No-fives 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 5.  
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 5.  


== [[Semaphore and Godzilla|Semaphore]] ==
== Temperaments with a 2.3.7 gene ==
Subgroup: 2.3.7
=== Archy ===
See [[Archytas clan #Archy]].  


[[Comma]]: 49/48
==== Suhajira ====
See [[Rastmic clan #Suhajira]].


[[Gencom]]: [2 8/7; 49/48]
==== Flutterpyth ====
Restricted to 2.3.7.11, this temperament is a no-5 restriction of 11-limit [[ultrapyth]]. This temperament was created to yield [[blackdye]] tunings where [[aberrisma|aberrisma-altered]] 3-limit thirds become tempered [[13/11]][[~]][[19/16]] and [[14/11]].


[[Gencom|Gencom mapping]]: [{{val|1 2 0 3}}, {{val|0 -2 0 -1}}]
Subgroup: 2.3.7.11.13


[[Mapping|Sval mapping]]: [{{val|1 2 3}}, {{val|0 -2 -1}}]
Comma list: 64/63, 364/363, 1078/1053


[[Tp tuning|POL2 generator]]: ~7/6 = 250.385
Mapping: {{mapping| 1 0 6 21 34 | 0 1 -2 -11 -19 }}


{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 67d, 91d }}
Optimal tunings:
* WE: ~2 = 1196.9412{{c}}, ~3/2 = 711.0195{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.0039{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 2.523 cents
{{Optimal ET sequence|legend=0| 32f, 69bf, 101beff }}


== Bleu ==
Badness (Sintel): 1.52
Subgroup: 2.3.7


[[Comma]]: 17496/16807
===== 2.3.7.11.13.19 subgroup =====
Subgroup: 2.3.7.11.13.19


[[Gencom]]: [2 54/49; 17496/16807]
Comma list: 64/63, 209/208, 343/342, 364/363


[[Gencom|Gencom mapping]]: [{{val|1 1 0 2}}, {{val|0 5 0 7}}]
Mapping: {{mapping| 1 0 6 21 34 17 | 0 1 -2 -11 -19 -8 }}


[[Mapping|Sval mapping]]: [{{val|1 1 2}}, {{val|0 5 7}}]
Optimal tunings:  
* WE: ~2 = 1197.4072{{c}}, ~3/2 = 711.2733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 712.9612{{c}}


[[Tp tuning|POL2 generator]]: ~54/49 = 139.848
{{Optimal ET sequence|legend=0| 32f, 69bf }}


{{Optimal ET sequence|legend=1| 9, 17, 43, 60d }}
Badness (Sintel): 1.28


[[Tp tuning #T2 tuning|RMS error]]: 1.917 cents
=== Semaphore ===
See [[Semaphoresmic clan #Semaphore]].  


=== 2.3.7.11 ===
=== Slendric ===
Subgroup: 2.3.7.11
See [[Gamelismic clan #Slendric]].  


[[Comma list]]: 99/98, 864/847
=== Slendroschismic ===
See [[5th-octave temperaments #Slendroschismic]].


[[Gencom]]: [2 12/11; 99/98 864/847]
=== Navy ===
This temperament is the common [[restriction]] of [[tsaharuk]] and [[quanic]].


[[Gencom|Gencom mapping]]: [{{val|1 1 0 2 3}}, {{val|0 5 0 7 4}}]
[[Subgroup]]: 2.3.7


[[Mapping|Sval mapping]]: [{{val|1 1 2 3}}, {{val|0 5 7 4}}]
[[Comma list]]: 282429536481/281974669312


[[Tp tuning|POL2 generator]]: ~12/11 = 140.005
{{Mapping|legend=2| 1 1 0| 0 5 24 }}
: mapping generators: ~2, ~243/224


{{Optimal ET sequence|legend=1| 9, 17, 43, 60d }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0302{{c}}, ~243/224 = 140.3698{{c}}
: [[error map]]: {{val| +0.030 -0.076 +0.050 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/224 = 140.3681{{c}}
: error map: {{val| 0.000 -0.115 +0.008 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.829 cents
{{Optimal ET sequence|legend=1| 17, 60, 77, 94, 171, 265, 436, 2351, 2787, 3223, 3659, 4095, 7754b }}


=== 2.3.7.11.13 ===
[[Badness]] (Sintel): 0.670
Subgroup: 2.3.7.11.13


[[Comma list]]: 78/77, 99/98, 144/143
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


[[Gencom]]: [2 12/11; 78/77 99/98 144/143]
Comma list: 1331/1323, 19712/19683


[[Gencom|Gencom mapping]]: [{{val|1 1 0 2 3 3}}, {{val|0 5 0 7 4 6}}]
Subgroup-val mapping: {{mapping| 1 1 0 1 | 0 5 24 21 }}


[[Mapping|Sval mapping]]: [{{val|1 1 2 3 3}}, {{val|0 5 7 4 6}}]
Optimal tunings:  
* WE: ~2 = 1200.1038{{c}}, ~88/81 = 140.4190{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.4133{{c}}


[[Tp tuning|POL2 generator]]: ~12/11 = 139.990
{{Optimal ET sequence|legend=0| 17, 60e, 77, 94 }}


{{Optimal ET sequence|legend=1| 17, 43, 60d }}
Badness (Sintel): 0.887


[[Tp tuning #T2 tuning|RMS error]]: 1.752 cents
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13


== Archy ==
Comma list: 352/351, 729/728, 1331/1323
Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives [[dominant]]" or "no-fives [[superpyth]]". The name comes from the fact that it tempers out 64/63, the Archytas comma.


Subgroup: 2.3.7
Subgroup-val mapping: {{mapping| 1 1 0 1 3 | 0 5 24 21 6 }}


[[Comma]]: 64/63
Optimal tunings:  
* WE: ~2 = 1199.8640{{c}}, ~13/12 = 140.4206{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4292{{c}}


[[Gencom]]: [2 3/2; 64/63]
{{Optimal ET sequence|legend=0| 17, 60e, 77, 94 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4}}, {{val|0 1 0 -2}}]
Badness (Sintel): 0.520


[[Mapping|Sval mapping]]: [{{val|1 2 2}}, {{val|0 -1 2}}]
=== Lee ===
[[Subgroup]]: 2.3.7


[[Tp tuning|POL2 generator]]: ~3/2 = 709.321
[[Comma list]]: 177147/175616


{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 27, 137bd }}
{{Mapping|legend=2| 1 0 -3 | 0 3 11 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.856 cents
{{Mapping|legend=3| 1 0 0 -3 | 0 3 0 11 }}
: mapping generators: ~2, ~81/56


=== Supra ===
[[Optimal tuning]]s:
Subgroup: 2.3.7.11
* [[WE]]: ~2 = 1200.2962{{c}}, ~81/56 = 633.6812{{c}}
: [[error map]]: {{val| +0.296 -0.912 +0.778 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~81/56 = 633.5658{{c}}
: error map: {{val| 0.000 -1.258 +0.398 }}


[[Comma list]]: 64/63, 99/98
{{Optimal ET sequence|legend=1| 17, 36, 89, 125, 161, 358, 519b }}


[[Gencom]]: [2 3/2; 64/63 99/98]
[[Badness]] (Sintel): 0.741


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 7}}, {{val|0 1 0 -2 -6}}]
=== Buzzard ===
See [[Buzzardsmic clan #Buzzard]].


[[Mapping|Sval mapping]]: [{{val|1 0 6 13}}, {{val|0 1 -2 -6}}]
=== Hemif ===
Hemif is the no-5 [[restriction]] of [[hemififths]], and the add-7 [[extension]] of [[namo]].


[[Tp tuning|POL2 generator]]: ~3/2 = 707.192
[[Subgroup]]: 2.3.7
 
{{Optimal ET sequence|legend=1| 5, 12, 17, 39d, 56d }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents
 
==== Supraphon ====
Subgroup: 2.3.7.11.13


[[Comma list]]: 64/63, 78/77, 99/98
[[Comma list]]: 1605632/1594323


[[Gencom]]: [2 3/2; 64/63 78/77 99/98]
{{Mapping|legend=2| 1 1 -1 | 0 2 13 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 7 9}}, {{val|0 1 0 -2 -6 -9}}]
{{Mapping|legend=3| 1 1 0 -1 | 0 2 0 13 }}
: mapping generators: ~2, ~2187/1792


[[Mapping|Sval mapping]]: [{{val|1 0 6 13 18}}, {{val|0 1 -2 -6 -9}}]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7303{{c}}, ~2187/1792 = 351.4056{{c}}
: [[error map]]: {{val| -0.270 +0.586 -0.284 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2187/1792 = 351.4569{{c}}
: error map: {{val| 0.000 +0.959 +0.114 }}


[[Tp tuning|POL2 generator]]: ~3/2 = 706.137
{{Optimal ET sequence|legend=1| 17, 41, 58, 99, 239, 338, 437, 775b, 1212bb }}


{{Optimal ET sequence|legend=1| 5f, 12f, 17, 22, 39d, 56d }}
[[Badness]] (Sintel): 0.901


[[Tp tuning #T2 tuning|RMS error]]: 2.095 cents
==== 2.3.7.11 ====
 
=== Suhajira ===
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


[[Comma list]]: 64/63, 243/242
Comma list: 243/242, 896/891


[[Gencom]]: [2 11/9; 64/63 243/242]
Subgroup-val mapping: {{mapping| 1 1 -1 2 | 0 2 13 5 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 2}}, {{val|0 2 0 -4 5}}]
Gencom mapping: {{mapping| 1 1 0 -1 2 | 0 2 0 13 5 }}


[[Mapping|Sval mapping]]: [{{val|1 1 4 2}}, {{val|0 2 -4 5}}]
Optimal tunings:  
* WE: ~2 = 1199.2633{{c}}, ~11/9 = 351.3189{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.4593{{c}}


[[Tp tuning|POL2 generator]]: ~11/9 = 353.958
{{Optimal ET sequence|legend=0| 17, 41, 58, 99e }}


{{Optimal ET sequence|legend=1| 7, 10, 17, 44e, 61de, 78de }}
Badness (Sintel): 0.409


[[Tp tuning #T2 tuning|RMS error]]: 1.968 cents
===== 2.3.7.11.13 =====
 
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


[[Comma list]]: 64/63, 78/77, 144/143
Comma list: 144/143, 243/242, 364/363


[[Gencom]]: [2 11/9; 64/63 78/77 144/143]
Sval mapping: {{mapping| 1 1 -1 2 4 | 0 2 13 5 -1 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 2 4}}, {{val|0 2 0 -4 5 -1}}]
Gencom mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 0 13 5 -1 }}


[[Mapping|Sval mapping]]: [{{val|1 1 4 2 4}}, {{val|0 2 -4 5 -1}}]
Optimal tunings:  
* WE: ~2 = 1198.7603{{c}}, ~11/9 = 351.3275{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.6042{{c}}


[[Tp tuning|POL2 generator]]: ~11/9 = 353.775
{{Optimal ET sequence|legend=0| 17, 41, 58, 331deeeffff }}


{{Optimal ET sequence|legend=1| 7, 10, 17, 44e, 61de, 78de }}
Badness (Sintel): 0.358


[[Tp tuning #T2 tuning|RMS error]]: 1.953 cents
===== Heartful =====
{{See also| Heartland }}


== Skwares ==
Related temperaments: [[bunya]].  
Subgroup: 2.3.7


[[Comma]]: 19683/19208
Subgroup: 2.3.7.11.19


[[Gencom]]: [2 9/7; 19683/19208]
Comma list: 243/242, 896/891, 1083/1078


[[Gencom|Gencom mapping]]: [{{val|1 3 6}}, {{val|0 -4 -9}}]
Subgroup-val mapping: {{mapping| 1 1 -1 2 0 | 0 4 26 10 29 }}
: mapping generators: ~2, ~21/19


[[Mapping|Sval mapping]]: [{{val|1 3 6}}, {{val|0 -4 -9}}]
Optimal tunings:  
* WE: ~2 = 1199.2636{{c}}, ~21/19 = 175.6963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.7665{{c}}


[[Tp tuning|POL2 generator]]: ~9/7 = 425.365
{{Optimal ET sequence|legend=0| 34dh, 41, 116e, 157e }}


{{Optimal ET sequence|legend=1| 14, 17, 31, 48, 79, 189b, 268bd, 347bd }}
Badness (Sintel): 0.984


[[Tp tuning #T2 tuning|RMS error]]: 1.149 cents
=== Hearts ===
{{See also| Heartland }}


Related temperament: [[Meantone family #Squares|squares]]
This temperament is the common [[restriction]] of [[monkey]] and [[sesquiquartififths]].


=== 2.3.7.11 ===
[[Subgroup]]: 2.3.7
Subgroup: 2.3.7.11


[[Comma list]]: 99/98, 243/242
[[Comma list]]: 34451725707/34359738368


[[Gencom]]: [2 9/7; 99/98 243/242]
{{Mapping|legend=2| 1 1 5 | 0 4 -15 }}
: mapping generators: ~2, ~567/512


[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7}}, {{val|0 -4 0 -9 -10}}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0845{{c}}, ~567/512 = 175.4449{{c}}
: [[error map]]: {{val| +0.085 -0.091 -0.076 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~567/512 = 175.4307{{c}}
: error map: {{val| 0.000 -0.232 -0.286 }}


[[Mapping|Sval mapping]]: [{{val|1 3 6 7}}, {{val|0 -4 -9 -10}}]
{{Optimal ET sequence|legend=1| 7, 27d, 34, 41, 89, 130, 171, 643, 814, 985, 1156 }}


[[Tp tuning|POL2 generator]]: ~9/7 = 425.244
[[Badness]] (Sintel): 0.959


{{Optimal ET sequence|legend=1| 14, 17, 31, 48, 79, 127, 206bde }}
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


[[Tp tuning #T2 tuning|RMS error]]: 1.099 cents
Comma list: 243/242, 65536/65219


==== 2.3.7.11.13 ====
Subgroup-val mapping: {{mapping| 1 1 5 2 | 0 4 -15 10 }}
Subgroup: 2.3.7.11.13


[[Comma list]]: 78/77, 99/98, 243/242
Optimal tunings:  
* WE: ~2 = 1199.8467{{c}}, ~256/231 = 175.3468{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/231 = 175.3691{{c}}


[[Gencom]]: [2 9/7; 78/77, 99/98, 243/242]
{{Optimal ET sequence|legend=0| 7, 34, 41, 89, 130, 349e, 479e }}


[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7 9}}, {{val|0 -4 0 -9 -10 -15}}]
Badness (Sintel): 0.801


[[Mapping|Sval mapping]]: [{{val|1 3 6 7 9}}, {{val|0 -4 -9 -10 -15}}]
==== 2.3.7.11.19 ====
Subgroup: 2.3.7.11.19


[[Tp tuning|POL2 generator]]: ~9/7 = 424.457
Comma list: 243/242, 513/512, 1083/1078


{{Optimal ET sequence|legend=1| 17, 48f, 65ef, 82d, 147df }}
Subgroup-val mapping: {{mapping| 1 1 5 2 6 | 0 4 -15 10 -12 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.769 cents
Optimal tunings:  
* WE: ~2 = 1199.9531{{c}}, ~21/19 = 175.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3417{{c}}


==== Skwairs ====
{{Optimal ET sequence|legend=0| 7, 34, 41, 89, 130, 219 }}
Subgroup: 2.3.7.11.13


[[Comma list]]: 99/98, 144/143, 243/242
Badness (Sintel): 0.529


[[Gencom]]: [2 9/7; 99/98, 144/143, 243/242]
=== Magi ===
This temperament is the no-5 [[restriction]] of [[magic]], tempering out the [[septimagic comma]].


[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7 3}}, {{val|0 -4 0 -9 -10 2}}]
[[Subgroup]]: 2.3.7


[[Mapping|Sval mapping]]: [{{val|1 3 6 7 3}}, {{val|0 -4 -9 -10 2}}]
[[Comma list]]: 537824/531441


[[Tp tuning|POL2 generator]]: ~9/7 = 424.702
{{Mapping|legend=2| 1 0 -1 | 0 5 12 }}
: mapping generators: ~2, ~243/196


{{Optimal ET sequence|legend=1| 14, 17, 31 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8224{{c}}, ~243/196 = 380.6043{{c}}
: [[error map]]: {{val| -0.178 +1.066 -1.397 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/196 = 380.6378{{c}}
: error map: {{val| 0.000 +1.234 -1.173 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.290 cents
{{Optimal ET sequence|legend=1| 19, 22, 41, 104, 145, 186, 331 }}


==== Byhearted ====
[[Badness]] (Sintel): 1.30
Subgroup: 2.3.7.11.19


[[Comma list]]: 99/98, 243/242, 363/361
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


[[Gencom]]: [209/147 21/19; 99/98 243/242 363/361]
Comma list: 896/891, 26411/26244


[[Mapping|Sval mapping]]: [{{val|2 2 3 4 5}}, {{val|0 4 9 10 12}}]
Subgroup-val mapping: {{mapping| 1 0 -1 6 | 0 5 12 -8 }}


[[Tp tuning|POL2 generator]]: ~21/19 = 174.735
Optimal tunings:  
* WE: ~2 = 1199.4843{{c}}, ~96/77 = 380.6040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~96/77 = 380.7490{{c}}


{{Optimal ET sequence|legend=1| 14, 34dh, 48, 110e, 158e }}
{{Optimal ET sequence|legend=0| 19, 22, 41, 63, 104 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.8727 cents
Badness (Sintel): 0.661


== Harrison ==
===== Caspar =====
Subgroup: 2.3.7
Subgroup: 2.3.7.11.13


[[Comma]]: [[59049/57344]]
Comma list: 144/143, 343/338, 729/728


[[Gencom]]: [2 3/2; 59049/57344]
Subgroup-val mapping: {{mapping| 1 0 -1 6 -2 | 0 5 12 -8 18 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -3}}, {{val|0 1 0 10}}]
Optimal tunings:  
* WE: ~2 = 1199.3353{{c}}, ~26/21 = 380.3206{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/21 = 380.5041{{c}}


[[Mapping|Sval mapping]]: [{{val|1 1 -3}}, {{val|0 1 10}}]
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


[[Tp tuning|POL2 generator]]: ~3/2 = 696.544
Badness (Sintel): 1.09


{{Optimal ET sequence|legend=1| 12, 19, 31, 112b, 143b, 174b }}
====== Twenothology ======
Subgroup: 2.3.7.11.13.29


[[Tp tuning #T2 tuning|RMS error]]: 1.226 cents
Comma list: 144/143, 232/231, 343/338, 729/728


Related temperament: [[Meantone family #Septimal meantone|meantone]]
Subgroup-val mapping: {{mapping| 1 0 -1 6 -2 2 | 0 5 12 -8 18 9 }}


== Leapfrog ==
Optimal tunings:
{{see also|Gentle region}}
* WE: ~2 = 1199.6175{{c}}, ~26/21 = 380.4049{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/21 = 380.5103{{c}}


Subgroup: 2.3.7
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


[[Comma list]]: 14680064/14348907
Badness (Sintel): 0.964


[[Gencom]]: [2 3/2; 14680064/14348907]
===== Melchior =====
Subgroup: 2.3.7.11.13


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6}}, {{val|0 1 0 15}}]
Comma list: 352/351, 364/363, 26411/26244


[[Mapping|Sval mapping]]: [{{val|1 0 -21}}, {{val|0 1 15}}]
Subgroup-val mapping: {{mapping| 1 0 -1 6 11 | 0 5 12 -8 -23 }}


[[Tp tuning|POL2 generator]]: ~3/2 = 704.721 cents
Optimal tunings:
* WE: ~2 = 1199.4887{{c}}, ~96/77 = 380.6034{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~96/77 = 380.7669{{c}}


{{Optimal ET sequence|legend=1| 17, 46, 63 }}
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63, 104 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6202 cents
Badness (Sintel): 0.710


Related temperaments: [[Hemifamity temperaments #Leapday|leapday]], [[Sensamagic clan #Leapweek|leapweek]], [[srutal]]
===== Balthazar =====
Subgroup: 2.3.7.11.13


=== 2.3.7.11 ===
Comma list: 169/168, 896/891, 26411/26244
Subgroup: 2.3.7.11


[[Comma list]]: 896/891, 1331/1323
Subgroup-val mapping: {{mapping| 1 0 -1 6 1 | 0 10 24 -16 17 }}
: mapping generators: ~2, ~143/128


[[Gencom]]: [2 3/2; 896/891 1331/1323]
Optimal tunings:  
* WE: ~2 = 1199.7322{{c}}, ~143/128 = 190.3647{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~143/128 = 190.4016{{c}}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3}}, {{val|0 1 0 15 11}}]
{{Optimal ET sequence|legend=0| 19, 44, 63, 145f }}


[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14}}, {{val|0 1 15 11}}]
Badness (Sintel): 1.82


[[Tp tuning|POL2 generator]]: ~3/2 = 704.753 cents
==== Hogwarts ====
Subgroup: 2.3.7.29


{{Optimal ET sequence|legend=1| 17, 46, 63 }}
Comma list: 784/783, 5887/5832


[[Tp tuning #T2 tuning|RMS error]]: 0.6047 cents
Subgroup-val mapping: {{mapping| 1 0 -1 2 | 0 5 12 9 }}


=== 2.3.7.11.13 ===
Optimal tunings:
Subgroup: 2.3.7.11.13
* WE: ~2 = 1200.1518{{c}}, ~36/29 = 380.6661{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/29 = 380.6375{{c}}


[[Comma list]]: 169/168, 352/351, 364/363
{{Optimal ET sequence|legend=0| 19, 22, 41, 145, 186j, 227j }}


[[Gencom]]: [2 3/2; 169/169 352/351 364/363]
Badness (Sintel): 0.424


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3 -1}}, {{val|0 1 0 15 11 8}}]
=== Skwares ===
{{Main| Squares }}


[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14 -9}}, {{val|0 1 15 11 8}}]
Skwares is the no-5 [[restriction]] of [[squares]].


[[Tp tuning|POL2 generator]]: ~3/2 = 704.745 cents
[[Subgroup]]: 2.3.7


{{Optimal ET sequence|legend=1| 17, 46, 63 }}
[[Comma list]]: 19683/19208


[[Tp tuning #T2 tuning|RMS error]]: 0.7541 cents
{{Mapping|legend=2| 1 -1 -3 | 0 4 9 }}


==== Skidoo ====
{{Mapping|legend=3| 1 -1 0 -3 | 0 4 0 9 }}
Subgroup: 2.3.7.11.13.23
: mapping generators: ~2, ~14/9


[[Comma list]]: 169/168, 208/207, 352/351, 364/363
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3703{{c}}, ~14/9 = 774.8736{{c}}
: [[error map]]: {{val| +0.370 -2.831 +3.925 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 774.6974{{c}}
: error map: {{val| 0.000 -3.166 +3.450 }}


[[Gencom]]: [2 3/2; 169/169 208/207 352/351 364/363]
{{Optimal ET sequence|legend=1| 14, 17, 31, 48, 79 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3 -1 0 0 1}}, {{val|0 1 0 15 11 8 0 0 6}}]
[[Badness]] (Sintel): 1.55


[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14 -9 -5}}, {{val|0 1 15 11 8 6}}]
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


[[Tp tuning|POL2 generator]]: ~3/2 = 704.729 cents
Comma list: 99/98, 243/242


{{Optimal ET sequence|legend=1| 17, 46, 63 }}
Subgroup-val mapping: {{mapping| 1 -1 -3 -3 | 0 4 9 10 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6265 cents
Gencom mapping: {{mapping| 1 -1 0 -3 -3 | 0 4 0 9 10 }}


===== 2.3.7.11.13.23.29 =====
Optimal tunings:
Subgroup: 2.3.7.11.13.23.29
* WE: ~2 = 1200.3726{{c}}, ~14/9 = 774.9970{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.8197{{c}}


[[Comma list]]: 169/168, 208/207, 232/231, 352/351, 364/363
{{Optimal ET sequence|legend=0| 14, 17, 31, 48, 79, 127 }}


[[Gencom]]: [2 3/2; 169/169 208/207 352/351 364/363]
Badness (Sintel): 0.405


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3 -1 0 0 1 -11}}, {{val|0 1 0 15 11 8 0 0 6 27}}]
===== 2.3.7.11.13 =====
Subgroup: 2.3.7.11.13


[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14 -9 -5 -38}}, {{val|0 1 15 11 8 6 27}}]
Comma list: 78/77, 99/98, 243/242


[[Tp tuning|POL2 generator]]: ~3/2 = 704.729 cents
Subgroup-val mapping: {{mapping| 1 -1 -3 -3 -6 | 0 4 9 10 15 }}


{{Optimal ET sequence|legend=1| 17, 46, 63 }}
Gencom mapping: {{mapping| 1 -1 0 -3 -3 -6 | 0 4 0 9 10 15}}


; Music
Optimal tunings:
* Suite for Harpsichord in A Locrian, tuning: Eb-G# in [[46edo|46EDO]] by [[User:IlL|IlL]] (in progress):
* WE: ~2 = 1199.3264{{c}}, ~14/9 = 775.1081{{c}}
** I. Prelude
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 775.4463{{c}}
** II. Allemande
 
** III. Courante
{{Optimal ET sequence|legend=0| 14f, 17, 48f }}
** [[:File:Locrian Suite Sarabande.mp3|IV. Sarabande]] ([[:File:Locrian Suite Sarabande Score.pdf|score]], [[:File:Locrian Suite Sarabande 17edo.mp3|17EDO version]])
** [[:File:Locrian Suite Menuet.mp3|V. Menuet and Trio]]
** [[:File:Locrian Suite Gavotte.mp3|VI. Gavotte I and II]]
** [[:File:Locrian Suite Gigue.mp3|VII. Gigue]]


== Doublehearted ==
Badness (Sintel): 0.587
Subgroup: 2.3.7


[[Comma list]]: 5764801/5668704
===== Skwairs =====
Subgroup: 2.3.7.11.13


[[Gencom]]: [2 343/324; 5764801/5668704]
Comma list: 99/98, 144/143, 243/242


[[Mapping|Sval mapping]]: [{{val|1 1 2}}, {{val|0 8 11}}]
Subgroup-val mapping: {{mapping| 1 -1 -3 -3 5 | 0 4 9 10 -2 }}


[[Tp tuning|POL2 generator]]: ~343/324 = 87.8304
Gencom mapping: {{mapping| 1 -1 0 -3 -3 5 | 0 4 0 9 10 -2 }}


{{Optimal ET sequence|legend=1| 14, 27, 41 }}
Optimal tunings:
* WE: ~2 = 1198.8812{{c}}, ~14/9 = 775.5748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 775.1930{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.5041 cents
{{Optimal ET sequence|legend=0| 14, 17, 31, 48, 65d, 113df }}


Related temperaments: [[Tetracot family|octacot]]
Badness (Sintel): 0.538


=== 2.3.7.11 ===
===== Byhearted =====
Subgroup: 2.3.7.11
This temperament is the restriction of [[weasel]] to the 2.3.7.11.19 subgroup.


[[Comma list]]: 243/242, 2401/2376
Subgroup: 2.3.7.11.19


[[Gencom]]: [2 22/21; 243/242 2401/2376]
Comma list: 99/98, 243/242, 363/361


[[Mapping|Sval mapping]]: [{{val|1 1 2 2}}, {{val|0 8 11 20}}]
Subgroup-val mapping: {{mapping| 2 2 3 4 5 | 0 4 9 10 12 }}
: mapping generators: ~209/147, ~21/19


[[Tp tuning|POL2 generator]]: ~22/21 = 87.6512
Optimal tunings:  
* WE: ~2 = 600.1836{{c}}, ~21/19 = 174.7882{{c}}
* CWE: ~2 = 600.0000{{c}}, ~21/19 = 174.7975{{c}}


{{Optimal ET sequence|legend=1| 14, 27e, 41, 96d, 137d, 178d }}
{{Optimal ET sequence|legend=0| 14, 34dh, 48, 110e }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7147 cents
Badness (Sintel): 0.893


Related temperaments: [[Tetracot family|octacot]]
=== Harrison ===
Harrison is the no-5 [[restriction]] of [[meantone]]. As such, there is little reason to consider this temperament in practice – since intervals of 5 in meantone are as accurate as intervals of 7, only simpler, they are always available by the time intervals of 7 are generated.


=== 2.3.7.11.19 ===
[[Subgroup]]: 2.3.7
Subgroup: 2.3.7.11.19


[[Comma list]]: 133/132, 243/242, 343/342
[[Comma list]]: [[59049/57344]]


[[Gencom]]: [2 19/18; 133/132 243/242 343/342]
{{Mapping|legend=2| 1 0 -13 | 0 1 10 }}


[[Mapping|Sval mapping]]: [{{val|1 1 2 2 3}}, {{val|0 8 11 20 17}}]
{{Mapping|legend=3| 1 0 0 -13 | 0 1 0 10 }}
: mapping generators: ~2, ~3


[[Tp tuning|POL2 generator]]: ~19/18 = 87.6684
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.5353{{c}}, ~3/2 = 697.4352{{c}}
: [[error map]]: {{val| +1.535 -2.984 +0.920 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.7289{{c}}
: error map: {{val| 0.000 -5.226 -1.537 }}


{{Optimal ET sequence|legend=1| 14, 27e, 41 }}
{{Optimal ET sequence|legend=1| 12, 19, 31, 112b, 143b, 174b }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7065 cents
[[Badness]] (Sintel): 2.35


Related temperaments: [[Tetracot family|octacot]]
=== Bleu ===
Bleu can be described as the {{nowrap| 8d & 9 }} temperament in the no-5 13-limit, and is the common [[restriction]] of [[progression]] and [[jerome]].


== Magi ==
[[Subgroup]]: 2.3.7
Subgroup: 2.3.7


[[Comma list]]: 537824/531441
[[Comma list]]: 17496/16807


[[Gencom]]: [2 243/196; 537824/531441]
{{Mapping|legend=2| 1 1 2 | 0 5 7 }}


[[Mapping|Sval mapping]]: [{{val|1 0 -1}}, {{val|0 5 12}}]
{{Mapping|legend=3| 1 1 0 2 | 0 5 0 7 }}
: mapping generators: ~2, ~54/49


[[Tp tuning|POL2 generator]]: ~243/196 = 380.661
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3538{{c}}, ~54/49 = 139.848{{c}}
: [[error map]]: {{val| -0.646 -3.736 +8.293 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~54/49 = 139.848{{c}}
: error map: {{val| 0.000 -3.270 +9.333 }}


{{Optimal ET sequence|legend=1| 19, 22, 41, 104, 145, 186 }}
{{Optimal ET sequence|legend=1| 8d, 9, 17, 43, 60d, 103d }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4277 cents
[[Badness]] (Sintel): 2.48


=== 2.3.7.11 ===
==== 2.3.7.11 subgroup ====
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


[[Comma list]]: 896/891, 26411/26244
Comma list: 99/98, 864/847


[[Gencom]]: [2 96/77; 896/891 26411/26244]
Subgroup-val mapping: {{mapping| 1 1 2 3 | 0 5 7 4 }}


[[Mapping|Sval mapping]]: [{{val|1 0 -1 6}}, {{val|0 5 12 -8}}]
Gencom mapping: {{mapping| 1 1 0 2 3 | 0 5 0 7 4 }}


[[Tp tuning|POL2 generator]]: ~96/77 = 380.768
Optimal tunings:  
* WE: ~2 = 1198.6613{{c}}, ~12/11 = 139.8489{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.7839{{c}}


{{Optimal ET sequence|legend=1| 19, 22, 41, 63, 104 }}
{{Optimal ET sequence|legend=0| 8d, 9, 17, 43, 60d }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents
Badness (Sintel): 0.624


==== Balthazar ====
==== 2.3.7.11.13 subgroup ====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


[[Comma list]]: 169/168, 896/891, 26411/26244
Comma list: 78/77, 99/98, 144/143


[[Gencom]]: [2 143/128; 169/168 896/891 26411/26244]
Subgroup-val mapping: {{mapping| 1 1 2 3 3 | 0 5 7 4 6 }}


[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 1}}, {{val|0 10 24 -16 17}}]
Gencom mapping: {{mapping| 1 1 0 2 3 3 | 0 5 0 7 4 6 }}


[[Tp tuning|POL2 generator]]: ~143/128 = 190.407
Optimal tunings:  
* WE: ~2 = 1198.9768{{c}}, ~13/12 = 139.8704{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8166{{c}}


{{Optimal ET sequence|legend=1| 19, 44, 63, 145f }}
{{Optimal ET sequence|legend=0| 8d, 9, 17, 43, 60d }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6937 cents
Badness (Sintel): 0.400


==== Caspar ====
; Music
Subgroup: 2.3.7.11.13
* ''On a Well Worn Riff'' (2011) by [[Chris Vaisvil]] – [https://www.chrisvaisvil.com/on-a-well-worn-riff-bleu-17/ blog] | [https://web.archive.org/web/20201127014513/http://micro.soonlabel.com/temperaments/Bleu/20131103_2-11-13-subgroup-Bleu17_a-well-worn-riff.mp3 play] – in Bleu[17]


[[Comma list]]: 144/143, 343/338, 729/728
=== Doublehearted ===
{{See also| Heartland }}


[[Gencom]]: [2 26/21; 144/143 343/338 729/728]
This temperament is the no-5 [[restriction]] of [[octacot]].


[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 -2}}, {{val|0 5 12 -8 18}}]
[[Subgroup]]: 2.3.7


[[Tp tuning|POL2 generator]]: ~26/21 = 380.531
[[Comma list]]: 5764801/5668704


{{Optimal ET sequence|legend=1| 19, 22f, 41 }}
{{Mapping|legend=2| 1 1 2 | 0 8 11 }}
: mapping generators: ~2, ~343/342


[[Tp tuning #T2 tuning|RMS error]]: 1.032 cents
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0000{{c}}, ~343/324 = 87.8431{{c}}
: [[error map]]: {{val| +0.174 +0.964 -2.204 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/324 = 87.8492{{c}}
: error map: {{val| 0.000 +0.838 -2.485 }}


==== Melchior ====
{{Optimal ET sequence|legend=1| 14, 27, 41 }}
Subgroup: 2.3.7.11.13


[[Comma list]]: 352/351, 364/363, 26411/26244
[[Badness]] (Sintel): 2.62


[[Gencom]]: [2 96/77; 352/351 364/363 26411/26244]
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 11}}, {{val|0 5 12 -8 -23}}]
Comma list: 243/242, 2401/2376


[[Tp tuning|POL2 generator]]: ~96/77 = 380.766
Subgroup-val mapping: {{mapping| 1 1 2 2 | 0 8 11 20 }}


{{Optimal ET sequence|legend=1| 19f, 22, 41, 63, 104 }}
Optimal tunings:
* WE: ~2 = 1200.4071{{c}}, ~22/21 = 87.6809{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 87.6902{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.3891 cents
{{Optimal ET sequence|legend=0| 14, 27e, 41, 96d, 137d, 178d }}


=== Hogwarts ===
Badness (Sintel): 0.815
Subgroup: 2.3.7.29


[[Comma list]]: 784/783, 5887/5832
==== 2.3.7.11.19 ====
Subgroup: 2.3.7.11.19


[[Gencom]]: [2 36/29; 784/783 5887/5832]
Comma list: 133/132, 243/242, 343/342


[[Mapping|Sval mapping]]: [{{val|1 0 -1 2}}, {{val|0 5 12 9}}]
Subgroup-val mapping: {{mapping| 1 1 2 2 3 | 0 8 11 20 17 }}


[[Tp tuning|POL2 generator]]: ~36/29 = 380.618
Optimal tunings:  
* WE: ~2 = 1200.6100{{c}}, ~19/18 = 87.7129{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/18 = 87.7285{{c}}


{{Optimal ET sequence|legend=1| 19, 22, 41, 145, 186j, 227j }}
{{Optimal ET sequence|legend=0| 14, 27e, 41, 137dh }}


==== Twenothology ====
Badness (Sintel): 0.560
Subgroup: 2.3.7.11.13.29


[[Comma list]]: 144/143, 232/231, 343/338, 729/728
=== Purpleheart ===
{{Main| Whitewood }}


[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 -2 2}}, {{val|0 5 12 -8 18 9}}]
[[Subgroup]]: 2.3.7


[[Tp tuning|POL2 generator]]: ~26/21 = 380.526
[[Comma list]]: 2187/2048


{{Optimal ET sequence|legend=1| 19, 22f, 41 }}
{{Mapping|legend=2| 7 11 0 | 0 0 1 }}
: mapping generators: ~9/8, ~7


== Lee ==
[[Optimal tuning]]s:
Subgroup: 2.3.7
* [[WE]]: ~9/8 = 172.1541{{c}}, ~7/4 = 958.5433{{c}} (~64/63 = 74.3805{{c}})
: [[error map]]: {{val| +5.079 -8.260 -0.124 }}
* [[CWE]]: ~9/8 = 171.4286{{c}}, ~7/4 = 959.2372{{c}} (~64/63 = 69.3373{{c}})
: error map: {{val| 0.000 -16.241 -9.589 }}


[[Comma]]: 177147/175616
{{Optimal ET sequence|legend=1| 7, 14, 35, 49bd }}


[[Gencom]]: [2 81/56; 177147/175616]
[[Badness]] (Sintel): 3.00


[[Gencom|Gencom mapping]]: [{{val|1 0 0 -3}}, {{val|0 3 0 11}}]
=== Chrysanthemum ===
This microtemperament extends [[no-threes subgroup temperaments #Amaranthine|amaranthine]] to prime 3 by tempering out [[43923/43904]], the [[chrysia]], to find 3 at 29 steps down on the chain of nearly pure [[7/4]]'s.


[[Mapping|Sval mapping]]: [{{val|1 0 -3}}, {{val|0 3 11}}]
[[Subgroup]]: 2.3.7


[[Tp tuning|POL2 generator]]: ~81/56 = 633.525
[[Comma list]]: {{monzo| 83 -1 -29 }}


{{Optimal ET sequence|legend=1| 17, 36, 89, 125, 161, 358, 519b }}
{{Mapping|legend=2| 1 -4 3 | 0 29 -1 }}
: mapping generators: ~2, ~8/7


[[Tp tuning #T2 tuning|RMS error]]: 0.3519 cents
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9871{{c}}, ~8/7 = 231.1001{{c}}
: [[error map]]: {{val| -0.013 +0.000 +0.035 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 231.1024{{c}}
: error map: {{val| 0.000 +0.014 +0.072 }}


== [[Slendric]] ==
{{Optimal ET sequence|legend=0| 26, 83, 109, 135, 566, 701, 836, 971, 1106, 2077, 5260, 7337, 9414d }}
Subgroup: 2.3.7


[[Comma]]: 1029/1024
[[Badness]] (Sintel): 3.06


[[Gencom]]: [2 8/7; 1029/1024]
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


[[Gencom|Gencom mapping]]: [{{val|1 1 0 3}}, {{val|0 3 0 -1}}]
Comma list: 43923/43904, 5767168/5764801


[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 3 -1}}]
Subgroup-val mapping: {{mapping| 1 -4 3 5 | 0 29 -1 -8 }}


[[Tp tuning|POL2 generator]]: ~8/7 = 233.688
Optimal tunings:  
* WE: ~2 = 1200.0050{{c}}, ~8/7 = 231.1024{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.1015{{c}}


{{Optimal ET sequence|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}
{{Optimal ET sequence|legend=0| 26, 83, 109, 135, 566, 701, 836, 971, 1807, 2778, 4585 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
Badness (Sintel): 0.324


=== [[Gamelismic clan #Baladic|Baladic]] ===
=== Leapfrog ===
Subgroup: 2.3.7.13
{{See also| Gentle region }}


[[Comma list]]: 169/168, 1029/1024
Leapfrog is generated by a [[3/2|perfect fifth]] and the [[interval class]] of [[7/1|7]] is found at +15 steps, as a double-augmented fifth (C–G𝄪). For this to work, it entails a fifth about 2–3 cents sharp of just; as a result the major third lands comfortably at a near-just [[14/11]] so that it can be extended to the [[2.3.7.11 subgroup]] via tempering out [[896/891]]. The minor third can then be identified with [[13/11]], tempering out [[352/351]] and [[364/363]], which implies [[169/168]] is tempered out as well in this case. Leapfrog is most naturally treated as such, in which it is very efficient.


[[Gencom]]: [91/64 8/7; 169/168 1029/1024]
A notable [[patent val|patent-val]] edo tuning not appearing in the [[optimal ET sequence]] is [[80edo]], which is approximately the just-13's tuning (as [[10edo]] is used as a [[consistent circle]] of [[~]][[16/13]]'s therein), with 13/8 still tuned slightly flat so qualifying a reasonable tuning for the 2.3.13 subgroup (as evidenced by appearing in the sequence for [[tetris]]).


[[Mapping|Sval mapping]]: [{{val|2 2 6 7}}, {{val|0 3 -1 1}}]
Strong extensions for prime 5 include [[leapday]] (29 & 46), [[leapweek]] (46 & 63), and [[leapmonth]] (63 & 80), all of which are more complex than vanilla leapfrog. A low-complexity low-accuracy extension is given by [[supermean]] (5de & 17c), where it is merged with [[meantone]]. [[Srutal]] (46 & 80), usually considered as a strong extension of [[diaschismic]], is a weak extension of leapfrog, and yet another weak extension is [[immune]] (29 & 63), which is in turn a strong extension of 5-limit [[immunity]].


[[Tp tuning|POL2 generator]]: ~8/7 = 233.6044
[[Subgroup]]: 2.3.7


{{Optimal ET sequence|legend=1| 10, 26, 36, 154…, 190…, 226…, 262… }}
[[Comma list]]: 14680064/14348907


[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents
{{Mapping|legend=2| 1 0 -21 | 0 1 15 }}


==== 2.3.7.13.17 ====
{{Mapping|legend=3| 1 0 0 -21 | 0 1 0 15 }}
Subgroup: 2.3.7.13.17
: mapping generators: ~2, ~3


[[Comma list]]: 169/168, 273/272, 289/288
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.1807{{c}}, ~3/2 = 704.2400{{c}}
: [[error map]]: {{val| -0.819 +1.466 -0.311 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.6600{{c}}
: error map: {{val| 0.000 +2.705 +1.074 }}


[[Gencom]]: [17/12 8/7; 169/168 273/272 289/288]
{{Optimal ET sequence|legend=1| 17, 46, 63, 235b, 298b, 361bd, 424bd, 487bbd }}


[[Mapping|Sval mapping]]: [{{val|2 2 6 7 7}}, {{val|0 3 -1 1 3}}]
[[Badness]] (Sintel): 4.33


[[Tp tuning|POL2 generator]]: ~8/7 = 233.6155
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


{{Optimal ET sequence|legend=1| 10, 26, 36, 154…, 190…, 226… }}
Comma list: 896/891, 1331/1323


[[Tp tuning #T2 tuning|RMS error]]: 0.5073 cents
Subgroup-val mapping: {{mapping| 1 0 -21 -14 | 0 1 15 11 }}


== Hemif ==
Gencom mapping: {{mapping| 1 0 0 -21 -14 | 0 1 0 15 11 }}
Subgroup: 2.3.7


[[Comma]]: 1605632/1594323
Optimal tunings:  
* WE: ~2 = 1199.2683{{c}}, ~3/2 = 704.3230{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.6926{{c}}


[[Gencom]]: [2 2187/1792; 1605632/1594323]
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1}}, {{val|0 2 0 13}}]
Badness (Sintel): 0.629


[[Mapping|Sval mapping]]: [{{val|1 1 -1}}, {{val|0 2 13}}]
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13


[[Tp tuning|POL2 generator]]: ~2187/1792 = 351.485
Comma list: 169/168, 352/351, 364/363


{{Optimal ET sequence|legend=1| 7, 17, 41, 58, 99 }}
Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 | 0 1 15 11 8 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2344 cents
Gencom mapping: {{mapping| 1 0 0 -21 -14 -9 | 0 1 0 15 11 8 }}


Related temperaments: [[Breedsmic temperaments #Hemififths|hemififths]], namo
Optimal tunings:  
* WE: ~2 = 1199.5654{{c}}, ~3/2 = 704.4898{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7084{{c}}


=== 2.3.7.11 ===
{{Optimal ET sequence|legend=0| 17, 46, 63 }}
Subgroup: 2.3.7.11


[[Comma list]]: 243/242, 896/891
Badness (Sintel): 0.436


[[Gencom]]: [2 11/9; 243/242 896/891]
===== Skidoo =====
Subgroup: 2.3.7.11.13.23


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1 2}}, {{val|0 2 0 13 5}}]
Comma list: 169/168, 208/207, 352/351, 364/363


[[Mapping|Sval mapping]]: [{{val|1 1 -1 2}}, {{val|0 2 13 5}}]
Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 -5 | 0 1 15 11 8 6 }}


[[Tp tuning|POL2 generator]]: ~11/9 = 351.535
Gencom mapping: {{mapping| 1 0 0 -21 -14 -9 0 0 -5 | 0 1 0 15 11 8 0 0 6 }}


{{Optimal ET sequence|legend=1| 7, 17, 41, 58, 99e }}
Optimal tunings:
* WE: ~2 = 1199.6639{{c}}, ~3/2 = 704.5315{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7021{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


==== 2.3.7.11.13 ====
Badness (Sintel): 0.356
Subgroup: 2.3.7.11.13


[[Comma list]]: 144/143, 243/242, 364/363
====== 2.3.7.11.13.23.29 ======
Subgroup: 2.3.7.11.13.23.29


[[Gencom]]: [2 11/9; 144/143 243/242 364/363]
Comma list: 169/168, 208/207, 232/231, 352/351, 364/363


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1 2 4}}, {{val|0 2 0 13 5 -1}}]
Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 -5 -38 | 0 1 15 11 8 6 27 }}


[[Mapping|Sval mapping]]: [{{val|1 1 -1 2 4}}, {{val|0 2 13 5 -1}}]
Gencom mapping: {{mapping| 1 0 0 -21 -14 -9 -5 0 0 -38 | 0 1 0 15 11 8 0 0 6 27 }}


[[Tp tuning|POL2 generator]]: ~11/9 = 351.691
Optimal tunings:  
* WE: ~2 = 1199.5755{{c}}, ~3/2 = 704.5533{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7750{{c}}


{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41, 58 }}
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7167 cents
Badness (Sintel): 0.441


==== Heartful ====
; Music
Subgroup: 2.3.7.11.19
* Suite for Harpsichord in A Locrian, tuning: Eb–G# in [[46edo]] by [[Inthar]] (in progress):
** I. Prelude
** II. Allemande
** III. Courante
** [[:File:Locrian Suite Sarabande.mp3|IV. Sarabande]] ([[:File:Locrian Suite Sarabande Score.pdf|score]], [[:File:Locrian Suite Sarabande 17edo.mp3|17edo version]])
** [[:File:Locrian Suite Menuet.mp3|V. Menuet and Trio]]
** [[:File:Locrian Suite Gavotte.mp3|VI. Gavotte I and II]]
** [[:File:Locrian Suite Gigue.mp3|VII. Gigue]]


[[Comma list]]: 243/242, 896/891, 1083/1078
=== Superslendric ===
In superslendric, eight [[8/7]]'s are equated to [[3/1]]. This relates it to [[8edt]].


[[Gencom]]: [2 21/19; 243/242 896/891 1083/1078]
[[Subgroup]]: 2.3.7


[[Mapping|Sval mapping]]: [{{val|1 1 -1 2 0}}, {{val|0 4 26 10 29}}]
[[Comma list]]: 17294403/16777216


[[Tp tuning|POL2 generator]]: ~21/19 = 175.804
{{Mapping|legend=2| 1 0 3 | 0 8 -1 }}
: mapping generators: ~2, ~8/7


{{Optimal ET sequence|legend=1| 34dh, 41, 116e, 157e }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.1628{{c}}, ~8/7 = 237.7287{{c}}
: [[error map]]: {{val| +1.163 -0.125 -3.066 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 237.5664{{c}}
: error map: {{val| 0.000 -1.424 -6.392 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5360 cents
{{Optimal ET sequence|legend=1| 5, …, 66b, 71b, 76, 81, 86, 91, 96d }}


Related temperaments: [[Tetracot family|bunya]]
[[Badness]] (Sintel): 6.15


== Hearts ==
=== Hectosaros leap week ===
Subgroup: 2.3.7
This temperament may be described as the {{nowrap| 320 & 1803 }} temperament, in the 2.3.7.13.17.19 on the basis of the fact that 1803 tropical years make up almost exactly 100 saros cycles.


[[Comma list]]: 34451725707/34359738368 (trila-quadzo comma)
[[Subgroup]]: 2.3.7


[[Gencom]]: [2 567/512; 34451725707/34359738368]
[[Comma list]]: {{monzo| -50 -746 439 }}


[[Mapping|Sval mapping]]: [{{val|1 1 5}}, {{val|0 4 -15}}]
{{Mapping|legend=2| 1 -126 -214 | 0 439 746 }}
: mapping generators: ~2, ~{{monzo| -16 -243 143 }}


[[Tp tuning|POL2 generator]]: ~567/512 = 175.433
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0010{{c}}, ~{{monzo| -16 -243 143 }} = 348.7520{{c}}
: [[error map]]: {{val| +0.001 +0.036 -0.067 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -16 -243 143 }} = 348.7517{{c}}
: error map: {{val| 0.000 +0.035 -0.068 }}


{{Optimal ET sequence|legend=1| 7, 27d, 34, 41, 89, 130, 171 }}
{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0529 cents
[[Badness]] (Sintel): 17.7 × 10<sup>3</sup>


Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquiquartififths]]
==== 2.3.7.13 subgroup ====
Subgroup: 2.3.7.13


=== 2.3.7.11 ===
Comma list: {{monzo| -42 -2 -5 16 }}, {{monzo| 10 -46 29 -5 }}
Subgroup: 2.3.7.11


[[Comma list]]: 243/242, 65536/65219
Subgroup-val mapping: {{mapping| 1 -126 -214 -80 | 0 439 746 288 }}


[[Gencom]]: [2 256/231; 243/242 65536/65219]
Optimal tunings:  
* WE: ~2 = 1200.0058{{c}}, ~{{monzo| 18 -9 8 -7 }} = 348.7534{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~{{monzo| 18 -9 8 -7 }} = 348.8517{{c}}


[[Mapping|Sval mapping]]: [{{val|1 1 5 2}}, {{val|0 4 -15 10}}]
{{Optimal ET sequence|legend=0| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689, 11255d }}  


[[Tp tuning|POL2 generator]]: ~256/231 = 175.369
Badness (Sintel): 53.2


{{Optimal ET sequence|legend=1| 7, 27de, 34, 41, 89, 130 }}
==== 2.3.7.13.17 subgroup ====
Subgroup: 2.3.7.13.17


[[Tp tuning #T2 tuning|RMS error]]: 0.3224 cents
Comma list: 39337984/39328497, {{monzo| 0 -14 7 4 -3 }}, {{monzo| -18 -24 14 -1 5 }}


Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquart]]
Subgroup-val mapping: {{mapping| 1 -126 -214 -80 -18 | 0 439 746 288 76 }}


=== 2.3.7.11.19 ===
Optimal tunings:
Subgroup: 2.3.7.11.19
* WE: ~2 = 1200.9870{{c}}, ~3757/3072 = 348.7480{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3757/3072 = 348.7517{{c}}


[[Comma list]]: 243/242, 513/512, 1083/1078
{{Optimal ET sequence|legend=0| 320, 1483bd, 1803, 2123 }}


[[Gencom]]: [2 21/19; 243/242 513/512 1083/1078]
Badness (Sintel): 13.4


[[Mapping|Sval mapping]]: [{{val|1 1 5 2 6}}, {{val|0 4 -15 10 -12}}]
==== 2.3.7.13.17.19 subgroup ====
Subgroup: 2.3.7.13.17.19


[[Tp tuning|POL2 generator]]: ~21/19 = 175.341
Comma list: 10081799/10077696, 10754912/10744731, 39337984/39328497, 480024727/480020256


{{Optimal ET sequence|legend=1| 7, 27deh, 34, 41, 89, 130, 219 }}
Subgroup-val mapping: {{mapping| 1 -126 -214 -80 -18 -171 | 0 439 746 288 76 603 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3121 cents
Optimal tunings:  
* WE: ~2 = 1200.9961{{c}}, ~3757/3072 = 348.7506{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3757/3072 = 348.7517{{c}}


Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquart]]
{{Optimal ET sequence|legend=0| 320, 1483bd, 1803, 2123 }}


== Hectosaros leap week ==
Badness (Sintel): 7.46
Defined as the 320 &amp; 1803 temperament, in the 2.3.7.13.17.19 on the basis of the fact that 1803 tropical years make up almost exactly 100 saros cycles.


[[Subgroup]]: 2.3.7
=== Heartland (rank 3) ===
{{Main| Heartland }}


[[Comma list]]: {{monzo|-50 -746 439}}
Heartland, with a generator of [[~]][[21/19]], is named for its tempering of the heartlandisma, [[3971/3969]]. Aside from the heartlandisma, the heartland temperament tempers out 243/242 (rastma) and 1083/1078 (bihendrixma), and slices the fifth in four (the number of chambers of the heart).


[[Mapping]]: [{{val|1 313 532}}, {{val|0 -439 -746}}]
[[Subgroup]]: 2.3.7.11.19


[[Optimal tuning]] ([[CTE]]): ~{{monzo|17 343 143}} = 851.248
[[Comma list]]: 243/242, 1083/1078


{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}
{{Mapping|legend=2| 1 1 0 2 1 | 0 4 0 10 3 | 0 0 1 0 1 }}
: mapping generators: ~2, ~21/19, ~7


[[Tp tuning #T2 tuning|RMS error]]: 0.0164 cents
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0983{{c}}, ~21/19 = 175.2856{{c}}, ~7/4 = 969.4578{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/19 = 175.2894{{c}}, ~7/4 = 969.5203{{c}}


=== 2.3.7.13 subgroup ===
{{Optimal ET sequence|legend=1| 14, 27e, 34dh, 41, 89, 130, 219 }}
Subgroup: 2.3.7.13


Comma list: {{monzo|-42  -2 -5 16}}, {{monzo|10 -46 29 -5}}
[[Badness]] (Sintel): 0.615


Mapping: [{{val|1 313 532 208}}, {{val|0 -439 -746 -288}}]
== Temperaments with a 2.3.11 gene ==
=== Neutral ===
See [[Rastmic clan #Neutral]].


Optimal tuning (CTE): ~1235079060111/755603996672 = 851.248
=== Io ===
Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]]. This equates very different intervals (for example, the generator itself represents both [[3/2]] and [[16/11]]), and as such some consider it to be an [[exotemperament]]. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament.


{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}
The name ''io'' was coined by [[User:CompactStar|CompactStar]] in 2024 based on the [[Kite's color notation|color name]] ''ilo'', prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.


=== 2.3.7.13.17 subgroup ===
[[Subgroup]]: 2.3.11
Subgroup: 2.3.7.13.17


Comma list: 39337984/39328497, {{monzo|0 -14  7  4 -3}}, {{monzo|-18 -24 14 -1  5}}
[[Comma list]]: 33/32


Mapping: [{{val|1 313 532 208 58}}, {{val|0 -439 -746 -288 -76}}]
{{Mapping|legend=2| 1 0 5 | 0 1 -1 }}
: mapping generators: ~2, ~3


Optimal tuning (CTE): ~6144/3757 = 851.248
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1206.6866{{c}}, ~3/2 = 691.7837{{c}}
: [[error map]]: {{val| +6.687 -3.485 -16.355 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 689.2066{{c}}
: error map: {{val| 0.000 -12.748 -40.525 }}


{{Optimal ET sequence|legend=1| 320, 1483bd, 1803, 2123 }}
{{Optimal ET sequence|legend=1| 2, 5, 7, 12e, 40ee, 47eee, 54beee, 61beeee }}


=== 2.3.7.13.17.19 subgroup ===
[[Badness]] (Sintel): 0.185
Subgroup: 2.3.7.13.17.19


Comma list: 10081799/10077696, 39337984/39328497, 10754912/10744731, 480024727/480020256
=== Alphaxenean ===
Alphaxenean tempers out the [[Alpharabian comma]] and equates a stack of four undecimal quartertones with the [[9/8|Pythagorean whole tone]]. It also divides the [[2/1|octave]] into two.


Mapping: [{{val|1 313 532 208 58 432}}, {{val|0 -439 -746 -288 -76 -603}}]
[[Subgroup]]: 2.3.11


Optimal tuning (CTE): ~6144/3757 = 851.248
[[Comma list]]: 131769/131072


{{Optimal ET sequence|legend=1| 320, 1483bd, 1803, 2123 }}
{{Mapping|legend=2| 2 1 8 | 0 2 -1 }}
: mapping generators: ~363/256, ~16/11


== Ennea ==
[[Optimal tuning]]s:
Subgroup: 2.3.7.11
* [[WE]]: ~363/256 = 600.1590{{c}}, ~16/11 = 650.8508{{c}}
: [[error map]]: {{val| +0.318 -0.094 -0.897 }}
* [[CWE]]: ~363/256 = 600.0000{{c}}, ~16/11 = 650.7321{{c}}
: error map: {{val| 0.000 -0.491 -2.050 }}


[[Comma list]]: 41503/41472, 43923/43904
{{Optimal ET sequence|legend=1| 22, 24, 94, 118, 142, 450e, 592e, 1326beeee }}


[[Gencom]]: [2 99/98; 41503/41472, 43923/43904]
[[Badness]] (Sintel): 0.395


[[Gencom|Gencom mapping]]: [{{val|1 14/9 0 25/9 31/9}}, {{val|0 2 0 2 1}}]
=== Infraug ===
[[Subgroup]]: 2.3.11


[[Mapping|Sval mapping]]: [{{val|9 0 11 24}}, {{val|0 2 2 1}}]
[[Comma list]]: 729/704


[[Tp tuning|POL2 generator]]: ~99/98 = 17.6258
{{Mapping|legend=2| 1 0 -6 | 0 1 6 }}
: mapping generators: ~2, ~3


{{Optimal ET sequence|legend=1| 54, 63, 72, 135, 342, 477, 1089, 1566 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.5969{{c}} ~3/2 = 692.7443{{c}}
: [[error map]]: {{val| +2.597 -6.614 +5.148 }}
* [[CWE]]: ~2 = 1200.0000{{c}} ~3/2 = 692.0116{{c}}
: error map: {{val| 0.000 -9.943 +0.752 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0383 cents
{{Optimal ET sequence|legend=1| 7, 19, 26, 33, 59b, 92b }}


== Parapyth (rank 3) ==
[[Badness]] (Sintel): 0.734
{{see also| Pentacircle temperaments #Parapyth }}


Subgroup: 2.3.7.11
==== 2.3.11.13 ====
Subgroup: 2.3.11.13


[[Comma list]]: 896/891
Comma list: 144/143, 729/704


[[Gencom]]: [2 3/2 28/27; 896/891]
Subgroup-val mapping: {{mapping| 1 0 -6 10 | 0 1 6 -4 }}


[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 }}, {{val| 0 1 0 3 -1 }}, {{val| 0 0 0 1 1 }}]
Optimal tunings:  
* WE: ~2 = 1202.1934{{c}}, ~3/2 = 692.8902{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 691.7585{{c}}


[[Sval]] [[mapping]]: [{{val| 1 0 0 7 }}, {{val| 0 1 0 -4 }}, {{val| 0 0 1 1 }}]
{{Optimal ET sequence|legend=0| 7, 19, 26, 59b }}


[[Tp tuning|POL2 tuning]]: ~3 = 1903.834, ~7 = 3369.872
Badness (Sintel): 0.725


{{Optimal ET sequence|legend=1| 17, 36, 41, 58, 63, 104 }}
==== Aerophore ====
Subgroup: 2.3.11.19


[[Tp tuning #T2 tuning|RMS error]]: 0.4149 cents
Comma list: 363/361, 729/704


=== 2.3.7.11.13 ===
Subgroup-val mapping: {{mapping| 1 0 -6 -6 | 0 2 12 13 }}
Subgroup: 2.3.7.11.13
: mapping generators: ~2, ~19/11


[[Comma list]]: 352/351, 364/363
Optimal tunings:  
* WE: ~2 = 1202.6380{{c}}, ~19/11 = 947.4782{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 945.7779{{c}}


The gencom below gives [[Margo Schulter]]'s favored basis
{{Optimal ET sequence|legend=1| 14, 19, 33 }}


[[Gencom]]: [2 3/2 28/27; 352/351 364/363]
Badness (Sintel): 1.59


[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 6 }}, {{val| 0 1 0 3 -1 -4 }}, {{val| 0 0 0 1 1 1 }}]
=== Paralimmal ===
[[Subgroup]]: 2.3.11


[[Sval]] [[mapping]]: [{{val| 1 0 0 7 12 }}, {{val| 0 1 0 -4 -7 }}, {{val| 0 0 1 1 1 }}]
[[Comma list]]: 4096/3993


[[Tp tuning|POL2 tuning]]: ~3 = 1903.856, ~7 = 3369.907
{{Mapping|legend=2| 1 0 4 | 0 3 -1 }}
: mapping generators: ~2, ~16/11


{{Optimal ET sequence|legend=1| 17, 41, 46, 58, 87, 104 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.9124{{c}}, ~16/11 = 634.1269{{c}}
: [[error map]]: {{val| -2.088 +0.426 +6.205 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/11 = 634.9546{{c}}
: error map: {{val| 0.000 +2.909 +13.727 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
{{Optimal ET sequence|legend=1| 2, 11b, 13, 15, 17 }}


== Neutral ==
[[Badness]] (Sintel): 0.984
Neutral can be thought of as the 2.3.11 version of either [[mohajira]] or [[maqamic]], as well as suhajira and ringo. Among other things, it is the temperament optimizing the [[neutral tetrad]].


Subgroup: 2.3.11
==== Huxley ====
{{Main| Huxley }}


[[Comma]]: 243/242
Huxley, the {{nowrap| 4 & 13 }} temperament in the 2.3.11.13 subgroup, extends [[lovecraft]]. Specifically it tunes the ~13/8 to exactly half of ~8/3.


[[Gencom]]: [2 11/9; 243/242]
Subgroup: 2.3.11.13


[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2}}, {{val|0 2 0 0 5}}]
Comma list: 512/507, 1352/1331


[[Mapping|Sval mapping]]: [{{val|1 1 2}}, {{val|0 2 5}}]
Subgroup-val mapping: {{mapping| 1 -3 5 6 | 0 6 -2 -3 }}
: mapping generators: ~2, ~22/13


[[Tp tuning|POL2 generator]]: ~11/9 = 350.525
Optimal tunings:  
* WE: ~2 = 1198.0036{{c}}, ~22/13 = 916.0595{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 917.5184{{c}}


{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41, 65, 89, 202, 291, 380 }}
{{Optimal ET sequence|legend=0| 4, 13, 17 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3021 cents
Badness (Sintel): 1.31


; Scales
=== Glaishur ===
* [[neutraltet7|Seven note albitonic scale]]
This temperament is the no-5 no-7 [[restriction]] of [[#Navy|navy]], as well as the add-11 [[extension]] of [[#Glacier|glacier]].
* [[neutraltet10|Ten note chromatic scale]]
* [[neutraltet17|Seventeen note mega chromatic scale]]


=== Namo ===
[[Subgroup]]: 2.3.11
Subgroup: 2.3.11.13


[[Comma list]]: 144/143, 243/242
[[Comma list]]: 10554638336/10460353203


[[Gencom]]: [2 11/9; 144/143 243/242]
{{Mapping|legend=2| 1 1 0 | 0 5 21 }}
: mapping generators: ~2, ~88/81


[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2 4}}, {{val|0 2 0 0 5 -1}}]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0000{{c}}, ~88/81 = 140.537{{c}}
: [[error map]]: {{val| -0.150 +0.493 -0.559 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~88/81 = 140.537{{c}}
: error map: {{val| 0.000 +0.662 -0.326 }}


[[Mapping|Sval mapping]]: [{{val|1 1 2 4}}, {{val|0 2 5 -1}}]
{{Optimal ET sequence|legend=1| 17, 60e, 77, 94, 111 }}


[[Tp tuning|POL2 generator]]: ~11/9 = 351.488
[[Badness]] (Sintel): 2.27


{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41 }}
==== 2.3.11.13 ====
Subgroup: 2.3.11.13


[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents
Comma list: 352/351, 531674/531441


== Heartland (rank 3) ==
Subgroup-val mapping: {{mapping| 1 1 0 3 | 0 5 21 6 }}
Heartland, with a generator of ~21/19, is named for its tempering of the heartlandisma, [[3971/3969]]. Aside from the heartlandisma, the heartland temperament tempers out 243/242 (rastma) and 1083/1078 (bihendrixma), and slices the fifth in four (the number of chambers of the heart).


Subgroup: 2.3.7.11.19
Optimal tunings:  
* WE: ~2 = 1200.0000{{c}}, ~13/12 = 140.537{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.537{{c}}


[[Comma list]]: 243/242, 1083/1078
{{Optimal ET sequence|legend=0| 17, 60e, 77, 94, 111, 239f, 350f }}


[[Gencom]]: [2 21/19 7; 243/242 1083/1078]
Badness (Sintel): 0.415


[[Mapping|Sval mapping]]: [{{val|1 1 0 2 1}}, {{val|0 4 0 10 3}}, {{val| 0 0 1 0 1 }}]
=== Profanity ===
Profanity identifies [[11/9]] with 2\7.


[[Tp tuning|POL2 generator]]: ~21/19 = 175.2713, ~7 = 3369.3784
[[Subgroup]]: 2.3.11


{{Optimal ET sequence|legend=1| 7, 14, 27e, 34dh, 41, 89, 130 }}
[[Comma list]]: 19487171/19131876


[[Tp tuning #T2 tuning|RMS error]]: 0.3066 cents
{{Mapping|legend=2| 7 0 2 | 0 1 2 }}
: mapping generators: ~1458/1331, ~3


== Reversed meantone ==
[[Optimal tuning]]s:
{{Main| Reversed meantone }}
* [[WE]]: ~1458/1331 = 171.4369{{c}}, ~3/2 = 702.9304{{c}}
: [[error map]]: {{val| +0.058 +1.033 -2.467 }}
* [[CWE]]: ~1458/1331 = 171.4286{{c}}, ~3/2 = 702.9442{{c}}
: error map: {{val| 0.000 -0.989 -2.572 }}


Subgroup: 2.3.41
{{Optimal ET sequence|legend=1| 7, … 49, 56, 63, 70 }}


[[Comma list]]: 82/81
[[Badness]] (Sintel): 3.03


[[Gencom]]: [2 4/3; 82/81]
== Temperaments with a 2.3.13 gene ==
=== Threedic ===
[[Subgroup]]: 2.3.13


[[Mapping|Sval mapping]]: [{{val|1 2 7}}, {{val|0 -1 -4}}]
[[Comma list]]: 2197/2187


[[Tp tuning|POL2 generator]]: ~4/3 = 494.509
{{Mapping|legend=2| 1 0 0 | 0 3 7 }}
: mapping generators: ~2, ~13/9


{{Optimal ET sequence|legend=1| 5, 12, 17 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0000, ~13/9 = 634.1729{{c}}
: [[error map]]: {{val| -0.000 +0.563 -1.318 }}
* [[CWE]]: ~2 = 1200.0000, ~13/9 = 634.1729{{c}}
: error map: {{val| 0.000 +0.564 -1.318 }}


=== 2.3.7.41 subgroup ===
{{Optimal ET sequence|legend=1| 15, 17, 36, 53, 70, 123, 193, 316, 755f }}
Subgroup: 2.3.7.41


[[Comma list]]: 64/63, 82/81
[[Badness]] (Sintel): 0.160


[[Gencom]]: [2 4/3; 64/63 82/81]
=== Ultraflat ===
Ultraflat is a diatonic-based [[exotemperament]] that makes [[27/26]] vanish, so [[13/8]] is a major sixth.


[[Mapping|Sval mapping]]: [{{val| 1 2 2 7 }}, {{val| 0 -1 2 -4 }}]
[[Subgroup]]: 2.3.13


[[POTE generator]]: ~4/3 = 490.0323
[[Comma list]]: 27/26


[[TOP tuning|TOP generator]]s: ~2 = 1197.2342, ~4/3 = 488.9029
{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}
: mapping generators: ~2, ~3


{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 49 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.6561{{c}}, ~3/2 = 686.9485{{c}}
: [[error map]]: {{val| +1.656 -13.350 +23.630 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 687.1143{{c}}
: error map: {{val| 0.000 -14.841 +20.815 }}


=== 2.3.7.11.41 subgroup ===
{{Optimal ET sequence|legend=1| 2, 5, 7 }}
Subgroup: 2.3.7.11.41


[[Comma list]]: 64/63, 82/81, 99/98
[[Badness]] (Sintel): 0.200


[[Gencom]]: [2 4/3; 64/63 82/81 99/98]
=== Superflat ===
Superflat is a less inaccurate cousin of ultraflat, with less flat fifths. It tempers out [[1053/1024]], so [[13/8]] is a minor sixth, and [[16/13]] is a major third. Superflat and ultraflat intersect in [[7edo]], where major sixths and minor sixths are not distinguished.


[[Mapping|Sval mapping]]: [{{val| 1 2 2 1 7 }}, {{val| 0 -1 2 6 -4 }}]
The more accurate tunings for this temperament are generated by a fifth at least as flat as those of [[flattone]], although often even flatter (such as [[40edo]]'s fifth). Superflat can be viewed as a 2.3.13 subgroup analogue of [[meantone]] and [[archy]]. Superflat diatonic scales have a character somewhere between neutral third scales (or [[3L 4s|mosh]]) and meantone diatonic scales.


[[POTE generator]]: ~4/3 = 492.1787
[[Subgroup]]: 2.3.13


[[TOP tuning|TOP generator]]s: ~2 = 1197.9683, ~4/3 = 491.3454
[[Comma list]]: 1053/1024


{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 39d }}
{{Mapping|legend=2| 1 0 10 | 0 1 -4 }}
: mapping generators: ~2, ~3


== Sematology ==
[[Optimal tuning]]s:
This temperament tempers out 4107/4096 and thus equates 2 [[37/32]]'s with [[4/3]].
* [[WE]]: ~2 = 1203.1291{{c}} ~3/2 = 695.6489{{c}}
: [[error map]]: {{val| +3.129 -3.177 -4.349 }}
* [[CWE]]: ~2 = 1200.0000{{c}} ~3/2 = 693.6081{{c}}
: error map: {{val| 0.000 -8.347 +14.960 }}


[[Subgroup]]: 2.3.37
{{Optimal ET sequence|legend=1| 5f, 7, 12, 19, 45f, 64f, 147bfff }}


[[Comma list]]: 4107/4096
[[Badness]] (Sintel): 0.610


[[Gencom]]: [2 37/32; 4107/4096]
=== Shoal ===
The 2.3.13.23-subgroup [[microtemperament]] is remarkable for containing not one but two [[superparticular]] intervals as small as [[3888/3887]] and [[12168/12167]]. Tempering out both of them gives us this rank-2 temperament where a sharp whole tone of [[26/23]] is the generator, two of which stack to a [[23/18]] supermajor third, and eight of which stack to a [[8/3]] perfect eleventh. [[17edo]] is a trivial tuning where 26/23 is equated to [[9/8]], tempering out the comma [[208/207]]. More accurate tunings of shoal create a 17-note [[mos]] scale, serving as a [[circulating temperament]] of 17edo, where 208/207 is the [[chroma]] between large and small steps.


[[Mapping]]: [{{val|1 1 5}}, {{val|0 -2 1}}]
[[Subgroup]]: 2.3.13


[[POTE generator]]: ~[[37/32]] = 249.075
[[Comma list]]: 816293376/815730721


{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53, 77, 130 }}
{{Mapping|legend=2| 1 -5 -7 | 0 8 13 }}
: mapping generators: ~2, ~3888/2197


=== 2.3.7.37 subgroup ===
[[Optimal tuning]]s:
[[Subgroup]]: 2.3.7.37
* [[WE]]: ~2 = 1199.9922{{c}}, ~3888/2197 = 987.7360{{c}}
: [[error map]]: {{val| -0.008 -0.028 +0.095 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3888/2197 = 987.7415{{c}}
: error map: {{val| 0.000 -0.023 +0.112 }}


[[Comma list]]: 4107/4096, 259/256
{{Optimal ET sequence|legend=1| 17, 79, 96, 113, 130, 147, 424, 571, 1289, 10883ff, 12172ff }}


[[Gencom]]: [2 37/32; 4107/4096 259/256]
Badness (Sintel): 0.135


[[Mapping]]: [{{val|1 1 1 5}}, {{val|0 -2 -1 1}}]
==== 2.3.13.23 ====
Subgroup: 2.3.13.23


[[POTE generator]]: ~[[37/32]] = 247.782
Comma list: 3888/3887, 12168/12167


{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53d }}
Subgroup-val mapping: {{mapping| 1 -5 -7 -7 | 0 8 13 14 }}


=== 2.3.5.37 subgroup ===
Optimal tunings:
It is difficult to extend sematology to include 5, due the 5th harmonic being quite high-complexity.
* WE: ~2 = 1199.9883{{c}}, ~23/13 = 987.7325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~23/13 = 987.7408{{c}}


[[Subgroup]]: 2.3.5.37
{{Optimal ET sequence|legend=0| 17, 79, 96, 113, 130, 147, 424, 571, 1289, 1860, 3149 }}


[[Comma list]]: 4107/4096, 17592186044416/17562397269605
Badness (Sintel): 0.0213


[[Gencom]]: [2 37/32; 4107/4096 17592186044416/17562397269605]
Scales:
* [[Shoal17]]


[[Mapping]]: [{{val|1 1 4 5}}, {{val|0 -2 -8 1}}]
; Music
* [https://www.youtube.com/watch?v=bwGjN56iMM0 ''Moody Improvisation in the Shoal Temperament''] by [[Budjarn Lambeth]] (2025)


[[POTE generator]]: ~[[37/32]] = 251.393
=== Glacier ===
This 2.3.13-subgroup gene is not nearly as good as shoal, but it can extend extremely well to other no-5 subgroups. It is the common [[restriction]] of [[#Bleu|bleu]] and [[#Navy|navy]]. It is very well represented in [[26edo]], where a nearly pure 13/12 can serve as the generator, but [[94edo]] provides a much better tuning.


{{Optimal ET sequence|legend=1| 5, 14c, 19, 43, 62 }}
[[Subgroup]]: 2.3.13


==== 2.3.5.7.37 subgroup ====
[[Comma list]]: 373248/371293
[[Subgroup]]: 2.3.5.7.37


[[Comma list]]: 4107/4096, 17592186044416/17562397269605, 259/256
{{Mapping|legend=2| 1 1 3 | 0 5 6 }}
: mapping generators: ~2, ~13/12


[[Gencom]]: [2 37/32; 4107/4096 17592186044416/17562397269605 259/256]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.8406{{c}}, ~13/12 = 140.3695{{c}}
: [[error map]]: {{val| -0.159 -0.267 +1.211 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/12 = 140.3605{{c}}
: error map: {{val| 0.000 -0.153 +1.635 }}


[[Mapping]]: [{{val|1 1 4 1 5}}, {{val|0 -2 -8 -1 1}}]
{{Optimal ET sequence|legend=1| 8, 9, 17, 60, 77, 94, 171, 265, 359f, 983ff }}


[[POTE generator]]: ~[[37/32]] = 251.204
[[Badness]] (Sintel): 0.383


{{Optimal ET sequence|legend=1| 5, 14c, 19 }}
== Temperaments with a higher-limit gene ==
=== Semitonic ===
[[Subgroup]]: 2.3.17


== Reversed mavila ==
[[Comma list]]: [[289/288]]
[[Subgroup]]: 2.3.37


[[Comma list]]: 81/74
{{Mapping|legend=2| 2 0 5 | 0 1 1 }}
: mapping generators: ~17/12, ~3


[[Gencom]]: [2 4/3; 81/74]
[[Optimal tuning]]s:  
* [[WE]]: ~17/12 = 600.1471{{c}}, ~3/2 = 701.9563{{c}} (~17/16 = 101.8091{{c}})
: [[error map]]: {{val| +0.294 +0.295 -1.969 }}
* [[CWE]]: ~17/12 = 600.0000{{c}}, ~3/2 = 702.0260{{c}} (~17/16 = 102.0260{{c}})
: error map: {{val| 0.000 +0.071 -2.929 }}


[[Mapping]]: [{{val|1 1 0}}, {{val|0 -1 12}}]
{{Optimal ET sequence|legend=1| 10, 12, 58, 70, 82, 94, 106, 118, 224g }}


[[POTE generator]]: ~4/3 = 521.397
[[Badness]] (Sintel): 0.0454


{{Optimal ET sequence|legend=1| 5l, 7l, 9, 16l }}
=== Boethian ===
Boethian is a [[5L 2s|diatonic-based]] temperament that makes [[513/512]] vanish, so that the major third (C–E) is ~[[24/19]] and the minor third (C–E♭) is ~[[19/16]]. As such, it functions as a 2.3.19-subgroup analogue of [[meantone]], though the small size of the comma puts it at [[schismic]] level of accuracy. In particular, the equal temperaments in the tuning spectrum up to 1/2-comma (flattened) boethian temperament (very close to [[12edo]]) are included in the schismic tuning spectrum in the 5-limit, so boethian intersects with schismic in the prime-5 infill extension thereof, called [[nestoria]], which also tempers out [[361/360]], the difference between 19/18 and 20/19 or between 19/15 and 24/19.


== Aerophore ==
[[Subgroup]]: 2.3.19
[[Subgroup]]: 2.3.11.19


[[Comma list]]: 363/361, 729/704
[[Comma list]]: 513/512


[[Mapping]]: [{{val|1 0 -6 -6}}, {{val|0 2 12 13}}]
{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}
: mapping generators: ~2, ~3


[[POTE generator]]: ~[[19/11]] = 945.4
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.2498{{c}}, ~3/2 = 701.4958{{c}}
: [[error map]]: {{val| +0.250 -0.209 -0.501 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.3445{{c}}
: error map: {{val| 0.000 -0.610 -1.547 }}


{{Optimal ET sequence|legend=1| 9eehh, 14, 19, 33 }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296, 1557bhhhh, 1853bhhhh }}


=== Semaerophore ===
[[Badness]] (Sintel): 0.0294
[[Subgroup]]: 2.3.7.11.19


[[Comma list]]: 49/48, 77/76, 729/704
=== Dog ===
Dog is based by [[2L 5s]] or [[7L 2s]] scale that makes [[81/76]] vanish, so [[19/16]] is a major third. It can be viewed as a 2.3.19-subgroup analogue of [[mavila]].


[[Mapping]]: [{{val|1 0 2 -6 -6}}, {{val|0 2 1 12 13}}]
[[Subgroup]]: 2.3.19


[[POTE generator]]: ~[[7/4]] = 944.667
[[Comma list]]: 81/76


{{Optimal ET sequence|legend=1| 9eehh, 14, 33d, 47deh }}
{{Mapping|legend=2| 1 0 -2 | 0 1 4 }}
: mapping generators: ~2, ~3


== Superflat aka tridecimal ==
[[Optimal tuning]]s:
Superflat temperament, or alternatively, tridecimal temperament, is a diatonic-based temperament that makes [[1053/1024]] vanish, so [[13/8]] is a minor sixth, and [[16/13]] is a major third. The more accurate tunings for this temperament are generated by a fifth at least as flat as those of [[flattone]], although often even flatter (such as [[40edo]]'s fifth). Superflat can be viewed as a 2.3.13 subgroup analogue of [[meantone]] and [[archy]]. Superflat diatonic scales have a character somewhere between neutral third scales (or [[3L 4s|mosh]]) and meantone diatonic scales.
* [[WE]]: ~2 = 1203.3813{{c}}, ~3/2 = 680.5089{{c}}
: [[error map]]: {{val| +3.381 -18.065 +31.285 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 680.5856{{c}}
: error map: {{val| 0.000 -21.369 +24.8295 }}


[[Subgroup]]: 2.3.13
{{Optimal ET sequence|legend=1| 2, 5h, 7, 16, 23 }}


[[Comma list]]: [[1053/1024]]
[[Badness]] (Sintel): 0.491


{{Mapping|legend=2|1 1 6|0 1 -4}}
=== Lipsett ===
Lipsett is a pleasantly melodic little temperament with a highly usable 5-tone and 9-tone mos. It is audibly similar to [[semaphore]] temperament, so it could be thought of as semaphore but for the 23rd harmonic instead of the 7th. It is named for Arthur Lipsett, the director of the Canadian short film ''21-87''. Leia's prison cell in ''Star Wars'' is numbered 2187, as a nod to the influence the film had on George Lucas.


[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.939
[[Subgroup]]: 2.3.23


[[Support]]ing [[ET]]s: {{EDOs|7, 19, 12, 5f, 26, 9f, 31f, 33, 17f, 45f, 16f, 40, 50f, 43f}}
[[Comma list]]: 2187/2116


=== 2.3.11.13 ===
{{Mapping|legend=2| 1 0 -1 | 0 2 7 }}
[[Subgroup]]: 2.3.11.13
: mapping generators: ~2, ~46/27


[[Comma list]]: [[144/143]], [[729/704]]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.5339{{c}}, ~46/27 = 948.5629{{c}}
: [[error map]]: {{val| +0.534 -4.829 +11.132 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~46/27 = 948.3272{{c}}
: error map: {{val| 0.000 -5.301 +10.016 }}


[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.247
{{Optimal ET sequence|legend=1| 5, 14, 19, 43, 62i, 81i }}


[[Support]]ing [[ET]]s: {{EDOs|7, 26, 19, 12, 33, 40, 45f, 47e, 59b, 73be, 85bf, 92b}}
[[Badness]] (Sintel): 0.801


== Ultraflat ==
=== Porpoise ===
Ultraflat is the much more inaccurate cousin of superflat, with even flatter fifths. [[27/26]] is tempered out rather than [[1053/1024]], so [[13/8]] is a major sixth. These temperamenets intersect in [[7edo]], where major sixths and minor sixths are not distinguished.
[[Subgroup]]: 2.3.29


[[Subgroup]]: 2.3.13
[[Comma list]]: 24576/24389


[[Comma list]]: [[27/26]]
{{Mapping|legend=2| 1 2 5 | 0 -3 -1 }}
: mapping generators: ~2, ~32/29


{{Mapping|legend=2|1 1 2|0 1 3}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5519{{c}}, ~32/29 = 165.7453{{c}}
: [[error map]]: {{val| -0.448 -0.087 +2.437 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~32/29 = 165.9004{{c}}
: error map: {{val| 0.000 +0.344 +4.522 }}


[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[3/2]] = 688.391
{{Optimal ET sequence|legend=1| 7, 22, 29, 94, 123, 152j, 275jj, 427jjj }}


{{Optimal ET sequence|legend=1| 5, 7 }}
[[Badness]] (Sintel): 0.317


== Boethian ==
=== Sematology ===
[[Subgroup]]: 2.3.19
This temperament tempers out [[4107/4096]] and thus equates a stack of two [[37/32]]'s with [[4/3]].


[[Comma list]]: [[513/512]]
[[Subgroup]]: 2.3.37


{{Mapping|legend=1| 1 0 9 | 0 1 - 3 }}
[[Comma list]]: 4107/4096


: mapping generators: ~2, ~3
{{Mapping|legend=2| 1 0 6 | 0 2 -1 }}
: mapping generators: ~2, ~64/37


[[Optimal tuning]] (CTE): ~2 = 1\1, ~3/2 = 701.3288
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2184{{c}}, ~64/37 = 950.9546{{c}}
: [[error map]]: {{val| +0.218 -0.046 -0.988 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/37 = 950.8250{{c}}
: error map: {{val| 0.000 -0.305 -2.169 }}


{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296, 1557bhhhh, 1853bhhhh }}
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53, 77, 130, 443l, 573ll, 703ll, 1536blllll }}


[[Badness]]: 0.000374
[[Badness]] (Sintel): 0.0690


== Porpoise ==
=== Reversed mavila ===
[[Subgroup]]: 2.3.29
[[Subgroup]]: 2.3.37


[[Comma list]]: 24576/24389
[[Comma list]]: 81/74


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


[[CTE tuning|CTE generator]]: ~32/29 = 166.067
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.9908{{c}}, ~3/2 = 676.4865{{c}}
: [[error map]]: {{val| +1.991 -23.478 +60.575 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 676.7603{{c}}
: error map: {{val| 0.000 -25.195 +55.697 }}


{{Optimal ET sequence|legend=1| 7, 22, 29, 94, 123, 152j, 275jj, 427jjj }}
{{Optimal ET sequence|legend=1| 2, 5l, 7l, 9, 16l }}


== Paralimmic ==
[[Badness]] (Sintel): 0.623
[[Subgroup]]: 2.3.11


[[Comma list]]: [[4096/3993]]
=== Reversed meantone ===
{{Main| Reversed meantone }}


{{Mapping|legend=2|1 0 4|0 3 -1}}
[[Subgroup]]: 2.3.41


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[16/11]] = 634.320
[[Comma list]]: 82/81


[[Support]]ing [[ET]]s: {{EDOs|17, 15, 19, 32, 13, 36, 21, 11b, 49, 53e, 23, 66e, 70e, 47b}}
{{Mapping|legend=2| 1 0 -1 | 0 1 4 }}
 
: mapping generators: ~2, ~3
== Threedic ==
[[Subgroup]]: 2.3.13
 
[[Comma list]]: [[2197/2187]]


{{Mapping|legend=2|1 0 0|0 3 7}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6907{{c}}, ~3/2 = 705.3096{{c}}
: [[error map]]: {{val| -0.309 +3.045 -8.752 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.2699{{c}}
: error map: {{val| 0.000 +3.315 -7.983 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/9]] = 634.173
{{Optimal ET sequence|legend=1| 5, 12, 17, 97m, 114m, 131m }}


[[Support]]ing [[ET]]s: {{EDOs|17, 53, 36, 19, 15, 21f, 13f, 32f, 11bf, 49f, 23f, 9bff, 7bff, 40f}}
[[Badness]] (Sintel): 0.0841


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 11:44, 29 May 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 5.

Temperaments with a 2.3.7 gene

Archy

See Archytas clan #Archy.

Suhajira

See Rastmic clan #Suhajira.

Flutterpyth

Restricted to 2.3.7.11, this temperament is a no-5 restriction of 11-limit ultrapyth. This temperament was created to yield blackdye tunings where aberrisma-altered 3-limit thirds become tempered 13/11~19/16 and 14/11.

Subgroup: 2.3.7.11.13

Comma list: 64/63, 364/363, 1078/1053

Mapping: [⟨1 0 6 21 34], ⟨0 1 -2 -11 -19]]

Optimal tunings:

  • WE: ~2 = 1196.9412 ¢, ~3/2 = 711.0195 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.0039 ¢

Optimal ET sequence: 32f, 69bf, 101beff

Badness (Sintel): 1.52

2.3.7.11.13.19 subgroup

Subgroup: 2.3.7.11.13.19

Comma list: 64/63, 209/208, 343/342, 364/363

Mapping: [⟨1 0 6 21 34 17], ⟨0 1 -2 -11 -19 -8]]

Optimal tunings:

  • WE: ~2 = 1197.4072 ¢, ~3/2 = 711.2733 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 712.9612 ¢

Optimal ET sequence: 32f, 69bf

Badness (Sintel): 1.28

Semaphore

See Semaphoresmic clan #Semaphore.

Slendric

See Gamelismic clan #Slendric.

Slendroschismic

See 5th-octave temperaments #Slendroschismic.

Navy

This temperament is the common restriction of tsaharuk and quanic.

Subgroup: 2.3.7

Comma list: 282429536481/281974669312

Subgroup-val mapping: [⟨1 1 0], ⟨0 5 24]]

mapping generators: ~2, ~243/224

Optimal tunings:

  • WE: ~2 = 1200.0302 ¢, ~243/224 = 140.3698 ¢
error map: ⟨+0.030 -0.076 +0.050]
  • CWE: ~2 = 1200.0000 ¢, ~243/224 = 140.3681 ¢
error map: ⟨0.000 -0.115 +0.008]

Optimal ET sequence: 17, 60, 77, 94, 171, 265, 436, 2351, 2787, 3223, 3659, 4095, 7754b

Badness (Sintel): 0.670

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 1331/1323, 19712/19683

Subgroup-val mapping: [⟨1 1 0 1], ⟨0 5 24 21]]

Optimal tunings:

  • WE: ~2 = 1200.1038 ¢, ~88/81 = 140.4190 ¢
  • CWE: ~2 = 1200.0000 ¢, ~88/81 = 140.4133 ¢

Optimal ET sequence: 17, 60e, 77, 94

Badness (Sintel): 0.887

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 352/351, 729/728, 1331/1323

Subgroup-val mapping: [⟨1 1 0 1 3], ⟨0 5 24 21 6]]

Optimal tunings:

  • WE: ~2 = 1199.8640 ¢, ~13/12 = 140.4206 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.4292 ¢

Optimal ET sequence: 17, 60e, 77, 94

Badness (Sintel): 0.520

Lee

Subgroup: 2.3.7

Comma list: 177147/175616

Subgroup-val mapping: [⟨1 0 -3], ⟨0 3 11]]

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

mapping generators: ~2, ~81/56

Optimal tunings:

  • WE: ~2 = 1200.2962 ¢, ~81/56 = 633.6812 ¢
error map: ⟨+0.296 -0.912 +0.778]
  • CWE: ~2 = 1200.0000 ¢, ~81/56 = 633.5658 ¢
error map: ⟨0.000 -1.258 +0.398]

Optimal ET sequence: 17, 36, 89, 125, 161, 358, 519b

Badness (Sintel): 0.741

Buzzard

See Buzzardsmic clan #Buzzard.

Hemif

Hemif is the no-5 restriction of hemififths, and the add-7 extension of namo.

Subgroup: 2.3.7

Comma list: 1605632/1594323

Subgroup-val mapping: [⟨1 1 -1], ⟨0 2 13]]

Gencom mapping: [⟨1 1 0 -1], ⟨0 2 0 13]]

mapping generators: ~2, ~2187/1792

Optimal tunings:

  • WE: ~2 = 1199.7303 ¢, ~2187/1792 = 351.4056 ¢
error map: ⟨-0.270 +0.586 -0.284]
  • CWE: ~2 = 1200.0000 ¢, ~2187/1792 = 351.4569 ¢
error map: ⟨0.000 +0.959 +0.114]

Optimal ET sequence: 17, 41, 58, 99, 239, 338, 437, 775b, 1212bb

Badness (Sintel): 0.901

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 896/891

Subgroup-val mapping: [⟨1 1 -1 2], ⟨0 2 13 5]]

Gencom mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5]]

Optimal tunings:

  • WE: ~2 = 1199.2633 ¢, ~11/9 = 351.3189 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.4593 ¢

Optimal ET sequence: 17, 41, 58, 99e

Badness (Sintel): 0.409

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 144/143, 243/242, 364/363

Sval mapping: [⟨1 1 -1 2 4], ⟨0 2 13 5 -1]]

Gencom mapping: [⟨1 1 0 -1 2 4], ⟨0 2 0 13 5 -1]]

Optimal tunings:

  • WE: ~2 = 1198.7603 ¢, ~11/9 = 351.3275 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.6042 ¢

Optimal ET sequence: 17, 41, 58, 331deeeffff

Badness (Sintel): 0.358

Heartful
See also: Heartland

Related temperaments: bunya.

Subgroup: 2.3.7.11.19

Comma list: 243/242, 896/891, 1083/1078

Subgroup-val mapping: [⟨1 1 -1 2 0], ⟨0 4 26 10 29]]

mapping generators: ~2, ~21/19

Optimal tunings:

  • WE: ~2 = 1199.2636 ¢, ~21/19 = 175.6963 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.7665 ¢

Optimal ET sequence: 34dh, 41, 116e, 157e

Badness (Sintel): 0.984

Hearts

See also: Heartland

This temperament is the common restriction of monkey and sesquiquartififths.

Subgroup: 2.3.7

Comma list: 34451725707/34359738368

Subgroup-val mapping: [⟨1 1 5], ⟨0 4 -15]]

mapping generators: ~2, ~567/512

Optimal tunings:

  • WE: ~2 = 1200.0845 ¢, ~567/512 = 175.4449 ¢
error map: ⟨+0.085 -0.091 -0.076]
  • CWE: ~2 = 1200.0000 ¢, ~567/512 = 175.4307 ¢
error map: ⟨0.000 -0.232 -0.286]

Optimal ET sequence: 7, 27d, 34, 41, 89, 130, 171, 643, 814, 985, 1156

Badness (Sintel): 0.959

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 65536/65219

Subgroup-val mapping: [⟨1 1 5 2], ⟨0 4 -15 10]]

Optimal tunings:

  • WE: ~2 = 1199.8467 ¢, ~256/231 = 175.3468 ¢
  • CWE: ~2 = 1200.0000 ¢, ~256/231 = 175.3691 ¢

Optimal ET sequence: 7, 34, 41, 89, 130, 349e, 479e

Badness (Sintel): 0.801

2.3.7.11.19

Subgroup: 2.3.7.11.19

Comma list: 243/242, 513/512, 1083/1078

Subgroup-val mapping: [⟨1 1 5 2 6], ⟨0 4 -15 10 -12]]

Optimal tunings:

  • WE: ~2 = 1199.9531 ¢, ~21/19 = 175.3344 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.3417 ¢

Optimal ET sequence: 7, 34, 41, 89, 130, 219

Badness (Sintel): 0.529

Magi

This temperament is the no-5 restriction of magic, tempering out the septimagic comma.

Subgroup: 2.3.7

Comma list: 537824/531441

Subgroup-val mapping: [⟨1 0 -1], ⟨0 5 12]]

mapping generators: ~2, ~243/196

Optimal tunings:

  • WE: ~2 = 1199.8224 ¢, ~243/196 = 380.6043 ¢
error map: ⟨-0.178 +1.066 -1.397]
  • CWE: ~2 = 1200.0000 ¢, ~243/196 = 380.6378 ¢
error map: ⟨0.000 +1.234 -1.173]

Optimal ET sequence: 19, 22, 41, 104, 145, 186, 331

Badness (Sintel): 1.30

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 896/891, 26411/26244

Subgroup-val mapping: [⟨1 0 -1 6], ⟨0 5 12 -8]]

Optimal tunings:

  • WE: ~2 = 1199.4843 ¢, ~96/77 = 380.6040 ¢
  • CWE: ~2 = 1200.0000 ¢, ~96/77 = 380.7490 ¢

Optimal ET sequence: 19, 22, 41, 63, 104

Badness (Sintel): 0.661

Caspar

Subgroup: 2.3.7.11.13

Comma list: 144/143, 343/338, 729/728

Subgroup-val mapping: [⟨1 0 -1 6 -2], ⟨0 5 12 -8 18]]

Optimal tunings:

  • WE: ~2 = 1199.3353 ¢, ~26/21 = 380.3206 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/21 = 380.5041 ¢

Optimal ET sequence: 19, 22f, 41

Badness (Sintel): 1.09

Twenothology

Subgroup: 2.3.7.11.13.29

Comma list: 144/143, 232/231, 343/338, 729/728

Subgroup-val mapping: [⟨1 0 -1 6 -2 2], ⟨0 5 12 -8 18 9]]

Optimal tunings:

  • WE: ~2 = 1199.6175 ¢, ~26/21 = 380.4049 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/21 = 380.5103 ¢

Optimal ET sequence: 19, 22f, 41

Badness (Sintel): 0.964

Melchior

Subgroup: 2.3.7.11.13

Comma list: 352/351, 364/363, 26411/26244

Subgroup-val mapping: [⟨1 0 -1 6 11], ⟨0 5 12 -8 -23]]

Optimal tunings:

  • WE: ~2 = 1199.4887 ¢, ~96/77 = 380.6034 ¢
  • CWE: ~2 = 1200.0000 ¢, ~96/77 = 380.7669 ¢

Optimal ET sequence: 19f, 22, 41, 63, 104

Badness (Sintel): 0.710

Balthazar

Subgroup: 2.3.7.11.13

Comma list: 169/168, 896/891, 26411/26244

Subgroup-val mapping: [⟨1 0 -1 6 1], ⟨0 10 24 -16 17]]

mapping generators: ~2, ~143/128

Optimal tunings:

  • WE: ~2 = 1199.7322 ¢, ~143/128 = 190.3647 ¢
  • CWE: ~2 = 1200.0000 ¢, ~143/128 = 190.4016 ¢

Optimal ET sequence: 19, 44, 63, 145f

Badness (Sintel): 1.82

Hogwarts

Subgroup: 2.3.7.29

Comma list: 784/783, 5887/5832

Subgroup-val mapping: [⟨1 0 -1 2], ⟨0 5 12 9]]

Optimal tunings:

  • WE: ~2 = 1200.1518 ¢, ~36/29 = 380.6661 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/29 = 380.6375 ¢

Optimal ET sequence: 19, 22, 41, 145, 186j, 227j

Badness (Sintel): 0.424

Skwares

Main article: Squares

Skwares is the no-5 restriction of squares.

Subgroup: 2.3.7

Comma list: 19683/19208

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

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

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1200.3703 ¢, ~14/9 = 774.8736 ¢
error map: ⟨+0.370 -2.831 +3.925]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 774.6974 ¢
error map: ⟨0.000 -3.166 +3.450]

Optimal ET sequence: 14, 17, 31, 48, 79

Badness (Sintel): 1.55

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 99/98, 243/242

Subgroup-val mapping: [⟨1 -1 -3 -3], ⟨0 4 9 10]]

Gencom mapping: [⟨1 -1 0 -3 -3], ⟨0 4 0 9 10]]

Optimal tunings:

  • WE: ~2 = 1200.3726 ¢, ~14/9 = 774.9970 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 774.8197 ¢

Optimal ET sequence: 14, 17, 31, 48, 79, 127

Badness (Sintel): 0.405

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 78/77, 99/98, 243/242

Subgroup-val mapping: [⟨1 -1 -3 -3 -6], ⟨0 4 9 10 15]]

Gencom mapping: [⟨1 -1 0 -3 -3 -6], ⟨0 4 0 9 10 15]]

Optimal tunings:

  • WE: ~2 = 1199.3264 ¢, ~14/9 = 775.1081 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 775.4463 ¢

Optimal ET sequence: 14f, 17, 48f

Badness (Sintel): 0.587

Skwairs

Subgroup: 2.3.7.11.13

Comma list: 99/98, 144/143, 243/242

Subgroup-val mapping: [⟨1 -1 -3 -3 5], ⟨0 4 9 10 -2]]

Gencom mapping: [⟨1 -1 0 -3 -3 5], ⟨0 4 0 9 10 -2]]

Optimal tunings:

  • WE: ~2 = 1198.8812 ¢, ~14/9 = 775.5748 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 775.1930 ¢

Optimal ET sequence: 14, 17, 31, 48, 65d, 113df

Badness (Sintel): 0.538

Byhearted

This temperament is the restriction of weasel to the 2.3.7.11.19 subgroup.

Subgroup: 2.3.7.11.19

Comma list: 99/98, 243/242, 363/361

Subgroup-val mapping: [⟨2 2 3 4 5], ⟨0 4 9 10 12]]

mapping generators: ~209/147, ~21/19

Optimal tunings:

  • WE: ~2 = 600.1836 ¢, ~21/19 = 174.7882 ¢
  • CWE: ~2 = 600.0000 ¢, ~21/19 = 174.7975 ¢

Optimal ET sequence: 14, 34dh, 48, 110e

Badness (Sintel): 0.893

Harrison

Harrison is the no-5 restriction of meantone. As such, there is little reason to consider this temperament in practice – since intervals of 5 in meantone are as accurate as intervals of 7, only simpler, they are always available by the time intervals of 7 are generated.

Subgroup: 2.3.7

Comma list: 59049/57344

Subgroup-val mapping: [⟨1 0 -13], ⟨0 1 10]]

Gencom mapping: [⟨1 0 0 -13], ⟨0 1 0 10]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1201.5353 ¢, ~3/2 = 697.4352 ¢
error map: ⟨+1.535 -2.984 +0.920]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.7289 ¢
error map: ⟨0.000 -5.226 -1.537]

Optimal ET sequence: 12, 19, 31, 112b, 143b, 174b

Badness (Sintel): 2.35

Bleu

Bleu can be described as the 8d & 9 temperament in the no-5 13-limit, and is the common restriction of progression and jerome.

Subgroup: 2.3.7

Comma list: 17496/16807

Subgroup-val mapping: [⟨1 1 2], ⟨0 5 7]]

Gencom mapping: [⟨1 1 0 2], ⟨0 5 0 7]]

mapping generators: ~2, ~54/49

Optimal tunings:

  • WE: ~2 = 1199.3538 ¢, ~54/49 = 139.848 ¢
error map: ⟨-0.646 -3.736 +8.293]
  • CWE: ~2 = 1200.0000 ¢, ~54/49 = 139.848 ¢
error map: ⟨0.000 -3.270 +9.333]

Optimal ET sequence: 8d, 9, 17, 43, 60d, 103d

Badness (Sintel): 2.48

2.3.7.11 subgroup

Subgroup: 2.3.7.11

Comma list: 99/98, 864/847

Subgroup-val mapping: [⟨1 1 2 3], ⟨0 5 7 4]]

Gencom mapping: [⟨1 1 0 2 3], ⟨0 5 0 7 4]]

Optimal tunings:

  • WE: ~2 = 1198.6613 ¢, ~12/11 = 139.8489 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 139.7839 ¢

Optimal ET sequence: 8d, 9, 17, 43, 60d

Badness (Sintel): 0.624

2.3.7.11.13 subgroup

Subgroup: 2.3.7.11.13

Comma list: 78/77, 99/98, 144/143

Subgroup-val mapping: [⟨1 1 2 3 3], ⟨0 5 7 4 6]]

Gencom mapping: [⟨1 1 0 2 3 3], ⟨0 5 0 7 4 6]]

Optimal tunings:

  • WE: ~2 = 1198.9768 ¢, ~13/12 = 139.8704 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.8166 ¢

Optimal ET sequence: 8d, 9, 17, 43, 60d

Badness (Sintel): 0.400

Music

Doublehearted

See also: Heartland

This temperament is the no-5 restriction of octacot.

Subgroup: 2.3.7

Comma list: 5764801/5668704

Subgroup-val mapping: [⟨1 1 2], ⟨0 8 11]]

mapping generators: ~2, ~343/342

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~343/324 = 87.8431 ¢
error map: ⟨+0.174 +0.964 -2.204]
  • CWE: ~2 = 1200.0000 ¢, ~343/324 = 87.8492 ¢
error map: ⟨0.000 +0.838 -2.485]

Optimal ET sequence: 14, 27, 41

Badness (Sintel): 2.62

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 2401/2376

Subgroup-val mapping: [⟨1 1 2 2], ⟨0 8 11 20]]

Optimal tunings:

  • WE: ~2 = 1200.4071 ¢, ~22/21 = 87.6809 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/21 = 87.6902 ¢

Optimal ET sequence: 14, 27e, 41, 96d, 137d, 178d

Badness (Sintel): 0.815

2.3.7.11.19

Subgroup: 2.3.7.11.19

Comma list: 133/132, 243/242, 343/342

Subgroup-val mapping: [⟨1 1 2 2 3], ⟨0 8 11 20 17]]

Optimal tunings:

  • WE: ~2 = 1200.6100 ¢, ~19/18 = 87.7129 ¢
  • CWE: ~2 = 1200.0000 ¢, ~19/18 = 87.7285 ¢

Optimal ET sequence: 14, 27e, 41, 137dh

Badness (Sintel): 0.560

Purpleheart

Main article: Whitewood

Subgroup: 2.3.7

Comma list: 2187/2048

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

mapping generators: ~9/8, ~7

Optimal tunings:

  • WE: ~9/8 = 172.1541 ¢, ~7/4 = 958.5433 ¢ (~64/63 = 74.3805 ¢)
error map: ⟨+5.079 -8.260 -0.124]
  • CWE: ~9/8 = 171.4286 ¢, ~7/4 = 959.2372 ¢ (~64/63 = 69.3373 ¢)
error map: ⟨0.000 -16.241 -9.589]

Optimal ET sequence: 7, 14, 35, 49bd

Badness (Sintel): 3.00

Chrysanthemum

This microtemperament extends amaranthine to prime 3 by tempering out 43923/43904, the chrysia, to find 3 at 29 steps down on the chain of nearly pure 7/4's.

Subgroup: 2.3.7

Comma list: [83 -1 -29⟩

Subgroup-val mapping: [⟨1 -4 3], ⟨0 29 -1]]

mapping generators: ~2, ~8/7

Optimal tunings:

  • WE: ~2 = 1199.9871 ¢, ~8/7 = 231.1001 ¢
error map: ⟨-0.013 +0.000 +0.035]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 231.1024 ¢
error map: ⟨0.000 +0.014 +0.072]

Optimal ET sequence: 26, 83, 109, 135, 566, 701, 836, 971, 1106, 2077, 5260, 7337, 9414d

Badness (Sintel): 3.06

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 43923/43904, 5767168/5764801

Subgroup-val mapping: [⟨1 -4 3 5], ⟨0 29 -1 -8]]

Optimal tunings:

  • WE: ~2 = 1200.0050 ¢, ~8/7 = 231.1024 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 231.1015 ¢

Optimal ET sequence: 26, 83, 109, 135, 566, 701, 836, 971, 1807, 2778, 4585

Badness (Sintel): 0.324

Leapfrog

See also: Gentle region

Leapfrog is generated by a perfect fifth and the interval class of 7 is found at +15 steps, as a double-augmented fifth (C–G𝄪). For this to work, it entails a fifth about 2–3 cents sharp of just; as a result the major third lands comfortably at a near-just 14/11 so that it can be extended to the 2.3.7.11 subgroup via tempering out 896/891. The minor third can then be identified with 13/11, tempering out 352/351 and 364/363, which implies 169/168 is tempered out as well in this case. Leapfrog is most naturally treated as such, in which it is very efficient.

A notable patent-val edo tuning not appearing in the optimal ET sequence is 80edo, which is approximately the just-13's tuning (as 10edo is used as a consistent circle of ~16/13's therein), with 13/8 still tuned slightly flat so qualifying a reasonable tuning for the 2.3.13 subgroup (as evidenced by appearing in the sequence for tetris).

Strong extensions for prime 5 include leapday (29 & 46), leapweek (46 & 63), and leapmonth (63 & 80), all of which are more complex than vanilla leapfrog. A low-complexity low-accuracy extension is given by supermean (5de & 17c), where it is merged with meantone. Srutal (46 & 80), usually considered as a strong extension of diaschismic, is a weak extension of leapfrog, and yet another weak extension is immune (29 & 63), which is in turn a strong extension of 5-limit immunity.

Subgroup: 2.3.7

Comma list: 14680064/14348907

Subgroup-val mapping: [⟨1 0 -21], ⟨0 1 15]]

Gencom mapping: [⟨1 0 0 -21], ⟨0 1 0 15]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.1807 ¢, ~3/2 = 704.2400 ¢
error map: ⟨-0.819 +1.466 -0.311]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.6600 ¢
error map: ⟨0.000 +2.705 +1.074]

Optimal ET sequence: 17, 46, 63, 235b, 298b, 361bd, 424bd, 487bbd

Badness (Sintel): 4.33

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 896/891, 1331/1323

Subgroup-val mapping: [⟨1 0 -21 -14], ⟨0 1 15 11]]

Gencom mapping: [⟨1 0 0 -21 -14], ⟨0 1 0 15 11]]

Optimal tunings:

  • WE: ~2 = 1199.2683 ¢, ~3/2 = 704.3230 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.6926 ¢

Optimal ET sequence: 17, 46, 63

Badness (Sintel): 0.629

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 169/168, 352/351, 364/363

Subgroup-val mapping: [⟨1 0 -21 -14 -9], ⟨0 1 15 11 8]]

Gencom mapping: [⟨1 0 0 -21 -14 -9], ⟨0 1 0 15 11 8]]

Optimal tunings:

  • WE: ~2 = 1199.5654 ¢, ~3/2 = 704.4898 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7084 ¢

Optimal ET sequence: 17, 46, 63

Badness (Sintel): 0.436

Skidoo

Subgroup: 2.3.7.11.13.23

Comma list: 169/168, 208/207, 352/351, 364/363

Subgroup-val mapping: [⟨1 0 -21 -14 -9 -5], ⟨0 1 15 11 8 6]]

Gencom mapping: [⟨1 0 0 -21 -14 -9 0 0 -5], ⟨0 1 0 15 11 8 0 0 6]]

Optimal tunings:

  • WE: ~2 = 1199.6639 ¢, ~3/2 = 704.5315 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7021 ¢

Optimal ET sequence: 17, 46, 63

Badness (Sintel): 0.356

2.3.7.11.13.23.29

Subgroup: 2.3.7.11.13.23.29

Comma list: 169/168, 208/207, 232/231, 352/351, 364/363

Subgroup-val mapping: [⟨1 0 -21 -14 -9 -5 -38], ⟨0 1 15 11 8 6 27]]

Gencom mapping: [⟨1 0 0 -21 -14 -9 -5 0 0 -38], ⟨0 1 0 15 11 8 0 0 6 27]]

Optimal tunings:

  • WE: ~2 = 1199.5755 ¢, ~3/2 = 704.5533 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7750 ¢

Optimal ET sequence: 17, 46, 63

Badness (Sintel): 0.441

Music

Superslendric

In superslendric, eight 8/7's are equated to 3/1. This relates it to 8edt.

Subgroup: 2.3.7

Comma list: 17294403/16777216

Subgroup-val mapping: [⟨1 0 3], ⟨0 8 -1]]

mapping generators: ~2, ~8/7

Optimal tunings:

  • WE: ~2 = 1201.1628 ¢, ~8/7 = 237.7287 ¢
error map: ⟨+1.163 -0.125 -3.066]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 237.5664 ¢
error map: ⟨0.000 -1.424 -6.392]

Optimal ET sequence: 5, …, 66b, 71b, 76, 81, 86, 91, 96d

Badness (Sintel): 6.15

Hectosaros leap week

This temperament may be described as the 320 & 1803 temperament, in the 2.3.7.13.17.19 on the basis of the fact that 1803 tropical years make up almost exactly 100 saros cycles.

Subgroup: 2.3.7

Comma list: [-50 -746 439⟩

Subgroup-val mapping: [⟨1 -126 -214], ⟨0 439 746]]

mapping generators: ~2, ~[-16 -243 143⟩

Optimal tunings:

  • WE: ~2 = 1200.0010 ¢, ~[-16 -243 143⟩ = 348.7520 ¢
error map: ⟨+0.001 +0.036 -0.067]
  • CWE: ~2 = 1200.0000 ¢, ~[-16 -243 143⟩ = 348.7517 ¢
error map: ⟨0.000 +0.035 -0.068]

Optimal ET sequence: 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689

Badness (Sintel): 17.7 × 103

2.3.7.13 subgroup

Subgroup: 2.3.7.13

Comma list: [-42 -2 -5 16⟩, [10 -46 29 -5⟩

Subgroup-val mapping: [⟨1 -126 -214 -80], ⟨0 439 746 288]]

Optimal tunings:

  • WE: ~2 = 1200.0058 ¢, ~[18 -9 8 -7⟩ = 348.7534 ¢
  • CWE: ~2 = 1200.0000 ¢, ~[18 -9 8 -7⟩ = 348.8517 ¢

Optimal ET sequence: 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689, 11255d

Badness (Sintel): 53.2

2.3.7.13.17 subgroup

Subgroup: 2.3.7.13.17

Comma list: 39337984/39328497, [0 -14 7 4 -3⟩, [-18 -24 14 -1 5⟩

Subgroup-val mapping: [⟨1 -126 -214 -80 -18], ⟨0 439 746 288 76]]

Optimal tunings:

  • WE: ~2 = 1200.9870 ¢, ~3757/3072 = 348.7480 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3757/3072 = 348.7517 ¢

Optimal ET sequence: 320, 1483bd, 1803, 2123

Badness (Sintel): 13.4

2.3.7.13.17.19 subgroup

Subgroup: 2.3.7.13.17.19

Comma list: 10081799/10077696, 10754912/10744731, 39337984/39328497, 480024727/480020256

Subgroup-val mapping: [⟨1 -126 -214 -80 -18 -171], ⟨0 439 746 288 76 603]]

Optimal tunings:

  • WE: ~2 = 1200.9961 ¢, ~3757/3072 = 348.7506 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3757/3072 = 348.7517 ¢

Optimal ET sequence: 320, 1483bd, 1803, 2123

Badness (Sintel): 7.46

Heartland (rank 3)

Main article: Heartland

Heartland, with a generator of ~21/19, is named for its tempering of the heartlandisma, 3971/3969. Aside from the heartlandisma, the heartland temperament tempers out 243/242 (rastma) and 1083/1078 (bihendrixma), and slices the fifth in four (the number of chambers of the heart).

Subgroup: 2.3.7.11.19

Comma list: 243/242, 1083/1078

Subgroup-val mapping: [⟨1 1 0 2 1], ⟨0 4 0 10 3], ⟨0 0 1 0 1]]

mapping generators: ~2, ~21/19, ~7

Optimal tunings:

  • WE: ~2 = 1200.0983 ¢, ~21/19 = 175.2856 ¢, ~7/4 = 969.4578 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.2894 ¢, ~7/4 = 969.5203 ¢

Optimal ET sequence: 14, 27e, 34dh, 41, 89, 130, 219

Badness (Sintel): 0.615

Temperaments with a 2.3.11 gene

Neutral

See Rastmic clan #Neutral.

Io

Io is a very low-complexity temperament which tempers out the undecimal quartertone 33/32. This equates very different intervals (for example, the generator itself represents both 3/2 and 16/11), and as such some consider it to be an exotemperament. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of flattone temperament.

The name io was coined by CompactStar in 2024 based on the color name ilo, prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.

Subgroup: 2.3.11

Comma list: 33/32

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

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1206.6866 ¢, ~3/2 = 691.7837 ¢
error map: ⟨+6.687 -3.485 -16.355]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 689.2066 ¢
error map: ⟨0.000 -12.748 -40.525]

Optimal ET sequence: 2, 5, 7, 12e, 40ee, 47eee, 54beee, 61beeee

Badness (Sintel): 0.185

Alphaxenean

Alphaxenean tempers out the Alpharabian comma and equates a stack of four undecimal quartertones with the Pythagorean whole tone. It also divides the octave into two.

Subgroup: 2.3.11

Comma list: 131769/131072

Subgroup-val mapping: [⟨2 1 8], ⟨0 2 -1]]

mapping generators: ~363/256, ~16/11

Optimal tunings:

  • WE: ~363/256 = 600.1590 ¢, ~16/11 = 650.8508 ¢
error map: ⟨+0.318 -0.094 -0.897]
  • CWE: ~363/256 = 600.0000 ¢, ~16/11 = 650.7321 ¢
error map: ⟨0.000 -0.491 -2.050]

Optimal ET sequence: 22, 24, 94, 118, 142, 450e, 592e, 1326beeee

Badness (Sintel): 0.395

Infraug

Subgroup: 2.3.11

Comma list: 729/704

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

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1202.5969 ¢ ~3/2 = 692.7443 ¢
error map: ⟨+2.597 -6.614 +5.148]
  • CWE: ~2 = 1200.0000 ¢ ~3/2 = 692.0116 ¢
error map: ⟨0.000 -9.943 +0.752]

Optimal ET sequence: 7, 19, 26, 33, 59b, 92b

Badness (Sintel): 0.734

2.3.11.13

Subgroup: 2.3.11.13

Comma list: 144/143, 729/704

Subgroup-val mapping: [⟨1 0 -6 10], ⟨0 1 6 -4]]

Optimal tunings:

  • WE: ~2 = 1202.1934 ¢, ~3/2 = 692.8902 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 691.7585 ¢

Optimal ET sequence: 7, 19, 26, 59b

Badness (Sintel): 0.725

Aerophore

Subgroup: 2.3.11.19

Comma list: 363/361, 729/704

Subgroup-val mapping: [⟨1 0 -6 -6], ⟨0 2 12 13]]

mapping generators: ~2, ~19/11

Optimal tunings:

  • WE: ~2 = 1202.6380 ¢, ~19/11 = 947.4782 ¢
  • CWE: ~2 = 1200.0000 ¢, ~19/11 = 945.7779 ¢

Optimal ET sequence: 14, 19, 33

Badness (Sintel): 1.59

Paralimmal

Subgroup: 2.3.11

Comma list: 4096/3993

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

mapping generators: ~2, ~16/11

Optimal tunings:

  • WE: ~2 = 1197.9124 ¢, ~16/11 = 634.1269 ¢
error map: ⟨-2.088 +0.426 +6.205]
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 634.9546 ¢
error map: ⟨0.000 +2.909 +13.727]

Optimal ET sequence: 2, 11b, 13, 15, 17

Badness (Sintel): 0.984

Huxley

Main article: Huxley

Huxley, the 4 & 13 temperament in the 2.3.11.13 subgroup, extends lovecraft. Specifically it tunes the ~13/8 to exactly half of ~8/3.

Subgroup: 2.3.11.13

Comma list: 512/507, 1352/1331

Subgroup-val mapping: [⟨1 -3 5 6], ⟨0 6 -2 -3]]

mapping generators: ~2, ~22/13

Optimal tunings:

  • WE: ~2 = 1198.0036 ¢, ~22/13 = 916.0595 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/13 = 917.5184 ¢

Optimal ET sequence: 4, 13, 17

Badness (Sintel): 1.31

Glaishur

This temperament is the no-5 no-7 restriction of navy, as well as the add-11 extension of glacier.

Subgroup: 2.3.11

Comma list: 10554638336/10460353203

Subgroup-val mapping: [⟨1 1 0], ⟨0 5 21]]

mapping generators: ~2, ~88/81

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~88/81 = 140.537 ¢
error map: ⟨-0.150 +0.493 -0.559]
  • CWE: ~2 = 1200.0000 ¢, ~88/81 = 140.537 ¢
error map: ⟨0.000 +0.662 -0.326]

Optimal ET sequence: 17, 60e, 77, 94, 111

Badness (Sintel): 2.27

2.3.11.13

Subgroup: 2.3.11.13

Comma list: 352/351, 531674/531441

Subgroup-val mapping: [⟨1 1 0 3], ⟨0 5 21 6]]

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~13/12 = 140.537 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.537 ¢

Optimal ET sequence: 17, 60e, 77, 94, 111, 239f, 350f

Badness (Sintel): 0.415

Profanity

Profanity identifies 11/9 with 2\7.

Subgroup: 2.3.11

Comma list: 19487171/19131876

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

mapping generators: ~1458/1331, ~3

Optimal tunings:

  • WE: ~1458/1331 = 171.4369 ¢, ~3/2 = 702.9304 ¢
error map: ⟨+0.058 +1.033 -2.467]
  • CWE: ~1458/1331 = 171.4286 ¢, ~3/2 = 702.9442 ¢
error map: ⟨0.000 -0.989 -2.572]

Optimal ET sequence: 7, … 49, 56, 63, 70

Badness (Sintel): 3.03

Temperaments with a 2.3.13 gene

Threedic

Subgroup: 2.3.13

Comma list: 2197/2187

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

mapping generators: ~2, ~13/9

Optimal tunings:

  • WE: ~2 = 1200.0000, ~13/9 = 634.1729 ¢
error map: ⟨-0.000 +0.563 -1.318]
  • CWE: ~2 = 1200.0000, ~13/9 = 634.1729 ¢
error map: ⟨0.000 +0.564 -1.318]

Optimal ET sequence: 15, 17, 36, 53, 70, 123, 193, 316, 755f

Badness (Sintel): 0.160

Ultraflat

Ultraflat is a diatonic-based exotemperament that makes 27/26 vanish, so 13/8 is a major sixth.

Subgroup: 2.3.13

Comma list: 27/26

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

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1201.6561 ¢, ~3/2 = 686.9485 ¢
error map: ⟨+1.656 -13.350 +23.630]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 687.1143 ¢
error map: ⟨0.000 -14.841 +20.815]

Optimal ET sequence: 2, 5, 7

Badness (Sintel): 0.200

Superflat

Superflat is a less inaccurate cousin of ultraflat, with less flat fifths. It tempers out 1053/1024, so 13/8 is a minor sixth, and 16/13 is a major third. Superflat and ultraflat intersect in 7edo, where major sixths and minor sixths are not distinguished.

The more accurate tunings for this temperament are generated by a fifth at least as flat as those of flattone, although often even flatter (such as 40edo's fifth). Superflat can be viewed as a 2.3.13 subgroup analogue of meantone and archy. Superflat diatonic scales have a character somewhere between neutral third scales (or mosh) and meantone diatonic scales.

Subgroup: 2.3.13

Comma list: 1053/1024

Subgroup-val mapping: [⟨1 0 10], ⟨0 1 -4]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1203.1291 ¢ ~3/2 = 695.6489 ¢
error map: ⟨+3.129 -3.177 -4.349]
  • CWE: ~2 = 1200.0000 ¢ ~3/2 = 693.6081 ¢
error map: ⟨0.000 -8.347 +14.960]

Optimal ET sequence: 5f, 7, 12, 19, 45f, 64f, 147bfff

Badness (Sintel): 0.610

Shoal

The 2.3.13.23-subgroup microtemperament is remarkable for containing not one but two superparticular intervals as small as 3888/3887 and 12168/12167. Tempering out both of them gives us this rank-2 temperament where a sharp whole tone of 26/23 is the generator, two of which stack to a 23/18 supermajor third, and eight of which stack to a 8/3 perfect eleventh. 17edo is a trivial tuning where 26/23 is equated to 9/8, tempering out the comma 208/207. More accurate tunings of shoal create a 17-note mos scale, serving as a circulating temperament of 17edo, where 208/207 is the chroma between large and small steps.

Subgroup: 2.3.13

Comma list: 816293376/815730721

Subgroup-val mapping: [⟨1 -5 -7], ⟨0 8 13]]

mapping generators: ~2, ~3888/2197

Optimal tunings:

  • WE: ~2 = 1199.9922 ¢, ~3888/2197 = 987.7360 ¢
error map: ⟨-0.008 -0.028 +0.095]
  • CWE: ~2 = 1200.0000 ¢, ~3888/2197 = 987.7415 ¢
error map: ⟨0.000 -0.023 +0.112]

Optimal ET sequence: 17, 79, 96, 113, 130, 147, 424, 571, 1289, 10883ff, 12172ff

Badness (Sintel): 0.135

2.3.13.23

Subgroup: 2.3.13.23

Comma list: 3888/3887, 12168/12167

Subgroup-val mapping: [⟨1 -5 -7 -7], ⟨0 8 13 14]]

Optimal tunings:

  • WE: ~2 = 1199.9883 ¢, ~23/13 = 987.7325 ¢
  • CWE: ~2 = 1200.0000 ¢, ~23/13 = 987.7408 ¢

Optimal ET sequence: 17, 79, 96, 113, 130, 147, 424, 571, 1289, 1860, 3149

Badness (Sintel): 0.0213

Scales:

Music

Glacier

This 2.3.13-subgroup gene is not nearly as good as shoal, but it can extend extremely well to other no-5 subgroups. It is the common restriction of bleu and navy. It is very well represented in 26edo, where a nearly pure 13/12 can serve as the generator, but 94edo provides a much better tuning.

Subgroup: 2.3.13

Comma list: 373248/371293

Subgroup-val mapping: [⟨1 1 3], ⟨0 5 6]]

mapping generators: ~2, ~13/12

Optimal tunings:

  • WE: ~2 = 1199.8406 ¢, ~13/12 = 140.3695 ¢
error map: ⟨-0.159 -0.267 +1.211]
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.3605 ¢
error map: ⟨0.000 -0.153 +1.635]

Optimal ET sequence: 8, 9, 17, 60, 77, 94, 171, 265, 359f, 983ff

Badness (Sintel): 0.383

Temperaments with a higher-limit gene

Semitonic

Subgroup: 2.3.17

Comma list: 289/288

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

mapping generators: ~17/12, ~3

Optimal tunings:

  • WE: ~17/12 = 600.1471 ¢, ~3/2 = 701.9563 ¢ (~17/16 = 101.8091 ¢)
error map: ⟨+0.294 +0.295 -1.969]
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 702.0260 ¢ (~17/16 = 102.0260 ¢)
error map: ⟨0.000 +0.071 -2.929]

Optimal ET sequence: 10, 12, 58, 70, 82, 94, 106, 118, 224g

Badness (Sintel): 0.0454

Boethian

Boethian is a diatonic-based temperament that makes 513/512 vanish, so that the major third (C–E) is ~24/19 and the minor third (C–E♭) is ~19/16. As such, it functions as a 2.3.19-subgroup analogue of meantone, though the small size of the comma puts it at schismic level of accuracy. In particular, the equal temperaments in the tuning spectrum up to 1/2-comma (flattened) boethian temperament (very close to 12edo) are included in the schismic tuning spectrum in the 5-limit, so boethian intersects with schismic in the prime-5 infill extension thereof, called nestoria, which also tempers out 361/360, the difference between 19/18 and 20/19 or between 19/15 and 24/19.

Subgroup: 2.3.19

Comma list: 513/512

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

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1200.2498 ¢, ~3/2 = 701.4958 ¢
error map: ⟨+0.250 -0.209 -0.501]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.3445 ¢
error map: ⟨0.000 -0.610 -1.547]

Optimal ET sequence: 5, 7, 12, 41, 53, 65, 77, 219, 296, 1557bhhhh, 1853bhhhh

Badness (Sintel): 0.0294

Dog

Dog is based by 2L 5s or 7L 2s scale that makes 81/76 vanish, so 19/16 is a major third. It can be viewed as a 2.3.19-subgroup analogue of mavila.

Subgroup: 2.3.19

Comma list: 81/76

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

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1203.3813 ¢, ~3/2 = 680.5089 ¢
error map: ⟨+3.381 -18.065 +31.285]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 680.5856 ¢
error map: ⟨0.000 -21.369 +24.8295]

Optimal ET sequence: 2, 5h, 7, 16, 23

Badness (Sintel): 0.491

Lipsett

Lipsett is a pleasantly melodic little temperament with a highly usable 5-tone and 9-tone mos. It is audibly similar to semaphore temperament, so it could be thought of as semaphore but for the 23rd harmonic instead of the 7th. It is named for Arthur Lipsett, the director of the Canadian short film 21-87. Leia's prison cell in Star Wars is numbered 2187, as a nod to the influence the film had on George Lucas.

Subgroup: 2.3.23

Comma list: 2187/2116

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

mapping generators: ~2, ~46/27

Optimal tunings:

  • WE: ~2 = 1200.5339 ¢, ~46/27 = 948.5629 ¢
error map: ⟨+0.534 -4.829 +11.132]
  • CWE: ~2 = 1200.0000 ¢, ~46/27 = 948.3272 ¢
error map: ⟨0.000 -5.301 +10.016]

Optimal ET sequence: 5, 14, 19, 43, 62i, 81i

Badness (Sintel): 0.801

Porpoise

Subgroup: 2.3.29

Comma list: 24576/24389

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

mapping generators: ~2, ~32/29

Optimal tunings:

  • WE: ~2 = 1199.5519 ¢, ~32/29 = 165.7453 ¢
error map: ⟨-0.448 -0.087 +2.437]
  • CWE: ~2 = 1200.0000 ¢, ~32/29 = 165.9004 ¢
error map: ⟨0.000 +0.344 +4.522]

Optimal ET sequence: 7, 22, 29, 94, 123, 152j, 275jj, 427jjj

Badness (Sintel): 0.317

Sematology

This temperament tempers out 4107/4096 and thus equates a stack of two 37/32's with 4/3.

Subgroup: 2.3.37

Comma list: 4107/4096

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

mapping generators: ~2, ~64/37

Optimal tunings:

  • WE: ~2 = 1200.2184 ¢, ~64/37 = 950.9546 ¢
error map: ⟨+0.218 -0.046 -0.988]
  • CWE: ~2 = 1200.0000 ¢, ~64/37 = 950.8250 ¢
error map: ⟨0.000 -0.305 -2.169]

Optimal ET sequence: 5, 14, 19, 24, 53, 77, 130, 443l, 573ll, 703ll, 1536blllll

Badness (Sintel): 0.0690

Reversed mavila

Subgroup: 2.3.37

Comma list: 81/74

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

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1201.9908 ¢, ~3/2 = 676.4865 ¢
error map: ⟨+1.991 -23.478 +60.575]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 676.7603 ¢
error map: ⟨0.000 -25.195 +55.697]

Optimal ET sequence: 2, 5l, 7l, 9, 16l

Badness (Sintel): 0.623

Reversed meantone

Main article: Reversed meantone

Subgroup: 2.3.41

Comma list: 82/81

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

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.6907 ¢, ~3/2 = 705.3096 ¢
error map: ⟨-0.309 +3.045 -8.752]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 705.2699 ¢
error map: ⟨0.000 +3.315 -7.983]

Optimal ET sequence: 5, 12, 17, 97m, 114m, 131m

Badness (Sintel): 0.0841

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