Valinorsmic clan: Difference between revisions

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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).

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]].

Discussed elsewhere are:

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* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]]

Considered below are manwe, mandos and ulmo besides the no-3 subgroup temperament~~, but we can begin with~~ the rank-4 temperament.

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)]].

== No-3 valinorsmic ==

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

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: mapping generators: ~2, ~11/9, ~5

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-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).
+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]].
 Discussed elsewhere are:
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 * ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]]
-Considered below are manwe, mandos and ulmo besides the no-3 subgroup temperament, but we can begin with the rank-4 temperament.
+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)]].
 == No-3 valinorsmic ==
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 {{Mapping|legend=1| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }}
+: mapping generators: ~2, ~3, ~5
 {{Optimal ET sequence|legend=1| 15, 31, 46, 77, 80, 111 }}
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 {{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 }}