Palingenetic chords

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.