Valinorsmic clan: Difference between revisions
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The '''valinorsmic clan''' of rank-3 | The '''valinorsmic clan''' of [[Rank-3 temperament|rank-3]] [[temperament]]s [[Tempering out|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]]. 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: | Discussed elsewhere are: | ||
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* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]] | * ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]] | ||
Considered below are manwe, mandos and ulmo besides the no-3 subgroup temperament | Considered below are manwe, mandos and ulmo besides the no-3 subgroup temperament. For the rank-4 valinorsmic temperament, see [[Rank-4 temperament #Valinorsmic (176/175)]]. | ||
== No-3 valinorsmic == | == No-3 valinorsmic == | ||
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{{Mapping|legend=1| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }} | {{Mapping|legend=1| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }} | ||
: mapping generators: ~2, ~3, ~5 | |||
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 80, 111 }} | {{Optimal ET sequence|legend=1| 15, 31, 46, 77, 80, 111 }} | ||
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{{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 5 5 | 0 0 1 -2 0 }} | {{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 5 5 | 0 0 1 -2 0 }} | ||
: mapping generators: ~2, ~11/9, ~5 | |||
{{Optimal ET sequence|legend=1| 7, 24, 31, 58, 89 }} | {{Optimal ET sequence|legend=1| 7, 24, 31, 58, 89 }} | ||
Revision as of 11:22, 9 September 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).
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. 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:
- 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. 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
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]]
- mapping generators: ~2, ~11/9, ~5
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