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 }}
 
{{Optimal ET sequence|legend=1| 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d }}
 
[[Badness]] (Sintel): 0.156
 
=== Augment ===
{{See also| Chromatic pairs #Augment }}


[[Sval]] [[mapping]]: [{{Val| 1 1 3 }}, {{Val| 0 7 -1 }}]
Augment is related to [[augmented]], but for 2.5.7 instead of 2.3.5.


Mapping generators: 2, ~8/7
[[Subgroup]]: 2.5.7


[[Gencom]] [[mapping]]: [{{Val| 1 0 1 3 }}, {{Val| 0 0 7 -1 }}]
[[Comma list]]: 128/125


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


[[Optimal tuning]] ([[POTE]]): ~8/7 = 226.910
{{Mapping|legend=3| 3 0 7 0 | 0 0 0 1 }}
: mapping generators: ~5/4, ~7


{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37 }}
[[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 }}


=== 2.5.7.11 subgroup ===
{{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


Sval mapping: [{{val| 1 1 3 1 }}, {{val| 0 7 -1 13 }}]
Subgroup-val mapping: {{mapping| 3 7 0 2 | 0 0 1 1 }}


Gencom: [2 8/7; 176/175 1310720/1294139]
Gencom mapping: {{mapping| 3 0 7 0 2 | 0 0 0 1 1 }}


Gencom mapping: [{{val| 1 0 1 3 1 }}, {{val| 0 0 7 -1 13 }}]
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 tuning (POTE): ~8/7 = 227.114
{{Optimal ET sequence|legend=0| 3, 6, 15, 21 }}


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
Badness (Sintel): 0.196


=== 2.5.7.11.13 subgroup ===
=== Frostburn ===
Subgroup: 2.5.7.11.13
Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].  


Comma list: 176/175, 640/637, 847/845
[[Subgroup]]: 2.5.7


Sval mapping: [{{val| 1 1 3 1 2 }}, {{val| 0 7 -1 13 9 }}]
[[Comma list]]: 78125/76832


Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]
{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
: mapping generators: ~2, ~28/25


Gencom mapping: [{{val| 1 0 1 3 1 2 }}, {{val| 0 0 7 -1 13 9 }}]
[[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 }}


Optimal tuning (POTE): ~8/7 = 227.108
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
[[Badness]] (Sintel): 0.886


=== 2.5.7.11.13.17 subgroup ===
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11.13.17
Subgroup: 2.5.7.11


Comma list: 176/175, 221/200, 640/637, 833/832
Comma list: 245/242, 625/616


Sval mapping: [{{val| 1 1 3 1 2 2 }}, {{val| 0 7 -1 13 9 11 }}]
Subgroup-val mapping: {{mapping| 1 3 4 5 | 0 -4 -7 -9 }}
: mapping generators: ~2, ~28/25


Gencom: [2 8/7; 176/175 221/200, 640/637, 833/832]
Optimal tunings:  
* WE: ~2 = 1200.6817{{c}}, ~28/25 = 205.0734{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 204.8199{{c}}


Gencom mapping: [{{val| 1 0 1 3 1 2 2 }}, {{val| 0 0 7 -1 13 9 11 }}]
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}


Optimal tuning (POTE): ~8/7 = 227.242
Badness (Sintel): 0.463


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
=== Mabilic ===
{{Main| Mabilic and trismegistus }}
{{See also| Chromatic pairs #Mabilic }}


== Didacus ==
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.
{{See also| Hemimean clan #Didacus }}
 
Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]


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


[[Comma list]]: 3136/3125
[[Comma list]]: 1071875/1048576


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


[[Gencom]]: [2 28/25; 3136/3125]
{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}
: mapping generators: ~2, ~175/128


[[Gencom]] [[mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 0 2 5 }}]
[[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]]): ~28/25 = 93.772
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd }}


{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
[[Badness]] (Sintel): 1.70


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
=== Huntington ===
{{See also| Chromatic pairs #Huntington }}


== Rainy ==
Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.  
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].


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


[[Comma list]]: [[2100875/2097152]]
[[Comma list]]: 40960000/40353607
 
{{Mapping|legend=2| 1 -4 0 | 0 9 4 }}


[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
{{Mapping|legend=3| 1 0 -4 0 | 0 0 9 4 }}
: mapping generators: ~2, ~80/49


[[Gencom]]: [2 256/245; 2100875/2097152]
[[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 }}


[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]
{{Optimal ET sequence|legend=0| 7c, 10, 27, 37, 84, 121 }}


Optimal tuning ([[POTE]]): ~256/245 = 77.205
Badness (Sintel): 1.87


{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
==== 2.5.7.13 subgroup ====
Subgroup: 2.5.7.13


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
Comma list: 640/637, 10985/10976


== Mercy ==
Subgroup-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}
{{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]].
Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
: mapping generators: ~2, ~13/8


[[Subgroup]]: 2.5.7
Optimal tunings:  
* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}


[[Comma list]]: 823543/819200
{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}


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


[[Gencom]]: [2 2744/2560; 823543/819200]
==== Silver ====
{{See also| Chromatic pairs #Silver }}


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


Optimal tuning ([[POTE]]): ~343/320 = 116.291
Subgroup: 2.5.7.13.17


{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
Comma list: 170/169, 640/637, 5525/5488


=== 2.5.7.13 ===
Subgroup-val mapping: {{mapping| 1 -4 0 3 9 | 0 9 4 1 -7 }}
[[Subgroup]]: 2.5.7.13


[[Comma list]]: 343/338, 640/637
Gencom mapping: {{mapping| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 }}, {{val| 0 -7 -2 -3 }}]
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; 343/338 640/637]
{{Optimal ET sequence|legend=0| 10, 27, 37, 47, 84, 131, 178g }}


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


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


{{Optimal ET sequence|legend=1| 10, 21, 31}}
Subgroup: 2.5.7.11


=== 2.5.7.13.17 ===
Comma list: 8589934592/8544921875, 30691800524/30517578125
[[Subgroup]]: 2.5.7.13.17


[[Comma list]]: 170/169, 224/221, 640/637
Subgroup-val mapping: {{mapping| 1 1 20 -49 | 0 3 -39 119 }}
: mapping generators: ~2, ~5120/3773


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]
Optimal tunings:  
* WE: ~2 = 1199.9115{{c}}, ~5120/3773 = 528.9650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5120/3773 = 529.0037{{c}}


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 }}, {{val| 0 0 -7 -2 -3 1 }}]
Badness (Sintel): 11.7


Optimal tuning ([[POTE]]): ~14/13 = 115.769
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


{{Optimal ET sequence|legend=1| 10, 21, 31}}
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125


=== 2.5.7.13.17.19 ===
Subgroup-val mapping: {{mapping| 1 1 20 -49 35 | 0 3 -39 119 -71 }}
[[Subgroup]]: 2.5.7.13.17.19


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
Optimal tunings:  
* WE: ~2 = 1199.9194{{c}}, ~1664/1225 = 528.9681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0036{{c}}


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 3 }}, {{val| 0 -7 -2 -3 1 13 }}]
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]
Badness (Sintel): 3.42


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


Optimal tuning ([[POTE]]): ~14/13 = 115.716
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251


{{Optimal ET sequence|legend=1| 10, 21, 31, 52f }}
Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}


== Pakkanen (rank 3) ==
Optimal tunings:
[[Subgroup]]: 2.5.7.11
* 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]]: 625/616
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875


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


: mapping generators: ~2, ~5, ~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]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
Badness (Sintel): 1.29


== Frostburn ==
=== French decimal ===
{{See also| Magic family #Quadrimage }}
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]]: 78125/76832
[[Comma list]]: {{monzo| 372 -159 -1 }}


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


: Sval mapping generators: ~2, ~28/25
[[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]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389
{{Optimal ET sequence|legend=1| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
[[Badness]] (Sintel): 148


=== 2.5.7.11 ===
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
Subgroup: 2.5.7.11


Comma list: 245/242, 625/616
Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}
 
Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}


{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
Optimal tunings:
* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}


: Sval mapping generators: ~2, ~28/25
{{Optimal ET sequence|legend=0| 264, 997e, 1261e, 1525, 1789 }}


Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
Badness (Sintel): 52.2


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


== Yer (rank 3) ==
Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
[[Subgroup]]: 2.11.13.17.19
 
Subgroup-val mapping: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}


[[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]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 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


{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}
=== 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 }}


{{Optimal ET sequence|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 }}


{{Optimal ET sequence|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 }}


Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
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 tuning (POTE): ~1664/1225 = 529.003¢
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e }}


{{Optimal ET sequence|legend=1| 93, 245e, 338, 431, 1386c }}
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


Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
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 }}
 
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}}


Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
{{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


Optimal tuning (POTE): ~1664/1225 = 529.003¢
[[Comma list]]: 1331/1280


{{Optimal ET sequence|legend=1| 93, 338, 431, 955c, 1386cg }}
{{Mapping|legend=2| 1 1 3 | 0 3 1 }}
: mapping generators: ~2, ~11/8


=== 2.5.7.11.13.17.19 subgroup ===
[[Optimal tuning]]s:
Subgroup: 2.5.7.11.13.17.19
* [[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 }}


Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}


Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
[[Badness]] (Sintel): 0.718


Optimal tuning (POTE): ~19/14 = 529.003¢
=== Sephiroth ===
{{See also| Chromatic pairs #Sephiroth }}


== Pure onzonic ==
Sephiroth is the no-7 [[restriction]] of [[muggles]].  
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.


Subgroup: 2.5.11.13
[[Subgroup]]: 2.5.11


Comma list: 6656/6655, {{monzo| -119 -46 15 47 }}
[[Comma list]]: 34375/32768


Mapping: [{{val| 1 74 3 74 }}, {{val| 0 -156 1 -153 }}]
{{Mapping|legend=2| 1 0 15 | 0 1 -5 }}


Optimal tuning (POTE): ~11/8 = 551.370
{{Mapping|legend=3| 1 0 0 0 15 | 0 0 1 0 -5 }}
: mapping generators: ~2, ~5


{{Optimal ET sequence|legend=1|37, 1789}}
[[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 }}


== Tricesimoprimal miracloid ==
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceee }}
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.


Subgroup: 2.5.7.11.19.29.31
[[Badness]] (Sintel): 1.85


Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
==== 2.5.11.13 subgroup ====
Subgroup: 2.5.11.13


Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]
Comma list: 65/64, 6875/6656


Optimal tuning (CTE): ~58/31 = 1084.628
Subgroup-val mapping: {{mapping| 1 0 15 6 | 0 1 -5 -1 }}


{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
Gencom mapping: {{mapping| 1 0 0 0 15 6 | 0 0 1 0 -5 -1 }}


== French decimal ==
Optimal tunings:
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.
* WE: ~2 = 1203.3825{{c}}, ~5/4 = 373.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 372.1519{{c}}


Subgroup: 2.5.7
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceeeff }}


Comma basis: {{monzo|372 -159 -1}}
Badness (Sintel): 0.410


Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}]
==== 2.5.11.13.17 subgroup ====
Subgroup: 2.5.11.13.17


Optimal tuning (CTE): ~5/4 = 386.360
Comma list: 65/64, 170/169, 221/220


{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
Subgroup-val mapping: {{mapping| 1 0 15 6 11 | 0 1 -5 -1 -3 }}


=== 2.5.7.11 subgroup ===
Gencom mapping: {{mapping| 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}}


Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29g, 129ccceeffgggg }}


Sval mapping: [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]
Badness (Sintel): 0.299


Optimal tuning (CTE): ~5/4 = 386.361
=== Trader ===
[[Subgroup]]: 2.5.13


{{Optimal ET sequence|legend=1|264, 733}}, ...
[[Comma list]]: 26/25


=== 2.5.7.11.13 subgroup ===
{{Mapping|legend=2| 1 2 3 | 0 1 2 }}
Subgroup: 2.5.7.11.13
: mapping generators, ~2, ~5/4


Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
[[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 }}


Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]
{{Optimal ET sequence|legend=1| 3, 20c, 23c, 26c }}


Optimal tuning (CTE): ~5/4 = 386.361
[[Badness]] (Sintel): 0.138


{{Optimal ET sequence|legend=1|1525, 1789}}, ...
=== Superquintal ===
[[Subgroup]]: 2.5.13


== Mabon ==
[[Comma list]]: 64000000/62748517
Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.


Subgroup: 2.9.7
{{Mapping|legend=2| 1 -2 0 | 0 7 6 }}
: mapping generators, ~2, ~20/13


Comma basis: 44957696/43046721
[[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 }}


Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]
{{Optimal ET sequence|legend=1| 8, 13, 21, 34, 81, 115 }}


Optimal tuning (CTE): ~729/448 = 870.792
[[Badness]] (Sintel): 1.93


{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...
== 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.


=== 2.9.7.11 subgroup ===
=== Amaranthine ===
Subgroup: 2.9.7.11
{{See also| No-fives subgroup temperaments #Chrysanthemum }}


Comma basis: 896/891, 1331/1296
Amaranthine is the high-accuracy 2.7.11-subgroup strong [[restriction]] of [[Gamelismic clan #11-limit 3|undecimal mothra]].


Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]
[[Subgroup]]: 2.7.11


Optimal tuning (CTE): ~16/11 = 870.966
[[Comma list]]: 5767168/5764801


{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}
{{Mapping|legend=2| 1 0 -19 | 0 1 8 }}
: mapping generators: ~2, ~7


== Bastille ==
[[Optimal tuning]]s:
{{Main|Bastille}}
* [[WE]]: ~2 = 1199.9846{{c}}, ~7/4 = 968.9078{{c}}
Described as the 2.5.7 subgroup 1407 & 1789 temperament, and named after an eponymous historical event which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Pure Bastille|pure bastille]].
: [[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 }}


Subgroup: 2.5.7
{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d, 12780dd }}


Comma list: {{Monzo|1426 -596 -15}}
Badness (Sintel): 0.0309


Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
=== Argument ===
Argument tempers out [[1372/1331]] in the 2.7.11 subgroup. It is the no-3 [[restriction]] of [[#Augment|augment]].


Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
[[Subgroup]]: 2.7.11


{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
[[Comma list]]: 1372/1331


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


[[Subgroup]]: 2.7.53
[[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 }}


[[Comma list]]: 1048576/1042139
{{Optimal ET sequence|legend=1| 6, 9, 15, 36, 51e, 66e }}


[[Gencom]]: [2 64/53; 1048576/1042139]
[[Badness]] (Sintel): 0.475


[[Mapping]]: [{{val|1 0 6}}, {{val|0 3 -1}}]]
=== Score ===
{{See also| Chromatic pairs #Score }}


[[POTE generator]]: ~64/53 = 323.034
Score is a low-accuracy [[extension]] of the unnamed 2.7.11-subgroup temperament tempering out 14641/14336.  


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
[[Subgroup]]: 2.7.11.13


== Movila ==
[[Comma list]]: 343/338, 847/832
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
{{Mapping|legend=2| 1 1 3 1 | 0 4 1 6 }}


[[Comma list]]: 1331/1280
{{Mapping|legend=3| 1 0 0 1 3 1 | 0 0 0 4 1 6 }}
: mapping generators: ~2, ~11/8


[[Mapping]]: [{{val|1 1 3}}, {{val|0 3 1}}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.5484{{c}}, ~11/8 = 540.7963{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 540.5091{{c}}


[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[11/8]] = 529.846
{{Optimal ET sequence|legend=1| 9, 11, 20 }}


{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
[[Badness]] (Sintel): 0.368


== Mavericks ==
=== Bossier ===
{{See also| Chromatic pairs #Bossier }}


[[Subgroup]]: 2.13.19
Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup, tempering out [[1573/1568]] and [[15488/15379]].  


[[Comma list]]: 47525504/47045881
[[Subgroup]]: 2.7.11


[[Mapping]]: [{{val|1 1 2}}, {{val|0 6 5}}]
[[Comma list]]: 214358881/210827008


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/19 = 539.886
{{Mapping|legend=2| 1 0 1 | 0 8 7 }}


{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}
{{Mapping|legend=3| 1 0 0 0 1 | 0 0 0 8 7 }}
: mapping generators: ~2, ~14/11


== Vengeance ==
[[Optimal tuning]]s:
''Main article: [[vengeance]]''
* [[WE]]: ~2 = 1200.1886{{c}}, ~14/11 = 421.2661{{c}}
Another lower-error replica of mavila, with the fifth being ~[[25/17]] instead of ~[[3/2]].
: [[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 }}


[[Subgroup]]: 2.5.17
{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 151 }}


[[Comma list]]: 78608/78125
[[Badness]] (Sintel): 1.73


{{Mapping|legend=2|1 1 1|0 3 7}}
==== 2.7.11.13 subgroup ====
Subgroup: 2.7.11.13


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[34/25]] = 529.095
Comma list: 1573/1568, 15488/15379


{{Optimal ET sequence|legend=1|7g, 9, 25, 34, 93, 127, 288, 415}}
Subgroup-val mapping: {{mapping| 1 0 1 3 | 0 8 7 2 }}


== Superquintal ==
Gencom mapping: {{mapping|| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
[[Subgroup]]: 2.5.13


[[Comma list]]: 64000000/62748517
Optimal tunings:  
* WE: ~2 = 1199.8668{{c}}, ~14/11 = 421.2623{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 421.2874{{c}}


{{Mapping|legend=2|1 5 6|0 -7 -6}}
{{Optimal ET sequence|legend=0| 17, 20, 37, 57, 94, 225 }}


: Mapping generators, ~2, ~13/10
Badness (Sintel): 0.307


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281
=== 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.  


{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}
[[Subgroup]]: 2.7.13


== Insect ==
[[Comma list]]: 28672/28561
[[Subgroup]]: 2.5.11


[[Comma list]]: 33275/32768
{{Mapping|legend=2| 1 0 3 | 0 4 1 }}


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


: Mapping generators, ~2, ~[[55/32]]
[[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 tuning]] ([[CTE]]): ~2 = 1\1, ~[[55/32]] = 928.032
{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104, 463f, 567f, 671ff, 775ff }}


{{Optimal ET sequence|legend=1|9, 13, 22, 97e, 119e, 141e, 163e, 304ceee}}
[[Badness]] (Sintel): 0.115


== Supraminor ==
=== Ultrakleismic ===
[[Subgroup]]: 2.7.17
[[Subgroup]]: 2.7.17


[[Comma list]]: 4913/4802
[[Comma list]]: 4913/4802


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


: 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 tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 324.446
{{Optimal ET sequence|legend=1| 4, 7, 11, 26, 37 }}


{{Optimal ET sequence|legend=1|4, 7, 11, 26, 37}}
[[Badness]] (Sintel): 0.460


== Countersupraminor ==
=== Counterultrakleismic ===
[[Subgroup]]: 2.7.17
[[Subgroup]]: 2.7.17


[[Comma list]]: 2024782584832/2015993900449
[[Comma list]]: 2024782584832/2015993900449


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


: 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 tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 336.858
{{Optimal ET sequence|legend=1| 7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g }}


{{Optimal ET sequence|legend=1|7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g}}
[[Badness]] (Sintel): 0.860


== Trader ==
=== Shipwreck ===
[[Subgroup]]: 2.5.13
[[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


[[Comma list]]: [[26/25]]
=== Yer (rank 3) ===
[[Subgroup]]: 2.11.13.17.19


{{Mapping|legend=2|1 2 3|0 1 2}}
[[Comma list]]: 209/208, 2057/2048


: Mapping generators, ~2, ~[[5/4]]
{{Mapping|legend=2| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
: mapping generators: ~2, ~11, ~13


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[5/4]] = 407.079
[[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}}
 
{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}


{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}
[[Badness]] (Sintel): 0.106


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]

Revision as of 09:20, 29 July 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of subgroup temperaments which omit the prime harmonic of 3.

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.
  • 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

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)3 (128/125, the lesser diesis) with (2/1)/(8/7)5 (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 ¢ sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7 ¢). 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

Subgroup-val mapping: [⟨1 2 3], ⟨0 5 -3]]

Gencom mapping: [⟨1 0 2 3], ⟨0 0 5 -3]]

mapping generators: ~2, ~256/245

Optimal tunings:

  • WE: ~2 = 1200.0939 ¢, ~256/245 = 77.2107 ¢
error map: ⟨+0.094 -0.072 -0.176]
  • CWE: ~2 = 1200.0000 ¢, ~256/245 = 77.2093 ¢
error map: ⟨0.000 -0.267 -0.454]

Optimal ET sequence: 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d

Badness (Sintel): 0.156

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

Comma list: 128/125

Subgroup-val mapping: [⟨3 7 0], ⟨0 0 1]]

Gencom mapping: [⟨3 0 7 0], ⟨0 0 0 1]]

mapping generators: ~5/4, ~7

Optimal tunings:

  • WE: ~5/4 = 399.0128 ¢, ~7/4 = 974.7085 ¢
error map: ⟨-2.962 +6.776 -0.040]
  • CWE: ~5/4 = 400.0000 ¢, ~7/4 = 974.3418 ¢
error map: ⟨0.000 +13.686 +5.516]

Optimal ET sequence: 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

Subgroup-val mapping: [⟨3 7 0 2], ⟨0 0 1 1]]

Gencom mapping: [⟨3 0 7 0 2], ⟨0 0 0 1 1]]

Optimal tunings:

  • WE: ~5/4 = 398.9239 ¢, ~7/4 = 969.1106 ¢
  • CWE: ~5/4 = 400.0000 ¢, ~7/4 = 968.4397 ¢

Optimal ET sequence: 3, 6, 15, 21

Badness (Sintel): 0.196

Frostburn

Frostburn is the common restriction of quadrimage and baldy.

Subgroup: 2.5.7

Comma list: 78125/76832

Subgroup-val mapping: [⟨1 3 4], ⟨0 -4 -7]]

mapping generators: ~2, ~28/25

Optimal tunings:

  • WE: ~2 = 1200.3462 ¢, ~28/25 = 204.3386 ¢
error map: ⟨+0.346 -2.630 +2.189]
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 204.2027 ¢
error map: ⟨0.000 -3.125 +1.755]

Optimal ET sequence: 6, 29, 35, 41, 47

Badness (Sintel): 0.886

2.5.7.11 subgroup

Subgroup: 2.5.7.11

Comma list: 245/242, 625/616

Subgroup-val mapping: [⟨1 3 4 5], ⟨0 -4 -7 -9]]

mapping generators: ~2, ~28/25

Optimal tunings:

  • WE: ~2 = 1200.6817 ¢, ~28/25 = 205.0734 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 204.8199 ¢

Optimal ET sequence: 6, 23de, 29, 35, 41

Badness (Sintel): 0.463

Mabilic

Main article: Mabilic and trismegistus
See also: Chromatic pairs § Mabilic

Mabilic is the no-3 restriction of armodue, semabila, and trismegistus. It is the 7 & 9 temperament in the 2.5.7 subgroup, and tempers out 1071875/1048576, the mabilisma.

Subgroup: 2.5.7

Comma list: 1071875/1048576

Subgroup-val mapping: [⟨1 1 5], ⟨0 3 -5]]

Gencom mapping: [⟨1 0 1 5], ⟨0 0 3 -5]]

mapping generators: ~2, ~175/128

Optimal tunings:

  • WE: ~2 = 1201.2543 ¢, ~175/128 = 527.7872 ¢
error map: ⟨+1.254 -1.698 -1.491]
  • CWE: ~2 = 1200.0000 ¢, ~175/128 = 527.2041 ¢
error map: ⟨0.000 -4.701 -4.846]

Optimal ET sequence: 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd

Badness (Sintel): 1.70

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

Subgroup-val mapping: [⟨1 -4 0], ⟨0 9 4]]

Gencom mapping: [⟨1 0 -4 0], ⟨0 0 9 4]]

mapping generators: ~2, ~80/49

Optimal tunings:

  • WE: ~2 = 1199.5781 ¢, ~80/49 = 842.6730 ¢
error map: ⟨-0.422 -0.569 +1.866]
  • CWE: ~2 = 1200.0000 ¢, ~80/49 = 842.9136 ¢
error map: ⟨0.000 -0.091 +2.828]

Optimal ET sequence: 7c, 10, 27, 37, 84, 121

Badness (Sintel): 1.87

2.5.7.13 subgroup

Subgroup: 2.5.7.13

Comma list: 640/637, 10985/10976

Subgroup-val mapping: [⟨1 -4 0 3], ⟨0 9 4 1]]

Gencom mapping: [⟨1 0 -4 0 0 3], ⟨0 0 9 4 0 1]]

mapping generators: ~2, ~13/8

Optimal tunings:

  • WE: ~2 = 1199.4788 ¢, ~13/8 = 842.6318 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 842.9447 ¢

Optimal ET sequence: 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf

Badness (Sintel): 0.319

Silver

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

Subgroup-val mapping: [⟨1 -4 0 3 9], ⟨0 9 4 1 -7]]

Gencom mapping: [⟨1 0 -4 0 0 3 9], ⟨0 0 9 4 0 1 -7]]

Optimal tunings:

  • WE: ~2 = 1200.0932 ¢, ~13/8 = 842.7764 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 842.7143 ¢

Optimal ET sequence: 10, 27, 37, 47, 84, 131, 178g

Badness (Sintel): 0.504

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

Subgroup-val mapping: [⟨1 1 20 -49], ⟨0 3 -39 119]]

mapping generators: ~2, ~5120/3773

Optimal tunings:

  • WE: ~2 = 1199.9115 ¢, ~5120/3773 = 528.9650 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5120/3773 = 529.0037 ¢

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 1 20 -49 35], ⟨0 3 -39 119 -71]]

Optimal tunings:

  • WE: ~2 = 1199.9194 ¢, ~1664/1225 = 528.9681 ¢
  • CWE: ~2 = 1200.0000 ¢, ~1664/1225 = 529.0036 ¢

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 1 20 -49 35 42], ⟨0 3 -39 119 -71 -86]]

Optimal tunings:

  • WE: ~2 = 1199.9054 ¢, ~1664/1225 = 528.9628 ¢
  • CWE: ~2 = 1200.0000 ¢, ~1664/1225 = 529.0046 ¢

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 1 20 -49 35 42 21], ⟨0 3 -39 119 -71 -86 -38]]

Optimal tunings:

  • WE: ~2 = 1199.9081 ¢, ~19/14 = 528.9639 ¢
  • CWE: ~2 = 1200.0000 ¢, ~19/14 = 529.0045 ¢

Optimal ET sequence: 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

Comma list: [372 -159 -1⟩

Subgroup-val mapping: [⟨1 0 372], ⟨0 1 -159]]

mapping generators: ~2, ~5

Optimal tunings:

  • WE: ~2 = 1199.9901 ¢, ~5/4 = 386.3562 ¢
error map: ⟨-0.010 +0.023 +0.000]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.3595 ¢
error map: ⟨0.000 +0.046 +0.019]

Optimal ET sequence: 205, 264, 733, 997, 2258, 3255, 7507, 10762

Badness (Sintel): 148

2.5.7.11 subgroup

Subgroup: 2.5.7.11

Comma list: [-49 8 17 -5⟩, [45 -27 10 -3⟩

Subgroup-val mapping: [⟨1 0 372 1255], ⟨0 1 -159 -539]]

Optimal tunings:

  • WE: ~2 = 1200.0130 ¢, ~5/4 = 386.3653 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.3611 ¢

Optimal ET sequence: 264, 997e, 1261e, 1525, 1789

Badness (Sintel): 52.2

2.5.7.11.13 subgroup

Subgroup: 2.5.7.11.13

Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625

Subgroup-val mapping: [⟨1 0 372 1255 -398], ⟨0 1 -159 -539 173]]

Optimal tunings:

  • WE: ~2 = 1200.0137 ¢, ~5/4 = 386.3655 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.3611 ¢

Optimal ET sequence: 261, 1261e, 1525, 1789

Badness (Sintel): 10.5

Bastille

Main article: Bastille

Bastille is described as the 2.5.7-subgroup 1407 & 1789 temperament, and named after an eponymous historical event which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include double bastille.

Subgroup: 2.5.7

Comma list: [1426 -596 -15⟩

Subgroup-val mapping: [⟨1 -4 254], ⟨0 15 -596]]

mapping generators: ~2, ~[-380 159 4⟩

Optimal tunings:

  • WE: ~2 = 1199.9911 ¢, ~[-380 159 4⟩ = 505.7532 ¢
error map: ⟨-0.009 +0.020 +0.001]
  • CWE: ~2 = 1200.0000 ¢, ~[-380 159 4⟩ = 505.7570 ¢
error map: ⟨0.000 +0.041 +0.018]

Optimal ET sequence: 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd

Badness (Sintel): 7.18 × 103

No-threes naiad (rank-3)

Todo: review

Devise the permanent name for this temperament; determine its status with respect to its extensions linked below.

See also: Wizardharry clan § Naiad, and Werckismic temperaments § Seminaiad

This temperament can be described as the 21 & 29 & 37 temperament in no-threes subgroups. It expands tridec and naiadec.

Subgroup: 2.5.7.11

Comma list: 5021863/5000000

Subgroup-val mapping: [⟨1 0 -2 3], ⟨0 1 1 1], ⟨0 0 4 -3]]

mapping generators: ~2, ~5, ~77/50

Optimal tunings:

  • WE: ~2 = 1200.0805 ¢, ~5/4 = 386.6593 ¢, ~77/50 = 745.4622 ¢
error map: ⟨+0.080 +0.507 -1.318 -0.643]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.7404 ¢, ~77/50 = 745.4102 ¢
error map: ⟨0.000 +0.427 -0.445 -0.808]

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 0 -2 3 2], ⟨0 1 1 1 1], ⟨0 0 4 -3 -1]]

Optimal tunings:

  • WE: ~2 = 1200.0343 ¢, ~5/4 = 386.6098 ¢, ~20/13 = 745.4658 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.6458 ¢, ~20/13 = 745.4431 ¢

Optimal ET sequence: 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

Subgroup-val 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 ¢, ~5/4 = 386.6701 ¢, ~17/11 = 745.3706 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.1074 ¢, ~17/11 = 745.0940 ¢

Optimal ET sequence: 16, 21, 29g, 37, 66g, 87g, 124g

Badness (Sintel): 0.438

Temperaments with a higher 2.5.p gene

Temperaments discussed elsewhere include:

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

Subgroup-val mapping: [⟨2 0 -7], ⟨0 1 3]]

Gencom mapping: [⟨2 0 4 0 5], ⟨0 0 1 0 3]]

mapping generators: ~125/88, ~5/4

Optimal tunings:

  • WE: ~125/88 = 600.1831 ¢, ~5/4 = 383.8848 ¢
error map: ⟨+0.366 -1.697 +1.252]
  • CWE: ~125/88 = 600.0000 ¢, ~5/4 = 383.9977 ¢
error map: ⟨0.000 -2.316 +0.675]

Optimal ET sequence: 6, 16, 22, 28, 50, 122, 172, 222, 394c

Badness (Sintel): 0.266

Insect

Subgroup: 2.5.11

Comma list: 33275/32768

Subgroup-val mapping: [⟨1 0 5], ⟨0 3 -2]]

mapping generators, ~2, ~55/32

Optimal tunings:

  • WE: ~2 = 1201.1238 ¢, ~55/32 = 928.5003 ¢
error map: ⟨+1.124 -0.813 -2.700]
  • CWE: ~2 = 1200.0000 ¢, ~55/32 = 927.7384 ¢
error map: ⟨0.000 -3.099 -6.975]

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 1 3], ⟨0 3 1]]

mapping generators: ~2, ~11/8

Optimal tunings:

  • WE: ~2 = 1203.0339 ¢, ~11/8 = 528.4296 ¢
error map: ⟨+3.034 +2.009 -13.787]
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 528.1575 ¢
error map: ⟨0.000 -1.841 -23.160]

Optimal ET sequence: 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

Comma list: 34375/32768

Subgroup-val mapping: [⟨1 0 15], ⟨0 1 -5]]

Gencom mapping: [⟨1 0 0 0 15], ⟨0 0 1 0 -5]]

mapping generators: ~2, ~5

Optimal tunings:

  • WE: ~2 = 1203.3290 ¢, ~5/4 = 373.6097 ¢
error map: ⟨+3.329 -6.046 -2.722]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 372.1586 ¢
error map: ⟨0.000 -14.155 -12.111]

Optimal ET sequence: 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: [⟨1 0 15 6], ⟨0 1 -5 -1]]

Gencom mapping: [⟨1 0 0 0 15 6], ⟨0 0 1 0 -5 -1]]

Optimal tunings:

  • WE: ~2 = 1203.3825 ¢, ~5/4 = 373.6318 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 372.1519 ¢

Optimal ET sequence: 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: [⟨1 0 15 6 11], ⟨0 1 -5 -1 -3]]

Gencom mapping: [⟨1 0 0 0 15 6 11], ⟨0 0 1 0 -5 -1 -3]]

Optimal tunings:

  • WE: ~2 = 1203.6741 ¢, ~5/4 = 373.3775 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 371.6773 ¢

Optimal ET sequence: 3, 10, 13, 16, 29g, 129ccceeffgggg

Badness (Sintel): 0.299

Trader

Subgroup: 2.5.13

Comma list: 26/25

Subgroup-val mapping: [⟨1 2 3], ⟨0 1 2]]

mapping generators, ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1198.0216 ¢, ~5/4 = 410.2152 ¢
error map: ⟨-1.978 +19.945 -26.033]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 408.9029 ¢
error map: ⟨0.000 +22.589 -22.722]

Optimal ET sequence: 3, 20c, 23c, 26c

Badness (Sintel): 0.138

Superquintal

Subgroup: 2.5.13

Comma list: 64000000/62748517

Subgroup-val mapping: [⟨1 -2 0], ⟨0 7 6]]

mapping generators, ~2, ~20/13

Optimal tunings:

  • WE: ~2 = 1199.5925 ¢, ~20/13 = 740.6286 ¢
error map: ⟨-0.408 -1.098 +3.244]
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 740.8058 ¢
error map: ⟨0.000 -0.673 +4.307]

Optimal ET sequence: 8, 13, 21, 34, 81, 115

Badness (Sintel): 1.93

No-threes no-fives subgroup temperaments

Temperaments discussed elsewhere include

Amaranthine

See also: No-fives subgroup temperaments § Chrysanthemum

Amaranthine is the high-accuracy 2.7.11-subgroup strong restriction of undecimal mothra.

Subgroup: 2.7.11

Comma list: 5767168/5764801

Subgroup-val mapping: [⟨1 0 -19], ⟨0 1 8]]

mapping generators: ~2, ~7

Optimal tunings:

  • WE: ~2 = 1199.9846 ¢, ~7/4 = 968.9078 ¢
error map: ⟨-0.015 +0.051 -0.010]
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 968.9174 ¢
error map: ⟨0.000 +0.091 +0.021]

Optimal ET sequence: 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.

Subgroup: 2.7.11

Comma list: 1372/1331

Subgroup-val mapping: [⟨3 0 2], ⟨0 1 1]]

mapping generators: ~14/11, ~7

Optimal tunings:

  • WE: ~14/11 = 399.8041 ¢, ~7/4 = 963.1666 ¢
error map: ⟨-0.588 -6.835 +10.281]
  • CWE: ~14/11 = 400.0000 ¢, ~4/4 = 962.7466 ¢
error map: ⟨0.000 -6.079 +11.429]

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 1 3 1], ⟨0 4 1 6]]

Gencom mapping: [⟨1 0 0 1 3 1], ⟨0 0 0 4 1 6]]

mapping generators: ~2, ~11/8

Optimal tunings:

  • WE: ~2 = 1201.5484 ¢, ~11/8 = 540.7963 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 540.5091 ¢

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 0 1], ⟨0 8 7]]

Gencom mapping: [⟨1 0 0 0 1], ⟨0 0 0 8 7]]

mapping generators: ~2, ~14/11

Optimal tunings:

  • WE: ~2 = 1200.1886 ¢, ~14/11 = 421.2661 ¢
error map: ⟨+0.189 +1.303 -2.266]
  • CWE: ~2 = 1200.0000 ¢, ~14/11 = 421.2365 ¢
error map: ⟨0.000 +1.066 -2.662]

Optimal ET sequence: 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: [⟨1 0 1 3], ⟨0 8 7 2]]

Gencom mapping: [⟨], ⟨1 0 0 0 1 3], ⟨0 0 0 8 7 2]]

Optimal tunings:

  • WE: ~2 = 1199.8668 ¢, ~14/11 = 421.2623 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/11 = 421.2874 ¢

Optimal ET sequence: 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, 71/4 of 842.2065 ¢, which is also the CTC (constrained Tenney–Chebyshevian) tuning.

Subgroup: 2.7.13

Comma list: 28672/28561

Subgroup-val mapping: [⟨1 0 3], ⟨0 4 1]]

Gencom mapping: [⟨1 0 0 0 0 3], ⟨0 0 0 4 0 1]]

mapping generators: ~2, ~13

Optimal tunings:

  • WE: ~2 = 1199.7827 ¢, ~13/8 = 842.1707 ¢
error map: ⟨-0.217 -0.143 +0.991]
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 842.2568 ¢
error map: ⟨0.000 +0.201 +1.729]

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 2 3], ⟨0 3 4]]

mapping generators, ~2, ~17/14

Optimal tunings:

  • WE: ~2 = 1200.1379 ¢, ~17/14 = 324.3440 ¢
error map: ⟨+0.138 +4.482 -7.166]
  • CWE: ~2 = 1200.000 ¢, ~17/14 = 324.3738 ¢
error map: ⟨0.000 +4.295 -7.460]

Optimal ET sequence: 4, 7, 11, 26, 37

Badness (Sintel): 0.460

Counterultrakleismic

Subgroup: 2.7.17

Comma list: 2024782584832/2015993900449

Subgroup-val mapping: [⟨1 0 1], ⟨0 10 11]]

mapping generators, ~2, ~17/14

Optimal tunings:

  • WE: ~2 = 1199.9723 ¢, ~17/14 = 336.8586 ¢
error map: ⟨-0.028 -0.240 +0.462]
  • CWE: ~2 = 1200.000 ¢, ~17/14 = 336.8621 ¢
error map: ⟨0.000 -0.205 +0.528]

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 0 6], ⟨0 3 -1]]]

mapping generators, ~2, ~64/53

Optimal tunings:

  • WE: ~2 = 1199.6967 ¢, ~64/53 = 323.1839 ¢
error map: ⟨-0.303 +0.119 +1.491]
  • CWE: ~2 = 1200.0000 ¢, ~64/53 = 323.1959 ¢
error map: ⟨0.000 +0.762 +3.300]

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 3 3], ⟨0 2 3]]

Gencom mapping: [⟨1 0 0 0 3 3], ⟨0 0 0 0 2 3]]

mapping generators, ~2, ~13/11

Optimal tunings:

  • WE: ~2 = 1199.5223 ¢, ~13/11 = 279.2064 ¢
error map: ⟨-0.478 +5.662 -4.341]
  • CWE: ~2 = 1200.0000 ¢, ~13/11 = 278.9918 ¢
error map: ⟨0.000 +6.666 -3.552]

Optimal ET sequence: 4, 9, 13, 30, 43, 73, 116e

Badness (Sintel): 0.175

Bluebirds

Not to be confused with Bluebird.
See also: Chromatic pairs § Bluebirds

Subgroup: 2.11.13

Comma list: 265837/262144

Subgroup-val mapping: [⟨1 0 6], ⟨0 3 -2]]

Gencom mapping: [⟨1 0 0 0 3 4], ⟨0 0 0 0 3 -2]]

mapping generators, ~2, ~143/128

Optimal tunings:

  • WE: ~2 = 1200.8795 ¢, ~143/128 = 182.5017 ¢
error map: ⟨+0.880 -1.174 -2.013]
  • CWE: ~2 = 1200.0000 ¢, ~143/128 = 182.4386 ¢
error map: ⟨0.000 -4.002 -5.405]

Optimal ET sequence: 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

Subgroup-val mapping: [⟨4 0 1], ⟨0 1 1]]

Gencom mapping: [⟨4 0 0 0 12 13], ⟨0 0 0 0 1 1]]

mapping generators, ~13/11, ~11

Optimal tunings:

  • WE: ~13/11 = 299.9728 ¢, ~11/8 = 546.6107 ¢
error map: ⟨-0.109 -5.033 +5.730]
  • CWE: ~13/11 = 300.0000 ¢, ~11/8 = 546.4664 ¢
error map: ⟨0.000 -4.852 +5.939]

Optimal ET sequence: 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 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

Subgroup-val mapping: [⟨1 1 8 9 11], ⟨0 4 -7 -8 -11]]

mapping generators: ~2, ~26/17

Optimal tunings:

  • WE: ~2 = 1200.4661 ¢, ~26/17 = 737.3256 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/17 = 737.0014 ¢

Optimal ET sequence: 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

Subgroup-val mapping: [⟨33 0 8 249 26], ⟨0 1 1 -1 1]]

mapping generators: ~247/242, ~11

Optimal tunings:

  • WE: ~247/242 = 36.3676 ¢, ~11/8 = 550.4246 ¢
  • CWE: ~247/242 = 36.3636 ¢, ~11/8 = 550.4187 ¢

Optimal ET sequence: 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 continued fraction convergent to 21/4 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 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

Subgroup-val mapping: [⟨4 0 7], ⟨0 1 1]]

mapping generators: ~44/37, ~11

Optimal tunings:

  • WE: ~44/37 = 300.0003 ¢, ~11/8 = 551.3211 ¢
error map: ⟨+0.001 +0.007 -0.017]
  • CWE: ~44/37 = 300.0000 ¢, ~11/8 = 551.3237 ¢
error map: ⟨0.000 +0.006 -0.020]

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 1 2], ⟨0 6 5]]

mapping generators: ~2, ~26/19

Optimal tunings:

  • WE: ~2 = 1199.8817 ¢, ~26/19 = 539.9150 ¢
error map: ⟨-0.118 -1.156 +1.825]
  • CWE: ~2 = 1200.0000 ¢, ~26/19 = 539.9280 ¢
error map: ⟨0.000 -0.960 +2.127]

Optimal ET sequence: 9, 11, 20

Badness (Sintel): 0.559

Yer (rank 3)

Subgroup: 2.11.13.17.19

Comma list: 209/208, 2057/2048

Subgroup-val mapping: [⟨1 0 0 11 4], ⟨0 1 0 -2 -1], ⟨0 0 1 0 1]]

mapping generators: ~2, ~11, ~13

Optimal tunings:

  • WE: ~2 = 1200.4447 ¢, ~11/8 = 548.4929 ¢, ~13/8 = 841.3613 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 548.2193 ¢, ~13/8 = 841.4707 ¢

Optimal ET sequence: 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh

Badness (Sintel): 0.106

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