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Marvel temperaments: Difference between revisions

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This page discusses some of the temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the [[Marvel family|marvel]] comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton, merman and marvo. Considered elsewhere are [[Meantone family #Septimal meantone|meantone]], [[Gamelismic clan #Miracle|miracle]], [[Magic family|magic]], [[Diaschismic family #Pajara|pajara]], [[orwell]], [[Kleismic family #Catakleismic|catakleismic]], [[Schismatic family #Garibaldi|garibaldi]], [[Augmented family #August|august]], [[Pythagorean family #Compton|compton]], [[Dicot family #Sharp|sharp]], [[Escapade family #Escapade|escapade]] and [[Pelogic family #Mavila|mavila]].
{{Technical data page}}
This page discusses miscellaneous [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] [[225/224]], the marvel comma or septimal kleisma.  


Since (5/4)<sup>2</sup> = 225/224 × 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
Temperaments considered in families and clans are:
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* [[Wizard]] (+118098/117649) → [[Stearnsmic clan #Wizard|Stearnsmic clan]]
* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]


The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.
Considered below are tritonic, septimin, merman, slender, triton, marvolo, decic, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, hendeca, gwazy, and tertiosec, in the order of increasing [[badness]].  


Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)
Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.


= Negri =
The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.
{{main| Negri }}


Negri tempers out the [[negri comma]] in the 5-limit, [[49/48]] and [[225/224]] in the 7-limit. It can be extended naturally to the 2.3.5.7.13 subgroup by adding [[91/90]] to the comma list; this will be discussed below under the title of negra.  
Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)


== 5-limit ==
== Tritonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


Period: 1\1
Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.


Optimal ([[POTE]]) generator: ~16/15 = 125.7549
[[Subgroup]]: 2.3.5.7


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
[[Comma list]]: 225/224, 50421/50000


Scales (Scala files):  
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Optimal tuning]]s:
<div style="line-height:1.6;">Technical data</div>
* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
<div class="mw-collapsible-content">
: [[error map]]: {{val| +1.354 -1.243 -0.027 -1.598 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}}
: error map: {{val| 0.000 -3.566 -2.769 -4.959 }}


Subgroup: 2.3.5
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


[[Comma list]]: 16875/16384
[[Badness]] (Sintel): 1.20


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Wedgie]]: {{wedgie| 4 -3 -14 }}
Comma list: 121/120, 225/224, 441/440


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
{{Mapping|legend=0| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


== 7-limit ==
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


Period: 1\1
Badness (Sintel): 0.782


Optimal ([[POTE]]) generator: ~15/14 = 125.608
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
Comma list: 105/104, 121/120, 196/195, 275/273


Scales (Scala files):
{{Mapping|legend=0| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


[[Comma list]]: 49/48, 225/224
Badness (Sintel): 0.950


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


[[Wedgie]]: {{wedgie| 4 -3 2 -14 -8 13 }}
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
{{Mapping|legend=0| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


=== Negra ===
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


This is the 2.3.5.7.13 extension of negri.  
Badness (Sintel): 0.973


Period: 1\1
==== 19-limit ====
 
Subgroup: 2.3.5.7.11.13.17.19
Optimal ([[POTE]]) generator: ~14/13 = 125.567
 
EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]


Scales (Scala files):  
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.13
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


[[Comma list]]: 49/48, 65/64, 91/90
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


[[Mapping]]: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -3 }}]
Badness (Sintel): 1.03


Gencom map: [{{val| 1 2 2 3 0 4 }}, {{val| 0 -4 3 -2 0 -3 }}]
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Gencom: [2 14/13; 49/48 65/64 91/90]
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Period: 1\1
Badness (Sintel): 1.04
 
Optimal ([[POTE]]) generator: ~15/14 = 126.474
 
EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
 
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">


=== Tritoni ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 45/44, 49/48, 56/55
Comma list: 225/224, 385/384, 27783/27500
 
Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
 
{{Val list|legend=1| 9, 10, 19 }}
 
[[Badness]]: 0.0262
 
</div></div>


=== 13-limit ===
{{Mapping|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~14/13 = 126.431
 
EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
 
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: 2.3.5.7.11.13


Comma list: 45/44, 49/48, 56/55, 78/77
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


[[POTE generator]]: ~14/13 = 126.431
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
Badness (Sintel): 1.50


{{Val list|legend=1| 9, 10, 19 }}
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


</div></div>
Septimin may be described as the {{nowrap| 41 & 50 }} temperament. It is generated by a septimal minor third ([[7/6]]), which gives rise to the name, but the generator can be taken to be the [[octave complement]], [[12/7]], such that eleven of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus eta-hendecacot. [[91edo]] may be recommended as a tuning.


== Negril ==
[[Subgroup]]: 2.3.5.7


Period: 1\1
[[Comma list]]: 225/224, 84035/82944


Optimal ([[POTE]]) generator: ~15/14 = 124.767
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


EDO generators: [[19edo|2\19]], [[29edo|3\29]]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}
: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 1.38
<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: 49/48, 100/99, 225/224
Comma list: 225/224, 245/242, 385/384


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
{{Mapping|legend=0| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


{{Val list|legend=1| 19, 29, 48d, 77cdd }}
Optimal tunings:
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


Badness: 0.0387
{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}


</div></div>
Badness (Sintel): 1.04


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


Period: 1\1
Comma list: 105/104, 144/143, 196/195, 245/242


Optimal ([[POTE]]) generator: ~14/13 = 124.716
{{Mapping|legend=0| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


EDO generators: [[19edo|2\19]], [[29edo|3\29]]
Optimal tunings:  
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.955
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Comma list: 49/48, 65/64, 91/90, 875/858
Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot.


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.


{{Val list|legend=1| 19, 29, 48df, 77cddf }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0244
[[Comma list]]: 225/224, 2500000/2470629


</div></div>
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


== Negric ==
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
: [[error map]]: {{val| +0.390 -0.435 -1.630 +1.634 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}}
: error map: {{val| 0.000 -1.104 -2.423 +0.657 }}


Period: 1\1
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


Optimal ([[POTE]]) generator: ~15/14 = 127.039
[[Badness]] (Sintel): 1.39
 
EDO generators: [[9edo|1\9]], [[19edo|2\19]]
 
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">


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


Comma list: 33/32, 49/48, 77/75
Comma list: 225/224, 441/440, 1344/1331
 
Mapping: [{{val| 1 2 2 3 3 }}, {{val| 0 -4 3 -2 4 }}]


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


Badness: 0.0306
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


=== 13-limit ===
Badness (Sintel): 1.20
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~14/13 = 127.039
 
EDO generators: [[9edo|1\9]], [[19edo|2\19]]
 
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: 33/32, 49/48, 65/64, 91/90
Comma list: 144/143, 225/224, 364/363, 441/440
 
Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
 
{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
 
Badness: 0.0202
 
</div></div>


== Negroni ==
{{Mapping|legend=0| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Optimal ([[POTE]]) generator: ~15/14 = 124.539
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


EDO generators: [[10edo|1\10]], [[19edo|2\19]], [[29edo|3\29]]
Badness (Sintel): 1.14
 
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">


=== Mermaid ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 55/54, 225/224
Comma list: 225/224, 385/384, 532400/531441


Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
{{Mapping|legend=0| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


{{Val list|legend=1| 10, 19e, 29, 77cddee }}
Optimal tunings:
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


Badness: 0.0353
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


</div></div>
Badness (Sintel): 1.46


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


Period: 1\1
Comma list: 225/224, 325/324, 385/384, 10648/10647


Optimal ([[POTE]]) generator: ~14/13 = 124.545
{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


EDO generators: [[10edo|1\10]], [[19edo|2\19]], [[29edo|3\29]]
Optimal tunings:  
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.47
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
== Slender ==
Slender tempers out the [[hewuermera comma]] in addition to the marvel comma, and may be described as the {{nowrap| 31 & 32 }} temperament. This temperament has a generator of [[49/48]], three of which equal marvel's [[16/15]][[~]][[15/14]], and ten generators give [[5/4]]. Its [[ploidacot]] is omega-13-cot.  


Comma list: 49/48, 55/54, 65/64, 91/90
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 10, 19e, 29, 77cddeef }}
[[Comma list]]: 225/224, 589824/588245


Badness: 0.0216
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


</div></div>
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
: [[error map]]: {{val| +0.382 -0.725 -1.295 +1.765 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}}
: error map: {{val| 0.000 -1.257 -2.235 +0.727 }}


== Wilsec ==
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Period: 1\1
[[Badness]] (Sintel): 1.44


Optimal ([[POTE]]) generator: ~11/8 = 537.186
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
Comma list: 225/224, 385/384, 1331/1323


Scales (Scala files):
{{Mapping|legend=0| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


Comma list: 49/48, 121/120, 225/224
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]
Badness (Sintel): 0.838
 
{{Val list|legend=1| 9, 20, 29, 38d, 67cdde }}
 
Badness: 0.0419
 
</div></div>


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


Period: 1\1
Comma list: 225/224, 275/273, 385/384, 1331/1323


Optimal ([[POTE]]) generator: ~11/8 = 537.208
{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.07
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 49/48, 65/64, 91/90, 121/120
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot.


{{Val list|legend=1| 9, 20, 29, 38df, 67cddef }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0252
[[Comma list]]: 225/224, 1029/1000


</div></div>
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


=== 17-limit ===
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}
: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}


Period: 1\1
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


Optimal ([[POTE]]) generator: ~11/8 = 537.230
[[Badness]] (Sintel): 1.50


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Scales (Scala files):  
Comma list: 45/44, 56/55, 1029/1000


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
Optimal tunings:  
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefg }}
Badness (Sintel): 1.51


Badness: 0.0218
== Marvolo ==
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 156250000/155649627


=== 19-limit ===
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


Period: 1\1
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7714{{c}}, ~21/20 = 83.4014{{c}}
: [[error map]]: {{val| +0.772 -0.820 -0.916 +0.381 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}}
: error map: {{val| 0.000 -2.139 -2.398 -1.362 }}


Optimal ([[POTE]]) generator: ~11/8 = 537.214
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
[[Badness]] (Sintel): 2.11


Scales (Scala files):  
=== 11-limit ===
Subgroup: 2.3.5.7.11


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 225/224, 441/440, 4000/3993
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
{{Mapping|legend=0| 1 2 1 1 2 | 0 -6 19 26 21 }}


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefgh }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.0168
Badness (Sintel): 0.958


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


= Passive =
Comma list: 169/168, 225/224, 364/363, 441/440
{{see also | Archytas clan #Passion }}


Period: 1\1
{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Optimal ([[POTE]]) generator: ~16/15 = 98.809
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


EDO generators: [[12edo|1\12]], [[13edo|1\13]]
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Scales (Scala files):  
Badness (Sintel): 0.887


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
=== 17-limit ===
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13.17
<div class="mw-collapsible-content">


Comma list: 225/224, 256/245
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
{{Mapping|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


{{Val list|legend=1| 12, 49dd }}
Optimal tunings:
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Badness: 0.0751
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


</div></div>
Badness (Sintel): 0.760


= Wizard =
=== 19-limit ===
== 5-limit ==
Subgroup: 2.3.5.7.11.13.17.19


Period: 1\2
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Optimal ([[POTE]]) generator: ~5/4 = 383.212
{{Mapping|legend=0| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.895
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 2197265625/2147483648
== Decic ==
{{Main| Decic }}


Mapping: [{{val| 2 1 5 }}, {{val| 0 6 -1 }}]
Named by [[Xenllium]] in 2021, decic tempers out 16807/16384, the [[cloudy comma]], and {{monzo| 11 -10 -10 10 }}, the [[linus comma]], in addition to the marvel comma. It may be described as the {{nowrap| 50 & 60 }} temperament, with a period of 1/10 octave and a [[ploidacot]] signature of decaploid monocot. It is [[support]]ed by [[10edo|10-]], [[50edo|50-]], and [[60edo]].


Mapping generators: ~42875/32768, ~9375/8192
It can be extended to the 11-, 13-, and 17-limit by adding [[385/384]], [[105/104]], and [[170/169]] to the comma list in this order.


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 548bc }}
[[Subgroup]]: 2.3.5.7


Badness: 0.3864
[[Comma list]]: 225/224, 16807/16384


</div></div>
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


== 7-limit ==
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 120.1837{{c}}, ~3/2 = 699.7654{{c}} (~49/48 = 21.3366{{c}})
: [[error map]]: {{val| +1.837 -0.353 -0.753 -3.683 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 698.8236{{c}} (~49/48 = 21.1764{{c}})
: error map: {{val| 0.000 -3.131 -5.137 -8.826 }}


Period: 1\2
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


Optimal ([[POTE]]) generator: ~5/4 = 383.256
[[Badness]] (Sintel): 2.26


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Scales (Scala files):  
Comma list: 225/224, 385/384, 3087/3025


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=0| 10 0 39 28 3 | 0 1 -1 0 2 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 118098/117649
Optimal tunings:  
* WE: ~15/14 = 120.1406{{c}}, ~3/2 = 697.6075{{c}} (~56/55 = 23.2360{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.0142{{c}} (~56/55 = 22.9858{{c}})


Mapping: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Mapping generators: ~1225/864, ~245/216
Badness (Sintel): 2.11


Wedgie: {{wedgie| 12 -2 20 -31 -2 52 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
Comma list: 105/104, 144/143, 196/195, 2200/2197


Badness: 0.0408
{{Mapping|legend=0| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


</div></div>
Optimal tunings:
* WE: ~15/14 = 120.1166{{c}}, ~3/2 = 697.6705{{c}} (~78/77 = 23.0289{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.1492{{c}} (~78/77 = 22.8508{{c}})


== 11-limit ==
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Period: 1\2
Badness (Sintel): 1.52


Optimal ([[POTE]]) generator: ~5/4 = 383.232
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


Scales (Scala files):
{{Mapping|legend=0| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~15/14 = 120.1262{{c}}, ~3/2 = 697.8185{{c}} (~78/77 = 22.9388{{c}})
<div class="mw-collapsible-content">
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.2757{{c}} (~78/77 = 22.7243{{c}})


Comma list: 225/224, 385/384, 4000/3993
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
Badness (Sintel): 1.28


Mapping generators: ~99/70, ~25/22
=== Splendecic ===
Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[fantastic]] rank-3 temperament, its name is a portmanteau of ''splendid'' and ''decic''.


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c }}
Subgroup: 2.3.5.7.11


Badness: 0.0185
Comma list: 225/224, 1617/1600, 2401/2376


</div></div>
{{Mapping|legend=0| 10 0 39 28 82 | 0 1 -1 0 -3 }}


=== Lizard ===
Optimal tunings:
* WE: ~15/14 = 120.1874{{c}}, ~3/2 = 699.6085{{c}} (~99/98 = 21.5156{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3531{{c}} (~99/98 = 21.6469{{c}})


Period: 1\2
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


Optimal ([[POTE]]) generator: ~5/4 = 383.389
Badness (Sintel): 1.98


EDO generators: [[50edo|16\50]], [[72edo|23\72]]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Scales (Scala files):  
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=0| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 351/350, 364/363, 385/384
Optimal tunings:  
* WE: ~15/14 = 120.1565{{c}}, ~3/2 = 699.2756{{c}} (~91/90 = 21.6631{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.2480{{c}} (~91/90 = 21.7520{{c}})


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


Mapping generators: ~99/70, ~25/22
Badness (Sintel): 1.57
 
{{Val list|legend=1| 22, 50, 72, 122, 194df }}
 
Badness: 0.0218
 
</div></div>


==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Period: 1\2
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374


Optimal ([[POTE]]) generator: ~5/4 = 383.381
{{Mapping|legend=0| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


EDO generators: [[50edo|16\50]], [[72edo|23\72]]
Optimal tunings:  
* WE: ~15/14 = 120.1571{{c}}, ~3/2 = 699.2892{{c}} (~91/90 = 21.6536{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3144{{c}} (~91/90 = 21.6856{{c}})


Scales (Scala files):
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.33
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
=== Prodecic ===
Prodecic (50e & 60e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the [[prodigy]] rank-3 temperament, its name is a portmanteau of ''prodigy'' and ''decic''.


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
Subgroup: 2.3.5.7.11


Mapping generators: ~17/12, ~17/15
Comma list: 225/224, 441/440, 5929/5832


{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
{{Mapping|legend=0| 10 0 39 28 -13 | 0 1 -1 0 3 }}


Badness: 0.0145
Optimal tunings:  
* WE: ~15/14 = 120.2024{{c}}, ~3/2 = 701.3908{{c}} (~55/54 = 19.8237{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.5235{{c}} (~55/54 = 19.4765{{c}})


</div></div>
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


==== 19-limit ====
Badness (Sintel): 2.20


Period: 1\2
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Optimal ([[POTE]]) generator: ~5/4 = 383.477
Comma list: 105/104, 196/195, 275/273, 5929/5832


EDO generators: [[50edo|16\50]], [[72edo|23\72]]
{{Mapping|legend=0| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


Scales (Scala files):  
Optimal tunings:
* WE: ~15/14 = 120.1654{{c}}, ~3/2 = 701.4683{{c}} (~91/90 = 19.5242{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.7175{{c}} (~91/90 = 19.2825{{c}})


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Badness (Sintel): 1.73


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Mapping generators: ~17/12, ~17/15
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


{{Val list|legend=1| 22h, 50, 72, 122g, 194dfg }}
{{Mapping|legend=0| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


Badness: 0.0157
Optimal tunings:  
* WE: ~15/14 = 120.1577{{c}}, ~3/2 = 701.3950{{c}} (~91/90 = 19.5514{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.6932{{c}} (~91/90 = 19.3068{{c}})


</div></div>
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


=== Gizzard ===
Badness (Sintel): 1.41


Period: 1\2
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


Optimal ([[POTE]]) generator: ~5/4 = 383.170
[[Comma list]]: 225/224, 40353607/40310784


EDO generators: [[72edo|23\72]], [[94edo|30\94]]
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


Scales (Scala files):  
[[Optimal tuning]]s:
* [[WE]]: ~2592/2401 = 133.4174{{c}}, ~12005/6912 = 950.7667{{c}} (~1728/1715 = 16.8452{{c}})
: [[error map]]: {{val| +0.756 -0.422 -1.395 +0.298 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}})
: error map: {{val| 0.000 -1.361 -3.277 -1.565 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 385/384, 325/324, 1573/1568
[[Badness]] (Sintel): 2.37


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping generators: ~99/70, ~25/22
Comma list: 225/224, 385/384, 12005/11979


{{Val list|legend=1| 72, 166, 238cf }}
{{Mapping|legend=0| 9 0 28 11 24 | 0 2 -1 2 1 }}


Badness: 0.0203
Optimal tunings:  
* WE: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}})
* CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}})


</div></div>
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


==== 17-limit ====
Badness (Sintel): 1.01


Period: 1\2
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Optimal ([[POTE]]) generator: ~5/4 = 383.175
Comma list: 169/168, 225/224, 364/363, 1716/1715


EDO generators: [[72edo|23\72]], [[94edo|30\94]]
{{Mapping|legend=0| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


Scales (Scala files):  
Optimal tunings:
* WE: ~14/13 = 133.4245{{c}}, ~26/15 = 950.9362{{c}} (~105/104 = 16.9650{{c}})
* CWE: ~14/13 = 133.3333{{c}}, ~26/15 = 950.4364{{c}} (~99/98 = 17.1031{{c}})


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Badness (Sintel): 0.922


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


Mapping generators: ~17/12, ~17/15
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 72, 166g, 238cfg }}
[[Comma list]]: 225/224, 781250000/771895089


Badness: 0.0136
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


</div></div>
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


==== 19-limit ====
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


Period: 1\2
[[Badness]] (Sintel): 2.44


Optimal ([[POTE]]) generator: ~5/4 = 383.138
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDO generators: [[72edo|23\72]], [[94edo|30\94]]
Comma list: 225/224, 385/384, 236328125/234365481


Scales (Scala files):
{{Mapping|legend=0| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Comma list: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
Badness (Sintel): 2.96


Mapping generators: ~17/12, ~17/15
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 72, 94, 166g }}
Comma list: 225/224, 325/324, 385/384, 831875/830466


Badness: 0.0148
{{Mapping|legend=0| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


== Mage ==
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Period: 1\2
Badness (Sintel): 2.16


Optimal ([[POTE]]) generator: ~5/4 = 383.124
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


Scales (Scala files):
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


Comma list: 99/98, 176/175, 1331/1296
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
Badness (Sintel): 1.96


Mapping generators: ~77/54, ~55/48
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


{{Val list|legend=1| 22, 50e, 72ee, 94ee }}
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


Badness: 0.0578
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


= Triton =
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
Comma list: 225/224, 1029/1000


[[POTE generator]]: ~7/5 = 568.865
Badness (Sintel): 1.71


Mapping: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Wedgie: {{wedgie| 3 -7 -8 -18 -21 1 }}
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


{{Val list|legend=1| 17d, 19, 78bd, 97bd }}
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Badness: 0.0592
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}
Comma list: 45/44, 56/55, 1029/1000


POTE generator: ~7/5 = 569.144
Badness (Sintel): 1.57


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Badness: 0.0457
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


= Tritonic =
Optimal tunings:
== 5-limit ==
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
Comma list: 553584375/536870912
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


POTE generator: ~45/32 = 580.219
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Mapping: [{{val| 1 4 -3 }}, {{val| 0 -5 11 }}]
Badness (Sintel): 1.50


{{Val list|legend=1| 29, 31, 60, 91 }}
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Badness: 0.3785
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


== 7-limit ==
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}
Comma list: 225/224, 50421/50000


POTE generator: ~7/5 = 580.286
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


Mapping: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Wedgie: {{wedgie| 5 -11 -12 -29 -33 3 }}
Badness (Sintel): 1.53


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
=== Gracecord ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 225/224, 441/440, 109375/107811
Comma list: 121/120, 225/224, 441/440


POTE generator: ~7/5 = 580.267
{{Mapping|legend=0| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


{{Val list|legend=1| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Badness: 0.0237
Badness (Sintel): 2.21


=== 13-limit ===
==== 13-limit ====
Comma list: 105/104, 121/120, 196/195, 275/273
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 580.108
Comma list: 225/224, 364/363, 441/440, 6125/6084


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
{{Mapping|legend=0| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


{{Val list|legend=1| 29, 31, 60e, 151cde }}
Optimal tunings:
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


Badness: 0.0230
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


== Tritoni ==
Badness (Sintel): 1.83
Comma list: 225/224, 385/384, 27783/27500


POTE generator: ~7/5 = 580.389
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


{{Val list|legend=1| 31, 91, 122 }}
{{Mapping|legend=0| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Badness: 0.0455
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


= Merman =
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
Comma list: 1121008359375/1099511627776


POTE generator: ~45/32 = 585.634
Badness (Sintel): 1.87


Mapping: [{{val| 1 5 -5 }}, {{val| 0 -7 15 }}]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


{{Val list|legend=1| 41, 84, 125, 209, 334c, 543bc }}
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


Badness: 0.4538
{{Mapping|legend=0| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


== 7-limit ==
Optimal tunings:
Comma list: 225/224, 2500000/2470629
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


POTE generator: ~7/5 = 585.585
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Mapping: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
Badness (Sintel): 1.68


Wedgie: {{wedgie| 7 -15 -16 -40 -45 5 }}
== Alphorn ==
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 41, 84, 125 }}
[[Comma list]]: 225/224, 5764801/5668704


Badness: 0.0551
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


== 11-limit ==
[[Optimal tuning]]s:
Comma list: 225/224, 441/440, 1344/1331
* [[WE]]: ~2 = 1201.3004{{c}}, ~35/24 = 644.4767{{c}}
: [[error map]]: {{val| +1.300 +0.569 -2.195 -2.043 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~35/24 = 643.8137{{c}}
: error map: {{val| 0.000 -0.936 -5.382 -4.924 }}


POTE generator: ~7/5 = 585.606
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
[[Badness]] (Sintel): 3.27


{{Val list|legend=1| 41, 84, 125e }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0364
Comma list: 225/224, 385/384, 12250/11979


== 13-limit ==
{{Mapping|legend=0| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}
Comma list: 144/143, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 585.657
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
Badness (Sintel): 2.43


Badness: 0.0275
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


= Septimin =
[[Subgroup]]: 2.3.5.7
Comma list: 35595703125/34359738368


POTE generator: ~75/64 = 263.658
[[Comma list]]: 225/224, 4194304/4117715


Mapping: [{{val| 1 4 1 }}, {{val| 0 -11 6 }}]
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


{{Val list|legend=1| 9, 32, 41, 91, 132, 223c, 355bc }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7642{{c}}, ~15/8 = 1086.5513{{c}}
: [[error map]]: {{val| -0.236 -0.856 -1.569 +4.041 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1086.7633{{c}}
: error map: {{val| 0.000 -0.506 -0.999 +4.701 }}


Badness: 0.6709
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


== 7-limit ==
[[Badness]] (Sintel): 3.57
Comma list: 225/224, 84035/82944


POTE generator: ~7/6 = 263.632
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
Comma list: 99/98, 176/175, 1310720/1294139


Wedgie: {{wedgie| 11 -6 10 -35 -15 40 }}
{{Mapping|legend=0| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


{{Val list|legend=1| 9, 32, 41, 91, 132 }}
Optimal tunings:
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}
Comma list: 225/224, 385/384, 2401/2376


POTE generator: ~7/6 = 263.634
Badness (Sintel): 2.82


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 9, 32, 41, 91, 223cdef }}
Comma list: 99/98, 176/175, 640/637, 847/845


== 13-limit ==
{{Mapping|legend=0| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}
Comma list: 105/104, 144/143, 196/195, 245/242


POTE generator: ~7/6 = 263.700
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


{{Val list|legend=1| 9, 41, 91 }}
Badness (Sintel): 1.88


= Slender =
=== Musneb ===
Comma list: 225/224, 589824/588245
Subgroup: 2.3.5.7.11


POTE generator: ~49/48 = 38.413
Comma list: 225/224, 385/384, 66550/64827


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
{{Mapping|legend=0| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


Wedgie: {{wedgie| 13 -10 6 -46 -27 42 }}
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


{{Val list|legend=1| 31, 94, 125 }}
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Badness: 0.0569
Badness (Sintel): 2.89


==11-limit==
== Untriton ==
Comma list: 225/224, 385/384, 1331/1323
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


POTE generator: ~49/48 = 38.387
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.  


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 31, 63, 94, 125 }}
[[Comma list]]: 225/224, 125000000/121060821


Badness: 25.342
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


== 13-limit ==
[[Optimal tuning]]s:
Comma list: 225/224, 275/273, 385/384, 1331/1323
* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
: [[error map]]: {{val| -0.172 +0.002 -2.533 +3.511 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}}
: error map: {{val| 0.000 +0.298 -2.181 +3.946 }}


POTE generator: ~49/48 = 38.314
{{Optimal ET sequence|legend=1| 51, 53 }}


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
[[Badness]] (Sintel): 3.64


{{Val list|legend=1| 31, 63, 94 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 25.913
Comma list: 121/120, 225/224, 22000/21609


= Marvo =
{{Mapping|legend=0| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}
{{see also| Gravity family }}


Comma list: 225/224, 78125000/78121827
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


[[POTE generator]]: ~27/20 = 516.694
{{Optimal ET sequence|legend=0| 51, 53 }}


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
Badness (Sintel): 2.46


Wedgie: {{wedgie| 6 17 46 13 56 59 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 7d, 65d, 72, 137, 209, 281, 569bcc }}
Comma list: 121/120, 225/224, 275/273, 1040/1029


Badness: 0.0976
{{Mapping|legend=0| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


== 11-limit ==
Optimal tunings:
Comma list: 225/224, 243/242, 4000/3993
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


POTE generator: ~27/20 = 516.699
{{Optimal ET sequence|legend=0| 51f, 53 }}


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
Badness (Sintel): 1.96


{{Val list|legend=1| 7d, 65d, 72, 281, 353c, 425bc, 497bc }}
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


Badness: 0.0317
[[Subgroup]]: 2.3.5.7


== 13-limit ==
[[Comma list]]: 225/224, 823543/800000
Comma list: 225/224, 243/242, 351/350, 1625/1617


POTE generator: ~27/20 = 516.730
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.1198{{c}}, ~32/21 = 745.4675{{c}}
: [[error map]]: {{val| +2.120 -1.227 +0.459 -4.423 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~32/21 = 744.1318{{c}}
: error map: {{val| 0.000 -4.769 -4.159 -10.144 }}


{{Val list|legend=1| 72, 137, 209, 281f, 490bcf }}
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


Badness: 0.0269
[[Badness]] (Sintel): 3.67


= Marvolo =
=== 11-limit ===
Comma list: 225/224, 156250000/155649627
Subgroup: 2.3.5.7.11


POTE generator: ~21/20 = 83.348
Comma list: 225/224, 245/242, 1617/1600


Mapping: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


Wedgie: {{wedgie| 6 -19 -26 -44 -58 -7 }}
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Badness: 0.0833
Badness (Sintel): 2.00


== 11-limit ==
=== 13-limit ===
Comma list: 225/224, 441/440, 4000/3993
Subgroup: 2.3.5.7.11.13


POTE generator: ~21/20 = 83.340
Comma list: 105/104, 196/195, 245/242, 1001/1000


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


{{Val list|legend=1| 29, 43, 72 }}
Optimal tunings:
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


Badness: 0.0290
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


== 13-limit ==
Badness (Sintel): 1.43
Comma list: 169/168, 225/224, 364/363, 441/440


POTE generator: ~21/20 = 83.330
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


{{Val list|legend=1| 29, 43, 72, 115f }}
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


Badness: 0.0215
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


= Amavil =
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}
== 5-limit (mabila) ==
Comma list: 268435456/263671875


POTE generator: ~512/375 = 529.6849
Badness (Sintel): 1.26


Mapping: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
[[Subgroup]]: 2.3.5.7


Badness: 0.2325
[[Comma list]]: 225/224, 9805926501/9765625000


== 7-limit ==
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
Comma list: 225/224, 17496/16807
: mapping generators: ~2, ~10000/9261


POTE generator: ~48/35 = 529.979
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.9803{{c}}, ~10000/9261 = 139.9522{{c}}
: [[error map]]: {{val| +0.980 -1.214 -0.313 -0.243 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10000/9261 = 139.8184{{c}}
: error map: {{val| 0.000 -2.863 -2.136 -2.287 }}


Mapping: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


Wedgie: {{wedgie| 10 -3 14 -28 -6 41 }}
[[Badness]] (Sintel): 3.81


{{Val list|legend=1| 9, 34d, 43, 77d }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.1096
Comma list: 225/224, 441/440, 43923/43750


== 11-limit ==
{{Mapping|legend=0| 1 1 5 7 8 | 0 5 -23 -36 -39 }}
Comma list: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


{{Val list|legend=1| 9, 34d, 43, 77de }}
Badness (Sintel): 1.74


Badness: 0.0426
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 13-limit ==
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
Comma list: 78/77, 99/98, 144/143, 176/175


POTE generator: ~15/11 = 529.951
{{Mapping|legend=0| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


{{Val list|legend=1| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 0.0258
Badness (Sintel): 1.35


= Enneaportent =
=== 17-limit ===
Comma list: 225/224, 40353607/40310784
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~5/4 = 383.165
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Mapping: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
{{Mapping|legend=0| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


Wedgie: {{wedgie| 18 -9 18 -56 -22 67 }}
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


{{Val list|legend=1| 9, 54, 63, 72, 495bcd }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 0.0937
Badness (Sintel): 1.17


== 11-limit ==
== Hendeca ==
Comma list: 225/224, 385/384, 12005/11979
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


POTE generator: ~5/4 = 383.146
Hendeca tempers out the same 5-limit comma as [[porwell temperaments #Hendecatonic|hendecatonic]], and has a period of 1/11 octave. However, in this temperament, nine periods represent [[7/4]], the same as [[keemic temperaments #Undeka|undeka]]. It can be tuned to [[22edo]] or [[33edo]] using the [[patent val]], or [[55edo]] using the 55d val. It was named by [[Xenllium]] in 2025 as a low-accuracy variant of hendecatonic.  


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 9, 54, 63, 72, 423cd, 495bcd }}
[[Comma list]]: 225/224, 122880/117649


Badness: 0.0304
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


== 13-limit ==
[[Optimal tuning]]s:
Comma list: 169/168, 225/224, 364/363, 1716/1715
* [[WE]]: ~16/15 = 108.9526{{c}}, ~3/2 = 702.4215{{c}}
: [[error map]]: {{val| -1.521 -1.055 -2.252 +8.705 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 703.2071{{c}}
: error map: {{val| 0.000 +1.252 +1.388 +12.992 }}


POTE generator: ~5/4 = 383.047
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
[[Badness]] (Sintel): 4.37


{{Val list|legend=1| 9, 54, 63, 72, 279cf }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0223
Comma list: 121/120, 225/224, 352/343


= Submajor =
{{Mapping|legend=0| 11 0 43 31 38 | 0 1 -1 0 0 }}
== 5-limit ==
Comma list: 69198046875/68719476736


POTE generator: ~10125/8192 = 362.321
Optimal tunings:  
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


Mapping: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
{{Optimal ET sequence|legend=0| 22, 55d }}


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
Badness (Sintel): 2.46


Badness: 0.1302
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


== 7-limit ==
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.
Comma list: 225/224, 51200/50421


POTE generator: ~49/40 = 362.255
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
[[Comma list]]: 225/224, 5971968/5764801


Wedgie: {{wedgie| 8 -11 -6 -36 -32 17 }}
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


{{Val list|legend=1| 10, 33, 43, 53 }}
[[Optimal tuning]]s:
* [[WE]]: ~2401/1728 = 599.7132{{c}}, ~35/32 = 162.5806{{c}}
: [[error map]]: {{val| -0.574 -1.597 -0.937 +5.510 }}
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~35/32 = 162.6388{{c}}
: error map: {{val| 0.000 -0.844 +0.492 +7.007 }}


Badness: 0.0605
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


== 11-limit ==
[[Badness]] (Sintel): 4.53
Comma list: 225/224, 385/384, 6655/6561


POTE generator: ~27/22 = 362.101
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
Comma list: 99/98, 176/175, 65536/65219


{{Val list|legend=1| 10, 53, 116, 169de, 285cde }}
{{Mapping|legend=0| 2 1 6 4 8 | 0 8 -5 6 -4 }}


Badness: 0.0506
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 22, 74, 96 }}
Comma list: 169/168, 225/224, 275/273, 385/384


POTE generator: ~16/13 = 362.105
Badness (Sintel): 2.26


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


{{Val list|legend=1| 10, 53, 116, 169de, 285cdef }}
Tertiosec may be described as the {{nowrap| 21 & 75 }} temperament. It was initially named ''tertiomar'' by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, but was changed to ''tertiosec'' in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.


Badness: 0.0277
[[Subgroup]]: 2.3.5.7


== Interpental ==
[[Comma list]]: 225/224, 14495514624/13841287201
Comma list: 99/98, 176/175, 51200/50421


POTE generator: ~49/40 = 362.418
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
[[Optimal tuning]]s:
* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
: [[error map]]: {{val| -0.523 -1.044 -1.549 +5.138 }}
* [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}}
: error map: {{val| 0.000 -0.284 -0.276 +6.592 }}


{{Val list|legend=1| 43, 53, 96, 149d }}
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


Badness: 0.0518
[[Badness]] (Sintel): 10.9


=== 13-limit ===
=== 11-limit ===
Comma list: 99/98, 169/168, 176/175, 640/637
Subgroup: 2.3.5.7.11
 
POTE generator: ~16/13 = 362.402


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
Comma list: 225/224, 3840/3773, 12005/11979


{{Val list|legend=1| 43, 53, 96, 149d }}
{{Mapping|legend=0| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
 
Badness: 0.0297
 
= Alphorn =
Comma list: 225/224, 5764801/5668704
 
POTE generator: ~48/35 = 556.221
 
Mapping: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
 
Wedgie: {{wedgie| 16 -5 22 -45 -10 65 }}
 
{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
 
Badness: 0.1293
 
== 11-limit ==
Comma list: 225/224, 385/384, 12250/11979


POTE generator: ~11/8 = 556.144
Optimal tunings:  
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


{{Val list|legend=1| 41, 315cde }}
Badness (Sintel): 5.74


Badness: 0.0735
== References ==


[[Category:Theory]]
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Temperament]]
[[Category:Temperament collections]]
[[Category:Marvel]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"