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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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