Valinorsmic clan - Revision history

FloraC: Re-organize

2026-05-14T19:58:41Z

Re-organize

@@ Line 5: / Line 5: @@
 == Valinor ==
-''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 [[Hemimean clan #tridecimal didacus|tridecimal didacus]].
+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 23: @@
 [[Badness]] (Sintel): 0.0793
 === Valimar ===
@@ Line 63: / Line 87: @@
 [[Badness]] (Sintel): 0.227
 == Manwe ==

FloraC: Complete data and misc. cleanup

2026-05-14T19:53:11Z

Complete data and misc. cleanup

← Older revision		Revision as of 19:53, 14 May 2026
Line 25:		Line 25:

	=== Valimar ===		=== 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 [[~~Orwell extensions\|~~winston]] and [[~~Valentine extensions\|~~valentino]]. This is also natural considering [[S-expression]]s, noting that 176/175 = S8/S10 is (nontrivially) equivalent to (~~S11S12~~)/(~~S14S15~~), 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]]).		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		[[Subgroup]]: 2.5.7.11.13
Line 40:		Line 40:
	: error map: {{val\| 0.000 +3.152 +2.691 -0.869 +1.607 }}		: error map: {{val\| 0.000 +3.152 +2.691 -0.869 +1.607 }}

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

	[[Badness]] (Sintel): 0.336		[[Badness]] (Sintel): 0.336

	=== Valarin ===		=== 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]], [[~~Valentine extensions\|~~lupercalia]] (an extension of [[valentine]]), and [[llywelyn]].		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		[[Subgroup]]: 2.5.7.11.13
Line 60:		Line 60:
	: error map: {{val\| 0.000 +2.875 +4.003 -0.110 +3.003 }}		: error map: {{val\| 0.000 +2.875 +4.003 -0.110 +3.003 }}

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

	[[Badness]] (Sintel): 0.227		[[Badness]] (Sintel): 0.227

Lériendil: /* Valinor */

2026-05-13T21:55:27Z

Valinor

← Older revision		Revision as of 21:55, 13 May 2026
Line 5:		Line 5:

	== Valinor ==		== Valinor ==
	''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]].		''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 [[Hemimean clan #tridecimal didacus\|tridecimal didacus]].

	[[Subgroup]]: 2.5.7.11		[[Subgroup]]: 2.5.7.11
Line 25:		Line 25:

	=== Valimar ===		=== 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 (S11S12)/(S14S15), 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]]).		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 [[Orwell extensions\|winston]] and [[Valentine extensions\|valentino]]. This is also natural considering [[S-expression]]s, noting that 176/175 = S8/S10 is (nontrivially) equivalent to (S11S12)/(S14S15), 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		[[Subgroup]]: 2.5.7.11.13
Line 45:		Line 45:

	=== Valarin ===		=== 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]].		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]], [[Valentine extensions\|lupercalia]] (an extension of [[valentine]]), and [[llywelyn]].

	[[Subgroup]]: 2.5.7.11.13		[[Subgroup]]: 2.5.7.11.13

Lériendil: added valimar and valarin following discussion of extensions on xengrove

2026-05-13T21:54:22Z

added valimar and valarin following discussion of extensions on xengrove

@@ Line 1: / Line 1: @@
 {{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
 === Overview to extensions ===
@@ Line 42: / Line 82: @@
 * ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]]
 Considered below are manwe and augenic.
 == Manwe ==

Lériendil: added description

2026-05-13T21:42:53Z

added description

← Older revision		Revision as of 21:42, 13 May 2026
Line 1:		Line 1:
	{{Technical data page}}		{{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 [[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]] [[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).

	For the rank-4 valinorsmic temperament, see [[Rank-4 temperament #Valinorsmic (176/175)]].		For the rank-4 valinorsmic temperament, see [[Rank-4 temperament #Valinorsmic (176/175)]].

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

	[[Subgroup]]: 2.5.7.11		[[Subgroup]]: 2.5.7.11

FloraC: + intro

2026-03-11T10:37:29Z

+ intro

← Older revision		Revision as of 10:37, 11 March 2026
Line 76:		Line 76:

	== Augenic ==		== Augenic ==
			{{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).

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11

FloraC: + error maps. Review optimal ET sequence. Unify precision

2026-03-11T10:26:34Z

+ error maps. Review optimal ET sequence. Unify precision

← Older revision		Revision as of 10:26, 11 March 2026
Line 13:		Line 13:

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~2 = 1199.~~311~~{{c}}, ~5/4 = 389.~~540~~{{c}}, ~7/4 = 971.~~553~~{{c}}		* [[WE]]: ~2 = 1199.3113{{c}}, ~5/4 = 389.5404{{c}}, ~7/4 = 971.5534{{c}}
	* [[CWE]]: ~2 = 1200.~~000~~{{c}}, ~5/4 = 389.~~393~~{{c}}, ~7/4 = 971.~~520~~{{c}}		: [[error map]]: {{val\| -0.689 +1.849 +1.350 -2.061 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 389.3935{{c}}, ~7/4 = 971.5196{{c}}
			: error map: {{val\| 0.000 +3.080 +2.694 -1.011 }}

	{{Optimal ET sequence\|legend=1\| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}		{{Optimal ET sequence\|legend=1\| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}

	[[Badness]] (Sintel): 0.~~079~~		[[Badness]] (Sintel): 0.0793

	=== Overview to extensions ===		=== Overview to 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. ~~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]].		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]].

	Discussed elsewhere are:		Discussed elsewhere are:
Line 49:		Line 51:

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~2 = 1199.~~360~~{{c}}, ~3/2 = 702.~~695~~{{c}}, ~5/4 = 389.~~312~~{{c}}		* [[WE]]: ~2 = 1199.3600{{c}}, ~3/2 = 702.6947{{c}}, ~5/4 = 389.3118{{c}}
	* [[CWE]]: ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~592~~{{c}}, ~5/4 = 389.~~376~~{{c}}		: [[error map]]: {{val\| -0.640 +0.100 +1.718 +1.467 -2.401 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.5916{{c}}, ~5/4 = 389.3756{{c}}
			: error map: {{val\| 0.000 +0.637 +3.062 +2.696 -1.045 }}

	{{Optimal ET sequence\|legend=1\| 15, 31, 46, 77, 80, 111 }}		{{Optimal ET sequence\|legend=1\| 15, 28de, 31, 46, 65d, 77, 80, 111, 237cd, 268cd }}

	[[Badness]] (Sintel): 0.816		[[Badness]] (Sintel): 0.816
Line 64:		Line 68:

	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1199.~~435~~{{c}}, ~3/2 = 702.~~506~~{{c}}, ~5/4 = 389.~~233~~{{c}}		* WE: ~2 = 1199.4350{{c}}, ~3/2 = 702.5061{{c}}, ~5/4 = 389.2327{{c}}
	* CWE: ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~703~~{{c}}, ~5/4 = 389.~~430~~{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7030{{c}}, ~5/4 = 389.4303{{c}}

	{{Optimal ET sequence\|legend=0\| 31, 46, 77, 80, 111 }}		{{Optimal ET sequence\|legend=0\| 31, 46, 65d, 77, 80, 111, 268cd }}

	Badness (Sintel): 1.~~073~~		Badness (Sintel): 1.07

	== Augenic ==		== Augenic ==
Line 80:		Line 84:

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~5/4 = 398.~~924~~{{c}}, ~3/2 = 705.~~145~~{{c}}, ~7/4 = 969.~~111~~{{c}}		* [[WE]]: ~5/4 = 398.9239{{c}}, ~3/2 = 705.1447{{c}}, ~7/4 = 969.1106{{c}}
	* [[CWE]]: ~5/4 = 400.~~000~~{{c}}, ~3/2 = 705.~~349~~{{c}}, ~7/4 = 968.~~440~~{{c}}		: [[error map]]: {{val\| -3.228 -0.039 +6.153 -6.172 +9.184 }}
			* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.3488{{c}}, ~7/4 = 968.4397{{c}}
			: error map: {{val\| 0.000 +3.394 +13.686 -0.386 +17.122 }}

	{{Optimal ET sequence\|legend=1\| 6, 9, 12, 15, 24, 27e }}		{{Optimal ET sequence\|legend=1\| 6, 9, 12, 15, 24, 27e, 51ce, 63cee }} *

			<nowiki/>* [[optimal patent val]]: [[36edo\|36]]

	[[Badness]] (Sintel): 0.735		[[Badness]] (Sintel): 0.735

FloraC: Sectioning & misc. cleanup. - redundant category

2026-03-11T10:13:19Z

Sectioning & misc. cleanup. - redundant category

← Older revision		Revision as of 10:13, 11 March 2026
Line 1:		Line 1:
	{{Technical data page}}		{{Technical data page}}
	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 '''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 ==
			[[Subgroup]]: 2.5.7.11

			[[Comma list]]: 176/175

			{{Mapping\|legend=2\| 1 0 0 -4 \| 0 1 0 2 \| 0 0 1 1 }}
			: mapping generators: ~2, ~5, ~7

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.311{{c}}, ~5/4 = 389.540{{c}}, ~7/4 = 971.553{{c}}
			* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 389.393{{c}}, ~7/4 = 971.520{{c}}

			{{Optimal ET sequence\|legend=1\| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}

			[[Badness]] (Sintel): 0.079

			=== Overview to 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. 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]].		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:		Discussed elsewhere are:
	* [[Zeus]] (+121/120) → ~~[[Porwell family #Zeus\|Porwell family]] and~~ [[Biyatismic clan #Zeus\|Biyatismic clan]]		* [[Zeus]] (+121/120) → [[Biyatismic clan #Zeus\|Biyatismic clan]]
	* ''[[Varda]]'' (+896/891) → [[Diaschismic rank-3 family #Varda\|Diaschismic rank-3 family]]		* ''[[Varda]]'' (+896/891) → [[Diaschismic rank-3 family #Varda\|Diaschismic rank-3 family]]
	* ''[[Nickel]]'' (+36/35 or 45/44) → [[Mint family #Nickel\|Mint family]]		* ''[[Nickel]]'' (+36/35 or 45/44) → [[Mint family #Nickel\|Mint family]]
Line 19:		Line 38:
	* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco\|Cataharry family]]		* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco\|Cataharry family]]

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

	~~== Valinor ==~~
	~~[[Subgroup]]: 2.5.7.11~~

	~~[[Comma list]]: 176/175~~

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

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

	~~[[Optimal tuning]]s:~~
	* [[WE]]: ~2 = 1199.311¢, ~5/4 = 389.540¢, ~7/4 = 971.553¢
	* [[CWE]]: ~2 = 1200.000¢, ~5/4 = 389.393¢, ~7/4 = 971.520¢

	~~{{Optimal ET sequence\|legend=1\| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}~~

	~~[[Badness]] (Sintel): 0.079~~

	== Manwe ==		== Manwe ==
Line 44:		Line 46:

	{{Mapping\|legend=1\| 1 0 0 12 8 \| 0 1 0 3 3 \| 0 0 1 -6 -4 }}		{{Mapping\|legend=1\| 1 0 0 12 8 \| 0 1 0 3 3 \| 0 0 1 -6 -4 }}

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

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~2 = 1199.~~360¢~~, ~3/2 = 702.~~695¢~~, ~5/4 = 389.~~312¢~~		* [[WE]]: ~2 = 1199.360{{c}}, ~3/2 = 702.695{{c}}, ~5/4 = 389.312{{c}}
	* [[CWE]]: ~2 = 1200.~~000¢~~, ~3/2 = 702.~~592¢~~, ~5/4 = 389.~~376¢~~		* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.592{{c}}, ~5/4 = 389.376{{c}}

	{{Optimal ET sequence\|legend=1\| 15, 31, 46, 77, 80, 111 }}		{{Optimal ET sequence\|legend=1\| 15, 31, 46, 77, 80, 111 }}
Line 63:		Line 64:

	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1199.~~435¢~~, ~3/2 = 702.~~506¢~~, ~5/4 = 389.~~233¢~~		* WE: ~2 = 1199.435{{c}}, ~3/2 = 702.506{{c}}, ~5/4 = 389.233{{c}}
	* CWE: ~2 = 1200.~~000¢~~, ~3/2 = 702.~~703¢~~, ~5/4 = 389.~~430¢~~		* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.703{{c}}, ~5/4 = 389.430{{c}}

	{{Optimal ET sequence\|legend=0\| 31, 46, 77, 80, 111 }}		{{Optimal ET sequence\|legend=0\| 31, 46, 77, 80, 111 }}
Line 76:		Line 77:

	{{Mapping\|legend=1\| 3 0 7 0 2 \| 0 1 0 0 0 \| 0 0 0 1 1 }}		{{Mapping\|legend=1\| 3 0 7 0 2 \| 0 1 0 0 0 \| 0 0 0 1 1 }}

	: mapping generators: ~5/4, ~3, ~7		: mapping generators: ~5/4, ~3, ~7

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~5/4 = 398.~~924¢~~, ~3/2 = 705.~~145¢~~, ~7/4 = 969.~~111¢~~		* [[WE]]: ~5/4 = 398.924{{c}}, ~3/2 = 705.145{{c}}, ~7/4 = 969.111{{c}}
	* [[CWE]]: ~5/4 = 400.~~000¢~~, ~3/2 = 705.~~349¢~~, ~7/4 = 968.~~440¢~~		* [[CWE]]: ~5/4 = 400.000{{c}}, ~3/2 = 705.349{{c}}, ~7/4 = 968.440{{c}}

	{{Optimal ET sequence\|legend=1\| 6, 9, 12, 15, 24, 27e }}		{{Optimal ET sequence\|legend=1\| 6, 9, 12, 15, 24, 27e }}
Line 89:		Line 89:
	[[Category:Temperament clans]]		[[Category:Temperament clans]]
	[[Category:Valinorsmic clan\| ]] <!-- main article -->		[[Category:Valinorsmic clan\| ]] <!-- main article -->
	~~[[Category:Valinorsmic\| ]] <!-- key article -->~~
	[[Category:Rank 3]]		[[Category:Rank 3]]

Xenllium: New temperament: augenic (9&12&15)

2026-03-01T07:42:44Z

New temperament: augenic (9&12&15)

← Older revision		Revision as of 07:42, 1 March 2026
Line 19:		Line 19:
	* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco\|Cataharry family]]		* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco\|Cataharry family]]

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

	== Valinor ==		== Valinor ==
Line 30:		Line 30:
	: sval mapping generators: ~2, ~5, ~7		: sval mapping generators: ~2, ~5, ~7

	[[Optimal tuning]] ([[~~POTE~~]]): ~2 = ~~1\1~~, ~5/4 = 389.~~7641~~, ~7/4 = ~~972~~.~~1113~~		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.311¢, ~5/4 = 389.540¢, ~7/4 = 971.553¢
			* [[CWE]]: ~2 = 1200.000¢, ~5/4 = 389.393¢, ~7/4 = 971.520¢

	{{Optimal ET sequence\|legend=1\| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}		{{Optimal ET sequence\|legend=1\| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}

	[[Badness]]: 1.~~73 × 10<sup>-6</sup>~~		[[Badness]] (Sintel): 0.079

	== Manwe ==		== Manwe ==
Line 44:		Line 46:

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

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.360¢, ~3/2 = 702.695¢, ~5/4 = 389.312¢
			* [[CWE]]: ~2 = 1200.000¢, ~3/2 = 702.592¢, ~5/4 = 389.376¢

	{{Optimal ET sequence\|legend=1\| 15, 31, 46, 77, 80, 111 }}		{{Optimal ET sequence\|legend=1\| 15, 31, 46, 77, 80, 111 }}

	[[Badness]]: 0.~~679 × 10<sup>-3</sup>~~		[[Badness]] (Sintel): 0.816

	=== 13-limit ===		=== 13-limit ===
	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.13

	Comma list: 176/175, 1331/1323, ~~351~~/~~350~~		Comma list: 176/175, 351/350, 1331/1323

			Mapping: {{mapping\| 1 0 0 12 8 13 \| 0 1 0 3 3 0 \| 0 0 1 -6 -4 -4 }}

			Optimal tunings:
			* WE: ~2 = 1199.435¢, ~3/2 = 702.506¢, ~5/4 = 389.233¢
			* CWE: ~2 = 1200.000¢, ~3/2 = 702.703¢, ~5/4 = 389.430¢

			{{Optimal ET sequence\|legend=0\| 31, 46, 77, 80, 111 }}

			Badness (Sintel): 1.073

			== Augenic ==
			[[Subgroup]]: 2.3.5.7.11

			[[Comma list]]: 56/55, 128/125

			{{Mapping\|legend=1\| 3 0 7 0 2 \| 0 1 0 0 0 \| 0 0 0 1 1 }}

			: mapping generators: ~5/4, ~3, ~7

	~~Mapping~~: ~~{{mapping\| 1 0 0 12 8 \| 0 1 0~~ 3 3 ~~\| 0 0 1 -6 -~~4 }}		[[Optimal tuning]]s:
			* [[WE]]: ~5/4 = 398.924¢, ~3/2 = 705.145¢, ~7/4 = 969.111¢
			* [[CWE]]: ~5/4 = 400.000¢, ~3/2 = 705.349¢, ~7/4 = 968.440¢

	{{Optimal ET sequence\|legend=1\| 31, 46, 77, 80, ~~111~~ }}		{{Optimal ET sequence\|legend=1\| 6, 9, 12, 15, 24, 27e }}

	Badness: 1.~~147 × 10<sup>-3</sup>~~		[[Badness]] (Sintel): 0.735

	[[Category:Temperament clans]]		[[Category:Temperament clans]]

FloraC: Text replacement - "Category:Temperament clans↵Category:Pages with mostly numerical content" to "Category:Temperament clans"

2025-11-27T12:20:36Z

Text replacement - "Category:Temperament clans↵Category:Pages with mostly numerical content" to "Category:Temperament clans"

← Older revision		Revision as of 12:20, 27 November 2025
Line 61:		Line 61:

	[[Category:Temperament clans]]		[[Category:Temperament clans]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Valinorsmic clan\| ]] <!-- main article -->		[[Category:Valinorsmic clan\| ]] <!-- main article -->
	[[Category:Valinorsmic\| ]] <!-- key article -->		[[Category:Valinorsmic\| ]] <!-- key article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]