Undecimal marvel chords

An undecimal marvel chord is an essentially tempered chord of undecimal (11-limit) marvel, with intervals in the tonality diamond of the 11-odd-limit. Since 225/224 is tempered out in undecimal marvel, 9-odd-limit marvel chords are undecimal marvel chords. Since 385/384 is tempered out, keenanismic chords are also undecimal marvel chords. Finally, since 540/539 is tempered out, swetismic chords are undecimal marvel chords. There are also undecimal marvel chords which are none of these, the essentially undecimal marvel chords.

These are four pairs of inversely related tetrads:

• 1-5/4-11/8-8/5 with steps 5/4-11/10-7/6-5/4 and 1-5/4-16/11-8/5 with steps 5/4-7/6-11/10-5/4;
• 1-9/8-8/5-7/4 with steps 9/8-10/7-12/11-8/7 and 1-9/8-9/7-7/5 with steps 9/8-8/7-12/11-10/7;
• 1-9/8-11/8-8/5 with steps 9/8-11/9-7/6-5/4 and 1-5/4-16/11-16/9 with steps 5/4-7/6-11/9-9/8;
• 1-5/4-8/5-7/4 with steps 5/4-9/7-12/11-8/7 and 1-5/4-10/7-14/9 with steps 5/4-8/7-12/11-9/7.

There are six pairs of inversely related essentially undecimal marvel pentads:

• 1-9/8-5/4-11/8-8/5 with steps 9/8-10/9-11/10-7/6-5/4 and 1-5/4-16/11-8/5-16/9 with steps 5/4-7/6-11/10-10/9-9/8;
• 1-5/4-7/5-8/5-7/4 with steps 5/4-9/8-8/7-12/11-8/7 and 1-9/8-7/5-8/5-7/4 with steps 9/8-5/4-8/7-12/11-8/7;
• 1-5/4-16/11-8/5-7/4 with steps 5/4-7/6-11/10-12/11-8/7 and 1-8/7-5/4-11/8-8/5 with steps 8/7-12/11-11/10-7/6-5/4;
• 1-5/4-11/8-8/5-7/4 with steps 5/4-11/10-7/6-12/11-8/7 and 1-8/7-5/4-16/11-8/5 with steps 8/7-12/11-7/6-11/10-5/4;
• 1-9/8-11/8-8/5-7/4 with steps 9/8-11/9-7/6-12/11-8/7 and 1-11/9-11/8-11/7-12/7 with steps 11/9-9/8-8/7-12/11-7/6;
• 1-9/8-5/4-8/5-7/4 with steps 9/8-10/9-9/7-12/11-8/7 and 1-9/7-10/7-8/5-11/6 with steps 9/7-10/9-9/8-8/7-12/11.

The three essentially undecimal marvel hexads are particularly interesting. One is palindromic:

• 1-9/8-5/4-7/5-8/5-7/4 with steps 9/8-10/9-9/8-8/7-12/11-8/7.

The two others are an inversive pair:

• 1-9/8-5/4-11/8-8/5-7/4 with steps 9/8-10/9-11/10-7/6-12/11-8/7 and 1-8/7-5/4-16/11-8/5-16/9 with steps 8/7-12/11-7/6-11/10-10/9-9/8.

The latter two have steps which are permutations of the steps of the 11-limit JI hexads.

The essentially undecimal marvel chords number tetrads: 8, pentads: 12, hexads: 3, for a total of 23.

Equal divisions with undecimal marvel chords include 19, 22, 31, 41, 50, 53, 72, 81, 94, 103, 166, 175, 197e, 269ce and 341ce.