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__FORCETOC__
{{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]], [[Magic_family|magic]], [[pajara]], [[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.


=Negri=
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
[[Comma|Comma]]s: 49/48, 225/224


[[POTE_tuning|POTE temperament]]: ~15/14 = 125.608
[[Subgroup]]: 2.3.5.7


[[Map|Map]]: [&lt;1 2 2 3|, &lt;0 -4 3 -2|]
[[Comma list]]: 225/224, 589824/588245


[[wedgie|Wedgie]]: &lt;&lt;4 -3 2 -14 -8 13||
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


EDOs: [[9edo|9]], [[10edo|10]], [[19edo|19]], [[86edo|86]]
[[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 }}


==11-limit==
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}
Commas: 45/44, 49/48, 56/55


[[POTE_tuning|POTE generator]]: ~15/14 = 126.474
[[Badness]] (Sintel): 1.44


Map: [&lt;1 2 2 3 4|, &lt;0 -4 3 -2 -5|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


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


[[Badness|Badness]]: 0.0262
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


==13-limit==
Optimal tunings:
Commas: 45/44, 49/48, 56/55, 78/77
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


[[POTE_tuning|POTE generator]]: ~14/13 = 126.431
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Map: [&lt;1 2 2 3 4 4|, &lt;0 -4 3 -2 -5 -3|]
Badness (Sintel): 0.838


EDOs: 9, 10, 19
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=Negril=
Comma list: 225/224, 275/273, 385/384, 1331/1323
Commas: 49/48, 100/99, 225/224


POTE generator: ~15/14 = 124.767
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Map: [&lt;1 2 2 3 2|, &lt;0 -4 3 -2 14|]
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


EDOs: 19, [[29edo|29]], [[48edo|48df]], [[77edo|77cdf]]
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Badness: 0.0387
Badness (Sintel): 1.07


==13-limit==
== Triton ==
Commas: 49/48, 65/64, 91/90, 875/858
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


POTE generator: ~15/14 = 124.716
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.  


Map: [&lt;1 2 2 3 2 4|, &lt;0 -4 3 -2 14 -3|]
[[Subgroup]]: 2.3.5.7


EDOs: 9, 10, 19, 29, 48, 77
[[Comma list]]: 225/224, 1029/1000


Badness: 0.0244
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


=Negric=
[[Optimal tuning]]s:
Commas: 33/32, 49/48, 77/75
* [[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 }}


POTE generator: ~15/14 = 127.039
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


Map: [&lt;1 2 2 3 3|, &lt;0 -4 3 -2 4|]
[[Badness]] (Sintel): 1.50


EDOs: 9, 19, [[85edo|85]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0306
Comma list: 45/44, 56/55, 1029/1000


==13-limit==
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
Commas: 33/32, 49/48, 65/64, 91/90


POTE generator: ~15/14 = 127.039
Optimal tunings:  
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


Map: [&lt;1 2 2 3 3 4|, &lt;0 -4 3 -2 4 -3|]
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


EDOs: 9, 19, 85
Badness (Sintel): 1.51


Badness: 0.0202
== Marvolo ==
[[Subgroup]]: 2.3.5.7


=Negroni=
[[Comma list]]: 225/224, 156250000/155649627
Commas: 49/48, 55/54, 225/224


POTE generator: ~15/14 = 124.539
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


Map: [&lt;1 2 2 3 5|, &lt;0 -4 3 -2 -15|]
[[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 }}


EDOs: 10, 19e, 29, 77cde
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


Badness: 0.0353
[[Badness]] (Sintel): 2.11


==13-limit==
=== 11-limit ===
Commas: 49/48, 55/54, 65/64, 91/90
Subgroup: 2.3.5.7.11


POTE generator: ~15/14 = 124.545
Comma list: 225/224, 441/440, 4000/3993


Map: [&lt;1 2 2 3 5 4|, &lt;0 -4 3 -2 -15 -3|]
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


EDOs: 10, 19e, 29, 77cdef
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


Badness: 0.0216
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


=Wizard=
Badness (Sintel): 0.958
Comma: 2197265625/2147483648


POTE generator: ~5/4 = 383.212
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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


EDOs: 22, 50, 72, 166, 238c, 310c, 548bc
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Badness: 0.3864
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


==7-limit==
{{Optimal ET sequence|legend=0| 29, 43, 72 }}
Commas: 225/224, 118098/117649


[[POTE_tuning|POTE temperament]]: 383.256
Badness (Sintel): 0.887


Map: [&lt;2 1 5 2|, &lt;0 6 -1 10|]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Wedgie: &lt;&lt;12 -2 20 -31 -2 52||
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


EDOs: [[22edo|22]], [[50edo|50]], [[72edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238c]], [[310edo|310c]], [[382edo|382c]]
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


Badness: 0.0408
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
Commas: 225/224, 385/384, 4000/3993


[[POTE_tuning|POTE temperament]]: 383.232
Badness (Sintel): 0.760


Map: [&lt;2 1 5 2 8|, &lt;0 6 -1 10 -3|]
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


EDOs: 22, 50, 72, 94, [[166edo|166]], [[238edo|238c]], [[310edo|310c]]
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Badness: 0.0185
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


==Lizard==
Optimal tunings:
Commas: 225/224, 351/350, 364/363, 385/384
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


[[POTE_tuning|POTE temperament]]: 383.389
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
Badness (Sintel): 0.895


EDOs: 22, 50, 72, 122, 194df
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


Badness: 0.0218
[[Comma list]]: 225/224, 40353607/40310784


===17-limit===
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
Commas: 221/220, 273/272, 289/288, 351/350, 375/374
: mapping generators: ~2592/2401, ~12005/6912


[[POTE_tuning|POTE temperament]]: 383.381
[[Optimal tuning]]s:
* [[WE]]: ~2592/2401 = 133.4174{{c}}, ~12005/6912 = 950.7667{{c}} (~1728/1715 = 16.8452{{c}})
: [[error map]]: {{val| +0.756 -0.422 -1.395 +0.298 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}})
: error map: {{val| 0.000 -1.361 -3.277 -1.565 }}


Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


EDOs: 22, 50, 72, 122g, 194dfg
[[Badness]] (Sintel): 2.37


Badness: 0.0145
=== 11-limit ===
Subgroup: 2.3.5.7.11


===19-limit===
Comma list: 225/224, 385/384, 12005/11979
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


[[POTE_tuning|POTE generator]]: ~5/4 = 383.477
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
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}})


EDOs: 22h, 50, 72, [[122edo|122g]], [[194edo|194dfg]]
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.0157
Badness (Sintel): 1.01


==Gizzard==
=== 13-limit ===
Commas: 225/224, 385/384, 325/324, 1573/1568
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 383.170
Comma list: 169/168, 225/224, 364/363, 1716/1715


Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


EDOs: [[22edo|22]], [[26edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238cf]]
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}})


Badness: 0.0203
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


===17-limit===
Badness (Sintel): 0.922
Commas: 225/224, 289/288, 325/324, 375/374, 385/384


POTE generator: ~5/4 = 383.175
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
[[Subgroup]]: 2.3.5.7


EDOs: 22f, 72, 94, [[166edo|166g]], [[238edo|238cfg]]
[[Comma list]]: 225/224, 781250000/771895089


Badness: 0.0136
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


===19-limit===
[[Optimal tuning]]s:
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399
* [[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 }}


POTE generator: ~5/4 = 383.138
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]
[[Badness]] (Sintel): 2.44


EDOs: 22f, 72, 94, 166g
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0148
Comma list: 225/224, 385/384, 236328125/234365481


==Mage==
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
Commas: 99/98, 176/175, 1331/1296


POTE generator: ~55/48 = 216.876
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


EDOs: 22, 72e, 94e
Badness (Sintel): 2.96


Badness: 0.0578
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=Triton=
Comma list: 225/224, 325/324, 385/384, 831875/830466
Commas: 225/224, 1029/1000


[[POTE_tuning|POTE temperament]]: ~7/5 = 568.865
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


EDOs: 17d, 19, 78bd, 97bd
Badness (Sintel): 2.16


Badness: 0.0592
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


==11-limit==
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
Commas: 45/44, 56/55, 1029/1000


POTE generator: ~7/5 = 569.144
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


Map: [&lt;1 0 6 7 4|, &lt;0 3 -7 -8 -1|]
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


EDOs: 19, 59bde, 78bde, 97bde
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.0457
Badness (Sintel): 1.96


=Tritonic=
==== 19-limit ====
Comnma: 553584375/536870912
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~45/32 = 580.219
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


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


EDOs: 29, 31, 60, 91
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


Badness: 0.3785
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


==7-limit==
Badness (Sintel): 1.71
Commas: 225/224, 50421/50000


[[POTE_tuning|POTE temperament]]: 580.286
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Map: [&lt;1 4 -3 -3|, &lt;0 -5 11 12|]
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


Wedgie: &lt;&lt;5 -11 -12 -29 -33 3||
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


EDOs: [[29edo|29]], [[31edo|31]], [[60edo|60]], [[91edo|91]], [[122edo|122]], 213bcd
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}
Commas: 121/120, 225/224, 441/440


[[POTE_tuning|POTE generator]]: ~7/5 = 580.267
Badness (Sintel): 1.57


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


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


Badness: 0.0237
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


==13-limit==
Optimal tunings:
Commas: 105/104, 121/120, 196/195, 275/273
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


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


Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
Badness (Sintel): 1.50


EDOs: 29, 31, 60e, 151cde
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Badness: 0.0230
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


==Tritoni==
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}
Commas: 225/224, 385/384, 27783/27500


POTE generator: ~7/5 = 580.389
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


EDOs: 31, 91, 122
Badness (Sintel): 1.53


Badness: 0.0455
=== Gracecord ===
Subgroup: 2.3.5.7.11


=Merman=
Comma list: 225/224, 441/440, 109375/107811
Comma: 1121008359375/1099511627776


POTE generator: ~45/32 = 585.634
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


Map: [&lt;1 5 -5|, &lt;0 -7 15|]
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


EDOs: 41, 84, 125, 209, 334c, 543bc
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Badness: 0.4538
Badness (Sintel): 2.21


==7-limit==
==== 13-limit ====
Commas: 225/224, 2500000/2470629
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 585.585
Comma list: 225/224, 364/363, 441/440, 6125/6084


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


Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


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


Badness: 0.0551
Badness (Sintel): 1.83


==11-limit==
==== 17-limit ====
Commas: 225/224, 441/440, 1344/1331
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~7/5 = 585.606
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


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


EDOs: 41, 84, 125e
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


Badness: 0.0364
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


==13-limit==
Badness (Sintel): 1.87
Commas: 144/143, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 585.657
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


EDOs: 41, 84, 125e, 209ef, 293ef
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


Badness: 0.0275
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


=Septimin=
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
Comma: 35595703125/34359738368


POTE generator: ~75/64 = 263.658
Badness (Sintel): 1.68


Map: [&lt;1 4 1|, &lt;0 -11 6|]
== Alphorn ==
[[Subgroup]]: 2.3.5.7


EDOs: 9, 32, 41, 91, 132, 223c, 355bc
[[Comma list]]: 225/224, 5764801/5668704


Badness: 0.6709
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


==7-limit==
[[Optimal tuning]]s:
Commas: 225/224, 84035/82944
* [[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 }}


[[POTE_tuning|POTE temperament]]: 263.632
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Map: [&lt;1 4 1 5|, &lt;0 -11 6 -10|]
[[Badness]] (Sintel): 3.27


Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]
Comma list: 225/224, 385/384, 12250/11979


==11-limit==
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}
Commas: 225/224, 385/384, 2401/2376


[[POTE_tuning|POTE temperament]]: 263.634
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


EDOs: 9, 32, 41, 91, 223cdef
Badness (Sintel): 2.43


==13-limit==
== Misneb ==
Commas: 105/104, 144/143, 196/195, 245/242
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


[[POTE_tuning|POTE temperament]]: 263.700
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
[[Comma list]]: 225/224, 4194304/4117715


EDOs: 9, 41, 91
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


=Slender=
[[Optimal tuning]]s:
Commas: 225/224, 589824/588245
* [[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_tuning|POTE temperament]]: ~49/48 = 38.413
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
[[Badness]] (Sintel): 3.57


Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 31, 94, 125
Comma list: 99/98, 176/175, 1310720/1294139


Badness: 0.0569
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


==11-limit==
Optimal tunings:
Commas: 225/224, 385/384, 1331/1323
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.387
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
Badness (Sintel): 2.82


EDOs: 31, 63, 94, 125
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 25.342
Comma list: 99/98, 176/175, 640/637, 847/845


==13-limit==
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}
Commas: 225/224, 275/273, 385/384, 1331/1323


POTE generator: ~49/48 = 38.314
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


EDOs: 31, 63, 94
Badness (Sintel): 1.88


Badness: 25.913
=== Musneb ===
Subgroup: 2.3.5.7.11


=Marvo=
Comma list: 225/224, 385/384, 66550/64827
Commas: 225/224, 78125000/78121827


[[POTE_tuning|POTE temperament]]: ~27/20 = 516.694
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


Wedgie: &lt;&lt;6, 17, 46, 13, 56 59||
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


EDOs: 65, 72
Badness (Sintel): 2.89


==11-limit==
== Untriton ==
Commas: 225/224, 243/242, 4000/3993
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


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


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
[[Subgroup]]: 2.3.5.7


EDOs: 65, 72, 281, 497
[[Comma list]]: 225/224, 125000000/121060821


==13-limit==
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
Commas: 243/242, 225/224, 351/350, 1625/1617
: mapping generators: ~2, ~10/7


POTE generator: ~27/20 = 516.730
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
: [[error map]]: {{val| -0.172 +0.002 -2.533 +3.511 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}}
: error map: {{val| 0.000 +0.298 -2.181 +3.946 }}


Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
{{Optimal ET sequence|legend=1| 51, 53 }}


EDOs: 72, 137, 209, 281f, 490bcf
[[Badness]] (Sintel): 3.64


Badness: 0.0269
=== 11-limit ===
Subgroup: 2.3.5.7.11


=Marvolo=
Comma list: 121/120, 225/224, 22000/21609
Commas: 225/224, 156250000/155649627


POTE generator: ~21/20 = 83.348
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


Map: [&lt;1 2 1 1|, &lt;0 -6 19 26|]
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||
{{Optimal ET sequence|legend=0| 51, 53 }}


EDOs: 29, 43, 72, 619bcd, 691bcd
Badness (Sintel): 2.46


Badness: 0.0833
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 121/120, 225/224, 275/273, 1040/1029
Commas: 225/224, 441/440, 4000/3993


POTE generator: ~21/20 = 83.340
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


Map: [&lt;1 2 1 1 2|, &lt;0 -6 19 26 21|]
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


EDOs: 29, 43, 72
{{Optimal ET sequence|legend=0| 51f, 53 }}


Badness: 0.0290
Badness (Sintel): 1.96


==13-limit==
== Naiadical ==
Commas: 169/168, 225/224, 364/363, 441/440
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


POTE generator: ~21/20 = 83.330
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
[[Comma list]]: 225/224, 823543/800000


EDOs: 29, 43, 72, 115f
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


Badness: 0.0215
[[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 }}


=Amavil=
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}
Commas: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
[[Badness]] (Sintel): 3.67


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
Comma list: 225/224, 245/242, 1617/1600


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


Badness: 0.1096
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
Commas: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
Badness (Sintel): 2.00


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 9, 34d, 43, 77de
Comma list: 105/104, 196/195, 245/242, 1001/1000


Badness: 0.0426
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


==13-limit==
Optimal tunings:
Commas: 78/77, 99/98, 144/143, 176/175
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


POTE generator: ~15/11 = 529.951
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
Badness (Sintel): 1.43


EDOs: 9, 34d, 43, 77de
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0258
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


=Enneaportent=
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}
Commas: 225/224, 40353607/40310784


POTE generator: ~5/4 = 383.165
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
Badness (Sintel): 1.26


EDOs: 9, 54, 63, 72, 495bcd
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


Badness: 0.0937
[[Subgroup]]: 2.3.5.7


==11-limit==
[[Comma list]]: 225/224, 9805926501/9765625000
Commas: 225/224, 385/384, 12005/11979


POTE generator: ~5/4 = 383.146
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
[[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 }}


EDOs: 9, 54, 63, 72, 423cd, 495bcd
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


Badness: 0.0304
[[Badness]] (Sintel): 3.81


==13-limit==
=== 11-limit ===
Commas: 169/168, 225/224, 364/363, 1716/1715
Subgroup: 2.3.5.7.11


POTE generator: ~5/4 = 383.047
Comma list: 225/224, 441/440, 43923/43750


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


EDOs: 9, 54, 63, 72, 279cf
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Badness: 0.0223
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


=Submajor=
Badness (Sintel): 1.74
Comma: 69198046875/68719476736


POTE generator: ~10125/8192 = 362.321
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 4 -1|, &lt;0 -8 11|]
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


EDOs: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


Badness: 0.1302
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


==7-limit==
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
Commas: 225/224, 51200/50421


POTE generator: ~49/40 = 362.255
Badness (Sintel): 1.35


Map: [&lt;1 4 -1 1|, &lt;0 -8 11 6|]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


EDOs: 10, 33, 43, 53
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


Badness: 0.0605
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
Commas: 225/224, 385/384, 6655/6561


POTE generator: ~27/22 = 362.101
Badness (Sintel): 1.17


Map: [&lt;1 4 -1 1 11|, &lt;0 -8 11 6 -25|]
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


EDOs: 10, 53, 116, 169de, 285cde
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


Badness: 0.0506
[[Subgroup]]: 2.3.5.7


==13-limit==
[[Comma list]]: 225/224, 5971968/5764801
Commas: 169/168, 225/224, 275/273, 385/384


POTE generator: ~16/13 = 362.105
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


Map: [&lt;1 4 -1 1 11 4|, &lt;0 -8 11 6 -25 -1|]
[[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 }}


EDOs: 10, 53, 116, 169de, 285cdef
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


Badness: 0.0277
[[Badness]] (Sintel): 4.53


==Interpental==
=== 11-limit ===
Commas: 99/98, 176/175, 51200/50421
Subgroup: 2.3.5.7.11


POTE generator: ~49/40 = 362.418
Comma list: 99/98, 176/175, 65536/65219


Map: [&lt;1 4 -1 1 -5|, &lt;0 -8 11 6 28|]
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


EDOs: 43, 53, 96, 149d
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


Badness: 0.0518
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


===13-limit===
Badness (Sintel): 2.26
Commas: 99/98, 169/168, 176/175, 640/637


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