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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]]
* [[Pythagorean family #Compton|compton]] ({225/224, 250047/250000}, pythagorean family)
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* [[Qintosec family|qintosec]] ({225/224, 2560000/2470629}, qintosec 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]]


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.  


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


Subgroup: 2.3.5
[[Comma list]]: 225/224, 118098/117649
 
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216


[[Comma list]]: 16875/16384
[[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 }}


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}


{{Multival|legend=1| 4 -3 -14 }}
[[Badness]] (Sintel): 1.03


[[POTE generator]]: ~16/15 = 125.7549
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
Comma list: 225/224, 385/384, 4000/3993


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


Subgroup: 2.3.5.7
Optimal tunings:  
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


[[Comma list]]: 49/48, 225/224
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
Badness (Sintel): 0.613


{{Multival|legend=1| 4 -3 2 -14 -8 13 }}
==== Lizard ====
Subgroup: 2.3.5.7.11.13


[[POTE generator]]: ~15/14 = 125.608
Comma list: 225/224, 351/350, 364/363, 385/384


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}


=== Negra ===
Optimal tunings:
This is the 2.3.5.7.13 extension of negri.  
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{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): 0.900


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


Gencom mapping: [{{val| 1 2 2 3 0 4 }}, {{val| 0 -4 3 -2 0 -3 }}]
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


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


POTE generator: ~14/13 = 125.567
Optimal tunings:  
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


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


== 11-limit ==
Badness (Sintel): 0.741


===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
Optimal tunings:
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
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: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}


POTE generator: ~15/14 = 126.474
Optimal tunings:  
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


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


Badness: 0.0262
Badness (Sintel): 1.91


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


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


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 }}


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


== Negril ==
[[Badness]] (Sintel): 1.20


=== 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: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
Optimal tunings:  
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


POTE generator: ~15/14 = 124.767
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


{{Val list|legend=1| 19, 29, 48d, 77cdd }}
Badness (Sintel): 0.782


Badness: 0.0387
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
Comma list: 105/104, 121/120, 196/195, 275/273
 
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


Subgroup: 2.3.5.7.11.13
Optimal tunings:  
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


Comma list: 49/48, 65/64, 91/90, 875/858
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
Badness (Sintel): 0.950


POTE generator: ~14/13 = 124.716
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 19, 29, 48df, 77cddf }}
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


Badness: 0.0244
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


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


Subgroup: 2.3.5.7.11
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Comma list: 33/32, 49/48, 77/75
Badness (Sintel): 0.973


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


POTE generator: ~15/14 = 127.039
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


Badness: 0.0306
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Subgroup: 2.3.5.7.11.13
Badness (Sintel): 1.03


Comma list: 33/32, 49/48, 65/64, 91/90
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


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


POTE generator: ~14/13 = 127.039
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
Optimal tunings:
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


Badness: 0.0202
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


== Negroni ==
Badness (Sintel): 1.04


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


Comma list: 49/48, 55/54, 225/224
Comma list: 225/224, 385/384, 27783/27500


Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


POTE generator: ~15/14 = 124.539
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


{{Val list|legend=1| 10, 19e, 29, 77cddee }}
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Badness: 0.0353
Badness (Sintel): 1.50


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


Subgroup: 2.3.5.7.11.13
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.  


Comma list: 49/48, 55/54, 65/64, 91/90
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
[[Comma list]]: 225/224, 84035/82944


POTE generator: ~14/13 = 124.545
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


{{Val list|legend=1| 10, 19e, 29, 77cddeef }}
[[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: 0.0216
{{Optimal ET sequence|legend=1| 41, 91, 132d }}


== Wilsec ==
[[Badness]] (Sintel): 1.38


=== 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, 245/242, 385/384


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


POTE generator: ~11/8 = 537.186
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| 9, 20, 29, 38d, 67cdde }}
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


Badness: 0.0419
Badness (Sintel): 1.04


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
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| 41, 91 }}
Badness (Sintel): 0.955
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''
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.


Subgroup: 2.3.5.7.11.13
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.  


Comma list: 49/48, 65/64, 91/90, 121/120
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
[[Comma list]]: 225/224, 2500000/2470629


POTE generator: ~11/8 = 537.208
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


{{Val list|legend=1| 9, 20, 29, 38df, 67cddef }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
: [[error map]]: {{val| +0.390 -0.435 -1.630 +1.634 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}}
: error map: {{val| 0.000 -1.104 -2.423 +0.657 }}


Badness: 0.0252
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


=== 17-limit ===
[[Badness]] (Sintel): 1.39


Subgroup: 2.3.5.7.11.13.17
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
Comma list: 225/224, 441/440, 1344/1331


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


POTE generator: ~11/8 = 537.230
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefg }}
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


Badness: 0.0218
Badness (Sintel): 1.20


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


Subgroup: 2.3.5.7.11.13.17.19
Comma list: 144/143, 225/224, 364/363, 441/440


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


POTE generator: ~11/8 = 537.214
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefgh }}
Badness (Sintel): 1.14


Badness: 0.0168
=== Mermaid ===
Subgroup: 2.3.5.7.11


= Passive =
Comma list: 225/224, 385/384, 532400/531441
{{see also | Archytas clan #Passion }}


Subgroup: 2.3.5.7
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


[[Comma list]]: 225/224, 256/245
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


[[POTE generator]]: ~16/15 = 98.809
Badness (Sintel): 1.46


{{Val list|legend=1| 12, 49dd }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.0751
Comma list: 225/224, 325/324, 385/384, 10648/10647


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


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
Optimal tunings:  
{{See also|Wizard}}
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


[[Comma list]]: 225/224, 118098/117649
Badness (Sintel): 1.47


[[Mapping]]: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
== 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.


Mapping generators: ~1225/864, ~245/216
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.


[[POTE generator]]: ~5/4 = 383.256
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
[[Comma list]]: 225/224, 589824/588245


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


[[Badness]]: 0.0408
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
: [[error map]]: {{val| +0.382 -0.725 -1.295 +1.765 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}}
: error map: {{val| 0.000 -1.257 -2.235 +0.727 }}


Scales: [[wizard22]]
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


== 11-limit ==
[[Badness]] (Sintel): 1.44


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


Comma list: 225/224, 385/384, 4000/3993
Comma list: 225/224, 385/384, 1331/1323
 
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
 
Optimal tunings:
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}
 
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
Badness (Sintel): 0.838


Mapping generators: ~99/70, ~25/22
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 383.232
Comma list: 225/224, 275/273, 385/384, 1331/1323


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c }}
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Badness: 0.0185
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Scales: [[wizard22]]
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


=== Lizard ===
Badness (Sintel): 1.07


Subgroup: 2.3.5.7.11.13
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


Comma list: 225/224, 351/350, 364/363, 385/384
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.


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


Mapping generators: ~99/70, ~25/22
[[Comma list]]: 225/224, 1029/1000


POTE generator: ~5/4 = 383.389
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


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


Badness: 0.0218
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


==== 17-limit ====
[[Badness]] (Sintel): 1.50


Subgroup: 2.3.5.7.11.13.17
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
Comma list: 45/44, 56/55, 1029/1000


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


Mapping generators: ~17/12, ~17/15
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


POTE generator: ~5/4 = 383.381
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
Badness (Sintel): 1.51


Badness: 0.0145
== Marvolo ==
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 156250000/155649627


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


Subgroup: 2.3.5.7.11.13.17.19
[[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 }}


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
[[Badness]] (Sintel): 2.11


Mapping generators: ~17/12, ~17/15
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~5/4 = 383.477
Comma list: 225/224, 441/440, 4000/3993


{{Val list|legend=1| 22h, 50, 72, 122g, 194dfg }}
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


Badness: 0.0157
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


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


=== Gizzard ===
Badness (Sintel): 0.958


=== 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 }}


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


Mapping generators: ~99/70, ~25/22
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


POTE generator: ~5/4 = 383.170
Badness (Sintel): 0.887


{{Val list|legend=1| 72, 166, 238cf }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0203
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


</div></div>
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


==== 17-limit ====
Optimal tunings:
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Subgroup: 2.3.5.7.11.13.17
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Badness (Sintel): 0.760


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Mapping generators: ~17/12, ~17/15
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


POTE generator: ~5/4 = 383.175
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


{{Val list|legend=1| 72, 166g, 238cfg }}
Optimal tunings:
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


Badness: 0.0136
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


</div></div>
Badness (Sintel): 0.895


==== 19-limit ====
== Decic ==
{{Main| Decic }}


Subgroup: 2.3.5.7.11.13.17.19
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]].  


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


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
[[Subgroup]]: 2.3.5.7


Mapping generators: ~17/12, ~17/15
[[Comma list]]: 225/224, 16807/16384


POTE generator: ~5/4 = 383.138
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


{{Val list|legend=1| 72, 94, 166g }}
[[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 }}


Badness: 0.0148
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


== Mage ==
[[Badness]] (Sintel): 2.26


=== 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: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}


Mapping generators: ~77/54, ~55/48
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}})


POTE generator: ~5/4 = 383.124
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


{{Val list|legend=1| 22, 50e, 72ee, 94ee }}
Badness (Sintel): 2.11


Badness: 0.0578
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


= Triton =
Comma list: 105/104, 144/143, 196/195, 2200/2197


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


Subgroup: 2.3.5.7
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}})


[[Comma list]]: 225/224, 1029/1000
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
Badness (Sintel): 1.52


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


[[POTE generator]]: ~7/5 = 568.865
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


{{Val list|legend=1| 19, 78bd, 97bd }}
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


[[Badness]]: 0.0592
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}})


== 11-limit ==
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Subgroup: 2.3.5.7.11
Badness (Sintel): 1.28


Comma list: 45/44, 56/55, 1029/1000
=== 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''.


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
Subgroup: 2.3.5.7.11


POTE generator: ~7/5 = 569.144
Comma list: 225/224, 1617/1600, 2401/2376


{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}


Badness: 0.0457
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}})


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


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
Badness (Sintel): 1.98


Subgroup: 2.3.5.7
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 225/224, 50421/50000
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


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


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


[[POTE generator]]: ~7/5 = 580.286
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
Badness (Sintel): 1.57


== 11-limit ==
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7.11
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374


Comma list: 121/120, 225/224, 441/440
Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 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 = 580.267
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


{{Val list|legend=1| 29, 31, 60e }}
Badness (Sintel): 1.33


Badness: 0.0237
=== 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''.


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


Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 441/440, 5929/5832


Comma list: 105/104, 121/120, 196/195, 275/273
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}


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


POTE generator: ~7/5 = 580.108
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


{{Val list|legend=1| 29, 31, 60e, 151cde }}
Badness (Sintel): 2.20


Badness: 0.0230
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


== Tritoni ==
Comma list: 105/104, 196/195, 275/273, 5929/5832


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


Comma list: 225/224, 385/384, 27783/27500
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}})


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


POTE generator: ~7/5 = 580.389
Badness (Sintel): 1.73


{{Val list|legend=1| 31, 91, 122, 153d }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0455
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


= Merman =
Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
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}})


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


[[Comma list]]: 225/224, 2500000/2470629
Badness (Sintel): 1.41


[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
[[Comma list]]: 225/224, 40353607/40310784


[[POTE generator]]: ~7/5 = 585.585
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


{{Val list|legend=1| 41, 84, 125 }}
[[Optimal tuning]]s:
* [[WE]]: ~2592/2401 = 133.4174{{c}}, ~12005/6912 = 950.7667{{c}} (~1728/1715 = 16.8452{{c}})
: [[error map]]: {{val| +0.756 -0.422 -1.395 +0.298 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}})
: error map: {{val| 0.000 -1.361 -3.277 -1.565 }}


[[Badness]]: 0.0551
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


== 11-limit ==
[[Badness]] (Sintel): 2.37


=== 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, 12005/11979


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


POTE generator: ~7/5 = 585.606
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}})


{{Val list|legend=1| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.0364
Badness (Sintel): 1.01
 
== 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: 169/168, 225/224, 364/363, 1716/1715


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


POTE generator: ~7/5 = 585.657
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}})


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.0275
Badness (Sintel): 0.922


= Septimin =
== Gracecordial ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 84035/82944
[[Comma list]]: 225/224, 781250000/771895089


[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


[[POTE generator]]: ~7/6 = 263.632
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


{{Val list|legend=1| 41, 91, 132 }}
[[Badness]] (Sintel): 2.44


[[Badness]]: 0.054502
=== 11-limit ===
 
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 2401/2376
Comma list: 225/224, 385/384, 236328125/234365481


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


POTE generator: ~7/6 = 263.634
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, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Badness: 0.031309
Badness (Sintel): 2.96


== 13-limit ==
==== 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: 225/224, 325/324, 385/384, 831875/830466
 
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


POTE generator: ~7/6 = 263.700
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Vals: {{Val list| 41, 91 }}
Badness (Sintel): 2.16


Badness: 0.023117
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


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


Subgroup: 2.3.5.7
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


[[Comma list]]: 225/224, 589824/588245
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
Badness (Sintel): 1.96


[[POTE generator]]: ~49/48 = 38.413
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


{{Val list|legend=1| 31, 94, 125 }}
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


[[Badness]]: 0.0569
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


== 11-limit ==
Optimal tunings:
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


Subgroup: 2.3.5.7.11
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Comma list: 225/224, 385/384, 1331/1323
Badness (Sintel): 1.71


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


POTE generator: ~49/48 = 38.387
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


{{Val list|legend=1| 31, 63, 94, 125 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Badness: 0.02534
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


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


Subgroup: 2.3.5.7.11.13
Badness (Sintel): 1.57


Comma list: 225/224, 275/273, 385/384, 1331/1323
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


POTE generator: ~49/48 = 38.314
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| 31, 63, 94 }}
Optimal tunings:
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Badness: 0.02591
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


= Marvo =
Badness (Sintel): 1.50
{{see also| Gravity family }}


Subgroup: 2.3.5.7
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Comma list: 225/224, 78125000/78121827
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


{{Multival|legend=1| 6 17 46 13 56 59 }}
Optimal tunings:
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


POTE generator: ~27/20 = 516.694
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


{{Val list|legend=1| 65d, 72, 137, 209, 281, 569bcc }}
Badness (Sintel): 1.53


Badness: 0.0976
=== Gracecord ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 225/224, 441/440, 109375/107811


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


Comma list: 225/224, 243/242, 4000/3993
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


POTE generator: ~27/20 = 516.699
Badness (Sintel): 2.21


{{Val list|legend=1| 65d, 72, 281, 353c, 425bc, 497bc }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0317
Comma list: 225/224, 364/363, 441/440, 6125/6084


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


Subgroup: 2.3.5.7.11.13
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


Comma list: 225/224, 243/242, 351/350, 1625/1617
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
Badness (Sintel): 1.83


POTE generator: ~27/20 = 516.730
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 65d, 72, 137, 209, 281f, 490bcf }}
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Badness: 0.0269
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


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


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


[[Comma list]]: 225/224, 156250000/155649627
Badness (Sintel): 1.87


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


[[POTE generator]]: ~21/20 = 83.348
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
Optimal tunings:
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


[[Badness]]: 0.0833
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


== 11-limit ==
Badness (Sintel): 1.68


Subgroup: 2.3.5.7.11
== Alphorn ==
[[Subgroup]]: 2.3.5.7


Comma list: 225/224, 441/440, 4000/3993
[[Comma list]]: 225/224, 5764801/5668704


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


POTE generator: ~21/20 = 83.340
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.3004{{c}}, ~35/24 = 644.4767{{c}}
: [[error map]]: {{val| +1.300 +0.569 -2.195 -2.043 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~35/24 = 643.8137{{c}}
: error map: {{val| 0.000 -0.936 -5.382 -4.924 }}


{{Val list|legend=1| 29, 43, 72 }}
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Badness: 0.0290
[[Badness]] (Sintel): 3.27


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


== 13-limit ==
Comma list: 225/224, 385/384, 12250/11979


Subgroup: 2.3.5.7.11.13
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


Comma list: 169/168, 225/224, 364/363, 441/440
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


POTE generator: ~21/20 = 83.330
Badness (Sintel): 2.43


{{Val list|legend=1| 29, 43, 72, 115f }}
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


Badness: 0.0215
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 4194304/4117715


= Amavil =
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
== 5-limit (mabila) ==
: mapping generators: ~2, ~15/8


Subgroup: 2.3.5
[[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 }}


[[Comma list]]: 268435456/263671875
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


[[Mapping]]: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
[[Badness]] (Sintel): 3.57


[[POTE generator]]: ~512/375 = 529.6849
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
Comma list: 99/98, 176/175, 1310720/1294139


[[Badness]]: 0.2325
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


== 7-limit ==
Optimal tunings:
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


[[Comma list]]: 225/224, 17496/16807
Badness (Sintel): 2.82


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
Comma list: 99/98, 176/175, 640/637, 847/845


[[POTE generator]]: ~48/35 = 529.979
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
Optimal tunings:
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


[[Badness]]: 0.1096
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


== 11-limit ==
Badness (Sintel): 1.88


=== Musneb ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 864/847
Comma list: 225/224, 385/384, 66550/64827
 
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
 
Optimal tunings:
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}
 
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
Badness (Sintel): 2.89


POTE generator: ~15/11 = 529.974
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


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


Badness: 0.0426
[[Subgroup]]: 2.3.5.7


== 13-limit ==
[[Comma list]]: 225/224, 125000000/121060821


Subgroup: 2.3.5.7.11.13
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 51, 53 }}


Comma list: 78/77, 99/98, 144/143, 176/175
[[Badness]] (Sintel): 3.64


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~15/11 = 529.951
Comma list: 121/120, 225/224, 22000/21609


{{Val list|legend=1| 9, 34d, 43, 77de }}
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


Badness: 0.0258
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


= Enneaportent =
{{Optimal ET sequence|legend=0| 51, 53 }}


Subgroup: 2.3.5.7
Badness (Sintel): 2.46


[[Comma list]]: 225/224, 40353607/40310784
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
Comma list: 121/120, 225/224, 275/273, 1040/1029


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


[[POTE generator]]: ~5/4 = 383.165
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


{{Val list|legend=1| 63, 72, 495bcd }}
{{Optimal ET sequence|legend=0| 51f, 53 }}


[[Badness]]: 0.0937
Badness (Sintel): 1.96


== 11-limit ==
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7


Comma list: 225/224, 385/384, 12005/11979
[[Comma list]]: 225/224, 823543/800000


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


POTE generator: ~5/4 = 383.146
[[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| 63, 72, 423cd, 495bcd }}
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


Badness: 0.0304
[[Badness]] (Sintel): 3.67


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


== 13-limit ==
Comma list: 225/224, 245/242, 1617/1600


Subgroup: 2.3.5.7.11.13
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


Comma list: 169/168, 225/224, 364/363, 1716/1715
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


POTE generator: ~5/4 = 383.047
Badness (Sintel): 2.00


{{Val list|legend=1| 63, 72, 279cf }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0223
Comma list: 105/104, 196/195, 245/242, 1001/1000


</div></div>
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


= Submajor =
Optimal tunings:
== 5-limit ==
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


Subgroup: 2.3.5
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


[[Comma list]]: 69198046875/68719476736
Badness (Sintel): 1.43


[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[POTE generator]]: ~10125/8192 = 362.321
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


[[Badness]]: 0.1302
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


== 7-limit ==
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Subgroup: 2.3.5.7
Badness (Sintel): 1.26


[[Comma list]]: 225/224, 51200/50421
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


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


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
[[Comma list]]: 225/224, 9805926501/9765625000


[[POTE generator]]: ~49/40 = 362.255
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


{{Val list|legend=1| 10, 33, 43, 53 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.9803{{c}}, ~10000/9261 = 139.9522{{c}}
: [[error map]]: {{val| +0.980 -1.214 -0.313 -0.243 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10000/9261 = 139.8184{{c}}
: error map: {{val| 0.000 -2.863 -2.136 -2.287 }}


[[Badness]]: 0.0605
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


== 11-limit ==
[[Badness]] (Sintel): 3.81


=== 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}}


{{Val list|legend=1| 53, 116, 169de, 285cde }}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.0506
Badness (Sintel): 1.74


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
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}}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
Badness (Sintel): 1.35
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
Optimal tunings:
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
 
Badness (Sintel): 1.17
 
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


Comma list: 169/168, 225/224, 275/273, 385/384
Hendeca tempers out the same 5-limit comma as [[porwell temperaments #Hendecatonic|hendecatonic]], and has a period of 1/11 octave. However, in this temperament, nine periods represent [[7/4]], the same as [[keemic temperaments #Undeka|undeka]]. It can be tuned to [[22edo]] or [[33edo]] using the [[patent val]], or [[55edo]] using the 55d val. It was named by [[Xenllium]] in 2025 as a low-accuracy variant of hendecatonic.


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~16/13 = 362.105
[[Comma list]]: 225/224, 122880/117649


{{Val list|legend=1| 53, 116, 169de, 285cdef }}
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


Badness: 0.0277
[[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 }}


== Interpental ==
[[Badness]] (Sintel): 4.37


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


Comma list: 99/98, 176/175, 51200/50421
Comma list: 121/120, 225/224, 352/343
 
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}
 
Optimal tunings:
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


POTE generator: ~49/40 = 362.418
{{Optimal ET sequence|legend=0| 22, 55d }}


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
Badness (Sintel): 2.46


POTE generator: ~49/40 = 362.418
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


{{Val list|legend=1| 43, 53, 96, 149d }}
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


Badness: 0.0518
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 5971968/5764801


=== 13-limit ===
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


Subgroup: 2.3.5.7.11.13
[[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 }}


Comma list: 99/98, 169/168, 176/175, 640/637
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


POTE generator: ~16/13 = 362.402
[[Badness]] (Sintel): 4.53


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~16/13 = 362.402
Comma list: 99/98, 176/175, 65536/65219


{{Val list|legend=1| 43, 53, 96, 149d }}
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


Badness: 0.0297
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


= Alphorn =
Badness (Sintel): 2.26


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


[[Comma list]]: 225/224, 5764801/5668704
Tertiosec may be described as the {{nowrap| 21 & 75 }} temperament. It was initially named ''tertiomar'' by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, but was changed to ''tertiosec'' in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.


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


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
[[Comma list]]: 225/224, 14495514624/13841287201


[[POTE generator]]: ~48/35 = 556.221
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
[[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 }}


[[Badness]]: 0.1293
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


== 11-limit ==
[[Badness]] (Sintel): 10.9


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


Comma list: 225/224, 385/384, 12250/11979
Comma list: 225/224, 3840/3773, 12005/11979
 
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
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: ~11/8 = 556.144
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


{{Val list|legend=1| 28d, 41, 315cde }}
Badness (Sintel): 5.74


Badness: 0.0735
== 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"