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Marvel temperaments: Difference between revisions

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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]]
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* ''[[Pelogic]]'', {21/20, 135/128} → [[Pelogic family #Pelogic]]
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[August]]'', {36/35, 128/125} → [[Augmented family #August]]
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Pajara]], {50/49, 64/63}, also in: [[Diaschismic family #Pajara|diaschismic family]]
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[Septimal meantone]]'', {81/80, 126/125} → [[Meantone family #Septimal meantone]]
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* [[Magic]], {225/224, 245/243}, also in: [[Magic family #Magic|magic family]]
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* [[Passive]], {225/224, 256/245} → [[Passion family #Passive]]
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* [[Miracle]], {225/224, 1029/1024}, also in: [[Gamelismic clan #Miracle|gamelismic clan]]
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Orwell]], {225/224, 1728/1715}, also in: [[Semicomma family #Orwell|semicomma family]]
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Garibaldi temperament|Garibaldi]], {225/224, 3125/3087}, also in: [[Schismatic family #Garibaldi|schismatic family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* [[Catakleismic]], {225/224, 4375/4374}, also in: [[Kleismic family #Catakleismic|kleismic family]]
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[Decic]]'', {225/224, 16807/16384} → [[Cloudy clan #Decic]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Escapade]]'', {225/224, 65625/65536} → [[Escapade family #Escapade]]
* ''[[Quintapole]]'' (+7812500/7411887) → [[Quintaleap family #Quintapole|Quintaleap family]]
* ''[[Fog]]'', {225/224, 156250/151263} → [[Misty family #Fog]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Compton]]'', {225/224, 250047/250000} → [[Compton family #Compton]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* ''[[Houborizic]]'' {225/224, 1250000/1240029} → [[Amity family #Houborizic]]
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Qintosec]]'', {225/224, 2560000/2470629} → [[Qintosec family #Qintosec]]
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Maquila]]'', {225/224, 30233088/28824005} → [[Maquila family #Maquila]]
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* ''[[Marvo]]'', {225/224, 78125000/78121827} → [[Gravity family #Marvo]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
* ''[[Decic]]'' (+16807/16384) → [[Cloudy clan #Decic|Cloudy clan]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Hendeca]]'' (+122880/117649) → [[11th-octave temperaments #Hendeca|11th-octave temperaments]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]


Considered below are negri, wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, submajor, alphorn, tertiosec, gwazy, gracecordial, and quintapole.  
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]].''
 
Wizard has a [[semi-octave]] period and is generated by an interval that can be treated as [[~]][[17/15]]. The semi-octave complement of this interval is ~[[5/4]]. Wizard can be described as {{nowrap| 22 & 72 }}. Its [[ploidacot]] is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.
 
[[Subgroup]]: 2.3.5.7


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


Subgroup: 2.3.5
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216


[[Comma list]]: 16875/16384
[[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 }}


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}


{{Multival|legend=1| 4 -3 -14 }}
[[Badness]] (Sintel): 1.03


[[POTE generator]]: ~16/15 = 125.7549
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
Comma list: 225/224, 385/384, 4000/3993


[[Badness]]: 0.086856
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}


=== 7-limit ===
Optimal tunings:
Subgroup: 2.3.5.7
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


[[Comma list]]: 49/48, 225/224
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
Badness (Sintel): 0.613


{{Multival|legend=1| 4 -3 2 -14 -8 13 }}
==== 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}}


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


Badness: 0.026190
Badness (Sintel): 0.955


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


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


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


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


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


Badness: 0.017833
Badness (Sintel): 0.837


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


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


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


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


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


Badness: 0.038679
Badness (Sintel): 0.694


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


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


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


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


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


Badness: 0.024383
Badness (Sintel): 0.901


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


Comma list: 33/32, 49/48, 77/75
Comma list: 99/98, 176/175, 1331/1296


Mapping: [{{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}}


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


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


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


POTE generator: ~14/13 = 127.039
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
: [[error map]]: {{val| +1.354 -1.243 -0.027 -1.598 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}}
: error map: {{val| 0.000 -3.566 -2.769 -4.959 }}


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


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


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


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


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


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


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


Badness: 0.035296
Badness (Sintel): 0.782


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


Comma list: 49/48, 55/54, 65/64, 91/90
Comma list: 105/104, 121/120, 196/195, 275/273
 
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Optimal tunings:  
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


POTE generator: ~14/13 = 124.545
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


Vals: {{Val list| 10, 19e, 29, 77cddeef }}
Badness (Sintel): 0.950


Badness: 0.021559
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


=== Wilsec ===
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
Subgroup: 2.3.5.7.11


Comma list: 49/48, 121/120, 225/224
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]
Optimal tunings:  
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


POTE generator: ~11/8 = 537.186
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Vals: {{Val list| 9, 20, 29, 38d, 67cdde }}
Badness (Sintel): 0.973


Badness: 0.041886
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


==== 13-limit ====
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 65/64, 91/90, 121/120
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


POTE generator: ~11/8 = 537.208
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Vals: {{Val list| 9, 20, 29, 38df, 67cddef }}
Badness (Sintel): 1.03


Badness: 0.025192
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


==== 17-limit ====
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
Subgroup: 2.3.5.7.11.13.17


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


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


Vals: {{Val list| 9, 20g, 29g, 38df, 67cddefg }}
Badness (Sintel): 1.04


Badness: 0.021778
=== Tritoni ===
Subgroup: 2.3.5.7.11


==== 19-limit ====
Comma list: 225/224, 385/384, 27783/27500
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


POTE generator: ~11/8 = 537.214
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Vals: {{Val list| 9, 20g, 29g, 38df, 67cddefgh }}
Badness (Sintel): 1.50


Badness: 0.016828
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 118098/117649
[[Comma list]]: 225/224, 84035/82944


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


Mapping generators: ~1225/864, ~245/216
[[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 }}


[[POTE generator]]: ~5/4 = 383.256
{{Optimal ET sequence|legend=1| 41, 91, 132d }}


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
[[Badness]] (Sintel): 1.38


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Badness]]: 0.040846
Comma list: 225/224, 245/242, 385/384


Scales: [[wizard22]]
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


=== 11-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


Comma list: 225/224, 385/384, 4000/3993
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
Badness (Sintel): 1.04


Mapping generators: ~99/70, ~25/22
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 383.232
Comma list: 105/104, 144/143, 196/195, 245/242


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


Badness: 0.018539
Optimal tunings:  
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


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


==== Lizard ====
Badness (Sintel): 0.955
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 351/350, 364/363, 385/384
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
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 generators: ~99/70, ~25/22
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.


POTE generator: ~5/4 = 383.389
[[Subgroup]]: 2.3.5.7


Vals: {{Val list| 22, 50, 72, 122, 194df }}
[[Comma list]]: 225/224, 2500000/2470629


Badness: 0.021781
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


===== 17-limit =====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13.17
* [[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 }}


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
[[Badness]] (Sintel): 1.39


Mapping generators: ~17/12, ~17/15
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~5/4 = 383.381
Comma list: 225/224, 441/440, 1344/1331


Vals: {{Val list| 22, 50, 72, 122g, 194dfg }}
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


Badness: 0.014536
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


===== 19-limit =====
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Badness (Sintel): 1.20


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping generators: ~17/12, ~17/15
Comma list: 144/143, 225/224, 364/363, 441/440


POTE generator: ~5/4 = 383.477
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Vals: {{Val list| 22h, 50, 72, 122g, 194dfg }}
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Badness: 0.015702
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


==== Gizzard ====
Badness (Sintel): 1.14
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 325/324, 385/384, 1573/1568
=== Mermaid ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
Comma list: 225/224, 385/384, 532400/531441


Mapping generators: ~99/70, ~25/22
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


POTE generator: ~5/4 = 383.170
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


Vals: {{Val list| 72, 166, 238cf }}
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


Badness: 0.020252
Badness (Sintel): 1.46


===== 17-limit =====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Comma list: 225/224, 325/324, 385/384, 10648/10647


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


Mapping generators: ~17/12, ~17/15
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 = 383.175
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


Vals: {{Val list| 72, 166g, 238cfg }}
Badness (Sintel): 1.47


Badness: 0.013624
== 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.  


===== 19-limit =====
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
[[Comma list]]: 225/224, 589824/588245


Mapping generators: ~17/12, ~17/15
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


POTE generator: ~5/4 = 383.138
[[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 }}


Vals: {{Val list| 72, 94, 166g }}
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Badness: 0.014810
[[Badness]] (Sintel): 1.44


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


Comma list: 99/98, 176/175, 1331/1296
Comma list: 225/224, 385/384, 1331/1323
 
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
 
Optimal tunings:
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}
 
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
 
Badness (Sintel): 0.838


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping generators: ~77/54, ~55/48
Comma list: 225/224, 275/273, 385/384, 1331/1323


POTE generator: ~5/4 = 383.124
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Vals: {{Val list| 22, 50e, 72ee, 94ee }}
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Badness: 0.057799
{{Optimal ET sequence|legend=0| 31, 63, 94 }}
 
Badness (Sintel): 1.07


== Triton ==
== Triton ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''
 
Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
 
: mapping generators: ~2, ~10/7
{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}


[[POTE generator]]: ~7/5 = 568.865
[[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 }}


{{Val list|legend=1| 2, 17d, 19, 78bd, 97bd }}
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


[[Badness]]: 0.059245
[[Badness]] (Sintel): 1.50


=== 11-limit ===
=== 11-limit ===
Line 403: Line 512:
Comma list: 45/44, 56/55, 1029/1000
Comma list: 45/44, 56/55, 1029/1000


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


POTE generator: ~7/5 = 569.144
Optimal tunings:  
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


Vals: {{Val list| 2, 17d, 19, 59bde, 78bde, 97bde }}
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


Badness: 0.045675
Badness (Sintel): 1.51


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


Subgroup: 2.3.5.7
[[Comma list]]: 225/224, 156250000/155649627
 
[[Comma list]]: 225/224, 50421/50000


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


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
[[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 = 580.286
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
[[Badness]] (Sintel): 2.11
 
[[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: 225/224, 441/440, 4000/3993


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


POTE generator: ~7/5 = 580.267
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


Vals: {{Val list| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.023659
Badness (Sintel): 0.958


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


Comma list: 105/104, 121/120, 196/195, 275/273
Comma list: 169/168, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
 
Optimal tunings:
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}
 
{{Optimal ET sequence|legend=0| 29, 43, 72 }}
 
Badness (Sintel): 0.887


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~7/5 = 580.108
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


Vals: {{Val list| 29, 31, 60e, 151cde }}
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


Badness: 0.022993
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


=== Tritoni ===
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 27783/27500
Badness (Sintel): 0.760


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~7/5 = 580.389
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Vals: {{Val list| 31, 91, 122, 153d }}
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Badness: 0.045456
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


== Merman ==
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''


Subgroup: 2.3.5.7
Badness (Sintel): 0.895


[[Comma list]]: 225/224, 2500000/2470629
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
[[Comma list]]: 225/224, 40353607/40310784


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


[[POTE generator]]: ~7/5 = 585.585
[[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, 84, 125 }}
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


[[Badness]]: 0.055078
[[Badness]] (Sintel): 2.37


=== 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, 385/384, 12005/11979


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


POTE generator: ~7/5 = 585.606
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}})


Vals: {{Val list| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.036383
Badness (Sintel): 1.01


=== 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, 1716/1715


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


POTE generator: ~7/5 = 585.657
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}})


Vals: {{Val list| 41, 84, 125e, 209ef, 293ef }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.027544
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}}


Vals: {{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: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
 
Optimal tunings:
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Badness (Sintel): 2.16


POTE generator: ~7/6 = 263.700
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Vals: {{Val list| 41, 91 }}
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


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


== Slender ==
Optimal tunings:
Slender (31 & 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.
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


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


[[Comma list]]: 225/224, 589824/588245
Badness (Sintel): 1.96


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


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


{{Val list|legend=1| 31, 94, 125 }}
Optimal tunings:
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


[[Badness]]: 0.056934
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


=== 11-limit ===
Badness (Sintel): 1.71
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 1331/1323
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


POTE generator: ~49/48 = 38.387
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Vals: {{Val list| 31, 63, 94, 125 }}
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Badness: 0.025342
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


=== 13-limit ===
Badness (Sintel): 1.57
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 275/273, 385/384, 1331/1323
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


POTE generator: ~49/48 = 38.314
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


Vals: {{Val list| 31, 63, 94 }}
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Badness: 0.025913
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


== Marvolo ==
Badness (Sintel): 1.50
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 156250000/155649627
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


[[POTE generator]]: ~21/20 = 83.348
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


[[Badness]]: 0.083338
Badness (Sintel): 1.53


=== 11-limit ===
=== Gracecord ===
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}}


Vals: {{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}}


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


Badness: 0.021470
Badness (Sintel): 1.83


== Amavil ==
==== 17-limit ====
=== 5-limit (mabila) ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5


[[Comma list]]: 268435456/263671875
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


[[Mapping]]: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


[[POTE generator]]: ~512/375 = 529.6849
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


[[Badness]]: 0.232481
Badness (Sintel): 1.87


=== 7-limit ===
==== 19-limit ====
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17.19


[[Comma list]]: 225/224, 17496/16807
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
Optimal tunings:
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


[[POTE generator]]: ~48/35 = 529.979
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
Badness (Sintel): 1.68


[[Badness]]: 0.109625
== Alphorn ==
[[Subgroup]]: 2.3.5.7


=== 11-limit ===
[[Comma list]]: 225/224, 5764801/5668704
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 864/847
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
[[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 }}


POTE generator: ~15/11 = 529.974
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Vals: {{Val list| 9, 34d, 43, 77de }}
[[Badness]] (Sintel): 3.27


Badness: 0.042649
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 225/224, 385/384, 12250/11979
Subgroup: 2.3.5.7.11.13


Comma list: 78/77, 99/98, 144/143, 176/175
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


POTE generator: ~15/11 = 529.951
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Vals: {{Val list| 9, 34d, 43, 77de }}
Badness (Sintel): 2.43


Badness: 0.025791
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


== Enneaportent ==
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 40353607/40310784
[[Comma list]]: 225/224, 4194304/4117715
 
[[Mapping]]: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


[[POTE generator]]: ~5/4 = 383.165
[[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, 63, 72, 495bcd }}
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


[[Badness]]: 0.093679
[[Badness]] (Sintel): 3.57


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


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


POTE generator: ~5/4 = 383.146
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


Vals: {{Val list| 9, 63, 72, 423cd, 495bcd }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.030426
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, 364/363, 1716/1715
Comma list: 99/98, 176/175, 640/637, 847/845
 
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


POTE generator: ~5/4 = 383.047
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Vals: {{Val list| 9, 63, 72, 279cf }}
Badness (Sintel): 1.88


Badness: 0.022322
=== Musneb ===
Subgroup: 2.3.5.7.11


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


[[Comma list]]: 69198046875/68719476736
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


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


[[POTE generator]]: ~10125/8192 = 362.321
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
Badness (Sintel): 2.89


[[Badness]]: 0.130236
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


=== 7-limit ===
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.  
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 51200/50421
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
[[Comma list]]: 225/224, 125000000/121060821


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


[[POTE generator]]: ~49/40 = 362.255
[[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| 10, 33, 43, 53 }}
{{Optimal ET sequence|legend=1| 51, 53 }}


[[Badness]]: 0.060533
[[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, 6655/6561
Comma list: 121/120, 225/224, 22000/21609


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


POTE generator: ~27/22 = 362.101
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


Vals: {{Val list| 10, 43e, 53, 116, 169de, 285cde }}
{{Optimal ET sequence|legend=0| 51, 53 }}


Badness: 0.050582
Badness (Sintel): 2.46


==== 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: 121/120, 225/224, 275/273, 1040/1029


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


POTE generator: ~16/13 = 362.105
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


Vals: {{Val list| 10, 43e, 53, 116, 169de, 285cdef }}
{{Optimal ET sequence|legend=0| 51f, 53 }}


Badness: 0.027689
Badness (Sintel): 1.96


=== Interpental ===
== Naiadical ==
Subgroup: 2.3.5.7.11
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.  


Comma list: 99/98, 176/175, 51200/50421
[[Subgroup]]: 2.3.5.7


POTE generator: ~49/40 = 362.418
[[Comma list]]: 225/224, 823543/800000


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


POTE generator: ~49/40 = 362.418
[[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 }}


Vals: {{Val list| 43, 53, 96, 149d }}
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


Badness: 0.051806
[[Badness]] (Sintel): 3.67


==== 13-limit ====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 99/98, 169/168, 176/175, 640/637
Comma list: 225/224, 245/242, 1617/1600


POTE generator: ~16/13 = 362.402
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


POTE generator: ~16/13 = 362.402
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Vals: {{Val list| 43, 53, 96, 149d }}
Badness (Sintel): 2.00


Badness: 0.029680
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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


[[Comma list]]: 225/224, 5764801/5668704
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


[[Mapping]]: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
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 -5 22 -45 -10 65 }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


[[POTE generator]]: ~48/35 = 556.221
Badness (Sintel): 1.43


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
=== 17-limit ===
 
Subgroup: 2.3.5.7.11.13.17
[[Badness]]: 0.129258
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 12250/11979
 
Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
 
POTE generator: ~11/8 = 556.144
 
Vals: {{Val list| 28d, 41, 315cde }}
 
Badness: 0.073459
 
== Tertiosec ==
Subgroup: 2.3.5


[[Comma list]]: {{monzo| -89 21 24 }}
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


[[Mapping]]: [{{val| 3 7 5 }}, {{val| 0 -8 7 }}]
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


[[POTE generator]]: ~16/15 = 112.281
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


{{Val list|legend=1| 21, 54, 75, 96, 171 }}
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


[[Badness]]: 0.851003
Badness (Sintel): 1.26


=== 7-limit ===
== Quintannic ==
Subgroup: 2.3.5.7
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} 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, 9805926501/9765625000


{{Multival|legend=1| 24 -21 6 -89 -58 73 }}
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


[[POTE generator]]: ~15/14 = 112.283
[[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| 21, 54, 75, 96, 171d }}
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


[[Badness]]: 0.431636
[[Badness]] (Sintel): 3.81


=== 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, 441/440, 43923/43750


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


POTE generator: ~15/14 = 112.171
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Vals: {{Val list| 21, 54, 75e }}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.173485
Badness (Sintel): 1.74


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


Subgroup: 2.3.5.7
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


[[Comma list]]: 225/224, 5971968/5764801
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


[[Mapping]]: [{{val| 2 1 6 4 }}, {{val| 0 8 -5 6 }}]
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


{{Multival|legend=1| 16 -10 12 -53 -26 56 }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


[[POTE generator]]: ~35/32 = 162.658
Badness (Sintel): 1.35


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


[[Badness]]: 0.178826
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


=== 11-limit ===
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 65536/65219
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


Mapping: [{{val| 2 1 6 4 8 }}, {{val| 0 8 -5 6 -4 }}]
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


POTE generator: ~11/10 = 162.592
Badness (Sintel): 1.17


Vals: {{Val list| 22, 74, 96, 118d }}
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Badness: 0.068410
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.  


== Gracecordial ==
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 781250000/771895089
[[Comma list]]: 225/224, 5971968/5764801


[[Mapping]]: [{{val| 1 0 34 63 }}, {{val| 0 1 -20 -38 }}]
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


{{Multival|legend=1| 1 -20 -38 -34 -63 -32 }}
[[Optimal tuning]]s:
* [[WE]]: ~2401/1728 = 599.7132{{c}}, ~35/32 = 162.5806{{c}}
: [[error map]]: {{val| -0.574 -1.597 -0.937 +5.510 }}
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~35/32 = 162.6388{{c}}
: error map: {{val| 0.000 -0.844 +0.492 +7.007 }}


[[POTE generator]]: ~3/2 = 700.824
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


{{Val list|legend=1| 12, 113, 125, 238c, 363c }}
[[Badness]] (Sintel): 4.53
 
[[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: 99/98, 176/175, 65536/65219


Mapping: [{{val| 1 0 34 63 -90 }}, {{val| 0 1 -20 -38 59 }}]
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


POTE generator: ~3/2 = 700.834
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


Vals: {{Val list| 12e, 113, 125, 238c }}
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


Badness: 0.089588
Badness (Sintel): 2.26


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


POTE generator: ~3/2 = 700.841
[[Subgroup]]: 2.3.5.7


Vals: {{Val list| 12e, 113, 125f, 238cf }}
[[Comma list]]: 225/224, 14495514624/13841287201


Badness: 0.052235
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


== Quintapole ==
[[Optimal tuning]]s:
: ''See also: [[16ed5/2 #16ed5/2 as a generator]] or [[42ed11 #42ed11 as a generator]]''
* [[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 }}


Aside from the marvel comma, the ''quintapole'' temperament (12&amp;85) tempers out 7812500/7411887 (sepru-atritriyo). In the 11-limit, it tempers out the ptolemisma (100/99) as well as 85184/84035 (trilo-aquinru-agu). It is so named for the following reasons - it has the same commas as the [[Marvel family #Appolo|apollo temperament]], and its generator is a semitone five of which gives a flat fourth (~4/3, about 495 cents). Proper pronunciation of the word "quintapole" is /'kwɪntəpəʊl/ or /'kwɪntəpoʊl/, like as "quint-a-pole". ''Not to be confused with [[Octagar temperaments #Quintupole|quint<u>'''u'''</u>pole]] temperament (12&amp;121).''
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 10.9
 
[[Comma list]]: 225/224, 7812500/7411887
 
[[Mapping]]: [{{val|1 2 1 1}}, {{val|0 -5 16 22}}]
 
[[POTE generator]]: ~21/20 = 98.994
 
{{Val list|legend=1| 12, 73c, 85, 97d }}
 
[[Badness]]: 0.192498


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


Comma list: 100/99, 225/224, 85184/84035
Comma list: 225/224, 3840/3773, 12005/11979
 
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


Mapping: [{{val|1 2 1 1 0}}, {{val|0 -5 16 22 42}}]
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: ~21/20 = 98.954
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


Vals: {{Val list| 12, 73ce, 85, 97d }}
Badness (Sintel): 5.74


Badness: 0.104353
== 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]]
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