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.  


== Temperaments with a 2.3.7 gene ==
== Temperaments with a 2.3.7 gene ==
=== Semaphore ===
=== Archy ===
See [[Semaphoresmic clan #Semaphore]].  
See [[Archytas clan #Archy]].  


=== Bleu ===
==== Suhajira ====
Bleu can be described as the 9 & 17 temperament in the no-5 13-limit.  
See [[Rastmic clan #Suhajira]].  


[[Subgroup]]: 2.3.7
==== 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]].


[[Comma list]]: 17496/16807
Subgroup: 2.3.7.11.13


{{Mapping|legend=2| 1 1 2 | 0 5 7 }}
Comma list: 64/63, 364/363, 1078/1053


{{Mapping|legend=3| 1 1 0 2 | 0 5 0 7 }}
Mapping: {{mapping| 1 0 6 21 34 | 0 1 -2 -11 -19 }}


: [[gencom]]: [2 54/49; 17496/16807]
Optimal tunings:  
* WE: ~2 = 1196.9412{{c}}, ~3/2 = 711.0195{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.0039{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~54/49 = 139.848
{{Optimal ET sequence|legend=0| 32f, 69bf, 101beff }}


{{Optimal ET sequence|legend=1| 9, 17, 43, 60d }}
Badness (Sintel): 1.52


[[Tp tuning #T2 tuning|RMS error]]: 1.917 cents
===== 2.3.7.11.13.19 subgroup =====
Subgroup: 2.3.7.11.13.19


==== 2.3.7.11 subgroup ====
Comma list: 64/63, 209/208, 343/342, 364/363
Subgroup: 2.3.7.11


Comma list: 99/98, 864/847
Mapping: {{mapping| 1 0 6 21 34 17 | 0 1 -2 -11 -19 -8 }}


Sval mapping: {{mapping| 1 1 2 3 | 0 5 7 4 }}
Optimal tunings:  
* WE: ~2 = 1197.4072{{c}}, ~3/2 = 711.2733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 712.9612{{c}}


Gencom mapping: {{mapping| 1 1 0 2 3 | 0 5 0 7 4 }}
{{Optimal ET sequence|legend=0| 32f, 69bf }}


: gencom: [2 12/11; 99/98 864/847]
Badness (Sintel): 1.28


Optimal tuning]] (POTE): ~2 = 1\1, ~12/11 = 140.005
=== Semaphore ===
See [[Semaphoresmic clan #Semaphore]].  


Optimal ET sequence: {{Optimal ET sequence| 9, 17, 43, 60d }}
=== Slendric ===
See [[Gamelismic clan #Slendric]].


RMS error: 1.829 cents
=== Slendroschismic ===
See [[5th-octave temperaments #Slendroschismic]].  


==== 2.3.7.11.13 subgroup ====
=== Navy ===
Subgroup: 2.3.7.11.13
This temperament is the common [[restriction]] of [[tsaharuk]] and [[quanic]].  


Comma list: 78/77, 99/98, 144/143
[[Subgroup]]: 2.3.7


Sval mapping: {{mapping| 1 1 2 3 3 | 0 5 7 4 6 }}
[[Comma list]]: 282429536481/281974669312


Gencom mapping: {{mapping| 1 1 0 2 3 3 | 0 5 0 7 4 6 }}
{{Mapping|legend=2| 1 1 0| 0 5 24 }}
: mapping generators: ~2, ~243/224


: gencom: [2 13/12; 78/77 99/98 144/143]
[[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 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.990
{{Optimal ET sequence|legend=1| 17, 60, 77, 94, 171, 265, 436, 2351, 2787, 3223, 3659, 4095, 7754b }}


Optimal ET sequence: {{Optimal ET sequence| 17, 43, 60d }}
[[Badness]] (Sintel): 0.670


RMS error: 1.752 cents
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


; Music
Comma list: 1331/1323, 19712/19683
* ''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]


=== Archy ===
Subgroup-val mapping: {{mapping| 1 1 0 1 | 0 5 24 21 }}
See [[Archytas clan #Archy]].


==== Supra ====
Optimal tunings:
See [[Archytas clan #Supra]].  
* WE: ~2 = 1200.1038{{c}}, ~88/81 = 140.4190{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.4133{{c}}


===== Supraphon =====
{{Optimal ET sequence|legend=0| 17, 60e, 77, 94 }}
See [[Archytas clan #Supraphon]].


==== Suhajira ====
Badness (Sintel): 0.887
See [[Rastmic clan #Suhajira]].  


==== Flutterpyth ====
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13.19
Subgroup: 2.3.7.11.13


Comma list: 64/63, 209/208, 343/342, 364/363
Comma list: 352/351, 729/728, 1331/1323


Mapping: {{mapping| 1 1 4 10 15 9 | 0 -1 -2 -11 -19 -8 }}
Subgroup-val mapping: {{mapping| 1 1 0 1 3 | 0 5 24 21 6 }}


Optimal tuning (CTE): ~3/2 = 713.459
Optimal tunings:
* WE: ~2 = 1199.8640{{c}}, ~13/12 = 140.4206{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4292{{c}}


Restricted to 2.3.7.11.13, this temperament is a no-5 restriction of 13-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.
{{Optimal ET sequence|legend=0| 17, 60e, 77, 94 }}


=== Skwares ===
Badness (Sintel): 0.520
{{Main| Squares }}
 
Skwares is the no-5 restriction of [[squares]].  


=== Lee ===
[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7


[[Comma list]]: 19683/19208
[[Comma list]]: 177147/175616


{{Mapping|legend=2| 1 3 6 | 0 -4 -9 }}
{{Mapping|legend=2| 1 0 -3 | 0 3 11 }}


{{Mapping|legend=3| 1 3 0 6 | 0 -4 0 -9 }}
{{Mapping|legend=3| 1 0 0 -3 | 0 3 0 11 }}
: mapping generators: ~2, ~81/56


: [[gencom]]: [2 9/7; 19683/19208]
[[Optimal tuning]]s:
* [[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 }}


[[Optimal tuning]] ([[POTE]]): ~9/7 = 425.365
{{Optimal ET sequence|legend=1| 17, 36, 89, 125, 161, 358, 519b }}


{{Optimal ET sequence|legend=1| 14, 17, 31, 48, 79, 189b, 268bd, 347bd }}
[[Badness]] (Sintel): 0.741


[[Tp tuning #T2 tuning|RMS error]]: 1.149 cents
=== Buzzard ===
See [[Buzzardsmic clan #Buzzard]].  


Related temperament: [[Meantone family #Squares|squares]]
=== Hemif ===
Hemif is the no-5 [[restriction]] of [[hemififths]], and the add-7 [[extension]] of [[namo]].


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


Comma list: 99/98, 243/242
[[Comma list]]: 1605632/1594323


Sval mapping: {{mapping| 1 3 6 7 | 0 -4 -9 -10 }}
{{Mapping|legend=2| 1 1 -1 | 0 2 13 }}


Gencom mapping: {{mapping| 1 3 0 6 7 | 0 -4 0 -9 -10 }}
{{Mapping|legend=3| 1 1 0 -1 | 0 2 0 13 }}
: mapping generators: ~2, ~2187/1792


: gencom: [2 9/7; 99/98 243/242]
[[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 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.244
{{Optimal ET sequence|legend=1| 17, 41, 58, 99, 239, 338, 437, 775b, 1212bb }}


Optimal ET sequence: {{Optimal ET sequence| 5, 8, 11, 14, 17, 31, 48, 79, 127, 206bcd }}
[[Badness]] (Sintel): 0.901


RMS error: 1.099 cents
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


===== 2.3.7.11.13 =====
Comma list: 243/242, 896/891
Subgroup: 2.3.7.11.13


Comma list: 78/77, 99/98, 243/242
Subgroup-val mapping: {{mapping| 1 1 -1 2 | 0 2 13 5 }}


Sval mapping: {{mapping| 1 3 6 7 9 | 0 -4 -9 -10 -15 }}
Gencom mapping: {{mapping| 1 1 0 -1 2 | 0 2 0 13 5 }}


Gencom mapping: {{mapping| 1 3 0 6 7 9 | 0 -4 0 -9 -10 -15}}
Optimal tunings:  
* WE: ~2 = 1199.2633{{c}}, ~11/9 = 351.3189{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.4593{{c}}


: gencom: [2 9/7; 78/77, 99/98, 243/242]
{{Optimal ET sequence|legend=0| 17, 41, 58, 99e }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 424.457
Badness (Sintel): 0.409


Optimal ET sequence: {{Optimal ET sequence| 17, 48f, 65ef, 82d, 147df }}
===== 2.3.7.11.13 =====
 
RMS error: 1.769 cents
 
===== Skwairs =====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


Comma list: 99/98, 144/143, 243/242
Comma list: 144/143, 243/242, 364/363


Sval mapping: {{mapping| 1 3 6 7 3 | 0 -4 -9 -10 2 }}
Sval mapping: {{mapping| 1 1 -1 2 4 | 0 2 13 5 -1 }}


Gencom mapping: {{mapping| 1 3 0 6 7 3 | 0 -4 0 -9 -10 2 }}
Gencom mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 0 13 5 -1 }}


: gencom: [2 9/7; 99/98, 144/143, 243/242]
Optimal tunings:  
* WE: ~2 = 1198.7603{{c}}, ~11/9 = 351.3275{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.6042{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 424.702
{{Optimal ET sequence|legend=0| 17, 41, 58, 331deeeffff }}


Optimal ET sequence: {{Optimal ET sequence| 14, 17, 31 }}
Badness (Sintel): 0.358


RMS error: 1.290 cents
===== Heartful =====
{{See also| Heartland }}


===== Byhearted =====
Related temperaments: [[bunya]].  
: ''For the full 19-limit version of this temperament, see [[Tetracot family #Byhearted]].''


Subgroup: 2.3.7.11.19
Subgroup: 2.3.7.11.19


Comma list: 99/98, 243/242, 363/361
Comma list: 243/242, 896/891, 1083/1078


Sval mapping: {{mapping| 2 2 3 4 5 | 0 4 9 10 12 }}
Subgroup-val mapping: {{mapping| 1 1 -1 2 0 | 0 4 26 10 29 }}
: mapping generators: ~2, ~21/19


: gencom: [209/147 21/19; 99/98 243/242 363/361]
Optimal tunings:  
* WE: ~2 = 1199.2636{{c}}, ~21/19 = 175.6963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.7665{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 174.735
{{Optimal ET sequence|legend=0| 34dh, 41, 116e, 157e }}


Optimal ET sequence: {{Optimal ET sequence| 14, 34dh, 48, 110e, 158e }}
Badness (Sintel): 0.984


RMS error: 0.8727 cents
=== Hearts ===
{{See also| Heartland }}


=== Harrison ===
This temperament is the common [[restriction]] of [[monkey]] and [[sesquiquartififths]].  
Subgroup: 2.3.7
 
[[Comma]]: [[59049/57344]]
 
[[Gencom]]: [2 3/2; 59049/57344]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 -3}}, {{val|0 1 0 10}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 -3}}, {{val|0 1 10}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 696.544
 
{{Optimal ET sequence|legend=1| 12, 19, 31, 112b, 143b, 174b }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.226 cents
 
Related temperament: [[Meantone family #Septimal meantone|septimal meantone]]
 
=== Leapfrog ===
{{Main| Leapday }}
{{See also| Gentle region }}
 
In regular [[13-limit]] [[leapday]], the mapping for prime 5 is very complex at +21 generator steps. Furthermore, adding prime 5 to rank-3 [[parapythic]] is arguably against the original vision of it as a 2.3.7.11.13-subgroup temperament, so avoiding prime 5 may be preferred for this reason also. This results in no-5's leapday, or leapfrog, which as aforementioned is much lower in badness, but it also allows more tunings to be used: a notable [[patent val]] 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]]). In other words, the only reason 80edo was "disqualified" from leapday is that the mapping for prime 5 constrains the tuning range which is naturally more flexible as a no-5's 13-limit temperament, which is also a sign of leapfrog being very efficient.
 
Other related temperaments include [[leapweek]] and [[srutal]].  


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


[[Comma list]]: 14680064/14348907
[[Comma list]]: 34451725707/34359738368


{{Mapping|legend=2| 1 0 -21 | 0 1 15 }}
{{Mapping|legend=2| 1 1 5 | 0 4 -15 }}
 
: mapping generators: ~2, ~567/512
{{Mapping|legend=3| 1 1 0 -6 | 0 1 0 15 }}
 
: [[gencom]]: [2 3/2; 14680064/14348907]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1\1, ~3/2 = 704.721
* [[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 }}


{{Optimal ET sequence|legend=1| 17, 46, 63 }}
{{Optimal ET sequence|legend=1| 7, 27d, 34, 41, 89, 130, 171, 643, 814, 985, 1156 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6202 cents
[[Badness]] (Sintel): 0.959


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


[[Comma list]]: 896/891, 1331/1323
Comma list: 243/242, 65536/65219


{{Mapping|legend=2| 1 0 -21 -14 | 0 1 15 11 }}
Subgroup-val mapping: {{mapping| 1 1 5 2 | 0 4 -15 10 }}


{{Mapping|legend=3| 1 1 0 -6 -3 | 0 1 0 15 11 }}
Optimal tunings:
* WE: ~2 = 1199.8467{{c}}, ~256/231 = 175.3468{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/231 = 175.3691{{c}}


: [[gencom]]: [2 3/2; 896/891 1331/1323]
{{Optimal ET sequence|legend=0| 7, 34, 41, 89, 130, 349e, 479e }}


[[Optimal tuning]]s:  
Badness (Sintel): 0.801
* [[POTE]]: ~3/2 = 704.753


{{Optimal ET sequence|legend=1| 17, 46, 63 }}
==== 2.3.7.11.19 ====
Subgroup: 2.3.7.11.19


[[Tp tuning #T2 tuning|RMS error]]: 0.6047 cents
Comma list: 243/242, 513/512, 1083/1078


==== 2.3.7.11.13 ====
Subgroup-val mapping: {{mapping| 1 1 5 2 6 | 0 4 -15 10 -12 }}
[[Subgroup]]: 2.3.7.11.13


[[Comma list]]: 169/168, 352/351, 364/363
Optimal tunings:  
* WE: ~2 = 1199.9531{{c}}, ~21/19 = 175.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3417{{c}}


{{Mapping|legend=2| 1 0 -21 -14 -9 | 0 1 15 11 8 }}
{{Optimal ET sequence|legend=0| 7, 34, 41, 89, 130, 219 }}


{{Mapping|legend=3| 1 1 0 -6 -3 -1 | 0 1 0 15 11 8 }}
Badness (Sintel): 0.529


: [[gencom]]: [2 3/2; 169/169 352/351 364/363]
=== Magi ===
This temperament is the no-5 [[restriction]] of [[magic]], tempering out the [[septimagic comma]].


[[Optimal tuning]]s:  
[[Subgroup]]: 2.3.7
* [[CTE]]: ~2 = 1\1, ~3/2 = 704.633
* [[POTE]]: ~2 = 1\1, ~3/2 = 704.745


{{Optimal ET sequence|legend=1| 17, 46, 63 }}
[[Comma list]]: 537824/531441


[[Tp tuning #T2 tuning|RMS error]]: 0.7541 cents
{{Mapping|legend=2| 1 0 -1 | 0 5 12 }}
: mapping generators: ~2, ~243/196


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


[[Comma list]]: 169/168, 208/207, 352/351, 364/363
{{Optimal ET sequence|legend=1| 19, 22, 41, 104, 145, 186, 331 }}


{{Mapping|legend=2| 1 0 -21 -14 -9 -5 | 0 1 15 11 8 6 }}
[[Badness]] (Sintel): 1.30


{{Mapping|legend=3| 1 1 0 -6 -3 -1 0 0 1 | 0 1 0 15 11 8 0 0 6 }}
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


: [[gencom]]: [2 3/2; 169/169 208/207 352/351 364/363]
Comma list: 896/891, 26411/26244


[[Optimal tuning]]s:  
Subgroup-val mapping: {{mapping| 1 0 -1 6 | 0 5 12 -8 }}
* [[POTE]]: ~2 = 1\1, ~3/2 = 704.729


{{Optimal ET sequence|legend=1| 17, 46, 63 }}
Optimal tunings:
* WE: ~2 = 1199.4843{{c}}, ~96/77 = 380.6040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~96/77 = 380.7490{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.6265 cents
{{Optimal ET sequence|legend=0| 19, 22, 41, 63, 104 }}


====== 2.3.7.11.13.23.29 ======
Badness (Sintel): 0.661
[[Subgroup]]: 2.3.7.11.13.23.29


[[Comma list]]: 169/168, 208/207, 232/231, 352/351, 364/363
===== Caspar =====
Subgroup: 2.3.7.11.13


{{Mapping|legend=2| 1 0 -21 -14 -9 -5 -38 | 0 1 15 11 8 6 27 }}
Comma list: 144/143, 343/338, 729/728


{{Mapping|legend=3| 1 1 0 -6 -3 -1 0 0 1 -11 | 0 1 0 15 11 8 0 0 6 27 }}
Subgroup-val mapping: {{mapping| 1 0 -1 6 -2 | 0 5 12 -8 18 }}


: [[gencom]]: [2 3/2; 169/169 208/207 352/351 364/363]
Optimal tunings:  
* WE: ~2 = 1199.3353{{c}}, ~26/21 = 380.3206{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/21 = 380.5041{{c}}


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
* [[POTE]]: ~2 = 1\1, ~3/2 = 704.729


{{Optimal ET sequence|legend=1| 17, 46, 63 }}
Badness (Sintel): 1.09


; Music
====== Twenothology ======
* Suite for Harpsichord in A Locrian, tuning: Eb-G# in [[46edo]] by [[Inthar]] (in progress):
Subgroup: 2.3.7.11.13.29
** 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: 144/143, 232/231, 343/338, 729/728
{{see also|Heartland}}


Subgroup: 2.3.7
Subgroup-val mapping: {{mapping| 1 0 -1 6 -2 2 | 0 5 12 -8 18 9 }}


[[Comma list]]: 5764801/5668704
Optimal tunings:  
* WE: ~2 = 1199.6175{{c}}, ~26/21 = 380.4049{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/21 = 380.5103{{c}}


[[Gencom]]: [2 343/324; 5764801/5668704]
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


[[Mapping|Sval mapping]]: [{{val|1 1 2}}, {{val|0 8 11}}]
Badness (Sintel): 0.964


[[Tp tuning|POL2 generator]]: ~343/324 = 87.8304
===== Melchior =====
Subgroup: 2.3.7.11.13


{{Optimal ET sequence|legend=1| 14, 27, 41 }}
Comma list: 352/351, 364/363, 26411/26244


[[Tp tuning #T2 tuning|RMS error]]: 0.5041 cents
Subgroup-val mapping: {{mapping| 1 0 -1 6 11 | 0 5 12 -8 -23 }}


Related temperaments: [[octacot]]
Optimal tunings:  
* WE: ~2 = 1199.4887{{c}}, ~96/77 = 380.6034{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~96/77 = 380.7669{{c}}


==== 2.3.7.11 ====
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63, 104 }}
Subgroup: 2.3.7.11


[[Comma list]]: 243/242, 2401/2376
Badness (Sintel): 0.710


[[Gencom]]: [2 22/21; 243/242 2401/2376]
===== Balthazar =====
Subgroup: 2.3.7.11.13


[[Mapping|Sval mapping]]: [{{val|1 1 2 2}}, {{val|0 8 11 20}}]
Comma list: 169/168, 896/891, 26411/26244


[[Tp tuning|POL2 generator]]: ~22/21 = 87.6512
Subgroup-val mapping: {{mapping| 1 0 -1 6 1 | 0 10 24 -16 17 }}
: mapping generators: ~2, ~143/128


{{Optimal ET sequence|legend=1| 14, 27e, 41, 96d, 137d, 178d }}
Optimal tunings:
* WE: ~2 = 1199.7322{{c}}, ~143/128 = 190.3647{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~143/128 = 190.4016{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.7147 cents
{{Optimal ET sequence|legend=0| 19, 44, 63, 145f }}


Related temperaments: [[octacot]]
Badness (Sintel): 1.82


==== 2.3.7.11.19 ====
==== Hogwarts ====
Subgroup: 2.3.7.11.19
Subgroup: 2.3.7.29


[[Comma list]]: 133/132, 243/242, 343/342
Comma list: 784/783, 5887/5832


[[Gencom]]: [2 19/18; 133/132 243/242 343/342]
Subgroup-val mapping: {{mapping| 1 0 -1 2 | 0 5 12 9 }}


[[Mapping|Sval mapping]]: [{{val|1 1 2 2 3}}, {{val|0 8 11 20 17}}]
Optimal tunings:  
* WE: ~2 = 1200.1518{{c}}, ~36/29 = 380.6661{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/29 = 380.6375{{c}}


[[Tp tuning|POL2 generator]]: ~19/18 = 87.6684
{{Optimal ET sequence|legend=0| 19, 22, 41, 145, 186j, 227j }}


{{Optimal ET sequence|legend=1| 14, 27e, 41 }}
Badness (Sintel): 0.424


[[Tp tuning #T2 tuning|RMS error]]: 0.7065 cents
=== Skwares ===
{{Main| Squares }}


Related temperaments: [[octacot]]
Skwares is the no-5 [[restriction]] of [[squares]].


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


[[Comma list]]: 537824/531441
[[Comma list]]: 19683/19208


[[Gencom]]: [2 243/196; 537824/531441]
{{Mapping|legend=2| 1 -1 -3 | 0 4 9 }}


[[Mapping|Sval mapping]]: [{{val|1 0 -1}}, {{val|0 5 12}}]
{{Mapping|legend=3| 1 -1 0 -3 | 0 4 0 9 }}
: mapping generators: ~2, ~14/9


[[Tp tuning|POL2 generator]]: ~243/196 = 380.661
[[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 }}


{{Optimal ET sequence|legend=1| 19, 22, 41, 104, 145, 186 }}
{{Optimal ET sequence|legend=1| 14, 17, 31, 48, 79 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4277 cents
[[Badness]] (Sintel): 1.55


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


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


[[Gencom]]: [2 96/77; 896/891 26411/26244]
Subgroup-val mapping: {{mapping| 1 -1 -3 -3 | 0 4 9 10 }}


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


[[Tp tuning|POL2 generator]]: ~96/77 = 380.768
Optimal tunings:  
* WE: ~2 = 1200.3726{{c}}, ~14/9 = 774.9970{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.8197{{c}}


{{Optimal ET sequence|legend=1| 19, 22, 41, 63, 104 }}
{{Optimal ET sequence|legend=0| 14, 17, 31, 48, 79, 127 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents
Badness (Sintel): 0.405


===== Balthazar =====
===== 2.3.7.11.13 =====
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, 243/242


[[Gencom]]: [2 143/128; 169/168 896/891 26411/26244]
Subgroup-val mapping: {{mapping| 1 -1 -3 -3 -6 | 0 4 9 10 15 }}


[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 1}}, {{val|0 10 24 -16 17}}]
Gencom mapping: {{mapping| 1 -1 0 -3 -3 -6 | 0 4 0 9 10 15}}


[[Tp tuning|POL2 generator]]: ~143/128 = 190.407
Optimal tunings:  
* WE: ~2 = 1199.3264{{c}}, ~14/9 = 775.1081{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 775.4463{{c}}


{{Optimal ET sequence|legend=1| 19, 44, 63, 145f }}
{{Optimal ET sequence|legend=0| 14f, 17, 48f }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6937 cents
Badness (Sintel): 0.587


===== Caspar =====
===== Skwairs =====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


[[Comma list]]: 144/143, 343/338, 729/728
Comma list: 99/98, 144/143, 243/242


[[Gencom]]: [2 26/21; 144/143 343/338 729/728]
Subgroup-val mapping: {{mapping| 1 -1 -3 -3 5 | 0 4 9 10 -2 }}


[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 -2}}, {{val|0 5 12 -8 18}}]
Gencom mapping: {{mapping| 1 -1 0 -3 -3 5 | 0 4 0 9 10 -2 }}


[[Tp tuning|POL2 generator]]: ~26/21 = 380.531
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| 19, 22f, 41 }}
{{Optimal ET sequence|legend=0| 14, 17, 31, 48, 65d, 113df }}


[[Tp tuning #T2 tuning|RMS error]]: 1.032 cents
Badness (Sintel): 0.538


===== Melchior =====
===== Byhearted =====
Subgroup: 2.3.7.11.13
This temperament is the restriction of [[weasel]] to the 2.3.7.11.19 subgroup.


[[Comma list]]: 352/351, 364/363, 26411/26244
Subgroup: 2.3.7.11.19


[[Gencom]]: [2 96/77; 352/351 364/363 26411/26244]
Comma list: 99/98, 243/242, 363/361


[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 11}}, {{val|0 5 12 -8 -23}}]
Subgroup-val mapping: {{mapping| 2 2 3 4 5 | 0 4 9 10 12 }}
: mapping generators: ~209/147, ~21/19


[[Tp tuning|POL2 generator]]: ~96/77 = 380.766
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| 19f, 22, 41, 63, 104 }}
{{Optimal ET sequence|legend=0| 14, 34dh, 48, 110e }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3891 cents
Badness (Sintel): 0.893


==== Hogwarts ====
=== Harrison ===
Subgroup: 2.3.7.29
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.  


[[Comma list]]: 784/783, 5887/5832
[[Subgroup]]: 2.3.7


[[Gencom]]: [2 36/29; 784/783 5887/5832]
[[Comma list]]: [[59049/57344]]


[[Mapping|Sval mapping]]: [{{val|1 0 -1 2}}, {{val|0 5 12 9}}]
{{Mapping|legend=2| 1 0 -13 | 0 1 10 }}


[[Tp tuning|POL2 generator]]: ~36/29 = 380.618
{{Mapping|legend=3| 1 0 0 -13 | 0 1 0 10 }}
: mapping generators: ~2, ~3


{{Optimal ET sequence|legend=1| 19, 22, 41, 145, 186j, 227j }}
[[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 }}


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


[[Comma list]]: 144/143, 232/231, 343/338, 729/728
[[Badness]] (Sintel): 2.35


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


[[Tp tuning|POL2 generator]]: ~26/21 = 380.526
[[Subgroup]]: 2.3.7


{{Optimal ET sequence|legend=1| 19, 22f, 41 }}
[[Comma list]]: 17496/16807


=== Lee ===
{{Mapping|legend=2| 1 1 2 | 0 5 7 }}
Subgroup: 2.3.7


[[Comma]]: 177147/175616
{{Mapping|legend=3| 1 1 0 2 | 0 5 0 7 }}
 
: mapping generators: ~2, ~54/49
[[Gencom]]: [2 81/56; 177147/175616]
 
[[Gencom|Gencom mapping]]: [{{val|1 0 0 -3}}, {{val|0 3 0 11}}]
 
[[Mapping|Sval mapping]]: [{{val|1 0 -3}}, {{val|0 3 11}}]
 
[[Tp tuning|POL2 generator]]: ~81/56 = 633.525
 
{{Optimal ET sequence|legend=1| 17, 36, 89, 125, 161, 358, 519b }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3519 cents
 
=== Buzzard ===
{{See also| Vulture family #Buzzard }}
 
[[Subgroup]]: 2.3.7
 
[[Comma list]]: 65536/64827
 
{{Mapping|legend=1| 1 0 4 | 0 4 -3 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~21/16 = 475.727
* [[WE]]: ~2 = 1199.3538{{c}}, ~54/49 = 139.848{{c}}
* [[CWE]]: ~2 = 1200.000, ~21/16 = 475.833
: [[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| 5, 33, 38, 43, 48, 53, 58 }}
{{Optimal ET sequence|legend=1| 8d, 9, 17, 43, 60d, 103d }}


[[Badness]]:  
[[Badness]] (Sintel): 2.48
* Smith: 0.0240
* Dirichlet: 0.824


=== [[Slendric]] ===
==== 2.3.7.11 subgroup ====
Subgroup: 2.3.7
Subgroup: 2.3.7.11


[[Comma]]: 1029/1024
Comma list: 99/98, 864/847


[[Gencom]]: [2 8/7; 1029/1024]
Subgroup-val mapping: {{mapping| 1 1 2 3 | 0 5 7 4 }}


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


[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 3 -1}}]
Optimal tunings:  
* WE: ~2 = 1198.6613{{c}}, ~12/11 = 139.8489{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.7839{{c}}


[[Tp tuning|POL2 generator]]: ~8/7 = 233.688
{{Optimal ET sequence|legend=0| 8d, 9, 17, 43, 60d }}


{{Optimal ET sequence|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}
Badness (Sintel): 0.624


[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
==== 2.3.7.11.13 subgroup ====
Subgroup: 2.3.7.11.13


==== [[Gamelismic clan #Baladic|Baladic]] ====
Comma list: 78/77, 99/98, 144/143
Subgroup: 2.3.7.13


[[Comma list]]: 169/168, 1029/1024
Subgroup-val mapping: {{mapping| 1 1 2 3 3 | 0 5 7 4 6 }}


[[Gencom]]: [91/64 8/7; 169/168 1029/1024]
Gencom mapping: {{mapping| 1 1 0 2 3 3 | 0 5 0 7 4 6 }}


[[Mapping|Sval mapping]]: [{{val|2 2 6 7}}, {{val|0 3 -1 1}}]
Optimal tunings:  
* WE: ~2 = 1198.9768{{c}}, ~13/12 = 139.8704{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8166{{c}}


[[Tp tuning|POL2 generator]]: ~8/7 = 233.6044
{{Optimal ET sequence|legend=0| 8d, 9, 17, 43, 60d }}


{{Optimal ET sequence|legend=1| 10, 26, 36, 154…, 190…, 226…, 262… }}
Badness (Sintel): 0.400


[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents
; Music
* ''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]


===== 2.3.7.13.17 =====
=== Doublehearted ===
[[Subgroup]]: 2.3.7.13.17
{{See also| Heartland }}
 
[[Comma list]]: 169/168, 273/272, 289/288
 
[[Gencom]]: [17/12 8/7; 169/168 273/272 289/288]
 
{{Mapping|legend=2| 2 2 6 7 7 | 0 3 -1 1 3 }}


[[Tp tuning|POL2 generator]]: ~8/7 = 233.6155
This temperament is the no-5 [[restriction]] of [[octacot]].  
 
{{Optimal ET sequence|legend=1| 10, 26, 36, 154…, 190…, 226… }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5073 cents
 
=== Hemif ===
Related temperaments: [[hemififths]], [[namo]]


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


[[Comma list]]: 1605632/1594323
[[Comma list]]: 5764801/5668704


{{Mapping|legend=2| 1 1 -1 | 0 2 13 }}
{{Mapping|legend=2| 1 1 2 | 0 8 11 }}
: mapping generators: ~2, ~343/342


{{Mapping|legend=3| 1 1 0 -1 | 0 2 0 13 }}
[[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 }}


: [[gencom]]: [2 2187/1792; 1605632/1594323]
{{Optimal ET sequence|legend=1| 14, 27, 41 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2187/1792 = 351.485
[[Badness]] (Sintel): 2.62
 
{{Optimal ET sequence|legend=1| 7, 17, 41, 58, 99 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2344 cents


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


Comma list: 243/242, 896/891
Comma list: 243/242, 2401/2376
 
Sval mapping: {{mapping| 1 1 -1 2 | 0 2 13 5 }}


Gencom mapping: {{mapping| 1 1 0 -1 2 | 0 2 0 13 5 }}
Subgroup-val mapping: {{mapping| 1 1 2 2 | 0 8 11 20 }}


: gencom: [2 11/9; 243/242 896/891]
Optimal tunings:  
* WE: ~2 = 1200.4071{{c}}, ~22/21 = 87.6809{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 87.6902{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.535
{{Optimal ET sequence|legend=0| 14, 27e, 41, 96d, 137d, 178d }}


Optimal ET sequence: {{Optimal ET sequence| 7, 17, 41, 58, 99e }}
Badness (Sintel): 0.815


[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents
==== 2.3.7.11.19 ====
Subgroup: 2.3.7.11.19


===== 2.3.7.11.13 =====
Comma list: 133/132, 243/242, 343/342
Subgroup: 2.3.7.11.13


Comma list: 144/143, 243/242, 364/363
Subgroup-val mapping: {{mapping| 1 1 2 2 3 | 0 8 11 20 17 }}


Sval mapping: {{mapping| 1 1 -1 2 4 | 0 2 13 5 -1 }}
Optimal tunings:  
* WE: ~2 = 1200.6100{{c}}, ~19/18 = 87.7129{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/18 = 87.7285{{c}}


Gencom mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 0 13 5 -1 }}
{{Optimal ET sequence|legend=0| 14, 27e, 41, 137dh }}


: gencom: [2 11/9; 144/143 243/242 364/363]
Badness (Sintel): 0.560


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.691
=== Purpleheart ===
{{Main| Whitewood }}


Optimal ET sequence: {{Optimal ET sequence| 7, 10, 17, 24, 41, 58 }}
[[Subgroup]]: 2.3.7


RMS error: 0.7167 cents
[[Comma list]]: 2187/2048


===== Heartful =====
{{Mapping|legend=2| 7 11 0 | 0 0 1 }}
{{See also| Heartland }}
: mapping generators: ~9/8, ~7


Related temperaments: [[bunya]]
[[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 }}


Subgroup: 2.3.7.11.19
{{Optimal ET sequence|legend=1| 7, 14, 35, 49bd }}


Comma list: 243/242, 896/891, 1083/1078
[[Badness]] (Sintel): 3.00


Sval mapping: {{mapping| 1 1 -1 2 0 | 0 4 26 10 29 }}
=== 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: [2 21/19; 243/242 896/891 1083/1078]
[[Subgroup]]: 2.3.7


Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.804
[[Comma list]]: {{monzo| 83 -1 -29 }}


Optimal ET sequence: {{Optimal ET sequence| 34dh, 41, 116e, 157e }}
{{Mapping|legend=2| 1 -4 3 | 0 29 -1 }}
: mapping generators: ~2, ~8/7


RMS error: 0.5360 cents
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.9871{{c}}, ~8/7 = 231.1001{{c}}
=== Hearts ===
: [[error map]]: {{val| -0.013 +0.000 +0.035 }}
{{see also|Heartland}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 231.1024{{c}}
 
: error map: {{val| 0.000 +0.014 +0.072 }}
Subgroup: 2.3.7
 
[[Comma list]]: 34451725707/34359738368 (trila-quadzo comma)
 
[[Gencom]]: [2 567/512; 34451725707/34359738368]
 
[[Mapping|Sval mapping]]: [{{val|1 1 5}}, {{val|0 4 -15}}]
 
[[Tp tuning|POL2 generator]]: ~567/512 = 175.433


{{Optimal ET sequence|legend=1| 7, 27d, 34, 41, 89, 130, 171 }}
{{Optimal ET sequence|legend=0| 26, 83, 109, 135, 566, 701, 836, 971, 1106, 2077, 5260, 7337, 9414d }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0529 cents
[[Badness]] (Sintel): 3.06
 
Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquiquartififths]]


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


[[Comma list]]: 243/242, 65536/65219
Comma list: 43923/43904, 5767168/5764801


[[Gencom]]: [2 256/231; 243/242 65536/65219]
Subgroup-val mapping: {{mapping| 1 -4 3 5 | 0 29 -1 -8 }}


[[Mapping|Sval mapping]]: [{{val|1 1 5 2}}, {{val|0 4 -15 10}}]
Optimal tunings:  
* WE: ~2 = 1200.0050{{c}}, ~8/7 = 231.1024{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.1015{{c}}


[[Tp tuning|POL2 generator]]: ~256/231 = 175.369
{{Optimal ET sequence|legend=0| 26, 83, 109, 135, 566, 701, 836, 971, 1807, 2778, 4585 }}


{{Optimal ET sequence|legend=1| 7, 27de, 34, 41, 89, 130 }}
Badness (Sintel): 0.324


[[Tp tuning #T2 tuning|RMS error]]: 0.3224 cents
=== Leapfrog ===
{{See also| Gentle region }}


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


==== 2.3.7.11.19 ====
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]]).  
Subgroup: 2.3.7.11.19


[[Comma list]]: 243/242, 513/512, 1083/1078
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]].


[[Gencom]]: [2 21/19; 243/242 513/512 1083/1078]
[[Subgroup]]: 2.3.7


[[Mapping|Sval mapping]]: [{{val|1 1 5 2 6}}, {{val|0 4 -15 10 -12}}]
[[Comma list]]: 14680064/14348907


[[Tp tuning|POL2 generator]]: ~21/19 = 175.341
{{Mapping|legend=2| 1 0 -21 | 0 1 15 }}


{{Optimal ET sequence|legend=1| 7, 27deh, 34, 41, 89, 130, 219 }}
{{Mapping|legend=3| 1 0 0 -21 | 0 1 0 15 }}
: mapping generators: ~2, ~3


[[Tp tuning #T2 tuning|RMS error]]: 0.3121 cents
[[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 }}


Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquart]]
{{Optimal ET sequence|legend=1| 17, 46, 63, 235b, 298b, 361bd, 424bd, 487bbd }}


=== Navy ===
[[Badness]] (Sintel): 4.33
[[Subgroup]]: 2.3.7


[[Comma list]]: 282429536481/281974669312
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


{{Mapping|legend=1|1 1 0|0 5 24}}
Comma list: 896/891, 1331/1323


[[Tp tuning|POL2 generator]]: ~243/224 = 140.366
Subgroup-val mapping: {{mapping| 1 0 -21 -14 | 0 1 15 11 }}


{{Optimal ET sequence|legend=1| 17, 60, 77, 94, 171, 265, 436 }}
Gencom mapping: {{mapping| 1 0 0 -21 -14 | 0 1 0 15 11 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0296 cents
Optimal tunings:  
* WE: ~2 = 1199.2683{{c}}, ~3/2 = 704.3230{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.6926{{c}}


Related temperaments: [[Schismatic family|tsaharuk]], [[Hemifamity temperaments|quanic]]
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


==== 2.3.7.11 ====
Badness (Sintel): 0.629
Subgroup: 2.3.7.11


[[Comma list]]: 1331/1323, 19712/19683
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13


{{Mapping|legend=1|1 1 0 1|0 5 24 21}}
Comma list: 169/168, 352/351, 364/363


[[Tp tuning|POL2 generator]]: ~88/81 = 140.407
Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 | 0 1 15 11 8 }}


{{Optimal ET sequence|legend=1| 17, 60e, 77, 94, 359e, 453ee, 547ee, 641ee }}
Gencom mapping: {{mapping| 1 0 0 -21 -14 -9 | 0 1 0 15 11 8 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3778 cents
Optimal tunings:  
* WE: ~2 = 1199.5654{{c}}, ~3/2 = 704.4898{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7084{{c}}


Related temperaments: [[Schismatic family|tsaharuk]], [[Hemifamity temperaments|quanic]]
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


==== 2.3.7.11.13 ====
Badness (Sintel): 0.436
Subgroup: 2.3.7.11.13


[[Comma list]]: 352/351, 729/728, 1331/1323
===== Skidoo =====
Subgroup: 2.3.7.11.13.23


{{Mapping|legend=1|1 1 0 1 3|0 5 24 21 6}}
Comma list: 169/168, 208/207, 352/351, 364/363


[[Tp tuning|POL2 generator]]: ~13/12 = 140.437
Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 -5 | 0 1 15 11 8 6 }}


{{Optimal ET sequence|legend=1| 17, 60e, 77, 94 }}
Gencom mapping: {{mapping| 1 0 0 -21 -14 -9 0 0 -5 | 0 1 0 15 11 8 0 0 6 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4044 cents
Optimal tunings:  
* WE: ~2 = 1199.6639{{c}}, ~3/2 = 704.5315{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7021{{c}}


Related temperaments: [[Schismatic family|tsaharuk]], [[Hemifamity temperaments|quanic]]
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


=== Slendrismic ===
Badness (Sintel): 0.356
{{see also| 5th-octave temperaments #Slendrismic }}
In slendrismic, the period (1＼5) is given a very accurate interpretation of [[147/128]] = ([[3/2]])/([[8/7]])<sup>2</sup> = [[8/7]] * [[1029/1024|1029/1024 = S7/S8]], which is a significant interval as it is the "harmonic 5edostep" in that it's a [[rooted]] (/2^n) interval that approximates 1＼5 very well. The generator is [[1029/1024]], the difference between [[8/7]] and [[147/128]] and therefore between 3/2 and (8/7)<sup>3</sup>. The temperament is named for the very "slender" generator as well as as a pun on "[[slendric]]" (which it shouldn't be confused with). One can consider this as a microtemperament counterpart to [[cloudy]], which equates them.


[[Subgroup]]: 2.3.7
====== 2.3.7.11.13.23.29 ======
Subgroup: 2.3.7.11.13.23.29


[[Comma list]]: 68719476736/68641485507
Comma list: 169/168, 208/207, 232/231, 352/351, 364/363


{{Mapping|legend=1|5 8 14|0 -2 1}}
Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 -5 -38 | 0 1 15 11 8 6 27 }}


[[Tp tuning|POL2 generator]]: ~1029/1024 = 8.9906
Gencom mapping: {{mapping| 1 0 0 -21 -14 -9 -5 0 0 -38 | 0 1 0 15 11 8 0 0 6 27 }}


{{Optimal ET sequence|legend=1| 130, 135, 265, 400, 935, 1335, 1735 }}
Optimal tunings:
* WE: ~2 = 1199.5755{{c}}, ~3/2 = 704.5533{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7750{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.0212 cents
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


Related temperaments: [[Pental family|hemipental]]
Badness (Sintel): 0.441


==== Pentadecoid ====
; Music
{{see also| 15th-octave temperaments #Pentadecoid }}
* Suite for Harpsichord in A Locrian, tuning: Eb–G# in [[46edo]] by [[Inthar]] (in progress):
In pentadecoid, the period (1＼15) is given an interpretation of [[22/21]], and four of them represent keenanismic minor third, [[77/64]] = ([[147/128]]) * ([[22/21]]). The temperament is named for an analog of its relating temperaments, [[decoid]] (130 &amp; 140) and [[decoid|triacontoid]] (120 &amp; 150). Triacontoid is a weak extension of pentadecoid, as well as [[Landscape microtemperaments|slendscape]] (255 &amp; 270).
** 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]]


[[Subgroup]]: 2.3.7.11
=== Superslendric ===
In superslendric, eight [[8/7]]'s are equated to [[3/1]]. This relates it to [[8edt]].


[[Comma list]]: 1362944/1361367, 1771561/1769472
[[Subgroup]]: 2.3.7


[[Mapping]]: {{mapping| 15 108 0 94 | 0 -2 1 -1 }}
[[Comma list]]: 17294403/16777216


: mapping generators: ~22/21, ~7
{{Mapping|legend=2| 1 0 3 | 0 8 -1 }}
: mapping generators: ~2, ~8/7


[[Optimal tuning]] ([[CTE]]): ~22/21 = 1＼15, ~8/7 = 231.0071 (~1029/1024 = 8.9929)
[[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 }}


{{Optimal ET sequence|legend=1| 120, 135, 660, 795, 930, 1065, 1200 }}
{{Optimal ET sequence|legend=1| 5, …, 66b, 71b, 76, 81, 86, 91, 96d }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0570 cents
[[Badness]] (Sintel): 6.15


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


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


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


[[Mapping]]: [{{val|1 313 532}}, {{val|0 -439 -746}}]
{{Mapping|legend=2| 1 -126 -214 | 0 439 746 }}
: mapping generators: ~2, ~{{monzo| -16 -243 143 }}


[[Optimal tuning]] ([[CTE]]): ~{{monzo|17 343 143}} = 851.248
[[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| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}
{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0164 cents
[[Badness]] (Sintel): 17.7 × 10<sup>3</sup>


==== 2.3.7.13 subgroup ====
==== 2.3.7.13 subgroup ====
Subgroup: 2.3.7.13
Subgroup: 2.3.7.13


Comma list: {{monzo|-42 -2 -5 16}}, {{monzo|10 -46 29 -5}}
Comma list: {{monzo| -42 -2 -5 16 }}, {{monzo| 10 -46 29 -5 }}
 
Subgroup-val mapping: {{mapping| 1 -126 -214 -80 | 0 439 746 288 }}


Mapping: [{{val|1 313 532 208}}, {{val|0 -439 -746 -288}}]
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}}


Optimal tuning (CTE): ~1235079060111/755603996672 = 851.248
{{Optimal ET sequence|legend=0| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689, 11255d }}


{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}
Badness (Sintel): 53.2


==== 2.3.7.13.17 subgroup ====
==== 2.3.7.13.17 subgroup ====
Subgroup: 2.3.7.13.17
Subgroup: 2.3.7.13.17


Comma list: 39337984/39328497, {{monzo|0 -14 7 4 -3}}, {{monzo|-18 -24 14 -1 5}}
Comma list: 39337984/39328497, {{monzo| 0 -14 7 4 -3 }}, {{monzo| -18 -24 14 -1 5 }}
 
Subgroup-val mapping: {{mapping| 1 -126 -214 -80 -18 | 0 439 746 288 76 }}


Mapping: [{{val|1 313 532 208 58}}, {{val|0 -439 -746 -288 -76}}]
Optimal tunings:  
* WE: ~2 = 1200.9870{{c}}, ~3757/3072 = 348.7480{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3757/3072 = 348.7517{{c}}


Optimal tuning (CTE): ~6144/3757 = 851.248
{{Optimal ET sequence|legend=0| 320, 1483bd, 1803, 2123 }}


{{Optimal ET sequence|legend=1| 320, 1483bd, 1803, 2123 }}
Badness (Sintel): 13.4


==== 2.3.7.13.17.19 subgroup ====
==== 2.3.7.13.17.19 subgroup ====
Subgroup: 2.3.7.13.17.19
Subgroup: 2.3.7.13.17.19


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


Mapping: [{{val|1 313 532 208 58 432}}, {{val|0 -439 -746 -288 -76 -603}}]
Subgroup-val mapping: {{mapping| 1 -126 -214 -80 -18 -171 | 0 439 746 288 76 603 }}


Optimal tuning (CTE): ~6144/3757 = 851.248
Optimal tunings:  
* WE: ~2 = 1200.9961{{c}}, ~3757/3072 = 348.7506{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3757/3072 = 348.7517{{c}}


{{Optimal ET sequence|legend=1| 320, 1483bd, 1803, 2123 }}
{{Optimal ET sequence|legend=0| 320, 1483bd, 1803, 2123 }}


=== Purpleheart ===
Badness (Sintel): 7.46
[[Subgroup]]: 2.3.7


[[Comma list]]: 2187/2048
=== Heartland (rank 3) ===
{{Main| Heartland }}


{{Mapping|legend=1| 7 11 0 | 0 0 1 }}
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 generators: ~9/8, ~7
[[Subgroup]]: 2.3.7.11.19


[[Optimal tuning]] ([[CTE]]): ~9/8 = 1\7, ~7/4 = 968.826 (~64/63 = 59.746)
[[Comma list]]: 243/242, 1083/1078


{{Optimal ET sequence|legend=1| 7, 14, 35, 49bd }}
{{Mapping|legend=2| 1 1 0 2 1 | 0 4 0 10 3 | 0 0 1 0 1 }}
: mapping generators: ~2, ~21/19, ~7


[[Badness]]: 0.0875
[[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}}


=== Ennea ===
{{Optimal ET sequence|legend=1| 14, 27e, 34dh, 41, 89, 130, 219 }}
Subgroup: 2.3.7.11


[[Comma list]]: 41503/41472, 43923/43904
[[Badness]] (Sintel): 0.615


[[Gencom]]: [2 99/98; 41503/41472, 43923/43904]
== Temperaments with a 2.3.11 gene ==
=== Neutral ===
See [[Rastmic clan #Neutral]].


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


[[Mapping|Sval mapping]]: [{{val|9 0 11 24}}, {{val|0 2 2 1}}]
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.


[[Tp tuning|POL2 generator]]: ~99/98 = 17.6258
[[Subgroup]]: 2.3.11


{{Optimal ET sequence|legend=1| 54, 63, 72, 135, 342, 477, 1089, 1566 }}
[[Comma list]]: 33/32


[[Tp tuning #T2 tuning|RMS error]]: 0.0383 cents
{{Mapping|legend=2| 1 0 5 | 0 1 -1 }}
: mapping generators: ~2, ~3


=== Superslendric ===
[[Optimal tuning]]s:
In superslendric, 8 [[8/7]]s are equated to [[3/1]] (related to [[8edt]]).
* [[WE]]: ~2 = 1206.6866{{c}}, ~3/2 = 691.7837{{c}}
Subgroup: 2.3.7
: [[error map]]: {{val| +6.687 -3.485 -16.355 }}
[[Comma list]]: 17294403/16777216
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 689.2066{{c}}
: error map: {{val| 0.000 -12.748 -40.525 }}


[[Gencom]]: [2 8/7; 17294403/16777216]
{{Optimal ET sequence|legend=1| 2, 5, 7, 12e, 40ee, 47eee, 54beee, 61beeee }}


{{mapping|legend=2|1 0 3|0 8 -1}}
[[Badness]] (Sintel): 0.185


[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[8/7]] = 237.712
=== 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.  


=== Parapyth (rank 3) ===
[[Subgroup]]: 2.3.11
{{see also| Pentacircle temperaments #Parapyth }}


Subgroup: 2.3.7.11
[[Comma list]]: 131769/131072


[[Comma list]]: 896/891
{{Mapping|legend=2| 2 1 8 | 0 2 -1 }}
: mapping generators: ~363/256, ~16/11


[[Gencom]]: [2 3/2 28/27; 896/891]
[[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 }}


[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 }}, {{val| 0 1 0 3 -1 }}, {{val| 0 0 0 1 1 }}]
{{Optimal ET sequence|legend=1| 22, 24, 94, 118, 142, 450e, 592e, 1326beeee }}


[[Sval]] [[mapping]]: [{{val| 1 0 0 7 }}, {{val| 0 1 0 -4 }}, {{val| 0 0 1 1 }}]
[[Badness]] (Sintel): 0.395


[[Tp tuning|POL2 tuning]]: ~3 = 1903.834, ~7 = 3369.872
=== Infraug ===
[[Subgroup]]: 2.3.11


{{Optimal ET sequence|legend=1| 17, 36, 41, 58, 63, 104 }}
[[Comma list]]: 729/704


[[Tp tuning #T2 tuning|RMS error]]: 0.4149 cents
{{Mapping|legend=2| 1 0 -6 | 0 1 6 }}
: mapping generators: ~2, ~3


==== 2.3.7.11.13 ====
[[Optimal tuning]]s:
Subgroup: 2.3.7.11.13
* [[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 }}


[[Comma list]]: 352/351, 364/363
{{Optimal ET sequence|legend=1| 7, 19, 26, 33, 59b, 92b }}


The gencom below gives [[Margo Schulter]]'s favored basis
[[Badness]] (Sintel): 0.734


[[Gencom]]: [2 3/2 28/27; 352/351 364/363]
==== 2.3.11.13 ====
Subgroup: 2.3.11.13


[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 6 }}, {{val| 0 1 0 3 -1 -4 }}, {{val| 0 0 0 1 1 1 }}]
Comma list: 144/143, 729/704


[[Sval]] [[mapping]]: [{{val| 1 0 0 7 12 }}, {{val| 0 1 0 -4 -7 }}, {{val| 0 0 1 1 1 }}]
Subgroup-val mapping: {{mapping| 1 0 -6 10 | 0 1 6 -4 }}


[[Tp tuning|POL2 tuning]]: ~3 = 1903.856, ~7 = 3369.907
Optimal tunings:  
* WE: ~2 = 1202.1934{{c}}, ~3/2 = 692.8902{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 691.7585{{c}}


{{Optimal ET sequence|legend=1| 17, 41, 46, 58, 87, 104 }}
{{Optimal ET sequence|legend=0| 7, 19, 26, 59b }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
Badness (Sintel): 0.725


=== Heartland (rank 3) ===
==== Aerophore ====
{{Main| Heartland }}
Subgroup: 2.3.11.19


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).
Comma list: 363/361, 729/704


Subgroup: 2.3.7.11.19
Subgroup-val mapping: {{mapping| 1 0 -6 -6 | 0 2 12 13 }}
: mapping generators: ~2, ~19/11


[[Comma list]]: 243/242, 1083/1078
Optimal tunings:  
* WE: ~2 = 1202.6380{{c}}, ~19/11 = 947.4782{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 945.7779{{c}}


[[Gencom]]: [2 21/19 7; 243/242 1083/1078]
{{Optimal ET sequence|legend=1| 14, 19, 33 }}


[[Mapping|Sval mapping]]: [{{val|1 1 0 2 1}}, {{val|0 4 0 10 3}}, {{val| 0 0 1 0 1 }}]
Badness (Sintel): 1.59
 
[[Tp tuning|POL2 generator]]: ~21/19 = 175.2713, ~7 = 3369.3784
 
{{Optimal ET sequence|legend=1| 7, 14, 27e, 34dh, 41, 89, 130 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3066 cents
 
== Temperaments with a 2.3.11 gene ==
=== 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 [[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.


=== Paralimmal ===
[[Subgroup]]: 2.3.11
[[Subgroup]]: 2.3.11


[[Comma list]]: 33/32
[[Comma list]]: 4096/3993


{{Mapping|legend=2| 1 0 5 | 0 1 -1 }}
{{Mapping|legend=2| 1 0 4 | 0 3 -1 }}
: mapping generators: ~2, ~16/11


: mapping generators: ~2, ~3
[[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 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 692.713
{{Optimal ET sequence|legend=1| 2, 11b, 13, 15, 17 }}


{{Optimal ET sequence|legend=1| 2, 5, 7, 12e }}
[[Badness]] (Sintel): 0.984


[[Badness]]: 0.185
==== Huxley ====
{{Main| Huxley }}


=== Paralimmal ===
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.  
[[Subgroup]]: 2.3.11


[[Comma list]]: [[4096/3993]]
Subgroup: 2.3.11.13


{{Mapping|legend=2| 1 0 4 | 0 3 -1 }}
Comma list: 512/507, 1352/1331


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[16/11]] = 634.320
Subgroup-val mapping: {{mapping| 1 -3 5 6 | 0 6 -2 -3 }}
: mapping generators: ~2, ~22/13


{{Optimal ET sequence|legend=1| 11b, 13, 15, 17 }}
Optimal tunings:
* WE: ~2 = 1198.0036{{c}}, ~22/13 = 916.0595{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 917.5184{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 1.237 cents
{{Optimal ET sequence|legend=0| 4, 13, 17 }}


=== Neutral ===
Badness (Sintel): 1.31
See [[Rastmic clan #Neutral]]


==== Namo ====
=== Glaishur ===
See [[Rastmic clan #Namo]]
This temperament is the no-5 no-7 [[restriction]] of [[#Navy|navy]], as well as the add-11 [[extension]] of [[#Glacier|glacier]].


=== Huxley ===
[[Subgroup]]: 2.3.11
{{Main| 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.
[[Comma list]]: 10554638336/10460353203


[[Subgroup]]: 2.3.11.13
{{Mapping|legend=2| 1 1 0 | 0 5 21 }}
: mapping generators: ~2, ~88/81


[[Comma list]]: 512/507, 1352/1331
[[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|legend=2| 1 3 3 3 | 0 -6 2 3 }}
{{Optimal ET sequence|legend=1| 17, 60e, 77, 94, 111 }}


: mapping generators: ~2, ~13/11
[[Badness]] (Sintel): 2.27


[[Optimal tuning]]s:
==== 2.3.11.13 ====
* [[CTE]]: ~2 = 1\1, ~13/11 = 282.726
Subgroup: 2.3.11.13
* [[CWE]]: ~2 = 1\1, ~13/11 = 282.482


{{Optimal ET sequence|legend=1| 4, 13, 17 }}
Comma list: 352/351, 531674/531441


[[Badness]]: 0.0263
Subgroup-val mapping: {{mapping| 1 1 0 3 | 0 5 21 6 }}


=== Aerophore ===
Optimal tunings:
[[Subgroup]]: 2.3.11.19
* WE: ~2 = 1200.0000{{c}}, ~13/12 = 140.537{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.537{{c}}


[[Comma list]]: 363/361, 729/704
{{Optimal ET sequence|legend=0| 17, 60e, 77, 94, 111, 239f, 350f }}


{{Mapping|legend=2| 1 0 -6 -6 | 0 2 12 13 }}
Badness (Sintel): 0.415


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~19/11 = 945.4
=== Profanity ===
Profanity identifies [[11/9]] with 2\7.


{{Optimal ET sequence|legend=1| 9eehh, 14, 19, 33 }}
[[Subgroup]]: 2.3.11


==== Semaerophore ====
[[Comma list]]: 19487171/19131876
[[Subgroup]]: 2.3.7.11.19


[[Comma list]]: 49/48, 77/76, 729/704
{{Mapping|legend=2| 7 0 2 | 0 1 2 }}
: mapping generators: ~1458/1331, ~3


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/4 = 944.667
{{Optimal ET sequence|legend=1| 7, … 49, 56, 63, 70 }}


{{Optimal ET sequence|legend=1| 9eehh, 14, 33d, 47deh }}
[[Badness]] (Sintel): 3.03


== Temperaments with a 2.3.13 gene ==
== Temperaments with a 2.3.13 gene ==
=== Superflat ===
=== Threedic ===
Superflat 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.
 
[[Subgroup]]: 2.3.13
[[Subgroup]]: 2.3.13


[[Comma list]]: [[1053/1024]]
[[Comma list]]: 2197/2187


{{Mapping|legend=2|1 1 6|0 1 -4}}
{{Mapping|legend=2| 1 0 0 | 0 3 7 }}
: mapping generators: ~2, ~13/9


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


{{Optimal ET sequence|legend=1|5f, 7, 12, 19, 45f, 64f, 147bfff }}
{{Optimal ET sequence|legend=1| 15, 17, 36, 53, 70, 123, 193, 316, 755f }}


[[Tp tuning #T2 tuning|RMS error]]: 1.591 cents
[[Badness]] (Sintel): 0.160


==== 2.3.11.13 ====
=== Ultraflat ===
[[Subgroup]]: 2.3.11.13
Ultraflat is a diatonic-based [[exotemperament]] that makes [[27/26]] vanish, so [[13/8]] is a major sixth.  


[[Comma list]]: [[144/143]], [[729/704]]
[[Subgroup]]: 2.3.13


[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.247
[[Comma list]]: 27/26


{{Optimal ET sequence|legend=1|7, 19, 26, 59b }}
{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}
: mapping generators: ~2, ~3


=== Ultraflat ===
[[Optimal tuning]]s:
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 temperaments intersect in [[7edo]], where major sixths and minor sixths are not distinguished.
* [[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 }}


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


[[Comma list]]: [[27/26]]
[[Badness]] (Sintel): 0.200


{{Mapping|legend=2|1 1 2|0 1 3}}
=== 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.


[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[3/2]] = 688.391
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.  


{{Optimal ET sequence|legend=1| 5, 7 }}
[[Tp tuning #T2 tuning|RMS error]]: 4.367 cents
=== Threedic ===
[[Subgroup]]: 2.3.13
[[Subgroup]]: 2.3.13


[[Comma list]]: [[2197/2187]]
[[Comma list]]: 1053/1024


{{Mapping|legend=2|1 0 0|0 3 7}}
{{Mapping|legend=2| 1 0 10 | 0 1 -4 }}
: mapping generators: ~2, ~3


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


{{Optimal ET sequence|legend=1|11bff, 13f, 15, 17, 36, 53, 70, 123, 193, 316, 755f }}
{{Optimal ET sequence|legend=1| 5f, 7, 12, 19, 45f, 64f, 147bfff }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2054 cents
[[Badness]] (Sintel): 0.610


=== Shoal ===
=== Shoal ===
The 2.3.13.23 subgroup 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 a 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.
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]]: 2.3.13.23
[[Subgroup]]: 2.3.13


[[Comma list]]: 3888/3887, 12168/12167
[[Comma list]]: 816293376/815730721


{{Mapping|legend=2|1 3 6 7|0 -8 -13 -14}}
{{Mapping|legend=2| 1 -5 -7 | 0 8 13 }}
: mapping generators: ~2, ~3888/2197


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~26/23 = 212.261
[[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 }}


{{Optimal ET sequence|legend=1|11fi, 17, 79, 96, 113, 130, 147, 424, 571, 1289, 1860, 3149 }}
{{Optimal ET sequence|legend=1| 17, 79, 96, 113, 130, 147, 424, 571, 1289, 10883ff, 12172ff }}


Badness (Dirichlet): 0.021
Badness (Sintel): 0.135


Scales:
==== 2.3.13.23 ====
* [[Shoal17]]
Subgroup: 2.3.13.23


Music:
Comma list: 3888/3887, 12168/12167
* [https://www.youtube.com/watch?v=bwGjN56iMM0 ''Moody Improvisation in the Shoal Temperament''] - [[Budjarn Lambeth]] (2025)


== Temperaments with a higher-limit gene ==
Subgroup-val mapping: {{mapping| 1 -5 -7 -7 | 0 8 13 14 }}
=== Semitonic ===
[[Subgroup]]: 2.3.17


[[Comma list]]: [[289/288]]
Optimal tunings:  
* WE: ~2 = 1199.9883{{c}}, ~23/13 = 987.7325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~23/13 = 987.7408{{c}}


{{Mapping|legend=2| 2 0 5 | 0 1 1 }}
{{Optimal ET sequence|legend=0| 17, 79, 96, 113, 130, 147, 424, 571, 1289, 1860, 3149 }}


: sval mapping generators: ~17/12, ~3
Badness (Sintel): 0.0213


: [[gencom]]: [17/12 3; 289/288]
Scales:
* [[Shoal17]]


[[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)  
; Music
* [https://www.youtube.com/watch?v=bwGjN56iMM0 ''Moody Improvisation in the Shoal Temperament''] by [[Budjarn Lambeth]] (2025)


{{Optimal ET sequence|legend=1| 12, 58, 70, 82, 94, 106, 118, 224g }}
=== 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.


[[Tp tuning #T2 tuning|RMS error]]: 0.2247 cents
[[Subgroup]]: 2.3.13


=== Gigapyth ===
[[Comma list]]: 373248/371293
[[Subgroup]]: 2.3.85


[[Comma list]]: 2.3.85 {{val|-40 1 6 }}
{{Mapping|legend=2| 1 1 3 | 0 5 6 }}
: mapping generators: ~2, ~13/12


{{mapping|legend=2| 1 4 6 | 0 -6 1 }}
[[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 generators: ~2, ~85/64
{{Optimal ET sequence|legend=1| 8, 9, 17, 60, 77, 94, 171, 265, 359f, 983ff }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~85/64 = 483.034
[[Badness]] (Sintel): 0.383


[[Support]]ing [[ET]]s: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159*
== Temperaments with a higher-limit gene ==
=== Semitonic ===
[[Subgroup]]: 2.3.17


<nowiki>*</nowiki>Wart for 85
[[Comma list]]: [[289/288]]


==== 2.3.7.85 subgroup ====
{{Mapping|legend=2| 2 0 5 | 0 1 1 }}
[[Subgroup]]: 2.3.7.85
: mapping generators: ~17/12, ~3


[[Comma list]]: 1029/1024, 7225/7203
[[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|legend=2| 1 4 2 6 | 0 -6 2 1}}
{{Optimal ET sequence|legend=1| 10, 12, 58, 70, 82, 94, 106, 118, 224g }}


: mapping generators: ~2, ~85/64
[[Badness]] (Sintel): 0.0454


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~85/64 = 483.031
=== 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.  


[[Support]]ing [[ET]]s: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159*
[[Subgroup]]: 2.3.19


<nowiki>*</nowiki>Wart for 85
[[Comma list]]: 513/512


=== Dog ===
{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}
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]].
: mapping generators: ~2, ~3


Subgroup: 2.3.19
[[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 }}


[[Comma list]]: 81/76
{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296, 1557bhhhh, 1853bhhhh }}


[[Gencom]]: [2 4/3; 81/76]
[[Badness]] (Sintel): 0.0294


[[Mapping|Sval mapping]]: [{{val|1 2 6}}, {{val|0 -1 -4}}]
=== 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]].
[[Tp tuning|POL2 generator]]: ~4/3 = 521.403
 
{{Optimal ET sequence|legend=1| 5h, 7, 16, 23 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 4.943 cents
 
=== Boethian ===
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
[[Subgroup]]: 2.3.19


[[Comma list]]: [[513/512]]
[[Comma list]]: 81/76


{{Mapping|legend=1| 1 0 9 | 0 1 -3 }}
{{Mapping|legend=2| 1 0 -2 | 0 1 4 }}
: mapping generators: ~2, ~3


: Mapping generators: ~2, ~3
[[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 }}


[[Optimal tuning]] (CTE): ~2 = 1\1, ~3/2 = 701.3288
{{Optimal ET sequence|legend=1| 2, 5h, 7, 16, 23 }}


{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296 }}
[[Badness]] (Sintel): 0.491
 
[[Badness]]: 0.000374


=== Lipsett ===
=== 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.
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
[[Subgroup]]: 2.3.23


[[Comma list]]: [[2187/2116]]
[[Comma list]]: 2187/2116


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


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~46/27 = 948.526
[[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, 14, 19, 43, 62i, 81i }}
{{Optimal ET sequence|legend=1| 5, 14, 19, 43, 62i, 81i }}


[[Badness]] (Smith): 8.998 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.801


=== Porpoise ===
=== Porpoise ===
Line 1,147: Line 1,248:
[[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 }}
[[Badness]] (Sintel): 0.317


=== Sematology ===
=== Sematology ===
This temperament tempers out 4107/4096 and thus equates 2 [[37/32]]'s with [[4/3]].
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 1,160: 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}}
[[POTE generator]]: ~[[37/32]] = 249.075
: [[error map]]: {{val| +0.218 -0.046 -0.988 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/37 = 950.8250{{c}}
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53, 77, 130 }}
: error map: {{val| 0.000 -0.305 -2.169 }}
 
==== 2.3.7.37 subgroup ====
[[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, 14, 19, 24, 53, 77, 130, 443l, 573ll, 703ll, 1536blllll }}


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


=== Reversed mavila ===
=== Reversed mavila ===
Line 1,214: Line 1,286:
[[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}}]
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.6907{{c}}, ~3/2 = 705.3096{{c}}
[[Tp tuning|POL2 generator]]: ~4/3 = 494.509
: [[error map]]: {{val| -0.309 +3.045 -8.752 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.2699{{c}}
{{Optimal ET sequence|legend=1| 5, 12, 17 }}
: error map: {{val| 0.000 +3.315 -7.983 }}
 
==== 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, 97m, 114m, 131m }}


{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 39d }}
[[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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