Palingenetic chords: Difference between revisions

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A '''~~palingenic~~ chord''' is an [[~~Dyadic chord|~~essentially tempered dyadic chord]] tempered by [[1701/1700]], the ~~palingenesis~~ comma~~. Currently, there are only a handful of palingenic chords known. Assuming we stick to the 27-odd-limit, we have two known triads and one known tetrad~~.

A '''palingenetic chord''' is an [[essentially tempered dyadic chord]] tempered by [[1701/1700]], the palingenetic comma.

~~The known palingenic triads~~ are:

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.

* 1-34/~~27-~~3~~/2 with steps 34/27~~-~~25/21~~-4/3, and ~~its inversions; dubbed the "palingenic major triad"~~

* 1-25/21-3/2 ~~with steps 25/21-34/27-4/3~~, ~~and its inversions; dubbed the "palingenic minor triad"~~

The ~~known palingenic tetrads are~~:

The most typical palingenetic triad is a palindrome since it identifies [[21/17]] by a stack of two [[10/9]]'s:

* 1-18/17-32/~~27-5/4~~ with steps ~~18/17-28~~/25-~~135~~/~~128~~-8/~~5, and its inversions~~

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

[[Category:17-limit]]

There is an inversely related pair which is even more squeezed:

[[Category:~~Chords~~]]

* 1-18/17-10/9 with steps 18/17-21/20-9/5 and its inverse

[[Category:~~Dyadic~~]]

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

[[Category:~~Palingenesis~~]]

[[Category:~~todo~~]]

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;

* 1-21/20-10/9-7/6 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

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

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.

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, dubbed the "palingenetic major triad", and its inverse

* 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 inverse

* 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 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:Triads]]

[[Category:Tetrads]]

[[Category:Pentads]]

[[Category:Palingenetic]]

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-A '''palingenic chord''' is an [[Dyadic chord|essentially tempered dyadic chord]] tempered by [[1701/1700]], the palingenesis comma.  Currently, there are only a handful of palingenic chords known.  Assuming we stick to the 27-odd-limit, we have two known triads and one known tetrad.
+A '''palingenetic chord''' is an [[essentially tempered dyadic chord]] tempered by [[1701/1700]], the palingenetic comma.
-The known palingenic triads are:
+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.
-* 1-34/27-3/2 with steps 34/27-25/21-4/3, and its inversions; dubbed the "palingenic major triad"
-* 1-25/21-3/2 with steps 25/21-34/27-4/3, and its inversions; dubbed the "palingenic minor triad"
-The known palingenic tetrads are:
+The most typical palingenetic triad is a palindrome since it identifies [[21/17]] by a stack of two [[10/9]]'s:
-* 1-18/17-32/27-5/4 with steps 18/17-28/25-135/128-8/5, and its inversions
+* 1-10/9-21/17 chord with steps 10/9-10/9-34/21.
-[[Category:17-limit]]
+There is an inversely related pair which is even more squeezed:
-[[Category:Chords]]
+* 1-18/17-10/9 with steps 18/17-21/20-9/5 and its inverse
-[[Category:Dyadic]]
+* 1-21/20-10/9 with steps 21/20-18/17-9/5.
-[[Category:Palingenesis]]
-[[Category:todo]]
+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;
+* 1-21/20-10/9-7/6 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
+* 1-18/17-10/9-21/17 with steps 18/17-21/20-10/9-34/21.
+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.
+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, dubbed the "palingenetic major triad", and its inverse
+* 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 inverse
+* 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 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:Triads]]
+[[Category:Tetrads]]
+[[Category:Pentads]]
+[[Category:Palingenetic]]