Jubilismic clan: Difference between revisions

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__FORCETOC__
{{Technical data page}}
The '''jubilismic clan''' tempers out the jubilisma, [[50/49]], which means [[7/5]] and [[10/7]] are identified and the [[octave]] is divided in two. Doublewide, lemba and diminished are discussed below; others in the clan are [[Diaschismic family #Pajara|pajara]], [[Dicot family #Decimal|decimal]], [[Meantone family #Injera|injera]], [[Trienstonic clan #Octokaidecal|octokaidecal]], [[Porcupine family #Hedgehog|hedgehog]], [[Pelogic family #Bipelog|bipelog]] and [[Augmented family #Hexe|hexe]], which are discussed elsewhere.
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.  


No-threes [[POTE_tuning|POTE generator]]: 380.840
== 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.  


No-threes map: [<2 0 1 1|, <0 0 1 1|]
[[Subgroup]]: 2.5.7


EDOs: {{EDOs|10, 12, 16, 22, 104}}
[[Comma list]]: 50/49


=Diminished=
{{Mapping|legend=2| 2 0 1 | 0 1 1 }}
<span style="display: block; text-align: right;">[[:de:Verminderte_Temperaturen|Deutsch]]</span>
{{see also|Dimipent family}}


Commas: 36/35, 50/49
: sval mapping generators: ~7/5, ~5


[[POTE_tuning|POTE generator]]: ~3/2 = 699.523
{{Mapping|legend=3| 2 0 0 1 | 0 0 1 1 }}


Map: [&lt;4 0 3 5|, &lt;0 1 1 1|]
[[Optimal tuning]]s:  
* [[WE]]: ~7/5 = 599.6673{{c}}, ~5/4 = 380.6287{{c}} (~8/7 = 219.0386{{c}})
: [[error map]]: {{val| -0.665 -7.016 +10.139 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~5/4 = 380.0086{{c}} (~8/7 = 219.9914{{c}})
: error map: {{val| 0.000 -6.305 +11.183 }}


EDOs: {{EDOs|4, 12}}
{{Optimal ET sequence|legend=1| 2, 4, 6, 16, 22, 60d }}


Badness: 0.0224
[[Badness]] (Sintel): 0.140


==11-limit==
=== Overview to extensions ===
Commas: 36/35, 50/49, 56/55
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]].


[[POTE_tuning|POTE generator]]: ~3/2 = 709.109
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.  


Map: [&lt;4 0 3 5 14|, &lt;0 1 1 1 0|]
Temperaments discussed elsewhere are:  
* [[Decimal]] (+25/24) → [[Dicot family #Decimal|Dicot family]]
* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]
* [[Pajara]] (+64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[Dubbla]]'' (+78125/73728) → [[Wesley family #Dubbla|Wesley family]]
* ''[[Injera]]'' (+81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[Octokaidecal]]'' (+28/27) → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]
* ''[[Bipelog]]'' (+135/128) → [[Mavila #Bipelog|Mavila family]]
* ''[[Hexe]]'' (+128/125) → [[Augmented family #Hexe|Augmented family]]
* ''[[Hedgehog]]'' (+250/243) → [[Porcupine family #Hedgehog|Porcupine family]]
* ''[[Crepuscular]]'' (+4375/4374) → [[Fifive family #Crepuscular|Fifive family]]
* ''[[Byhearted]]'' (+19683/19208) → [[Tetracot family #Byhearted|Tetracot family]]


EDOs: {{EDOs|4, 8, 12, 44}}
Considered below are lemba, astrology, walid, antikythera, doublewide, elvis, comic, and bipyth.


Badness: 0.0221
== Lemba ==
{{Main| Lemba }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lemba]].''


==13-limit==
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.
Commas: 36/35, 40/39, 50/49, 66/65


POTE generator: ~3/2 = 713.773
[[Subgroup]]: 2.3.5.7


Map: [&lt;4 0 3 5 14 15|, &lt;0 1 1 1 0 0|]
[[Comma list]]: 50/49, 525/512


EDOs: {{EDOs|4, 8d, 12f, 20cdef}}
{{Mapping|legend=1| 2 2 5 6 | 0 3 -1 -1 }}


Badness: 0.0195
: mapping generators: ~7/5, ~8/7


==Demolished==
[[Optimal tuning]]s:
Commas: 36/35, 45/44, 50/49
* [[WE]]: ~7/5 = 601.4623{{c}}, ~8/7 = 232.6544{{c}}
: [[error map]]: {{val| +2.925 -1.067 -11.656 +7.294 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~8/7 = 232.2655{{c}}
: error map: {{val| 0.000 -5.158 -18.579 -1.091 }}


POTE generator: ~3/2 = 689.881
{{Optimal ET sequence|legend=1| 10, 16, 26, 36c, 62c }}


Map: [&lt;4 0 3 5 -5|, &lt;0 1 1 1 3|]
[[Badness]] (Sintel): 1.57


EDOs: {{EDOs|12, 28, 40de}}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0266
Comma list: 45/44, 50/49, 385/384


==Hemidim==
Mapping: {{mapping| 2 2 5 6 5 | 0 3 -1 -1 5 }}
Commas: 36/35, 50/49, 125/121


POTE generator: ~12/11 = 101.679
Optimal tunings:  
* WE: ~7/5 = 601.1769{{c}}, ~8/7 = 231.4273{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~8/7 = 231.1781{{c}}


Map: [&lt;4 1 4 6 6|, &lt;0 2 2 2 3|]
{{Optimal ET sequence|legend=0| 10, 16, 26 }}


EDOs: {{EDOs|12, 20b}}
Badness (Sintel): 1.37


Badness: 0.0550
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=Doublewide=
Comma list: 45/44, 50/49, 65/64, 78/77
Commas: 50/49, 875/864


[[POTE_tuning|POTE generator]]: ~6/5 = 325.719
Mapping: {{mapping| 2 2 5 6 5 7 | 0 3 -1 -1 5 1 }}


Map: [&lt;2 1 3 4|, &lt;0 4 3 3|]
Optimal tunings:  
* WE: ~7/5 = 601.1939{{c}}, ~8/7 = 231.4261{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~8/7 = 231.1617{{c}}


EDOs: {{EDOs|18, 22, 26, 48, 70c}}
{{Optimal ET sequence|legend=0| 10, 16, 26 }}


Badness: 0.0435
Badness (Sintel): 1.05


==11-limit==
== Astrology ==
Commas: 50/49, 99/98, 875/864
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.


[[POTE_tuning|POTE generator]]: ~6/5 = 325.548
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 1 3 4 8|, &lt;0 4 3 3 -2|]
[[Comma list]]: 50/49, 3125/3072


EDOs: {{EDOs|18, 22, 48, 70], 118cd}}
{{Mapping|legend=1| 2 0 4 5 | 0 5 1 1 }}


Badness: 0.0321
: mapping geenerators: ~7/5, ~5/4


==Fleetwood==
[[Optimal tuning]]s:
Commas: 50/49, 55/54, 176/175
* [[WE]]: ~7/5 = 599.6999{{c}}, ~5/4 = 380.3881{{c}} (~8/7 = 219.3119{{c}})
: [[error map]]: {{val| -0.600 -0.015 -7.126 +10.062 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~5/4 = 380.5123{{c}} (~8/7 = 219.4877{{c}})
: error map: {{val| 0.000 +0.606 -5.801 +11.686 }}


POTE generator: ~6/5 = 327.038
{{Optimal ET sequence|legend=1| 6, 16, 22, 60d }}


Map: [&lt;2 1 3 4 2|, &lt;0 4 3 3 9|]
[[Badness]] (Sintel): 2.09


EDOs: {{EDOs|22, 26e}}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0352
Comma list: 50/49, 121/120, 176/175


===13-limit===
Mapping: {{mapping| 2 0 4 5 5 | 0 5 1 1 3 }}
Commas: 50/49, 55/54, 65/63, 176/175


POTE generator: ~6/5 = 327.841
Optimal tunings:
* WE: ~7/5 = 600.0538{{c}}, ~5/4 = 380.5640{{c}} (~8/7 = 219.4897{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~5/4 = 380.5419{{c}} (~8/7 = 219.4581{{c}})


Map: [&lt;2 1 3 4 2 3|, &lt;0 4 3 3 9 8|]
{{Optimal ET sequence|legend=0| 6, 16, 22 }}


EDOs: {{EDOs|22, 84bdf}}
Badness (Sintel): 1.29


Badness: 0.0318
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


==Cavalier==
Comma list: 50/49, 65/64, 78/77, 121/120
Commas: 45/44, 50/49, 875/864


POTE generator: ~6/5 = 323.427
Mapping: {{mapping| 2 0 4 5 5 8 | 0 5 1 1 3 -1 }}


Map: [&lt;2 1 3 4 1|, &lt;0 4 3 3 11|]
Optimal tunings:  
* WE: ~7/5 = 600.7886{{c}}, ~5/4 = 380.2857{{c}} (~8/7 = 220.5028{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~5/4 = 379.9119{{c}} (~8/7 = 220.0881{{c}})


EDOs: {{EDOs|26}}
{{Optimal ET sequence|legend=0| 6, 16, 22, 38f }}


Badness: 0.0529
Badness (Sintel): 1.42


===13-limit===
; Music
Commas: 45/44, 50/49, 78/77, 325/324
* [https://soundcloud.com/joelgranttaylor/astrology-percussion-quintet ''Astrology Percussion Quintet No 1'']{{dead link}} [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/AstrologyPercQuintet1_c.mp3 play]{{dead link}} by [[Joel Taylor]]


POTE generator: ~6/5 = 323.396
==== Horoscope ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 1 3 4 1 2|, &lt;0 4 3 3 11 10|]
Comma list: 50/49, 66/65, 105/104, 121/120


EDOs: {{EDOs|26}}
Mapping: {{mapping| 2 0 4 5 5 3 | 0 5 1 1 3 7 }}


Badness: 0.0350
Optimal tunings:  
* WE: ~7/5 = 599.8927{{c}}, ~5/4 = 379.7688{{c}} (~8/7 = 220.1239{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~5/4 = 379.8117{{c}} (~8/7 = 220.1883{{c}})


=Lemba=
{{Optimal ET sequence|legend=0| 6f, 16, 22f, 38 }}
{{see also|Gamelismic clan #Lemba}}


Commas: 50/49, 525/512
Badness (Sintel): 1.46


[[POTE_tuning|POTE generator]]: ~8/7 = 232.089
== 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.


Map: [&lt;2 2 5 6|, &lt;0 3 -1 -1|]
[[Subgroup]]: 2.3.5.7


EDOs: {{EDOs|10, 16, 26, 62}}
[[Comma list]]: 16/15, 50/49


Badness: 0.0622
{{Mapping|legend=1| 2 0 8 9 | 0 1 -1 -1 }}


==11-limit==
: mapping generators: ~7/5, ~3
Commas: 45/44, 50/49, 385/384


POTE generator: ~8/7 = 230.974
[[Optimal tuning]]s:  
* [[WE]]: ~7/5 = 589.0384{{c}}, ~3/2 = 735.7242{{c}} (~15/14 = 146.6857{{c}})
: [[error map]]: {{val| -21.923 +11.846 +12.193 +18.719 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 750.4026{{c}} (~15/14 = 150.4026{{c}})
: error map: {{val| 0.000 +48.448 +63.284 +80.771 }}


Map: [&lt;2 2 5 6 5|, &lt;0 3 -1 -1 5|]
{{Optimal ET sequence|legend=1| 2, 6, 8d }}


EDOs: {{EDOs|10, 16, 26}}
[[Badness]] (Sintel): 1.24


Badness: 0.0416
=== 11-limit ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 16/15, 22/21, 50/49
Commas: 45/44, 50/49, 65/64, 78/77


POTE generator: ~8/7 = 230.966
Mapping: {{mapping| 2 0 8 9 7 | 0 1 -1 -1 0 }}


Map: [&lt;2 2 5 6 5 7|, &lt;0 3 -1 -1 5 1|]
Optimal tunings:  
* WE: ~7/5 = 589.7684{{c}}, ~3/2 = 736.9708{{c}} (~12/11 = 147.2023{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 750.5221{{c}} (~12/11 = 150.5221{{c}})


EDOs: {{EDOs|10, 16, 26}}
{{Optimal ET sequence|legend=0| 2, 6, 8d }}


Badness: 0.0255
Badness (Sintel): 0.965


=Vigintiduo=
== Antikythera ==
Commas: 50/49, 64/63, 245/243
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]].


POTE generator: ~11/8 = 557.563
[[Subgroup]]: 2.9.5.7


Map: [&lt;22 35 51 62 0|, &lt;0 0 0 0 1|]
[[Comma list]]: 50/49, 64/63


EDOs: {{EDOs|22, 66de, 88bde, 110bd, 198bcdde}}
{{Mapping|legend=2| 2 0 11 12 | 0 1 -1 -1 }}


Badness: 0.0484
: mapping generators: ~7/5, ~9


=Vigin=
{{Mapping|legend=3| 2 3 5 6 | 0 1/2 -1 -1 }}
Commas: 50/49, 55/54, 64/63, 99/98


POTE generator: ~13/8 = 844.624
: [[gencom]]: [7/5 8/7; 50/49 64/63]


Map: [&lt;22 35 51 62 76 0|, &lt;0 0 0 0 0 1|]
[[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 }}


EDOs: {{EDOs|22, 44}}
{{Optimal ET sequence|legend=1| 2, 4, 6, 16, 22, 28 }}


Badness: 0.0298
[[Badness]] (Sintel): 0.253


=Duodecim=
== Doublewide ==
Commas: 36/35, 50/49, 64/63
: ''For the 5-limit version, see [[Superpyth–22 equivalence continuum #Doublewide (5-limit)]].''


POTE generator: ~11/8 = 565.023
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.


Map: [&lt;12 19 28 34 0|, &lt;0 0 0 0 1|]
[[Subgroup]]: 2.3.5.7


EDOs: {{EDOs|12, 24d, 36d}}
[[Comma list]]: 50/49, 875/864


Badness: 0.030536
{{Mapping|legend=1| 2 1 3 4 | 0 4 3 3 }}


=Crepuscular=
: mapping generators: ~7/5, ~6/5
Commas: 50/49, 4375/4374


POTE generator: ~27/25 = 140.349
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 600.0365{{c}}, ~6/5 = 325.7389{{c}} (~7/6 = 274.2975{{c}})
: [[error map]]: {{val| -2.303 +2.864 -5.756 +10.580 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~6/5 = 325.7353{{c}} (~7/6 = 274.2647{{c}})
: error map: {{val| 0.000 +10.778 -1.001 +16.487 }}


Map: [&lt;2 2 3 4|, &lt;0 5 7 7|]
{{Optimal ET sequence|legend=1| 4, 14bd, 18, 22, 48 }}


Wedgie: &lt;&lt;10 14 14 -1 -6 -7||
[[Badness]] (Sintel): 1.10


EDOs: {{EDOs|26, 34d, 60d, 94d}}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0867
Comma list: 50/49, 99/98, 385/384


==11-limit==
Mapping: {{mapping| 2 1 3 4 8 | 0 4 3 3 -2 }}
Commas: 50/49, 99/98, 1944/1925


POTE generator: ~12/11 = 140.587
Optimal tunings:  
* WE: ~7/5 = 600.1818{{c}}, ~6/5 = 325.6434{{c}} (~7/6 = 274.5384{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~6/5 = 325.5854{{c}} (~7/6 = 274.4146{{c}})


Map: [&lt;2 2 3 4 6|, &lt;0 5 7 7 4|]
{{Optimal ET sequence|legend=0| 4, 18, 22, 48 }}


EDOs: {{EDOs|26, 34d, 60d, 94de}}
Badness (Sintel): 1.06


Badness: 0.0408
=== Fleetwood ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 50/49, 55/54, 176/175
Commas: 50/49, 78/77, 99/98, 144/143


POTE generator: ~12/11 = 140.554
Mapping: {{mapping| 2 1 3 4 2 | 0 4 3 3 9 }}


Map: [&lt;2 2 3 4 6 6|, &lt;0 5 7 7 4 6|]
Optimal tunings:  
* WE: ~7/5 = 599.6049{{c}}, ~6/5 = 326.8229{{c}} (~7/6 = 272.7819{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~6/5 = 326.8890{{c}} (~7/6 = 273.1110{{c}})


EDOs: {{EDOs|26, 34d, 60d, 94de}}
{{Optimal ET sequence|legend=0| 4e, …, 18e, 22 }}


Badness: 0.0244
Badness (Sintel): 1.16


=Bipyth=
==== 13-limit ====
Commas: 50/49, 20480/19683
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 709.437
Comma list: 50/49, 55/54, 65/63, 176/175


Map: [&lt;2 0 -24 -23|, &lt;0 1 9 9|]
Mapping: {{mapping| 2 1 3 4 2 3 | 0 4 3 3 9 8 }}


Wedgie: &lt;&lt;2 18 18 24 23 -9||
Optimal tunings:  
* WE: ~7/5 = 599.5482{{c}}, ~6/5 = 327.5939{{c}} (~7/6 = 271.9543{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~6/5 = 327.6706{{c}} (~7/6 = 272.3294{{c}})


Badness: 0.1650
{{Optimal ET sequence|legend=0| 4ef, …, 18e, 22 }}


==11-limit==
Badness (Sintel): 1.32
Commas: 50/49, 121/120, 896/891


POTE generator: ~3/2 = 709.310
=== Cavalier ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 0 -24 -23 -9|, &lt;0 1 9 9 5|]
Comma list: 45/44, 50/49, 875/864


EDOs: {{EDOs|22}}
Mapping: {{mapping| 2 1 3 4 1 | 0 4 3 3 11 }}


Badness: 0.0709
Optimal tunings:  
* WE: ~7/5 = 600.9467{{c}}, ~6/5 = 323.9369{{c}} (~7/6 = 277.0098{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~6/5 = 323.7272{{c}} (~7/6 = 276.2728{{c}})


=Elvis=
{{Optimal ET sequence|legend=0| 4e, 22e, 26 }}
Comma: 36905625/33554432


POTE generator: ~45/32 = 554.546
Badness (Sintel): 1.75


Map: [&lt;2 1 10|, &lt;0 2 -5|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: {{EDOs|24c, 26, 80bc, 106bc, 132bc}}
Comma list: 45/44, 50/49, 78/77, 325/324


Badness: 0.8840
Mapping: {{mapping| 2 1 3 4 1 2 | 0 4 3 3 11 10 }}


==7-limit==
Optimal tunings:
Commas: 50/49, 8505/8192
* WE: ~7/5 = 600.9537{{c}}, ~6/5 = 323.9097{{c}} (~7/6 = 277.0440{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~6/5 = 323.6876{{c}} (~7/6 = 276.3124{{c}})


POTE generator: ~45/32 = 553.721
{{Optimal ET sequence|legend=0| 4ef, 22ef, 26 }}


Map: [&lt;2 1 10 11|, &lt;0 2 -5 -5|]
Badness (Sintel): 1.45


Wedgie: &lt;&lt;4 -10 -10 -25 -27 5||
== Elvis ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Elvis]].''


EDOs: {{EDOs|24c, 26}}
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.


Badness: 0.1415
[[Subgroup]]: 2.3.5.7


==11-limit==
[[Comma list]]: 50/49, 8505/8192
Commas: 45/44, 50/49, 1344/1331


POTE generator: ~11/8 = 553.882
{{Mapping|legend=1| 2 1 10 11 | 0 2 -5 -5 }}


Map: [&lt;2 1 10 11 8|, &lt;0 2 -5 -5 -1|]
: mapping generators: ~7/5, ~64/45


EDOs: {{EDOs|24c, 26}}
[[Optimal tuning]]s:  
* [[WE]]: ~7/5 = 601.6846{{c}}, ~64/45 = 648.0937{{c}} (~64/63 = 46.4091{{c}})
: [[error map]]: {{val| +3.369 -4.083 -9.936 +9.236 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~64/45 = 646.0539{{c}} (~64/63 = 46.0539{{c}})
: error map: {{val| 0.000 -9.847 -16.583 +0.904 }}


Badness: 0.0632
{{Optimal ET sequence|legend=1| 2, 24c, 26 }}


==13-limit==
[[Badness]] (Sintel): 3.58
Commas: 45/44, 50/49, 78/77, 1053/1024


POTE generator: ~11/8 = 553.892
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 1 10 11 8 16|, &lt;0 2 -5 -5 -1 -8|]
Comma list: 45/44, 50/49, 1344/1331


EDOs: {{EDOs|26}}
Mapping: {{mapping| 2 1 10 11 8 | 0 2 -5 -5 -1 }}


Badness: 0.0440
Optimal tunings:  
* WE: ~7/5 = 601.2186{{c}}, ~16/11 = 647.4300{{c}} (~56/55 = 46.2114{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 645.9681{{c}} (~56/55 = 45.9681{{c}})


[[Category:Theory]]
{{Optimal ET sequence|legend=0| 2, 24c, 26 }}
[[Category:Jubilismic]]
 
[[Category:Clan]]
Badness (Sintel): 2.09
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 45/44, 50/49, 78/77, 1053/1024
 
Mapping: {{mapping| 2 1 10 11 8 16 | 0 2 -5 -5 -1 -8 }}
 
Optimal tunings:
* WE: ~7/5 = 601.2206{{c}}, ~16/11 = 647.4219{{c}} (~56/55 = 46.2013{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 645.9362{{c}} (~56/55 = 45.9362{{c}})
 
{{Optimal ET sequence|legend=0| 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|legend=1| 2 1 -3 -2 | 0 2 7 7 }}
 
: mapping generators: ~7/5, ~40/27
 
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 598.9554{{c}}, ~40/27 = 653.5596{{c}} (~28/27 = 54.6042{{c}})
: [[error map]]: {{val| +3.369 -4.083 -9.936 +9.236 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~40/27 = 654.3329{{c}} (~28/27 = 54.3329{{c}})
: error map: {{val| 0.000 -9.847 -16.583 +0.904 }}
 
{{Optimal ET sequence|legend=1| 2cd, …, 20cd, 22 }}
 
[[Badness]] (Sintel): 2.14
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 99/98, 2662/2625
 
Mapping: {{mapping| 2 1 -3 -2 -4 | 0 2 7 7 10 }}
 
Optimal tunings:
* WE: ~7/5 = 598.8161{{c}}, ~22/15 = 653.8909{{c}} (~28/27 = 55.0747{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~22/15 = 654.7898{{c}} (~28/27 = 54.7898{{c}})
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 1 -3 -2 -4 3 | 0 2 7 7 10 4 }}
 
Optimal tunings:
* WE: ~7/5 = 600.1030{{c}}, ~22/15 = 654.5470{{c}} (~28/27 = 54.4440{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~22/15 = 654.4665{{c}} (~28/27 = 54.4665{{c}})
 
{{Optimal ET sequence|legend=0| 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|legend=1| 2 0 -24 -23 | 0 1 9 9 }}
 
: mapping generators: ~7/5, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 598.7533{{c}}, ~3/2 = 707.9630{{c}} (~15/14 = 109.2098{{c}})
: [[error map]]: {{val| +3.369 -4.083 -9.936 +9.236 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 709.1579{{c}} (~15/14 = 109.1579{{c}})
: error map: {{val| 0.000 -9.847 -16.583 +0.904 }}
 
{{Optimal ET sequence|legend=1| 10cd, 12cd, 22 }}
 
[[Badness]] (Sintel): 4.18
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 121/120, 896/891
 
Mapping: {{mapping| 2 0 -24 -23 -9 | 0 1 9 9 5 }}
 
Optimal tunings:
* WE: ~7/5 = 599.2296{{c}}, ~3/2 = 708.3992{{c}} (~15/14 = 109.1697{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 709.1395{{c}} (~15/14 = 109.1395{{c}})
 
{{Optimal ET sequence|legend=0| 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|legend=1| 16 0 37 45 | 0 1 0 0 }}
 
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
 
{{Optimal ET sequence|legend=1| 16, 32, 48 }}
 
[[Badness]] (Sintel): 6.73
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 385/384, 1331/1323
 
Mapping: {{mapping| 16 0 37 45 30 | 0 1 0 0 1 }}
 
Optimal tunings:
* WE: ~22/21 = 75.0000{{c}}, ~3/2 = 700.7810{{c}} (~45/44 = 25.3476{{c}})
* CWE: ~22/21 = 75.0000{{c}}, ~3/2 = 700.6780{{c}} (~45/44 = 25.6780{{c}})
 
{{Optimal ET sequence|legend=0| 16, 32, 48 }}
 
Badness (Sintel): 3.07
 
== Notes ==
 
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Jubilismic clan| ]] <!-- main article -->
[[Category:Jubilismic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Todo:review]]

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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