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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]]
* [[Wizard]] (+118098/117649) → [[Stearnsmic clan #Wizard|Stearnsmic clan]]
* ''[[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]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]


Considered below are negri, wizard, tritonic, septimin, slender, triton, merman, marvo, marvolo, amavil, enneaportent, submajor, and alphorn.  
Considered below are 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 =
== Tritonic ==
{{main| Negri }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


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.  
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16384
[[Comma list]]: 225/224, 50421/50000
 
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
[[Badness]] (Sintel): 1.20


{{Multival|legend=1| 4 -3 -14 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~16/15 = 125.7549
Comma list: 121/120, 225/224, 441/440
 
{{Mapping|legend=0| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}
 
Optimal tunings:
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}
 
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
 
Badness (Sintel): 0.782
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 121/120, 196/195, 275/273
 
{{Mapping|legend=0| 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


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


[[Badness]]: 0.086856
{{Mapping|legend=0| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


== 7-limit ==
Optimal tunings:
Subgroup: 2.3.5.7
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


[[Comma list]]: 49/48, 225/224
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
Badness (Sintel): 0.973


{{Multival|legend=1| 4 -3 2 -14 -8 13 }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


[[POTE generator]]: ~15/14 = 125.608
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


[[Badness]]: 0.026483
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


=== Negra ===
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
This is the 2.3.5.7.13 extension of negri.


Subgroup: 2.3.5.7.13
Badness (Sintel): 1.03


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


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


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


[[Gencom]]: [2 14/13; 49/48 65/64 91/90]
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


POTE generator: ~14/13 = 125.567
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
Badness (Sintel): 1.04


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


Comma list: 45/44, 49/48, 56/55
Comma list: 225/224, 385/384, 27783/27500
 
{{Mapping|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
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 = 126.474
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Vals: {{Val list| 9, 10, 19 }}
Badness (Sintel): 1.50


Badness: 0.026190
== 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
 
[[Subgroup]]: 2.3.5.7


Comma list: 45/44, 49/48, 56/55, 78/77
[[Comma list]]: 225/224, 84035/82944


Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


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


Vals: {{Val list| 9, 10, 19 }}
{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}


Badness: 0.017833
[[Badness]] (Sintel): 1.38


== 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: 225/224, 245/242, 385/384


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
{{Mapping|legend=0| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


POTE generator: ~15/14 = 124.767
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| 19, 29, 48d, 77cdd }}
{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}


Badness: 0.038679
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, 65/64, 91/90, 875/858
Comma list: 105/104, 144/143, 196/195, 245/242


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
{{Mapping|legend=0| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


POTE generator: ~14/13 = 124.716
Optimal tunings:  
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


Vals: {{Val list| 19, 29, 48df, 77cddf }}
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}


Badness: 0.024383
Badness (Sintel): 0.955


== Negric ==
== 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.
 
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 2500000/2470629
 
{{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: 33/32, 49/48, 77/75
Comma list: 225/224, 441/440, 1344/1331


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


POTE generator: ~15/14 = 127.039
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, 19e }}
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


Badness: 0.030617
Badness (Sintel): 1.20


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


Comma list: 33/32, 49/48, 65/64, 91/90
Comma list: 144/143, 225/224, 364/363, 441/440


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


POTE generator: ~14/13 = 127.039
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, 19e }}
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Badness: 0.020205
Badness (Sintel): 1.14


== Negroni ==
=== Mermaid ===
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, 532400/531441


Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
{{Mapping|legend=0| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


POTE generator: ~15/14 = 124.539
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| 10, 19e, 29, 77cddee }}
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


Badness: 0.035296
Badness (Sintel): 1.46


=== 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: 225/224, 325/324, 385/384, 10648/10647
 
{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
 
Optimal tunings:
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


POTE generator: ~14/13 = 124.545
Badness (Sintel): 1.47


Vals: {{Val list| 10, 19e, 29, 77cddeef }}
== Slender ==
Slender tempers out the [[hewuermera comma]] in addition to the marvel comma, and may be described as the {{nowrap| 31 & 32 }} temperament. This temperament has a generator of [[49/48]], three of which equal marvel's [[16/15]][[~]][[15/14]], and ten generators give [[5/4]]. Its [[ploidacot]] is omega-13-cot.


Badness: 0.021559
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  


== Wilsec ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 589824/588245
 
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}
 
[[Badness]] (Sintel): 1.44
 
=== 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, 385/384, 1331/1323


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


POTE generator: ~11/8 = 537.186
Optimal tunings:  
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


Vals: {{Val list| 9, 20, 29, 38d, 67cdde }}
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Badness: 0.041886
Badness (Sintel): 0.838


=== 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: 225/224, 275/273, 385/384, 1331/1323


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


POTE generator: ~11/8 = 537.208
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Vals: {{Val list| 9, 20, 29, 38df, 67cddef }}
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Badness: 0.025192
Badness (Sintel): 1.07


=== 17-limit ===
== Triton ==
Subgroup: 2.3.5.7.11.13.17
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''
 
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.  


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
[[Comma list]]: 225/224, 1029/1000


POTE generator: ~11/8 = 537.230
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


Vals: {{Val list| 9, 20g, 29g, 38df, 67cddefg }}
[[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: 0.021778
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


=== 19-limit ===
[[Badness]] (Sintel): 1.50
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
Comma list: 45/44, 56/55, 1029/1000


POTE generator: ~11/8 = 537.214
{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


Vals: {{Val list| 9, 20g, 29g, 38df, 67cddefgh }}
Optimal tunings:  
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


Badness: 0.016828
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


= Passive =
Badness (Sintel): 1.51
{{see also | Archytas clan #Passion }}


Subgroup: 2.3.5.7
== Marvolo ==
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 256/245
[[Comma list]]: 225/224, 156250000/155649627


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


[[POTE generator]]: ~16/15 = 98.809
[[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 }}


{{Val list|legend=1| 1, 11, 12, 49dd }}
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


[[Badness]]: 0.075055
[[Badness]] (Sintel): 2.11


= Wizard =
=== 11-limit ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
Subgroup: 2.3.5.7.11
{{See also|Wizard}}


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


[[Comma list]]: 225/224, 118098/117649
{{Mapping|legend=0| 1 2 1 1 2 | 0 -6 19 26 21 }}


[[Mapping]]: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


Mapping generators: ~1225/864, ~245/216
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


[[POTE generator]]: ~5/4 = 383.256
Badness (Sintel): 0.958


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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


[[Badness]]: 0.040846
{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Scales: [[wizard22]]
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 29, 43, 72 }}
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 4000/3993
Badness (Sintel): 0.887


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Mapping generators: ~99/70, ~25/22
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


POTE generator: ~5/4 = 383.232
{{Mapping|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


Vals: {{Val list| 22, 50, 72, 166, 238c, 310c }}
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Badness: 0.018539
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Scales: [[wizard22]]
Badness (Sintel): 0.760


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


Comma list: 225/224, 351/350, 364/363, 385/384
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
{{Mapping|legend=0| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Mapping generators: ~99/70, ~25/22
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 = 383.389
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Vals: {{Val list| 22, 50, 72, 122, 194df }}
Badness (Sintel): 0.895


Badness: 0.021781
== Decic ==
{{Main| Decic }}


==== 17-limit ====
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]].  
Subgroup: 2.3.5.7.11.13.17


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
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 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
[[Subgroup]]: 2.3.5.7


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


POTE generator: ~5/4 = 383.381
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


Vals: {{Val list| 22, 50, 72, 122g, 194dfg }}
[[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: 0.014536
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


==== 19-limit ====
[[Badness]] (Sintel): 2.26
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, 385/384, 3087/3025


Mapping generators: ~17/12, ~17/15
{{Mapping|legend=0| 10 0 39 28 3 | 0 1 -1 0 2 }}


POTE generator: ~5/4 = 383.477
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}})


Vals: {{Val list| 22h, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Badness: 0.015702
Badness (Sintel): 2.11


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


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
{{Mapping|legend=0| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


Mapping generators: ~99/70, ~25/22
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}})


POTE generator: ~5/4 = 383.170
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Vals: {{Val list| 72, 166, 238cf }}
Badness (Sintel): 1.52
 
Badness: 0.020252


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


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
{{Mapping|legend=0| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


Mapping generators: ~17/12, ~17/15
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: ~5/4 = 383.175
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Vals: {{Val list| 72, 166g, 238cfg }}
Badness (Sintel): 1.28


Badness: 0.013624
=== 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''.


Subgroup: 2.3.5.7.11


==== 19-limit ====
Comma list: 225/224, 1617/1600, 2401/2376
Subgroup: 2.3.5.7.11.13.17.19
 
{{Mapping|legend=0| 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
 
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
{{Mapping|legend=0| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


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


Mapping generators: ~17/12, ~17/15
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


POTE generator: ~5/4 = 383.138
Badness (Sintel): 1.57


Vals: {{Val list| 72, 94, 166g }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.014810
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374


== Mage ==
{{Mapping|legend=0| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 1331/1296
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}})


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


Mapping generators: ~77/54, ~55/48
Badness (Sintel): 1.33


POTE generator: ~5/4 = 383.124
=== 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''.


Vals: {{Val list| 22, 50e, 72ee, 94ee }}
Subgroup: 2.3.5.7.11


Badness: 0.057799
Comma list: 225/224, 441/440, 5929/5832


= Triton =
{{Mapping|legend=0| 10 0 39 28 -13 | 0 1 -1 0 3 }}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''


Subgroup: 2.3.5.7
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}})


[[Comma list]]: 225/224, 1029/1000
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
Badness (Sintel): 2.20


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[POTE generator]]: ~7/5 = 568.865
Comma list: 105/104, 196/195, 275/273, 5929/5832


{{Val list|legend=1| 2, 17d, 19, 78bd, 97bd }}
{{Mapping|legend=0| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


[[Badness]]: 0.059245
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}})


== 11-limit ==
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
Subgroup: 2.3.5.7.11


Comma list: 45/44, 56/55, 1029/1000
Badness (Sintel): 1.73


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~7/5 = 569.144
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


Vals: {{Val list| 2, 17d, 19, 59bde, 78bde, 97bde }}
{{Mapping|legend=0| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


Badness: 0.045675
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}})


= Tritonic =
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''


Subgroup: 2.3.5.7
Badness (Sintel): 1.41


[[Comma list]]: 225/224, 50421/50000
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
[[Comma list]]: 225/224, 40353607/40310784


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


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


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


[[Badness]]: 0.047578
[[Badness]] (Sintel): 2.37


== 11-limit ==
=== 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, 385/384, 12005/11979


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
{{Mapping|legend=0| 9 0 28 11 24 | 0 2 -1 2 1 }}


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


Vals: {{Val list| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.023659
Badness (Sintel): 1.01


=== 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: 169/168, 225/224, 364/363, 1716/1715


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
{{Mapping|legend=0| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


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


Vals: {{Val list| 29, 31, 60e, 151cde }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.022993
Badness (Sintel): 0.922


== Tritoni ==
== Gracecordial ==
Subgroup: 2.3.5.7.11
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


Comma list: 225/224, 385/384, 27783/27500
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
[[Comma list]]: 225/224, 781250000/771895089


POTE generator: ~7/5 = 580.389
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


Vals: {{Val list| 31, 91, 122, 153d }}
[[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 }}


Badness: 0.045456
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


= Merman =
[[Badness]] (Sintel): 2.44
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''


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


[[Comma list]]: 225/224, 2500000/2470629
Comma list: 225/224, 385/384, 236328125/234365481


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


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
Optimal tunings:
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


[[POTE generator]]: ~7/5 = 585.585
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


{{Val list|legend=1| 41, 84, 125 }}
Badness (Sintel): 2.96


[[Badness]]: 0.055078
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


== 11-limit ==
Comma list: 225/224, 325/324, 385/384, 831875/830466
Subgroup: 2.3.5.7.11


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


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


POTE generator: ~7/5 = 585.606
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Vals: {{Val list| 41, 84, 125e }}
Badness (Sintel): 2.16


Badness: 0.036383
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


== 13-limit ==
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
Subgroup: 2.3.5.7.11.13


Comma list: 144/143, 225/224, 364/363, 441/440
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


POTE generator: ~7/5 = 585.657
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Vals: {{Val list| 41, 84, 125e, 209ef, 293ef }}
Badness (Sintel): 1.96


Badness: 0.027544
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


= Septimin =
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''


Subgroup: 2.3.5.7
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


[[Comma list]]: 225/224, 84035/82944
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
Badness (Sintel): 1.71


[[POTE generator]]: ~7/6 = 263.632
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


{{Val list|legend=1| 41, 91, 132d }}
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


[[Badness]]: 0.054502
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


== 11-limit ==
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Comma list: 225/224, 385/384, 2401/2376
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
Badness (Sintel): 1.57


POTE generator: ~7/6 = 263.634
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Vals: {{Val list| 41, 91, 223cdef }}
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Badness: 0.031309
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


== 13-limit ==
Optimal tunings:
Subgroup: 2.3.5.7.11.13
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Comma list: 105/104, 144/143, 196/195, 245/242
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Badness (Sintel): 1.50


POTE generator: ~7/6 = 263.700
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Vals: {{Val list| 41, 91 }}
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


Badness: 0.023117
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


= Slender =
Optimal tunings:
Subgroup: 2.3.5.7
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


[[Comma list]]: 225/224, 589824/588245
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
Badness (Sintel): 1.53


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
=== Gracecord ===
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~49/48 = 38.413
Comma list: 225/224, 441/440, 109375/107811


{{Val list|legend=1| 31, 94, 125 }}
{{Mapping|legend=0| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


[[Badness]]: 0.056934
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 1331/1323
Badness (Sintel): 2.21


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~49/48 = 38.387
Comma list: 225/224, 364/363, 441/440, 6125/6084


Vals: {{Val list| 31, 63, 94, 125 }}
{{Mapping|legend=0| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Badness: 0.025342
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 275/273, 385/384, 1331/1323
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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
{{Mapping|legend=0| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


POTE generator: ~21/20 = 83.330
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


Vals: {{Val list| 29, 43, 72, 115f }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.021470
Badness (Sintel): 1.88


= Amavil =
=== Musneb ===
== 5-limit (Mabila) ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5


[[Comma list]]: 268435456/263671875
Comma list: 225/224, 385/384, 66550/64827


[[Mapping]]: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
{{Mapping|legend=0| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


[[POTE generator]]: ~512/375 = 529.6849
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


[[Badness]]: 0.232481
Badness (Sintel): 2.89


== 7-limit ==
== Untriton ==
Subgroup: 2.3.5.7
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


[[Comma list]]: 225/224, 17496/16807
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
[[Comma list]]: 225/224, 125000000/121060821


[[POTE generator]]: ~48/35 = 529.979
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
[[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 }}


[[Badness]]: 0.109625
{{Optimal ET sequence|legend=1| 51, 53 }}


== 11-limit ==
[[Badness]] (Sintel): 3.64
 
=== 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


POTE generator: ~16/13 = 362.105
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
 
Badness (Sintel): 1.35
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Vals: {{Val list| 10, 43e, 53, 116, 169de, 285cdef }}
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Badness: 0.027689
{{Mapping|legend=0| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


== Interpental ==
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


Comma list: 99/98, 176/175, 51200/50421
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


POTE generator: ~49/40 = 362.418
Badness (Sintel): 1.17


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


POTE generator: ~49/40 = 362.418
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.  


Vals: {{Val list| 43, 53, 96, 149d }}
[[Subgroup]]: 2.3.5.7


Badness: 0.051806
[[Comma list]]: 225/224, 122880/117649


=== 13-limit ===
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
Subgroup: 2.3.5.7.11.13
: mapping generators: ~16/15, ~3


Comma list: 99/98, 169/168, 176/175, 640/637
[[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 }}


POTE generator: ~16/13 = 362.402
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
[[Badness]] (Sintel): 4.37


POTE generator: ~16/13 = 362.402
=== 11-limit ===
Subgroup: 2.3.5.7.11


Vals: {{Val list| 43, 53, 96, 149d }}
Comma list: 121/120, 225/224, 352/343


Badness: 0.029680
{{Mapping|legend=0| 11 0 43 31 38 | 0 1 -1 0 0 }}


= Alphorn =
Optimal tunings:
Subgroup: 2.3.5.7
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


[[Comma list]]: 225/224, 5764801/5668704
{{Optimal ET sequence|legend=0| 22, 55d }}


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


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


[[POTE generator]]: ~48/35 = 556.221
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.  


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.129258
[[Comma list]]: 225/224, 5971968/5764801


== 11-limit ==
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
Subgroup: 2.3.5.7.11
: mapping generators: ~2401/1728, ~35/32


Comma list: 225/224, 385/384, 12250/11979
[[Optimal tuning]]s:  
* [[WE]]: ~2401/1728 = 599.7132{{c}}, ~35/32 = 162.5806{{c}}
: [[error map]]: {{val| -0.574 -1.597 -0.937 +5.510 }}
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~35/32 = 162.6388{{c}}
: error map: {{val| 0.000 -0.844 +0.492 +7.007 }}


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


POTE generator: ~11/8 = 556.144
[[Badness]] (Sintel): 4.53


Vals: {{Val list| 28d, 41, 315cde }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.073459
Comma list: 99/98, 176/175, 65536/65219


= Tertiosec =
{{Mapping|legend=0| 2 1 6 4 8 | 0 8 -5 6 -4 }}
Subgroup: 2.3.5


[[Comma list]]: {{monzo| -89 21 24 }}
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


[[Mapping]]: [{{val| 3 7 5 }}, {{val| 0 -8 7 }}]
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


[[POTE generator]]: ~16/15 = 112.281
Badness (Sintel): 2.26


{{Val list|legend=1| 21, 54, 75, 96, 171 }}
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


[[Badness]]: 0.851003
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>.  


== 7-limit ==
[[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|legend=0| 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 collection]]
[[Category:Marvel]]
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Temperament collections]]
[[Category:Catalogs of rank-2 temperaments]]
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