Valinorsmic clan: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
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 '''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 ==
[[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 = 1199.311{{c}}, ~5/4 = 389.540{{c}}, ~7/4 = 971.553{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 389.393{{c}}, ~7/4 = 971.520{{c}}
{{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}
[[Badness]] (Sintel): 0.079
=== 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. 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]].  
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]].  


Discussed elsewhere are:  
Discussed elsewhere are:  
* [[Zeus]] (+121/120) → [[Porwell family #Zeus|Porwell family]] and [[Biyatismic clan #Zeus|Biyatismic clan]]
* [[Zeus]] (+121/120) → [[Biyatismic clan #Zeus|Biyatismic clan]]
* ''[[Varda]]'' (+896/891) → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]
* ''[[Varda]]'' (+896/891) → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]
* ''[[Nickel]]'' (+36/35 or 45/44) → [[Mint family #Nickel|Mint family]]
* ''[[Nickel]]'' (+36/35 or 45/44) → [[Mint family #Nickel|Mint family]]
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* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]]
* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]]


Considered below are manwe and augenic, besides the no-3 subgroup temperament. For the rank-4 valinorsmic temperament, see [[Rank-4 temperament #Valinorsmic (176/175)]].
Considered below are manwe and augenic.  
 
== Valinor ==
[[Subgroup]]: 2.5.7.11
 
[[Comma list]]: 176/175
 
{{Mapping|legend=2| 1 0 0 -4 | 0 1 0 2 | 0 0 1 1 }}
 
: sval mapping generators: ~2, ~5, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.311¢, ~5/4 = 389.540¢, ~7/4 = 971.553¢
* [[CWE]]: ~2 = 1200.000¢, ~5/4 = 389.393¢, ~7/4 = 971.520¢
 
{{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}
 
[[Badness]] (Sintel): 0.079


== Manwe ==
== Manwe ==
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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
: mapping generators: ~2, ~3, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.360¢, ~3/2 = 702.695¢, ~5/4 = 389.312¢
* [[WE]]: ~2 = 1199.360{{c}}, ~3/2 = 702.695{{c}}, ~5/4 = 389.312{{c}}
* [[CWE]]: ~2 = 1200.000¢, ~3/2 = 702.592¢, ~5/4 = 389.376¢
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.592{{c}}, ~5/4 = 389.376{{c}}


{{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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Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.435¢, ~3/2 = 702.506¢, ~5/4 = 389.233¢
* WE: ~2 = 1199.435{{c}}, ~3/2 = 702.506{{c}}, ~5/4 = 389.233{{c}}
* CWE: ~2 = 1200.000¢, ~3/2 = 702.703¢, ~5/4 = 389.430¢
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.703{{c}}, ~5/4 = 389.430{{c}}


{{Optimal ET sequence|legend=0| 31, 46, 77, 80, 111 }}
{{Optimal ET sequence|legend=0| 31, 46, 77, 80, 111 }}
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{{Mapping|legend=1| 3 0 7 0 2 | 0 1 0 0 0 | 0 0 0 1 1 }}
{{Mapping|legend=1| 3 0 7 0 2 | 0 1 0 0 0 | 0 0 0 1 1 }}
: mapping generators: ~5/4, ~3, ~7
: mapping generators: ~5/4, ~3, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~5/4 = 398.924¢, ~3/2 = 705.145¢, ~7/4 = 969.111¢
* [[WE]]: ~5/4 = 398.924{{c}}, ~3/2 = 705.145{{c}}, ~7/4 = 969.111{{c}}
* [[CWE]]: ~5/4 = 400.000¢, ~3/2 = 705.349¢, ~7/4 = 968.440¢
* [[CWE]]: ~5/4 = 400.000{{c}}, ~3/2 = 705.349{{c}}, ~7/4 = 968.440{{c}}


{{Optimal ET sequence|legend=1| 6, 9, 12, 15, 24, 27e }}
{{Optimal ET sequence|legend=1| 6, 9, 12, 15, 24, 27e }}
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[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Valinorsmic clan| ]] <!-- main article -->
[[Category:Valinorsmic clan| ]] <!-- main article -->
[[Category:Valinorsmic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 10:13, 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

Optimal tunings:

  • WE: ~2 = 1199.311 ¢, ~5/4 = 389.540 ¢, ~7/4 = 971.553 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 389.393 ¢, ~7/4 = 971.520 ¢

Optimal ET sequence: 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd

Badness (Sintel): 0.079

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. 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.

Discussed elsewhere are:

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

Optimal tunings:

  • WE: ~2 = 1199.360 ¢, ~3/2 = 702.695 ¢, ~5/4 = 389.312 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.592 ¢, ~5/4 = 389.376 ¢

Optimal ET sequence: 15, 31, 46, 77, 80, 111

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 = 1199.435 ¢, ~3/2 = 702.506 ¢, ~5/4 = 389.233 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.703 ¢, ~5/4 = 389.430 ¢

Optimal ET sequence: 31, 46, 77, 80, 111

Badness (Sintel): 1.073

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

Optimal tunings:

  • WE: ~5/4 = 398.924 ¢, ~3/2 = 705.145 ¢, ~7/4 = 969.111 ¢
  • CWE: ~5/4 = 400.000 ¢, ~3/2 = 705.349 ¢, ~7/4 = 968.440 ¢

Optimal ET sequence: 6, 9, 12, 15, 24, 27e

Badness (Sintel): 0.735

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