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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]]
* ''[[Quintapole]]'' (+7812500/7411887) → [[Quintaleap family #Quintapole|Quintaleap 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]]
* ''[[Decic]]'' (+16807/16384) → [[Cloudy clan #Decic|Cloudy clan]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Hendeca]]'' (+122880/117649) → [[11th-octave temperaments #Hendeca|11th-octave temperaments]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
* ''[[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]]
 
Considered below are wizard, tritonic, septimin, merman, slender, triton, marvolo, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, gwazy, and tertiosec, in the order of increasing [[badness]].
 
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.


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.
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.


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.)
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}}.)


= Negri =
== Wizard ==
{{main| Negri }}
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''


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.  
Wizard has a [[semi-octave]] period and is generated by an interval that can be treated as [[~]][[17/15]]. The semi-octave complement of this interval is ~[[5/4]]. Wizard can be described as {{nowrap| 22 & 72 }}. Its [[ploidacot]] is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.  


== 5-limit ==
[[Subgroup]]: 2.3.5.7


Period: 1\1
[[Comma list]]: 225/224, 118098/117649


Optimal ([[POTE]]) generator: ~16/15 = 125.7549
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
[[Optimal tuning]]s:  
* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}
: error map: {{val| 0.000 -1.169 -3.111 -0.849 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 1.03
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Comma list]]: 16875/16384
Comma list: 225/224, 385/384, 4000/3993


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}


[[Wedgie]]: {{wedgie| 4 -3 -14 }}
Optimal tunings:  
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


</div></div>
Badness (Sintel): 0.613


== 7-limit ==
==== Lizard ====
Subgroup: 2.3.5.7.11.13


Period: 1\1
Comma list: 225/224, 351/350, 364/363, 385/384


Optimal ([[POTE]]) generator: ~15/14 = 125.608
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
Optimal tunings:  
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.900
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 49/48, 225/224
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}


[[Wedgie]]: {{wedgie| 4 -3 2 -14 -8 13 }}
Optimal tunings:  
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


</div></div>
Badness (Sintel): 0.741


=== Negra ===
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


This is the 2.3.5.7.13 extension of negri.
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


Period: 1\1
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}


Optimal ([[POTE]]) generator: ~14/13 = 125.567
Optimal tunings:  
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


Scales (Scala files):  
Badness (Sintel): 0.955


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
==== Gizzard ====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13
<div class="mw-collapsible-content">
 
Comma list: 225/224, 325/324, 385/384, 1573/1568
 
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}


Subgroup: 2.3.5.7.13
Optimal tunings:  
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}


[[Comma list]]: 49/48, 65/64, 91/90
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}


[[Mapping]]: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -3 }}]
Badness (Sintel): 0.837


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


Gencom: [2 14/13; 49/48 65/64 91/90]
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}


</div></div>
Optimal tunings:
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}


Period: 1\1
Badness (Sintel): 0.694


Optimal ([[POTE]]) generator: ~15/14 = 126.474
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594


Scales (Scala files):  
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
<div class="mw-collapsible-content">
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


{{Optimal ET sequence|legend=0| 72, 94, 166g }}
Badness (Sintel): 0.901
=== Mage ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 45/44, 49/48, 56/55
Comma list: 99/98, 176/175, 1331/1296
 
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
Optimal tunings:  
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


{{Val list|legend=1| 9, 10, 19 }}
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


[[Badness]]: 0.0262
Badness (Sintel): 1.91


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


=== 13-limit ===
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.


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


Optimal ([[POTE]]) generator: ~14/13 = 126.431
[[Comma list]]: 225/224, 50421/50000


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


Scales (Scala files):  
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
[[Badness]] (Sintel): 1.20


Comma list: 45/44, 49/48, 56/55, 78/77
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~14/13 = 126.431
Comma list: 121/120, 225/224, 441/440


Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


{{Val list|legend=1| 9, 10, 19 }}
Optimal tunings:
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


== Negril ==
Badness (Sintel): 0.782


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


Optimal ([[POTE]]) generator: ~15/14 = 124.767
Comma list: 105/104, 121/120, 196/195, 275/273


EDO generators: [[19edo|2\19]], [[29edo|3\29]]
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 29, 31, 60e }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Badness (Sintel): 0.950


Comma list: 49/48, 100/99, 225/224
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


{{Val list|legend=1| 19, 29, 48d, 77cdd }}
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


Badness: 0.0387
Optimal tunings:  
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


=== 13-limit ===
Badness (Sintel): 0.973


Period: 1\1
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Optimal ([[POTE]]) generator: ~14/13 = 124.716
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


EDO generators: [[19edo|2\19]], [[29edo|3\29]]
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Badness (Sintel): 1.03


Comma list: 49/48, 65/64, 91/90, 875/858
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


{{Val list|legend=1| 19, 29, 48df, 77cddf }}
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


Badness: 0.0244
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


== Negric ==
Badness (Sintel): 1.04


Period: 1\1
=== Tritoni ===
Subgroup: 2.3.5.7.11


Optimal ([[POTE]]) generator: ~15/14 = 127.039
Comma list: 225/224, 385/384, 27783/27500


EDO generators: [[9edo|1\9]], [[19edo|2\19]]
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Badness (Sintel): 1.50


Comma list: 33/32, 49/48, 77/75
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


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


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0306
[[Comma list]]: 225/224, 84035/82944


</div></div>
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


=== 13-limit ===
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}
: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}


Period: 1\1
{{Optimal ET sequence|legend=1| 41, 91, 132d }}


Optimal ([[POTE]]) generator: ~14/13 = 127.039
[[Badness]] (Sintel): 1.38


EDO generators: [[9edo|1\9]], [[19edo|2\19]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Scales (Scala files):  
Comma list: 225/224, 245/242, 385/384


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
<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 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


Comma list: 33/32, 49/48, 65/64, 91/90
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
Badness (Sintel): 1.04


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0202
Comma list: 105/104, 144/143, 196/195, 245/242


</div></div>
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


== Negroni ==
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


Period: 1\1
{{Optimal ET sequence|legend=0| 41, 91 }}


Optimal ([[POTE]]) generator: ~15/14 = 124.539
Badness (Sintel): 0.955


EDO generators: [[10edo|1\10]], [[19edo|2\19]], [[29edo|3\29]]
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Scales (Scala files):
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.


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.  
<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


Comma list: 49/48, 55/54, 225/224
[[Comma list]]: 225/224, 2500000/2470629


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


{{Val list|legend=1| 10, 19e, 29, 77cddee }}
[[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 }}


Badness: 0.0353
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


</div></div>
[[Badness]] (Sintel): 1.39


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


Period: 1\1
Comma list: 225/224, 441/440, 1344/1331


Optimal ([[POTE]]) generator: ~14/13 = 124.545
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


EDO generators: [[10edo|1\10]], [[19edo|2\19]], [[29edo|3\29]]
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: 49/48, 55/54, 65/64, 91/90
Comma list: 144/143, 225/224, 364/363, 441/440


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


{{Val list|legend=1| 10, 19e, 29, 77cddeef }}
Optimal tunings:
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Badness: 0.0216
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


</div></div>
Badness (Sintel): 1.14


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


Period: 1\1
Comma list: 225/224, 385/384, 532400/531441


Optimal ([[POTE]]) generator: ~11/8 = 537.186
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.46
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 49/48, 121/120, 225/224
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]
Comma list: 225/224, 325/324, 385/384, 10648/10647


{{Val list|legend=1| 9, 20, 29, 38d, 67cdde }}
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


Badness: 0.0419
Optimal tunings:  
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


=== 13-limit ===
Badness (Sintel): 1.47


Period: 1\1
== 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.


Optimal ([[POTE]]) generator: ~11/8 = 537.208
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  


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


Scales (Scala files):  
[[Comma list]]: 225/224, 589824/588245


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
<div style="line-height:1.6;">Technical data</div>
: mapping generators: ~2, ~49/48
<div class="mw-collapsible-content">


Comma list: 49/48, 65/64, 91/90, 121/120
[[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 }}


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


{{Val list|legend=1| 9, 20, 29, 38df, 67cddef }}
[[Badness]] (Sintel): 1.44


Badness: 0.0252
=== 11-limit ===
Subgroup: 2.3.5.7.11


</div></div>
Comma list: 225/224, 385/384, 1331/1323


=== 17-limit ===
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


Optimal ([[POTE]]) generator: ~11/8 = 537.230
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
Badness (Sintel): 0.838


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: 225/224, 275/273, 385/384, 1331/1323
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefg }}
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Badness: 0.0218
Badness (Sintel): 1.07


</div></div>
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


=== 19-limit ===
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.


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


Optimal ([[POTE]]) generator: ~11/8 = 537.214
[[Comma list]]: 225/224, 1029/1000


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


Scales (Scala files):  
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
[[Badness]] (Sintel): 1.50


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefgh }}
Comma list: 45/44, 56/55, 1029/1000


Badness: 0.0168
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


= Passive =
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
{{see also | Archytas clan #Passion }}


Comma list: 225/224, 256/245
Badness (Sintel): 1.51


POTE generator: ~16/15 = 98.809
== Marvolo ==
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
[[Comma list]]: 225/224, 156250000/155649627


{{Val list|legend=1| 1, 11, 12, 49d }}
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


Badness: 0.0751
[[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 }}


= Wizard =
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}
== 5-limit ==
Comma list: 2197265625/2147483648


POTE generator: ~5/4 = 383.212
[[Badness]] (Sintel): 2.11


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


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 548bc }}
Comma list: 225/224, 441/440, 4000/3993


Badness: 0.3864
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


== 7-limit ==
Optimal tunings:
Comma list: 225/224, 118098/117649
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


[[POTE generator]]: ~5/4 = 383.256
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Mapping: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
Badness (Sintel): 0.958


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


{{Val list|legend=1| 22, 50, 72, 94, 166, 238c, 310c, 382c }}
Comma list: 169/168, 225/224, 364/363, 441/440


Badness: 0.0408
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


== 11-limit ==
Optimal tunings:
Comma list: 225/224, 385/384, 4000/3993
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


[[POTE generator]]: ~5/4 = 383.232
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
Badness (Sintel): 0.887


{{Val list|legend=1| 22, 50, 72, 94, 166, 238c, 310c }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0185
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


=== Lizard ===
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
Comma list: 225/224, 351/350, 364/363, 385/384


[[POTE generator]]: ~5/4 = 383.389
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


{{Val list|legend=1| 22, 50, 72, 122, 194df }}
Badness (Sintel): 0.760


Badness: 0.0218
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


==== 17-limit ====
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


[[POTE generator]]: ~5/4 = 383.381
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Badness: 0.0145
Badness (Sintel): 0.895


==== 19-limit ====
== Enneaportent ==
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~5/4 = 383.477
[[Comma list]]: 225/224, 40353607/40310784


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


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


Badness: 0.0157
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


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


POTE generator: ~5/4 = 383.170
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
Comma list: 225/224, 385/384, 12005/11979


{{Val list|legend=1| 22, 72, 94, 166, 238cf }}
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


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


==== 17-limit ====
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384


POTE generator: ~5/4 = 383.175
Badness (Sintel): 1.01


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 22f, 72, 94, 166g, 238cfg }}
Comma list: 169/168, 225/224, 364/363, 1716/1715


Badness: 0.0136
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


==== 19-limit ====
Optimal tunings:
Comma list: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399
* 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}})


POTE generator: ~5/4 = 383.138
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
Badness (Sintel): 0.922


{{Val list|legend=1| 22f, 72, 94, 166g }}
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


Badness: 0.0148
[[Subgroup]]: 2.3.5.7


== Mage ==
[[Comma list]]: 225/224, 781250000/771895089
Comma list: 99/98, 176/175, 1331/1296


POTE generator: ~55/48 = 216.876
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
[[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 = 1200.3333{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


{{Val list|legend=1| 22, 72e, 94e }}
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


Badness: 0.0578
[[Badness]] (Sintel): 2.44


= Triton =
=== 11-limit ===
Comma list: 225/224, 1029/1000
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~7/5 = 568.865
Comma list: 225/224, 385/384, 236328125/234365481


Mapping: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


Wedgie: {{wedgie| 3 -7 -8 -18 -21 1 }}
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


{{Val list|legend=1| 17d, 19, 78bd, 97bd }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Badness: 0.0592
Badness (Sintel): 2.96


==11-limit==
==== 13-limit ====
Comma list: 45/44, 56/55, 1029/1000
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 569.144
Comma list: 225/224, 325/324, 385/384, 831875/830466


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
Optimal tunings:
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


Badness: 0.0457
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


= Tritonic =
Badness (Sintel): 2.16
== 5-limit ==
Comma list: 553584375/536870912


POTE generator: ~45/32 = 580.219
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 4 -3 }}, {{val| 0 -5 11 }}]
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


{{Val list|legend=1| 29, 31, 60, 91 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


Badness: 0.3785
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


== 7-limit ==
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
Comma list: 225/224, 50421/50000


POTE generator: ~7/5 = 580.286
Badness (Sintel): 1.96


Mapping: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Wedgie: {{wedgie| 5 -11 -12 -29 -33 3 }}
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


== 11-limit ==
Optimal tunings:
Comma list: 121/120, 225/224, 441/440
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


POTE generator: ~7/5 = 580.267
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
Badness (Sintel): 1.71


{{Val list|legend=1| 29, 31, 60e }}
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Badness: 0.0237
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


=== 13-limit ===
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
Comma list: 105/104, 121/120, 196/195, 275/273


POTE generator: ~7/5 = 580.108
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 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


{{Val list|legend=1| 29, 31, 60e, 151cde }}
Badness (Sintel): 1.57


Badness: 0.0230
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


== Tritoni ==
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
Comma list: 225/224, 385/384, 27783/27500


POTE generator: ~7/5 = 580.389
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
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| 31, 91, 122 }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Badness: 0.0455
Badness (Sintel): 1.50


= Merman =
==== 31-limit ====
Comma list: 1121008359375/1099511627776
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


POTE generator: ~45/32 = 585.634
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


Mapping: [{{val| 1 5 -5 }}, {{val| 0 -7 15 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


{{Val list|legend=1| 41, 84, 125, 209, 334c, 543bc }}
Optimal tunings:
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


Badness: 0.4538
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


== 7-limit ==
Badness (Sintel): 1.53
Comma list: 225/224, 2500000/2470629


POTE generator: ~7/5 = 585.585
=== Gracecord ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
Comma list: 225/224, 441/440, 109375/107811


Wedgie: {{wedgie| 7 -15 -16 -40 -45 5 }}
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


{{Val list|legend=1| 41, 84, 125 }}
Optimal tunings:
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


Badness: 0.0551
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


== 11-limit ==
Badness (Sintel): 2.21
Comma list: 225/224, 441/440, 1344/1331


POTE generator: ~7/5 = 585.606
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
Comma list: 225/224, 364/363, 441/440, 6125/6084


{{Val list|legend=1| 41, 84, 125e }}
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Badness: 0.0364
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}
Comma list: 144/143, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 585.657
Badness (Sintel): 1.83


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Badness: 0.0275
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


= Septimin =
Optimal tunings:
Comma list: 35595703125/34359738368
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


POTE generator: ~75/64 = 263.658
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Mapping: [{{val| 1 4 1 }}, {{val| 0 -11 6 }}]
Badness (Sintel): 1.87


{{Val list|legend=1| 9, 32, 41, 91, 132, 223c, 355bc }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.6709
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


== 7-limit ==
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}
Comma list: 225/224, 84035/82944


POTE generator: ~7/6 = 263.632
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


Mapping: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Wedgie: {{wedgie| 11 -6 10 -35 -15 40 }}
Badness (Sintel): 1.68


{{Val list|legend=1| 9, 32, 41, 91, 132 }}
== Alphorn ==
[[Subgroup]]: 2.3.5.7


== 11-limit ==
[[Comma list]]: 225/224, 5764801/5668704
Comma list: 225/224, 385/384, 2401/2376


POTE generator: ~7/6 = 263.634
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
[[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 }}


{{Val list|legend=1| 9, 32, 41, 91, 223cdef }}
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


== 13-limit ==
[[Badness]] (Sintel): 3.27
Comma list: 105/104, 144/143, 196/195, 245/242


POTE generator: ~7/6 = 263.700
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Comma list: 225/224, 385/384, 12250/11979


{{Val list|legend=1| 9, 41, 91 }}
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


= Slender =
Optimal tunings:
Comma list: 225/224, 589824/588245
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


POTE generator: ~49/48 = 38.413
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
Badness (Sintel): 2.43


Wedgie: {{wedgie| 13 -10 6 -46 -27 42 }}
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


{{Val list|legend=1| 31, 94, 125 }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0569
[[Comma list]]: 225/224, 4194304/4117715


==11-limit==
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
Comma list: 225/224, 385/384, 1331/1323
: mapping generators: ~2, ~15/8


POTE generator: ~49/48 = 38.387
[[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 }}


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


{{Val list|legend=1| 31, 63, 94, 125 }}
[[Badness]] (Sintel): 3.57


Badness: 25.342
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 13-limit ==
Comma list: 99/98, 176/175, 1310720/1294139
Comma list: 225/224, 275/273, 385/384, 1331/1323


POTE generator: ~49/48 = 38.314
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


{{Val list|legend=1| 31, 63, 94 }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 25.913
Badness (Sintel): 2.82


= Marvo =
==== 13-limit ====
{{see also| Gravity family }}
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 78125000/78121827
Comma list: 99/98, 176/175, 640/637, 847/845


[[POTE generator]]: ~27/20 = 516.694
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


Wedgie: {{wedgie| 6 17 46 13 56 59 }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


{{Val list|legend=1| 7d, 65d, 72, 137, 209, 281, 569bcc }}
Badness (Sintel): 1.88


Badness: 0.0976
=== Musneb ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 225/224, 385/384, 66550/64827
Comma list: 225/224, 243/242, 4000/3993


POTE generator: ~27/20 = 516.699
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


{{Val list|legend=1| 7d, 65d, 72, 281, 353c, 425bc, 497bc }}
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Badness: 0.0317
Badness (Sintel): 2.89


== 13-limit ==
== Untriton ==
Comma list: 225/224, 243/242, 351/350, 1625/1617
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


POTE generator: ~27/20 = 516.730
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 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 72, 137, 209, 281f, 490bcf }}
[[Comma list]]: 225/224, 125000000/121060821


Badness: 0.0269
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


= Marvolo =
[[Optimal tuning]]s:
Comma list: 225/224, 156250000/155649627
* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
: [[error map]]: {{val| -0.172 +0.002 -2.533 +3.511 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}}
: error map: {{val| 0.000 +0.298 -2.181 +3.946 }}


POTE generator: ~21/20 = 83.348
{{Optimal ET sequence|legend=1| 51, 53 }}


Mapping: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
[[Badness]] (Sintel): 3.64


Wedgie: {{wedgie| 6 -19 -26 -44 -58 -7 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
Comma list: 121/120, 225/224, 22000/21609


Badness: 0.0833
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


== 11-limit ==
Optimal tunings:
Comma list: 225/224, 441/440, 4000/3993
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


POTE generator: ~21/20 = 83.340
{{Optimal ET sequence|legend=0| 51, 53 }}


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
Badness (Sintel): 2.46


{{Val list|legend=1| 29, 43, 72 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0290
Comma list: 121/120, 225/224, 275/273, 1040/1029


== 13-limit ==
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}
Comma list: 169/168, 225/224, 364/363, 441/440


POTE generator: ~21/20 = 83.330
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
{{Optimal ET sequence|legend=0| 51f, 53 }}


{{Val list|legend=1| 29, 43, 72, 115f }}
Badness (Sintel): 1.96


Badness: 0.0215
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.  


= Amavil =
[[Subgroup]]: 2.3.5.7
== 5-limit (mabila) ==
Comma list: 268435456/263671875


POTE generator: ~512/375 = 529.6849
[[Comma list]]: 225/224, 823543/800000


Mapping: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


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


Badness: 0.2325
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


== 7-limit ==
[[Badness]] (Sintel): 3.67
Comma list: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
Comma list: 225/224, 245/242, 1617/1600


Wedgie: {{wedgie| 10 -3 14 -28 -6 41 }}
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


{{Val list|legend=1| 9, 34d, 43, 77d }}
Optimal tunings:
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


Badness: 0.1096
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


== 11-limit ==
Badness (Sintel): 2.00
Comma list: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
Comma list: 105/104, 196/195, 245/242, 1001/1000


{{Val list|legend=1| 9, 34d, 43, 77de }}
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


Badness: 0.0426
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
Comma list: 78/77, 99/98, 144/143, 176/175


POTE generator: ~15/11 = 529.951
Badness (Sintel): 1.43


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 9, 34d, 43, 77de }}
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


Badness: 0.0258
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


= Enneaportent =
Optimal tunings:
Comma list: 225/224, 40353607/40310784
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


POTE generator: ~5/4 = 383.165
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Mapping: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
Badness (Sintel): 1.26


Wedgie: {{wedgie| 18 -9 18 -56 -22 67 }}
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


{{Val list|legend=1| 9, 54, 63, 72, 495bcd }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0937
[[Comma list]]: 225/224, 9805926501/9765625000


== 11-limit ==
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
Comma list: 225/224, 385/384, 12005/11979
: mapping generators: ~2, ~10000/9261


POTE generator: ~5/4 = 383.146
[[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| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


{{Val list|legend=1| 9, 54, 63, 72, 423cd, 495bcd }}
[[Badness]] (Sintel): 3.81


Badness: 0.0304
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 13-limit ==
Comma list: 225/224, 441/440, 43923/43750
Comma list: 169/168, 225/224, 364/363, 1716/1715


POTE generator: ~5/4 = 383.047
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


{{Val list|legend=1| 9, 54, 63, 72, 279cf }}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.0223
Badness (Sintel): 1.74


= Submajor =
=== 13-limit ===
== 5-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 69198046875/68719476736


POTE generator: ~10125/8192 = 362.321
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


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


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
Optimal tunings:
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


Badness: 0.1302
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


== 7-limit ==
Badness (Sintel): 1.35
Comma list: 225/224, 51200/50421


POTE generator: ~49/40 = 362.255
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Wedgie: {{wedgie| 8 -11 -6 -36 -32 17 }}
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


{{Val list|legend=1| 10, 33, 43, 53 }}
Optimal tunings:
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


Badness: 0.0605
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


== 11-limit ==
Badness (Sintel): 1.17
Comma list: 225/224, 385/384, 6655/6561


POTE generator: ~27/22 = 362.101
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


{{Val list|legend=1| 10, 53, 116, 169de, 285cde }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0506
[[Comma list]]: 225/224, 5971968/5764801


=== 13-limit ===
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
Comma list: 169/168, 225/224, 275/273, 385/384
: mapping generators: ~2401/1728, ~35/32


POTE generator: ~16/13 = 362.105
[[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 }}


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


{{Val list|legend=1| 10, 53, 116, 169de, 285cdef }}
[[Badness]] (Sintel): 4.53


Badness: 0.0277
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Interpental ==
Comma list: 99/98, 176/175, 65536/65219
Comma list: 99/98, 176/175, 51200/50421


POTE generator: ~49/40 = 362.418
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


{{Val list|legend=1| 43, 53, 96, 149d }}
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


Badness: 0.0518
Badness (Sintel): 2.26


=== 13-limit ===
== Tertiosec ==
Comma list: 99/98, 169/168, 176/175, 640/637
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


POTE generator: ~16/13 = 362.402
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>.  


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 43, 53, 96, 149d }}
[[Comma list]]: 225/224, 14495514624/13841287201


Badness: 0.0297
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


= Alphorn =
[[Optimal tuning]]s:
Comma list: 225/224, 5764801/5668704
* [[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 }}


POTE generator: ~48/35 = 556.221
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


Mapping: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
[[Badness]] (Sintel): 10.9


Wedgie: {{wedgie| 16 -5 22 -45 -10 65 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
Comma list: 225/224, 3840/3773, 12005/11979


Badness: 0.1293
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
 
== 11-limit ==
Comma list: 225/224, 385/384, 12250/11979


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


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


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


Badness: 0.0735
== References ==


[[Category:Theory]]
[[Category:Temperament collections]]
[[Category:Temperament]]
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Marvel]]
[[Category:Rank 2]]
[[Category:Rank 2]]
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