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 15: Line 16:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 1\2, ~5/4 = 379.21
* [[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 = 1\2, ~5/4 = 380.840
* [[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]]: 0.00190
[[Badness]] (Sintel): 0.140


=== Overview to extensions ===
=== 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]].  
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 splits the ~7/5 period into a further two. Pajara slices the ~7/4 into two, with antikythera being every other step thereof. Injera slices the ~5/1 into four. Hedgehog slices the ~7/1 into five. Crepuscular slices the ~7/4 into seven.  
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.  


Lemba, astrology, and doublewide are discussed below; others in the clan are  
Temperaments discussed elsewhere are:
* [[Diminished]] → [[Dimipent family #Diminished|Dimipent family]]
* [[Decimal]] (+25/24) → [[Dicot family #Decimal|Dicot family]]
* [[Pajara]] → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]
* [[Decimal]] → [[Dicot family #Decimal|Dicot family]]
* [[Pajara]] (+64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Injera]] → [[Meantone family #Injera|Meantone family]]
* ''[[Dubbla]]'' (+78125/73728) → [[Wesley family #Dubbla|Wesley family]]
* [[Octokaidecal]] → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]
* ''[[Injera]]'' (+81/80) → [[Meantone family #Injera|Meantone family]]
* [[Hedgehog]] → [[Porcupine family #Hedgehog|Porcupine family]]
* ''[[Octokaidecal]]'' (+28/27) → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]
* [[Dubbla]] → [[Wesley family #Dubbla|Wesley family]]
* ''[[Bipelog]]'' (+135/128) → [[Mavila #Bipelog|Mavila family]]
* [[Bipelog]] → [[Pelogic family #Bipelog|Pelogic family]]
* ''[[Hexe]]'' (+128/125) → [[Augmented family #Hexe|Augmented family]]
* [[Crepuscular]] → [[Fifive family #Crepuscular|Fifive family]]
* ''[[Hedgehog]]'' (+250/243) → [[Porcupine family #Hedgehog|Porcupine family]]
* [[Hexe]] → [[Augmented family #Hexe|Augmented family]]
* ''[[Crepuscular]]'' (+4375/4374) → [[Fifive family #Crepuscular|Fifive family]]
* [[Byhearted]] → [[Tetracot family #Byhearted|Tetracot family]]
* ''[[Byhearted]]'' (+19683/19208) → [[Tetracot family #Byhearted|Tetracot family]]


which are discussed elsewhere.
Considered below are lemba, astrology, walid, antikythera, 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 = 1\2, ~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 = 1\2, ~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]]: 0.062208
[[Badness]] (Sintel): 1.57


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~8/7 = 231.997
* WE: ~7/5 = 601.1769{{c}}, ~8/7 = 231.4273{{c}}
* POTE: ~7/5 = 1\2, ~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: 0.041563
Badness (Sintel): 1.37


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~8/7 = 232.100
* WE: ~7/5 = 601.1939{{c}}, ~8/7 = 231.4261{{c}}
* POTE: ~7/5 = 1\2, ~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: 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.  
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 109:


: 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 = 1\2, ~8/7 = 380.355
* [[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 = 1\2, ~5/4 = 380.578
* [[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]]: 0.082673
[[Badness]] (Sintel): 2.09


=== 11-limit ===
=== 11-limit ===
Line 131: Line 128:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~5/4 = 380.588
* WE: ~7/5 = 600.0538{{c}}, ~5/4 = 380.5640{{c}} (~8/7 = 219.4897{{c}})
* POTE: ~7/5 = 1\2, ~5/4 = 380.530
* 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: 0.039151
Badness (Sintel): 1.29


==== 13-limit ====
==== 13-limit ====
Line 146: Line 143:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~5/4 = 380.449
* WE: ~7/5 = 600.7886{{c}}, ~5/4 = 380.2857{{c}} (~8/7 = 220.5028{{c}})
* POTE: ~7/5 = 1\2, ~5/4 = 379.787
* 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: 0.034376
Badness (Sintel): 1.42


; Music
; Music
Line 164: Line 161:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~5/4 = 379.762
* WE: ~7/5 = 599.8927{{c}}, ~5/4 = 379.7688{{c}} (~8/7 = 220.1239{{c}})
* POTE: ~7/5 = 1\2, ~5/4 = 379.837
* 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: 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 178:


: 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 = 1\2, ~3/2 = 754.204
* [[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 = 1\2, ~3/2 = 749.415
* [[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]]: 0.048978
[[Badness]] (Sintel): 1.24


=== 11-limit ===
=== 11-limit ===
Line 200: Line 197:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~3/2 = 754.205
* WE: ~7/5 = 589.7684{{c}}, ~3/2 = 736.9708{{c}} (~12/11 = 147.2023{{c}})
* POTE: ~7/5 = 1\2, ~3/2 = 749.756
* 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: 0.029193
Badness (Sintel): 0.965


== Antikythera ==
== Antikythera ==
Line 223: Line 220:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 1\2, ~9/8 = 216.711
* [[WE]]: ~7/5 = 598.8483{{c}}, ~9/8 = 213.6844{{c}}
: [[error map]]: {{val| 0.000 +12.801 +3.025 +14.463 }}
: [[error map]]: {{val| -2.303 +2.864 -5.756 +10.580 }}
* [[POTE]]: ~7/5 = 1\2, ~9/8 = 214.095
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~9/8 = 214.6875{{c}}
: error map: {{val| 0.000 +10.815 -0.409 +17.079 }}
: error map: {{val| 0.000 +10.778 -1.001 +16.487 }}


{{Optimal ET sequence|legend=1| 2, 4, 6, 16, 22, 28 }}
{{Optimal ET sequence|legend=1| 2, 4, 6, 16, 22, 28 }}


[[Tp tuning #T2 tuning|RMS error]]: 2.572 cents
[[Badness]] (Sintel): 0.253
 
[[Badness]]: 0.00501


== Doublewide ==
== Doublewide ==
=== 5-limit ===
: ''For the 5-limit version, see [[Superpyth–22 equivalence continuum #Doublewide (5-limit)]].''
''Note: the 5-limit temperament is only stored here temporarily. [[Xenharmonic Wiki:WikiProject TempClean|TempClean]] intends to reorganize clan pages such that "in-law" temperaments that do not directly fall from the clan's gene will no longer be recorded directly on these pages.''
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 390625/373248
 
{{Mapping|legend=1| 2 1 3 | 0 4 3 }}
 
: mapping generators: ~625/432, ~6/5
 
[[Optimal tuning]]s:
* [[CTE]]: ~625/432 = 1\2, ~6/5 = 326.170
: [[error map]]: {{val| 0.000 +2.727 -7.802 }}
* [[CWE]]: ~625/432 = 1\2, ~6/5 = 325.815
: error map: {{val| 0.000 +1.304 -8.869 }}


{{Optimal ET sequence|legend=1| 4, 14b, 18, 22, 48, 70c, 118c }}
Doublewide is generated by a sharply tuned ~6/5 minor third, four of which and a semi-octave period give the 3rd harmonic. It may be described as the {{nowrap| 22 & 26 }} temperament; 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.


[[Badness]]:
* Smith: 0.227
* Dirichlet: 5.32
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 266: Line 241:


: 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 = 1\2, ~6/5 = 325.769
* [[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 = 1\2, ~6/5 = 325.719
* [[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]]: 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 = 1\2, ~6/5 = 325.719
* WE: ~7/5 = 600.1818{{c}}, ~6/5 = 325.6434{{c}} (~7/6 = 274.5384{{c}})
* POTE: ~7/5 = 1\2, ~6/5 = 325.545
* 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: 0.032058
Badness (Sintel): 1.06


=== Fleetwood ===
=== Fleetwood ===
Line 302: Line 275:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~6/5 = 326.684
* WE: ~7/5 = 599.6049{{c}}, ~6/5 = 326.8229{{c}} (~7/6 = 272.7819{{c}})
* POTE: ~7/5 = 1\2, ~6/5 = 327.038
* 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: 0.035202
Badness (Sintel): 1.16


==== 13-limit ====
==== 13-limit ====
Line 317: Line 290:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~6/5 = 327.450
* WE: ~7/5 = 599.5482{{c}}, ~6/5 = 327.5939{{c}} (~7/6 = 271.9543{{c}})
* POTE: ~7/5 = 1\2, ~6/5 = 327.841
* 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: 0.031835
Badness (Sintel): 1.32


=== Cavalier ===
=== Cavalier ===
Line 332: Line 305:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~6/5 = 324.238
* WE: ~7/5 = 600.9467{{c}}, ~6/5 = 323.9369{{c}} (~7/6 = 277.0098{{c}})
* POTE: ~7/5 = 1\2, ~6/5 = 323.427
* 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: 0.052899
Badness (Sintel): 1.75


==== 13-limit ====
==== 13-limit ====
Line 347: Line 320:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~6/5 = 324.187
* WE: ~7/5 = 600.9537{{c}}, ~6/5 = 323.9097{{c}} (~7/6 = 277.0440{{c}})
* POTE: ~7/5 = 1\2, ~6/5 = 323.396
* 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: 0.035040
Badness (Sintel): 1.45


== Elvis ==
== Elvis ==
: ''For the 5-limit version of this temperament, 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 364: Line 339:


: 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 = 1\2, ~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 = 1\2, ~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]]: 0.141473
[[Badness]] (Sintel): 3.58


=== 11-limit ===
=== 11-limit ===
Line 385: Line 358:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~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 = 1\2, ~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: 0.063212
Badness (Sintel): 2.09


=== 13-limit ===
=== 13-limit ===
Line 400: Line 373:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~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 = 1\2, ~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: 0.043997
Badness (Sintel): 1.82


== Comic ==
== Comic ==
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Comic]].''
: ''For the 5-limit version, see [[Superpyth–22 equivalence continuum #Comic (5-limit)]].''
 
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 417: Line 392:


: 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 = 1\2, ~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 = 1\2, ~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]]: 0.084395
[[Badness]] (Sintel): 2.14


=== 11-limit ===
=== 11-limit ===
Line 438: Line 411:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~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 = 1\2, ~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: 0.045052
Badness (Sintel): 1.49


=== 13-limit ===
=== 13-limit ===
Line 453: Line 426:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~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 = 1\2, ~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: 0.041470
Badness (Sintel): 1.71


== Bipyth ==
== Bipyth ==
{{See also| Archytas clan #Superpyth }}
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 470: Line 443:


: 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 = 1\2, ~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 = 1\2, ~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]]: 0.165033
[[Badness]] (Sintel): 4.18


=== 11-limit ===
=== 11-limit ===
Line 491: Line 462:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~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 = 1\2, ~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: 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 504: Line 477:


{{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 = 1\16, ~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 = 1\16, ~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]]: 0.265972
[[Badness]] (Sintel): 6.73


=== 11-limit ===
=== 11-limit ===
Line 525: Line 496:


Optimal tunings:  
Optimal tunings:  
* CTE: ~22/21 = 1\16, ~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 = 1\16, ~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: 0.092774
Badness (Sintel): 3.07


== Notes ==
== Notes ==


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Jubilismic clan| ]] <!-- main article -->
[[Category:Jubilismic clan| ]] <!-- main article -->
[[Category:Jubilismic| ]] <!-- key article -->
[[Category:Jubilismic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 12:38, 21 August 2025

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

Sval mapping: [⟨2 0 1], ⟨0 1 1]]

sval mapping generators: ~7/5, ~5

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

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, antikythera, 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

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

Antikythera

Named by Gene Ward Smith in 2011[1], antikythera is every other step of pajara.

Subgroup: 2.9.5.7

Comma list: 50/49, 64/63

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

mapping generators: ~7/5, ~9

Gencom mapping: [⟨2 3 5 6], ⟨0 1/2 -1 -1]]

gencom: [7/5 8/7; 50/49 64/63]

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

Doublewide

For the 5-limit version, see Superpyth–22 equivalence continuum #Doublewide (5-limit).

Doublewide is generated by a sharply tuned ~6/5 minor third, four of which and a semi-octave period give the 3rd harmonic. It may be described as the 22 & 26 temperament; 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 Superpyth–22 equivalence continuum #Comic (5-limit).

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

Notes

  1. ↑ Yahoo! Tuning Group | Antikythera
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