Valinorsmic clan

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The valinorsmic clan of rank-3 temperaments tempers out the valinorsma, 176/175, which equates 48/35 with 15/11 (rather than 11/8, which is what the keenanisma 385/384 does).

For the rank-4 valinorsmic temperament, see Rank-4 temperament #Valinorsmic (176/175).

Valinor

Subgroup: 2.5.7.11

Comma list: 176/175

Subgroup-val mapping: [⟨1 0 0 -4], ⟨0 1 0 2], ⟨0 0 1 1]]

mapping generators: ~2, ~5, ~7

Optimal tunings:

WE: ~2 = 1199.311 ¢, ~5/4 = 389.540 ¢, ~7/4 = 971.553 ¢
CWE: ~2 = 1200.000 ¢, ~5/4 = 389.393 ¢, ~7/4 = 971.520 ¢

Optimal ET sequence: 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd

Badness (Sintel): 0.079

Overview to extensions

The second comma in the comma list determines how we extend the no-3 subgroup temperament to include the harmonic 3. Zeus adds 121/120 as well as 385/384, and shares the same lattice structure as no-3 valinorsmic. Julius, a.k.a. varda, adds 896/891, slicing the first generator in two with a semi-octave period. Nickel adds 36/35 as well as 45/44. Ares adds 64/63. Minerva adds 99/98. Thrush adds 126/125. These slice the last generator in two. Guanyin adds 540/539, slicing the first generator in three. Manwe adds 1331/1323, slicing the last generator in three. Clio adds 81/80, slicing the first generator in four. Lono adds 5120/5103, slicing the last generator in six. Mandos adds 243/242. Shrusus adds 245/243. These slice the last generator in five. Ulmo adds 2200/2187, slicing the last generator in seven. Finally, draco adds 19683/19600, slicing the last generator in nine. Most of these have neat extensions to the 13-limit via tempering out both 351/350 and 352/351.