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{{Technical data page}}
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 3.  
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 3.  


== Llywelyn ==
== Overview by mapping of 5 ==
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]].
* 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.
* Insect has a ~55/32 generator, three of which give the ~5/1.
* Frostburn has a ~28/25 generator, four of which give the ~8/5.
Others have a more complex mapping of 5.
 
== Temperaments with a 2.5.7 gene ==
Temperaments discussed elsewhere include
* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus|Hemimean clan]]
* Sidewalk ([[823543/800000]]) → [[2023/2000 #Sidewalk]]
 
=== Augment ===
{{See also| Chromatic pairs #Augment }}
 
Augment is related to [[augmented]], but for 2.5.7 instead of 2.3.5.
 
[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7


[[Comma list]]: 4194304/4117715
[[Comma list]]: 128/125
 
{{Mapping|legend=3| 3 7 0 | 0 0 1 }}
 
{{Mapping|legend=5| 3 0 7 0 | 0 0 0 1 }}
: mapping generators: ~5/4, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 399.0128{{c}}, ~7/4 = 974.7085{{c}}
: [[error map]]: {{val| -2.962 +6.776 -0.040 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~7/4 = 974.3418{{c}}
: error map: {{val| 0.000 +13.686 +5.516 }}
 
{{Optimal ET sequence|legend=1| 3, 6, 15, 21, 27, 102ccd, 129ccd }}
 
[[Badness]] (Sintel): 0.296
 
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
 
Comma list: 56/55, 128/125
 
{{Mapping|legend=2| 3 7 0 2 | 0 0 1 1 }}
 
{{Mapping|legend=4| 3 0 7 0 2 | 0 0 0 1 1 }}
 
Optimal tunings:
* WE: ~5/4 = 398.9239{{c}}, ~7/4 = 969.1106{{c}}
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 968.4397{{c}}
 
{{Optimal ET sequence|legend=0| 3, 6, 15, 21 }}
 
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


[[Sval]] [[mapping]]: [{{val| 1 1 3 }}, {{val| 0 7 -1 }}]
[[Comma list]]: 1071875/1048576


[[Gencom]]: [2 8/7; 4194304/4117715]
{{Mapping|legend=3| 1 1 5 | 0 3 -5 }}


[[Gencom]] [[mapping]]: [{{val| 1 0 1 3 }}, {{val| 0 0 7 -1 }}]
{{Mapping|legend=5| 1 0 1 5 | 0 0 3 -5 }}
: mapping generators: ~2, ~175/128


Optimal tuning ([[POTE]]): ~8/7 = 226.910
[[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 }}


{{Val list|legend=1| 16, 37 }}
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5391 cents
[[Badness]] (Sintel): 1.70


=== 2.5.7.11 ===
=== Ostara ===
[[Subgroup]]: 2.5.7.11
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.  


[[Comma list]]: 176/175, 1310720/1294139
Subgroup: 2.5.7.11


[[Sval]] [[mapping]]: [{{val| 1 1 3 1 }}, {{val| 0 7 -1 13 }}]
Comma list: 8589934592/8544921875, 30691800524/30517578125


[[Gencom]]: [2 8/7; 176/175 1310720/1294139]
{{Mapping|legend=2| 1 1 20 -49 | 0 3 -39 119 }}
: mapping generators: ~2, ~5120/3773


[[Gencom]] [[mapping]]: [{{val| 1 0 1 3 1 }}, {{val| 0 0 7 -1 13 }}]
Optimal tunings:  
* WE: ~2 = 1199.9115{{c}}, ~5120/3773 = 528.9650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5120/3773 = 529.0037{{c}}


Optimal tuning ([[POTE]]): ~8/7 = 227.114
{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}


{{Val list|legend=1| 16, 21, 37 }}
Badness (Sintel): 11.7


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


[[Comma list]]: 176/175, 640/637, 1304576/1294139
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 }}


[[Sval]] [[mapping]]: [{{val| 1 1 3 1 2 }}, {{val| 0 7 -1 13 9 }}]
Badness (Sintel): 3.42


[[Gencom]]: [2 8/7; 176/175 640/637, 1304576/1294139]
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


[[Gencom]] [[mapping]]: [{{val| 1 0 1 3 1 2 }}, {{val| 0 0 7 -1 13 9 }}]
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251


Optimal tuning ([[POTE]]): ~8/7 = 227.108
{{Mapping|legend=2| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}


{{Val list|legend=1| 16, 21, 37 }}
Optimal tunings:
* WE: ~2 = 1199.9054{{c}}, ~1664/1225 = 528.9628{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0046{{c}}


=== 2.5.7.11.13.17 ===
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
Subgroup: 2.5.7.11.13.17
 
Badness (Sintel): 1.99


[[Comma list]]: 176/175, 221/200, 640/637, 833/832
==== 2.5.7.11.13.17.19 subgroup ====
Subgroup: 2.5.7.11.13.17.19


[[Sval]] [[mapping]]: [{{val| 1 1 3 1 2 2 }}, {{val| 0 7 -1 13 9 11 }}]
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875


[[Gencom]]: [2 8/7; 176/175 221/200, 640/637, 833/832]
{{Mapping|legend=2| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}


[[Gencom]] [[mapping]]: [{{val| 1 0 1 3 1 2 2 }}, {{val| 0 0 7 -1 13 9 11 }}]
Optimal tunings:  
* WE: ~2 = 1199.9081{{c}}, ~19/14 = 528.9639{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/14 = 529.0045{{c}}


Optimal tuning ([[POTE]]): ~8/7 = 227.242
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


{{Val list|legend=1| 16, 21, 37 }}
Badness (Sintel): 1.29


== Didacus ==
=== Frostburn ===
{{See also| Hemimean clan #Didacus }}
{{Redirect|Frostburn|the software developer|Lumi Pakkanen}}


Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].


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


[[Comma list]]: 3136/3125
[[Comma list]]: 78125/76832


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


[[Gencom]]: [2 28/25; 3136/3125]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3462{{c}}, ~28/25 = 204.3386{{c}}
: [[error map]]: {{val| +0.346 -2.630 +2.189 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/25 = 204.2027{{c}}
: error map: {{val| 0.000 -3.125 +1.755 }}


[[Gencom]] [[mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 0 2 5 }}]
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


Optimal tuning ([[POTE]]): ~28/25 = 93.772
[[Badness]] (Sintel): 0.886


{{Val list|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
Comma list: 245/242, 625/616


== Rainy ==
{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].
: mapping generators: ~2, ~28/25
 
Optimal tunings:
* WE: ~2 = 1200.6817{{c}}, ~28/25 = 205.0734{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 204.8199{{c}}
 
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
 
Badness (Sintel): 0.463
 
=== 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
[[Subgroup]]: 2.5.7
Line 91: Line 193:
[[Comma list]]: [[2100875/2097152]]
[[Comma list]]: [[2100875/2097152]]


[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
{{Mapping|legend=3| 1 2 3 | 0 5 -3 }}
 
{{Mapping|legend=5| 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
 
=== Llywelyn a.k.a. shoe ===
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]].
 
; Scales
* [[Shoe5]], [[Shoe6]], [[Shoe11]], [[Shoe16]]
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 4194304/4117715
 
{{Mapping|legend=3| 1 1 3 | 0 7 -1 }}
 
{{Mapping|legend=5| 1 0 1 3 | 0 0 7 -1 }}
: mapping generators: ~2, ~8/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1974{{c}}, ~8/7 = 226.7586{{c}}
: [[error map]]: {{val| -0.803 +0.194 +2.008 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.8388{{c}}
: 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| 16, 21, 37, 206cd, 243cd, …, 354ccdd }}
 
Badness (Sintel): 0.651
 
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13
 
Comma list: 176/175, 640/637, 847/845
 
{{Mapping|legend=2| 1 1 3 1 2 | 0 7 -1 13 9 }}
 
{{Mapping|legend=4| 1 0 1 3 1 2 | 0 0 7 -1 13 9 }}
 
Optimal tunings:
* WE: ~2 = 1199.1643{{c}}, ~8/7 = 226.9495{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.0315{{c}}
 
{{Optimal ET sequence|legend=0| 16, 21, 37, 206cdf, 243cdf, …, 354ccddff }}
 
Badness (Sintel): 0.319


[[Gencom]]: [2 256/245; 2100875/2097152]
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]
Comma list: 176/175, 221/220, 640/637, 847/845


Optimal tuning ([[POTE]]): ~256/245 = 77.205
{{Mapping|legend=2| 1 1 3 1 2 2 | 0 7 -1 13 9 11 }}


{{Val list|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
{{Mapping|legend=4| 1 0 1 3 1 2 2 | 0 0 7 -1 13 9 11 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
Optimal tunings:  
* WE: ~2 = 1199.3162{{c}}, ~8/7 = 227.1125{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.1774{{c}}


== Mercy ==
{{Optimal ET sequence|legend=0| 16, 21, 37, 95 }}
{{See also| Quince clan #Mercy }}


Two generators make an [[8/7]]; seven generators make an [[8/5]]. Mercy can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]], and is the no-threes or elevens version of [[miracle]].
Badness (Sintel): 0.458
 
=== Mercy ===
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]].  


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7
Line 112: Line 291:
[[Comma list]]: 823543/819200
[[Comma list]]: 823543/819200


[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
{{Mapping|legend=3| 1 3 3 | 0 -7 -2 }}
 
{{Mapping|legend=5| 1 0 3 3 | 0 0 -7 -2 }}
: mapping generators: ~2, ~343/320
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
 
[[Badness]] (Sintel): 0.428
 
==== 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]].
 
Subgroup: 2.5.7.13
 
Comma list: 343/338, 640/637
 
{{Mapping|legend=2| 1 3 3 4 | 0 -7 -2 -3 }}
 
{{Mapping|legend=4| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.2166{{c}}, ~14/13 = 116.0181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 116.2099{{c}}
 
{{Optimal ET sequence|legend=0| 10, 21, 31 }}
 
Badness (Sintel): 0.546
 
==== 2.5.7.13.17 subgroup ====
Subgroup: 2.5.7.13.17
 
Comma list: 170/169, 224/221, 343/338
 
{{Mapping|legend=2| 1 3 3 4 4 | 0 -7 -2 -3 1 }}
 
{{Mapping|legend=4| 1 0 3 3 0 4 4 | 0 0 -7 -2 0 -3 1 }}
 
Optimal tunings:
* WE: ~2 = 1198.5476{{c}}, ~14/13 = 115.6294{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.9132{{c}}
 
{{Optimal ET sequence|legend=0| 10, 21, 31, 83fg }}
 
Badness (Sintel): 0.451
 
==== 2.5.7.13.17.19 subgroup ====
Subgroup: 2.5.7.13.17.19
 
Comma list: 170/169, 224/221, 343/338, 476/475
 
{{Mapping|legend=2| 1 3 3 4 4 3 | 0 -7 -2 -3 1 13 }}
 
{{Mapping|legend=4| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}
 
Optimal tunings:
* WE: ~2 = 1198.4839{{c}}, ~14/13 = 115.5700{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.7299{{c}}
 
{{Optimal ET sequence|legend=0| 10h, 21, 31, 52f, 83fg }}
 
Badness (Sintel): 0.759
 
=== Huntington ===
{{See also| Chromatic pairs #Huntington }}
 
Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 40960000/40353607
 
{{Mapping|legend=3| 1 -4 0 | 0 9 4 }}
 
{{Mapping|legend=5| 1 0 -4 0 | 0 0 9 4 }}
: mapping generators: ~2, ~80/49
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5781{{c}}, ~80/49 = 842.6730{{c}}
: [[error map]]: {{val| -0.422 -0.569 +1.866 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9136{{c}}
: error map: {{val| 0.000 -0.091 +2.828 }}
 
{{Optimal ET sequence|legend=0| 7c, 10, 27, 37, 84, 121 }}
 
Badness (Sintel): 1.87


[[Gencom]]: [2 2744/2560; 823543/819200]
==== 2.5.7.13 subgroup ====
Subgroup: 2.5.7.13


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
Comma list: 640/637, 10985/10976


Optimal tuning ([[POTE]]): ~343/320 = 116.291
{{Mapping|legend=2| 1 -4 0 3 | 0 9 4 1 }}


{{Val list|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
{{Mapping|legend=4| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
: mapping generators: ~2, ~13/8


=== 2.5.7.13 ===
Optimal tunings:
[[Subgroup]]: 2.5.7.13
* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}


[[Comma list]]: 343/338, 640/637
{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 }}, {{val| 0 -7 -2 -3 }}]
Badness (Sintel): 0.319


[[Gencom]]: [2 14/13; 343/338 640/637]
==== Silver ====
{{See also| Chromatic pairs #Silver }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 }}, {{val| 0 0 -7 -2 -3 }}]
Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.


Optimal tuning ([[POTE]]): ~14/13 = 116.094
Subgroup: 2.5.7.13.17


{{Val list|legend=1| 10, 21, 31}}
Comma list: 170/169, 640/637, 5525/5488


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


[[Comma list]]: 170/169, 224/221, 640/637
{{Mapping|legend=4| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]
Optimal tunings:  
* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
{{Optimal ET sequence|legend=0| 10, 27, 37, 47, 84, 131, 178g }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 }}, {{val| 0 0 -7 -2 -3 1 }}]
Badness (Sintel): 0.504


Optimal tuning ([[POTE]]): ~14/13 = 115.769
=== Bastille ===
{{Main| Bastille }}


{{Val list|legend=1| 10, 21, 31}}
Bastille is described as the 2.5.7-subgroup 1407 & 1789 temperament, and named after an [[Wikipedia: Storming of the Bastille|eponymous historical event]] which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins #Double bastille|double bastille]].


=== 2.5.7.13.17.19 ===
[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7.13.17.19


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
[[Comma list]]: {{monzo| 1426 -596 -15 }}


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 3 }}, {{val| 0 -7 -2 -3 1 13 }}]
{{Mapping|legend=3| 1 -4 254 | 0 15 -596 }}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9911{{c}}, ~{{monzo| -380 159 4 }} = 505.7532{{c}}
: [[error map]]: {{val| -0.009 +0.020 +0.001 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -380 159 4 }} = 505.7570{{c}}
: error map: {{val| 0.000 +0.041 +0.018 }}


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
{{Optimal ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}


Optimal tuning ([[POTE]]): ~14/13 = 115.716
[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>


{{Val list|legend=1| 10, 21, 31, 52f }}
=== 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 }}
 
This temperament can be described as the 21 & 29 & 37 temperament in no-threes subgroups. It expands [[Subgroup temperaments #Tridec|tridec]] and [[Subgroup temperaments #Naiadec|naiadec]].


== Frostburn (rank 3) ==
[[Subgroup]]: 2.5.7.11
[[Subgroup]]: 2.5.7.11


[[Comma list]]: 625/616
[[Comma list]]: 5021863/5000000
 
{{Mapping|legend=3| 1 0 -2 3 | 0 1 1 1 | 0 0 4 -3 }}
: mapping generators: ~2, ~5, ~77/50
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0805{{c}}, ~5/4 = 386.6593{{c}}, ~77/50 = 745.4622{{c}}
: [[error map]]: {{val| +0.080 +0.507 -1.318 -0.643 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.7404{{c}}, ~77/50 = 745.4102{{c}}
: error map: {{val| 0.000 +0.427 -0.445 -0.808 }}
 
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 388, 425, 652e, 689e, 1077de }}
 
[[Badness]] (Sintel): 1.86
 
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13
 
Comma list: 847/845, 1001/1000
 
{{Mapping|legend=2| 1 0 -2 3 2 | 0 1 1 1 1 | 0 0 4 -3 -1 }}
 
Optimal tunings:
* WE: ~2 = 1200.0343{{c}}, ~5/4 = 386.6098{{c}}, ~20/13 = 745.4658{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.6458{{c}}, ~20/13 = 745.4431{{c}}
 
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e }}
 
Badness (Sintel): 0.179
 
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17
 
Comma list: 170/169, 221/220, 847/845
 
{{Mapping|legend=2| 1 0 -2 3 2 3 | 0 1 1 1 1 1 | 0 0 4 -3 -1 -2 }}
 
Optimal tunings:
* WE: ~2 = 1200.4068{{c}}, ~5/4 = 386.6701{{c}}, ~17/11 = 745.3706{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.1074{{c}}, ~17/11 = 745.0940{{c}}
 
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 66g, 87g, 124g }}
 
Badness (Sintel): 0.438
 
== Temperaments with a higher 2.5.''p'' gene ==
Temperaments discussed elsewhere include:
* 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


Optimal tuning ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
Comma list: 65/64, 170/169, 221/220


{{Val list|legend=1| 13, 16, 22, 28, 35, 41, 47, 57, 63, 98c }}
{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}


== Xeimtionic ==
{{Mapping|legend=4| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}
[[Subgroup]]: 2.5.7.11
 
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 ===
{{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 [[jubilismic clan #Astrology|astrology]] and [[stearnsmic clan #Wizard|wizard]].
 
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 15625/15488
 
{{Mapping|legend=3| 2 0 -7 | 0 1 3 }}
 
{{Mapping|legend=5| 2 0 4 0 5 | 0 0 1 0 3 }}
: mapping generators: ~125/88, ~5/4
 
[[Optimal tuning]]s:
* [[WE]]: ~125/88 = 600.1831{{c}}, ~5/4 = 383.8848{{c}}
: [[error map]]: {{val| +0.366 -1.697 +1.252 }}
* [[CWE]]: ~125/88 = 600.0000{{c}}, ~5/4 = 383.9977{{c}}
: error map: {{val| 0.000 -2.316 +0.675 }}
 
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222, 394c }}
 
[[Badness]] (Sintel): 0.266
 
=== Insect ===
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 33275/32768
 
{{Mapping|legend=3| 1 0 5 | 0 3 -2 }}
: mapping generators, ~2, ~55/32
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.1238{{c}}, ~55/32 = 928.5003{{c}}
: [[error map]]: {{val| +1.124 -0.813 -2.700 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~55/32 = 927.7384{{c}}
: error map: {{val| 0.000 -3.099 -6.975 }}
 
{{Optimal ET sequence|legend=1| 9, 13, 22, 97e, 119e, 141e, 163e, 304ceee }}
 
[[Badness]] (Sintel): 0.564
 
=== Movila ===
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
 
[[Comma list]]: 1331/1280
 
{{Mapping|legend=3| 1 1 3 | 0 3 1 }}
: mapping generators: ~2, ~11/8
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.0339{{c}}, ~11/8 = 528.4296{{c}}
: [[error map]]: {{val| +3.034 +2.009 -13.787 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 528.1575{{c}}
: error map: {{val| 0.000 -1.841 -23.160 }}
 
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
 
[[Badness]] (Sintel): 0.718
 
=== Trader ===
[[Subgroup]]: 2.5.13
 
[[Comma list]]: 26/25
 
{{Mapping|legend=3| 1 0 -1 | 0 1 2 }}
: mapping generators, ~2, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.0216{{c}}, ~5/4 = 410.2152{{c}}
: [[error map]]: {{val| -1.978 +19.945 -26.033 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 408.9029{{c}}
: error map: {{val| 0.000 +22.589 -22.722 }}
 
{{Optimal ET sequence|legend=1| 3, 20c, 23c, 26c }}
 
[[Badness]] (Sintel): 0.138
 
=== Superquintal ===
[[Subgroup]]: 2.5.13
 
[[Comma list]]: 64000000/62748517
 
{{Mapping|legend=3| 1 -2 0 | 0 7 6 }}
: mapping generators, ~2, ~20/13
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5925{{c}}, ~20/13 = 740.6286{{c}}
: [[error map]]: {{val| -0.408 -1.098 +3.244 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~20/13 = 740.8058{{c}}
: error map: {{val| 0.000 -0.673 +4.307 }}
 
{{Optimal ET sequence|legend=1| 8, 13, 21, 34, 81, 115 }}
 
[[Badness]] (Sintel): 1.93
 
== No-threes no-fives subgroup temperaments ==
Temperaments discussed elsewhere include
* Orgone → [[Orgonia #Orgone|Orgonia]]
* 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]]
* 37-11-commatic (rank-1) → [[37th-octave temperaments #37-11-commatic (rank-1)|37th-octave temperaments]]
* etc.
 
=== Amaranthine ===
{{See also| No-fives subgroup temperaments #Chrysanthemum }}
 
Amaranthine is the high-accuracy 2.7.11-subgroup strong [[restriction]] of [[Gamelismic clan #11-limit 3|undecimal mothra]].
 
[[Subgroup]]: 2.7.11
 
[[Comma list]]: 5767168/5764801
 
{{Mapping|legend=3| 1 0 -19 | 0 1 8 }}
: mapping generators: ~2, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9846{{c}}, ~7/4 = 968.9078{{c}}
: [[error map]]: {{val| -0.015 +0.051 -0.010 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 968.9174{{c}}
: error map: {{val| 0.000 +0.091 +0.021 }}
 
{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d, 12780dd }}
 
Badness (Sintel): 0.0309
 
=== Argument ===
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
 
[[Comma list]]: 1372/1331
 
{{Mapping|legend=3| 3 0 2 | 0 1 1 }}
: mapping generators: ~14/11, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~14/11 = 399.8041{{c}}, ~7/4 = 963.1666{{c}}
: [[error map]]: {{val| -0.588 -6.835 +10.281 }}
* [[CWE]]: ~14/11 = 400.0000{{c}}, ~4/4 = 962.7466{{c}}
: error map: {{val| 0.000 -6.079 +11.429 }}
 
{{Optimal ET sequence|legend=1| 6, 9, 15, 36, 51e, 66e }}
 
[[Badness]] (Sintel): 0.475
 
=== 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 }}
 
Score is an extension of scaevola that maps [[13/8]] to +6 generators.
 
Subgroup: 2.7.11.13
 
Comma list: 343/338, 847/832
 
{{Mapping|legend=2| 1 1 3 1 | 0 4 1 6 }}
 
{{Mapping|legend=4| 1 0 0 1 3 1 | 0 0 0 4 1 6 }}
 
Optimal tunings:
* WE: ~2 = 1201.5484{{c}}, ~11/8 = 540.7963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 540.5091{{c}}
 
{{Optimal ET sequence|legend=0| 9, 11, 20 }}
 
Badness (Sintel): 0.368
 
=== Bossier ===
{{See also| Chromatic pairs #Bossier }}
 
Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup, tempering out [[1573/1568]] and [[15488/15379]].
 
[[Subgroup]]: 2.7.11
 
[[Comma list]]: 214358881/210827008
 
{{Mapping|legend=3| 1 0 1 | 0 8 7 }}
 
{{Mapping|legend=5| 1 0 0 0 1 | 0 0 0 8 7 }}
: mapping generators: ~2, ~14/11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1886{{c}}, ~14/11 = 421.2661{{c}}
: [[error map]]: {{val| +0.189 +1.303 -2.266 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/11 = 421.2365{{c}}
: error map: {{val| 0.000 +1.066 -2.662 }}
 
{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 151 }}
 
[[Badness]] (Sintel): 1.73
 
==== 2.7.11.13 subgroup ====
Subgroup: 2.7.11.13
 
Comma list: 1573/1568, 15488/15379
 
{{Mapping|legend=2| 1 0 1 3 | 0 8 7 2 }}
 
{{Mapping|legend=4| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
 
Optimal tunings:
* WE: ~2 = 1199.8668{{c}}, ~14/11 = 421.2623{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 421.2874{{c}}
 
{{Optimal ET sequence|legend=0| 17, 20, 37, 57, 94, 225 }}
 
Badness (Sintel): 0.307
 
=== Voltage ===
Voltage is the 3 & 7 temperament in the 2.7.13 subgroup. Among the notable tunings is pure-7 tuning, 7<sup>1/4</sup> of 842.2065{{c}}, which is also the CTC (constrained Tenney–Chebyshevian) tuning.
 
[[Subgroup]]: 2.7.13
 
[[Comma list]]: 28672/28561
 
{{Mapping|legend=3| 1 0 3 | 0 4 1 }}
 
{{Mapping|legend=5| 1 0 0 0 0 3 | 0 0 0 4 0 1 }}
: mapping generators: ~2, ~13
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7827{{c}}, ~13/8 = 842.1707{{c}}
: [[error map]]: {{val| -0.217 -0.143 +0.991 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/8 = 842.2568{{c}}
: error map: {{val| 0.000 +0.201 +1.729 }}
 
{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104, 463f, 567f, 671ff, 775ff }}
 
[[Badness]] (Sintel): 0.115
 
=== Ultrakleismic ===
[[Subgroup]]: 2.7.17
 
[[Comma list]]: 4913/4802
 
{{Mapping|legend=3| 1 2 3 | 0 3 4 }}
: mapping generators, ~2, ~17/14
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1379{{c}}, ~17/14 = 324.3440{{c}}
: [[error map]]: {{val| +0.138 +4.482 -7.166 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 324.3738{{c}}
: error map: {{val| 0.000 +4.295 -7.460 }}
 
{{Optimal ET sequence|legend=1| 4, 7, 11, 26, 37 }}
 
[[Badness]] (Sintel): 0.460
 
=== Counterultrakleismic ===
[[Subgroup]]: 2.7.17
 
[[Comma list]]: 2024782584832/2015993900449
 
{{Mapping|legend=3| 1 0 1 | 0 10 11 }}
: mapping generators, ~2, ~17/14
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9723{{c}}, ~17/14 = 336.8586{{c}}
: [[error map]]: {{val| -0.028 -0.240 +0.462 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 336.8621{{c}}
: error map: {{val| 0.000 -0.205 +0.528 }}
 
{{Optimal ET sequence|legend=1| 7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g }}
 
[[Badness]] (Sintel): 0.860
 
=== Shipwreck ===
[[Subgroup]]: 2.7.53
 
[[Comma list]]: 1048576/1042139
 
{{Mapping|legend=3| 1 0 6 | 0 3 -1 }}]
: mapping generators, ~2, ~64/53
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6967{{c}}, ~64/53 = 323.1839{{c}}
: [[error map]]: {{val| -0.303 +0.119 +1.491 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/53 = 323.1959{{c}}
: error map: {{val| 0.000 +0.762 +3.300 }}
 
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
 
[[Badness]] (Sintel): 0.224
 
=== Lovecraft ===
{{See also | Chromatic pairs #Lovecraft }}
 
Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, tempers out [[1352/1331]], and is generated by ~13/11. Two generator steps give ~11/8 and three generator steps give ~13/8.
 
[[Subgroup]]: 2.11.13
 
[[Comma list]]: 1352/1331
 
{{Mapping|legend=3| 1 1 0 | 0 2 3 }}
 
{{Mapping|legend=5| 1 0 0 0 1 0 | 0 0 0 0 2 3 }}
: mapping generators: ~2, ~26/11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5223{{c}}, ~13/11 = 279.2064{{c}}
: [[error map]]: {{val| -0.478 +5.662 -4.341 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/11 = 278.9918{{c}}
: error map: {{val| 0.000 +6.666 -3.552 }}
 
{{Optimal ET sequence|legend=1| 4, 9, 13, 30, 43, 73, 116e }}
 
[[Badness]] (Sintel): 0.175
 
=== Bluebirds ===
{{Distinguish| Bluebird }}
{{See also| Chromatic pairs #Bluebirds }}
 
[[Subgroup]]: 2.11.13
 
[[Comma list]]: 265837/262144
 
{{Mapping|legend=3| 1 0 6 | 0 3 -2 }}
 
{{Mapping|legend=5| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}
: mapping generators, ~2, ~143/128
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8795{{c}}, ~143/128 = 182.5017{{c}}
: [[error map]]: {{val| +0.880 -1.174 -2.013 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~143/128 = 182.4386{{c}}
: error map: {{val| 0.000 -4.002 -5.405 }}
 
{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125f, 204ef, 329eeff }}
 
[[Badness]] (Sintel): 0.451
 
=== Blackbirds ===
{{See also | Chromatic pairs #Blackbirds }}
 
Blackbirds is a fairly straightforward temperament. It simply equates ~13/11 to 1/4 of the octave with a generator for prime 11 or 13.
 
[[Subgroup]]: 2.11.13
 
[[Comma list]]: 29282/28561
 
{{Mapping|legend=3| 4 0 1 | 0 1 1 }}
 
{{Mapping|legend=5| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
: mapping generators, ~13/11, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~13/11 = 299.9728{{c}}, ~11/8 = 546.6107{{c}}
: [[error map]]: {{val| -0.109 -5.033 +5.730 }}
* [[CWE]]: ~13/11 = 300.0000{{c}}, ~11/8 = 546.4664{{c}}
: error map: {{val| 0.000 -4.852 +5.939 }}


[[Comma list]]: 245/242, 625/616
{{Optimal ET sequence|legend=1| 4, 12e, 16, 20, 24, 44, 68 }}


Optimal tuning ([[TE tuning|TE]]): ~2/1 = 1200.6817, ~28/25 = 205.0745
[[Badness]] (Sintel): 0.668


{{Val list|legend=1| 29, 35, 41, 47, 88e }}
=== Yamablu ===
Yamablu, with a generator of ~26/17, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). It extends the 2.11.13-subgroup temperament tempering out 556573090931/549755813888. The [[Kite's Genchain mode numbering|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].


== Yer (rank 3) ==
[[Subgroup]]: 2.11.13.17.19
[[Subgroup]]: 2.11.13.17.19


[[Comma list]]: 209/208, 2057/2048
[[Comma list]]: 209/208, 2057/2048, 83521/83486
 
{{Mapping|legend=3| 1 1 8 9 11 | 0 4 -7 -8 -11 }}
: mapping generators: ~2, ~26/17


[[Sval]] [[mapping]]: [{{val| 1 0 0 11 4 }}, {{val| 0 1 0 -2 -1 }}, {{val| 0 0 1 0 1 }}]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4661{{c}}, ~26/17 = 737.3256{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/17 = 737.0014{{c}}


Optimal tuning ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
{{Optimal ET sequence|legend=1| 13, 44, 57, 70, 127, 197eh }}


{{Val list|legend=1| 13, 24, 33, 37, 46, 57, 70, 127 }}
[[Badness]] (Sintel): 0.386


== Yamablu ==
=== Arsenic ===
Yamablu, with a generator of ~17/13, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The [[Kite's Method of Naming Rank-2 Scales using Mode Numbers|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].
Arsenic was named by [[Xenllium]] in 2026 after the 33rd chemical element.  


[[Subgroup]]: 2.11.13.17.19
[[Subgroup]]: 2.11.13.17.19


[[Comma list]]: 209/208, 2057/2048, 83521/83486
[[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 ===
[[Subgroup]]: 2.13.19
 
[[Comma list]]: 47525504/47045881
 
{{Mapping|legend=3| 1 1 2 | 0 6 5 }}
: mapping generators: ~2, ~26/19
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8817{{c}}, ~26/19 = 539.9150{{c}}
: [[error map]]: {{val| -0.118 -1.156 +1.825 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/19 = 539.9280{{c}}
: error map: {{val| 0.000 -0.960 +2.127 }}
 
{{Optimal ET sequence|legend=1| 9, 11, 20 }}
 
[[Badness]] (Sintel): 0.559
 
=== Yer (rank 3) ===
[[Subgroup]]: 2.11.13.17.19


[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]
[[Comma list]]: 209/208, 2057/2048


Optimal tuning ([[POTE]]): ~17/13 = 462.9606
{{Mapping|legend=3| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
: mapping generators: ~2, ~11, ~13


{{Val list|legend=1| 13, 44, 57, 70}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4447{{c}}, ~11/8 = 548.4929{{c}}, ~13/8 = 841.3613{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 548.2193{{c}}, ~13/8 = 841.4707{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}


[[Category:Temperament collections]]
[[Badness]] (Sintel): 0.106
[[Category:Subgroup temperaments]]
[[Category:Rank 2]]


{{Todo| cleanup | review }}
[[Category:Catalogs of subgroup temperaments]]
[[Category:Catalogs of rank-2 temperaments]]
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