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Marvel temperaments: Difference between revisions

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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)
* ''[[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]]
* ''[[Compton]]'' → [[Compton family #Compton|Compton family]] (+250047/250000, generated by the classical major third with a 1/12-octave period)
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Raccoon]]'' → [[Vavoom family #Raccoon|Vavoom family]] (+41943040/40353607, twelfth sliced in seventeen)
* ''[[Hendeca]]'' (+122880/117649) → [[11th-octave temperaments #Hendeca|11th-octave temperaments]]
* ''[[Maquila]]'' → [[Maquila family #Septimal maquila|Maquila family]] (+30233088/28824005, seven octaves and a fifth sliced in seventeen)
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
* ''[[Gammy]]'' → [[Gammic family #Gammy|Gammic family]] (+94143178827/91913281250, fifth sliced in twenty)
* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]


Considered below are wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, submajor, 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.


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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== Wizard ==
== Wizard ==
{{Main| Wizard }}
{{Main| Wizard }}
: ''For the 5-limit version of this temperament, 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 &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.  
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
Line 49: Line 52:


{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216
: 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]]: ~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, ~245/216 = 216.744
* [[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, ~25/22 = 216.900
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* POTE: ~99/70 = 600.000, ~25/22 = 216.768
* 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, ~25/22 = 216.703
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* POTE: ~55/39 = 600.000, ~25/22 = 216.711
* 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 =====
Line 102: Line 102:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~17/15 = 216.748
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* POTE: ~17/12 = 600.000, ~17/15 = 216.619
* 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 =====
Line 117: Line 117:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~17/15 = 216.677
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* POTE: ~17/12 = 600.000, ~17/15 = 216.523
* 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 ====
Line 132: Line 132:


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000, ~25/22 = 216.928
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* POTE: ~99/70 = 600.000, ~25/22 = 216.830
* 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, ~17/15 = 216.943
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* POTE: ~17/12 = 600.000, ~17/15 = 216.825
* 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, ~17/15 = 216.918
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* POTE: ~17/12 = 600.000, ~17/15 = 216.862
* 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, ~55/48 = 217.247
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* POTE: ~77/54 = 600.000, ~55/48 = 216.876
* 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 [[High badness 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
Line 191: Line 193:
[[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


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 580.286
: [[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]]: 0.047578
[[Badness]] (Sintel): 1.20


=== 11-limit ===
=== 11-limit ===
Line 206: Line 211:
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 }}


{{Multival|legend=1| 5 -11 -12 -3 -29 -33 -22 3 31 33 }}
Optimal tunings:
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.267
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


{{Optimal ET sequence|legend=1| 29, 31, 60e }}
Badness (Sintel): 0.782
 
Badness: 0.023659


==== 13-limit ====
==== 13-limit ====
Line 221: Line 226:
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 = 1\1, ~7/5 = 580.108
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=1| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


Badness: 0.022993
Badness (Sintel): 0.950


==== 17-limit ====
==== 17-limit ====
Line 234: Line 241:
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 = 1\1, ~7/5 = 580.055
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=1| 29g, 31, 60e }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness: ?
Badness (Sintel): 0.973


==== 19-limit ====
==== 19-limit ====
Line 247: Line 256:
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 = 1\1, ~7/5 = 580.026
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=1| 29g, 31, 60e }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness: ?
Badness (Sintel): 1.03


==== 23-limit ====
==== 23-limit ====
Line 260: Line 271:
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 = 1\1, ~7/5 = 580.009
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=1| 29g, 31, 60e }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness: ?
Badness (Sintel): 1.04


=== Tritoni ===
=== Tritoni ===
Line 273: Line 286:
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 }}


{{Multival|legend=1| 5 -11 -12 28 -29 -33 27 3 103 120 }}
Optimal tunings:
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.389
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


{{Optimal ET sequence|legend=1| 31, 91, 122, 153d }}
Badness (Sintel): 1.50


Badness: 0.045456
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 290: Line 305:
[[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


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/6 = 263.632
: [[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]]: 0.054502
[[Badness]] (Sintel): 1.38


=== 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, 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 = 1\1, ~7/6 = 263.634
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=1| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


Badness: 0.031309
Badness (Sintel): 1.04


=== 13-limit ===
=== 13-limit ===
Line 318: Line 338:
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 = 1\1, ~7/6 = 263.700
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=1| 41, 91 }}
{{Optimal ET sequence|legend=0| 41, 91 }}


Badness: 0.023117
Badness (Sintel): 0.955


== Merman ==
== Merman ==
: ''For the 5-limit version of this temperament, see [[High badness 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 333: Line 359:
[[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


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 585.585
: [[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]]: 0.055078
[[Badness]] (Sintel): 1.39


=== 11-limit ===
=== 11-limit ===
Line 348: Line 377:
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 = 1\1, ~7/5 = 585.606
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| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


Badness: 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 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 tuning (POTE): ~2 = 1\1, ~7/5 = 585.657
Badness (Sintel): 1.14


{{Optimal ET sequence|legend=1| 41, 84, 125e, 209ef, 293ef }}
=== Mermaid ===
Subgroup: 2.3.5.7.11


Badness: 0.027544
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 &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.
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 377: Line 442:


{{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


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
[[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, ~49/48 = 38.413
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


{{Optimal ET sequence|legend=1| 31, 94, 125 }}
[[Badness]] (Sintel): 1.44
 
[[Badness]]: 0.056934


=== 11-limit ===
=== 11-limit ===
Line 393: Line 461:
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 = 1\1, ~49/48 = 38.387
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| 31, 63, 94, 125 }}
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Badness: 0.025342
Badness (Sintel): 0.838


=== 13-limit ===
=== 13-limit ===
Line 406: Line 476:
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 = 1\1, ~49/48 = 38.314
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=1| 31, 63, 94 }}
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Badness: 0.025913
Badness (Sintel): 1.07


== Triton ==
== Triton ==
: ''For the 5-limit version of this temperament, see [[High badness 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 420: Line 494:


{{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


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 568.865
: [[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]]: 0.059245
[[Badness]] (Sintel): 1.50


=== 11-limit ===
=== 11-limit ===
Line 436: Line 513:
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 = 1\1, ~7/5 = 569.144
Optimal tunings:
 
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
{{Optimal ET sequence|legend=1| 2, 17d, 19, 59bde, 78bde, 97bde }}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


Badness: 0.045675
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


== Submajor ==
Badness (Sintel): 1.51
[[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
 
[[Comma list]]: 225/224, 51200/50421
 
{{Mapping|legend=1| 1 4 -1 1 | 0 -8 11 6 }}
 
{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 362.255
 
{{Optimal ET sequence|legend=1| 10, 33, 43, 53 }}
 
[[Badness]]: 0.060533
 
==== 2.3.5.7.13 subgroup ====
{{ 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]].
 
Subgroup: 2.3.5.7.13
 
Comma list: 169/168, 225/224, 640/637
 
Mapping: {{mapping| 1 4 -1 1 4 | 0 -8 11 6 -1 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~16/13 = 362.242
 
Badness (Dirichlet): 0.847
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 6655/6561
 
Mapping: {{mapping| 1 4 -1 1 11 | 0 -8 11 6 -25 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~27/22 = 362.101
 
{{Optimal ET sequence|legend=1| 10, 43e, 53, 116, 169de, 285cde }}
 
Badness: 0.050582
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 275/273, 385/384
 
Mapping: {{mapping| 1 4 -1 1 11 4 | 0 -8 11 6 -25 -1 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.105
 
{{Optimal ET sequence|legend=1| 10, 43e, 53, 116, 169de, 285cdef }}
 
Badness: 0.027689
 
=== Interpental ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 51200/50421
 
Mapping: {{mapping| 1 4 -1 1 -5 | 0 -8 11 6 28 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 362.418
 
{{Optimal ET sequence|legend=1| 43, 53, 96, 149d }}
 
Badness: 0.051806
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 99/98, 169/168, 176/175, 640/637
 
Mapping: {{mapping| 1 4 -1 1 -5 4 | 0 -8 11 6 28 -1 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.402
 
{{Optimal ET sequence|legend=1| 43, 53, 96, 149d }}
 
Badness: 0.029680


== Marvolo ==
== Marvolo ==
Line 543: Line 527:


{{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


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
[[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, ~21/20 = 83.348
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


{{Optimal ET sequence|legend=1| 29, 43, 72, 619bcd, 691bcd }}
[[Badness]] (Sintel): 2.11
 
[[Badness]]: 0.083338


=== 11-limit ===
=== 11-limit ===
Line 559: Line 546:
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 = 1\1, ~21/20 = 83.340
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| 29, 43, 72 }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.028965
Badness (Sintel): 0.958


=== 13-limit ===
=== 13-limit ===
Line 572: Line 561:
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 = 1\1, ~21/20 = 83.330
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| 29, 43, 72, 115f }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.021470
Badness (Sintel): 0.887


=== 17-limit ===
=== 17-limit ===
Line 585: Line 576:
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 = 1\1, ~21/20 = 83.330
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=1| 29g, 43, 72 }}
Badness (Sintel): 0.760


=== 19-limit ===
=== 19-limit ===
Line 596: Line 591:
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 = 1\1, ~21/20 = 83.330
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=1| 29g, 43, 72 }}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
 
Badness (Sintel): 0.895


== Enneaportent ==
== Enneaportent ==
Line 606: Line 605:


{{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}})
[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 1\9, ~12005/6912 = 950.1680 (~1728/1715 = 16.8347)
: [[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]]: 0.093679
[[Badness]] (Sintel): 2.37


=== 11-limit ===
=== 11-limit ===
Line 624: Line 624:
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 = 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 637: Line 639:
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 = 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 655:


{{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}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.824
: [[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]]: 0.096279
[[Badness]] (Sintel): 2.44


=== 11-limit ===
=== 11-limit ===
Line 667: Line 674:
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 = 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 680: Line 689:
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 = 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 693: Line 704:
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 = 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 706: Line 719:
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 = 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 719: Line 734:
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 = 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 732: Line 749:
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 = 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 745: Line 764:
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 = 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 758: Line 779:
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 = 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 771: Line 794:
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 = 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 784: Line 809:
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 = 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 797: Line 824:
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 = 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 837:
[[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 855:
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 = 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 872:
[[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 890:
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 = 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 905:
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 = 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 881: Line 922:
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 = 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 939:
[[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 957:
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 = 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 972:
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 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


{{Optimal ET sequence|legend=1| 51f, 53 }}
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


Badness: 0.047441
[[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=0| 21, 29g, 50, 79dg }}
 
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 1,055:


{{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 952: Line 1,074:
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 = 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 965: Line 1,089:
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 = 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 978: Line 1,104:
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 = 1\1, ~13/12 = 139.815
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=1| 43, 60e, 103 }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 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 992: Line 1,122:


{{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 1,008: Line 1,141:
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 = 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 1,158:
[[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 1,176:
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 tunings:
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}


Optimal tuning (POTE): ~44/35 = 1\3, ~15/14 = 112.171
{{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:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Marvel| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"