Keenanismic chords: Difference between revisions
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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: | 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; | * 1-5/4-12/7 with steps 5/4-11/8-7/6, and its inverse | ||
* 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; | * 1-7/6-8/5 with steps 7/6-11/8-5/4; | ||
* 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. | * 1-6/5-7/4 with steps 6/5-16/11-8/7, and its inverse | ||
* 1-6/5-11/8 with steps 6/5-8/7-16/11; | |||
* 1-8/5-7/4 with steps 8/5-12/11-8/7, and its inverse | |||
* 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: | Keenanismic tetrads are especially interesting; there are fifteen in total. These include three palindromic tetrads: | ||
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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 | 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; | * 1-5/4-3/2-12/7 with steps 5/4-6/5-8/7-7/6, and its inverse | ||
* 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. | * 1-6/5-3/2-7/4 with steps 6/5-5/4-7/6-8/7; | ||
* 1-5/4-10/7-12/7 with steps 5/4-8/7-6/5-7/6, and its inverse | |||
* 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 | 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; | * 1-11/8-3/2-12/7 with steps 11/8-12/11-8/7-7/6, and its inverse | ||
* 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. | * 1-12/11-3/2-7/4 with steps 12/11-11/8-7/6-8/7; | ||
* 1-7/6-8/5-11/6 with steps 7/6-11/8-8/7-12/11, and its inverse | |||
* 1-11/8-8/5-7/4 with steps 11/8-7/6-12/11-8/7. | |||
Another pair of inversely related tetrads are | 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. | * 1-6/5-11/8-3/2 with steps 6/5-8/7-12/11-4/3, and its inverse | ||
* 1-6/5-8/5-7/4 with steps 6/5-4/3-12/11-8/7. | |||
Finally, there is the pair | 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. | * 1-8/7-5/4-11/8 with steps 8/7-12/11-11/10-16/11, and its inverse | ||
* 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 [[Tablets #The keenanismic tablet|keenanismic tablet]] section of the article on tablets. | 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 [[Tablets #The keenanismic tablet|keenanismic tablet]] section of the article on tablets. | ||
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Finally, there are six keenanismic pentads coming in three inverse pairs. These are | 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; | * 1-5/4-11/8-3/2-12/7 with steps 5/4-11/10-12/11-8/7-7/6, and its inverse | ||
* 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 | 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; | * 1-6/5-11/8-3/2-7/4 with steps 6/5-8/7-12/11-7/6-8/7, and its inverse | ||
* 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. | * 1-12/11-5/4-3/2-12/7 with steps 12/11-8/7-6/5-8/7-7/6; | ||
* 1-6/5-11/8-3/2-12/7 with steps 6/5-8/7-12/11-8/7-7/6, and its inverse | |||
* 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. | The count of 11-limit chords is triads: 6, tetrads: 15, pentads: 6, for a total of 27. | ||
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[[Category:11-odd-limit]] | [[Category:11-odd-limit]] | ||
[[Category:Essentially tempered chords]] | |||
[[Category:Triads]] | |||
[[Category:Tetrads]] | |||
[[Category:Pentads]] | |||
[[Category:Keenanismic]] | [[Category:Keenanismic]] | ||