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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:  
* ''[[Sharp]]'', {25/24, 28/27} → [[Dicot family #Sharp|Dicot family]]
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* ''[[Pelogic]]'', {21/20, 135/128} → [[Pelogic family #Pelogic|Pelogic family]]
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[August]]'', {36/35, 128/125} → [[Augmented family #August|Augmented family]]
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Pajara]], {50/49, 64/63} → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Meantone]], {81/80, 126/125} → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* [[Magic]], {225/224, 245/243} → [[Magic family #Magic|Magic family]]
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* ''[[Passive]]'', {225/224, 256/245} → [[Passion family #Passive|Passion family]]
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* [[Miracle]], {225/224, 1029/1024} → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Orwell]], {225/224, 1728/1715} → [[Semicomma family #Orwell|Semicomma family]]
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Garibaldi temperament|Garibaldi]], {225/224, 3125/3087} → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* [[Catakleismic]], {225/224, 4375/4374} → [[Kleismic family #Catakleismic|Kleismic family]]
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[Decic]]'', {225/224, 16807/16384} → [[Cloudy clan #Decic|Cloudy clan]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Escapade]]'', {225/224, 65625/65536} → [[Escapade family #Escapade|Escapade family]]
* ''[[Quintapole]]'' (+7812500/7411887) → [[Quintaleap family #Quintapole|Quintaleap family]]
* ''[[Fog]]'', {225/224, 156250/151263} → [[Misty family #Fog|Misty family]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Compton]]'', {225/224, 250047/250000} → [[Compton family #Compton|Compton family]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* ''[[Betic]]'', {225/224, 1071875/1062882} → [[Sycamore family #Betic|Sycamore family]]
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Houborizic]]'' {225/224, 1250000/1240029} → [[Amity family #Houborizic|Amity family]]
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Qintosec]]'', {225/224, 2560000/2470629} → [[Qintosec family #Qintosec|Qintosec family]]
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* ''[[Quintapole]]'', {225/224, 7812500/7411887} → [[Quintaleap family #Quintapole|Quintaleap family]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Maquila]]'', {225/224, 30233088/28824005} → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
* ''[[Marvo]]'', {225/224, 78125000/78121827} → [[Gravity family #Marvo|Gravity 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, marvolo, amavil, enneaportent, submajor, alphorn, tertiosec, gwazy, and gracecordial.  
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 }}
 
Badness (Sintel): 0.613
 
==== Lizard ====
Subgroup: 2.3.5.7.11.13


[[POTE generator]]: ~15/14 = 125.608
Comma list: 225/224, 351/350, 364/363, 385/384


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}


[[Badness]]: 0.026483
Optimal tunings:  
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


==== Negra ====
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
This is the 2.3.5.7.13 extension of negri.


Subgroup: 2.3.5.7.13
Badness (Sintel): 0.900


[[Comma list]]: 49/48, 65/64, 91/90
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Sval]] [[mapping]]: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -3 }}]
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


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


[[Gencom]]: [2 14/13; 49/48 65/64 91/90]
Optimal tunings:  
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


[[POTE generator]]: ~14/13 = 125.567
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
Badness (Sintel): 0.741


=== 11-limit ===
===== 19-limit =====
Subgroup: 2.3.5.7.11
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}}


Optimal GPV sequence: {{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}}


Optimal GPV sequence: {{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}}


Optimal GPV sequence: {{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}}


Optimal GPV sequence: {{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 }}
 
Optimal tunings:
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


Mapping: [{{val| 1 2 2 3 3 }}, {{val| 0 -4 3 -2 4 }}]
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


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


Optimal GPV sequence: {{Val list| 9, 19e }}
== Tritonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


Badness: 0.030617
Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.  


==== 13-limit ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13


Comma list: 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 }}


Optimal GPV sequence: {{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}}


Optimal GPV sequence: {{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}}


Optimal GPV sequence: {{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}}


Optimal GPV sequence: {{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}}


Optimal GPV sequence: {{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}}


Optimal GPV sequence: {{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}}


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


Badness: 0.016828
Badness (Sintel): 1.50


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


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, 118098/117649
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
[[Comma list]]: 225/224, 84035/82944
 
Mapping generators: ~1225/864, ~245/216
 
[[POTE generator]]: ~5/4 = 383.256


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
[[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]]: 0.040846
{{Optimal ET sequence|legend=1| 41, 91, 132d }}


Scales: [[wizard22]]
[[Badness]] (Sintel): 1.38


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


Comma list: 225/224, 385/384, 4000/3993
Comma list: 225/224, 245/242, 385/384


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


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


POTE generator: ~5/4 = 383.232
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


Optimal GPV sequence: {{Val list| 22, 50, 72, 166, 238c, 310c }}
Badness (Sintel): 1.04


Badness: 0.018539
=== 13-limit ===
 
Scales: [[wizard22]]
 
==== Lizard ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 351/350, 364/363, 385/384
Comma list: 105/104, 144/143, 196/195, 245/242


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


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


POTE generator: ~5/4 = 383.389
{{Optimal ET sequence|legend=0| 41, 91 }}


Optimal GPV sequence: {{Val list| 22, 50, 72, 122, 194df }}
Badness (Sintel): 0.955


Badness: 0.021781
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


===== 17-limit =====
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.  
Subgroup: 2.3.5.7.11.13.17


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
[[Subgroup]]: 2.3.5.7


Mapping generators: ~17/12, ~17/15
[[Comma list]]: 225/224, 2500000/2470629


POTE generator: ~5/4 = 383.381
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


Optimal GPV sequence: {{Val list| 22, 50, 72, 122g, 194dfg }}
[[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 }}


Badness: 0.014536
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


===== 19-limit =====
[[Badness]] (Sintel): 1.39
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
Comma list: 225/224, 441/440, 1344/1331


Mapping generators: ~17/12, ~17/15
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


POTE generator: ~5/4 = 383.477
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


Optimal GPV sequence: {{Val list| 22h, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


Badness: 0.015702
Badness (Sintel): 1.20


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


Comma list: 225/224, 325/324, 385/384, 1573/1568
Comma list: 144/143, 225/224, 364/363, 441/440


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


POTE generator: ~5/4 = 383.170
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Optimal GPV sequence: {{Val list| 72, 166, 238cf }}
Badness (Sintel): 1.14


Badness: 0.020252
=== Mermaid ===
Subgroup: 2.3.5.7.11


===== 17-limit =====
Comma list: 225/224, 385/384, 532400/531441
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


Mapping generators: ~17/12, ~17/15
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


POTE generator: ~5/4 = 383.175
Badness (Sintel): 1.46


Optimal GPV sequence: {{Val list| 72, 166g, 238cfg }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.013624
Comma list: 225/224, 325/324, 385/384, 10648/10647


===== 19-limit =====
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
Optimal tunings:  
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


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


POTE generator: ~5/4 = 383.138
== 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.  


Optimal GPV sequence: {{Val list| 72, 94, 166g }}
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.


Badness: 0.014810
[[Subgroup]]: 2.3.5.7


=== Mage ===
[[Comma list]]: 225/224, 589824/588245
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 1331/1296
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


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


Mapping generators: ~77/54, ~55/48
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


POTE generator: ~5/4 = 383.124
[[Badness]] (Sintel): 1.44


Optimal GPV sequence: {{Val list| 22, 50e, 72ee, 94ee }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.057799
Comma list: 225/224, 385/384, 1331/1323


== Triton ==
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''


Subgroup: 2.3.5.7
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, 1029/1000
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
Badness (Sintel): 0.838


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[POTE generator]]: ~7/5 = 568.865
Comma list: 225/224, 275/273, 385/384, 1331/1323


{{Val list|legend=1| 2, 17d, 19, 78bd, 97bd }}
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


[[Badness]]: 0.059245
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


=== 11-limit ===
{{Optimal ET sequence|legend=0| 31, 63, 94 }}
Subgroup: 2.3.5.7.11


Comma list: 45/44, 56/55, 1029/1000
Badness (Sintel): 1.07


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


POTE generator: ~7/5 = 569.144
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.  


Optimal GPV sequence: {{Val list| 2, 17d, 19, 59bde, 78bde, 97bde }}
[[Subgroup]]: 2.3.5.7


Badness: 0.045675
[[Comma list]]: 225/224, 1029/1000
 
== Tritonic ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 225/224, 50421/50000


[[Mapping]]: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
[[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 }}


[[POTE generator]]: ~7/5 = 580.286
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
[[Badness]] (Sintel): 1.50
 
[[Badness]]: 0.047578


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


Comma list: 121/120, 225/224, 441/440
Comma list: 45/44, 56/55, 1029/1000


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


POTE generator: ~7/5 = 580.267
Optimal tunings:  
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


Optimal GPV sequence: {{Val list| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


Badness: 0.023659
Badness (Sintel): 1.51


==== 13-limit ====
== Marvolo ==
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7


Comma list: 105/104, 121/120, 196/195, 275/273
[[Comma list]]: 225/224, 156250000/155649627


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


POTE generator: ~7/5 = 580.108
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7714{{c}}, ~21/20 = 83.4014{{c}}
: [[error map]]: {{val| +0.772 -0.820 -0.916 +0.381 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}}
: error map: {{val| 0.000 -2.139 -2.398 -1.362 }}


Optimal GPV sequence: {{Val list| 29, 31, 60e, 151cde }}
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


Badness: 0.022993
[[Badness]] (Sintel): 2.11


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


Comma list: 225/224, 385/384, 27783/27500
Comma list: 225/224, 441/440, 4000/3993


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


POTE generator: ~7/5 = 580.389
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


Optimal GPV sequence: {{Val list| 31, 91, 122, 153d }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.045456
Badness (Sintel): 0.958


== Merman ==
=== 13-limit ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
Subgroup: 2.3.5.7.11.13


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


[[Comma list]]: 225/224, 2500000/2470629
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


[[POTE generator]]: ~7/5 = 585.585
Badness (Sintel): 0.887


{{Val list|legend=1| 41, 84, 125 }}
=== 17-limit ===
 
Subgroup: 2.3.5.7.11.13.17
[[Badness]]: 0.055078
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 225/224, 441/440, 1344/1331
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


POTE generator: ~7/5 = 585.606
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Optimal GPV sequence: {{Val list| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Badness: 0.036383
Badness (Sintel): 0.760


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


Comma list: 144/143, 225/224, 364/363, 441/440
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


POTE generator: ~7/5 = 585.657
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


Optimal GPV sequence: {{Val list| 41, 84, 125e, 209ef, 293ef }}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Badness: 0.027544
Badness (Sintel): 0.895


== Untriton ==
== Enneaportent ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Untriton]].''
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7
[[Comma list]]: 225/224, 40353607/40310784


[[Comma list]]: 225/224, 125000000/121060821
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
 
: mapping generators: ~2592/2401, ~12005/6912
[[Mapping]]: [{{Val|1 6 -7 -7}}, {{Val|0 -9 19 20}}]
 
{{Multival|legend=1|9 -19 -20 -51 -57 7}}


[[POTE generator]]: ~7/5 = 588.641
[[Optimal tuning]]s:  
* [[WE]]: ~2592/2401 = 133.4174{{c}}, ~12005/6912 = 950.7667{{c}} (~1728/1715 = 16.8452{{c}})
: [[error map]]: {{val| +0.756 -0.422 -1.395 +0.298 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}})
: error map: {{val| 0.000 -1.361 -3.277 -1.565 }}


{{Val list|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


[[Badness]]: 0.143976
[[Badness]] (Sintel): 2.37


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


Comma list: 121/120, 225/224, 22000/21609
Comma list: 225/224, 385/384, 12005/11979


Mapping: [{{Val|1 6 -7 -7 1}}, {{Val|0 -9 19 20 5}}]
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


POTE generator: ~7/5 = 588.626
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}})


Optimal GPV sequence: {{Val list| 51, 53 }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.074295
Badness (Sintel): 1.01


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


Comma list: 121/120, 225/224, 275/273, 1040/1029
Comma list: 169/168, 225/224, 364/363, 1716/1715


Mapping: [{{Val|1 6 -7 -7 1 -12}}, {{Val|0 -9 19 20 5 32}}]
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


POTE generator: ~7/5 = 588.654
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}})


Optimal GPV sequence: {{Val list| 51f, 53 }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.047441
Badness (Sintel): 0.922


== Septimin ==
== Gracecordial ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 84035/82944
[[Comma list]]: 225/224, 781250000/771895089
 
[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


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


{{Val list|legend=1| 41, 91, 132d }}
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


[[Badness]]: 0.054502
[[Badness]] (Sintel): 2.44


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


Comma list: 225/224, 385/384, 2401/2376
Comma list: 225/224, 385/384, 236328125/234365481


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


POTE generator: ~7/6 = 263.634
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Optimal GPV sequence: {{Val list| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Badness: 0.031309
Badness (Sintel): 2.96


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


Comma list: 105/104, 144/143, 196/195, 245/242
Comma list: 225/224, 325/324, 385/384, 831875/830466


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


POTE generator: ~7/6 = 263.700
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


Optimal GPV sequence: {{Val list| 41, 91 }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.023117
Badness (Sintel): 2.16


== Misneb ==
==== 17-limit ====
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Misneb]].''
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, 4194304/4117715
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


[[Mapping]]: [{{Val|1 3 1 3}}, {{Val|0 -15 14 -2}}]
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|15 -14 2 -57 -39 44}}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


[[POTE generator]]: ~16/15 = 113.235
Badness (Sintel): 1.96


{{Val list|legend=1| 21, 32, 53 }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


[[Badness]]: 0.140970
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


Comma list: 99/98, 176/175, 1310720/1294139
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


Mapping: [{{Val|1 3 1 3 1}}, {{Val|0 -15 14 -2 26}}]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


POTE generator: ~16/15 = 113.323
Badness (Sintel): 1.71


Optimal GPV sequence: {{Val list| 21, 32e, 53, 127, 180de }}
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Badness: 0.085390
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


==== 13-limit ====
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
Subgroup: 2.3.5.7.11.13


Comma list: 99/98, 176/175, 640/637, 847/845
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Mapping: [{{Val|1 3 1 3 1 2}}, {{Val|0 -15 14 -2 26 18}}]
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


POTE generator: ~16/15 = 113.323
Badness (Sintel): 1.57


Optimal GPV sequence: {{Val list| 21, 32e, 53, 127, 180de }}
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Badness: 0.045569
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


=== Musneb ===
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 66550/64827
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Mapping: [{{Val|1 3 1 3 6}}, {{Val|0 -15 14 -2 -27}}]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


POTE generator: ~16/15 = 113.142
Badness (Sintel): 1.50


Optimal GPV sequence: {{Val list| 32, 53, 191de, 244cddee, 297cddee }}
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Badness: 0.087333
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


== Slender ==
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}
Slender (31&amp;32) tempers out the [[Hewuermera temperaments|hewuermera comma]] in addition to the marvel comma, this has a generator [[49/48]], 3 of which equal marvel's 16/15 - 15/14, and 10 generators is 5/4.


Subgroup: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


[[Comma list]]: 225/224, 589824/588245
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
Badness (Sintel): 1.53


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
=== Gracecord ===
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~49/48 = 38.413
Comma list: 225/224, 441/440, 109375/107811


{{Val list|legend=1| 31, 94, 125 }}
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


[[Badness]]: 0.056934
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


=== 11-limit ===
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 1331/1323
Badness (Sintel): 2.21


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~49/48 = 38.387
Comma list: 225/224, 364/363, 441/440, 6125/6084


Optimal GPV sequence: {{Val list| 31, 63, 94, 125 }}
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Badness: 0.025342
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 275/273, 385/384, 1331/1323
Badness (Sintel): 1.83


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~49/48 = 38.314
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Optimal GPV sequence: {{Val list| 31, 63, 94 }}
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Badness: 0.025913
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


== Marvolo ==
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 156250000/155649627
Badness (Sintel): 1.87


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


[[POTE generator]]: ~21/20 = 83.348
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
Optimal tunings:
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


[[Badness]]: 0.083338
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


=== 11-limit ===
Badness (Sintel): 1.68
Subgroup: 2.3.5.7.11


Comma list: 225/224, 441/440, 4000/3993
== Alphorn ==
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
[[Comma list]]: 225/224, 5764801/5668704


POTE generator: ~21/20 = 83.340
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


Optimal GPV sequence: {{Val list| 29, 43, 72 }}
[[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 }}


Badness: 0.028965
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


=== 13-limit ===
[[Badness]] (Sintel): 3.27
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 225/224, 364/363, 441/440
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
Comma list: 225/224, 385/384, 12250/11979


POTE generator: ~21/20 = 83.330
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


Optimal GPV sequence: {{Val list| 29, 43, 72, 115f }}
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Badness: 0.021470
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


== Amavil ==
Badness (Sintel): 2.43
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Mabila]].''


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


[[Comma list]]: 225/224, 17496/16807
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
[[Comma list]]: 225/224, 4194304/4117715


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


[[POTE generator]]: ~48/35 = 529.979
[[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 }}


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


[[Badness]]: 0.109625
[[Badness]] (Sintel): 3.57


=== 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: 99/98, 176/175, 1310720/1294139


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


POTE generator: ~15/11 = 529.974
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


Optimal GPV sequence: {{Val list| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.042649
Badness (Sintel): 2.82


=== 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: 99/98, 176/175, 640/637, 847/845


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


POTE generator: ~15/11 = 529.951
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


Optimal GPV sequence: {{Val list| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.025791
Badness (Sintel): 1.88


== Enneaportent ==
=== Musneb ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 225/224, 40353607/40310784
Comma list: 225/224, 385/384, 66550/64827


[[Mapping]]: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
Optimal tunings:
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


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


{{Val list|legend=1| 9, 63, 72, 495bcd }}
Badness (Sintel): 2.89


[[Badness]]: 0.093679
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


=== 11-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: 2.3.5.7.11


Comma list: 225/224, 385/384, 12005/11979
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
[[Comma list]]: 225/224, 125000000/121060821


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


Optimal GPV sequence: {{Val list| 9, 63, 72, 423cd, 495bcd }}
[[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 }}


Badness: 0.030426
{{Optimal ET sequence|legend=1| 51, 53 }}


=== 13-limit ===
[[Badness]] (Sintel): 3.64
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
Comma list: 121/120, 225/224, 22000/21609


POTE generator: ~5/4 = 383.047
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


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


Badness: 0.022322
{{Optimal ET sequence|legend=0| 51, 53 }}


== Submajor ==
Badness (Sintel): 2.46
Subgroup: 2.3.5


[[Comma list]]: 69198046875/68719476736
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
Comma list: 121/120, 225/224, 275/273, 1040/1029


[[POTE generator]]: ~10125/8192 = 362.321
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
Optimal tunings:
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


[[Badness]]: 0.130236
{{Optimal ET sequence|legend=0| 51f, 53 }}


=== 7-limit ===
Badness (Sintel): 1.96
Subgroup: 2.3.5.7


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


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


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


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


{{Val list|legend=1| 10, 33, 43, 53 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.1198{{c}}, ~32/21 = 745.4675{{c}}
: [[error map]]: {{val| +2.120 -1.227 +0.459 -4.423 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~32/21 = 744.1318{{c}}
: error map: {{val| 0.000 -4.769 -4.159 -10.144 }}


[[Badness]]: 0.060533
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}
 
[[Badness]] (Sintel): 3.67


=== 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, 245/242, 1617/1600


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


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


Optimal GPV sequence: {{Val list| 10, 43e, 53, 116, 169de, 285cde }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Badness: 0.050582
Badness (Sintel): 2.00


==== 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: 105/104, 196/195, 245/242, 1001/1000


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


POTE generator: ~16/13 = 362.105
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


Optimal GPV sequence: {{Val list| 10, 43e, 53, 116, 169de, 285cdef }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Badness: 0.027689
Badness (Sintel): 1.43


=== Interpental ===
=== 17-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 99/98, 176/175, 51200/50421
 
POTE generator: ~49/40 = 362.418
 
Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
 
POTE generator: ~49/40 = 362.418
 
Optimal GPV sequence: {{Val list| 43, 53, 96, 149d }}
 
Badness: 0.051806
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 99/98, 169/168, 176/175, 640/637


POTE generator: ~16/13 = 362.402
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


POTE generator: ~16/13 = 362.402
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


Optimal GPV sequence: {{Val list| 43, 53, 96, 149d }}
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Badness: 0.029680
Badness (Sintel): 1.26


== Alphorn ==
== Quintannic ==
Subgroup: 2.3.5.7
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.  


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


[[Mapping]]: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
[[Comma list]]: 225/224, 9805926501/9765625000


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


[[POTE generator]]: ~48/35 = 556.221
[[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 }}


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


[[Badness]]: 0.129258
[[Badness]] (Sintel): 3.81


=== 11-limit ===
=== 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, 441/440, 43923/43750


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


POTE generator: ~11/8 = 556.144
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Optimal GPV sequence: {{Val list| 28d, 41, 315cde }}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.073459
Badness (Sintel): 1.74


== Tertiosec ==
=== 13-limit ===
Subgroup: 2.3.5
Subgroup: 2.3.5.7.11.13


[[Comma list]]: {{monzo| -89 21 24 }}
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


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


[[POTE generator]]: ~16/15 = 112.281
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| 21, 54, 75, 96, 171 }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


[[Badness]]: 0.851003
Badness (Sintel): 1.35


=== 7-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 225/224, 14495514624/13841287201
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


[[Mapping]]: [{{val| 3 7 5 9 }}, {{val| 0 -8 7 -2 }}]
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


{{Multival|legend=1| 24 -21 6 -89 -58 73 }}
Optimal tunings:
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


[[POTE generator]]: ~15/14 = 112.283
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


{{Val list|legend=1| 21, 54, 75, 96, 171d }}
Badness (Sintel): 1.17


[[Badness]]: 0.431636
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


=== 11-limit ===
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.  
Subgroup: 2.3.5.7.11


Comma list: 225/224, 3840/3773, 12005/11979
[[Subgroup]]: 2.3.5.7
 
Mapping: [{{val| 3 7 5 9 9 }}, {{val| 0 -8 7 -2 5 }}]
 
POTE generator: ~15/14 = 112.171
 
Optimal GPV sequence: {{Val list| 21, 54, 75e }}
 
Badness: 0.173485
 
== Gwazy ==
{{see also|Very high accuracy temperaments #Kwazy}}
 
Subgroup: 2.3.5.7


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


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


{{Multival|legend=1| 16 -10 12 -53 -26 56 }}
[[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 }}


[[POTE generator]]: ~35/32 = 162.658
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


{{Val list|legend=1| 22, 74, 96, 118d }}
[[Badness]] (Sintel): 4.53
 
[[Badness]]: 0.178826


=== 11-limit ===
=== 11-limit ===
Line 997: Line 1,140:
Comma list: 99/98, 176/175, 65536/65219
Comma list: 99/98, 176/175, 65536/65219


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


POTE generator: ~11/10 = 162.592
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


Optimal GPV sequence: {{Val list| 22, 74, 96, 118d }}
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


Badness: 0.068410
Badness (Sintel): 2.26


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


[[Comma list]]: 225/224, 781250000/771895089
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 0 34 63 }}, {{val| 0 1 -20 -38 }}]
[[Comma list]]: 225/224, 14495514624/13841287201


{{Multival|legend=1| 1 -20 -38 -34 -63 -32 }}
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


[[POTE generator]]: ~3/2 = 700.824
[[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 }}


{{Val list|legend=1| 12, 113, 125, 238c, 363c }}
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


[[Badness]]: 0.096279
[[Badness]] (Sintel): 10.9


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


Comma list: 225/224, 385/384, 236328125/234365481
Comma list: 225/224, 3840/3773, 12005/11979


Mapping: [{{val| 1 0 34 63 -90 }}, {{val| 0 1 -20 -38 59 }}]
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
 
POTE generator: ~3/2 = 700.834
 
Optimal GPV sequence: {{Val list| 12e, 113, 125, 238c }}
 
Badness: 0.089588
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 831875/830466


Mapping: [{{val| 1 0 34 63 -90 -66 }}, {{val| 0 1 -20 -38 59 44 }}]
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: ~3/2 = 700.841
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


Optimal GPV sequence: {{Val list| 12e, 113, 125f, 238cf }}
Badness (Sintel): 5.74


Badness: 0.052235
== 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"