Island chords

From Xenharmonic Wiki
Jump to navigation Jump to search

Island chords are essentially tempered chords tempered by the island comma, 676/675.

There are 9 triads, 37 tetrads, 51 pentads, 29 hexads and 6 heptads as 2.3.5.13 subgroup 15-odd-limit essentially tempered chords.

For triads, there are one palindromic chord and four pairs of chords in inverse relationship.

The palindromic triad consists of two semifourths and one perfect fifth, splitting a fourth in two:

  • 1–15/13–4/3 with steps of 15/13, 15/13, 3/2.

The inversely related pairs of chords are

  • 1–5/4–13/9 with steps of 5/4, 15/13, 18/13, and its inverse
  • 1–15/13–13/9 with steps of 15/13, 5/4, 18/13;
  • 1–13/10–18/13 with steps of 13/10, 16/15, 13/9, and its inverse
  • 1–16/15–18/13 with steps of 16/15, 13/10, 13/9;
  • 1–15/13–13/10 with steps of 15/13, 9/8, 20/13, and its inverse
  • 1–9/8–13/10 with steps of 9/8, 15/13, 20/13;
  • 1–13/12–15/13 with steps of 13/12, 16/15, 26/15, and its inverse
  • 1–16/15–15/13 with steps of 13/12, 16/15, 26/15.

For tetrads, there are seven palindromic chords and fifteen pairs of chords in inverse relationship.

One of the palindromic tetrads consists of three semifourths and one semisixth,

  • 1–13/10–3/2–26/15 with steps of 13/10, 15/13, 15/13, 15/13.

Aside from above, the following palindromic tetrad also contains a barbados triad (otonal 1–13/10–3/2 chord) and its inversion (utonal 1–15/13–3/2 chord),

  • 1–15/13–13/10–3/2 with steps of 15/13, 9/8, 15/13, 4/3.

The rest five palindromic tetrads are

  • 1–15/13–13/9–5/3 with steps of 15/13, 5/4, 15/13, 6/5;
  • 1–10/9–13/9–20/13 with steps of 10/9, 13/10, 16/15, 13/10;
  • 1–15/13–5/4–13/9 with steps of 15/13, 13/12, 15/13, 18/13;
  • 1–13/12–15/13–5/4 with steps of 13/12, 16/15, 13/12, 8/5;
  • 1–16/15–15/13–16/13 with steps of 16/15, 13/12, 16/15, 13/8.

The inversely related pairs of chords are

  • 1–9/8–13/10–13/8 with steps of 9/8, 15/13, 5/4, 16/13, and its inverse
  • 1–5/4–13/9–13/8 with steps of 5/4, 15/13, 9/8, 16/13;
  • 1–5/4–3/2–26/15 with steps of 5/4, 6/5, 15/13, 15/13, and its inverse
  • 1–6/5–3/2–26/15 with steps of 6/5, 5/4, 15/13, 15/13;
  • 1–5/4–13/9–20/13 with steps of 5/4, 15/13, 16/15, 13/10, and its inverse
  • 1–5/4–13/8–26/15 with steps of 5/4, 13/10, 16/15, 15/13;
  • 1–13/10–3/2–15/8 with steps of 13/10, 15/13, 5/4, 16/15, and its inverse
  • 1–15/13–3/2–8/5 with steps of 15/13, 13/10, 16/15, 5/4;
  • 1–13/10–18/13–3/2 with steps of 13/10, 16/15, 13/12, 4/3, and its inverse
  • 1–13/12–15/13–3/2 with steps of 13/12, 16/15, 13/10, 4/3;
  • 1–15/13–4/3–3/2 with steps of 15/13, 15/13, 9/8, 4/3, and its inverse
  • 1–9/8–13/10–3/2 with steps of 9/8, 15/13, 15/13, 4/3;
  • 1–18/13–3/2–8/5 with steps of 18/13, 13/12, 16/15, 5/4, and its inverse
  • 1–13/12–3/2–15/8 with steps of 13/12, 18/13, 5/4, 16/15;
  • 1–15/13–13/10–13/9 with steps of 15/13, 9/8, 10/9, 18/13, and its inverse
  • 1–10/9–5/4–13/9 with steps of 10/9, 9/8, 15/13, 18/13;
  • 1–18/13–3/2–26/15 with steps of 18/13, 13/12, 15/13, 15/13, and its inverse
  • 1–13/12–3/2–26/15 with steps of 13/12, 18/13, 15/13, 15/13;
  • 1–6/5–13/10–18/13 with steps of 6/5, 13/12, 16/15, 13/9, and its inverse
  • 1–16/15–15/13–18/13 with steps of 16/15, 13/12, 6/5, 13/9;
  • 1–15/13–13/10–18/13 with steps of 15/13, 9/8, 16/15, 13/9, and its inverse
  • 1–16/15–6/5–18/13 with steps of 16/15, 9/8, 15/13, 13/9;
  • 1–9/8–13/10–18/13 with steps of 9/8, 15/13, 16/15, 13/9, and its inverse
  • 1–16/15–16/13–18/13 with steps of 16/15, 15/13, 9/8, 13/9;
  • 1–15/13–16/13–4/3 with steps of 15/13, 16/15, 13/12, 3/2, and its inverse
  • 1–13/12–15/13–4/3 with steps of 13/12, 16/15, 15/13, 3/2;
  • 1–15/13–5/4–4/3 with steps of 15/13, 13/12, 16/15, 3/2, and its inverse
  • 1–16/15–15/13–4/3 with steps of 16/15, 13/12, 15/13, 3/2;
  • 1–9/8–6/5–13/10 with steps of 9/8, 16/15, 13/12, 20/13, and its inverse
  • 1–13/12–15/13–13/10 with steps of 13/12, 16/15, 9/8, 20/13.

For pentads, there are one palindromic chord and twenty-five pairs of chords in inverse relationship.

The palindromic pentad consists of four semifourths and one whole tone,

  • 1–9/8–13/10–3/2–26/15 with steps of 9/8, 15/13, 15/13, 15/13, 15/13.

The inversely related pairs of chords are

  • 1–6/5–18/13–3/2–26/15 with steps of 6/5, 15/13, 13/12, 15/13, 15/13, and its inverse
  • 1–13/12–5/4–3/2–26/15 with steps of 13/12, 15/13, 6/5, 15/13, 15/13;
  • 1–9/8–5/4–3/2–26/15 with steps of 9/8, 10/9, 6/5, 15/13, 15/13, and its inverse
  • 1–6/5–4/3–3/2–26/15 with steps of 6/5, 10/9, 9/8, 15/13, 15/13;
  • 1–6/5–18/13–3/2–26/15 with steps of 6/5, 13/12, 15/13, 15/13, 15/13, and its inverse
  • 1–15/13–5/4–3/2–26/15 with steps of 15/13, 13/12, 6/5, 15/13, 15/13;
  • 1–9/8–18/13–3/2–26/15 with steps of 9/8, 16/13, 13/12, 15/13, 15/13, and its inverse
  • 1–9/8–13/10–3/2–13/8 with steps of 9/8, 15/13, 15/13, 13/12, 16/13;
  • 1–5/4–3/2–8/5–26/15 with steps of 5/4, 6/5, 16/15, 13/12, 15/13, and its inverse
  • 1–6/5–3/2–26/15–15/8 with steps of 6/5, 5/4, 15/13, 13/12, 16/15;
  • 1–5/4–3/2–13/8–26/15 with steps of 5/4, 6/5, 13/12, 16/15, 15/13, and its inverse
  • 1–6/5–3/2–26/15–24/13 with steps of 6/5, 5/4, 15/13, 16/15, 13/12;
  • 1–15/13–13/9–5/3–15/8 with steps of 15/13, 5/4, 15/13, 9/8, 16/15, and its inverse
  • 1–9/8–13/10–13/8–15/8 with steps of 9/8, 15/13, 5/4, 15/13, 16/15;
  • 1–15/13–18/13–3/2–8/5 with steps of 15/13, 6/5, 13/12, 16/15, 5/4, and its inverse
  • 1–13/12–13/10–3/2–15/8 with steps of 13/12, 6/5, 15/13, 5/4, 16/15;
  • 1–5/4–4/3–3/2–26/15 with steps of 5/4, 16/15, 9/8, 15/13, 15/13, and its inverse
  • 1–9/8–6/5–3/2–26/15 with steps of 9/8, 16/15, 5/4, 15/13, 15/13;
  • 1–15/13–13/10–3/2–15/8 with steps of 15/13, 9/8, 15/13, 5/4, 16/15, and its inverse
  • 1–15/13–13/10–3/2–8/5 with steps of 15/13, 9/8, 15/13, 16/15, 5/4;
  • 1–13/10–18/13–3/2–8/5 with steps of 13/10, 16/15, 13/12, 16/15, 5/4, and its inverse
  • 1–13/12–15/13–3/2–15/8 with steps of 13/12, 16/15, 13/10, 5/4, 16/15;
  • 1–13/10–18/13–3/2–9/5 with steps of 13/10, 16/15, 13/12, 6/5, 10/9, and its inverse
  • 1–13/12–15/13–3/2–5/3 with steps of 13/12, 16/15, 13/10, 10/9, 6/5;
  • 1–9/8–5/4–13/8–26/15 with steps of 9/8, 10/9, 13/10, 16/15, 15/13, and its inverse
  • 1–15/13–16/13–8/5–16/9 with steps of 15/13, 16/15, 13/10, 10/9, 9/8;
  • 1–13/10–3/2–13/8–26/15 with steps of 13/10, 15/13, 13/12, 16/15, 15/13, and its inverse
  • 1–13/10–3/2–8/5–26/15 with steps of 13/10, 15/13, 16/15, 13/12, 15/13;
  • 1–13/10–3/2–26/15–15/8 with steps of 13/10, 15/13, 15/13, 13/12, 16/15, and its inverse
  • 1–13/10–18/13–3/2–26/15 with steps of 13/10, 16/15, 13/12, 15/13, 15/13;
  • 1–13/10–3/2–13/8–15/8 with steps of 13/10, 15/13, 13/12, 15/13, 16/15, and its inverse
  • 1–15/13–3/2–8/5–24/13 with steps of 15/13, 13/10, 16/15, 15/13, 13/12;
  • 1–6/5–13/10–18/13–3/2 with steps of 6/5, 13/12, 16/15, 13/12, 4/3, and its inverse
  • 1–13/12–15/13–5/4–3/2 with steps of 13/12, 16/15, 13/12, 6/5, 4/3;
  • 1–15/13–5/4–4/3–3/2 with steps of 15/13, 13/12, 16/15, 9/8, 4/3, and its inverse
  • 1–9/8–6/5–13/10–3/2 with steps of 9/8, 16/15, 13/12, 15/13, 4/3;
  • 1–9/8–13/10–18/13–3/2 with steps of 9/8, 15/13, 16/15, 13/12, 4/3, and its inverse
  • 1–13/12–15/13–4/3–3/2 with steps of 13/12, 16/15, 15/13, 9/8, 4/3;
  • 1–15/13–13/10–18/13–3/2 with steps of 15/13, 9/8, 16/15, 13/12, 4/3, and its inverse
  • 1–13/12–15/13–13/10–3/2 with steps of 13/12, 16/15, 9/8, 15/13, 4/3;
  • 1–18/13–3/2–8/5–26/15 with steps of 18/13, 13/12, 16/15, 13/12, 15/13, and its inverse
  • 1–13/12–3/2–26/15–15/8 with steps of 13/12, 18/13, 15/13, 13/12, 16/15;
  • 1–18/13–3/2–8/5–9/5 with steps of 18/13, 13/12, 16/15, 9/8, 10/9, and its inverse
  • 1–13/12–3/2–5/3–15/8 with steps of 13/12, 18/13, 10/9, 9/8, 16/15;
  • 1–13/12–3/2–13/8–15/8 with steps of 13/12, 18/13, 13/12, 15/13, 16/15, and its inverse
  • 1–13/12–3/2–13/8–26/15 with steps of 13/12, 18/13, 13/12, 16/15, 15/13;
  • 1–9/8–6/5–13/10–18/13 with steps of 9/8, 16/15, 13/12, 16/15, 13/9, and its inverse
  • 1–16/15–15/13–16/13–18/13 with steps of 16/15, 13/12, 16/15, 9/8, 13/9;
  • 1–13/12–15/13–5/4–4/3 with steps of 13/12, 16/15, 13/12, 16/15, 3/2, and its inverse
  • 1–16/15–15/13–16/13–4/3 with steps of 16/15, 13/12, 16/15, 13/12, 3/2.

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

  • 1–13/12–5/4–3/2–26/15–15/8 with steps of 13/12, 15/13, 6/5, 15/13, 13/12, 16/15;
  • 1–13/12–15/13–5/4–3/2–5/3 with steps of 13/12, 16/15, 13/12, 6/5, 10/9, 6/5;
  • 1–13/12–3/2–13/8–26/15–15/8 with steps of 13/12, 18/13, 13/12, 16/15, 13/12, 16/15.

The inversely related pairs of chords are

  • 1–15/13–5/4–4/3–3/2–26/15 with steps of 15/13, 13/12, 16/15, 9/8, 15/13, 15/13, and its inverse
  • 1–9/8–6/5–13/10–3/2–26/15 with steps of 9/8, 16/15, 13/12, 15/13, 15/13, 15/13;
  • 1–9/8–13/10–3/2–13/8–26/15 with steps of 9/8, 15/13, 15/13, 13/12, 16/15, 15/13, and its inverse
  • 1–9/8–13/10–18/13–3/2–26/15 with steps of 9/8, 15/13, 16/15, 13/12, 15/13, 15/13;
  • 1–9/8–6/5–18/13–3/2–26/15 with steps of 9/8, 16/15, 15/13, 13/12, 15/13, 15/13, and its inverse
  • 1–13/12–5/4–4/3–3/2–26/15 with steps of 13/12, 15/13, 16/15, 9/8, 15/13, 15/13;
  • 1–6/5–4/3–3/2–8/5–26/15 with steps of 6/5, 10/9, 9/8, 16/15, 13/12, 15/13, and its inverse
  • 1–9/8–5/4–3/2–26/15–15/8 with steps of 9/8, 10/9, 6/5, 15/13, 13/12, 16/15;
  • 1–6/5–13/10–18/13–3/2–26/15 with steps of 6/5, 13/12, 16/15, 13/12, 15/13, 15/13, and its inverse
  • 1–13/12–15/13–5/4–3/2–26/15 with steps of 13/12, 16/15, 13/12, 6/5, 15/13, 15/13;
  • 1–6/5–18/13–3/2–26/15–24/13 with steps of 6/5, 15/13, 13/12, 15/13, 16/15, 13/12, and its inverse
  • 1–13/12–5/4–3/2–13/8–26/15 with steps of 13/12, 15/13, 6/5, 13/12, 16/15, 15/13;
  • 1–6/5–13/10–3/2–8/5–26/15 with steps of 6/5, 13/12, 15/13, 16/15, 13/12, 15/13, and its inverse
  • 1–15/13–5/4–3/2–26/15–15/8 with steps of 15/13, 13/12, 6/5, 15/13, 13/12, 16/15;
  • 1–6/5–4/3–3/2–26/15–24/13 with steps of 6/5, 10/9, 9/8, 15/13, 16/15, 13/12, and its inverse
  • 1–9/8–5/4–3/2–8/5–26/15 with steps of 9/8, 10/9, 6/5, 13/12, 16/15, 15/13;
  • 1–5/4–3/2–13/8–26/15–15/8 with steps of 5/4, 6/5, 13/12, 16/15, 13/12, 16/15, and its inverse
  • 1–6/5–3/2–8/5–26/15–24/13 with steps of 6/5, 5/4, 16/15, 13/12, 16/15, 13/12;
  • 1–15/13–13/10–18/13–3/2–8/5 with steps of 15/13, 9/8, 16/15, 13/12, 16/15, 5/4, and its inverse
  • 1–13/12–15/13–13/10–3/2–15/8 with steps of 13/12, 16/15, 9/8, 15/13, 5/4, 16/15;
  • 1–13/10–18/13–3/2–8/5–26/15 with steps of 13/10, 16/15, 13/12, 16/15, 13/12, 15/13, and its inverse
  • 1–13/12–15/13–3/2–26/15–15/8 with steps of 13/12, 16/15, 13/10, 15/13, 13/12, 16/15;
  • 1–13/10–18/13–3/2–8/5–9/5 with steps of 13/10, 16/15, 13/12, 16/15, 9/8, 10/9, and its inverse
  • 1–13/12–15/13–3/2–5/3–15/8 with steps of 13/12, 16/15, 13/10, 10/9, 9/8, 16/15;
  • 1–9/8–6/5–13/10–18/13–3/2 with steps of 9/8, 16/15, 13/12, 16/15, 13/12, 4/3, and its inverse
  • 1–13/12–15/13–5/4–4/3–3/2 with steps of 13/12, 16/15, 13/12, 16/15, 9/8, 4/3.

Finally, there are three pairs of heptads in inverse relationship:

  • 1–9/8–6/5–13/10–18/13–3/2–26/15 with steps of 9/8, 16/15, 13/12, 16/15, 13/12, 15/13, 15/13, and its inverse
  • 1–13/12–15/13–5/4–4/3–3/2–26/15 with steps of 13/12, 16/15, 13/12, 16/15, 9/8, 15/13, 15/13;
  • 1–13/12–5/4–3/2–13/8–26/15–15/8 with steps of 13/12, 15/13, 6/5, 13/12, 16/15, 13/12, 16/15, and its inverse
  • 1–13/12–15/13–5/4–3/2–26/15–15/8 with steps of 13/12, 16/15, 13/12, 6/5, 15/13, 13/12, 16/15;
  • 1–13/12–15/13–5/4–4/3–3/2–5/3 with steps of 13/12, 16/15, 13/12, 16/15, 9/8, 10/9, 6/5, and its inverse
  • 1–13/12–15/13–5/4–3/2–5/3–15/8 with steps of 13/12, 16/15, 13/12, 6/5, 10/9, 9/8, 16/15.

Equal temperaments with island chords include 10, 15, 19, 24, 29, 34, 43, 53, 58, 72, 77, 87, 111, 130, 140, 164, 183 and 217.

See also

Retrieved from "https://en.xen.wiki/index.php?title=Island_chords&oldid=192093"