Quadrantonismic chords

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Quadrantonismic chords are essentially tempered dyadic chords tempered by the quadrantonisma, 1156/1155.

Quadrantonismic chords are of pattern 2 in the 2.3.5.7.11.17 subgroup 17-odd-limit, meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures.

For triads, there are three pairs of chords in inverse relationship:

  • 1-17/11-7/4 with steps 17/11-17/15-8/7 and its inverse
  • 1-17/15-7/4 with steps 17/15-17/11-8/7;
  • 1-17/11-15/8 with steps 17/11-17/14-16/15 and its inverse
  • 1-17/14-15/8 with steps 17/14-17/11-15/8;
  • 1-17/14-11/8 with steps 17/14-17/15-16/11 and its inverse
  • 1-17/15-11/8 with steps 17/15-17/14-16/11.

For tetrads, there are three palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are

  • 1-17/15-17/11-7/4 with steps 17/15-15/11-17/15-8/7;
  • 1-17/14-17/11-15/8 with steps 17/14-14/11-17/14-16/15;
  • 1-17/15-17/14-11/8 with steps 17/15-15/14-17/15-16/11.

The inversely related pairs of chords are

  • 1-17/11-7/4-15/8 with steps 17/11-17/15-15/14-16/15 and its inverse
  • 1-17/11-28/17-30/17 with steps 17/11-16/15-15/14-17/15;
  • 1-17/15-11/8-7/4 with steps 17/15-17/14-14/11-8/7 and its inverse
  • 1-17/14-11/8-11/7 with steps 17/14-17/15-8/7-14/11;
  • 1-17/14-11/8-15/8 with steps 17/14-17/15-15/11-16/15 and its inverse
  • 1-17/15-11/8-22/15 with steps 17/15-17/14-16/15-15/11;
  • 1-17/16-17/15-7/4 with steps 17/16-16/15-17/11-8/7 and its inverse
  • 1-17/11-28/17-7/4 with steps 17/11-16/15-17/16-8/7;
  • 1-17/15-22/17-11/8 with steps 17/15-8/7-17/16-16/11 and its inverse
  • 1-17/16-17/14-11/8 with steps 17/16-8/7-17/15-16/11;
  • 1-17/14-22/17-11/8 with steps 17/14-16/15-17/16-16/11 and its inverse
  • 1-17/16-17/15-11/8 with steps 17/16-16/15-17/14-16/11.

For pentads, there are three pairs of chords in inverse relationship:

  • 1-17/15-17/14-22/17-11/8 with steps 17/15-15/14-16/15-17/16-16/11 and its inverse
  • 1-17/16-17/15-17/14-11/8 with steps 17/16-16/15-15/14-17/15-16/11;
  • 1-17/16-17/14-11/8-15/8 with steps 17/16-8/7-17/15-15/11-16/15 and its inverse
  • 1-17/15-22/17-11/8-22/15 with steps 17/15-8/7-17/16-16/15-15/11;
  • 1-17/16-17/15-11/8-7/4 with steps 17/16-16/15-17/14-14/11-8/7 and its inverse
  • 1-17/14-22/17-11/8-11/7 with steps 17/14-16/15-17/16-8/7-14/11.

If we are willing to go to the 21-odd-limit, There are four additional pairs of triads of inverse relationship:

  • 1-5/4-17/11 with steps 5/4-21/17-22/17 and its inverse
  • 1-5/4-34/21 with steps 5/4-22/17-21/17;
  • 1-21/16-17/11 with steps 21/16-20/17-22/17 and its inverse
  • 1-21/16-17/10 with steps 21/16-22/17-20/17;
  • 1-11/8-17/10 with steps 11/8-21/17-20/17 and its inverse
  • 1-11/8-34/21 with steps 11/8-20/17-21/17;
  • 1-12/11-30/17 with steps 12/11-34/21-17/15 and its inverse
  • 1-12/11-21/17 with steps 12/11-17/15-34/21.

They can be extended to the following palindromic tetrads:

  • 1-5/4-17/11-34/21 with steps 5/4-21/17-22/21-21/17;
  • 1-21/16-17/11-17/10 with steps 21/16-20/17-11/10-20/17;
  • 1-11/8-34/21-17/10 with steps 11/8-20/17-21/20-20/17;
  • 1-12/11-21/17-30/17 with steps 12/11-17/15-10/7-17/15.

As well as the following additional pairs of triads of inverse relationship:

  • 1-17/14-3/2-30/17 with steps 17/14-21/17-20/17-17/15 and its inverse
  • 1-21/17-3/2-17/10 with steps 21/17-17/14-17/15-20/17;
  • 1-17/14-3/2-15/8 with steps 17/14-21/17-5/4-16/15 and its inverse
  • 1-21/17-3/2-8/5 with steps 21/17-17/14-16/15-5/4;
  • 1-21/16-3/2-17/10 with steps 21/16-8/7-17/15-20/17 and its inverse
  • 1-8/7-3/2-30/17 with steps 8/7-21/16-20/17-17/15;
  • 1-17/14-11/8-3/2 with steps 17/14-17/15-12/11-4/3 and its inverse
  • 1-12/11-21/17-3/2 with steps 12/11-17/15-17/14-4/3;
  • 1-11/8-3/2-17/10 with steps 11/8-12/11-17/15-20/17 and its inverse
  • 1-12/11-3/2-30/17 with step 12/11-11/8-20/17-17/15;
  • 1-5/4-17/11-7/4 with steps 5/4-21/17-17/15-8/7 and its inverse
  • 1-21/17-17/11-30/17 with steps 21/17-5/4-8/7-17/15;
  • 1-21/16-17/11-15/8 with steps 21/16-20/17-17/14-16/15 and its inverse
  • 1-20/17-17/11-28/17 with steps 20/17-21/16-16/15-17/14;
  • 1-5/4-21/16-17/11 with steps 5/4-21/20-20/17-22/17 and its inverse
  • 1-20/17-21/17-17/11 with steps 20/17-21/20-5/4-22/17;
  • 1-17/14-11/8-17/10 with steps 17/14-17/15-21/17-20/17 and its inverse
  • 1-17/15-11/8-34/21 with steps 17/15-17/14-20/17-21/17;
  • 1-5/4-11/8-34/21 with steps 5/4-11/10-20/17-21/17 and its inverse
  • 1-11/10-11/8-17/10 with steps 11/10-5/4-21/17-20/17;
  • 1-21/16-11/8-17/10 with steps 21/16-22/21-21/17-20/17 and its inverse
  • 1-22/21-11/8-34/21 with steps 22/21-21/16-20/17-21/17.

For pentads, there are

  • 1-17/14-3/2-30/17-15/8 with steps 17/14-21/17-20/17-17/16-16/15 and its inverse
  • 1-21/17-3/2-8/5-17/10 with steps 21/17-17/14-16/15-17/16-20/17;
  • 1-21/17-21/16-3/2-17/10 with steps 21/17-17/16-8/7-17/15-20/17 and its inverse
  • 1-8/7-17/14-3/2-30/17 with steps 8/7-17/16-21/17-20/17-17/15;
  • 1-17/14-11/8-3/2-15/8 with steps 17/14-17/15-12/11-5/4-16/15 and its inverse
  • 1-12/11-21/17-3/2-8/5 with steps 12/11-17/15-17/14-16/15-5/4;
  • 1-21/16-11/8-3/2-17/10 with steps 21/16-22/21-12/11-17/15-20/17 and its inverse
  • 1-12/11-8/7-3/2-30/17 with steps 12/11-22/21-21/16-20/17-17/15;
  • 1-17/14-11/8-3/2-17/10 with steps 17/14-17/15-12/11-17/15-20/17 and its inverse
  • 1-12/11-21/17-3/2-30/17 with steps 12/11-17/15-17/14-20/17-17/15.

Equal temperaments with quadrantonismic chords include 22, 26, 43, 46, 50, 68, 72, 89, 94, 111, 118, 121, 140, 183, 239, 311, 400, 422 and 494.

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