Marveltwin chords

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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 consists of a stack of two 6/5 minor thirds followed by an 18/13, 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 triad consists of a stack of two 10/9 whole tones followed by a 16/13, that is,

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

as well as 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.

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