Valinorsmic clan: Difference between revisions

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== 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

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=== 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]].

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

Considered below are manwe and augenic.

=== Valimar ===

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]]).

[[Subgroup]]: 2.5.7.11.13

[[Comma list]]: 176/175, 1573/1568

: mapping generators: ~2, ~5, ~7

[[Optimal tuning]]s:

* [[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 }}

{{Optimal ET sequence|legend=1| 6, 9, 15f, 22f, 28, 31, 37, 231cd, 268cd }}

[[Badness]] (Sintel): 0.336

=== 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]].

[[Subgroup]]: 2.5.7.11.13

[[Comma list]]: 176/175, 640/637

: mapping generators: ~2, ~5, ~7

[[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 }}

{{Optimal ET sequence|legend=1| 6, 10e, 15, 16, 21, 31, 37, 206cdf, 243cdf, 280cdf, 317cddf, 354ccddff }}

[[Badness]] (Sintel): 0.227

== Manwe ==

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 == 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
@@ Line 23: / Line 25: @@
 === 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]].
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 * ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]]
 Considered below are manwe and augenic.
+=== Valimar ===
+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]]).
+[[Subgroup]]: 2.5.7.11.13
+[[Comma list]]: 176/175, 1573/1568
+{{Mapping|legend=2| 1 0 0 -4 13 | 0 1 0 2 -4 | 0 0 1 1 0 }}
+: mapping generators: ~2, ~5, ~7
+[[Optimal tuning]]s:
+* [[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 }}
+{{Optimal ET sequence|legend=1| 6, 9, 15f, 22f, 28, 31, 37, 231cd, 268cd }}
+[[Badness]] (Sintel): 0.336
+=== 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]].
+[[Subgroup]]: 2.5.7.11.13
+[[Comma list]]: 176/175, 640/637
+{{Mapping|legend=2| 1 0 0 -4 7 | 0 1 0 2 1 | 0 0 1 1 -2 }}
+: mapping generators: ~2, ~5, ~7
+[[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 }}
+{{Optimal ET sequence|legend=1| 6, 10e, 15, 16, 21, 31, 37, 206cdf, 243cdf, 280cdf, 317cddf, 354ccddff }}
+[[Badness]] (Sintel): 0.227
 == Manwe ==