Valinorsmic chords

A valinorsmic chord is a valinorsma (176/175) tempered essentially tempered chord.

Valinorsmic chords are of pattern 1a in the 2.5.7.11 subgroup 11-odd-limit, meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures.

There are three valinorsmic triads: the valinorsmic version of the augmented triad,

• 1–5/4–8/5 with steps 5/4, 14/11, 5/4,

and an inversely related pair,

• 1–8/5–7/4 with steps 8/5, 11/10, 8/7, and its inverse
• 1–8/7–5/4 with steps 8/7, 11/10, 8/5.

Valinorsmic tetrads are six in number, consisting of two palindromic chords and two pairs of inversely related chords. The palindromes are

• 1–11/10–5/4–7/4 with steps 11/10, 8/7, 7/5, 8/7, and its inverse
• 1–11/10–5/4–11/8 with steps 11/10, 8/7, 11/10, 16/11.

The rest are

• 1–5/4–8/5–7/4 with steps 5/4, 14/11, 11/10, 8/7, and its inverse
• 1–5/4–10/7–11/7 with steps 5/4, 8/7, 11/10, 14/11;
• 1–5/4–11/8–11/7 with steps 5/4, 11/10, 8/7, 14/11, and its inverse
• 1–11/10–11/8–7/4 with steps 11/10, 5/4, 14/11, 8/7.

Finally, there are the two inversely related pentads,

• 1–11/10–5/4–8/5–7/4 with steps 11/10, 8/7, 14/11, 11/10, 8/7, and its inverse
• 1–11/10–5/4–11/8–7/4 with steps 11/10, 8/7, 11/10, 14/11, 8/7.

If we are willing to go to the 15-odd-limit, we get four alternative, valinorsmic, temperings of the four keenanismic tetrads which have steps which are permutations of the otonal/utonal 7-limit tetrads. The reason for the similarity between valinorsmic and keenanismic tetrads is that they both temper 48/35, valinorsmic to 15/11 and keenanismic to 11/8. Another valinorsmic tetrad has steps 11/9, 6/5, 8/7, 6/5, leading to 1–11/9–22/15–5/3 chords.

Equal temperaments with valinorsmic chords include 22, 31, 43, 46, 53, 58, 68, 80, 89, 111 with 111edo giving the optimal patent val.