Valinorsmic clan: Difference between revisions
m Removing from Category:Regular temperament theory using Cat-a-lot |
Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
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[[POTE generator]]s: ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113 | [[POTE generator]]s: ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113 | ||
{{ | {{Optimal ET sequence|legend=1| 22, 31, 46, 53, 58, 80, 111 }} | ||
[[Badness]]: 0.0186 × 10<sup>-6</sup> | [[Badness]]: 0.0186 × 10<sup>-6</sup> | ||
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[[POTE generator]]s: ~5/4 = 389.7641, ~7/4 = 972.1113 | [[POTE generator]]s: ~5/4 = 389.7641, ~7/4 = 972.1113 | ||
{{ | {{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }} | ||
[[Badness]]: 1.73 × 10<sup>-6</sup> | [[Badness]]: 1.73 × 10<sup>-6</sup> | ||
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[[Mapping]]: [{{val| 1 0 0 12 8 }}, {{val| 0 1 0 3 3 }}, {{val| 0 0 1 -6 -4 }}] | [[Mapping]]: [{{val| 1 0 0 12 8 }}, {{val| 0 1 0 3 3 }}, {{val| 0 0 1 -6 -4 }}] | ||
{{ | {{Optimal ET sequence|legend=1| 15, 31, 46, 77, 80, 111 }} | ||
[[Badness]]: 0.679 × 10<sup>-3</sup> | [[Badness]]: 0.679 × 10<sup>-3</sup> | ||
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Mapping: [{{val| 1 0 0 12 8 }}, {{val| 0 1 0 3 3 }}, {{val| 0 0 1 -6 -4 }}] | Mapping: [{{val| 1 0 0 12 8 }}, {{val| 0 1 0 3 3 }}, {{val| 0 0 1 -6 -4 }}] | ||
{{Optimal ET sequence|legend=1| 31, 46, 77, 80, 111 }} | |||
Badness: 1.147 × 10<sup>-3</sup> | Badness: 1.147 × 10<sup>-3</sup> | ||
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[[Mapping]]: [{{val| 1 1 0 6 2 }}, {{val| 0 2 0 5 5 }}, {{val| 0 0 1 -2 0 }}] | [[Mapping]]: [{{val| 1 1 0 6 2 }}, {{val| 0 2 0 5 5 }}, {{val| 0 0 1 -2 0 }}] | ||
{{ | {{Optimal ET sequence|legend=1| 7, 24, 31, 58, 89 }} | ||
[[Badness]]: 0.751 × 10<sup>-3</sup> | [[Badness]]: 0.751 × 10<sup>-3</sup> | ||
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Mapping: [{{val| 1 1 0 6 2 4 }}, {{val| 0 2 0 5 5 -1 }}, {{val| 0 0 1 -2 0 0 }}] | Mapping: [{{val| 1 1 0 6 2 4 }}, {{val| 0 2 0 5 5 -1 }}, {{val| 0 0 1 -2 0 0 }}] | ||
{{Optimal ET sequence|legend=1| 7, 24, 31, 58 }} | |||
Badness: 0.923 × 10<sup>-3</sup> | Badness: 0.923 × 10<sup>-3</sup> | ||
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[[Mapping]]: [{{val| 1 0 0 1, -3 }}, {{val| 0 1 0 7 7 }}, {{val| 0 0 1 -4 -2 }}] | [[Mapping]]: [{{val| 1 0 0 1, -3 }}, {{val| 0 1 0 7 7 }}, {{val| 0 0 1 -4 -2 }}] | ||
{{ | {{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd }} | ||
[[Badness]]: 1.23 × 10<sup>-3</sup> | [[Badness]]: 1.23 × 10<sup>-3</sup> | ||
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Mapping: [{{val| 1 0 0 1, -3 2 }}, {{val| 0 1 0 7 7 4 }}, {{val| 0 0 1 -4 -2 -2 }}] | Mapping: [{{val| 1 0 0 1, -3 2 }}, {{val| 0 1 0 7 7 4 }}, {{val| 0 0 1 -4 -2 -2 }}] | ||
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 133d, 179, 312cdf }} | |||
Badness: 0.932 × 10<sup>-3</sup> | Badness: 0.932 × 10<sup>-3</sup> | ||
Revision as of 18:22, 7 May 2023
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 besides the no-3 subgroup temperament, but we can begin with the rank-4 temperament.
Valinorsmic
Subgroup: 2.3.5.7.11
Mapping: [⟨1 0 0 0 -4], ⟨0 1 0 0 0], ⟨0 0 1 0 2], ⟨0 0 0 1 1]]
Mapping generators: ~2, ~3, ~5, ~7
POTE generators: ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113
Optimal ET sequence: 22, 31, 46, 53, 58, 80, 111
Badness: 0.0186 × 10-6
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
POTE generators: ~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