Xenharmonic Wiki

Marvel temperaments: Difference between revisions

← Older edit
VisualWikitext
Formatting and some cleanup
Overthink (talk | contribs)
autopatrolled
2,970 edits
 
(134 intermediate revisions by 15 users not shown)
Line 1: Line 1:
__FORCETOC__
{{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
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
 
Optimal tunings:
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
Badness (Sintel): 0.973
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
 
Optimal tunings:
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
Badness (Sintel): 1.03
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}
 
Optimal tunings:
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


=Negri=
Badness (Sintel): 1.04
{{main|Negri}}


==5-limit==
=== Tritoni ===
Note that this is technically not a Marvel temperament until the 7-limit, but also placed here for reference.  
Subgroup: 2.3.5.7.11


[[Comma|Comma]]s: 16875/16384
Comma list: 225/224, 385/384, 27783/27500


[[POTE_tuning|POTE temperament]]: ~15/14 = 125.7549
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


[[Map|Map]]: [&lt;1 2 2|, &lt;0 -4 3|]
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


[[wedgie|Wedgie]]: &lt;&lt;4 -3 -14||
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


EDOs: {{EDOs|9, 10, 19, 86}}
Badness (Sintel): 1.50


==7-limit==
== Septimin ==
[[Comma|Comma]]s: 49/48, 225/224
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


[[POTE_tuning|POTE temperament]]: ~15/14 = 125.608
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.  


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


[[wedgie|Wedgie]]: &lt;&lt;4 -3 2 -14 -8 13||
[[Comma list]]: 225/224, 84035/82944


EDOs: {{EDOs|9, 10, 19, 86}}
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


==11-limit==
[[Optimal tuning]]s:
Commas: 45/44, 49/48, 56/55
* [[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 }}


[[POTE_tuning|POTE generator]]: ~15/14 = 126.474
{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}


Map: [&lt;1 2 2 3 4|, &lt;0 -4 3 -2 -5|]
[[Badness]] (Sintel): 1.38


EDOs: {{EDOs|9, 10, 19}}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Badness|Badness]]: 0.0262
Comma list: 225/224, 245/242, 385/384
 
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
 
Optimal tunings:
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}
 
{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}
 
Badness (Sintel): 1.04


=== 13-limit ===
=== 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| 9, 32f, 41, 91 }}
 
Badness (Sintel): 0.955
 
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''
 
Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot.
 
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 2500000/2470629
 
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


[[POTE_tuning|POTE generator]]: ~14/13 = 126.431
[[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 }}


Map: [&lt;1 2 2 3 4 4|, &lt;0 -4 3 -2 -5 -3|]
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


EDOs: {{EDOs|9, 10, 19}}
[[Badness]] (Sintel): 1.39


== Negril ==
=== 11-limit ===
Commas: 49/48, 100/99, 225/224
Subgroup: 2.3.5.7.11


POTE generator: ~15/14 = 124.767
Comma list: 225/224, 441/440, 1344/1331


Map: [&lt;1 2 2 3 2|, &lt;0 -4 3 -2 14|]
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


EDOs: {{EDOs|19, 29, 48df, 77cdf}}
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


Badness: 0.0387
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
 
Badness (Sintel): 1.20
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
 
Optimal tunings:
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
 
Badness (Sintel): 1.14
 
=== Mermaid ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 532400/531441
 
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
 
Optimal tunings:
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}
 
Badness (Sintel): 1.46
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 10648/10647
 
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
 
Optimal tunings:
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
 
Badness (Sintel): 1.47
 
== Slender ==
Slender tempers out the [[hewuermera comma]] in addition to the marvel comma, and may be described as the {{nowrap| 31 & 32 }} temperament. This temperament has a generator of [[49/48]], three of which equal marvel's [[16/15]][[~]][[15/14]], and ten generators give [[5/4]]. Its [[ploidacot]] is omega-13-cot.
 
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 589824/588245
 
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
: [[error map]]: {{val| +0.382 -0.725 -1.295 +1.765 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}}
: error map: {{val| 0.000 -1.257 -2.235 +0.727 }}
 
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}
 
[[Badness]] (Sintel): 1.44
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 1331/1323
 
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
 
Optimal tunings:
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}
 
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
 
Badness (Sintel): 0.838


=== 13-limit ===
=== 13-limit ===
Commas: 49/48, 65/64, 91/90, 875/858
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 275/273, 385/384, 1331/1323
 
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
 
Optimal tunings:
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}
 
{{Optimal ET sequence|legend=0| 31, 63, 94 }}
 
Badness (Sintel): 1.07
 
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''
 
Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 1029/1000
 
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}
: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}
 
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
 
[[Badness]] (Sintel): 1.50
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 56/55, 1029/1000
 
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
 
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}
 
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
 
Badness (Sintel): 1.51
 
== Marvolo ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 156250000/155649627
 
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


POTE generator: ~15/14 = 124.716
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7714{{c}}, ~21/20 = 83.4014{{c}}
: [[error map]]: {{val| +0.772 -0.820 -0.916 +0.381 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}}
: error map: {{val| 0.000 -2.139 -2.398 -1.362 }}


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


EDOs: {{EDOs|9, 10, 19, 29, 48, 77}}
[[Badness]] (Sintel): 2.11


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


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


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


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


EDOs: {{EDOs|9, 19, 85}}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.0306
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}}


POTE generator: ~15/14 = 127.039
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Map: [&lt;1 2 2 3 3 4|, &lt;0 -4 3 -2 4 -3|]
Badness (Sintel): 0.887


EDOs: {{EDOs|9, 19, 85}}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0202
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


== Negroni ==
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
Commas: 49/48, 55/54, 225/224


POTE generator: ~15/14 = 124.539
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Map: [&lt;1 2 2 3 5|, &lt;0 -4 3 -2 -15|]
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


EDOs: {{EDOs|10, 19e, 29, 77cde}}
Badness (Sintel): 0.760


Badness: 0.0353
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


=== 13-limit ===
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
Commas: 49/48, 55/54, 65/64, 91/90


POTE generator: ~15/14 = 124.545
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Map: [&lt;1 2 2 3 5 4|, &lt;0 -4 3 -2 -15 -3|]
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


EDOs: {{EDOs|10, 19e, 29, 77cdef}}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Badness: 0.0216
Badness (Sintel): 0.895


=Wizard=
== Decic ==
Comma: 2197265625/2147483648
{{Main| Decic }}


POTE generator: ~5/4 = 383.212
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]].  


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


EDOs: {{EDOs|22, 50, 72, 166, 238c, 310c, 548bc}}
[[Subgroup]]: 2.3.5.7


Badness: 0.3864
[[Comma list]]: 225/224, 16807/16384


==7-limit==
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
Commas: 225/224, 118098/117649
: mapping generators: ~15/14, ~3


[[POTE_tuning|POTE temperament]]: 383.256
[[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 }}


Map: [&lt;2 1 5 2|, &lt;0 6 -1 10|]
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


Wedgie: &lt;&lt;12 -2 20 -31 -2 52||
[[Badness]] (Sintel): 2.26


EDOs: {{EDOs|22, 50, 72, 94, 166, 238c, 310c, 382c}}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0408
Comma list: 225/224, 385/384, 3087/3025


==11-limit==
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}
Commas: 225/224, 385/384, 4000/3993


[[POTE_tuning|POTE temperament]]: 383.232
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}})


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


EDOs: {{EDOs|22, 50, 72, 94, 166, 238c, 310c}}
Badness (Sintel): 2.11


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


=== Lizard ===
Comma list: 105/104, 144/143, 196/195, 2200/2197
Commas: 225/224, 351/350, 364/363, 385/384


[[POTE_tuning|POTE temperament]]: 383.389
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
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}})


EDOs: {{EDOs|22, 50, 72, 122, 194df}}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Badness: 0.0218
Badness (Sintel): 1.52


==== 17-limit ====
==== 17-limit ====
Commas: 221/220, 273/272, 289/288, 351/350, 375/374
Subgroup: 2.3.5.7.11.13.17


[[POTE_tuning|POTE temperament]]: 383.381
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


EDOs: {{EDOs|22, 50, 72, 122g, 194dfg}}
Optimal tunings:  
* WE: ~15/14 = 120.1262{{c}}, ~3/2 = 697.8185{{c}} (~78/77 = 22.9388{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.2757{{c}} (~78/77 = 22.7243{{c}})


Badness: 0.0145
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


==== 19-limit ====
Badness (Sintel): 1.28
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


[[POTE_tuning|POTE generator]]: ~5/4 = 383.477
=== Splendecic ===
Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[fantastic]] rank-3 temperament, its name is a portmanteau of ''splendid'' and ''decic''.


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
Subgroup: 2.3.5.7.11


EDOs: {{EDOs|22h, 50, 72, 122g, 194dfg}}
Comma list: 225/224, 1617/1600, 2401/2376


Badness: 0.0157
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}


=== Gizzard ===
Optimal tunings:
Commas: 225/224, 385/384, 325/324, 1573/1568
* 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: ~5/4 = 383.170
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]
Badness (Sintel): 1.98


EDOs: {{EDOs|22, 72, 94, 166, 238cf}}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0203
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


==== 17-limit ====
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}
Commas: 225/224, 289/288, 325/324, 375/374, 385/384


POTE generator: ~5/4 = 383.175
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}})


Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


EDOs: {{EDOs|22f, 72, 94, 166g, 238cfg}}
Badness (Sintel): 1.57


Badness: 0.0136
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


==== 19-limit ====
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399


POTE generator: ~5/4 = 383.138
Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]
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}})


EDOs: {{EDOs|22f, 72, 94, 166g}}
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


Badness: 0.0148
Badness (Sintel): 1.33


== Mage ==
=== Prodecic ===
Commas: 99/98, 176/175, 1331/1296
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''.


POTE generator: ~55/48 = 216.876
Subgroup: 2.3.5.7.11


Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
Comma list: 225/224, 441/440, 5929/5832


EDOs: {{EDOs|22, 72e, 94e}}
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}


Badness: 0.0578
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}})


=Triton=
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
Commas: 225/224, 1029/1000


[[POTE_tuning|POTE temperament]]: ~7/5 = 568.865
Badness (Sintel): 2.20


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
Comma list: 105/104, 196/195, 275/273, 5929/5832


EDOs: {{EDOs|17d, 19, 78bd, 97bd}}
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


Badness: 0.0592
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}})


==11-limit==
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
Commas: 45/44, 56/55, 1029/1000


POTE generator: ~7/5 = 569.144
Badness (Sintel): 1.73


Map: [&lt;1 0 6 7 4|, &lt;0 3 -7 -8 -1|]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


EDOs: {{EDOs|19, 59bde, 78bde, 97bde}}
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


Badness: 0.0457
Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


= Tritonic =
Optimal tunings:
== 5-limit ==
* WE: ~15/14 = 120.1577{{c}}, ~3/2 = 701.3950{{c}} (~91/90 = 19.5514{{c}})
Comnma: 553584375/536870912
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.6932{{c}} (~91/90 = 19.3068{{c}})


POTE generator: ~45/32 = 580.219
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Map: [&lt;1 4 -3|, &lt;0 -5 11|]
Badness (Sintel): 1.41


EDOs: {{EDOs|29, 31, 60, 91}}
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


Badness: 0.3785
[[Comma list]]: 225/224, 40353607/40310784


== 7-limit ==
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
Commas: 225/224, 50421/50000
: mapping generators: ~2592/2401, ~12005/6912


[[POTE_tuning|POTE temperament]]: 580.286
[[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 }}


Map: [&lt;1 4 -3 -3|, &lt;0 -5 11 12|]
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


Wedgie: &lt;&lt;5 -11 -12 -29 -33 3||
[[Badness]] (Sintel): 2.37


EDOs: {{EDOs|29, 31, 60, 91, 122, 213bcd}}
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 225/224, 385/384, 12005/11979
Commas: 121/120, 225/224, 441/440


[[POTE_tuning|POTE generator]]: ~7/5 = 580.267
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
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}})


EDOs: {{EDOs|29, 31, 60e}}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.0237
Badness (Sintel): 1.01


=== 13-limit ===
=== 13-limit ===
Commas: 105/104, 121/120, 196/195, 275/273
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 580.108
Comma list: 169/168, 225/224, 364/363, 1716/1715


Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


EDOs: {{EDOs|29, 31, 60e, 151cde}}
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}})


Badness: 0.0230
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


== Tritoni ==
Badness (Sintel): 0.922
Commas: 225/224, 385/384, 27783/27500


POTE generator: ~7/5 = 580.389
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
[[Subgroup]]: 2.3.5.7


EDOs: {{EDOs|31, 91, 122}}
[[Comma list]]: 225/224, 781250000/771895089


Badness: 0.0455
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


=Merman=
[[Optimal tuning]]s:
Comma: 1121008359375/1099511627776
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


POTE generator: ~45/32 = 585.634
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


Map: [&lt;1 5 -5|, &lt;0 -7 15|]
[[Badness]] (Sintel): 2.44


EDOs: {{EDOs|41, 84, 125, 209, 334c, 543bc}}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.4538
Comma list: 225/224, 385/384, 236328125/234365481


==7-limit==
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
Commas: 225/224, 2500000/2470629


POTE generator: ~7/5 = 585.585
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Map: [&lt;1 5 -5 -5|, &lt;0 -7 15 16|]
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
Badness (Sintel): 2.96


EDOs: {{EDOs|41, 84, 125}}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0551
Comma list: 225/224, 325/324, 385/384, 831875/830466


==11-limit==
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
Commas: 225/224, 441/440, 1344/1331


POTE generator: ~7/5 = 585.606
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


Map: [&lt;1 5 -5 -5 2|, &lt;0 -7 15 16 3]]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


EDOs: {{EDOs|41, 84, 125e}}
Badness (Sintel): 2.16


Badness: 0.0364
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


==13-limit==
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
Commas: 144/143, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 585.657
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


Map: [&lt;1 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


EDOs: {{EDOs|41, 84, 125e, 209ef, 293ef}}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.0275
Badness (Sintel): 1.96


=Septimin=
==== 19-limit ====
Comma: 35595703125/34359738368
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~75/64 = 263.658
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


Map: [&lt;1 4 1|, &lt;0 -11 6|]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


EDOs: {{EDOs|9, 32, 41, 91, 132, 223c, 355bc}}
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


Badness: 0.6709
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


==7-limit==
Badness (Sintel): 1.71
Commas: 225/224, 84035/82944


[[POTE_tuning|POTE temperament]]: 263.632
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Map: [&lt;1 4 1 5|, &lt;0 -11 6 -10|]
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


EDOs: {{EDOs|9, 32, 41, 91, 132}}
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}
Commas: 225/224, 385/384, 2401/2376


[[POTE_tuning|POTE temperament]]: 263.634
Badness (Sintel): 1.57


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


EDOs: {{EDOs|9, 32, 41, 91, 223cdef}}
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


==13-limit==
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}
Commas: 105/104, 144/143, 196/195, 245/242


[[POTE_tuning|POTE temperament]]: 263.700
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


EDOs: {{EDOs|9, 41, 91}}
Badness (Sintel): 1.50


=Slender=
==== 31-limit ====
Commas: 225/224, 589824/588245
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.413
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


EDOs: {{EDOs|31, 94, 125}}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Badness: 0.0569
Badness (Sintel): 1.53


==11-limit==
=== Gracecord ===
Commas: 225/224, 385/384, 1331/1323
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.387
Comma list: 225/224, 441/440, 109375/107811


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


EDOs: {{EDOs|31, 63, 94, 125}}
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


Badness: 25.342
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


==13-limit==
Badness (Sintel): 2.21
Commas: 225/224, 275/273, 385/384, 1331/1323


POTE generator: ~49/48 = 38.314
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
Comma list: 225/224, 364/363, 441/440, 6125/6084


EDOs: {{EDOs|31, 63, 94}}
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Badness: 25.913
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


=Marvo=
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}
Commas: 225/224, 78125000/78121827


[[POTE_tuning|POTE generator]]: ~27/20 = 516.694
Badness (Sintel): 1.83


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Wedgie: &lt;&lt;6 17 46 13 56 59||
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


EDOs: {{EDOs|7d, 65d, 72, 137, 209, 281, 569bcc}}
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Badness: 0.0976
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
Commas: 225/224, 243/242, 4000/3993


POTE generator: ~27/20 = 516.699
Badness (Sintel): 1.87


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


EDOs: {{EDOs|7d, 65d, 72, 281, 353c, 425bc, 497bc}}
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


Badness: 0.0317
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


==13-limit==
Optimal tunings:
Commas: 225/224, 243/242, 351/350, 1625/1617
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


POTE generator: ~27/20 = 516.730
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
Badness (Sintel): 1.68


EDOs: {{EDOs|72, 137, 209, 281f, 490bcf}}
== Alphorn ==
[[Subgroup]]: 2.3.5.7


Badness: 0.0269
[[Comma list]]: 225/224, 5764801/5668704


=Marvolo=
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
Commas: 225/224, 156250000/155649627
: mapping generators: ~2, ~35/24


POTE generator: ~21/20 = 83.348
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.3004{{c}}, ~35/24 = 644.4767{{c}}
: [[error map]]: {{val| +1.300 +0.569 -2.195 -2.043 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~35/24 = 643.8137{{c}}
: error map: {{val| 0.000 -0.936 -5.382 -4.924 }}


Map: [&lt;1 2 1 1|, &lt;0 -6 19 26|]
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||
[[Badness]] (Sintel): 3.27


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


Badness: 0.0833
Comma list: 225/224, 385/384, 12250/11979


==11-limit==
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}
Commas: 225/224, 441/440, 4000/3993


POTE generator: ~21/20 = 83.340
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Map: [&lt;1 2 1 1 2|, &lt;0 -6 19 26 21|]
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


EDOs: {{EDOs|29, 43, 72}}
Badness (Sintel): 2.43


Badness: 0.0290
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 169/168, 225/224, 364/363, 441/440


POTE generator: ~21/20 = 83.330
[[Comma list]]: 225/224, 4194304/4117715


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


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


Badness: 0.0215
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


=Amavil=
[[Badness]] (Sintel): 3.57
Commas: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
Comma list: 99/98, 176/175, 1310720/1294139


Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


EDOs: {{EDOs|9, 34d, 43, 77d}}
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


Badness: 0.1096
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


==11-limit==
Badness (Sintel): 2.82
Commas: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
Comma list: 99/98, 176/175, 640/637, 847/845


EDOs: {{EDOs|9, 34d, 43, 77de}}
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


Badness: 0.0426
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}
Commas: 78/77, 99/98, 144/143, 176/175


POTE generator: ~15/11 = 529.951
Badness (Sintel): 1.88


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


EDOs: {{EDOs|9, 34d, 43, 77de}}
Comma list: 225/224, 385/384, 66550/64827


Badness: 0.0258
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


=Enneaportent=
Optimal tunings:
Commas: 225/224, 40353607/40310784
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


POTE generator: ~5/4 = 383.165
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
Badness (Sintel): 2.89


Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


EDOs: {{EDOs|9, 54, 63, 72, 495bcd}}
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.


Badness: 0.0937
[[Subgroup]]: 2.3.5.7


==11-limit==
[[Comma list]]: 225/224, 125000000/121060821
Commas: 225/224, 385/384, 12005/11979


POTE generator: ~5/4 = 383.146
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
[[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 }}


EDOs: {{EDOs|9, 54, 63, 72, 423cd, 495bcd}}
{{Optimal ET sequence|legend=1| 51, 53 }}


Badness: 0.0304
[[Badness]] (Sintel): 3.64


==13-limit==
=== 11-limit ===
Commas: 169/168, 225/224, 364/363, 1716/1715
Subgroup: 2.3.5.7.11


POTE generator: ~5/4 = 383.047
Comma list: 121/120, 225/224, 22000/21609


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


EDOs: {{EDOs|9, 54, 63, 72, 279cf}}
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


Badness: 0.0223
{{Optimal ET sequence|legend=0| 51, 53 }}


= Submajor =
Badness (Sintel): 2.46
== 5-limit ==
Comma: 69198046875/68719476736


POTE generator: ~10125/8192 = 362.321
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 225/224, 275/273, 1040/1029
 
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}
 
Optimal tunings:
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}
 
{{Optimal ET sequence|legend=0| 51f, 53 }}


Map: [&lt;1 4 -1|, &lt;0 -8 11|]
Badness (Sintel): 1.96


EDOs: {{EDOs|10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


Badness: 0.1302
[[Subgroup]]: 2.3.5.7


== 7-limit ==
[[Comma list]]: 225/224, 823543/800000
Commas: 225/224, 51200/50421


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


Map: [&lt;1 4 -1 1|, &lt;0 -8 11 6|]
[[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 }}


Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


EDOs: {{EDOs|10, 33, 43, 53}}
[[Badness]] (Sintel): 3.67


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


== 11-limit ==
Comma list: 225/224, 245/242, 1617/1600
Commas: 225/224, 385/384, 6655/6561


POTE generator: ~27/22 = 362.101
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


Map: [&lt;1 4 -1 1 11|, &lt;0 -8 11 6 -25|]
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


EDOs: {{EDOs|10, 53, 116, 169de, 285cde}}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Badness: 0.0506
Badness (Sintel): 2.00


=== 13-limit ===
=== 13-limit ===
Commas: 169/168, 225/224, 275/273, 385/384
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 196/195, 245/242, 1001/1000
 
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}
 
Optimal tunings:
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}
 
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
 
Badness (Sintel): 1.43
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242
 
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}
 
Optimal tunings:
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}
 
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}
 
Badness (Sintel): 1.26
 
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 9805926501/9765625000
 
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


POTE generator: ~16/13 = 362.105
[[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 }}


Map: [&lt;1 4 -1 1 11 4|, &lt;0 -8 11 6 -25 -1|]
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


EDOs: {{EDOs|10, 53, 116, 169de, 285cdef}}
[[Badness]] (Sintel): 3.81


Badness: 0.0277
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Interpental ==
Comma list: 225/224, 441/440, 43923/43750
Commas: 99/98, 176/175, 51200/50421


POTE generator: ~49/40 = 362.418
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


Map: [&lt;1 4 -1 1 -5|, &lt;0 -8 11 6 28|]
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


EDOs: {{EDOs|43, 53, 96, 149d}}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.0518
Badness (Sintel): 1.74


=== 13-limit ===
=== 13-limit ===
Commas: 99/98, 169/168, 176/175, 640/637
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
 
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
 
Optimal tunings:
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}
 
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
 
Badness (Sintel): 1.35
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
 
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
 
Optimal tunings:
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}
 
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
 
Badness (Sintel): 1.17
 
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''
 
Hendeca tempers out the same 5-limit comma as [[porwell temperaments #Hendecatonic|hendecatonic]], and has a period of 1/11 octave. However, in this temperament, nine periods represent [[7/4]], the same as [[keemic temperaments #Undeka|undeka]]. It can be tuned to [[22edo]] or [[33edo]] using the [[patent val]], or [[55edo]] using the 55d val. It was named by [[Xenllium]] in 2025 as a low-accuracy variant of hendecatonic.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 122880/117649
 
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~16/15 = 108.9526{{c}}, ~3/2 = 702.4215{{c}}
: [[error map]]: {{val| -1.521 -1.055 -2.252 +8.705 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 703.2071{{c}}
: error map: {{val| 0.000 +1.252 +1.388 +12.992 }}
 
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}
 
[[Badness]] (Sintel): 4.37
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 225/224, 352/343
 
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}
 
Optimal tunings:
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}
 
{{Optimal ET sequence|legend=0| 22, 55d }}
 
Badness (Sintel): 2.46
 
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
 
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 5971968/5764801
 
{{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]]
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"