Palingenetic chords: Difference between revisions

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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:

* 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:

* 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:Triad]]

[[Category:Tetrad]]

[[Category:Pentad]]

[[Category:Palingenetic]]

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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:
-* 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:
-* 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:Triad]]
+[[Category:Tetrad]]
+[[Category:Pentad]]
 [[Category:Palingenetic]]
-{{todo|review|expand}}
+{{Todo| review | expand }}