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This is a collection of rank-3 temperaments tempering out the valinorsma, [[176/175]], which equates 48/35 with 15/11 (rather than 11/8, which is what the keenanisma 385/384 does).  
{{Technical data page}}
The '''valinorsmic clan''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[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).
 
For the rank-4 valinorsmic temperament, see [[Rank-4 temperament #Valinorsmic (176/175)]].
 
== Valinor ==
Valinor is the [[2.5.7.11 subgroup|2.5.7.11-subgroup]] temperament defined by tempering out 176/175.
 
[[Subgroup]]: 2.5.7.11
 
[[Comma list]]: 176/175
 
{{Mapping|legend=2| 1 0 0 -4 | 0 1 0 2 | 0 0 1 1 }}
: mapping generators: ~2, ~5, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3113{{c}}, ~5/4 = 389.5404{{c}}, ~7/4 = 971.5534{{c}}
: [[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 }}
 
[[Badness]] (Sintel): 0.0793
 
=== Overview to extensions ===
==== Subgroup extensions ====
When extending to prime 13, two strong extensions are considerable: ''valimar'' and ''valarin'', both considered below; these intersect in [[Hemimean clan #tridecimal didacus|tridecimal didacus]].
 
==== Full 11-limit 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:  
Discussed elsewhere are:  
* ''[[nickel]]'', {36/35, 45/44} → [[Mint family #Nickel]]
* [[Zeus]] (+121/120) → [[Biyatismic clan #Zeus|Biyatismic clan]]
* [[ares]], {64/63, 100/99} → [[Archytas family #Ares]]
* ''[[Varda]]'' (+896/891) → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]
* ''[[clio]]'', {81/80, 176/175} → [[Didymus rank three family #Clio]]
* ''[[Nickel]]'' (+36/35 or 45/44) → [[Mint family #Nickel|Mint family]]
* ''[[minerva]]'', {99/98, 176/175} → [[Marvel family #Minerva]]
* [[Ares]] (+64/63 or 100/99) → [[Archytas family #Ares|Archytas family]]
* ''[[zeus]]'', {121/120, 176/175} → [[Porwell family #Zeus]]
* ''[[Minerva]]'' (+99/98) → [[Marvel family #Minerva|Marvel family]]
* ''[[thrush]]'', {126/125, 176/175} → [[Starling family #Thrush]]
* [[Thrush]] (+126/125) → [[Starling family #Thrush|Starling family]]
* ''[[shrusus]]'', {176/175, 245/243} → [[Sensamagic family #Shrusus]]
* ''[[Guanyin]]'' (+540/539) → [[Orwellismic family #Guanyin|Orwellismic family]]
* ''[[guanyin]]'', {176/175, 540/539} → [[Orwellismic family #Guanyin]]
* ''[[Clio]]'' (+81/80) → [[Didymus rank-3 family #Clio|Didymus rank-3 family]]
* ''[[julius]]'' or ''[[varda]]'', {176/175, 896/891} → [[Diaschismic rank three family #Julius aka varda]]
* ''[[Lono]]'' (+5120/5103) → [[Hemifamity family #Lono|Hemifamity family]]
* ''[[lono]]'', {176/175, 5120/5103} → [[Hemifamity family #Lono]]
* ''[[Mandos]]'' (+243/242) → [[Rastmic rank-3 clan #Mandos|Rastmic rank-3 clan]]
* ''[[draco]]'', {176/175, 19683/19600} → [[Cataharry family #Draco]]
* ''[[Shrusus]]'' (+245/243) → [[Sensamagic family #Shrusus|Sensamagic family]]
* ''[[Ulmo]]'' (+2200/2187) → [[Ragismic family #Ulmo|Ragismic family]]
* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]]


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


== Valinorsmic ==
=== Valimar ===
Subgroup: 2.3.5.7.11
This extension to the no-threes 13-limit is motivated by extending the valinorsmic chain of 5/4's, where two form 11/7, so that four form 16/13. This logic is supported by rank-2 temperaments such as tridecimal didacus and the 13-limit [[magus]]/[[amigo]] extensions, as well as [[winston]] and [[valentino]]. This is also natural considering [[S-expression]]s, noting that 176/175 (S8/S10) is nontrivially equivalent to (S11⋅S12)/(S14⋅S15), and this extension tempers out [[1573/1568]] (S11/S14), and [[3584/3575]] (S12/S15). It remains well-tuned in equal temperaments with optimal valinorsmic 5/4's (such as [[37edo|37]], [[40edo|40]], and [[43edo|43]]).


[[Comma list]]: [[176/175]]
[[Subgroup]]: 2.5.7.11.13


[[Mapping]]: [{{val| 1 0 0 0 -4 }}, {{val| 0 1 0 0 0 }}, {{val| 0 0 1 0 2 }}, {{val| 0 0 0 1 1 }}]
[[Comma list]]: 176/175, 1573/1568


{{Val list|legend=1| 22, 31, 46, 53, 58, 80, 111 }}
{{Mapping|legend=2| 1 0 0 -4 13 | 0 1 0 2 -4 | 0 0 1 1 0 }}
: mapping generators: ~2, ~5, ~7


== Manwe ==
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11
* [[WE]]: ~2 = 1199.433{{c}}, ~5/4 = 389.226{{c}}, ~7/4 = 971.559{{c}}
: [[error map]]: {{val| -0.567 +1.779 +1.600 -2.440 -0.267 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 389.466{{c}}, ~7/4 = 971.517{{c}}
: error map: {{val| 0.000 +3.152 +2.691 -0.869 +1.607 }}


[[Comma list]]: 176/175, 1331/1323
{{Optimal ET sequence|legend=1| 6, 9, 15f, 22f, 28, 31, 37, 231cd, 268cd }}


[[Mapping]]: [{{val| 1 0 0 12 8 }}, {{val| 0 1 0 3 3 }}, {{val| 0 0 1 -6 -4 }}]
[[Badness]] (Sintel): 0.336


{{Val list|legend=1| 15, 31, 46, 77, 80, 111 }}
=== Valarin ===
This extension equates two, ideally slightly flattened, [[8/7]]'s to [[13/10]], which is also natural given valinorsmic's tuning tendency towards sharpening [[7/4]]. This extension is supported by rank-2 temperaments including tridecimal didacus, [[orwell extensions|tridecimal orwell]], [[lupercalia]] (an extension of [[valentine]]), and [[llywelyn]].


[[Badness]]: 0.679 × 10<sup>-3</sup>
[[Subgroup]]: 2.5.7.11.13


=== 13-limit ===
[[Comma list]]: 176/175, 640/637
Subgroup: 2.3.5.7.11.13


Comma list: 176/175, 1331/1323, 351/350
{{Mapping|legend=2| 1 0 0 -4 7 | 0 1 0 2 1 | 0 0 1 1 -2 }}
: mapping generators: ~2, ~5, ~7


Mapping: [{{val| 1 0 0 12 8 }}, {{val| 0 1 0 3 3 }}, {{val| 0 0 1 -6 -4 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.167{{c}}, ~5/4 = 389.474{{c}}, ~7/4 = 972.185{{c}}
: [[error map]]: {{val| -0.833 +1.495 +1.694 -1.851 +0.413 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 389.189{{c}}, ~7/4 = 972.829{{c}}
: error map: {{val| 0.000 +2.875 +4.003 -0.110 +3.003 }}


Vals: {{val list| 31, 46, 77, 80, 111 }}
{{Optimal ET sequence|legend=1| 6, 10e, 15, 16, 21, 31, 37, 206cdf, 243cdf, 280cdf, 317cddf, 354ccddff }}


Badness: 1.147 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.227


== Mandos ==
== Manwe ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 176/175, 243/242
[[Comma list]]: 176/175, 1331/1323
 
{{Mapping|legend=1| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }}
: mapping generators: ~2, ~3, ~5


[[Mapping]]: [{{val| 1 1 0 6 2 }}, {{val| 0 2 0 5 5 }}, {{val| 0 0 1 -2 0 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.3600{{c}}, ~3/2 = 702.6947{{c}}, ~5/4 = 389.3118{{c}}
: [[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 }}


{{Val list|legend=1| 7, 24, 31, 58, 89 }}
{{Optimal ET sequence|legend=1| 15, 28de, 31, 46, 65d, 77, 80, 111, 237cd, 268cd }}


[[Badness]]: 0.751 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.816


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 144/143, 176/175, 243/242
Comma list: 176/175, 351/350, 1331/1323


Mapping: [{{val| 1 1 0 6 2 4 }}, {{val| 0 2 0 5 5 -1 }}, {{val| 0 0 1 -2 0 0 }}]
Mapping: {{mapping| 1 0 0 12 8 13 | 0 1 0 3 3 0 | 0 0 1 -6 -4 -4 }}


Vals: {{val list| 7, 24, 31, 58 }}
Optimal tunings:  
* WE: ~2 = 1199.4350{{c}}, ~3/2 = 702.5061{{c}}, ~5/4 = 389.2327{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7030{{c}}, ~5/4 = 389.4303{{c}}


Badness: 0.923 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 31, 46, 65d, 77, 80, 111, 268cd }}


== Ulmo ==
Badness (Sintel): 1.07
Subgroup: 2.3.5.7.11


[[Comma list]]: 176/175, 2200/2187
== Augenic ==
{{Distinguish| Augene }}


[[Mapping]]: [{{val| 1 0 0 1, -3 }}, {{val| 0 1 0 7 7 }}, {{val| 0 0 1 -4 -2 }}]
Named by [[Xenllium]] in 2026, augenic is closely related to [[augene]]. It tempers out the [[augmented comma]] but with a free generator for [[7/1|7]], and then extends it to the 11-limit through the identity [[128/125]] = ([[56/55]])⋅(176/175).


{{Val list|legend=1| 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd }}
[[Subgroup]]: 2.3.5.7.11


[[Badness]]: 1.23 × 10<sup>-3</sup>
[[Comma list]]: 56/55, 128/125


=== 13-limit ===
{{Mapping|legend=1| 3 0 7 0 2 | 0 1 0 0 0 | 0 0 0 1 1 }}
Subgroup: 2.3.5.7.11.13
: mapping generators: ~5/4, ~3, ~7


Comma list: 169/168, 176/175, 325/324
[[Optimal tuning]]s:  
* [[WE]]: ~5/4 = 398.9239{{c}}, ~3/2 = 705.1447{{c}}, ~7/4 = 969.1106{{c}}
: [[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 }}


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| 6, 9, 12, 15, 24, 27e, 51ce, 63cee }} *


Vals: {{val list| 7, 39, 46, 53, 80, 133d, 179, 312cdf }}
<nowiki/>* [[optimal patent val]]: [[36edo|36]]


Badness: 0.932 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.735


[[Category:Regular temperament theory]]
[[Category:Temperament clans]]
[[Category:Temperament collections]]
[[Category:Valinorsmic clan| ]] <!-- main article -->
[[Category:Valinorsmic temperaments| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
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