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Ursulismic chords

Ursulismic chords are essentially tempered chords tempered by the ursulisma, 375/374.

There are 8 triads, 28 tetrads, 26 pentads and 7 hexads as 2.3.5.11.17 subgroup 17-odd-limit essentially tempered chords.

The ursulismic triads include the following inversely related pairs:

  • 1–5/4–22/15 with steps of 5/4, 20/17, 15/11, and its inverse
  • 1–20/17–22/15 with steps of 20/17, 5/4, 15/11;
  • 1–6/5–15/11 with steps of 6/5, 17/15, 22/15, and its inverse
  • 1–17/15–15/11 with steps of 17/15, 6/5, 22/15;
  • 1–17/15–5/4 with steps of 17/15, 11/10, 8/5, and its inverse
  • 1–11/10–5/4 with steps of 11/10, 17/15, 8/5;
  • 1–11/10–20/17 with steps of 11/10, 16/15, 17/10, and its inverse
  • 1–16/15–20/17 with steps of 16/15, 11/10, 17/10.

For tetrads, there are eight palindromic chords and ten pairs of chords in inverse relationship. The palindromic chords are

  • 1–6/5–15/11–18/11 with steps of 6/5, 17/15, 6/5, 11/9;
  • 1–20/17–22/15–8/5 with steps of 20/17, 5/4, 12/11, 5/4;
  • 1–20/17–5/4–22/15 with steps of 20/17, 17/16, 20/17, 15/11;
  • 1–17/15–5/4–17/12 with steps of 17/15, 11/10, 17/15, 24/17;
  • 1–11/10–5/4–11/8 with steps of 11/10, 17/15, 11/10, 16/11;
  • 1–17/15–6/5–15/11 with steps of 17/15, 18/17, 17/15, 22/15;
  • 1–11/10–20/17–22/17 with steps of 11/10, 16/15, 11/10, 17/11;
  • 1–17/15–15/11–17/11 with steps of 17/15, 6/5, 17/15, 22/17.

The inversely related pairs of chords are

  • 1–5/4–3/2–17/10 with steps of 5/4, 6/5, 17/15, 20/17, and its inverse
  • 1–6/5–3/2–30/17 with steps of 6/5, 5/4, 20/17, 17/15;
  • 1–20/17–24/17–8/5 with steps of 20/17, 6/5, 17/15, 5/4, and its inverse
  • 1–17/15–15/11–8/5 with steps of 17/15, 6/5, 20/17, 5/4;
  • 1–6/5–15/11–3/2 with steps of 6/5, 17/15, 11/10, 4/3, and its inverse
  • 1–11/10–5/4–3/2 with steps of 11/10, 17/15, 6/5, 4/3;
  • 1–15/11–3/2–8/5 with steps of 15/11, 11/10, 16/15, 5/4, and its inverse
  • 1–11/10–3/2–15/8 with steps of 11/10, 15/11, 5/4, 16/15;
  • 1–5/4–11/8–22/15 with steps of 5/4, 11/10, 16/15, 15/11, and its inverse
  • 1–16/15–20/17–22/15 with steps of 16/15, 11/10, 5/4, 15/11;
  • 1–20/17–22/17–22/15 with steps of 20/17, 11/10, 17/15, 15/11, and its inverse
  • 1–17/15–5/4–22/15 with steps of 17/15, 11/10, 20/17, 15/11;
  • 1–15/11–3/2–17/10 with steps of 15/11, 11/10, 17/15, 20/17, and its inverse
  • 1–11/10–3/2–30/17 with steps of 11/10, 15/11, 20/17, 17/15;
  • 1–17/15–5/4–15/11 with steps of 17/15, 11/10, 12/11, 22/15, and its inverse
  • 1–12/11–6/5–15/11 with steps of 12/11, 11/10, 17/15, 22/15;
  • 1–17/15–5/4–4/3 with steps of 17/15, 11/10, 16/15, 3/2, and its inverse
  • 1–16/15–20/17–4/3 with steps of 16/15, 11/10, 17/15, 3/2;
  • 1–11/10–20/17–5/4 with steps of 11/10, 16/15, 17/16, 8/5, and its inverse
  • 1–17/16–17/15–5/4 with steps of 17/16, 16/15, 11/10, 8/5.

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

  • 1–6/5–15/11–3/2–17/10 with steps of 6/5, 17/15, 11/10, 17/15, 20/17, and its inverse
  • 1–11/10–5/4–3/2–30/17 with steps of 11/10, 17/15, 6/5, 20/17, 17/15;
  • 1–6/5–24/17–3/2–30/17 with steps of 6/5, 20/17, 17/16, 20/17, 17/15, and its inverse
  • 1–17/16–5/4–3/2–17/10 with steps of 17/16, 20/17, 6/5, 17/15, 20/17;
  • 1–6/5–15/11–3/2–30/17 with steps of 6/5, 17/15, 11/10, 20/17, 17/15, and its inverse
  • 1–11/10–5/4–3/2–17/10 with steps of 11/10, 17/15, 6/5, 17/15, 20/17;
  • 1–6/5–15/11–3/2–18/11 with steps of 6/5, 17/15, 11/10, 12/11, 11/9, and its inverse
  • 1–11/10–5/4–3/2–11/6 with steps of 11/10, 17/15, 6/5, 11/9, 12/11;
  • 1–5/4–3/2–17/10–15/8 with steps of 5/4, 6/5, 17/15, 11/10, 16/15, and its inverse
  • 1–6/5–3/2–8/5–30/17 with steps of 6/5, 5/4, 16/15, 11/10, 17/15;
  • 1–5/4–17/12–3/2–17/10 with steps of 5/4, 17/15, 18/17, 17/15, 20/17, and its inverse
  • 1–18/17–6/5–3/2–30/17 with steps of 18/17, 17/15, 5/4, 20/17, 17/15;
  • 1–5/4–15/11–3/2–17/10 with steps of 5/4, 12/11, 11/10, 17/15, 20/17, and its inverse
  • 1–11/10–6/5–3/2–30/17 with steps of 11/10, 12/11, 5/4, 20/17, 17/15;
  • 1–11/10–11/8–3/2–15/8 with steps of 11/10, 5/4, 12/11, 5/4, 16/15, and its inverse
  • 1–12/11–15/11–3/2–8/5 with steps of 12/11, 5/4, 11/10, 16/15, 5/4;
  • 1–15/11–3/2–8/5–17/10 with steps of 15/11, 11/10, 16/15, 17/16, 20/17, and its inverse
  • 1–11/10–3/2–30/17–15/8 with steps of 11/10, 15/11, 20/17, 17/16, 16/15;
  • 1–15/11–3/2–8/5–30/17 with steps of 15/11, 11/10, 16/15, 11/10, 17/15, and its inverse
  • 1–11/10–3/2–17/10–15/8 with steps of 11/10, 15/11, 17/15, 11/10, 16/15;
  • 1–15/11–3/2–17/10–15/8 with steps of 15/11, 11/10, 17/15, 11/10, 16/15, and its inverse
  • 1–11/10–3/2–8/5–30/17 with steps of 11/10, 15/11, 16/15, 11/10, 17/15;
  • 1–24/17–3/2–8/5–30/17 with steps of 24/17, 17/16, 16/15, 11/10, 17/15, and its inverse
  • 1–17/16–3/2–17/10–15/8 with steps of 17/16, 24/17, 17/15, 11/10, 16/15;
  • 1–11/10–5/4–11/8–3/2 with steps of 11/10, 17/15, 11/10, 12/11, 4/3, and its inverse
  • 1–12/11–6/5–15/11–3/2 with steps of 12/11, 11/10, 17/15, 11/10, 4/3.

For hexads, there are three palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are

  • 1–17/16–5/4–17/12–3/2–17/10 with steps of 17/16, 20/17, 17/15, 18/17, 17/15, 20/17;
  • 1–11/10–5/4–3/2–17/10–15/8 with steps of 11/10, 17/15, 6/5, 17/15, 11/10, 16/15;
  • 1–11/10–5/4–11/8–3/2–11/6 with steps of 11/10, 17/15, 11/10, 12/11, 11/9, 12/11.

The inversely related pairs of chords are

  • 1–6/5–24/17–3/2–8/5–30/17 with steps of 6/5, 20/17, 17/16, 16/15, 11/10, 17/15, and its inverse
  • 1–17/16–5/4–3/2–17/10–15/8 with steps of 17/16, 20/17, 6/5, 17/15, 11/10, 16/15;
  • 1–5/4–15/11–3/2–17/10–15/8 with steps of 5/4, 12/11, 11/10, 17/15, 11/10, 16/15, and its inverse
  • 1–11/10–6/5–3/2–8/5–30/17 with steps of 11/10, 12/11, 5/4, 16/15, 11/10, 17/15.

Equal temperaments with ursulismic chords include 9, 12, 16, 22, 31, 34, 38, 65, 72, 87, 94, 137 and 159.

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