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{{Technical data page}}
The valinorsmic rank three temperaments temper 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 '''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).  


== Rank four valinorsmic temperament  ==
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]]. 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 neat extensions to the 13-limit via tempering out both [[351/350]] and [[352/351]].
Comma: 176/175


Map: [<1 0 0 0 -4|, <0 1 0 0 0|, <0 0 1 0 2|, <0 0 0 1 1|]
Discussed elsewhere are:  
* [[Zeus]] (+121/120) → [[Porwell family #Zeus|Porwell family]] and [[Biyatismic clan #Zeus|Biyatismic clan]]
* ''[[Julius]]'' or ''[[varda]]'' (+896/891) → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]]
* ''[[Nickel]]'' (+36/35 or 45/44) → [[Mint family #Nickel|Mint family]]
* [[Ares]] (+64/63 or 100/99) → [[Archytas family #Ares|Archytas family]]
* ''[[Minerva]]'' (+99/98) → [[Marvel family #Minerva|Marvel family]]
* [[Thrush]] (+126/125) → [[Starling family #Thrush|Starling family]]
* ''[[Guanyin]]'' (+540/539) → [[Orwellismic family #Guanyin|Orwellismic family]]
* ''[[Clio]]'' (+81/80) → [[Didymus rank three family #Clio|Didymus rank-3 family]]
* ''[[Lono]]'' (+5120/5103) → [[Hemifamity family #Lono|Hemifamity family]]
* ''[[Mandos]]'' (+243/242) → [[Rastmic rank three clan #Mandos|Rastmic rank-3 clan]]
* ''[[Shrusus]]'' (+245/243) → [[Sensamagic family #Shrusus|Sensamagic family]]
* ''[[Ulmo]]'' (+2200/2187) → [[Ragismic family #Ulmo|Ragismic family]]
* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]]


EDOs: 22, 31, 46, 53, 58, 80, 111
Considered below is manwe, besides the no-3 subgroup temperament. For the rank-4 valinorsmic temperament, see [[Rank-4 temperament #Valinorsmic (176/175)]].


== Manwe  ==
== No-3 valinorsmic ==
Commas: 176/175, 1331/1323
[[Subgroup]]: 2.5.7.11


Map: [<1 0 0 12 8|, <0 1 0 3 3|, <0 0 1 -6 -4|]
[[Comma list]]: 176/175


EDOs: 15, 31, 46, 77, 80, 111
{{Mapping|legend=2| 1 0 0 -4 | 0 1 0 2 | 0 0 1 1 }}


Badness: 0.000679
: sval mapping generators: ~2, ~5, ~7


=== 13-limit  ===
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 389.7641, ~7/4 = 972.1113
Commas: 176/175, 1331/1323, 351/350


Map: [<1 0 0 12 8|, <0 1 0 3 3|, <0 0 1 -6 -4|]
{{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}


EDOs: 31, 46, 77, 80, 111
[[Badness]]: 1.73 × 10<sup>-6</sup>


Badness: 0.001147
== Manwe ==
[[Subgroup]]: 2.3.5.7.11


== Varda  ==
[[Comma list]]: 176/175, 1331/1323
Commas: 176/175, 896/891


Map: [&lt;2 0 11 0 14|, &lt;0 1 -2 0 -4|, &lt;0 0 0 1 1|]
{{Mapping|legend=1| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }}


EDOs: 12, 22, 46, 58, 80, 126
: mapping generators: ~2, ~3, ~5


Badness: 0.000728
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 80, 111 }}


=== 13-limit  ===
[[Badness]]: 0.679 × 10<sup>-3</sup>
Commas: 176/175, 352/351, 364/363


Map: [&lt;2 0 11 0 14 24|, &lt;0 1 -2 0 -4 -7|, &lt;0 0 0 1 1 1|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 22, 46, 58, 80, 138cde, 242cde, 322bcde
Comma list: 176/175, 1331/1323, 351/350


Badness: 0.000970
Mapping: {{mapping| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }}


== Mandos  ==
{{Optimal ET sequence|legend=1| 31, 46, 77, 80, 111 }}
Commas: 176/175, 243/242


Map: [&lt;1 1 0 6 2|, &lt;0 2 0 5 5|, &lt;0 0 1 -2 0|]
Badness: 1.147 × 10<sup>-3</sup>


EDOs: 7, 24, 31, 58, 89
[[Category:Temperament clans]]
 
[[Category:Pages with mostly numerical content]]
Badness: 0.000751
[[Category:Valinorsmic clan| ]] <!-- main article -->
 
[[Category:Valinorsmic| ]] <!-- key article -->
=== 13-limit  ===
Commas: 144/143, 176/175, 243/242
 
Map: [&lt;1 1 0 6 2 4|, &lt;0 2 0 5 5 -1|, &lt;0 0 1 -2 0 0|]
 
EDOs: 7, 24, 31, 58
 
Badness: 0.000923
 
== Ulmo  ==
Commas: 176/175, 2200/2187
 
Map: [&lt;1 0 0 1, -3|, &lt;0 1 0 7 7|, &lt;0 0 1 -4 -2|]
 
EDOs: 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd
 
Badness: 0.00123
 
=== 13-limit  ===
Commas: 169/168, 176/175, 325/324
 
Map: [&lt;1 0 0 1, -3 2|, &lt;0 1 0 7 7 4|, &lt;0 0 1 -4 -2 -2|]
 
EDOs: 7, 39, 46, 53, 80, 133d, 179, 312cdf
 
Badness: 0.000932
 
[[Category:Theory]]
[[Category:Temperament]]
[[Category:Valinorsmic]]
[[Category:Rank 3]]
[[Category:Rank 3]]
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