Valinorsmic clan: Difference between revisions

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The '''valinorsmic clan''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the valinorsma, [[176/175]], which equates ~~a stack of two~~ [[5/4]]s with [[11/~~7]]; ideally, 5/4 is slightly sharpened to make this equivalence. The valinorsma can also be seen as equating [[35/32]] with [[~~11~~/10~~]] (rather than [[12/11]], 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 ==

''Valinor'' is the 2.5.7.11 temperament defined by tempering out 176/175.

''Valinor'' is the 2.5.7.11 temperament defined by tempering out 176/175. When extending to prime 13, two strong extensions are considerable: ''valimar'' and ''valarin''; these intersect in [[tridecimal didacus]].

[[Subgroup]]: 2.5.7.11

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[[Badness]] (Sintel): 0.0793

=== Valimar ===

This extension to the no-threes 13-limit is motivated by extending the valinorsmic chain of 5/4s, where two form 11/7, so that four form 16/13. This logic is supported by rank-2s 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 S11/S14 = [[1573/1568]], and S12/S15 = [[3584/3575]]. It remains well-tuned in ETs with optimal valinorsmic 5/4s (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 }}

[[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, [[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 }}

[[Badness]] (Sintel): 0.227

=== Overview to extensions ===

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

Considered below are manwe and augenic.

== Manwe ==

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 {{Technical data page}}
-The '''valinorsmic clan''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the valinorsma, [[176/175]], which equates a stack of two [[5/4]]s with [[11/7]]; ideally, 5/4 is slightly sharpened to make this equivalence. The valinorsma can also be seen as equating [[35/32]] with [[11/10]] (rather than [[12/11]], 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 ==
-''Valinor'' is the 2.5.7.11 temperament defined by tempering out 176/175.
+''Valinor'' is the 2.5.7.11 temperament defined by tempering out 176/175. When extending to prime 13, two strong extensions are considerable: ''valimar'' and ''valarin''; these intersect in [[tridecimal didacus]].
 [[Subgroup]]: 2.5.7.11
@@ Line 23: / Line 23: @@
 [[Badness]] (Sintel): 0.0793
+=== Valimar ===
+This extension to the no-threes 13-limit is motivated by extending the valinorsmic chain of 5/4s, where two form 11/7, so that four form 16/13. This logic is supported by rank-2s 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 S11/S14 = [[1573/1568]], and S12/S15 = [[3584/3575]]. It remains well-tuned in ETs with optimal valinorsmic 5/4s (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| TBD }}
+[[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, [[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| TBD }}
+[[Badness]] (Sintel): 0.227
 === Overview to extensions ===
@@ Line 42: / Line 82: @@
 * ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]]
 Considered below are manwe and augenic.
 == Manwe ==