Valinorsmic clan: Difference between revisions

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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]].

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:

* ''[[Zeus]]'' (+121/120) → [[Porwell family #Zeus|Porwell family]]

* ''[[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]]

* ''[[Clio]]'' (+81/80) → [[Didymus rank three family #Clio|Didymus rank-3 family]]

* ''[[Minerva]]'' (+99/98) → [[Marvel family #Minerva|Marvel family]]

* ''[[Zeus]]'' (+121/120) → [[Porwell family #Zeus|Porwell family]]

* ''[[Thrush]]'' (+126/125) → [[Starling family #Thrush|Starling family]]

* ''[[Shrusus]]'' (+245/243) → [[Sensamagic family #Shrusus|Sensamagic family]]

* ''[[Guanyin]]'' (+540/539) → [[Orwellismic family #Guanyin|Orwellismic family]]

* ''[[~~Julius]]'' or ''[[varda~~]]'' (+~~896~~/~~891~~) → [[~~Diaschismic~~ rank three family #~~Julius aka varda~~|~~Diaschismic~~ rank-3 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]]

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)]].

Considered below is manwe, besides the no-3 subgroup temperament. For the rank-4 valinorsmic temperament, see [[Rank-4 temperament #Valinorsmic (176/175)]].

== No-3 valinorsmic ==

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Badness: 1.147 × 10-3

~~== Mandos ==~~

~~[[Subgroup]]: 2.3.5.7.11~~

~~[[Comma list]]: 176/175, 243/242~~

~~{{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 }}~~

~~[[Badness]]: 0.751 × 10-3~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 144/143, 176/175, 243/242~~

~~Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 0 5 5 -1 | 0 0 1 -2 0 0 }}~~

~~{{Optimal ET sequence|legend=1| 7, 24, 31, 58 }}~~

~~Badness: 0.923 × 10-3~~

~~== Ulmo ==~~

~~[[Subgroup]]: 2.3.5.7.11~~

~~[[Comma list]]: 176/175, 2200/2187~~

~~{{Mapping|legend=1| 1 0 0 1 -3 | 0 1 0 7 7 | 0 0 1 -4 -2 }}~~

~~{{Optimal ET sequence|legend=1| 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: {{mapping| 1 0 0 1 -3 2 | 0 1 0 7 7 4 | 0 0 1 -4 -2 -2 }}~~

~~{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 133d, 179, 312cdf }}~~

~~Badness: 0.932 × 10-3~~

[[Category:Temperament clans]]

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 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]].
+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:
+* ''[[Zeus]]'' (+121/120) → [[Porwell family #Zeus|Porwell family]]
+* ''[[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]]
-* ''[[Clio]]'' (+81/80) → [[Didymus rank three family #Clio|Didymus rank-3 family]]
 * ''[[Minerva]]'' (+99/98) → [[Marvel family #Minerva|Marvel family]]
-* ''[[Zeus]]'' (+121/120) → [[Porwell family #Zeus|Porwell family]]
 * ''[[Thrush]]'' (+126/125) → [[Starling family #Thrush|Starling family]]
-* ''[[Shrusus]]'' (+245/243) → [[Sensamagic family #Shrusus|Sensamagic family]]
 * ''[[Guanyin]]'' (+540/539) → [[Orwellismic family #Guanyin|Orwellismic family]]
-* ''[[Julius]]'' or ''[[varda]]'' (+896/891) → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 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]]
-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)]].
+Considered below is manwe, besides the no-3 subgroup temperament. For the rank-4 valinorsmic temperament, see [[Rank-4 temperament #Valinorsmic (176/175)]].
 == No-3 valinorsmic ==
@@ Line 56: / Line 58: @@
 Badness: 1.147 × 10<sup>-3</sup>
-== Mandos ==
-[[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: 176/175, 243/242
-{{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 }}
-[[Badness]]: 0.751 × 10<sup>-3</sup>
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 144/143, 176/175, 243/242
-Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 0 5 5 -1 | 0 0 1 -2 0 0 }}
-{{Optimal ET sequence|legend=1| 7, 24, 31, 58 }}
-Badness: 0.923 × 10<sup>-3</sup>
-== Ulmo ==
-[[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: 176/175, 2200/2187
-{{Mapping|legend=1| 1 0 0 1 -3 | 0 1 0 7 7 | 0 0 1 -4 -2 }}
-{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd }}
-[[Badness]]: 1.23 × 10<sup>-3</sup>
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 176/175, 325/324
-Mapping: {{mapping| 1 0 0 1 -3 2 | 0 1 0 7 7 4 | 0 0 1 -4 -2 -2 }}
-{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 133d, 179, 312cdf }}
-Badness: 0.932 × 10<sup>-3</sup>
 [[Category:Temperament clans]]