Marveltwin chords

A marveltwin chord is a 13-odd-limit essentially tempered dyadic chord tempered by the marveltwin comma, 325/324.

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 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 10/9-10/9-13/8,

as well as those built by

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

Each of these has multiple ways to extend to tetrads, such as using the perfect fifth as an anchor:

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

A palindrome exists as

• 1-3/2-5/3-9/5 with steps 3/2-10/9-13/12-10/9.

In the 15-odd-limit we may have

• 1-6/5-13/9-26/15 with steps 6/5-6/5-6/5-15/13.

The marveltwin tonestack is a hexad where each interval belongs to the 13-odd-limit consonance set:

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

Equal temperaments with marveltwin triads 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.