Jubilismic clan: Difference between revisions

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{{Technical data page}}
The '''jubilismic clan''' tempers out the jubilisma, [[50/49]], which means [[7/5]] and [[10/7]] are both equated to the 600-cent tritone and the [[octave]] is divided in two.  
The '''jubilismic clan''' tempers out the jubilisma, [[50/49]], which means [[7/5]] and [[10/7]] are both equated to the 600-cent tritone and the [[octave]] is divided in two.  


== Jubilic ==
== Jubilic ==
The head of this clan, jubilic, is generated by [[~]][[5/4]]. That and a semioctave give ~[[7/4]].  
The head of this clan, jubilic, is generated by [[~]][[5/4]]. That and a semioctave give ~[[7/4]]. As such, a reasonable tuning would tune the 5/4 flat and 7/4 sharp.  


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7
Line 10: Line 11:
{{Mapping|legend=2| 2 0 1 | 0 1 1 }}
{{Mapping|legend=2| 2 0 1 | 0 1 1 }}


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


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000, ~5/4 = 379.210 (~8/7 = 220.890)
* [[WE]]: ~7/5 = 599.6673{{c}}, ~5/4 = 380.6287{{c}} (~8/7 = 219.0386{{c}})
: [[error map]]: {{val| 0.000 -7.104 +10.384 }}
: [[error map]]: {{val| -0.665 -7.016 +10.139 }}
* [[POTE]]: ~7/5 = 600.000, ~5/4 = 380.840 (~8/7 = 219.160)
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~5/4 = 380.0086{{c}} (~8/7 = 219.9914{{c}})
: error map: {{val| 0.000 -5.474 +12.014 }}
: error map: {{val| 0.000 -6.305 +11.183 }}


{{Optimal ET sequence|legend=1| 2, 4, 6, 16, 22, 60d }}
{{Optimal ET sequence|legend=1| 2, 4, 6, 16, 22, 60d }}


[[Badness]] (Smith): 0.00190
[[Badness]] (Sintel): 0.140


=== Overview to extensions ===
=== Overview to extensions ===
Line 31: Line 32:
Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* [[Decimal]] (+25/24) → [[Dicot family #Decimal|Dicot family]]
* [[Decimal]] (+25/24) → [[Dicot family #Decimal|Dicot family]]
* [[Diminished (temperament)|Diminished]] (+36/35) → [[Dimipent family #Diminished|Dimipent family]]
* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]
* [[Pajara]] (+64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Pajara]] (+64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[Dubbla]]'' (+78125/73728) → [[Wesley family #Dubbla|Wesley family]]
* ''[[Dubbla]]'' (+78125/73728) → [[Wesley family #Dubbla|Wesley family]]
Line 40: Line 41:
* ''[[Hedgehog]]'' (+250/243) → [[Porcupine family #Hedgehog|Porcupine family]]
* ''[[Hedgehog]]'' (+250/243) → [[Porcupine family #Hedgehog|Porcupine family]]
* ''[[Crepuscular]]'' (+4375/4374) → [[Fifive family #Crepuscular|Fifive family]]
* ''[[Crepuscular]]'' (+4375/4374) → [[Fifive family #Crepuscular|Fifive family]]
* ''[[Byhearted]]'' (+19683/19208) → [[Tetracot family #Byhearted|Tetracot family]]
* ''[[Weasel]]'' (+19683/19208) → [[Tetracot family #Byhearted|Tetracot family]]


Considered below are lemba, astrology, walid, antikythera, doublewide, elvis, comic, and bipyth.
Considered below are lemba, astrology, walid, doublewide, elvis, comic, and bipyth.


== Lemba ==
== Lemba ==
Line 48: Line 49:
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lemba]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lemba]].''


Lemba tempers out 1029/1024, the gamelisma, and a stack of three ~8/7 generators gives an approximate perfect fifth.  
Lemba tempers out 1029/1024, the gamelisma, and a stack of three ~8/7 generators gives an approximate perfect fifth. It may be described as the {{nowrap| 10 & 16 }} temperament; its [[ploidacot]] is diploid tricot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 57: Line 58:


: mapping generators: ~7/5, ~8/7
: mapping generators: ~7/5, ~8/7
{{Multival|legend=1| 6 -2 -2 -17 -20 1 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000, ~8/7 = 232.927
* [[WE]]: ~7/5 = 601.4623{{c}}, ~8/7 = 232.6544{{c}}
: [[error map]]: {{val| 0.000 -3.175 -19.241 -1.753 }}
: [[error map]]: {{val| +2.925 -1.067 -11.656 +7.294 }}
* [[POTE]]: ~7/5 = 600.000, ~8/7 = 232.089
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~8/7 = 232.2655{{c}}
: error map: {{val| 0.000 -5.689 -18.402 -0.915 }}
: error map: {{val| 0.000 -5.158 -18.579 -1.091 }}


{{Optimal ET sequence|legend=1| 10, 16, 26, 62c }}
{{Optimal ET sequence|legend=1| 10, 16, 26, 36c, 62c }}


[[Badness]] (Smith): 0.062208
[[Badness]] (Sintel): 1.57


=== 11-limit ===
=== 11-limit ===
Line 78: Line 77:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~8/7 = 231.997
* WE: ~7/5 = 601.1769{{c}}, ~8/7 = 231.4273{{c}}
* POTE: ~7/5 = 600.000, ~8/7 = 230.974
* CWE: ~7/5 = 600.0000{{c}}, ~8/7 = 231.1781{{c}}


{{Optimal ET sequence|legend=0| 10, 16, 26 }}
{{Optimal ET sequence|legend=0| 10, 16, 26 }}


Badness (Smith): 0.041563
Badness (Sintel): 1.37


=== 13-limit ===
=== 13-limit ===
Line 93: Line 92:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~8/7 = 232.100
* WE: ~7/5 = 601.1939{{c}}, ~8/7 = 231.4261{{c}}
* POTE: ~7/5 = 600.000, ~8/7 = 230.966
* CWE: ~7/5 = 600.0000{{c}}, ~8/7 = 231.1617{{c}}


{{Optimal ET sequence|legend=0| 10, 16, 26 }}
{{Optimal ET sequence|legend=0| 10, 16, 26 }}


Badness (Smith): 0.025477
Badness (Sintel): 1.05


== Astrology ==
== Astrology ==
Astrology tempers out 3125/3072, the magic comma, and a stack of five ~5/4 generators gives an approximate harmonic 3.  
[[:de:Magische_Temperaturen#Astrology|Deutsch]]
 
{{see also| Magic family }}
 
Astrology tempers out 3125/3072, the magic comma, and a stack of five ~5/4 generators gives an approximate harmonic 3. It may be described as the {{nowrap| 16 & 22 }} temperament; its ploidacot is diploid pentacot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 110: Line 113:


: mapping geenerators: ~7/5, ~5/4
: mapping geenerators: ~7/5, ~5/4
{{Multival|legend=1| 10 2 2 -20 -25 -1 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000, ~5/4 = 380.355 (~8/7 = 219.645)
* [[WE]]: ~7/5 = 599.6999{{c}}, ~5/4 = 380.3881{{c}} (~8/7 = 219.3119{{c}})
: [[error map]]: {{val| 0.000 -0.180 -5.959 +11.529 }}
: [[error map]]: {{val| -0.600 -0.015 -7.126 +10.062 }}
* [[POTE]]: ~7/5 = 600.000, ~5/4 = 380.578 (~8/7 = 219.422)
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~5/4 = 380.5123{{c}} (~8/7 = 219.4877{{c}})
: error map: {{val| 0.000 +0.937 -5.735 +11.752 }}
: error map: {{val| 0.000 +0.606 -5.801 +11.686 }}


{{Optimal ET sequence|legend=1| 6, 16, 22, 60d }}
{{Optimal ET sequence|legend=1| 6, 16, 22, 60d }}


[[Badness]] (Smith): 0.082673
[[Badness]] (Sintel): 2.09


=== 11-limit ===
=== 11-limit ===
Line 131: Line 132:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~5/4 = 380.588 (~8/7 = 219.412)
* WE: ~7/5 = 600.0538{{c}}, ~5/4 = 380.5640{{c}} (~8/7 = 219.4897{{c}})
* POTE: ~7/5 = 600.000, ~5/4 = 380.530 (~8/7 = 219.470)
* CWE: ~7/5 = 600.0000{{c}}, ~5/4 = 380.5419{{c}} (~8/7 = 219.4581{{c}})


{{Optimal ET sequence|legend=0| 6, 16, 22 }}
{{Optimal ET sequence|legend=0| 6, 16, 22 }}


Badness (Smith): 0.039151
Badness (Sintel): 1.29


==== 13-limit ====
==== 13-limit ====
Line 146: Line 147:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~5/4 = 380.449 (~8/7 = 219.551)
* WE: ~7/5 = 600.7886{{c}}, ~5/4 = 380.2857{{c}} (~8/7 = 220.5028{{c}})
* POTE: ~7/5 = 600.000, ~5/4 = 379.787 (~8/7 = 220.213)
* CWE: ~7/5 = 600.0000{{c}}, ~5/4 = 379.9119{{c}} (~8/7 = 220.0881{{c}})


{{Optimal ET sequence|legend=0| 6, 16, 22, 38f }}
{{Optimal ET sequence|legend=0| 6, 16, 22, 38f }}


Badness (Smith): 0.034376
Badness (Sintel): 1.42


; Music
; Music
Line 164: Line 165:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~5/4 = 379.762 (~8/7 = 220.238)
* WE: ~7/5 = 599.8927{{c}}, ~5/4 = 379.7688{{c}} (~8/7 = 220.1239{{c}})
* POTE: ~7/5 = 600.000, ~5/4 = 379.837 (~8/7 = 220.163)
* CWE: ~7/5 = 600.0000{{c}}, ~5/4 = 379.8117{{c}} (~8/7 = 220.1883{{c}})


{{Optimal ET sequence|legend=0| 16, 22f, 38 }}
{{Optimal ET sequence|legend=0| 6f, 16, 22f, 38 }}


Badness (Smith): 0.035284
Badness (Sintel): 1.46


== Walid ==
== Walid ==
This low-accuracy extension tempers out 16/15, so the perfect fifth stands in for ~8/5 as in [[father]]. Its ploidacot is diploid monocot.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 179: Line 182:


: mapping generators: ~7/5, ~3
: mapping generators: ~7/5, ~3
{{Multival|legend=1| 2 -2 -2 -8 -9 1 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000, ~3/2 = 754.204 (~15/14 = 154.204)
* [[WE]]: ~7/5 = 589.0384{{c}}, ~3/2 = 735.7242{{c}} (~15/14 = 146.6857{{c}})
: [[error map]]: {{val| 0.000 +52.249 +59.482 +76.970 }}
: [[error map]]: {{val| -21.923 +11.846 +12.193 +18.719 }}
* [[POTE]]: ~7/5 = 600.000, ~3/2 = 749.415 (~15/14 = 149.415)
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 750.4026{{c}} (~15/14 = 150.4026{{c}})
: error map: {{val| 0.000 +47.460 +64.271 +81.759 }}
: error map: {{val| 0.000 +48.448 +63.284 +80.771 }}


{{Optimal ET sequence|legend=1| 2, 6, 8d }}
{{Optimal ET sequence|legend=1| 2, 6, 8d }}


[[Badness]] (Smith): 0.048978
[[Badness]] (Sintel): 1.24


=== 11-limit ===
=== 11-limit ===
Line 200: Line 201:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~3/2 = 754.205 (~15/14 = 154.205)
* WE: ~7/5 = 589.7684{{c}}, ~3/2 = 736.9708{{c}} (~12/11 = 147.2023{{c}})
* POTE: ~7/5 = 600.000, ~3/2 = 749.756 (~15/14 = 149.756)
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 750.5221{{c}} (~12/11 = 150.5221{{c}})


{{Optimal ET sequence|legend=0| 2, 6, 8d }}
{{Optimal ET sequence|legend=0| 2, 6, 8d }}


Badness (Smith): 0.029193
Badness (Sintel): 0.965


== Antikythera ==
== Doublewide ==
Named by [[Gene Ward Smith]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101481.html Yahoo! Tuning Group | ''Antikythera'']</ref>, antikythera is every other step of [[pajara]].  
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Doublewide]].''
 
[[Subgroup]]: 2.9.5.7


[[Comma list]]: 50/49, 64/63
Doublewide tempers out not only the keema, [[875/864]], but the orwellisma, [[1728/1715]], and may be described as the {{nowrap| 22 & 26 }} temperament. It is the unique temperament that equates the classical chroma ([[25/24]]), the large septimal diesis ([[49/48]]), and the interval between the classical and septimal thirds ([[36/35]]). It is generated by a sharply tuned [[~]][[6/5]] minor third, four of which and a [[semi-octave]] period give the [[3/1|3rd]] [[harmonic]], so its [[ploidacot]] is diploid alpha-tetracot. An [[11-limit]] extension is immediately available by identifying two generator steps as ~[[16/11]]. [[48edo]] makes for an excellent tuning.  
 
{{Mapping|legend=2| 2 0 11 12 | 0 1 -1 -1 }}
 
: mapping generators: ~7/5, ~9
 
{{Mapping|legend=3| 2 3 5 6 | 0 1/2 -1 -1 }}
 
: [[gencom]]: [7/5 8/7; 50/49 64/63]
 
[[Optimal tuning]]s:
* [[CTE]]: ~7/5 = 600.000, ~9/8 = 216.711
: [[error map]]: {{val| 0.000 +12.801 +3.025 +14.463 }}
* [[POTE]]: ~7/5 = 600.000, ~9/8 = 214.095
: error map: {{val| 0.000 +10.815 -0.409 +17.079 }}
 
{{Optimal ET sequence|legend=1| 2, 4, 6, 16, 22, 28 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 2.572 cents
 
[[Badness]] (Smith): 0.00501
 
== Doublewide ==
: ''For the 5-limit version, see [[Superpyth–22 equivalence continuum]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 244: Line 220:


: mapping generators: ~7/5, ~6/5
: mapping generators: ~7/5, ~6/5
{{Multival|legend=1| 8 6 6 -9 -13 -3 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000, ~6/5 = 325.769 (~7/6 = 274.231)
* [[WE]]: ~7/5 = 600.0365{{c}}, ~6/5 = 325.7389{{c}} (~7/6 = 274.2975{{c}})
: [[error map]]: {{val| 0.000 +1.120 -9.007 +8.481 }}
: [[error map]]: {{val| -2.303 +2.864 -5.756 +10.580 }}
* [[POTE]]: ~7/5 = 600.000, ~6/5 = 325.719 (~7/6 = 274.281)
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~6/5 = 325.7353{{c}} (~7/6 = 274.2647{{c}})
: error map: {{val| 0.000 +0.921 -9.156 +8.331 }}
: error map: {{val| 0.000 +10.778 -1.001 +16.487 }}


{{Optimal ET sequence|legend=1| 4, 14bd, 18, 22, 48 }}
{{Optimal ET sequence|legend=1| 4, 14bd, 18, 22, 48 }}


[[Badness]] (Smith): 0.043462
[[Badness]] (Sintel): 1.10


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 50/49, 99/98, 875/864
Comma list: 50/49, 99/98, 385/384


Mapping: {{mapping| 2 1 3 4 8 | 0 4 3 3 -2 }}
Mapping: {{mapping| 2 1 3 4 8 | 0 4 3 3 -2 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~6/5 = 325.719 (~7/6 = 274.281)
* WE: ~7/5 = 600.1818{{c}}, ~6/5 = 325.6434{{c}} (~7/6 = 274.5384{{c}})
* POTE: ~7/5 = 600.000, ~6/5 = 325.545 (~7/6 = 274.455)
* CWE: ~7/5 = 600.0000{{c}}, ~6/5 = 325.5854{{c}} (~7/6 = 274.4146{{c}})


{{Optimal ET sequence|legend=0| 4, 18, 22, 48 }}
{{Optimal ET sequence|legend=0| 4, 18, 22, 48 }}


Badness (Smith): 0.032058
Badness (Sintel): 1.06


=== Fleetwood ===
=== Fleetwood ===
Line 280: Line 254:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~6/5 = 326.684 (~7/6 = 273.316)
* WE: ~7/5 = 599.6049{{c}}, ~6/5 = 326.8229{{c}} (~7/6 = 272.7819{{c}})
* POTE: ~7/5 = 600.000, ~6/5 = 327.038 (~7/6 = 272.962)
* CWE: ~7/5 = 600.0000{{c}}, ~6/5 = 326.8890{{c}} (~7/6 = 273.1110{{c}})


{{Optimal ET sequence|legend=0| 4e, …, 18e, 22 }}
{{Optimal ET sequence|legend=0| 4e, …, 18e, 22 }}


Badness (Smith): 0.035202
Badness (Sintel): 1.16


==== 13-limit ====
==== 13-limit ====
Line 295: Line 269:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~6/5 = 327.450 (~7/6 = 272.540)
* WE: ~7/5 = 599.5482{{c}}, ~6/5 = 327.5939{{c}} (~7/6 = 271.9543{{c}})
* POTE: ~7/5 = 600.000, ~6/5 = 327.841 (~7/6 = 272.159)
* CWE: ~7/5 = 600.0000{{c}}, ~6/5 = 327.6706{{c}} (~7/6 = 272.3294{{c}})


{{Optimal ET sequence|legend=0| 4ef, …, 18e, 22 }}
{{Optimal ET sequence|legend=0| 4ef, …, 18e, 22 }}


Badness (Smith): 0.031835
Badness (Sintel): 1.32


=== Cavalier ===
=== Cavalier ===
Line 310: Line 284:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~6/5 = 324.238 (~7/6 = 275.762)
* WE: ~7/5 = 600.9467{{c}}, ~6/5 = 323.9369{{c}} (~7/6 = 277.0098{{c}})
* POTE: ~7/5 = 600.000, ~6/5 = 323.427 (~7/6 = 276.573)
* CWE: ~7/5 = 600.0000{{c}}, ~6/5 = 323.7272{{c}} (~7/6 = 276.2728{{c}})


{{Optimal ET sequence|legend=0| 4e, 22e, 26 }}
{{Optimal ET sequence|legend=0| 4e, 22e, 26 }}


Badness (Smith): 0.052899
Badness (Sintel): 1.75


==== 13-limit ====
==== 13-limit ====
Line 325: Line 299:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~6/5 = 324.187 (~7/6 = 275.813)
* WE: ~7/5 = 600.9537{{c}}, ~6/5 = 323.9097{{c}} (~7/6 = 277.0440{{c}})
* POTE: ~7/5 = 600.000, ~6/5 = 323.396 (~7/6 = 276.604)
* CWE: ~7/5 = 600.0000{{c}}, ~6/5 = 323.6876{{c}} (~7/6 = 276.3124{{c}})


{{Optimal ET sequence|legend=0| 4ef, 22ef, 26 }}
{{Optimal ET sequence|legend=0| 4ef, 22ef, 26 }}


Badness (Smith): 0.035040
Badness (Sintel): 1.45


== Elvis ==
== Elvis ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Elvis]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Elvis]].''
Elvis is generated by a ptolemaic diminished fifth, tuned sharp such that two generators and a semi-octave period give the 3rd harmonic. Its ploidacot is diploid alpha-dicot. [[26edo]] makes for an obvious tuning.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 342: Line 318:


: mapping generators: ~7/5, ~64/45
: mapping generators: ~7/5, ~64/45
{{Multival|legend=1| 4 -10 -10 -25 -27 5 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000, ~64/45 = 645.313 (~64/63 = 45.313)
* [[WE]]: ~7/5 = 601.6846{{c}}, ~64/45 = 648.0937{{c}} (~64/63 = 46.4091{{c}})
: [[error map]]: {{val| 0.000 -11.329 -12.879 +4.609 }}
: [[error map]]: {{val| +3.369 -4.083 -9.936 +9.236 }}
* [[POTE]]: ~7/5 = 600.000, ~64/45 = 646.279 (~64/63 = 46.279)
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~64/45 = 646.0539{{c}} (~64/63 = 46.0539{{c}})
: error map: {{val| 0.000 -9.397 -17.710 -0.222 }}
: error map: {{val| 0.000 -9.847 -16.583 +0.904 }}


{{Optimal ET sequence|legend=1| 2, 24c, 26 }}
{{Optimal ET sequence|legend=1| 2, 24c, 26 }}


[[Badness]] (Smith): 0.141473
[[Badness]] (Sintel): 3.58


=== 11-limit ===
=== 11-limit ===
Line 363: Line 337:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~16/11 = 645.343 (~56/55 = 45.343)
* WE: ~7/5 = 601.2186{{c}}, ~16/11 = 647.4300{{c}} (~56/55 = 46.2114{{c}})
* POTE: ~7/5 = 600.000, ~16/11 = 646.118 (~56/55 = 46.118)
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 645.9681{{c}} (~56/55 = 45.9681{{c}})


{{Optimal ET sequence|legend=0| 2, 24c, 26 }}
{{Optimal ET sequence|legend=0| 2, 24c, 26 }}


Badness (Smith): 0.063212
Badness (Sintel): 2.09


=== 13-limit ===
=== 13-limit ===
Line 378: Line 352:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~16/11 = 645.208 (~56/55 = 45.208)
* WE: ~7/5 = 601.2206{{c}}, ~16/11 = 647.4219{{c}} (~56/55 = 46.2013{{c}})
* POTE: ~7/5 = 600.000, ~16/11 = 646.108 (~56/55 = 46.108)
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 645.9362{{c}} (~56/55 = 45.9362{{c}})


{{Optimal ET sequence|legend=0| 2f, 24cf, 26 }}
{{Optimal ET sequence|legend=0| 2f, 24cf, 26 }}


Badness (Smith): 0.043997
Badness (Sintel): 1.82


== Comic ==
== Comic ==
: ''For the 5-limit version, see [[Superpyth–22 equivalence continuum]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Comic]].''
 
Comic is generated by a grave fifth, tuned flat such that two generators and a semi-octave period give the 3rd harmonic. Its ploidacot is diploid alpha-dicot. [[22edo]] makes for an obvious tuning.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 395: Line 371:


: mapping generators: ~7/5, ~40/27
: mapping generators: ~7/5, ~40/27
{{Multival|legend=1| 4 14 14 13 11 -7 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000, ~40/27 = 653.872 (~28/27 = 53.872)
* [[WE]]: ~7/5 = 598.9554{{c}}, ~40/27 = 653.5596{{c}} (~28/27 = 54.6042{{c}})
: [[error map]]: {{val| 0.000 +5.789 -9.211 +8.277 }}
: [[error map]]: {{val| +3.369 -4.083 -9.936 +9.236 }}
* [[POTE]]: ~7/5 = 600.000, ~40/27 = 654.699 (~28/27 = 54.699)
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~40/27 = 654.3329{{c}} (~28/27 = 54.3329{{c}})
: error map: {{val| 0.000 +7.444 -3.417 +14.070 }}
: error map: {{val| 0.000 -9.847 -16.583 +0.904 }}


{{Optimal ET sequence|legend=1| 2cd, …, 20cd, 22 }}
{{Optimal ET sequence|legend=1| 2cd, …, 20cd, 22 }}


[[Badness]] (Smith): 0.084395
[[Badness]] (Sintel): 2.14


=== 11-limit ===
=== 11-limit ===
Line 416: Line 390:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~22/15 = 654.289 (~28/27 = 54.289)
* WE: ~7/5 = 598.8161{{c}}, ~22/15 = 653.8909{{c}} (~28/27 = 55.0747{{c}})
* POTE: ~7/5 = 600.000, ~22/15 = 655.184 (~28/27 = 55.184)
* CWE: ~7/5 = 600.0000{{c}}, ~22/15 = 654.7898{{c}} (~28/27 = 54.7898{{c}})


{{Optimal ET sequence|legend=0| 2cde, …, 20cde, 22 }}
{{Optimal ET sequence|legend=0| 2cde, …, 20cde, 22 }}


Badness (Smith): 0.045052
Badness (Sintel): 1.49


=== 13-limit ===
=== 13-limit ===
Line 431: Line 405:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~22/15 = 654.547 (~28/27 = 54.547)
* WE: ~7/5 = 600.1030{{c}}, ~22/15 = 654.5470{{c}} (~28/27 = 54.4440{{c}})
* POTE: ~7/5 = 600.000, ~22/15 = 654.435 (~28/27 = 54.435)
* CWE: ~7/5 = 600.0000{{c}}, ~22/15 = 654.4665{{c}} (~28/27 = 54.4665{{c}})


{{Optimal ET sequence|legend=0| 2cde, 20cde, 22 }}
{{Optimal ET sequence|legend=0| 2cde, 20cde, 22 }}


Badness (Smith): 0.041470
Badness (Sintel): 1.71


== Bipyth ==
== Bipyth ==
: ''For the 5-limit version, see [[Superpyth–22 equivalence continuum]].''
Bipyth tempers out the 5-limit [[superpyth comma]], 20480/19683, making it an alternative extension of 5-limit [[superpyth]]. Its ploidacot is diploid monocot.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 448: Line 422:


: mapping generators: ~7/5, ~3
: mapping generators: ~7/5, ~3
{{Multival|legend=1| 2 18 18 24 23 -9 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000, ~3/2 = 708.695 (~15/14 = 108.695)
* [[WE]]: ~7/5 = 598.7533{{c}}, ~3/2 = 707.9630{{c}} (~15/14 = 109.2098{{c}})
: [[error map]]: {{val| 0.000 +6.740 -8.058 +9.430 }}
: [[error map]]: {{val| +3.369 -4.083 -9.936 +9.236 }}
* [[POTE]]: ~7/5 = 600.000, ~3/2 = 709.437 (~15/14 = 109.437)
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 709.1579{{c}} (~15/14 = 109.1579{{c}})
: error map: {{val| 0.000 +7.482 -1.379 +16.109 }}
: error map: {{val| 0.000 -9.847 -16.583 +0.904 }}


{{Optimal ET sequence|legend=1| 10cd, 12cd, 22 }}
{{Optimal ET sequence|legend=1| 10cd, 12cd, 22 }}


[[Badness]] (Smith): 0.165033
[[Badness]] (Sintel): 4.18


=== 11-limit ===
=== 11-limit ===
Line 469: Line 441:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~3/2 = 708.813 (~15/14 = 108.813)
* WE: ~7/5 = 599.2296{{c}}, ~3/2 = 708.3992{{c}} (~15/14 = 109.1697{{c}})
* POTE: ~7/5 = 600.000, ~3/2 = 709.310 (~15/14 = 109.310)
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 709.1395{{c}} (~15/14 = 109.1395{{c}})


{{Optimal ET sequence|legend=0| 10cd, 12cde, 22 }}
{{Optimal ET sequence|legend=0| 10cd, 12cde, 22 }}


Badness (Smith): 0.070910
Badness (Sintel): 2.34


== Sedecic ==
== Sedecic ==
Sedecic has 1/16-octave period and may be thought of as 16edo with an independent generator for prime 3. Its ploidacot is 16-ploid monocot.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 482: Line 456:


{{Mapping|legend=1| 16 0 37 45 | 0 1 0 0 }}
{{Mapping|legend=1| 16 0 37 45 | 0 1 0 0 }}
{{Multival|legend=1| 16 0 0 -37 -45 0 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~128/125 = 75.000, ~3/2 = 701.955 (~525/512 = 26.955)
* [[WE]]: ~128/125 = 75.0539{{c}}, ~3/2 = 701.0578{{c}} (~525/512 = 25.5726{{c}})
: [[error map]]: {{val| 0.000 0.000 -11.314 +6.174 }}
: [[error map]]: {{val| 0.000 0.000 -11.314 +6.174 }}
* [[POTE]]: ~128/125 = 75.000, ~3/2 = 700.554 (~525/512 = 25.554)
* [[CWE]]: ~128/125 = 75.0000{{c}}, ~3/2 = 700.8957{{c}} (~525/512 = 25.8957{{c}})
: error map: {{val| 0.000 -1.401 -11.314 +6.174 }}
: error map: {{val| 0.000 -1.401 -11.314 +6.174 }}


{{Optimal ET sequence|legend=1| 16, 32, 48 }}
{{Optimal ET sequence|legend=1| 16, 32, 48 }}


[[Badness]] (Smith): 0.265972
[[Badness]] (Sintel): 6.73


=== 11-limit ===
=== 11-limit ===
Line 503: Line 475:


Optimal tunings:  
Optimal tunings:  
* CTE: ~22/21 = 75.000, ~3/2 = 701.844 (~45/44 = 26.844)
* WE: ~22/21 = 75.0000{{c}}, ~3/2 = 700.7810{{c}} (~45/44 = 25.3476{{c}})
* POTE: ~22/21 = 75.000, ~3/2 = 700.331 (~45/44 = 25.331)
* CWE: ~22/21 = 75.0000{{c}}, ~3/2 = 700.6780{{c}} (~45/44 = 25.6780{{c}})


{{Optimal ET sequence|legend=0| 16, 32, 48 }}
{{Optimal ET sequence|legend=0| 16, 32, 48 }}


Badness (Smith): 0.092774
Badness (Sintel): 3.07
 
== Subgroup extensions ==
=== Antikythera (2.9.5.7) ===
Antikythera is every other step of [[pajara]]. It was allegedly named by [[Keenan Pepper]] in 2011 after the {{w|Antikythera mechanism}} for its association with [[astrology]] and [[machine]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101481.html Yahoo! Tuning Group | ''Antikythera'']</ref>.
 
[[Subgroup]]: 2.9.5.7
 
[[Comma list]]: 50/49, 64/63
 
{{Mapping|legend=2| 2 0 11 12 | 0 1 -1 -1 }}
 
{{Mapping|legend=3| 2 0 11 12 | 0 1/2 -1 -1 }}
 
: mapping generators: ~7/5, ~9
 
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 598.8483{{c}}, ~9/8 = 213.6844{{c}}
: [[error map]]: {{val| -2.303 +2.864 -5.756 +10.580 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~9/8 = 214.6875{{c}}
: error map: {{val| 0.000 +10.778 -1.001 +16.487 }}
 
{{Optimal ET sequence|legend=1| 2, 4, 6, 16, 22, 28 }}
 
[[Badness]] (Sintel): 0.253


== Notes ==
== References ==


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Jubilismic clan| ]] <!-- main article -->
[[Category:Jubilismic clan| ]] <!-- main article -->
[[Category:Jubilismic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 10:24, 14 March 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.

The jubilismic clan tempers out the jubilisma, 50/49, which means 7/5 and 10/7 are both equated to the 600-cent tritone and the octave is divided in two.

Jubilic

The head of this clan, jubilic, is generated by ~5/4. That and a semioctave give ~7/4. As such, a reasonable tuning would tune the 5/4 flat and 7/4 sharp.

Subgroup: 2.5.7

Comma list: 50/49

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

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

mapping generators: ~7/5, ~5

Optimal tunings:

  • WE: ~7/5 = 599.6673 ¢, ~5/4 = 380.6287 ¢ (~8/7 = 219.0386 ¢)
error map: ⟨-0.665 -7.016 +10.139]
  • CWE: ~7/5 = 600.0000 ¢, ~5/4 = 380.0086 ¢ (~8/7 = 219.9914 ¢)
error map: ⟨0.000 -6.305 +11.183]

Optimal ET sequence: 2, 4, 6, 16, 22, 60d

Badness (Sintel): 0.140

Overview to extensions

Lemba finds the perfect fifth three steps away by tempering out 1029/1024. Astrology, five steps away by tempering out 3125/3072. Decimal, two steps away by tempering out 25/24 and 49/48. Walid merges ~5/4 and ~4/3 by tempering out 16/15.

Diminished adds 36/35 and splits the ~7/5 period in a further two. Pajara adds 64/63 and slices the ~7/4 in two, with antikythera being every other step thereof. Dubbla adds 78125/73728 and slices the ~5/4 in two. Injera adds 81/80 and slices the ~5/1 in four. Octokaidecal adds 28/27. Bipelog adds 135/128. Those splits the generator into three in various ways. Hexe adds 128/125 and slices the period in three. Hedgehog adds 250/243. Elvis adds 8505/8192. Those slice the generator in five. Comic adds 2240/2187. Crepuscular adds 4375/4374. Those slice the generator in seven. Byhearted adds 19683/19208. Bipyth adds 20480/19683. Those slice the generator in nine.

Temperaments discussed elsewhere are:

Considered below are lemba, astrology, walid, doublewide, elvis, comic, and bipyth.

Lemba

Main article: Lemba
For the 5-limit version, see Miscellaneous 5-limit temperaments #Lemba.

Lemba tempers out 1029/1024, the gamelisma, and a stack of three ~8/7 generators gives an approximate perfect fifth. It may be described as the 10 & 16 temperament; its ploidacot is diploid tricot.

Subgroup: 2.3.5.7

Comma list: 50/49, 525/512

Mapping: [⟨2 2 5 6], ⟨0 3 -1 -1]]

mapping generators: ~7/5, ~8/7

Optimal tunings:

  • WE: ~7/5 = 601.4623 ¢, ~8/7 = 232.6544 ¢
error map: ⟨+2.925 -1.067 -11.656 +7.294]
  • CWE: ~7/5 = 600.0000 ¢, ~8/7 = 232.2655 ¢
error map: ⟨0.000 -5.158 -18.579 -1.091]

Optimal ET sequence: 10, 16, 26, 36c, 62c

Badness (Sintel): 1.57

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 385/384

Mapping: [⟨2 2 5 6 5], ⟨0 3 -1 -1 5]]

Optimal tunings:

  • WE: ~7/5 = 601.1769 ¢, ~8/7 = 231.4273 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~8/7 = 231.1781 ¢

Optimal ET sequence: 10, 16, 26

Badness (Sintel): 1.37

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 50/49, 65/64, 78/77

Mapping: [⟨2 2 5 6 5 7], ⟨0 3 -1 -1 5 1]]

Optimal tunings:

  • WE: ~7/5 = 601.1939 ¢, ~8/7 = 231.4261 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~8/7 = 231.1617 ¢

Optimal ET sequence: 10, 16, 26

Badness (Sintel): 1.05

Astrology

Deutsch

See also: Magic family

Astrology tempers out 3125/3072, the magic comma, and a stack of five ~5/4 generators gives an approximate harmonic 3. It may be described as the 16 & 22 temperament; its ploidacot is diploid pentacot.

Subgroup: 2.3.5.7

Comma list: 50/49, 3125/3072

Mapping: [⟨2 0 4 5], ⟨0 5 1 1]]

mapping geenerators: ~7/5, ~5/4

Optimal tunings:

  • WE: ~7/5 = 599.6999 ¢, ~5/4 = 380.3881 ¢ (~8/7 = 219.3119 ¢)
error map: ⟨-0.600 -0.015 -7.126 +10.062]
  • CWE: ~7/5 = 600.0000 ¢, ~5/4 = 380.5123 ¢ (~8/7 = 219.4877 ¢)
error map: ⟨0.000 +0.606 -5.801 +11.686]

Optimal ET sequence: 6, 16, 22, 60d

Badness (Sintel): 2.09

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 121/120, 176/175

Mapping: [⟨2 0 4 5 5], ⟨0 5 1 1 3]]

Optimal tunings:

  • WE: ~7/5 = 600.0538 ¢, ~5/4 = 380.5640 ¢ (~8/7 = 219.4897 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~5/4 = 380.5419 ¢ (~8/7 = 219.4581 ¢)

Optimal ET sequence: 6, 16, 22

Badness (Sintel): 1.29

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 65/64, 78/77, 121/120

Mapping: [⟨2 0 4 5 5 8], ⟨0 5 1 1 3 -1]]

Optimal tunings:

  • WE: ~7/5 = 600.7886 ¢, ~5/4 = 380.2857 ¢ (~8/7 = 220.5028 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~5/4 = 379.9119 ¢ (~8/7 = 220.0881 ¢)

Optimal ET sequence: 6, 16, 22, 38f

Badness (Sintel): 1.42

Music

Horoscope

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 66/65, 105/104, 121/120

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

Optimal tunings:

  • WE: ~7/5 = 599.8927 ¢, ~5/4 = 379.7688 ¢ (~8/7 = 220.1239 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~5/4 = 379.8117 ¢ (~8/7 = 220.1883 ¢)

Optimal ET sequence: 6f, 16, 22f, 38

Badness (Sintel): 1.46

Walid

This low-accuracy extension tempers out 16/15, so the perfect fifth stands in for ~8/5 as in father. Its ploidacot is diploid monocot.

Subgroup: 2.3.5.7

Comma list: 16/15, 50/49

Mapping: [⟨2 0 8 9], ⟨0 1 -1 -1]]

mapping generators: ~7/5, ~3

Optimal tunings:

  • WE: ~7/5 = 589.0384 ¢, ~3/2 = 735.7242 ¢ (~15/14 = 146.6857 ¢)
error map: ⟨-21.923 +11.846 +12.193 +18.719]
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 750.4026 ¢ (~15/14 = 150.4026 ¢)
error map: ⟨0.000 +48.448 +63.284 +80.771]

Optimal ET sequence: 2, 6, 8d

Badness (Sintel): 1.24

11-limit

Subgroup: 2.3.5.7.11

Comma list: 16/15, 22/21, 50/49

Mapping: [⟨2 0 8 9 7], ⟨0 1 -1 -1 0]]

Optimal tunings:

  • WE: ~7/5 = 589.7684 ¢, ~3/2 = 736.9708 ¢ (~12/11 = 147.2023 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 750.5221 ¢ (~12/11 = 150.5221 ¢)

Optimal ET sequence: 2, 6, 8d

Badness (Sintel): 0.965

Doublewide

For the 5-limit version, see Miscellaneous 5-limit temperaments #Doublewide.

Doublewide tempers out not only the keema, 875/864, but the orwellisma, 1728/1715, and may be described as the 22 & 26 temperament. It is the unique temperament that equates the classical chroma (25/24), the large septimal diesis (49/48), and the interval between the classical and septimal thirds (36/35). It is generated by a sharply tuned ~6/5 minor third, four of which and a semi-octave period give the 3rd harmonic, so its ploidacot is diploid alpha-tetracot. An 11-limit extension is immediately available by identifying two generator steps as ~16/11. 48edo makes for an excellent tuning.

Subgroup: 2.3.5.7

Comma list: 50/49, 875/864

Mapping: [⟨2 1 3 4], ⟨0 4 3 3]]

mapping generators: ~7/5, ~6/5

Optimal tunings:

  • WE: ~7/5 = 600.0365 ¢, ~6/5 = 325.7389 ¢ (~7/6 = 274.2975 ¢)
error map: ⟨-2.303 +2.864 -5.756 +10.580]
  • CWE: ~7/5 = 600.0000 ¢, ~6/5 = 325.7353 ¢ (~7/6 = 274.2647 ¢)
error map: ⟨0.000 +10.778 -1.001 +16.487]

Optimal ET sequence: 4, 14bd, 18, 22, 48

Badness (Sintel): 1.10

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 99/98, 385/384

Mapping: [⟨2 1 3 4 8], ⟨0 4 3 3 -2]]

Optimal tunings:

  • WE: ~7/5 = 600.1818 ¢, ~6/5 = 325.6434 ¢ (~7/6 = 274.5384 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~6/5 = 325.5854 ¢ (~7/6 = 274.4146 ¢)

Optimal ET sequence: 4, 18, 22, 48

Badness (Sintel): 1.06

Fleetwood

Subgroup: 2.3.5.7.11

Comma list: 50/49, 55/54, 176/175

Mapping: [⟨2 1 3 4 2], ⟨0 4 3 3 9]]

Optimal tunings:

  • WE: ~7/5 = 599.6049 ¢, ~6/5 = 326.8229 ¢ (~7/6 = 272.7819 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~6/5 = 326.8890 ¢ (~7/6 = 273.1110 ¢)

Optimal ET sequence: 4e, …, 18e, 22

Badness (Sintel): 1.16

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 55/54, 65/63, 176/175

Mapping: [⟨2 1 3 4 2 3], ⟨0 4 3 3 9 8]]

Optimal tunings:

  • WE: ~7/5 = 599.5482 ¢, ~6/5 = 327.5939 ¢ (~7/6 = 271.9543 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~6/5 = 327.6706 ¢ (~7/6 = 272.3294 ¢)

Optimal ET sequence: 4ef, …, 18e, 22

Badness (Sintel): 1.32

Cavalier

Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 875/864

Mapping: [⟨2 1 3 4 1], ⟨0 4 3 3 11]]

Optimal tunings:

  • WE: ~7/5 = 600.9467 ¢, ~6/5 = 323.9369 ¢ (~7/6 = 277.0098 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~6/5 = 323.7272 ¢ (~7/6 = 276.2728 ¢)

Optimal ET sequence: 4e, 22e, 26

Badness (Sintel): 1.75

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 50/49, 78/77, 325/324

Mapping: [⟨2 1 3 4 1 2], ⟨0 4 3 3 11 10]]

Optimal tunings:

  • WE: ~7/5 = 600.9537 ¢, ~6/5 = 323.9097 ¢ (~7/6 = 277.0440 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~6/5 = 323.6876 ¢ (~7/6 = 276.3124 ¢)

Optimal ET sequence: 4ef, 22ef, 26

Badness (Sintel): 1.45

Elvis

For the 5-limit version, see Miscellaneous 5-limit temperaments #Elvis.

Elvis is generated by a ptolemaic diminished fifth, tuned sharp such that two generators and a semi-octave period give the 3rd harmonic. Its ploidacot is diploid alpha-dicot. 26edo makes for an obvious tuning.

Subgroup: 2.3.5.7

Comma list: 50/49, 8505/8192

Mapping: [⟨2 1 10 11], ⟨0 2 -5 -5]]

mapping generators: ~7/5, ~64/45

Optimal tunings:

  • WE: ~7/5 = 601.6846 ¢, ~64/45 = 648.0937 ¢ (~64/63 = 46.4091 ¢)
error map: ⟨+3.369 -4.083 -9.936 +9.236]
  • CWE: ~7/5 = 600.0000 ¢, ~64/45 = 646.0539 ¢ (~64/63 = 46.0539 ¢)
error map: ⟨0.000 -9.847 -16.583 +0.904]

Optimal ET sequence: 2, 24c, 26

Badness (Sintel): 3.58

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 1344/1331

Mapping: [⟨2 1 10 11 8], ⟨0 2 -5 -5 -1]]

Optimal tunings:

  • WE: ~7/5 = 601.2186 ¢, ~16/11 = 647.4300 ¢ (~56/55 = 46.2114 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~16/11 = 645.9681 ¢ (~56/55 = 45.9681 ¢)

Optimal ET sequence: 2, 24c, 26

Badness (Sintel): 2.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 50/49, 78/77, 1053/1024

Mapping: [⟨2 1 10 11 8 16], ⟨0 2 -5 -5 -1 -8]]

Optimal tunings:

  • WE: ~7/5 = 601.2206 ¢, ~16/11 = 647.4219 ¢ (~56/55 = 46.2013 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~16/11 = 645.9362 ¢ (~56/55 = 45.9362 ¢)

Optimal ET sequence: 2f, 24cf, 26

Badness (Sintel): 1.82

Comic

For the 5-limit version, see Miscellaneous 5-limit temperaments #Comic.

Comic is generated by a grave fifth, tuned flat such that two generators and a semi-octave period give the 3rd harmonic. Its ploidacot is diploid alpha-dicot. 22edo makes for an obvious tuning.

Subgroup: 2.3.5.7

Comma list: 50/49, 2240/2187

Mapping: [⟨2 1 -3 -2], ⟨0 2 7 7]]

mapping generators: ~7/5, ~40/27

Optimal tunings:

  • WE: ~7/5 = 598.9554 ¢, ~40/27 = 653.5596 ¢ (~28/27 = 54.6042 ¢)
error map: ⟨+3.369 -4.083 -9.936 +9.236]
  • CWE: ~7/5 = 600.0000 ¢, ~40/27 = 654.3329 ¢ (~28/27 = 54.3329 ¢)
error map: ⟨0.000 -9.847 -16.583 +0.904]

Optimal ET sequence: 2cd, …, 20cd, 22

Badness (Sintel): 2.14

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 99/98, 2662/2625

Mapping: [⟨2 1 -3 -2 -4], ⟨0 2 7 7 10]]

Optimal tunings:

  • WE: ~7/5 = 598.8161 ¢, ~22/15 = 653.8909 ¢ (~28/27 = 55.0747 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~22/15 = 654.7898 ¢ (~28/27 = 54.7898 ¢)

Optimal ET sequence: 2cde, …, 20cde, 22

Badness (Sintel): 1.49

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 65/63, 99/98, 968/945

Mapping: [⟨2 1 -3 -2 -4 3], ⟨0 2 7 7 10 4]]

Optimal tunings:

  • WE: ~7/5 = 600.1030 ¢, ~22/15 = 654.5470 ¢ (~28/27 = 54.4440 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~22/15 = 654.4665 ¢ (~28/27 = 54.4665 ¢)

Optimal ET sequence: 2cde, 20cde, 22

Badness (Sintel): 1.71

Bipyth

Bipyth tempers out the 5-limit superpyth comma, 20480/19683, making it an alternative extension of 5-limit superpyth. Its ploidacot is diploid monocot.

Subgroup: 2.3.5.7

Comma list: 50/49, 20480/19683

Mapping: [⟨2 0 -24 -23], ⟨0 1 9 9]]

mapping generators: ~7/5, ~3

Optimal tunings:

  • WE: ~7/5 = 598.7533 ¢, ~3/2 = 707.9630 ¢ (~15/14 = 109.2098 ¢)
error map: ⟨+3.369 -4.083 -9.936 +9.236]
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 709.1579 ¢ (~15/14 = 109.1579 ¢)
error map: ⟨0.000 -9.847 -16.583 +0.904]

Optimal ET sequence: 10cd, 12cd, 22

Badness (Sintel): 4.18

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 121/120, 896/891

Mapping: [⟨2 0 -24 -23 -9], ⟨0 1 9 9 5]]

Optimal tunings:

  • WE: ~7/5 = 599.2296 ¢, ~3/2 = 708.3992 ¢ (~15/14 = 109.1697 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 709.1395 ¢ (~15/14 = 109.1395 ¢)

Optimal ET sequence: 10cd, 12cde, 22

Badness (Sintel): 2.34

Sedecic

Sedecic has 1/16-octave period and may be thought of as 16edo with an independent generator for prime 3. Its ploidacot is 16-ploid monocot.

Subgroup: 2.3.5.7

Comma list: 50/49, 546875/524288

Mapping: [⟨16 0 37 45], ⟨0 1 0 0]]

Optimal tunings:

  • WE: ~128/125 = 75.0539 ¢, ~3/2 = 701.0578 ¢ (~525/512 = 25.5726 ¢)
error map: ⟨0.000 0.000 -11.314 +6.174]
  • CWE: ~128/125 = 75.0000 ¢, ~3/2 = 700.8957 ¢ (~525/512 = 25.8957 ¢)
error map: ⟨0.000 -1.401 -11.314 +6.174]

Optimal ET sequence: 16, 32, 48

Badness (Sintel): 6.73

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 385/384, 1331/1323

Mapping: [⟨16 0 37 45 30], ⟨0 1 0 0 1]]

Optimal tunings:

  • WE: ~22/21 = 75.0000 ¢, ~3/2 = 700.7810 ¢ (~45/44 = 25.3476 ¢)
  • CWE: ~22/21 = 75.0000 ¢, ~3/2 = 700.6780 ¢ (~45/44 = 25.6780 ¢)

Optimal ET sequence: 16, 32, 48

Badness (Sintel): 3.07

Subgroup extensions

Antikythera (2.9.5.7)

Antikythera is every other step of pajara. It was allegedly named by Keenan Pepper in 2011 after the Antikythera mechanism for its association with astrology and machine[1].

Subgroup: 2.9.5.7

Comma list: 50/49, 64/63

Subgroup-val mapping: [⟨2 0 11 12], ⟨0 1 -1 -1]]

Gencom mapping: [⟨2 0 11 12], ⟨0 1/2 -1 -1]]

mapping generators: ~7/5, ~9

Optimal tunings:

  • WE: ~7/5 = 598.8483 ¢, ~9/8 = 213.6844 ¢
error map: ⟨-2.303 +2.864 -5.756 +10.580]
  • CWE: ~7/5 = 600.0000 ¢, ~9/8 = 214.6875 ¢
error map: ⟨0.000 +10.778 -1.001 +16.487]

Optimal ET sequence: 2, 4, 6, 16, 22, 28

Badness (Sintel): 0.253

References

  1. ↑ Yahoo! Tuning Group | Antikythera
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