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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]]
* ''[[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


=Negri=
Comma list: 105/104, 144/143, 196/195, 2200/2197
[[Comma|Comma]]s: 49/48, 225/224


[[POTE_tuning|POTE temperament]]: ~15/14 = 125.608
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


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


[[wedgie|Wedgie]]: &lt;&lt;4 -3 2 -14 -8 13||
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


EDOs: [[9edo|9]], [[10edo|10]], [[19edo|19]], [[86edo|86]]
Badness (Sintel): 1.52


==11-limit==
==== 17-limit ====
Commas: 45/44, 49/48, 56/55
Subgroup: 2.3.5.7.11.13.17


[[POTE_tuning|POTE generator]]: ~15/14 = 126.474
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


Map: [&lt;1 2 2 3 4|, &lt;0 -4 3 -2 -5|]
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


EDOs: 9, 10, 19
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}})


[[Badness|Badness]]: 0.0262
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


==13-limit==
Badness (Sintel): 1.28
Commas: 45/44, 49/48, 56/55, 78/77


[[POTE_tuning|POTE generator]]: ~14/13 = 126.431
=== Splendecic ===
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''.


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


EDOs: 9, 10, 19
Comma list: 225/224, 1617/1600, 2401/2376


=Negril=
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}
Commas: 49/48, 100/99, 225/224


POTE generator: ~15/14 = 124.767
Optimal tunings:
* 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}})


Map: [&lt;1 2 2 3 2|, &lt;0 -4 3 -2 14|]
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


EDOs: 19, [[29edo|29]], [[48edo|48df]], [[77edo|77cdf]]
Badness (Sintel): 1.98


Badness: 0.0387
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 105/104, 196/195, 1001/1000, 1188/1183
Commas: 49/48, 65/64, 91/90, 875/858


POTE generator: ~15/14 = 124.716
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


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


EDOs: 9, 10, 19, 29, 48, 77
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


Badness: 0.0244
Badness (Sintel): 1.57


=Negric=
==== 17-limit ====
Commas: 33/32, 49/48, 77/75
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~15/14 = 127.039
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374


Map: [&lt;1 2 2 3 3|, &lt;0 -4 3 -2 4|]
Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


EDOs: 9, 19, [[85edo|85]]
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}})


Badness: 0.0306
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


==13-limit==
Badness (Sintel): 1.33
Commas: 33/32, 49/48, 65/64, 91/90


POTE generator: ~15/14 = 127.039
=== Prodecic ===
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''.


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


EDOs: 9, 19, 85
Comma list: 225/224, 441/440, 5929/5832


Badness: 0.0202
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}


=Negroni=
Optimal tunings:
Commas: 49/48, 55/54, 225/224
* 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}})


POTE generator: ~15/14 = 124.539
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Map: [&lt;1 2 2 3 5|, &lt;0 -4 3 -2 -15|]
Badness (Sintel): 2.20


EDOs: 10, 19e, 29, 77cde
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0353
Comma list: 105/104, 196/195, 275/273, 5929/5832


==13-limit==
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}
Commas: 49/48, 55/54, 65/64, 91/90


POTE generator: ~15/14 = 124.545
Optimal tunings:
* 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}})


Map: [&lt;1 2 2 3 5 4|, &lt;0 -4 3 -2 -15 -3|]
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


EDOs: 10, 19e, 29, 77cdef
Badness (Sintel): 1.73


Badness: 0.0216
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


=Wizard=
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288
Comma: 2197265625/2147483648


POTE generator: ~5/4 = 383.212
Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


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


EDOs: 22, 50, 72, 166, 238c, 310c, 548bc
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness: 0.3864
Badness (Sintel): 1.41


==7-limit==
== Enneaportent ==
Commas: 225/224, 118098/117649
[[Subgroup]]: 2.3.5.7


[[POTE_tuning|POTE temperament]]: 383.256
[[Comma list]]: 225/224, 40353607/40310784


Map: [&lt;2 1 5 2|, &lt;0 6 -1 10|]
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


Wedgie: &lt;&lt;12 -2 20 -31 -2 52||
[[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 }}


EDOs: [[22edo|22]], [[50edo|50]], [[72edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238c]], [[310edo|310c]], [[382edo|382c]]
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


Badness: 0.0408
[[Badness]] (Sintel): 2.37


==11-limit==
=== 11-limit ===
Commas: 225/224, 385/384, 4000/3993
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE temperament]]: 383.232
Comma list: 225/224, 385/384, 12005/11979


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


EDOs: 22, 50, 72, 94, [[166edo|166]], [[238edo|238c]], [[310edo|310c]]
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}})


Badness: 0.0185
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


==Lizard==
Badness (Sintel): 1.01
Commas: 225/224, 351/350, 364/363, 385/384


[[POTE_tuning|POTE temperament]]: 383.389
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
Comma list: 169/168, 225/224, 364/363, 1716/1715


EDOs: 22, 50, 72, 122, 194df
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


Badness: 0.0218
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}})


===17-limit===
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
Commas: 221/220, 273/272, 289/288, 351/350, 375/374


[[POTE_tuning|POTE temperament]]: 383.381
Badness (Sintel): 0.922


Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


EDOs: 22, 50, 72, 122g, 194dfg
[[Subgroup]]: 2.3.5.7


Badness: 0.0145
[[Comma list]]: 225/224, 781250000/771895089


===19-limit===
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
: mapping generators: ~2, ~3


[[POTE_tuning|POTE generator]]: ~5/4 = 383.477
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


EDOs: 22h, 50, 72, [[122edo|122g]], [[194edo|194dfg]]
[[Badness]] (Sintel): 2.44


Badness: 0.0157
=== 11-limit ===
Subgroup: 2.3.5.7.11


==Gizzard==
Comma list: 225/224, 385/384, 236328125/234365481
Commas: 225/224, 385/384, 325/324, 1573/1568


POTE generator: ~5/4 = 383.170
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


EDOs: [[22edo|22]], [[26edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238cf]]
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Badness: 0.0203
Badness (Sintel): 2.96


===17-limit===
==== 13-limit ====
Commas: 225/224, 289/288, 325/324, 375/374, 385/384
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 383.175
Comma list: 225/224, 325/324, 385/384, 831875/830466


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


EDOs: 22f, 72, 94, [[166edo|166g]], [[238edo|238cfg]]
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


Badness: 0.0136
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


===19-limit===
Badness (Sintel): 2.16
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399


POTE generator: ~5/4 = 383.138
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


EDOs: 22f, 72, 94, 166g
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


Badness: 0.0148
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


==Mage==
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
Commas: 99/98, 176/175, 1331/1296


POTE generator: ~55/48 = 216.876
Badness (Sintel): 1.96


Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


EDOs: 22, 72e, 94e
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


Badness: 0.0578
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


=Triton=
Optimal tunings:
Commas: 225/224, 1029/1000
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


[[POTE_tuning|POTE temperament]]: ~7/5 = 568.865
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
Badness (Sintel): 1.71


Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


EDOs: 17d, 19, 78bd, 97bd
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


Badness: 0.0592
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


==11-limit==
Optimal tunings:
Commas: 45/44, 56/55, 1029/1000
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


POTE generator: ~7/5 = 569.144
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


Map: [&lt;1 0 6 7 4|, &lt;0 3 -7 -8 -1|]
Badness (Sintel): 1.57


EDOs: 19, 59bde, 78bde, 97bde
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Badness: 0.0457
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


=Tritonic=
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}
Comnma: 553584375/536870912


POTE generator: ~45/32 = 580.219
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


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


EDOs: 29, 31, 60, 91
Badness (Sintel): 1.50


Badness: 0.3785
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


==7-limit==
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
Commas: 225/224, 50421/50000


[[POTE_tuning|POTE temperament]]: 580.286
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


Map: [&lt;1 4 -3 -3|, &lt;0 -5 11 12|]
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


Wedgie: &lt;&lt;5 -11 -12 -29 -33 3||
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


EDOs: [[29edo|29]], [[31edo|31]], [[60edo|60]], [[91edo|91]], [[122edo|122]], 213bcd
Badness (Sintel): 1.53


==11-limit==
=== Gracecord ===
Commas: 121/120, 225/224, 441/440
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE generator]]: ~7/5 = 580.267
Comma list: 225/224, 441/440, 109375/107811


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


EDOs: 29, 31, 60e
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


Badness: 0.0237
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


==13-limit==
Badness (Sintel): 2.21
Commas: 105/104, 121/120, 196/195, 275/273


POTE generator: ~7/5 = 580.108
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
Comma list: 225/224, 364/363, 441/440, 6125/6084


EDOs: 29, 31, 60e, 151cde
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Badness: 0.0230
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


==Tritoni==
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}
Commas: 225/224, 385/384, 27783/27500


POTE generator: ~7/5 = 580.389
Badness (Sintel): 1.83


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


EDOs: 31, 91, 122
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Badness: 0.0455
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


=Merman=
Optimal tunings:
Comma: 1121008359375/1099511627776
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


POTE generator: ~45/32 = 585.634
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Map: [&lt;1 5 -5|, &lt;0 -7 15|]
Badness (Sintel): 1.87


EDOs: 41, 84, 125, 209, 334c, 543bc
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.4538
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


==7-limit==
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}
Commas: 225/224, 2500000/2470629


POTE generator: ~7/5 = 585.585
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


Map: [&lt;1 5 -5 -5|, &lt;0 -7 15 16|]
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
Badness (Sintel): 1.68


EDOs: 41, 84, 125
== Alphorn ==
[[Subgroup]]: 2.3.5.7


Badness: 0.0551
[[Comma list]]: 225/224, 5764801/5668704


==11-limit==
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
Commas: 225/224, 441/440, 1344/1331
: mapping generators: ~2, ~35/24


POTE generator: ~7/5 = 585.606
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.3004{{c}}, ~35/24 = 644.4767{{c}}
: [[error map]]: {{val| +1.300 +0.569 -2.195 -2.043 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~35/24 = 643.8137{{c}}
: error map: {{val| 0.000 -0.936 -5.382 -4.924 }}


Map: [&lt;1 5 -5 -5 2|, &lt;0 -7 15 16 3]]
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


EDOs: 41, 84, 125e
[[Badness]] (Sintel): 3.27


Badness: 0.0364
=== 11-limit ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 225/224, 385/384, 12250/11979
Commas: 144/143, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 585.657
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


Map: [&lt;1 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


EDOs: 41, 84, 125e, 209ef, 293ef
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Badness: 0.0275
Badness (Sintel): 2.43


=Septimin=
== Misneb ==
Comma: 35595703125/34359738368
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


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


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


EDOs: 9, 32, 41, 91, 132, 223c, 355bc
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


Badness: 0.6709
[[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 }}


==7-limit==
{{Optimal ET sequence|legend=1| 21, 32, 53 }}
Commas: 225/224, 84035/82944


[[POTE_tuning|POTE temperament]]: 263.632
[[Badness]] (Sintel): 3.57


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


Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
Comma list: 99/98, 176/175, 1310720/1294139


EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


==11-limit==
Optimal tunings:
Commas: 225/224, 385/384, 2401/2376
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


[[POTE_tuning|POTE temperament]]: 263.634
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
Badness (Sintel): 2.82


EDOs: 9, 32, 41, 91, 223cdef
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 99/98, 176/175, 640/637, 847/845
Commas: 105/104, 144/143, 196/195, 245/242


[[POTE_tuning|POTE temperament]]: 263.700
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


EDOs: 9, 41, 91
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


=Slender=
Badness (Sintel): 1.88
Commas: 225/224, 589824/588245


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.413
=== Musneb ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
Comma list: 225/224, 385/384, 66550/64827


Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


EDOs: 31, 94, 125
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


Badness: 0.0569
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


==11-limit==
Badness (Sintel): 2.89
Commas: 225/224, 385/384, 1331/1323


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.387
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
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.


EDOs: 31, 63, 94, 125
[[Subgroup]]: 2.3.5.7


Badness: 25.342
[[Comma list]]: 225/224, 125000000/121060821


==13-limit==
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
Commas: 225/224, 275/273, 385/384, 1331/1323
: mapping generators: ~2, ~10/7


POTE generator: ~49/48 = 38.314
[[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 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
{{Optimal ET sequence|legend=1| 51, 53 }}


EDOs: 31, 63, 94
[[Badness]] (Sintel): 3.64


Badness: 25.913
=== 11-limit ===
Subgroup: 2.3.5.7.11


=Marvo=
Comma list: 121/120, 225/224, 22000/21609
Commas: 225/224, 78125000/78121827


[[POTE_tuning|POTE temperament]]: ~27/20 = 516.694
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
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, 17, 46, 13, 56 59||
{{Optimal ET sequence|legend=0| 51, 53 }}


EDOs: 65, 72
Badness (Sintel): 2.46


==11-limit==
=== 13-limit ===
Commas: 225/224, 243/242, 4000/3993
Subgroup: 2.3.5.7.11.13


[[POTE_tuning|POTE temperament]]: ~27/20 = 516.699
Comma list: 121/120, 225/224, 275/273, 1040/1029


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


EDOs: 65, 72, 281, 497
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 51f, 53 }}
Commas: 243/242, 225/224, 351/350, 1625/1617


POTE generator: ~27/20 = 516.730
Badness (Sintel): 1.96


Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


EDOs: 72, 137, 209, 281f, 490bcf
[[Subgroup]]: 2.3.5.7


Badness: 0.0269
[[Comma list]]: 225/224, 823543/800000


=Marvolo=
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
Commas: 225/224, 156250000/155649627
: mapping generators: ~2, ~32/21


POTE generator: ~21/20 = 83.348
[[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 }}


Map: [&lt;1 2 1 1|, &lt;0 -6 19 26|]
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||
[[Badness]] (Sintel): 3.67


EDOs: 29, 43, 72, 619bcd, 691bcd
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0833
Comma list: 225/224, 245/242, 1617/1600


==11-limit==
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}
Commas: 225/224, 441/440, 4000/3993


POTE generator: ~21/20 = 83.340
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


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


EDOs: 29, 43, 72
Badness (Sintel): 2.00


Badness: 0.0290
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 105/104, 196/195, 245/242, 1001/1000
Commas: 169/168, 225/224, 364/363, 441/440


POTE generator: ~21/20 = 83.330
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


EDOs: 29, 43, 72, 115f
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Badness: 0.0215
Badness (Sintel): 1.43


=Amavil=
=== 17-limit ===
Commas: 225/224, 17496/16807
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~48/35 = 529.979
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


EDOs: 9, 34d, 43, 77d
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Badness: 0.1096
Badness (Sintel): 1.26


==11-limit==
== Quintannic ==
Commas: 99/98, 176/175, 864/847
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


POTE generator: ~15/11 = 529.974
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
[[Comma list]]: 225/224, 9805926501/9765625000


EDOs: 9, 34d, 43, 77de
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


Badness: 0.0426
[[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 }}


==13-limit==
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}
Commas: 78/77, 99/98, 144/143, 176/175


POTE generator: ~15/11 = 529.951
[[Badness]] (Sintel): 3.81


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 9, 34d, 43, 77de
Comma list: 225/224, 441/440, 43923/43750


Badness: 0.0258
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


=Enneaportent=
Optimal tunings:
Commas: 225/224, 40353607/40310784
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


POTE generator: ~5/4 = 383.165
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
Badness (Sintel): 1.74


Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 9, 54, 63, 72, 495bcd
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


Badness: 0.0937
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


==11-limit==
Optimal tunings:
Commas: 225/224, 385/384, 12005/11979
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


POTE generator: ~5/4 = 383.146
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
Badness (Sintel): 1.35


EDOs: 9, 54, 63, 72, 423cd, 495bcd
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0304
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


==13-limit==
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
Commas: 169/168, 225/224, 364/363, 1716/1715


POTE generator: ~5/4 = 383.047
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


EDOs: 9, 54, 63, 72, 279cf
Badness (Sintel): 1.17


Badness: 0.0223
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


=Submajor=
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.
Comma: 69198046875/68719476736


POTE generator: ~10125/8192 = 362.321
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 4 -1|, &lt;0 -8 11|]
[[Comma list]]: 225/224, 122880/117649


EDOs: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


Badness: 0.1302
[[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 }}


==7-limit==
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}
Commas: 225/224, 51200/50421


POTE generator: ~49/40 = 362.255
[[Badness]] (Sintel): 4.37


Map: [&lt;1 4 -1 1|, &lt;0 -8 11 6|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
Comma list: 121/120, 225/224, 352/343


EDOs: 10, 33, 43, 53
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}


Badness: 0.0605
Optimal tunings:  
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 22, 55d }}
Commas: 225/224, 385/384, 6655/6561


POTE generator: ~27/22 = 362.101
Badness (Sintel): 2.46


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]]
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