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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 ==
== Overview by mapping of 5 ==
Subgroup: 2.5.7
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.


[[Comma]]: 4194304/4117715
== 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]]


[[Gencom]]: [2 8/7; 4194304/4117715]
=== 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.


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3}}, {{val|0 0 7 -1}}]
A highly notable tuning of rainy not shown here is [[311edo]], which is 140 + 171 so tuned between them.


[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 7 -1}}]
[[Subgroup]]: 2.5.7


[[Tp tuning|POL2 generator]]: ~8/7 = 226.910
[[Comma list]]: [[2100875/2097152]]


{{Val list|legend=1| 16, 37 }}
{{Mapping|legend=2| 1 2 3 | 0 5 -3 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5391 cents
{{Mapping|legend=3| 1 0 2 3 | 0 0 5 -3 }}
: mapping generators: ~2, ~256/245


=== 2.5.7.11 ===
[[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 }}
 
Augment is related to [[augmented]], but for 2.5.7 instead of 2.3.5.
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 128/125
 
{{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]]: 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 }}


[[Gencom]]: [2 8/7; 176/175 1310720/1294139]
Badness (Sintel): 0.196


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3 1}}, {{val|0 0 7 -1 13}}]
=== Frostburn ===
Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].


[[Mapping|Sval mapping]]: [{{val|1 1 3 1}}, {{val|0 7 -1 13}}]
[[Subgroup]]: 2.5.7


[[Tp tuning|POL2 generator]]: ~8/7 = 227.114
[[Comma list]]: 78125/76832


{{Val list|legend=1| 16, 21, 37 }}
{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
: mapping generators: ~2, ~28/25


=== 2.5.7.11.13 ===
[[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 ET sequence|legend=1| 6, 29, 35, 41, 47 }}
 
[[Badness]] (Sintel): 0.886
 
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
Subgroup: 2.5.7.11


[[Comma]]: 176/175, 640/639, 1308672/1294139
Comma list: 245/242, 625/616
 
Subgroup-val mapping: {{mapping| 1 3 4 5 | 0 -4 -7 -9 }}
: mapping generators: ~2, ~28/25
 
Optimal tunings:
* WE: ~2 = 1200.6817{{c}}, ~28/25 = 205.0734{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 204.8199{{c}}
 
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
 
Badness (Sintel): 0.463
 
=== Mabilic ===
{{Main| Mabilic and trismegistus }}
{{See also| Chromatic pairs #Mabilic }}
 
Mabilic is the no-3 [[restriction]] of [[mavila family #Armodue|armodue]], [[mabila family #Semabila|semabila]], and [[magic family #Trismegistus|trismegistus]]. It is the 7 & 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma.
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 1071875/1048576
 
{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}
 
{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}
: mapping generators: ~2, ~175/128
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2543{{c}}, ~175/128 = 527.7872{{c}}
: [[error map]]: {{val| +1.254 -1.698 -1.491 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~175/128 = 527.2041{{c}}
: error map: {{val| 0.000 -4.701 -4.846 }}
 
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd }}
 
[[Badness]] (Sintel): 1.70
 
=== Huntington ===
{{See also| Chromatic pairs #Huntington }}
 
Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 40960000/40353607
 
{{Mapping|legend=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 }}
 
Badness (Sintel): 1.87
 
==== 2.5.7.13 subgroup ====
Subgroup: 2.5.7.13
 
Comma list: 640/637, 10985/10976
 
Subgroup-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}
 
Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
: mapping generators: ~2, ~13/8
 
Optimal tunings:
* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -4 0 3 9 | 0 9 4 1 -7 }}
 
Gencom mapping: {{mapping| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}
 
Optimal tunings:
* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 20 -49 | 0 3 -39 119 }}
: mapping generators: ~2, ~5120/3773
 
Optimal tunings:
* WE: ~2 = 1199.9115{{c}}, ~5120/3773 = 528.9650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5120/3773 = 529.0037{{c}}
 
{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}


[[Gencom]]: [2 8/7; 176/175 640/639, 1308672/1294139]
Badness (Sintel): 11.7


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3 1 2}}, {{val|0 0 7 -1 13 9}}]
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


[[Mapping|Sval mapping]]: [{{val|1 1 3 1 2}}, {{val|0 7 -1 13 9}}]
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125


[[Tp tuning|POL2 generator]]: ~8/7 = 227.108
Subgroup-val mapping: {{mapping| 1 1 20 -49 35 | 0 3 -39 119 -71 }}


{{Val list|legend=1| 16, 21, 37 }}
Optimal tunings:
* WE: ~2 = 1199.9194{{c}}, ~1664/1225 = 528.9681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0036{{c}}


=== 2.5.7.11.13.17 ===
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}
 
Badness (Sintel): 3.42
 
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17
 
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
 
Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}
 
Optimal tunings:
* WE: ~2 = 1199.9054{{c}}, ~1664/1225 = 528.9628{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0046{{c}}
 
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
 
Badness (Sintel): 1.99
 
==== 2.5.7.11.13.17.19 subgroup ====
Subgroup: 2.5.7.11.13.17.19
 
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
 
Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}
 
Optimal tunings:
* WE: ~2 = 1199.9081{{c}}, ~19/14 = 528.9639{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/14 = 529.0045{{c}}
 
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
 
Badness (Sintel): 1.29
 
=== 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]]: {{monzo| 372 -159 -1 }}
 
{{Mapping|legend=2| 1 0 372 | 0 1 -159 }}
: mapping generators: ~2, ~5
 
[[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 ET sequence|legend=1| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}
 
[[Badness]] (Sintel): 148
 
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
Subgroup: 2.5.7.11


[[Comma]]: 176/175, 221/200, 640/639, 833/832
Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}
 
Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}
 
Optimal tunings:
* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}
 
Optimal tunings:
* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}
 
{{Optimal ET sequence|legend=0| 261, 1261e, 1525, 1789 }}
 
Badness (Sintel): 10.5


[[Gencom]]: [2 8/7; 176/175 221/200, 640/639, 833/832]
=== Bastille ===
{{Main| Bastille }}


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3 1 2 2}}, {{val|0 0 7 -1 13 9 11}}]
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]].


[[Mapping|Sval mapping]]: [{{val|1 1 3 1 2 2}}, {{val|0 7 -1 13 9 11}}]
[[Subgroup]]: 2.5.7


[[Tp tuning|POL2 generator]]: ~8/7 = 227.242
[[Comma list]]: {{monzo| 1426 -596 -15 }}


{{Val list|legend=1| 16, 21, 37 }}
{{Mapping|legend=2| 1 -4 254 | 0 15 -596 }}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}


== Didacus ==
[[Optimal tuning]]s:
{{See also| Hemimean clan #Didacus }}
* [[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 }}


Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
{{Optimal ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}


Subgroup: 2.5.7
[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>


[[Comma]]: 3136/3125
=== No-threes naiad (rank-3) ===
{{Todo|inline=1| review | comment = Devise the permanent name for this temperament; determine its status with respect to its extensions linked below. }}
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}


[[Gencom]]: [2 28/25; 3136/3125]
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]].


[[Gencom|Gencom mapping]]: [{{val|1 0 2 2}}, {{val|0 0 2 5}}]
[[Subgroup]]: 2.5.7.11


[[Mapping|Sval mapping]]: [{{val|1 2 2}}, {{val|0 2 5}}]
[[Comma list]]: 5021863/5000000


[[Tp tuning|POL2 generator]]: ~28/25 = 93.772
{{Mapping|legend=2| 1 0 -2 3 | 0 1 1 1 | 0 0 4 -3 }}
: mapping generators: ~2, ~5, ~77/50


{{Val list|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
[[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 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 388, 425, 652e, 689e, 1077de }}


== Rainy ==
[[Badness]] (Sintel): 1.86
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].


Subgroup: 2.5.7
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


[[Comma]]: [[2100875/2097152]]
Comma list: 847/845, 1001/1000


[[Gencom]]: [2 256/245; 2100875/2097152]
Subgroup-val mapping: {{mapping| 1 0 -2 3 2 | 0 1 1 1 1 | 0 0 4 -3 -1 }}


[[Gencom|Gencom mapping]]: [{{val|1 0 2 3 }}, {{val|0 0 5 -3}}]
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|Sval mapping]]: [{{val|1 2 3 }}, {{val|0 5 -3}}]
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e }}


[[Tp tuning|POL2 generator]]: ~256/245 = 77.205
Badness (Sintel): 0.179


{{Val list|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
Comma list: 170/169, 221/220, 847/845


== Mercy ==
Subgroup-val mapping: {{mapping| 1 0 -2 3 2 3 | 0 1 1 1 1 1 | 0 0 4 -3 -1 -2 }}
{{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]].
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}}


Subgroup: 2.5.7
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 66g, 87g, 124g }}


[[Comma list]]: 823543/819200
Badness (Sintel): 0.438


[[Gencom]]: [2 2744/2560; 823543/819200]
== Temperaments with a higher 2.5.''p'' gene ==
Temperaments discussed elsewhere include:
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]


[[Gencom|Gencom mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
=== Wizz ===
{{See also| Chromatic pairs #Wizz }}


[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, tempers out [[15625/15488]], and is the common [[restriction]] of [[astrology]] and [[wizard]].


[[Tp tuning|POL2 generator]]: ~343/320 = 116.291
[[Subgroup]]: 2.5.11


{{Val list|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
[[Comma list]]: 15625/15488


=== 2.5.7.13 ===
{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}
Subgroup: 2.5.7.13
 
{{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 common [[restriction]] of [[muggles]] and [[submerged]].
 
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 34375/32768
 
{{Mapping|legend=2| 1 0 15 | 0 1 -5 }}
 
{{Mapping|legend=3| 1 0 0 0 15 | 0 0 1 0 -5 }}
: mapping generators: ~2, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.3290{{c}}, ~5/4 = 373.6097{{c}}
: [[error map]]: {{val| +3.329 -6.046 -2.722 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 372.1586{{c}}
: error map: {{val| 0.000 -14.155 -12.111 }}
 
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceee }}
 
[[Badness]] (Sintel): 1.85
 
==== 2.5.11.13 subgroup ====
Subgroup: 2.5.11.13
 
Comma list: 65/64, 6875/6656
 
Subgroup-val mapping: {{mapping| 1 0 15 6 | 0 1 -5 -1 }}
 
Gencom mapping: {{mapping| 1 0 0 0 15 6 | 0 0 1 0 -5 -1 }}
 
Optimal tunings:
* WE: ~2 = 1203.3825{{c}}, ~5/4 = 373.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 372.1519{{c}}
 
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceeeff }}
 
Badness (Sintel): 0.410
 
==== 2.5.11.13.17 subgroup ====
Subgroup: 2.5.11.13.17
 
Comma list: 65/64, 170/169, 221/220
 
Subgroup-val mapping: {{mapping| 1 0 15 6 11 | 0 1 -5 -1 -3 }}
 
Gencom mapping: {{mapping| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}
 
Optimal tunings:
* WE: ~2 = 1203.6741{{c}}, ~5/4 = 373.3775{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 371.6773{{c}}
 
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29g, 129ccceeffgggg }}
 
Badness (Sintel): 0.299
 
=== Trader ===
[[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|legend=2| 1 0 1 | 0 10 11 }}
: mapping generators, ~2, ~17/14
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9723{{c}}, ~17/14 = 336.8586{{c}}
: [[error map]]: {{val| -0.028 -0.240 +0.462 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 336.8621{{c}}
: error map: {{val| 0.000 -0.205 +0.528 }}
 
{{Optimal ET sequence|legend=1| 7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g }}
 
[[Badness]] (Sintel): 0.860
 
=== Shipwreck ===
[[Subgroup]]: 2.7.53
 
[[Comma list]]: 1048576/1042139
 
{{Mapping|legend=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


[[Comma list]]: 343/338, 640/637
=== 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.


[[Gencom]]: [2 14/13; 343/338 640/637]
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.


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4}}, {{val|0 0 -7 -2 -3}}]
[[Subgroup]]: 2.11.37


[[Mapping|Sval mapping]]: [{{val|1 3 3 4}}, {{val|0 -7 -2 -3}}]
[[Comma list]]: 1874161/1874048


[[Tp tuning|POL2 generator]]: ~14/13 = 116.094
{{Mapping|legend=2| 4 0 7 | 0 1 1 }}
: mapping generators: ~44/37, ~11


{{Val list|legend=1| 10, 21, 31}}
[[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 }}


=== 2.5.7.13.17 ===
{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 76, 100, 124, 148, 616, 764, 912, 1060, 3328, 4388, 5448 }}
Subgroup: 2.5.7.13.17


[[Comma list]]: 170/169, 224/221, 640/637
[[Badness]] (Sintel): 0.00188


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
=== Mavericks ===
[[Subgroup]]: 2.13.19


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4 4}}, {{val|0 0 -7 -2 -3 1}}]
[[Comma list]]: 47525504/47045881


[[Mapping|Sval mapping]]: [{{val|1 3 3 4 4}}, {{val|0 -7 -2 -3 1}}]
{{Mapping|legend=2| 1 1 2 | 0 6 5 }}
: mapping generators: ~2, ~26/19


[[Tp tuning|POL2 generator]]: ~14/13 = 115.769
[[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 }}


{{Val list|legend=1| 10, 21, 31}}
{{Optimal ET sequence|legend=1| 9, 11, 20 }}


=== 2.5.7.13.17.19 ===
[[Badness]] (Sintel): 0.559
Subgroup: 2.5.7.13.17.19


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
=== Yer (rank 3) ===
[[Subgroup]]: 2.11.13.17.19


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
[[Comma list]]: 209/208, 2057/2048


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4 4 3}}, {{val|0 0 -7 -2 -3 1 13}}]
{{Mapping|legend=2| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
: mapping generators: ~2, ~11, ~13


[[Mapping|Sval mapping]]: [{{val|1 3 3 4 4 3}}, {{val|0 -7 -2 -3 1 13}}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.4447{{c}}, ~11/8 = 548.4929{{c}}, ~13/8 = 841.3613{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 548.2193{{c}}, ~13/8 = 841.4707{{c}}


[[Tp tuning|POL2 generator]]: ~14/13 = 115.716
{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}


{{Val list|legend=1| 10, 21, 31, 52f}}
[[Badness]] (Sintel): 0.106


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Rank 2]]
{{Todo| cleanup | review }}
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