Xenharmonic Wiki

Valinorsmic clan: Difference between revisions

← Older edit
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
m +cat, +link
Re-organize
 
(34 intermediate revisions by 7 users not shown)
Line 1: Line 1:
__FORCETOC__
{{Technical data page}}
The valinorsmic rank three temperaments temper 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).  


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


Map: [<1 0 0 0 -4|, <0 1 0 0 0|, <0 0 1 0 2|, <0 0 0 1 1|]
== Valinor ==
Valinor is the [[2.5.7.11 subgroup|2.5.7.11-subgroup]] temperament defined by tempering out 176/175.


EDOs: 22, 31, 46, 53, 58, 80, 111
[[Subgroup]]: 2.5.7.11


=Manwe=
[[Comma list]]: 176/175
Commas: 176/175, 1331/1323


Map: [<1 0 0 12 8|, <0 1 0 3 3|, <0 0 1 -6 -4|]
{{Mapping|legend=2| 1 0 0 -4 | 0 1 0 2 | 0 0 1 1 }}
: mapping generators: ~2, ~5, ~7


EDOs: 15, 31, 46, 77, 80, 111
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.3113{{c}}, ~5/4 = 389.5404{{c}}, ~7/4 = 971.5534{{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 }}


Badness: 0.000679
{{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}


==13-limit==
[[Badness]] (Sintel): 0.0793
Commas: 176/175, 1331/1323, 351/350


Map: [<1 0 0 12 8|, <0 1 0 3 3|, <0 0 1 -6 -4|]
=== 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]].


EDOs: 31, 46, 77, 80, 111
==== 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]].


Badness: 0.001147
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]]


=Varda=
Considered below are manwe and augenic.
Commas: 176/175, 896/891


Map: [<2 0 11 0 14|, <0 1 -2 0 -4|, <0 0 0 1 1|]
=== 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]]).


EDOs: 12, 22, 46, 58, 80, 126
[[Subgroup]]: 2.5.7.11.13


Badness: 0.000728
[[Comma list]]: 176/175, 1573/1568


==13-limit==
{{Mapping|legend=2| 1 0 0 -4 13 | 0 1 0 2 -4 | 0 0 1 1 0 }}
Commas: 176/175, 352/351, 364/363
: mapping generators: ~2, ~5, ~7


Map: [<2 0 11 0 14 24|, <0 1 -2 0 -4 -7|, <0 0 0 1 1 1|]
[[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 }}


EDOs: 22, 46, 58, 80, 138cde, 242cde, 322bcde
{{Optimal ET sequence|legend=1| 6, 9, 15f, 22f, 28, 31, 37, 231cd, 268cd }}


Badness: 0.000970
[[Badness]] (Sintel): 0.336


=Mandos=
=== Valarin ===
Commas: 176/175, 243/242
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]].


Map: [<1 1 0 6 2|, <0 2 0 5 5|, <0 0 1 -2 0|]
[[Subgroup]]: 2.5.7.11.13


EDOs: 7, 24, 31, 58, 89
[[Comma list]]: 176/175, 640/637


Badness: 0.000751
{{Mapping|legend=2| 1 0 0 -4 7 | 0 1 0 2 1 | 0 0 1 1 -2 }}
: mapping generators: ~2, ~5, ~7


==13-limit==
[[Optimal tuning]]s:
Commas: 144/143, 176/175, 243/242
* [[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 }}


Map: [<1 1 0 6 2 4|, <0 2 0 5 5 -1|, <0 0 1 -2 0 0|]
{{Optimal ET sequence|legend=1| 6, 10e, 15, 16, 21, 31, 37, 206cdf, 243cdf, 280cdf, 317cddf, 354ccddff }}


EDOs: 7, 24, 31, 58
[[Badness]] (Sintel): 0.227


Badness: 0.000923
== Manwe ==
[[Subgroup]]: 2.3.5.7.11


=Ulmo=
[[Comma list]]: 176/175, 1331/1323
Commas: 176/175, 2200/2187


Map: [<1 0 0 1, -3|, <0 1 0 7 7|, <0 0 1 -4 -2|]
{{Mapping|legend=1| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }}
: mapping generators: ~2, ~3, ~5


EDOs: 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.3600{{c}}, ~3/2 = 702.6947{{c}}, ~5/4 = 389.3118{{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 }}


Badness: 0.00123
{{Optimal ET sequence|legend=1| 15, 28de, 31, 46, 65d, 77, 80, 111, 237cd, 268cd }}


==13-limit==
[[Badness]] (Sintel): 0.816
Commas: 169/168, 176/175, 325/324


Map: [<1 0 0 1, -3 2|, <0 1 0 7 7 4|, <0 0 1 -4 -2 -2|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 7, 39, 46, 53, 80, 133d, 179, 312cdf
Comma list: 176/175, 351/350, 1331/1323


Badness: 0.000932
Mapping: {{mapping| 1 0 0 12 8 13 | 0 1 0 3 3 0 | 0 0 1 -6 -4 -4 }}


[[Category:Valinorsmic]]
Optimal tunings:
* WE: ~2 = 1199.4350{{c}}, ~3/2 = 702.5061{{c}}, ~5/4 = 389.2327{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7030{{c}}, ~5/4 = 389.4303{{c}}
 
{{Optimal ET sequence|legend=0| 31, 46, 65d, 77, 80, 111, 268cd }}
 
Badness (Sintel): 1.07
 
== 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
 
[[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
 
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 398.9239{{c}}, ~3/2 = 705.1447{{c}}, ~7/4 = 969.1106{{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, 51ce, 63cee }} *
 
<nowiki/>* [[optimal patent val]]: [[36edo|36]]
 
[[Badness]] (Sintel): 0.735
 
[[Category:Temperament clans]]
[[Category:Valinorsmic clan| ]] <!-- main article -->
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Valinorsmic_clan"