Valinorsmic clan: Difference between revisions

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

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

* ''[[Lono]]'' (+5120/5103) → [[~~Hemifamity~~ family #Lono|~~Hemifamity~~ family]]

* ''[[Lono]]'' (+5120/5103) → [[Aberschismic family #Lono|Aberschismic family]]

* ''[[Mandos]]'' (+243/242) → [[Rastmic rank-3 clan #Mandos|Rastmic rank-3 clan]]

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

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Considered below are manwe and augenic.

=== ~~Valimar~~ ===

==== Subgroup extensions ====

~~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]]).

When extending to the 2.5.7.11.13 subgroup, two strong extensions are considerable: ''valimar'' and ''valarin'', considered in [[#Subgroup extensions_2|#Subgroup extensions]]; they intersect in [[hemimean clan #tridecimal didacus|tridecimal didacus]].

~~[[Subgroup]]:~~ 2.5.7.11.13

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

~~{{Mapping|legend=3| 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=3| 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 ==

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Named by [[Xenllium]] in 2026, augenic is closely related to [[augene]]. It tempers out the [[augmented comma]] but with a free generator for [[7/1|7]], and then extends it to the 11-limit through the identity [[128/125]] = ([[56/55]])⋅(176/175).

Named by [[Xenllium]] in 2026, augenic is closely related to [[augmented family #Septimal augmented (augene)|augene]]. It tempers out the [[augmented comma]] but with a free generator for [[7/1|7]], and then extends it to the 11-limit through the identity [[128/125]] = ([[56/55]])⋅(176/175).

[[Subgroup]]: 2.3.5.7.11

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[[Badness]] (Sintel): 0.735

== Subgroup extensions ==

=== Valimar (2.5.7.11.13) ===

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 [[sensamagic clan #Magus|magus]]/[[starling temperaments #Amigo|amigo]] extensions, as well as [[semicomma family #Winston|winston]] and [[gamelismic clan #Valentino|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 (2.5.7.11.13) ===

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, [[semicomma family #Orwell|tridecimal orwell]], [[gamelismic clan #Lupercalia|lupercalia]] (an extension of [[gamelismic clan #valentine|valentine]]), and [[no-threes subgroup temperaments #Llywelyn a.k.a. shoe|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

[[Category:Valinorsmic clan| ]]

@@ Line 25: / 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]].
@@ Line 40: / Line 37: @@
 * ''[[Guanyin]]'' (+540/539) → [[Orwellismic family #Guanyin|Orwellismic family]]
 * ''[[Clio]]'' (+81/80) → [[Didymus rank-3 family #Clio|Didymus rank-3 family]]
-* ''[[Lono]]'' (+5120/5103) → [[Hemifamity family #Lono|Hemifamity family]]
+* ''[[Lono]]'' (+5120/5103) → [[Aberschismic family #Lono|Aberschismic family]]
 * ''[[Mandos]]'' (+243/242) → [[Rastmic rank-3 clan #Mandos|Rastmic rank-3 clan]]
 * ''[[Shrusus]]'' (+245/243) → [[Sensamagic family #Shrusus|Sensamagic family]]
@@ Line 48: / Line 45: @@
 Considered below are manwe and augenic.
-=== Valimar ===
+==== Subgroup extensions ====
-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]]).
+When extending to the 2.5.7.11.13 subgroup, two strong extensions are considerable: ''valimar'' and ''valarin'', considered in [[#Subgroup extensions_2|#Subgroup extensions]]; they intersect in [[hemimean clan #tridecimal didacus|tridecimal didacus]].
-[[Subgroup]]: 2.5.7.11.13
-[[Comma list]]: 176/175, 1573/1568
-{{Mapping|legend=3| 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=3| 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 ==
@@ Line 124: / Line 84: @@
 {{Distinguish| Augene }}
-Named by [[Xenllium]] in 2026, augenic is closely related to [[augene]]. It tempers out the [[augmented comma]] but with a free generator for [[7/1|7]], and then extends it to the 11-limit through the identity [[128/125]] = ([[56/55]])⋅(176/175).
+Named by [[Xenllium]] in 2026, augenic is closely related to [[augmented family #Septimal augmented (augene)|augene]]. It tempers out the [[augmented comma]] but with a free generator for [[7/1|7]], and then extends it to the 11-limit through the identity [[128/125]] = ([[56/55]])⋅(176/175).
 [[Subgroup]]: 2.3.5.7.11
@@ Line 144: / Line 104: @@
 [[Badness]] (Sintel): 0.735
+== Subgroup extensions ==
+=== Valimar (2.5.7.11.13) ===
+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 [[sensamagic clan #Magus|magus]]/[[starling temperaments #Amigo|amigo]] extensions, as well as [[semicomma family #Winston|winston]] and [[gamelismic clan #Valentino|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=3| 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 (2.5.7.11.13) ===
+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, [[semicomma family #Orwell|tridecimal orwell]], [[gamelismic clan #Lupercalia|lupercalia]] (an extension of [[gamelismic clan #valentine|valentine]]), and [[no-threes subgroup temperaments #Llywelyn a.k.a. shoe|llywelyn]].
+[[Subgroup]]: 2.5.7.11.13
+[[Comma list]]: 176/175, 640/637
+{{Mapping|legend=3| 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
 [[Category:Valinorsmic clan| ]] <!-- main article -->

Valinorsmic clan: Difference between revisions

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