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No-fives subgroup temperaments: Difference between revisions

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{{Technical data page}}
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 5.  
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 5.  


== 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, 60c }}
[[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


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


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


: gencom: [2 9/7; 78/77, 99/98, 243/242]
Optimal tunings:  
* WE: ~2 = 1199.2633{{c}}, ~11/9 = 351.3189{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.4593{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 424.457
{{Optimal ET sequence|legend=0| 17, 41, 58, 99e }}


Optimal ET sequence: {{Optimal ET sequence| 17, 48f, 65ef, 82d, 147df }}
Badness (Sintel): 0.409


RMS error: 1.769 cents
===== 2.3.7.11.13 =====
 
===== 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 -25 generator steps, which is also in the opposite direction of how all other primes in the [[13-limit]] are reached. 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, 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 this case is also a sign of leapday 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=3| 1 1 0 -6 | 0 1 0 15 }}


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


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


[[Gencom|Gencom mapping]]: [{{val|1 0 0 -3}}, {{val|0 3 0 11}}]
{{Optimal ET sequence|legend=1| 8d, 9, 17, 43, 60d, 103d }}


[[Mapping|Sval mapping]]: [{{val|1 0 -3}}, {{val|0 3 11}}]
[[Badness]] (Sintel): 2.48


[[Tp tuning|POL2 generator]]: ~81/56 = 633.525
==== 2.3.7.11 subgroup ====
Subgroup: 2.3.7.11


{{Optimal ET sequence|legend=1| 17, 36, 89, 125, 161, 358, 519b }}
Comma list: 99/98, 864/847


[[Tp tuning #T2 tuning|RMS error]]: 0.3519 cents
Subgroup-val mapping: {{mapping| 1 1 2 3 | 0 5 7 4 }}


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


[[Comma]]: 1029/1024
Optimal tunings:  
* WE: ~2 = 1198.6613{{c}}, ~12/11 = 139.8489{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.7839{{c}}


[[Gencom]]: [2 8/7; 1029/1024]
{{Optimal ET sequence|legend=0| 8d, 9, 17, 43, 60d }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 3}}, {{val|0 3 0 -1}}]
Badness (Sintel): 0.624


[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 3 -1}}]
==== 2.3.7.11.13 subgroup ====
Subgroup: 2.3.7.11.13


[[Tp tuning|POL2 generator]]: ~8/7 = 233.688
Comma list: 78/77, 99/98, 144/143


{{Optimal ET sequence|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}
Subgroup-val mapping: {{mapping| 1 1 2 3 3 | 0 5 7 4 6 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
Gencom mapping: {{mapping| 1 1 0 2 3 3 | 0 5 0 7 4 6 }}


==== [[Gamelismic clan #Baladic|Baladic]] ====
Optimal tunings:
Subgroup: 2.3.7.13
* WE: ~2 = 1198.9768{{c}}, ~13/12 = 139.8704{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8166{{c}}


[[Comma list]]: 169/168, 1029/1024
{{Optimal ET sequence|legend=0| 8d, 9, 17, 43, 60d }}


[[Gencom]]: [91/64 8/7; 169/168 1029/1024]
Badness (Sintel): 0.400


[[Mapping|Sval mapping]]: [{{val|2 2 6 7}}, {{val|0 3 -1 1}}]
; 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]


[[Tp tuning|POL2 generator]]: ~8/7 = 233.6044
=== Doublehearted ===
{{See also| Heartland }}


{{Optimal ET sequence|legend=1| 10, 26, 36, 154…, 190…, 226…, 262… }}
This temperament is the no-5 [[restriction]] of [[octacot]].


[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents
[[Subgroup]]: 2.3.7


===== 2.3.7.13.17 =====
[[Comma list]]: 5764801/5668704
[[Subgroup]]: 2.3.7.13.17


[[Comma list]]: 169/168, 273/272, 289/288
{{Mapping|legend=2| 1 1 2 | 0 8 11 }}
: mapping generators: ~2, ~343/342


[[Gencom]]: [17/12 8/7; 169/168 273/272 289/288]
[[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 }}


{{Mapping|legend=2| 2 2 6 7 7 | 0 3 -1 1 3 }}
{{Optimal ET sequence|legend=1| 14, 27, 41 }}


[[Tp tuning|POL2 generator]]: ~8/7 = 233.6155
[[Badness]] (Sintel): 2.62


{{Optimal ET sequence|legend=1| 10, 26, 36, 154…, 190…, 226… }}
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


[[Tp tuning #T2 tuning|RMS error]]: 0.5073 cents
Comma list: 243/242, 2401/2376


=== Hemif ===
Subgroup-val mapping: {{mapping| 1 1 2 2 | 0 8 11 20 }}
Related temperaments: [[hemififths]], [[namo]]


[[Subgroup]]: 2.3.7
Optimal tunings:  
* WE: ~2 = 1200.4071{{c}}, ~22/21 = 87.6809{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 87.6902{{c}}


[[Comma list]]: 1605632/1594323
{{Optimal ET sequence|legend=0| 14, 27e, 41, 96d, 137d, 178d }}


{{Mapping|legend=2| 1 1 -1 | 0 2 13 }}
Badness (Sintel): 0.815


{{Mapping|legend=3| 1 1 0 -1 | 0 2 0 13 }}
==== 2.3.7.11.19 ====
Subgroup: 2.3.7.11.19


: [[gencom]]: [2 2187/1792; 1605632/1594323]
Comma list: 133/132, 243/242, 343/342


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2187/1792 = 351.485
Subgroup-val mapping: {{mapping| 1 1 2 2 3 | 0 8 11 20 17 }}


{{Optimal ET sequence|legend=1| 7, 17, 41, 58, 99 }}
Optimal tunings:
* WE: ~2 = 1200.6100{{c}}, ~19/18 = 87.7129{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/18 = 87.7285{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.2344 cents
{{Optimal ET sequence|legend=0| 14, 27e, 41, 137dh }}


==== 2.3.7.11 ====
Badness (Sintel): 0.560
Subgroup: 2.3.7.11


Comma list: 243/242, 896/891
=== Purpleheart ===
{{Main| Whitewood }}


Sval mapping: {{mapping| 1 1 -1 2 | 0 2 13 5 }}
[[Subgroup]]: 2.3.7


Gencom mapping: {{mapping| 1 1 0 -1 2 | 0 2 0 13 5 }}
[[Comma list]]: 2187/2048


: gencom: [2 11/9; 243/242 896/891]
{{Mapping|legend=2| 7 11 0 | 0 0 1 }}
: mapping generators: ~9/8, ~7


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.535
[[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 }}


Optimal ET sequence: {{Optimal ET sequence| 7, 17, 41, 58, 99e }}
{{Optimal ET sequence|legend=1| 7, 14, 35, 49bd }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents
[[Badness]] (Sintel): 3.00


===== 2.3.7.11.13 =====
=== Chrysanthemum ===
Subgroup: 2.3.7.11.13
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.


Comma list: 144/143, 243/242, 364/363
[[Subgroup]]: 2.3.7


Sval mapping: {{mapping| 1 1 -1 2 4 | 0 2 13 5 -1 }}
[[Comma list]]: {{monzo| 83 -1 -29 }}


Gencom mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 0 13 5 -1 }}
{{Mapping|legend=2| 1 -4 3 | 0 29 -1 }}
: mapping generators: ~2, ~8/7


: gencom: [2 11/9; 144/143 243/242 364/363]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9871{{c}}, ~8/7 = 231.1001{{c}}
: [[error map]]: {{val| -0.013 +0.000 +0.035 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 231.1024{{c}}
: error map: {{val| 0.000 +0.014 +0.072 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.691
{{Optimal ET sequence|legend=0| 26, 83, 109, 135, 566, 701, 836, 971, 1106, 2077, 5260, 7337, 9414d }}


Optimal ET sequence: {{Optimal ET sequence| 7, 10, 17, 24, 41, 58 }}
[[Badness]] (Sintel): 3.06


RMS error: 0.7167 cents
==== 2.3.7.11 ====
Subgroup: 2.3.7.11


===== Heartful =====
Comma list: 43923/43904, 5767168/5764801
{{See also| Heartland }}


Related temperaments: [[bunya]]
Subgroup-val mapping: {{mapping| 1 -4 3 5 | 0 29 -1 -8 }}


Subgroup: 2.3.7.11.19
Optimal tunings:  
* WE: ~2 = 1200.0050{{c}}, ~8/7 = 231.1024{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.1015{{c}}


Comma list: 243/242, 896/891, 1083/1078
{{Optimal ET sequence|legend=0| 26, 83, 109, 135, 566, 701, 836, 971, 1807, 2778, 4585 }}


Sval mapping: {{mapping| 1 1 -1 2 0 | 0 4 26 10 29 }}
Badness (Sintel): 0.324


: gencom: [2 21/19; 243/242 896/891 1083/1078]
=== Leapfrog ===
{{See also| Gentle region }}


Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.804
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.


Optimal ET sequence: {{Optimal ET sequence| 34dh, 41, 116e, 157e }}
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]]).


RMS error: 0.5360 cents
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]].  


=== Hearts ===
[[Subgroup]]: 2.3.7
{{see also|Heartland}}


Subgroup: 2.3.7
[[Comma list]]: 14680064/14348907


[[Comma list]]: 34451725707/34359738368 (trila-quadzo comma)
{{Mapping|legend=2| 1 0 -21 | 0 1 15 }}


[[Gencom]]: [2 567/512; 34451725707/34359738368]
{{Mapping|legend=3| 1 0 0 -21 | 0 1 0 15 }}
: mapping generators: ~2, ~3


[[Mapping|Sval mapping]]: [{{val|1 1 5}}, {{val|0 4 -15}}]
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.1807{{c}}, ~3/2 = 704.2400{{c}}
[[Tp tuning|POL2 generator]]: ~567/512 = 175.433
: [[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 }}


{{Optimal ET sequence|legend=1| 7, 27d, 34, 41, 89, 130, 171 }}
{{Optimal ET sequence|legend=1| 17, 46, 63, 235b, 298b, 361bd, 424bd, 487bbd }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0529 cents
[[Badness]] (Sintel): 4.33
 
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: 896/891, 1331/1323


[[Gencom]]: [2 256/231; 243/242 65536/65219]
Subgroup-val mapping: {{mapping| 1 0 -21 -14 | 0 1 15 11 }}


[[Mapping|Sval mapping]]: [{{val|1 1 5 2}}, {{val|0 4 -15 10}}]
Gencom mapping: {{mapping| 1 0 0 -21 -14 | 0 1 0 15 11 }}


[[Tp tuning|POL2 generator]]: ~256/231 = 175.369
Optimal tunings:  
* WE: ~2 = 1199.2683{{c}}, ~3/2 = 704.3230{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.6926{{c}}


{{Optimal ET sequence|legend=1| 7, 27de, 34, 41, 89, 130 }}
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3224 cents
Badness (Sintel): 0.629


Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquart]]
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13


==== 2.3.7.11.19 ====
Comma list: 169/168, 352/351, 364/363
Subgroup: 2.3.7.11.19


[[Comma list]]: 243/242, 513/512, 1083/1078
Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 | 0 1 15 11 8 }}


[[Gencom]]: [2 21/19; 243/242 513/512 1083/1078]
Gencom mapping: {{mapping| 1 0 0 -21 -14 -9 | 0 1 0 15 11 8 }}


[[Mapping|Sval mapping]]: [{{val|1 1 5 2 6}}, {{val|0 4 -15 10 -12}}]
Optimal tunings:  
* WE: ~2 = 1199.5654{{c}}, ~3/2 = 704.4898{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7084{{c}}


[[Tp tuning|POL2 generator]]: ~21/19 = 175.341
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


{{Optimal ET sequence|legend=1| 7, 27deh, 34, 41, 89, 130, 219 }}
Badness (Sintel): 0.436


[[Tp tuning #T2 tuning|RMS error]]: 0.3121 cents
===== Skidoo =====
Subgroup: 2.3.7.11.13.23


Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquart]]
Comma list: 169/168, 208/207, 352/351, 364/363


=== Navy ===
Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 -5 | 0 1 15 11 8 6 }}
[[Subgroup]]: 2.3.7


[[Comma list]]: 282429536481/281974669312
Gencom mapping: {{mapping| 1 0 0 -21 -14 -9 0 0 -5 | 0 1 0 15 11 8 0 0 6 }}


{{Mapping|legend=1|1 1 0|0 5 24}}
Optimal tunings:
* WE: ~2 = 1199.6639{{c}}, ~3/2 = 704.5315{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7021{{c}}


[[Tp tuning|POL2 generator]]: ~243/224 = 140.366
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


{{Optimal ET sequence|legend=1| 17, 60, 77, 94, 171, 265, 436 }}
Badness (Sintel): 0.356


[[Tp tuning #T2 tuning|RMS error]]: 0.0296 cents
====== 2.3.7.11.13.23.29 ======
Subgroup: 2.3.7.11.13.23.29


Related temperaments: [[Schismatic family|tsaharuk]], [[Hemifamity temperaments|quanic]]
Comma list: 169/168, 208/207, 232/231, 352/351, 364/363


==== 2.3.7.11 ====
Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 -5 -38 | 0 1 15 11 8 6 27 }}
Subgroup: 2.3.7.11


[[Comma list]]: 1331/1323, 19712/19683
Gencom mapping: {{mapping| 1 0 0 -21 -14 -9 -5 0 0 -38 | 0 1 0 15 11 8 0 0 6 27 }}


{{Mapping|legend=1|1 1 0 1|0 5 24 21}}
Optimal tunings:
* WE: ~2 = 1199.5755{{c}}, ~3/2 = 704.5533{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7750{{c}}


[[Tp tuning|POL2 generator]]: ~88/81 = 140.407
{{Optimal ET sequence|legend=0| 17, 46, 63 }}


{{Optimal ET sequence|legend=1| 17, 60e, 77, 94, 359e, 453ee, 547ee, 641ee }}
Badness (Sintel): 0.441


[[Tp tuning #T2 tuning|RMS error]]: 0.3778 cents
; Music
 
* Suite for Harpsichord in A Locrian, tuning: Eb–G# in [[46edo]] by [[Inthar]] (in progress):
Related temperaments: [[Schismatic family|tsaharuk]], [[Hemifamity temperaments|quanic]]
** I. Prelude
 
** II. Allemande
==== 2.3.7.11.13 ====
** III. Courante
Subgroup: 2.3.7.11.13
** [[: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]]
[[Comma list]]: 352/351, 729/728, 1331/1323
** [[:File:Locrian Suite Gavotte.mp3|VI. Gavotte I and II]]
** [[:File:Locrian Suite Gigue.mp3|VII. Gigue]]


{{Mapping|legend=1|1 1 0 1 3|0 5 24 21 6}}
=== Superslendric ===
 
In superslendric, eight [[8/7]]'s are equated to [[3/1]]. This relates it to [[8edt]].
[[Tp tuning|POL2 generator]]: ~13/12 = 140.437
 
{{Optimal ET sequence|legend=1| 17, 60e, 77, 94 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.4044 cents
 
Related temperaments: [[Schismatic family|tsaharuk]], [[Hemifamity temperaments|quanic]]
 
=== Slendrismic ===
{{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
[[Subgroup]]: 2.3.7


[[Comma list]]: 68719476736/68641485507
[[Comma list]]: 17294403/16777216


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


[[Tp tuning|POL2 generator]]: ~1029/1024 = 8.9906
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1201.1628{{c}}, ~8/7 = 237.7287{{c}}
{{Optimal ET sequence|legend=1| 130, 135, 265, 400, 935, 1335, 1735 }}
: [[error map]]: {{val| +1.163 -0.125 -3.066 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 237.5664{{c}}
: error map: {{val| 0.000 -1.424 -6.392 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0212 cents
{{Optimal ET sequence|legend=1| 5, …, 66b, 71b, 76, 81, 86, 91, 96d }}


Related temperaments: [[Pental family|hemipental]]
[[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 }}


Mapping: [{{val|1 313 532 208}}, {{val|0 -439 -746 -288}}]
Subgroup-val mapping: {{mapping| 1 -126 -214 -80 | 0 439 746 288 }}


Optimal tuning (CTE): ~1235079060111/755603996672 = 851.248
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 ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}  
{{Optimal ET sequence|legend=0| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689, 11255d }}  
 
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


=== Parapyth (rank 3) ===
[[Optimal tuning]]s:
{{see also| Pentacircle temperaments #Parapyth }}
* [[WE]]: ~2 = 1206.6866{{c}}, ~3/2 = 691.7837{{c}}
: [[error map]]: {{val| +6.687 -3.485 -16.355 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 689.2066{{c}}
: error map: {{val| 0.000 -12.748 -40.525 }}


Subgroup: 2.3.7.11
{{Optimal ET sequence|legend=1| 2, 5, 7, 12e, 40ee, 47eee, 54beee, 61beeee }}


[[Comma list]]: 896/891
[[Badness]] (Sintel): 0.185


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


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


[[Sval]] [[mapping]]: [{{val| 1 0 0 7 }}, {{val| 0 1 0 -4 }}, {{val| 0 0 1 1 }}]
[[Comma list]]: 131769/131072


[[Tp tuning|POL2 tuning]]: ~3 = 1903.834, ~7 = 3369.872
{{Mapping|legend=2| 2 1 8 | 0 2 -1 }}
: mapping generators: ~363/256, ~16/11


{{Optimal ET sequence|legend=1| 17, 36, 41, 58, 63, 104 }}
[[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 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4149 cents
{{Optimal ET sequence|legend=1| 22, 24, 94, 118, 142, 450e, 592e, 1326beeee }}


==== 2.3.7.11.13 ====
[[Badness]] (Sintel): 0.395
Subgroup: 2.3.7.11.13


[[Comma list]]: 352/351, 364/363
=== Infraug ===
[[Subgroup]]: 2.3.11


The gencom below gives [[Margo Schulter]]'s favored basis
[[Comma list]]: 729/704


[[Gencom]]: [2 3/2 28/27; 352/351 364/363]
{{Mapping|legend=2| 1 0 -6 | 0 1 6 }}
: mapping generators: ~2, ~3


[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 6 }}, {{val| 0 1 0 3 -1 -4 }}, {{val| 0 0 0 1 1 1 }}]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.5969{{c}} ~3/2 = 692.7443{{c}}
: [[error map]]: {{val| +2.597 -6.614 +5.148 }}
* [[CWE]]: ~2 = 1200.0000{{c}} ~3/2 = 692.0116{{c}}
: error map: {{val| 0.000 -9.943 +0.752 }}


[[Sval]] [[mapping]]: [{{val| 1 0 0 7 12 }}, {{val| 0 1 0 -4 -7 }}, {{val| 0 0 1 1 1 }}]
{{Optimal ET sequence|legend=1| 7, 19, 26, 33, 59b, 92b }}


[[Tp tuning|POL2 tuning]]: ~3 = 1903.856, ~7 = 3369.907
[[Badness]] (Sintel): 0.734


{{Optimal ET sequence|legend=1| 17, 41, 46, 58, 87, 104 }}
==== 2.3.11.13 ====
Subgroup: 2.3.11.13


[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
Comma list: 144/143, 729/704
 
=== Heartland (rank 3) ===
{{Main| Heartland }}


Heartland, with a generator of ~21/19, is named for its tempering of the heartlandisma, [[3971/3969]]. Aside from the heartlandisma, the heartland temperament tempers out 243/242 (rastma) and 1083/1078 (bihendrixma), and slices the fifth in four (the number of chambers of the heart).
Subgroup-val mapping: {{mapping| 1 0 -6 10 | 0 1 6 -4 }}


Subgroup: 2.3.7.11.19
Optimal tunings:  
* WE: ~2 = 1202.1934{{c}}, ~3/2 = 692.8902{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 691.7585{{c}}


[[Comma list]]: 243/242, 1083/1078
{{Optimal ET sequence|legend=0| 7, 19, 26, 59b }}


[[Gencom]]: [2 21/19 7; 243/242 1083/1078]
Badness (Sintel): 0.725


[[Mapping|Sval mapping]]: [{{val|1 1 0 2 1}}, {{val|0 4 0 10 3}}, {{val| 0 0 1 0 1 }}]
==== Aerophore ====
Subgroup: 2.3.11.19


[[Tp tuning|POL2 generator]]: ~21/19 = 175.2713, ~7 = 3369.3784
Comma list: 363/361, 729/704


{{Optimal ET sequence|legend=1| 7, 14, 27e, 34dh, 41, 89, 130 }}
Subgroup-val mapping: {{mapping| 1 0 -6 -6 | 0 2 12 13 }}
: mapping generators: ~2, ~19/11


[[Tp tuning #T2 tuning|RMS error]]: 0.3066 cents
Optimal tunings:  
* WE: ~2 = 1202.6380{{c}}, ~19/11 = 947.4782{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 945.7779{{c}}


== Temperaments with a 2.3.11 gene ==
{{Optimal ET sequence|legend=1| 14, 19, 33 }}
=== Io aka undecimal ===
Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]], and with a generator representing both [[3/2]] and [[16/11]]. It may be considered an [[exotemperament]] by some definitions and not one by others. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament.


The name Io was coined by [[User:CompactStar|CompactStar]] in 2024 based on the [[color notation|color name]] ilo, prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.
Badness (Sintel): 1.59


=== 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 aka tridecimal ===
=== Threedic ===
Superflat temperament, or alternatively, tridecimal temperament, is a diatonic-based temperament that makes [[1053/1024]] vanish, so [[13/8]] is a minor sixth, and [[16/13]] is a major third. The more accurate tunings for this temperament are generated by a fifth at least as flat as those of [[flattone]], although often even flatter (such as [[40edo]]'s fifth). Superflat can be viewed as a 2.3.13 subgroup analogue of [[meantone]] and [[archy]]. Superflat diatonic scales have a character somewhere between neutral third scales (or [[3L 4s|mosh]]) and meantone diatonic scales.
[[Subgroup]]: 2.3.13


[[Subgroup]]: 2.3.13
[[Comma list]]: 2197/2187
 
{{Mapping|legend=2| 1 0 0 | 0 3 7 }}
: mapping generators: ~2, ~13/9
 
[[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| 15, 17, 36, 53, 70, 123, 193, 316, 755f }}


[[Comma list]]: [[1053/1024]]
[[Badness]] (Sintel): 0.160


{{Mapping|legend=2|1 1 6|0 1 -4}}
=== Ultraflat ===
Ultraflat is a diatonic-based [[exotemperament]] that makes [[27/26]] vanish, so [[13/8]] is a major sixth.


[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.939
[[Subgroup]]: 2.3.13


{{Optimal ET sequence|legend=1|5f, 7, 12, 19, 45f, 64f, 147bfff }}
[[Comma list]]: 27/26


[[Tp tuning #T2 tuning|RMS error]]: 1.591 cents
{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}
: mapping generators: ~2, ~3


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


[[Comma list]]: [[144/143]], [[729/704]]
{{Optimal ET sequence|legend=1| 2, 5, 7 }}


[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.247
[[Badness]] (Sintel): 0.200


{{Optimal ET sequence|legend=1|7, 19, 26, 59b }}
=== 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.


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


[[Subgroup]]: 2.3.13
[[Subgroup]]: 2.3.13


[[Comma list]]: [[27/26]]
[[Comma list]]: 1053/1024
 
{{Mapping|legend=2| 1 0 10 | 0 1 -4 }}
: mapping generators: ~2, ~3


{{Mapping|legend=2|1 1 2|0 1 3}}
[[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 tuning]] (CTE): ~2 = 1/1, ~[[3/2]] = 688.391
{{Optimal ET sequence|legend=1| 5f, 7, 12, 19, 45f, 64f, 147bfff }}


{{Optimal ET sequence|legend=1| 5, 7 }}
[[Badness]] (Sintel): 0.610


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


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


[[Comma list]]: [[2197/2187]]
[[Comma list]]: 816293376/815730721


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


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/9]] = 634.173
[[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|11bff, 13f, 15, 17, 36, 53, 70, 123, 193, 316, 755f }}
{{Optimal ET sequence|legend=1| 17, 79, 96, 113, 130, 147, 424, 571, 1289, 10883ff, 12172ff }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2054 cents
Badness (Sintel): 0.135


== Temperaments with a higher-limit gene ==
==== 2.3.13.23 ====
=== Semitonic ===
Subgroup: 2.3.13.23
[[Subgroup]]: 2.3.17


[[Comma list]]: [[289/288]]
Comma list: 3888/3887, 12168/12167


{{Mapping|legend=2| 2 0 5 | 0 1 1 }}
Subgroup-val mapping: {{mapping| 1 -5 -7 -7 | 0 8 13 14 }}


: sval mapping generators: ~17/12, ~3
Optimal tunings:  
* WE: ~2 = 1199.9883{{c}}, ~23/13 = 987.7325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~23/13 = 987.7408{{c}}


: [[gencom]]: [17/12 3; 289/288]
{{Optimal ET sequence|legend=0| 17, 79, 96, 113, 130, 147, 424, 571, 1289, 1860, 3149 }}


[[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)
Badness (Sintel): 0.0213


{{Optimal ET sequence|legend=1| 12, 58, 70, 82, 94, 106, 118, 224g }}
Scales:
* [[Shoal17]]


[[Tp tuning #T2 tuning|RMS error]]: 0.2247 cents
; Music
* [https://www.youtube.com/watch?v=bwGjN56iMM0 ''Moody Improvisation in the Shoal Temperament''] by [[Budjarn Lambeth]] (2025)


=== Gigapyth ===
=== Glacier ===
[[Subgroup]]: 2.3.85
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.


[[Comma list]]: 2.3.85 {{val|-40 1 6 }}
[[Subgroup]]: 2.3.13


{{mapping|legend=2| 1 4 6 | 0 -6 1 }}
[[Comma list]]: 373248/371293


: mapping generators: ~2, ~85/64
{{Mapping|legend=2| 1 1 3 | 0 5 6 }}
: mapping generators: ~2, ~13/12


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


[[Support]]ing [[ET]]s: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159*
{{Optimal ET sequence|legend=1| 8, 9, 17, 60, 77, 94, 171, 265, 359f, 983ff }}


<nowiki>*</nowiki>Wart for 85
[[Badness]] (Sintel): 0.383


==== 2.3.7.85 subgroup ====
== Temperaments with a higher-limit gene ==
[[Subgroup]]: 2.3.7.85
=== Semitonic ===
[[Subgroup]]: 2.3.17


[[Comma list]]: 1029/1024, 7225/7203
[[Comma list]]: [[289/288]]


{{mapping|legend=2| 1 4 2 6 | 0 -6 2 1}}
{{Mapping|legend=2| 2 0 5 | 0 1 1 }}
: mapping generators: ~17/12, ~3


: mapping generators: ~2, ~85/64
[[Optimal tuning]]s:  
* [[WE]]: ~17/12 = 600.1471{{c}}, ~3/2 = 701.9563{{c}} (~17/16 = 101.8091{{c}})
: [[error map]]: {{val| +0.294 +0.295 -1.969 }}
* [[CWE]]: ~17/12 = 600.0000{{c}}, ~3/2 = 702.0260{{c}} (~17/16 = 102.0260{{c}})
: error map: {{val| 0.000 +0.071 -2.929 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~85/64 = 483.031
{{Optimal ET sequence|legend=1| 10, 12, 58, 70, 82, 94, 106, 118, 224g }}


[[Support]]ing [[ET]]s: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159*
[[Badness]] (Sintel): 0.0454


<nowiki>*</nowiki>Wart for 85
=== 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.


=== Dog ===
[[Subgroup]]: 2.3.19
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]].


Subgroup: 2.3.19
[[Comma list]]: 513/512


[[Comma list]]: 81/76
{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}
: mapping generators: ~2, ~3


[[Gencom]]: [2 4/3; 81/76]
[[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 }}


[[Mapping|Sval mapping]]: [{{val|1 2 6}}, {{val|0 -1 -4}}]
{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296, 1557bhhhh, 1853bhhhh }}


[[Tp tuning|POL2 generator]]: ~4/3 = 521.403
[[Badness]] (Sintel): 0.0294


{{Optimal ET sequence|legend=1| 5h, 7, 16, 23 }}
=== 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 #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 1,078: 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 1,091: 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 =====
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53, 77, 130, 443l, 573ll, 703ll, 1536blllll }}
[[Subgroup]]: 2.3.5.7.37


[[Comma list]]: 4107/4096, 17592186044416/17562397269605, 259/256
[[Badness]] (Sintel): 0.0690
 
[[Gencom]]: [2 37/32; 4107/4096 17592186044416/17562397269605 259/256]
 
[[Mapping]]: [{{val|1 1 4 1 5}}, {{val|0 -2 -8 -1 1}}]
 
[[POTE generator]]: ~[[37/32]] = 251.204
 
{{Optimal ET sequence|legend=1| 5, 14c, 19 }}


=== Reversed mavila ===
=== Reversed mavila ===
Line 1,145: 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]]
Retrieved from "https://en.xen.wiki/w/No-fives_subgroup_temperaments"