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.
 
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 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]]

Latest revision as of 15:44, 26 July 2023

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

Palingenetic chords are of 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.

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 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 12, 46, 53, 58, 60, 72, 99, 111, 118, 171, 183, 243, 270, 289, 354, 400, 472 and 571.

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