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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]]
* ''[[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]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* [[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, decic, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, hendeca, 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| 9, 23, 32, 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| 9, 23, 32, 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| 9, 32f, 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 589824/588245
 
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
: [[error map]]: {{val| +0.382 -0.725 -1.295 +1.765 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}}
: error map: {{val| 0.000 -1.257 -2.235 +0.727 }}
 
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}
 
[[Badness]] (Sintel): 1.44
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 1331/1323
 
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
 
Optimal tunings:
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}
 
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
 
Badness (Sintel): 0.838
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 275/273, 385/384, 1331/1323
 
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
 
Optimal tunings:
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}
 
{{Optimal ET sequence|legend=0| 31, 63, 94 }}
 
Badness (Sintel): 1.07
 
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''
 
Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 1029/1000
 
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}
: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}
 
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
 
[[Badness]] (Sintel): 1.50
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 56/55, 1029/1000
 
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
 
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}
 
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
 
Badness (Sintel): 1.51
 
== Marvolo ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 156250000/155649627
 
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7714{{c}}, ~21/20 = 83.4014{{c}}
: [[error map]]: {{val| +0.772 -0.820 -0.916 +0.381 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}}
: error map: {{val| 0.000 -2.139 -2.398 -1.362 }}
 
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}
 
[[Badness]] (Sintel): 2.11
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 4000/3993
 
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}
 
Optimal tunings:
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}
 
{{Optimal ET sequence|legend=0| 29, 43, 72 }}
 
Badness (Sintel): 0.958
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
 
Optimal tunings:
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}
 
{{Optimal ET sequence|legend=0| 29, 43, 72 }}
 
Badness (Sintel): 0.887
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
 
Optimal tunings:
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
 
Badness (Sintel): 0.760
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
 
Optimal tunings:
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
 
Badness (Sintel): 0.895
 
== Decic ==
{{Main| Decic }}
 
Named by [[Xenllium]] in 2021, decic tempers out 16807/16384, the [[cloudy comma]], and {{monzo| 11 -10 -10 10 }}, the [[linus comma]], in addition to the marvel comma. It may be described as the {{nowrap| 50 & 60 }} temperament, with a period of 1/10 octave and a [[ploidacot]] signature of decaploid monocot. It is [[support]]ed by [[10edo|10-]], [[50edo|50-]], and [[60edo]].
 
It can be extended to the 11-, 13-, and 17-limit by adding [[385/384]], [[105/104]], and [[170/169]] to the comma list in this order.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 16807/16384
 
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 120.1837{{c}}, ~3/2 = 699.7654{{c}} (~49/48 = 21.3366{{c}})
: [[error map]]: {{val| +1.837 -0.353 -0.753 -3.683 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 698.8236{{c}} (~49/48 = 21.1764{{c}})
: error map: {{val| 0.000 -3.131 -5.137 -8.826 }}
 
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}
 
[[Badness]] (Sintel): 2.26
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 3087/3025
 
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}
 
Optimal tunings:
* WE: ~15/14 = 120.1406{{c}}, ~3/2 = 697.6075{{c}} (~56/55 = 23.2360{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.0142{{c}} (~56/55 = 22.9858{{c}})
 
{{Optimal ET sequence|legend=0| 10, 40, 50 }}
 
Badness (Sintel): 2.11
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 196/195, 2200/2197


=Negri=
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}
[[Comma|Comma]]s: 49/48, 225/224


[[POTE_tuning|POTE temperament]]: ~15/14 = 125.608
Optimal tunings:
* WE: ~15/14 = 120.1166{{c}}, ~3/2 = 697.6705{{c}} (~78/77 = 23.0289{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.1492{{c}} (~78/77 = 22.8508{{c}})


[[Map|Map]]: [&lt;1 2 2 3|, &lt;0 -4 3 -2|]
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


[[wedgie|Wedgie]]: &lt;&lt;4 -3 2 -14 -8 13||
Badness (Sintel): 1.52


EDOs: [[9edo|9]], [[10edo|10]], [[19edo|19]], [[86edo|86]]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


==11-limit==
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220
Commas: 45/44, 49/48, 56/55


[[POTE_tuning|POTE generator]]: ~15/14 = 126.474
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


Map: [&lt;1 2 2 3 4|, &lt;0 -4 3 -2 -5|]
Optimal tunings:  
* WE: ~15/14 = 120.1262{{c}}, ~3/2 = 697.8185{{c}} (~78/77 = 22.9388{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.2757{{c}} (~78/77 = 22.7243{{c}})


EDOs: 9, 10, 19
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


[[Badness|Badness]]: 0.0262
Badness (Sintel): 1.28


==13-limit==
=== Splendecic ===
Commas: 45/44, 49/48, 56/55, 78/77
Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[fantastic]] rank-3 temperament, its name is a portmanteau of ''splendid'' and ''decic''.


[[POTE_tuning|POTE generator]]: ~14/13 = 126.431
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 2 3 4 4|, &lt;0 -4 3 -2 -5 -3|]
Comma list: 225/224, 1617/1600, 2401/2376


EDOs: 9, 10, 19
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}


=Negril=
Optimal tunings:
Commas: 49/48, 100/99, 225/224
* WE: ~15/14 = 120.1874{{c}}, ~3/2 = 699.6085{{c}} (~99/98 = 21.5156{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3531{{c}} (~99/98 = 21.6469{{c}})


POTE generator: ~15/14 = 124.767
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


Map: [&lt;1 2 2 3 2|, &lt;0 -4 3 -2 14|]
Badness (Sintel): 1.98


EDOs: 19, [[29edo|29]], [[48edo|48df]], [[77edo|77cdf]]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0387
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


==13-limit==
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}
Commas: 49/48, 65/64, 91/90, 875/858


POTE generator: ~15/14 = 124.716
Optimal tunings:
* WE: ~15/14 = 120.1565{{c}}, ~3/2 = 699.2756{{c}} (~91/90 = 21.6631{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.2480{{c}} (~91/90 = 21.7520{{c}})


Map: [&lt;1 2 2 3 2 4|, &lt;0 -4 3 -2 14 -3|]
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


EDOs: 9, 10, 19, 29, 48, 77
Badness (Sintel): 1.57


Badness: 0.0244
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


=Negric=
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374
Commas: 33/32, 49/48, 77/75


POTE generator: ~15/14 = 127.039
Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


Map: [&lt;1 2 2 3 3|, &lt;0 -4 3 -2 4|]
Optimal tunings:  
* WE: ~15/14 = 120.1571{{c}}, ~3/2 = 699.2892{{c}} (~91/90 = 21.6536{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3144{{c}} (~91/90 = 21.6856{{c}})


EDOs: 9, 19, [[85edo|85]]
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


Badness: 0.0306
Badness (Sintel): 1.33


==13-limit==
=== Prodecic ===
Commas: 33/32, 49/48, 65/64, 91/90
Prodecic (50e & 60e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the [[prodigy]] rank-3 temperament, its name is a portmanteau of ''prodigy'' and ''decic''.


POTE generator: ~15/14 = 127.039
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 2 3 3 4|, &lt;0 -4 3 -2 4 -3|]
Comma list: 225/224, 441/440, 5929/5832


EDOs: 9, 19, 85
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}


Badness: 0.0202
Optimal tunings:  
* WE: ~15/14 = 120.2024{{c}}, ~3/2 = 701.3908{{c}} (~55/54 = 19.8237{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.5235{{c}} (~55/54 = 19.4765{{c}})


=Negroni=
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
Commas: 49/48, 55/54, 225/224


POTE generator: ~15/14 = 124.539
Badness (Sintel): 2.20


Map: [&lt;1 2 2 3 5|, &lt;0 -4 3 -2 -15|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 10, 19e, 29, 77cde
Comma list: 105/104, 196/195, 275/273, 5929/5832


Badness: 0.0353
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


==13-limit==
Optimal tunings:
Commas: 49/48, 55/54, 65/64, 91/90
* WE: ~15/14 = 120.1654{{c}}, ~3/2 = 701.4683{{c}} (~91/90 = 19.5242{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.7175{{c}} (~91/90 = 19.2825{{c}})


POTE generator: ~15/14 = 124.545
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Map: [&lt;1 2 2 3 5 4|, &lt;0 -4 3 -2 -15 -3|]
Badness (Sintel): 1.73


EDOs: 10, 19e, 29, 77cdef
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0216
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


=Wizard=
Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}
Comma: 2197265625/2147483648


POTE generator: ~5/4 = 383.212
Optimal tunings:  
* WE: ~15/14 = 120.1577{{c}}, ~3/2 = 701.3950{{c}} (~91/90 = 19.5514{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.6932{{c}} (~91/90 = 19.3068{{c}})


Map: [&lt;2 1 5|, &lt;0 6 -1|]
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


EDOs: 22, 50, 72, 166, 238c, 310c, 548bc
Badness (Sintel): 1.41


Badness: 0.3864
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


==7-limit==
[[Comma list]]: 225/224, 40353607/40310784
Commas: 225/224, 118098/117649


[[POTE_tuning|POTE temperament]]: 383.256
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


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


Wedgie: &lt;&lt;12 -2 20 -31 -2 52||
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


EDOs: [[22edo|22]], [[50edo|50]], [[72edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238c]], [[310edo|310c]], [[382edo|382c]]
[[Badness]] (Sintel): 2.37


Badness: 0.0408
=== 11-limit ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 225/224, 385/384, 12005/11979
Commas: 225/224, 385/384, 4000/3993


[[POTE_tuning|POTE temperament]]: 383.232
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


Map: [&lt;2 1 5 2 8|, &lt;0 6 -1 10 -3|]
Optimal tunings:  
* WE: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}})
* CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}})


EDOs: 22, 50, 72, 94, [[166edo|166]], [[238edo|238c]], [[310edo|310c]]
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.0185
Badness (Sintel): 1.01


==Lizard==
=== 13-limit ===
Commas: 225/224, 351/350, 364/363, 385/384
Subgroup: 2.3.5.7.11.13


[[POTE_tuning|POTE temperament]]: 383.389
Comma list: 169/168, 225/224, 364/363, 1716/1715


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


EDOs: 22, 50, 72, 122, 194df
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: 0.0218
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


===17-limit===
Badness (Sintel): 0.922
Commas: 221/220, 273/272, 289/288, 351/350, 375/374


[[POTE_tuning|POTE temperament]]: 383.381
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


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


EDOs: 22, 50, 72, 122g, 194dfg
[[Comma list]]: 225/224, 781250000/771895089


Badness: 0.0145
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


===19-limit===
[[Optimal tuning]]s:
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


[[POTE_tuning|POTE generator]]: ~5/4 = 383.477
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
[[Badness]] (Sintel): 2.44


EDOs: 22h, 50, 72, [[122edo|122g]], [[194edo|194dfg]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0157
Comma list: 225/224, 385/384, 236328125/234365481


==Gizzard==
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
Commas: 225/224, 385/384, 325/324, 1573/1568


POTE generator: ~5/4 = 383.170
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 8 -2|, &lt;0 6 -1 10 -3 26|]
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


EDOs: [[22edo|22]], [[26edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238cf]]
Badness (Sintel): 2.96


Badness: 0.0203
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


===17-limit===
Comma list: 225/224, 325/324, 385/384, 831875/830466
Commas: 225/224, 289/288, 325/324, 375/374, 385/384


POTE generator: ~5/4 = 383.175
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


EDOs: 22f, 72, 94, [[166edo|166g]], [[238edo|238cfg]]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.0136
Badness (Sintel): 2.16


===19-limit===
==== 17-limit ====
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~5/4 = 383.138
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


EDOs: 22f, 72, 94, 166g
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


Badness: 0.0148
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


==Mage==
Badness (Sintel): 1.96
Commas: 99/98, 176/175, 1331/1296


POTE generator: ~55/48 = 216.876
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


EDOs: 22, 72e, 94e
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


Badness: 0.0578
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


=Triton=
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
Commas: 225/224, 1029/1000


[[POTE_tuning|POTE temperament]]: ~7/5 = 568.865
Badness (Sintel): 1.71


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


EDOs: 17d, 19, 78bd, 97bd
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Badness: 0.0592
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: 45/44, 56/55, 1029/1000


POTE generator: ~7/5 = 569.144
Badness (Sintel): 1.57


Map: [&lt;1 0 6 7 4|, &lt;0 3 -7 -8 -1|]
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


EDOs: 19, 59bde, 78bde, 97bde
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Badness: 0.0457
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


=Tritonic=
Optimal tunings:
Comnma: 553584375/536870912
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


POTE generator: ~45/32 = 580.219
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


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


EDOs: 29, 31, 60, 91
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Badness: 0.3785
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


==7-limit==
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, 50421/50000


[[POTE_tuning|POTE temperament]]: 580.286
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|, &lt;0 -5 11 12|]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Wedgie: &lt;&lt;5 -11 -12 -29 -33 3||
Badness (Sintel): 1.53


EDOs: [[29edo|29]], [[31edo|31]], [[60edo|60]], [[91edo|91]], [[122edo|122]], 213bcd
=== Gracecord ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 225/224, 441/440, 109375/107811
Commas: 121/120, 225/224, 441/440


[[POTE_tuning|POTE generator]]: ~7/5 = 580.267
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


EDOs: 29, 31, 60e
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Badness: 0.0237
Badness (Sintel): 2.21


==13-limit==
==== 13-limit ====
Commas: 105/104, 121/120, 196/195, 275/273
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 580.108
Comma list: 225/224, 364/363, 441/440, 6125/6084


Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


EDOs: 29, 31, 60e, 151cde
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


Badness: 0.0230
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


==Tritoni==
Badness (Sintel): 1.83
Commas: 225/224, 385/384, 27783/27500


POTE generator: ~7/5 = 580.389
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


EDOs: 31, 91, 122
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Badness: 0.0455
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


=Merman=
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
Comma: 1121008359375/1099511627776


POTE generator: ~45/32 = 585.634
Badness (Sintel): 1.87


Map: [&lt;1 5 -5|, &lt;0 -7 15|]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


EDOs: 41, 84, 125, 209, 334c, 543bc
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


Badness: 0.4538
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


==7-limit==
Optimal tunings:
Commas: 225/224, 2500000/2470629
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


POTE generator: ~7/5 = 585.585
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Map: [&lt;1 5 -5 -5|, &lt;0 -7 15 16|]
Badness (Sintel): 1.68


Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
== Alphorn ==
[[Subgroup]]: 2.3.5.7


EDOs: 41, 84, 125
[[Comma list]]: 225/224, 5764801/5668704


Badness: 0.0551
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


==11-limit==
[[Optimal tuning]]s:
Commas: 225/224, 441/440, 1344/1331
* [[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 generator: ~7/5 = 585.606
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Map: [&lt;1 5 -5 -5 2|, &lt;0 -7 15 16 3]]
[[Badness]] (Sintel): 3.27


EDOs: 41, 84, 125e
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0364
Comma list: 225/224, 385/384, 12250/11979


==13-limit==
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}
Commas: 144/143, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 585.657
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 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


EDOs: 41, 84, 125e, 209ef, 293ef
Badness (Sintel): 2.43


Badness: 0.0275
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


=Septimin=
[[Subgroup]]: 2.3.5.7
Comma: 35595703125/34359738368


POTE generator: ~75/64 = 263.658
[[Comma list]]: 225/224, 4194304/4117715


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


EDOs: 9, 32, 41, 91, 132, 223c, 355bc
[[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 }}


Badness: 0.6709
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


==7-limit==
[[Badness]] (Sintel): 3.57
Commas: 225/224, 84035/82944


[[POTE_tuning|POTE temperament]]: 263.632
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 4 1 5|, &lt;0 -11 6 -10|]
Comma list: 99/98, 176/175, 1310720/1294139


Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]
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, 2401/2376


[[POTE_tuning|POTE temperament]]: 263.634
Badness (Sintel): 2.82


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 9, 32, 41, 91, 223cdef
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: 105/104, 144/143, 196/195, 245/242


[[POTE_tuning|POTE temperament]]: 263.700
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 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


EDOs: 9, 41, 91
Badness (Sintel): 1.88


=Slender=
=== Musneb ===
Commas: 225/224, 589824/588245
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.413
Comma list: 225/224, 385/384, 66550/64827


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


EDOs: 31, 94, 125
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Badness: 0.0569
Badness (Sintel): 2.89


==11-limit==
== Untriton ==
Commas: 225/224, 385/384, 1331/1323
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.387
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 2 2 3 4|, &lt;0 -13 10 -6 -17|]
[[Subgroup]]: 2.3.5.7


EDOs: 31, 63, 94, 125
[[Comma list]]: 225/224, 125000000/121060821


Badness: 25.342
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


==13-limit==
[[Optimal tuning]]s:
Commas: 225/224, 275/273, 385/384, 1331/1323
* [[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: ~49/48 = 38.314
{{Optimal ET sequence|legend=1| 51, 53 }}


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
[[Badness]] (Sintel): 3.64


EDOs: 31, 63, 94
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 25.913
Comma list: 121/120, 225/224, 22000/21609


=Marvo=
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}
Commas: 225/224, 78125000/78121827


[[POTE_tuning|POTE temperament]]: ~27/20 = 516.694
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 5 12 29|, &lt;0 -6 -17 -46|]
{{Optimal ET sequence|legend=0| 51, 53 }}


Wedgie: &lt;&lt;6, 17, 46, 13, 56 59||
Badness (Sintel): 2.46


EDOs: 65, 72
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 121/120, 225/224, 275/273, 1040/1029
Commas: 225/224, 243/242, 4000/3993


[[POTE_tuning|POTE temperament]]: ~27/20 = 516.699
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


EDOs: 65, 72, 281, 497
{{Optimal ET sequence|legend=0| 51f, 53 }}


==13-limit==
Badness (Sintel): 1.96
Commas: 243/242, 225/224, 351/350, 1625/1617


POTE generator: ~27/20 = 516.730
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.  


Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
[[Subgroup]]: 2.3.5.7


EDOs: 72, 137, 209, 281f, 490bcf
[[Comma list]]: 225/224, 823543/800000


Badness: 0.0269
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


=Marvolo=
[[Optimal tuning]]s:
Commas: 225/224, 156250000/155649627
* [[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 }}


POTE generator: ~21/20 = 83.348
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


Map: [&lt;1 2 1 1|, &lt;0 -6 19 26|]
[[Badness]] (Sintel): 3.67


Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 29, 43, 72, 619bcd, 691bcd
Comma list: 225/224, 245/242, 1617/1600


Badness: 0.0833
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


==11-limit==
Optimal tunings:
Commas: 225/224, 441/440, 4000/3993
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


POTE generator: ~21/20 = 83.340
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Map: [&lt;1 2 1 1 2|, &lt;0 -6 19 26 21|]
Badness (Sintel): 2.00


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


Badness: 0.0290
Comma list: 105/104, 196/195, 245/242, 1001/1000


==13-limit==
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}
Commas: 169/168, 225/224, 364/363, 441/440


POTE generator: ~21/20 = 83.330
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


EDOs: 29, 43, 72, 115f
Badness (Sintel): 1.43


Badness: 0.0215
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


=Amavil=
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242
Commas: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


EDOs: 9, 34d, 43, 77d
Badness (Sintel): 1.26


Badness: 0.1096
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.  


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
[[Comma list]]: 225/224, 9805926501/9765625000


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


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


Badness: 0.0426
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


==13-limit==
[[Badness]] (Sintel): 3.81
Commas: 78/77, 99/98, 144/143, 176/175


POTE generator: ~15/11 = 529.951
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
Comma list: 225/224, 441/440, 43923/43750


EDOs: 9, 34d, 43, 77de
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


Badness: 0.0258
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


=Enneaportent=
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}
Commas: 225/224, 40353607/40310784


POTE generator: ~5/4 = 383.165
Badness (Sintel): 1.74


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


EDOs: 9, 54, 63, 72, 495bcd
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


Badness: 0.0937
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
Commas: 225/224, 385/384, 12005/11979


POTE generator: ~5/4 = 383.146
Badness (Sintel): 1.35


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


EDOs: 9, 54, 63, 72, 423cd, 495bcd
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Badness: 0.0304
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


==13-limit==
Optimal tunings:
Commas: 169/168, 225/224, 364/363, 1716/1715
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


POTE generator: ~5/4 = 383.047
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
Badness (Sintel): 1.17


EDOs: 9, 54, 63, 72, 279cf
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


Badness: 0.0223
Hendeca tempers out the same 5-limit comma as [[porwell temperaments #Hendecatonic|hendecatonic]], and has a period of 1/11 octave. However, in this temperament, nine periods represent [[7/4]], the same as [[keemic temperaments #Undeka|undeka]]. It can be tuned to [[22edo]] or [[33edo]] using the [[patent val]], or [[55edo]] using the 55d val. It was named by [[Xenllium]] in 2025 as a low-accuracy variant of hendecatonic.  


=Submajor=
[[Subgroup]]: 2.3.5.7
Comma: 69198046875/68719476736


POTE generator: ~10125/8192 = 362.321
[[Comma list]]: 225/224, 122880/117649


Map: [&lt;1 4 -1|, &lt;0 -8 11|]
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


EDOs: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc
[[Optimal tuning]]s:  
* [[WE]]: ~16/15 = 108.9526{{c}}, ~3/2 = 702.4215{{c}}
: [[error map]]: {{val| -1.521 -1.055 -2.252 +8.705 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 703.2071{{c}}
: error map: {{val| 0.000 +1.252 +1.388 +12.992 }}


Badness: 0.1302
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}


==7-limit==
[[Badness]] (Sintel): 4.37
Commas: 225/224, 51200/50421


POTE generator: ~49/40 = 362.255
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 4 -1 1|, &lt;0 -8 11 6|]
Comma list: 121/120, 225/224, 352/343


Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}


EDOs: 10, 33, 43, 53
Optimal tunings:  
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


Badness: 0.0605
{{Optimal ET sequence|legend=0| 22, 55d }}


==11-limit==
Badness (Sintel): 2.46
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]]
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