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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


= Negri =
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
{{main| Negri }}


== 5-limit ==
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
Note that this is technically not a marvel temperament until the 7-limit, but also placed here for reference.


[[Comma list]]: 16875/16384
Optimal tunings:  
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


[[POTE generator]]: ~16/15 = 125.7549
{{Optimal ET sequence|legend=0| 72, 94, 166g }}


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
Badness (Sintel): 0.901


[[Wedgie]]: {{wedgie| 4 -3 -14 }}
=== Mage ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
Comma list: 99/98, 176/175, 1331/1296


== 7-limit ==
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
[[Comma list]]: 49/48, 225/224


[[POTE generator]]: ~15/14 = 125.608
Optimal tunings:  
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


[[Wedgie]]: {{wedgie| 4 -3 2 -14 -8 13 }}
Badness (Sintel): 1.91


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
== Tritonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


== 11-limit ==
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.
Comma list: 45/44, 49/48, 56/55


[[POTE generator]]: ~15/14 = 126.474
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
[[Comma list]]: 225/224, 50421/50000


{{Val list|legend=1| 9, 10, 19 }}
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


[[Badness]]: 0.0262
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
: [[error map]]: {{val| +1.354 -1.243 -0.027 -1.598 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}}
: error map: {{val| 0.000 -3.566 -2.769 -4.959 }}
 
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
 
[[Badness]] (Sintel): 1.20
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 225/224, 441/440
 
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}
 
Optimal tunings:
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}
 
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
 
Badness (Sintel): 0.782
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 121/120, 196/195, 275/273
 
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}
 
Optimal tunings:
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}
 
{{Optimal ET sequence|legend=0| 29, 31, 60e }}
 
Badness (Sintel): 0.950
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
 
Optimal tunings:
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
Badness (Sintel): 0.973
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
 
Optimal tunings:
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
Badness (Sintel): 1.03
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}
 
Optimal tunings:
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
Badness (Sintel): 1.04
 
=== Tritoni ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 27783/27500
 
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}
 
Optimal tunings:
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}
 
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
 
Badness (Sintel): 1.50
 
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''
 
Septimin may be described as the {{nowrap| 41 & 50 }} temperament. It is generated by a septimal minor third ([[7/6]]), which gives rise to the name, but the generator can be taken to be the [[octave complement]], [[12/7]], such that eleven of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus eta-hendecacot. [[91edo]] may be recommended as a tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 84035/82944
 
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}
: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}
 
{{Optimal ET sequence|legend=1| 41, 91, 132d }}
 
[[Badness]] (Sintel): 1.38
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 245/242, 385/384
 
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
 
Optimal tunings:
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}
 
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}
 
Badness (Sintel): 1.04


=== 13-limit ===
=== 13-limit ===
Comma list: 45/44, 49/48, 56/55, 78/77
Subgroup: 2.3.5.7.11.13


[[POTE generator]]: ~14/13 = 126.431
Comma list: 105/104, 144/143, 196/195, 245/242


Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


{{Val list|legend=1| 9, 10, 19 }}
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


== Negril ==
{{Optimal ET sequence|legend=0| 41, 91 }}
Comma list: 49/48, 100/99, 225/224


POTE generator: ~15/14 = 124.767
Badness (Sintel): 0.955


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


{{Val list|legend=1| 19, 29, 48d, 77cdd }}
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.


Badness: 0.0387
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 ===
=== 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


POTE generator: ~15/14 = 124.716
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


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


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


Badness: 0.0244
Badness (Sintel): 1.07


== Negric ==
== Triton ==
Comma list: 33/32, 49/48, 77/75
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


POTE generator: ~15/14 = 127.039
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.  


Mapping: [{{val| 1 2 2 3 3 }}, {{val| 0 -4 3 -2 4 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
[[Comma list]]: 225/224, 1029/1000


Badness: 0.0306
{{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 ===
=== 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


POTE generator: ~15/14 = 127.039
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.0202
Badness (Sintel): 0.887


== Negroni ==
=== 17-limit ===
Comma list: 49/48, 55/54, 225/224
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~15/14 = 124.539
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


{{Val list|legend=1| 10, 19e, 29, 77cddee }}
Optimal tunings:
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Badness: 0.0353
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


=== 13-limit ===
Badness (Sintel): 0.760
Comma list: 49/48, 55/54, 65/64, 91/90
 
=== 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


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, 77cddeef }}
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


POTE generator: ~5/4 = 383.175
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


{{Val list|legend=1| 22f, 72, 94, 166g, 238cfg }}
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}})


Badness: 0.0136
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


==== 19-limit ====
Badness (Sintel): 1.41
Comma list: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399


POTE generator: ~5/4 = 383.138
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
[[Comma list]]: 225/224, 40353607/40310784


{{Val list|legend=1| 22f, 72, 94, 166g }}
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


Badness: 0.0148
[[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 }}


== Mage ==
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}
Comma list: 99/98, 176/175, 1331/1296


POTE generator: ~55/48 = 216.876
[[Badness]] (Sintel): 2.37


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 22, 72e, 94e }}
Comma list: 225/224, 385/384, 12005/11979


Badness: 0.0578
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


= Triton =
Optimal tunings:
Comma list: 225/224, 1029/1000
* 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}})


[[POTE generator]]: ~7/5 = 568.865
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Mapping: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
Badness (Sintel): 1.01


Wedgie: {{wedgie| 3 -7 -8 -18 -21 1 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 17d, 19, 78bd, 97bd }}
Comma list: 169/168, 225/224, 364/363, 1716/1715


Badness: 0.0592
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


==11-limit==
Optimal tunings:
Comma list: 45/44, 56/55, 1029/1000
* 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}})


POTE generator: ~7/5 = 569.144
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
Badness (Sintel): 0.922


{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


Badness: 0.0457
[[Subgroup]]: 2.3.5.7


= Tritonic =
[[Comma list]]: 225/224, 781250000/771895089
== 5-limit ==
Comma list: 553584375/536870912


POTE generator: ~45/32 = 580.219
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


Mapping: [{{val| 1 4 -3 }}, {{val| 0 -5 11 }}]
[[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 }}


{{Val list|legend=1| 29, 31, 60, 91 }}
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


Badness: 0.3785
[[Badness]] (Sintel): 2.44


== 7-limit ==
=== 11-limit ===
Comma list: 225/224, 50421/50000
Subgroup: 2.3.5.7.11


POTE generator: ~7/5 = 580.286
Comma list: 225/224, 385/384, 236328125/234365481


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


Wedgie: {{wedgie| 5 -11 -12 -29 -33 3 }}
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


== 11-limit ==
Badness (Sintel): 2.96
Comma list: 121/120, 225/224, 441/440


POTE generator: ~7/5 = 580.267
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
Comma list: 225/224, 325/324, 385/384, 831875/830466


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


Badness: 0.0237
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


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


POTE generator: ~7/5 = 580.108
Badness (Sintel): 2.16


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 29, 31, 60e, 151cde }}
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


Badness: 0.0230
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


== Tritoni ==
Optimal tunings:
Comma list: 225/224, 385/384, 27783/27500
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


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


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
Badness (Sintel): 1.96


{{Val list|legend=1| 31, 91, 122 }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0455
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


= Merman =
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}
Comma list: 1121008359375/1099511627776


POTE generator: ~45/32 = 585.634
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


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


{{Val list|legend=1| 41, 84, 125, 209, 334c, 543bc }}
Badness (Sintel): 1.71


Badness: 0.4538
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


== 7-limit ==
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528
Comma list: 225/224, 2500000/2470629


POTE generator: ~7/5 = 585.585
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Mapping: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Wedgie: {{wedgie| 7 -15 -16 -40 -45 5 }}
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


{{Val list|legend=1| 41, 84, 125 }}
Badness (Sintel): 1.57


Badness: 0.0551
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


== 11-limit ==
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
Comma list: 225/224, 441/440, 1344/1331


POTE generator: ~7/5 = 585.606
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


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


Badness: 0.0364
Badness (Sintel): 1.50


== 13-limit ==
==== 31-limit ====
Comma list: 144/143, 225/224, 364/363, 441/440
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


POTE generator: ~7/5 = 585.657
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
Optimal tunings:
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


Badness: 0.0275
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


= Septimin =
Badness (Sintel): 1.53
Comma list: 35595703125/34359738368


POTE generator: ~75/64 = 263.658
=== Gracecord ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 4 1 }}, {{val| 0 -11 6 }}]
Comma list: 225/224, 441/440, 109375/107811


{{Val list|legend=1| 9, 32, 41, 91, 132, 223c, 355bc }}
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


Badness: 0.6709
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


== 7-limit ==
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}
Comma list: 225/224, 84035/82944


POTE generator: ~7/6 = 263.632
Badness (Sintel): 2.21


Mapping: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Wedgie: {{wedgie| 11 -6 10 -35 -15 40 }}
Comma list: 225/224, 364/363, 441/440, 6125/6084


{{Val list|legend=1| 9, 32, 41, 91, 132 }}
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


== 11-limit ==
Optimal tunings:
Comma list: 225/224, 385/384, 2401/2376
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


POTE generator: ~7/6 = 263.634
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
Badness (Sintel): 1.83


{{Val list|legend=1| 9, 32, 41, 91, 223cdef }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


== 13-limit ==
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
Comma list: 105/104, 144/143, 196/195, 245/242


POTE generator: ~7/6 = 263.700
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


{{Val list|legend=1| 9, 41, 91 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


= Slender =
Badness (Sintel): 1.87
Comma list: 225/224, 589824/588245


POTE generator: ~49/48 = 38.413
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


Wedgie: {{wedgie| 13 -10 6 -46 -27 42 }}
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


{{Val list|legend=1| 31, 94, 125 }}
Optimal tunings:
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


Badness: 0.0569
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


==11-limit==
Badness (Sintel): 1.68
Comma list: 225/224, 385/384, 1331/1323


POTE generator: ~49/48 = 38.387
== Alphorn ==
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
[[Comma list]]: 225/224, 5764801/5668704


{{Val list|legend=1| 31, 63, 94, 125 }}
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


Badness: 25.342
[[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 }}


== 13-limit ==
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}
Comma list: 225/224, 275/273, 385/384, 1331/1323


POTE generator: ~49/48 = 38.314
[[Badness]] (Sintel): 3.27


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 31, 63, 94 }}
Comma list: 225/224, 385/384, 12250/11979


Badness: 25.913
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


= Marvo =
Optimal tunings:
{{see also| Gravity family }}
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


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


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


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


Wedgie: {{wedgie| 6 17 46 13 56 59 }}
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 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:
Comma list: 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 }}


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
[[Badness]] (Sintel): 3.57


{{Val list|legend=1| 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 }}
Comma list: 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}}


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


{{Val list|legend=1| 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
Comma list: 225/224, 156250000/155649627


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


Mapping: [{{val| 1 2 1 1 }}, {{val| 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: {{wedgie| 6 -19 -26 -44 -58 -7 }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


{{Val list|legend=1| 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
Comma list: 225/224, 441/440, 4000/3993


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


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 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}}


{{Val list|legend=1| 29, 43, 72 }}
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Badness: 0.0290
Badness (Sintel): 2.89


== 13-limit ==
== Untriton ==
Comma list: 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.  


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 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 list: 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 }}


Mapping: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
[[Badness]] (Sintel): 3.64


{{Val list|legend=1| 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 }}
Comma list: 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}}


Mapping: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
{{Optimal ET sequence|legend=0| 51, 53 }}


Wedgie: {{wedgie| 10 -3 14 -28 -6 41 }}
Badness (Sintel): 2.46


{{Val list|legend=1| 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 }}
Comma list: 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}}


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


{{Val list|legend=1| 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
Comma list: 78/77, 99/98, 144/143, 176/175


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


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


{{Val list|legend=1| 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
Comma list: 225/224, 40353607/40310784


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


Mapping: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
Comma list: 225/224, 245/242, 1617/1600


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


{{Val list|legend=1| 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
Comma list: 225/224, 385/384, 12005/11979


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


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
Comma list: 105/104, 196/195, 245/242, 1001/1000


{{Val list|legend=1| 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 }}
Comma list: 169/168, 225/224, 364/363, 1716/1715


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


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 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 list: 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 }}


Mapping: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
Badness (Sintel): 1.26


{{Val list|legend=1| 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
Comma list: 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


Mapping: [{{val| 1 4 -1 1 }}, {{val| 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: {{wedgie| 8 -11 -6 -36 -32 17 }}
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


{{Val list|legend=1| 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
Comma list: 225/224, 385/384, 6655/6561


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


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 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}}


{{Val list|legend=1| 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 ===
Comma list: 169/168, 225/224, 275/273, 385/384
Subgroup: 2.3.5.7.11.13


POTE generator: ~16/13 = 362.105
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


{{Val list|legend=1| 10, 53, 116, 169de, 285cdef }}
Optimal tunings:
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


Badness: 0.0277
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


== Interpental ==
Badness (Sintel): 1.35
Comma list: 99/98, 176/175, 51200/50421


POTE generator: ~49/40 = 362.418
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


{{Val list|legend=1| 43, 53, 96, 149d }}
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


Badness: 0.0518
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
Comma list: 99/98, 169/168, 176/175, 640/637
 
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
 
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}
 
[[Badness]] (Sintel): 4.53
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 65536/65219
 
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}
 
Optimal tunings:
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}
 
{{Optimal ET sequence|legend=0| 22, 74, 96 }}
 
Badness (Sintel): 2.26


POTE generator: ~16/13 = 362.402
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 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>.


{{Val list|legend=1| 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 }}
Comma list: 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 }}


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


Wedgie: {{wedgie| 16 -5 22 -45 -10 65 }}
[[Badness]] (Sintel): 10.9


{{Val list|legend=1| 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 }}
Comma list: 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}}


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


{{Val list|legend=1| 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]]
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"