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Marvel temperaments: Difference between revisions

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


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


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


= Negri =
The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.
{{main| Negri }}
 
Negri tempers out the [[negri comma]] in the 5-limit, [[49/48]] and [[225/224]] in the 7-limit. It can be extended naturally to the 2.3.5.7.13 subgroup by adding [[91/90]] to the comma list; this will be discussed below under the title of negra.
 
== 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;">
<div style="line-height:1.6;">Technical data</div>
<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;">
<div style="line-height:1.6;">Technical data</div>
<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;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
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 }}]


Gencom map: [{{val| 1 2 2 3 0 4 }}, {{val| 0 -4 3 -2 0 -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: [2 14/13; 49/48 65/64 91/90]
== Wizard ==
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
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.


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


== 11-limit ==
[[Comma list]]: 225/224, 118098/117649
Period: 1\1


Optimal ([[POTE]]) generator: ~15/14 = 126.474
{{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">


=== 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


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


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
Optimal tunings:  
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.613
<div style="line-height:1.6;">Technical data</div>
<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 =====
Period: 1\1
Subgroup: 2.3.5.7.11.13.17


Optimal ([[POTE]]) generator: ~15/14 = 124.767
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


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


Scales (Scala files):  
Optimal tunings:  
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Badness (Sintel): 0.741


Comma list: 49/48, 100/99, 225/224
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


{{Val list|legend=1| 19, 29, 48d, 77cdd }}
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}


Badness: 0.0387
Optimal tunings:  
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


=== 13-limit ===
Badness (Sintel): 0.955
Period: 1\1
 
Optimal ([[POTE]]) generator: ~14/13 = 124.716
 
EDO generators: [[19edo|2\19]], [[29edo|3\29]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


==== Gizzard ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 65/64, 91/90, 875/858
Comma list: 225/224, 325/324, 385/384, 1573/1568


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


{{Val list|legend=1| 19, 29, 48df, 77cddf }}
Optimal tunings:
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}


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


</div></div>
Badness (Sintel): 0.837


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


Optimal ([[POTE]]) generator: ~15/14 = 127.039
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384


EDO generators: [[9edo|1\9]], [[19edo|2\19]]
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Badness (Sintel): 0.694


Comma list: 33/32, 49/48, 77/75
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 2 2 3 3 }}, {{val| 0 -4 3 -2 4 }}]
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
 
{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
 
Badness: 0.0306
 
</div></div>
 
=== 13-limit ===
Period: 1\1
 
Optimal ([[POTE]]) generator: ~14/13 = 127.039
 
EDO generators: [[9edo|1\9]], [[19edo|2\19]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 33/32, 49/48, 65/64, 91/90


Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
Optimal tunings:
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


Badness: 0.0202
{{Optimal ET sequence|legend=0| 72, 94, 166g }}


</div></div>
Badness (Sintel): 0.901
 
== 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">


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


Comma list: 49/48, 55/54, 225/224
Comma list: 99/98, 176/175, 1331/1296
 
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>
 
=== 13-limit ===
Period: 1\1


Optimal ([[POTE]]) generator: ~14/13 = 124.545
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}


EDO generators: [[10edo|1\10]], [[19edo|2\19]], [[29edo|3\29]]
Optimal tunings:  
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.91
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 49/48, 55/54, 65/64, 91/90
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.


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


{{Val list|legend=1| 10, 19e, 29, 77cddeef }}
[[Comma list]]: 225/224, 50421/50000


Badness: 0.0216
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


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


== Wilsec ==
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
Period: 1\1


Optimal ([[POTE]]) generator: ~11/8 = 537.186
[[Badness]] (Sintel): 1.20
 
EDO generators: [[29edo|13\29]], [[38edo|17\38]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 49/48, 121/120, 225/224
Comma list: 121/120, 225/224, 441/440
 
Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]


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


Badness: 0.0419
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 }}


=== 13-limit ===
Badness (Sintel): 0.782
Period: 1\1
 
Optimal ([[POTE]]) generator: ~11/8 = 537.208
 
EDO generators: [[29edo|13\29]], [[38edo|17\38]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 49/48, 65/64, 91/90, 121/120
Comma list: 105/104, 121/120, 196/195, 275/273


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


{{Val list|legend=1| 9, 20, 29, 38df, 67cddef }}
Optimal tunings:
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


Badness: 0.0252
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


</div></div>
Badness (Sintel): 0.950
 
=== 17-limit ===
Period: 1\1
 
Optimal ([[POTE]]) generator: ~11/8 = 537.230
 
EDO generators: [[29edo|13\29]], [[38edo|17\38]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefg }}
Optimal tunings:
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


Badness: 0.0218
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


</div></div>
Badness (Sintel): 0.973
 
=== 19-limit ===
Period: 1\1
 
Optimal ([[POTE]]) generator: ~11/8 = 537.214
 
EDO generators: [[29edo|13\29]], [[38edo|17\38]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefgh }}
Optimal tunings:
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


Badness: 0.0168
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


</div></div>
Badness (Sintel): 1.03


= Passive =
==== 23-limit ====
{{see also | Archytas clan #Passion }}
Subgroup: 2.3.5.7.11.13.17.19.23


Period: 1\1
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


Optimal ([[POTE]]) generator: ~16/15 = 98.809
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


EDO generators: [[12edo|1\12]], [[13edo|1\13]]
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{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): 1.04
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
=== Tritoni ===
Subgroup: 2.3.5.7.11


Comma list: 225/224, 256/245
Comma list: 225/224, 385/384, 27783/27500
 
Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
 
{{Val list|legend=1| 12, 49dd }}
 
Badness: 0.0751
 
</div></div>
 
= Wizard =
 
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~5/4 = 383.256
 
EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
 
Scales (Scala files): [[wizard22]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7
 
Comma list: 225/224, 118098/117649


Mapping: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Mapping generators: ~1225/864, ~245/216
Optimal tunings:
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


Wedgie: {{wedgie| 12 -2 20 -31 -2 52 }}
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
Badness (Sintel): 1.50


Badness: 0.0408
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


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


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


Period: 1\2
[[Comma list]]: 225/224, 84035/82944


Optimal ([[POTE]]) generator: ~5/4 = 383.232
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}
: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}


Scales (Scala files): [[wizard22]]
{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 1.38
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 225/224, 385/384, 4000/3993
Comma list: 225/224, 245/242, 385/384


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c }}
{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}


Badness: 0.0185
Badness (Sintel): 1.04
 
</div></div>
 
=== Lizard ===
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~5/4 = 383.389
 
EDO generators: [[50edo|16\50]], [[72edo|23\72]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 225/224, 351/350, 364/363, 385/384
Comma list: 105/104, 144/143, 196/195, 245/242


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


{{Val list|legend=1| 22, 50, 72, 122, 194df }}
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}


Badness: 0.0218
Badness (Sintel): 0.955


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


==== 17-limit ====
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.


Period: 1\2
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.


Optimal ([[POTE]]) generator: ~5/4 = 383.381
[[Subgroup]]: 2.3.5.7


EDO generators: [[50edo|16\50]], [[72edo|23\72]]
[[Comma list]]: 225/224, 2500000/2470629


Scales (Scala files):  
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: 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 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
<div class="mw-collapsible-content">
: [[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 }}


Subgroup: 2.3.5.7.11.13.17
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
[[Badness]] (Sintel): 1.39


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


Mapping generators: ~17/12, ~17/15
Comma list: 225/224, 441/440, 1344/1331


{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


Badness: 0.0145
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


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


==== 19-limit ====
Badness (Sintel): 1.20


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


Optimal ([[POTE]]) generator: ~5/4 = 383.477
Comma list: 144/143, 225/224, 364/363, 441/440


EDO generators: [[50edo|16\50]], [[72edo|23\72]]
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17.19
Badness (Sintel): 1.14


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
=== Mermaid ===
 
Subgroup: 2.3.5.7.11
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 ===
Comma list: 225/224, 385/384, 532400/531441


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


Optimal ([[POTE]]) generator: ~5/4 = 383.170
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


EDO generators: [[72edo|23\72]], [[94edo|30\94]]
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


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


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


Comma list: 225/224, 325/324, 385/384, 1573/1568
Comma list: 225/224, 325/324, 385/384, 10648/10647
 
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">


Subgroup: 2.3.5.7.11.13.17
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Optimal tunings:  
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


Mapping generators: ~17/12, ~17/15
Badness (Sintel): 1.47


{{Val list|legend=1| 72, 166g, 238cfg }}
== 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: 0.0136
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  


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


==== 19-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 = 383.138
[[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: [[72edo|23\72]], [[94edo|30\94]]
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Scales (Scala files):
[[Badness]] (Sintel): 1.44
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
 
Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
 
Mapping generators: ~17/12, ~17/15
 
{{Val list|legend=1| 72, 94, 166g }}
 
Badness: 0.0148
 
</div></div>
 
== Mage ==
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~5/4 = 383.124
 
EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 99/98, 176/175, 1331/1296
Comma list: 225/224, 385/384, 1331/1323


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


Mapping generators: ~77/54, ~55/48
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, 50e, 72ee, 94ee }}
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Badness: 0.0578
Badness (Sintel): 0.838


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


= Triton =
Comma list: 225/224, 275/273, 385/384, 1331/1323
 
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''
 
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">


Subgroup: 2.3.5.7
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Comma list: 225/224, 1029/1000
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Mapping: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Wedgie: {{wedgie| 3 -7 -8 -18 -21 1 }}
Badness (Sintel): 1.07


{{Val list|legend=1| 19, 78bd, 97bd }}
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


Badness: 0.0592
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.  


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


== 11-limit ==
[[Comma list]]: 225/224, 1029/1000


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


Optimal ([[POTE]]) generator: ~7/5 = 569.144
[[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 }}


EDO generators: [[19edo|9\19]], [[21edo|10\21]]
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


Scales (Scala files):
[[Badness]] (Sintel): 1.50
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 45/44, 56/55, 1029/1000
Comma list: 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 ==
[[Subgroup]]: 2.3.5.7


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
[[Comma list]]: 225/224, 156250000/155649627


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


Optimal ([[POTE]]) generator: ~7/5 = 580.286
[[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 }}


EDO generators: [[29edo|14\29]], [[31edo|15\31]], [[60edo|29\60]]
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


Scales (Scala files):  
[[Badness]] (Sintel): 2.11


<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">


Subgroup: 2.3.5.7
Comma list: 225/224, 441/440, 4000/3993


Comma list: 225/224, 50421/50000
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


Mapping: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


Wedgie: {{wedgie| 5 -11 -12 -29 -33 3 }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
Badness (Sintel): 0.958


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


== 11-limit ==
Comma list: 169/168, 225/224, 364/363, 441/440


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


Optimal ([[POTE]]) generator: ~7/5 = 580.267
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


EDO generators: [[29edo|14\29]], [[31edo|15\31]], [[60edo|29\60]]
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


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


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


Subgroup: 2.3.5.7.11
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


Comma list: 121/120, 225/224, 441/440
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


{{Val list|legend=1| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Badness: 0.0237
Badness (Sintel): 0.760


</div></div>
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
=== 13-limit ===
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/5 = 580.108
 
EDO generators: [[29edo|14\29]], [[31edo|15\31]], [[60edo|29\60]]


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


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


Subgroup: 2.3.5.7.11.13
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


Comma list: 105/104, 121/120, 196/195, 275/273
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
Badness (Sintel): 0.895


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


Badness: 0.0230
Named by [[Xenllium]] in 2021, decic tempers out 16807/16384, the [[cloudy comma]], and {{monzo| 11 -10 -10 10 }}, the [[linus comma]], in addition to the marvel comma. It may be described as the {{nowrap| 50 & 60 }} temperament, with a period of 1/10 octave and a [[ploidacot]] signature of decaploid monocot. It is [[support]]ed by [[10edo|10-]], [[50edo|50-]], and [[60edo]].  


</div></div>
It can be extended to the 11-, 13-, and 17-limit by adding [[385/384]], [[105/104]], and [[170/169]] to the comma list in this order.


== Tritoni ==
[[Subgroup]]: 2.3.5.7


Period: 1\1
[[Comma list]]: 225/224, 16807/16384


Optimal ([[POTE]]) generator: ~7/5 = 580.389
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


EDO generators: [[31edo|15\31]], [[91edo|44\91]], [[122edo|59\122]]
[[Optimal tuning]]s:  
* [[WE]]: ~15/14 = 120.1837{{c}}, ~3/2 = 699.7654{{c}} (~49/48 = 21.3366{{c}})
: [[error map]]: {{val| +1.837 -0.353 -0.753 -3.683 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 698.8236{{c}} (~49/48 = 21.1764{{c}})
: error map: {{val| 0.000 -3.131 -5.137 -8.826 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 2.26
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 225/224, 385/384, 27783/27500
Comma list: 225/224, 385/384, 3087/3025


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}


{{Val list|legend=1| 31, 91, 122, 153d }}
Optimal tunings:
* WE: ~15/14 = 120.1406{{c}}, ~3/2 = 697.6075{{c}} (~56/55 = 23.2360{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.0142{{c}} (~56/55 = 22.9858{{c}})


Badness: 0.0455
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


</div></div>
Badness (Sintel): 2.11


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


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
Comma list: 105/104, 144/143, 196/195, 2200/2197


Period: 1\1
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


Optimal ([[POTE]]) generator: ~7/5 = 585.585
Optimal tunings:
* WE: ~15/14 = 120.1166{{c}}, ~3/2 = 697.6705{{c}} (~78/77 = 23.0289{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.1492{{c}} (~78/77 = 22.8508{{c}})


EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Scales (Scala files):  
Badness (Sintel): 1.52


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
==== 17-limit ====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13.17
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7
 
Comma list: 225/224, 2500000/2470629
 
Mapping: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
 
Wedgie: {{wedgie| 7 -15 -16 -40 -45 5 }}
 
{{Val list|legend=1| 41, 84, 125 }}
 
Badness: 0.0551
 
</div></div>


== 11-limit ==
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


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


Optimal ([[POTE]]) generator: ~7/5 = 585.606
Optimal tunings:
* WE: ~15/14 = 120.1262{{c}}, ~3/2 = 697.8185{{c}} (~78/77 = 22.9388{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.2757{{c}} (~78/77 = 22.7243{{c}})


EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Scales (Scala files):  
Badness (Sintel): 1.28


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
=== Splendecic ===
<div style="line-height:1.6;">Technical data</div>
Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[fantastic]] rank-3 temperament, its name is a portmanteau of ''splendid'' and ''decic''.
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 441/440, 1344/1331
Comma list: 225/224, 1617/1600, 2401/2376
 
Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
 
{{Val list|legend=1| 41, 84, 125e }}
 
Badness: 0.0364
 
</div></div>
 
== 13-limit ==
 
Period: 1\1


Optimal ([[POTE]]) generator: ~7/5 = 585.657
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}


EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
Optimal tunings:  
* WE: ~15/14 = 120.1874{{c}}, ~3/2 = 699.6085{{c}} (~99/98 = 21.5156{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3531{{c}} (~99/98 = 21.6469{{c}})


Scales (Scala files):
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.98
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 144/143, 225/224, 364/363, 441/440
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
Optimal tunings:
* WE: ~15/14 = 120.1565{{c}}, ~3/2 = 699.2756{{c}} (~91/90 = 21.6631{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.2480{{c}} (~91/90 = 21.7520{{c}})


Badness: 0.0275
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


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


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


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374


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


Optimal ([[POTE]]) generator: ~7/6 = 263.632
Optimal tunings:
* WE: ~15/14 = 120.1571{{c}}, ~3/2 = 699.2892{{c}} (~91/90 = 21.6536{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3144{{c}} (~91/90 = 21.6856{{c}})


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
=== Prodecic ===
<div style="line-height:1.6;">Technical data</div>
Prodecic (50e & 60e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the [[prodigy]] rank-3 temperament, its name is a portmanteau of ''prodigy'' and ''decic''.
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7
 
Comma list: 225/224, 84035/82944
 
Mapping: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
 
Wedgie: {{wedgie| 11 -6 10 -35 -15 40 }}
 
{{Val list|legend=1| 41, 91, 132 }}
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/6 = 263.634
 
EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 2401/2376
Comma list: 225/224, 441/440, 5929/5832
 
Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
 
{{Val list|legend=1| 41, 91, 223cdef }}
 
</div></div>
 
== 13-limit ==
 
Period: 1\1


Optimal ([[POTE]]) generator: ~7/6 = 263.700
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
Optimal tunings:  
* WE: ~15/14 = 120.2024{{c}}, ~3/2 = 701.3908{{c}} (~55/54 = 19.8237{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.5235{{c}} (~55/54 = 19.4765{{c}})


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 2.20
<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: 105/104, 144/143, 196/195, 245/242
Comma list: 105/104, 196/195, 275/273, 5929/5832


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


{{Val list|legend=1| 41, 91 }}
Optimal tunings:
* WE: ~15/14 = 120.1654{{c}}, ~3/2 = 701.4683{{c}} (~91/90 = 19.5242{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.7175{{c}} (~91/90 = 19.2825{{c}})


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


= Slender =
Badness (Sintel): 1.73


Period: 1\1
==== 17-limit ====
 
Subgroup: 2.3.5.7.11.13.17
Optimal ([[POTE]]) generator: ~49/48 = 38.413
 
EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


Comma list: 225/224, 589824/588245
Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
Optimal tunings:  
* WE: ~15/14 = 120.1577{{c}}, ~3/2 = 701.3950{{c}} (~91/90 = 19.5514{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.6932{{c}} (~91/90 = 19.3068{{c}})


Wedgie: {{wedgie| 13 -10 6 -46 -27 42 }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


{{Val list|legend=1| 31, 94, 125 }}
Badness (Sintel): 1.41


Badness: 0.0569
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


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


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


Period: 1\1
[[Optimal tuning]]s:  
* [[WE]]: ~2592/2401 = 133.4174{{c}}, ~12005/6912 = 950.7667{{c}} (~1728/1715 = 16.8452{{c}})
: [[error map]]: {{val| +0.756 -0.422 -1.395 +0.298 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}})
: error map: {{val| 0.000 -1.361 -3.277 -1.565 }}


Optimal ([[POTE]]) generator: ~49/48 = 38.387
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
[[Badness]] (Sintel): 2.37
 
Scales (Scala files):  
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 225/224, 385/384, 1331/1323
Comma list: 225/224, 385/384, 12005/11979


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


{{Val list|legend=1| 31, 63, 94, 125 }}
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}})


Badness: 0.02534
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


</div></div>
Badness (Sintel): 1.01
 
== 13-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~49/48 = 38.314
 
EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 225/224, 275/273, 385/384, 1331/1323
Comma list: 169/168, 225/224, 364/363, 1716/1715


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


{{Val list|legend=1| 31, 63, 94 }}
Optimal tunings:
* WE: ~14/13 = 133.4245{{c}}, ~26/15 = 950.9362{{c}} (~105/104 = 16.9650{{c}})
* CWE: ~14/13 = 133.3333{{c}}, ~26/15 = 950.4364{{c}} (~99/98 = 17.1031{{c}})


Badness: 0.02591
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


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


= Marvo =
== Gracecordial ==
{{see also| Gravity family }}
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


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


Optimal ([[POTE]]) generator: ~27/20 = 516.694
[[Comma list]]: 225/224, 781250000/771895089


EDO generators: [[65edo|28\65]], [[72edo|31\72]], [[137edo|59\137]]
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


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


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


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 2.44


Comma list: 225/224, 78125000/78121827
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
Comma list: 225/224, 385/384, 236328125/234365481


Wedgie: {{wedgie| 6 17 46 13 56 59 }}
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


{{Val list|legend=1| 65d, 72, 137, 209, 281, 569bcc }}
Optimal tunings:
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Badness: 0.0976
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


</div></div>
Badness (Sintel): 2.96


== 11-limit ==
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Period: 1\1
Comma list: 225/224, 325/324, 385/384, 831875/830466


Optimal ([[POTE]]) generator: ~27/20 = 516.699
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


EDO generators: [[65edo|28\65]], [[72edo|31\72]], [[137edo|59\137]]
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 2.16
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 243/242, 4000/3993
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


{{Val list|legend=1| 65d, 72, 281, 353c, 425bc, 497bc }}
Optimal tunings:
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


Badness: 0.0317
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


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


== 13-limit ==
==== 19-limit ====
 
Subgroup: 2.3.5.7.11.13.17.19
Period: 1\1
 
Optimal ([[POTE]]) generator: ~27/20 = 516.730
 
EDO generators: [[65edo|28\65]], [[72edo|31\72]], [[137edo|59\137]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 243/242, 351/350, 1625/1617


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


{{Val list|legend=1| 65d, 72, 137, 209, 281f, 490bcf }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


Badness: 0.0269
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


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


= Marvolo =
Badness (Sintel): 1.71


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


Optimal ([[POTE]]) generator: ~21/20 = 83.348
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


EDO generators: [[29edo|2\29]], [[43edo|3\43]], [[72edo|5\72]]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


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


Subgroup: 2.3.5.7
Badness (Sintel): 1.57


Comma list: 225/224, 156250000/155649627
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Mapping: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Wedgie: {{wedgie| 6 -19 -26 -44 -58 -7 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
Optimal tunings:
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Badness: 0.0833
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


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


== 11-limit ==
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


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


Optimal ([[POTE]]) generator: ~21/20 = 83.340
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


EDO generators: [[29edo|2\29]], [[43edo|3\43]], [[72edo|5\72]]
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.53
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 225/224, 441/440, 4000/3993
Comma list: 225/224, 441/440, 109375/107811
 
Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
 
{{Val list|legend=1| 29, 43, 72 }}
 
Badness: 0.0290
 
</div></div>
 
== 13-limit ==
 
Period: 1\1


Optimal ([[POTE]]) generator: ~21/20 = 83.330
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


EDO generators: [[29edo|2\29]], [[43edo|3\43]], [[72edo|5\72]]
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 2.21
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 169/168, 225/224, 364/363, 441/440
Comma list: 225/224, 364/363, 441/440, 6125/6084


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


{{Val list|legend=1| 29, 43, 72, 115f }}
Optimal tunings:
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


Badness: 0.0215
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


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


= Amavil =
==== 17-limit ====
== 5-limit (mabila) ==
Subgroup: 2.3.5.7.11.13.17


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


Optimal ([[POTE]]) generator: ~512/375 = 529.6849
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


EDO generators: [[34edo|15\34]], [[43edo|19\43]], [[77edo|34\77]], [[111edo|49\111]]
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


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


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


Subgroup: 2.3.5
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 268435456/263671875
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


Mapping: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
Optimal tunings:
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


Badness: 0.2325
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


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


== 7-limit ==
== Alphorn ==
[[Subgroup]]: 2.3.5.7


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


Optimal ([[POTE]]) generator: ~48/35 = 529.979
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


EDO generators: [[34edo|15\34]], [[43edo|19\43]]
[[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">


Subgroup: 2.3.5.7
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 225/224, 17496/16807
Comma list: 225/224, 385/384, 12250/11979


Mapping: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


Wedgie: {{wedgie| 10 -3 14 -28 -6 41 }}
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Badness: 0.1096
Badness (Sintel): 2.43


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


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


Period: 1\1
[[Comma list]]: 225/224, 4194304/4117715


Optimal ([[POTE]]) generator: ~15/11 = 529.974
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


EDO generators: [[34edo|15\34]], [[43edo|19\43]]
[[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 3.57
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 99/98, 176/175, 864/847
Comma list: 99/98, 176/175, 1310720/1294139
 
Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
 
{{Val list|legend=1| 9, 34d, 43, 77de }}
 
Badness: 0.0426
 
</div></div>
 
== 13-limit ==


Period: 1\1
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


Optimal ([[POTE]]) generator: ~15/11 = 529.951
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


EDO generators: [[34edo|15\34]], [[43edo|19\43]]
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Scales (Scala files):  
Badness (Sintel): 2.82
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 78/77, 99/98, 144/143, 176/175
Comma list: 99/98, 176/175, 640/637, 847/845


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


{{Val list|legend=1| 9, 34d, 43, 77de }}
Optimal tunings:
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


Badness: 0.0258
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


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


= Enneaportent =
=== Musneb ===
 
Subgroup: 2.3.5.7.11
Period: 1\9
 
Optimal ([[POTE]]) generator: ~5/4 = 383.165
 
EDO generators: [[63edo|20\63]], [[72edo|23\72]], [[81edo|26\81]]
 
Scales (Scala files):


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 225/224, 385/384, 66550/64827
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


Comma list: 225/224, 40353607/40310784
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


Mapping: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Wedgie: {{wedgie| 18 -9 18 -56 -22 67 }}
Badness (Sintel): 2.89


{{Val list|legend=1| 63, 72, 495bcd }}
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


Badness: 0.0937
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.  


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


== 11-limit ==
[[Comma list]]: 225/224, 125000000/121060821


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


Optimal ([[POTE]]) generator: ~5/4 = 383.146
[[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 }}


EDO generators: [[63edo|20\63]], [[72edo|23\72]], [[81edo|26\81]]
{{Optimal ET sequence|legend=1| 51, 53 }}


Scales (Scala files):  
[[Badness]] (Sintel): 3.64
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 225/224, 385/384, 12005/11979
Comma list: 121/120, 225/224, 22000/21609
 
Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]


{{Val list|legend=1| 63, 72, 423cd, 495bcd }}
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


Badness: 0.0304
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


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


== 13-limit ==
Badness (Sintel): 2.46
 
Period: 1\9
 
Optimal ([[POTE]]) generator: ~5/4 = 383.047
 
EDO generators: [[63edo|20\63]], [[72edo|23\72]], [[81edo|26\81]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 169/168, 225/224, 364/363, 1716/1715
Comma list: 121/120, 225/224, 275/273, 1040/1029
 
Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
 
{{Val list|legend=1| 63, 72, 279cf }}
 
Badness: 0.0223
 
</div></div>
 
= Submajor =
== 5-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~10125/8192 = 362.321
 
EDO generators: [[43edo|13\43]], [[53edo|16\53]], [[63edo|19\63]], [[96edo|29\96]], [[116edo|35\116]]


Scales (Scala files):  
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


<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 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


Subgroup: 2.3.5
{{Optimal ET sequence|legend=0| 51f, 53 }}


Comma list: 69198046875/68719476736
Badness (Sintel): 1.96


Mapping: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
[[Subgroup]]: 2.3.5.7


Badness: 0.1302
[[Comma list]]: 225/224, 823543/800000


</div></div>
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


== 7-limit ==
[[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 }}


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


Optimal ([[POTE]]) generator: ~49/40 = 362.255
[[Badness]] (Sintel): 3.67
 
EDO generators: [[43edo|13\43]], [[53edo|16\53]], [[63edo|19\63]], [[96edo|29\96]], [[116edo|35\116]]
 
Scales (Scala files):  
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7
 
Comma list: 225/224, 51200/50421
 
Mapping: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
 
Wedgie: {{wedgie| 8 -11 -6 -36 -32 17 }}
 
{{Val list|legend=1| 10, 33, 43, 53 }}
 
Badness: 0.0605
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~27/22 = 362.101
 
EDO generators: [[53edo|16\53]], [[63edo|19\63]], [[116edo|35\116]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 225/224, 385/384, 6655/6561
Comma list: 225/224, 245/242, 1617/1600


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


{{Val list|legend=1| 53, 116, 169de, 285cde }}
Optimal tunings:
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


Badness: 0.0506
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


</div></div>
Badness (Sintel): 2.00


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 196/195, 245/242, 1001/1000


Period: 1\1
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


Optimal ([[POTE]]) generator: ~16/13 = 362.105
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


EDO generators: [[53edo|16\53]], [[63edo|19\63]], [[116edo|35\116]]
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Scales (Scala files):  
Badness (Sintel): 1.43


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


Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


Comma list: 169/168, 225/224, 275/273, 385/384
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


{{Val list|legend=1| 53, 116, 169de, 285cdef }}
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Badness: 0.0277
Badness (Sintel): 1.26


</div></div>
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


== Interpental ==
[[Subgroup]]: 2.3.5.7


Period: 1\1
[[Comma list]]: 225/224, 9805926501/9765625000


Optimal ([[POTE]]) generator: ~49/40 = 362.418
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


EDO generators: [[43edo|13\43]], [[53edo|16\53]], [[96edo|29\96]]
[[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 3.81
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 99/98, 176/175, 51200/50421
Comma list: 225/224, 441/440, 43923/43750


POTE generator: ~49/40 = 362.418
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
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| 43, 53, 96, 149d }}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.0518
Badness (Sintel): 1.74
 
</div></div>


=== 13-limit ===
=== 13-limit ===
Period: 1\1
Optimal ([[POTE]]) generator: ~16/13 = 362.402
EDO generators: [[43edo|13\43]], [[53edo|16\53]], [[96edo|29\96]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 99/98, 169/168, 176/175, 640/637
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


POTE generator: ~16/13 = 362.402
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


{{Val list|legend=1| 43, 53, 96, 149d }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 0.0297
Badness (Sintel): 1.35


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


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


Optimal ([[POTE]]) generator: ~48/35 = 556.221
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


EDO generators: [[28edo|13\28]], [[41edo|19\41]]
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


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


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


Subgroup: 2.3.5.7
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


Comma list: 225/224, 5764801/5668704
Hendeca tempers out the same 5-limit comma as [[porwell temperaments #Hendecatonic|hendecatonic]], and has a period of 1/11 octave. However, in this temperament, nine periods represent [[7/4]], the same as [[keemic temperaments #Undeka|undeka]]. It can be tuned to [[22edo]] or [[33edo]] using the [[patent val]], or [[55edo]] using the 55d val. It was named by [[Xenllium]] in 2025 as a low-accuracy variant of hendecatonic.


Mapping: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
[[Subgroup]]: 2.3.5.7


Wedgie: {{wedgie| 16 -5 22 -45 -10 65 }}
[[Comma list]]: 225/224, 122880/117649


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


Badness: 0.1293
[[Optimal tuning]]s:  
* [[WE]]: ~16/15 = 108.9526{{c}}, ~3/2 = 702.4215{{c}}
: [[error map]]: {{val| -1.521 -1.055 -2.252 +8.705 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 703.2071{{c}}
: error map: {{val| 0.000 +1.252 +1.388 +12.992 }}


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


== 11-limit ==
[[Badness]] (Sintel): 4.37
Period: 1\1


Optimal ([[POTE]]) generator: ~11/8 = 556.144
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
EDO generators: [[28edo|13\28]], [[41edo|19\41]]


Scales (Scala files):  
Comma list: 121/120, 225/224, 352/343


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Optimal tunings:  
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


Comma list: 225/224, 385/384, 12250/11979
{{Optimal ET sequence|legend=0| 22, 55d }}


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
Badness (Sintel): 2.46


{{Val list|legend=1| 28d, 41, 315cde }}
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Badness: 0.0735
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.  


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


= Decic (10&amp;50) =
[[Comma list]]: 225/224, 5971968/5764801
Period: 1\10


Optimal ([[POTE]]) generator: ~3/2 = 698.696
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


EDO generators: [[10edo|0\10]], [[50edo|1\50]]
[[Optimal tuning]]s:  
* [[WE]]: ~2401/1728 = 599.7132{{c}}, ~35/32 = 162.5806{{c}}
: [[error map]]: {{val| -0.574 -1.597 -0.937 +5.510 }}
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~35/32 = 162.6388{{c}}
: error map: {{val| 0.000 -0.844 +0.492 +7.007 }}


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 4.53
<div style="line-height:1.6;">Technical data</div>


<div class="mw-collapsible-content">
=== 11-limit ===
Subgroup: 2.3.5.7.11


Subgroup: 2.3.5.7
Comma list: 99/98, 176/175, 65536/65219


Comma list: 225/224, 16807/16384
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


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


Wedgie: {{wedgie| 10 -10 0 -39 -28 28 }}
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


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


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


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


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


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


EDO generators: [[10edo|0\10]], [[50edo|1\50]]
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


Scales (Scala files):  
[[Optimal tuning]]s:  
* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
: [[error map]]: {{val| -0.523 -1.044 -1.549 +5.138 }}
* [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}}
: error map: {{val| 0.000 -0.284 -0.276 +6.592 }}


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


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


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


Comma list: 225/224, 385/384, 3087/3025
Comma list: 225/224, 3840/3773, 12005/11979
 
Mapping: [{{val| 10 0 39 28 3 }}, {{val| 0 1 -1 0 2 }}]


{{Val list|legend=1| 10, 50, 60 }}
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
 
Badness: 0.063900
 
</div></div>
 
==13-limit==
Period: 1\10
 
Optimal (POTE) generator: ~3/2 = 696.993
 
EDO generators: [[10edo|0\10]], [[50edo|1\50]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
 
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 196/195, 2200/2197


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


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


Badness: 0.036880
Badness (Sintel): 5.74


</div></div>
== 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"