Valinorsmic clan: Difference between revisions

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== 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 subgroup|2.5.7.11-subgroup]] temperament defined by tempering out 176/175.

[[Subgroup]]: 2.5.7.11

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[[Badness]] (Sintel): 0.0793

=== 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]].

Discussed elsewhere are:

* [[Zeus]] (+121/120) → [[Biyatismic clan #Zeus|Biyatismic clan]]

* ''[[Varda]]'' (+896/891) → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]

* ''[[Nickel]]'' (+36/35 or 45/44) → [[Mint family #Nickel|Mint family]]

* [[Ares]] (+64/63 or 100/99) → [[Archytas family #Ares|Archytas family]]

* ''[[Minerva]]'' (+99/98) → [[Marvel family #Minerva|Marvel family]]

* [[Thrush]] (+126/125) → [[Starling family #Thrush|Starling family]]

* ''[[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]]

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

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

* ''[[Ulmo]]'' (+2200/2187) → [[Ragismic family #Ulmo|Ragismic family]]

* ''[[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 5/4s, 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]]).

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

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: 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, [[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]], [[lupercalia]] (an extension of [[valentine]]), and [[llywelyn]].

[[Subgroup]]: 2.5.7.11.13

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: 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

~~=== 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. 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:~~