Valinorsmic clan

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).

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.

Discussed elsewhere are:

• Zeus (+121/120) → Porwell family and Biyatismic clan
• Julius or varda (+896/891) → Diaschismic rank-3 family
• Nickel (+36/35 or 45/44) → Mint family
• Ares (+64/63 or 100/99) → Archytas family
• Minerva (+99/98) → Marvel family
• Thrush (+126/125) → Starling family
• Guanyin (+540/539) → Orwellismic family
• Clio (+81/80) → Didymus rank-3 family
• Lono (+5120/5103) → Hemifamity family
• Mandos (+243/242) → Rastmic rank-3 clan
• Shrusus (+245/243) → Sensamagic family
• Ulmo (+2200/2187) → Ragismic family
• Draco (+19683/19600) → Cataharry family

Considered below is manwe, besides the no-3 subgroup temperament. For the rank-4 valinorsmic temperament, see Rank-4 temperament #Valinorsmic (176/175).

No-3 valinorsmic

Subgroup: 2.5.7.11

Comma list: 176/175

Sval mapping: [⟨1 0 0 -4], ⟨0 1 0 2], ⟨0 0 1 1]]

sval mapping generators: ~2, ~5, ~7

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 389.7641, ~7/4 = 972.1113

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

Badness: 1.73 × 10^-6

Manwe

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1331/1323

Mapping: [⟨1 0 0 12 8], ⟨0 1 0 3 3], ⟨0 0 1 -6 -4]]

mapping generators: ~2, ~3, ~5

Optimal ET sequence: 15, 31, 46, 77, 80, 111

Badness: 0.679 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 1331/1323, 351/350

Mapping: [⟨1 0 0 12 8], ⟨0 1 0 3 3], ⟨0 0 1 -6 -4]]

Optimal ET sequence: 31, 46, 77, 80, 111

Badness: 1.147 × 10^-3