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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


Comma list: 49/48, 65/64, 91/90, 875/858
Badness (Sintel): 0.955


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


{{Val list|legend=1| 19, 29, 48df, 77cddf }}
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.


Badness: 0.0244
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.  


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


== Negric ==
[[Comma list]]: 225/224, 2500000/2470629
Period: 1\1


Optimal ([[POTE]]) generator: ~15/14 = 127.039
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


EDO generators: [[9edo|1\9]], [[19edo|2\19]]
[[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 1.39
<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
 
</div></div>
 
=== 13-limit ===
Period: 1\1


Optimal ([[POTE]]) generator: ~14/13 = 127.039
{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


EDO generators: [[9edo|1\9]], [[19edo|2\19]]
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.20
<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
{{Mapping|legend=0| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


== Negroni ==
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
Period: 1\1


Optimal ([[POTE]]) generator: ~15/14 = 124.539
Badness (Sintel): 1.14
 
EDO generators: [[10edo|1\10]], [[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">


=== 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]]


Scales (Scala files):  
Comma list: 225/224, 325/324, 385/384, 10648/10647


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Optimal tunings:  
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


Comma list: 49/48, 55/54, 65/64, 91/90
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Badness (Sintel): 1.47


{{Val list|legend=1| 10, 19e, 29, 77cddeef }}
== 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.


Badness: 0.0216
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  


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


== Wilsec ==
[[Comma list]]: 225/224, 589824/588245
Period: 1\1


Optimal ([[POTE]]) generator: ~11/8 = 537.186
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


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


Scales (Scala files):
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 1.44
<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 ==
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


Period: 1\2
[[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 }}


Optimal ([[POTE]]) generator: ~5/4 = 383.232
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
[[Badness]] (Sintel): 2.26
 
Scales (Scala files): [[wizard22]]
 
<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|legend=0| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


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


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


Badness: 0.0218
Badness (Sintel): 1.52
 
</div></div>


==== 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


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
Comma list: 225/224, 1617/1600, 2401/2376


Mapping generators: ~17/12, ~17/15
{{Mapping|legend=0| 10 0 39 28 82 | 0 1 -1 0 -3 }}


{{Val list|legend=1| 22h, 50, 72, 122g, 194dfg }}
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}})


Badness: 0.0157
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


</div></div>
Badness (Sintel): 1.98
 
=== 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 generators: ~99/70, ~25/22
{{Mapping|legend=0| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


{{Val list|legend=1| 72, 166, 238cf }}
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}})


Badness: 0.0203
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


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


==== 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 ====


Period: 1\2
{{Mapping|legend=0| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


Optimal ([[POTE]]) generator: ~5/4 = 383.138
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}})


EDO generators: [[72edo|23\72]], [[94edo|30\94]]
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


Scales (Scala files):  
Badness (Sintel): 1.33


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
=== Prodecic ===
<div style="line-height:1.6;">Technical data</div>
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="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: 225/224, 1029/1000


Mapping: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


Wedgie: {{wedgie| 3 -7 -8 -18 -21 1 }}
{{Mapping|legend=0| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


{{Val list|legend=1| 19, 78bd, 97bd }}
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}})


Badness: 0.0592
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


</div></div>
Badness (Sintel): 1.41


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


Period: 1\1
[[Comma list]]: 225/224, 40353607/40310784


Optimal ([[POTE]]) generator: ~7/5 = 569.144
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


EDO generators: [[19edo|9\19]], [[21edo|10\21]]
[[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 2.37
<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]].''
{{Mapping|legend=0| 9 0 28 11 24 | 0 2 -1 2 1 }}


Period: 1\1
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}})


Optimal ([[POTE]]) generator: ~7/5 = 580.286
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


EDO generators: [[29edo|14\29]], [[31edo|15\31]], [[60edo|29\60]]
Badness (Sintel): 1.01
 
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, 50421/50000
 
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 }}]
{{Mapping|legend=0| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


{{Val list|legend=1| 29, 31, 60e, 151cde }}
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}})


Badness: 0.0230
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


</div></div>
Badness (Sintel): 0.922


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


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


Optimal ([[POTE]]) generator: ~7/5 = 580.389
[[Comma list]]: 225/224, 781250000/771895089


EDO generators: [[31edo|15\31]], [[91edo|44\91]], [[122edo|59\122]]
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


Scales (Scala files):  
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
[[Badness]] (Sintel): 2.44
 
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: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
 
Wedgie: {{wedgie| 7 -15 -16 -40 -45 5 }}
 
{{Val list|legend=1| 41, 84, 125 }}
 
Badness: 0.0551
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/5 = 585.606
 
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">


=== 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 }}]
 
{{Val list|legend=1| 41, 84, 125e }}


Badness: 0.0364
{{Mapping|legend=0| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Period: 1\1
Badness (Sintel): 2.96
 
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, 273/272, 324/323, 325/324, 385/384, 1445/1444


Comma list: 225/224, 84035/82944
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


Mapping: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


Wedgie: {{wedgie| 11 -6 10 -35 -15 40 }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


{{Val list|legend=1| 41, 91, 132 }}
Badness (Sintel): 1.71


</div></div>
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


== 11-limit ==
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


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


Optimal ([[POTE]]) generator: ~7/6 = 263.634
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


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


Scales (Scala files):  
Badness (Sintel): 1.57


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
==== 29-limit ====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13.17.19.23.29
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Comma list: 225/224, 385/384, 2401/2376
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


{{Val list|legend=1| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


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


== 13-limit ==
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Period: 1\1
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


Optimal ([[POTE]]) generator: ~7/6 = 263.700
{{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 }}


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.53
<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, 245/242
 
Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
 
{{Val list|legend=1| 41, 91 }}
 
</div></div>
 
= 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 }}]
 
{{Val list|legend=1| 31, 63, 94, 125 }}
 
Badness: 0.02534


</div></div>
{{Mapping|legend=0| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


== 13-limit ==
Optimal tunings:
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


Period: 1\1
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Optimal ([[POTE]]) generator: ~49/48 = 38.314
Badness (Sintel): 2.21
 
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 }}]


{{Val list|legend=1| 31, 63, 94 }}
{{Mapping|legend=0| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Badness: 0.02591
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


= Marvo =
Badness (Sintel): 1.83
{{see also| Gravity family }}


Period: 1\1
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Optimal ([[POTE]]) generator: ~27/20 = 516.694
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


EDO generators: [[65edo|28\65]], [[72edo|31\72]], [[137edo|59\137]]
{{Mapping|legend=0| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
Badness (Sintel): 1.87


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


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
Comma list: 225/224, 385/384, 66550/64827


Optimal ([[POTE]]) generator: ~512/375 = 529.6849
{{Mapping|legend=0| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


EDO generators: [[34edo|15\34]], [[43edo|19\43]], [[77edo|34\77]], [[111edo|49\111]]
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 2.89
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


Comma list: 268435456/263671875
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 6 1 }}, {{val| 0 -10 3 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
[[Comma list]]: 225/224, 125000000/121060821


Badness: 0.2325
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


</div></div>
[[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 }}


== 7-limit ==
{{Optimal ET sequence|legend=1| 51, 53 }}


Period: 1\1
[[Badness]] (Sintel): 3.64
 
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, 17496/16807
 
Mapping: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
 
Wedgie: {{wedgie| 10 -3 14 -28 -6 41 }}
 
{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
 
Badness: 0.1096
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~15/11 = 529.974
 
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">


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


{{Val list|legend=1| 9, 34d, 43, 77de }}
Optimal tunings:
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


Badness: 0.0258
{{Optimal ET sequence|legend=0| 51f, 53 }}


</div></div>
Badness (Sintel): 1.96


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


Period: 1\9
[[Subgroup]]: 2.3.5.7


Optimal ([[POTE]]) generator: ~5/4 = 383.165
[[Comma list]]: 225/224, 823543/800000


EDO generators: [[63edo|20\63]], [[72edo|23\72]], [[81edo|26\81]]
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


Scales (Scala files):  
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 3.67
 
Comma list: 225/224, 40353607/40310784
 
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 }}]
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


{{Val list|legend=1| 63, 72, 423cd, 495bcd }}
Optimal tunings:
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


Badness: 0.0304
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


</div></div>
Badness (Sintel): 2.00
 
== 13-limit ==
 
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
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


Optimal ([[POTE]]) generator: ~10125/8192 = 362.321
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


EDO generators: [[43edo|13\43]], [[53edo|16\53]], [[63edo|19\63]], [[96edo|29\96]], [[116edo|35\116]]
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.26
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.  


Comma list: 69198046875/68719476736
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
[[Comma list]]: 225/224, 9805926501/9765625000


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


Badness: 0.1302
[[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 }}


</div></div>
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


== 7-limit ==
[[Badness]] (Sintel): 3.81
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~49/40 = 362.255
 
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: 2.3.5.7
 
Comma list: 225/224, 51200/50421
 
Mapping: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
 
Wedgie: {{wedgie| 8 -11 -6 -36 -32 17 }}
 
{{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


Optimal ([[POTE]]) generator: ~49/40 = 362.418
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


EDO generators: [[43edo|13\43]], [[53edo|16\53]], [[96edo|29\96]]
{{Mapping|legend=0| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Badness (Sintel): 1.17


Comma list: 99/98, 176/175, 51200/50421
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


POTE generator: ~49/40 = 362.418
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| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 43, 53, 96, 149d }}
[[Comma list]]: 225/224, 122880/117649


Badness: 0.0518
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


</div></div>
[[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 }}


=== 13-limit ===
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}


Period: 1\1
[[Badness]] (Sintel): 4.37


Optimal ([[POTE]]) generator: ~16/13 = 362.402
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
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
Comma list: 121/120, 225/224, 352/343


EDO generators: [[28edo|13\28]], [[41edo|19\41]]
{{Mapping|legend=0| 11 0 43 31 38 | 0 1 -1 0 0 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 22, 55d }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
Badness (Sintel): 2.46


Comma list: 225/224, 5764801/5668704
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Mapping: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


Wedgie: {{wedgie| 16 -5 22 -45 -10 65 }}
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
[[Comma list]]: 225/224, 5971968/5764801


Badness: 0.1293
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


</div></div>
[[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 }}


== 11-limit ==
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}
Period: 1\1


Optimal ([[POTE]]) generator: ~11/8 = 556.144
[[Badness]] (Sintel): 4.53
 
EDO generators: [[28edo|13\28]], [[41edo|19\41]]
 
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, 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


</div></div>
{{Mapping|legend=0| 2 1 6 4 8 | 0 8 -5 6 -4 }}


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


== 7-limit ==
{{Optimal ET sequence|legend=0| 22, 74, 96 }}
Period: 1\10


Optimal ([[POTE]]) generator: ~3/2 = 698.696
Badness (Sintel): 2.26


EDO generators: [[50edo|1\50]], [[60edo|1\60]]
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


Scales (Scala files):  
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>.


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[[Subgroup]]: 2.3.5.7
<div style="line-height:1.6;">Technical data</div>
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Subgroup: 2.3.5.7
[[Comma list]]: 225/224, 14495514624/13841287201


[[Comma list]]: 225/224, 16807/16384
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


[[Mapping]]: [{{val| 10 0 39 28 }}, {{val| 0 1 -1 0 }}]
[[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 }}


Mapping generators: ~15/14, ~3
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


{{Multival|legend=1| 10 -10 0 -39 -28 28 }}
[[Badness]] (Sintel): 10.9
 
[[Tuning ranges]]:
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* nice range: ~3/2 = [693.129, 702.512]
* strict range: ~3/2 = [696.000, 700.000]
 
{{Val list|legend=1| 10, 50, 60, 110d }}
 
[[Badness]]: 0.089135
 
</div></div>
 
== 11-limit ==
Period: 1\10
 
Optimal (POTE) generator: ~3/2 = 696.791
 
EDO generators: [[50edo|1\50]], [[60edo|1\60]]
 
Scales (Scala files):
 
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<div style="line-height:1.6;">Technical data</div>
 
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=== 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 }}]
 
Mapping generators: ~15/14, ~3
 
[[Tuning ranges]]:
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* nice range: ~3/2 = [689.363, 702.512]
* strict range: ~3/2 = [696.000, 700.000]
 
{{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):
 
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<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 }}]
 
Mapping generators: ~14/13, ~3
 
[[Tuning ranges]]:
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* nice range: ~3/2 = [689.363, 702.512]
* strict range: ~3/2 = [696.000, 700.000]
 
{{Val list|legend=1| 10, 50, 60 }}
 
Badness: 0.036880
 
</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):
 
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<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 }}]


Mapping generators: ~14/13, ~3
{{Mapping|legend=0| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


[[Tuning ranges]]:  
Optimal tunings:  
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* nice range: ~3/2 = [689.363, 704.955]
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}
* strict range: ~3/2 = [696.000, 700.000]


{{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"