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No-threes subgroup temperaments: Difference between revisions

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→Temperaments with a 2.5.7 gene: move pakkanen to the comma page as this page is mainly for rank-2
Move mercy back here
 
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Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
* For no-fives, see [[#No-threes no-fives subgroup temperaments]].
* For no-fives, see [[#No-threes no-fives subgroup temperaments]].
* French decimal and trader have a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.
* Trader has a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.
* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
* Insect has a ~55/32 generator, three of which give the ~5/1.
* Insect has a ~55/32 generator, three of which give the ~5/1.
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* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus|Hemimean clan]]
* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus|Hemimean clan]]
* Mercy ([[823543/819200]]) → [[Quince clan #Mercy|Quince clan]]
* Llywelyn a.k.a. shoe ([[4194304/4117715]]) → [[Llywelynsmic clan #Llywelyn a.k.a. shoe|Llywelynsmic clan]]
* Sidewalk ([[823543/800000]]) → [[2023/2000 #Sidewalk]]
* Sidewalk ([[823543/800000]]) → [[2023/2000 #Sidewalk]]
=== Rainy ===
In rainy, three generators make an [[8/7]]; five generators make a [[5/4]]. It is the no-3's [[restriction]] of [[tertiaseptal]] (and [[valentine]]), notable theoretically as it equates ([[2/1]])/([[5/4]])<sup>3</sup> (128/125, the lesser diesis) with ([[2/1]])/([[8/7]])<sup>5</sup> (the 2.7-subgroup [[cloudy comma]], which is similar to the 2.5-subgroup lesser diesis in that tempering it out tunes the 8/7 about 8.8{{cent}} sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7{{cent}}). By tempering out their difference, stacked 5's and stacked 7's become easier to navigate, using the general-purpose diesis to simplify clusters.
A highly notable tuning of rainy not shown here is [[311edo]], which is 140 + 171 so tuned between them.
[[Subgroup]]: 2.5.7
[[Comma list]]: [[2100875/2097152]]
{{Mapping|legend=2| 1 2 3 | 0 5 -3 }}
{{Mapping|legend=3| 1 0 2 3 | 0 0 5 -3 }}
: mapping generators: ~2, ~256/245
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0939{{c}}, ~256/245 = 77.2107{{c}}
: [[error map]]: {{val| +0.094 -0.072 -0.176 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/245 = 77.2093{{c}}
: error map: {{val| 0.000 -0.267 -0.454 }}
{{Optimal ET sequence|legend=1| 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d }}
[[Badness]] (Sintel): 0.156


=== Augment ===
=== Augment ===
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[[Comma list]]: 128/125
[[Comma list]]: 128/125


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


{{Mapping|legend=3| 3 0 7 0 | 0 0 0 1 }}
{{Mapping|legend=5| 3 0 7 0 | 0 0 0 1 }}
: mapping generators: ~5/4, ~7
: mapping generators: ~5/4, ~7


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


Subgroup-val mapping: {{mapping| 3 7 0 2 | 0 0 1 1 }}
{{Mapping|legend=2| 3 7 0 2 | 0 0 1 1 }}


Gencom mapping: {{mapping| 3 0 7 0 2 | 0 0 0 1 1 }}
{{Mapping|legend=4| 3 0 7 0 2 | 0 0 0 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 83: Line 57:


Badness (Sintel): 0.196
Badness (Sintel): 0.196
=== Mabilic ===
{{Main| Mabilic and trismegistus }}
{{See also| Chromatic pairs #Mabilic }}
Mabilic is the no-3 [[restriction]] of [[mavila family #Armodue|armodue]] (7 & 16), [[magic family #Trismegistus|trismegistus]] (16 & 25), and [[mabila family #Semabila|semabila]] (25 & 34). It is the 7 & 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma. [[Weak extension]]s of mabilic include [[archytas clan #Sixix|sixix]] (7 & 25), [[vishnu family #Vishnean|vishnean]] (16 & 34), [[marvel temperaments #Septimin|septimin]] (9 & 41) and [[tetracot family #Monkey|monkey]] (7 & 34).
[[Subgroup]]: 2.5.7
[[Comma list]]: 1071875/1048576
{{Mapping|legend=3| 1 1 5 | 0 3 -5 }}
{{Mapping|legend=5| 1 0 1 5 | 0 0 3 -5 }}
: mapping generators: ~2, ~175/128
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2543{{c}}, ~175/128 = 527.7872{{c}}
: [[error map]]: {{val| +1.254 -1.698 -1.491 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~175/128 = 527.2041{{c}}
: error map: {{val| 0.000 -4.701 -4.846 }}
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd }}
[[Badness]] (Sintel): 1.70
=== Ostara ===
Ostara is a temperament that is derived from 93 & 524 solar calendar leap rule scale, interpreted in general no-3's 19-limit. It is a weak extension of the unnamed 2.5.7-subgroup 28 & 31 temperament, which tempers out 8589934592/8544921875.
Subgroup: 2.5.7.11
Comma list: 8589934592/8544921875, 30691800524/30517578125
{{Mapping|legend=2| 1 1 20 -49 | 0 3 -39 119 }}
: mapping generators: ~2, ~5120/3773
Optimal tunings:
* WE: ~2 = 1199.9115{{c}}, ~5120/3773 = 528.9650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5120/3773 = 529.0037{{c}}
{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}
Badness (Sintel): 11.7
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
{{Mapping|legend=2| 1 1 20 -49 35 | 0 3 -39 119 -71 }}
Optimal tunings:
* WE: ~2 = 1199.9194{{c}}, ~1664/1225 = 528.9681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0036{{c}}
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}
Badness (Sintel): 3.42
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
{{Mapping|legend=2| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}
Optimal tunings:
* WE: ~2 = 1199.9054{{c}}, ~1664/1225 = 528.9628{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0046{{c}}
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
Badness (Sintel): 1.99
==== 2.5.7.11.13.17.19 subgroup ====
Subgroup: 2.5.7.11.13.17.19
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
{{Mapping|legend=2| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}
Optimal tunings:
* WE: ~2 = 1199.9081{{c}}, ~19/14 = 528.9639{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/14 = 529.0045{{c}}
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
Badness (Sintel): 1.29


=== Frostburn ===
=== Frostburn ===
{{Redirect|Frostburn|the software developer|Lumi Pakkanen}}
Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].  
Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].  


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[[Comma list]]: 78125/76832
[[Comma list]]: 78125/76832


{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
{{Mapping|legend=3| 1 3 4 | 0 -4 -7 }}
: mapping generators: ~2, ~28/25
: mapping generators: ~2, ~28/25


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Comma list: 245/242, 625/616
Comma list: 245/242, 625/616


Subgroup-val mapping: {{mapping| 1 3 4 5 | 0 -4 -7 -9 }}
{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
: mapping generators: ~2, ~28/25
: mapping generators: ~2, ~28/25


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Badness (Sintel): 0.463
Badness (Sintel): 0.463


=== Mabilic ===
=== Rainy ===
{{Main| Mabilic and trismegistus }}
In rainy, three generators make an [[8/7]]; five generators make a [[5/4]]. It is the no-3's [[restriction]] of [[tertiaseptal]] (and [[valentine]]), notable theoretically as it equates ([[2/1]])/([[5/4]])<sup>3</sup> (128/125, the lesser diesis) with ([[2/1]])/([[8/7]])<sup>5</sup> (the 2.7-subgroup [[cloudy comma]], which is similar to the 2.5-subgroup lesser diesis in that tempering it out tunes the 8/7 about 8.8{{cent}} sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7{{cent}}). By tempering out their difference, stacked 5's and stacked 7's become easier to navigate, using the general-purpose diesis to simplify clusters.
{{See also| Chromatic pairs #Mabilic }}


Mabilic is the no-3 [[restriction]] of [[mavila family #Armodue|armodue]], [[mabila family #Semabila|semabila]], and [[magic family #Trismegistus|trismegistus]]. It is the 7 & 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma.
A highly notable tuning of rainy not shown here is [[311edo]], which is 140 + 171 so tuned between them.


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7


[[Comma list]]: 1071875/1048576
[[Comma list]]: [[2100875/2097152]]


{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}
{{Mapping|legend=3| 1 2 3 | 0 5 -3 }}


{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}
{{Mapping|legend=5| 1 0 2 3 | 0 0 5 -3 }}
: mapping generators: ~2, ~175/128
: mapping generators: ~2, ~256/245


[[Optimal tuning]]s:
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.2543{{c}}, ~175/128 = 527.7872{{c}}
* [[WE]]: ~2 = 1200.0939{{c}}, ~256/245 = 77.2107{{c}}
: [[error map]]: {{val| +1.254 -1.698 -1.491 }}
: [[error map]]: {{val| +0.094 -0.072 -0.176 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~175/128 = 527.2041{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/245 = 77.2093{{c}}
: error map: {{val| 0.000 -4.701 -4.846 }}
: error map: {{val| 0.000 -0.267 -0.454 }}


{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd }}
{{Optimal ET sequence|legend=1| 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d }}


[[Badness]] (Sintel): 1.70
[[Badness]] (Sintel): 0.156


=== Huntington ===
=== Llywelyn a.k.a. shoe ===
{{See also| Chromatic pairs #Huntington }}
Llywelyn a.k.a. shoe is the common [[restriction]] of [[gamelismic clan #Gorgo|gorgo]] (16 & 21), [[breedsmic temperaments #Gorgik|gorgik]] (21 & 37), and [[gariboh clan #Semaja|semaja]] (16 & 37). It can be described as 16 & 21, and tempers out the [[llywelynsma]]. Its generator is a flattened [[~]][[8/7]] whole tone, seven of which give the interval class of [[5/1|5]]. Its [[ploidacot]] is alpha-heptaseph. [[Weak extension]]s to the full 7-limit include [[marvel temperaments #Misneb|misneb]] (21 & 32), which splits 8/7 into two for [[15/14]]~[[16/15]] each, [[archytas clan #Ultrapyth|ultrapyth]] (32 & 37), which splits [[16/7]] into two for a [[3/2|perfect fifth]], and [[buzzardsmic clan #Septimal buzzard|buzzard]] (53 & 58), which splits 16/7 into three, each for [[21/16]].


Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.
; Scales
* [[Shoe5]], [[Shoe6]], [[Shoe11]], [[Shoe16]]


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7


[[Comma list]]: 40960000/40353607
[[Comma list]]: 4194304/4117715


{{Mapping|legend=2| 1 -4 0 | 0 9 4 }}
{{Mapping|legend=3| 1 1 3 | 0 7 -1 }}


{{Mapping|legend=3| 1 0 -4 0 | 0 0 9 4 }}
{{Mapping|legend=5| 1 0 1 3 | 0 0 7 -1 }}
: mapping generators: ~2, ~80/49
: mapping generators: ~2, ~8/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.5781{{c}}, ~80/49 = 842.6730{{c}}
* [[WE]]: ~2 = 1199.1974{{c}}, ~8/7 = 226.7586{{c}}
: [[error map]]: {{val| -0.422 -0.569 +1.866 }}
: [[error map]]: {{val| -0.803 +0.194 +2.008 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9136{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.8388{{c}}
: error map: {{val| 0.000 -0.091 +2.828 }}
: error map: {{val| 0.000 +1.558 +4.335 }}
 
{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37, 164d }}
 
[[Badness]] (Sintel): 2.06
 
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
 
Comma list: 176/175, 1310720/1294139
 
{{Mapping|legend=2| 1 1 3 1 | 0 7 -1 13 }}
 
{{Mapping|legend=4| 1 0 1 3 1 | 0 0 7 -1 13 }}
 
Optimal tunings:
* WE: ~2 = 1199.1784{{c}}, ~8/7 = 226.9582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.0438{{c}}


{{Optimal ET sequence|legend=0| 7c, 10, 27, 37, 84, 121 }}
{{Optimal ET sequence|legend=0| 16, 21, 37, 206cd, 243cd, …, 354ccdd }}


Badness (Sintel): 1.87
Badness (Sintel): 0.651


==== 2.5.7.13 subgroup ====
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.13
Subgroup: 2.5.7.11.13


Comma list: 640/637, 10985/10976
Comma list: 176/175, 640/637, 847/845


Subgroup-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}
{{Mapping|legend=2| 1 1 3 1 2 | 0 7 -1 13 9 }}


Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
{{Mapping|legend=4| 1 0 1 3 1 2 | 0 0 7 -1 13 9 }}
: mapping generators: ~2, ~13/8


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
* WE: ~2 = 1199.1643{{c}}, ~8/7 = 226.9495{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.0315{{c}}


{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}
{{Optimal ET sequence|legend=0| 16, 21, 37, 206cdf, 243cdf, …, 354ccddff }}


Badness (Sintel): 0.319
Badness (Sintel): 0.319


==== Silver ====
==== 2.5.7.11.13.17 subgroup ====
{{See also| Chromatic pairs #Silver }}
Subgroup: 2.5.7.11.13.17
 
Comma list: 176/175, 221/220, 640/637, 847/845


Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.
{{Mapping|legend=2| 1 1 3 1 2 2 | 0 7 -1 13 9 11 }}


Subgroup: 2.5.7.13.17
{{Mapping|legend=4| 1 0 1 3 1 2 2 | 0 0 7 -1 13 9 11 }}


Comma list: 170/169, 640/637, 5525/5488
Optimal tunings:  
* WE: ~2 = 1199.3162{{c}}, ~8/7 = 227.1125{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.1774{{c}}


Subgroup-val mapping: {{mapping| 1 -4 0 3 9 | 0 9 4 1 -7 }}
{{Optimal ET sequence|legend=0| 16, 21, 37, 95 }}


Gencom mapping: {{mapping| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}
Badness (Sintel): 0.458


Optimal tunings:
=== Mercy ===
* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}
Named by {{u|KingHyperio}} in 2021, mercy is the common restriction of [[gamelismic clan #Miracle|miracle]] and [[porwell temperaments #Countermiracle|countermiracle]]. It tempers out the [[quince comma]]. In this temperament, two generators make [[~]][[8/7]], five make ~[[7/5]], and seven make ~[[8/5]]. The [[ploidacot]] for this temperament is omega-heptaseph. [[Weak extension]]s of this temperament include [[starling temperaments #Cassablanca|cassablanca]], [[octagar temperaments #Slendi|slendi]], [[ragismic microtemperaments #Quincy|quincy]], [[septischismic clan #Cotoneum|cotoneum]], and [[varunismic temperaments #Octowerck|octowerck]].  
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}


{{Optimal ET sequence|legend=0| 10, 27, 37, 47, 84, 131, 178g }}
[[Subgroup]]: 2.5.7


Badness (Sintel): 0.504
[[Comma list]]: 823543/819200


=== Ostara ===
{{Mapping|legend=3| 1 3 3 | 0 -7 -2 }}
Ostara is a temperament that is derived from 93 & 524 solar calendar leap rule scale, interpreted in general no-3's 19-limit. It is a weak extension of the unnamed 2.5.7-subgroup 28 & 31 temperament, which tempers out 8589934592/8544921875.


Subgroup: 2.5.7.11
{{Mapping|legend=5| 1 0 3 3 | 0 0 -7 -2 }}
: mapping generators: ~2, ~343/320


Comma list: 8589934592/8544921875, 30691800524/30517578125
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.2169{{c}}, ~343/320 = 116.3116{{c}}
: [[error map]]: {{val| +0.217 +0.156 -0.798 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/320 = 116.2576{{c}}
: error map: {{val| 0.000 -0.117 -1.341 }}


Subgroup-val mapping: {{mapping| 1 1 20 -49 | 0 3 -39 119 }}
{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
: mapping generators: ~2, ~5120/3773


Optimal tunings:  
[[Badness]] (Sintel): 0.428
* WE: ~2 = 1199.9115{{c}}, ~5120/3773 = 528.9650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5120/3773 = 529.0037{{c}}


{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}
==== 2.5.7.13 subgroup ====
Presented here is a low-accuracy extension, which may be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]].


Badness (Sintel): 11.7
Subgroup: 2.5.7.13


==== 2.5.7.11.13 subgroup ====
Comma list: 343/338, 640/637
Subgroup: 2.5.7.11.13


Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
{{Mapping|legend=2| 1 3 3 4 | 0 -7 -2 -3 }}


Subgroup-val mapping: {{mapping| 1 1 20 -49 35 | 0 3 -39 119 -71 }}
{{Mapping|legend=4| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9194{{c}}, ~1664/1225 = 528.9681{{c}}
* WE: ~2 = 1199.2166{{c}}, ~14/13 = 116.0181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0036{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 116.2099{{c}}
 
{{Optimal ET sequence|legend=0| 10, 21, 31 }}


{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}
Badness (Sintel): 0.546


Badness (Sintel): 3.42
==== 2.5.7.13.17 subgroup ====
Subgroup: 2.5.7.13.17


==== 2.5.7.11.13.17 subgroup ====
Comma list: 170/169, 224/221, 343/338
Subgroup: 2.5.7.11.13.17


Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
{{Mapping|legend=2| 1 3 3 4 4 | 0 -7 -2 -3 1 }}


Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}
{{Mapping|legend=4| 1 0 3 3 0 4 4 | 0 0 -7 -2 0 -3 1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9054{{c}}, ~1664/1225 = 528.9628{{c}}
* WE: ~2 = 1198.5476{{c}}, ~14/13 = 115.6294{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0046{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.9132{{c}}
 
{{Optimal ET sequence|legend=0| 10, 21, 31, 83fg }}


{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
Badness (Sintel): 0.451


Badness (Sintel): 1.99
==== 2.5.7.13.17.19 subgroup ====
Subgroup: 2.5.7.13.17.19


==== 2.5.7.11.13.17.19 subgroup ====
Comma list: 170/169, 224/221, 343/338, 476/475
Subgroup: 2.5.7.11.13.17.19


Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
{{Mapping|legend=2| 1 3 3 4 4 3 | 0 -7 -2 -3 1 13 }}


Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}
{{Mapping|legend=4| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9081{{c}}, ~19/14 = 528.9639{{c}}
* WE: ~2 = 1198.4839{{c}}, ~14/13 = 115.5700{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/14 = 529.0045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.7299{{c}}
 
{{Optimal ET sequence|legend=0| 10h, 21, 31, 52f, 83fg }}


{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
Badness (Sintel): 0.759


Badness (Sintel): 1.29
=== Huntington ===
{{See also| Chromatic pairs #Huntington }}


=== French decimal ===
Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.  
French decimal is conceived upon the fact that 1789edo has an excellent 5/4, and uses it as the generator. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, a 1525 & 1789 temperament is obtained.


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7


[[Comma list]]: {{monzo| 372 -159 -1 }}
[[Comma list]]: 40960000/40353607
 
{{Mapping|legend=3| 1 -4 0 | 0 9 4 }}


{{Mapping|legend=2| 1 0 372 | 0 1 -159 }}
{{Mapping|legend=5| 1 0 -4 0 | 0 0 9 4 }}
: mapping generators: ~2, ~5
: mapping generators: ~2, ~80/49


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9901{{c}}, ~5/4 = 386.3562{{c}}
* [[WE]]: ~2 = 1199.5781{{c}}, ~80/49 = 842.6730{{c}}
: [[error map]]: {{val| -0.010 +0.023 +0.000 }}
: [[error map]]: {{val| -0.422 -0.569 +1.866 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.3595{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9136{{c}}
: error map: {{val| 0.000 +0.046 +0.019 }}
: error map: {{val| 0.000 -0.091 +2.828 }}
 
{{Optimal ET sequence|legend=0| 7c, 10, 27, 37, 84, 121 }}


{{Optimal ET sequence|legend=1| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}
Badness (Sintel): 1.87


[[Badness]] (Sintel): 148
==== 2.5.7.13 subgroup ====
Subgroup: 2.5.7.13


==== 2.5.7.11 subgroup ====
Comma list: 640/637, 10985/10976
Subgroup: 2.5.7.11


Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}
{{Mapping|legend=2| 1 -4 0 3 | 0 9 4 1 }}


Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}
{{Mapping|legend=4| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
: mapping generators: ~2, ~13/8


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}
* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}


{{Optimal ET sequence|legend=0| 264, 997e, 1261e, 1525, 1789 }}
{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}


Badness (Sintel): 52.2
Badness (Sintel): 0.319


==== 2.5.7.11.13 subgroup ====
==== Silver ====
Subgroup: 2.5.7.11.13
{{See also| Chromatic pairs #Silver }}
 
Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.
 
Subgroup: 2.5.7.13.17
 
Comma list: 170/169, 640/637, 5525/5488


Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
{{Mapping|legend=2| 1 -4 0 3 9 | 0 9 4 1 -7 }}


Subgroup-val mapping: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}
{{Mapping|legend=4| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}
* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}


{{Optimal ET sequence|legend=0| 261, 1261e, 1525, 1789 }}
{{Optimal ET sequence|legend=0| 10, 27, 37, 47, 84, 131, 178g }}


Badness (Sintel): 10.5
Badness (Sintel): 0.504


=== Bastille ===
=== Bastille ===
Line 330: Line 431:
[[Comma list]]: {{monzo| 1426 -596 -15 }}
[[Comma list]]: {{monzo| 1426 -596 -15 }}


{{Mapping|legend=2| 1 -4 254 | 0 15 -596 }}
{{Mapping|legend=3| 1 -4 254 | 0 15 -596 }}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}


Line 342: Line 443:


[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>
[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>
=== Tricesimoprimal miracloid ===
{{See also| Tricesimoprimal miracloid/Eliora's approach }}
{{Todo|inline=1| complete section | comment = Add missing data from 2.5.7 to 2.5.7.11.19.29. }}
Tricesimoprimal miracloid is described as the 52 & 1789 temperament in the 2.5.7.11.19.29.31 subgroup, with harmonics specifically selected for 52edo and 1789edo. Its generator is [[31/29]], which is also close to the secor. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable [[well temperament]] for 52edo.
==== 2.5.7.11.19.29.31 subgroup ====
Subgroup: 2.5.7.11.19.29.31
Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
Subgroup-val mapping: {{mapping| 1 -42 -2 -15 -12 -61 -61 | 0 461 50 192 169 685 686 }}
Optimal tunings:
* WE: ~2 = 1200.0079{{c}}, ~31/29 = 115.3723{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/29 = 115.3716{{c}}
{{Optimal ET sequence|legend=0| 52, 1737, 1789 }}
Badness (Sintel): 4.51


=== No-threes naiad (rank-3) ===
=== No-threes naiad (rank-3) ===
{{Todo|inline=1| review | comment = Devise the permanent name for this temperament; determine its status with respect to its extensions linked below. }}
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}


Line 373: Line 454:
[[Comma list]]: 5021863/5000000
[[Comma list]]: 5021863/5000000


{{Mapping|legend=2| 1 0 -2 3 | 0 1 1 1 | 0 0 4 -3 }}
{{Mapping|legend=3| 1 0 -2 3 | 0 1 1 1 | 0 0 4 -3 }}
: mapping generators: ~2, ~5, ~77/50
: mapping generators: ~2, ~5, ~77/50


Line 391: Line 472:
Comma list: 847/845, 1001/1000
Comma list: 847/845, 1001/1000


Subgroup-val mapping: {{mapping| 1 0 -2 3 2 | 0 1 1 1 1 | 0 0 4 -3 -1 }}
{{Mapping|legend=2| 1 0 -2 3 2 | 0 1 1 1 1 | 0 0 4 -3 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 406: Line 487:
Comma list: 170/169, 221/220, 847/845
Comma list: 170/169, 221/220, 847/845


Subgroup-val mapping: {{mapping| 1 0 -2 3 2 3 | 0 1 1 1 1 1 | 0 0 4 -3 -1 -2 }}
{{Mapping|legend=2| 1 0 -2 3 2 3 | 0 1 1 1 1 1 | 0 0 4 -3 -1 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 419: Line 500:
Temperaments discussed elsewhere include:
Temperaments discussed elsewhere include:
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]
=== Sephiroth ===
{{See also| Chromatic pairs #Sephiroth }}
Sephiroth is the common [[restriction]] of [[magic family #Muggles|muggles]] and [[avicennmic temperaments #Submerged|submerged]].
[[Subgroup]]: 2.5.11
[[Comma list]]: 34375/32768
{{Mapping|legend=3| 1 0 15 | 0 1 -5 }}
{{Mapping|legend=5| 1 0 0 0 15 | 0 0 1 0 -5 }}
: mapping generators: ~2, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.3290{{c}}, ~5/4 = 373.6097{{c}}
: [[error map]]: {{val| +3.329 -6.046 -2.722 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 372.1586{{c}}
: error map: {{val| 0.000 -14.155 -12.111 }}
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceee }}
[[Badness]] (Sintel): 1.85
==== 2.5.11.13 subgroup ====
Subgroup: 2.5.11.13
Comma list: 65/64, 6875/6656
{{Mapping|legend=2| 1 0 15 6 | 0 1 -5 -1 }}
{{Mapping|legend=4| 1 0 0 0 15 6 | 0 0 1 0 -5 -1 }}
Optimal tunings:
* WE: ~2 = 1203.3825{{c}}, ~5/4 = 373.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 372.1519{{c}}
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceeeff }}
Badness (Sintel): 0.410
==== 2.5.11.13.17 subgroup ====
Subgroup: 2.5.11.13.17
Comma list: 65/64, 170/169, 221/220
{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}
{{Mapping|legend=4| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}
Optimal tunings:
* WE: ~2 = 1203.6741{{c}}, ~5/4 = 373.3775{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 371.6773{{c}}
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29g, 129ccceeffgggg }}
Badness (Sintel): 0.299


=== Wizz ===
=== Wizz ===
{{See also| Chromatic pairs #Wizz }}
{{See also| Chromatic pairs #Wizz }}


Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, tempers out [[15625/15488]], and is the common [[restriction]] of [[astrology]] and [[wizard]].  
Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, tempers out [[15625/15488]], and is the common [[restriction]] of [[jubilismic clan #Astrology|astrology]] and [[stearnsmic clan #Wizard|wizard]].  


[[Subgroup]]: 2.5.11
[[Subgroup]]: 2.5.11
Line 429: Line 568:
[[Comma list]]: 15625/15488
[[Comma list]]: 15625/15488


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


{{Mapping|legend=3| 2 0 4 0 5 | 0 0 1 0 3 }}
{{Mapping|legend=5| 2 0 4 0 5 | 0 0 1 0 3 }}
: mapping generators: ~125/88, ~5/4
: mapping generators: ~125/88, ~5/4


Line 449: Line 588:
[[Comma list]]: 33275/32768
[[Comma list]]: 33275/32768


{{Mapping|legend=2| 1 0 5 | 0 3 -2 }}
{{Mapping|legend=3| 1 0 5 | 0 3 -2 }}
: mapping generators, ~2, ~55/32
: mapping generators, ~2, ~55/32


Line 463: Line 602:


=== Movila ===
=== Movila ===
This temperament has a structure very similar to [[mavila]] but is somewhat more accurate because the generator is a flat [[11/8]] rather than a sharp [[4/3]]. The major third is still ~[[5/4]], but the minor third is now ~[[64/55]] instead of ~[[6/5]].
This temperament has a structure very similar to [[mavila family #Mavila|mavila]] but is somewhat more accurate because the generator is a flat [[11/8]] rather than a sharp [[4/3]]. The major third is still ~[[5/4]], but the minor third is now ~[[64/55]] instead of ~[[6/5]].


[[Subgroup]]: 2.5.11
[[Subgroup]]: 2.5.11
Line 469: Line 608:
[[Comma list]]: 1331/1280
[[Comma list]]: 1331/1280


{{Mapping|legend=2| 1 1 3 | 0 3 1 }}
{{Mapping|legend=3| 1 1 3 | 0 3 1 }}
: mapping generators: ~2, ~11/8
: mapping generators: ~2, ~11/8


Line 481: Line 620:


[[Badness]] (Sintel): 0.718
[[Badness]] (Sintel): 0.718
=== Sephiroth ===
{{See also| Chromatic pairs #Sephiroth }}
Sephiroth is the no-7 [[restriction]] of [[muggles]].
[[Subgroup]]: 2.5.11
[[Comma list]]: 34375/32768
{{Mapping|legend=2| 1 0 15 | 0 1 -5 }}
{{Mapping|legend=3| 1 0 0 0 15 | 0 0 1 0 -5 }}
: mapping generators: ~2, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.3290{{c}}, ~5/4 = 373.6097{{c}}
: [[error map]]: {{val| +3.329 -6.046 -2.722 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 372.1586{{c}}
: error map: {{val| 0.000 -14.155 -12.111 }}
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceee }}
[[Badness]] (Sintel): 1.85
==== 2.5.11.13 subgroup ====
Subgroup: 2.5.11.13
Comma list: 65/64, 6875/6656
Subgroup-val mapping: {{mapping| 1 0 15 6 | 0 1 -5 -1 }}
Gencom mapping: {{mapping| 1 0 0 0 15 6 | 0 0 1 0 -5 -1 }}
Optimal tunings:
* WE: ~2 = 1203.3825{{c}}, ~5/4 = 373.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 372.1519{{c}}
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceeeff }}
Badness (Sintel): 0.410
==== 2.5.11.13.17 subgroup ====
Subgroup: 2.5.11.13.17
Comma list: 65/64, 170/169, 221/220
Subgroup-val mapping: {{mapping| 1 0 15 6 11 | 0 1 -5 -1 -3 }}
Gencom mapping: {{mapping| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}
Optimal tunings:
* WE: ~2 = 1203.6741{{c}}, ~5/4 = 373.3775{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 371.6773{{c}}
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29g, 129ccceeffgggg }}
Badness (Sintel): 0.299


=== Trader ===
=== Trader ===
Line 545: Line 626:
[[Comma list]]: 26/25
[[Comma list]]: 26/25


{{Mapping|legend=2| 1 2 3 | 0 1 2 }}
{{Mapping|legend=3| 1 0 -1 | 0 1 2 }}
: mapping generators, ~2, ~5/4
: mapping generators, ~2, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 563: Line 644:
[[Comma list]]: 64000000/62748517
[[Comma list]]: 64000000/62748517


{{Mapping|legend=2| 1 -2 0 | 0 7 6 }}
{{Mapping|legend=3| 1 -2 0 | 0 7 6 }}
: mapping generators, ~2, ~20/13
: mapping generators, ~2, ~20/13


Line 579: Line 660:
Temperaments discussed elsewhere include
Temperaments discussed elsewhere include
* Orgone → [[Orgonia #Orgone|Orgonia]]
* Orgone → [[Orgonia #Orgone|Orgonia]]
* Berylic → [[4th-octave temperaments #Berylic|4th-octave temperaments]]
* 21-23-commatic → [[21st-octave temperaments #21-23-commatic|21st-octave temperaments]]
* 21-23-commatic → [[21st-octave temperaments #21-23-commatic|21st-octave temperaments]]
* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic|31st-octave temperaments]]
* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic|31st-octave temperaments]]
Line 594: Line 674:
[[Comma list]]: 5767168/5764801
[[Comma list]]: 5767168/5764801


{{Mapping|legend=2| 1 0 -19 | 0 1 8 }}
{{Mapping|legend=3| 1 0 -19 | 0 1 8 }}
: mapping generators: ~2, ~7
: mapping generators: ~2, ~7


Line 608: Line 688:


=== Argument ===
=== Argument ===
Argument tempers out [[1372/1331]] in the 2.7.11 subgroup. It is the no-3 [[restriction]] of [[#Augment|augment]].  
Argument tempers out [[1372/1331]] in the 2.7.11 subgroup, equating [[14/11]] to a step of [[3edo]]. It is the no-3 [[restriction]] of [[#Augment|augment]].  


[[Subgroup]]: 2.7.11
[[Subgroup]]: 2.7.11
Line 614: Line 694:
[[Comma list]]: 1372/1331
[[Comma list]]: 1372/1331


{{Mapping|legend=2| 3 0 2 | 0 1 1 }}
{{Mapping|legend=3| 3 0 2 | 0 1 1 }}
: mapping generators: ~14/11, ~7
: mapping generators: ~14/11, ~7


Line 627: Line 707:
[[Badness]] (Sintel): 0.475
[[Badness]] (Sintel): 0.475


=== Score ===
=== Scaevola ===
Scaevola is the 2.7.11-subgroup temperament tempering out 14641/14336, with a generator of [[11/8]] and mapping +4 generators to [[7/4]]. Two strong extensions are [[#Score|score]] to 2.7.11.13 and [[subgroup temperaments #Joan|joan]] to 2.9.7.11.
 
Scaevola has a structure that is very similar to [[mavila family #Mavila|mavila]], but whereas mavila has a "fifth" of [[3/2]], "major third" of [[5/4]], and "minor third" of [[6/5]], scaevola has a "fifth" of [[16/11]], "major third" of [[14/11]], and "minor third" of [[8/7]], substituting mavila's 4:5:6 for 11:14:16. Essentially, Scaevola can be seen as a counterpart of mavila for fifths even flatter than [[9edo]], such as [[11edo]]'s fifth.
 
It was named by [[CompactStar]] in 2026, referring to ''{{w|Scaevola}}'' (the genus of flowering plants), which is meant to sound reminiscent of ''score'' and ''mavila'' (the score and joan temperaments were actually named prior to this temperament).
 
[[Subgroup]]: 2.7.11
 
[[Comma list]]: 14641/14336
 
{{Mapping|legend=3| 1 1 3 | 0 4 1 }}
 
{{Mapping|legend=5| 1 0 0 1 3 | 0 0 0 4 1 }}
: mapping generators: ~2, ~11/8
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2479{{c}}, ~11/8 = 542.1162{{c}}
: [[error map]]: {{val| +1.248 +0.887 -5.458 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 541.9403{{c}}
: error map: {{val| 0.000 -1.065 -9.378 }}
 
{{Optimal ET sequence|legend=1| 2, 7d, 9, 11, 20, 31, 51, 82e, 113e, 195eee }}
 
[[Badness]] (Sintel): 0.578
 
==== Score ====
{{See also| Chromatic pairs #Score }}
{{See also| Chromatic pairs #Score }}


Score is a low-accuracy [[extension]] of the unnamed 2.7.11-subgroup temperament tempering out 14641/14336.  
Score is an extension of scaevola that maps [[13/8]] to +6 generators.


[[Subgroup]]: 2.7.11.13
Subgroup: 2.7.11.13


[[Comma list]]: 343/338, 847/832
Comma list: 343/338, 847/832


{{Mapping|legend=2| 1 1 3 1 | 0 4 1 6 }}
{{Mapping|legend=2| 1 1 3 1 | 0 4 1 6 }}


{{Mapping|legend=3| 1 0 0 1 3 1 | 0 0 0 4 1 6 }}
{{Mapping|legend=4| 1 0 0 1 3 1 | 0 0 0 4 1 6 }}
: mapping generators: ~2, ~11/8


[[Optimal tuning]]s:  
Optimal tunings:  
* [[WE]]: ~2 = 1201.5484{{c}}, ~11/8 = 540.7963{{c}}
* WE: ~2 = 1201.5484{{c}}, ~11/8 = 540.7963{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 540.5091{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 540.5091{{c}}


{{Optimal ET sequence|legend=1| 9, 11, 20 }}
{{Optimal ET sequence|legend=0| 9, 11, 20 }}


[[Badness]] (Sintel): 0.368
Badness (Sintel): 0.368


=== Bossier ===
=== Bossier ===
Line 658: Line 763:
[[Comma list]]: 214358881/210827008
[[Comma list]]: 214358881/210827008


{{Mapping|legend=2| 1 0 1 | 0 8 7 }}
{{Mapping|legend=3| 1 0 1 | 0 8 7 }}


{{Mapping|legend=3| 1 0 0 0 1 | 0 0 0 8 7 }}
{{Mapping|legend=5| 1 0 0 0 1 | 0 0 0 8 7 }}
: mapping generators: ~2, ~14/11
: mapping generators: ~2, ~14/11


Line 678: Line 783:
Comma list: 1573/1568, 15488/15379
Comma list: 1573/1568, 15488/15379


Subgroup-val mapping: {{mapping| 1 0 1 3 | 0 8 7 2 }}
{{Mapping|legend=2| 1 0 1 3 | 0 8 7 2 }}


Gencom mapping: {{mapping|| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
{{Mapping|legend=4| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}


Optimal tunings:  
Optimal tunings:  
Line 697: Line 802:
[[Comma list]]: 28672/28561
[[Comma list]]: 28672/28561


{{Mapping|legend=2| 1 0 3 | 0 4 1 }}
{{Mapping|legend=3| 1 0 3 | 0 4 1 }}


{{Mapping|legend=3| 1 0 0 0 0 3 | 0 0 0 4 0 1 }}
{{Mapping|legend=5| 1 0 0 0 0 3 | 0 0 0 4 0 1 }}
: mapping generators: ~2, ~13
: mapping generators: ~2, ~13


Line 717: Line 822:
[[Comma list]]: 4913/4802
[[Comma list]]: 4913/4802


{{Mapping|legend=2| 1 2 3 | 0 3 4 }}
{{Mapping|legend=3| 1 2 3 | 0 3 4 }}
: mapping generators, ~2, ~17/14
: mapping generators, ~2, ~17/14


Line 735: Line 840:
[[Comma list]]: 2024782584832/2015993900449
[[Comma list]]: 2024782584832/2015993900449


{{Mapping|legend=2| 1 0 1 | 0 10 11 }}
{{Mapping|legend=3| 1 0 1 | 0 10 11 }}
: mapping generators, ~2, ~17/14
: mapping generators, ~2, ~17/14


Line 753: Line 858:
[[Comma list]]: 1048576/1042139
[[Comma list]]: 1048576/1042139


{{Mapping|legend=2| 1 0 6 | 0 3 -1 }}]
{{Mapping|legend=3| 1 0 6 | 0 3 -1 }}]
: mapping generators, ~2, ~64/53
: mapping generators, ~2, ~64/53


Line 775: Line 880:
[[Comma list]]: 1352/1331
[[Comma list]]: 1352/1331


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


{{Mapping|legend=3| 1 0 0 0 3 3 | 0 0 0 0 2 3 }}
{{Mapping|legend=5| 1 0 0 0 1 0 | 0 0 0 0 2 3 }}
: mapping generators, ~2, ~13/11
: mapping generators: ~2, ~26/11


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 798: Line 903:
[[Comma list]]: 265837/262144
[[Comma list]]: 265837/262144


{{Mapping|legend=2| 1 0 6 | 0 3 -2 }}
{{Mapping|legend=3| 1 0 6 | 0 3 -2 }}


{{Mapping|legend=3| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}
{{Mapping|legend=5| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}
: mapping generators, ~2, ~143/128
: mapping generators, ~2, ~143/128


Line 822: Line 927:
[[Comma list]]: 29282/28561
[[Comma list]]: 29282/28561


{{Mapping|legend=2| 4 0 1 | 0 1 1 }}
{{Mapping|legend=3| 4 0 1 | 0 1 1 }}


{{Mapping|legend=3| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
{{Mapping|legend=5| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
: mapping generators, ~13/11, ~11
: mapping generators, ~13/11, ~11


Line 844: Line 949:
[[Comma list]]: 209/208, 2057/2048, 83521/83486
[[Comma list]]: 209/208, 2057/2048, 83521/83486


{{Mapping|legend=2| 1 1 8 9 11 | 0 4 -7 -8 -11 }}
{{Mapping|legend=3| 1 1 8 9 11 | 0 4 -7 -8 -11 }}
: mapping generators: ~2, ~26/17
: mapping generators: ~2, ~26/17


Line 854: Line 959:


[[Badness]] (Sintel): 0.386
[[Badness]] (Sintel): 0.386
=== Arsenic ===
Arsenic was named by [[Xenllium]] in 2026 after the 33rd chemical element.
[[Subgroup]]: 2.11.13.17.19
[[Comma list]]: 2432/2431, 7086244/7054567, 33698267/33554432
{{Mapping|legend=3| 33 0 8 249 26 | 0 1 1 -1 1 }}
: mapping generators: ~247/242, ~11
[[Optimal tuning]]s:
* [[WE]]: ~247/242 = 36.3676{{c}}, ~11/8 = 550.4246{{c}}
* [[CWE]]: ~247/242 = 36.3636{{c}}, ~11/8 = 550.4187{{c}}
{{Optimal ET sequence|legend=1| 33, 165, 198, 231, 495h }}
[[Badness]] (Sintel): 2.30
=== Berylic ===
Berylic tempers out the [[berylisma]] in the 2.11.37 subgroup, representing the fact that [[44/37]] is a {{w|continued fraction}} convergent to 2<sup>1/4</sup> the fourth root of 2. Beryllic is an example of a temperament which has an astronomically low [[badness]], being a very high-accuracy [[microtemperament]] with low-to-average [[complexity]] for the harmonics in its [[subgroup]]. This also makes it simultaneously supported by edo systems as low as [[16edo]] and up into the tens of thousands. The tradeoff with this temperament, not captured within the metric of badness, is that it is defined within an obscure subgroup, 2.11.37.
If one wishes to explore harmony in this temperament, a great way is to use the 8-note [[4L 4s]] [[mos]], and use the [[32:37:44]] triad and its inversion [[296:352:407|1/(44:37:32)]] as the root chords. However, the consonance of the 37th harmonic is questionable.
[[Subgroup]]: 2.11.37
[[Comma list]]: 1874161/1874048
{{Mapping|legend=3| 4 0 7 | 0 1 1 }}
: mapping generators: ~44/37, ~11
[[Optimal tuning]]s:
* [[WE]]: ~44/37 = 300.0003{{c}}, ~11/8 = 551.3211{{c}}
: [[error map]]: {{val| +0.001 +0.007 -0.017 }}
* [[CWE]]: ~44/37 = 300.0000{{c}}, ~11/8 = 551.3237{{c}}
: error map: {{val| 0.000 +0.006 -0.020 }}
{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 76, 100, 124, 148, 616, 764, 912, 1060, 3328, 4388, 5448 }}
[[Badness]] (Sintel): 0.00188


=== Mavericks ===
=== Mavericks ===
Line 860: Line 1,005:
[[Comma list]]: 47525504/47045881
[[Comma list]]: 47525504/47045881


{{Mapping|legend=2| 1 1 2 | 0 6 5 }}
{{Mapping|legend=3| 1 1 2 | 0 6 5 }}
: mapping generators: ~2, ~26/19
: mapping generators: ~2, ~26/19


Line 878: Line 1,023:
[[Comma list]]: 209/208, 2057/2048
[[Comma list]]: 209/208, 2057/2048


{{Mapping|legend=2| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
{{Mapping|legend=3| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
: mapping generators: ~2, ~11, ~13
: mapping generators: ~2, ~11, ~13


Line 889: Line 1,034:
[[Badness]] (Sintel): 0.106
[[Badness]] (Sintel): 0.106


[[Category:Temperament collections]]
[[Category:Catalogs of subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/No-threes_subgroup_temperaments"