Keenanismic chords
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Original Wikitext content:
A //keenanismic chord// is an 11-limit [[Dyadic chord|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 a palindromic tetrads, have all step sizes larger than 9/8, and it is these five which are featured in the [[Tablets#x4et tablets-The keenanismic tablet|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. Equal temperaments with keenanismic tetrads include 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212, 284 and 402de.
Original HTML content:
<html><head><title>keenanismic chords</title></head><body>A <em>keenanismic chord</em> is an 11-limit <a class="wiki_link" href="/Dyadic%20chord">essentially tempered chord</a> 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. <br /> <br /> 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 a palindromic tetrads, have all step sizes larger than 9/8, and it is these five which are featured in the <a class="wiki_link" href="/Tablets#x4et tablets-The keenanismic tablet">keenanismic tablet</a> section of the article on tablets.<br /> <br /> 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.<br /> <br /> Equal temperaments with keenanismic tetrads include 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212, 284 and 402de.</body></html>