Xenharmonic Wiki

Marvel temperaments: Difference between revisions

← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,012 edits
Undo (see talk)
Tag: Undo
 
(81 intermediate revisions by 15 users not shown)
Line 1: Line 1:
This page discusses miscellaneous rank-2 temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the marvel comma or septimal kleisma.  
{{Technical data page}}
This page discusses miscellaneous [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] [[225/224]], the marvel comma or septimal kleisma.  


Temperaments considered in families and clans are:  
Temperaments considered in families and clans are:  
* [[Dicot family #Sharp|sharp]] ({25/24, 28/27}, dicot family)
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* [[Pelogic family #Pelogic|pelogic]] ({21/20, 135/128}, pelogic family)
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Augmented family #August|august]] ({36/35, 128/125}, augmented family)
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Diaschismic family #Pajara|pajara]] ({50/49, 64/63}, diaschismic family / jubilismic clan / archytas clan)
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Meantone family #Septimal meantone|meantone]] ({81/80, 126/125}, meantone family)
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* [[Magic family|magic]] ({225/224, 245/243}, magic family)
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* [[Gamelismic clan #Miracle|miracle]] ({225/224, 1029/1024}, gamelismic clan)
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* [[Semicomma family #Orwell|orwell]] ({225/224, 1728/1715}, semicomma family)
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Schismatic family #Garibaldi|garibaldi]] ({225/224, 3125/3087}, schismatic family)
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Kleismic family #Catakleismic|catakleismic]] ({225/224, 4375/4374}, kleismic family)
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* [[Cloudy clan #Decic|decic]] ({225/224, 16807/16384}, cloudy clan)
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* [[Escapade family #Escapade|escapade]] ({225/224, 65625/65536}, escapade family)
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* [[Pythagorean family #Compton|compton]] ({225/224, 250047/250000}, pythagorean family)
* ''[[Quintapole]]'' (+7812500/7411887) → [[Quintaleap family #Quintapole|Quintaleap family]]
* [[Qintosec family|qintosec]] ({225/224, 2560000/2470629}, qintosec 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 negri, wizard, tritonic, septimin, slender, triton, merman, marvo, marvolo, amavil, enneaportent, submajor, and alphorn.  
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 (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.
Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.


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


Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)
Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)


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


Negri tempers out the [[negri comma]] in the 5-limit, [[49/48]] and [[225/224]] in the 7-limit. It can be extended naturally to the 2.3.5.7.13 subgroup by adding [[91/90]] to the comma list; this will be discussed below under the title of negra.  
Wizard has a [[semi-octave]] period and is generated by an interval that can be treated as [[~]][[17/15]]. The semi-octave complement of this interval is ~[[5/4]]. Wizard can be described as {{nowrap| 22 & 72 }}. Its [[ploidacot]] is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.  


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16384
[[Comma list]]: 225/224, 118098/117649


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216


{{Multival|legend=1| 4 -3 -14 }}
[[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 }}


[[POTE generator]]: ~16/15 = 125.7549
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
[[Badness]] (Sintel): 1.03


[[Badness]]: 0.086856
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 7-limit ==
Comma list: 225/224, 385/384, 4000/3993
Subgroup: 2.3.5.7


[[Comma list]]: 49/48, 225/224
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
Optimal tunings:  
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


{{Multival|legend=1| 4 -3 2 -14 -8 13 }}
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


[[POTE generator]]: ~15/14 = 125.608
Badness (Sintel): 0.613


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
==== Lizard ====
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.026483
Comma list: 225/224, 351/350, 364/363, 385/384


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


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


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


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


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


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


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


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


== 11-limit ==
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
Subgroup: 2.3.5.7.11
 
Badness (Sintel): 0.741
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 45/44, 49/48, 56/55
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}


POTE generator: ~15/14 = 126.474
Optimal tunings:  
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


Vals: {{Val list| 9, 10, 19 }}
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


Badness: 0.026190
Badness (Sintel): 0.955


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


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


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


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


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


Badness: 0.017833
Badness (Sintel): 0.837


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


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


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


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


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


Badness: 0.038679
Badness (Sintel): 0.694


=== 13-limit ===
===== 19-limit =====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 49/48, 65/64, 91/90, 875/858
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}


POTE generator: ~14/13 = 124.716
Optimal tunings:  
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


Vals: {{Val list| 19, 29, 48df, 77cddf }}
{{Optimal ET sequence|legend=0| 72, 94, 166g }}


Badness: 0.024383
Badness (Sintel): 0.901


== Negric ==
=== Mage ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 33/32, 49/48, 77/75
Comma list: 99/98, 176/175, 1331/1296
 
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}


Mapping: [{{val| 1 2 2 3 3 }}, {{val| 0 -4 3 -2 4 }}]
Optimal tunings:  
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


POTE generator: ~15/14 = 127.039
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


Vals: {{Val list| 9, 19e }}
Badness (Sintel): 1.91


Badness: 0.030617
== Tritonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


=== 13-limit ===
Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.
Subgroup: 2.3.5.7.11.13
 
[[Subgroup]]: 2.3.5.7


Comma list: 33/32, 49/48, 65/64, 91/90
[[Comma list]]: 225/224, 50421/50000


Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


POTE generator: ~14/13 = 127.039
[[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 }}


Vals: {{Val list| 9, 19e }}
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


Badness: 0.020205
[[Badness]] (Sintel): 1.20


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


Comma list: 49/48, 55/54, 225/224
Comma list: 121/120, 225/224, 441/440


Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


POTE generator: ~15/14 = 124.539
Optimal tunings:  
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


Vals: {{Val list| 10, 19e, 29, 77cddee }}
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


Badness: 0.035296
Badness (Sintel): 0.782


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


Comma list: 49/48, 55/54, 65/64, 91/90
Comma list: 105/104, 121/120, 196/195, 275/273


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


POTE generator: ~14/13 = 124.545
Optimal tunings:  
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


Vals: {{Val list| 10, 19e, 29, 77cddeef }}
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


Badness: 0.021559
Badness (Sintel): 0.950


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


Comma list: 49/48, 121/120, 225/224
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


POTE generator: ~11/8 = 537.186
Optimal tunings:  
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


Vals: {{Val list| 9, 20, 29, 38d, 67cdde }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness: 0.041886
Badness (Sintel): 0.973


=== 13-limit ===
==== 19-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 49/48, 65/64, 91/90, 121/120
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


POTE generator: ~11/8 = 537.208
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


Vals: {{Val list| 9, 20, 29, 38df, 67cddef }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness: 0.025192
Badness (Sintel): 1.03


=== 17-limit ===
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


POTE generator: ~11/8 = 537.230
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


Vals: {{Val list| 9, 20g, 29g, 38df, 67cddefg }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness: 0.021778
Badness (Sintel): 1.04


=== 19-limit ===
=== Tritoni ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
Comma list: 225/224, 385/384, 27783/27500


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


POTE generator: ~11/8 = 537.214
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


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


Badness: 0.016828
Badness (Sintel): 1.50


= Passive =
== Septimin ==
{{see also | Archytas clan #Passion }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


Subgroup: 2.3.5.7
Septimin may be described as the {{nowrap| 41 & 50 }} temperament. It is generated by a septimal minor third ([[7/6]]), which gives rise to the name, but the generator can be taken to be the [[octave complement]], [[12/7]], such that eleven of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus eta-hendecacot. [[91edo]] may be recommended as a tuning.  


[[Comma list]]: 225/224, 256/245
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
[[Comma list]]: 225/224, 84035/82944


[[POTE generator]]: ~16/15 = 98.809
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


{{Val list|legend=1| 12, 49dd }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}
: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}


[[Badness]]: 0.0751
{{Optimal ET sequence|legend=1| 41, 91, 132d }}


= Wizard =
[[Badness]] (Sintel): 1.38


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
=== 11-limit ===
{{See also|Wizard}}
Subgroup: 2.3.5.7.11


Subgroup: 2.3.5.7
Comma list: 225/224, 245/242, 385/384


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


[[Mapping]]: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


Mapping generators: ~1225/864, ~245/216
Badness (Sintel): 1.04


[[POTE generator]]: ~5/4 = 383.256
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
Comma list: 105/104, 144/143, 196/195, 245/242


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


[[Badness]]: 0.0408
Optimal tunings:  
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


Scales: [[wizard22]]
{{Optimal ET sequence|legend=0| 41, 91 }}


== 11-limit ==
Badness (Sintel): 0.955


Subgroup: 2.3.5.7.11
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


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


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.


Mapping generators: ~99/70, ~25/22
[[Subgroup]]: 2.3.5.7


POTE generator: ~5/4 = 383.232
[[Comma list]]: 225/224, 2500000/2470629


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


Badness: 0.0185
[[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 }}


Scales: [[wizard22]]
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


=== Lizard ===
[[Badness]] (Sintel): 1.39


Subgroup: 2.3.5.7.11.13
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 225/224, 351/350, 364/363, 385/384
Comma list: 225/224, 441/440, 1344/1331


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


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


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


{{Val list|legend=1| 22, 50, 72, 122, 194df }}
Badness (Sintel): 1.20


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


==== 17-limit ====
Comma list: 144/143, 225/224, 364/363, 441/440


Subgroup: 2.3.5.7.11.13.17
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Mapping generators: ~17/12, ~17/15
Badness (Sintel): 1.14


POTE generator: ~5/4 = 383.381
=== Mermaid ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
Comma list: 225/224, 385/384, 532400/531441


Badness: 0.0145
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


==== 19-limit ====
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


Subgroup: 2.3.5.7.11.13.17.19
Badness (Sintel): 1.46


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
Comma list: 225/224, 325/324, 385/384, 10648/10647


Mapping generators: ~17/12, ~17/15
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


POTE generator: ~5/4 = 383.477
Optimal tunings:  
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


{{Val list|legend=1| 22h, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


Badness: 0.0157
Badness (Sintel): 1.47


</div></div>
== 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.


=== Gizzard ===
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.


Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7


Comma list: 225/224, 325/324, 385/384, 1573/1568
[[Comma list]]: 225/224, 589824/588245


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


Mapping generators: ~99/70, ~25/22
[[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 }}


POTE generator: ~5/4 = 383.170
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


{{Val list|legend=1| 72, 166, 238cf }}
[[Badness]] (Sintel): 1.44


Badness: 0.0203
=== 11-limit ===
Subgroup: 2.3.5.7.11


</div></div>
Comma list: 225/224, 385/384, 1331/1323


==== 17-limit ====
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


Subgroup: 2.3.5.7.11.13.17
Optimal tunings:  
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
Badness (Sintel): 0.838


Mapping generators: ~17/12, ~17/15
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 383.175
Comma list: 225/224, 275/273, 385/384, 1331/1323


{{Val list|legend=1| 72, 166g, 238cfg }}
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Badness: 0.0136
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


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


==== 19-limit ====
Badness (Sintel): 1.07


Subgroup: 2.3.5.7.11.13.17.19
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot.


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


Mapping generators: ~17/12, ~17/15
[[Comma list]]: 225/224, 1029/1000


POTE generator: ~5/4 = 383.138
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


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


Badness: 0.0148
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


== Mage ==
[[Badness]] (Sintel): 1.50


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


Comma list: 99/98, 176/175, 1331/1296
Comma list: 45/44, 56/55, 1029/1000


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


Mapping generators: ~77/54, ~55/48
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


POTE generator: ~5/4 = 383.124
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


{{Val list|legend=1| 22, 50e, 72ee, 94ee }}
Badness (Sintel): 1.51


Badness: 0.0578
== Marvolo ==
[[Subgroup]]: 2.3.5.7


= Triton =
[[Comma list]]: 225/224, 156250000/155649627


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


Subgroup: 2.3.5.7
[[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 }}


[[Comma list]]: 225/224, 1029/1000
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
[[Badness]] (Sintel): 2.11


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~7/5 = 568.865
Comma list: 225/224, 441/440, 4000/3993


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


[[Badness]]: 0.0592
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


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


Subgroup: 2.3.5.7.11
Badness (Sintel): 0.958


Comma list: 45/44, 56/55, 1029/1000
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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


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


{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
Optimal tunings:
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


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


= Tritonic =
Badness (Sintel): 0.887


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


[[Comma list]]: 225/224, 50421/50000
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


[[Mapping]]: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


[[POTE generator]]: ~7/5 = 580.286
Badness (Sintel): 0.760


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


== 11-limit ==
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Comma list: 121/120, 225/224, 441/440
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


POTE generator: ~7/5 = 580.267
Badness (Sintel): 0.895


{{Val list|legend=1| 29, 31, 60e }}
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


Badness: 0.0237
[[Comma list]]: 225/224, 40353607/40310784


=== 13-limit ===
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


Subgroup: 2.3.5.7.11.13
[[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 }}


Comma list: 105/104, 121/120, 196/195, 275/273
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
[[Badness]] (Sintel): 2.37


POTE generator: ~7/5 = 580.108
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 29, 31, 60e, 151cde }}
Comma list: 225/224, 385/384, 12005/11979


Badness: 0.0230
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


== Tritoni ==
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}})


Subgroup: 2.3.5.7.11
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Comma list: 225/224, 385/384, 27783/27500
Badness (Sintel): 1.01


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 580.389
Comma list: 169/168, 225/224, 364/363, 1716/1715
 
{{Val list|legend=1| 31, 91, 122, 153d }}


Badness: 0.0455
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


= Merman =
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}})


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Subgroup: 2.3.5.7
Badness (Sintel): 0.922


[[Comma list]]: 225/224, 2500000/2470629
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
[[Comma list]]: 225/224, 781250000/771895089


[[POTE generator]]: ~7/5 = 585.585
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


{{Val list|legend=1| 41, 84, 125 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


[[Badness]]: 0.0551
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


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


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


Comma list: 225/224, 441/440, 1344/1331
Comma list: 225/224, 385/384, 236328125/234365481


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


POTE generator: ~7/5 = 585.606
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


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


Badness: 0.0364
Badness (Sintel): 2.96
 
== 13-limit ==


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


Comma list: 144/143, 225/224, 364/363, 441/440
Comma list: 225/224, 325/324, 385/384, 831875/830466


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


POTE generator: ~7/5 = 585.657
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.0275
Badness (Sintel): 2.16


= Septimin =
==== 17-limit ====
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
Subgroup: 2.3.5.7.11.13.17


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


[[Comma list]]: 225/224, 84035/82944
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


[[POTE generator]]: ~7/6 = 263.632
Badness (Sintel): 1.96


{{Val list|legend=1| 41, 91, 132 }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


[[Badness]]: 0.054502
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


== 11-limit ==
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}
Subgroup: 2.3.5.7.11
 
Optimal tunings:
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


Comma list: 225/224, 385/384, 2401/2376
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
Badness (Sintel): 1.71


POTE generator: ~7/6 = 263.634
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Vals: {{Val list| 41, 91, 223cdef }}
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


Badness: 0.031309
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


== 13-limit ==
Optimal tunings:
Subgroup: 2.3.5.7.11.13
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Comma list: 105/104, 144/143, 196/195, 245/242
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Badness (Sintel): 1.57


POTE generator: ~7/6 = 263.700
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Vals: {{Val list| 41, 91 }}
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Badness: 0.023117
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


= Slender =
Optimal tunings:
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


[[Comma list]]: 225/224, 589824/588245
Badness (Sintel): 1.50


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


[[POTE generator]]: ~49/48 = 38.413
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


{{Val list|legend=1| 31, 94, 125 }}
Optimal tunings:
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


[[Badness]]: 0.0569
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


== 11-limit ==
Badness (Sintel): 1.53


=== Gracecord ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 1331/1323
Comma list: 225/224, 441/440, 109375/107811


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


POTE generator: ~49/48 = 38.387
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


{{Val list|legend=1| 31, 63, 94, 125 }}
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Badness: 0.02534
Badness (Sintel): 2.21
 
== 13-limit ==


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


Comma list: 225/224, 275/273, 385/384, 1331/1323
Comma list: 225/224, 364/363, 441/440, 6125/6084


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


POTE generator: ~49/48 = 38.314
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


{{Val list|legend=1| 31, 63, 94 }}
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Badness: 0.02591
Badness (Sintel): 1.83


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


Subgroup: 2.3.5.7
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Comma list: 225/224, 78125000/78121827
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


{{Multival|legend=1| 6 17 46 13 56 59 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


POTE generator: ~27/20 = 516.694
Badness (Sintel): 1.87


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


Badness: 0.0976
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


== 11-limit ==
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


Subgroup: 2.3.5.7.11
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


Comma list: 225/224, 243/242, 4000/3993
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
Badness (Sintel): 1.68


POTE generator: ~27/20 = 516.699
== Alphorn ==
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 65d, 72, 281, 353c, 425bc, 497bc }}
[[Comma list]]: 225/224, 5764801/5668704


Badness: 0.0317
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


== 13-limit ==
[[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 }}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Comma list: 225/224, 243/242, 351/350, 1625/1617
[[Badness]] (Sintel): 3.27


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~27/20 = 516.730
Comma list: 225/224, 385/384, 12250/11979


{{Val list|legend=1| 65d, 72, 137, 209, 281f, 490bcf }}
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


Badness: 0.0269
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


= Marvolo =
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Subgroup: 2.3.5.7
Badness (Sintel): 2.43


[[Comma list]]: 225/224, 156250000/155649627
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


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


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
[[Comma list]]: 225/224, 4194304/4117715


[[POTE generator]]: ~21/20 = 83.348
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
[[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]]: 0.0833
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


== 11-limit ==
[[Badness]] (Sintel): 3.57


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


Comma list: 225/224, 441/440, 4000/3993
Comma list: 99/98, 176/175, 1310720/1294139
 
Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
 
POTE generator: ~21/20 = 83.340


{{Val list|legend=1| 29, 43, 72 }}
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


Badness: 0.0290
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


== 13-limit ==
Badness (Sintel): 2.82


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


Comma list: 169/168, 225/224, 364/363, 441/440
Comma list: 99/98, 176/175, 640/637, 847/845


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


POTE generator: ~21/20 = 83.330
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


{{Val list|legend=1| 29, 43, 72, 115f }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.0215
Badness (Sintel): 1.88


</div></div>
=== Musneb ===
 
Subgroup: 2.3.5.7.11
= Amavil =
== 5-limit (mabila) ==


Subgroup: 2.3.5
Comma list: 225/224, 385/384, 66550/64827


[[Comma list]]: 268435456/263671875
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


[[Mapping]]: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


[[POTE generator]]: ~512/375 = 529.6849
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
Badness (Sintel): 2.89


[[Badness]]: 0.2325
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


== 7-limit ==
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.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 17496/16807
[[Comma list]]: 225/224, 125000000/121060821


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
[[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 }}


[[POTE generator]]: ~48/35 = 529.979
{{Optimal ET sequence|legend=1| 51, 53 }}


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
[[Badness]] (Sintel): 3.64
 
[[Badness]]: 0.1096
 
== 11-limit ==


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


Comma list: 99/98, 176/175, 864/847
Comma list: 121/120, 225/224, 22000/21609


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


POTE generator: ~15/11 = 529.974
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


{{Val list|legend=1| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=0| 51, 53 }}


Badness: 0.0426
Badness (Sintel): 2.46
 
== 13-limit ==


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


Comma list: 78/77, 99/98, 144/143, 176/175
Comma list: 121/120, 225/224, 275/273, 1040/1029


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


POTE generator: ~15/11 = 529.951
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


{{Val list|legend=1| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=0| 51f, 53 }}


Badness: 0.0258
Badness (Sintel): 1.96


= Enneaportent =
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 40353607/40310784
[[Comma list]]: 225/224, 823543/800000


[[Mapping]]: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
[[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 }}


[[POTE generator]]: ~5/4 = 383.165
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


{{Val list|legend=1| 63, 72, 495bcd }}
[[Badness]] (Sintel): 3.67
 
[[Badness]]: 0.0937
 
== 11-limit ==


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


Comma list: 225/224, 385/384, 12005/11979
Comma list: 225/224, 245/242, 1617/1600


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


POTE generator: ~5/4 = 383.146
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


{{Val list|legend=1| 63, 72, 423cd, 495bcd }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Badness: 0.0304
Badness (Sintel): 2.00
 
</div></div>
 
== 13-limit ==


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


Comma list: 169/168, 225/224, 364/363, 1716/1715
Comma list: 105/104, 196/195, 245/242, 1001/1000


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


POTE generator: ~5/4 = 383.047
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


{{Val list|legend=1| 63, 72, 279cf }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Badness: 0.0223
Badness (Sintel): 1.43


</div></div>
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


= Submajor =
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242
== 5-limit ==


Subgroup: 2.3.5
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


[[Comma list]]: 69198046875/68719476736
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


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


[[POTE generator]]: ~10125/8192 = 362.321
Badness (Sintel): 1.26


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


[[Badness]]: 0.1302
[[Subgroup]]: 2.3.5.7


== 7-limit ==
[[Comma list]]: 225/224, 9805926501/9765625000


Subgroup: 2.3.5.7
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


[[Comma list]]: 225/224, 51200/50421
[[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 }}


[[Mapping]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
[[Badness]] (Sintel): 3.81
 
[[POTE generator]]: ~49/40 = 362.255
 
{{Val list|legend=1| 10, 33, 43, 53 }}
 
[[Badness]]: 0.0605
 
== 11-limit ==


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


Comma list: 225/224, 385/384, 6655/6561
Comma list: 225/224, 441/440, 43923/43750


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


POTE generator: ~27/22 = 362.101
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


{{Val list|legend=1| 53, 116, 169de, 285cde }}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.0506
Badness (Sintel): 1.74


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


Comma list: 169/168, 225/224, 275/273, 385/384
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


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


POTE generator: ~16/13 = 362.105
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


{{Val list|legend=1| 53, 116, 169de, 285cdef }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 0.0277
Badness (Sintel): 1.35


</div></div>
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== Interpental ==
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


Comma list: 99/98, 176/175, 51200/50421
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


POTE generator: ~49/40 = 362.418
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
Badness (Sintel): 1.17


POTE generator: ~49/40 = 362.418
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


{{Val list|legend=1| 43, 53, 96, 149d }}
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


Badness: 0.0518
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 5971968/5764801


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


Subgroup: 2.3.5.7.11.13
[[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 }}


Comma list: 99/98, 169/168, 176/175, 640/637
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


POTE generator: ~16/13 = 362.402
[[Badness]] (Sintel): 4.53


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~16/13 = 362.402
Comma list: 99/98, 176/175, 65536/65219


{{Val list|legend=1| 43, 53, 96, 149d }}
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


Badness: 0.0297
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


= Alphorn =
Badness (Sintel): 2.26


Subgroup: 2.3.5.7
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


[[Comma list]]: 225/224, 5764801/5668704
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>.


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


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
[[Comma list]]: 225/224, 14495514624/13841287201


[[POTE generator]]: ~48/35 = 556.221
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
[[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 }}


[[Badness]]: 0.1293
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


== 11-limit ==
[[Badness]] (Sintel): 10.9


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


Comma list: 225/224, 385/384, 12250/11979
Comma list: 225/224, 3840/3773, 12005/11979
 
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
Optimal tunings:  
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}


POTE generator: ~11/8 = 556.144
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


{{Val list|legend=1| 28d, 41, 315cde }}
Badness (Sintel): 5.74


Badness: 0.0735
== References ==


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