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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:  
* ''[[Sharp]]'', {25/24, 28/27} → [[Dicot family #Sharp]]
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* ''[[Pelogic]]'', {21/20, 135/128} → [[Pelogic family #Pelogic]]
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[August]]'', {36/35, 128/125} → [[Augmented family #August]]
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Pajara]], {50/49, 64/63}, also in: [[Diaschismic family #Pajara|diaschismic family]]
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[Septimal meantone]]'', {81/80, 126/125} → [[Meantone family #Septimal meantone]]
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* [[Magic]], {225/224, 245/243}, also in: [[Magic family #Magic|magic family]]
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* [[Miracle]], {225/224, 1029/1024}, also in: [[Gamelismic clan #Miracle|gamelismic clan]]
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* [[Orwell]], {225/224, 1728/1715}, also in: [[Semicomma family #Orwell|semicomma family]]
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Garibaldi temperament|Garibaldi]], {225/224, 3125/3087}, also in: [[Schismatic family #Garibaldi|schismatic family]]
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Catakleismic]], {225/224, 4375/4374}, also in: [[Kleismic family #Catakleismic|kleismic family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* ''[[Decic]]'', {225/224, 16807/16384} → [[Cloudy clan #Decic]]
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[Escapade]]'', {225/224, 65625/65536} → [[Escapade family #Escapade]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Fog]]'', {225/224, 156250/151263} → [[Misty family #Fog]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Compton]]'', {225/224, 250047/250000} → [[Compton family #Compton]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* ''[[Houborizic]]'' {225/224, 1250000/1240029} → [[Amity family #Houborizic]]
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Qintosec]]'', {225/224, 2560000/2470629} → [[Qintosec family #Qintosec]]
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Maquila]]'', {225/224, 30233088/28824005} → [[Maquila family #Maquila]]
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* ''[[Marvo]]'', {225/224, 78125000/78121827} → [[Gravity family #Marvo]]
* [[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, passive, wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, submajor, alphorn, tertiosec, gwazy, gracecordial, and quintapole.  
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]].''
 
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.


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.
[[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
 
Comma list: 121/120, 225/224, 441/440


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


=== 7-limit ===
Optimal tunings:
Subgroup: 2.3.5.7
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


[[Comma list]]: 49/48, 225/224
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
Badness (Sintel): 0.782


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


[[POTE generator]]: ~15/14 = 125.608
Comma list: 105/104, 121/120, 196/195, 275/273


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


[[Badness]]: 0.026483
Optimal tunings:  
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


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


Subgroup: 2.3.5.7.13
Badness (Sintel): 0.950


[[Comma list]]: 49/48, 65/64, 91/90
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


[[Sval]] [[mapping]]: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -3 }}]
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


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


[[Gencom]]: [2 14/13; 49/48 65/64 91/90]
Optimal tunings:  
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


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


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
Badness (Sintel): 0.973


=== 11-limit ===
==== 19-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 45/44, 49/48, 56/55
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


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


POTE generator: ~15/14 = 126.474
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


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


Badness: 0.026190
Badness (Sintel): 1.03


==== 13-limit ====
==== 23-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19.23


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}}


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


Badness: 0.017833
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|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


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


Vals: {{Val list| 19, 29, 48d, 77cdd }}
Badness (Sintel): 1.50


Badness: 0.038679
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


==== 13-limit ====
Septimin may be described as the {{nowrap| 41 & 50 }} temperament. It is generated by a septimal minor third ([[7/6]]), which gives rise to the name, but the generator can be taken to be the [[octave complement]], [[12/7]], such that eleven of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus eta-hendecacot. [[91edo]] may be recommended as a tuning.  
Subgroup: 2.3.5.7.11.13


Comma list: 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


Vals: {{Val list| 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: 0.024383
{{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}}


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


Badness: 0.030617
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|legend=0| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
 
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}
 
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}
 
Badness (Sintel): 0.955
 
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''
 
Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot.
 
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.


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


POTE generator: ~14/13 = 127.039
[[Comma list]]: 225/224, 2500000/2470629


Vals: {{Val list| 9, 19e }}
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


Badness: 0.020205
[[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 }}


=== Negroni ===
{{Optimal ET sequence|legend=1| 41, 84, 125 }}
 
[[Badness]] (Sintel): 1.39
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 55/54, 225/224
Comma list: 225/224, 441/440, 1344/1331


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


POTE generator: ~15/14 = 124.539
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


Vals: {{Val list| 10, 19e, 29, 77cddee }}
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


Badness: 0.035296
Badness (Sintel): 1.20


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


Comma list: 49/48, 55/54, 65/64, 91/90
Comma list: 144/143, 225/224, 364/363, 441/440


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


POTE generator: ~14/13 = 124.545
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Vals: {{Val list| 10, 19e, 29, 77cddeef }}
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Badness: 0.021559
Badness (Sintel): 1.14


=== Wilsec ===
=== Mermaid ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 121/120, 225/224
Comma list: 225/224, 385/384, 532400/531441


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


POTE generator: ~11/8 = 537.186
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


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


Badness: 0.041886
Badness (Sintel): 1.46


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


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


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


POTE generator: ~11/8 = 537.208
Optimal tunings:  
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


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


Badness: 0.025192
Badness (Sintel): 1.47


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


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~11/8 = 537.230
[[Comma list]]: 225/224, 589824/588245


Vals: {{Val list| 9, 20g, 29g, 38df, 67cddefg }}
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


Badness: 0.021778
[[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 }}


==== 19-limit ====
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
[[Badness]] (Sintel): 1.44


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~11/8 = 537.214
Comma list: 225/224, 385/384, 1331/1323


Vals: {{Val list| 9, 20g, 29g, 38df, 67cddefgh }}
{{Mapping|legend=0| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


Badness: 0.016828
Optimal tunings:  
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


== Passive ==
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
Passive (1&amp;11) tempers out the same 5-limit comma as the [[Archytas clan #Passion|passion temperament]] (12&amp;49), but with the marvel (225/224) rather than the archytas (64/63) tempered out.


Subgroup: 2.3.5.7
Badness (Sintel): 0.838


[[Comma list]]: 225/224, 256/245
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
Comma list: 225/224, 275/273, 385/384, 1331/1323


[[POTE generator]]: ~16/15 = 98.809
{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


{{Val list|legend=1| 1, 11, 12, 49dd }}
Optimal tunings:
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


[[Badness]]: 0.075055
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


== Wizard ==
Badness (Sintel): 1.07
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
{{See also|Wizard}}


Subgroup: 2.3.5.7
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


[[Comma list]]: 225/224, 118098/117649
Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot.


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


Mapping generators: ~1225/864, ~245/216
[[Comma list]]: 225/224, 1029/1000
 
[[POTE generator]]: ~5/4 = 383.256


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}
: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}


[[Badness]]: 0.040846
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


Scales: [[wizard22]]
[[Badness]] (Sintel): 1.50


=== 11-limit ===
=== 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}}
 
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
 
Badness (Sintel): 1.51


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


Mapping generators: ~99/70, ~25/22
[[Comma list]]: 225/224, 156250000/155649627


POTE generator: ~5/4 = 383.232
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


Vals: {{Val list| 22, 50, 72, 166, 238c, 310c }}
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7714{{c}}, ~21/20 = 83.4014{{c}}
: [[error map]]: {{val| +0.772 -0.820 -0.916 +0.381 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}}
: error map: {{val| 0.000 -2.139 -2.398 -1.362 }}


Badness: 0.018539
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


Scales: [[wizard22]]
[[Badness]] (Sintel): 2.11


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


Comma list: 225/224, 351/350, 364/363, 385/384
Comma list: 225/224, 441/440, 4000/3993


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


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


POTE generator: ~5/4 = 383.389
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Vals: {{Val list| 22, 50, 72, 122, 194df }}
Badness (Sintel): 0.958


Badness: 0.021781
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


===== 17-limit =====
Comma list: 169/168, 225/224, 364/363, 441/440
Subgroup: 2.3.5.7.11.13.17


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
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 = 383.381
Badness (Sintel): 0.887


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


Badness: 0.014536
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


===== 19-limit =====
{{Mapping|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Mapping generators: ~17/12, ~17/15
Badness (Sintel): 0.760


POTE generator: ~5/4 = 383.477
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Vals: {{Val list| 22h, 50, 72, 122g, 194dfg }}
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Badness: 0.015702
{{Mapping|legend=0| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


==== Gizzard ====
Optimal tunings:
Subgroup: 2.3.5.7.11.13
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


Comma list: 225/224, 325/324, 385/384, 1573/1568
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
Badness (Sintel): 0.895


Mapping generators: ~99/70, ~25/22
== Decic ==
{{Main| Decic }}


POTE generator: ~5/4 = 383.170
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]].  


Vals: {{Val list| 72, 166, 238cf }}
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.


Badness: 0.020252
[[Subgroup]]: 2.3.5.7


===== 17-limit =====
[[Comma list]]: 225/224, 16807/16384
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


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


Mapping generators: ~17/12, ~17/15
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


POTE generator: ~5/4 = 383.175
[[Badness]] (Sintel): 2.26


Vals: {{Val list| 72, 166g, 238cfg }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.013624
Comma list: 225/224, 385/384, 3087/3025


===== 19-limit =====
{{Mapping|legend=0| 10 0 39 28 3 | 0 1 -1 0 2 }}
Subgroup: 2.3.5.7.11.13.17.19


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


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Mapping generators: ~17/12, ~17/15
Badness (Sintel): 2.11


POTE generator: ~5/4 = 383.138
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 72, 94, 166g }}
Comma list: 105/104, 144/143, 196/195, 2200/2197


Badness: 0.014810
{{Mapping|legend=0| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


=== Mage ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~15/14 = 120.1166{{c}}, ~3/2 = 697.6705{{c}} (~78/77 = 23.0289{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.1492{{c}} (~78/77 = 22.8508{{c}})


Comma list: 99/98, 176/175, 1331/1296
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
Badness (Sintel): 1.52


Mapping generators: ~77/54, ~55/48
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~5/4 = 383.124
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


Vals: {{Val list| 22, 50e, 72ee, 94ee }}
{{Mapping|legend=0| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


Badness: 0.057799
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}})


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


Subgroup: 2.3.5.7
Badness (Sintel): 1.28


[[Comma list]]: 225/224, 1029/1000
=== Splendecic ===
Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[fantastic]] rank-3 temperament, its name is a portmanteau of ''splendid'' and ''decic''.


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


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
Comma list: 225/224, 1617/1600, 2401/2376


[[POTE generator]]: ~7/5 = 568.865
{{Mapping|legend=0| 10 0 39 28 82 | 0 1 -1 0 -3 }}


{{Val list|legend=1| 2, 17d, 19, 78bd, 97bd }}
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}})


[[Badness]]: 0.059245
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


=== 11-limit ===
Badness (Sintel): 1.98
Subgroup: 2.3.5.7.11


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 }}


Vals: {{Val list| 2, 17d, 19, 59bde, 78bde, 97bde }}
Optimal tunings:  
* WE: ~15/14 = 120.1565{{c}}, ~3/2 = 699.2756{{c}} (~91/90 = 21.6631{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.2480{{c}} (~91/90 = 21.7520{{c}})


Badness: 0.045675
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


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


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


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


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


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
Optimal tunings:
* WE: ~15/14 = 120.1571{{c}}, ~3/2 = 699.2892{{c}} (~91/90 = 21.6536{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3144{{c}} (~91/90 = 21.6856{{c}})


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


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


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


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


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


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


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


Badness: 0.023659
Badness (Sintel): 2.20


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


Comma list: 105/104, 121/120, 196/195, 275/273
Comma list: 105/104, 196/195, 275/273, 5929/5832


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
{{Mapping|legend=0| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


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


Vals: {{Val list| 29, 31, 60e, 151cde }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness: 0.022993
Badness (Sintel): 1.73


=== Tritoni ===
==== 17-limit ====
Subgroup: 2.3.5.7.11
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}})


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


Badness: 0.045456
Badness (Sintel): 1.41


== Merman ==
== Enneaportent ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
[[Subgroup]]: 2.3.5.7


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


[[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]]: 0.055078


=== 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}})


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


Badness: 0.036383
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}})


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


Badness: 0.027544
Badness (Sintel): 0.922


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


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


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


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


{{Val list|legend=1| 41, 91, 132d }}
[[Badness]] (Sintel): 2.44
 
[[Badness]]: 0.054502


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


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


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


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


Badness: 0.031309
Badness (Sintel): 2.96


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


Comma list: 105/104, 144/143, 196/195, 245/242
Comma list: 225/224, 325/324, 385/384, 831875/830466
 
{{Mapping|legend=0| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
 
Optimal tunings:
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Badness (Sintel): 2.16


POTE generator: ~7/6 = 263.700
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Vals: {{Val list| 41, 91 }}
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


Badness: 0.023117
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


== Slender ==
Optimal tunings:
Slender (31 & 32) tempers out the [[Hewuermera temperaments|hewuermera comma]] in addition to the marvel comma, this has a generator [[49/48]], 3 of which equal marvel's 16/15 - 15/14, and 10 generators is 5/4.
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


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


[[Comma list]]: 225/224, 589824/588245
Badness (Sintel): 1.96


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


[[POTE generator]]: ~49/48 = 38.413
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


{{Val list|legend=1| 31, 94, 125 }}
Optimal tunings:
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


[[Badness]]: 0.056934
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


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


Comma list: 225/224, 385/384, 1331/1323
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


POTE generator: ~49/48 = 38.387
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Vals: {{Val list| 31, 63, 94, 125 }}
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Badness: 0.025342
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


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


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


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


POTE generator: ~49/48 = 38.314
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


Vals: {{Val list| 31, 63, 94 }}
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Badness: 0.025913
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


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


[[Comma list]]: 225/224, 156250000/155649627
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
{{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 }}


[[POTE generator]]: ~21/20 = 83.348
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


[[Badness]]: 0.083338
Badness (Sintel): 1.53


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


Comma list: 225/224, 441/440, 4000/3993
Comma list: 225/224, 441/440, 109375/107811


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
{{Mapping|legend=0| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


POTE generator: ~21/20 = 83.340
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


Vals: {{Val list| 29, 43, 72 }}
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Badness: 0.028965
Badness (Sintel): 2.21


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


Comma list: 169/168, 225/224, 364/363, 441/440
Comma list: 225/224, 364/363, 441/440, 6125/6084


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
{{Mapping|legend=0| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


POTE generator: ~21/20 = 83.330
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


Vals: {{Val list| 29, 43, 72, 115f }}
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Badness: 0.021470
Badness (Sintel): 1.83


== Amavil ==
==== 17-limit ====
=== 5-limit (mabila) ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5


[[Comma list]]: 268435456/263671875
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


[[Mapping]]: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
{{Mapping|legend=0| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


[[POTE generator]]: ~512/375 = 529.6849
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


[[Badness]]: 0.232481
Badness (Sintel): 1.87


=== 7-limit ===
==== 19-limit ====
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17.19


[[Comma list]]: 225/224, 17496/16807
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
{{Mapping|legend=0| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
Optimal tunings:
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


[[POTE generator]]: ~48/35 = 529.979
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
Badness (Sintel): 1.68


[[Badness]]: 0.109625
== Alphorn ==
[[Subgroup]]: 2.3.5.7


=== 11-limit ===
[[Comma list]]: 225/224, 5764801/5668704
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 864/847
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
[[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 }}


POTE generator: ~15/11 = 529.974
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Vals: {{Val list| 9, 34d, 43, 77de }}
[[Badness]] (Sintel): 3.27


Badness: 0.042649
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 225/224, 385/384, 12250/11979
Subgroup: 2.3.5.7.11.13
 
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 }}]
{{Mapping|legend=0| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


POTE generator: ~15/11 = 529.951
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Vals: {{Val list| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Badness: 0.025791
Badness (Sintel): 2.43


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


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


[[Mapping]]: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
[[Comma list]]: 225/224, 4194304/4117715


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


[[POTE generator]]: ~5/4 = 383.165
[[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, 63, 72, 495bcd }}
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


[[Badness]]: 0.093679
[[Badness]] (Sintel): 3.57


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


Comma list: 225/224, 385/384, 12005/11979
Comma list: 99/98, 176/175, 1310720/1294139


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
{{Mapping|legend=0| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


POTE generator: ~5/4 = 383.146
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


Vals: {{Val list| 9, 63, 72, 423cd, 495bcd }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.030426
Badness (Sintel): 2.82


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


Comma list: 169/168, 225/224, 364/363, 1716/1715
Comma list: 99/98, 176/175, 640/637, 847/845


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
{{Mapping|legend=0| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


POTE generator: ~5/4 = 383.047
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


Vals: {{Val list| 9, 63, 72, 279cf }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.022322
Badness (Sintel): 1.88


== Submajor ==
=== Musneb ===
Subgroup: 2.3.5
Subgroup: 2.3.5.7.11


[[Comma list]]: 69198046875/68719476736
Comma list: 225/224, 385/384, 66550/64827


[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
{{Mapping|legend=0| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


[[POTE generator]]: ~10125/8192 = 362.321
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


[[Badness]]: 0.130236
Badness (Sintel): 2.89


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


[[Comma list]]: 225/224, 51200/50421
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.


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


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
[[Comma list]]: 225/224, 125000000/121060821


[[POTE generator]]: ~49/40 = 362.255
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


{{Val list|legend=1| 10, 33, 43, 53 }}
[[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]]: 0.060533
{{Optimal ET sequence|legend=1| 51, 53 }}
 
[[Badness]] (Sintel): 3.64


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


Comma list: 225/224, 385/384, 6655/6561
Comma list: 121/120, 225/224, 22000/21609


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
{{Mapping|legend=0| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


POTE generator: ~27/22 = 362.101
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


Vals: {{Val list| 10, 43e, 53, 116, 169de, 285cde }}
{{Optimal ET sequence|legend=0| 51, 53 }}


Badness: 0.050582
Badness (Sintel): 2.46


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


Comma list: 169/168, 225/224, 275/273, 385/384
Comma list: 121/120, 225/224, 275/273, 1040/1029


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
{{Mapping|legend=0| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


POTE generator: ~16/13 = 362.105
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


Vals: {{Val list| 10, 43e, 53, 116, 169de, 285cdef }}
{{Optimal ET sequence|legend=0| 51f, 53 }}


Badness: 0.027689
Badness (Sintel): 1.96


=== Interpental ===
== Naiadical ==
Subgroup: 2.3.5.7.11
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.  


Comma list: 99/98, 176/175, 51200/50421
[[Subgroup]]: 2.3.5.7


POTE generator: ~49/40 = 362.418
[[Comma list]]: 225/224, 823543/800000


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


POTE generator: ~49/40 = 362.418
[[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 }}


Vals: {{Val list| 43, 53, 96, 149d }}
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


Badness: 0.051806
[[Badness]] (Sintel): 3.67


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


Comma list: 99/98, 169/168, 176/175, 640/637
Comma list: 225/224, 245/242, 1617/1600


POTE generator: ~16/13 = 362.402
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


POTE generator: ~16/13 = 362.402
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Vals: {{Val list| 43, 53, 96, 149d }}
Badness (Sintel): 2.00


Badness: 0.029680
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== Alphorn ==
Comma list: 105/104, 196/195, 245/242, 1001/1000
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 5764801/5668704
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


[[Mapping]]: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


[[POTE generator]]: ~48/35 = 556.221
Badness (Sintel): 1.43


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Badness]]: 0.129258
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


=== 11-limit ===
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 12250/11979
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


POTE generator: ~11/8 = 556.144
Badness (Sintel): 1.26


Vals: {{Val list| 28d, 41, 315cde }}
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


Badness: 0.073459
[[Subgroup]]: 2.3.5.7


== Tertiosec ==
[[Comma list]]: 225/224, 9805926501/9765625000
Subgroup: 2.3.5


[[Comma list]]: {{monzo| -89 21 24 }}
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


[[Mapping]]: [{{val| 3 7 5 }}, {{val| 0 -8 7 }}]
[[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 }}


[[POTE generator]]: ~16/15 = 112.281
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


{{Val list|legend=1| 21, 54, 75, 96, 171 }}
[[Badness]] (Sintel): 3.81


[[Badness]]: 0.851003
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 7-limit ===
Comma list: 225/224, 441/440, 43923/43750
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 14495514624/13841287201
{{Mapping|legend=0| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


[[Mapping]]: [{{val| 3 7 5 9 }}, {{val| 0 -8 7 -2 }}]
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


{{Multival|legend=1| 24 -21 6 -89 -58 73 }}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


[[POTE generator]]: ~15/14 = 112.283
Badness (Sintel): 1.74


{{Val list|legend=1| 21, 54, 75, 96, 171d }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.431636
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


=== 11-limit ===
{{Mapping|legend=0| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
Subgroup: 2.3.5.7.11


Comma list: 225/224, 3840/3773, 12005/11979
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


Mapping: [{{val| 3 7 5 9 9 }}, {{val| 0 -8 7 -2 5 }}]
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


POTE generator: ~15/14 = 112.171
Badness (Sintel): 1.35


Vals: {{Val list| 21, 54, 75e }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.173485
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


== Gwazy ==
{{Mapping|legend=0| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
{{see also|Very high accuracy temperaments #Kwazy}}


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, 5971968/5764801
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


[[Mapping]]: [{{val| 2 1 6 4 }}, {{val| 0 8 -5 6 }}]
Badness (Sintel): 1.17


{{Multival|legend=1| 16 -10 12 -53 -26 56 }}
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


[[POTE generator]]: ~35/32 = 162.658
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| 22, 74, 96, 118d }}
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.178826
[[Comma list]]: 225/224, 122880/117649


=== 11-limit ===
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
Subgroup: 2.3.5.7.11
: mapping generators: ~16/15, ~3


Comma list: 99/98, 176/175, 65536/65219
[[Optimal tuning]]s:  
* [[WE]]: ~16/15 = 108.9526{{c}}, ~3/2 = 702.4215{{c}}
: [[error map]]: {{val| -1.521 -1.055 -2.252 +8.705 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 703.2071{{c}}
: error map: {{val| 0.000 +1.252 +1.388 +12.992 }}


Mapping: [{{val| 2 1 6 4 8 }}, {{val| 0 8 -5 6 -4 }}]
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}


POTE generator: ~11/10 = 162.592
[[Badness]] (Sintel): 4.37


Vals: {{Val list| 22, 74, 96, 118d }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.068410
Comma list: 121/120, 225/224, 352/343


== Gracecordial ==
{{Mapping|legend=0| 11 0 43 31 38 | 0 1 -1 0 0 }}
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 781250000/771895089
Optimal tunings:  
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


[[Mapping]]: [{{val| 1 0 34 63 }}, {{val| 0 1 -20 -38 }}]
{{Optimal ET sequence|legend=0| 22, 55d }}


{{Multival|legend=1| 1 -20 -38 -34 -63 -32 }}
Badness (Sintel): 2.46


[[POTE generator]]: ~3/2 = 700.824
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


{{Val list|legend=1| 12, 113, 125, 238c, 363c }}
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


[[Badness]]: 0.096279
[[Subgroup]]: 2.3.5.7


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


Comma list: 225/224, 385/384, 236328125/234365481
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


Mapping: [{{val| 1 0 34 63 -90 }}, {{val| 0 1 -20 -38 59 }}]
[[Optimal tuning]]s:
* [[WE]]: ~2401/1728 = 599.7132{{c}}, ~35/32 = 162.5806{{c}}
: [[error map]]: {{val| -0.574 -1.597 -0.937 +5.510 }}
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~35/32 = 162.6388{{c}}
: error map: {{val| 0.000 -0.844 +0.492 +7.007 }}


POTE generator: ~3/2 = 700.834
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


Vals: {{Val list| 12e, 113, 125, 238c }}
[[Badness]] (Sintel): 4.53


Badness: 0.089588
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 99/98, 176/175, 65536/65219
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 325/324, 385/384, 831875/830466
{{Mapping|legend=0| 2 1 6 4 8 | 0 8 -5 6 -4 }}


Mapping: [{{val| 1 0 34 63 -90 -66 }}, {{val| 0 1 -20 -38 59 44 }}]
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


POTE generator: ~3/2 = 700.841
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


Vals: {{Val list| 12e, 113, 125f, 238cf }}
Badness (Sintel): 2.26


Badness: 0.052235
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


== Quintapole ==
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>.
: ''See also: [[16ed5/2 #16ed5/2 as a generator]] or [[42ed11 #42ed11 as a generator]]''


Aside from the marvel comma, the ''quintapole'' temperament (12&amp;85) tempers out 7812500/7411887 (sepru-atritriyo). In the 11-limit, it tempers out the ptolemisma (100/99) as well as 85184/84035 (trilo-aquinru-agu). It is so named for the following reasons - it has the same commas as the [[Marvel family #Appolo|apollo temperament]], and its generator is a semitone five of which gives a flat fourth (~4/3, about 495 cents). Proper pronunciation of the word "quintapole" is /'kwɪntəpəʊl/ or /'kwɪntəpoʊl/, like as "quint-a-pole". ''Not to be confused with [[Octagar temperaments #Quintupole|quint<u>'''u'''</u>pole]] temperament (12&amp;121).''
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7
[[Comma list]]: 225/224, 14495514624/13841287201
 
[[Comma list]]: 225/224, 7812500/7411887


[[Mapping]]: [{{val|1 2 1 1}}, {{val|0 -5 16 22}}]
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


[[POTE generator]]: ~21/20 = 98.994
[[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| 12, 73c, 85, 97d }}
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


[[Badness]]: 0.192498
[[Badness]] (Sintel): 10.9


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


Comma list: 100/99, 225/224, 85184/84035
Comma list: 225/224, 3840/3773, 12005/11979
 
{{Mapping|legend=0| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


Mapping: [{{val|1 2 1 1 0}}, {{val|0 -5 16 22 42}}]
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: ~21/20 = 98.954
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


Vals: {{Val list| 12, 73ce, 85, 97d }}
Badness (Sintel): 5.74


Badness: 0.104353
== References ==


[[Category:Regular temperament theory]]
[[Category:Temperament collection]]
[[Category:Marvel]]
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Temperament collections]]
[[Category:Catalogs of rank-2 temperaments]]
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