Keenanismic chords

A keenanismic chord is an 11-odd-limit essentially tempered chord which is tempered by the keenanisma, 385/384. There are six keenanismic triads, consisting of three pairs of chords in inverse relationship: 1-5/4-12/7 with steps 5/4-11/8-7/6 and 1-7/6-8/5 with steps 7/6-11/8-5/4; 1-6/5-7/4 with steps 6/5-16/11-8/7 and 1-6/5-11/8 with steps 6/5-8/7-16/11; and 1-8/5-7/4 with steps 8/5-12/11-8/7 and 1-8/5-11/6 with steps 8/5-8/7-12/11.

Keenanismic tetrads are especially interesting; there are fifteen in total. These include three palindromic tetrads: 1-5/4-11/8-12/7 with steps 5/4-11/10-5/4-7/6; 1-6/5-11/8-7/4 with steps 6/5-8/7-14/11-8/7; and 1-7/5-8/5-7/4 with steps 7/5-8/7-12/11-8/7. Then there are four remarkable tetrads in two inversely related pairs which are the remaining (circular) permutations of the intervals 5/4, 6/5, 7/6, 8/7 making up the otonal and utonal tetrad after those two chords have been removed. These are 1-5/4-3/2-12/7 with steps 5/4-6/5-8/7-7/6 and 1-6/5-3/2-7/4 with steps 6/5-5/4-7/6-8/7; and 1-5/4-10/7-12/7 with steps 5/4-8/7-6/5-7/6 and 1-5/4-16/11-7/4 with steps 5/4-7/6-6/5-8/7. Another set of four tetrads coming in two inverse pairs are 1-11/8-3/2-12/7 with steps 11/8-12/11-8/7-7/6 and 1-12/11-3/2-7/4 with steps 12/11-11/8-7/6-8/7; and 1-7/6-8/5-11/6 with steps 7/6-11/8-8/7-12/11 and 1-11/8-8/5-7/4 with steps 11/8-7/6-12/11-8/7. Another pair of inversely related tetrads are 1-6/5-11/8-3/2 with steps 6/5-8/7-12/11-4/3 and 1-6/5-8/5-7/4 with steps 6/5-4/3-12/11-8/7. Finally, there is the pair 1-8/7-5/4-11/8 with steps 8/7-12/11-11/10-16/11 and 1-16/11-8/5-7/4 with steps 16/11-11/10-12/11-8/7. Five of these tetrads, the four which permute the steps of the otonal tetrad and one of the palindromic tetrads, have all step sizes larger than 9/8, and it is these five which are featured in the keenanismic tablet section of the article on tablets.

Finally, there are six keenanismic pentads coming in three inverse pairs. These are 1-5/4-11/8-3/2-12/7 with steps 5/4-11/10-12/11-8/7-7/6 and 1-8/7-5/4-11/8-12/7 with steps 8/7-12/11-11/10-5/4-7/6; and the pair of pairs 1-6/5-11/8-3/2-7/4 with steps 6/5-8/7-12/11-7/6-8/7 and 1-12/11-5/4-3/2-12/7 with steps 12/11-8/7-6/5-8/7-7/6 plus 1-6/5-11/8-3/2-12/7 with steps 6/5-8/7-12/11-8/7-7/6 and 1-12/11-5/4-3/2-7/4 with steps 12/11-8/7-6/5-7/6-8/7.

The count of 11-limit chords is triads: 6, tetrads: 15, pentads: 6, for a total of 27.

If we are willing to go as high as the 15 limit, there are many more keenanismic chords, all of which have a step of size 16/15. There are six additional keenanismic triads in the 15 limit, 32 additional keenanismic tetrads and 44 additional pentads.

Equal temperaments with keenanismic tetrads include 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212, 284 and 402de.