Xenharmonic Wiki

Marvel temperaments: Difference between revisions

← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,019 edits
Tags: Mobile edit Mobile web edit
Undo (see talk)
Tag: Undo
 
(129 intermediate revisions by 16 users not shown)
Line 1: Line 1:
__FORCETOC__
{{Technical data page}}
This page discusses some of the temperaments tempering out |-5 2 2 -1> =[[225/224| 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|meantone]], [[miracle]], [[Magic_family|magic]], [[pajara]], [[orwell]], [[Kleismic_family|catakleismic]], [[Garibaldi_temperament|garibaldi]], [[Augmented_family|august]], [[Pythagorean_family|compton]], [[Dicot_family|sharp]], [[Escapade_family|escapade]] and [[Pelogic_family|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)^2 = 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]]
* ''[[Quintapole]]'' (+7812500/7411887) → [[Quintaleap family #Quintapole|Quintaleap 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]]
* ''[[Decic]]'' (+16807/16384) → [[Cloudy clan #Decic|Cloudy clan]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Hendeca]]'' (+122880/117649) → [[11th-octave temperaments #Hendeca|11th-octave temperaments]]
* [[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, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, gwazy, and tertiosec, in the order of increasing [[badness]].
 
Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.


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


Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4)=64/45. Setting this equal to 10/7 gives (10/7)/(64/45)=225/224.)
Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)
 
== Wizard ==
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''
 
Wizard has a [[semi-octave]] period and is generated by an interval that can be treated as [[~]][[17/15]]. The semi-octave complement of this interval is ~[[5/4]]. Wizard can be described as {{nowrap| 22 & 72 }}. Its [[ploidacot]] is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 118098/117649
 
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216
 
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}
: error map: {{val| 0.000 -1.169 -3.111 -0.849 }}
 
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}
 
[[Badness]] (Sintel): 1.03
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 4000/3993
 
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}
 
Optimal tunings:
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
 
Badness (Sintel): 0.613
 
==== Lizard ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 351/350, 364/363, 385/384
 
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
 
Optimal tunings:
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
 
Badness (Sintel): 0.900
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
 
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
 
Badness (Sintel): 0.741
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
 
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}
 
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
 
Badness (Sintel): 0.955
 
==== Gizzard ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 1573/1568
 
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
 
Optimal tunings:
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
 
Badness (Sintel): 0.837
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
 
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
 
Badness (Sintel): 0.694
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
 
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}
 
{{Optimal ET sequence|legend=0| 72, 94, 166g }}
 
Badness (Sintel): 0.901
 
=== Mage ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 1331/1296
 
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
 
Optimal tunings:
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
 
Badness (Sintel): 1.91
 
== Tritonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''
 
Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 50421/50000
 
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
: [[error map]]: {{val| +1.354 -1.243 -0.027 -1.598 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}}
: error map: {{val| 0.000 -3.566 -2.769 -4.959 }}
 
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
 
[[Badness]] (Sintel): 1.20
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 225/224, 441/440
 
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}
 
Optimal tunings:
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}
 
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
 
Badness (Sintel): 0.782
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 121/120, 196/195, 275/273
 
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}
 
Optimal tunings:
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}
 
{{Optimal ET sequence|legend=0| 29, 31, 60e }}
 
Badness (Sintel): 0.950
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
 
Optimal tunings:
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
Badness (Sintel): 0.973
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
 
Optimal tunings:
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
Badness (Sintel): 1.03
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}
 
Optimal tunings:
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
Badness (Sintel): 1.04
 
=== Tritoni ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 27783/27500
 
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}
 
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 ===
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 }}


=Negri=
Badness (Sintel): 1.47
[[Comma|Comma]]s: 49/48, 225/224


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


[[Map|Map]]: [&lt;1 2 2 3|, &lt;0 -4 3 -2|]
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.


[[wedgie|Wedgie]]: &lt;&lt;4 -3 2 -14 -8 13||
[[Subgroup]]: 2.3.5.7


EDOs: [[9edo|9]], [[10edo|10]], [[19edo|19]], [[86edo|86]]
[[Comma list]]: 225/224, 589824/588245


==11-limit==
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
Commas: 45/44, 49/48, 56/55
: mapping generators: ~2, ~49/48


[[POTE_tuning|POTE generator]]: ~15/14 = 126.474
[[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 4|, &lt;0 -4 3 -2 -5|]
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


EDOs: 9, 10, 19
[[Badness]] (Sintel): 1.44


[[Badness|Badness]]: 0.0262
=== 11-limit ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 225/224, 385/384, 1331/1323
Commas: 45/44, 49/48, 56/55, 78/77


[[POTE_tuning|POTE generator]]: ~14/13 = 126.431
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


Map: [&lt;1 2 2 3 4 4|, &lt;0 -4 3 -2 -5 -3|]
Optimal tunings:  
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


EDOs: 9, 10, 19
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


=Negril=
Badness (Sintel): 0.838
Commas: 49/48, 100/99, 225/224


POTE generator: ~15/14 = 124.767
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 2 3 2|, &lt;0 -4 3 -2 14|]
Comma list: 225/224, 275/273, 385/384, 1331/1323


EDOs: 19, [[29edo|29]], [[48edo|48df]], [[77edo|77cdf]]
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Badness: 0.0387
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 31, 63, 94 }}
Commas: 49/48, 65/64, 91/90, 875/858


POTE generator: ~15/14 = 124.716
Badness (Sintel): 1.07


Map: [&lt;1 2 2 3 2 4|, &lt;0 -4 3 -2 14 -3|]
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


EDOs: 9, 10, 19, 29, 48, 77
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.


Badness: 0.0244
[[Subgroup]]: 2.3.5.7


=Negric=
[[Comma list]]: 225/224, 1029/1000
Commas: 33/32, 49/48, 77/75


POTE generator: ~15/14 = 127.039
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


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


EDOs: 9, 19, [[85edo|85]]
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


Badness: 0.0306
[[Badness]] (Sintel): 1.50


==13-limit==
=== 11-limit ===
Commas: 33/32, 49/48, 65/64, 91/90
Subgroup: 2.3.5.7.11


POTE generator: ~15/14 = 127.039
Comma list: 45/44, 56/55, 1029/1000


Map: [&lt;1 2 2 3 3 4|, &lt;0 -4 3 -2 4 -3|]
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


EDOs: 9, 19, 85
Optimal tunings:  
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


Badness: 0.0202
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


=Negroni=
Badness (Sintel): 1.51
Commas: 49/48, 55/54, 225/224


POTE generator: ~15/14 = 124.539
== Marvolo ==
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 2 3 5|, &lt;0 -4 3 -2 -15|]
[[Comma list]]: 225/224, 156250000/155649627


EDOs: 10, 19e, 29, 77cde
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


Badness: 0.0353
[[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 }}


==13-limit==
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}
Commas: 49/48, 55/54, 65/64, 91/90


POTE generator: ~15/14 = 124.545
[[Badness]] (Sintel): 2.11


Map: [&lt;1 2 2 3 5 4|, &lt;0 -4 3 -2 -15 -3|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 10, 19e, 29, 77cdef
Comma list: 225/224, 441/440, 4000/3993


Badness: 0.0216
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


=Wizard=
Optimal tunings:
Comma: 2197265625/2147483648
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


POTE generator: ~5/4 = 383.212
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Map: [&lt;2 1 5|, &lt;0 6 -1|]
Badness (Sintel): 0.958


EDOs: 22, 50, 72, 166, 238c, 310c, 548bc
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.3864
Comma list: 169/168, 225/224, 364/363, 441/440


==7-limit==
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
Commas: 225/224, 118098/117649


[[POTE_tuning|POTE temperament]]: 383.256
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


Map: [&lt;2 1 5 2|, &lt;0 6 -1 10|]
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Wedgie: &lt;&lt;12 -2 20 -31 -2 52||
Badness (Sintel): 0.887


EDOs: [[22edo|22]], [[50edo|50]], [[72edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238c]], [[310edo|310c]], [[382edo|382c]]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0408
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


==11-limit==
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
Commas: 225/224, 385/384, 4000/3993


[[POTE_tuning|POTE temperament]]: 383.232
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Map: [&lt;2 1 5 2 8|, &lt;0 6 -1 10 -3|]
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


EDOs: 22, 50, 72, 94, [[166edo|166]], [[238edo|238c]], [[310edo|310c]]
Badness (Sintel): 0.760


Badness: 0.0185
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


==Lizard==
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
Commas: 225/224, 351/350, 364/363, 385/384


[[POTE_tuning|POTE temperament]]: 383.389
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


EDOs: 22, 50, 72, 122, 194df
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Badness: 0.0218
Badness (Sintel): 0.895


===17-limit===
== Enneaportent ==
Commas: 221/220, 273/272, 289/288, 351/350, 375/374
[[Subgroup]]: 2.3.5.7


[[POTE_tuning|POTE temperament]]: 383.381
[[Comma list]]: 225/224, 40353607/40310784


Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


EDOs: 22, 50, 72, 122g, 194dfg
[[Optimal tuning]]s:  
* [[WE]]: ~2592/2401 = 133.4174{{c}}, ~12005/6912 = 950.7667{{c}} (~1728/1715 = 16.8452{{c}})
: [[error map]]: {{val| +0.756 -0.422 -1.395 +0.298 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}})
: error map: {{val| 0.000 -1.361 -3.277 -1.565 }}


Badness: 0.0145
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


===19-limit===
[[Badness]] (Sintel): 2.37
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


[[POTE_tuning|POTE generator]]: ~5/4 = 383.477
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
Comma list: 225/224, 385/384, 12005/11979


EDOs: 22h, 50, 72, [[122edo|122g]], [[194edo|194dfg]]
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


Badness: 0.0157
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}})


==Gizzard==
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
Commas: 225/224, 385/384, 325/324, 1573/1568


POTE generator: ~5/4 = 383.170
Badness (Sintel): 1.01


Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: [[22edo|22]], [[26edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238cf]]
Comma list: 169/168, 225/224, 364/363, 1716/1715


Badness: 0.0203
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


===17-limit===
Optimal tunings:
Commas: 225/224, 289/288, 325/324, 375/374, 385/384
* 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: ~5/4 = 383.175
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
Badness (Sintel): 0.922


EDOs: 22f, 72, 94, [[166edo|166g]], [[238edo|238cfg]]
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


Badness: 0.0136
[[Subgroup]]: 2.3.5.7


===19-limit===
[[Comma list]]: 225/224, 781250000/771895089
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399


POTE generator: ~5/4 = 383.138
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]
[[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: 22f, 72, 94, 166g
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


Badness: 0.0148
[[Badness]] (Sintel): 2.44


==Mage==
=== 11-limit ===
Commas: 99/98, 176/175, 1331/1296
Subgroup: 2.3.5.7.11


POTE generator: ~55/48 = 216.876
Comma list: 225/224, 385/384, 236328125/234365481


Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


EDOs: 22, 72e, 94e
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Badness: 0.0578
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


=Triton=
Badness (Sintel): 2.96
Commas: 225/224, 1029/1000


[[POTE_tuning|POTE temperament]]: ~7/5 = 568.865
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
Comma list: 225/224, 325/324, 385/384, 831875/830466


Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


EDOs: 17d, 19, 78bd, 97bd
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


Badness: 0.0592
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


==11-limit==
Badness (Sintel): 2.16
Commas: 45/44, 56/55, 1029/1000


POTE generator: ~7/5 = 569.144
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 0 6 7 4|, &lt;0 3 -7 -8 -1|]
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


EDOs: 19, 59bde, 78bde, 97bde
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


Badness: 0.0457
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


=Tritonic=
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
Comnma: 553584375/536870912


POTE generator: ~45/32 = 580.219
Badness (Sintel): 1.96


Map: [&lt;1 4 -3|, &lt;0 -5 11|]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


EDOs: 29, 31, 60, 91
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


Badness: 0.3785
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


==7-limit==
Optimal tunings:
Commas: 225/224, 50421/50000
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


[[POTE_tuning|POTE temperament]]: 580.286
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Map: [&lt;1 4 -3 -3|, &lt;0 -5 11 12|]
Badness (Sintel): 1.71


Wedgie: &lt;&lt;5 -11 -12 -29 -33 3||
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


EDOs: [[29edo|29]], [[31edo|31]], [[60edo|60]], [[91edo|91]], [[122edo|122]], 213bcd
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


==11-limit==
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
Commas: 121/120, 225/224, 441/440


[[POTE_tuning|POTE generator]]: ~7/5 = 580.267
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


EDOs: 29, 31, 60e
Badness (Sintel): 1.57


Badness: 0.0237
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


==13-limit==
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
Commas: 105/104, 121/120, 196/195, 275/273


POTE generator: ~7/5 = 580.108
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 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


EDOs: 29, 31, 60e, 151cde
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Badness: 0.0230
Badness (Sintel): 1.50


==Tritoni==
==== 31-limit ====
Commas: 225/224, 385/384, 27783/27500
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


POTE generator: ~7/5 = 580.389
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
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: 31, 91, 122
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


Badness: 0.0455
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


=Merman=
Badness (Sintel): 1.53
Comma: 1121008359375/1099511627776


POTE generator: ~45/32 = 585.634
=== Gracecord ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 5 -5|, &lt;0 -7 15|]
Comma list: 225/224, 441/440, 109375/107811


EDOs: 41, 84, 125, 209, 334c, 543bc
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


Badness: 0.4538
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, 2500000/2470629


POTE generator: ~7/5 = 585.585
Badness (Sintel): 2.21


Map: [&lt;1 5 -5 -5|, &lt;0 -7 15 16|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
Comma list: 225/224, 364/363, 441/440, 6125/6084


EDOs: 41, 84, 125
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Badness: 0.0551
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}
Commas: 225/224, 441/440, 1344/1331


POTE generator: ~7/5 = 585.606
Badness (Sintel): 1.83


Map: [&lt;1 5 -5 -5 2|, &lt;0 -7 15 16 3]]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


EDOs: 41, 84, 125e
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Badness: 0.0364
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


==13-limit==
Optimal tunings:
Commas: 144/143, 225/224, 364/363, 441/440
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


POTE generator: ~7/5 = 585.657
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Map: [&lt;1 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
Badness (Sintel): 1.87


EDOs: 41, 84, 125e, 209ef, 293ef
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0275
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


=Septimin=
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}
Comma: 35595703125/34359738368


POTE generator: ~75/64 = 263.658
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


Map: [&lt;1 4 1|, &lt;0 -11 6|]
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


EDOs: 9, 32, 41, 91, 132, 223c, 355bc
Badness (Sintel): 1.68


Badness: 0.6709
== Alphorn ==
[[Subgroup]]: 2.3.5.7


==7-limit==
[[Comma list]]: 225/224, 5764801/5668704
Commas: 225/224, 84035/82944


[[POTE_tuning|POTE temperament]]: 263.632
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


Map: [&lt;1 4 1 5|, &lt;0 -11 6 -10|]
[[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 }}


Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]
[[Badness]] (Sintel): 3.27


==11-limit==
=== 11-limit ===
Commas: 225/224, 385/384, 2401/2376
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE temperament]]: 263.634
Comma list: 225/224, 385/384, 12250/11979


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


EDOs: 9, 32, 41, 91, 223cdef
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}
Commas: 105/104, 144/143, 196/195, 245/242


[[POTE_tuning|POTE temperament]]: 263.700
Badness (Sintel): 2.43


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


EDOs: 9, 41, 91
[[Subgroup]]: 2.3.5.7


=Slender=
[[Comma list]]: 225/224, 4194304/4117715
Commas: 225/224, 589824/588245


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.413
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
[[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 }}


Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


EDOs: 31, 94, 125
[[Badness]] (Sintel): 3.57


Badness: 0.0569
=== 11-limit ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 99/98, 176/175, 1310720/1294139
Commas: 225/224, 385/384, 1331/1323


[[POTE_tuning|POTE temperament]]: ~49/48 = 38.387
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


EDOs: 31, 63, 94, 125
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 25.342
Badness (Sintel): 2.82


==13-limit==
==== 13-limit ====
Commas: 225/224, 275/273, 385/384, 1331/1323
Subgroup: 2.3.5.7.11.13


POTE generator: ~49/48 = 38.314
Comma list: 99/98, 176/175, 640/637, 847/845


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


EDOs: 31, 63, 94
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


Badness: 25.913
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


=Marvo=
Badness (Sintel): 1.88
Commas: 225/224, 78125000/78121827


[[POTE_tuning|POTE generator]]: ~27/20 = 516.694
=== Musneb ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
Comma list: 225/224, 385/384, 66550/64827


Wedgie: &lt;&lt;6 17 46 13 56 59||
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


EDOs: 7d, 65d, 72, 137, 209, 281, 569bcc
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


Badness: 0.0976
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


==11-limit==
Badness (Sintel): 2.89
Commas: 225/224, 243/242, 4000/3993


POTE generator: ~27/20 = 516.699
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
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: 7d, 65d, 72, 281, 353c, 425bc, 497bc
[[Subgroup]]: 2.3.5.7


Badness: 0.0317
[[Comma list]]: 225/224, 125000000/121060821


==13-limit==
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
Commas: 225/224, 243/242, 351/350, 1625/1617
: mapping generators: ~2, ~10/7


POTE generator: ~27/20 = 516.730
[[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: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
{{Optimal ET sequence|legend=1| 51, 53 }}


EDOs: 72, 137, 209, 281f, 490bcf
[[Badness]] (Sintel): 3.64


Badness: 0.0269
=== 11-limit ===
Subgroup: 2.3.5.7.11


=Marvolo=
Comma list: 121/120, 225/224, 22000/21609
Commas: 225/224, 156250000/155649627


POTE generator: ~21/20 = 83.348
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


Map: [&lt;1 2 1 1|, &lt;0 -6 19 26|]
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;6 -19 -26 -44 -58 -7||
{{Optimal ET sequence|legend=0| 51, 53 }}


EDOs: 29, 43, 72, 619bcd, 691bcd
Badness (Sintel): 2.46


Badness: 0.0833
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 121/120, 225/224, 275/273, 1040/1029
Commas: 225/224, 441/440, 4000/3993


POTE generator: ~21/20 = 83.340
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


Map: [&lt;1 2 1 1 2|, &lt;0 -6 19 26 21|]
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


EDOs: 29, 43, 72
{{Optimal ET sequence|legend=0| 51f, 53 }}


Badness: 0.0290
Badness (Sintel): 1.96


==13-limit==
== Naiadical ==
Commas: 169/168, 225/224, 364/363, 441/440
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


POTE generator: ~21/20 = 83.330
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
[[Comma list]]: 225/224, 823543/800000


EDOs: 29, 43, 72, 115f
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


Badness: 0.0215
[[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 }}


=Amavil=
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}
Commas: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
[[Badness]] (Sintel): 3.67


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
Comma list: 225/224, 245/242, 1617/1600


EDOs: 9, 34d, 43, 77d
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


Badness: 0.1096
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: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
Badness (Sintel): 2.00


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 9, 34d, 43, 77de
Comma list: 105/104, 196/195, 245/242, 1001/1000


Badness: 0.0426
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


==13-limit==
Optimal tunings:
Commas: 78/77, 99/98, 144/143, 176/175
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


POTE generator: ~15/11 = 529.951
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
Badness (Sintel): 1.43


EDOs: 9, 34d, 43, 77de
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0258
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


=Enneaportent=
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}
Commas: 225/224, 40353607/40310784


POTE generator: ~5/4 = 383.165
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
Badness (Sintel): 1.26


EDOs: 9, 54, 63, 72, 495bcd
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


Badness: 0.0937
[[Subgroup]]: 2.3.5.7


==11-limit==
[[Comma list]]: 225/224, 9805926501/9765625000
Commas: 225/224, 385/384, 12005/11979


POTE generator: ~5/4 = 383.146
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
[[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: 9, 54, 63, 72, 423cd, 495bcd
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


Badness: 0.0304
[[Badness]] (Sintel): 3.81


==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: 225/224, 441/440, 43923/43750


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


EDOs: 9, 54, 63, 72, 279cf
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Badness: 0.0223
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


=Submajor=
Badness (Sintel): 1.74
Comma: 69198046875/68719476736


POTE generator: ~10125/8192 = 362.321
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 4 -1|, &lt;0 -8 11|]
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


EDOs: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


Badness: 0.1302
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


==7-limit==
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
Commas: 225/224, 51200/50421


POTE generator: ~49/40 = 362.255
Badness (Sintel): 1.35


Map: [&lt;1 4 -1 1|, &lt;0 -8 11 6|]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


EDOs: 10, 33, 43, 53
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


Badness: 0.0605
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
Commas: 225/224, 385/384, 6655/6561


POTE generator: ~27/22 = 362.101
Badness (Sintel): 1.17


Map: [&lt;1 4 -1 1 11|, &lt;0 -8 11 6 -25|]
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


EDOs: 10, 53, 116, 169de, 285cde
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


Badness: 0.0506
[[Subgroup]]: 2.3.5.7


==13-limit==
[[Comma list]]: 225/224, 5971968/5764801
Commas: 169/168, 225/224, 275/273, 385/384


POTE generator: ~16/13 = 362.105
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


Map: [&lt;1 4 -1 1 11 4|, &lt;0 -8 11 6 -25 -1|]
[[Optimal tuning]]s:  
* [[WE]]: ~2401/1728 = 599.7132{{c}}, ~35/32 = 162.5806{{c}}
: [[error map]]: {{val| -0.574 -1.597 -0.937 +5.510 }}
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~35/32 = 162.6388{{c}}
: error map: {{val| 0.000 -0.844 +0.492 +7.007 }}


EDOs: 10, 53, 116, 169de, 285cdef
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


Badness: 0.0277
[[Badness]] (Sintel): 4.53


==Interpental==
=== 11-limit ===
Commas: 99/98, 176/175, 51200/50421
Subgroup: 2.3.5.7.11


POTE generator: ~49/40 = 362.418
Comma list: 99/98, 176/175, 65536/65219


Map: [&lt;1 4 -1 1 -5|, &lt;0 -8 11 6 28|]
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


EDOs: 43, 53, 96, 149d
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


Badness: 0.0518
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


===13-limit===
Badness (Sintel): 2.26
Commas: 99/98, 169/168, 176/175, 640/637


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


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


EDOs: 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"