/20-18/17-9/5.

	They can be extended to the following tetrads, with two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are		They can be extended to the following tetrads, with two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are
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	The inversely related pairs of chords are		The inversely related pairs of chords are
	* 1-10/9-20/17-21/17 with steps 10/9-18/17-21/20-34/21 and its inverse 1-21/20-10/9-21/17 with steps 21/20-18/17-10/9-34/21;		* 1-10/9-20/17-21/17 with steps 10/9-18/17-21/20-34/21 and its inverse
	* 1-10/9-7/6-21/17 with steps 10/9-21/20-18/17-34/21 and its inverse 1-18/17-10/9-21/17 with steps 18/17-21/20-10/9-34/21.		* 1-21/20-10/9-21/17 with steps 21/20-18/17-10/9-34/21;
			* 1-10/9-7/6-21/17 with steps 10/9-21/20-18/17-34/21 and its inverse
			* 1-18/17-10/9-21/17 with steps 18/17-21/20-10/9-34/21.

	Then there are two inversely related pentads:		Then there are two inversely related pentads:
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	* 1-28/25-32/27-40/27 with steps 28/25-18/17-5/4-27/20		* 1-28/25-32/27-40/27 with steps 28/25-18/17-5/4-27/20

	Equal temperaments with palingenetic chords include {{Optimal ET sequence\|12, 46, 53, 58, 60, 72, 99, 111, 118, 171, 183, 243, 270, 289, 354, 400, 472 and 571}}.		Equal temperaments with palingenetic chords include {{Optimal ET sequence\| 12, 46, 53, 58, 60, 72, 99, 111, 118, 171, 183, 243, 270, 289, 354, 400, 472 and 571 }}.

	[[Category:21-odd-limit]]		[[Category:21-odd-limit]]

Xenllium at 03:27, 21 July 2023

2023-07-21T03:27:51Z

← Older revision		Revision as of 03:27, 21 July 2023
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	A '''palingenetic chord''' is an [[essentially tempered dyadic chord]] tempered by [[1701/1700]], the palingenetic comma~~. Currently, there are only a handful of palingenetic chords known~~.		A '''palingenetic chord''' is an [[essentially tempered dyadic chord]] tempered by [[1701/1700]], the palingenetic comma.

	The most typical palingenetic triad is a palindrome in the 2.3.5.7.17 [[subgroup]] [[21-odd-limit]] since it identifies [[21/17]] by a stack of two [[10/9]]'s:		The most typical palingenetic triad is a palindrome in the 2.3.5.7.17 [[subgroup]] [[21-odd-limit]] since it identifies [[21/17]] by a stack of two [[10/9]]'s:
	* 1-10/9-21/17 with steps 10/9-10/9-34/21.		* 1-10/9-21/17 chord with steps 10/9-10/9-34/21.

	There is an inversely related pair which is even more squeezed:		There is an inversely related pair which is even more squeezed:
	* 1-18/17-10/9 with steps 18/17-21/20-9/5, and its inverse		* 1-18/17-10/9 with steps 18/17-21/20-9/5 and its inverse 1-21/20-10/9 with steps 21/20-18/17-9/5.
	* 1-21/20-10/9 with steps 21/20-18/17-9/5.

	They can be extended to the following ~~inversely related~~ tetrads, ~~all~~ of ~~which seem to be based largely on a sort~~ of ~~secundal harmony:~~		They can be extended to the following tetrads, with two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are
	* 1-10/9-20/17-21/17 with steps 10/9-18/17-21/20-34/21, and its inverse		* 1-18/17-10/9-20/17 chord with steps 18/17-21/10-18/17-17/10;
	* 1-21/20-10/9-21/17 with steps 21/20-18/17-10/9-34/21;		* 1-21/20-10/9-7/6 chord with steps 21/20-18/17-21/10-12/7.
	* 1-10/9-7/6-21/17 with steps 10/9-21/20-18/17-34/21, and its inverse
	* 1-18/17-10/9-21/17 with steps 18/17-21/20-10/9-34/21.		The inversely related pairs of chords are
			* 1-10/9-20/17-21/17 with steps 10/9-18/17-21/20-34/21 and its inverse 1-21/20-10/9-21/17 with steps 21/20-18/17-10/9-34/21;
			* 1-10/9-7/6-21/17 with steps 10/9-21/20-18/17-34/21 and its inverse 1-18/17-10/9-21/17 with steps 18/17-21/20-10/9-34/21.

	Then there are two inversely related pentads:		Then there are two inversely related pentads:
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	* 1-18/17-32/27-8/5 with steps 18/17-28/25-27/20-5/4, and its inverse		* 1-18/17-32/27-8/5 with steps 18/17-28/25-27/20-5/4, and its inverse
	* 1-28/25-32/27-40/27 with steps 28/25-18/17-5/4-27/20		* 1-28/25-32/27-40/27 with steps 28/25-18/17-5/4-27/20

			Equal temperaments with palingenetic chords include {{Optimal ET sequence\|12, 46, 53, 58, 60, 72, 99, 111, 118, 171, 183, 243, 270, 289, 354, 400, 472 and 571}}.

	[[Category:21-odd-limit]]		[[Category:21-odd-limit]]
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	[[Category:Pentads]]		[[Category:Pentads]]
	[[Category:Palingenetic]]		[[Category:Palingenetic]]

	~~{{Todo\| review \| expand }}~~

FloraC: Recategorize

2023-05-23T11:43:52Z

Recategorize

← Older revision		Revision as of 11:43, 23 May 2023
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	[[Category:21-odd-limit]]		[[Category:21-odd-limit]]
	[[Category:Essentially tempered chords]]		[[Category:Essentially tempered chords]]
	[[Category:~~Triad~~]]		[[Category:Triads]]
	[[Category:~~Tetrad~~]]		[[Category:Tetrads]]
	[[Category:~~Pentad~~]]		[[Category:Pentads]]
	[[Category:Palingenetic]]		[[Category:Palingenetic]]

	{{Todo\| review \| expand }}		{{Todo\| review \| expand }}

FloraC: Rework

2023-05-16T06:52:20Z

Rework

← Older revision		Revision as of 06:52, 16 May 2023
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	A '''palingenetic chord''' is an [[~~Dyadic chord\|~~essentially tempered dyadic chord]] tempered by [[1701/1700]], the palingenetic comma. Currently, there are only a handful of palingenetic chords known.		A '''palingenetic chord''' is an [[essentially tempered dyadic chord]] tempered by [[1701/1700]], the palingenetic comma. Currently, there are only a handful of palingenetic chords known.

	~~There~~ is a 21-odd-limit ~~palingenetic triad~~:		The most typical palingenetic triad is a palindrome in the 2.3.5.7.17 [[subgroup]] [[21-odd-limit]] since it identifies [[21/17]] by a stack of two [[10/9]]'s:
	* 1-10/9-9/5 with steps 10/9-34/21~~-10/9~~.		* 1-10/9-21/17 with steps 10/9-10/9-34/21.

	~~Assuming~~ we ~~stick to~~ the 27-odd-limit, we have ~~two~~ known triads and ~~one~~ known ~~tetrad~~.		There is an inversely related pair which is even more squeezed:
			* 1-18/17-10/9 with steps 18/17-21/20-9/5, and its inverse
			* 1-21/20-10/9 with steps 21/20-18/17-9/5.

			They can be extended to the following inversely related tetrads, all of which seem to be based largely on a sort of secundal harmony:
			* 1-10/9-20/17-21/17 with steps 10/9-18/17-21/20-34/21, and its inverse
			* 1-21/20-10/9-21/17 with steps 21/20-18/17-10/9-34/21;
			* 1-10/9-7/6-21/17 with steps 10/9-21/20-18/17-34/21, and its inverse
			* 1-18/17-10/9-21/17 with steps 18/17-21/20-10/9-34/21.

			Then there are two inversely related pentads:
			* 1-18/17-10/9-20/17-21/17 with steps 18/17-21/20-18/17-21/20-34/21, and its inverse
			* 1-21/20-10/9-7/6-21/17 with steps 21/20-18/17-21/20-18/17-34/21.

			If we allow the 27-odd-limit, we have four more known triads and two more known tetrads.

	The known 27-odd-limit palingenetic triads are:		The known 27-odd-limit palingenetic triads are:
	* 1-34/27-3/2 with steps 34/27-25/21-4/3, ~~and its inversions;~~ dubbed the "palingenetic major triad"		* 1-34/27-3/2 with steps 34/27-25/21-4/3, dubbed the "palingenetic major triad", and its inverse
	* 1-25/21-3/2 with steps 25/21-34/27-4/3, ~~and its inversions;~~ dubbed the "palingenetic minor triad"		* 1-25/21-3/2 with steps 25/21-34/27-4/3, dubbed the "palingenetic minor triad";
	* 1-18/17-32/27 with steps 18/17-28/25-27/16, and its ~~inversions~~		* 1-18/17-32/27 with steps 18/17-28/25-27/16, and its inverse
	* 1-28/25-32/27 with steps 28/25-18/17-27/16~~, and its inversions~~		* 1-28/25-32/27 with steps 28/25-18/17-27/16.

	The known 27-odd-limit palingenetic tetrads are:		The known 27-odd-limit palingenetic tetrads are:
	* 1-18/17-32/27-8/5 with steps 18/17-28/25-27/20-5/4, and its ~~inversions~~		* 1-18/17-32/27-8/5 with steps 18/17-28/25-27/20-5/4, and its inverse
	* 1-28/25-32/27-40/27 with steps 28/25-18/17-5/4-27/20~~, and its inversions~~		* 1-28/25-32/27-40/27 with steps 28/25-18/17-5/4-27/20

	[[Category:17-limit]]		[[Category:21-odd-limit]]
	[[Category:Essentially tempered chords]]		[[Category:Essentially tempered chords]]
			[[Category:Triad]]
			[[Category:Tetrad]]
			[[Category:Pentad]]
	[[Category:Palingenetic]]		[[Category:Palingenetic]]

	{{~~todo~~\|review\|expand}}		{{Todo\| review \| expand }}

FloraC: -redundant category

2023-04-28T07:27:11Z

-redundant category

← Older revision		Revision as of 07:27, 28 April 2023
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	[[Category:17-limit]]		[[Category:17-limit]]
	[[Category:~~Chords]]~~		[[Category:Essentially tempered chords]]
	~~[[Category:Dyadic~~]]
	[[Category:Palingenetic]]		[[Category:Palingenetic]]

	{{todo\|review\|expand}}		{{todo\|review\|expand}}

Aura at 01:49, 13 May 2022

2022-05-13T01:49:27Z

← Older revision		Revision as of 01:49, 13 May 2022
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	The known 27-odd-limit palingenetic tetrads are:		The known 27-odd-limit palingenetic tetrads are:
	* 1-18/17-32/27-8/5 with steps 18/17-28/25-27/20-5/4, and its inversions		* 1-18/17-32/27-8/5 with steps 18/17-28/25-27/20-5/4, and its inversions
	* 1-28/25-32/27 with steps 28/25-18/17-5/4-27/20, and its inversions		* 1-28/25-32/27-40/27 with steps 28/25-18/17-5/4-27/20, and its inversions

	[[Category:17-limit]]		[[Category:17-limit]]

Aura: it's about time I corrected some of the stuff on here...

2022-05-13T01:44:26Z

it's about time I corrected some of the stuff on here...

← Older revision		Revision as of 01:44, 13 May 2022
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	* 1-34/27-3/2 with steps 34/27-25/21-4/3, and its inversions; dubbed the "palingenetic major triad"		* 1-34/27-3/2 with steps 34/27-25/21-4/3, and its inversions; dubbed the "palingenetic major triad"
	* 1-25/21-3/2 with steps 25/21-34/27-4/3, and its inversions; dubbed the "palingenetic minor triad"		* 1-25/21-3/2 with steps 25/21-34/27-4/3, and its inversions; dubbed the "palingenetic minor triad"
			* 1-18/17-32/27 with steps 18/17-28/25-27/16, and its inversions
			* 1-28/25-32/27 with steps 28/25-18/17-27/16, and its inversions

	The known 27-odd-limit palingenetic tetrads are:		The known 27-odd-limit palingenetic tetrads are:
	* 1-18/17-32/27-5/4 with steps 18/17-28/25-~~135~~/~~128~~-8/5, and its inversions		* 1-18/17-32/27-8/5 with steps 18/17-28/25-27/20-5/4, and its inversions
			* 1-28/25-32/27 with steps 28/25-18/17-5/4-27/20, and its inversions

	[[Category:17-limit]]		[[Category:17-limit]]

FloraC: +21-odd-limit triad

2021-02-15T16:14:50Z

+21-odd-limit triad

← Older revision		Revision as of 16:14, 15 February 2021
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	A '''palingenetic chord''' is an [[Dyadic chord\|essentially tempered dyadic chord]] tempered by [[1701/1700]], the palingenetic comma. Currently, there are only a handful of palingenetic chords known~~. Assuming we stick to the 27-odd-limit, we have two known triads and one known tetrad~~.		A '''palingenetic chord''' is an [[Dyadic chord\|essentially tempered dyadic chord]] tempered by [[1701/1700]], the palingenetic comma. Currently, there are only a handful of palingenetic chords known.

	The known palingenetic triads are:		There is a 21-odd-limit palingenetic triad:
			* 1-10/9-9/5 with steps 10/9-34/21-10/9.

			Assuming we stick to the 27-odd-limit, we have two known triads and one known tetrad.

			The known 27-odd-limit palingenetic triads are:
	* 1-34/27-3/2 with steps 34/27-25/21-4/3, and its inversions; dubbed the "palingenetic major triad"		* 1-34/27-3/2 with steps 34/27-25/21-4/3, and its inversions; dubbed the "palingenetic major triad"
	* 1-25/21-3/2 with steps 25/21-34/27-4/3, and its inversions; dubbed the "palingenetic minor triad"		* 1-25/21-3/2 with steps 25/21-34/27-4/3, and its inversions; dubbed the "palingenetic minor triad"

	The known palingenetic tetrads are:		The known 27-odd-limit palingenetic tetrads are:
	* 1-18/17-32/27-5/4 with steps 18/17-28/25-135/128-8/5, and its inversions		* 1-18/17-32/27-5/4 with steps 18/17-28/25-135/128-8/5, and its inversions

← Older revision		Revision as of 15:44, 26 July 2023
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	A '''palingenetic chord''' is an [[essentially tempered dyadic chord]] tempered by [[1701/1700]], the palingenetic comma.		A '''palingenetic chord''' is an [[essentially tempered dyadic chord]] tempered by [[1701/1700]], the palingenetic comma.

	~~The most typical palingenetic triad is a palindrome~~ in the 2.3.5.7.17 [[subgroup]] [[21-odd-limit]] since it identifies [[21/17]] by a stack of two [[10/9]]'s:		Palingenetic chords are of [[Dyadic chord/Pattern of essentially tempered chords\|pattern 1a]] in the 2.3.5.7.17 [[subgroup]] [[21-odd-limit]], meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures.

			The most typical palingenetic triad is a palindrome since it identifies [[21/17]] by a stack of two [[10/9]]'s:
	* 1-10/9-21/17 chord with steps 10/9-10/9-34/21.		* 1-10/9-21/17 chord with steps 10/9-10/9-34/21.

Line 8:		Line 10:
	* 1-21/20-10/9 with steps 21/20-18/17-9/5.		* 1-21/20-10/9 with steps 21/20-18/17-9/5.

	They can be extended to the following tetrads, with two palindromic chords and two pairs of chords in inverse relationship. The palindromic ~~chords~~ are		They can be extended to the following tetrads, with two palindromic chords and two pairs of chords in inverse relationship. The palindromic tetrads are
	* 1-18/17-10/9-20/17 chord with steps 18/17-21/20-18/17-17/10;		* 1-18/17-10/9-20/17 chord with steps 18/17-21/20-18/17-17/10;
	* 1-21/20-10/9-7/6 chord with steps 21/20-18/17-21/20-12/7.		* 1-21/20-10/9-7/6 chord with steps 21/20-18/17-21/20-12/7.

	The inversely related pairs of ~~chords~~ are		The inversely related pairs of tetrads are
	* 1-10/9-20/17-21/17 with steps 10/9-18/17-21/20-34/21 and its inverse		* 1-10/9-20/17-21/17 with steps 10/9-18/17-21/20-34/21 and its inverse
	* 1-21/20-10/9-21/17 with steps 21/20-18/17-10/9-34/21;		* 1-21/20-10/9-21/17 with steps 21/20-18/17-10/9-34/21;
Line 18:		Line 20:
	* 1-18/17-10/9-21/17 with steps 18/17-21/20-10/9-34/21.		* 1-18/17-10/9-21/17 with steps 18/17-21/20-10/9-34/21.

	Then there ~~are two~~ inversely related pentads:		Then there is an inversely related pair of pentads:
	* 1-18/17-10/9-20/17-21/17 with steps 18/17-21/20-18/17-21/20-34/21, and its inverse		* 1-18/17-10/9-20/17-21/17 with steps 18/17-21/20-18/17-21/20-34/21, and its inverse
	* 1-21/20-10/9-7/6-21/17 with steps 21/20-18/17-21/20-18/17-34/21.		* 1-21/20-10/9-7/6-21/17 with steps 21/20-18/17-21/20-18/17-34/21.