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


Since (5/4)<sup>2</sup> = 225/224 × 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
Temperaments considered in families and clans are:
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* [[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 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 {{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: 45/44, 49/48, 56/55, 78/77
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


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


POTE generator: ~14/13 = 126.431
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


{{Val list|legend=1| 9, 10, 19 }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


== Negril ==
Badness (Sintel): 1.04


=== 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 }}
 
Badness: 0.0353
 
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 55/54, 65/64, 91/90


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


POTE generator: ~14/13 = 124.545
[[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 }}


{{Val list|legend=1| 10, 19e, 29, 77cddeef }}
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


Badness: 0.0216
[[Badness]] (Sintel): 1.39
 
== Wilsec ==


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


Comma list: 49/48, 121/120, 225/224
Comma list: 225/224, 441/440, 1344/1331
 
Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]


POTE generator: ~11/8 = 537.186
{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


{{Val list|legend=1| 9, 20, 29, 38d, 67cdde }}
Optimal tunings:
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


Badness: 0.0419
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


=== 13-limit ===
Badness (Sintel): 1.20


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


Comma list: 49/48, 65/64, 91/90, 121/120
Comma list: 144/143, 225/224, 364/363, 441/440


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


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


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


Badness: 0.0252
Badness (Sintel): 1.14


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


Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 385/384, 532400/531441


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
{{Mapping|legend=0| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
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 = 537.230
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


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


Badness: 0.0218
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 19-limit ===
Comma list: 225/224, 325/324, 385/384, 10648/10647


Subgroup: 2.3.5.7.11.13.17.19
{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


Comma list: 49/48, 65/64, 77/76, 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 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


POTE generator: ~11/8 = 537.214
Badness (Sintel): 1.47


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefgh }}
== 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: 0.0168
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  


= Passive =
[[Subgroup]]: 2.3.5.7
{{see also | Archytas clan #Passion }}


Subgroup: 2.3.5.7
[[Comma list]]: 225/224, 589824/588245


[[Comma list]]: 225/224, 256/245
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
[[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]]: ~16/15 = 98.809
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


{{Val list|legend=1| 12, 49dd }}
[[Badness]] (Sintel): 1.44


[[Badness]]: 0.0751
=== 11-limit ===
Subgroup: 2.3.5.7.11


= Wizard =
Comma list: 225/224, 385/384, 1331/1323


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
{{Mapping|legend=0| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


Subgroup: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


[[Comma list]]: 225/224, 118098/117649
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


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


Mapping generators: ~1225/864, ~245/216
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[POTE generator]]: ~5/4 = 383.256
Comma list: 225/224, 275/273, 385/384, 1331/1323


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
Optimal tunings:
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


[[Badness]]: 0.0408
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Scales: [[wizard22]]
Badness (Sintel): 1.07


== 11-limit ==
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


Subgroup: 2.3.5.7.11
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: 225/224, 385/384, 4000/3993
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
[[Comma list]]: 225/224, 1029/1000


Mapping generators: ~99/70, ~25/22
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


POTE generator: ~5/4 = 383.232
[[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 }}


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c }}
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


Badness: 0.0185
[[Badness]] (Sintel): 1.50


Scales: [[wizard22]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


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


Subgroup: 2.3.5.7.11.13
{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


Comma list: 225/224, 351/350, 364/363, 385/384
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 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


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


POTE generator: ~5/4 = 383.389
== Marvolo ==
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 22, 50, 72, 122, 194df }}
[[Comma list]]: 225/224, 156250000/155649627


Badness: 0.0218
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


==== 17-limit ====
[[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 }}


Subgroup: 2.3.5.7.11.13.17
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
[[Badness]] (Sintel): 2.11


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


Mapping generators: ~17/12, ~17/15
Comma list: 225/224, 441/440, 4000/3993


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


{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
Optimal tunings:
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


Badness: 0.0145
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


</div></div>
Badness (Sintel): 0.958


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


Subgroup: 2.3.5.7.11.13.17.19
Comma list: 169/168, 225/224, 364/363, 441/440


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


Mapping generators: ~17/12, ~17/15
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


POTE generator: ~5/4 = 383.477
Badness (Sintel): 0.887


{{Val list|legend=1| 22h, 50, 72, 122g, 194dfg }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0157
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


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


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


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Comma list: 225/224, 325/324, 385/384, 1573/1568
Badness (Sintel): 0.760


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
Mapping generators: ~99/70, ~25/22
 
POTE generator: ~5/4 = 383.170


{{Val list|legend=1| 72, 166, 238cf }}
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Badness: 0.0203
{{Mapping|legend=0| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


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


==== 17-limit ====
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Subgroup: 2.3.5.7.11.13.17
Badness (Sintel): 0.895


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
== Decic ==
{{Main| Decic }}


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


Mapping generators: ~17/12, ~17/15
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.


POTE generator: ~5/4 = 383.175
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 72, 166g, 238cfg }}
[[Comma list]]: 225/224, 16807/16384


Badness: 0.0136
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


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


==== 19-limit ====
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


Subgroup: 2.3.5.7.11.13.17.19
[[Badness]] (Sintel): 2.26


Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 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 = 383.138
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}})


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


Badness: 0.0148
Badness (Sintel): 2.11


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


Subgroup: 2.3.5.7.11
Comma list: 105/104, 144/143, 196/195, 2200/2197


Comma list: 99/98, 176/175, 1331/1296
{{Mapping|legend=0| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


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


Mapping generators: ~77/54, ~55/48
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


POTE generator: ~5/4 = 383.124
Badness (Sintel): 1.52


{{Val list|legend=1| 22, 50e, 72ee, 94ee }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0578
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


= Triton =
{{Mapping|legend=0| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


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


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


[[Comma list]]: 225/224, 1029/1000
Badness (Sintel): 1.28


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
=== 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''.


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~7/5 = 568.865
Comma list: 225/224, 1617/1600, 2401/2376


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


[[Badness]]: 0.0592
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}})


== 11-limit ==
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


Subgroup: 2.3.5.7.11
Badness (Sintel): 1.98


Comma list: 45/44, 56/55, 1029/1000
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


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


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


{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
Optimal tunings:
* WE: ~15/14 = 120.1565{{c}}, ~3/2 = 699.2756{{c}} (~91/90 = 21.6631{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.2480{{c}} (~91/90 = 21.7520{{c}})


Badness: 0.0457
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


= Tritonic =
Badness (Sintel): 1.57


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374


[[Comma list]]: 225/224, 50421/50000
{{Mapping|legend=0| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


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


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


[[POTE generator]]: ~7/5 = 580.286
Badness (Sintel): 1.33


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
=== Prodecic ===
 
Prodecic (50e & 60e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the [[prodigy]] rank-3 temperament, its name is a portmanteau of ''prodigy'' and ''decic''.
== 11-limit ==


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


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


POTE generator: ~7/5 = 580.267
Optimal tunings:  
* WE: ~15/14 = 120.2024{{c}}, ~3/2 = 701.3908{{c}} (~55/54 = 19.8237{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.5235{{c}} (~55/54 = 19.4765{{c}})


{{Val list|legend=1| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness: 0.0237
Badness (Sintel): 2.20
 
=== 13-limit ===


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


Comma list: 105/104, 121/120, 196/195, 275/273
Comma list: 105/104, 196/195, 275/273, 5929/5832
 
Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]


POTE generator: ~7/5 = 580.108
{{Mapping|legend=0| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


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


Badness: 0.0230
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


== Tritoni ==
Badness (Sintel): 1.73


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


Comma list: 225/224, 385/384, 27783/27500
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
{{Mapping|legend=0| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


POTE generator: ~7/5 = 580.389
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}})


{{Val list|legend=1| 31, 91, 122, 153d }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness: 0.0455
Badness (Sintel): 1.41


= Merman =
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
[[Comma list]]: 225/224, 40353607/40310784
 
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 2500000/2470629
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


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


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


[[POTE generator]]: ~7/5 = 585.585
[[Badness]] (Sintel): 2.37
 
{{Val list|legend=1| 41, 84, 125 }}
 
[[Badness]]: 0.0551
 
== 11-limit ==


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


Comma list: 225/224, 441/440, 1344/1331
Comma list: 225/224, 385/384, 12005/11979


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


POTE generator: ~7/5 = 585.606
Optimal tunings:  
* WE: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}})
* CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}})


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


Badness: 0.0364
Badness (Sintel): 1.01
 
== 13-limit ==


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


Comma list: 144/143, 225/224, 364/363, 441/440
Comma list: 169/168, 225/224, 364/363, 1716/1715


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


POTE generator: ~7/5 = 585.657
Optimal tunings:  
* WE: ~14/13 = 133.4245{{c}}, ~26/15 = 950.9362{{c}} (~105/104 = 16.9650{{c}})
* CWE: ~14/13 = 133.3333{{c}}, ~26/15 = 950.4364{{c}} (~99/98 = 17.1031{{c}})


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


Badness: 0.0275
Badness (Sintel): 0.922


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


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
[[Subgroup]]: 2.3.5.7
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 225/224, 84035/82944


[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
[[Comma list]]: 225/224, 781250000/771895089


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


[[POTE generator]]: ~7/6 = 263.632
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


{{Val list|legend=1| 41, 91, 132 }}
{{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, 385/384, 2401/2376
Comma list: 225/224, 385/384, 236328125/234365481


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


POTE generator: ~7/6 = 263.634
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


{{Val list|legend=1| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


== 13-limit ==
Badness (Sintel): 2.96


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


Comma list: 105/104, 144/143, 196/195, 245/242
Comma list: 225/224, 325/324, 385/384, 831875/830466


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
{{Mapping|legend=0| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


POTE generator: ~7/6 = 263.700
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


{{Val list|legend=1| 41, 91 }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


= Slender =
Badness (Sintel): 2.16


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


[[Comma list]]: 225/224, 589824/588245
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
Optimal tunings:
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


[[POTE generator]]: ~49/48 = 38.413
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


{{Val list|legend=1| 31, 94, 125 }}
Badness (Sintel): 1.96


[[Badness]]: 0.0569
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


== 11-limit ==
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


Subgroup: 2.3.5.7.11
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


Comma list: 225/224, 385/384, 1331/1323
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 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


POTE generator: ~49/48 = 38.387
Badness (Sintel): 1.71


{{Val list|legend=1| 31, 63, 94, 125 }}
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Badness: 0.02534
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


== 13-limit ==
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Subgroup: 2.3.5.7.11.13
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Comma list: 225/224, 275/273, 385/384, 1331/1323
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
Badness (Sintel): 1.57


POTE generator: ~49/48 = 38.314
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


{{Val list|legend=1| 31, 63, 94 }}
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Badness: 0.02591
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


= Marvo =
Optimal tunings:
{{see also| Gravity family }}
* 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, 78125000/78121827
Badness (Sintel): 1.50


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


{{Multival|legend=1| 6 17 46 13 56 59 }}
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


POTE generator: ~27/20 = 516.694
{{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| 65d, 72, 137, 209, 281, 569bcc }}
Optimal tunings:
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


Badness: 0.0976
{{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, 243/242, 4000/3993
Comma list: 225/224, 441/440, 109375/107811


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
{{Mapping|legend=0| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


POTE generator: ~27/20 = 516.699
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


{{Val list|legend=1| 65d, 72, 281, 353c, 425bc, 497bc }}
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Badness: 0.0317
Badness (Sintel): 2.21
 
== 13-limit ==


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


Comma list: 225/224, 243/242, 351/350, 1625/1617
Comma list: 225/224, 364/363, 441/440, 6125/6084


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
{{Mapping|legend=0| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


POTE generator: ~27/20 = 516.730
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| 65d, 72, 137, 209, 281f, 490bcf }}
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Badness: 0.0269
Badness (Sintel): 1.83


= Marvolo =
==== 17-limit ====
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, 156250000/155649627
{{Mapping|legend=0| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
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 -19 -26 -44 -58 -7 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


[[POTE generator]]: ~21/20 = 83.348
Badness (Sintel): 1.87


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


[[Badness]]: 0.0833
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, 441/440, 4000/3993
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
Badness (Sintel): 1.68


POTE generator: ~21/20 = 83.340
== Alphorn ==
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 29, 43, 72 }}
[[Comma list]]: 225/224, 5764801/5668704


Badness: 0.0290
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


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


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


Subgroup: 2.3.5.7.11.13
[[Badness]] (Sintel): 3.27


Comma list: 169/168, 225/224, 364/363, 441/440
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
Comma list: 225/224, 385/384, 12250/11979


POTE generator: ~21/20 = 83.330
{{Mapping|legend=0| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


{{Val list|legend=1| 29, 43, 72, 115f }}
Optimal tunings:
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Badness: 0.0215
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


</div></div>
Badness (Sintel): 2.43


= Amavil =
== Misneb ==
== 5-limit (mabila) ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 268435456/263671875
[[Comma list]]: 225/224, 4194304/4117715


[[Mapping]]: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


[[POTE generator]]: ~512/375 = 529.6849
[[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| 9, 25, 34, 77, 111, 145, 256c }}
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


[[Badness]]: 0.2325
[[Badness]] (Sintel): 3.57


== 7-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11


Subgroup: 2.3.5.7
Comma list: 99/98, 176/175, 1310720/1294139


[[Comma list]]: 225/224, 17496/16807
{{Mapping|legend=0| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


[[POTE generator]]: ~48/35 = 529.979
Badness (Sintel): 2.82


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
==== 13-limit ====
 
Subgroup: 2.3.5.7.11.13
[[Badness]]: 0.1096


== 11-limit ==
Comma list: 99/98, 176/175, 640/637, 847/845


Subgroup: 2.3.5.7.11
{{Mapping|legend=0| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


Comma list: 99/98, 176/175, 864/847
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 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


POTE generator: ~15/11 = 529.974
Badness (Sintel): 1.88


{{Val list|legend=1| 9, 34d, 43, 77de }}
=== Musneb ===
 
Subgroup: 2.3.5.7.11
Badness: 0.0426
 
== 13-limit ==
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 78/77, 99/98, 144/143, 176/175


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
Comma list: 225/224, 385/384, 66550/64827


POTE generator: ~15/11 = 529.951
{{Mapping|legend=0| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


{{Val list|legend=1| 9, 34d, 43, 77de }}
Optimal tunings:
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


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


= Enneaportent =
Badness (Sintel): 2.89


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


[[Comma list]]: 225/224, 40353607/40310784
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| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
[[Comma list]]: 225/224, 125000000/121060821


[[POTE generator]]: ~5/4 = 383.165
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


{{Val list|legend=1| 63, 72, 495bcd }}
[[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]]: 0.0937
{{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: 225/224, 385/384, 12005/11979
Comma list: 121/120, 225/224, 22000/21609


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
{{Mapping|legend=0| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


POTE generator: ~5/4 = 383.146
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


{{Val list|legend=1| 63, 72, 423cd, 495bcd }}
{{Optimal ET sequence|legend=0| 51, 53 }}


Badness: 0.0304
Badness (Sintel): 2.46
 
</div></div>
 
== 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: 121/120, 225/224, 275/273, 1040/1029


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
{{Mapping|legend=0| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


POTE generator: ~5/4 = 383.047
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


{{Val list|legend=1| 63, 72, 279cf }}
{{Optimal ET sequence|legend=0| 51f, 53 }}


Badness: 0.0223
Badness (Sintel): 1.96


</div></div>
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


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


Subgroup: 2.3.5
[[Comma list]]: 225/224, 823543/800000


[[Comma list]]: 69198046875/68719476736
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


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


[[POTE generator]]: ~10125/8192 = 362.321
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
[[Badness]] (Sintel): 3.67


[[Badness]]: 0.1302
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 7-limit ==
Comma list: 225/224, 245/242, 1617/1600


Subgroup: 2.3.5.7
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


[[Comma list]]: 225/224, 51200/50421
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


[[Mapping]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
Badness (Sintel): 2.00


[[POTE generator]]: ~49/40 = 362.255
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 10, 33, 43, 53 }}
Comma list: 105/104, 196/195, 245/242, 1001/1000


[[Badness]]: 0.0605
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


== 11-limit ==
Optimal tunings:
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


Subgroup: 2.3.5.7.11
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Comma list: 225/224, 385/384, 6655/6561
Badness (Sintel): 1.43


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~27/22 = 362.101
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


{{Val list|legend=1| 53, 116, 169de, 285cde }}
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


Badness: 0.0506
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Subgroup: 2.3.5.7.11.13
Badness (Sintel): 1.26


Comma list: 169/168, 225/224, 275/273, 385/384
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


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


POTE generator: ~16/13 = 362.105
[[Comma list]]: 225/224, 9805926501/9765625000


{{Val list|legend=1| 53, 116, 169de, 285cdef }}
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


Badness: 0.0277
[[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 }}


</div></div>
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


== Interpental ==
[[Badness]] (Sintel): 3.81


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


Comma list: 99/98, 176/175, 51200/50421
Comma list: 225/224, 441/440, 43923/43750
 
POTE generator: ~49/40 = 362.418
 
Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]


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


{{Val list|legend=1| 43, 53, 96, 149d }}
Optimal tunings:
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Badness: 0.0518
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


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


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


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


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


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


POTE generator: ~16/13 = 362.402
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


{{Val list|legend=1| 43, 53, 96, 149d }}
Badness (Sintel): 1.35


Badness: 0.0297
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


</div></div>
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


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


Subgroup: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


[[Comma list]]: 225/224, 5764801/5668704
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


[[Mapping]]: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
Badness (Sintel): 1.17


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


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


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


[[Badness]]: 0.1293
[[Comma list]]: 225/224, 122880/117649


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


Subgroup: 2.3.5.7.11
[[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 }}


Comma list: 225/224, 385/384, 12250/11979
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
[[Badness]] (Sintel): 4.37


POTE generator: ~11/8 = 556.144
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 28d, 41, 315cde }}
Comma list: 121/120, 225/224, 352/343


Badness: 0.0735
{{Mapping|legend=0| 11 0 43 31 38 | 0 1 -1 0 0 }}


= Decic =
Optimal tunings:
The ''decic'' temperament (10&amp;50, named by [[User:Xenllium|Xenllium]]) tempers out the [[cloudy comma]], 16807/16384 and the [[225/224|marvel comma]], 225/224 in the 7-limit. It can be extended naturally to the 11-, 13-, and 17-limit by adding 385/384, 105/104, and 170/169 to the comma list in this order.
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 22, 55d }}


[[Comma list]]: 225/224, 16807/16384
Badness (Sintel): 2.46


[[Mapping]]: [{{val| 10 0 39 28 }}, {{val| 0 1 -1 0 }}]
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Mapping generators: ~15/14, ~3
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


{{Multival|legend=1| 10 -10 0 -39 -28 28 }}
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~3/2 = 698.696
[[Comma list]]: 225/224, 5971968/5764801


[[Tuning ranges]]:
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
: mapping generators: ~2401/1728, ~35/32
* nice range: ~3/2 = [693.129, 702.512]
* strict range: ~3/2 = [696.000, 700.000]


{{Val list|legend=1| 10, 50, 60, 110d, 170cdd }}
[[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 }}


[[Badness]]: 0.089135
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


== 11-limit ==
[[Badness]] (Sintel): 4.53


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


Comma list: 225/224, 385/384, 3087/3025
Comma list: 99/98, 176/175, 65536/65219
 
Mapping: [{{val| 10 0 39 28 3 }}, {{val| 0 1 -1 0 2 }}]
 
Mapping generators: ~15/14, ~3


POTE generator: ~3/2 = 696.791
{{Mapping|legend=0| 2 1 6 4 8 | 0 8 -5 6 -4 }}


Tuning ranges:  
Optimal tunings:  
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* nice range: ~3/2 = [689.363, 702.512]
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}
* strict range: ~3/2 = [696.000, 700.000]


{{Val list|legend=1| 10, 50, 160bcdd, 210bcddde, 260bbcddde, 310bbcdddde }}
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


Badness: 0.063900
Badness (Sintel): 2.26


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


Subgroup: 2.3.5.7.11.13
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: 105/104, 144/143, 196/195, 2200/2197
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 10 0 39 28 3 37 }}, {{val| 0 1 -1 0 2 0 }}]
[[Comma list]]: 225/224, 14495514624/13841287201


Mapping generators: ~14/13, ~3
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


POTE generator: ~3/2 = 696.993
[[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 }}


Tuning ranges:
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* nice range: ~3/2 = [689.363, 702.512]
* strict range: ~3/2 = [696.000, 700.000]


{{Val list|legend=1| 10, 50, 60 }}
[[Badness]] (Sintel): 10.9


Badness: 0.036880
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
== 17-limit ==
 
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


Mapping: [{{val| 10 0 39 28 3 37 25 }}, {{val| 0 1 -1 0 2 0 1 }}]
Comma list: 225/224, 3840/3773, 12005/11979


Mapping generators: ~14/13, ~3
{{Mapping|legend=0| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


POTE generator: ~3/2 = 697.085
Optimal tunings:  
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}


Tuning ranges:
{{Optimal ET sequence|legend=0| 21, 54, 75e }}
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* nice range: ~3/2 = [689.363, 704.955]
* strict range: ~3/2 = [696.000, 700.000]


{{Val list|legend=1| 10, 50, 60 }}
Badness (Sintel): 5.74


Badness: 0.025064
== 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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