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No-threes subgroup temperaments: Difference between revisions

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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 aka shoe ==
== Overview by mapping of 5 ==
{{See also| Chromatic pairs #Shoe }}
Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
{{See also| Llywelyn clan #Llywelyn aka shoe }}
* 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.
* 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]]
* 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]]
 
=== 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


[[Comma list]]: 4194304/4117715
[[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 }}


[[Sval]] [[mapping]]: [{{Val| 1 1 3 }}, {{Val| 0 7 -1 }}]
{{Optimal ET sequence|legend=1| 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d }}


Mapping generators: 2, ~8/7
[[Badness]] (Sintel): 0.156


[[Gencom]] [[mapping]]: [{{Val| 1 0 1 3 }}, {{Val| 0 0 7 -1 }}]
=== Augment ===
{{See also| Chromatic pairs #Augment }}


[[Gencom]]: [2 8/7; 4194304/4117715]
Augment is related to [[augmented]], but for 2.5.7 instead of 2.3.5.


[[Optimal tuning]] ([[POTE]]): ~8/7 = 226.910
[[Subgroup]]: 2.5.7


{{Val list|legend=1| 5, 11c, 16, 21, 37 }}
[[Comma list]]: 128/125


=== 2.5.7.11 subgroup ===
{{Mapping|legend=2| 3 7 0 | 0 0 1 }}
 
{{Mapping|legend=3| 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
Subgroup: 2.5.7.11


Comma list: 176/175, 1310720/1294139
Comma list: 56/55, 128/125
 
Subgroup-val mapping: {{mapping| 3 7 0 2 | 0 0 1 1 }}
 
Gencom mapping: {{mapping| 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
 
=== Frostburn ===
Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 78125/76832


Sval mapping: [{{val| 1 1 3 1 }}, {{val| 0 7 -1 13 }}]
{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
: mapping generators: ~2, ~28/25


Gencom: [2 8/7; 176/175 1310720/1294139]
[[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 1 3 1 }}, {{val| 0 0 7 -1 13 }}]
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


Optimal tuning (POTE): ~8/7 = 227.114
[[Badness]] (Sintel): 0.886


Optimal GPV sequence: {{Val list| 16, 21, 37 }}
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11


=== 2.5.7.11.13 subgroup ===
Comma list: 245/242, 625/616
Subgroup: 2.5.7.11.13


Comma list: 176/175, 640/637, 847/845
Subgroup-val mapping: {{mapping| 1 3 4 5 | 0 -4 -7 -9 }}
: mapping generators: ~2, ~28/25


Sval mapping: [{{val| 1 1 3 1 2 }}, {{val| 0 7 -1 13 9 }}]
Optimal tunings:  
* WE: ~2 = 1200.6817{{c}}, ~28/25 = 205.0734{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 204.8199{{c}}


Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}


Gencom mapping: [{{val| 1 0 1 3 1 2 }}, {{val| 0 0 7 -1 13 9 }}]
Badness (Sintel): 0.463


Optimal tuning (POTE): ~8/7 = 227.108
=== Mabilic ===
{{Main| Mabilic and trismegistus }}
{{See also| Chromatic pairs #Mabilic }}


Optimal GPV sequence: {{Val list| 16, 21, 37 }}
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.


=== 2.5.7.11.13.17 subgroup ===
[[Subgroup]]: 2.5.7
Subgroup: 2.5.7.11.13.17


Comma list: 176/175, 221/200, 640/637, 833/832
[[Comma list]]: 1071875/1048576


Sval mapping: [{{val| 1 1 3 1 2 2 }}, {{val| 0 7 -1 13 9 11 }}]
{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}


Gencom: [2 8/7; 176/175 221/200, 640/637, 833/832]
{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}
: mapping generators: ~2, ~175/128


Gencom mapping: [{{val| 1 0 1 3 1 2 2 }}, {{val| 0 0 7 -1 13 9 11 }}]
[[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 tuning (POTE): ~8/7 = 227.242
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd }}


Optimal GPV sequence: {{Val list| 16, 21, 37 }}
[[Badness]] (Sintel): 1.70


== Didacus ==
=== Huntington ===
{{See also| Hemimean clan #Didacus }}
{{See also| Chromatic pairs #Huntington }}


Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.


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


[[Comma list]]: 3136/3125
[[Comma list]]: 40960000/40353607
 
{{Mapping|legend=2| 1 -4 0 | 0 9 4 }}
 
{{Mapping|legend=3| 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 }}


[[Sval]] [[mapping]]: [{{val| 1 2 2 }}, {{val| 0 2 5 }}]
Badness (Sintel): 1.87


[[Gencom]]: [2 28/25; 3136/3125]
==== 2.5.7.13 subgroup ====
Subgroup: 2.5.7.13


[[Gencom]] [[mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 0 2 5 }}]
Comma list: 640/637, 10985/10976


Optimal tuning ([[POTE]]): ~28/25 = 93.772
Subgroup-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}


{{Val list|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
: mapping generators: ~2, ~13/8


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
Optimal tunings:  
* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}


== Rainy ==
{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].


[[Subgroup]]: 2.5.7
Badness (Sintel): 0.319


[[Comma list]]: [[2100875/2097152]]
==== Silver ====
{{See also| Chromatic pairs #Silver }}


[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.


[[Gencom]]: [2 256/245; 2100875/2097152]
Subgroup: 2.5.7.13.17


[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]
Comma list: 170/169, 640/637, 5525/5488


Optimal tuning ([[POTE]]): ~256/245 = 77.205
Subgroup-val mapping: {{mapping| 1 -4 0 3 9 | 0 9 4 1 -7 }}


{{Val list|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
Gencom mapping: {{mapping| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
Optimal tunings:  
* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}


== Mercy ==
{{Optimal ET sequence|legend=0| 10, 27, 37, 47, 84, 131, 178g }}
{{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): 0.504


[[Subgroup]]: 2.5.7
=== 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.


[[Comma list]]: 823543/819200
Subgroup: 2.5.7.11


[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
Comma list: 8589934592/8544921875, 30691800524/30517578125


[[Gencom]]: [2 2744/2560; 823543/819200]
Subgroup-val mapping: {{mapping| 1 1 20 -49 | 0 3 -39 119 }}
: mapping generators: ~2, ~5120/3773


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
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]]): ~343/320 = 116.291
{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}


{{Val list|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
Badness (Sintel): 11.7


=== 2.5.7.13 ===
==== 2.5.7.11.13 subgroup ====
[[Subgroup]]: 2.5.7.13
Subgroup: 2.5.7.11.13


[[Comma list]]: 343/338, 640/637
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 }}, {{val| 0 -7 -2 -3 }}]
Subgroup-val mapping: {{mapping| 1 1 20 -49 35 | 0 3 -39 119 -71 }}


[[Gencom]]: [2 14/13; 343/338 640/637]
Optimal tunings:  
* WE: ~2 = 1199.9194{{c}}, ~1664/1225 = 528.9681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0036{{c}}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 }}, {{val| 0 0 -7 -2 -3 }}]
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}


Optimal tuning ([[POTE]]): ~14/13 = 116.094
Badness (Sintel): 3.42


{{Val list|legend=1| 10, 21, 31}}
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


=== 2.5.7.13.17 ===
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
[[Subgroup]]: 2.5.7.13.17


[[Comma list]]: 170/169, 224/221, 640/637
Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]
Optimal tunings:  
* WE: ~2 = 1199.9054{{c}}, ~1664/1225 = 528.9628{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0046{{c}}


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 }}, {{val| 0 0 -7 -2 -3 1 }}]
Badness (Sintel): 1.99


Optimal tuning ([[POTE]]): ~14/13 = 115.769
==== 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


{{Val list|legend=1| 10, 21, 31}}
Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}


=== 2.5.7.13.17.19 ===
Optimal tunings:
[[Subgroup]]: 2.5.7.13.17.19
* WE: ~2 = 1199.9081{{c}}, ~19/14 = 528.9639{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/14 = 529.0045{{c}}


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
 
Badness (Sintel): 1.29
 
=== French decimal ===
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


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 3 }}, {{val| 0 -7 -2 -3 1 13 }}]
[[Comma list]]: {{monzo| 372 -159 -1 }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]
{{Mapping|legend=2| 1 0 372 | 0 1 -159 }}
: mapping generators: ~2, ~5


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9901{{c}}, ~5/4 = 386.3562{{c}}
: [[error map]]: {{val| -0.010 +0.023 +0.000 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.3595{{c}}
: error map: {{val| 0.000 +0.046 +0.019 }}


Optimal tuning ([[POTE]]): ~14/13 = 115.716
{{Optimal ET sequence|legend=1| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}


{{Val list|legend=1| 10, 21, 31, 52f }}
[[Badness]] (Sintel): 148


== Pakkanen (rank 3) ==
==== 2.5.7.11 subgroup ====
[[Subgroup]]: 2.5.7.11
Subgroup: 2.5.7.11


[[Comma list]]: 625/616
Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}


Optimal tuning ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}


{{Val list|legend=1| 13, 16, 22, 28, 35, 41, 47, 57, 63, 98c }}
Optimal tunings:
* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}


== Frostburn ==
{{Optimal ET sequence|legend=0| 264, 997e, 1261e, 1525, 1789 }}
[[Subgroup]]: 2.5.7.11


[[Comma list]]: 245/242, 625/616
Badness (Sintel): 52.2


Optimal tuning ([[TE tuning|TE]]): ~2/1 = 1200.6817, ~28/25 = 205.0745
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


{{Val list|legend=1| 29, 35, 41, 47, 88e }}
Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625


== Yer (rank 3) ==
Subgroup-val mapping: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}
[[Subgroup]]: 2.11.13.17.19


[[Comma list]]: 209/208, 2057/2048
Optimal tunings:  
* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}


[[Sval]] [[mapping]]: [{{val| 1 0 0 11 4 }}, {{val| 0 1 0 -2 -1 }}, {{val| 0 0 1 0 1 }}]
{{Optimal ET sequence|legend=0| 261, 1261e, 1525, 1789 }}


Optimal tuning ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
Badness (Sintel): 10.5


{{Val list|legend=1| 13, 24, 33, 37, 46, 57, 70, 127 }}
=== Bastille ===
{{Main| Bastille }}


== Yamablu ==
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]].
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]].


[[Subgroup]]: 2.11.13.17.19
[[Subgroup]]: 2.5.7


[[Comma list]]: 209/208, 2057/2048, 83521/83486
[[Comma list]]: {{monzo| 1426 -596 -15 }}


[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]
{{Mapping|legend=2| 1 -4 254 | 0 15 -596 }}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}


Optimal tuning ([[POTE]]): ~17/13 = 462.9606
[[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 }}


{{Val list|legend=1| 13, 44, 57, 70}}
{{Optimal ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>


== Ostara ==
=== No-threes naiad (rank-3) ===
'''Ostara''' is a temperament that is derived from 93 & 524 solar calendar leap rule scale. It was initially defined by taking the 2.7.13.17.19 subgroup, but it can also be intepreted in general no-threes 19-limit.
{{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 }}


Ostara can also refer to a collection of temperaments which temper out 16807/16796.
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]].


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


[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200
[[Comma list]]: 5021863/5000000
 
{{Mapping|legend=2| 1 0 -2 3 | 0 1 1 1 | 0 0 4 -3 }}
: mapping generators: ~2, ~5, ~77/50


[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]
[[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 tuning]] ([[POTE]]): ~5120/3773 = 529.003¢
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 388, 425, 652e, 689e, 1077de }}


{{Val list|legend=1| 93, 431, 338, 524 }}
[[Badness]] (Sintel): 1.86


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


Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
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 }}
 
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}}


Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e }}


Optimal tuning (POTE): ~1664/1225 = 529.003¢
Badness (Sintel): 0.179


=== 2.5.7.11.13.17 subgroup ===
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17
Subgroup: 2.5.7.11.13.17


Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
Comma list: 170/169, 221/220, 847/845


Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
Subgroup-val mapping: {{mapping| 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]]
 
=== 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]].
 
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 15625/15488
 
{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}
 
{{Mapping|legend=3| 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=2| 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]] 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=2| 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
 
=== Sephiroth ===
{{See also| Chromatic pairs #Sephiroth }}
 
Sephiroth is the no-7 [[restriction]] of [[muggles]].
 
[[Subgroup]]: 2.5.11


Optimal tuning (POTE): ~1664/1225 = 529.003¢
[[Comma list]]: 34375/32768


=== 2.5.7.11.13.17.19 subgroup ===
{{Mapping|legend=2| 1 0 15 | 0 1 -5 }}
Subgroup: 2.5.7.11.13.17.19


Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
{{Mapping|legend=3| 1 0 0 0 15 | 0 0 1 0 -5 }}
: mapping generators: ~2, ~5


Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
[[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 tuning (POTE): ~19/14 = 529.003¢
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceee }}


== Pure onzonic ==
[[Badness]] (Sintel): 1.85
The 2.5.11.13 subgroup primarily contains temperaments developed for 1789edo, since it tempers out the jacobin comma 6656/6655, for which 2.5.11.13 is the subgroup, and the year 1789 is hallmark for the French revolution.


==== 2.5.11.13 subgroup ====
Subgroup: 2.5.11.13
Subgroup: 2.5.11.13


Comma list: 6656/6655, {{monzo| -119 -46 15 47 }}
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 ===
[[Subgroup]]: 2.5.13
 
[[Comma list]]: 26/25
 
{{Mapping|legend=2| 1 2 3 | 0 1 2 }}
: mapping generators, ~2, ~5/4
 
[[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=2| 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=2| 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. It is the no-3 [[restriction]] of [[#Augment|augment]].
 
[[Subgroup]]: 2.7.11
 
[[Comma list]]: 1372/1331
 
{{Mapping|legend=2| 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
 
=== Score ===
{{See also| Chromatic pairs #Score }}
 
Score is a low-accuracy [[extension]] of the unnamed 2.7.11-subgroup temperament tempering out 14641/14336.
 
[[Subgroup]]: 2.7.11.13
 
[[Comma list]]: 343/338, 847/832
 
{{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 generators: ~2, ~11/8
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.5484{{c}}, ~11/8 = 540.7963{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 540.5091{{c}}
 
{{Optimal ET sequence|legend=1| 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=2| 1 0 1 | 0 8 7 }}
 
{{Mapping|legend=3| 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
 
Subgroup-val mapping: {{mapping| 1 0 1 3 | 0 8 7 2 }}
 
Gencom mapping: {{mapping|| 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=2| 1 0 3 | 0 4 1 }}
 
{{Mapping|legend=3| 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=2| 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: [{{val| 1 74 3 74 }}, {{val| 0 -156 1 -153 }}]
{{Mapping|legend=2| 1 0 1 | 0 10 11 }}
: mapping generators, ~2, ~17/14


Optimal tuning (POTE): ~11/8 = 551.370
[[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 GPV sequence: 37, 1789
{{Optimal ET sequence|legend=1| 7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g }}


== Tricesimoprimal miracloid ==
[[Badness]] (Sintel): 0.860
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. Since it is conceived as the temperament in the above specific subgroup, it makes no sense to name it for smaller subgroups. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable [[well temperament]] for 52edo.
 
=== Shipwreck ===
[[Subgroup]]: 2.7.53
 
[[Comma list]]: 1048576/1042139
 
{{Mapping|legend=2| 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=2| 1 3 3 | 0 2 3 }}
 
{{Mapping|legend=3| 1 0 0 0 3 3 | 0 0 0 0 2 3 }}
: mapping generators, ~2, ~13/11
 
[[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=2| 1 0 6 | 0 3 -2 }}
 
{{Mapping|legend=3| 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=2| 4 0 1 | 0 1 1 }}
 
{{Mapping|legend=3| 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 }}
 
{{Optimal ET sequence|legend=1| 4, 12e, 16, 20, 24, 44, 68 }}
 
[[Badness]] (Sintel): 0.668
 
=== 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]].
 
[[Subgroup]]: 2.11.13.17.19
 
[[Comma list]]: 209/208, 2057/2048, 83521/83486
 
{{Mapping|legend=2| 1 1 8 9 11 | 0 4 -7 -8 -11 }}
: mapping generators: ~2, ~26/17
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4661{{c}}, ~26/17 = 737.3256{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/17 = 737.0014{{c}}
 
{{Optimal ET sequence|legend=1| 13, 44, 57, 70, 127, 197eh }}
 
[[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=2| 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=2| 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=2| 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
 
[[Comma list]]: 209/208, 2057/2048


Subgroup: 2.5.7.11.19.29.31
{{Mapping|legend=2| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
: mapping generators: ~2, ~11, ~13


Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
[[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}}


Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]
{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}


Optimal tuning (CTE): ~58/31 = 1084.628
[[Badness]] (Sintel): 0.106


Vals: {{EDOs| 52, 1737, 1789 }}, ...
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
Retrieved from "https://en.xen.wiki/w/No-threes_subgroup_temperaments"