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).
Discussed elsewhere are:
- Nickel (+36/35 or 45/44) → Mint family
- Ares (+64/63 or 100/99) → Archytas family
- Clio (+81/80) → Didymus rank-3 family
- Minerva (+99/98) → Marvel family
- Zeus (+121/120) → Porwell family
- Thrush (+126/125) → Starling family
- Shrusus (+245/243) → Sensamagic family
- Guanyin (+540/539) → Orwellismic family
- Julius or varda (+896/891) → Diaschismic rank-3 family
- Lono (+5120/5103) → Hemifamity family
- Draco (+19683/19600) → Cataharry family
Considered below are manwe, mandos and ulmo besides the no-3 subgroup temperament, but we can begin with the rank-4 temperament.
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]]
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
Mandos
Subgroup: 2.3.5.7.11
Comma list: 176/175, 243/242
Mapping: [⟨1 1 0 6 2], ⟨0 2 0 5 5], ⟨0 0 1 -2 0]]
Optimal ET sequence: 7, 24, 31, 58, 89
Badness: 0.751 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 144/143, 176/175, 243/242
Mapping: [⟨1 1 0 6 2 4], ⟨0 2 0 5 5 -1], ⟨0 0 1 -2 0 0]]
Optimal ET sequence: 7, 24, 31, 58
Badness: 0.923 × 10-3
Ulmo
Subgroup: 2.3.5.7.11
Comma list: 176/175, 2200/2187
Mapping: [⟨1 0 0 1 -3], ⟨0 1 0 7 7], ⟨0 0 1 -4 -2]]
Optimal ET sequence: 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd
Badness: 1.23 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 176/175, 325/324
Mapping: [⟨1 0 0 1 -3 2], ⟨0 1 0 7 7 4], ⟨0 0 1 -4 -2 -2]]
Optimal ET sequence: 7, 39, 46, 53, 80, 133d, 179, 312cdf
Badness: 0.932 × 10-3