Jubilismic clan: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
m Update the keys
→Subgroup extensions: correct source of name
 
(55 intermediate revisions by 8 users not shown)
Line 1: Line 1:
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.  
{{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.  


== Jubilic ==
== Jubilic ==
The head of this clan, jubilic, is generated by [[~]][[5/4]]. That and a semioctave gives ~[[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 8: Line 9:
[[Comma list]]: 50/49
[[Comma list]]: 50/49


[[Sval]] [[mapping]]: [{{val| 2 0 1 }}, {{val| 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 }}


[[Gencom]] [[mapping]]: [{{val| 2 0 0 1 }}, {{val| 0 0 1 1 }}]
: mapping generators: ~7/5, ~5


[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~5/4 = 380.840
[[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 }}


{{Val list|legend=1| 2, 4, 6, 16, 22, 60d, 82d, 104dd }}
{{Optimal ET sequence|legend=1| 2, 4, 6, 16, 22, 60d }}
 
[[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]]. Pajara slices the ~7/4 into two. Injera slices the ~5/1 into four. Hedgehog slices the ~7/1 into five.  
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.  


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


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


== Lemba ==
== Lemba ==
{{Main| Lemba }}
{{Main| 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 {{nowrap| 10 & 16 }} temperament; its [[ploidacot]] is diploid tricot.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 41: Line 55:
[[Comma list]]: 50/49, 525/512
[[Comma list]]: 50/49, 525/512


[[Mapping]]: [{{val| 2 2 5 6 }}, {{val| 0 3 -1 -1 }}]
{{Mapping|legend=1| 2 2 5 6 | 0 3 -1 -1 }}


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


[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~8/7 = 232.089
[[Optimal tuning]]s:
* [[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 }}


{{Val list|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 56: Line 74:
Comma list: 45/44, 50/49, 385/384
Comma list: 45/44, 50/49, 385/384


Mapping: [{{val| 2 2 5 6 5 }}, {{val| 0 3 -1 -1 5 }}]
Mapping: {{mapping| 2 2 5 6 5 | 0 3 -1 -1 5 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~8/7 = 230.974
Optimal tunings:
* WE: ~7/5 = 601.1769{{c}}, ~8/7 = 231.4273{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~8/7 = 231.1781{{c}}


Optimal GPV sequence: {{Val list| 10, 16, 26 }}
{{Optimal ET sequence|legend=0| 10, 16, 26 }}


Badness: 0.041563
Badness (Sintel): 1.37


=== 13-limit ===
=== 13-limit ===
Line 69: Line 89:
Comma list: 45/44, 50/49, 65/64, 78/77
Comma list: 45/44, 50/49, 65/64, 78/77


Mapping: [{{val| 2 2 5 6 5 7 }}, {{val| 0 3 -1 -1 5 1 }}]
Mapping: {{mapping| 2 2 5 6 5 7 | 0 3 -1 -1 5 1 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~8/7 = 230.966
Optimal tunings:
* WE: ~7/5 = 601.1939{{c}}, ~8/7 = 231.4261{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~8/7 = 231.1617{{c}}


Optimal GPV sequence: {{Val list| 10, 16, 26 }}
{{Optimal ET sequence|legend=0| 10, 16, 26 }}


Badness: 0.025477
Badness (Sintel): 1.05


== Diminished ==
== Astrology ==
<div style="float: right">[[:de:Verminderte Temperaturen|Deutsch]]</div>
[[:de:Magische_Temperaturen#Astrology|Deutsch]]
{{See also| Dimipent family #Diminished }}
 
{{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


[[Comma list]]: 36/35, 50/49
[[Comma list]]: 50/49, 3125/3072


[[Mapping]]: [{{val| 4 0 3 5 }}, {{val| 0 1 1 1 }}]
{{Mapping|legend=1| 2 0 4 5 | 0 5 1 1 }}


Mapping generators: ~6/5, ~3
: mapping geenerators: ~7/5, ~5/4


[[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~3/2 = 699.523
[[Optimal tuning]]s:
* [[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 }}


{{Val list|legend=1| 4, 8d, 12 }}
{{Optimal ET sequence|legend=1| 6, 16, 22, 60d }}


[[Badness]]: 0.022401
[[Badness]] (Sintel): 2.09
 
Scales: [[diminished12]]


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


Comma list: 36/35, 50/49, 56/55
Comma list: 50/49, 121/120, 176/175
 
Mapping: [{{val| 4 0 3 5 14 }}, {{val| 0 1 1 1 0 }}]


Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 709.109
Mapping: {{mapping| 2 0 4 5 5 | 0 5 1 1 3 }}


Optimal GPV sequence: {{Val list| 4, 8d, 12, 32cddee, 44cddeee }}
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}})


Badness: 0.022132
{{Optimal ET sequence|legend=0| 6, 16, 22 }}


Scales: [[diminished12]]
Badness (Sintel): 1.29


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 36/35, 40/39, 50/49, 66/65
Comma list: 50/49, 65/64, 78/77, 121/120


Mapping: [{{val| 4 0 3 5 14 15 }}, {{val| 0 1 1 1 0 0 }}]
Mapping: {{mapping| 2 0 4 5 5 8 | 0 5 1 1 3 -1 }}


Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 713.773
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}})


Optimal GPV sequence: {{Val list| 4, 8d, 12f, 20cdef }}
{{Optimal ET sequence|legend=0| 6, 16, 22, 38f }}


Badness: 0.019509
Badness (Sintel): 1.42


Scales: [[diminished12]]
; Music
* [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]]


=== Demolished ===
==== Horoscope ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 36/35, 45/44, 50/49
Comma list: 50/49, 66/65, 105/104, 121/120


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


Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 689.881
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}})


Optimal GPV sequence: {{Val list| 12, 28, 40de }}
{{Optimal ET sequence|legend=0| 6f, 16, 22f, 38 }}


Badness: 0.026574
Badness (Sintel): 1.46


=== Cohedim ===
== Walid ==
This temperament has been documented in Graham Breed's temperament finder as ''hemidim'', the same name as [[Dimipent family #Hemidim|11-limit 4e&amp;24 and 13-limit 4ef&amp;24]]. For 11-limit 8bce&amp;12 temperament, ''cohedim'' arguably makes more sense.
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: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7


Comma list: 36/35, 50/49, 125/121
[[Comma list]]: 16/15, 50/49


Mapping: [{{val| 4 1 4 6 6 }}, {{val| 0 2 2 2 3 }}]
{{Mapping|legend=1| 2 0 8 9 | 0 1 -1 -1 }}


Mapping generators: ~6/5, ~11/7
: mapping generators: ~7/5, ~3


Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 101.679
[[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 }}


Optimal GPV sequence: {{Val list| 8bce, 12 }}
{{Optimal ET sequence|legend=1| 2, 6, 8d }}


Badness: 0.054965
[[Badness]] (Sintel): 1.24


==== 13-limit ====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 36/35, 50/49, 66/65, 125/121
Comma list: 16/15, 22/21, 50/49


Mapping: [{{val| 4 1 4 6 6 7 }}, {{val| 0 2 2 2 3 3 }}]
Mapping: {{mapping| 2 0 8 9 7 | 0 1 -1 -1 0 }}


Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 102.299
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}})


Optimal GPV sequence: {{Val list| 8bcef, 12f }}
{{Optimal ET sequence|legend=0| 2, 6, 8d }}


Badness: 0.041707
Badness (Sintel): 0.965


== Doublewide ==
== 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 {{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.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 50/49, 875/864
[[Comma list]]: 50/49, 875/864


[[Mapping]]: [{{val| 2 1 3 4 }}, {{val| 0 4 3 3 }}]
{{Mapping|legend=1| 2 1 3 4 | 0 4 3 3 }}


[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~6/5 = 325.719
: mapping generators: ~7/5, ~6/5


{{Val list|legend=1| 4, 14bd, 18, 22, 48, 70c }}
[[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 }}


[[Badness]]: 0.043462
{{Optimal ET sequence|legend=1| 4, 14bd, 18, 22, 48 }}
 
[[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: [{{val| 2 1 3 4 8 }}, {{val| 0 4 3 3 -2 }}]
Mapping: {{mapping| 2 1 3 4 8 | 0 4 3 3 -2 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~6/5 = 325.545
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}})


Optimal GPV sequence: {{Val list| 4, 14bd, 18, 22, 48, 70c, 118cd }}
{{Optimal ET sequence|legend=0| 4, 18, 22, 48 }}


Badness: 0.032058
Badness (Sintel): 1.06


=== Fleetwood ===
=== Fleetwood ===
Line 201: Line 251:
Comma list: 50/49, 55/54, 176/175
Comma list: 50/49, 55/54, 176/175


Mapping: [{{val| 2 1 3 4 2 }}, {{val| 0 4 3 3 9 }}]
Mapping: {{mapping| 2 1 3 4 2 | 0 4 3 3 9 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~6/5 = 327.038
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}})


Optimal GPV sequence: {{Val list| 4e, 18e, 22 }}
{{Optimal ET sequence|legend=0| 4e, …, 18e, 22 }}


Badness: 0.035202
Badness (Sintel): 1.16


==== 13-limit ====
==== 13-limit ====
Line 214: Line 266:
Comma list: 50/49, 55/54, 65/63, 176/175
Comma list: 50/49, 55/54, 65/63, 176/175


Mapping: [{{val| 2 1 3 4 2 3 }}, {{val| 0 4 3 3 9 8 }}]
Mapping: {{mapping| 2 1 3 4 2 3 | 0 4 3 3 9 8 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~6/5 = 327.841
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}})


Optimal GPV sequence: {{Val list| 4ef, 18e, 22, 84bddf }}
{{Optimal ET sequence|legend=0| 4ef, …, 18e, 22 }}


Badness: 0.031835
Badness (Sintel): 1.32


=== Cavalier ===
=== Cavalier ===
Line 227: Line 281:
Comma list: 45/44, 50/49, 875/864
Comma list: 45/44, 50/49, 875/864


Mapping: [{{val| 2 1 3 4 1 }}, {{val| 0 4 3 3 11 }}]
Mapping: {{mapping| 2 1 3 4 1 | 0 4 3 3 11 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~6/5 = 323.427
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}})


Optimal GPV sequence: {{Val list| 22e, 26 }}
{{Optimal ET sequence|legend=0| 4e, 22e, 26 }}


Badness: 0.052899
Badness (Sintel): 1.75


==== 13-limit ====
==== 13-limit ====
Line 240: Line 296:
Comma list: 45/44, 50/49, 78/77, 325/324
Comma list: 45/44, 50/49, 78/77, 325/324


Mapping: [{{val| 2 1 3 4 1 2 }}, {{val| 0 4 3 3 11 10 }}]
Mapping: {{mapping| 2 1 3 4 1 2 | 0 4 3 3 11 10 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~6/5 = 323.396
Optimal tunings:
* 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}})


Optimal GPV sequence: {{Val list| 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 [[High badness 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 255: Line 315:
[[Comma list]]: 50/49, 8505/8192
[[Comma list]]: 50/49, 8505/8192


[[Mapping]]: [{{val| 2 1 10 11 }}, {{val| 0 2 -5 -5 }}]
{{Mapping|legend=1| 2 1 10 11 | 0 2 -5 -5 }}


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


[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~45/32 = 553.721
[[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 }}


{{Val list|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 270: Line 334:
Comma list: 45/44, 50/49, 1344/1331
Comma list: 45/44, 50/49, 1344/1331


Mapping: [{{val| 2 1 10 11 8 }}, {{val| 0 2 -5 -5 -1 }}]
Mapping: {{mapping| 2 1 10 11 8 | 0 2 -5 -5 -1 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 553.882
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}})


Optimal GPV sequence: {{Val list| 2, 24c, 26 }}
{{Optimal ET sequence|legend=0| 2, 24c, 26 }}


Badness: 0.063212
Badness (Sintel): 2.09


=== 13-limit ===
=== 13-limit ===
Line 283: Line 349:
Comma list: 45/44, 50/49, 78/77, 1053/1024
Comma list: 45/44, 50/49, 78/77, 1053/1024


Mapping: [{{val| 2 1 10 11 8 16 }}, {{val| 0 2 -5 -5 -1 -8 }}]
Mapping: {{mapping| 2 1 10 11 8 16 | 0 2 -5 -5 -1 -8 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 553.892
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 GPV sequence: {{Val list| 2f, 24cf, 26 }}
{{Optimal ET sequence|legend=0| 2f, 24cf, 26 }}


Badness: 0.043997
Badness (Sintel): 1.82


== Crepuscular ==
== Comic ==
{{See also| Fifive family #Crepuscular }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Comic]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 50/49, 4375/4374
 
[[Mapping]]: [{{val| 2 2 3 4 }}, {{val| 0 5 7 7 }}]
 
{{Multival|legend=1|10 14 14 -1 -6 -7}}
 
[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~27/25 = 140.349
 
{{Val list|legend=1| 8d, 26, 34d, 60d }}
 
[[Badness]]: 0.086669
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 99/98, 864/847
 
Mapping: [{{val| 2 2 3 4 6 }}, {{val| 0 5 7 7 4 }}]
 
Optimal tuning (POTE): ~7/5 = 1\2, ~12/11 = 140.587
 
Optimal GPV sequence: {{Val list| 8d, 26, 34d, 60d }}
 
Badness: 0.040758
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 78/77, 99/98, 144/143
 
Mapping: [{{val| 2 2 3 4 6 6 }}, {{val| 0 5 7 7 4 6 }}]
 
Optimal tuning (POTE): ~7/5 = 1\2, ~12/11 = 140.554
 
Optimal GPV sequence: {{Val list| 8d, 26, 34d, 60d }}
 
Badness: 0.024368


== 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.  
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Comic]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 341: Line 368:
[[Comma list]]: 50/49, 2240/2187
[[Comma list]]: 50/49, 2240/2187


[[Mapping]]: [{{val| 2 1 -3 -2 }}, {{val| 0 2 7 7 }}]
{{Mapping|legend=1| 2 1 -3 -2 | 0 2 7 7 }}


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


[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~81/80 = 54.699
[[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 356: Line 387:
Comma list: 50/49, 99/98, 2662/2625
Comma list: 50/49, 99/98, 2662/2625


Mapping: [{{val| 2 1 -3 -2 -4 }}, {{val| 0 2 7 7 10 }}]
Mapping: {{mapping| 2 1 -3 -2 -4 | 0 2 7 7 10 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~81/80 = 55.184
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 GPV sequence: {{Val list| 20cde, 22 }}
{{Optimal ET sequence|legend=0| 2cde, …, 20cde, 22 }}


Badness: 0.045052
Badness (Sintel): 1.49


=== 13-limit ===
=== 13-limit ===
Line 369: Line 402:
Comma list: 50/49, 65/63, 99/98, 968/945
Comma list: 50/49, 65/63, 99/98, 968/945


Mapping: [{{val| 2 1 -3 -2 -4 3 }}, {{val| 0 2 7 7 10 4 }}]
Mapping: {{mapping| 2 1 -3 -2 -4 3 | 0 2 7 7 10 4 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~81/80 = 54.435
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 GPV sequence: {{Val list| 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 384: Line 419:
[[Comma list]]: 50/49, 20480/19683
[[Comma list]]: 50/49, 20480/19683


[[Mapping]]: [{{val| 2 0 -24 -23 }}, {{val| 0 1 9 9 }}]
{{Mapping|legend=1| 2 0 -24 -23 | 0 1 9 9 }}


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


[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~3/2 = 709.437
[[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 }}


{{Val list|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 399: Line 438:
Comma list: 50/49, 121/120, 896/891
Comma list: 50/49, 121/120, 896/891


Mapping: [{{val| 2 0 -24 -23 -9 }}, {{val| 0 1 9 9 5 }}]
Mapping: {{mapping| 2 0 -24 -23 -9 | 0 1 9 9 5 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 709.310
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 GPV sequence: {{Val list| 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


[[Comma list]]: 50/49, 546875/524288
[[Comma list]]: 50/49, 546875/524288


[[Mapping]]: [{{val| 16 0 37 45 }}, {{val| 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:
* [[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 tuning]] ([[POTE]]): ~128/125 = 1\16, ~3/2 = 700.554
{{Optimal ET sequence|legend=1| 16, 32, 48 }}


{{Val list|legend=1| 16, 32, 48 }}
[[Badness]] (Sintel): 6.73
 
[[Badness]]: 0.265972


=== 11-limit ===
=== 11-limit ===
Line 427: Line 472:
Comma list: 50/49, 385/384, 1331/1323
Comma list: 50/49, 385/384, 1331/1323


Mapping: [{{val| 16 0 37 45 30 }}, {{val| 0 1 0 0 1 }}]
Mapping: {{mapping| 16 0 37 45 30 | 0 1 0 0 1 }}
 
Optimal tuning (POTE): ~22/21 = 1\16, ~3/2 = 700.331
 
Optimal GPV sequence: {{Val list| 16, 32, 48 }}
 
Badness: 0.092774
 
== Duodecim ==
{{See also| Compton family #Duodecim }}
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 36/35, 50/49, 64/63
 
[[Mapping]]: [{{val| 12 19 28 34 0 }}, {{val| 0 0 0 0 1 }}]
 
[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\12, ~11/8 = 565.023
 
{{Val list|legend=1| 12, 24d, 36d }}


[[Badness]]: 0.030536
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}})


== Vigintiduo ==
{{Optimal ET sequence|legend=0| 16, 32, 48 }}
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 50/49, 64/63, 245/243
Badness (Sintel): 3.07


[[Mapping]]: [{{val| 22 35 51 62 0 }}, {{val| 0 0 0 0 1 }}]
== 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>.


[[Optimal tuning]] ([[POTE]]): ~36/35 = 1\22, ~11/8 = 557.563
[[Subgroup]]: 2.9.5.7


{{Val list|legend=1| 22, 66de, 88bde, 110bd, 198bcdde }}
[[Comma list]]: 50/49, 64/63


[[Badness]]: 0.048372
{{Mapping|legend=2| 2 0 11 12 | 0 1 -1 -1 }}


== Vigin ==
{{Mapping|legend=3| 2 0 11 12 | 0 1/2 -1 -1 }}
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 50/49, 55/54, 64/63, 99/98
: mapping generators: ~7/5, ~9


[[Mapping]]: [{{val| 22 35 51 62 76 0 }}, {{val| 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 }}


[[Optimal tuning]] ([[POTE]]): ~33/32 = 1\22, ~13/8 = 844.624
{{Optimal ET sequence|legend=1| 2, 4, 6, 16, 22, 28 }}


{{Val list|legend=1| 22, 44 }}
[[Badness]] (Sintel): 0.253


[[Badness]]: 0.029849
== References ==


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Jubilismic clan| ]] <!-- main article -->
[[Category:Jubilismic clan| ]] <!-- main 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
Retrieved from "https://en.xen.wiki/index.php?title=Jubilismic_clan&oldid=226036"