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This page discusses miscellaneous rank-2 temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the marvel comma or septimal kleisma.  
{{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.  


Temperaments considered in families and clans are:  
Temperaments considered in families and clans are:  
* [[Dicot family #Sharp|sharp]] ({25/24, 28/27}, dicot family)
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* [[Pelogic family #Pelogic|pelogic]] ({21/20, 135/128}, pelogic family)
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Augmented family #August|august]] ({36/35, 128/125}, augmented family)
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Diaschismic family #Pajara|pajara]] ({50/49, 64/63}, diaschismic family / jubilismic clan / archytas clan)
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Meantone family #Septimal meantone|meantone]] ({81/80, 126/125}, meantone family)
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* [[Magic family|magic]] ({225/224, 245/243}, magic family)
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* [[Gamelismic clan #Miracle|miracle]] ({225/224, 1029/1024}, gamelismic clan)
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* [[Semicomma family #Orwell|orwell]] ({225/224, 1728/1715}, semicomma family)
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Schismatic family #Garibaldi|garibaldi]] ({225/224, 3125/3087}, schismatic family)
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Kleismic family #Catakleismic|catakleismic]] ({225/224, 4375/4374}, kleismic family)
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* [[Cloudy clan #Decic|decic]] ({225/224, 16807/16384}, cloudy clan)
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* [[Escapade family #Escapade|escapade]] ({225/224, 65625/65536}, escapade family)
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* [[Misty family #Fog|fog]] ({225/224, 156250/151263}, misty family)
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* [[Pythagorean family #Compton|compton]] ({225/224, 250047/250000}, pythagorean family)
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* [[Qintosec family|qintosec]] ({225/224, 2560000/2470629}, qintosec 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]]


Considered below are negri, wizard, tritonic, septimin, slender, triton, merman, marvo, marvolo, amavil, enneaportent, submajor, and alphorn.  
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]].  


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.
Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.


The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.
The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.


Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)
Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)


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


Negri tempers out the [[negri comma]] in the 5-limit, [[49/48]] and [[225/224]] in the 7-limit. It can be extended naturally to the 2.3.5.7.13 subgroup by adding [[91/90]] to the comma list; this will be discussed below under the title of negra.  
Wizard has a [[semi-octave]] period and is generated by an interval that can be treated as [[~]][[17/15]]. The semi-octave complement of this interval is ~[[5/4]]. Wizard can be described as {{nowrap| 22 & 72 }}. Its [[ploidacot]] is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.  


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16384
[[Comma list]]: 225/224, 118098/117649
 
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216
 
[[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 ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}
 
[[Badness]] (Sintel): 1.03
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 4000/3993
 
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
Optimal tunings:  
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


{{Multival|legend=1| 4 -3 -14 }}
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


[[POTE generator]]: ~16/15 = 125.7549
Badness (Sintel): 0.613


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
==== Lizard ====
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.086856
Comma list: 225/224, 351/350, 364/363, 385/384


== 7-limit ==
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
Subgroup: 2.3.5.7


[[Comma list]]: 49/48, 225/224
Optimal tunings:  
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


{{Multival|legend=1| 4 -3 2 -14 -8 13 }}
Badness (Sintel): 0.900


[[POTE generator]]: ~15/14 = 125.608
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


[[Badness]]: 0.026483
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}


=== Negra ===
Optimal tunings:
This is the 2.3.5.7.13 extension of negri.  
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


Subgroup: 2.3.5.7.13
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


[[Comma list]]: 49/48, 65/64, 91/90
Badness (Sintel): 0.741


[[Sval]] [[Mapping]]: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -3 }}]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


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


[[Gencom]]: [2 14/13; 49/48 65/64 91/90]
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}


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


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


== 11-limit ==
Badness (Sintel): 0.955
 
==== Gizzard ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 1573/1568
 
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
 
Optimal tunings:
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
 
Badness (Sintel): 0.837
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
 
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
 
Badness (Sintel): 0.694
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
 
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}
 
{{Optimal ET sequence|legend=0| 72, 94, 166g }}
 
Badness (Sintel): 0.901
 
=== Mage ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


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


POTE generator: ~15/14 = 126.474
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


Vals: {{Val list| 9, 10, 19 }}
Badness (Sintel): 1.91


Badness: 0.026190
== Tritonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


=== 13-limit ===
Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.
Subgroup: 2.3.5.7.11.13
 
[[Subgroup]]: 2.3.5.7


Comma list: 45/44, 49/48, 56/55, 78/77
[[Comma list]]: 225/224, 50421/50000


Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


POTE generator: ~14/13 = 126.431
[[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 }}


Vals: {{Val list| 9, 10, 19 }}
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


Badness: 0.017833
[[Badness]] (Sintel): 1.20


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


Comma list: 49/48, 100/99, 225/224
Comma list: 121/120, 225/224, 441/440


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


POTE generator: ~15/14 = 124.767
Optimal tunings:  
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


Vals: {{Val list| 19, 29, 48d, 77cdd }}
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


Badness: 0.038679
Badness (Sintel): 0.782


=== 13-limit ===
==== 13-limit ====
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: 105/104, 121/120, 196/195, 275/273
 
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}
 
Optimal tunings:
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}
 
{{Optimal ET sequence|legend=0| 29, 31, 60e }}
 
Badness (Sintel): 0.950
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
 
Optimal tunings:
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
Badness (Sintel): 0.973
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
 
Optimal tunings:
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
Badness (Sintel): 1.03
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


POTE generator: ~14/13 = 124.716
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


Vals: {{Val list| 19, 29, 48df, 77cddf }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness: 0.024383
Badness (Sintel): 1.04


== Negric ==
=== Tritoni ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 33/32, 49/48, 77/75
Comma list: 225/224, 385/384, 27783/27500
 
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Mapping: [{{val| 1 2 2 3 3 }}, {{val| 0 -4 3 -2 4 }}]
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


POTE generator: ~15/14 = 127.039
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Vals: {{Val list| 9, 19e }}
Badness (Sintel): 1.50


Badness: 0.030617
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


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


Comma list: 33/32, 49/48, 65/64, 91/90
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
[[Comma list]]: 225/224, 84035/82944


POTE generator: ~14/13 = 127.039
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


Vals: {{Val list| 9, 19e }}
[[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 }}


Badness: 0.020205
{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}


== Negroni ==
[[Badness]] (Sintel): 1.38
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 55/54, 225/224
Comma list: 225/224, 245/242, 385/384


Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


POTE generator: ~15/14 = 124.539
Optimal tunings:  
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


Vals: {{Val list| 10, 19e, 29, 77cddee }}
{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}


Badness: 0.035296
Badness (Sintel): 1.04


=== 13-limit ===
=== 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, 144/143, 196/195, 245/242
 
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
 
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}
 
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}
 
Badness (Sintel): 0.955
 
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot.


POTE generator: ~14/13 = 124.545
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.  


Vals: {{Val list| 10, 19e, 29, 77cddeef }}
[[Subgroup]]: 2.3.5.7


Badness: 0.021559
[[Comma list]]: 225/224, 2500000/2470629


== Wilsec ==
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 41, 84, 125 }}
 
[[Badness]] (Sintel): 1.39
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 121/120, 225/224
Comma list: 225/224, 441/440, 1344/1331


Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


POTE generator: ~11/8 = 537.186
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


Vals: {{Val list| 9, 20, 29, 38d, 67cdde }}
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


Badness: 0.041886
Badness (Sintel): 1.20


=== 13-limit ===
==== 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: 144/143, 225/224, 364/363, 441/440


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


POTE generator: ~11/8 = 537.208
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Vals: {{Val list| 9, 20, 29, 38df, 67cddef }}
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Badness: 0.025192
Badness (Sintel): 1.14


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


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
Comma list: 225/224, 385/384, 532400/531441


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


POTE generator: ~11/8 = 537.230
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


Vals: {{Val list| 9, 20g, 29g, 38df, 67cddefg }}
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


Badness: 0.021778
Badness (Sintel): 1.46


=== 19-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
Comma list: 225/224, 325/324, 385/384, 10648/10647


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


POTE generator: ~11/8 = 537.214
Optimal tunings:  
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


Vals: {{Val list| 9, 20g, 29g, 38df, 67cddefgh }}
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


Badness: 0.016828
Badness (Sintel): 1.47


= Passive =
== Slender ==
{{see also | Archytas clan #Passion }}
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.


Subgroup: 2.3.5.7
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  


[[Comma list]]: 225/224, 256/245
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
[[Comma list]]: 225/224, 589824/588245


[[POTE generator]]: ~16/15 = 98.809
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


{{Val list|legend=1| 1, 11, 12, 49dd }}
[[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 }}


[[Badness]]: 0.075055
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


= Wizard =
[[Badness]] (Sintel): 1.44
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
{{See also|Wizard}}


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


[[Comma list]]: 225/224, 118098/117649
Comma list: 225/224, 385/384, 1331/1323


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


Mapping generators: ~1225/864, ~245/216
Optimal tunings:
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


[[POTE generator]]: ~5/4 = 383.256
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
Badness (Sintel): 0.838


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.040846
Comma list: 225/224, 275/273, 385/384, 1331/1323


Scales: [[wizard22]]
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


== 11-limit ==
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Comma list: 225/224, 385/384, 4000/3993
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


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


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


POTE generator: ~5/4 = 383.232
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.  


Vals: {{Val list| 22, 50, 72, 166, 238c, 310c }}
[[Subgroup]]: 2.3.5.7


Badness: 0.018539
[[Comma list]]: 225/224, 1029/1000


Scales: [[wizard22]]
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


=== Lizard ===
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13
* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}
: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}


Comma list: 225/224, 351/350, 364/363, 385/384
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
[[Badness]] (Sintel): 1.50


Mapping generators: ~99/70, ~25/22
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~5/4 = 383.389
Comma list: 45/44, 56/55, 1029/1000


Vals: {{Val list| 22, 50, 72, 122, 194df }}
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


Badness: 0.021781
Optimal tunings:  
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


==== 17-limit ====
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
Badness (Sintel): 1.51


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
== Marvolo ==
[[Subgroup]]: 2.3.5.7


Mapping generators: ~17/12, ~17/15
[[Comma list]]: 225/224, 156250000/155649627


POTE generator: ~5/4 = 383.381
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


Vals: {{Val list| 22, 50, 72, 122g, 194dfg }}
[[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 }}


Badness: 0.014536
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


==== 19-limit ====
[[Badness]] (Sintel): 2.11
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
=== 11-limit ===
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 }}]
Comma list: 225/224, 441/440, 4000/3993


Mapping generators: ~17/12, ~17/15
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


POTE generator: ~5/4 = 383.477
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


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


Badness: 0.015702
Badness (Sintel): 0.958


=== Gizzard ===
=== 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: 169/168, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
 
Optimal tunings:
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}
 
{{Optimal ET sequence|legend=0| 29, 43, 72 }}
 
Badness (Sintel): 0.887
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


Mapping generators: ~99/70, ~25/22
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


POTE generator: ~5/4 = 383.170
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Vals: {{Val list| 72, 166, 238cf }}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Badness: 0.020252
Badness (Sintel): 0.760


==== 17-limit ====
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Mapping generators: ~17/12, ~17/15
Optimal tunings:
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


POTE generator: ~5/4 = 383.175
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Vals: {{Val list| 72, 166g, 238cfg }}
Badness (Sintel): 0.895


Badness: 0.013624
== Decic ==
{{Main| Decic }}


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]].


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


Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
[[Comma list]]: 225/224, 16807/16384


Mapping generators: ~17/12, ~17/15
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


POTE generator: ~5/4 = 383.138
[[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 }}


Vals: {{Val list| 72, 94, 166g }}
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


Badness: 0.014810
[[Badness]] (Sintel): 2.26


== Mage ==
=== 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, 3087/3025
 
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
Badness (Sintel): 2.11


Mapping generators: ~77/54, ~55/48
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 383.124
Comma list: 105/104, 144/143, 196/195, 2200/2197


Vals: {{Val list| 22, 50e, 72ee, 94ee }}
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


Badness: 0.057799
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}})


= Triton =
{{Optimal ET sequence|legend=0| 10, 40, 50 }}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''


Subgroup: 2.3.5.7
Badness (Sintel): 1.52


[[Comma list]]: 225/224, 1029/1000
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
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}})


[[POTE generator]]: ~7/5 = 568.865
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


{{Val list|legend=1| 2, 17d, 19, 78bd, 97bd }}
Badness (Sintel): 1.28


[[Badness]]: 0.059245
=== Splendecic ===
Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[fantastic]] rank-3 temperament, its name is a portmanteau of ''splendid'' and ''decic''.


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


Comma list: 45/44, 56/55, 1029/1000
Comma list: 225/224, 1617/1600, 2401/2376
 
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}
 
Badness (Sintel): 1.98
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


POTE generator: ~7/5 = 569.144
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


Vals: {{Val list| 2, 17d, 19, 59bde, 78bde, 97bde }}
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: 0.045675
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


= Tritonic =
Badness (Sintel): 1.57
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''


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


[[Comma list]]: 225/224, 50421/50000
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374


[[Mapping]]: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
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}})


[[POTE generator]]: ~7/5 = 580.286
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
Badness (Sintel): 1.33


[[Badness]]: 0.047578
=== Prodecic ===
Prodecic (50e & 60e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the [[prodigy]] rank-3 temperament, its name is a portmanteau of ''prodigy'' and ''decic''.


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


Comma list: 121/120, 225/224, 441/440
Comma list: 225/224, 441/440, 5929/5832


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}


POTE generator: ~7/5 = 580.267
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}})


Vals: {{Val list| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness: 0.023659
Badness (Sintel): 2.20


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


Comma list: 105/104, 121/120, 196/195, 275/273
Comma list: 105/104, 196/195, 275/273, 5929/5832
 
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}
 
Optimal tunings:
* WE: ~15/14 = 120.1654{{c}}, ~3/2 = 701.4683{{c}} (~91/90 = 19.5242{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.7175{{c}} (~91/90 = 19.2825{{c}})
 
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
 
Badness (Sintel): 1.73
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288
 
Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
 
Badness (Sintel): 1.41
 
== Enneaportent ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 40353607/40310784


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


POTE generator: ~7/5 = 580.108
[[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 }}


Vals: {{Val list| 29, 31, 60e, 151cde }}
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


Badness: 0.022993
[[Badness]] (Sintel): 2.37


== Tritoni ==
=== 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, 12005/11979
 
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}
 
Optimal tunings:
* WE: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}})
* CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}})
 
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
 
Badness (Sintel): 1.01
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
Comma list: 169/168, 225/224, 364/363, 1716/1715


POTE generator: ~7/5 = 580.389
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


Vals: {{Val list| 31, 91, 122, 153d }}
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: 0.045456
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


= Merman =
Badness (Sintel): 0.922
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''


Subgroup: 2.3.5.7
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


[[Comma list]]: 225/224, 2500000/2470629
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
[[Comma list]]: 225/224, 781250000/771895089


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


[[POTE generator]]: ~7/5 = 585.585
[[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 }}


{{Val list|legend=1| 41, 84, 125 }}
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


[[Badness]]: 0.055078
[[Badness]] (Sintel): 2.44


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


Comma list: 225/224, 441/440, 1344/1331
Comma list: 225/224, 385/384, 236328125/234365481


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


POTE generator: ~7/5 = 585.606
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Vals: {{Val list| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Badness: 0.036383
Badness (Sintel): 2.96


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


Comma list: 144/143, 225/224, 364/363, 441/440
Comma list: 225/224, 325/324, 385/384, 831875/830466
 
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
 
Optimal tunings:
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Badness (Sintel): 2.16


POTE generator: ~7/5 = 585.657
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Vals: {{Val list| 41, 84, 125e, 209ef, 293ef }}
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


Badness: 0.027544
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


= Septimin =
Optimal tunings:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


[[Comma list]]: 225/224, 84035/82944
Badness (Sintel): 1.96


[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


[[POTE generator]]: ~7/6 = 263.632
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


{{Val list|legend=1| 41, 91, 132d }}
Optimal tunings:
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


[[Badness]]: 0.054502
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


== 11-limit ==
Badness (Sintel): 1.71
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 2401/2376
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


POTE generator: ~7/6 = 263.634
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Vals: {{Val list| 41, 91, 223cdef }}
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Badness: 0.031309
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


== 13-limit ==
Badness (Sintel): 1.57
Subgroup: 2.3.5.7.11.13


Comma list: 105/104, 144/143, 196/195, 245/242
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


POTE generator: ~7/6 = 263.700
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


Vals: {{Val list| 41, 91 }}
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Badness: 0.023117
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


= Slender =
Badness (Sintel): 1.50
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 589824/588245
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


[[POTE generator]]: ~49/48 = 38.413
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


{{Val list|legend=1| 31, 94, 125 }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


[[Badness]]: 0.056934
Badness (Sintel): 1.53


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


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


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


POTE generator: ~49/48 = 38.387
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


Vals: {{Val list| 31, 63, 94, 125 }}
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Badness: 0.025342
Badness (Sintel): 2.21


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


Comma list: 225/224, 275/273, 385/384, 1331/1323
Comma list: 225/224, 364/363, 441/440, 6125/6084
 
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}
 
Optimal tunings:
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}
 
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}
 
Badness (Sintel): 1.83


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~49/48 = 38.314
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Vals: {{Val list| 31, 63, 94 }}
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Badness: 0.025913
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


= Marvo =
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
{{see also| Gravity family }}


Subgroup: 2.3.5.7
Badness (Sintel): 1.87


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


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


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


[[POTE generator]]: ~27/20 = 516.694
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| 65d, 72, 137, 209, 281, 569bcc }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


[[Badness]]: 0.097627
Badness (Sintel): 1.68


== 11-limit ==
== Alphorn ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7


Comma list: 225/224, 243/242, 4000/3993
[[Comma list]]: 225/224, 5764801/5668704


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


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


Vals: {{Val list| 65d, 72, 281, 353c, 425bc, 497bc }}
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Badness: 0.031685
[[Badness]] (Sintel): 3.27


== 13-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 225/224, 243/242, 351/350, 1625/1617
Comma list: 225/224, 385/384, 12250/11979


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


POTE generator: ~27/20 = 516.730
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Vals: {{Val list| 65d, 72, 137, 209, 281f, 490bcf }}
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Badness: 0.026882
Badness (Sintel): 2.43


= Marvolo =
== Misneb ==
Subgroup: 2.3.5.7
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


[[Comma list]]: 225/224, 156250000/155649627
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
[[Comma list]]: 225/224, 4194304/4117715


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


[[POTE generator]]: ~21/20 = 83.348
[[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 }}


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


[[Badness]]: 0.083338
[[Badness]] (Sintel): 3.57


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


Comma list: 225/224, 441/440, 4000/3993
Comma list: 99/98, 176/175, 1310720/1294139


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


POTE generator: ~21/20 = 83.340
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


Vals: {{Val list| 29, 43, 72 }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.028965
Badness (Sintel): 2.82


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


Comma list: 169/168, 225/224, 364/363, 441/440
Comma list: 99/98, 176/175, 640/637, 847/845
 
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


POTE generator: ~21/20 = 83.330
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Vals: {{Val list| 29, 43, 72, 115f }}
Badness (Sintel): 1.88


Badness: 0.021470
=== Musneb ===
Subgroup: 2.3.5.7.11


= Amavil =
Comma list: 225/224, 385/384, 66550/64827
== 5-limit (Mabila) ==
Subgroup: 2.3.5


[[Comma list]]: 268435456/263671875
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


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


[[POTE generator]]: ~512/375 = 529.6849
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
Badness (Sintel): 2.89


[[Badness]]: 0.232481
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


== 7-limit ==
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.  
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 17496/16807
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
[[Comma list]]: 225/224, 125000000/121060821


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


[[POTE generator]]: ~48/35 = 529.979
[[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 }}


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
{{Optimal ET sequence|legend=1| 51, 53 }}


[[Badness]]: 0.109625
[[Badness]] (Sintel): 3.64


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


Comma list: 99/98, 176/175, 864/847
Comma list: 121/120, 225/224, 22000/21609


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


POTE generator: ~15/11 = 529.974
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


Vals: {{Val list| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=0| 51, 53 }}


Badness: 0.042649
Badness (Sintel): 2.46


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


Comma list: 78/77, 99/98, 144/143, 176/175
Comma list: 121/120, 225/224, 275/273, 1040/1029


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


POTE generator: ~15/11 = 529.951
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


Vals: {{Val list| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=0| 51f, 53 }}


Badness: 0.025791
Badness (Sintel): 1.96


= Enneaportent =
== Naiadical ==
Subgroup: 2.3.5.7
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.  


[[Comma list]]: 225/224, 40353607/40310784
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
[[Comma list]]: 225/224, 823543/800000


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


[[POTE generator]]: ~5/4 = 383.165
[[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 }}


{{Val list|legend=1| 9, 63, 72, 495bcd }}
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


[[Badness]]: 0.093679
[[Badness]] (Sintel): 3.67


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


Comma list: 225/224, 385/384, 12005/11979
Comma list: 225/224, 245/242, 1617/1600


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


POTE generator: ~5/4 = 383.146
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


Vals: {{Val list| 9, 63, 72, 423cd, 495bcd }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Badness: 0.030426
Badness (Sintel): 2.00


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


Comma list: 169/168, 225/224, 364/363, 1716/1715
Comma list: 105/104, 196/195, 245/242, 1001/1000


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


POTE generator: ~5/4 = 383.047
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


Vals: {{Val list| 9, 63, 72, 279cf }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Badness: 0.022322
Badness (Sintel): 1.43


= Submajor =
=== 17-limit ===
Subgroup: 2.3.5
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 69198046875/68719476736
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


[[POTE generator]]: ~10125/8192 = 362.321
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| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


[[Badness]]: 0.130236
Badness (Sintel): 1.26


== 7-limit ==
== Quintannic ==
Subgroup: 2.3.5.7
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.  


[[Comma list]]: 225/224, 51200/50421
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
[[Comma list]]: 225/224, 9805926501/9765625000


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


[[POTE generator]]: ~49/40 = 362.255
[[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 }}


{{Val list|legend=1| 10, 33, 43, 53 }}
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


[[Badness]]: 0.060533
[[Badness]] (Sintel): 3.81


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


Comma list: 225/224, 385/384, 6655/6561
Comma list: 225/224, 441/440, 43923/43750


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


POTE generator: ~27/22 = 362.101
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Vals: {{Val list| 10, 43e, 53, 116, 169de, 285cde }}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.050582
Badness (Sintel): 1.74


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


Comma list: 169/168, 225/224, 275/273, 385/384
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
 
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
 
Optimal tunings:
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


POTE generator: ~16/13 = 362.105
Badness (Sintel): 1.35


Vals: {{Val list| 10, 43e, 53, 116, 169de, 285cdef }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.027689
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


== Interpental ==
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 51200/50421
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


POTE generator: ~49/40 = 362.418
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
Badness (Sintel): 1.17


POTE generator: ~49/40 = 362.418
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


Vals: {{Val list| 43, 53, 96, 149d }}
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.


Badness: 0.051806
[[Subgroup]]: 2.3.5.7


=== 13-limit ===
[[Comma list]]: 225/224, 122880/117649
Subgroup: 2.3.5.7.11.13


Comma list: 99/98, 169/168, 176/175, 640/637
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


POTE generator: ~16/13 = 362.402
[[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 }}


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}


POTE generator: ~16/13 = 362.402
[[Badness]] (Sintel): 4.37


Vals: {{Val list| 43, 53, 96, 149d }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.029680
Comma list: 121/120, 225/224, 352/343


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


[[Comma list]]: 225/224, 5764801/5668704
Optimal tunings:  
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


[[Mapping]]: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
{{Optimal ET sequence|legend=0| 22, 55d }}


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
Badness (Sintel): 2.46


[[POTE generator]]: ~48/35 = 556.221
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


[[Badness]]: 0.129258
[[Subgroup]]: 2.3.5.7


== 11-limit ==
[[Comma list]]: 225/224, 5971968/5764801
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 12250/11979
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
[[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 }}


POTE generator: ~11/8 = 556.144
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


Vals: {{Val list| 28d, 41, 315cde }}
[[Badness]] (Sintel): 4.53


Badness: 0.073459
=== 11-limit ===
Subgroup: 2.3.5.7.11


= Tertiosec =
Comma list: 99/98, 176/175, 65536/65219
Subgroup: 2.3.5


[[Comma list]]: {{monzo| -89 21 24 }}
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


[[Mapping]]: [{{val| 3 7 5 }}, {{val| 0 -8 7 }}]
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


[[POTE generator]]: ~16/15 = 112.281
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


{{Val list|legend=1| 21, 54, 75, 96, 171 }}
Badness (Sintel): 2.26


[[Badness]]: 0.851003
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


== 7-limit ==
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>.
Subgroup: 2.3.5.7
 
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 3 7 5 9 }}, {{val| 0 -8 7 -2 }}]
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


{{Multival|legend=1| 24 -21 6 -89 -58 73 }}
[[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 }}


[[POTE generator]]: ~15/14 = 112.283
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


{{Val list|legend=1| 21, 54, 75, 96, 171d }}
[[Badness]] (Sintel): 10.9


[[Badness]]: 0.431636
=== 11-limit ===
 
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 3840/3773, 12005/11979
Comma list: 225/224, 3840/3773, 12005/11979


Mapping: [{{val| 3 7 5 9 9 }}, {{val| 0 -8 7 -2 5 }}]
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
 
Optimal tunings:
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}


POTE generator: ~15/14 = 112.171
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


Vals: {{Val list| 21, 54, 75e }}
Badness (Sintel): 5.74


Badness: 0.173485
== References ==


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