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, 45/44} → Mint family
• Ares, {64/63, 100/99} → Archytas family
• Clio, {81/80, 176/175} → Didymus rank-3 family
• Minerva, {99/98, 176/175} → Marvel family
• Zeus, {121/120, 176/175} → Porwell family
• Thrush, {126/125, 176/175} → Starling family
• Shrusus, {176/175, 245/243} → Sensamagic family
• Guanyin, {176/175, 540/539} → Orwellismic family
• Julius or varda, {176/175, 896/891} → Diaschismic rank-3 family
• Lono, {176/175, 5120/5103} → Hemifamity family
• Draco, {176/175, 19683/19600} → Cataharry family

Considered below are manwe, mandos and ulmo, but we can begin with the rank-4 temperament.

Valinorsmic

Subgroup: 2.3.5.7.11

Comma list: 176/175

Mapping: [⟨1 0 0 0 -4], ⟨0 1 0 0 0], ⟨0 0 1 0 2], ⟨0 0 0 1 1]]

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]]

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]]

Vals: Template:Val list

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]]

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]]

Vals: Template:Val list

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]]

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]]

Vals: Template:Val list

Badness: 0.932 × 10^-3