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{{Technical data page}}
This page discusses some of the temperaments tempering out |-5 2 2 -1> =[[225/224| 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|meantone]], [[Miracle|miracle]], [[Magic_family|magic]], [[pajara|pajara]], [[Orwell|orwell]], [[Kleismic_family|catakleismic]], [[Garibaldi_temperament|garibaldi]], [[Augmented_family|august]], [[Pythagorean_family|compton]], [[Dicot_family|sharp]], [[Escapade_family|escapade]] and [[Pelogic_family|mavila]].
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)^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.
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 }}
 
Optimal tunings:
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
 
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 }}
 
Optimal tunings:
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
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: {{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{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
Badness (Sintel): 1.04
 
=== Tritoni ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 27783/27500
 
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}
 
Optimal tunings:
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}
 
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
 
Badness (Sintel): 1.50
 
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''
 
Septimin may be described as the {{nowrap| 41 & 50 }} temperament. It is generated by a septimal minor third ([[7/6]]), which gives rise to the name, but the generator can be taken to be the [[octave complement]], [[12/7]], such that eleven of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus eta-hendecacot. [[91edo]] may be recommended as a tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 84035/82944
 
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}
: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}
 
{{Optimal ET sequence|legend=1| 41, 91, 132d }}
 
[[Badness]] (Sintel): 1.38
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 245/242, 385/384
 
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
 
Optimal tunings:
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}
 
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}
 
Badness (Sintel): 1.04
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 196/195, 245/242
 
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
 
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}
 
{{Optimal ET sequence|legend=0| 41, 91 }}
 
Badness (Sintel): 0.955
 
== 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
 
[[Comma list]]: 225/224, 2500000/2470629
 
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
: [[error map]]: {{val| +0.390 -0.435 -1.630 +1.634 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}}
: error map: {{val| 0.000 -1.104 -2.423 +0.657 }}
 
{{Optimal ET sequence|legend=1| 41, 84, 125 }}
 
[[Badness]] (Sintel): 1.39
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 1344/1331
 
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
 
Badness (Sintel): 1.20
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 225/224, 364/363, 441/440
 
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 }}
 
Badness (Sintel): 1.14
 
=== Mermaid ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 532400/531441
 
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
 
Optimal tunings:
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}
 
Badness (Sintel): 1.46
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 10648/10647
 
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
 
Optimal tunings:
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
 
Badness (Sintel): 1.47
 
== Slender ==
Slender 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.


=Negri=
[[Subgroup]]: 2.3.5.7
[[Comma|Comma]]s: 49/48, 225/224


[[POTE_tuning|POTE temperament]]: ~15/14 = 125.608
[[Comma list]]: 225/224, 589824/588245


[[Map|Map]]: [&lt;1 2 2 3|, &lt;0 -4 3 -2|]
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


[[wedgie|Wedgie]]: &lt;&lt;4 -3 2 -14 -8 13||
[[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 }}


EDOs: [[9edo|9]], [[10edo|10]], [[19edo|19]], [[86edo|86]]
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


==11-limit==
[[Badness]] (Sintel): 1.44
Commas: 45/44, 49/48, 56/55


[[POTE_tuning|POTE generator]]: ~15/14 = 126.474
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 2 3 4|, &lt;0 -4 3 -2 -5|]
Comma list: 225/224, 385/384, 1331/1323


EDOs: 9, 10, 19
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


[[Badness|Badness]]: 0.0262
Optimal tunings:  
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
Commas: 45/44, 49/48, 56/55, 78/77


[[POTE_tuning|POTE generator]]: ~14/13 = 126.431
Badness (Sintel): 0.838


Map: [&lt;1 2 2 3 4 4|, &lt;0 -4 3 -2 -5 -3|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 9, 10, 19
Comma list: 225/224, 275/273, 385/384, 1331/1323


=Negril=
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
Commas: 49/48, 100/99, 225/224


POTE generator: ~15/14 = 124.767
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Map: [&lt;1 2 2 3 2|, &lt;0 -4 3 -2 14|]
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


EDOs: 19, [[29edo|29]], [[48edo|48df]], [[77edo|77cdf]]
Badness (Sintel): 1.07


Badness: 0.0387
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


==13-limit==
Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot.
Commas: 49/48, 65/64, 91/90, 875/858


POTE generator: ~15/14 = 124.716
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 2 3 2 4|, &lt;0 -4 3 -2 14 -3|]
[[Comma list]]: 225/224, 1029/1000


EDOs: 9, 10, 19, 29, 48, 77
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


Badness: 0.0244
[[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 }}


=Negric=
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
Commas: 33/32, 49/48, 77/75


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


Map: [&lt;1 2 2 3 3|, &lt;0 -4 3 -2 4|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 9, 19, [[85edo|85]]
Comma list: 45/44, 56/55, 1029/1000


Badness: 0.0306
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


==13-limit==
Optimal tunings:
Commas: 33/32, 49/48, 65/64, 91/90
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


POTE generator: ~15/14 = 127.039
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


Map: [&lt;1 2 2 3 3 4|, &lt;0 -4 3 -2 4 -3|]
Badness (Sintel): 1.51


EDOs: 9, 19, 85
== Marvolo ==
[[Subgroup]]: 2.3.5.7


Badness: 0.0202
[[Comma list]]: 225/224, 156250000/155649627


=Negroni=
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
Commas: 49/48, 55/54, 225/224
: mapping generators: ~2, ~21/20


POTE generator: ~15/14 = 124.539
[[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 }}


Map: [&lt;1 2 2 3 5|, &lt;0 -4 3 -2 -15|]
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


EDOs: 10, 19e, 29, 77cde
[[Badness]] (Sintel): 2.11


Badness: 0.0353
=== 11-limit ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 225/224, 441/440, 4000/3993
Commas: 49/48, 55/54, 65/64, 91/90


POTE generator: ~15/14 = 124.545
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


Map: [&lt;1 2 2 3 5 4|, &lt;0 -4 3 -2 -15 -3|]
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


EDOs: 10, 19e, 29, 77cdef
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.0216
Badness (Sintel): 0.958


=Wizard=
=== 13-limit ===
Comma: 2197265625/2147483648
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 383.212
Comma list: 169/168, 225/224, 364/363, 441/440


Map: [&lt;2 1 5|, &lt;0 6 -1|]
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


EDOs: 22, 50, 72, 166, 238c, 310c, 548bc
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


Badness: 0.3864
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


==7-limit==
Badness (Sintel): 0.887
Commas: 225/224, 118098/117649


[[POTE_tuning|POTE temperament]]: 383.256
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 1 5 2|, &lt;0 6 -1 10|]
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


Wedgie: &lt;&lt;12 -2 20 -31 -2 52||
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


EDOs: [[22edo|22]], [[50edo|50]], [[72edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238c]], [[310edo|310c]], [[382edo|382c]]
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Badness: 0.0408
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


==11-limit==
Badness (Sintel): 0.760
Commas: 225/224, 385/384, 4000/3993


[[POTE_tuning|POTE temperament]]: 383.232
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;2 1 5 2 8|, &lt;0 6 -1 10 -3|]
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


EDOs: 22, 50, 72, 94, [[166edo|166]], [[238edo|238c]], [[310edo|310c]]
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Badness: 0.0185
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


==Lizard==
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
Commas: 225/224, 351/350, 364/363, 385/384


[[POTE_tuning|POTE temperament]]: 383.389
Badness (Sintel): 0.895


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


EDOs: 22, 50, 72, 122, 194df
[[Comma list]]: 225/224, 40353607/40310784


Badness: 0.0218
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


===17-limit===
[[Optimal tuning]]s:
Commas: 221/220, 273/272, 289/288, 351/350, 375/374
* [[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 }}


[[POTE_tuning|POTE temperament]]: 383.381
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
[[Badness]] (Sintel): 2.37


EDOs: 22, 50, 72, 122g, 194dfg
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0145
Comma list: 225/224, 385/384, 12005/11979


===19-limit===
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


[[POTE_tuning|POTE generator]]: ~5/4 = 383.477
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}})


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


EDOs: 22h, 50, 72, [[122edo|122g]], [[194edo|194dfg]]
Badness (Sintel): 1.01


Badness: 0.0157
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==Gizzard==
Comma list: 169/168, 225/224, 364/363, 1716/1715
Commas: 225/224, 385/384, 325/324, 1573/1568


POTE generator: ~5/4 = 383.170
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]
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}})


EDOs: [[22edo|22]], [[26edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238cf]]
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.0203
Badness (Sintel): 0.922


===17-limit===
== Gracecordial ==
Commas: 225/224, 289/288, 325/324, 375/374, 385/384
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


POTE generator: ~5/4 = 383.175
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
[[Comma list]]: 225/224, 781250000/771895089


EDOs: 22f, 72, 94, [[166edo|166g]], [[238edo|238cfg]]
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


Badness: 0.0136
[[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 }}


===19-limit===
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399


POTE generator: ~5/4 = 383.138
[[Badness]] (Sintel): 2.44


Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 22f, 72, 94, 166g
Comma list: 225/224, 385/384, 236328125/234365481


Badness: 0.0148
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


==Mage==
Optimal tunings:
Commas: 99/98, 176/175, 1331/1296
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


POTE generator: ~55/48 = 216.876
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
Badness (Sintel): 2.96


EDOs: 22, 72e, 94e
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0578
Comma list: 225/224, 325/324, 385/384, 831875/830466


=Triton=
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
Commas: 225/224, 1029/1000


[[POTE_tuning|POTE temperament]]: ~7/5 = 568.865
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
Badness (Sintel): 2.16


EDOs: 17d, 19, 78bd, 97bd
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0592
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


==11-limit==
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
Commas: 45/44, 56/55, 1029/1000


POTE generator: ~7/5 = 569.144
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


Map: [&lt;1 0 6 7 4|, &lt;0 3 -7 -8 -1|]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


EDOs: 19, 59bde, 78bde, 97bde
Badness (Sintel): 1.96


Badness: 0.0457
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


=Tritonic=
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
Comnma: 553584375/536870912


POTE generator: ~45/32 = 580.219
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


Map: [&lt;1 4 -3|, &lt;0 -5 11|]
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


EDOs: 29, 31, 60, 91
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.3785
Badness (Sintel): 1.71


==7-limit==
==== 23-limit ====
Commas: 225/224, 50421/50000
Subgroup: 2.3.5.7.11.13.17.19.23


[[POTE_tuning|POTE temperament]]: 580.286
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


Map: [&lt;1 4 -3 -3|, &lt;0 -5 11 12|]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Wedgie: &lt;&lt;5 -11 -12 -29 -33 3||
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


EDOs: [[29edo|29]], [[31edo|31]], [[60edo|60]], [[91edo|91]], [[122edo|122]], 213bcd
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


==11-limit==
Badness (Sintel): 1.57
Commas: 121/120, 225/224, 441/440


[[POTE_tuning|POTE generator]]: ~7/5 = 580.267
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


EDOs: 29, 31, 60e
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


Badness: 0.0237
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}
Commas: 105/104, 121/120, 196/195, 275/273


POTE generator: ~7/5 = 580.108
Badness (Sintel): 1.50


Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


EDOs: 29, 31, 60e, 151cde
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


Badness: 0.0230
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


==Tritoni==
Optimal tunings:
Commas: 225/224, 385/384, 27783/27500
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


POTE generator: ~7/5 = 580.389
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
Badness (Sintel): 1.53


EDOs: 31, 91, 122
=== Gracecord ===
Subgroup: 2.3.5.7.11


Badness: 0.0455
Comma list: 225/224, 441/440, 109375/107811


=Merman=
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}
Comma: 1121008359375/1099511627776


POTE generator: ~45/32 = 585.634
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


Map: [&lt;1 5 -5|, &lt;0 -7 15|]
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


EDOs: 41, 84, 125, 209, 334c, 543bc
Badness (Sintel): 2.21


Badness: 0.4538
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


==7-limit==
Comma list: 225/224, 364/363, 441/440, 6125/6084
Commas: 225/224, 2500000/2470629


POTE generator: ~7/5 = 585.585
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Map: [&lt;1 5 -5 -5|, &lt;0 -7 15 16|]
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


EDOs: 41, 84, 125
Badness (Sintel): 1.83


Badness: 0.0551
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


==11-limit==
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
Commas: 225/224, 441/440, 1344/1331


POTE generator: ~7/5 = 585.606
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Map: [&lt;1 5 -5 -5 2|, &lt;0 -7 15 16 3]]
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


EDOs: 41, 84, 125e
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Badness: 0.0364
Badness (Sintel): 1.87


==13-limit==
==== 19-limit ====
Commas: 144/143, 225/224, 364/363, 441/440
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~7/5 = 585.657
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


Map: [&lt;1 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


EDOs: 41, 84, 125e, 209ef, 293ef
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


Badness: 0.0275
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


=Septimin=
Badness (Sintel): 1.68
Comma: 35595703125/34359738368


POTE generator: ~75/64 = 263.658
== Alphorn ==
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 4 1|, &lt;0 -11 6|]
[[Comma list]]: 225/224, 5764801/5668704


EDOs: 9, 32, 41, 91, 132, 223c, 355bc
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


Badness: 0.6709
[[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 }}


==7-limit==
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}
Commas: 225/224, 84035/82944


[[POTE_tuning|POTE temperament]]: 263.632
[[Badness]] (Sintel): 3.27


Map: [&lt;1 4 1 5|, &lt;0 -11 6 -10|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
Comma list: 225/224, 385/384, 12250/11979


EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


==11-limit==
Optimal tunings:
Commas: 225/224, 385/384, 2401/2376
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


[[POTE_tuning|POTE temperament]]: 263.634
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
Badness (Sintel): 2.43


EDOs: 9, 32, 41, 91, 223cdef
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 105/104, 144/143, 196/195, 245/242


[[POTE_tuning|POTE temperament]]: 263.700
[[Comma list]]: 225/224, 4194304/4117715


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


EDOs: 9, 41, 91
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7642{{c}}, ~15/8 = 1086.5513{{c}}
: [[error map]]: {{val| -0.236 -0.856 -1.569 +4.041 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1086.7633{{c}}
: error map: {{val| 0.000 -0.506 -0.999 +4.701 }}


=Slender=
{{Optimal ET sequence|legend=1| 21, 32, 53 }}
Commas: 225/224, 589824/588245


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.413
[[Badness]] (Sintel): 3.57


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
Comma list: 99/98, 176/175, 1310720/1294139


EDOs: 31, 94, 125
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


Badness: 0.0569
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}
Commas: 225/224, 385/384, 1331/1323


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.387
Badness (Sintel): 2.82


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 31, 63, 94, 125
Comma list: 99/98, 176/175, 640/637, 847/845


Badness: 25.342
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


==13-limit==
Optimal tunings:
Commas: 225/224, 275/273, 385/384, 1331/1323
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


POTE generator: ~49/48 = 38.314
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
Badness (Sintel): 1.88


EDOs: 31, 63, 94
=== Musneb ===
Subgroup: 2.3.5.7.11


Badness: 25.913
Comma list: 225/224, 385/384, 66550/64827


=Marvo=
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
Commas: 225/224, 78125000/78121827


[[POTE_tuning|POTE temperament]]: ~27/20 = 516.694
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Wedgie: &lt;&lt;6, 17, 46, 13, 56 59||
Badness (Sintel): 2.89


EDOs: 65, 72
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


==11-limit==
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.
Commas: 225/224, 243/242, 4000/3993


[[POTE_tuning|POTE temperament]]: ~27/20 = 516.699
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
[[Comma list]]: 225/224, 125000000/121060821


EDOs: 65, 72, 281, 497
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


==13-limit==
[[Optimal tuning]]s:
Commas: 243/242, 225/224, 351/350, 1625/1617
* [[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 }}


POTE generator: ~27/20 = 516.730
{{Optimal ET sequence|legend=1| 51, 53 }}


Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
[[Badness]] (Sintel): 3.64


EDOs: 72, 137, 209, 281f, 490bcf
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0269
Comma list: 121/120, 225/224, 22000/21609


=Marvolo=
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}
Commas: 225/224, 156250000/155649627


POTE generator: ~21/20 = 83.348
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 2 1 1|, &lt;0 -6 19 26|]
{{Optimal ET sequence|legend=0| 51, 53 }}


Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||
Badness (Sintel): 2.46


EDOs: 29, 43, 72, 619bcd, 691bcd
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0833
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: 225/224, 441/440, 4000/3993


POTE generator: ~21/20 = 83.340
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 2 1 1 2|, &lt;0 -6 19 26 21|]
{{Optimal ET sequence|legend=0| 51f, 53 }}


EDOs: 29, 43, 72
Badness (Sintel): 1.96


Badness: 0.0290
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.  


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 169/168, 225/224, 364/363, 441/440


POTE generator: ~21/20 = 83.330
[[Comma list]]: 225/224, 823543/800000


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


EDOs: 29, 43, 72, 115f
[[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: 0.0215
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


=Amavil=
[[Badness]] (Sintel): 3.67
Commas: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
Comma list: 225/224, 245/242, 1617/1600


Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


EDOs: 9, 34d, 43, 77d
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


Badness: 0.1096
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


==11-limit==
Badness (Sintel): 2.00
Commas: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
Comma list: 105/104, 196/195, 245/242, 1001/1000


EDOs: 9, 34d, 43, 77de
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


Badness: 0.0426
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: 78/77, 99/98, 144/143, 176/175


POTE generator: ~15/11 = 529.951
Badness (Sintel): 1.43


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


EDOs: 9, 34d, 43, 77de
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


Badness: 0.0258
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


=Enneaportent=
Optimal tunings:
Commas: 225/224, 40353607/40310784
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


POTE generator: ~5/4 = 383.165
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
Badness (Sintel): 1.26


Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


EDOs: 9, 54, 63, 72, 495bcd
[[Subgroup]]: 2.3.5.7


Badness: 0.0937
[[Comma list]]: 225/224, 9805926501/9765625000


==11-limit==
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
Commas: 225/224, 385/384, 12005/11979
: mapping generators: ~2, ~10000/9261


POTE generator: ~5/4 = 383.146
[[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 }}


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


EDOs: 9, 54, 63, 72, 423cd, 495bcd
[[Badness]] (Sintel): 3.81


Badness: 0.0304
=== 11-limit ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 225/224, 441/440, 43923/43750
Commas: 169/168, 225/224, 364/363, 1716/1715


POTE generator: ~5/4 = 383.047
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


EDOs: 9, 54, 63, 72, 279cf
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.0223
Badness (Sintel): 1.74


=Submajor=
=== 13-limit ===
Comma: 69198046875/68719476736
Subgroup: 2.3.5.7.11.13


POTE generator: ~10125/8192 = 362.321
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


Map: [&lt;1 4 -1|, &lt;0 -8 11|]
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


EDOs: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


Badness: 0.1302
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


==7-limit==
Badness (Sintel): 1.35
Commas: 225/224, 51200/50421


POTE generator: ~49/40 = 362.255
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 4 -1 1|, &lt;0 -8 11 6|]
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


EDOs: 10, 33, 43, 53
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


Badness: 0.0605
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


==11-limit==
Badness (Sintel): 1.17
Commas: 225/224, 385/384, 6655/6561


POTE generator: ~27/22 = 362.101
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Map: [&lt;1 4 -1 1 11|, &lt;0 -8 11 6 -25|]
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


EDOs: 10, 53, 116, 169de, 285cde
[[Subgroup]]: 2.3.5.7


Badness: 0.0506
[[Comma list]]: 225/224, 5971968/5764801


==13-limit==
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
Commas: 169/168, 225/224, 275/273, 385/384
: mapping generators: ~2401/1728, ~35/32


POTE generator: ~16/13 = 362.105
[[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 }}


Map: [&lt;1 4 -1 1 11 4|, &lt;0 -8 11 6 -25 -1|]
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


EDOs: 10, 53, 116, 169de, 285cdef
[[Badness]] (Sintel): 4.53


Badness: 0.0277
=== 11-limit ===
Subgroup: 2.3.5.7.11


==Interpental==
Comma list: 99/98, 176/175, 65536/65219
Commas: 99/98, 176/175, 51200/50421


POTE generator: ~49/40 = 362.418
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


Map: [&lt;1 4 -1 1 -5|, &lt;0 -8 11 6 28|]
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


EDOs: 43, 53, 96, 149d
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


Badness: 0.0518
Badness (Sintel): 2.26


===13-limit===
== Tertiosec ==
Commas: 99/98, 169/168, 176/175, 640/637
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


POTE generator: ~16/13 = 362.402
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>.  


Map: [&lt;1 4 -1 1 -5 4|, &lt;0 -8 11 6 28 -1|]
[[Subgroup]]: 2.3.5.7


EDOs: 43, 53, 96, 149d
[[Comma list]]: 225/224, 14495514624/13841287201


Badness: 0.0297
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


=Alphorn=
[[Optimal tuning]]s:
Commas: 225/224, 5764801/5668704
* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
: [[error map]]: {{val| -0.523 -1.044 -1.549 +5.138 }}
* [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}}
: error map: {{val| 0.000 -0.284 -0.276 +6.592 }}


POTE generator: ~48/35 = 556.221
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


Map: [&lt;1 9 0 13|, &lt;0 -16 5 -22|]
[[Badness]] (Sintel): 10.9


Wedgie: &lt;&lt;16 -5 22 -45 -10 65||
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 28d, 41, 151cd, 192cd, 233cd
Comma list: 225/224, 3840/3773, 12005/11979


Badness: 0.1293
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


==11-limit==
Optimal tunings:
Commas: 225/224, 385/384, 12250/11979
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}


POTE generator: ~11/8 = 556.144
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


Map: [&lt;1 9 0 13 3|, &lt;0 -16 5 -22 1|]
Badness (Sintel): 5.74


EDOs: 41, 315cde
== References ==


Badness: 0.0735
[[Category:Temperament collections]]
[[Category:marvel]]
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:overview]]
[[Category:Rank 2]]
[[Category:rank_2]]
[[Category:temperaments]]
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"