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Marvel temperaments: Difference between revisions

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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)
* ''[[Quintapole]]'' (+7812500/7411887) → [[Quintaleap family #Quintapole|Quintaleap family]]
* ''[[Qintosec]]'' → [[Quintosec family #Qintosec|Quintosec family]] (+2560000/2470629, generated by the classical minor second with a 1/5-octave period)
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* [[Miracle]] → [[Gamelismic clan #Miracle|Gamelismic clan]] (+1029/1024, fifth sliced in six)
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* [[Catakleismic]] → [[Kleismic family #Catakleismic|Kleismic family]] (+4375/4374, twelfth sliced in six)
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Marvo]]'' → [[Gravity family #Marvo|Gravity family]] (+78125000/78121827, two octaves and a fifth sliced in six)
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* [[Orwell]] → [[Semicomma family #Orwell|Semicomma family]] (+1728/1715, twelfth sliced in seven)
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* ''[[Snipes]]''  → [[Wesley family #Snipes|Wesley family]] (+6125/5832, two octaves and a fourth sliced in seven)
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Submajor]]'' → [[Buzzardsmic clan #Submajor|Buzzardsmic clan]] (+65536/64827, two octaves and a fourth sliced in eight)
* ''[[Snipes]]'' (+6125/5832) → [[Wesley family #Snipes|Wesley family]]
* ''[[Escapist]]'' → [[Escapade family #Escapist|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)
* ''[[Decic]]'' (+16807/16384) → [[Cloudy clan #Decic|Cloudy clan]]
* ''[[Betic]]'' → [[Sycamore family #Betic|Sycamore family]] (+1071875/1062882, fifth sliced in eleven)
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Hendeca]]'' → [[11th-octave temperaments #Hendeca|11th-octave temperaments]] (+122880/117649, generated by the fifth with a 1/11-octave period)
* ''[[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)
* ''[[Hendeca]]'' (+122880/117649) → [[11th-octave temperaments #Hendeca|11th-octave temperaments]]
* ''[[Raccoon]]'' → [[Vavoom family #Raccoon|Vavoom family]] (+41943040/40353607, twelfth sliced in seventeen)
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
* ''[[Maquila]]'' → [[Maquila family #Septimal maquila|Maquila family]] (+30233088/28824005, seven octaves and a fifth sliced in seventeen)
* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
* ''[[Gammy]]'' → [[Gammic family #Gammy|Gammic family]] (+94143178827/91913281250, fifth sliced in twenty)
* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]


Considered below are wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, alphorn, tertiosec, gwazy, and gracecordial.  
Considered below are wizard, tritonic, septimin, merman, slender, triton, marvolo, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, 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.
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: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''
: ''For the 5-limit version, 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 & 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.  
Wizard has a [[semi-octave]] period and is generated by an interval that can be treated as [[~]][[17/15]]. The semi-octave complement of this interval is ~[[5/4]]. Wizard can be described as {{nowrap| 22 & 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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~1225/864 = 600.000{{c}}, ~245/216 = 216.919{{c}}
* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
: [[error map]]: {{val| 0.000 -0.443 -3.232 +0.361 }}
: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}
* [[POTE]]: ~1225/864 = 600.000{{c}}, ~245/216 = 216.744{{c}}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}
: error map: {{val| 0.000 -1.492 -3.057 -1.388 }}
: error map: {{val| 0.000 -1.169 -3.111 -0.849 }}


{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}


[[Badness]] (Smith): 0.040846
[[Badness]] (Sintel): 1.03


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.900{{c}}
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.768{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


Badness (Smith): 0.018539
Badness (Sintel): 0.613


==== Lizard ====
==== Lizard ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 600.000{{c}}, ~25/22 = 216.703{{c}}
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* POTE: ~55/39 = 600.000{{c}}, ~25/22 = 216.711{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


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


Badness (Smith): 0.021781
Badness (Sintel): 0.900


===== 17-limit =====
===== 17-limit =====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.748{{c}}
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.619{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


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


Badness (Smith): 0.014536
Badness (Sintel): 0.741


===== 19-limit =====
===== 19-limit =====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.677{{c}}
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.523{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


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


Badness (Smith): 0.015702
Badness (Sintel): 0.955


==== Gizzard ====
==== Gizzard ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.928{{c}}
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.830{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}


{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}


Badness (Smith): 0.020252
Badness (Sintel): 0.837


===== 17-limit =====
===== 17-limit =====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.943{{c}}
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.825{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}


{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}


Badness (Smith): 0.013624
Badness (Sintel): 0.694


===== 19-limit =====
===== 19-limit =====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.918{{c}}
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.862{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


{{Optimal ET sequence|legend=0| 72, 94, 166g }}
{{Optimal ET sequence|legend=0| 72, 94, 166g }}


Badness (Smith): 0.014810
Badness (Sintel): 0.901


=== Mage ===
=== Mage ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~77/54 = 600.000{{c}}, ~55/48 = 217.247{{c}}
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* POTE: ~77/54 = 600.000{{c}}, ~55/48 = 216.876{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


Badness (Smith): 0.057799
Badness (Sintel): 1.91


== Tritonic ==
== Tritonic ==
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Tritonic]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''
 
Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 225/224, 50421/50000
[[Comma list]]: 225/224, 50421/50000


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 580.286{{c}}
[[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 }}


{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


[[Badness]] (Smith): 0.047578
[[Badness]] (Sintel): 1.20


=== 11-limit ===
=== 11-limit ===
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Comma list: 121/120, 225/224, 441/440
Comma list: 121/120, 225/224, 441/440


Mapping: {{mapping| 1 4 -3 -3 2 | 0 -5 11 12 3 }}
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.267{{c}}
Optimal tunings:
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


{{Optimal ET sequence|legend=0| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


Badness (Smith): 0.023659
Badness (Sintel): 0.782


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


Mapping: {{mapping| 1 4 -3 -3 2 -5 | 0 -5 11 12 3 18 }}
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.108{{c}}
Optimal tunings:
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


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


Badness (Smith): 0.022993
Badness (Sintel): 0.950


==== 17-limit ====
==== 17-limit ====
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Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 | 0 -5 11 12 3 18 25 }}
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.055{{c}}
Optimal tunings:
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


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


Badness (Smith): 0.019093
Badness (Sintel): 0.973


==== 19-limit ====
==== 19-limit ====
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Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


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


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.026{{c}}
Optimal tunings:
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


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


Badness (Smith): 0.017010
Badness (Sintel): 1.03


==== 23-limit ====
==== 23-limit ====
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Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


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


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.009{{c}}
Optimal tunings:
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


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


Badness (Smith): 0.014471
Badness (Sintel): 1.04


=== Tritoni ===
=== Tritoni ===
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Comma list: 225/224, 385/384, 27783/27500
Comma list: 225/224, 385/384, 27783/27500


Mapping: {{mapping| 1 4 -3 -3 17 | 0 -5 11 12 -28 }}
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.389{{c}}
Optimal tunings:
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Badness (Smith): 0.045456
Badness (Sintel): 1.50


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


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/6 = 263.632{{c}}
[[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 ET sequence|legend=1| 41, 91, 132d }}
{{Optimal ET sequence|legend=1| 41, 91, 132d }}


[[Badness]] (Smith): 0.054502
[[Badness]] (Sintel): 1.38


=== 11-limit ===
=== 11-limit ===
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Comma list: 225/224, 245/242, 385/384
Comma list: 225/224, 245/242, 385/384


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


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/6 = 263.634{{c}}
Optimal tunings:
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


Badness (Smith): 0.031309
Badness (Sintel): 1.04


=== 13-limit ===
=== 13-limit ===
Line 310: Line 339:
Comma list: 105/104, 144/143, 196/195, 245/242
Comma list: 105/104, 144/143, 196/195, 245/242


Mapping: {{mapping| 1 4 1 5 5 7 | 0 -11 6 -10 -7 -15 }}
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/6 = 263.700{{c}}
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


{{Optimal ET sequence|legend=0| 41, 91 }}
{{Optimal ET sequence|legend=0| 41, 91 }}


Badness (Smith): 0.023117
Badness (Sintel): 0.955


== Merman ==
== Merman ==
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Merman]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''
 
Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot.
 
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 325: Line 360:
[[Comma list]]: 225/224, 2500000/2470629
[[Comma list]]: 225/224, 2500000/2470629


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 585.585{{c}}
[[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| 41, 84, 125 }}
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


[[Badness]] (Smith): 0.055078
[[Badness]] (Sintel): 1.39


=== 11-limit ===
=== 11-limit ===
Line 338: Line 378:
Comma list: 225/224, 441/440, 1344/1331
Comma list: 225/224, 441/440, 1344/1331


Mapping: {{mapping| 1 5 -5 -5 2 | 0 -7 15 16 3 }}
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.606{{c}}
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=0| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


Badness (Smith): 0.036383
Badness (Sintel): 1.20


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


Comma list: 144/143, 225/224, 364/363, 441/440
Comma list: 144/143, 225/224, 364/363, 441/440


Mapping: {{mapping| 1 5 -5 -5 2 12 | 0 -7 15 16 3 -17 }}
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.657{{c}}
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=0| 41, 84, 125e, 209ef, 293ef }}
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Badness (Smith): 0.027544
Badness (Sintel): 1.14
 
=== Mermaid ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 532400/531441
 
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
 
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=0| 41, 84e, 125, 166 }}
 
Badness (Sintel): 1.46
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 10648/10647
 
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
 
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=0| 41, 84ef, 125f, 166 }}
 
Badness (Sintel): 1.47


== Slender ==
== Slender ==
Slender ({{nowrap| 31 & 32 }}) tempers out the [[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.
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.
 
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 367: Line 443:


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/48 = 38.413{{c}}
[[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 ET sequence|legend=1| 31, 94, 125 }}
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


[[Badness]] (Smith): 0.056934
[[Badness]] (Sintel): 1.44


=== 11-limit ===
=== 11-limit ===
Line 381: Line 462:
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 38.387{{c}}
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=0| 31, 63, 94, 125 }}
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Badness (Smith): 0.025342
Badness (Sintel): 0.838


=== 13-limit ===
=== 13-limit ===
Line 394: Line 477:
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 38.314{{c}}
Optimal tunings:
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


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


Badness (Smith): 0.025913
Badness (Sintel): 1.07


== Triton ==
== Triton ==
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Stump]].''
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''
 
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
[[Subgroup]]: 2.3.5.7
Line 408: Line 495:


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 568.865{{c}}
[[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 ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


[[Badness]] (Smith): 0.059245
[[Badness]] (Sintel): 1.50


=== 11-limit ===
=== 11-limit ===
Line 422: Line 514:
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 569.144{{c}}
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


{{Optimal ET sequence|legend=0| 2, 17d, 19, 59bde, 78bde, 97bde }}
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


Badness (Smith): 0.045675
Badness (Sintel): 1.51


== Marvolo ==
== Marvolo ==
Line 434: Line 528:


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~21/20 = 83.348{{c}}
[[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 ET sequence|legend=1| 29, 43, 72, 619bcd, 691bcd }}
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


[[Badness]] (Smith): 0.083338
[[Badness]] (Sintel): 2.11


=== 11-limit ===
=== 11-limit ===
Line 448: Line 547:
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.340{{c}}
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=0| 29, 43, 72 }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness (Smith): 0.028965
Badness (Sintel): 0.958


=== 13-limit ===
=== 13-limit ===
Line 461: Line 562:
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.330{{c}}
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=0| 29, 43, 72, 115f }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness (Smith): 0.021470
Badness (Sintel): 0.887


=== 17-limit ===
=== 17-limit ===
Line 474: Line 577:
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.330{{c}}
Optimal tunings:
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


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


Badness (Smith): 0.014924
Badness (Sintel): 0.760


=== 19-limit ===
=== 19-limit ===
Line 487: Line 592:
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.330{{c}}
Optimal tunings:
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


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


Badness (Smith): 0.014722
Badness (Sintel): 0.895


== Enneaportent ==
== Enneaportent ==
Line 499: Line 606:


{{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


[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.1680{{c}} (~1728/1715 = 16.8347{{c}})
[[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 ET sequence|legend=1| 9, 63, 72, 495bcd }}
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


[[Badness]] (Smith): 0.093679
[[Badness]] (Sintel): 2.37


=== 11-limit ===
=== 11-limit ===
Line 514: Line 625:
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


Optimal tuning (POTE): ~121/112 = 133.3333{{c}}, ~210/121 = 950.1873{{c}} (~99/98 = 16.8540{{c}})
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=0| 9, 63, 72, 423cd, 495bcd }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness (Smith): 0.030426
Badness (Sintel): 1.01


=== 13-limit ===
=== 13-limit ===
Line 527: Line 640:
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


Optimal tuning (POTE): ~14/13 = 133.3333{{c}}, ~26/15 = 950.2867{{c}} (~105/104 = 16.9534{{c}})
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=0| 9, 63, 72, 279cf }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness (Smith): 0.022322
Badness (Sintel): 0.922


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 541: Line 656:


{{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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 700.824{{c}}
[[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 = 1200.3333{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


{{Optimal ET sequence|legend=1| 12, 113, 125, 238c, 363c }}
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


[[Badness]] (Smith): 0.096279
[[Badness]] (Sintel): 2.44


=== 11-limit ===
=== 11-limit ===
Line 555: Line 675:
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.834{{c}}
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=0| 12e, 101cde, 113, 125, 238c }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Badness (Smith): 0.089588
Badness (Sintel): 2.96


==== 13-limit ====
==== 13-limit ====
Line 568: Line 690:
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.841{{c}}
Optimal tunings:
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


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


Badness (Smith): 0.052235
Badness (Sintel): 2.16


==== 17-limit ====
==== 17-limit ====
Line 581: Line 705:
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.841{{c}}
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=0| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness (Smith): 0.038565
Badness (Sintel): 1.96


==== 19-limit ====
==== 19-limit ====
Line 594: Line 720:
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.842{{c}}
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=0| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness (Smith): 0.028165
Badness (Sintel): 1.71


==== 23-limit ====
==== 23-limit ====
Line 607: Line 735:
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.843{{c}}
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=0| 12e, 101cde, 113, 238cfi }}
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


Badness (Smith): 0.021879
Badness (Sintel): 1.57


==== 29-limit ====
==== 29-limit ====
Line 620: Line 750:
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.842{{c}}
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=0| 12e, 101cde, 113, 125f, 238cfi }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Badness (Smith): 0.018011
Badness (Sintel): 1.50


==== 31-limit ====
==== 31-limit ====
Line 633: Line 765:
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: {{mapping| 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 = 1200.000{{c}}, ~3/2 = 700.838{{c}}
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=0| 12e, 101cdek, 113, 125f, 238cfi }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Badness (Smith): 0.016007
Badness (Sintel): 1.53


=== Gracecord ===
=== Gracecord ===
Line 646: Line 780:
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.885{{c}}
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=0| 12, 101cd, 113 }}
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Badness (Smith): 0.066964
Badness (Sintel): 2.21


==== 13-limit ====
==== 13-limit ====
Line 659: Line 795:
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.890{{c}}
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=0| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Badness (Smith): 0.044196
Badness (Sintel): 1.83


==== 17-limit ====
==== 17-limit ====
Line 672: Line 810:
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.870{{c}}
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=0| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Badness (Smith): 0.036637
Badness (Sintel): 1.87


==== 19-limit ====
==== 19-limit ====
Line 685: Line 825:
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.866{{c}}
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=0| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Badness (Smith): 0.027559
Badness (Sintel): 1.68


== Alphorn ==
== Alphorn ==
Line 696: Line 838:
[[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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~48/35 = 556.221{{c}}
[[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 ET sequence|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


[[Badness]] (Smith): 0.129258
[[Badness]] (Sintel): 3.27


=== 11-limit ===
=== 11-limit ===
Line 709: Line 856:
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: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 556.144{{c}}
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=0| 28d, 41, 315cde }}
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Badness (Smith): 0.073459
Badness (Sintel): 2.43


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 724: Line 873:
[[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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~16/15 = 113.235{{c}}
[[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 737: Line 891:
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: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 113.323{{c}}
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=0| 21, 32e, 53, 127, 180de }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness (Smith): 0.085390
Badness (Sintel): 2.82


==== 13-limit ====
==== 13-limit ====
Line 750: Line 906:
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: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 113.323{{c}}
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=0| 21, 32e, 53, 127, 180de }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness (Smith): 0.045569
Badness (Sintel): 1.88


=== Musneb ===
=== Musneb ===
Line 765: Line 923:
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 113.142{{c}}
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=0| 32, 53, 191de, 244cddee, 297cddee }}
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Badness (Smith): 0.087333
Badness (Sintel): 2.89


== Untriton ==
== Untriton ==
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit 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 778: Line 940:
[[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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 588.641{{c}}
[[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 ET sequence|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}
{{Optimal ET sequence|legend=1| 51, 53 }}


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


=== 11-limit ===
=== 11-limit ===
Line 791: Line 958:
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: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 588.626{{c}}
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=0| 51, 53 }}
{{Optimal ET sequence|legend=0| 51, 53 }}


Badness (Smith): 0.074295
Badness (Sintel): 2.46


=== 13-limit ===
=== 13-limit ===
Line 804: Line 973:
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: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 588.654{{c}}
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 }}
{{Optimal ET sequence|legend=0| 51f, 53 }}


Badness (Smith): 0.047441
Badness (Sintel): 1.96


== Naiadical ==
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


{{Mapping|legend=1| 1 5 -3 -1 | 0 -9 14 10 }}
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1202.120{{c}}, ~21/16 = 456.652{{c}}
* [[WE]]: ~2 = 1202.1198{{c}}, ~32/21 = 745.4675{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~21/16 = 455.868{{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 }}
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


[[Badness]] (Sintel): 3.667
[[Badness]] (Sintel): 3.67


=== 11-limit ===
=== 11-limit ===
Line 832: Line 1,008:
Comma list: 225/224, 245/242, 1617/1600
Comma list: 225/224, 245/242, 1617/1600


Mapping: {{Mapping| 1 5 -3 -1 -3 | 0 -9 14 10 17 }}
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.901, ~21/16{{c}} = 456.514{{c}}
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.000, ~21/16{{c}} = 455.822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


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


Badness (Sintel): 1.998
Badness (Sintel): 2.00


=== 13-limit ===
=== 13-limit ===
Line 847: Line 1,023:
Comma list: 105/104, 196/195, 245/242, 1001/1000
Comma list: 105/104, 196/195, 245/242, 1001/1000


Mapping: {{Mapping| 1 5 -3 -1 -3 -2 | 0 -9 14 10 17 15 }}
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.786{{c}}, ~13/10 = 456.452{{c}}
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.000{{c}}, ~13/10 = 455.807{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


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


Badness (Sintel): 1.430
Badness (Sintel): 1.43


=== 17-limit ===
=== 17-limit ===
Line 862: Line 1,038:
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


Mapping: {{Mapping| 1 5 -3 -1 -3 -2 -2 | 0 -9 14 10 17 15 16 }}
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.921{{c}}, ~13/10 = 456.523{{c}}
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.000{{c}}, ~13/10 = 455.833{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


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


Badness (Sintel): 1.260
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 878: Line 1,056:


{{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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~10000/9261 = 139.838{{c}}
[[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 892: Line 1,075:
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~320/297 = 139.827{{c}}
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=0| 43, 60e, 103 }}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness (Smith): 0.052590
Badness (Sintel): 1.74


=== 13-limit ===
=== 13-limit ===
Line 905: Line 1,090:
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 139.812{{c}}
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=0| 43, 60e, 103 }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness (Smith): 0.032730
Badness (Sintel): 1.35


=== 17-limit ===
=== 17-limit ===
Line 918: Line 1,105:
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 139.815{{c}}
Optimal tunings:
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


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


Badness (Smith): 0.023038
Badness (Sintel): 1.17


== 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 932: Line 1,123:


{{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


[[Optimal tuning]] ([[POTE]]): ~2401/1728 = 600.000{{c}}, ~35/32 = 162.658{{c}}
[[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 946: Line 1,142:
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


Optimal tuning (POTE): ~363/256 = 600.000{{c}}, ~11/10 = 162.592{{c}}
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=0| 22, 74, 96, 118d }}
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


Badness (Smith): 0.068410
Badness (Sintel): 2.26


== Tertiosec ==
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit 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 959: Line 1,159:
[[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


[[Optimal tuning]] ([[POTE]]): ~3072/2401 = 400.000{{c}}, ~15/14 = 112.283{{c}}
[[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 972: Line 1,177:
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: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~15/14 = 112.171{{c}}
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=0| 21, 54, 75e }}


Badness (Smith): 0.173485
Badness (Sintel): 5.74
 
== References ==


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