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


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


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
[[Subgroup]]: 2.3.5.7


Scales (Scala files):  
[[Comma list]]: 225/224, 50421/50000


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
<div style="line-height:1.6;">Technical data</div>
: mapping generators: ~2, ~10/7
<div class="mw-collapsible-content">


Subgroup: 2.3.5
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
: [[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 }}


[[Comma list]]: 16875/16384
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
[[Badness]] (Sintel): 1.20


[[Wedgie]]: {{wedgie| 4 -3 -14 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
Comma list: 121/120, 225/224, 441/440


</div></div>
{{Mapping|legend=0| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


== 7-limit ==
Optimal tunings:
Period: 1\1
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


Optimal ([[POTE]]) generator: ~15/14 = 125.608
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
Badness (Sintel): 0.782


Scales (Scala files):  
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 105/104, 121/120, 196/195, 275/273
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
{{Mapping|legend=0| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


[[Comma list]]: 49/48, 225/224
Optimal tunings:  
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


[[Wedgie]]: {{wedgie| 4 -3 2 -14 -8 13 }}
Badness (Sintel): 0.950


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


</div></div>
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


=== Negra ===
{{Mapping|legend=0| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
This is the 2.3.5.7.13 extension of negri.


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


Optimal (POTE) generator: ~14/13 = 125.567
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
Badness (Sintel): 0.973


Scales (Scala files):  
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.13
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


[[Comma list]]: 49/48, 65/64, 91/90
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


[[Mapping]]: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -3 }}]
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


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


Gencom: [2 14/13; 49/48 65/64 91/90]
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


</div></div>
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


== 11-limit ==
Optimal tunings:
Period: 1\1
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


Optimal ([[POTE]]) generator: ~15/14 = 126.474
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
Badness (Sintel): 1.04
 
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 }}]
{{Mapping|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


{{Val list|legend=1| 9, 10, 19 }}
Optimal tunings:
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


[[Badness]]: 0.0262
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


</div></div>
Badness (Sintel): 1.50


=== 13-limit ===
== Septimin ==
Period: 1\1
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


Optimal ([[POTE]]) generator: ~14/13 = 126.431
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.  


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
[[Subgroup]]: 2.3.5.7


Scales (Scala files):  
[[Comma list]]: 225/224, 84035/82944


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
<div style="line-height:1.6;">Technical data</div>
: mapping generators: ~2, ~12/7
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
Comma list: 45/44, 49/48, 56/55, 78/77
: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}
[[POTE generator]]: ~14/13 = 126.431
: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}


Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}


{{Val list|legend=1| 9, 10, 19 }}
[[Badness]] (Sintel): 1.38
 
</div></div>
 
== Negril ==
Period: 1\1
 
Optimal ([[POTE]]) generator: ~15/14 = 124.767
 
EDO generators: [[19edo|2\19]], [[29edo|3\29]]
 
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: 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 ===
Period: 1\1
Subgroup: 2.3.5.7.11.13


Optimal ([[POTE]]) generator: ~14/13 = 124.716
Comma list: 105/104, 144/143, 196/195, 245/242


EDO generators: [[19edo|2\19]], [[29edo|3\29]]
{{Mapping|legend=0| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Badness (Sintel): 0.955


Comma list: 49/48, 65/64, 91/90, 875/858
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


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


{{Val list|legend=1| 19, 29, 48df, 77cddf }}
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.


Badness: 0.0244
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 2500000/2470629


== Negric ==
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
Period: 1\1
: mapping generators: ~2, ~10/7


Optimal ([[POTE]]) generator: ~15/14 = 127.039
[[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 }}


EDO generators: [[9edo|1\9]], [[19edo|2\19]]
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


Scales (Scala files):
[[Badness]] (Sintel): 1.39
 
<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… }}


Badness: 0.0306
{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


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


=== 13-limit ===
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
Period: 1\1


Optimal ([[POTE]]) generator: ~14/13 = 127.039
Badness (Sintel): 1.20
 
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 ==
Period: 1\1


Optimal ([[POTE]]) generator: ~15/14 = 124.539
{{Mapping|legend=0| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


EDO generators: [[10edo|1\10]], [[19edo|2\19]], [[29edo|3\29]]
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.14
<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 ====
Period: 1\1
Subgroup: 2.3.5.7.11.13
 
Optimal ([[POTE]]) generator: ~14/13 = 124.545


EDO generators: [[10edo|1\10]], [[19edo|2\19]], [[29edo|3\29]]
Comma list: 225/224, 325/324, 385/384, 10648/10647


Scales (Scala files):
{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


<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 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


Comma list: 49/48, 55/54, 65/64, 91/90
Badness (Sintel): 1.47


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


{{Val list|legend=1| 10, 19e, 29, 77cddeef }}
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.


Badness: 0.0216
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 589824/588245


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


Optimal ([[POTE]]) generator: ~11/8 = 537.186
[[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 }}


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Scales (Scala files):
[[Badness]] (Sintel): 1.44
 
<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: 49/48, 121/120, 225/224
Comma list: 225/224, 385/384, 1331/1323


Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]
{{Mapping|legend=0| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


{{Val list|legend=1| 9, 20, 29, 38d, 67cdde }}
Optimal tunings:
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


Badness: 0.0419
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


</div></div>
Badness (Sintel): 0.838


=== 13-limit ===
=== 13-limit ===
Period: 1\1
Optimal ([[POTE]]) generator: ~11/8 = 537.208
EDO generators: [[29edo|13\29]], [[38edo|17\38]]
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
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 65/64, 91/90, 121/120
Comma list: 225/224, 275/273, 385/384, 1331/1323


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


{{Val list|legend=1| 9, 20, 29, 38df, 67cddef }}
Optimal tunings:
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Badness: 0.0252
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


</div></div>
Badness (Sintel): 1.07


=== 17-limit ===
== Triton ==
Period: 1\1
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


Optimal ([[POTE]]) generator: ~11/8 = 537.230
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.  


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
[[Subgroup]]: 2.3.5.7


Scales (Scala files):  
[[Comma list]]: 225/224, 1029/1000


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
<div style="line-height:1.6;">Technical data</div>
: mapping generators: ~2, ~10/7
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17
[[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 }}


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
[[Badness]] (Sintel): 1.50


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefg }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0218
Comma list: 45/44, 56/55, 1029/1000


</div></div>
{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


=== 19-limit ===
Optimal tunings:
Period: 1\1
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


Optimal ([[POTE]]) generator: ~11/8 = 537.214
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
Badness (Sintel): 1.51


Scales (Scala files):  
== Marvolo ==
[[Subgroup]]: 2.3.5.7


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 225/224, 156250000/155649627
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17.19
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
[[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 }}


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefgh }}
[[Badness]] (Sintel): 2.11


Badness: 0.0168
=== 11-limit ===
Subgroup: 2.3.5.7.11


</div></div>
Comma list: 225/224, 441/440, 4000/3993


= Passive =
{{Mapping|legend=0| 1 2 1 1 2 | 0 -6 19 26 21 }}
{{see also | Archytas clan #Passion }}


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


Optimal ([[POTE]]) generator: ~16/15 = 98.809
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


EDO generators: [[12edo|1\12]], [[13edo|1\13]]
Badness (Sintel): 0.958


Scales (Scala files):  
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 169/168, 225/224, 364/363, 441/440
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Comma list: 225/224, 256/245
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


{{Val list|legend=1| 12, 49dd }}
Badness (Sintel): 0.887


Badness: 0.0751
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


</div></div>
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


= Wizard =
{{Mapping|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Period: 1\2
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Optimal ([[POTE]]) generator: ~5/4 = 383.256
Badness (Sintel): 0.760


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Scales (Scala files): [[wizard22]]
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=0| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


Comma list: 225/224, 118098/117649
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Mapping: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
Badness (Sintel): 0.895


Mapping generators: ~1225/864, ~245/216
== Decic ==
{{Main| Decic }}


Wedgie: {{wedgie| 12 -2 20 -31 -2 52 }}
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]].


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
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.


Badness: 0.0408
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 16807/16384
 
== 11-limit ==


Period: 1\2
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


Optimal ([[POTE]]) generator: ~5/4 = 383.232
[[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 }}


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


Scales (Scala files): [[wizard22]]
[[Badness]] (Sintel): 2.26
 
<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: 225/224, 385/384, 4000/3993
Comma list: 225/224, 385/384, 3087/3025


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
{{Mapping|legend=0| 10 0 39 28 3 | 0 1 -1 0 2 }}


Mapping generators: ~99/70, ~25/22
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}})


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Badness: 0.0185
Badness (Sintel): 2.11
 
</div></div>
 
=== Lizard ===
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~5/4 = 383.389
 
EDO generators: [[50edo|16\50]], [[72edo|23\72]]
 
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: 225/224, 351/350, 364/363, 385/384
Comma list: 105/104, 144/143, 196/195, 2200/2197
 
Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]


Mapping generators: ~99/70, ~25/22
{{Mapping|legend=0| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


{{Val list|legend=1| 22, 50, 72, 122, 194df }}
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}})


Badness: 0.0218
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


</div></div>
Badness (Sintel): 1.52


==== 17-limit ====
==== 17-limit ====
Period: 1\2
Optimal ([[POTE]]) generator: ~5/4 = 383.381
EDO generators: [[50edo|16\50]], [[72edo|23\72]]
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: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
{{Mapping|legend=0| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


Mapping generators: ~17/12, ~17/15
Optimal tunings:
* WE: ~15/14 = 120.1262{{c}}, ~3/2 = 697.8185{{c}} (~78/77 = 22.9388{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.2757{{c}} (~78/77 = 22.7243{{c}})


{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Badness: 0.0145
Badness (Sintel): 1.28


</div></div>
=== 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''.


==== 19-limit ====
Subgroup: 2.3.5.7.11
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~5/4 = 383.477
 
EDO generators: [[50edo|16\50]], [[72edo|23\72]]
 
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: 2.3.5.7.11.13.17.19


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Comma list: 225/224, 1617/1600, 2401/2376


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
{{Mapping|legend=0| 10 0 39 28 82 | 0 1 -1 0 -3 }}


Mapping generators: ~17/12, ~17/15
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}})


{{Val list|legend=1| 22h, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


Badness: 0.0157
Badness (Sintel): 1.98
 
</div></div>
 
=== Gizzard ===
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~5/4 = 383.170
 
EDO generators: [[72edo|23\72]], [[94edo|30\94]]
 
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: 225/224, 325/324, 385/384, 1573/1568
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
{{Mapping|legend=0| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


Mapping generators: ~99/70, ~25/22
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}})


{{Val list|legend=1| 72, 166, 238cf }}
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


Badness: 0.0203
Badness (Sintel): 1.57
 
</div></div>


==== 17-limit ====
==== 17-limit ====
Period: 1\2
Optimal ([[POTE]]) generator: ~5/4 = 383.175
EDO generators: [[72edo|23\72]], [[94edo|30\94]]
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: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374
 
Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
 
Mapping generators: ~17/12, ~17/15
 
{{Val list|legend=1| 72, 166g, 238cfg }}
 
Badness: 0.0136
 
</div></div>


==== 19-limit ====
{{Mapping|legend=0| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


Period: 1\2
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}})


Optimal ([[POTE]]) generator: ~5/4 = 383.138
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


EDO generators: [[72edo|23\72]], [[94edo|30\94]]
Badness (Sintel): 1.33


Scales (Scala files):
=== 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''.
<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: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
 
Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
 
Mapping generators: ~17/12, ~17/15
 
{{Val list|legend=1| 72, 94, 166g }}
 
Badness: 0.0148
 
</div></div>
 
== Mage ==
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~5/4 = 383.124
 
EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
 
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
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 1331/1296
Comma list: 225/224, 441/440, 5929/5832


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
{{Mapping|legend=0| 10 0 39 28 -13 | 0 1 -1 0 3 }}


Mapping generators: ~77/54, ~55/48
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}})


{{Val list|legend=1| 22, 50e, 72ee, 94ee }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness: 0.0578
Badness (Sintel): 2.20


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


= Triton =
Comma list: 105/104, 196/195, 275/273, 5929/5832


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''
{{Mapping|legend=0| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


Period: 1\1
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}})


Optimal ([[POTE]]) generator: ~7/5 = 568.865
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


EDO generators: [[19edo|9\19]], [[21edo|10\21]]
Badness (Sintel): 1.73


Scales (Scala files):
==== 17-limit ====
 
Subgroup: 2.3.5.7.11.13.17
<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
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


Comma list: 225/224, 1029/1000
{{Mapping|legend=0| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


Mapping: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
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}})


Wedgie: {{wedgie| 3 -7 -8 -18 -21 1 }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


{{Val list|legend=1| 19, 78bd, 97bd }}
Badness (Sintel): 1.41


Badness: 0.0592
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 40353607/40310784


== 11-limit ==
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


Period: 1\1
[[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 }}


Optimal ([[POTE]]) generator: ~7/5 = 569.144
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


EDO generators: [[19edo|9\19]], [[21edo|10\21]]
[[Badness]] (Sintel): 2.37
 
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: 45/44, 56/55, 1029/1000
Comma list: 225/224, 385/384, 12005/11979
 
Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
 
{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
 
Badness: 0.0457
 
</div></div>
 
= Tritonic =
 
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/5 = 580.286
 
EDO generators: [[29edo|14\29]], [[31edo|15\31]], [[60edo|29\60]]


Scales (Scala files):
{{Mapping|legend=0| 9 0 28 11 24 | 0 2 -1 2 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: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}})
<div class="mw-collapsible-content">
* CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}})


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Comma list: 225/224, 50421/50000
Badness (Sintel): 1.01
 
Mapping: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
 
Wedgie: {{wedgie| 5 -11 -12 -29 -33 3 }}
 
{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/5 = 580.267
 
EDO generators: [[29edo|14\29]], [[31edo|15\31]], [[60edo|29\60]]
 
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
 
Comma list: 121/120, 225/224, 441/440
 
Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
 
{{Val list|legend=1| 29, 31, 60e }}
 
Badness: 0.0237
 
</div></div>


=== 13-limit ===
=== 13-limit ===
Period: 1\1
Optimal ([[POTE]]) generator: ~7/5 = 580.108
EDO generators: [[29edo|14\29]], [[31edo|15\31]], [[60edo|29\60]]
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
Subgroup: 2.3.5.7.11.13


Comma list: 105/104, 121/120, 196/195, 275/273
Comma list: 169/168, 225/224, 364/363, 1716/1715
 
Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
 
{{Val list|legend=1| 29, 31, 60e, 151cde }}
 
Badness: 0.0230
 
</div></div>
 
== Tritoni ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/5 = 580.389
 
EDO generators: [[31edo|15\31]], [[91edo|44\91]], [[122edo|59\122]]
 
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
 
Comma list: 225/224, 385/384, 27783/27500
 
Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
 
{{Val list|legend=1| 31, 91, 122, 153d }}
 
Badness: 0.0455
 
</div></div>
 
= Merman =
 
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/5 = 585.585
 
EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
 
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


Comma list: 225/224, 2500000/2470629
{{Mapping|legend=0| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


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


Wedgie: {{wedgie| 7 -15 -16 -40 -45 5 }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


{{Val list|legend=1| 41, 84, 125 }}
Badness (Sintel): 0.922


Badness: 0.0551
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


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


== 11-limit ==
[[Comma list]]: 225/224, 781250000/771895089


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


Optimal ([[POTE]]) generator: ~7/5 = 585.606
[[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 }}


EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


Scales (Scala files):  
[[Badness]] (Sintel): 2.44
 
<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: 225/224, 441/440, 1344/1331
Comma list: 225/224, 385/384, 236328125/234365481


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
{{Mapping|legend=0| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


{{Val list|legend=1| 41, 84, 125e }}
Optimal tunings:
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Badness: 0.0364
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


</div></div>
Badness (Sintel): 2.96
 
== 13-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/5 = 585.657
 
EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
 
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: 144/143, 225/224, 364/363, 441/440
Comma list: 225/224, 325/324, 385/384, 831875/830466


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
{{Mapping|legend=0| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
Optimal tunings:
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


Badness: 0.0275
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


</div></div>
Badness (Sintel): 2.16


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


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


Period: 1\1
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


Optimal ([[POTE]]) generator: ~7/6 = 263.632
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Scales (Scala files):  
Badness (Sintel): 1.96


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
==== 19-limit ====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13.17.19
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7
 
Comma list: 225/224, 84035/82944
 
Mapping: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
 
Wedgie: {{wedgie| 11 -6 10 -35 -15 40 }}
 
{{Val list|legend=1| 41, 91, 132 }}


</div></div>
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


== 11-limit ==
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


Optimal ([[POTE]]) generator: ~7/6 = 263.634
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
Badness (Sintel): 1.71


Scales (Scala files):  
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Comma list: 225/224, 385/384, 2401/2376
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


{{Val list|legend=1| 41, 91, 223cdef }}
Badness (Sintel): 1.57


</div></div>
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


== 13-limit ==
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


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


Optimal ([[POTE]]) generator: ~7/6 = 263.700
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Scales (Scala files):  
Badness (Sintel): 1.50


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
==== 31-limit ====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


Comma list: 105/104, 144/143, 196/195, 245/242
{{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 }}


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


{{Val list|legend=1| 41, 91 }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


</div></div>
Badness (Sintel): 1.53
 
= Slender =
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~49/48 = 38.413
 
EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
 
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
 
Comma list: 225/224, 589824/588245
 
Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
 
Wedgie: {{wedgie| 13 -10 6 -46 -27 42 }}
 
{{Val list|legend=1| 31, 94, 125 }}
 
Badness: 0.0569
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~49/48 = 38.387
 
EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
 
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">


=== Gracecord ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 1331/1323
Comma list: 225/224, 441/440, 109375/107811


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
{{Mapping|legend=0| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


{{Val list|legend=1| 31, 63, 94, 125 }}
Optimal tunings:
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


Badness: 0.02534
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


</div></div>
Badness (Sintel): 2.21
 
== 13-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~49/48 = 38.314
 
EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
 
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: 225/224, 275/273, 385/384, 1331/1323
Comma list: 225/224, 364/363, 441/440, 6125/6084


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
{{Mapping|legend=0| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


{{Val list|legend=1| 31, 63, 94 }}
Optimal tunings:
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


Badness: 0.02591
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


</div></div>
Badness (Sintel): 1.83


= Marvo =
==== 17-limit ====
{{see also| Gravity family }}
Subgroup: 2.3.5.7.11.13.17


Period: 1\1
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Optimal ([[POTE]]) generator: ~27/20 = 516.694
{{Mapping|legend=0| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


EDO generators: [[65edo|28\65]], [[72edo|31\72]], [[137edo|59\137]]
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.87
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
==== 19-limit ====
 
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 225/224, 78125000/78121827


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


Wedgie: {{wedgie| 6 17 46 13 56 59 }}
{{Mapping|legend=0| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


{{Val list|legend=1| 65d, 72, 137, 209, 281, 569bcc }}
Optimal tunings:
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


Badness: 0.0976
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


</div></div>
Badness (Sintel): 1.68


== 11-limit ==
== Alphorn ==
[[Subgroup]]: 2.3.5.7


Period: 1\1
[[Comma list]]: 225/224, 5764801/5668704


Optimal ([[POTE]]) generator: ~27/20 = 516.699
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


EDO generators: [[65edo|28\65]], [[72edo|31\72]], [[137edo|59\137]]
[[Optimal tuning]]s:  
* [[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 3.27
<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: 225/224, 243/242, 4000/3993
Comma list: 225/224, 385/384, 12250/11979


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
{{Mapping|legend=0| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


{{Val list|legend=1| 65d, 72, 281, 353c, 425bc, 497bc }}
Optimal tunings:
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Badness: 0.0317
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


</div></div>
Badness (Sintel): 2.43


== 13-limit ==
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


Period: 1\1
[[Subgroup]]: 2.3.5.7


Optimal ([[POTE]]) generator: ~27/20 = 516.730
[[Comma list]]: 225/224, 4194304/4117715


EDO generators: [[65edo|28\65]], [[72edo|31\72]], [[137edo|59\137]]
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


Scales (Scala files):  
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 21, 32, 53 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
[[Badness]] (Sintel): 3.57
 
Comma list: 225/224, 243/242, 351/350, 1625/1617
 
Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
 
{{Val list|legend=1| 65d, 72, 137, 209, 281f, 490bcf }}
 
Badness: 0.0269
 
</div></div>
 
= Marvolo =
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~21/20 = 83.348
 
EDO generators: [[29edo|2\29]], [[43edo|3\43]], [[72edo|5\72]]
 
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
 
Comma list: 225/224, 156250000/155649627
 
Mapping: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
 
Wedgie: {{wedgie| 6 -19 -26 -44 -58 -7 }}
 
{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
 
Badness: 0.0833
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~21/20 = 83.340
 
EDO generators: [[29edo|2\29]], [[43edo|3\43]], [[72edo|5\72]]
 
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: 225/224, 441/440, 4000/3993
Comma list: 99/98, 176/175, 1310720/1294139


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
{{Mapping|legend=0| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


{{Val list|legend=1| 29, 43, 72 }}
Optimal tunings:
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


Badness: 0.0290
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


</div></div>
Badness (Sintel): 2.82
 
== 13-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~21/20 = 83.330
 
EDO generators: [[29edo|2\29]], [[43edo|3\43]], [[72edo|5\72]]
 
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: 169/168, 225/224, 364/363, 441/440
Comma list: 99/98, 176/175, 640/637, 847/845


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
{{Mapping|legend=0| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


{{Val list|legend=1| 29, 43, 72, 115f }}
Optimal tunings:
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


Badness: 0.0215
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


</div></div>
Badness (Sintel): 1.88


= Amavil =
=== Musneb ===
== 5-limit (mabila) ==
Subgroup: 2.3.5.7.11
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~512/375 = 529.6849
 
EDO generators: [[34edo|15\34]], [[43edo|19\43]], [[77edo|34\77]], [[111edo|49\111]]
 
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
 
Comma list: 268435456/263671875
 
Mapping: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
 
{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
 
Badness: 0.2325
 
</div></div>
 
== 7-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~48/35 = 529.979
 
EDO generators: [[34edo|15\34]], [[43edo|19\43]]
 
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
Comma list: 225/224, 385/384, 66550/64827


Comma list: 225/224, 17496/16807
{{Mapping|legend=0| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


Mapping: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


Wedgie: {{wedgie| 10 -3 14 -28 -6 41 }}
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
Badness (Sintel): 2.89


Badness: 0.1096
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


</div></div>
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.


== 11-limit ==
[[Subgroup]]: 2.3.5.7


Period: 1\1
[[Comma list]]: 225/224, 125000000/121060821


Optimal ([[POTE]]) generator: ~15/11 = 529.974
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


EDO generators: [[34edo|15\34]], [[43edo|19\43]]
[[Optimal tuning]]s:  
* [[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 51, 53 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 3.64
<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: 99/98, 176/175, 864/847
Comma list: 121/120, 225/224, 22000/21609


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
{{Mapping|legend=0| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


{{Val list|legend=1| 9, 34d, 43, 77de }}
Optimal tunings:
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


Badness: 0.0426
{{Optimal ET sequence|legend=0| 51, 53 }}


</div></div>
Badness (Sintel): 2.46
 
== 13-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~15/11 = 529.951
 
EDO generators: [[34edo|15\34]], [[43edo|19\43]]
 
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: 78/77, 99/98, 144/143, 176/175
Comma list: 121/120, 225/224, 275/273, 1040/1029
 
Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]


{{Val list|legend=1| 9, 34d, 43, 77de }}
{{Mapping|legend=0| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


Badness: 0.0258
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 51f, 53 }}


= Enneaportent =
Badness (Sintel): 1.96


Period: 1\9
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


Optimal ([[POTE]]) generator: ~5/4 = 383.165
[[Subgroup]]: 2.3.5.7


EDO generators: [[63edo|20\63]], [[72edo|23\72]], [[81edo|26\81]]
[[Comma list]]: 225/224, 823543/800000


Scales (Scala files):  
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


<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 = 1202.1198{{c}}, ~32/21 = 745.4675{{c}}
<div class="mw-collapsible-content">
: [[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 }}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


Comma list: 225/224, 40353607/40310784
[[Badness]] (Sintel): 3.67
 
Mapping: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
 
Wedgie: {{wedgie| 18 -9 18 -56 -22 67 }}
 
{{Val list|legend=1| 63, 72, 495bcd }}
 
Badness: 0.0937
 
</div></div>
 
== 11-limit ==
 
Period: 1\9
 
Optimal ([[POTE]]) generator: ~5/4 = 383.146
 
EDO generators: [[63edo|20\63]], [[72edo|23\72]], [[81edo|26\81]]
 
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: 225/224, 385/384, 12005/11979
Comma list: 225/224, 245/242, 1617/1600
 
Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]


{{Val list|legend=1| 63, 72, 423cd, 495bcd }}
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


Badness: 0.0304
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


== 13-limit ==
Badness (Sintel): 2.00
 
Period: 1\9
 
Optimal ([[POTE]]) generator: ~5/4 = 383.047
 
EDO generators: [[63edo|20\63]], [[72edo|23\72]], [[81edo|26\81]]
 
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: 169/168, 225/224, 364/363, 1716/1715
Comma list: 105/104, 196/195, 245/242, 1001/1000


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


{{Val list|legend=1| 63, 72, 279cf }}
Optimal tunings:
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


Badness: 0.0223
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


</div></div>
Badness (Sintel): 1.43


= Submajor =
=== 17-limit ===
== 5-limit ==
Subgroup: 2.3.5.7.11.13.17
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~10125/8192 = 362.321
 
EDO generators: [[43edo|13\43]], [[53edo|16\53]], [[63edo|19\63]], [[96edo|29\96]], [[116edo|35\116]]
 
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
 
Comma list: 69198046875/68719476736
 
Mapping: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
 
{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}


Badness: 0.1302
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


</div></div>
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


== 7-limit ==
Optimal tunings:
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


Period: 1\1
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Optimal ([[POTE]]) generator: ~49/40 = 362.255
Badness (Sintel): 1.26


EDO generators: [[43edo|13\43]], [[53edo|16\53]], [[63edo|19\63]], [[96edo|29\96]], [[116edo|35\116]]
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


Scales (Scala files):  
[[Subgroup]]: 2.3.5.7


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 225/224, 9805926501/9765625000
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


Comma list: 225/224, 51200/50421
[[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 4 -1 1 }}, {{val| 0 -8 11 6 }}]
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


Wedgie: {{wedgie| 8 -11 -6 -36 -32 17 }}
[[Badness]] (Sintel): 3.81
 
{{Val list|legend=1| 10, 33, 43, 53 }}
 
Badness: 0.0605
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~27/22 = 362.101
 
EDO generators: [[53edo|16\53]], [[63edo|19\63]], [[116edo|35\116]]
 
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: 225/224, 385/384, 6655/6561
Comma list: 225/224, 441/440, 43923/43750


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
{{Mapping|legend=0| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


{{Val list|legend=1| 53, 116, 169de, 285cde }}
Optimal tunings:
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Badness: 0.0506
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


</div></div>
Badness (Sintel): 1.74


=== 13-limit ===
=== 13-limit ===
Period: 1\1
Optimal ([[POTE]]) generator: ~16/13 = 362.105
EDO generators: [[53edo|16\53]], [[63edo|19\63]], [[116edo|35\116]]
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
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 225/224, 275/273, 385/384
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
{{Mapping|legend=0| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


{{Val list|legend=1| 53, 116, 169de, 285cdef }}
Optimal tunings:
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


Badness: 0.0277
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


</div></div>
Badness (Sintel): 1.35


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


Period: 1\1
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Optimal ([[POTE]]) generator: ~49/40 = 362.418
{{Mapping|legend=0| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


EDO generators: [[43edo|13\43]], [[53edo|16\53]], [[96edo|29\96]]
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.17
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


Comma list: 99/98, 176/175, 51200/50421
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.


POTE generator: ~49/40 = 362.418
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
[[Comma list]]: 225/224, 122880/117649


{{Val list|legend=1| 43, 53, 96, 149d }}
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


Badness: 0.0518
[[Optimal tuning]]s:  
* [[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 }}


</div></div>
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}


=== 13-limit ===
[[Badness]] (Sintel): 4.37


Period: 1\1
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
Optimal ([[POTE]]) generator: ~16/13 = 362.402
 
EDO generators: [[43edo|13\43]], [[53edo|16\53]], [[96edo|29\96]]
 
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: 99/98, 169/168, 176/175, 640/637
 
POTE generator: ~16/13 = 362.402
 
Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
 
{{Val list|legend=1| 43, 53, 96, 149d }}
 
Badness: 0.0297
 
</div></div>
 
= Alphorn =
Period: 1\1
 
Optimal ([[POTE]]) generator: ~48/35 = 556.221
 
EDO generators: [[28edo|13\28]], [[41edo|19\41]]
 
Scales (Scala files):


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 121/120, 225/224, 352/343
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
{{Mapping|legend=0| 11 0 43 31 38 | 0 1 -1 0 0 }}


Comma list: 225/224, 5764801/5668704
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 9 0 13 }}, {{val| 0 -16 5 -22 }}]
{{Optimal ET sequence|legend=0| 22, 55d }}


Wedgie: {{wedgie| 16 -5 22 -45 -10 65 }}
Badness (Sintel): 2.46


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Badness: 0.1293
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.  


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


== 11-limit ==
[[Comma list]]: 225/224, 5971968/5764801
Period: 1\1


Optimal ([[POTE]]) generator: ~11/8 = 556.144
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


EDO generators: [[28edo|13\28]], [[41edo|19\41]]
[[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 4.53
<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: 225/224, 385/384, 12250/11979
Comma list: 99/98, 176/175, 65536/65219
 
Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
 
{{Val list|legend=1| 28d, 41, 315cde }}


Badness: 0.0735
{{Mapping|legend=0| 2 1 6 4 8 | 0 8 -5 6 -4 }}
 
</div></div>
 
= Decic =
The ''decic'' temperament (10&amp;50, named by [[User:Xenllium|Xenllium]]) tempers out the [[cloudy comma]], 16807/16384 and the [[225/224|marvel comma]], 225/224 in the 7-limit. It can be extended naturally to the 11-, 13-, and 17-limit by adding 385/384, 105/104, and 170/169 to the comma list in this order.
 
== 7-limit ==
Period: 1\10
 
Optimal ([[POTE]]) generator: ~3/2 = 698.696
 
EDO generators: [[50edo|1\50]], [[60edo|1\60]]
 
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
 
[[Comma list]]: 225/224, 16807/16384


[[Mapping]]: [{{val| 10 0 39 28 }}, {{val| 0 1 -1 0 }}]
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


{{Multival|legend=1| 10 -10 0 -39 -28 28 }}
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


{{Val list|legend=1| 10, 50, 60, 110d }}
Badness (Sintel): 2.26


[[Badness]]: 0.089135
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


</div></div>
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>.


== 11-limit ==
[[Subgroup]]: 2.3.5.7
Period: 1\10


Optimal (POTE) generator: ~3/2 = 696.791
[[Comma list]]: 225/224, 14495514624/13841287201


EDO generators: [[50edo|1\50]], [[60edo|1\60]]
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


Scales (Scala files):  
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}
<div style="line-height:1.6;">Technical data</div>


<div class="mw-collapsible-content">
[[Badness]] (Sintel): 10.9


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


Comma list: 225/224, 385/384, 3087/3025
Comma list: 225/224, 3840/3773, 12005/11979
 
Mapping: [{{val| 10 0 39 28 3 }}, {{val| 0 1 -1 0 2 }}]
 
{{Val list|legend=1| 10, 50, 60 }}
 
Badness: 0.063900
 
</div></div>
 
== 13-limit ==
Period: 1\10
 
Optimal (POTE) generator: ~3/2 = 696.993
 
EDO generators: [[50edo|1\50]], [[60edo|1\60]]
 
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: 105/104, 144/143, 196/195, 2200/2197
 
Mapping: [{{val| 10 0 39 28 3 37 }}, {{val| 0 1 -1 0 2 0 }}]
 
{{Val list|legend=1| 10, 50, 60 }}


Badness: 0.036880
{{Mapping|legend=0| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
 
</div></div>
 
== 17-limit ==
Period: 1\10
 
Optimal (POTE) generator: ~3/2 = 697.085
 
EDO generators: [[50edo|1\50]], [[60edo|1\60]]
 
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: 2.3.5.7.11.13.17
 
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


Mapping: [{{val| 10 0 39 28 3 37 25 }}, {{val| 0 1 -1 0 2 0 1 }}]
Optimal tunings:  
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}


{{Val list|legend=1| 10, 50, 60 }}
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


Badness: 0.025064
Badness (Sintel): 5.74


</div></div>
== 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"