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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]]


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 wizard, tritonic, septimin, merman, slender, triton, marvolo, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, 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.
 
== 5-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~16/15 = 125.7549
 
EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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<div class="mw-collapsible-content">
 
Subgroup: 2.3.5
 
[[Comma list]]: 16875/16384
 
[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
 
[[Wedgie]]: {{wedgie| 4 -3 -14 }}
 
{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
 
</div></div>
 
== 7-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~15/14 = 125.608
 
EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 49/48, 225/224
 
[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
 
[[Wedgie]]: {{wedgie| 4 -3 2 -14 -8 13 }}
 
{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
 
</div></div>
 
=== Negra ===
 
This is the 2.3.5.7.13 extension of negri.
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~14/13 = 125.567
 
EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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Subgroup: 2.3.5.7.13
 
[[Comma list]]: 49/48, 65/64, 91/90


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


Gencom map: [{{val| 1 2 2 3 0 4 }}, {{val| 0 -4 3 -2 0 -3 }}]
== Wizard ==
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''


Gencom: [2 14/13; 49/48 65/64 91/90]
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.


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 118098/117649


== 11-limit ==
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216


Period: 1\1
[[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 }}


Optimal ([[POTE]]) generator: ~15/14 = 126.474
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
[[Badness]] (Sintel): 1.03
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 45/44, 49/48, 56/55
Comma list: 225/224, 385/384, 4000/3993
 
Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
 
{{Val list|legend=1| 9, 10, 19 }}
 
[[Badness]]: 0.0262
 
</div></div>
 
=== 13-limit ===


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


Optimal ([[POTE]]) generator: ~14/13 = 126.431
Optimal tunings:  
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


Scales (Scala files):  
Badness (Sintel): 0.613
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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<div class="mw-collapsible-content">


==== Lizard ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 49/48, 56/55, 78/77
Comma list: 225/224, 351/350, 364/363, 385/384


[[POTE generator]]: ~14/13 = 126.431
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}


Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
Optimal tunings:  
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


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


</div></div>
Badness (Sintel): 0.900


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


Period: 1\1
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


Optimal ([[POTE]]) generator: ~15/14 = 124.767
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}


EDO generators: [[19edo|2\19]], [[29edo|3\29]]
Optimal tunings:  
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{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): 0.741
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 49/48, 100/99, 225/224
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}


{{Val list|legend=1| 19, 29, 48d, 77cdd }}
Optimal tunings:
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


Badness: 0.0387
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


</div></div>
Badness (Sintel): 0.955


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


Period: 1\1
Comma list: 225/224, 325/324, 385/384, 1573/1568


Optimal ([[POTE]]) generator: ~14/13 = 124.716
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}


EDO generators: [[19edo|2\19]], [[29edo|3\29]]
Optimal tunings:  
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.837
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Comma list: 49/48, 65/64, 91/90, 875/858
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}


{{Val list|legend=1| 19, 29, 48df, 77cddf }}
Optimal tunings:
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}


Badness: 0.0244
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}


</div></div>
Badness (Sintel): 0.694


== Negric ==
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Period: 1\1
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594


Optimal ([[POTE]]) generator: ~15/14 = 127.039
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}


EDO generators: [[9edo|1\9]], [[19edo|2\19]]
Optimal tunings:  
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 72, 94, 166g }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.901
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== Mage ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 33/32, 49/48, 77/75
Comma list: 99/98, 176/175, 1331/1296


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


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
Optimal tunings:
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


Badness: 0.0306
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


</div></div>
Badness (Sintel): 1.91


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


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


Optimal ([[POTE]]) generator: ~14/13 = 127.039
[[Subgroup]]: 2.3.5.7


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


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


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


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


Comma list: 33/32, 49/48, 65/64, 91/90
[[Badness]] (Sintel): 1.20
 
Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
 
{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
 
Badness: 0.0202
 
</div></div>
 
== Negroni ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~15/14 = 124.539
 
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">


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


Comma list: 49/48, 55/54, 225/224
Comma list: 121/120, 225/224, 441/440
 
Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
 
{{Val list|legend=1| 10, 19e, 29, 77cddee }}
 
Badness: 0.0353


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


=== 13-limit ===
Optimal tunings:
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


Period: 1\1
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


Optimal ([[POTE]]) generator: ~14/13 = 124.545
Badness (Sintel): 0.782
 
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">


==== 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: 105/104, 121/120, 196/195, 275/273


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


{{Val list|legend=1| 10, 19e, 29, 77cddeef }}
Optimal tunings:
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


Badness: 0.0216
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


</div></div>
Badness (Sintel): 0.950


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


Period: 1\1
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


Optimal ([[POTE]]) generator: ~11/8 = 537.186
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
Optimal tunings:  
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.973
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 49/48, 121/120, 225/224
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


{{Val list|legend=1| 9, 20, 29, 38d, 67cdde }}
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


Badness: 0.0419
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


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


=== 13-limit ===
Badness (Sintel): 1.03


Period: 1\1
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Optimal ([[POTE]]) generator: ~11/8 = 537.208
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{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">


Comma list: 49/48, 65/64, 91/90, 121/120
Badness (Sintel): 1.04


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


{{Val list|legend=1| 9, 20, 29, 38df, 67cddef }}
Comma list: 225/224, 385/384, 27783/27500


Badness: 0.0252
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


=== 17-limit ===
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Period: 1\1
Badness (Sintel): 1.50


Optimal ([[POTE]]) generator: ~11/8 = 537.230
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


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


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 225/224, 84035/82944
<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|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


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


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefg }}
{{Optimal ET sequence|legend=1| 41, 91, 132d }}


Badness: 0.0218
[[Badness]] (Sintel): 1.38


</div></div>
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 19-limit ===
Comma list: 225/224, 245/242, 385/384


Period: 1\1
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


Optimal ([[POTE]]) generator: ~11/8 = 537.214
Optimal tunings:  
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


Scales (Scala files):  
Badness (Sintel): 1.04


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
=== 13-limit ===
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13
<div class="mw-collapsible-content">


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
Comma list: 105/104, 144/143, 196/195, 245/242


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefgh }}
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


Badness: 0.0168
{{Optimal ET sequence|legend=0| 41, 91 }}


</div></div>
Badness (Sintel): 0.955


= Passive =
== Merman ==
{{see also | Archytas clan #Passion }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Period: 1\1
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.


Optimal ([[POTE]]) generator: ~16/15 = 98.809
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.  


EDO generators: [[12edo|1\12]], [[13edo|1\13]]
[[Subgroup]]: 2.3.5.7


Scales (Scala files):  
[[Comma list]]: 225/224, 2500000/2470629


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


Comma list: 225/224, 256/245
[[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 }}


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


{{Val list|legend=1| 12, 49dd }}
[[Badness]] (Sintel): 1.39


Badness: 0.0751
=== 11-limit ===
Subgroup: 2.3.5.7.11


</div></div>
Comma list: 225/224, 441/440, 1344/1331


= Wizard =
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
== 5-limit ==


Period: 1\2
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


Optimal ([[POTE]]) generator: ~5/4 = 383.212
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
Badness (Sintel): 1.20


Scales (Scala files): [[wizard22]]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 144/143, 225/224, 364/363, 441/440
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 2197265625/2147483648
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Mapping: [{{val| 2 1 5 }}, {{val| 0 6 -1 }}]
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Mapping generators: ~42875/32768, ~9375/8192
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 548bc }}
Badness (Sintel): 1.14


Badness: 0.3864
=== Mermaid ===
Subgroup: 2.3.5.7.11


</div></div>
Comma list: 225/224, 385/384, 532400/531441


== 7-limit ==
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


Period: 1\2
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


Optimal ([[POTE]]) generator: ~5/4 = 383.256
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
Badness (Sintel): 1.46


Scales (Scala files): [[wizard22]]
==== 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, 325/324, 385/384, 10648/10647
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 118098/117649
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


Mapping: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
Optimal tunings:  
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


Mapping generators: ~1225/864, ~245/216
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


Wedgie: {{wedgie| 12 -2 20 -31 -2 52 }}
Badness (Sintel): 1.47


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
== 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: 0.0408
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  


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


== 11-limit ==
[[Comma list]]: 225/224, 589824/588245


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


Optimal ([[POTE]]) generator: ~5/4 = 383.232
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
: [[error map]]: {{val| +0.382 -0.725 -1.295 +1.765 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}}
: error map: {{val| 0.000 -1.257 -2.235 +0.727 }}


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Scales (Scala files): [[wizard22]]
[[Badness]] (Sintel): 1.44


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
=== 11-limit ===
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11
<div class="mw-collapsible-content">


Comma list: 225/224, 385/384, 4000/3993
Comma list: 225/224, 385/384, 1331/1323


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


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c }}
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Badness: 0.0185
Badness (Sintel): 0.838


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


=== Lizard ===
Comma list: 225/224, 275/273, 385/384, 1331/1323
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~5/4 = 383.389
 
EDO generators: [[50edo|16\50]], [[72edo|23\72]]


Scales (Scala files):  
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Comma list: 225/224, 351/350, 364/363, 385/384
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
Badness (Sintel): 1.07


Mapping generators: ~99/70, ~25/22
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


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


Badness: 0.0218
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 1029/1000


==== 17-limit ====
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


Period: 1\2
[[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 }}


Optimal ([[POTE]]) generator: ~5/4 = 383.381
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


EDO generators: [[50edo|16\50]], [[72edo|23\72]]
[[Badness]] (Sintel): 1.50


Scales (Scala files):
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
<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">
 
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
 
Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
 
Mapping generators: ~17/12, ~17/15
 
{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
 
Badness: 0.0145
 
</div></div>
 
==== 19-limit ====
 
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">
 
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 }}]
 
Mapping generators: ~17/12, ~17/15
 
{{Val list|legend=1| 22h, 50, 72, 122g, 194dfg }}
 
Badness: 0.0157
 
</div></div>
 
=== Gizzard ===
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~5/4 = 383.170
 
EDO generators: [[72edo|23\72]], [[94edo|30\94]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 225/224, 385/384, 325/324, 1573/1568
 
Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
 
Mapping generators: ~99/70, ~25/22
 
{{Val list|legend=1| 72, 166, 238cf }}
 
Badness: 0.0203
 
</div></div>
 
==== 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">
 
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
 
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
 
Optimal ([[POTE]]) generator: ~5/4 = 383.138
 
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">
 
Comma list: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399
 
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">
 
Comma list: 99/98, 176/175, 1331/1296
 
Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
 
Mapping generators: ~77/54, ~55/48
 
{{Val list|legend=1| 22, 50e, 72ee, 94ee }}
 
Badness: 0.0578
 
</div></div>
 
= Triton =
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/5 = 568.865
 
EDO generators: [[19edo|9\19]], [[21edo|10\21]]
 
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">
 
Comma list: 225/224, 1029/1000
 
Mapping: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
 
Wedgie: {{wedgie| 3 -7 -8 -18 -21 1 }}
 
{{Val list|legend=1| 19, 78bd, 97bd }}
 
Badness: 0.0592
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/5 = 569.144
 
EDO generators: [[19edo|9\19]], [[21edo|10\21]]
 
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">


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


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


Badness: 0.0457
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


</div></div>
Badness (Sintel): 1.51


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


Period: 1\1
[[Comma list]]: 225/224, 156250000/155649627


Optimal ([[POTE]]) generator: ~45/32 = 580.219
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


EDO generators: [[29edo|14\29]], [[31edo|15\31]], [[60edo|29\60]]
[[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 2.11
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 553584375/536870912
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 4 -3 }}, {{val| 0 -5 11 }}]
Comma list: 225/224, 441/440, 4000/3993
 
{{Val list|legend=1| 29, 31, 60, 91 }}
 
Badness: 0.3785
 
</div></div>
 
== 7-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/5 = 580.286
 
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">
 
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">
 
Comma list: 121/120, 225/224, 441/440


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


{{Val list|legend=1| 29, 31, 60e }}
Optimal tunings:
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


Badness: 0.0237
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


</div></div>
Badness (Sintel): 0.958


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


Period: 1\1
Comma list: 169/168, 225/224, 364/363, 441/440


Optimal ([[POTE]]) generator: ~7/5 = 580.108
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


EDO generators: [[29edo|14\29]], [[31edo|15\31]], [[60edo|29\60]]
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.887
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 105/104, 121/120, 196/195, 275/273
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


{{Val list|legend=1| 29, 31, 60e, 151cde }}
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


Badness: 0.0230
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


== Tritoni ==
Badness (Sintel): 0.760


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


Optimal ([[POTE]]) generator: ~7/5 = 580.389
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


EDO generators: [[31edo|15\31]], [[91edo|44\91]], [[122edo|59\122]]
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 385/384, 27783/27500
Badness (Sintel): 0.895


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 31, 91, 122, 153d }}
[[Comma list]]: 225/224, 40353607/40310784


Badness: 0.0455
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


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


= Merman =
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}
== 5-limit ==


Period: 1\1
[[Badness]] (Sintel): 2.37


Optimal ([[POTE]]) generator: ~45/32 = 585.634
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
Comma list: 225/224, 385/384, 12005/11979


Scales (Scala files):  
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}})
<div class="mw-collapsible-content">
* CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}})


Comma list: 1121008359375/1099511627776
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Mapping: [{{val| 1 5 -5 }}, {{val| 0 -7 15 }}]
Badness (Sintel): 1.01


{{Val list|legend=1| 41, 84, 125, 209, 334c, 543bc }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.4538
Comma list: 169/168, 225/224, 364/363, 1716/1715


</div></div>
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


== 7-limit ==
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}})


Period: 1\1
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Optimal ([[POTE]]) generator: ~7/5 = 585.585
Badness (Sintel): 0.922


EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 225/224, 781250000/771895089
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 2500000/2470629
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


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


Wedgie: {{wedgie| 7 -15 -16 -40 -45 5 }}
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


{{Val list|legend=1| 41, 84, 125 }}
[[Badness]] (Sintel): 2.44


Badness: 0.0551
=== 11-limit ===
Subgroup: 2.3.5.7.11


</div></div>
Comma list: 225/224, 385/384, 236328125/234365481


== 11-limit ==
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Optimal ([[POTE]]) generator: ~7/5 = 585.606
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
Badness (Sintel): 2.96


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, 325/324, 385/384, 831875/830466
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 441/440, 1344/1331
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


{{Val list|legend=1| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.0364
Badness (Sintel): 2.16


</div></div>
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


== 13-limit ==
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


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


Optimal ([[POTE]]) generator: ~7/5 = 585.657
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|20\41]], [[43edo|21\43]], [[84edo|41\84]]
{{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">


Comma list: 144/143, 225/224, 364/363, 441/440
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
Optimal tunings:
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


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


</div></div>
Badness (Sintel): 1.71


= Septimin =
==== 23-limit ====
== 5-limit ==
Subgroup: 2.3.5.7.11.13.17.19.23


Period: 1\1
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


Optimal ([[POTE]]) generator: ~75/64 = 263.658
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


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


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


Comma list: 35595703125/34359738368
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Mapping: [{{val| 1 4 1 }}, {{val| 0 -11 6 }}]
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


{{Val list|legend=1| 9, 23b, 32, 41, 91, 132, 223c, 355bc }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


Badness: 0.6709
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


== 7-limit ==
Badness (Sintel): 1.50


Period: 1\1
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Optimal ([[POTE]]) generator: ~7/6 = 263.632
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 84035/82944
Badness (Sintel): 1.53


Mapping: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
=== Gracecord ===
Subgroup: 2.3.5.7.11


Wedgie: {{wedgie| 11 -6 10 -35 -15 40 }}
Comma list: 225/224, 441/440, 109375/107811


{{Val list|legend=1| 41, 91, 132 }}
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Period: 1\1
Badness (Sintel): 2.21


Optimal ([[POTE]]) generator: ~7/6 = 263.634
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
Comma list: 225/224, 364/363, 441/440, 6125/6084


Scales (Scala files):  
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


Comma list: 225/224, 385/384, 2401/2376
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
Badness (Sintel): 1.83


{{Val list|legend=1| 41, 91, 223cdef }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


</div></div>
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


== 13-limit ==
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


Optimal ([[POTE]]) generator: ~7/6 = 263.700
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
Badness (Sintel): 1.87


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: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 105/104, 144/143, 196/195, 245/242
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


{{Val list|legend=1| 41, 91 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


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


= Slender =
== Alphorn ==
[[Subgroup]]: 2.3.5.7


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


Optimal ([[POTE]]) generator: ~49/48 = 38.413
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
[[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">


Comma list: 225/224, 589824/588245
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
Comma list: 225/224, 385/384, 12250/11979


Wedgie: {{wedgie| 13 -10 6 -46 -27 42 }}
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


{{Val list|legend=1| 31, 94, 125 }}
Optimal tunings:
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Badness: 0.0569
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


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


== 11-limit ==
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


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


Optimal ([[POTE]]) generator: ~49/48 = 38.387
[[Comma list]]: 225/224, 4194304/4117715


EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
{{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">


Comma list: 225/224, 385/384, 1331/1323
[[Badness]] (Sintel): 3.57


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 31, 63, 94, 125 }}
Comma list: 99/98, 176/175, 1310720/1294139


Badness: 0.02534
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Period: 1\1
Badness (Sintel): 2.82


Optimal ([[POTE]]) generator: ~49/48 = 38.314
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
Comma list: 99/98, 176/175, 640/637, 847/845


Scales (Scala files):  
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


Comma list: 225/224, 275/273, 385/384, 1331/1323
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
Badness (Sintel): 1.88


{{Val list|legend=1| 31, 63, 94 }}
=== Musneb ===
Subgroup: 2.3.5.7.11


Badness: 0.02591
Comma list: 225/224, 385/384, 66550/64827


</div></div>
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


= Marvo =
Optimal tunings:
{{see also| Gravity family }}
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


Comma list: 225/224, 78125000/78121827
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


[[POTE generator]]: ~27/20 = 516.694
Badness (Sintel): 2.89


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


Wedgie: {{wedgie| 6 17 46 13 56 59 }}
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.


{{Val list|legend=1| 7d, 65d, 72, 137, 209, 281, 569bcc }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0976
[[Comma list]]: 225/224, 125000000/121060821


== 11-limit ==
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
Comma list: 225/224, 243/242, 4000/3993
: mapping generators: ~2, ~10/7


POTE generator: ~27/20 = 516.699
[[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 }}


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
{{Optimal ET sequence|legend=1| 51, 53 }}


{{Val list|legend=1| 7d, 65d, 72, 281, 353c, 425bc, 497bc }}
[[Badness]] (Sintel): 3.64


Badness: 0.0317
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 13-limit ==
Comma list: 121/120, 225/224, 22000/21609
Comma list: 225/224, 243/242, 351/350, 1625/1617


POTE generator: ~27/20 = 516.730
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


{{Val list|legend=1| 72, 137, 209, 281f, 490bcf }}
{{Optimal ET sequence|legend=0| 51, 53 }}


Badness: 0.0269
Badness (Sintel): 2.46


= Marvolo =
=== 13-limit ===
Comma list: 225/224, 156250000/155649627
Subgroup: 2.3.5.7.11.13


POTE generator: ~21/20 = 83.348
Comma list: 121/120, 225/224, 275/273, 1040/1029


Mapping: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


Wedgie: {{wedgie| 6 -19 -26 -44 -58 -7 }}
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
{{Optimal ET sequence|legend=0| 51f, 53 }}


Badness: 0.0833
Badness (Sintel): 1.96


== 11-limit ==
== Naiadical ==
Comma list: 225/224, 441/440, 4000/3993
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


POTE generator: ~21/20 = 83.340
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
[[Comma list]]: 225/224, 823543/800000


{{Val list|legend=1| 29, 43, 72 }}
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


Badness: 0.0290
[[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 }}


== 13-limit ==
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}
Comma list: 169/168, 225/224, 364/363, 441/440


POTE generator: ~21/20 = 83.330
[[Badness]] (Sintel): 3.67


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 29, 43, 72, 115f }}
Comma list: 225/224, 245/242, 1617/1600


Badness: 0.0215
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


= Amavil =
Optimal tunings:
== 5-limit (mabila) ==
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
Comma list: 268435456/263671875
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


POTE generator: ~512/375 = 529.6849
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Mapping: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
Badness (Sintel): 2.00


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.2325
Comma list: 105/104, 196/195, 245/242, 1001/1000


== 7-limit ==
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}
Comma list: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


Mapping: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Wedgie: {{wedgie| 10 -3 14 -28 -6 41 }}
Badness (Sintel): 1.43


{{Val list|legend=1| 9, 34d, 43, 77d }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.1096
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


== 11-limit ==
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}
Comma list: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


{{Val list|legend=1| 9, 34d, 43, 77de }}
Badness (Sintel): 1.26


Badness: 0.0426
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.  


== 13-limit ==
[[Subgroup]]: 2.3.5.7
Comma list: 78/77, 99/98, 144/143, 176/175


POTE generator: ~15/11 = 529.951
[[Comma list]]: 225/224, 9805926501/9765625000


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


{{Val list|legend=1| 9, 34d, 43, 77de }}
[[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 }}


Badness: 0.0258
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


= Enneaportent =
[[Badness]] (Sintel): 3.81
Comma list: 225/224, 40353607/40310784


POTE generator: ~5/4 = 383.165
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
Comma list: 225/224, 441/440, 43923/43750


Wedgie: {{wedgie| 18 -9 18 -56 -22 67 }}
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


{{Val list|legend=1| 9, 54, 63, 72, 495bcd }}
Optimal tunings:
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Badness: 0.0937
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


== 11-limit ==
Badness (Sintel): 1.74
Comma list: 225/224, 385/384, 12005/11979


POTE generator: ~5/4 = 383.146
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


{{Val list|legend=1| 9, 54, 63, 72, 423cd, 495bcd }}
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


Badness: 0.0304
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
Comma list: 169/168, 225/224, 364/363, 1716/1715


POTE generator: ~5/4 = 383.047
Badness (Sintel): 1.35


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 9, 54, 63, 72, 279cf }}
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Badness: 0.0223
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


= Submajor =
Optimal tunings:
== 5-limit ==
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
Comma list: 69198046875/68719476736
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


POTE generator: ~10125/8192 = 362.321
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Mapping: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
Badness (Sintel): 1.17


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Badness: 0.1302
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.  


== 7-limit ==
[[Subgroup]]: 2.3.5.7
Comma list: 225/224, 51200/50421


POTE generator: ~49/40 = 362.255
[[Comma list]]: 225/224, 5971968/5764801


Mapping: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


Wedgie: {{wedgie| 8 -11 -6 -36 -32 17 }}
[[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 }}


{{Val list|legend=1| 10, 33, 43, 53 }}
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


Badness: 0.0605
[[Badness]] (Sintel): 4.53


== 11-limit ==
=== 11-limit ===
Comma list: 225/224, 385/384, 6655/6561
Subgroup: 2.3.5.7.11


POTE generator: ~27/22 = 362.101
Comma list: 99/98, 176/175, 65536/65219


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


{{Val list|legend=1| 10, 53, 116, 169de, 285cde }}
Optimal tunings:
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


Badness: 0.0506
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


=== 13-limit ===
Badness (Sintel): 2.26
Comma list: 169/168, 225/224, 275/273, 385/384


POTE generator: ~16/13 = 362.105
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
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>.


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


Badness: 0.0277
[[Comma list]]: 225/224, 14495514624/13841287201


== Interpental ==
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
Comma list: 99/98, 176/175, 51200/50421
: mapping generators: ~3072/2401, ~2048/1715


POTE generator: ~49/40 = 362.418
[[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: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


{{Val list|legend=1| 43, 53, 96, 149d }}
[[Badness]] (Sintel): 10.9


Badness: 0.0518
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 225/224, 3840/3773, 12005/11979
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 }}
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
 
Badness: 0.0297
 
= Alphorn =
Comma list: 225/224, 5764801/5668704
 
POTE generator: ~48/35 = 556.221
 
Mapping: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
 
Wedgie: {{wedgie| 16 -5 22 -45 -10 65 }}
 
{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
 
Badness: 0.1293
 
== 11-limit ==
Comma list: 225/224, 385/384, 12250/11979


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


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


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


Badness: 0.0735
== References ==


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