Valinorsmic chords

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A valinorsmic chord is a valinorsma (176/175) tempered essentially tempered chord in the subgroup in the 11 odd limit. There are three valinorsmic triads: the valinorsmic version of the augmented triad, 1-5/4-8/5 with steps 5/4-14/11-5/4, and an inversely related pair, 1-8/5-7/4 with steps 8/5-11/10-8/7 and 1-8/7-5/4 with steps 8/7-11/10-8/5. Valinorsmic tetrads are six in number, consisting of two palindromic chords and a pair of pairs of inversely related chords. The palindromes are 1-11/10-5/4-7/4 with steps 11/10-8/7-7/5-8/7 and 1-11/10-5/4-11/8 with steps 11/10-8/7-11/10-16/11. The rest are 1-5/4-8/5-7/4 with steps 5/4-14/11-11/10-8/7 and 1-5/4-10/7-11/7 with steps 5/4-8/7-11/10-14/11, plus 1-5/4-11/8-11/7 with steps 5/4-11/10-8/7-14/11 and 1-11/10-11/8-7/4 with steps 11/10-5/4-14/11-8/7. Finally, there are the two inversely related pentads, 1-11/10-5/4-8/5-7/4 with steps 11/10-8/7-14/11-11/10-8/7 and 1-11/10-5/4-11/8-7/4 with steps 11/10-8/7-11/10-14/11-8/7.

The count is triads: 3, tetrads: 6, pentads: 2; 11 in total.

If we are willing to go to the 15 odd limit, we get four alternative, valinorsmic, temperings of the four keenanismic tetrads which have steps which are permutations of the otonal/utonal 7-limit tetrads. The reason for the similarity between valinorsmic and keenanismic tetrads is that they both temper 48/35, valinorsmic to 15/11 and keenanismic to 11/8. Another valinorsmic tetrad has steps 11/9-6/5-8/7-6/5, leading to 1-11/9-22/15-5/3 chords.

Equal temperaments with valinorsmic chords include 22, 31, 43, 46, 53, 58, 68, 80, 89, 111, 268cd, 326cd and 437cd.