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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]], [[Qintosec family|qintosec]] 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}}.)


= Negri =
== Wizard ==
{{main| Negri }}
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''


Negri tempers out the [[negri comma]] in the 5-limit, [[49/48]] and [[225/224]] in the 7-limit. It can be extended naturally to the 2.3.5.7.13 subgroup by adding [[91/90]] to the comma list; this will be discussed below under the title of negra.  
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.  


== 5-limit ==
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5
[[Comma list]]: 225/224, 118098/117649


[[Comma list]]: 16875/16384
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216


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


{{Multival|legend=1| 4 -3 -14 }}
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}


[[POTE generator]]: ~16/15 = 125.7549
[[Badness]] (Sintel): 1.03


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 7-limit ==
Comma list: 225/224, 385/384, 4000/3993


Subgroup: 2.3.5.7
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}


[[Comma list]]: 49/48, 225/224
Optimal tunings:  
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


{{Multival|legend=1| 4 -3 2 -14 -8 13 }}
Badness (Sintel): 0.613


[[POTE generator]]: ~15/14 = 125.608
==== Lizard ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
Comma list: 225/224, 351/350, 364/363, 385/384


=== Negra ===
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
This is the 2.3.5.7.13 extension of negri.


Subgroup: 2.3.5.7.13
Optimal tunings:  
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


[[Comma list]]: 49/48, 65/64, 91/90
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


[[Sval]] [[Mapping]]: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -3 }}]
Badness (Sintel): 0.900


Gencom mapping: [{{val| 1 2 2 3 0 4 }}, {{val| 0 -4 3 -2 0 -3 }}]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Gencom]]: [2 14/13; 49/48 65/64 91/90]
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


POTE generator: ~14/13 = 125.567
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
Optimal tunings:
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


Subgroup: 2.3.5.7.11
Badness (Sintel): 0.741


Comma list: 45/44, 49/48, 56/55
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


POTE generator: ~15/14 = 126.474
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}


{{Val list|legend=1| 9, 10, 19 }}
Optimal tunings:
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


Badness: 0.0262
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


</div></div>
Badness (Sintel): 0.955
 
=== 13-limit ===


==== Gizzard ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 49/48, 56/55, 78/77
Comma list: 225/224, 325/324, 385/384, 1573/1568


Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}


POTE generator: ~14/13 = 126.431
Optimal tunings:  
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}


{{Val list|legend=1| 9, 10, 19 }}
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}


== Negril ==
Badness (Sintel): 0.837


Subgroup: 2.3.5.7.11
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Comma list: 49/48, 100/99, 225/224
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}


POTE generator: ~15/14 = 124.767
Optimal tunings:  
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}


{{Val list|legend=1| 19, 29, 48d, 77cdd }}
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}


Badness: 0.0387
Badness (Sintel): 0.694


=== 13-limit ===
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594


Comma list: 49/48, 65/64, 91/90, 875/858
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
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: ~14/13 = 124.716
{{Optimal ET sequence|legend=0| 72, 94, 166g }}


{{Val list|legend=1| 19, 29, 48df, 77cddf }}
Badness (Sintel): 0.901


Badness: 0.0244
=== Mage ===
Subgroup: 2.3.5.7.11


== Negric ==
Comma list: 99/98, 176/175, 1331/1296


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}


Comma list: 33/32, 49/48, 77/75
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 3 }}, {{val| 0 -4 3 -2 4 }}]
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


POTE generator: ~15/14 = 127.039
Badness (Sintel): 1.91


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
== Tritonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


Badness: 0.0306
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.  


=== 13-limit ===
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7.11.13
[[Comma list]]: 225/224, 50421/50000


Comma list: 33/32, 49/48, 65/64, 91/90
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


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


POTE generator: ~14/13 = 127.039
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
[[Badness]] (Sintel): 1.20


Badness: 0.0202
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Negroni ==
Comma list: 121/120, 225/224, 441/440


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


Comma list: 49/48, 55/54, 225/224
Optimal tunings:  
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


POTE generator: ~15/14 = 124.539
Badness (Sintel): 0.782


{{Val list|legend=1| 10, 19e, 29, 77cddee }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0353
Comma list: 105/104, 121/120, 196/195, 275/273


=== 13-limit ===
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


Subgroup: 2.3.5.7.11.13
Optimal tunings:  
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


Comma list: 49/48, 55/54, 65/64, 91/90
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Badness (Sintel): 0.950


POTE generator: ~14/13 = 124.545
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 10, 19e, 29, 77cddeef }}
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


Badness: 0.0216
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


== Wilsec ==
Optimal tunings:
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


Subgroup: 2.3.5.7.11
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Comma list: 49/48, 121/120, 225/224
Badness (Sintel): 0.973


Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~11/8 = 537.186
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


{{Val list|legend=1| 9, 20, 29, 38d, 67cdde }}
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


Badness: 0.0419
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Subgroup: 2.3.5.7.11.13
Badness (Sintel): 1.03


Comma list: 49/48, 65/64, 91/90, 121/120
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


POTE generator: ~11/8 = 537.208
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


{{Val list|legend=1| 9, 20, 29, 38df, 67cddef }}
Optimal tunings:
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


Badness: 0.0252
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


=== 17-limit ===
Badness (Sintel): 1.04


Subgroup: 2.3.5.7.11.13.17
=== Tritoni ===
Subgroup: 2.3.5.7.11


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
Comma list: 225/224, 385/384, 27783/27500


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


POTE generator: ~11/8 = 537.230
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefg }}
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Badness: 0.0218
Badness (Sintel): 1.50


=== 19-limit ===
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


Subgroup: 2.3.5.7.11.13.17.19
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.  


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
[[Comma list]]: 225/224, 84035/82944


POTE generator: ~11/8 = 537.214
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefgh }}
[[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 }}


Badness: 0.0168
{{Optimal ET sequence|legend=1| 41, 91, 132d }}


= Passive =
[[Badness]] (Sintel): 1.38
{{see also | Archytas clan #Passion }}


Subgroup: 2.3.5.7
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Comma list]]: 225/224, 256/245
Comma list: 225/224, 245/242, 385/384


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


[[POTE generator]]: ~16/15 = 98.809
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| 12, 49dd }}
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


[[Badness]]: 0.0751
Badness (Sintel): 1.04


= Wizard =
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
Comma list: 105/104, 144/143, 196/195, 245/242


Subgroup: 2.3.5.7
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


[[Comma list]]: 225/224, 118098/117649
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


[[Mapping]]: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Mapping generators: ~1225/864, ~245/216
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.


[[POTE generator]]: ~5/4 = 383.256
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.  


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
[[Comma list]]: 225/224, 2500000/2470629


[[Badness]]: 0.0408
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


Scales: [[wizard22]]
[[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 }}


== 11-limit ==
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


[[Badness]] (Sintel): 1.39
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 4000/3993
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 }}


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
Badness (Sintel): 1.20


Mapping generators: ~99/70, ~25/22
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 383.232
Comma list: 144/143, 225/224, 364/363, 441/440


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c }}
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Badness: 0.0185
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Scales: [[wizard22]]
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


=== Lizard ===
Badness (Sintel): 1.14


Subgroup: 2.3.5.7.11.13
=== Mermaid ===
Subgroup: 2.3.5.7.11


Comma list: 225/224, 351/350, 364/363, 385/384
Comma list: 225/224, 385/384, 532400/531441


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


POTE generator: ~5/4 = 383.389
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


{{Val list|legend=1| 22, 50, 72, 122, 194df }}
Badness (Sintel): 1.46


Badness: 0.0218
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


==== 17-limit ====
Comma list: 225/224, 325/324, 385/384, 10648/10647


Subgroup: 2.3.5.7.11.13.17
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
Optimal tunings:  
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


Mapping generators: ~17/12, ~17/15
Badness (Sintel): 1.47


POTE generator: ~5/4 = 383.381
== 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.  


{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.


Badness: 0.0145
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 589824/588245


==== 19-limit ====
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


Subgroup: 2.3.5.7.11.13.17.19
[[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 }}


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
[[Badness]] (Sintel): 1.44


Mapping generators: ~17/12, ~17/15
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~5/4 = 383.477
Comma list: 225/224, 385/384, 1331/1323


{{Val list|legend=1| 22h, 50, 72, 122g, 194dfg }}
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


Badness: 0.0157
Optimal tunings:  
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


=== Gizzard ===
Badness (Sintel): 0.838


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 325/324, 385/384, 1573/1568
Comma list: 225/224, 275/273, 385/384, 1331/1323


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


POTE generator: ~5/4 = 383.170
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


{{Val list|legend=1| 72, 166, 238cf }}
Badness (Sintel): 1.07


Badness: 0.0203
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


</div></div>
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.


==== 17-limit ====
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7.11.13.17
[[Comma list]]: 225/224, 1029/1000


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


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


Mapping generators: ~17/12, ~17/15
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


POTE generator: ~5/4 = 383.175
[[Badness]] (Sintel): 1.50


{{Val list|legend=1| 72, 166g, 238cfg }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0136
Comma list: 45/44, 56/55, 1029/1000


</div></div>
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


==== 19-limit ====
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


Subgroup: 2.3.5.7.11.13.17.19
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
Badness (Sintel): 1.51


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
== Marvolo ==
[[Subgroup]]: 2.3.5.7


Mapping generators: ~17/12, ~17/15
[[Comma list]]: 225/224, 156250000/155649627


POTE generator: ~5/4 = 383.138
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


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


Badness: 0.0148
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


== Mage ==
[[Badness]] (Sintel): 2.11


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 1331/1296
Comma list: 225/224, 441/440, 4000/3993


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


Mapping generators: ~77/54, ~55/48
Optimal tunings:
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


POTE generator: ~5/4 = 383.124
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


{{Val list|legend=1| 22, 50e, 72ee, 94ee }}
Badness (Sintel): 0.958


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


= Triton =
Comma list: 169/168, 225/224, 364/363, 441/440


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Subgroup: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


[[Comma list]]: 225/224, 1029/1000
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
Badness (Sintel): 0.887


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[POTE generator]]: ~7/5 = 568.865
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


{{Val list|legend=1| 19, 78bd, 97bd }}
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


[[Badness]]: 0.0592
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Subgroup: 2.3.5.7.11
Badness (Sintel): 0.760


Comma list: 45/44, 56/55, 1029/1000
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 569.144
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
Optimal tunings:
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


Badness: 0.0457
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


= Tritonic =
Badness (Sintel): 0.895


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
== Decic ==
{{Main| Decic }}


Subgroup: 2.3.5.7
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]].  


[[Comma list]]: 225/224, 50421/50000
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.


[[Mapping]]: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
[[Comma list]]: 225/224, 16807/16384


[[POTE generator]]: ~7/5 = 580.286
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
[[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 }}


== 11-limit ==
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


[[Badness]] (Sintel): 2.26
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 225/224, 441/440
Comma list: 225/224, 385/384, 3087/3025
 
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}
 
Optimal tunings:
* WE: ~15/14 = 120.1406{{c}}, ~3/2 = 697.6075{{c}} (~56/55 = 23.2360{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.0142{{c}} (~56/55 = 22.9858{{c}})
 
{{Optimal ET sequence|legend=0| 10, 40, 50 }}
 
Badness (Sintel): 2.11
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
Comma list: 105/104, 144/143, 196/195, 2200/2197


POTE generator: ~7/5 = 580.267
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


{{Val list|legend=1| 29, 31, 60e }}
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}})


Badness: 0.0237
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


=== 13-limit ===
Badness (Sintel): 1.52


Subgroup: 2.3.5.7.11.13
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Comma list: 105/104, 121/120, 196/195, 275/273
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


POTE generator: ~7/5 = 580.108
Optimal tunings:  
* WE: ~15/14 = 120.1262{{c}}, ~3/2 = 697.8185{{c}} (~78/77 = 22.9388{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.2757{{c}} (~78/77 = 22.7243{{c}})


{{Val list|legend=1| 29, 31, 60e, 151cde }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Badness: 0.0230
Badness (Sintel): 1.28


== Tritoni ==
=== 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''.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 27783/27500
Comma list: 225/224, 1617/1600, 2401/2376
 
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}


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


POTE generator: ~7/5 = 580.389
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


{{Val list|legend=1| 31, 91, 122, 153d }}
Badness (Sintel): 1.98


Badness: 0.0455
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


= Merman =
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


Subgroup: 2.3.5.7
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}})


[[Comma list]]: 225/224, 2500000/2470629
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}
 
Badness (Sintel): 1.57
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


[[POTE generator]]: ~7/5 = 585.585
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}})


{{Val list|legend=1| 41, 84, 125 }}
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


[[Badness]]: 0.0551
Badness (Sintel): 1.33


== 11-limit ==
=== 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''.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 441/440, 1344/1331
Comma list: 225/224, 441/440, 5929/5832
 
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}
 
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| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


POTE generator: ~7/5 = 585.606
Badness (Sintel): 2.20


{{Val list|legend=1| 41, 84, 125e }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0364
Comma list: 105/104, 196/195, 275/273, 5929/5832


== 13-limit ==
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


Subgroup: 2.3.5.7.11.13
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}})


Comma list: 144/143, 225/224, 364/363, 441/440
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Badness (Sintel): 1.73


POTE generator: ~7/5 = 585.657
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


Badness: 0.0275
Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


= Septimin =
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}})


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Subgroup: 2.3.5.7
Badness (Sintel): 1.41


[[Comma list]]: 225/224, 84035/82944
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
[[Comma list]]: 225/224, 40353607/40310784


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


[[POTE generator]]: ~7/6 = 263.632
[[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 }}


{{Val list|legend=1| 41, 91, 132 }}
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


== 11-limit ==
[[Badness]] (Sintel): 2.37


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 2401/2376
Comma list: 225/224, 385/384, 12005/11979


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


POTE generator: ~7/6 = 263.634
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}})


{{Val list|legend=1| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


== 13-limit ==
Badness (Sintel): 1.01


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 105/104, 144/143, 196/195, 245/242
Comma list: 169/168, 225/224, 364/363, 1716/1715
 
Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]


POTE generator: ~7/6 = 263.700
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


{{Val list|legend=1| 41, 91 }}
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}})


= Slender =
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Subgroup: 2.3.5.7
Badness (Sintel): 0.922


[[Comma list]]: 225/224, 589824/588245
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
[[Comma list]]: 225/224, 781250000/771895089


[[POTE generator]]: ~49/48 = 38.413
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


{{Val list|legend=1| 31, 94, 125 }}
[[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 }}


[[Badness]]: 0.0569
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


== 11-limit ==
[[Badness]] (Sintel): 2.44


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 1331/1323
Comma list: 225/224, 385/384, 236328125/234365481
 
Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]


POTE generator: ~49/48 = 38.387
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


{{Val list|legend=1| 31, 63, 94, 125 }}
Optimal tunings:
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Badness: 0.02534
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


== 13-limit ==
Badness (Sintel): 2.96


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 275/273, 385/384, 1331/1323
Comma list: 225/224, 325/324, 385/384, 831875/830466


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


POTE generator: ~49/48 = 38.314
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


{{Val list|legend=1| 31, 63, 94 }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.02591
Badness (Sintel): 2.16


= Marvo =
==== 17-limit ====
{{see also| Gravity family }}
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


Comma list: 225/224, 78125000/78121827
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


{{Multival|legend=1| 6 17 46 13 56 59 }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


POTE generator: ~27/20 = 516.694
Badness (Sintel): 1.96


{{Val list|legend=1| 65d, 72, 137, 209, 281, 569bcc }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0976
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


== 11-limit ==
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


Subgroup: 2.3.5.7.11
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Comma list: 225/224, 243/242, 4000/3993
Badness (Sintel): 1.71


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


POTE generator: ~27/20 = 516.699
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


{{Val list|legend=1| 65d, 72, 281, 353c, 425bc, 497bc }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Badness: 0.0317
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


Subgroup: 2.3.5.7.11.13
Badness (Sintel): 1.57


Comma list: 225/224, 243/242, 351/350, 1625/1617
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


POTE generator: ~27/20 = 516.730
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


{{Val list|legend=1| 65d, 72, 137, 209, 281f, 490bcf }}
Optimal tunings:
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Badness: 0.0269
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


= Marvolo =
Badness (Sintel): 1.50


Subgroup: 2.3.5.7
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


[[Comma list]]: 225/224, 156250000/155649627
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
Optimal tunings:
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


[[POTE generator]]: ~21/20 = 83.348
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
Badness (Sintel): 1.53


[[Badness]]: 0.0833
=== Gracecord ===
Subgroup: 2.3.5.7.11


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


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


Comma list: 225/224, 441/440, 4000/3993
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


POTE generator: ~21/20 = 83.340
Badness (Sintel): 2.21


{{Val list|legend=1| 29, 43, 72 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0290
Comma list: 225/224, 364/363, 441/440, 6125/6084


</div></div>
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


== 13-limit ==
Optimal tunings:
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Comma list: 169/168, 225/224, 364/363, 441/440
Badness (Sintel): 1.83


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~21/20 = 83.330
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


{{Val list|legend=1| 29, 43, 72, 115f }}
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Badness: 0.0215
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


= Amavil =
Badness (Sintel): 1.87
== 5-limit (mabila) ==


Subgroup: 2.3.5
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


[[Comma list]]: 268435456/263671875
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


[[Mapping]]: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


[[POTE generator]]: ~512/375 = 529.6849
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


[[Badness]]: 0.2325
Badness (Sintel): 1.68


== 7-limit ==
== Alphorn ==
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7
[[Comma list]]: 225/224, 5764801/5668704


[[Comma list]]: 225/224, 17496/16807
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
[[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 }}


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


[[POTE generator]]: ~48/35 = 529.979
[[Badness]] (Sintel): 3.27


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Badness]]: 0.1096
Comma list: 225/224, 385/384, 12250/11979


== 11-limit ==
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


Subgroup: 2.3.5.7.11
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Comma list: 99/98, 176/175, 864/847
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
Badness (Sintel): 2.43


POTE generator: ~15/11 = 529.974
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


{{Val list|legend=1| 9, 34d, 43, 77de }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0426
[[Comma list]]: 225/224, 4194304/4117715


== 13-limit ==
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


Subgroup: 2.3.5.7.11.13
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7642{{c}}, ~15/8 = 1086.5513{{c}}
: [[error map]]: {{val| -0.236 -0.856 -1.569 +4.041 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1086.7633{{c}}
: error map: {{val| 0.000 -0.506 -0.999 +4.701 }}


Comma list: 78/77, 99/98, 144/143, 176/175
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
[[Badness]] (Sintel): 3.57


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


{{Val list|legend=1| 9, 34d, 43, 77de }}
Comma list: 99/98, 176/175, 1310720/1294139


Badness: 0.0258
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


= Enneaportent =
Optimal tunings:
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


[[Comma list]]: 225/224, 40353607/40310784
Badness (Sintel): 2.82


[[Mapping]]: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
Comma list: 99/98, 176/175, 640/637, 847/845


[[POTE generator]]: ~5/4 = 383.165
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


{{Val list|legend=1| 63, 72, 495bcd }}
Optimal tunings:
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


[[Badness]]: 0.0937
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


== 11-limit ==
Badness (Sintel): 1.88


=== Musneb ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 12005/11979
Comma list: 225/224, 385/384, 66550/64827
 
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
 
Optimal tunings:
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


POTE generator: ~5/4 = 383.146
Badness (Sintel): 2.89


{{Val list|legend=1| 63, 72, 423cd, 495bcd }}
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


Badness: 0.0304
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.  


</div></div>
[[Subgroup]]: 2.3.5.7


== 13-limit ==
[[Comma list]]: 225/224, 125000000/121060821


Subgroup: 2.3.5.7.11.13
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


Comma list: 169/168, 225/224, 364/363, 1716/1715
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
: [[error map]]: {{val| -0.172 +0.002 -2.533 +3.511 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}}
: error map: {{val| 0.000 +0.298 -2.181 +3.946 }}


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
{{Optimal ET sequence|legend=1| 51, 53 }}


POTE generator: ~5/4 = 383.047
[[Badness]] (Sintel): 3.64


{{Val list|legend=1| 63, 72, 279cf }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0223
Comma list: 121/120, 225/224, 22000/21609


</div></div>
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


= Submajor =
Optimal tunings:
== 5-limit ==
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


Subgroup: 2.3.5
{{Optimal ET sequence|legend=0| 51, 53 }}


[[Comma list]]: 69198046875/68719476736
Badness (Sintel): 2.46


[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[POTE generator]]: ~10125/8192 = 362.321
Comma list: 121/120, 225/224, 275/273, 1040/1029


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


[[Badness]]: 0.1302
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


== 7-limit ==
{{Optimal ET sequence|legend=0| 51f, 53 }}


Subgroup: 2.3.5.7
Badness (Sintel): 1.96


[[Comma list]]: 225/224, 51200/50421
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


[[Mapping]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
[[Comma list]]: 225/224, 823543/800000


[[POTE generator]]: ~49/40 = 362.255
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


{{Val list|legend=1| 10, 33, 43, 53 }}
[[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]]: 0.0605
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


== 11-limit ==
[[Badness]] (Sintel): 3.67


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 6655/6561
Comma list: 225/224, 245/242, 1617/1600


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


POTE generator: ~27/22 = 362.101
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


{{Val list|legend=1| 53, 116, 169de, 285cde }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Badness: 0.0506
Badness (Sintel): 2.00


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 196/195, 245/242, 1001/1000


Subgroup: 2.3.5.7.11.13
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


Comma list: 169/168, 225/224, 275/273, 385/384
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


POTE generator: ~16/13 = 362.105
Badness (Sintel): 1.43


{{Val list|legend=1| 53, 116, 169de, 285cdef }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0277
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


</div></div>
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


== Interpental ==
Optimal tunings:
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


Subgroup: 2.3.5.7.11
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}
 
Badness (Sintel): 1.26


Comma list: 99/98, 176/175, 51200/50421
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


POTE generator: ~49/40 = 362.418
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
[[Comma list]]: 225/224, 9805926501/9765625000


POTE generator: ~49/40 = 362.418
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


{{Val list|legend=1| 43, 53, 96, 149d }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.9803{{c}}, ~10000/9261 = 139.9522{{c}}
: [[error map]]: {{val| +0.980 -1.214 -0.313 -0.243 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10000/9261 = 139.8184{{c}}
: error map: {{val| 0.000 -2.863 -2.136 -2.287 }}


Badness: 0.0518
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


</div></div>
[[Badness]] (Sintel): 3.81


=== 13-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11


Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 441/440, 43923/43750


Comma list: 99/98, 169/168, 176/175, 640/637
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


POTE generator: ~16/13 = 362.402
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


POTE generator: ~16/13 = 362.402
Badness (Sintel): 1.74


{{Val list|legend=1| 43, 53, 96, 149d }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0297
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


</div></div>
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


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


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


[[Comma list]]: 225/224, 5764801/5668704
Badness (Sintel): 1.35


[[Mapping]]: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


[[POTE generator]]: ~48/35 = 556.221
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
Optimal tunings:
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


[[Badness]]: 0.1293
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


== 11-limit ==
Badness (Sintel): 1.17


Subgroup: 2.3.5.7.11
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


Comma list: 225/224, 385/384, 12250/11979
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.


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~11/8 = 556.144
[[Comma list]]: 225/224, 122880/117649


{{Val list|legend=1| 28d, 41, 315cde }}
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


Badness: 0.0735
[[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 }}


= Decic =
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}
The ''decic'' temperament (10&amp;50, named by [[User:Xenllium|Xenllium]]) tempers out the [[cloudy comma]], 16807/16384 and the [[225/224|marvel comma]], 225/224 in the 7-limit. It can be extended naturally 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
[[Badness]] (Sintel): 4.37


[[Comma list]]: 225/224, 16807/16384
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val| 10 0 39 28 }}, {{val| 0 1 -1 0 }}]
Comma list: 121/120, 225/224, 352/343


Mapping generators: ~15/14, ~3
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}


{{Multival|legend=1| 10 -10 0 -39 -28 28 }}
Optimal tunings:
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


[[POTE generator]]: ~3/2 = 698.696
{{Optimal ET sequence|legend=0| 22, 55d }}


[[Tuning ranges]]:
Badness (Sintel): 2.46
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* nice range: ~3/2 = [693.129, 702.512]
* strict range: ~3/2 = [696.000, 700.000]


{{Val list|legend=1| 10, 50, 60, 110d }}
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


[[Badness]]: 0.089135
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.  


== 11-limit ==
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7.11
[[Comma list]]: 225/224, 5971968/5764801


Comma list: 225/224, 385/384, 3087/3025
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


Mapping: [{{val| 10 0 39 28 3 }}, {{val| 0 1 -1 0 2 }}]
[[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 }}


Mapping generators: ~15/14, ~3
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


POTE generator: ~3/2 = 696.791
[[Badness]] (Sintel): 4.53


Tuning ranges:
=== 11-limit ===
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
Subgroup: 2.3.5.7.11
* nice range: ~3/2 = [689.363, 702.512]
* strict range: ~3/2 = [696.000, 700.000]


{{Val list|legend=1| 10, 50, 60 }}
Comma list: 99/98, 176/175, 65536/65219


Badness: 0.063900
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


== 13-limit ==
Optimal tunings:
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


Comma list: 105/104, 144/143, 196/195, 2200/2197
Badness (Sintel): 2.26


Mapping: [{{val| 10 0 39 28 3 37 }}, {{val| 0 1 -1 0 2 0 }}]
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


Mapping generators: ~14/13, ~3
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>.


POTE generator: ~3/2 = 696.993
[[Subgroup]]: 2.3.5.7


Tuning ranges:
[[Comma list]]: 225/224, 14495514624/13841287201
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* nice range: ~3/2 = [689.363, 702.512]
* strict range: ~3/2 = [696.000, 700.000]


{{Val list|legend=1| 10, 50, 60 }}
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


Badness: 0.036880
[[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 }}


== 17-limit ==
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


Subgroup: 2.3.5.7.11.13.17
[[Badness]] (Sintel): 10.9


Comma list: 105/104, 144/143, 170/169, 196/195, 221/220
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 10 0 39 28 3 37 25 }}, {{val| 0 1 -1 0 2 0 1 }}]
Comma list: 225/224, 3840/3773, 12005/11979


Mapping generators: ~14/13, ~3
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


POTE generator: ~3/2 = 697.085
Optimal tunings:  
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}


Tuning ranges:
{{Optimal ET sequence|legend=0| 21, 54, 75e }}
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* nice range: ~3/2 = [689.363, 704.955]
* strict range: ~3/2 = [696.000, 700.000]


{{Val list|legend=1| 10, 50, 60 }}
Badness (Sintel): 5.74


Badness: 0.025064
== References ==


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