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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:  
* ''[[Pelogic]]'' → [[Mavila family #Pelogic|Mavila family]] (+21/20 or 135/128, generated by the fifth with 5/4 mapped to the m3)
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* [[Meantone]] → [[Meantone family #Septimal meantone|Meantone family]] (+81/80 or 126/125, generated by the fifth with 5/4 mapped to the M3)
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Garibaldi]] → [[Schismatic family #Garibaldi|Schismatic family]] (+3125/3087, generated by the fifth with 5/4 mapped to the d4)
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Pajara]] → [[Diaschismic family #Pajara|Diaschismic family]] (+50/49 or 64/63, generated by the fifth with a semioctave period)
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[Sharpie]]'' → [[Dicot family #Sharpie|Dicot family]] (+25/24 or 28/27, fifth sliced in two)
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* ''[[Immune]]'' → [[Immunity family #Immune|Immunity family]] (+781250/750141, twelfth sliced in two)
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* ''[[August]]'' → [[Augmented family #August|Augmented family]] (+36/35 or 128/125, generated by the fifth with a 1/3-octave period)
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* ''[[Fog]]'' → [[Misty family #Fog|Misty family]] (+156250/151263, generated by the fifth with a 1/3-octave period)
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Negri]] → [[Slendro clan #Negri|Slendro clan]] (+49/48, fourth sliced in four)
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Magic]] → [[Magic family #Magic|Magic family]] (+245/243, twelfth sliced in five)
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* ''[[Passive]]'' → [[Passion family #Passive|Passion family]] (+256/245, fourth sliced in five)
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[Quintapole]]'' → [[Quintaleap family #Quintapole|Quintaleap family]] (+7812500/7411887, fourth sliced in five)
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Houborizic]]'' → [[Amity family #Houborizic|Amity family]] (+1250000/1240029, eleventh sliced in five)
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Qintosec]]'' → [[Quintosec family #Qintosec|Quintosec family]] (+2560000/2470629, generated by the classical minor second with a 1/5-octave period)
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* [[Miracle]] → [[Gamelismic clan #Miracle|Gamelismic clan]] (+1029/1024, fifth sliced in six)
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* [[Catakleismic]] → [[Kleismic family #Catakleismic|Kleismic family]] (+4375/4374, twelfth sliced in six)
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Marvo]]'' → [[Gravity family #Marvo|Gravity family]] (+78125000/78121827, two octaves and a fifth sliced in six)
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* [[Orwell]] → [[Semicomma family #Orwell|Semicomma family]] (+1728/1715, twelfth sliced in seven)
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Snipes]]''  → [[Wesley family #Snipes|Wesley family]] (+6125/5832, two octaves and a fourth sliced in seven)
* ''[[Snipes]]'' (+6125/5832) → [[Wesley family #Snipes|Wesley family]]
* ''[[Escapist]]'' → [[Escapade family #Escapade|Escapade family]] (+65625/65536, fourth sliced in nine)
* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
* ''[[Decic]]'' → [[Cloudy clan #Decic|Cloudy clan]] (+16807/16384, generated by the fifth with a 1/10-octave period)
* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
* ''[[Amavil]]'' → [[Mabila family #Amavil|Mabila family]] (+17496/16807, four octaves and a fourth sliced in ten)
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' → [[Sycamore family #Betic|Sycamore family]] (+1071875/1062882, fifth sliced in eleven)
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Compton]]'' → [[Compton family #Compton|Compton family]] (+250047/250000, generated by the classical major third with a 1/12-octave period)
* [[Wizard]] (+118098/117649) → [[Stearnsmic clan #Wizard|Stearnsmic clan]]
* ''[[Raccoon]]'' → [[Vavoom family #Raccoon|Vavoom family]] (+41943040/40353607, twelfth sliced in seventeen)
* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
* ''[[Maquila]]'' → [[Maquila family #Septimal maquila|Maquila family]] (+30233088/28824005, seven octaves and a fifth sliced in seventeen)
* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Gammy]]'' → [[Gammic family #Gammy|Gammic family]] (+94143178827/91913281250, fifth sliced in twenty)
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]


Considered below are 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 {{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.
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.
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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}}.)
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}}.)


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


Wizard has a semi-octave period and is generated by an interval that is best treated as ~17/15. The semi-octave complement of this interval is ~5/4. Wizard can be described as {{nowrap|22 &amp; 72}}. Its ploidacot is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.  
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]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 118098/117649
[[Comma list]]: 225/224, 50421/50000


{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
 
: mapping generators: ~2, ~10/7
: mapping generators: ~1225/864, ~245/216
 
{{Multival|legend=1| 12 -2 20 -31 -2 52 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~1225/864 = 600.000, ~245/216 = 216.919
* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
: [[error map]]: {{val| 0.000 -0.443 -3.232 +0.361 }}
: [[error map]]: {{val| +1.354 -1.243 -0.027 -1.598 }}
* [[POTE]]: ~1225/864 = 600.000, ~245/216 = 216.744
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}}
: error map: {{val| 0.000 -1.492 -3.057 -1.388 }}
: error map: {{val| 0.000 -3.566 -2.769 -4.959 }}


{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


[[Badness]] (Smith): 0.040846
[[Badness]] (Sintel): 1.20


=== 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: 121/120, 225/224, 441/440


Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}
{{Mapping|legend=0| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000, ~25/22 = 216.900
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* POTE: ~99/70 = 600.000, ~25/22 = 216.768
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


Badness (Smith): 0.018539
Badness (Sintel): 0.782


==== Lizard ====
==== 13-limit ====
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: 105/104, 121/120, 196/195, 275/273


Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
{{Mapping|legend=0| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 600.000, ~25/22 = 216.703
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* POTE: ~55/39 = 600.000, ~25/22 = 216.711
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


{{Optimal ET sequence|legend=0| 22, 50, 72, 122, 194df }}
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


Badness (Smith): 0.021781
Badness (Sintel): 0.950


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


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
{{Mapping|legend=0| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~17/15 = 216.748
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* POTE: ~17/12 = 600.000, ~17/15 = 216.619
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


{{Optimal ET sequence|legend=0| 22, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness (Smith): 0.014536
Badness (Sintel): 0.973


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


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~17/15 = 216.677
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* POTE: ~17/12 = 600.000, ~17/15 = 216.523
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness (Smith): 0.015702
Badness (Sintel): 1.03


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


Comma list: 225/224, 325/324, 385/384, 1573/1568
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000, ~25/22 = 216.928
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* POTE: ~99/70 = 600.000, ~25/22 = 216.830
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness (Smith): 0.020252
Badness (Sintel): 1.04


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


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Comma list: 225/224, 385/384, 27783/27500


Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
{{Mapping|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~17/15 = 216.943
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* POTE: ~17/12 = 600.000, ~17/15 = 216.825
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Badness (Smith): 0.013624
Badness (Sintel): 1.50


===== 19-limit =====
== Septimin ==
Subgroup: 2.3.5.7.11.13.17.19
: ''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: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
[[Comma list]]: 225/224, 84035/82944


Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


Optimal tunings:  
[[Optimal tuning]]s:  
* CTE: ~17/12 = 600.000, ~17/15 = 216.918
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
* POTE: ~17/12 = 600.000, ~17/15 = 216.862
: [[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 ET sequence|legend=0| 72, 94, 166g }}
{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}


Badness (Smith): 0.014810
[[Badness]] (Sintel): 1.38


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


Comma list: 99/98, 176/175, 1331/1296
Comma list: 225/224, 245/242, 385/384


Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
{{Mapping|legend=0| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~77/54 = 600.000, ~55/48 = 217.247
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* POTE: ~77/54 = 600.000, ~55/48 = 216.876
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}


Badness (Smith): 0.057799
Badness (Sintel): 1.04


== Tritonic ==
=== 13-limit ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
Subgroup: 2.3.5.7.11.13


[[Subgroup]]: 2.3.5.7
Comma list: 105/104, 144/143, 196/195, 245/242


[[Comma list]]: 225/224, 50421/50000
{{Mapping|legend=0| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


{{Mapping|legend=1| 1 4 -3 -3 | 0 -5 11 12 }}
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 580.286
Badness (Sintel): 0.955


{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


[[Badness]]: 0.047578
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.  


=== 11-limit ===
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.  
Subgroup: 2.3.5.7.11


Comma list: 121/120, 225/224, 441/440
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 4 -3 -3 2 | 0 -5 11 12 3 }}
[[Comma list]]: 225/224, 2500000/2470629


{{Multival|legend=1| 5 -11 -12 -3 -29 -33 -22 3 31 33 }}
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.267
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
: [[error map]]: {{val| +0.390 -0.435 -1.630 +1.634 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}}
: error map: {{val| 0.000 -1.104 -2.423 +0.657 }}


{{Optimal ET sequence|legend=1| 29, 31, 60e }}
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


Badness: 0.023659
[[Badness]] (Sintel): 1.39


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


Comma list: 105/104, 121/120, 196/195, 275/273
Comma list: 225/224, 441/440, 1344/1331


Mapping: {{mapping| 1 4 -3 -3 2 -5 | 0 -5 11 12 3 18 }}
{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.108
Optimal tunings:
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


{{Optimal ET sequence|legend=1| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


Badness: 0.022993
Badness (Sintel): 1.20


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


Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
Comma list: 144/143, 225/224, 364/363, 441/440


Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 | 0 -5 11 12 3 18 25 }}
{{Mapping|legend=0| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.055
Optimal tunings:
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


{{Optimal ET sequence|legend=1| 29g, 31, 60e }}
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Badness: ?
Badness (Sintel): 1.14


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


Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
Comma list: 225/224, 385/384, 532400/531441


Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 -3 | 0 -5 11 12 3 18 25 15 }}
{{Mapping|legend=0| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.026
Optimal tunings:
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


{{Optimal ET sequence|legend=1| 29g, 31, 60e }}
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


Badness: ?
Badness (Sintel): 1.46


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


Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
Comma list: 225/224, 325/324, 385/384, 10648/10647


Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 -3 5 | 0 -5 11 12 3 18 25 15 -1 }}
{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.009
Optimal tunings:
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


{{Optimal ET sequence|legend=1| 29g, 31, 60e }}
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


Badness: ?
Badness (Sintel): 1.47


=== Tritoni ===
== Slender ==
Subgroup: 2.3.5.7.11
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: 225/224, 385/384, 27783/27500


Mapping: {{mapping| 1 4 -3 -3 17 | 0 -5 11 12 -28 }}
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  
 
{{Multival|legend=1| 5 -11 -12 28 -29 -33 27 3 103 120 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.389
 
{{Optimal ET sequence|legend=1| 31, 91, 122, 153d }}
 
Badness: 0.045456
 
== Septimin ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 84035/82944
[[Comma list]]: 225/224, 589824/588245


{{Mapping|legend=1| 1 4 1 5 | 0 -11 6 -10 }}
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
: [[error map]]: {{val| +0.382 -0.725 -1.295 +1.765 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}}
: error map: {{val| 0.000 -1.257 -2.235 +0.727 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/6 = 263.632
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


{{Optimal ET sequence|legend=1| 41, 91, 132d }}
[[Badness]] (Sintel): 1.44
 
[[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, 1331/1323


Mapping: {{mapping| 1 4 1 5 5 | 0 -11 6 -10 -7 }}
{{Mapping|legend=0| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 263.634
Optimal tunings:
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


{{Optimal ET sequence|legend=1| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Badness: 0.031309
Badness (Sintel): 0.838


=== 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, 275/273, 385/384, 1331/1323
 
{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Mapping: {{mapping| 1 4 1 5 5 7 | 0 -11 6 -10 -7 -15 }}
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 263.700
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


{{Optimal ET sequence|legend=1| 41, 91 }}
Badness (Sintel): 1.07


Badness: 0.023117
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


== Merman ==
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.  
: ''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, 2500000/2470629
[[Comma list]]: 225/224, 1029/1000


{{Mapping|legend=1| 1 5 -5 -5 | 0 -7 15 16 }}
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 585.585
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


{{Optimal ET sequence|legend=1| 41, 84, 125 }}
[[Badness]] (Sintel): 1.50
 
[[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: 45/44, 56/55, 1029/1000


Mapping: {{mapping| 1 5 -5 -5 2 | 0 -7 15 16 3 }}
{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.606
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=1| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


Badness: 0.036383
Badness (Sintel): 1.51
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 5 -5 -5 2 12 | 0 -7 15 16 3 -17 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.657
 
{{Optimal ET sequence|legend=1| 41, 84, 125e, 209ef, 293ef }}
 
Badness: 0.027544
 
== Slender ==
Slender ({{nowrap|31 &amp; 32}}) tempers out the [[Hewuermera temperaments|hewuermera comma]] in addition to the marvel comma. This temperament has a generator of [[49/48]], 3 of which equal marvel's 16/15~15/14, and 10 generators is 5/4.


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


[[Comma list]]: 225/224, 589824/588245
[[Comma list]]: 225/224, 156250000/155649627


{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 38.413
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


{{Optimal ET sequence|legend=1| 31, 94, 125 }}
[[Badness]] (Sintel): 2.11
 
[[Badness]]: 0.056934


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


Comma list: 225/224, 385/384, 1331/1323
Comma list: 225/224, 441/440, 4000/3993


Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
{{Mapping|legend=0| 1 2 1 1 2 | 0 -6 19 26 21 }}


Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 38.387
Optimal tunings:
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


{{Optimal ET sequence|legend=1| 31, 63, 94, 125 }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.025342
Badness (Sintel): 0.958


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


Comma list: 225/224, 275/273, 385/384, 1331/1323
Comma list: 169/168, 225/224, 364/363, 441/440


Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 38.314
Optimal tunings:
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


{{Optimal ET sequence|legend=1| 31, 63, 94 }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.025913
Badness (Sintel): 0.887


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


[[Subgroup]]: 2.3.5.7
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


[[Comma list]]: 225/224, 1029/1000
{{Mapping|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
Optimal tunings:
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 568.865
Badness (Sintel): 0.760


{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


[[Badness]]: 0.059245
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


=== 11-limit ===
{{Mapping|legend=0| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
Subgroup: 2.3.5.7.11


Comma list: 45/44, 56/55, 1029/1000
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 569.144
Badness (Sintel): 0.895


{{Optimal ET sequence|legend=1| 2, 17d, 19, 59bde, 78bde, 97bde }}
== Decic ==
{{Main| Decic }}


Badness: 0.045675
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]].  


== Submajor ==
It can be extended to the 11-, 13-, and 17-limit by adding [[385/384]], [[105/104]], and [[170/169]] to the comma list in this order.
[[Subgroup]]: 2.3.5


[[Comma list]]: 69198046875/68719476736
{{Mapping|legend=1| 1 4 -1 | 0 -8 11 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10125/8192 = 362.321
{{Optimal ET sequence|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
[[Badness]]: 0.130236
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 51200/50421
[[Comma list]]: 225/224, 16807/16384


{{Mapping|legend=1| 1 4 -1 1 | 0 -8 11 6 }}
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 362.255
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


{{Optimal ET sequence|legend=1| 10, 33, 43, 53 }}
[[Badness]] (Sintel): 2.26


[[Badness]]: 0.060533
=== 11-limit ===
Subgroup: 2.3.5.7.11


==== 2.3.5.7.13 subgroup ====
Comma list: 225/224, 385/384, 3087/3025
{{ See also | Greater tendoneutralic }}


This temperament naturally comes about from a structure in EDOs like [[43edo|43]] and [[53edo|53]] where two flattened ~[[13/8]] intervals reach the [[No-fives subgroup temperaments#Buzzard|buzzard]] generator of ~[[21/16]], two of which produce a semitritave (that can here be equated to [[26/15]], providing a mapping of 5 significantly less complex than the [[vulture]] mapping), and two of those finally reach [[3/1]].
{{Mapping|legend=0| 10 0 39 28 3 | 0 1 -1 0 2 }}


Subgroup: 2.3.5.7.13
Optimal tunings:  
* WE: ~15/14 = 120.1406{{c}}, ~3/2 = 697.6075{{c}} (~56/55 = 23.2360{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.0142{{c}} (~56/55 = 22.9858{{c}})


Comma list: 169/168, 225/224, 640/637
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Mapping: {{mapping| 1 4 -1 1 4 | 0 -8 11 6 -1 }}
Badness (Sintel): 2.11


Optimal tuning (CTE): ~2 = 1\1, ~16/13 = 362.242
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness (Dirichlet): 0.847
Comma list: 105/104, 144/143, 196/195, 2200/2197


=== 11-limit ===
{{Mapping|legend=0| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 6655/6561
Optimal tunings:  
* WE: ~15/14 = 120.1166{{c}}, ~3/2 = 697.6705{{c}} (~78/77 = 23.0289{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.1492{{c}} (~78/77 = 22.8508{{c}})


Mapping: {{mapping| 1 4 -1 1 11 | 0 -8 11 6 -25 }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/22 = 362.101
Badness (Sintel): 1.52


{{Optimal ET sequence|legend=1| 10, 43e, 53, 116, 169de, 285cde }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.050582
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


==== 13-limit ====
{{Mapping|legend=0| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 225/224, 275/273, 385/384
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}})


Mapping: {{mapping| 1 4 -1 1 11 4 | 0 -8 11 6 -25 -1 }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.105
Badness (Sintel): 1.28


{{Optimal ET sequence|legend=1| 10, 43e, 53, 116, 169de, 285cdef }}
=== 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''.


Badness: 0.027689
=== Interpental ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 51200/50421
Comma list: 225/224, 1617/1600, 2401/2376


Mapping: {{mapping| 1 4 -1 1 -5 | 0 -8 11 6 28 }}
{{Mapping|legend=0| 10 0 39 28 82 | 0 1 -1 0 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 362.418
Optimal tunings:
* WE: ~15/14 = 120.1874{{c}}, ~3/2 = 699.6085{{c}} (~99/98 = 21.5156{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3531{{c}} (~99/98 = 21.6469{{c}})


{{Optimal ET sequence|legend=1| 43, 53, 96, 149d }}
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


Badness: 0.051806
Badness (Sintel): 1.98


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


Comma list: 99/98, 169/168, 176/175, 640/637
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


Mapping: {{mapping| 1 4 -1 1 -5 4 | 0 -8 11 6 28 -1 }}
{{Mapping|legend=0| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.402
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 ET sequence|legend=1| 43, 53, 96, 149d }}
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


Badness: 0.029680
Badness (Sintel): 1.57


== Marvolo ==
==== 17-limit ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 225/224, 156250000/155649627
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374


{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
{{Mapping|legend=0| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
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 tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 83.348
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


{{Optimal ET sequence|legend=1| 29, 43, 72, 619bcd, 691bcd }}
Badness (Sintel): 1.33


[[Badness]]: 0.083338
=== 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: 225/224, 441/440, 4000/3993
Comma list: 225/224, 441/440, 5929/5832


Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}
{{Mapping|legend=0| 10 0 39 28 -13 | 0 1 -1 0 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.340
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}})


{{Optimal ET sequence|legend=1| 29, 43, 72 }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness: 0.028965
Badness (Sintel): 2.20


=== 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: 105/104, 196/195, 275/273, 5929/5832


Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
{{Mapping|legend=0| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.330
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}})


{{Optimal ET sequence|legend=1| 29, 43, 72, 115f }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness: 0.021470
Badness (Sintel): 1.73


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


Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
{{Mapping|legend=0| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.330
Optimal tunings:
 
* WE: ~15/14 = 120.1577{{c}}, ~3/2 = 701.3950{{c}} (~91/90 = 19.5514{{c}})
{{Optimal ET sequence|legend=1| 29g, 43, 72 }}
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.6932{{c}} (~91/90 = 19.3068{{c}})
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.330
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


{{Optimal ET sequence|legend=1| 29g, 43, 72 }}
Badness (Sintel): 1.41


== Enneaportent ==
== Enneaportent ==
Line 606: Line 624:


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


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 1\9, ~12005/6912 = 950.1680 (~1728/1715 = 16.8347)
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


{{Optimal ET sequence|legend=1| 9, 63, 72, 495bcd }}
[[Badness]] (Sintel): 2.37
 
[[Badness]]: 0.093679


=== 11-limit ===
=== 11-limit ===
Line 622: Line 641:
Comma list: 225/224, 385/384, 12005/11979
Comma list: 225/224, 385/384, 12005/11979


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


Optimal tuning (POTE): ~121/112 = 1\9, ~210/121 = 950.1873 (~99/98 = 16.8540)
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 ET sequence|legend=1| 9, 63, 72, 423cd, 495bcd }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.030426
Badness (Sintel): 1.01


=== 13-limit ===
=== 13-limit ===
Line 635: Line 656:
Comma list: 169/168, 225/224, 364/363, 1716/1715
Comma list: 169/168, 225/224, 364/363, 1716/1715


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


Optimal tuning (POTE): ~14/13 = 1\9, ~26/15 = 950.2867 (~105/104 = 16.9534)
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 ET sequence|legend=1| 9, 63, 72, 279cf }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.022322
Badness (Sintel): 0.922


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 651: Line 674:


{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


{{Multival|legend=1| 1 -20 -38 -34 -63 -32 }}
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.824
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


{{Optimal ET sequence|legend=1| 12, 113, 125, 238c, 363c }}
[[Badness]] (Sintel): 2.44
 
[[Badness]]: 0.096279


=== 11-limit ===
=== 11-limit ===
Line 665: Line 691:
Comma list: 225/224, 385/384, 236328125/234365481
Comma list: 225/224, 385/384, 236328125/234365481


Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
{{Mapping|legend=0| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.834
Optimal tunings:
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125, 238c }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Badness: 0.089588
Badness (Sintel): 2.96


==== 13-limit ====
==== 13-limit ====
Line 678: Line 706:
Comma list: 225/224, 325/324, 385/384, 831875/830466
Comma list: 225/224, 325/324, 385/384, 831875/830466


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.841
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=1| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.052235
Badness (Sintel): 2.16


==== 17-limit ====
==== 17-limit ====
Line 691: Line 721:
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.841
Optimal tunings:
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.038565
Badness (Sintel): 1.96


==== 19-limit ====
==== 19-limit ====
Line 704: Line 736:
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.842
Optimal tunings:
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.028165
Badness (Sintel): 1.71


==== 23-limit ====
==== 23-limit ====
Line 717: Line 751:
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.843
Optimal tunings:
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 238cfi }}
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


Badness: 0.021879
Badness (Sintel): 1.57


==== 29-limit ====
==== 29-limit ====
Line 730: Line 766:
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.842
Optimal tunings:
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cfi }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Badness: 0.018011
Badness (Sintel): 1.50


==== 31-limit ====
==== 31-limit ====
Line 743: Line 781:
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.838
Optimal tunings:
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


{{Optimal ET sequence|legend=1| 12e, 101cdek, 113, 125f, 238cfi }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Badness: 0.016007
Badness (Sintel): 1.53


=== Gracecord ===
=== Gracecord ===
Line 756: Line 796:
Comma list: 225/224, 441/440, 109375/107811
Comma list: 225/224, 441/440, 109375/107811


Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}
{{Mapping|legend=0| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.885
Optimal tunings:
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


{{Optimal ET sequence|legend=1| 12, 101cd, 113 }}
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Badness: 0.066964
Badness (Sintel): 2.21


==== 13-limit ====
==== 13-limit ====
Line 769: Line 811:
Comma list: 225/224, 364/363, 441/440, 6125/6084
Comma list: 225/224, 364/363, 441/440, 6125/6084


Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}
{{Mapping|legend=0| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.890
Optimal tunings:
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Badness: 0.044196
Badness (Sintel): 1.83


==== 17-limit ====
==== 17-limit ====
Line 782: Line 826:
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}
{{Mapping|legend=0| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.870
Optimal tunings:
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Badness: 0.036637
Badness (Sintel): 1.87


==== 19-limit ====
==== 19-limit ====
Line 795: Line 841:
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.866
Optimal tunings:
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Badness: 0.027559
Badness (Sintel): 1.68


== Alphorn ==
== Alphorn ==
Line 808: Line 856:
[[Comma list]]: 225/224, 5764801/5668704
[[Comma list]]: 225/224, 5764801/5668704


{{Mapping|legend=1| 1 9 0 13 | 0 -16 5 -22 }}
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 556.221
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


{{Optimal ET sequence|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
[[Badness]] (Sintel): 3.27
 
[[Badness]]: 0.129258


=== 11-limit ===
=== 11-limit ===
Line 823: Line 874:
Comma list: 225/224, 385/384, 12250/11979
Comma list: 225/224, 385/384, 12250/11979


Mapping: {{mapping| 1 9 0 13 3 | 0 -16 5 -22 1 }}
{{Mapping|legend=0| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 556.144
Optimal tunings:
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


{{Optimal ET sequence|legend=1| 28d, 41, 315cde }}
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Badness: 0.073459
Badness (Sintel): 2.43


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 838: Line 891:
[[Comma list]]: 225/224, 4194304/4117715  
[[Comma list]]: 225/224, 4194304/4117715  


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


{{Multival|legend=1| 15 -14 2 -57 -39 44 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.7642{{c}}, ~15/8 = 1086.5513{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/15 = 113.235
: [[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 }}


{{Optimal ET sequence|legend=1| 21, 32, 53 }}
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


[[Badness]]: 0.140970
[[Badness]] (Sintel): 3.57


=== 11-limit ===
=== 11-limit ===
Line 853: Line 909:
Comma list: 99/98, 176/175, 1310720/1294139
Comma list: 99/98, 176/175, 1310720/1294139


Mapping: {{mapping| 1 3 1 3 1 | 0 -15 14 -2 26 }}
{{Mapping|legend=0| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 113.323
Optimal tunings:
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


{{Optimal ET sequence|legend=1| 21, 32e, 53, 127, 180de }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.085390
Badness (Sintel): 2.82


==== 13-limit ====
==== 13-limit ====
Line 866: Line 924:
Comma list: 99/98, 176/175, 640/637, 847/845
Comma list: 99/98, 176/175, 640/637, 847/845


Mapping: {{mapping| 1 3 1 3 1 2 | 0 -15 14 -2 26 18 }}
{{Mapping|legend=0| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 113.323
Optimal tunings:
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


{{Optimal ET sequence|legend=1| 21, 32e, 53, 127, 180de }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.045569
Badness (Sintel): 1.88


=== Musneb ===
=== Musneb ===
Line 879: Line 939:
Comma list: 225/224, 385/384, 66550/64827
Comma list: 225/224, 385/384, 66550/64827


Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
{{Mapping|legend=0| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 113.142
Optimal tunings:
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


{{Optimal ET sequence|legend=1| 32, 53, 191de, 244cddee, 297cddee }}
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Badness: 0.087333
Badness (Sintel): 2.89


== Untriton ==
== Untriton ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Untriton]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''
 
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
[[Subgroup]]: 2.3.5.7
Line 894: Line 958:
[[Comma list]]: 225/224, 125000000/121060821
[[Comma list]]: 225/224, 125000000/121060821


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


{{Multival|legend=1| 9 -19 -20 -51 -57 7 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
: [[error map]]: {{val| -0.172 +0.002 -2.533 +3.511 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}}
: error map: {{val| 0.000 +0.298 -2.181 +3.946 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 588.641
{{Optimal ET sequence|legend=1| 51, 53 }}
 
{{Optimal ET sequence|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}


[[Badness]]: 0.143976
[[Badness]] (Sintel): 3.64


=== 11-limit ===
=== 11-limit ===
Line 909: Line 976:
Comma list: 121/120, 225/224, 22000/21609
Comma list: 121/120, 225/224, 22000/21609


Mapping: {{mapping| 1 6 -7 -7 1 | 0 -9 19 20 5 }}
{{Mapping|legend=0| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 588.626
Optimal tunings:
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


{{Optimal ET sequence|legend=1| 51, 53 }}
{{Optimal ET sequence|legend=0| 51, 53 }}


Badness: 0.074295
Badness (Sintel): 2.46


=== 13-limit ===
=== 13-limit ===
Line 922: Line 991:
Comma list: 121/120, 225/224, 275/273, 1040/1029
Comma list: 121/120, 225/224, 275/273, 1040/1029


Mapping: {{mapping| 1 6 -7 -7 1 -12 | 0 -9 19 20 5 32 }}
{{Mapping|legend=0| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}
 
Optimal tunings:
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}
 
{{Optimal ET sequence|legend=0| 51f, 53 }}
 
Badness (Sintel): 1.96
 
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.
 
[[Subgroup]]: 2.3.5.7


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 588.654
[[Comma list]]: 225/224, 823543/800000
 
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21
 
[[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 ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}
 
[[Badness]] (Sintel): 3.67
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 245/242, 1617/1600
 
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}
 
Optimal tunings:
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}
 
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
 
Badness (Sintel): 2.00
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 196/195, 245/242, 1001/1000
 
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}
 
Optimal tunings:
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}
 
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
 
Badness (Sintel): 1.43
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242
 
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}
 
Optimal tunings:
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


{{Optimal ET sequence|legend=1| 51f, 53 }}
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Badness: 0.047441
Badness (Sintel): 1.26


== Quintannic ==
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 936: Line 1,074:


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


{{Multival|legend=1| 5 -23 -36 -48 -71 -19 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.9803{{c}}, ~10000/9261 = 139.9522{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10000/9261 = 139.838
: [[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 }}


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


[[Badness]]: 0.150565
[[Badness]] (Sintel): 3.81


=== 11-limit ===
=== 11-limit ===
Line 950: Line 1,091:
Comma list: 225/224, 441/440, 43923/43750
Comma list: 225/224, 441/440, 43923/43750


Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}
{{Mapping|legend=0| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


Optimal tuning (POTE): ~2 = 1\1, ~320/297 = 139.827
Optimal tunings:
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


{{Optimal ET sequence|legend=1| 43, 60e, 103 }}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.052590
Badness (Sintel): 1.74


=== 13-limit ===
=== 13-limit ===
Line 963: Line 1,106:
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


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


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.812
Optimal tunings:
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


{{Optimal ET sequence|legend=1| 43, 60e, 103 }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 0.032730
Badness (Sintel): 1.35


=== 17-limit ===
=== 17-limit ===
Line 976: Line 1,121:
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
{{Mapping|legend=0| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
 
Optimal tunings:
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.815
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


{{Optimal ET sequence|legend=1| 43, 60e, 103 }}
Badness (Sintel): 1.17


Badness: 0.023038
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''
 
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, 122880/117649
 
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}
 
[[Badness]] (Sintel): 4.37
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 225/224, 352/343
 
{{Mapping|legend=0| 11 0 43 31 38 | 0 1 -1 0 0 }}
 
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 ET sequence|legend=0| 22, 55d }}
 
Badness (Sintel): 2.46


== Gwazy ==
== Gwazy ==
{{See also| Very high accuracy temperaments #Kwazy }}
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
 
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 992: Line 1,179:


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


{{Multival|legend=1| 16 -10 12 -53 -26 56 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2401/1728 = 599.7132{{c}}, ~35/32 = 162.5806{{c}}
[[Optimal tuning]] ([[POTE]]): ~2401/1728 = 1\2, ~35/32 = 162.658
: [[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 }}


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


[[Badness]]: 0.178826
[[Badness]] (Sintel): 4.53


=== 11-limit ===
=== 11-limit ===
Line 1,006: Line 1,196:
Comma list: 99/98, 176/175, 65536/65219
Comma list: 99/98, 176/175, 65536/65219


Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}
{{Mapping|legend=0| 2 1 6 4 8 | 0 8 -5 6 -4 }}


Optimal tuning (POTE): ~363/256 = 1\2, ~11/10 = 162.592
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 ET sequence|legend=1| 22, 74, 96, 118d }}
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


Badness: 0.068410
Badness (Sintel): 2.26


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,021: Line 1,215:
[[Comma list]]: 225/224, 14495514624/13841287201
[[Comma list]]: 225/224, 14495514624/13841287201


{{Mapping|legend=1| 3 7 5 9 | 0 -8 7 -2 }}
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


{{Multival|legend=1| 24 -21 6 -89 -58 73 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
[[Optimal tuning]] ([[POTE]]): ~3072/2401 = 1\3, ~15/14 = 112.283
: [[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 }}


{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


[[Badness]]: 0.431636
[[Badness]] (Sintel): 10.9


=== 11-limit ===
=== 11-limit ===
Line 1,036: Line 1,233:
Comma list: 225/224, 3840/3773, 12005/11979
Comma list: 225/224, 3840/3773, 12005/11979


Mapping: {{mapping| 3 7 5 9 9 | 0 -8 7 -2 5 }}
{{Mapping|legend=0| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~15/14 = 112.171
Optimal tunings:
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}
 
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


{{Optimal ET sequence|legend=1| 21, 54, 75e }}
Badness (Sintel): 5.74


Badness: 0.173485
== References ==


[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Marvel| ]] <!-- key article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"