Marveltwin chords

From Xenharmonic Wiki
(Redirected from Marveltwin triad)
Jump to navigation Jump to search

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.

Retrieved from "https://en.xen.wiki/index.php?title=Marveltwin_chords&oldid=127884"