Marveltwin chords

Marveltwin chords are essentially tempered chords tempered by the marveltwin comma, 325/324.

There are 6 triads, 18 tetrads, 14 pentads and 3 hexads as 2.3.5.13 subgroup 13-odd-limit essentially tempered chords.

One marveltwin triad is a palindrome since it identifies 13/9 by a stack of two 6/5 minor thirds, that is,

• 1–6/5–13/9 with steps of 6/5, 6/5, 18/13.

It makes for an instructive comparison with the septimal diminished triad, 1–6/5–10/7 in starling, and might also be called a tridecimal diminished triad.

Another palindromic triad consists of 16/13 found as a stack of two 10/9 whole tones, that is,

• 1–10/9–16/13 with steps of 10/9, 10/9, 13/8.

We also have those built by

• 1–5/4–18/13 with steps of 5/4, 10/9, 13/9, and its inverse
• 1–10/9–18/13 with steps of 10/9, 5/4, 13/9;
• 1–5/3–24/13 with steps of 5/3, 10/9, 13/12, and its inverse
• 1–10/9–24/13 with steps of 10/9, 5/3, 13/12.

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

• 1–10/9–18/13–20/13 with steps of 10/9, 5/4, 10/9, 13/10;
• 1–10/9–5/4–18/13 with steps of 10/9, 9/8, 10/9, 13/9;
• 1–10/9–6/5–4/3 with steps of 10/9, 13/12, 10/9, 3/2;
• 1–13/12–6/5–13/10 with steps of 13/12, 10/9, 13/12, 20/13.

The inversely related pairs of chords are

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

For pentads, there are two palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are

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

The inversely related pairs of chords are

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

For hexads, there are one palindromic chord and a pair of chords in inverse relationship. The palindromic chord is

• 1–5/4–18/13–3/2–5/3–9/5 with steps of 5/4, 10/9, 13/12, 10/9, 13/12, 10/9.

The inversely related pair of chords is

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

Equal temperaments with marveltwin chords include 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 246, 299 and 333, with 333edo giving the optimal patent val.