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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]]
* [[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.  


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


Subgroup: 2.3.5
[[Comma list]]: 225/224, 50421/50000


[[Comma list]]: 16875/16384
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


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


{{Multival|legend=1| 4 -3 -14 }}
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


[[POTE generator]]: ~16/15 = 125.7549
[[Badness]] (Sintel): 1.20


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 7-limit ==
Comma list: 121/120, 225/224, 441/440


Subgroup: 2.3.5.7
{{Mapping|legend=0| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


[[Comma list]]: 49/48, 225/224
Optimal tunings:  
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


{{Multival|legend=1| 4 -3 2 -14 -8 13 }}
Badness (Sintel): 0.782


[[POTE generator]]: ~15/14 = 125.608
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
Comma list: 105/104, 121/120, 196/195, 275/273


=== Negra ===
{{Mapping|legend=0| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}
This is the 2.3.5.7.13 extension of negri.


Subgroup: 2.3.5.7.13
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
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


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


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


[[Gencom]]: [2 14/13; 49/48 65/64 91/90]
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


POTE generator: ~14/13 = 125.567
{{Mapping|legend=0| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
Optimal tunings:
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Subgroup: 2.3.5.7.11
Badness (Sintel): 0.973


Comma list: 45/44, 49/48, 56/55
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


POTE generator: ~15/14 = 126.474
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


{{Val list|legend=1| 9, 10, 19 }}
Optimal tunings:
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


Badness: 0.0262
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


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


=== 13-limit ===
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Subgroup: 2.3.5.7.11.13
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


Comma list: 45/44, 49/48, 56/55, 78/77
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
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 = 126.431
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


{{Val list|legend=1| 9, 10, 19 }}
Badness (Sintel): 1.04
 
== Negril ==


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


Comma list: 49/48, 100/99, 225/224
Comma list: 225/224, 385/384, 27783/27500


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
{{Mapping|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


POTE generator: ~15/14 = 124.767
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| 19, 29, 48d, 77cdd }}
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Badness: 0.0387
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, 65/64, 91/90, 875/858
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
[[Comma list]]: 225/224, 84035/82944


POTE generator: ~14/13 = 124.716
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


{{Val list|legend=1| 19, 29, 48df, 77cddf }}
[[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: 0.0244
{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}


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


=== 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, 245/242, 385/384


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


POTE generator: ~15/14 = 127.039
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, 19e, 47…, 66…, 85… }}
{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}


Badness: 0.0306
Badness (Sintel): 1.04


=== 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: 105/104, 144/143, 196/195, 245/242


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


POTE generator: ~14/13 = 127.039
Optimal tunings:  
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}


Badness: 0.0202
Badness (Sintel): 0.955


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


Subgroup: 2.3.5.7.11
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.  


Comma list: 49/48, 55/54, 225/224
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.


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


POTE generator: ~15/14 = 124.539
[[Comma list]]: 225/224, 2500000/2470629


{{Val list|legend=1| 10, 19e, 29, 77cddee }}
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


Badness: 0.0353
[[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 }}


=== 13-limit ===
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


Subgroup: 2.3.5.7.11.13
[[Badness]] (Sintel): 1.39


Comma list: 49/48, 55/54, 65/64, 91/90
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Comma list: 225/224, 441/440, 1344/1331


POTE generator: ~14/13 = 124.545
{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


{{Val list|legend=1| 10, 19e, 29, 77cddeef }}
Optimal tunings:
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


Badness: 0.0216
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


== Wilsec ==
Badness (Sintel): 1.20


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


Comma list: 49/48, 121/120, 225/224
Comma list: 144/143, 225/224, 364/363, 441/440


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


POTE generator: ~11/8 = 537.186
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


{{Val list|legend=1| 9, 20, 29, 38d, 67cdde }}
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Badness: 0.0419
Badness (Sintel): 1.14


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


Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 385/384, 532400/531441


Comma list: 49/48, 65/64, 91/90, 121/120
{{Mapping|legend=0| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


POTE generator: ~11/8 = 537.208
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


{{Val list|legend=1| 9, 20, 29, 38df, 67cddef }}
Badness (Sintel): 1.46


Badness: 0.0252
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 17-limit ===
Comma list: 225/224, 325/324, 385/384, 10648/10647


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


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
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 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


POTE generator: ~11/8 = 537.230
Badness (Sintel): 1.47


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefg }}
== 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: 0.0218
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  


=== 19-limit ===
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7.11.13.17.19
[[Comma list]]: 225/224, 589824/588245


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


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


POTE generator: ~11/8 = 537.214
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefgh }}
[[Badness]] (Sintel): 1.44


Badness: 0.0168
=== 11-limit ===
Subgroup: 2.3.5.7.11


= Passive =
Comma list: 225/224, 385/384, 1331/1323
{{see also | Archytas clan #Passion }}


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


[[Comma list]]: 225/224, 256/245
Optimal tunings:  
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


[[POTE generator]]: ~16/15 = 98.809
Badness (Sintel): 0.838


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


[[Badness]]: 0.0751
Comma list: 225/224, 275/273, 385/384, 1331/1323


= Wizard =
{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
Optimal tunings:  
{{See also|Wizard}}
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


[[Comma list]]: 225/224, 118098/117649
Badness (Sintel): 1.07


[[Mapping]]: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


Mapping generators: ~1225/864, ~245/216
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.


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


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
[[Comma list]]: 225/224, 1029/1000


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


[[Badness]]: 0.0408
[[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 }}


Scales: [[wizard22]]
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


== 11-limit ==
[[Badness]] (Sintel): 1.50


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


Comma list: 225/224, 385/384, 4000/3993
Comma list: 45/44, 56/55, 1029/1000
 
{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
 
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


Mapping generators: ~99/70, ~25/22
Badness (Sintel): 1.51


POTE generator: ~5/4 = 383.232
== Marvolo ==
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c }}
[[Comma list]]: 225/224, 156250000/155649627


Badness: 0.0185
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


Scales: [[wizard22]]
[[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 }}


=== Lizard ===
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}
 
[[Badness]] (Sintel): 2.11
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 441/440, 4000/3993


Comma list: 225/224, 351/350, 364/363, 385/384
{{Mapping|legend=0| 1 2 1 1 2 | 0 -6 19 26 21 }}


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -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: ~99/70, ~25/22
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


POTE generator: ~5/4 = 383.389
Badness (Sintel): 0.958


{{Val list|legend=1| 22, 50, 72, 122, 194df }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0218
Comma list: 169/168, 225/224, 364/363, 441/440


==== 17-limit ====
{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Subgroup: 2.3.5.7.11.13.17
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
Badness (Sintel): 0.887


Mapping generators: ~17/12, ~17/15
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~5/4 = 383.381
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
{{Mapping|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


Badness: 0.0145
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


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


==== 19-limit ====
Badness (Sintel): 0.760


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


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


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


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


POTE generator: ~5/4 = 383.477
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


{{Val list|legend=1| 22h, 50, 72, 122g, 194dfg }}
Badness (Sintel): 0.895


Badness: 0.0157
== Decic ==
{{Main| Decic }}


</div></div>
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]].


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


Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7


Comma list: 225/224, 325/324, 385/384, 1573/1568
[[Comma list]]: 225/224, 16807/16384


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


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


POTE generator: ~5/4 = 383.170
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


{{Val list|legend=1| 72, 166, 238cf }}
[[Badness]] (Sintel): 2.26


Badness: 0.0203
=== 11-limit ===
Subgroup: 2.3.5.7.11


</div></div>
Comma list: 225/224, 385/384, 3087/3025


==== 17-limit ====
{{Mapping|legend=0| 10 0 39 28 3 | 0 1 -1 0 2 }}


Subgroup: 2.3.5.7.11.13.17
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}})


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
Badness (Sintel): 2.11


Mapping generators: ~17/12, ~17/15
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 383.175
Comma list: 105/104, 144/143, 196/195, 2200/2197


{{Val list|legend=1| 72, 166g, 238cfg }}
{{Mapping|legend=0| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


Badness: 0.0136
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}})


</div></div>
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


==== 19-limit ====
Badness (Sintel): 1.52


Subgroup: 2.3.5.7.11.13.17.19
==== 17-limit ====
 
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


Mapping generators: ~17/12, ~17/15
{{Mapping|legend=0| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


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


{{Val list|legend=1| 72, 94, 166g }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Badness: 0.0148
Badness (Sintel): 1.28


== Mage ==
=== 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
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 1331/1296
Comma list: 225/224, 1617/1600, 2401/2376


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
{{Mapping|legend=0| 10 0 39 28 82 | 0 1 -1 0 -3 }}


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


POTE generator: ~5/4 = 383.124
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


{{Val list|legend=1| 22, 50e, 72ee, 94ee }}
Badness (Sintel): 1.98


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


= Triton =
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''
{{Mapping|legend=0| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


Subgroup: 2.3.5.7
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}})


[[Comma list]]: 225/224, 1029/1000
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
Badness (Sintel): 1.57


{{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, 170/169, 196/195, 289/288, 375/374


{{Val list|legend=1| 19, 78bd, 97bd }}
{{Mapping|legend=0| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


[[Badness]]: 0.0592
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}})


== 11-limit ==
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


Subgroup: 2.3.5.7.11
Badness (Sintel): 1.33


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


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, 441/440, 5929/5832


{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
{{Mapping|legend=0| 10 0 39 28 -13 | 0 1 -1 0 3 }}


Badness: 0.0457
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}})


= Tritonic =
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
Badness (Sintel): 2.20


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, 275/273, 5929/5832


[[Mapping]]: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
{{Mapping|legend=0| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


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


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


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
Badness (Sintel): 1.73


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


Subgroup: 2.3.5.7.11
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


Comma list: 121/120, 225/224, 441/440
{{Mapping|legend=0| 10 0 39 28 -13 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 = 120.1577{{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}})


POTE generator: ~7/5 = 580.267
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


{{Val list|legend=1| 29, 31, 60e }}
Badness (Sintel): 1.41


Badness: 0.0237
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


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


Subgroup: 2.3.5.7.11.13
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


Comma list: 105/104, 121/120, 196/195, 275/273
[[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 }}


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


POTE generator: ~7/5 = 580.108
[[Badness]] (Sintel): 2.37


{{Val list|legend=1| 29, 31, 60e, 151cde }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0230
Comma list: 225/224, 385/384, 12005/11979


== Tritoni ==
{{Mapping|legend=0| 9 0 28 11 24 | 0 2 -1 2 1 }}


Subgroup: 2.3.5.7.11
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}})


Comma list: 225/224, 385/384, 27783/27500
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
Badness (Sintel): 1.01


POTE generator: ~7/5 = 580.389
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 31, 91, 122, 153d }}
Comma list: 169/168, 225/224, 364/363, 1716/1715


Badness: 0.0455
{{Mapping|legend=0| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


= Merman =
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}})


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Subgroup: 2.3.5.7
Badness (Sintel): 0.922


[[Comma list]]: 225/224, 2500000/2470629
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


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


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
[[Comma list]]: 225/224, 781250000/771895089


[[POTE generator]]: ~7/5 = 585.585
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


{{Val list|legend=1| 41, 84, 125 }}
[[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]]: 0.0551
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


== 11-limit ==
[[Badness]] (Sintel): 2.44


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


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


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


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


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


Badness: 0.0364
Badness (Sintel): 2.96


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


Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 325/324, 385/384, 831875/830466


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


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
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 = 585.657
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
Badness (Sintel): 2.16


Badness: 0.0275
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


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


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


Subgroup: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


[[Comma list]]: 225/224, 84035/82944
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
Badness (Sintel): 1.96


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


[[POTE generator]]: ~7/6 = 263.632
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


{{Val list|legend=1| 41, 91, 132 }}
{{Mapping|legend=0| 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, 2401/2376
Badness (Sintel): 1.71


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


POTE generator: ~7/6 = 263.634
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


{{Val list|legend=1| 41, 91, 223cdef }}
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


== 13-limit ==
Optimal tunings:
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


Comma list: 105/104, 144/143, 196/195, 245/242
Badness (Sintel): 1.57


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


POTE generator: ~7/6 = 263.700
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


{{Val list|legend=1| 41, 91 }}
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


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


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


[[Comma list]]: 225/224, 589824/588245
Badness (Sintel): 1.50


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


[[POTE generator]]: ~49/48 = 38.413
{{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 }}


{{Val list|legend=1| 31, 94, 125 }}
Optimal tunings:
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


[[Badness]]: 0.0569
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


== 11-limit ==
Badness (Sintel): 1.53


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


Comma list: 225/224, 385/384, 1331/1323
Comma list: 225/224, 441/440, 109375/107811
 
Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]


POTE generator: ~49/48 = 38.387
{{Mapping|legend=0| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


{{Val list|legend=1| 31, 63, 94, 125 }}
Optimal tunings:
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


Badness: 0.02534
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


== 13-limit ==
Badness (Sintel): 2.21


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


Comma list: 225/224, 275/273, 385/384, 1331/1323
Comma list: 225/224, 364/363, 441/440, 6125/6084


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
{{Mapping|legend=0| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


POTE generator: ~49/48 = 38.314
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


{{Val list|legend=1| 31, 63, 94 }}
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Badness: 0.02591
Badness (Sintel): 1.83


= Marvo =
==== 17-limit ====
{{see also| Gravity family }}
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Comma list: 225/224, 78125000/78121827
{{Mapping|legend=0| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


{{Multival|legend=1| 6 17 46 13 56 59 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


POTE generator: ~27/20 = 516.694
Badness (Sintel): 1.87


{{Val list|legend=1| 65d, 72, 137, 209, 281, 569bcc }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0976
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


== 11-limit ==
{{Mapping|legend=0| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


Subgroup: 2.3.5.7.11
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


Comma list: 225/224, 243/242, 4000/3993
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
Badness (Sintel): 1.68


POTE generator: ~27/20 = 516.699
== Alphorn ==
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 65d, 72, 281, 353c, 425bc, 497bc }}
[[Comma list]]: 225/224, 5764801/5668704


Badness: 0.0317
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


== 13-limit ==
[[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 }}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Comma list: 225/224, 243/242, 351/350, 1625/1617
[[Badness]] (Sintel): 3.27


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~27/20 = 516.730
Comma list: 225/224, 385/384, 12250/11979


{{Val list|legend=1| 65d, 72, 137, 209, 281f, 490bcf }}
{{Mapping|legend=0| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


Badness: 0.0269
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


= Marvolo =
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Subgroup: 2.3.5.7
Badness (Sintel): 2.43


[[Comma list]]: 225/224, 156250000/155649627
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
[[Comma list]]: 225/224, 4194304/4117715


[[POTE generator]]: ~21/20 = 83.348
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
[[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 }}


[[Badness]]: 0.0833
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


== 11-limit ==
[[Badness]] (Sintel): 3.57


=== 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}}


{{Val list|legend=1| 29, 43, 72 }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.0290
Badness (Sintel): 2.82


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


== 13-limit ==
Comma list: 99/98, 176/175, 640/637, 847/845


Subgroup: 2.3.5.7.11.13
{{Mapping|legend=0| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


Comma list: 169/168, 225/224, 364/363, 441/440
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


POTE generator: ~21/20 = 83.330
Badness (Sintel): 1.88


{{Val list|legend=1| 29, 43, 72, 115f }}
=== Musneb ===
Subgroup: 2.3.5.7.11


Badness: 0.0215
Comma list: 225/224, 385/384, 66550/64827


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


= Amavil =
Optimal tunings:
== 5-limit (mabila) ==
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


Subgroup: 2.3.5
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


[[Comma list]]: 268435456/263671875
Badness (Sintel): 2.89


[[Mapping]]: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


[[POTE generator]]: ~512/375 = 529.6849
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.  


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.2325
[[Comma list]]: 225/224, 125000000/121060821


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


Subgroup: 2.3.5.7
[[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 }}


[[Comma list]]: 225/224, 17496/16807
{{Optimal ET sequence|legend=1| 51, 53 }}


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
[[Badness]] (Sintel): 3.64


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~48/35 = 529.979
Comma list: 121/120, 225/224, 22000/21609


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
{{Mapping|legend=0| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


[[Badness]]: 0.1096
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 51, 53 }}


Subgroup: 2.3.5.7.11
Badness (Sintel): 2.46


Comma list: 99/98, 176/175, 864/847
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
Comma list: 121/120, 225/224, 275/273, 1040/1029


POTE generator: ~15/11 = 529.974
{{Mapping|legend=0| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


{{Val list|legend=1| 9, 34d, 43, 77de }}
Optimal tunings:
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


Badness: 0.0426
{{Optimal ET sequence|legend=0| 51f, 53 }}


== 13-limit ==
Badness (Sintel): 1.96


Subgroup: 2.3.5.7.11.13
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.  


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


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
[[Comma list]]: 225/224, 823543/800000


POTE generator: ~15/11 = 529.951
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


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


Badness: 0.0258
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


= Enneaportent =
[[Badness]] (Sintel): 3.67


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


[[Comma list]]: 225/224, 40353607/40310784
Comma list: 225/224, 245/242, 1617/1600


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


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
Optimal tunings:
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


[[POTE generator]]: ~5/4 = 383.165
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


{{Val list|legend=1| 63, 72, 495bcd }}
Badness (Sintel): 2.00


[[Badness]]: 0.0937
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 11-limit ==
Comma list: 105/104, 196/195, 245/242, 1001/1000


Subgroup: 2.3.5.7.11
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


Comma list: 225/224, 385/384, 12005/11979
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


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


POTE generator: ~5/4 = 383.146
Badness (Sintel): 1.43


{{Val list|legend=1| 63, 72, 423cd, 495bcd }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0304
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


</div></div>
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


== 13-limit ==
Optimal tunings:
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Comma list: 169/168, 225/224, 364/363, 1716/1715
Badness (Sintel): 1.26


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


POTE generator: ~5/4 = 383.047
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 63, 72, 279cf }}
[[Comma list]]: 225/224, 9805926501/9765625000


Badness: 0.0223
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


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


= Submajor =
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}
== 5-limit ==


Subgroup: 2.3.5
[[Badness]] (Sintel): 3.81


[[Comma list]]: 69198046875/68719476736
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
Comma list: 225/224, 441/440, 43923/43750


[[POTE generator]]: ~10125/8192 = 362.321
{{Mapping|legend=0| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
Optimal tunings:
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


[[Badness]]: 0.1302
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


== 7-limit ==
Badness (Sintel): 1.74


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


[[Comma list]]: 225/224, 51200/50421
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


[[Mapping]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
{{Mapping|legend=0| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
Optimal tunings:
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


[[POTE generator]]: ~49/40 = 362.255
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


{{Val list|legend=1| 10, 33, 43, 53 }}
Badness (Sintel): 1.35


[[Badness]]: 0.0605
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== 11-limit ==
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


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


Comma list: 225/224, 385/384, 6655/6561
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


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


POTE generator: ~27/22 = 362.101
Badness (Sintel): 1.17


{{Val list|legend=1| 53, 116, 169de, 285cde }}
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


Badness: 0.0506
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.  


=== 13-limit ===
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7.11.13
[[Comma list]]: 225/224, 122880/117649


Comma list: 169/168, 225/224, 275/273, 385/384
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
[[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 = 362.105
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}


{{Val list|legend=1| 53, 116, 169de, 285cdef }}
[[Badness]] (Sintel): 4.37


Badness: 0.0277
=== 11-limit ===
Subgroup: 2.3.5.7.11


</div></div>
Comma list: 121/120, 225/224, 352/343


== Interpental ==
{{Mapping|legend=0| 11 0 43 31 38 | 0 1 -1 0 0 }}


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


Comma list: 99/98, 176/175, 51200/50421
{{Optimal ET sequence|legend=0| 22, 55d }}


POTE generator: ~49/40 = 362.418
Badness (Sintel): 2.46


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


POTE generator: ~49/40 = 362.418
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| 43, 53, 96, 149d }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0518
[[Comma list]]: 225/224, 5971968/5764801


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


=== 13-limit ===
[[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 }}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


Comma list: 99/98, 169/168, 176/175, 640/637
[[Badness]] (Sintel): 4.53


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


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
Comma list: 99/98, 176/175, 65536/65219


POTE generator: ~16/13 = 362.402
{{Mapping|legend=0| 2 1 6 4 8 | 0 8 -5 6 -4 }}


{{Val list|legend=1| 43, 53, 96, 149d }}
Optimal tunings:
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


Badness: 0.0297
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


</div></div>
Badness (Sintel): 2.26


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


Subgroup: 2.3.5.7
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>.  


[[Comma list]]: 225/224, 5764801/5668704
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
[[Comma list]]: 225/224, 14495514624/13841287201


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


[[POTE generator]]: ~48/35 = 556.221
[[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 }}


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


[[Badness]]: 0.1293
[[Badness]] (Sintel): 10.9


== 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, 385/384, 12250/11979
{{Mapping|legend=0| 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:Marvel temperaments| ]] <!-- main article -->
[[Category:Temperament collection]]
[[Category:Temperament collections]]
[[Category:Marvel]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
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