Keenanismic chords

A keenanismic chord is an essentially tempered chord which is tempered by the keenanisma, 385/384.

Keenanismic chords are of 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 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, 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.

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