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Marvel temperaments: Difference between revisions

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


Since (5/4)<sup>2</sup> = 225/224 × 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
Temperaments considered in families and clans are:
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[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 }}


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


== 5-limit ==
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
Note that this is technically not a marvel temperament until the 7-limit, but also placed here for reference.


[[Comma list]]: 16875/16384
Badness (Sintel): 1.50


[[POTE generator]]: ~15/14 = 125.7549
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
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.


[[Wedgie]]: {{wedgie| 4 -3 -14 }}
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 9, 10, 19, 86 }}
[[Comma list]]: 225/224, 84035/82944


== 7-limit ==
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
[[Comma list]]: 49/48, 225/224
: mapping generators: ~2, ~12/7


[[POTE generator]]: ~15/14 = 125.608
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}
: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}


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


[[Wedgie]]: {{wedgie| 4 -3 2 -14 -8 13 }}
[[Badness]] (Sintel): 1.38


{{Val list|legend=1| 9, 10, 19, 86 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 225/224, 245/242, 385/384
Comma list: 45/44, 49/48, 56/55


[[POTE generator]]: ~15/14 = 126.474
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


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


{{Val list|legend=1| 9, 10, 19 }}
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


[[Badness]]: 0.0262
Badness (Sintel): 1.04


=== 13-limit ===
=== 13-limit ===
Comma list: 45/44, 49/48, 56/55, 78/77
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 196/195, 245/242
 
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
 
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}
 
{{Optimal ET sequence|legend=0| 41, 91 }}
 
Badness (Sintel): 0.955
 
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''
 
Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot.
 
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 2500000/2470629
 
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
: [[error map]]: {{val| +0.390 -0.435 -1.630 +1.634 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}}
: error map: {{val| 0.000 -1.104 -2.423 +0.657 }}
 
{{Optimal ET sequence|legend=1| 41, 84, 125 }}
 
[[Badness]] (Sintel): 1.39
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 1344/1331
 
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
 
Badness (Sintel): 1.20
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
 
Optimal tunings:
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
 
Badness (Sintel): 1.14
 
=== Mermaid ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 532400/531441
 
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
 
Optimal tunings:
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}
 
Badness (Sintel): 1.46
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 10648/10647
 
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
 
Optimal tunings:
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
 
Badness (Sintel): 1.47
 
== Slender ==
Slender tempers out the [[hewuermera comma]] in addition to the marvel comma, and may be described as the {{nowrap| 31 & 32 }} temperament. This temperament has a generator of [[49/48]], three of which equal marvel's [[16/15]][[~]][[15/14]], and ten generators give [[5/4]]. Its [[ploidacot]] is omega-13-cot.
 
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
 
[[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 }}


[[POTE generator]]: ~14/13 = 126.431
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
[[Badness]] (Sintel): 1.44


{{Val list|legend=1| 9, 10, 19 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


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


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


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
Optimal tunings:  
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


{{Val list|legend=1| 19, 29, 48df, 77cdf }}
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Badness: 0.0387
Badness (Sintel): 0.838


=== 13-limit ===
=== 13-limit ===
Comma list: 49/48, 65/64, 91/90, 875/858
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


POTE generator: ~15/14 = 124.716
[[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 }}


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


{{Val list|legend=1| 9, 10, 19, 29, 48, 77 }}
[[Badness]] (Sintel): 2.11


Badness: 0.0244
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Negric ==
Comma list: 225/224, 441/440, 4000/3993
Comma list: 33/32, 49/48, 77/75


POTE generator: ~15/14 = 127.039
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


Mapping: [{{val| 1 2 2 3 3 }}, {{val| 0 -4 3 -2 4 }}]
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


{{Val list|legend=1| 9, 19, 85 }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.0306
Badness (Sintel): 0.958


=== 13-limit ===
=== 13-limit ===
Comma list: 33/32, 49/48, 65/64, 91/90
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 }}


POTE generator: ~15/14 = 127.039
Badness (Sintel): 0.895


Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
== Decic ==
{{Main| Decic }}


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


Badness: 0.0202
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.


== Negroni ==
[[Subgroup]]: 2.3.5.7
Comma list: 49/48, 55/54, 225/224


POTE generator: ~15/14 = 124.539
[[Comma list]]: 225/224, 16807/16384


Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


{{Val list|legend=1| 10, 19e, 29, 77cde }}
[[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 }}


Badness: 0.0353
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


=== 13-limit ===
[[Badness]] (Sintel): 2.26
Comma list: 49/48, 55/54, 65/64, 91/90


POTE generator: ~15/14 = 124.545
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Comma list: 225/224, 385/384, 3087/3025


{{Val list|legend=1| 10, 19e, 29, 77cdef }}
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}


Badness: 0.0216
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}})


= Passive =
{{Optimal ET sequence|legend=0| 10, 40, 50 }}
{{see also | Archytas clan #Passion }}


Comma list: 225/224, 256/245
Badness (Sintel): 2.11


POTE generator: ~16/15 = 98.809
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
Comma list: 105/104, 144/143, 196/195, 2200/2197


{{Val list|legend=1| 1, 11, 12, 49d }}
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


Badness: 0.0751
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}})


= Wizard =
{{Optimal ET sequence|legend=0| 10, 40, 50 }}
== 5-limit ==
Comma list: 2197265625/2147483648


POTE generator: ~5/4 = 383.212
Badness (Sintel): 1.52


Mapping: [{{val| 2 1 5 }}, {{val| 0 6 -1 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 548bc }}
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


Badness: 0.3864
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


== 7-limit ==
Optimal tunings:
Comma list: 225/224, 118098/117649
* 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}})


[[POTE generator]]: ~5/4 = 383.256
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Mapping: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
Badness (Sintel): 1.28


Wedgie: {{wedgie| 12 -2 20 -31 -2 52 }}
=== 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''.


{{Val list|legend=1| 22, 50, 72, 94, 166, 238c, 310c, 382c }}
Subgroup: 2.3.5.7.11


Badness: 0.0408
Comma list: 225/224, 1617/1600, 2401/2376


== 11-limit ==
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}
Comma list: 225/224, 385/384, 4000/3993


[[POTE generator]]: ~5/4 = 383.232
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}})


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


{{Val list|legend=1| 22, 50, 72, 94, 166, 238c, 310c }}
Badness (Sintel): 1.98


Badness: 0.0185
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== Lizard ===
Comma list: 105/104, 196/195, 1001/1000, 1188/1183
Comma list: 225/224, 351/350, 364/363, 385/384


[[POTE generator]]: ~5/4 = 383.389
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
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}})


{{Val list|legend=1| 22, 50, 72, 122, 194df }}
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


Badness: 0.0218
Badness (Sintel): 1.57


==== 17-limit ====
==== 17-limit ====
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374
 
Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


[[POTE generator]]: ~5/4 = 383.381
Badness (Sintel): 1.33


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


{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
Subgroup: 2.3.5.7.11


Badness: 0.0145
Comma list: 225/224, 441/440, 5929/5832


==== 19-limit ====
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


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


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


{{Val list|legend=1| 22h, 50, 72, 122g, 194dfg }}
Badness (Sintel): 2.20


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


=== Gizzard ===
Comma list: 105/104, 196/195, 275/273, 5929/5832
Comma list: 225/224, 385/384, 325/324, 1573/1568


POTE generator: ~5/4 = 383.170
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


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


{{Val list|legend=1| 22, 72, 94, 166, 238cf }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness: 0.0203
Badness (Sintel): 1.73


==== 17-limit ====
==== 17-limit ====
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


POTE generator: ~5/4 = 383.175
Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


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


{{Val list|legend=1| 22f, 72, 94, 166g, 238cfg }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness: 0.0136
Badness (Sintel): 1.41


==== 19-limit ====
== Enneaportent ==
Comma list: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399
[[Subgroup]]: 2.3.5.7


POTE generator: ~5/4 = 383.138
[[Comma list]]: 225/224, 40353607/40310784


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


{{Val list|legend=1| 22f, 72, 94, 166g }}
[[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 }}


Badness: 0.0148
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


== Mage ==
[[Badness]] (Sintel): 2.37
Comma list: 99/98, 176/175, 1331/1296


POTE generator: ~55/48 = 216.876
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
Comma list: 225/224, 385/384, 12005/11979


{{Val list|legend=1| 22, 72e, 94e }}
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


Badness: 0.0578
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}})


= Triton =
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
Comma list: 225/224, 1029/1000


[[POTE generator]]: ~7/5 = 568.865
Badness (Sintel): 1.01


Mapping: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Wedgie: {{wedgie| 3 -7 -8 -18 -21 1 }}
Comma list: 169/168, 225/224, 364/363, 1716/1715


{{Val list|legend=1| 17d, 19, 78bd, 97bd }}
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


Badness: 0.0592
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}})


==11-limit==
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
Comma list: 45/44, 56/55, 1029/1000


POTE generator: ~7/5 = 569.144
Badness (Sintel): 0.922


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0457
[[Comma list]]: 225/224, 781250000/771895089


= Tritonic =
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
== 5-limit ==
: mapping generators: ~2, ~3
Comma list: 553584375/536870912


POTE generator: ~45/32 = 580.219
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


Mapping: [{{val| 1 4 -3 }}, {{val| 0 -5 11 }}]
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


{{Val list|legend=1| 29, 31, 60, 91 }}
[[Badness]] (Sintel): 2.44


Badness: 0.3785
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 7-limit ==
Comma list: 225/224, 385/384, 236328125/234365481
Comma list: 225/224, 50421/50000


POTE generator: ~7/5 = 580.286
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


Mapping: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Wedgie: {{wedgie| 5 -11 -12 -29 -33 3 }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
Badness (Sintel): 2.96


== 11-limit ==
==== 13-limit ====
Comma list: 121/120, 225/224, 441/440
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 580.267
Comma list: 225/224, 325/324, 385/384, 831875/830466


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


{{Val list|legend=1| 29, 31, 60e }}
Optimal tunings:
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


Badness: 0.0237
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


=== 13-limit ===
Badness (Sintel): 2.16
Comma list: 105/104, 121/120, 196/195, 275/273


POTE generator: ~7/5 = 580.108
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


{{Val list|legend=1| 29, 31, 60e, 151cde }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


Badness: 0.0230
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


== Tritoni ==
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
Comma list: 225/224, 385/384, 27783/27500


POTE generator: ~7/5 = 580.389
Badness (Sintel): 1.96


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


{{Val list|legend=1| 31, 91, 122 }}
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


Badness: 0.0455
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


= Merman =
Optimal tunings:
Comma list: 1121008359375/1099511627776
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


POTE generator: ~45/32 = 585.634
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Mapping: [{{val| 1 5 -5 }}, {{val| 0 -7 15 }}]
Badness (Sintel): 1.71


{{Val list|legend=1| 41, 84, 125, 209, 334c, 543bc }}
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Badness: 0.4538
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


== 7-limit ==
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
Comma list: 225/224, 2500000/2470629


POTE generator: ~7/5 = 585.585
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Mapping: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


Wedgie: {{wedgie| 7 -15 -16 -40 -45 5 }}
Badness (Sintel): 1.57


{{Val list|legend=1| 41, 84, 125 }}
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Badness: 0.0551
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


== 11-limit ==
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}
Comma list: 225/224, 441/440, 1344/1331


POTE generator: ~7/5 = 585.606
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


{{Val list|legend=1| 41, 84, 125e }}
Badness (Sintel): 1.50


Badness: 0.0364
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


== 13-limit ==
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
Comma list: 144/143, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 585.657
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Badness: 0.0275
Badness (Sintel): 1.53


= Septimin =
=== Gracecord ===
Comma list: 35595703125/34359738368
Subgroup: 2.3.5.7.11


POTE generator: ~75/64 = 263.658
Comma list: 225/224, 441/440, 109375/107811


Mapping: [{{val| 1 4 1 }}, {{val| 0 -11 6 }}]
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


{{Val list|legend=1| 9, 32, 41, 91, 132, 223c, 355bc }}
Optimal tunings:
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


Badness: 0.6709
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


== 7-limit ==
Badness (Sintel): 2.21
Comma list: 225/224, 84035/82944


POTE generator: ~7/6 = 263.632
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
Comma list: 225/224, 364/363, 441/440, 6125/6084


Wedgie: {{wedgie| 11 -6 10 -35 -15 40 }}
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


{{Val list|legend=1| 9, 32, 41, 91, 132 }}
Optimal tunings:
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}
Comma list: 225/224, 385/384, 2401/2376


POTE generator: ~7/6 = 263.634
Badness (Sintel): 1.83


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 9, 32, 41, 91, 223cdef }}
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


== 13-limit ==
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}
Comma list: 105/104, 144/143, 196/195, 245/242


POTE generator: ~7/6 = 263.700
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


{{Val list|legend=1| 9, 41, 91 }}
Badness (Sintel): 1.87


=Slender=
==== 19-limit ====
Commas: 225/224, 589824/588245
Subgroup: 2.3.5.7.11.13.17.19


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.413
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


EDOs: {{EDOs|31, 94, 125}}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Badness: 0.0569
Badness (Sintel): 1.68


==11-limit==
== Alphorn ==
Commas: 225/224, 385/384, 1331/1323
[[Subgroup]]: 2.3.5.7


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.387
[[Comma list]]: 225/224, 5764801/5668704


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


EDOs: {{EDOs|31, 63, 94, 125}}
[[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 }}


Badness: 25.342
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


==13-limit==
[[Badness]] (Sintel): 3.27
Commas: 225/224, 275/273, 385/384, 1331/1323


POTE generator: ~49/48 = 38.314
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
Comma list: 225/224, 385/384, 12250/11979


EDOs: {{EDOs|31, 63, 94}}
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


Badness: 25.913
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


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


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


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


Wedgie: &lt;&lt;6 17 46 13 56 59||
[[Subgroup]]: 2.3.5.7


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


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


==11-limit==
[[Optimal tuning]]s:
Commas: 225/224, 243/242, 4000/3993
* [[WE]]: ~2 = 1199.7642{{c}}, ~15/8 = 1086.5513{{c}}
: [[error map]]: {{val| -0.236 -0.856 -1.569 +4.041 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1086.7633{{c}}
: error map: {{val| 0.000 -0.506 -0.999 +4.701 }}


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


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
[[Badness]] (Sintel): 3.57


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


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


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


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


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


EDOs: {{EDOs|72, 137, 209, 281f, 490bcf}}
Badness (Sintel): 2.82


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


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


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


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


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


EDOs: {{EDOs|29, 43, 72, 619bcd, 691bcd}}
Badness (Sintel): 1.88


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


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


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


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


EDOs: {{EDOs|29, 43, 72}}
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Badness: 0.0290
Badness (Sintel): 2.89


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


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


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
[[Subgroup]]: 2.3.5.7


EDOs: {{EDOs|29, 43, 72, 115f}}
[[Comma list]]: 225/224, 125000000/121060821


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


= Amavil =
[[Optimal tuning]]s:
== 5-limit (mabila) ==
* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
Comma: 268435456/263671875
: [[error map]]: {{val| -0.172 +0.002 -2.533 +3.511 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}}
: error map: {{val| 0.000 +0.298 -2.181 +3.946 }}


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


EDOs: {{EDOs| 9, 34d, 43, 77de }}
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1202.1198{{c}}, ~32/21 = 745.4675{{c}}
: [[error map]]: {{val| +2.120 -1.227 +0.459 -4.423 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~32/21 = 744.1318{{c}}
: error map: {{val| 0.000 -4.769 -4.159 -10.144 }}


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


Badness: 0.1302
[[Subgroup]]: 2.3.5.7


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


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


Map: [&lt;1 4 -1 1|, &lt;0 -8 11 6|]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.9803{{c}}, ~10000/9261 = 139.9522{{c}}
: [[error map]]: {{val| +0.980 -1.214 -0.313 -0.243 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10000/9261 = 139.8184{{c}}
: error map: {{val| 0.000 -2.863 -2.136 -2.287 }}


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


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


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


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


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


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


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


Badness: 0.0506
Badness (Sintel): 1.74


=== 13-limit ===
=== 13-limit ===
Commas: 169/168, 225/224, 275/273, 385/384
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
 
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
 
Optimal tunings:
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}
 
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
 
Badness (Sintel): 1.35
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
 
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
 
Optimal tunings:
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}
 
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
 
Badness (Sintel): 1.17
 
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''
 
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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 122880/117649
 
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}
 
[[Badness]] (Sintel): 4.37
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 225/224, 352/343
 
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}
 
Optimal tunings:
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}
 
{{Optimal ET sequence|legend=0| 22, 55d }}
 
Badness (Sintel): 2.46
 
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
 
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 5971968/5764801


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


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


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


POTE generator: ~48/35 = 556.221
[[Optimal tuning]]s:
* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
: [[error map]]: {{val| -0.523 -1.044 -1.549 +5.138 }}
* [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}}
: error map: {{val| 0.000 -0.284 -0.276 +6.592 }}


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


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


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


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


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


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


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


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


Badness: 0.0735
== References ==


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