Palingenetic chords: Difference between revisions

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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:

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.

* 1-10/9-21/17 with steps 10/9-10/9-34/21.

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.

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.

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 tetrads 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 18/17-21/20-18/17-17/10;

* 1-21/20-10/9-7/6 chord with steps 21/20-18/17-21/20-12/7.

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

* 1-21/20-10/9-21/17 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

* 1-10/9-7/6-21/17 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 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

* 1-21/20-10/9-7/6-21/17 with steps 21/20-18/17-21/20-18/17-34/21.

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* 1-18/17-32/27-8/5 with steps 18/17-28/25-27/20-5/4, and its inverse

* 1-28/25-32/27-40/27 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:Essentially tempered chords]]

[[Category:~~Triad~~]]

[[Category:Triads]]

[[Category:~~Tetrad~~]]

[[Category:Tetrads]]

[[Category:~~Pentad~~]]

[[Category:Pentads]]

[[Category:Palingenetic]]

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

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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:
+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.
-* 1-10/9-21/17 with steps 10/9-10/9-34/21.
+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.
 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.
-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 tetrads 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 18/17-21/20-18/17-17/10;
+* 1-21/20-10/9-7/6 chord with steps 21/20-18/17-21/20-12/7.
+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
 * 1-21/20-10/9-21/17 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
+* 1-10/9-7/6-21/17 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 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
 * 1-21/20-10/9-7/6-21/17 with steps 21/20-18/17-21/20-18/17-34/21.
@@ Line 29: / Line 35: @@
 * 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
+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:Essentially tempered chords]]
-[[Category:Triad]]
+[[Category:Triads]]
-[[Category:Tetrad]]
+[[Category:Tetrads]]
-[[Category:Pentad]]
+[[Category:Pentads]]
 [[Category:Palingenetic]]
-{{Todo| review | expand }}