View source for Keenanismic chords

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: 
* 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 

* 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 
* 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 
* 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 
* 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.

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, including six additional triads, 33 additional tetrads and 46 additional pentads. There are also hexads, such as 1–5/4–11/8–3/2–12/7–15/8 (steps 5/4, 11/10, 12/11, 8/7, 12/11, 16/15) and its inverse 1–12/11–5/4–15/11–3/2–15/8 (steps 12/11, 8/7, 12/11, 11/10, 5/4, 16/15).

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 chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Keenanismic]]