Valinorsmic clan: Difference between revisions
Sectioning & misc. cleanup. - redundant category |
+ error maps. Review optimal ET sequence. Unify precision |
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1199. | * [[WE]]: ~2 = 1199.3113{{c}}, ~5/4 = 389.5404{{c}}, ~7/4 = 971.5534{{c}} | ||
* [[CWE]]: ~2 = 1200. | : [[error map]]: {{val| -0.689 +1.849 +1.350 -2.061 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 389.3935{{c}}, ~7/4 = 971.5196{{c}} | |||
: error map: {{val| 0.000 +3.080 +2.694 -1.011 }} | |||
{{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }} | {{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }} | ||
[[Badness]] (Sintel): 0. | [[Badness]] (Sintel): 0.0793 | ||
=== Overview to extensions === | === Overview to extensions === | ||
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. | 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. 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 natural extensions to the [[13-limit]] via tempering out both [[351/350]] and [[352/351]]. | ||
Discussed elsewhere are: | Discussed elsewhere are: | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1199. | * [[WE]]: ~2 = 1199.3600{{c}}, ~3/2 = 702.6947{{c}}, ~5/4 = 389.3118{{c}} | ||
* [[CWE]]: ~2 = 1200. | : [[error map]]: {{val| -0.640 +0.100 +1.718 +1.467 -2.401 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.5916{{c}}, ~5/4 = 389.3756{{c}} | |||
: error map: {{val| 0.000 +0.637 +3.062 +2.696 -1.045 }} | |||
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 80, 111 }} | {{Optimal ET sequence|legend=1| 15, 28de, 31, 46, 65d, 77, 80, 111, 237cd, 268cd }} | ||
[[Badness]] (Sintel): 0.816 | [[Badness]] (Sintel): 0.816 | ||
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Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199. | * WE: ~2 = 1199.4350{{c}}, ~3/2 = 702.5061{{c}}, ~5/4 = 389.2327{{c}} | ||
* CWE: ~2 = 1200. | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7030{{c}}, ~5/4 = 389.4303{{c}} | ||
{{Optimal ET sequence|legend=0| 31, 46, 77, 80, 111 }} | {{Optimal ET sequence|legend=0| 31, 46, 65d, 77, 80, 111, 268cd }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.07 | ||
== Augenic == | == Augenic == | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~5/4 = 398. | * [[WE]]: ~5/4 = 398.9239{{c}}, ~3/2 = 705.1447{{c}}, ~7/4 = 969.1106{{c}} | ||
* [[CWE]]: ~5/4 = 400. | : [[error map]]: {{val| -3.228 -0.039 +6.153 -6.172 +9.184 }} | ||
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.3488{{c}}, ~7/4 = 968.4397{{c}} | |||
: error map: {{val| 0.000 +3.394 +13.686 -0.386 +17.122 }} | |||
{{Optimal ET sequence|legend=1| 6, 9, 12, 15, 24, 27e }} | {{Optimal ET sequence|legend=1| 6, 9, 12, 15, 24, 27e, 51ce, 63cee }} * | ||
<nowiki/>* [[optimal patent val]]: [[36edo|36]] | |||
[[Badness]] (Sintel): 0.735 | [[Badness]] (Sintel): 0.735 | ||
Revision as of 10:26, 11 March 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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).
For the rank-4 valinorsmic temperament, see Rank-4 temperament #Valinorsmic (176/175).
Valinor
Subgroup: 2.5.7.11
Comma list: 176/175
Subgroup-val mapping: [⟨1 0 0 -4], ⟨0 1 0 2], ⟨0 0 1 1]]
- mapping generators: ~2, ~5, ~7
- WE: ~2 = 1199.3113 ¢, ~5/4 = 389.5404 ¢, ~7/4 = 971.5534 ¢
- error map: ⟨-0.689 +1.849 +1.350 -2.061]
- CWE: ~2 = 1200.0000 ¢, ~5/4 = 389.3935 ¢, ~7/4 = 971.5196 ¢
- error map: ⟨0.000 +3.080 +2.694 -1.011]
Optimal ET sequence: 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd
Badness (Sintel): 0.0793
Overview to extensions
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. 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 natural extensions to the 13-limit via tempering out both 351/350 and 352/351.
Discussed elsewhere are:
- Zeus (+121/120) → Biyatismic clan
- 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 are manwe and augenic.
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
- WE: ~2 = 1199.3600 ¢, ~3/2 = 702.6947 ¢, ~5/4 = 389.3118 ¢
- error map: ⟨-0.640 +0.100 +1.718 +1.467 -2.401]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.5916 ¢, ~5/4 = 389.3756 ¢
- error map: ⟨0.000 +0.637 +3.062 +2.696 -1.045]
Optimal ET sequence: 15, 28de, 31, 46, 65d, 77, 80, 111, 237cd, 268cd
Badness (Sintel): 0.816
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350, 1331/1323
Mapping: [⟨1 0 0 12 8 13], ⟨0 1 0 3 3 0], ⟨0 0 1 -6 -4 -4]]
Optimal tunings:
- WE: ~2 = 1199.4350 ¢, ~3/2 = 702.5061 ¢, ~5/4 = 389.2327 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.7030 ¢, ~5/4 = 389.4303 ¢
Optimal ET sequence: 31, 46, 65d, 77, 80, 111, 268cd
Badness (Sintel): 1.07
Augenic
Subgroup: 2.3.5.7.11
Comma list: 56/55, 128/125
Mapping: [⟨3 0 7 0 2], ⟨0 1 0 0 0], ⟨0 0 0 1 1]]
- mapping generators: ~5/4, ~3, ~7
- WE: ~5/4 = 398.9239 ¢, ~3/2 = 705.1447 ¢, ~7/4 = 969.1106 ¢
- error map: ⟨-3.228 -0.039 +6.153 -6.172 +9.184]
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 705.3488 ¢, ~7/4 = 968.4397 ¢
- error map: ⟨0.000 +3.394 +13.686 -0.386 +17.122]
Optimal ET sequence: 6, 9, 12, 15, 24, 27e, 51ce, 63cee *
Badness (Sintel): 0.735