Marvel temperaments: Difference between revisions

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This page discusses some of the temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the [[Marvel_family|marvel]] comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton, merman and marvo. Considered elsewhere are [[Meantone family #Septimal meantone|meantone]], [[Gamelismic clan #Miracle|miracle]], [[Magic family|magic]], [[Diaschismic family #Pajara|pajara]], [[orwell]], [[Kleismic family #Catakleismic|catakleismic]], [[Schismatic family #Garibaldi|garibaldi]], [[Augmented family #August|august]], [[Pythagorean family #Compton|compton]], [[Dicot family #Sharp|sharp]], [[Escapade family #Escapade|escapade]] and [[Pelogic family #Mavila|mavila]].
{{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.  


Since (5/4)<sup>2</sup> = 225/224 × 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
Temperaments considered in families and clans are:
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Quintapole]]'' (+7812500/7411887) → [[Quintaleap family #Quintapole|Quintaleap family]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
* ''[[Decic]]'' (+16807/16384) → [[Cloudy clan #Decic|Cloudy clan]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Hendeca]]'' (+122880/117649) → [[11th-octave temperaments #Hendeca|11th-octave temperaments]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
* ''[[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, 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.


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


Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)
Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)
 
== Wizard ==
{{Main| 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 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
 
[[Comma list]]: 225/224, 118098/117649
 
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216
 
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}
: error map: {{val| 0.000 -1.169 -3.111 -0.849 }}
 
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}
 
[[Badness]] (Sintel): 1.03
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 4000/3993
 
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}
 
Optimal tunings:
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
 
Badness (Sintel): 0.613
 
==== Lizard ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 351/350, 364/363, 385/384
 
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}


= Negri =
Optimal tunings:
{{main| Negri }}
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


== 5-limit ==
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
Note that this is technically not a marvel temperament until the 7-limit, but also placed here for reference.


[[Comma list]]: 16875/16384
Badness (Sintel): 0.900


[[POTE generator]]: ~15/14 = 125.7549
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


[[Wedgie]]: {{wedgie| 4 -3 -14 }}
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}


{{Val list|legend=1| 9, 10, 19, 86 }}
Optimal tunings:
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


== 7-limit ==
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
[[Comma list]]: 49/48, 225/224


[[POTE generator]]: ~15/14 = 125.608
Badness (Sintel): 0.741


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


[[Wedgie]]: {{wedgie| 4 -3 2 -14 -8 13 }}
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


{{Val list|legend=1| 9, 10, 19, 86 }}
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}


== 11-limit ==
Optimal tunings:
Comma list: 45/44, 49/48, 56/55
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


[[POTE generator]]: ~15/14 = 126.474
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
Badness (Sintel): 0.955


{{Val list|legend=1| 9, 10, 19 }}
==== Gizzard ====
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.0262
Comma list: 225/224, 325/324, 385/384, 1573/1568


=== 13-limit ===
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
Comma list: 45/44, 49/48, 56/55, 78/77
 
Optimal tunings:
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
 
Badness (Sintel): 0.837
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
 
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
 
Badness (Sintel): 0.694
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
 
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}
 
{{Optimal ET sequence|legend=0| 72, 94, 166g }}
 
Badness (Sintel): 0.901
 
=== Mage ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 1331/1296
 
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
 
Optimal tunings:
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
 
Badness (Sintel): 1.91
 
== 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
 
[[Comma list]]: 225/224, 50421/50000
 
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7
 
[[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 }}
 
[[Badness]] (Sintel): 1.20
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 225/224, 441/440
 
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}
 
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, 91e, 213bcdeee }}
 
Badness (Sintel): 0.782
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 121/120, 196/195, 275/273
 
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}
 
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 }}
 
Badness (Sintel): 0.950
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
 
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 }}
 
Badness (Sintel): 0.973
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


[[POTE generator]]: ~14/13 = 126.431
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


{{Val list|legend=1| 9, 10, 19 }}
Badness (Sintel): 1.03


== Negril ==
==== 23-limit ====
Comma list: 49/48, 100/99, 225/224
Subgroup: 2.3.5.7.11.13.17.19.23


POTE generator: ~15/14 = 124.767
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


{{Val list|legend=1| 19, 29, 48df, 77cdf }}
Optimal tunings:
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


Badness: 0.0387
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


=== 13-limit ===
Badness (Sintel): 1.04
Comma list: 49/48, 65/64, 91/90, 875/858


POTE generator: ~15/14 = 124.716
=== Tritoni ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
Comma list: 225/224, 385/384, 27783/27500


{{Val list|legend=1| 9, 10, 19, 29, 48, 77 }}
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Badness: 0.0244
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


== Negric ==
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
Comma list: 33/32, 49/48, 77/75


POTE generator: ~15/14 = 127.039
Badness (Sintel): 1.50


Mapping: [{{val| 1 2 2 3 3 }}, {{val| 0 -4 3 -2 4 }}]
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


{{Val list|legend=1| 9, 19, 85 }}
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.


Badness: 0.0306
[[Subgroup]]: 2.3.5.7


=== 13-limit ===
[[Comma list]]: 225/224, 84035/82944
Comma list: 33/32, 49/48, 65/64, 91/90
 
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


POTE generator: ~15/14 = 127.039
[[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 }}


Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
{{Optimal ET sequence|legend=1| 41, 91, 132d }}


{{Val list|legend=1| 9, 19, 85 }}
[[Badness]] (Sintel): 1.38


Badness: 0.0202
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Negroni ==
Comma list: 225/224, 245/242, 385/384
Comma list: 49/48, 55/54, 225/224


POTE generator: ~15/14 = 124.539
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
Optimal tunings:  
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


{{Val list|legend=1| 10, 19e, 29, 77cde }}
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


Badness: 0.0353
Badness (Sintel): 1.04


=== 13-limit ===
=== 13-limit ===
Comma list: 49/48, 55/54, 65/64, 91/90
Subgroup: 2.3.5.7.11.13


POTE generator: ~15/14 = 124.545
Comma list: 105/104, 144/143, 196/195, 245/242


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


{{Val list|legend=1| 10, 19e, 29, 77cdef }}
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


Badness: 0.0216
{{Optimal ET sequence|legend=0| 41, 91 }}


= Passive =
Badness (Sintel): 0.955
{{see also | Archytas clan #Passion }}


Comma list: 225/224, 256/245
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


POTE generator: ~16/15 = 98.809
Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot.  


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


{{Val list|legend=1| 1, 11, 12, 49d }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0751
[[Comma list]]: 225/224, 2500000/2470629


= Wizard =
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
== 5-limit ==
: mapping generators: ~2, ~10/7
Comma list: 2197265625/2147483648


POTE generator: ~5/4 = 383.212
[[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 }}


Mapping: [{{val| 2 1 5 }}, {{val| 0 6 -1 }}]
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 548bc }}
[[Badness]] (Sintel): 1.39


Badness: 0.3864
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 7-limit ==
Comma list: 225/224, 441/440, 1344/1331
Comma list: 225/224, 118098/117649


[[POTE generator]]: ~5/4 = 383.256
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


Mapping: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


Wedgie: {{wedgie| 12 -2 20 -31 -2 52 }}
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


{{Val list|legend=1| 22, 50, 72, 94, 166, 238c, 310c, 382c }}
Badness (Sintel): 1.20


Badness: 0.0408
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


== 11-limit ==
Comma list: 144/143, 225/224, 364/363, 441/440
Comma list: 225/224, 385/384, 4000/3993


[[POTE generator]]: ~5/4 = 383.232
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


{{Val list|legend=1| 22, 50, 72, 94, 166, 238c, 310c }}
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Badness: 0.0185
Badness (Sintel): 1.14


=== Lizard ===
=== Mermaid ===
Comma list: 225/224, 351/350, 364/363, 385/384
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~5/4 = 383.389
Comma list: 225/224, 385/384, 532400/531441


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


{{Val list|legend=1| 22, 50, 72, 122, 194df }}
Optimal tunings:
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


Badness: 0.0218
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


==== 17-limit ====
Badness (Sintel): 1.46
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


[[POTE generator]]: ~5/4 = 383.381
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
Comma list: 225/224, 325/324, 385/384, 10648/10647


{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


Badness: 0.0145
Optimal tunings:  
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


==== 19-limit ====
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


[[POTE generator]]: ~5/4 = 383.477
Badness (Sintel): 1.47


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


{{Val list|legend=1| 22h, 50, 72, 122g, 194dfg }}
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.


Badness: 0.0157
[[Subgroup]]: 2.3.5.7


=== Gizzard ===
[[Comma list]]: 225/224, 589824/588245
Comma list: 225/224, 385/384, 325/324, 1573/1568


POTE generator: ~5/4 = 383.170
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


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


{{Val list|legend=1| 22, 72, 94, 166, 238cf }}
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Badness: 0.0203
[[Badness]] (Sintel): 1.44


==== 17-limit ====
=== 11-limit ===
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Subgroup: 2.3.5.7.11


POTE generator: ~5/4 = 383.175
Comma list: 225/224, 385/384, 1331/1323


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


{{Val list|legend=1| 22f, 72, 94, 166g, 238cfg }}
Optimal tunings:
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


Badness: 0.0136
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


==== 19-limit ====
Badness (Sintel): 0.838
Comma list: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399


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


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
Comma list: 225/224, 275/273, 385/384, 1331/1323


{{Val list|legend=1| 22f, 72, 94, 166g }}
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Badness: 0.0148
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


== Mage ==
{{Optimal ET sequence|legend=0| 31, 63, 94 }}
Comma list: 99/98, 176/175, 1331/1296


POTE generator: ~55/48 = 216.876
Badness (Sintel): 1.07


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


{{Val list|legend=1| 22, 72e, 94e }}
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.


Badness: 0.0578
[[Subgroup]]: 2.3.5.7


= Triton =
[[Comma list]]: 225/224, 1029/1000
Comma list: 225/224, 1029/1000


[[POTE generator]]: ~7/5 = 568.865
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


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


Wedgie: {{wedgie| 3 -7 -8 -18 -21 1 }}
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


{{Val list|legend=1| 17d, 19, 78bd, 97bd }}
[[Badness]] (Sintel): 1.50


Badness: 0.0592
=== 11-limit ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 45/44, 56/55, 1029/1000
Comma list: 45/44, 56/55, 1029/1000


POTE generator: ~7/5 = 569.144
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
 
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}
 
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
 
Badness (Sintel): 1.51
 
== Marvolo ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 156250000/155649627
 
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20
 
[[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, 619bbccd, 691bbccd }}
 
[[Badness]] (Sintel): 2.11
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 4000/3993
 
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}
 
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 }}
 
Badness (Sintel): 0.958
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
 
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 }}
 
Badness (Sintel): 0.887
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
Optimal tunings:
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Badness: 0.0457
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


= Tritonic =
Badness (Sintel): 0.760
== 5-limit ==
Comma list: 553584375/536870912


POTE generator: ~45/32 = 580.219
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 4 -3 }}, {{val| 0 -5 11 }}]
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


{{Val list|legend=1| 29, 31, 60, 91 }}
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Badness: 0.3785
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


== 7-limit ==
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
Comma list: 225/224, 50421/50000


POTE generator: ~7/5 = 580.286
Badness (Sintel): 0.895


Mapping: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


Wedgie: {{wedgie| 5 -11 -12 -29 -33 3 }}
[[Comma list]]: 225/224, 40353607/40310784


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


== 11-limit ==
[[Optimal tuning]]s:
Comma list: 121/120, 225/224, 441/440
* [[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, 54, 63, 72, 495bccd, 567bcccd }}
 
[[Badness]] (Sintel): 2.37
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 12005/11979


POTE generator: ~7/5 = 580.267
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


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


{{Val list|legend=1| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.0237
Badness (Sintel): 1.01


=== 13-limit ===
=== 13-limit ===
Comma list: 105/104, 121/120, 196/195, 275/273
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 364/363, 1716/1715
 
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}
 
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, 54, 63, 72 }}
 
Badness (Sintel): 0.922
 
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 781250000/771895089
 
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3
 
[[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 }}
 
[[Badness]] (Sintel): 2.44
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 236328125/234365481
 
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
 
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, 113, 125, 238c }}
 
Badness (Sintel): 2.96
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 831875/830466
 
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
 
Optimal tunings:
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


POTE generator: ~7/5 = 580.108
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
Badness (Sintel): 2.16


{{Val list|legend=1| 29, 31, 60e, 151cde }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0230
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


== Tritoni ==
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
Comma list: 225/224, 385/384, 27783/27500


POTE generator: ~7/5 = 580.389
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


{{Val list|legend=1| 31, 91, 122 }}
Badness (Sintel): 1.96


Badness: 0.0455
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


= Merman =
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
Comma list: 1121008359375/1099511627776


POTE generator: ~45/32 = 585.634
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


Mapping: [{{val| 1 5 -5 }}, {{val| 0 -7 15 }}]
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


{{Val list|legend=1| 41, 84, 125, 209, 334c, 543bc }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.4538
Badness (Sintel): 1.71


== 7-limit ==
==== 23-limit ====
Comma list: 225/224, 2500000/2470629
Subgroup: 2.3.5.7.11.13.17.19.23


POTE generator: ~7/5 = 585.585
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


Mapping: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Wedgie: {{wedgie| 7 -15 -16 -40 -45 5 }}
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


{{Val list|legend=1| 41, 84, 125 }}
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


Badness: 0.0551
Badness (Sintel): 1.57


== 11-limit ==
==== 29-limit ====
Comma list: 225/224, 441/440, 1344/1331
Subgroup: 2.3.5.7.11.13.17.19.23.29


POTE generator: ~7/5 = 585.606
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


{{Val list|legend=1| 41, 84, 125e }}
Optimal tunings:
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Badness: 0.0364
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


== 13-limit ==
Badness (Sintel): 1.50
Comma list: 144/143, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 585.657
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


Badness: 0.0275
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


= Septimin =
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}
Comma list: 35595703125/34359738368


POTE generator: ~75/64 = 263.658
Badness (Sintel): 1.53


Mapping: [{{val| 1 4 1 }}, {{val| 0 -11 6 }}]
=== Gracecord ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 9, 32, 41, 91, 132, 223c, 355bc }}
Comma list: 225/224, 441/440, 109375/107811


Badness: 0.6709
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


== 7-limit ==
Optimal tunings:
Comma list: 225/224, 84035/82944
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


POTE generator: ~7/6 = 263.632
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Mapping: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
Badness (Sintel): 2.21


Wedgie: {{wedgie| 11 -6 10 -35 -15 40 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 9, 32, 41, 91, 132 }}
Comma list: 225/224, 364/363, 441/440, 6125/6084


== 11-limit ==
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}
Comma list: 225/224, 385/384, 2401/2376


POTE generator: ~7/6 = 263.634
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


{{Val list|legend=1| 9, 32, 41, 91, 223cdef }}
Badness (Sintel): 1.83


== 13-limit ==
==== 17-limit ====
Comma list: 105/104, 144/143, 196/195, 245/242
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


POTE generator: ~7/6 = 263.700
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


{{Val list|legend=1| 9, 41, 91 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


= Slender =
Badness (Sintel): 1.87
Comma list: 225/224, 589824/588245


POTE generator: ~49/48 = 38.413
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


Wedgie: {{wedgie| 13 -10 6 -46 -27 42 }}
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


{{Val list|legend=1| 31, 94, 125 }}
Optimal tunings:
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


Badness: 0.0569
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


==11-limit==
Badness (Sintel): 1.68
Comma list: 225/224, 385/384, 1331/1323


POTE generator: ~49/48 = 38.387
== Alphorn ==
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
[[Comma list]]: 225/224, 5764801/5668704


{{Val list|legend=1| 31, 63, 94, 125 }}
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


Badness: 25.342
[[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 }}


== 13-limit ==
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}
Comma list: 225/224, 275/273, 385/384, 1331/1323


POTE generator: ~49/48 = 38.314
[[Badness]] (Sintel): 3.27


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 31, 63, 94 }}
Comma list: 225/224, 385/384, 12250/11979


Badness: 25.913
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


= Marvo =
Optimal tunings:
{{see also| Gravity family }}
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Comma list: 225/224, 78125000/78121827
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


[[POTE generator]]: ~27/20 = 516.694
Badness (Sintel): 2.43


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


Wedgie: {{wedgie| 6 17 46 13 56 59 }}
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 7d, 65d, 72, 137, 209, 281, 569bcc }}
[[Comma list]]: 225/224, 4194304/4117715


Badness: 0.0976
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


== 11-limit ==
[[Optimal tuning]]s:
Comma list: 225/224, 243/242, 4000/3993
* [[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 }}


POTE generator: ~27/20 = 516.699
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
[[Badness]] (Sintel): 3.57


{{Val list|legend=1| 7d, 65d, 72, 281, 353c, 425bc, 497bc }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0317
Comma list: 99/98, 176/175, 1310720/1294139


== 13-limit ==
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}
Comma list: 225/224, 243/242, 351/350, 1625/1617


POTE generator: ~27/20 = 516.730
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


{{Val list|legend=1| 72, 137, 209, 281f, 490bcf }}
Badness (Sintel): 2.82


Badness: 0.0269
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


= Marvolo =
Comma list: 99/98, 176/175, 640/637, 847/845
Comma list: 225/224, 156250000/155649627


POTE generator: ~21/20 = 83.348
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


Mapping: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


Wedgie: {{wedgie| 6 -19 -26 -44 -58 -7 }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
Badness (Sintel): 1.88


Badness: 0.0833
=== Musneb ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 225/224, 385/384, 66550/64827
Comma list: 225/224, 441/440, 4000/3993


POTE generator: ~21/20 = 83.340
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


{{Val list|legend=1| 29, 43, 72 }}
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Badness: 0.0290
Badness (Sintel): 2.89


== 13-limit ==
== Untriton ==
Comma list: 169/168, 225/224, 364/363, 441/440
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


POTE generator: ~21/20 = 83.330
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.  


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 29, 43, 72, 115f }}
[[Comma list]]: 225/224, 125000000/121060821


Badness: 0.0215
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


= Amavil =
[[Optimal tuning]]s:
== 5-limit (mabila) ==
* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
Comma: 268435456/263671875
: [[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 }}


POTE generator: ~512/375 = 529.6849
{{Optimal ET sequence|legend=1| 51, 53 }}


Map: [<1 6 1|, <0 -10 3|]
[[Badness]] (Sintel): 3.64


EDOs: {{EDOs| 9, 25, 34, 77, 111, 145, 256c }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.2325
Comma list: 121/120, 225/224, 22000/21609


== 7-limit ==
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}
Commas: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
{{Optimal ET sequence|legend=0| 51, 53 }}


Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
Badness (Sintel): 2.46


EDOs: {{EDOs|9, 34d, 43, 77d}}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.1096
Comma list: 121/120, 225/224, 275/273, 1040/1029


== 11-limit ==
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}
Commas: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
{{Optimal ET sequence|legend=0| 51f, 53 }}


EDOs: {{EDOs| 9, 34d, 43, 77de }}
Badness (Sintel): 1.96


Badness: 0.0426
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.  


== 13-limit ==
[[Subgroup]]: 2.3.5.7
Commas: 78/77, 99/98, 144/143, 176/175


POTE generator: ~15/11 = 529.951
[[Comma list]]: 225/224, 823543/800000


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


EDOs: {{EDOs| 9, 34d, 43, 77de }}
[[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 }}


Badness: 0.0258
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


=Enneaportent=
[[Badness]] (Sintel): 3.67
Commas: 225/224, 40353607/40310784


POTE generator: ~5/4 = 383.165
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
Comma list: 225/224, 245/242, 1617/1600


Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


EDOs: {{EDOs|9, 54, 63, 72, 495bcd}}
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


Badness: 0.0937
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


==11-limit==
Badness (Sintel): 2.00
Commas: 225/224, 385/384, 12005/11979


POTE generator: ~5/4 = 383.146
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
Comma list: 105/104, 196/195, 245/242, 1001/1000


EDOs: {{EDOs|9, 54, 63, 72, 423cd, 495bcd}}
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


Badness: 0.0304
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
Commas: 169/168, 225/224, 364/363, 1716/1715


POTE generator: ~5/4 = 383.047
Badness (Sintel): 1.43


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


EDOs: {{EDOs|9, 54, 63, 72, 279cf}}
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


Badness: 0.0223
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


= Submajor =
Optimal tunings:
== 5-limit ==
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
Comma: 69198046875/68719476736
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


POTE generator: ~10125/8192 = 362.321
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Map: [&lt;1 4 -1|, &lt;0 -8 11|]
Badness (Sintel): 1.26


EDOs: {{EDOs|10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


Badness: 0.1302
[[Subgroup]]: 2.3.5.7


== 7-limit ==
[[Comma list]]: 225/224, 9805926501/9765625000
Commas: 225/224, 51200/50421


POTE generator: ~49/40 = 362.255
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


Map: [&lt;1 4 -1 1|, &lt;0 -8 11 6|]
[[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 }}


Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


EDOs: {{EDOs|10, 33, 43, 53}}
[[Badness]] (Sintel): 3.81


Badness: 0.0605
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 225/224, 441/440, 43923/43750
Commas: 225/224, 385/384, 6655/6561


POTE generator: ~27/22 = 362.101
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


Map: [&lt;1 4 -1 1 11|, &lt;0 -8 11 6 -25|]
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


EDOs: {{EDOs|10, 53, 116, 169de, 285cde}}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.0506
Badness (Sintel): 1.74


=== 13-limit ===
=== 13-limit ===
Commas: 169/168, 225/224, 275/273, 385/384
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
 
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
 
Optimal tunings:
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


POTE generator: ~16/13 = 362.105
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Map: [&lt;1 4 -1 1 11 4|, &lt;0 -8 11 6 -25 -1|]
Badness (Sintel): 1.35


EDOs: {{EDOs|10, 53, 116, 169de, 285cdef}}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0277
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


== Interpental ==
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
Commas: 99/98, 176/175, 51200/50421


POTE generator: ~49/40 = 362.418
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


Map: [&lt;1 4 -1 1 -5|, &lt;0 -8 11 6 28|]
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


EDOs: {{EDOs|43, 53, 96, 149d}}
Badness (Sintel): 1.17


Badness: 0.0518
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


=== 13-limit ===
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.
Commas: 99/98, 169/168, 176/175, 640/637
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 5971968/5764801
 
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32
 
[[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 }}
 
[[Badness]] (Sintel): 4.53
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 65536/65219
 
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}
 
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 }}
 
Badness (Sintel): 2.26


POTE generator: ~16/13 = 362.402
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


Map: [&lt;1 4 -1 1 -5 4|, &lt;0 -8 11 6 28 -1|]
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>.


EDOs: {{EDOs|43, 53, 96, 149d}}
[[Subgroup]]: 2.3.5.7


Badness: 0.0297
[[Comma list]]: 225/224, 14495514624/13841287201


=Alphorn=
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
Commas: 225/224, 5764801/5668704
: mapping generators: ~3072/2401, ~2048/1715


POTE generator: ~48/35 = 556.221
[[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 }}


Map: [&lt;1 9 0 13|, &lt;0 -16 5 -22|]
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


Wedgie: &lt;&lt;16 -5 22 -45 -10 65||
[[Badness]] (Sintel): 10.9


EDOs: 28d, 41, 151cd, 192cd, 233cd
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.1293
Comma list: 225/224, 3840/3773, 12005/11979


==11-limit==
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
Commas: 225/224, 385/384, 12250/11979


POTE generator: ~11/8 = 556.144
Optimal tunings:  
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}


Map: [&lt;1 9 0 13 3|, &lt;0 -16 5 -22 1|]
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


EDOs: 41, 315cde
Badness (Sintel): 5.74


Badness: 0.0735
== References ==


[[Category:Theory]]
[[Category:Temperament collections]]
[[Category:Temperament]]
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Marvel]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 10:20, 2 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This page discusses miscellaneous rank-2 temperaments tempering out 225/224, the marvel comma or septimal kleisma.

Temperaments considered in families and clans are:

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 (5/4)2 = (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.

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

Wizard

Main article: 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 can be treated as ~17/15. The semi-octave complement of this interval is ~5/4. Wizard can be described as 22 & 72. Its ploidacot is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the perfect twelfth. 72edo, 94edo, and especially 166edo are good tunings for it.

Subgroup: 2.3.5.7

Comma list: 225/224, 118098/117649

Mapping: [⟨2 1 5 2], ⟨0 6 -1 10]]

mapping generators: ~1225/864, ~245/216

Optimal tunings:

  • WE: ~1225/864 = 600.3438 ¢, ~245/216 = 216.8680 ¢
error map: ⟨+0.688 -0.403 -1.463 +0.541]
  • CWE: ~1225/864 = 600.0000 ¢, ~245/216 = 216.7977 ¢
error map: ⟨0.000 -1.169 -3.111 -0.849]

Optimal ET sequence: 22, 50, 72, 238c, 310c, 382c, 454bccd

Badness (Sintel): 1.03

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 4000/3993

Mapping: [⟨2 1 5 2 8], ⟨0 6 -1 10 -3]]

Optimal tunings:

  • WE: ~99/70 = 600.3051 ¢, ~25/22 = 216.8782 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~25/22 = 216.7961 ¢

Optimal ET sequence: 22, 50, 72, 166, 238c, 310c

Badness (Sintel): 0.613

Lizard

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 364/363, 385/384

Mapping: [⟨2 1 5 2 8 11], ⟨0 6 -1 10 -3 -10]]

Optimal tunings:

  • WE: ~55/39 = 600.4824 ¢, ~25/22 = 216.7852 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~25/22 = 216.6247 ¢

Optimal ET sequence: 22, 50, 72

Badness (Sintel): 0.900

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 273/272, 289/288, 351/350, 375/374

Mapping: [⟨2 1 5 2 8 11 6], ⟨0 6 -1 10 -3 -10 6]]

Optimal tunings:

  • WE: ~17/12 = 600.5032 ¢, ~17/15 = 216.8002 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.6361 ¢

Optimal ET sequence: 22, 50, 72

Badness (Sintel): 0.741

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342

Mapping: [⟨2 1 5 2 8 11 6 2], ⟨0 6 -1 10 -3 -10 6 18]]

Optimal tunings:

  • WE: ~17/12 = 600.4698 ¢, ~17/15 = 216.6925 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.5434 ¢

Optimal ET sequence: 22h, 50, 72, 122g, 194dfg

Badness (Sintel): 0.955

Gizzard

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 1573/1568

Mapping: [⟨2 1 5 2 8 -2], ⟨0 6 -1 10 -3 26]]

Optimal tunings:

  • WE: ~99/70 = 600.2896 ¢, ~25/22 = 216.9343 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~25/22 = 216.8501 ¢

Optimal ET sequence: 22f, 72, 166, 238cf

Badness (Sintel): 0.837

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨2 1 5 2 8 -2 6], ⟨0 6 -1 10 -3 26 6]]

Optimal tunings:

  • WE: ~17/12 = 600.3227 ¢, ~17/15 = 216.9414 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.8469 ¢

Optimal ET sequence: 22f, 72, 166g, 238cfg

Badness (Sintel): 0.694

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨2 1 5 2 8 -2 6 15], ⟨0 6 -1 10 -3 26 6 -18]]

Optimal tunings:

  • WE: ~17/12 = 600.2637 ¢, ~17/15 = 216.9570 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.8687 ¢

Optimal ET sequence: 72, 94, 166g

Badness (Sintel): 0.901

Mage

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1331/1296

Mapping: [⟨2 1 5 2 4], ⟨0 6 -1 10 8]]

Optimal tunings:

  • WE: ~77/54 = 600.6486 ¢, ~55/48 = 217.1099 ¢
  • CWE: ~77/54 = 600.0000 ¢, ~55/48 = 216.9841 ¢

Optimal ET sequence: 22, 50e, 72ee

Badness (Sintel): 1.91

Tritonic

For the 5-limit version, see Miscellaneous 5-limit temperaments #Tritonic.

Tritonic tempers out 50421/50000 and may be described as the 29 & 31 temperament. It splits the 6th harmonic into five generators of ~10/7 tritones, 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

Comma list: 225/224, 50421/50000

Mapping: [⟨1 -1 8 9], ⟨0 5 -11 -12]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1201.3539 ¢, ~10/7 = 620.4131 ¢
error map: ⟨+1.354 -1.243 -0.027 -1.598]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6778 ¢
error map: ⟨0.000 -3.566 -2.769 -4.959]

Optimal ET sequence: 29, 31, 60, 91, 122, 213bcd

Badness (Sintel): 1.20

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 8 9 5], ⟨0 5 -11 -12 -3]]

Optimal tunings:

  • WE: ~2 = 1201.7116 ¢, ~10/7 = 620.6166 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6890 ¢

Optimal ET sequence: 29, 31, 60e, 91e, 213bcdeee

Badness (Sintel): 0.782

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 8 9 5 13], ⟨0 5 -11 -12 -3 -18]]

Optimal tunings:

  • WE: ~2 = 1201.5355 ¢, ~10/7 = 620.6855 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.8469 ¢

Optimal ET sequence: 29, 31, 60e

Badness (Sintel): 0.950

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 121/120, 154/153, 196/195, 273/272

Mapping: [⟨1 -1 8 9 5 13 17], ⟨0 5 -11 -12 -3 -18 -25]]

Optimal tunings:

  • WE: ~2 = 1201.5260 ¢, ~10/7 = 620.7330 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.8986 ¢

Optimal ET sequence: 29g, 31, 60e

Badness (Sintel): 0.973

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 -1 8 9 5 13 17 12], ⟨0 5 -11 -12 -3 -18 -25 -15]]

Optimal tunings:

  • WE: ~2 = 1201.3100 ¢, ~10/7 = 620.6509 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.9328 ¢

Optimal ET sequence: 29g, 31, 60e

Badness (Sintel): 1.03

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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

Mapping: [⟨1 -1 8 9 5 13 17 12 4], ⟨0 5 -11 -12 -3 -18 -25 -15 1]]

Optimal tunings:

  • WE: ~2 = 1201.4074 ¢, ~10/7 = 620.7185 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.9548 ¢

Optimal ET sequence: 29g, 31, 60e

Badness (Sintel): 1.04

Tritoni

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 8 9 -11], ⟨0 5 -11 -12 28]]

Optimal tunings:

  • WE: ~2 = 1201.0888 ¢, ~10/7 = 620.1733 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6146 ¢

Optimal ET sequence: 31, 91, 122, 153d

Badness (Sintel): 1.50

Septimin

For the 5-limit version, see Miscellaneous 5-limit temperaments #Septimin.

Septimin may be described as the 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 reduced give the perfect fifth; its ploidacot is thus eta-hendecacot. 91edo may be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 225/224, 84035/82944

Mapping: [⟨1 -7 7 -5], ⟨0 11 -6 10]]

mapping generators: ~2, ~12/7

Optimal tunings:

  • WE: ~2 = 1201.2452 ¢, ~12/7 = 937.3394 ¢
error map: ⟨+1.245 +0.062 -1.633 -1.658]
  • CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.4036 ¢
error map: ⟨0.000 -1.516 -4.735 -4.790]

Optimal ET sequence: 41, 91, 132d

Badness (Sintel): 1.38

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/242, 385/384

Mapping: [⟨1 -7 7 -5 -2], ⟨0 11 -6 10 7]]

Optimal tunings:

  • WE: ~2 = 1200.8059 ¢, ~12/7 = 936.9952 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.3906 ¢

Optimal ET sequence: 41, 91, 223cdef

Badness (Sintel): 1.04

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195, 245/242

Mapping: [⟨1 -7 7 -5 -2 -8], ⟨0 11 -6 10 7 15]]

Optimal tunings:

  • WE: ~2 = 1200.5990 ¢, ~12/7 = 936.7670 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.3196 ¢

Optimal ET sequence: 41, 91

Badness (Sintel): 0.955

Merman

For the 5-limit version, see Miscellaneous 5-limit temperaments #Merman.

Merman may be described as the 41 & 43 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, seven generator steps give the interval class of 3. The ploidacot for this temperament is gamma-heptacot.

The name was likely derived from Triton, which was in turn derived from tritonic.

Subgroup: 2.3.5.7

Comma list: 225/224, 2500000/2470629

Mapping: [⟨1 -2 10 11], ⟨0 7 -15 -16]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1200.3898 ¢, ~10/7 = 614.6413 ¢
error map: ⟨+0.390 -0.435 -1.630 +1.634]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4073 ¢
error map: ⟨0.000 -1.104 -2.423 +0.657]

Optimal ET sequence: 41, 84, 125

Badness (Sintel): 1.39

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 1344/1331

Mapping: [⟨1 -2 10 11 5], ⟨0 7 -15 -16 -3]]

Optimal tunings:

  • WE: ~2 = 1199.9578 ¢, ~10/7 = 614.3720 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.3943 ¢

Optimal ET sequence: 41, 84, 125e

Badness (Sintel): 1.20

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -2 10 11 5 -5], ⟨0 7 -15 -16 -3 17]]

Optimal tunings:

  • WE: ~2 = 1199.7422 ¢, ~10/7 = 614.2110 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.3442 ¢

Optimal ET sequence: 41, 84, 125e, 209ef, 293ef

Badness (Sintel): 1.14

Mermaid

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 532400/531441

Mapping: [⟨1 -2 10 11 -16], ⟨0 7 -15 -16 38]]

Optimal tunings:

  • WE: ~2 = 1199.4973 ¢, ~10/7 = 614.7004 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4470 ¢

Optimal ET sequence: 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: [⟨1 -2 10 11 22 32], ⟨0 7 -15 -16 38 58]]

Optimal tunings:

  • WE: ~2 = 1200.5126 ¢, ~10/7 = 614.7152 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4562 ¢

Optimal ET sequence: 41, 84ef, 125f, 166

Badness (Sintel): 1.47

Slender

Slender tempers out the hewuermera comma in addition to the marvel comma, and may be described as the 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

Comma list: 225/224, 589824/588245

Mapping: [⟨1 2 2 3], ⟨0 -13 10 -6]]

mapping generators: ~2, ~49/48

Optimal tunings:

  • WE: ~2 = 1200.3816 ¢, ~49/48 = 38.4256 ¢
error map: ⟨+0.382 -0.725 -1.295 +1.765]
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.4079 ¢
error map: ⟨0.000 -1.257 -2.235 +0.727]

Optimal ET sequence: 31, 94, 125, 406c

Badness (Sintel): 1.44

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 1331/1323

Mapping: [⟨1 2 2 3 4], ⟨0 -13 10 -6 -17]]

Optimal tunings:

  • WE: ~2 = 1199.4983 ¢, ~49/48 = 38.4030 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.3775 ¢

Optimal ET sequence: 31, 63, 94, 125

Badness (Sintel): 0.838

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 385/384, 1331/1323

Mapping: [⟨1 2 2 3 4 3], ⟨0 -13 10 -6 -17 22]]

Optimal tunings:

  • WE: ~2 = 1200.1728 ¢, ~49/48 = 38.3192 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.3129 ¢

Optimal ET sequence: 31, 63, 94

Badness (Sintel): 1.07

Triton

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Stump.

Triton may be described as the 19 & 21 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, three generator steps give the interval class of 3. The ploidacot for this temperament is alpha-tricot.

Subgroup: 2.3.5.7

Comma list: 225/224, 1029/1000

Mapping: [⟨1 0 6 7], ⟨0 3 -7 -8]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1203.3828 ¢, ~10/7 = 632.9137 ¢
error map: ⟨+3.383 -3.214 +3.587 -8.457]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 630.9827 ¢
error map: ⟨0.000 -9.007 -3.192 -16.687]

Optimal ET sequence: 2, 17d, 19, 78bd, 97bd

Badness (Sintel): 1.50

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55, 1029/1000

Mapping: [⟨1 0 6 7 4], ⟨0 3 -7 -8 -1]]

Optimal tunings:

  • WE: ~2 = 1201.3875 ¢, ~10/7 = 631.5852 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 630.8007 ¢

Optimal ET sequence: 2, 17d, 19

Badness (Sintel): 1.51

Marvolo

Subgroup: 2.3.5.7

Comma list: 225/224, 156250000/155649627

Mapping: [⟨1 2 1 1], ⟨0 -6 19 26]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • WE: ~2 = 1200.7714 ¢, ~21/20 = 83.4014 ¢
error map: ⟨+0.772 -0.820 -0.916 +0.381]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3640 ¢
error map: ⟨0.000 -2.139 -2.398 -1.362]

Optimal ET sequence: 29, 43, 72, 619bbccd, 691bbccd

Badness (Sintel): 2.11

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 1 1 2], ⟨0 -6 19 26 21]]

Optimal tunings:

  • WE: ~2 = 1200.7075 ¢, ~21/20 = 83.3888 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3564 ¢

Optimal ET sequence: 29, 43, 72

Badness (Sintel): 0.958

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 1 1 2 3], ⟨0 -6 19 26 21 10]]

Optimal tunings:

  • WE: ~2 = 1200.9467 ¢, ~21/20 = 83.3956 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3516 ¢

Optimal ET sequence: 29, 43, 72

Badness (Sintel): 0.887

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 2 1 1 2 3 2], ⟨0 -6 19 26 21 10 30]]

Optimal tunings:

  • WE: ~2 = 1200.9606 ¢, ~21/20 = 83.4030 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3594 ¢

Optimal ET sequence: 29g, 43, 72

Badness (Sintel): 0.760

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 2 1 1 2 3 2 3], ⟨0 -6 19 26 21 10 30 18]]

Optimal tunings:

  • WE: ~2 = 1200.7625 ¢, ~21/20 = 83.3895 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3551 ¢

Optimal ET sequence: 29g, 43, 72

Badness (Sintel): 0.895

Enneaportent

Subgroup: 2.3.5.7

Comma list: 225/224, 40353607/40310784

Mapping: [⟨9 0 28 11], ⟨0 2 -1 2]]

mapping generators: ~2592/2401, ~12005/6912

Optimal tunings:

  • WE: ~2592/2401 = 133.4174 ¢, ~12005/6912 = 950.7667 ¢ (~1728/1715 = 16.8452 ¢)
error map: ⟨+0.756 -0.422 -1.395 +0.298]
  • CWE: ~2592/2401 = 133.3333 ¢, ~12005/6912 = 950.2969 ¢ (~1728/1715 = 16.9636 ¢)
error map: ⟨0.000 -1.361 -3.277 -1.565]

Optimal ET sequence: 9, 54, 63, 72, 495bccd, 567bcccd

Badness (Sintel): 2.37

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 12005/11979

Mapping: [⟨9 0 28 11 24], ⟨0 2 -1 2 1]]

Optimal tunings:

  • WE: ~121/112 = 133.4071 ¢, ~210/121 = 950.7131 ¢ (~99/98 = 16.8633 ¢)
  • CWE: ~121/112 = 133.3333 ¢, ~210/121 = 950.2994 ¢ (~99/98 = 16.9661 ¢)

Optimal ET sequence: 9, 54, 63, 72

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 364/363, 1716/1715

Mapping: [⟨9 0 28 11 24 19], ⟨0 2 -1 2 1 2]]

Optimal tunings:

  • WE: ~14/13 = 133.4245 ¢, ~26/15 = 950.9362 ¢ (~105/104 = 16.9650 ¢)
  • CWE: ~14/13 = 133.3333 ¢, ~26/15 = 950.4364 ¢ (~99/98 = 17.1031 ¢)

Optimal ET sequence: 9, 54, 63, 72

Badness (Sintel): 0.922

Gracecordial

For the 5-limit version, see Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit).

Subgroup: 2.3.5.7

Comma list: 225/224, 781250000/771895089

Mapping: [⟨1 0 34 63], ⟨0 1 -20 -38]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1200.4904 ¢, ~3/2 = 701.1103 ¢
error map: ⟨+0.490 -0.354 -1.655 +1.241]
  • CWE: ~2 = 1200.3333 ¢, ~3/2 = 700.8112 ¢
error map: ⟨0.000 -1.144 -2.537 +0.349]

Optimal ET sequence: 12, …, 113, 125, 238c, 363c

Badness (Sintel): 2.44

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 34 63 -90], ⟨0 1 -20 -38 59]]

Optimal tunings:

  • WE: ~2 = 1200.5571 ¢, ~3/2 = 701.1589 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8328 ¢

Optimal ET sequence: 12e, 113, 125, 238c

Badness (Sintel): 2.96

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 831875/830466

Mapping: [⟨1 0 34 63 -90 -66], ⟨0 1 -20 -38 59 44]]

Optimal tunings:

  • WE: ~2 = 1200.6282 ¢, ~3/2 = 701.2080 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8421 ¢

Optimal ET sequence: 12e, 113, 125f, 238cf

Badness (Sintel): 2.16

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 34 63 -90 -66 -7], ⟨0 1 -20 -38 59 44 7]]

Optimal tunings:

  • WE: ~2 = 1200.5058 ¢, ~3/2 = 701.1360 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8414 ¢

Optimal ET sequence: 12e, 113, 125f, 238cf

Badness (Sintel): 1.96

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444

Mapping: [⟨1 0 34 63 -90 -66 -7 9], ⟨0 1 -20 -38 59 44 7 -3]]

Optimal tunings:

  • WE: ~2 = 1200.4418 ¢, ~3/2 = 701.0999 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8425 ¢

Optimal ET sequence: 12e, 113, 125f, 238cf

Badness (Sintel): 1.71

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528

Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43], ⟨0 1 -20 -38 59 44 7 -3 30]]

Optimal tunings:

  • WE: ~2 = 1200.4641 ¢, ~3/2 = 701.1145 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8444 ¢

Optimal ET sequence: 12e, 113, 238cfi

Badness (Sintel): 1.57

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493

Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43 -49], ⟨0 1 -20 -38 59 44 7 -3 30 34]]

Optimal tunings:

  • WE: ~2 = 1200.4400 ¢, ~3/2 = 701.0986 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8428 ¢

Optimal ET sequence: 12e, 113, 125f, 238cfi

Badness (Sintel): 1.50

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493

Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43 -49 -79], ⟨0 1 -20 -38 59 44 7 -3 30 34 53]]

Optimal tunings:

  • WE: ~2 = 1200.4178 ¢, ~3/2 = 701.0822 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8396 ¢

Optimal ET sequence: 12e, 113, 125f, 238cfi

Badness (Sintel): 1.53

Gracecord

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 34 63 89], ⟨0 1 -20 -38 -54]]

Optimal tunings:

  • WE: ~2 = 1200.6064 ¢, ~3/2 = 701.2398 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8718 ¢

Optimal ET sequence: 12, …, 101cd, 113

Badness (Sintel): 2.21

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 34 63 89 113], ⟨0 1 -20 -38 -54 -69]]

Optimal tunings:

  • WE: ~2 = 1200.6225 ¢, ~3/2 = 701.2539 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8781 ¢

Optimal ET sequence: 12f, …, 101cdf, 113

Badness (Sintel): 1.83

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989

Mapping: [⟨1 0 34 63 89 113 -7], ⟨0 1 -20 -38 -54 -69 7]]

Optimal tunings:

  • WE: ~2 = 1200.3308 ¢, ~3/2 = 701.0632 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8654 ¢

Optimal ET sequence: 12f, 101cdf, 113

Badness (Sintel): 1.87

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 0 34 63 89 113 -7 9], ⟨0 1 -20 -38 -54 -69 7 -3]]

Optimal tunings:

  • WE: ~2 = 1200.2658 ¢, ~3/2 = 701.0213 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8629 ¢

Optimal ET sequence: 12f, 101cdf, 113

Badness (Sintel): 1.68

Alphorn

Subgroup: 2.3.5.7

Comma list: 225/224, 5764801/5668704

Mapping: [⟨1 -7 5 -9], ⟨0 16 -5 22]]

mapping generators: ~2, ~35/24

Optimal tunings:

  • WE: ~2 = 1201.3004 ¢, ~35/24 = 644.4767 ¢
error map: ⟨+1.300 +0.569 -2.195 -2.043]
  • CWE: ~2 = 1200.3333 ¢, ~35/24 = 643.8137 ¢
error map: ⟨0.000 -0.936 -5.382 -4.924]

Optimal ET sequence: 13d, 28d, 41, 151cd, 192cdd, 233ccdd

Badness (Sintel): 3.27

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 12250/11979

Mapping: [⟨1 -7 5 -9 4], ⟨0 16 -5 22 -1]]

Optimal tunings:

  • WE: ~2 = 1200.5123 ¢, ~16/11 = 644.1307 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 643.8662 ¢

Optimal ET sequence: 13d, 28d, 41

Badness (Sintel): 2.43

Misneb

For the 5-limit version, see Miscellaneous 5-limit temperaments #Misneb.

Subgroup: 2.3.5.7

Comma list: 225/224, 4194304/4117715

Mapping: [⟨1 -12 15 1], ⟨0 15 -14 2]]

mapping generators: ~2, ~15/8

Optimal tunings:

  • WE: ~2 = 1199.7642 ¢, ~15/8 = 1086.5513 ¢
error map: ⟨-0.236 -0.856 -1.569 +4.041]
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.7633 ¢
error map: ⟨0.000 -0.506 -0.999 +4.701]

Optimal ET sequence: 21, 32, 53

Badness (Sintel): 3.57

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1310720/1294139

Mapping: [⟨1 -12 15 1 27], ⟨0 15 -14 2 -26]]

Optimal tunings:

  • WE: ~2 = 1200.1654 ¢, ~15/8 = 1086.8269 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.6766 ¢

Optimal ET sequence: 21, 32e, 53, 127

Badness (Sintel): 2.82

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 640/637, 847/845

Mapping: [⟨1 -12 15 1 27 20], ⟨0 15 -14 2 -26 -18]]

Optimal tunings:

  • WE: ~2 = 1200.1687 ¢, ~15/8 = 1086.8295 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.6757 ¢

Optimal ET sequence: 21, 32e, 53, 127

Badness (Sintel): 1.88

Musneb

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 66550/64827

Mapping: [⟨1 3 1 3 6], ⟨0 -15 14 -2 -27]]

Optimal tunings:

  • WE: ~2 = 1200.0839 ¢, ~15/8 = 1086.9343 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.8593 ¢

Optimal ET sequence: 21e, 32, 53

Badness (Sintel): 2.89

Untriton

For the 5-limit version, see Miscellaneous 5-limit temperaments #Untriton.

Named by Petr Pařízek in 2011[1], untriton may be described as the 51 & 53 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, nine generator steps give the interval class of 3. The ploidacot for this temperament is delta-enneacot.

Subgroup: 2.3.5.7

Comma list: 225/224, 125000000/121060821

Mapping: [⟨1 -3 12 13], ⟨0 9 -19 -20]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1199.8275 ¢, ~10/7 = 611.2710 ¢
error map: ⟨-0.172 +0.002 -2.533 +3.511]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3614 ¢
error map: ⟨0.000 +0.298 -2.181 +3.946]

Optimal ET sequence: 51, 53

Badness (Sintel): 3.64

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 22000/21609

Mapping: [⟨1 -3 12 13 6], ⟨0 9 -19 -20 -5]]

Optimal tunings:

  • WE: ~2 = 1200.3591 ¢, ~10/7 = 611.5569 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3690 ¢

Optimal ET sequence: 51, 53

Badness (Sintel): 2.46

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 225/224, 275/273, 1040/1029

Mapping: [⟨1 -3 12 13 6 20], ⟨0 9 -19 -20 -5 -32]]

Optimal tunings:

  • WE: ~2 = 1200.4078 ¢, ~10/7 = 611.5536 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3392 ¢

Optimal ET sequence: 51f, 53

Badness (Sintel): 1.96

Naiadical

Named by Xenllium in 2026, naiadical may be described as the 21 & 29 temperament.

Subgroup: 2.3.5.7

Comma list: 225/224, 823543/800000

Mapping: [⟨1 -4 11 9], ⟨0 9 -14 -10]]

mapping generators: ~2, ~32/21

Optimal tunings:

  • WE: ~2 = 1202.1198 ¢, ~32/21 = 745.4675 ¢
error map: ⟨+2.120 -1.227 +0.459 -4.423]
  • CWE: ~2 = 1200.0000 ¢, ~32/21 = 744.1318 ¢
error map: ⟨0.000 -4.769 -4.159 -10.144]

Optimal ET sequence: 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: [⟨1 -4 11 9 14], ⟨0 9 -14 -10 -17]]

Optimal tunings:

  • WE: ~2 = 1201.9008 ¢, ~21/16 = 745.3867 ¢
  • CWE: ~2 = 1200.0000 ¢, ~32/21 = 744.1777 ¢

Optimal ET sequence: 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: [⟨1 -4 11 9 14 13], ⟨0 9 -14 -10 -17 -15]]

Optimal tunings:

  • WE: ~2 = 1201.7863 ¢, ~20/13 = 745.3344 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.1931 ¢

Optimal ET sequence: 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: [⟨1 -4 11 9 14 13 14], ⟨0 9 -14 -10 -17 -15 -16]]

Optimal tunings:

  • WE: ~2 = 1201.9208 ¢, ~20/13 = 745.3976 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.1669 ¢

Optimal ET sequence: 21, 29g, 50, 79dg

Badness (Sintel): 1.26

Quintannic

Named by Scott Dakota, quintannic may be described as the 43 & 60 temperament.

Subgroup: 2.3.5.7

Comma list: 225/224, 9805926501/9765625000

Mapping: [⟨1 1 5 7], ⟨0 5 -23 -36]]

mapping generators: ~2, ~10000/9261

Optimal tunings:

  • WE: ~2 = 1200.9803 ¢, ~10000/9261 = 139.9522 ¢
error map: ⟨+0.980 -1.214 -0.313 -0.243]
  • CWE: ~2 = 1200.0000 ¢, ~10000/9261 = 139.8184 ¢
error map: ⟨0.000 -2.863 -2.136 -2.287]

Optimal ET sequence: 43, 60, 103, 266bcd, 369bcd

Badness (Sintel): 3.81

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 43923/43750

Mapping: [⟨1 1 5 7 8], ⟨0 5 -23 -36 -39]]

Optimal tunings:

  • WE: ~2 = 1201.0031 ¢, ~320/297 = 139.9435 ¢
  • CWE: ~2 = 1200.0000 ¢, ~320/297 = 139.8053 ¢

Optimal ET sequence: 43, 60e, 103, 369bcdeee, 472bbcddeee

Badness (Sintel): 1.74

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 5 7 8 3], ⟨0 5 -23 -36 -39 6]]

Optimal tunings:

  • WE: ~2 = 1200.8354 ¢, ~13/12 = 139.9095 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.7997 ¢

Optimal ET sequence: 43, 60e, 103

Badness (Sintel): 1.35

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 1 5 7 8 3 7], ⟨0 5 -23 -36 -39 6 -25]]

Optimal tunings:

  • WE: ~2 = 1200.7402 ¢, ~13/12 = 139.9015 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.8038 ¢

Optimal ET sequence: 43, 60e, 103

Badness (Sintel): 1.17

Gwazy

For the 5-limit version, see Very high accuracy temperaments #Kwazy.

Named by Petr Pařízek in 2011[1], gwazy may be described as the 22 & 74 temperament.

Subgroup: 2.3.5.7

Comma list: 225/224, 5971968/5764801

Mapping: [⟨2 1 6 4], ⟨0 8 -5 6]]

mapping generators: ~2401/1728, ~35/32

Optimal tunings:

  • WE: ~2401/1728 = 599.7132 ¢, ~35/32 = 162.5806 ¢
error map: ⟨-0.574 -1.597 -0.937 +5.510]
  • CWE: ~2401/1728 = 600.0000 ¢, ~35/32 = 162.6388 ¢
error map: ⟨0.000 -0.844 +0.492 +7.007]

Optimal ET sequence: 22, 74, 96, 118d

Badness (Sintel): 4.53

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 65536/65219

Mapping: [⟨2 1 6 4 8], ⟨0 8 -5 6 -4]]

Optimal tunings:

  • WE: ~363/256 = 599.8517 ¢, ~11/10 = 162.5518 ¢
  • CWE: ~363/256 = 600.0000 ¢, ~11/10 = 162.5863 ¢

Optimal ET sequence: 22, 74, 96

Badness (Sintel): 2.26

Tertiosec

For the 5-limit version, see Miscellaneous 5-limit temperaments #Tertiosec.

Tertiosec may be described as the 21 & 75 temperament. It was initially named tertiomar by Petr Pařízek in 2011[1], but was changed to tertiosec in 2012[2].

Subgroup: 2.3.5.7

Comma list: 225/224, 14495514624/13841287201

Mapping: [⟨3 -1 12 7], ⟨0 8 -7 2]]

mapping generators: ~3072/2401, ~2048/1715

Optimal tunings:

  • WE: ~3072/2401 = 399.8257 ¢, ~2048/1715 = 287.5920 ¢
error map: ⟨-0.523 -1.044 -1.549 +5.138]
  • CWE: ~3072/2401 = 400.0000 ¢, ~2048/1715 = 287.7088 ¢
error map: ⟨0.000 -0.284 -0.276 +6.592]

Optimal ET sequence: 21, 54, 75, 96, 171d

Badness (Sintel): 10.9

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 3840/3773, 12005/11979

Mapping: [⟨3 -1 12 7 14], ⟨0 8 -7 2 -5]]

Optimal tunings:

  • WE: ~44/35 = 399.6550 ¢, ~33/28 = 287.5803 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~33/28 = 287.8224 ¢

Optimal ET sequence: 21, 54, 75e

Badness (Sintel): 5.74

References

  1. ↑ 1.0 1.1 1.2 Yahoo! Tuning Group | Suggested names for the unclasified temperaments
  2. ↑ Yahoo! Tuning Group | 2D temperament names, part I -- reclassified temperaments from message #101780
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