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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|Dicot family]]
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* ''[[Pelogic]]'', {21/20, 135/128} → [[Pelogic family #Pelogic|Pelogic family]]
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[August]]'', {36/35, 128/125} → [[Augmented family #August|Augmented family]]
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Pajara]], {50/49, 64/63} → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Meantone]], {81/80, 126/125} → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* [[Magic]], {225/224, 245/243} → [[Magic family #Magic|Magic family]]
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* ''[[Passive]]'', {225/224, 256/245} → [[Passion family #Passive|Passion family]]
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* [[Miracle]], {225/224, 1029/1024} → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Orwell]], {225/224, 1728/1715} → [[Semicomma family #Orwell|Semicomma family]]
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Garibaldi temperament|Garibaldi]], {225/224, 3125/3087} → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* [[Catakleismic]], {225/224, 4375/4374} → [[Kleismic family #Catakleismic|Kleismic family]]
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[Decic]]'', {225/224, 16807/16384} → [[Cloudy clan #Decic|Cloudy clan]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Escapade]]'', {225/224, 65625/65536} → [[Escapade family #Escapade|Escapade family]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Fog]]'', {225/224, 156250/151263} → [[Misty family #Fog|Misty family]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* ''[[Compton]]'', {225/224, 250047/250000} → [[Compton family #Compton|Compton family]]
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Betic]]'', {225/224, 1071875/1062882} → [[Sycamore family #Betic|Sycamore family]]
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Houborizic]]'' {225/224, 1250000/1240029} → [[Amity family #Houborizic|Amity family]]
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* ''[[Qintosec]]'', {225/224, 2560000/2470629} → [[Qintosec family #Qintosec|Qintosec family]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Quintapole]]'', {225/224, 7812500/7411887} → [[Quintaleap family #Quintapole|Quintaleap family]]
* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
* ''[[Maquila]]'', {225/224, 30233088/28824005} → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
* ''[[Marvo]]'', {225/224, 78125000/78121827} → [[Gravity family #Marvo|Gravity family]]
* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* [[Wizard]] (+118098/117649) → [[Stearnsmic clan #Wizard|Stearnsmic clan]]
* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]


Considered below are negri, wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, submajor, alphorn, tertiosec, gwazy, and gracecordial.  
Considered below are tritonic, septimin, merman, slender, triton, marvolo, decic, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, hendeca, gwazy, and tertiosec, in the order of increasing [[badness]].  


Since (5/4)<sup>2</sup> = 225/224 × 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.


The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.
The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.


Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)
Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)


== Negri ==
== Tritonic ==
{{main| Negri }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


Negri tempers out the [[negri comma]] in the 5-limit, [[49/48]] and [[225/224]] in the 7-limit. It can be extended naturally to the 2.3.5.7.13 subgroup by adding [[91/90]] to the comma list; this will be discussed below under the title of negra.  
Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.  


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16384
[[Comma list]]: 225/224, 50421/50000


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


{{Multival|legend=1| 4 -3 -14 }}
[[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 }}


[[POTE generator]]: ~16/15 = 125.7549
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
[[Badness]] (Sintel): 1.20


[[Badness]]: 0.086856
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 7-limit ===
Comma list: 121/120, 225/224, 441/440
Subgroup: 2.3.5.7


[[Comma list]]: 49/48, 225/224
{{Mapping|legend=0| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


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


{{Multival|legend=1| 4 -3 2 -14 -8 13 }}
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


[[POTE generator]]: ~15/14 = 125.608
Badness (Sintel): 0.782


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.026483
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
 
==== 19-limit ====
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}}


Optimal GPV sequence: {{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}}


Optimal GPV sequence: {{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: [{{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}}


Optimal GPV sequence: {{Val list| 19, 29, 48d, 77cdd }}
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Badness: 0.038679
Badness (Sintel): 1.50


==== 13-limit ====
== Septimin ==
Subgroup: 2.3.5.7.11.13
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''
 
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


Comma list: 49/48, 65/64, 91/90, 875/858
[[Comma list]]: 225/224, 84035/82944


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


POTE generator: ~14/13 = 124.716
[[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 }}


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


Badness: 0.024383
[[Badness]] (Sintel): 1.38


=== Negric ===
=== 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}}


Optimal GPV sequence: {{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: [{{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}}


Optimal GPV sequence: {{Val list| 9, 19e }}
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}


Badness: 0.020205
Badness (Sintel): 0.955


=== Negroni ===
== Merman ==
Subgroup: 2.3.5.7.11
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Comma list: 49/48, 55/54, 225/224
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.


Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.


POTE generator: ~15/14 = 124.539
[[Subgroup]]: 2.3.5.7


Optimal GPV sequence: {{Val list| 10, 19e, 29, 77cddee }}
[[Comma list]]: 225/224, 2500000/2470629


Badness: 0.035296
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


==== 13-limit ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13
* [[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 }}


Comma list: 49/48, 55/54, 65/64, 91/90
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
[[Badness]] (Sintel): 1.39


POTE generator: ~14/13 = 124.545
=== 11-limit ===
 
Optimal GPV sequence: {{Val list| 10, 19e, 29, 77cddeef }}
 
Badness: 0.021559
 
=== Wilsec ===
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 }}]
{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


POTE generator: ~11/8 = 537.186
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


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


Badness: 0.041886
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, 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}}


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


Badness: 0.025192
Badness (Sintel): 1.14


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


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
Comma list: 225/224, 385/384, 532400/531441


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


POTE generator: ~11/8 = 537.230
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


Optimal GPV sequence: {{Val list| 9, 20g, 29g, 38df, 67cddefg }}
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


Badness: 0.021778
Badness (Sintel): 1.46


==== 19-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
Comma list: 225/224, 325/324, 385/384, 10648/10647


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


POTE generator: ~11/8 = 537.214
Optimal tunings:  
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


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


Badness: 0.016828
Badness (Sintel): 1.47


== Wizard ==
== Slender ==
{{Main|Wizard}}
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.  
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''


Subgroup: 2.3.5.7
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  


[[Comma list]]: 225/224, 118098/117649
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
[[Comma list]]: 225/224, 589824/588245


Mapping generators: ~1225/864, ~245/216
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


[[POTE generator]]: ~5/4 = 383.256
[[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 }}


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
[[Badness]] (Sintel): 1.44
 
[[Badness]]: 0.040846
 
Scales: [[wizard22]]


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


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


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


POTE generator: ~5/4 = 383.232
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Optimal GPV sequence: {{Val list| 22, 50, 72, 166, 238c, 310c }}
Badness (Sintel): 0.838


Badness: 0.018539
=== 13-limit ===
 
Scales: [[wizard22]]
 
==== Lizard ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 351/350, 364/363, 385/384
Comma list: 225/224, 275/273, 385/384, 1331/1323


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


POTE generator: ~5/4 = 383.389
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Optimal GPV sequence: {{Val list| 22, 50, 72, 122, 194df }}
Badness (Sintel): 1.07


Badness: 0.021781
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


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


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
[[Comma list]]: 225/224, 1029/1000


Mapping generators: ~17/12, ~17/15
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


POTE generator: ~5/4 = 383.381
[[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 }}


Optimal GPV sequence: {{Val list| 22, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


Badness: 0.014536
[[Badness]] (Sintel): 1.50


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


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Comma list: 45/44, 56/55, 1029/1000


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


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


POTE generator: ~5/4 = 383.477
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


Optimal GPV sequence: {{Val list| 22h, 50, 72, 122g, 194dfg }}
Badness (Sintel): 1.51


Badness: 0.015702
== Marvolo ==
[[Subgroup]]: 2.3.5.7


==== Gizzard ====
[[Comma list]]: 225/224, 156250000/155649627
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 325/324, 385/384, 1573/1568
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


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


Mapping generators: ~99/70, ~25/22
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


POTE generator: ~5/4 = 383.170
[[Badness]] (Sintel): 2.11


Optimal GPV sequence: {{Val list| 72, 166, 238cf }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.020252
Comma list: 225/224, 441/440, 4000/3993


===== 17-limit =====
{{Mapping|legend=0| 1 2 1 1 2 | 0 -6 19 26 21 }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Mapping generators: ~17/12, ~17/15
Badness (Sintel): 0.958


POTE generator: ~5/4 = 383.175
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Optimal GPV sequence: {{Val list| 72, 166g, 238cfg }}
Comma list: 169/168, 225/224, 364/363, 441/440


Badness: 0.013624
{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


===== 19-limit =====
Optimal tunings:
Subgroup: 2.3.5.7.11.13.17.19
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
Badness (Sintel): 0.887


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


POTE generator: ~5/4 = 383.138
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


Optimal GPV sequence: {{Val list| 72, 94, 166g }}
{{Mapping|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


Badness: 0.014810
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


=== Mage ===
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 1331/1296
Badness (Sintel): 0.760


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
Mapping generators: ~77/54, ~55/48


POTE generator: ~5/4 = 383.124
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Optimal GPV sequence: {{Val list| 22, 50e, 72ee, 94ee }}
{{Mapping|legend=0| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Badness: 0.057799
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


== Triton ==
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''


Subgroup: 2.3.5.7
Badness (Sintel): 0.895


[[Comma list]]: 225/224, 1029/1000
== Decic ==
{{Main| Decic }}


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


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
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]]: ~7/5 = 568.865
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 2, 17d, 19, 78bd, 97bd }}
[[Comma list]]: 225/224, 16807/16384


[[Badness]]: 0.059245
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


=== 11-limit ===
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11
* [[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 }}


Comma list: 45/44, 56/55, 1029/1000
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
[[Badness]] (Sintel): 2.26
 
POTE generator: ~7/5 = 569.144
 
Optimal GPV sequence: {{Val list| 2, 17d, 19, 59bde, 78bde, 97bde }}
 
Badness: 0.045675
 
== Tritonic ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 225/224, 50421/50000
 
[[Mapping]]: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
 
{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
 
[[POTE generator]]: ~7/5 = 580.286
 
{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
 
[[Badness]]: 0.047578


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


Comma list: 121/120, 225/224, 441/440
Comma list: 225/224, 385/384, 3087/3025


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


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


Optimal GPV sequence: {{Val list| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Badness: 0.023659
Badness (Sintel): 2.11


==== 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, 144/143, 196/195, 2200/2197


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


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


Optimal GPV sequence: {{Val list| 29, 31, 60e, 151cde }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Badness: 0.022993
Badness (Sintel): 1.52


=== 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, 144/143, 170/169, 196/195, 221/220


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
{{Mapping|legend=0| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


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


Optimal GPV sequence: {{Val list| 31, 91, 122, 153d }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Badness: 0.045456
Badness (Sintel): 1.28


== Merman ==
=== Splendecic ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[fantastic]] rank-3 temperament, its name is a portmanteau of ''splendid'' and ''decic''.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 225/224, 2500000/2470629
Comma list: 225/224, 1617/1600, 2401/2376


[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
{{Mapping|legend=0| 10 0 39 28 82 | 0 1 -1 0 -3 }}


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
Optimal tunings:
* WE: ~15/14 = 120.1874{{c}}, ~3/2 = 699.6085{{c}} (~99/98 = 21.5156{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3531{{c}} (~99/98 = 21.6469{{c}})


[[POTE generator]]: ~7/5 = 585.585
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


{{Val list|legend=1| 41, 84, 125 }}
Badness (Sintel): 1.98


[[Badness]]: 0.055078
==== 13-limit ====
 
Subgroup: 2.3.5.7.11.13
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 225/224, 441/440, 1344/1331
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
{{Mapping|legend=0| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


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


Optimal GPV sequence: {{Val list| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


Badness: 0.036383
Badness (Sintel): 1.57


=== 13-limit ===
==== 17-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17


Comma list: 144/143, 225/224, 364/363, 441/440
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
{{Mapping|legend=0| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


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


Optimal GPV sequence: {{Val list| 41, 84, 125e, 209ef, 293ef }}
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


Badness: 0.027544
Badness (Sintel): 1.33


== Untriton ==
=== Prodecic ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Untriton]].''
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''.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 225/224, 125000000/121060821
Comma list: 225/224, 441/440, 5929/5832


[[Mapping]]: [{{Val|1 6 -7 -7}}, {{Val|0 -9 19 20}}]
{{Mapping|legend=0| 10 0 39 28 -13 | 0 1 -1 0 3 }}


{{Multival|legend=1|9 -19 -20 -51 -57 7}}
Optimal tunings:
* WE: ~15/14 = 120.2024{{c}}, ~3/2 = 701.3908{{c}} (~55/54 = 19.8237{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.5235{{c}} (~55/54 = 19.4765{{c}})


[[POTE generator]]: ~7/5 = 588.641
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


{{Val list|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}
Badness (Sintel): 2.20


[[Badness]]: 0.143976
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 11-limit ===
Comma list: 105/104, 196/195, 275/273, 5929/5832
Subgroup: 2.3.5.7.11


Comma list: 121/120, 225/224, 22000/21609
{{Mapping|legend=0| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


Mapping: [{{Val|1 6 -7 -7 1}}, {{Val|0 -9 19 20 5}}]
Optimal tunings:  
* WE: ~15/14 = 120.1654{{c}}, ~3/2 = 701.4683{{c}} (~91/90 = 19.5242{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.7175{{c}} (~91/90 = 19.2825{{c}})


POTE generator: ~7/5 = 588.626
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Optimal GPV sequence: {{Val list| 51, 53 }}
Badness (Sintel): 1.73


Badness: 0.074295
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


=== 13-limit ===
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 225/224, 275/273, 1040/1029
{{Mapping|legend=0| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


Mapping: [{{Val|1 6 -7 -7 1 -12}}, {{Val|0 -9 19 20 5 32}}]
Optimal tunings:  
* WE: ~15/14 = 120.1577{{c}}, ~3/2 = 701.3950{{c}} (~91/90 = 19.5514{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.6932{{c}} (~91/90 = 19.3068{{c}})


POTE generator: ~7/5 = 588.654
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Optimal GPV sequence: {{Val list| 51f, 53 }}
Badness (Sintel): 1.41


Badness: 0.047441
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


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


Subgroup: 2.3.5.7
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


[[Comma list]]: 225/224, 84035/82944
[[Optimal tuning]]s:  
* [[WE]]: ~2592/2401 = 133.4174{{c}}, ~12005/6912 = 950.7667{{c}} (~1728/1715 = 16.8452{{c}})
: [[error map]]: {{val| +0.756 -0.422 -1.395 +0.298 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}})
: error map: {{val| 0.000 -1.361 -3.277 -1.565 }}


[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
[[Badness]] (Sintel): 2.37
 
[[POTE generator]]: ~7/6 = 263.632
 
{{Val list|legend=1| 41, 91, 132d }}
 
[[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, 12005/11979


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
{{Mapping|legend=0| 9 0 28 11 24 | 0 2 -1 2 1 }}


POTE generator: ~7/6 = 263.634
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}})


Optimal GPV sequence: {{Val list| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.031309
Badness (Sintel): 1.01


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


Comma list: 105/104, 144/143, 196/195, 245/242
Comma list: 169/168, 225/224, 364/363, 1716/1715


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
{{Mapping|legend=0| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


POTE generator: ~7/6 = 263.700
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}})


Optimal GPV sequence: {{Val list| 41, 91 }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.023117
Badness (Sintel): 0.922


== Misneb ==
== Gracecordial ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Misneb]].''
: ''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, 4194304/4117715
[[Comma list]]: 225/224, 781250000/771895089
 
[[Mapping]]: [{{Val|1 3 1 3}}, {{Val|0 -15 14 -2}}]


{{Multival|legend=1|15 -14 2 -57 -39 44}}
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


[[POTE generator]]: ~16/15 = 113.235
[[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| 21, 32, 53 }}
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


[[Badness]]: 0.140970
[[Badness]] (Sintel): 2.44


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


Comma list: 99/98, 176/175, 1310720/1294139
Comma list: 225/224, 385/384, 236328125/234365481


Mapping: [{{Val|1 3 1 3 1}}, {{Val|0 -15 14 -2 26}}]
{{Mapping|legend=0| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


POTE generator: ~16/15 = 113.323
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Optimal GPV sequence: {{Val list| 21, 32e, 53, 127, 180de }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Badness: 0.085390
Badness (Sintel): 2.96


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


Comma list: 99/98, 176/175, 640/637, 847/845
Comma list: 225/224, 325/324, 385/384, 831875/830466


Mapping: [{{Val|1 3 1 3 1 2}}, {{Val|0 -15 14 -2 26 18}}]
{{Mapping|legend=0| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


POTE generator: ~16/15 = 113.323
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


Optimal GPV sequence: {{Val list| 21, 32e, 53, 127, 180de }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.045569
Badness (Sintel): 2.16


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


Comma list: 225/224, 385/384, 66550/64827
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


Mapping: [{{Val|1 3 1 3 6}}, {{Val|0 -15 14 -2 -27}}]
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


POTE generator: ~16/15 = 113.142
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


Optimal GPV sequence: {{Val list| 32, 53, 191de, 244cddee, 297cddee }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.087333
Badness (Sintel): 1.96


== Slender ==
==== 19-limit ====
Slender (31&amp;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.
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


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


[[Comma list]]: 225/224, 589824/588245
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 }}, {{val| 0 -13 10 -6 }}]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
Badness (Sintel): 1.71


[[POTE generator]]: ~49/48 = 38.413
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


{{Val list|legend=1| 31, 94, 125 }}
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


[[Badness]]: 0.056934
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


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


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


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


POTE generator: ~49/48 = 38.387
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Optimal GPV sequence: {{Val list| 31, 63, 94, 125 }}
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Badness: 0.025342
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


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


Comma list: 225/224, 275/273, 385/384, 1331/1323
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
Badness (Sintel): 1.50


POTE generator: ~49/48 = 38.314
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Optimal GPV sequence: {{Val list| 31, 63, 94 }}
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


Badness: 0.025913
{{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 }}


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


[[Comma list]]: 225/224, 156250000/155649627
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
Badness (Sintel): 1.53


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
=== Gracecord ===
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~21/20 = 83.348
Comma list: 225/224, 441/440, 109375/107811


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
{{Mapping|legend=0| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


[[Badness]]: 0.083338
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


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


Comma list: 225/224, 441/440, 4000/3993
Badness (Sintel): 2.21


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~21/20 = 83.340
Comma list: 225/224, 364/363, 441/440, 6125/6084


Optimal GPV sequence: {{Val list| 29, 43, 72 }}
{{Mapping|legend=0| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Badness: 0.028965
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


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


Comma list: 169/168, 225/224, 364/363, 441/440
Badness (Sintel): 1.83


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~21/20 = 83.330
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Optimal GPV sequence: {{Val list| 29, 43, 72, 115f }}
{{Mapping|legend=0| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Badness: 0.021470
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


== Amavil ==
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Mabila]].''


Subgroup: 2.3.5.7
Badness (Sintel): 1.87


[[Comma list]]: 225/224, 17496/16807
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
{{Mapping|legend=0| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


[[POTE generator]]: ~48/35 = 529.979
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


[[Badness]]: 0.109625
Badness (Sintel): 1.68


=== 11-limit ===
== Alphorn ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7


Comma list: 99/98, 176/175, 864/847
[[Comma list]]: 225/224, 5764801/5668704


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


POTE generator: ~15/11 = 529.974
[[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 }}


Optimal GPV sequence: {{Val list| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Badness: 0.042649
[[Badness]] (Sintel): 3.27


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


Comma list: 78/77, 99/98, 144/143, 176/175
Comma list: 225/224, 385/384, 12250/11979


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}}


Optimal GPV sequence: {{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}}


Optimal GPV sequence: {{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}}


Optimal GPV sequence: {{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|legend=0| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


[[POTE generator]]: ~10125/8192 = 362.321
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
Badness (Sintel): 2.89


[[Badness]]: 0.130236
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


=== 7-limit ===
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.  
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 51200/50421
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
[[Comma list]]: 225/224, 125000000/121060821


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


[[POTE generator]]: ~49/40 = 362.255
[[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 }}


{{Val list|legend=1| 10, 33, 43, 53 }}
{{Optimal ET sequence|legend=1| 51, 53 }}


[[Badness]]: 0.060533
[[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}}


Optimal GPV sequence: {{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}}


Optimal GPV sequence: {{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 }}


Optimal GPV sequence: {{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 }}


Optimal GPV sequence: {{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


Optimal GPV sequence: {{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
 
=== 7-limit ===
Subgroup: 2.3.5.7
 
[[Comma list]]: 225/224, 14495514624/13841287201
 
[[Mapping]]: [{{val| 3 7 5 9 }}, {{val| 0 -8 7 -2 }}]
 
{{Multival|legend=1| 24 -21 6 -89 -58 73 }}
 
[[POTE generator]]: ~15/14 = 112.283
 
{{Val list|legend=1| 21, 54, 75, 96, 171d }}
 
[[Badness]]: 0.431636


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


Comma list: 225/224, 3840/3773, 12005/11979
Comma list: 225/224, 441/440, 43923/43750


Mapping: [{{val| 3 7 5 9 9 }}, {{val| 0 -8 7 -2 5 }}]
{{Mapping|legend=0| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


POTE generator: ~15/14 = 112.171
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Optimal GPV sequence: {{Val list| 21, 54, 75e }}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.173485
Badness (Sintel): 1.74


== Gwazy ==
=== 13-limit ===
{{see also|Very high accuracy temperaments #Kwazy}}
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5.7
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


[[Comma list]]: 225/224, 5971968/5764801
{{Mapping|legend=0| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


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


{{Multival|legend=1| 16 -10 12 -53 -26 56 }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


[[POTE generator]]: ~35/32 = 162.658
Badness (Sintel): 1.35


{{Val list|legend=1| 22, 74, 96, 118d }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Badness]]: 0.178826
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


=== 11-limit ===
{{Mapping|legend=0| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 65536/65219


Mapping: [{{val| 2 1 6 4 8 }}, {{val| 0 8 -5 6 -4 }}]
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


POTE generator: ~11/10 = 162.592
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Optimal GPV sequence: {{Val list| 22, 74, 96, 118d }}
Badness (Sintel): 1.17


Badness: 0.068410
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


== Gracecordial ==
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.  
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 781250000/771895089
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 0 34 63 }}, {{val| 0 1 -20 -38 }}]
[[Comma list]]: 225/224, 122880/117649


{{Multival|legend=1| 1 -20 -38 -34 -63 -32 }}
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


[[POTE generator]]: ~3/2 = 700.824
[[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 }}


{{Val list|legend=1| 12, 113, 125, 238c, 363c }}
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}


[[Badness]]: 0.096279
[[Badness]] (Sintel): 4.37


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


Comma list: 225/224, 385/384, 236328125/234365481
Comma list: 121/120, 225/224, 352/343


Mapping: [{{val| 1 0 34 63 -90 }}, {{val| 0 1 -20 -38 59 }}]
{{Mapping|legend=0| 11 0 43 31 38 | 0 1 -1 0 0 }}


POTE generator: ~3/2 = 700.834
Optimal tunings:  
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


Optimal GPV sequence: {{Val list| 12e, 113, 125, 238c }}
{{Optimal ET sequence|legend=0| 22, 55d }}


Badness: 0.089588
Badness (Sintel): 2.46


==== 13-limit ====
== Gwazy ==
Subgroup: 2.3.5.7.11.13
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Comma list: 225/224, 325/324, 385/384, 831875/830466
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


Mapping: [{{val| 1 0 34 63 -90 -66 }}, {{val| 0 1 -20 -38 59 44 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~3/2 = 700.841
[[Comma list]]: 225/224, 5971968/5764801


Optimal GPV sequence: {{Val list| 12e, 113, 125f, 238cf }}
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


Badness: 0.052235
[[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 }}


==== 17-limit ====
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
[[Badness]] (Sintel): 4.53


Mapping: [{{val| 1 0 34 63 -90 -66 -7 }}, {{val| 0 1 -20 -38 59 44 7 }}]
=== 11-limit ===
 
POTE generator: ~3/2 = 700.841
 
Optimal GPV sequence: {{Val list| 12e, 113, 125f, 238cf }}
 
Badness: 0.038565
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
 
Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 }}, {{val| 0 1 -20 -38 59 44 7 3 }}]
 
POTE generator: ~3/2 = 700.842
 
Optimal GPV sequence: {{Val list| 12e, 113, 125f, 238cf }}
 
Badness: 0.028165
 
=== Gracecord ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 441/440, 109375/107811
Comma list: 99/98, 176/175, 65536/65219


Mapping: [{{val| 1 0 34 63 89 }}, {{val| 0 1 -20 -38 -54 }}]
{{Mapping|legend=0| 2 1 6 4 8 | 0 8 -5 6 -4 }}


POTE generator: ~3/2 = 700.885
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


Optimal GPV sequence: {{Val list| 12, 101cd, 113 }}
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


Badness: 0.066964
Badness (Sintel): 2.26


==== 13-limit ====
== Tertiosec ==
Subgroup: 2.3.5.7.11.13
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


Comma list: 225/224, 364/363, 441/440, 6125/6084
Tertiosec may be described as the {{nowrap| 21 & 75 }} temperament. It was initially named ''tertiomar'' by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, but was changed to ''tertiosec'' in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.


Mapping: [{{val| 1 0 34 63 89 113 }}, {{val| 0 1 -20 -38 -54 -69 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~3/2 = 700.890
[[Comma list]]: 225/224, 14495514624/13841287201
 
Optimal GPV sequence: {{Val list| 12f, 101cdf, 113 }}
 
Badness: 0.044196
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


Mapping: [{{val| 1 0 34 63 89 113 -7 }}, {{val| 0 1 -20 -38 -54 -69 7 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
: [[error map]]: {{val| -0.523 -1.044 -1.549 +5.138 }}
* [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}}
: error map: {{val| 0.000 -0.284 -0.276 +6.592 }}


POTE generator: ~3/2 = 700.870
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


Optimal GPV sequence: {{Val list| 12f, 101cdf, 113 }}
[[Badness]] (Sintel): 10.9


Badness: 0.036637
=== 11-limit ===
Subgroup: 2.3.5.7.11


==== 19-limit ====
Comma list: 225/224, 3840/3773, 12005/11979
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663
{{Mapping|legend=0| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


Mapping: [{{val| 1 0 34 63 89 113 -7 9 }}, {{val| 0 1 -20 -38 -54 -69 7 3 }}]
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: ~3/2 = 700.866
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


Optimal GPV sequence: {{Val list| 12e, 101cdf, 113 }}
Badness (Sintel): 5.74


Badness: 0.027559
== 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]]
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"