Jubilismic clan: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Databoxes for diminished to lemba
→Subgroup extensions: correct source of name
 
(80 intermediate revisions by 11 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. 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.
{{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.  


No-threes [[POTE generator]]: ~5/4 = 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 mapping: [{{val| 2 0 1 1 }}, {{val| 0 0 1 1 }}]
[[Subgroup]]: 2.5.7


{{Val list|legend=1| 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}}


Period: 1\4
{{Mapping|legend=3| 2 0 0 1 | 0 0 1 1 }}


Optimal ([[POTE]]) generator: ~3/2 = 699.523
: mapping generators: ~7/5, ~5


EDO generators: [[8edo|5\8]], [[12edo|7\12]], [[16edo|9\16]]
[[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 }}


Scales (Scala files): [[diminished12]]
{{Optimal ET sequence|legend=1| 2, 4, 6, 16, 22, 60d }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 0.140
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
=== 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]].  


Comma list: 36/35, 50/49
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.


Mapping: [{{val| 4 0 3 5 }}, {{val| 0 1 1 1 }}]
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]]
* ''[[Weasel]]'' (+19683/19208) → [[Tetracot family #Byhearted|Tetracot family]]


Mapping generators: ~6/5, ~3
Considered below are lemba, astrology, walid, doublewide, elvis, comic, and bipyth.


{{Val list|legend=1| 4, 8d, 12 }}
== Lemba ==
{{Main| Lemba }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lemba]].''


Badness: 0.0224
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.  


</div></div>
[[Subgroup]]: 2.3.5.7


== 11-limit ==
[[Comma list]]: 50/49, 525/512


Period: 1\4
{{Mapping|legend=1| 2 2 5 6 | 0 3 -1 -1 }}


Optimal ([[POTE]]) generator: ~3/2 = 709.109
: mapping generators: ~7/5, ~8/7


EDO generators: [[12edo|7\12]], [[20edo|12\20]]
[[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 }}


Scales (Scala files): [[diminished12]]
{{Optimal ET sequence|legend=1| 10, 16, 26, 36c, 62c }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 1.57
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 36/35, 50/49, 56/55
Comma list: 45/44, 50/49, 385/384


Mapping: [{{val| 4 0 3 5 14 }}, {{val| 0 1 1 1 0 }}]
Mapping: {{mapping| 2 2 5 6 5 | 0 3 -1 -1 5 }}


Mapping generators: ~6/5, ~3
Optimal tunings:
* WE: ~7/5 = 601.1769{{c}}, ~8/7 = 231.4273{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~8/7 = 231.1781{{c}}


{{Val list|legend=1| 4, 8d, 12, 32cddee, 44cddeee }}
{{Optimal ET sequence|legend=0| 10, 16, 26 }}


Badness: 0.0221
Badness (Sintel): 1.37
 
</div></div>


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


Period: 1\4
Comma list: 45/44, 50/49, 65/64, 78/77
 
Optimal ([[POTE]]) generator: ~3/2 = 713.773
 
EDO generators: [[12edo|7\12]], [[20edo|12\20]]
 
Scales (Scala files): [[diminished12]]


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 2 2 5 6 5 7 | 0 3 -1 -1 5 1 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Optimal tunings:  
* WE: ~7/5 = 601.1939{{c}}, ~8/7 = 231.4261{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~8/7 = 231.1617{{c}}


Comma list: 36/35, 40/39, 50/49, 66/65
{{Optimal ET sequence|legend=0| 10, 16, 26 }}


Mapping: [{{val| 4 0 3 5 14 15 }}, {{val| 0 1 1 1 0 0 }}]
Badness (Sintel): 1.05


Mapping generators: ~6/5, ~3
== Astrology ==
[[:de:Magische_Temperaturen#Astrology|Deutsch]]


{{Val list|legend=1| 4, 8d, 12f, 20cdef }}
{{see also| Magic family }}


Badness: 0.0195
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.  


</div></div>
[[Subgroup]]: 2.3.5.7


== Demolished ==
[[Comma list]]: 50/49, 3125/3072


Period: 1\4
{{Mapping|legend=1| 2 0 4 5 | 0 5 1 1 }}


Optimal ([[POTE]]) generator: ~3/2 = 689.881
: mapping geenerators: ~7/5, ~5/4


EDO generators: [[12edo|7\12]], [[16edo|9\16]], [[28edo|14\28]]
[[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 6, 16, 22, 60d }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 2.09
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 36/35, 45/44, 50/49
Comma list: 50/49, 121/120, 176/175


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


Mapping generators: ~6/5, ~3
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}})


{{Val list|legend=1| 12, 28, 40de }}
{{Optimal ET sequence|legend=0| 6, 16, 22 }}


Badness: 0.0266
Badness (Sintel): 1.29


</div></div>
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


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


Period: 1\4
Mapping: {{mapping| 2 0 4 5 5 8 | 0 5 1 1 3 -1 }}


Optimal ([[POTE]]) generator: ~12/11 = 101.679
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}})


EDO generators: [[12edo|7\12]]
{{Optimal ET sequence|legend=0| 6, 16, 22, 38f }}


Scales (Scala files):  
Badness (Sintel): 1.42


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
; Music
<div style="line-height:1.6;">Technical data</div>
* [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]]
<div class="mw-collapsible-content">


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


Comma list: 36/35, 50/49, 125/121
Comma list: 50/49, 66/65, 105/104, 121/120


Mapping: [{{val| 4 1 4 6 6 }}, {{val| 0 2 2 2 3 }}]
Mapping: {{mapping| 2 0 4 5 5 3 | 0 5 1 1 3 7 }}


Mapping generators: ~6/5, ~11/7
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}})


{{Val list|legend=1| 12 }}
{{Optimal ET sequence|legend=0| 6f, 16, 22f, 38 }}


Badness: 0.0550
Badness (Sintel): 1.46


</div></div>
== 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.


= Doublewide =
[[Subgroup]]: 2.3.5.7


Period: 1\2
[[Comma list]]: 16/15, 50/49


Optimal ([[POTE]]) generator: ~6/5 = 325.719
{{Mapping|legend=1| 2 0 8 9 | 0 1 -1 -1 }}


EDO generators: [[22edo|6\22]], [[26edo|7\26]], [[48edo|13\48]]
: mapping generators: ~7/5, ~3


Scales (Scala files):  
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 2, 6, 8d }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 1.24


Comma list: 50/49, 875/864
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 1 3 4 }}, {{val| 0 4 3 3 }}]
Comma list: 16/15, 22/21, 50/49


{{Val list|legend=1| 4, 14bd, 18, 22, 48, 70c }}
Mapping: {{mapping| 2 0 8 9 7 | 0 1 -1 -1 0 }}


Badness: 0.0435
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}})


</div></div>
{{Optimal ET sequence|legend=0| 2, 6, 8d }}


== 11-limit ==
Badness (Sintel): 0.965


Period: 1\2
== Doublewide ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Doublewide]].''


Optimal ([[POTE]]) generator: ~6/5 = 325.548
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.  


EDO generators: [[22edo|6\22]], [[26edo|7\26]], [[48edo|13\48]]
[[Subgroup]]: 2.3.5.7


Scales (Scala files):  
[[Comma list]]: 50/49, 875/864


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=1| 2 1 3 4 | 0 4 3 3 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
: mapping generators: ~7/5, ~6/5


Comma list: 50/49, 99/98, 875/864
[[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 }}


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


{{Val list|legend=1| 4, 14bd, 18, 22, 48, 70c, 118cd }}
[[Badness]] (Sintel): 1.10


Badness: 0.0321
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
</div></div>
 
== Fleetwood ==


Period: 1\2
Comma list: 50/49, 99/98, 385/384


Optimal ([[POTE]]) generator: ~6/5 = 327.038
Mapping: {{mapping| 2 1 3 4 8 | 0 4 3 3 -2 }}


EDO generators: [[22edo|6\22]], [[26edo|7\26]]
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}})


Scales (Scala files):
{{Optimal ET sequence|legend=0| 4, 18, 22, 48 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.06
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== Fleetwood ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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 }}
 
{{Val list|legend=1| 22 }}
 
Badness: 0.0352
 
</div></div>
 
=== 13-limit ===
 
Period: 1\2


Optimal ([[POTE]]) generator: ~6/5 = 327.841
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}})


EDO generators: [[18edo|5\18]], [[22edo|6\22]]
{{Optimal ET sequence|legend=0| 4e, …, 18e, 22 }}


Scales (Scala files):
Badness (Sintel): 1.16
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


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 }}
 
{{Val list|legend=1| 18e, 22, 84bddf }}
 
Badness: 0.0318
 
</div></div>
 
== Cavalier ==
 
Period: 1\2


Optimal ([[POTE]]) generator: ~6/5 = 323.427
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}})


EDO generators: [[22edo|6\22]], [[26edo|7\26]]
{{Optimal ET sequence|legend=0| 4ef, …, 18e, 22 }}


Scales (Scala files):
Badness (Sintel): 1.32
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== Cavalier ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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 }}
 
{{Val list|legend=1| 22e, 26 }}
 
Badness: 0.0529
 
</div></div>
 
=== 13-limit ===
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~6/5 = 323.396


EDO generators: [[22edo|6\22]], [[26edo|7\26]]
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}})


Scales (Scala files):
{{Optimal ET sequence|legend=0| 4e, 22e, 26 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.75
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


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 }}
 
{{Val list|legend=1| 22ef, 26 }}
 
Badness: 0.0350
 
</div></div>
 
= Lemba =
{{see also|Gamelismic clan #Lemba}}
{{see also|Lemba}}
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~8/7 = 232.089
 
EDO generator: [[10edo|2\10]], [[16edo|3\16]], [[26edo|5\26]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
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}})


Comma list: 50/49, 525/512
{{Optimal ET sequence|legend=0| 4ef, 22ef, 26 }}


Mapping: [{{val| 2 2 5 6 }}, {{val| 0 3 -1 -1 }}]
Badness (Sintel): 1.45


{{Val list|legend=1| 10, 16, 26, 62 }}
== Elvis ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Elvis]].''


Badness: 0.0622
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.  


</div></div>
[[Subgroup]]: 2.3.5.7


== 11-limit ==
[[Comma list]]: 50/49, 8505/8192


Period: 1\2
{{Mapping|legend=1| 2 1 10 11 | 0 2 -5 -5 }}


Optimal ([[POTE]]) generator: ~8/7 = 230.974
: mapping generators: ~7/5, ~64/45


EDO generators: [[10edo|2\10]], [[16edo|3\16]], [[26edo|5\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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 2, 24c, 26 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 3.58
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 45/44, 50/49, 385/384
Comma list: 45/44, 50/49, 1344/1331


Mapping: [{{val| 2 2 5 6 5 }}, {{val| 0 3 -1 -1 5 }}]
Mapping: {{mapping| 2 1 10 11 8 | 0 2 -5 -5 -1 }}


{{Val list|legend=1| 10, 16, 26 }}
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}})


Badness: 0.0416
{{Optimal ET sequence|legend=0| 2, 24c, 26 }}


</div></div>
Badness (Sintel): 2.09
 
== 13-limit ==
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~8/7 = 230.966
 
EDO generators: [[10edo|2\10]], [[16edo|3\16]], [[26edo|5\26]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 45/44, 50/49, 65/64, 78/77
Comma list: 45/44, 50/49, 78/77, 1053/1024


Mapping: [{{val| 2 2 5 6 5 7 }}, {{val| 0 3 -1 -1 5 1 }}]
Mapping: {{mapping| 2 1 10 11 8 16 | 0 2 -5 -5 -1 -8 }}


{{Val list|legend=1| 10, 16, 26 }}
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}})


Badness: 0.0255
{{Optimal ET sequence|legend=0| 2f, 24cf, 26 }}


</div></div>
Badness (Sintel): 1.82


= Crepuscular =
== Comic ==
{{see also|Fifive family #Crepuscular}}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Comic]].''


Comma list: 50/49, 4375/4374
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.


POTE generator: ~27/25 = 140.349
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 2 2 3 4 }}, {{val| 0 5 7 7 }}]
[[Comma list]]: 50/49, 2240/2187


{{Multival|legend=1| 10 14 14 -1 -6 -7 }}
{{Mapping|legend=1| 2 1 -3 -2 | 0 2 7 7 }}


{{Val list|legend=1| 26, 34d, 60d, 94d }}
: mapping generators: ~7/5, ~40/27


Badness: 0.0867
[[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 }}


== 11-limit ==
{{Optimal ET sequence|legend=1| 2cd, …, 20cd, 22 }}
Comma list: 50/49, 99/98, 1944/1925


POTE generator: ~12/11 = 140.587
[[Badness]] (Sintel): 2.14


Mapping: [{{val| 2 2 3 4 6 }}, {{val| 0 5 7 7 4 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 26, 34d, 60d, 94de }}
Comma list: 50/49, 99/98, 2662/2625


Badness: 0.0408
Mapping: {{mapping| 2 1 -3 -2 -4 | 0 2 7 7 10 }}


== 13-limit ==
Optimal tunings:
Comma list: 50/49, 78/77, 99/98, 144/143
* 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}})


POTE generator: ~12/11 = 140.554
{{Optimal ET sequence|legend=0| 2cde, …, 20cde, 22 }}


Mapping: [{{val| 2 2 3 4 6 6 }}, {{val| 0 5 7 7 4 6 }}]
Badness (Sintel): 1.49


{{Val list|legend=1| 26, 34d, 60d, 94de }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0244
Comma list: 50/49, 65/63, 99/98, 968/945


= Bipyth =
Mapping: {{mapping| 2 1 -3 -2 -4 3 | 0 2 7 7 10 4 }}
Comma list: 50/49, 20480/19683


POTE generator: ~3/2 = 709.437
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}})


Mapping: [{{val| 2 0 -24 -23 }}, {{val| 0 1 9 9 }}]
{{Optimal ET sequence|legend=0| 2cde, 20cde, 22 }}


{{Multival|legend=1| 2 18 18 24 23 -9 }}
Badness (Sintel): 1.71


Badness: 0.1650
== 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.  


== 11-limit ==
[[Subgroup]]: 2.3.5.7
Comma list: 50/49, 121/120, 896/891


POTE generator: ~3/2 = 709.310
[[Comma list]]: 50/49, 20480/19683


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


{{Val list|legend=1| 22 }}
: mapping generators: ~7/5, ~3


Badness: 0.0709
[[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 }}


= Elvis =
{{Optimal ET sequence|legend=1| 10cd, 12cd, 22 }}
== 5-limit ==
Comma list: 36905625/33554432


POTE generator: ~45/32 = 554.546
[[Badness]] (Sintel): 4.18


Mapping: [{{val| 2 1 10 }}, {{val| 0 2 -5 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 24c, 26, 80bc, 106bc, 132bc }}
Comma list: 50/49, 121/120, 896/891


Badness: 0.8840
Mapping: {{mapping| 2 0 -24 -23 -9 | 0 1 9 9 5 }}


== 7-limit ==
Optimal tunings:
Comma list: 50/49, 8505/8192
* 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}})


POTE generator: ~45/32 = 553.721
{{Optimal ET sequence|legend=0| 10cd, 12cde, 22 }}


Mapping: [{{val| 2 1 10 11 }}, {{val| 0 2 -5 -5 }}]
Badness (Sintel): 2.34


{{Multival|legend=1| 4 -10 -10 -25 -27 5 }}
== 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.


{{Val list|legend=1| 24c, 26 }}
[[Subgroup]]: 2.3.5.7


Badness: 0.1415
[[Comma list]]: 50/49, 546875/524288


== 11-limit ==
{{Mapping|legend=1| 16 0 37 45 | 0 1 0 0 }}
Comma list: 45/44, 50/49, 1344/1331


POTE generator: ~11/8 = 553.882
[[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 }}


Mapping: [{{val| 2 1 10 11 8 }}, {{val| 0 2 -5 -5 -1 }}]
{{Optimal ET sequence|legend=1| 16, 32, 48 }}


{{Val list|legend=1| 24c, 26 }}
[[Badness]] (Sintel): 6.73


Badness: 0.0632
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
== 13-limit ==
Comma list: 45/44, 50/49, 78/77, 1053/1024
 
POTE generator: ~11/8 = 553.892
 
Mapping: [{{val| 2 1 10 11 8 16 }}, {{val| 0 2 -5 -5 -1 -8 }}]
 
{{Val list|legend=1| 26 }}
 
Badness: 0.0440
 
= Comic =
== 5-limit ==
Comma list: 5120000/4782969
 
POTE generator: ~81/80 = 55.382
 
Mapping: [{{val| 2 1 -3 }}, {{val| 0 2 7 }}]
 
{{Val list|legend=1| 22, 86b, 108b, 130b }}
 
Badness: 0.4912
 
== 7-limit ==
Comma list: 50/49, 2240/2187
 
POTE generator: ~81/80 = 54.699
 
Mapping: [{{val| 2 1 -3 -2 }}, {{val| 0 2 7 7 }}]
 
Wedgie: &lt;&lt;4 14 14 13 11 -7||
 
{{Val list|legend=1| 22 }}
 
Badness: 0.0844
 
== 11-limit ==
Comma list: 50/49, 99/98, 2662/2625
 
POTE generator: ~81/80 = 55.184
 
Mapping: [{{val| 2 1 -3 -2 -4 }}, {{val| 0 2 7 7 10 }}]
 
{{Val list|legend=1| 22 }}
 
Badness: 0.0451
 
== 13-limit ==
Comma list: 50/49, 65/63, 99/98, 968/945
 
POTE generator: ~81/80 = 54.435
 
Mapping: [{{val| 2 1 -3 -2 -4 3 }}, {{val| 0 2 7 7 10 4 }}]
 
{{Val list|legend=1| 22 }}


Badness: 0.0415
Comma list: 50/49, 385/384, 1331/1323


= Duodecim =
Mapping: {{mapping| 16 0 37 45 30 | 0 1 0 0 1 }}
Comma list: 36/35, 50/49, 64/63


POTE generator: ~11/8 = 565.023
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}})


Mapping: [{{val| 12 19 28 34 0 }}, {{val| 0 0 0 0 1 }}]
{{Optimal ET sequence|legend=0| 16, 32, 48 }}


{{Val list|legend=1| 12, 24d, 36d }}
Badness (Sintel): 3.07


Badness: 0.030536
== 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>.  


= Vigintiduo =
[[Subgroup]]: 2.9.5.7
Comma list: 50/49, 64/63, 245/243


POTE generator: ~11/8 = 557.563
[[Comma list]]: 50/49, 64/63


Mapping: [{{val| 22 35 51 62 0 }}, {{val| 0 0 0 0 1 }}]
{{Mapping|legend=2| 2 0 11 12 | 0 1 -1 -1 }}


{{Val list|legend=1| 22, 66de, 88bde, 110bd, 198bcdde }}
{{Mapping|legend=3| 2 0 11 12 | 0 1/2 -1 -1 }}


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


= Vigin =
[[Optimal tuning]]s:
Comma list: 50/49, 55/54, 64/63, 99/98
* [[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 }}


POTE generator: ~13/8 = 844.624
{{Optimal ET sequence|legend=1| 2, 4, 6, 16, 22, 28 }}


Mapping: [{{val| 22 35 51 62 76 0 }}, {{val| 0 0 0 0 0 1 }}]
[[Badness]] (Sintel): 0.253


{{Val list|legend=1| 22, 44 }}
== References ==


Badness: 0.0298
[[Category:Temperament clans]]
 
[[Category:Jubilismic clan| ]] <!-- main article -->
[[Category:Theory]]
[[Category:Temperament clan]]
[[Category:Jubilismic]]
[[Category:Rank 2]]
[[Category:Rank 2]]
{{todo| review | improve readability }}

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"