Keenanismic chords: Difference between revisions
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A '''keenanismic chord''' is an | A '''keenanismic chord''' is an [[essentially tempered chord]] which is tempered by the keenanisma, [[385/384]]. | ||
* | Keenanismic chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 2]] in the [[11-odd-limit]], meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures. | ||
* | |||
* | 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 its inverse | ||
* | * 1–5/4–16/11 with steps 5/4, 7/6, 11/8; | ||
* | * 1–8/7–11/8 with steps 8/7, 6/5, 16/11, 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 | |||
* 1–12/11–7/4 with steps 12/11, 8/5, 8/7. | |||
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: | ||
* 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; | ||
* | * 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 | 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 its inverse | ||
* | * 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 its inverse | |||
* | * 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 its inverse | |||
* | * 1–12/11–5/4–3/2 with steps 12/11, 8/7, 6/5, 4/3. | ||
* | |||
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 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. | ||
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 its inverse | |||
* | * 1–12/11–6/5–3/2–7/4 with steps 12/11, 11/10, 5/4, 7/6, 8/7; | ||
* | * 1–6/5–11/8–3/2–7/4 with steps 6/5, 8/7, 12/11, 7/6, 8/7, and its inverse | ||
* | * 1–12/11–5/4–3/2–12/7 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. | ||
* | |||
If we are willing to go as high as the [[15-odd-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-odd-limit, 32 additional keenanismic tetrads and 44 additional pentads. | If we are willing to go as high as the [[15-odd-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-odd-limit, 32 additional keenanismic tetrads and 44 additional pentads. | ||
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Equal temperaments with keenanismic tetrads include {{EDOs| 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212, 284 and 402de }}. | Equal temperaments with keenanismic tetrads include {{EDOs| 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212, 284 and 402de }}. | ||
[[Category:11-odd-limit]] | [[Category:11-odd-limit chords]] | ||
[[Category:Essentially tempered chords]] | [[Category:Essentially tempered chords]] | ||
[[Category:Triads]] | [[Category:Triads]] | ||