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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]]: 225/224, 118098/117649


[[Comma list]]: 16875/16384
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
[[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 }}


{{Multival|legend=1| 4 -3 -14 }}
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}


[[POTE generator]]: ~16/15 = 125.7549
[[Badness]] (Sintel): 1.03


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Badness]]: 0.086856
Comma list: 225/224, 385/384, 4000/3993


=== 7-limit ===
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}
Subgroup: 2.3.5.7


[[Comma list]]: 49/48, 225/224
Optimal tunings:  
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


{{Multival|legend=1| 4 -3 2 -14 -8 13 }}
Badness (Sintel): 0.613


[[POTE generator]]: ~15/14 = 125.608
==== Lizard ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
Comma list: 225/224, 351/350, 364/363, 385/384


[[Badness]]: 0.026483
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}


==== Negra ====
Optimal tunings:
This is the 2.3.5.7.13 extension of negri.  
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


Subgroup: 2.3.5.7.13
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


[[Comma list]]: 49/48, 65/64, 91/90
Badness (Sintel): 0.900


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


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


[[Gencom]]: [2 14/13; 49/48 65/64 91/90]
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}


[[POTE generator]]: ~14/13 = 125.567
Optimal tunings:  
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


=== 11-limit ===
Badness (Sintel): 0.741
Subgroup: 2.3.5.7.11
 
===== 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}}


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


POTE generator: ~15/14 = 127.039
Optimal tunings:  
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


Optimal GPV sequence: {{Val list| 9, 19e }}
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


Badness: 0.030617
Badness (Sintel): 1.91


==== 13-limit ====
== Tritonic ==
Subgroup: 2.3.5.7.11.13
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


Comma list: 33/32, 49/48, 65/64, 91/90
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.


Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~14/13 = 127.039
[[Comma list]]: 225/224, 50421/50000
 
Optimal GPV sequence: {{Val list| 9, 19e }}


Badness: 0.020205
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


=== Negroni ===
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11
* [[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 }}


Comma list: 49/48, 55/54, 225/224
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
[[Badness]] (Sintel): 1.20


POTE generator: ~15/14 = 124.539
=== 11-limit ===
 
Optimal GPV sequence: {{Val list| 10, 19e, 29, 77cddee }}
 
Badness: 0.035296
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 55/54, 65/64, 91/90
 
Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
 
POTE generator: ~14/13 = 124.545
 
Optimal GPV sequence: {{Val list| 10, 19e, 29, 77cddeef }}
 
Badness: 0.021559
 
=== Wilsec ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


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


POTE generator: ~11/8 = 537.186
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| 9, 20, 29, 38d, 67cdde }}
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


Badness: 0.041886
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, 65/64, 91/90, 121/120
Comma list: 105/104, 121/120, 196/195, 275/273


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


POTE generator: ~11/8 = 537.208
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| 9, 20, 29, 38df, 67cddef }}
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


Badness: 0.025192
Badness (Sintel): 0.950


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


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


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 | 0 5 -11 -12 -3 -18 -25 }}


POTE generator: ~11/8 = 537.230
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, 20g, 29g, 38df, 67cddefg }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness: 0.021778
Badness (Sintel): 0.973


==== 19-limit ====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


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 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


POTE generator: ~11/8 = 537.214
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, 20g, 29g, 38df, 67cddefgh }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness: 0.016828
Badness (Sintel): 1.03


== Wizard ==
==== 23-limit ====
{{Main|Wizard}}
Subgroup: 2.3.5.7.11.13.17.19.23
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''


Subgroup: 2.3.5.7
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


[[Comma list]]: 225/224, 118098/117649
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


[[Mapping]]: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


Mapping generators: ~1225/864, ~245/216
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


[[POTE generator]]: ~5/4 = 383.256
Badness (Sintel): 1.04


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
=== Tritoni ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
Comma list: 225/224, 385/384, 27783/27500


[[Badness]]: 0.040846
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Scales: [[wizard22]]
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


=== 11-limit ===
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 4000/3993
Badness (Sintel): 1.50


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


Mapping generators: ~99/70, ~25/22
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.


POTE generator: ~5/4 = 383.232
[[Subgroup]]: 2.3.5.7


Optimal GPV sequence: {{Val list| 22, 50, 72, 166, 238c, 310c }}
[[Comma list]]: 225/224, 84035/82944


Badness: 0.018539
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


Scales: [[wizard22]]
[[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 }}


==== Lizard ====
{{Optimal ET sequence|legend=1| 41, 91, 132d }}
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 351/350, 364/363, 385/384
[[Badness]] (Sintel): 1.38


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping generators: ~99/70, ~25/22
Comma list: 225/224, 245/242, 385/384


POTE generator: ~5/4 = 383.389
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


Optimal GPV sequence: {{Val list| 22, 50, 72, 122, 194df }}
Optimal tunings:  
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


Badness: 0.021781
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


===== 17-limit =====
Badness (Sintel): 1.04
Subgroup: 2.3.5.7.11.13.17


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
Comma list: 105/104, 144/143, 196/195, 245/242


Mapping generators: ~17/12, ~17/15
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


POTE generator: ~5/4 = 383.381
Optimal tunings:  
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


Optimal GPV sequence: {{Val list| 22, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 41, 91 }}


Badness: 0.014536
Badness (Sintel): 0.955


===== 19-limit =====
== Merman ==
Subgroup: 2.3.5.7.11.13.17.19
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
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 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.


Mapping generators: ~17/12, ~17/15
[[Subgroup]]: 2.3.5.7


POTE generator: ~5/4 = 383.477
[[Comma list]]: 225/224, 2500000/2470629


Optimal GPV sequence: {{Val list| 22h, 50, 72, 122g, 194dfg }}
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


Badness: 0.015702
[[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 }}


==== Gizzard ====
{{Optimal ET sequence|legend=1| 41, 84, 125 }}
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 325/324, 385/384, 1573/1568
[[Badness]] (Sintel): 1.39


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping generators: ~99/70, ~25/22
Comma list: 225/224, 441/440, 1344/1331


POTE generator: ~5/4 = 383.170
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


Optimal GPV sequence: {{Val list| 72, 166, 238cf }}
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


Badness: 0.020252
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


===== 17-limit =====
Badness (Sintel): 1.20
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
Comma list: 144/143, 225/224, 364/363, 441/440


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


POTE generator: ~5/4 = 383.175
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Optimal GPV sequence: {{Val list| 72, 166g, 238cfg }}
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Badness: 0.013624
Badness (Sintel): 1.14


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


Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
Comma list: 225/224, 385/384, 532400/531441


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


Mapping generators: ~17/12, ~17/15
Optimal tunings:
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


POTE generator: ~5/4 = 383.138
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


Optimal GPV sequence: {{Val list| 72, 94, 166g }}
Badness (Sintel): 1.46


Badness: 0.014810
==== 13-limit ====
 
Subgroup: 2.3.5.7.11.13
=== Mage ===
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 1331/1296
Comma list: 225/224, 325/324, 385/384, 10648/10647


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


Mapping generators: ~77/54, ~55/48
Optimal tunings:
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


POTE generator: ~5/4 = 383.124
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


Optimal GPV sequence: {{Val list| 22, 50e, 72ee, 94ee }}
Badness (Sintel): 1.47


Badness: 0.057799
== 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.  


== Triton ==
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 1029/1000
[[Comma list]]: 225/224, 589824/588245


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
[[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]]: ~7/5 = 568.865
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


{{Val list|legend=1| 2, 17d, 19, 78bd, 97bd }}
[[Badness]] (Sintel): 1.44
 
[[Badness]]: 0.059245


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


Comma list: 45/44, 56/55, 1029/1000
Comma list: 225/224, 385/384, 1331/1323


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


POTE generator: ~7/5 = 569.144
Optimal tunings:  
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


Optimal GPV sequence: {{Val list| 2, 17d, 19, 59bde, 78bde, 97bde }}
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Badness: 0.045675
Badness (Sintel): 0.838


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


Subgroup: 2.3.5.7
Comma list: 225/224, 275/273, 385/384, 1331/1323


[[Comma list]]: 225/224, 50421/50000
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


[[Mapping]]: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


[[POTE generator]]: ~7/5 = 580.286
Badness (Sintel): 1.07


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


[[Badness]]: 0.047578
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.  


=== 11-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11


Comma list: 121/120, 225/224, 441/440
[[Comma list]]: 225/224, 1029/1000


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


POTE generator: ~7/5 = 580.267
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}
: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}


Optimal GPV sequence: {{Val list| 29, 31, 60e }}
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


Badness: 0.023659
[[Badness]] (Sintel): 1.50


==== 13-limit ====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 121/120, 196/195, 275/273
 
Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
 
POTE generator: ~7/5 = 580.108
 
Optimal GPV sequence: {{Val list| 29, 31, 60e, 151cde }}
 
Badness: 0.022993
 
=== Tritoni ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 27783/27500
Comma list: 45/44, 56/55, 1029/1000


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


POTE generator: ~7/5 = 580.389
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| 31, 91, 122, 153d }}
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


Badness: 0.045456
Badness (Sintel): 1.51


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


Subgroup: 2.3.5.7
[[Comma list]]: 225/224, 156250000/155649627
 
[[Comma list]]: 225/224, 2500000/2470629


[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
[[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 }}


[[POTE generator]]: ~7/5 = 585.585
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


{{Val list|legend=1| 41, 84, 125 }}
[[Badness]] (Sintel): 2.11
 
[[Badness]]: 0.055078


=== 11-limit ===
=== 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, 441/440, 4000/3993


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


POTE generator: ~7/5 = 585.606
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| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.036383
Badness (Sintel): 0.958


=== 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: 169/168, 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 | 0 -6 19 26 21 10 }}


POTE generator: ~7/5 = 585.657
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


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


Badness: 0.027544
Badness (Sintel): 0.887


== Untriton ==
=== 17-limit ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Untriton]].''
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, 125000000/121060821
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


[[Mapping]]: [{{Val|1 6 -7 -7}}, {{Val|0 -9 19 20}}]
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|9 -19 -20 -51 -57 7}}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


[[POTE generator]]: ~7/5 = 588.641
Badness (Sintel): 0.760


{{Val list|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


[[Badness]]: 0.143976
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


=== 11-limit ===
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
Subgroup: 2.3.5.7.11


Comma list: 121/120, 225/224, 22000/21609
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 6 -7 -7 1}}, {{Val|0 -9 19 20 5}}]
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


POTE generator: ~7/5 = 588.626
Badness (Sintel): 0.895


Optimal GPV sequence: {{Val list| 51, 53 }}
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


Badness: 0.074295
[[Comma list]]: 225/224, 40353607/40310784
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 225/224, 275/273, 1040/1029
 
Mapping: [{{Val|1 6 -7 -7 1 -12}}, {{Val|0 -9 19 20 5 32}}]
 
POTE generator: ~7/5 = 588.654
 
Optimal GPV sequence: {{Val list| 51f, 53 }}
 
Badness: 0.047441
 
== Septimin ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 225/224, 84035/82944
 
[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


[[POTE generator]]: ~7/6 = 263.632
[[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| 41, 91, 132d }}
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


[[Badness]]: 0.054502
[[Badness]] (Sintel): 2.37


=== 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, 12005/11979


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


POTE generator: ~7/6 = 263.634
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| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.031309
Badness (Sintel): 1.01


=== 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: 169/168, 225/224, 364/363, 1716/1715


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


POTE generator: ~7/6 = 263.700
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| 41, 91 }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.023117
Badness (Sintel): 0.922


== Misneb ==
== Gracecordial ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Misneb]].''
: ''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, 4194304/4117715
[[Comma list]]: 225/224, 781250000/771895089


[[Mapping]]: [{{Val|1 3 1 3}}, {{Val|0 -15 14 -2}}]
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


{{Multival|legend=1|15 -14 2 -57 -39 44}}
[[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 }}


[[POTE generator]]: ~16/15 = 113.235
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


{{Val list|legend=1| 21, 32, 53 }}
[[Badness]] (Sintel): 2.44
 
[[Badness]]: 0.140970


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


Comma list: 99/98, 176/175, 1310720/1294139
Comma list: 225/224, 385/384, 236328125/234365481


Mapping: [{{Val|1 3 1 3 1}}, {{Val|0 -15 14 -2 26}}]
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


POTE generator: ~16/15 = 113.323
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| 21, 32e, 53, 127, 180de }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Badness: 0.085390
Badness (Sintel): 2.96


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


Comma list: 99/98, 176/175, 640/637, 847/845
Comma list: 225/224, 325/324, 385/384, 831875/830466


Mapping: [{{Val|1 3 1 3 1 2}}, {{Val|0 -15 14 -2 26 18}}]
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


POTE generator: ~16/15 = 113.323
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| 21, 32e, 53, 127, 180de }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.045569
Badness (Sintel): 2.16


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


Comma list: 225/224, 385/384, 66550/64827
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


Mapping: [{{Val|1 3 1 3 6}}, {{Val|0 -15 14 -2 -27}}]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


POTE generator: ~16/15 = 113.142
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


Optimal GPV sequence: {{Val list| 32, 53, 191de, 244cddee, 297cddee }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.087333
Badness (Sintel): 1.96


== Slender ==
==== 19-limit ====
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: 2.3.5.7.11.13.17.19


Subgroup: 2.3.5.7
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


[[Comma list]]: 225/224, 589824/588245
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


[[POTE generator]]: ~49/48 = 38.413
Badness (Sintel): 1.71


{{Val list|legend=1| 31, 94, 125 }}
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


[[Badness]]: 0.056934
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


=== 11-limit ===
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 1331/1323
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 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


POTE generator: ~49/48 = 38.387
Badness (Sintel): 1.57


Optimal GPV sequence: {{Val list| 31, 63, 94, 125 }}
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Badness: 0.025342
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 275/273, 385/384, 1331/1323
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


POTE generator: ~49/48 = 38.314
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Optimal GPV sequence: {{Val list| 31, 63, 94 }}
Badness (Sintel): 1.50


Badness: 0.025913
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


== Marvolo ==
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 156250000/155649627
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


[[POTE generator]]: ~21/20 = 83.348
Badness (Sintel): 1.53


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
=== Gracecord ===
 
[[Badness]]: 0.083338
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 441/440, 4000/3993
Comma list: 225/224, 441/440, 109375/107811


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


POTE generator: ~21/20 = 83.340
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


Optimal GPV sequence: {{Val list| 29, 43, 72 }}
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Badness: 0.028965
Badness (Sintel): 2.21


=== 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, 441/440
Comma list: 225/224, 364/363, 441/440, 6125/6084


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


POTE generator: ~21/20 = 83.330
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


Optimal GPV sequence: {{Val list| 29, 43, 72, 115f }}
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Badness: 0.021470
Badness (Sintel): 1.83


== Amavil ==
==== 17-limit ====
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Mabila]].''
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, 17496/16807
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
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| 10 -3 14 -28 -6 41 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


[[POTE generator]]: ~48/35 = 529.979
Badness (Sintel): 1.87


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


[[Badness]]: 0.109625
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


Comma list: 99/98, 176/175, 864/847
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


POTE generator: ~15/11 = 529.974
Badness (Sintel): 1.68


Optimal GPV sequence: {{Val list| 9, 34d, 43, 77de }}
== Alphorn ==
[[Subgroup]]: 2.3.5.7


Badness: 0.042649
[[Comma list]]: 225/224, 5764801/5668704
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 78/77, 99/98, 144/143, 176/175
 
Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
 
POTE generator: ~15/11 = 529.951
 
Optimal GPV sequence: {{Val list| 9, 34d, 43, 77de }}
 
Badness: 0.025791


== Enneaportent ==
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
Subgroup: 2.3.5.7
: mapping generators: ~2, ~35/24


[[Comma list]]: 225/224, 40353607/40310784
[[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 }}


[[Mapping]]: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
[[Badness]] (Sintel): 3.27
 
[[POTE generator]]: ~5/4 = 383.165
 
{{Val list|legend=1| 9, 63, 72, 495bcd }}
 
[[Badness]]: 0.093679


=== 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, 385/384, 12250/11979


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


POTE generator: ~5/4 = 383.146
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Optimal GPV sequence: {{Val list| 9, 63, 72, 423cd, 495bcd }}
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Badness: 0.030426
Badness (Sintel): 2.43


=== 13-limit ===
== Misneb ==
Subgroup: 2.3.5.7.11.13
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''
 
Comma list: 169/168, 225/224, 364/363, 1716/1715


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~5/4 = 383.047
[[Comma list]]: 225/224, 4194304/4117715
 
Optimal GPV sequence: {{Val list| 9, 63, 72, 279cf }}
 
Badness: 0.022322
 
== Submajor ==
Subgroup: 2.3.5
 
[[Comma list]]: 69198046875/68719476736
 
[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
 
[[POTE generator]]: ~10125/8192 = 362.321
 
{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
 
[[Badness]]: 0.130236
 
=== 7-limit ===
Subgroup: 2.3.5.7
 
[[Comma list]]: 225/224, 51200/50421


[[Mapping]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
[[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 }}


[[POTE generator]]: ~49/40 = 362.255
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


{{Val list|legend=1| 10, 33, 43, 53 }}
[[Badness]] (Sintel): 3.57
 
[[Badness]]: 0.060533


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


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


POTE generator: ~27/22 = 362.101
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| 10, 43e, 53, 116, 169de, 285cde }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.050582
Badness (Sintel): 2.82


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


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


POTE generator: ~16/13 = 362.105
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| 10, 43e, 53, 116, 169de, 285cdef }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.027689
Badness (Sintel): 1.88


=== Interpental ===
=== Musneb ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 51200/50421
Comma list: 225/224, 385/384, 66550/64827


POTE generator: ~49/40 = 362.418
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


POTE generator: ~49/40 = 362.418
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Optimal GPV sequence: {{Val list| 43, 53, 96, 149d }}
Badness (Sintel): 2.89


Badness: 0.051806
== Untriton ==
 
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Comma list: 99/98, 169/168, 176/175, 640/637
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.


POTE generator: ~16/13 = 362.402
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
[[Comma list]]: 225/224, 125000000/121060821
 
POTE generator: ~16/13 = 362.402
 
Optimal GPV sequence: {{Val list| 43, 53, 96, 149d }}
 
Badness: 0.029680
 
== Alphorn ==
Subgroup: 2.3.5.7
 
[[Comma list]]: 225/224, 5764801/5668704
 
[[Mapping]]: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


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


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
{{Optimal ET sequence|legend=1| 51, 53 }}


[[Badness]]: 0.129258
[[Badness]] (Sintel): 3.64


=== 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: 121/120, 225/224, 22000/21609


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


POTE generator: ~11/8 = 556.144
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


Optimal GPV sequence: {{Val list| 28d, 41, 315cde }}
{{Optimal ET sequence|legend=0| 51, 53 }}


Badness: 0.073459
Badness (Sintel): 2.46


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


[[Comma list]]: {{monzo| -89 21 24 }}
Comma list: 121/120, 225/224, 275/273, 1040/1029


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


[[POTE generator]]: ~16/15 = 112.281
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| 21, 54, 75, 96, 171 }}
{{Optimal ET sequence|legend=0| 51f, 53 }}


[[Badness]]: 0.851003
Badness (Sintel): 1.96


=== 7-limit ===
== Naiadical ==
Subgroup: 2.3.5.7
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.  


[[Comma list]]: 225/224, 14495514624/13841287201
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 3 7 5 9 }}, {{val| 0 -8 7 -2 }}]
[[Comma list]]: 225/224, 823543/800000


{{Multival|legend=1| 24 -21 6 -89 -58 73 }}
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


[[POTE generator]]: ~15/14 = 112.283
[[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 }}


{{Val list|legend=1| 21, 54, 75, 96, 171d }}
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


[[Badness]]: 0.431636
[[Badness]] (Sintel): 3.67


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


Comma list: 225/224, 3840/3773, 12005/11979
Comma list: 225/224, 245/242, 1617/1600


Mapping: [{{val| 3 7 5 9 9 }}, {{val| 0 -8 7 -2 5 }}]
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


POTE generator: ~15/14 = 112.171
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| 21, 54, 75e }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Badness: 0.173485
Badness (Sintel): 2.00


== Gwazy ==
=== 13-limit ===
{{see also|Very high accuracy temperaments #Kwazy}}
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5.7
Comma list: 105/104, 196/195, 245/242, 1001/1000


[[Comma list]]: 225/224, 5971968/5764801
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


[[Mapping]]: [{{val| 2 1 6 4 }}, {{val| 0 8 -5 6 }}]
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


{{Multival|legend=1| 16 -10 12 -53 -26 56 }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


[[POTE generator]]: ~35/32 = 162.658
Badness (Sintel): 1.43


{{Val list|legend=1| 22, 74, 96, 118d }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Badness]]: 0.178826
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


=== 11-limit ===
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 65536/65219
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


Mapping: [{{val| 2 1 6 4 8 }}, {{val| 0 8 -5 6 -4 }}]
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


POTE generator: ~11/10 = 162.592
Badness (Sintel): 1.26


Optimal GPV sequence: {{Val list| 22, 74, 96, 118d }}
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


Badness: 0.068410
[[Subgroup]]: 2.3.5.7


== Gracecordial ==
[[Comma list]]: 225/224, 9805926501/9765625000
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 781250000/771895089
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


[[Mapping]]: [{{val| 1 0 34 63 }}, {{val| 0 1 -20 -38 }}]
[[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 }}


{{Multival|legend=1| 1 -20 -38 -34 -63 -32 }}
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


[[POTE generator]]: ~3/2 = 700.824
[[Badness]] (Sintel): 3.81
 
{{Val list|legend=1| 12, 113, 125, 238c, 363c }}
 
[[Badness]]: 0.096279


=== 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, 441/440, 43923/43750


Mapping: [{{val| 1 0 34 63 -90 }}, {{val| 0 1 -20 -38 59 }}]
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


POTE generator: ~3/2 = 700.834
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| 12e, 101cde, 113, 125, 238c }}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.089588
Badness (Sintel): 1.74


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


Comma list: 225/224, 325/324, 385/384, 831875/830466
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


Mapping: [{{val| 1 0 34 63 -90 -66 }}, {{val| 0 1 -20 -38 59 44 }}]
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


POTE generator: ~3/2 = 700.841
Optimal tunings:
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 0.052235
Badness (Sintel): 1.35


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


Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Mapping: [{{val| 1 0 34 63 -90 -66 -7 }}, {{val| 0 1 -20 -38 59 44 7 }}]
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


POTE generator: ~3/2 = 700.841
Optimal tunings:
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 0.038565
Badness (Sintel): 1.17


==== 19-limit ====
== Gwazy ==
Subgroup: 2.3.5.7.11.13.17.19
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 }}, {{val| 0 1 -20 -38 59 44 7 -3 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~3/2 = 700.842
[[Comma list]]: 225/224, 5971968/5764801
 
Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cf }}
 
Badness: 0.028165


==== 23-limit ====
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
Subgroup: 2.3.5.7.11.13.17.19.23
: mapping generators: ~2401/1728, ~35/32


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


Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 -43 }}, {{val| 0 1 -20 -38 59 44 7 -3 30 }}]
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


POTE generator: ~3/2 = 700.843
[[Badness]] (Sintel): 4.53


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 238cfi }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.021879
Comma list: 99/98, 176/175, 65536/65219


==== 29-limit ====
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}
Subgroup: 2.3.5.7.11.13.17.19.23.29


Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 -43 -49 }}, {{val| 0 1 -20 -38 59 44 7 -3 30 34 }}]
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


POTE generator: ~3/2 = 700.842
Badness (Sintel): 2.26


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cfi }}
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


Badness: 0.018011
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>.  


==== 31-limit ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
[[Comma list]]: 225/224, 14495514624/13841287201


Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 }}, {{val| 0 1 -20 -38 59 44 7 -3 30 34 53 }}]
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


POTE generator: ~3/2 = 700.838
[[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 }}


Optimal GPV sequence: {{Val list| 12e, 101cdek, 113, 125f, 238cfi }}
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


Badness: 0.016007
[[Badness]] (Sintel): 10.9


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


Comma list: 225/224, 441/440, 109375/107811
Comma list: 225/224, 3840/3773, 12005/11979
 
Mapping: [{{val| 1 0 34 63 89 }}, {{val| 0 1 -20 -38 -54 }}]
 
POTE generator: ~3/2 = 700.885
 
Optimal GPV sequence: {{Val list| 12, 101cd, 113 }}
 
Badness: 0.066964
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 364/363, 441/440, 6125/6084
 
Mapping: [{{val| 1 0 34 63 89 113 }}, {{val| 0 1 -20 -38 -54 -69 }}]
 
POTE generator: ~3/2 = 700.890
 
Optimal GPV sequence: {{Val list| 12f, 101cdf, 113 }}
 
Badness: 0.044196
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
 
Mapping: [{{val| 1 0 34 63 89 113 -7 }}, {{val| 0 1 -20 -38 -54 -69 7 }}]


POTE generator: ~3/2 = 700.870
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
 
Optimal GPV sequence: {{Val list| 12f, 101cdf, 113 }}
 
Badness: 0.036637
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


Mapping: [{{val| 1 0 34 63 89 113 -7 9 }}, {{val| 0 1 -20 -38 -54 -69 7 -3 }}]
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 = 700.866
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


Optimal GPV sequence: {{Val list| 12f, 101cdf, 113 }}
Badness (Sintel): 5.74


Badness: 0.027559
== 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"