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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 }}
[[Comma list]]: 1605632/1594323


[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents
{{Mapping|legend=2| 1 1 -1 | 0 2 13 }}


==== Supraphon ====
{{Mapping|legend=3| 1 1 0 -1 | 0 2 0 13 }}
Subgroup: 2.3.7.11.13
: mapping generators: ~2, ~2187/1792


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


[[Gencom]]: [2 3/2; 64/63 78/77 99/98]
{{Optimal ET sequence|legend=1| 17, 41, 58, 99, 239, 338, 437, 775b, 1212bb }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 7 9}}, {{val|0 1 0 -2 -6 -9}}]
[[Badness]] (Sintel): 0.901


[[Mapping|Sval mapping]]: [{{val|1 0 6 13 18}}, {{val|0 1 -2 -6 -9}}]
==== 2.3.7.11 ====
 
[[Tp tuning|POL2 generator]]: ~3/2 = 706.137
 
{{Optimal ET sequence|legend=1| 5f, 12f, 17, 22, 39d, 56d }}
 
[[Tp tuning #T2 tuning|RMS error]]: 2.095 cents
 
==== Flutterpyth ====
Subgroup: 2.3.7.11.13.19
 
Comma list: 64/63, 209/208, 343/342, 364/363
 
Mapping: {{mapping| 1 1 4 10 15 9 | 0 -1 -2 -11 -19 -8 }}
 
Optimal tuning (CTE): ~3/2 = 713.459
 
=== 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
: <small>''For the full 19-limit version of this temperament, see [[Tetracot family #Byhearted]].''</small>


Subgroup: 2.3.7.11.19
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


[[Comma list]]: 99/98, 243/242, 363/361
Comma list: 896/891, 26411/26244


[[Gencom]]: [209/147 21/19; 99/98 243/242 363/361]
Subgroup-val mapping: {{mapping| 1 0 -1 6 | 0 5 12 -8 }}


[[Mapping|Sval mapping]]: [{{val|2 2 3 4 5}}, {{val|0 4 9 10 12}}]
Optimal tunings:  
* WE: ~2 = 1199.4843{{c}}, ~96/77 = 380.6040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~96/77 = 380.7490{{c}}


[[Tp tuning|POL2 generator]]: ~21/19 = 174.735
{{Optimal ET sequence|legend=0| 19, 22, 41, 63, 104 }}


{{Optimal ET sequence|legend=1| 14, 34dh, 48, 110e, 158e }}
Badness (Sintel): 0.661


[[Tp tuning #T2 tuning|RMS error]]: 0.8727 cents
===== Caspar =====
Subgroup: 2.3.7.11.13


== Harrison ==
Comma list: 144/143, 343/338, 729/728
Subgroup: 2.3.7


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


[[Gencom]]: [2 3/2; 59049/57344]
Optimal tunings:  
* WE: ~2 = 1199.3353{{c}}, ~26/21 = 380.3206{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/21 = 380.5041{{c}}


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


[[Mapping|Sval mapping]]: [{{val|1 1 -3}}, {{val|0 1 10}}]
Badness (Sintel): 1.09


[[Tp tuning|POL2 generator]]: ~3/2 = 696.544
====== Twenothology ======
Subgroup: 2.3.7.11.13.29


{{Optimal ET sequence|legend=1| 12, 19, 31, 112b, 143b, 174b }}
Comma list: 144/143, 232/231, 343/338, 729/728


[[Tp tuning #T2 tuning|RMS error]]: 1.226 cents
Subgroup-val mapping: {{mapping| 1 0 -1 6 -2 2 | 0 5 12 -8 18 9 }}


Related temperament: [[Meantone family #Septimal meantone|meantone]]
Optimal tunings:  
* WE: ~2 = 1199.6175{{c}}, ~26/21 = 380.4049{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/21 = 380.5103{{c}}


== Leapfrog ==
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
{{see also|Gentle region}}


Subgroup: 2.3.7
Badness (Sintel): 0.964


[[Comma list]]: 14680064/14348907
===== Melchior =====
Subgroup: 2.3.7.11.13


[[Gencom]]: [2 3/2; 14680064/14348907]
Comma list: 352/351, 364/363, 26411/26244


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


[[Mapping|Sval mapping]]: [{{val|1 0 -21}}, {{val|0 1 15}}]
Optimal tunings:  
* WE: ~2 = 1199.4887{{c}}, ~96/77 = 380.6034{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~96/77 = 380.7669{{c}}


[[Tp tuning|POL2 generator]]: ~3/2 = 704.721 cents
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63, 104 }}


{{Optimal ET sequence|legend=1| 17, 46, 63 }}
Badness (Sintel): 0.710


[[Tp tuning #T2 tuning|RMS error]]: 0.6202 cents
===== Balthazar =====
Subgroup: 2.3.7.11.13


Related temperaments: [[Hemifamity temperaments #Leapday|leapday]], [[Sensamagic clan #Leapweek|leapweek]], [[srutal]]
Comma list: 169/168, 896/891, 26411/26244


=== 2.3.7.11 ===
Subgroup-val mapping: {{mapping| 1 0 -1 6 1 | 0 10 24 -16 17 }}
Subgroup: 2.3.7.11
: mapping generators: ~2, ~143/128


[[Comma list]]: 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]]: [2 3/2; 896/891 1331/1323]
{{Optimal ET sequence|legend=0| 19, 44, 63, 145f }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3}}, {{val|0 1 0 15 11}}]
Badness (Sintel): 1.82


[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14}}, {{val|0 1 15 11}}]
==== Hogwarts ====
Subgroup: 2.3.7.29


[[Tp tuning|POL2 generator]]: ~3/2 = 704.753 cents
Comma list: 784/783, 5887/5832


{{Optimal ET sequence|legend=1| 17, 46, 63 }}
Subgroup-val mapping: {{mapping| 1 0 -1 2 | 0 5 12 9 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6047 cents
Optimal tunings:  
* WE: ~2 = 1200.1518{{c}}, ~36/29 = 380.6661{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/29 = 380.6375{{c}}


=== 2.3.7.11.13 ===
{{Optimal ET sequence|legend=0| 19, 22, 41, 145, 186j, 227j }}
Subgroup: 2.3.7.11.13


[[Comma list]]: 169/168, 352/351, 364/363
Badness (Sintel): 0.424


[[Gencom]]: [2 3/2; 169/169 352/351 364/363]
=== Skwares ===
{{Main| Squares }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3 -1}}, {{val|0 1 0 15 11 8}}]
Skwares is the no-5 [[restriction]] of [[squares]].


[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14 -9}}, {{val|0 1 15 11 8}}]
[[Subgroup]]: 2.3.7


[[Tp tuning|POL2 generator]]: ~3/2 = 704.745 cents
[[Comma list]]: 19683/19208


{{Optimal ET sequence|legend=1| 17, 46, 63 }}
{{Mapping|legend=2| 1 -1 -3 | 0 4 9 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7541 cents
{{Mapping|legend=3| 1 -1 0 -3 | 0 4 0 9 }}
: mapping generators: ~2, ~14/9


==== Skidoo ====
[[Optimal tuning]]s:
Subgroup: 2.3.7.11.13.23
* [[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 }}


[[Comma list]]: 169/168, 208/207, 352/351, 364/363
{{Optimal ET sequence|legend=1| 14, 17, 31, 48, 79 }}


[[Gencom]]: [2 3/2; 169/169 208/207 352/351 364/363]
[[Badness]] (Sintel): 1.55


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3 -1 0 0 1}}, {{val|0 1 0 15 11 8 0 0 6}}]
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


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


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


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


[[Tp tuning #T2 tuning|RMS error]]: 0.6265 cents
Optimal tunings:  
* WE: ~2 = 1200.3726{{c}}, ~14/9 = 774.9970{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.8197{{c}}


===== 2.3.7.11.13.23.29 =====
{{Optimal ET sequence|legend=0| 14, 17, 31, 48, 79, 127 }}
Subgroup: 2.3.7.11.13.23.29


[[Comma list]]: 169/168, 208/207, 232/231, 352/351, 364/363
Badness (Sintel): 0.405


[[Gencom]]: [2 3/2; 169/169 208/207 352/351 364/363]
===== 2.3.7.11.13 =====
 
Subgroup: 2.3.7.11.13
[[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}}]
 
[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14 -9 -5 -38}}, {{val|0 1 15 11 8 6 27}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 704.729 cents
 
{{Optimal ET sequence|legend=1| 17, 46, 63 }}
 
; Music
* Suite for Harpsichord in A Locrian, tuning: Eb-G# in [[46edo|46EDO]] by [[User:IlL|IlL]] (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]]


== Doublehearted ==
Comma list: 78/77, 99/98, 243/242
{{see also|Heartland}}


Subgroup: 2.3.7
Subgroup-val mapping: {{mapping| 1 -1 -3 -3 -6 | 0 4 9 10 15 }}


[[Comma list]]: 5764801/5668704
Gencom mapping: {{mapping| 1 -1 0 -3 -3 -6 | 0 4 0 9 10 15}}


[[Gencom]]: [2 343/324; 5764801/5668704]
Optimal tunings:  
* WE: ~2 = 1199.3264{{c}}, ~14/9 = 775.1081{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 775.4463{{c}}


[[Mapping|Sval mapping]]: [{{val|1 1 2}}, {{val|0 8 11}}]
{{Optimal ET sequence|legend=0| 14f, 17, 48f }}


[[Tp tuning|POL2 generator]]: ~343/324 = 87.8304
Badness (Sintel): 0.587


{{Optimal ET sequence|legend=1| 14, 27, 41 }}
===== Skwairs =====
 
Subgroup: 2.3.7.11.13
[[Tp tuning #T2 tuning|RMS error]]: 0.5041 cents
 
Related temperaments: [[Tetracot family|octacot]]
 
=== 2.3.7.11 ===
Subgroup: 2.3.7.11


[[Comma list]]: 243/242, 2401/2376
Comma list: 99/98, 144/143, 243/242


[[Gencom]]: [2 22/21; 243/242 2401/2376]
Subgroup-val mapping: {{mapping| 1 -1 -3 -3 5 | 0 4 9 10 -2 }}


[[Mapping|Sval mapping]]: [{{val|1 1 2 2}}, {{val|0 8 11 20}}]
Gencom mapping: {{mapping| 1 -1 0 -3 -3 5 | 0 4 0 9 10 -2 }}


[[Tp tuning|POL2 generator]]: ~22/21 = 87.6512
Optimal tunings:  
* WE: ~2 = 1198.8812{{c}}, ~14/9 = 775.5748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 775.1930{{c}}


{{Optimal ET sequence|legend=1| 14, 27e, 41, 96d, 137d, 178d }}
{{Optimal ET sequence|legend=0| 14, 17, 31, 48, 65d, 113df }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7147 cents
Badness (Sintel): 0.538


Related temperaments: [[Tetracot family|octacot]]
===== Byhearted =====
This temperament is the restriction of [[weasel]] to the 2.3.7.11.19 subgroup.


=== 2.3.7.11.19 ===
Subgroup: 2.3.7.11.19
Subgroup: 2.3.7.11.19


[[Comma list]]: 133/132, 243/242, 343/342
Comma list: 99/98, 243/242, 363/361


[[Gencom]]: [2 19/18; 133/132 243/242 343/342]
Subgroup-val mapping: {{mapping| 2 2 3 4 5 | 0 4 9 10 12 }}
: mapping generators: ~209/147, ~21/19


[[Mapping|Sval mapping]]: [{{val|1 1 2 2 3}}, {{val|0 8 11 20 17}}]
Optimal tunings:  
* WE: ~2 = 600.1836{{c}}, ~21/19 = 174.7882{{c}}
* CWE: ~2 = 600.0000{{c}}, ~21/19 = 174.7975{{c}}


[[Tp tuning|POL2 generator]]: ~19/18 = 87.6684
{{Optimal ET sequence|legend=0| 14, 34dh, 48, 110e }}


{{Optimal ET sequence|legend=1| 14, 27e, 41 }}
Badness (Sintel): 0.893


[[Tp tuning #T2 tuning|RMS error]]: 0.7065 cents
=== 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.  


Related temperaments: [[Tetracot family|octacot]]
[[Subgroup]]: 2.3.7


== Magi ==
[[Comma list]]: [[59049/57344]]
Subgroup: 2.3.7


[[Comma list]]: 537824/531441
{{Mapping|legend=2| 1 0 -13 | 0 1 10 }}


[[Gencom]]: [2 243/196; 537824/531441]
{{Mapping|legend=3| 1 0 0 -13 | 0 1 0 10 }}
: mapping generators: ~2, ~3


[[Mapping|Sval mapping]]: [{{val|1 0 -1}}, {{val|0 5 12}}]
[[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 }}


[[Tp tuning|POL2 generator]]: ~243/196 = 380.661
{{Optimal ET sequence|legend=1| 12, 19, 31, 112b, 143b, 174b }}


{{Optimal ET sequence|legend=1| 19, 22, 41, 104, 145, 186 }}
[[Badness]] (Sintel): 2.35


[[Tp tuning #T2 tuning|RMS error]]: 0.4277 cents
=== 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]].  


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


[[Comma list]]: 896/891, 26411/26244
[[Comma list]]: 17496/16807


[[Gencom]]: [2 96/77; 896/891 26411/26244]
{{Mapping|legend=2| 1 1 2 | 0 5 7 }}


[[Mapping|Sval mapping]]: [{{val|1 0 -1 6}}, {{val|0 5 12 -8}}]
{{Mapping|legend=3| 1 1 0 2 | 0 5 0 7 }}
: mapping generators: ~2, ~54/49


[[Tp tuning|POL2 generator]]: ~96/77 = 380.768
[[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, 63, 104 }}
{{Optimal ET sequence|legend=1| 8d, 9, 17, 43, 60d, 103d }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents
[[Badness]] (Sintel): 2.48


==== Balthazar ====
==== 2.3.7.11 subgroup ====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11


[[Comma list]]: 169/168, 896/891, 26411/26244
Comma list: 99/98, 864/847


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


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


[[Tp tuning|POL2 generator]]: ~143/128 = 190.407
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, 44, 63, 145f }}
{{Optimal ET sequence|legend=0| 8d, 9, 17, 43, 60d }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6937 cents
Badness (Sintel): 0.624


==== Caspar ====
==== 2.3.7.11.13 subgroup ====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


[[Comma list]]: 144/143, 343/338, 729/728
Comma list: 78/77, 99/98, 144/143


[[Gencom]]: [2 26/21; 144/143 343/338 729/728]
Subgroup-val mapping: {{mapping| 1 1 2 3 3 | 0 5 7 4 6 }}


[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 -2}}, {{val|0 5 12 -8 18}}]
Gencom mapping: {{mapping| 1 1 0 2 3 3 | 0 5 0 7 4 6 }}


[[Tp tuning|POL2 generator]]: ~26/21 = 380.531
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, 22f, 41 }}
{{Optimal ET sequence|legend=0| 8d, 9, 17, 43, 60d }}


[[Tp tuning #T2 tuning|RMS error]]: 1.032 cents
Badness (Sintel): 0.400


==== Melchior ====
; 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]]: 352/351, 364/363, 26411/26244
=== Doublehearted ===
{{See also| Heartland }}


[[Gencom]]: [2 96/77; 352/351 364/363 26411/26244]
This temperament is the no-5 [[restriction]] of [[octacot]].


[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 11}}, {{val|0 5 12 -8 -23}}]
[[Subgroup]]: 2.3.7


[[Tp tuning|POL2 generator]]: ~96/77 = 380.766
[[Comma list]]: 5764801/5668704


{{Optimal ET sequence|legend=1| 19f, 22, 41, 63, 104 }}
{{Mapping|legend=2| 1 1 2 | 0 8 11 }}
: mapping generators: ~2, ~343/342


[[Tp tuning #T2 tuning|RMS error]]: 0.3891 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 }}


=== Hogwarts ===
{{Optimal ET sequence|legend=1| 14, 27, 41 }}
Subgroup: 2.3.7.29


[[Comma list]]: 784/783, 5887/5832
[[Badness]] (Sintel): 2.62


[[Gencom]]: [2 36/29; 784/783 5887/5832]
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


[[Mapping|Sval mapping]]: [{{val|1 0 -1 2}}, {{val|0 5 12 9}}]
Comma list: 243/242, 2401/2376


[[Tp tuning|POL2 generator]]: ~36/29 = 380.618
Subgroup-val mapping: {{mapping| 1 1 2 2 | 0 8 11 20 }}


{{Optimal ET sequence|legend=1| 19, 22, 41, 145, 186j, 227j }}
Optimal tunings:
* WE: ~2 = 1200.4071{{c}}, ~22/21 = 87.6809{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 87.6902{{c}}


==== Twenothology ====
{{Optimal ET sequence|legend=0| 14, 27e, 41, 96d, 137d, 178d }}
Subgroup: 2.3.7.11.13.29


[[Comma list]]: 144/143, 232/231, 343/338, 729/728
Badness (Sintel): 0.815


[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 -2 2}}, {{val|0 5 12 -8 18 9}}]
==== 2.3.7.11.19 ====
Subgroup: 2.3.7.11.19


[[Tp tuning|POL2 generator]]: ~26/21 = 380.526
Comma list: 133/132, 243/242, 343/342


{{Optimal ET sequence|legend=1| 19, 22f, 41 }}
Subgroup-val mapping: {{mapping| 1 1 2 2 3 | 0 8 11 20 17 }}


== Lee ==
Optimal tunings:
Subgroup: 2.3.7
* WE: ~2 = 1200.6100{{c}}, ~19/18 = 87.7129{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/18 = 87.7285{{c}}


[[Comma]]: 177147/175616
{{Optimal ET sequence|legend=0| 14, 27e, 41, 137dh }}


[[Gencom]]: [2 81/56; 177147/175616]
Badness (Sintel): 0.560


[[Gencom|Gencom mapping]]: [{{val|1 0 0 -3}}, {{val|0 3 0 11}}]
=== Purpleheart ===
{{Main| Whitewood }}


[[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]]: 2187/2048


{{Optimal ET sequence|legend=1| 17, 36, 89, 125, 161, 358, 519b }}
{{Mapping|legend=2| 7 11 0 | 0 0 1 }}
: mapping generators: ~9/8, ~7


[[Tp tuning #T2 tuning|RMS error]]: 0.3519 cents
[[Optimal tuning]]s:
* [[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 }}


== [[Slendric]] ==
{{Optimal ET sequence|legend=1| 7, 14, 35, 49bd }}
Subgroup: 2.3.7


[[Comma]]: 1029/1024
[[Badness]] (Sintel): 3.00


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


[[Gencom|Gencom mapping]]: [{{val|1 1 0 3}}, {{val|0 3 0 -1}}]
[[Subgroup]]: 2.3.7


[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 3 -1}}]
[[Comma list]]: {{monzo| 83 -1 -29 }}


[[Tp tuning|POL2 generator]]: ~8/7 = 233.688
{{Mapping|legend=2| 1 -4 3 | 0 29 -1 }}
: mapping generators: ~2, ~8/7


{{Optimal ET sequence|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}
[[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 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
{{Optimal ET sequence|legend=0| 26, 83, 109, 135, 566, 701, 836, 971, 1106, 2077, 5260, 7337, 9414d }}


=== [[Gamelismic clan #Baladic|Baladic]] ===
[[Badness]] (Sintel): 3.06
Subgroup: 2.3.7.13


[[Comma list]]: 169/168, 1029/1024
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


[[Gencom]]: [91/64 8/7; 169/168 1029/1024]
Comma list: 43923/43904, 5767168/5764801


[[Mapping|Sval mapping]]: [{{val|2 2 6 7}}, {{val|0 3 -1 1}}]
Subgroup-val mapping: {{mapping| 1 -4 3 5 | 0 29 -1 -8 }}


[[Tp tuning|POL2 generator]]: ~8/7 = 233.6044
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| 10, 26, 36, 154…, 190…, 226…, 262… }}
{{Optimal ET sequence|legend=0| 26, 83, 109, 135, 566, 701, 836, 971, 1807, 2778, 4585 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents
Badness (Sintel): 0.324


==== 2.3.7.13.17 ====
=== Leapfrog ===
Subgroup: 2.3.7.13.17
{{See also| Gentle region }}


[[Comma list]]: 169/168, 273/272, 289/288
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]]: [17/12 8/7; 169/168 273/272 289/288]
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 7}}, {{val|0 3 -1 1 3}}]
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 = 233.6155
[[Subgroup]]: 2.3.7


{{Optimal ET sequence|legend=1| 10, 26, 36, 154…, 190…, 226… }}
[[Comma list]]: 14680064/14348907
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5073 cents
 
== Hemif ==
Subgroup: 2.3.7


[[Comma]]: 1605632/1594323
{{Mapping|legend=2| 1 0 -21 | 0 1 15 }}


[[Gencom]]: [2 2187/1792; 1605632/1594323]
{{Mapping|legend=3| 1 0 0 -21 | 0 1 0 15 }}
: mapping generators: ~2, ~3


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


[[Mapping|Sval mapping]]: [{{val|1 1 -1}}, {{val|0 2 13}}]
{{Optimal ET sequence|legend=1| 17, 46, 63, 235b, 298b, 361bd, 424bd, 487bbd }}


[[Tp tuning|POL2 generator]]: ~2187/1792 = 351.485
[[Badness]] (Sintel): 4.33


{{Optimal ET sequence|legend=1| 7, 17, 41, 58, 99 }}
==== 2.3.7.11 ====
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2344 cents
 
Related temperaments: [[Breedsmic temperaments #Hemififths|hemififths]], namo
 
=== 2.3.7.11 ===
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


[[Comma list]]: 243/242, 896/891
Comma list: 896/891, 1331/1323
 
[[Gencom]]: [2 11/9; 243/242 896/891]


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1 2}}, {{val|0 2 0 13 5}}]
Subgroup-val mapping: {{mapping| 1 0 -21 -14 | 0 1 15 11 }}


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


[[Tp tuning|POL2 generator]]: ~11/9 = 351.535
Optimal tunings:  
* WE: ~2 = 1199.2683{{c}}, ~3/2 = 704.3230{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.6926{{c}}


{{Optimal ET sequence|legend=1| 7, 17, 41, 58, 99e }}
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents
Badness (Sintel): 0.629


==== 2.3.7.11.13 ====
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


[[Comma list]]: 144/143, 243/242, 364/363
Comma list: 169/168, 352/351, 364/363


[[Gencom]]: [2 11/9; 144/143 243/242 364/363]
Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 | 0 1 15 11 8 }}


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


[[Mapping|Sval mapping]]: [{{val|1 1 -1 2 4}}, {{val|0 2 13 5 -1}}]
Optimal tunings:  
* WE: ~2 = 1199.5654{{c}}, ~3/2 = 704.4898{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7084{{c}}


[[Tp tuning|POL2 generator]]: ~11/9 = 351.691
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41, 58 }}
Badness (Sintel): 0.436


[[Tp tuning #T2 tuning|RMS error]]: 0.7167 cents
===== Skidoo =====
Subgroup: 2.3.7.11.13.23


==== Heartful ====
Comma list: 169/168, 208/207, 352/351, 364/363
{{see also|Heartland}}


Subgroup: 2.3.7.11.19
Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 -5 | 0 1 15 11 8 6 }}


[[Comma list]]: 243/242, 896/891, 1083/1078
Gencom mapping: {{mapping| 1 0 0 -21 -14 -9 0 0 -5 | 0 1 0 15 11 8 0 0 6 }}


[[Gencom]]: [2 21/19; 243/242 896/891 1083/1078]
Optimal tunings:  
* WE: ~2 = 1199.6639{{c}}, ~3/2 = 704.5315{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7021{{c}}


[[Mapping|Sval mapping]]: [{{val|1 1 -1 2 0}}, {{val|0 4 26 10 29}}]
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


[[Tp tuning|POL2 generator]]: ~21/19 = 175.804
Badness (Sintel): 0.356


{{Optimal ET sequence|legend=1| 34dh, 41, 116e, 157e }}
====== 2.3.7.11.13.23.29 ======
Subgroup: 2.3.7.11.13.23.29


[[Tp tuning #T2 tuning|RMS error]]: 0.5360 cents
Comma list: 169/168, 208/207, 232/231, 352/351, 364/363


Related temperaments: [[Tetracot family|bunya]]
Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 -5 -38 | 0 1 15 11 8 6 27 }}


== Hearts ==
Gencom mapping: {{mapping| 1 0 0 -21 -14 -9 -5 0 0 -38 | 0 1 0 15 11 8 0 0 6 27 }}
{{see also|Heartland}}


Subgroup: 2.3.7
Optimal tunings:  
* WE: ~2 = 1199.5755{{c}}, ~3/2 = 704.5533{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7750{{c}}


[[Comma list]]: 34451725707/34359738368 (trila-quadzo comma)
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


[[Gencom]]: [2 567/512; 34451725707/34359738368]
Badness (Sintel): 0.441


[[Mapping|Sval mapping]]: [{{val|1 1 5}}, {{val|0 4 -15}}]
; Music
* 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]]


[[Tp tuning|POL2 generator]]: ~567/512 = 175.433
=== Superslendric ===
In superslendric, eight [[8/7]]'s are equated to [[3/1]]. This relates it to [[8edt]].


{{Optimal ET sequence|legend=1| 7, 27d, 34, 41, 89, 130, 171 }}
[[Subgroup]]: 2.3.7


[[Tp tuning #T2 tuning|RMS error]]: 0.0529 cents
[[Comma list]]: 17294403/16777216


Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquiquartififths]]
{{Mapping|legend=2| 1 0 3 | 0 8 -1 }}
: mapping generators: ~2, ~8/7


=== 2.3.7.11 ===
[[Optimal tuning]]s:
Subgroup: 2.3.7.11
* [[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 }}


[[Comma list]]: 243/242, 65536/65219
{{Optimal ET sequence|legend=1| 5, …, 66b, 71b, 76, 81, 86, 91, 96d }}


[[Gencom]]: [2 256/231; 243/242 65536/65219]
[[Badness]] (Sintel): 6.15


[[Mapping|Sval mapping]]: [{{val|1 1 5 2}}, {{val|0 4 -15 10}}]
=== Hectosaros leap week ===
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.


[[Tp tuning|POL2 generator]]: ~256/231 = 175.369
[[Subgroup]]: 2.3.7


{{Optimal ET sequence|legend=1| 7, 27de, 34, 41, 89, 130 }}
[[Comma list]]: {{monzo| -50 -746 439 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3224 cents
{{Mapping|legend=2| 1 -126 -214 | 0 439 746 }}
: mapping generators: ~2, ~{{monzo| -16 -243 143 }}


Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquart]]
[[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 }}


=== 2.3.7.11.19 ===
{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}
Subgroup: 2.3.7.11.19


[[Comma list]]: 243/242, 513/512, 1083/1078
[[Badness]] (Sintel): 17.7 × 10<sup>3</sup>


[[Gencom]]: [2 21/19; 243/242 513/512 1083/1078]
==== 2.3.7.13 subgroup ====
Subgroup: 2.3.7.13


[[Mapping|Sval mapping]]: [{{val|1 1 5 2 6}}, {{val|0 4 -15 10 -12}}]
Comma list: {{monzo| -42 -2 -5 16 }}, {{monzo| 10 -46 29 -5 }}


[[Tp tuning|POL2 generator]]: ~21/19 = 175.341
Subgroup-val mapping: {{mapping| 1 -126 -214 -80 | 0 439 746 288 }}


{{Optimal ET sequence|legend=1| 7, 27deh, 34, 41, 89, 130, 219 }}
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}}


[[Tp tuning #T2 tuning|RMS error]]: 0.3121 cents
{{Optimal ET sequence|legend=0| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689, 11255d }}


Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquart]]
Badness (Sintel): 53.2


== Navy ==
==== 2.3.7.13.17 subgroup ====
[[Subgroup]]: 2.3.7
Subgroup: 2.3.7.13.17


[[Comma list]]: 282429536481/281974669312
Comma list: 39337984/39328497, {{monzo| 0 -14 7 4 -3 }}, {{monzo| -18 -24 14 -1 5 }}


{{Mapping|legend=1|1 1 0|0 5 24}}
Subgroup-val mapping: {{mapping| 1 -126 -214 -80 -18 | 0 439 746 288 76 }}


[[Tp tuning|POL2 generator]]: ~243/224 = 140.366
Optimal tunings:  
* WE: ~2 = 1200.9870{{c}}, ~3757/3072 = 348.7480{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3757/3072 = 348.7517{{c}}


{{Optimal ET sequence|legend=1| 17, 60, 77, 94, 171, 265, 436 }}
{{Optimal ET sequence|legend=0| 320, 1483bd, 1803, 2123 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0296 cents
Badness (Sintel): 13.4


Related temperaments: [[Schismatic family|tsaharuk]], [[Hemifamity temperaments|quanic]]
==== 2.3.7.13.17.19 subgroup ====
Subgroup: 2.3.7.13.17.19


=== 2.3.7.11 ===
Comma list: 10081799/10077696, 10754912/10744731, 39337984/39328497, 480024727/480020256
Subgroup: 2.3.7.11


[[Comma list]]: 1331/1323, 19712/19683
Subgroup-val mapping: {{mapping| 1 -126 -214 -80 -18 -171 | 0 439 746 288 76 603 }}


{{Mapping|legend=1|1 1 0 1|0 5 24 21}}
Optimal tunings:
* WE: ~2 = 1200.9961{{c}}, ~3757/3072 = 348.7506{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3757/3072 = 348.7517{{c}}


[[Tp tuning|POL2 generator]]: ~88/81 = 140.407
{{Optimal ET sequence|legend=0| 320, 1483bd, 1803, 2123 }}


{{Optimal ET sequence|legend=1| 17, 60e, 77, 94, 359e, 453ee, 547ee, 641ee }}
Badness (Sintel): 7.46


[[Tp tuning #T2 tuning|RMS error]]: 0.3778 cents
=== Heartland (rank 3) ===
{{Main| Heartland }}


Related temperaments: [[Schismatic family|tsaharuk]], [[Hemifamity temperaments|quanic]]
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).


=== 2.3.7.11.13 ===
[[Subgroup]]: 2.3.7.11.19
Subgroup: 2.3.7.11.13


[[Comma list]]: 352/351, 729/728, 1331/1323
[[Comma list]]: 243/242, 1083/1078


{{Mapping|legend=1|1 1 0 1 3|0 5 24 21 6}}
{{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|POL2 generator]]: ~13/12 = 140.437
[[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}}


{{Optimal ET sequence|legend=1| 17, 60e, 77, 94 }}
{{Optimal ET sequence|legend=1| 14, 27e, 34dh, 41, 89, 130, 219 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4044 cents
[[Badness]] (Sintel): 0.615


Related temperaments: [[Schismatic family|tsaharuk]], [[Hemifamity temperaments|quanic]]
== Temperaments with a 2.3.11 gene ==
=== Neutral ===
See [[Rastmic clan #Neutral]].


== Slendrismic ==
=== Io ===
{{see also| 5th-octave temperaments #Slendrismic }}
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.


[[Subgroup]]: 2.3.7
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.


[[Comma list]]: 68719476736/68641485507
[[Subgroup]]: 2.3.11


{{Mapping|legend=1|5 8 14|0 -2 1}}
[[Comma list]]: 33/32


[[Tp tuning|POL2 generator]]: ~1029/1024 = 8.9906
{{Mapping|legend=2| 1 0 5 | 0 1 -1 }}
: mapping generators: ~2, ~3


{{Optimal ET sequence|legend=1| 130, 135, 265, 400, 935, 1335, 1735 }}
[[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 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0212 cents
{{Optimal ET sequence|legend=1| 2, 5, 7, 12e, 40ee, 47eee, 54beee, 61beeee }}


Related temperaments: [[Pental family|hemipental]]
[[Badness]] (Sintel): 0.185


== Hectosaros leap week ==
=== Alphaxenean ===
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.
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.  


[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.11


[[Comma list]]: {{monzo|-50 -746 439}}
[[Comma list]]: 131769/131072


[[Mapping]]: [{{val|1 313 532}}, {{val|0 -439 -746}}]
{{Mapping|legend=2| 2 1 8 | 0 2 -1 }}
: mapping generators: ~363/256, ~16/11


[[Optimal tuning]] ([[CTE]]): ~{{monzo|17 343 143}} = 851.248
[[Optimal tuning]]s:
* [[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 }}


{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}
{{Optimal ET sequence|legend=1| 22, 24, 94, 118, 142, 450e, 592e, 1326beeee }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0164 cents
[[Badness]] (Sintel): 0.395


=== 2.3.7.13 subgroup ===
=== Infraug ===
Subgroup: 2.3.7.13
[[Subgroup]]: 2.3.11


Comma list: {{monzo|-42 -2 -5 16}}, {{monzo|10 -46 29 -5}}
[[Comma list]]: 729/704


Mapping: [{{val|1 313 532 208}}, {{val|0 -439 -746 -288}}]
{{Mapping|legend=2| 1 0 -6 | 0 1 6 }}
: mapping generators: ~2, ~3


Optimal tuning (CTE): ~1235079060111/755603996672 = 851.248
[[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 }}


{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}  
{{Optimal ET sequence|legend=1| 7, 19, 26, 33, 59b, 92b }}


=== 2.3.7.13.17 subgroup ===
[[Badness]] (Sintel): 0.734
Subgroup: 2.3.7.13.17


Comma list: 39337984/39328497, {{monzo|0 -14 7 4 -3}}, {{monzo|-18 -24 14 -1 5}}
==== 2.3.11.13 ====
Subgroup: 2.3.11.13


Mapping: [{{val|1 313 532 208 58}}, {{val|0 -439 -746 -288 -76}}]
Comma list: 144/143, 729/704


Optimal tuning (CTE): ~6144/3757 = 851.248
Subgroup-val mapping: {{mapping| 1 0 -6 10 | 0 1 6 -4 }}


{{Optimal ET sequence|legend=1| 320, 1483bd, 1803, 2123 }}
Optimal tunings:
* WE: ~2 = 1202.1934{{c}}, ~3/2 = 692.8902{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 691.7585{{c}}


=== 2.3.7.13.17.19 subgroup ===
{{Optimal ET sequence|legend=0| 7, 19, 26, 59b }}
Subgroup: 2.3.7.13.17.19


Comma list: 10081799/10077696, 39337984/39328497, 10754912/10744731, 480024727/480020256
Badness (Sintel): 0.725


Mapping: [{{val|1 313 532 208 58 432}}, {{val|0 -439 -746 -288 -76 -603}}]
==== Aerophore ====
Subgroup: 2.3.11.19


Optimal tuning (CTE): ~6144/3757 = 851.248
Comma list: 363/361, 729/704


{{Optimal ET sequence|legend=1| 320, 1483bd, 1803, 2123 }}
Subgroup-val mapping: {{mapping| 1 0 -6 -6 | 0 2 12 13 }}
: mapping generators: ~2, ~19/11


== Ennea ==
Optimal tunings:
Subgroup: 2.3.7.11
* WE: ~2 = 1202.6380{{c}}, ~19/11 = 947.4782{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 945.7779{{c}}


[[Comma list]]: 41503/41472, 43923/43904
{{Optimal ET sequence|legend=1| 14, 19, 33 }}


[[Gencom]]: [2 99/98; 41503/41472, 43923/43904]
Badness (Sintel): 1.59


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


[[Mapping|Sval mapping]]: [{{val|9 0 11 24}}, {{val|0 2 2 1}}]
[[Comma list]]: 4096/3993


[[Tp tuning|POL2 generator]]: ~99/98 = 17.6258
{{Mapping|legend=2| 1 0 4 | 0 3 -1 }}
: mapping generators: ~2, ~16/11


{{Optimal ET sequence|legend=1| 54, 63, 72, 135, 342, 477, 1089, 1566 }}
[[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]]: 0.0383 cents
{{Optimal ET sequence|legend=1| 2, 11b, 13, 15, 17 }}


== Parapyth (rank 3) ==
[[Badness]] (Sintel): 0.984
{{see also| Pentacircle temperaments #Parapyth }}


Subgroup: 2.3.7.11
==== Huxley ====
{{Main| Huxley }}


[[Comma list]]: 896/891
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 3/2 28/27; 896/891]
Subgroup: 2.3.11.13
 
[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 }}, {{val| 0 1 0 3 -1 }}, {{val| 0 0 0 1 1 }}]
 
[[Sval]] [[mapping]]: [{{val| 1 0 0 7 }}, {{val| 0 1 0 -4 }}, {{val| 0 0 1 1 }}]


[[Tp tuning|POL2 tuning]]: ~3 = 1903.834, ~7 = 3369.872
Comma list: 512/507, 1352/1331


{{Optimal ET sequence|legend=1| 17, 36, 41, 58, 63, 104 }}
Subgroup-val mapping: {{mapping| 1 -3 5 6 | 0 6 -2 -3 }}
: mapping generators: ~2, ~22/13


[[Tp tuning #T2 tuning|RMS error]]: 0.4149 cents
Optimal tunings:  
* WE: ~2 = 1198.0036{{c}}, ~22/13 = 916.0595{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 917.5184{{c}}


=== 2.3.7.11.13 ===
{{Optimal ET sequence|legend=0| 4, 13, 17 }}
Subgroup: 2.3.7.11.13


[[Comma list]]: 352/351, 364/363
Badness (Sintel): 1.31


The gencom below gives [[Margo Schulter]]'s favored basis
=== Glaishur ===
This temperament is the no-5 no-7 [[restriction]] of [[#Navy|navy]], as well as the add-11 [[extension]] of [[#Glacier|glacier]].


[[Gencom]]: [2 3/2 28/27; 352/351 364/363]
[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 6 }}, {{val| 0 1 0 3 -1 -4 }}, {{val| 0 0 0 1 1 1 }}]
[[Sval]] [[mapping]]: [{{val| 1 0 0 7 12 }}, {{val| 0 1 0 -4 -7 }}, {{val| 0 0 1 1 1 }}]
[[Tp tuning|POL2 tuning]]: ~3 = 1903.856, ~7 = 3369.907
{{Optimal ET sequence|legend=1| 17, 41, 46, 58, 87, 104 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
== Paralimmic ==
[[Subgroup]]: 2.3.11
[[Subgroup]]: 2.3.11


[[Comma list]]: [[4096/3993]]
[[Comma list]]: 10554638336/10460353203


{{Mapping|legend=2|1 0 4|0 3 -1}}
{{Mapping|legend=2| 1 1 0 | 0 5 21 }}
: mapping generators: ~2, ~88/81


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[16/11]] = 634.320
[[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 }}


{{Optimal ET sequence|legend=1|11b, 13, 15, 17 }}
{{Optimal ET sequence|legend=1| 17, 60e, 77, 94, 111 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.237 cents
[[Badness]] (Sintel): 2.27


== Neutral ==
==== 2.3.11.13 ====
Neutral can be thought of as the 2.3.11 version of either [[mohajira]] or [[neutrominant]], as well as suhajira and ringo. Among other things, it is the temperament optimizing the [[neutral tetrad]].
Subgroup: 2.3.11.13


Subgroup: 2.3.11
Comma list: 352/351, 531674/531441


[[Comma]]: 243/242
Subgroup-val mapping: {{mapping| 1 1 0 3 | 0 5 21 6 }}


[[Gencom]]: [2 11/9; 243/242]
Optimal tunings:  
* WE: ~2 = 1200.0000{{c}}, ~13/12 = 140.537{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.537{{c}}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2}}, {{val|0 2 0 0 5}}]
{{Optimal ET sequence|legend=0| 17, 60e, 77, 94, 111, 239f, 350f }}


[[Mapping|Sval mapping]]: [{{val|1 1 2}}, {{val|0 2 5}}]
Badness (Sintel): 0.415


[[Tp tuning|POL2 generator]]: ~11/9 = 350.525
=== Profanity ===
Profanity identifies [[11/9]] with 2\7.


{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41, 65, 89, 202, 291, 380 }}
[[Subgroup]]: 2.3.11


[[Tp tuning #T2 tuning|RMS error]]: 0.3021 cents
[[Comma list]]: 19487171/19131876


; Scales
{{Mapping|legend=2| 7 0 2 | 0 1 2 }}
* [[neutraltet7|Seven note albitonic scale]]
: mapping generators: ~1458/1331, ~3
* [[neutraltet10|Ten note chromatic scale]]
* [[neutraltet17|Seventeen note mega chromatic scale]]


=== Namo ===
[[Optimal tuning]]s:
Subgroup: 2.3.11.13
* [[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 }}


[[Comma list]]: 144/143, 243/242
{{Optimal ET sequence|legend=1| 7, … 49, 56, 63, 70 }}


[[Gencom]]: [2 11/9; 144/143 243/242]
[[Badness]] (Sintel): 3.03


[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2 4}}, {{val|0 2 0 0 5 -1}}]
== Temperaments with a 2.3.13 gene ==
=== Threedic ===
[[Subgroup]]: 2.3.13


[[Mapping|Sval mapping]]: [{{val|1 1 2 4}}, {{val|0 2 5 -1}}]
[[Comma list]]: 2197/2187


[[Tp tuning|POL2 generator]]: ~11/9 = 351.488
{{Mapping|legend=2| 1 0 0 | 0 3 7 }}
: mapping generators: ~2, ~13/9


{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41 }}
[[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 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents
{{Optimal ET sequence|legend=1| 15, 17, 36, 53, 70, 123, 193, 316, 755f }}


== Heartland (rank 3) ==
[[Badness]] (Sintel): 0.160
{{Main| 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).  
=== Ultraflat ===
Ultraflat is a diatonic-based [[exotemperament]] that makes [[27/26]] vanish, so [[13/8]] is a major sixth.  


Subgroup: 2.3.7.11.19
[[Subgroup]]: 2.3.13


[[Comma list]]: 243/242, 1083/1078
[[Comma list]]: 27/26


[[Gencom]]: [2 21/19 7; 243/242 1083/1078]
{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}
: mapping generators: ~2, ~3


[[Mapping|Sval mapping]]: [{{val|1 1 0 2 1}}, {{val|0 4 0 10 3}}, {{val| 0 0 1 0 1 }}]
[[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 }}


[[Tp tuning|POL2 generator]]: ~21/19 = 175.2713, ~7 = 3369.3784
{{Optimal ET sequence|legend=1| 2, 5, 7 }}


{{Optimal ET sequence|legend=1| 7, 14, 27e, 34dh, 41, 89, 130 }}
[[Badness]] (Sintel): 0.200


[[Tp tuning #T2 tuning|RMS error]]: 0.3066 cents
=== 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.  


== Aerophore ==
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.  
[[Subgroup]]: 2.3.11.19


[[Comma list]]: 363/361, 729/704
[[Subgroup]]: 2.3.13
 
[[Mapping]]: [{{val|1 0 -6 -6}}, {{val|0 2 12 13}}]
 
[[POTE generator]]: ~[[19/11]] = 945.4
 
{{Optimal ET sequence|legend=1| 9eehh, 14, 19, 33 }}


=== Semaerophore ===
[[Comma list]]: 1053/1024
[[Subgroup]]: 2.3.7.11.19


[[Comma list]]: 49/48, 77/76, 729/704
{{Mapping|legend=2| 1 0 10 | 0 1 -4 }}
: mapping generators: ~2, ~3


[[Mapping]]: [{{val|1 0 2 -6 -6}}, {{val|0 2 1 12 13}}]
[[Optimal tuning]]s:  
* [[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 }}


[[POTE generator]]: ~[[7/4]] = 944.667
{{Optimal ET sequence|legend=1| 5f, 7, 12, 19, 45f, 64f, 147bfff }}


{{Optimal ET sequence|legend=1| 9eehh, 14, 33d, 47deh }}
[[Badness]] (Sintel): 0.610


== Superflat aka tridecimal ==
=== Shoal ===
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.
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
[[Subgroup]]: 2.3.13


[[Comma list]]: [[1053/1024]]
[[Comma list]]: 816293376/815730721
 
{{Mapping|legend=2|1 1 6|0 1 -4}}


[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.939
{{Mapping|legend=2| 1 -5 -7 | 0 8 13 }}
: mapping generators: ~2, ~3888/2197


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


[[Tp tuning #T2 tuning|RMS error]]: 1.591 cents
{{Optimal ET sequence|legend=1| 17, 79, 96, 113, 130, 147, 424, 571, 1289, 10883ff, 12172ff }}


=== 2.3.11.13 ===
Badness (Sintel): 0.135
[[Subgroup]]: 2.3.11.13


[[Comma list]]: [[144/143]], [[729/704]]
==== 2.3.13.23 ====
Subgroup: 2.3.13.23


[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.247
Comma list: 3888/3887, 12168/12167


{{Optimal ET sequence|legend=1|7, 19, 26, 59b }}
Subgroup-val mapping: {{mapping| 1 -5 -7 -7 | 0 8 13 14 }}


== Ultraflat ==
Optimal tunings:
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.
* WE: ~2 = 1199.9883{{c}}, ~23/13 = 987.7325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~23/13 = 987.7408{{c}}


[[Subgroup]]: 2.3.13
{{Optimal ET sequence|legend=0| 17, 79, 96, 113, 130, 147, 424, 571, 1289, 1860, 3149 }}


[[Comma list]]: [[27/26]]
Badness (Sintel): 0.0213


{{Mapping|legend=2|1 1 2|0 1 3}}
Scales:
* [[Shoal17]]


[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[3/2]] = 688.391
; Music
 
* [https://www.youtube.com/watch?v=bwGjN56iMM0 ''Moody Improvisation in the Shoal Temperament''] by [[Budjarn Lambeth]] (2025)
{{Optimal ET sequence|legend=1| 5, 7 }}


[[Tp tuning #T2 tuning|RMS error]]: 4.367 cents
=== 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.


== Threedic ==
[[Subgroup]]: 2.3.13
[[Subgroup]]: 2.3.13


[[Comma list]]: [[2197/2187]]
[[Comma list]]: 373248/371293


{{Mapping|legend=2|1 0 0|0 3 7}}
{{Mapping|legend=2| 1 1 3 | 0 5 6 }}
: mapping generators: ~2, ~13/12


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/9]] = 634.173
[[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 }}


{{Optimal ET sequence|legend=1|11bff, 13f, 15, 17, 36, 53, 70, 123, 193, 316, 755f }}
{{Optimal ET sequence|legend=1| 8, 9, 17, 60, 77, 94, 171, 265, 359f, 983ff }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2054 cents
[[Badness]] (Sintel): 0.383


== Semitonic ==
== Temperaments with a higher-limit gene ==
=== Semitonic ===
[[Subgroup]]: 2.3.17
[[Subgroup]]: 2.3.17


Line 1,027: Line 1,171:


{{Mapping|legend=2| 2 0 5 | 0 1 1 }}
{{Mapping|legend=2| 2 0 5 | 0 1 1 }}
: mapping generators: ~17/12, ~3
[[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 }}
{{Optimal ET sequence|legend=1| 10, 12, 58, 70, 82, 94, 106, 118, 224g }}


: sval mapping generators: ~17/12, ~3
[[Badness]] (Sintel): 0.0454


: [[gencom]]: [17/12 3; 289/288]
=== 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.


[[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)
[[Subgroup]]: 2.3.19


{{Optimal ET sequence|legend=1| 12, 58, 70, 82, 94, 106, 118, 224g }}
[[Comma list]]: 513/512


[[Tp tuning #T2 tuning|RMS error]]: 0.2247 cents
{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}
: mapping generators: ~2, ~3


== Dog ==
[[Optimal tuning]]s:
The dog temperament 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 [[Pelogic family|mavila]].
* [[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 }}


Subgroup: 2.3.19
{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296, 1557bhhhh, 1853bhhhh }}


[[Comma list]]: 81/76
[[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]].


[[Gencom]]: [2 4/3; 81/76]
[[Subgroup]]: 2.3.19


[[Mapping|Sval mapping]]: [{{val|1 2 6}}, {{val|0 -1 -4}}]
[[Comma list]]: 81/76


[[Tp tuning|POL2 generator]]: ~4/3 = 521.403
{{Mapping|legend=2| 1 0 -2 | 0 1 4 }}
: mapping generators: ~2, ~3


{{Optimal ET sequence|legend=1| 5h, 7, 16, 23 }}
[[Optimal tuning]]s:
* [[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 }}


[[Tp tuning #T2 tuning|RMS error]]: 4.943 cents
{{Optimal ET sequence|legend=1| 2, 5h, 7, 16, 23 }}


== Boethian ==
[[Badness]] (Sintel): 0.491
Boethian is a diatonic-based temperament that makes [[513/512]] vanish, so [[19/16]] is a minor third. It can be viewed as a 2.3.19 subgroup analogue of [[Schismatic family|schismic temperament]].


[[Subgroup]]: 2.3.19
=== 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.


[[Comma list]]: [[513/512]]
[[Subgroup]]: 2.3.23


{{Mapping|legend=1| 1 0 9 | 0 1 -3 }}
[[Comma list]]: 2187/2116


: Mapping generators: ~2, ~3
{{Mapping|legend=2| 1 0 -1 | 0 2 7 }}
: mapping generators: ~2, ~46/27


[[Optimal tuning]] (CTE): ~2 = 1\1, ~3/2 = 701.3288
[[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 ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296 }}
{{Optimal ET sequence|legend=1| 5, 14, 19, 43, 62i, 81i }}


[[Badness]]: 0.000374
[[Badness]] (Sintel): 0.801


== Porpoise ==
=== Porpoise ===
[[Subgroup]]: 2.3.29
[[Subgroup]]: 2.3.29


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


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


[[CTE tuning|CTE generator]]: ~32/29 = 166.067
[[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 ET sequence|legend=1| 7, 22, 29, 94, 123, 152j, 275jj, 427jjj }}
{{Optimal ET sequence|legend=1| 7, 22, 29, 94, 123, 152j, 275jj, 427jjj }}


== Sematology ==
[[Badness]] (Sintel): 0.317
This temperament tempers out 4107/4096 and thus equates 2 [[37/32]]'s with [[4/3]].
 
=== Sematology ===
This temperament tempers out [[4107/4096]] and thus equates a stack of two [[37/32]]'s with [[4/3]].


[[Subgroup]]: 2.3.37
[[Subgroup]]: 2.3.37
Line 1,090: Line 1,268:
[[Comma list]]: 4107/4096
[[Comma list]]: 4107/4096


[[Gencom]]: [2 37/32; 4107/4096]
{{Mapping|legend=2| 1 0 6 | 0 2 -1 }}
: mapping generators: ~2, ~64/37


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


[[POTE generator]]: ~[[37/32]] = 249.075
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53, 77, 130, 443l, 573ll, 703ll, 1536blllll }}


{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53, 77, 130 }}
[[Badness]] (Sintel): 0.0690


=== 2.3.7.37 subgroup ===
=== Reversed mavila ===
[[Subgroup]]: 2.3.7.37
 
[[Comma list]]: 4107/4096, 259/256
 
[[Gencom]]: [2 37/32; 4107/4096 259/256]
 
[[Mapping]]: [{{val|1 1 1 5}}, {{val|0 -2 -1 1}}]
 
[[POTE generator]]: ~[[37/32]] = 247.782
 
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53d }}
 
=== 2.3.5.37 subgroup ===
It is difficult to extend sematology to include 5, due the 5th harmonic being quite high-complexity.
 
[[Subgroup]]: 2.3.5.37
 
[[Comma list]]: 4107/4096, 17592186044416/17562397269605
 
[[Gencom]]: [2 37/32; 4107/4096 17592186044416/17562397269605]
 
[[Mapping]]: [{{val|1 1 4 5}}, {{val|0 -2 -8 1}}]
 
[[POTE generator]]: ~[[37/32]] = 251.393
 
{{Optimal ET sequence|legend=1| 5, 14c, 19, 43, 62 }}
 
==== 2.3.5.7.37 subgroup ====
[[Subgroup]]: 2.3.5.7.37
 
[[Comma list]]: 4107/4096, 17592186044416/17562397269605, 259/256
 
[[Gencom]]: [2 37/32; 4107/4096 17592186044416/17562397269605 259/256]
 
[[Mapping]]: [{{val|1 1 4 1 5}}, {{val|0 -2 -8 -1 1}}]
 
[[POTE generator]]: ~[[37/32]] = 251.204
 
{{Optimal ET sequence|legend=1| 5, 14c, 19 }}
 
== Reversed mavila ==
[[Subgroup]]: 2.3.37
[[Subgroup]]: 2.3.37


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


[[Gencom]]: [2 4/3; 81/74]
{{Mapping|legend=2| 1 0 -1 | 0 1 4 }}
: mapping generators: ~2, ~3


[[Mapping]]: [{{val|1 1 0}}, {{val|0 -1 12}}]
[[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 }}


[[POTE generator]]: ~4/3 = 521.397
{{Optimal ET sequence|legend=1| 2, 5l, 7l, 9, 16l }}


{{Optimal ET sequence|legend=1| 5l, 7l, 9, 16l }}
[[Badness]] (Sintel): 0.623


== Reversed meantone ==
=== Reversed meantone ===
{{Main| Reversed meantone }}
{{Main| Reversed meantone }}


Subgroup: 2.3.41
[[Subgroup]]: 2.3.41


[[Comma list]]: 82/81
[[Comma list]]: 82/81


[[Gencom]]: [2 4/3; 82/81]
{{Mapping|legend=2| 1 0 -1 | 0 1 4 }}
 
: mapping generators: ~2, ~3
[[Mapping|Sval mapping]]: [{{val|1 2 7}}, {{val|0 -1 -4}}]
 
[[Tp tuning|POL2 generator]]: ~4/3 = 494.509
 
{{Optimal ET sequence|legend=1| 5, 12, 17 }}
 
=== 2.3.7.41 subgroup ===
Subgroup: 2.3.7.41
 
[[Comma list]]: 64/63, 82/81
 
[[Gencom]]: [2 4/3; 64/63 82/81]
 
[[Mapping|Sval mapping]]: [{{val| 1 2 2 7 }}, {{val| 0 -1 2 -4 }}]
 
[[POTE generator]]: ~4/3 = 490.0323
 
[[TOP tuning|TOP generator]]s: ~2 = 1197.2342, ~4/3 = 488.9029
 
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 49 }}
 
=== 2.3.7.11.41 subgroup ===
Subgroup: 2.3.7.11.41
 
[[Comma list]]: 64/63, 82/81, 99/98
 
[[Gencom]]: [2 4/3; 64/63 82/81 99/98]
 
[[Mapping|Sval mapping]]: [{{val| 1 2 2 1 7 }}, {{val| 0 -1 2 6 -4 }}]
 
[[POTE generator]]: ~4/3 = 492.1787
 
[[TOP tuning|TOP generator]]s: ~2 = 1197.9683, ~4/3 = 491.3454
 
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 39d }}
 
== Purpleheart ==
[[Subgroup]]: 2.3.7
 
[[Comma list]]: 2187/2048
 
{{Mapping|legend=1| 7 11 0 | 0 0 1 }}
 
: mapping generators: ~9/8, ~7
 
[[Optimal tuning]] ([[CTE]]): ~9/8 = 1\7, ~7/4 = 968.826 (~64/63 = 59.746)
 
{{Optimal ET sequence|legend=1| 7, 14, 35, 49bd }}
 
[[Badness]]: 0.0875
 
== Io aka undecimal ==
Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]], and with a generator representing both [[3/2]] and [[16/11]]. It may be considered an [[exotemperament]] by some definitions and not one by others. 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 [[User:CompactStar|CompactStar]] in 2024 based on the [[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.
 
[[Subgroup]]: 2.3.11
 
[[Comma list]]: 33/32
 
{{Mapping|legend=2| 1 1 4 | 0 1 -1 }}
 
: mapping generators: ~2/1, ~3/2
 
[[Optimal tuning]] ([[CTE]]): ~2/1 = 1\1, ~3/2 = 692.713
 
{{Optimal ET sequence|legend=1| 2, 5, 7, 12e }}
 
[[Badness]]: 0.185
 
== Lipsett ==
Lipsett temperament is a pleasantly melodic little temperament with a highly useable 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 off 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]]


{{Mapping|legend=2| 1 2 6 | 0 -2 -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 ([[TE]]): 2/1 = 1200.5469, ~27/23 = 251.9737
{{Optimal ET sequence|legend=1| 5, 12, 17, 97m, 114m, 131m }}


[[Support]]ing [[ET]]s: {{EDOs|5, 9, 14, 19, 24, 43, 62, 81, 100}}
[[Badness]] (Sintel): 0.0841


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/No-fives_subgroup_temperaments"