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Marvel temperaments: Difference between revisions

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{{Technical data page}}
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]].
This page discusses miscellaneous [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] [[225/224]], the marvel comma or septimal kleisma.  


Since (5/4)<sup>2</sup> = 225/224 × 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
Temperaments considered in families and clans are:
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]
 
Considered below are wizard, tritonic, septimin, merman, slender, triton, marvolo, decic, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, hendeca, gwazy, and tertiosec, in the order of increasing [[badness]].
 
Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.


The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.
The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.


Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)
Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)
 
== Wizard ==
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''
 
Wizard has a [[semi-octave]] period and is generated by an interval that can be treated as [[~]][[17/15]]. The semi-octave complement of this interval is ~[[5/4]]. Wizard can be described as {{nowrap| 22 & 72 }}. Its [[ploidacot]] is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 118098/117649
 
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216
 
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}
: error map: {{val| 0.000 -1.169 -3.111 -0.849 }}
 
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}
 
[[Badness]] (Sintel): 1.03
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 4000/3993
 
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}
 
Optimal tunings:
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
 
Badness (Sintel): 0.613
 
==== Lizard ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 351/350, 364/363, 385/384
 
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
 
Optimal tunings:
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
 
Badness (Sintel): 0.900
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
 
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
 
Badness (Sintel): 0.741
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
 
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}
 
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
 
Badness (Sintel): 0.955
 
==== Gizzard ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 1573/1568
 
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
 
Optimal tunings:
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
 
Badness (Sintel): 0.837
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
 
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
 
Badness (Sintel): 0.694
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
 
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}
 
{{Optimal ET sequence|legend=0| 72, 94, 166g }}
 
Badness (Sintel): 0.901
 
=== Mage ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 1331/1296
 
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
 
Optimal tunings:
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
 
Badness (Sintel): 1.91
 
== Tritonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''
 
Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 50421/50000
 
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
: [[error map]]: {{val| +1.354 -1.243 -0.027 -1.598 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}}
: error map: {{val| 0.000 -3.566 -2.769 -4.959 }}
 
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
 
[[Badness]] (Sintel): 1.20
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 225/224, 441/440
 
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}
 
Optimal tunings:
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}
 
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
 
Badness (Sintel): 0.782
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 121/120, 196/195, 275/273
 
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}
 
Optimal tunings:
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}
 
{{Optimal ET sequence|legend=0| 29, 31, 60e }}
 
Badness (Sintel): 0.950
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


=Negri=
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
{{main|Negri}}


==5-limit==
Optimal tunings:
Note that this is technically not a Marvel temperament until the 7-limit, but also placed here for reference.  
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


[[Comma|Comma]]s: 16875/16384
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


[[POTE_tuning|POTE temperament]]: ~15/14 = 125.7549
Badness (Sintel): 0.973


[[Map|Map]]: [&lt;1 2 2|, &lt;0 -4 3|]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


[[wedgie|Wedgie]]: &lt;&lt;4 -3 -14||
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


EDOs: {{EDOs|9, 10, 19, 86}}
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


==7-limit==
Optimal tunings:
[[Comma|Comma]]s: 49/48, 225/224
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


[[POTE_tuning|POTE temperament]]: ~15/14 = 125.608
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


[[Map|Map]]: [&lt;1 2 2 3|, &lt;0 -4 3 -2|]
Badness (Sintel): 1.03


[[wedgie|Wedgie]]: &lt;&lt;4 -3 2 -14 -8 13||
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


EDOs: {{EDOs|9, 10, 19, 86}}
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


==11-limit==
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}
Commas: 45/44, 49/48, 56/55


[[POTE_tuning|POTE generator]]: ~15/14 = 126.474
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


Map: [&lt;1 2 2 3 4|, &lt;0 -4 3 -2 -5|]
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


EDOs: {{EDOs|9, 10, 19}}
Badness (Sintel): 1.04


[[Badness|Badness]]: 0.0262
=== 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 ===
Commas: 45/44, 49/48, 56/55, 78/77
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 196/195, 245/242
 
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
 
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}
 
{{Optimal ET sequence|legend=0| 41, 91 }}
 
Badness (Sintel): 0.955
 
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''
 
Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot.
 
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 2500000/2470629
 
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
: [[error map]]: {{val| +0.390 -0.435 -1.630 +1.634 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}}
: error map: {{val| 0.000 -1.104 -2.423 +0.657 }}
 
{{Optimal ET sequence|legend=1| 41, 84, 125 }}
 
[[Badness]] (Sintel): 1.39
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 1344/1331
 
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
 
Badness (Sintel): 1.20
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
 
Optimal tunings:
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
 
Badness (Sintel): 1.14
 
=== Mermaid ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 532400/531441
 
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
 
Optimal tunings:
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}
 
Badness (Sintel): 1.46
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 10648/10647
 
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
 
Optimal tunings:
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
 
Badness (Sintel): 1.47
 
== Slender ==
Slender tempers out the [[hewuermera comma]] in addition to the marvel comma, and may be described as the {{nowrap| 31 & 32 }} temperament. This temperament has a generator of [[49/48]], three of which equal marvel's [[16/15]][[~]][[15/14]], and ten generators give [[5/4]]. Its [[ploidacot]] is omega-13-cot.


[[POTE_tuning|POTE generator]]: ~14/13 = 126.431
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  


Map: [&lt;1 2 2 3 4 4|, &lt;0 -4 3 -2 -5 -3|]
[[Subgroup]]: 2.3.5.7


EDOs: {{EDOs|9, 10, 19}}
[[Comma list]]: 225/224, 589824/588245


== Negril ==
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
Commas: 49/48, 100/99, 225/224
: mapping generators: ~2, ~49/48


POTE generator: ~15/14 = 124.767
[[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 }}


Map: [&lt;1 2 2 3 2|, &lt;0 -4 3 -2 14|]
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


EDOs: {{EDOs|19, 29, 48df, 77cdf}}
[[Badness]] (Sintel): 1.44


Badness: 0.0387
=== 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 ===
Commas: 49/48, 65/64, 91/90, 875/858
Subgroup: 2.3.5.7.11.13


POTE generator: ~15/14 = 124.716
Comma list: 225/224, 275/273, 385/384, 1331/1323


Map: [&lt;1 2 2 3 2 4|, &lt;0 -4 3 -2 14 -3|]
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


EDOs: {{EDOs|9, 10, 19, 29, 48, 77}}
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Badness: 0.0244
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


== Negric ==
Badness (Sintel): 1.07
Commas: 33/32, 49/48, 77/75


POTE generator: ~15/14 = 127.039
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


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


EDOs: {{EDOs|9, 19, 85}}
[[Subgroup]]: 2.3.5.7


Badness: 0.0306
[[Comma list]]: 225/224, 1029/1000
 
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}
: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}
 
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
 
[[Badness]] (Sintel): 1.50
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 56/55, 1029/1000
 
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
 
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}
 
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
 
Badness (Sintel): 1.51
 
== Marvolo ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 156250000/155649627
 
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7714{{c}}, ~21/20 = 83.4014{{c}}
: [[error map]]: {{val| +0.772 -0.820 -0.916 +0.381 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}}
: error map: {{val| 0.000 -2.139 -2.398 -1.362 }}
 
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}
 
[[Badness]] (Sintel): 2.11
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 4000/3993
 
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}
 
Optimal tunings:
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}
 
{{Optimal ET sequence|legend=0| 29, 43, 72 }}
 
Badness (Sintel): 0.958


=== 13-limit ===
=== 13-limit ===
Commas: 33/32, 49/48, 65/64, 91/90
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
 
Optimal tunings:
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}
 
{{Optimal ET sequence|legend=0| 29, 43, 72 }}
 
Badness (Sintel): 0.887
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
 
Optimal tunings:
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
 
Badness (Sintel): 0.760
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
 
Optimal tunings:
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
 
Badness (Sintel): 0.895


POTE generator: ~15/14 = 127.039
== Decic ==
{{Main| Decic }}


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


EDOs: {{EDOs|9, 19, 85}}
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.


Badness: 0.0202
[[Subgroup]]: 2.3.5.7


== Negroni ==
[[Comma list]]: 225/224, 16807/16384
Commas: 49/48, 55/54, 225/224


POTE generator: ~15/14 = 124.539
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


Map: [&lt;1 2 2 3 5|, &lt;0 -4 3 -2 -15|]
[[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 }}


EDOs: {{EDOs|10, 19e, 29, 77cde}}
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


Badness: 0.0353
[[Badness]] (Sintel): 2.26


=== 13-limit ===
=== 11-limit ===
Commas: 49/48, 55/54, 65/64, 91/90
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 3087/3025
 
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}
 
Optimal tunings:
* WE: ~15/14 = 120.1406{{c}}, ~3/2 = 697.6075{{c}} (~56/55 = 23.2360{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.0142{{c}} (~56/55 = 22.9858{{c}})
 
{{Optimal ET sequence|legend=0| 10, 40, 50 }}
 
Badness (Sintel): 2.11


POTE generator: ~15/14 = 124.545
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 2 3 5 4|, &lt;0 -4 3 -2 -15 -3|]
Comma list: 105/104, 144/143, 196/195, 2200/2197


EDOs: {{EDOs|10, 19e, 29, 77cdef}}
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


Badness: 0.0216
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 }}
Comma: 2197265625/2147483648


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


Map: [&lt;2 1 5|, &lt;0 6 -1|]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


EDOs: {{EDOs|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:
Commas: 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_tuning|POTE temperament]]: 383.256
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Map: [&lt;2 1 5 2|, &lt;0 6 -1 10|]
Badness (Sintel): 1.28


Wedgie: &lt;&lt;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''.


EDOs: {{EDOs|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 }}
Commas: 225/224, 385/384, 4000/3993


[[POTE_tuning|POTE temperament]]: 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}})


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


EDOs: {{EDOs|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
Commas: 225/224, 351/350, 364/363, 385/384


[[POTE_tuning|POTE temperament]]: 383.389
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


Map: [&lt;2 1 5 2 8 11|, &lt;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}})


EDOs: {{EDOs|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 ====
Commas: 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_tuning|POTE temperament]]: 383.381
Badness (Sintel): 1.33


Map: [&lt;2 1 5 2 8 11 6|, &lt;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''.


EDOs: {{EDOs|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 }}
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


[[POTE_tuning|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}})


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


EDOs: {{EDOs|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
Commas: 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 }}


Map: [&lt;2 1 5 2 8 -2|, &lt;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}})


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


Badness: 0.0203
Badness (Sintel): 1.73


==== 17-limit ====
==== 17-limit ====
Commas: 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


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


EDOs: {{EDOs|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
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399


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


Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]
[[Comma list]]: 225/224, 40353607/40310784


EDOs: {{EDOs|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 }}
Commas: 99/98, 176/175, 1331/1296


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


Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: {{EDOs|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:
Commas: 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_tuning|POTE temperament]]: ~7/5 = 568.865
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
Badness (Sintel): 1.01


Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: {{EDOs|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:
Commas: 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 }}


Map: [&lt;1 0 6 7 4|, &lt;0 3 -7 -8 -1|]
Badness (Sintel): 0.922


EDOs: {{EDOs|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 ==
Comnma: 553584375/536870912


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


Map: [&lt;1 4 -3|, &lt;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 }}


EDOs: {{EDOs|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 ===
Commas: 225/224, 50421/50000
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE temperament]]: 580.286
Comma list: 225/224, 385/384, 236328125/234365481


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


Wedgie: &lt;&lt;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}}


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


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


[[POTE_tuning|POTE generator]]: ~7/5 = 580.267
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
Comma list: 225/224, 325/324, 385/384, 831875/830466


EDOs: {{EDOs|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 }}
Commas: 105/104, 121/120, 196/195, 275/273


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


Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


EDOs: {{EDOs|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:
Commas: 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 }}


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
Badness (Sintel): 1.96


EDOs: {{EDOs|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: 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}}


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


EDOs: {{EDOs|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
Commas: 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 }}


Map: [&lt;1 5 -5 -5|, &lt;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: &lt;&lt;7 -15 -16 -40 -45 5||
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


EDOs: {{EDOs|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
Commas: 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 }}


Map: [&lt;1 5 -5 -5 2|, &lt;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}}


EDOs: {{EDOs|41, 84, 125e}}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Badness: 0.0364
Badness (Sintel): 1.50


==13-limit==
==== 31-limit ====
Commas: 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


Map: [&lt;1 5 -5 -5 2 12|, &lt;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 }}


EDOs: {{EDOs|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: 35595703125/34359738368


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


Map: [&lt;1 4 1|, &lt;0 -11 6|]
Comma list: 225/224, 441/440, 109375/107811


EDOs: {{EDOs|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 }}
Commas: 225/224, 84035/82944


[[POTE_tuning|POTE temperament]]: 263.632
Badness (Sintel): 2.21


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


Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
Comma list: 225/224, 364/363, 441/440, 6125/6084


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


==11-limit==
Optimal tunings:
Commas: 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_tuning|POTE temperament]]: 263.634
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
Badness (Sintel): 1.83


EDOs: {{EDOs|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
Commas: 105/104, 144/143, 196/195, 245/242


[[POTE_tuning|POTE temperament]]: 263.700
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Map: [&lt;1 4 1 5 5 7|, &lt;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}}


EDOs: {{EDOs|9, 41, 91}}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


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


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.413
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


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


EDOs: {{EDOs|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
Commas: 225/224, 385/384, 1331/1323


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.387
== Alphorn ==
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
[[Comma list]]: 225/224, 5764801/5668704


EDOs: {{EDOs|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 }}
Commas: 225/224, 275/273, 385/384, 1331/1323


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


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: {{EDOs|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:
Commas: 225/224, 78125000/78121827
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


[[POTE_tuning|POTE generator]]: ~27/20 = 516.694
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
Badness (Sintel): 2.43


Wedgie: &lt;&lt;6 17 46 13 56 59||
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


EDOs: {{EDOs|7d, 65d, 72, 137, 209, 281, 569bcc}}
[[Subgroup]]: 2.3.5.7


Badness: 0.0976
[[Comma list]]: 225/224, 4194304/4117715


==11-limit==
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
Commas: 225/224, 243/242, 4000/3993
: mapping generators: ~2, ~15/8


POTE generator: ~27/20 = 516.699
[[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 }}


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


EDOs: {{EDOs|7d, 65d, 72, 281, 353c, 425bc, 497bc}}
[[Badness]] (Sintel): 3.57


Badness: 0.0317
=== 11-limit ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 99/98, 176/175, 1310720/1294139
Commas: 225/224, 243/242, 351/350, 1625/1617


POTE generator: ~27/20 = 516.730
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


EDOs: {{EDOs|72, 137, 209, 281f, 490bcf}}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.0269
Badness (Sintel): 2.82


=Marvolo=
==== 13-limit ====
Commas: 225/224, 156250000/155649627
Subgroup: 2.3.5.7.11.13


POTE generator: ~21/20 = 83.348
Comma list: 99/98, 176/175, 640/637, 847/845


Map: [&lt;1 2 1 1|, &lt;0 -6 19 26|]
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


EDOs: {{EDOs|29, 43, 72, 619bcd, 691bcd}}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.0833
Badness (Sintel): 1.88


==11-limit==
=== Musneb ===
Commas: 225/224, 441/440, 4000/3993
Subgroup: 2.3.5.7.11


POTE generator: ~21/20 = 83.340
Comma list: 225/224, 385/384, 66550/64827


Map: [&lt;1 2 1 1 2|, &lt;0 -6 19 26 21|]
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


EDOs: {{EDOs|29, 43, 72}}
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


Badness: 0.0290
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


==13-limit==
Badness (Sintel): 2.89
Commas: 169/168, 225/224, 364/363, 441/440


POTE generator: ~21/20 = 83.330
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.


EDOs: {{EDOs|29, 43, 72, 115f}}
[[Subgroup]]: 2.3.5.7


Badness: 0.0215
[[Comma list]]: 225/224, 125000000/121060821


= Amavil =
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
== 5-limit (mabila) ==
: mapping generators: ~2, ~10/7
Comma: 268435456/263671875


POTE generator: ~512/375 = 529.6849
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
: [[error map]]: {{val| -0.172 +0.002 -2.533 +3.511 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}}
: error map: {{val| 0.000 +0.298 -2.181 +3.946 }}


Map: [<1 6 1|, <0 -10 3|]
{{Optimal ET sequence|legend=1| 51, 53 }}


EDOs: {{EDOs| 9, 25, 34, 77, 111, 145, 256c }}
[[Badness]] (Sintel): 3.64


Badness: 0.2325
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 7-limit ==
Comma list: 121/120, 225/224, 22000/21609
Commas: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
{{Optimal ET sequence|legend=0| 51, 53 }}


EDOs: {{EDOs|9, 34d, 43, 77d}}
Badness (Sintel): 2.46


Badness: 0.1096
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 11-limit ==
Comma list: 121/120, 225/224, 275/273, 1040/1029
Commas: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


EDOs: {{EDOs| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=0| 51f, 53 }}


Badness: 0.0426
Badness (Sintel): 1.96


== 13-limit ==
== Naiadical ==
Commas: 78/77, 99/98, 144/143, 176/175
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


POTE generator: ~15/11 = 529.951
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
[[Comma list]]: 225/224, 823543/800000


EDOs: {{EDOs| 9, 34d, 43, 77de }}
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


Badness: 0.0258
[[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 }}


=Enneaportent=
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}
Commas: 225/224, 40353607/40310784


POTE generator: ~5/4 = 383.165
[[Badness]] (Sintel): 3.67


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
Comma list: 225/224, 245/242, 1617/1600


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


Badness: 0.0937
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
Commas: 225/224, 385/384, 12005/11979


POTE generator: ~5/4 = 383.146
Badness (Sintel): 2.00


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: {{EDOs|9, 54, 63, 72, 423cd, 495bcd}}
Comma list: 105/104, 196/195, 245/242, 1001/1000


Badness: 0.0304
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


==13-limit==
Optimal tunings:
Commas: 169/168, 225/224, 364/363, 1716/1715
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


POTE generator: ~5/4 = 383.047
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
Badness (Sintel): 1.43


EDOs: {{EDOs|9, 54, 63, 72, 279cf}}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0223
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


= Submajor =
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}
== 5-limit ==
Comma: 69198046875/68719476736


POTE generator: ~10125/8192 = 362.321
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


Map: [&lt;1 4 -1|, &lt;0 -8 11|]
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


EDOs: {{EDOs|10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
Badness (Sintel): 1.26


Badness: 0.1302
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.  


== 7-limit ==
[[Subgroup]]: 2.3.5.7
Commas: 225/224, 51200/50421


POTE generator: ~49/40 = 362.255
[[Comma list]]: 225/224, 9805926501/9765625000


Map: [&lt;1 4 -1 1|, &lt;0 -8 11 6|]
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
[[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 }}


EDOs: {{EDOs|10, 33, 43, 53}}
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


Badness: 0.0605
[[Badness]] (Sintel): 3.81


== 11-limit ==
=== 11-limit ===
Commas: 225/224, 385/384, 6655/6561
Subgroup: 2.3.5.7.11


POTE generator: ~27/22 = 362.101
Comma list: 225/224, 441/440, 43923/43750


Map: [&lt;1 4 -1 1 11|, &lt;0 -8 11 6 -25|]
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


EDOs: {{EDOs|10, 53, 116, 169de, 285cde}}
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Badness: 0.0506
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}
 
Badness (Sintel): 1.74


=== 13-limit ===
=== 13-limit ===
Commas: 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


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


EDOs: {{EDOs|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
Commas: 99/98, 176/175, 51200/50421


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


Map: [&lt;1 4 -1 1 -5|, &lt;0 -8 11 6 28|]
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


EDOs: {{EDOs|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 }}
Commas: 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]].''


Map: [&lt;1 4 -1 1 -5 4|, &lt;0 -8 11 6 28 -1|]
Tertiosec may be described as the {{nowrap| 21 & 75 }} temperament. It was initially named ''tertiomar'' by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, but was changed to ''tertiosec'' in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.


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


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


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


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


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


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


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


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


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


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


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


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


Badness: 0.0735
== References ==


[[Category:marvel]]
[[Category:Temperament collections]]
[[Category:overview]]
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:rank_2]]
[[Category:Rank 2]]
[[Category:temperament]]
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