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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]]
* ''[[Snipes]]'', {225/224, 6125/5832} → [[Wesley family #Snipes|Wesley family]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Decic]]'', {225/224, 16807/16384} → [[Cloudy clan #Decic|Cloudy clan]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Amavil]]'', {225/224, 17496/16807} → [[Mabila family #Amavil|Mabila family]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* ''[[Escapade]]'', {225/224, 65625/65536} → [[Escapade family #Escapade|Escapade family]]
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Fog]]'', {225/224, 156250/151263} → [[Misty family #Fog|Misty family]]
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Compton]]'', {225/224, 250047/250000} → [[Compton family #Compton|Compton family]]
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* ''[[Immune]]'', {225/224, 781250/750141} → [[Immunity family #Immune|Immunity family]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Betic]]'', {225/224, 1071875/1062882} → [[Sycamore family #Betic|Sycamore family]]
* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
* ''[[Houborizic]]'' {225/224, 1250000/1240029} → [[Amity family #Houborizic|Amity family]]
* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
* ''[[Qintosec]]'', {225/224, 2560000/2470629} → [[Qintosec family #Qintosec|Qintosec family]]
* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
* ''[[Quintapole]]'', {225/224, 7812500/7411887} → [[Quintaleap family #Quintapole|Quintaleap family]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Maquila]]'', {225/224, 30233088/28824005} → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Marvo]]'', {225/224, 78125000/78121827} → [[Gravity family #Marvo|Gravity family]]
* [[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, decic, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, hendeca, gwazy, and tertiosec, in the order of increasing [[badness]].  


Since (5/4)<sup>2</sup> = 225/224 × 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.


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


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


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


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


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5.7


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


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


{{Multival|legend=1| 4 -3 -14 }}
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}
: error map: {{val| 0.000 -1.169 -3.111 -0.849 }}


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


Comma list: 45/44, 49/48, 56/55
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


POTE generator: ~15/14 = 126.474
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}


Optimal GPV sequence: {{Val list| 9, 10, 19 }}
Optimal tunings:  
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


Badness: 0.026190
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


==== 13-limit ====
Badness (Sintel): 0.955
 
==== 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


Scales: [[wizard22]]
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


=== 11-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


Comma list: 225/224, 385/384, 4000/3993
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
Badness (Sintel): 1.50


Mapping generators: ~99/70, ~25/22
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


POTE generator: ~5/4 = 383.232
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.  


Optimal GPV sequence: {{Val list| 22, 50, 72, 166, 238c, 310c }}
[[Subgroup]]: 2.3.5.7


Badness: 0.018539
[[Comma list]]: 225/224, 84035/82944


Scales: [[wizard22]]
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


==== Lizard ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13
* [[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 }}


Comma list: 225/224, 351/350, 364/363, 385/384
{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
[[Badness]] (Sintel): 1.38


Mapping generators: ~99/70, ~25/22
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~5/4 = 383.389
Comma list: 225/224, 245/242, 385/384


Optimal GPV sequence: {{Val list| 22, 50, 72, 122, 194df }}
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


Badness: 0.021781
Optimal tunings:  
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


===== 17-limit =====
{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
Badness (Sintel): 1.04


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


Mapping generators: ~17/12, ~17/15
Comma list: 105/104, 144/143, 196/195, 245/242


POTE generator: ~5/4 = 383.381
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


Optimal GPV sequence: {{Val list| 22, 50, 72, 122g, 194dfg }}
Optimal tunings:  
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


Badness: 0.014536
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}


===== 19-limit =====
Badness (Sintel): 0.955
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
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: ~17/12, ~17/15
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.


POTE generator: ~5/4 = 383.477
[[Subgroup]]: 2.3.5.7


Optimal GPV sequence: {{Val list| 22h, 50, 72, 122g, 194dfg }}
[[Comma list]]: 225/224, 2500000/2470629


Badness: 0.015702
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


==== Gizzard ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13
* [[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: 225/224, 325/324, 385/384, 1573/1568
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
[[Badness]] (Sintel): 1.39


Mapping generators: ~99/70, ~25/22
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~5/4 = 383.170
Comma list: 225/224, 441/440, 1344/1331


Optimal GPV sequence: {{Val list| 72, 166, 238cf }}
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


Badness: 0.020252
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


===== 17-limit =====
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Badness (Sintel): 1.20


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
==== 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 = 383.175
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Optimal GPV sequence: {{Val list| 72, 166g, 238cfg }}
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Badness: 0.013624
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


===== 19-limit =====
Badness (Sintel): 1.14
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
=== Mermaid ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
Comma list: 225/224, 385/384, 532400/531441


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


POTE generator: ~5/4 = 383.138
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


Optimal GPV sequence: {{Val list| 72, 94, 166g }}
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


Badness: 0.014810
Badness (Sintel): 1.46


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


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.  


== Tritonic ==
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 #Tritonic]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 50421/50000
[[Comma list]]: 225/224, 589824/588245
 
[[Mapping]]: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


[[POTE generator]]: ~7/5 = 580.286
[[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 }}


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


[[Badness]]: 0.047578
[[Badness]] (Sintel): 1.44


=== 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, 385/384, 1331/1323


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


POTE generator: ~7/5 = 580.267
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| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Badness: 0.023659
Badness (Sintel): 0.838


==== 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: 225/224, 275/273, 385/384, 1331/1323


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


POTE generator: ~7/5 = 580.108
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Optimal GPV sequence: {{Val list| 29, 31, 60e, 151cde }}
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Badness: 0.022993
Badness (Sintel): 1.07


=== Tritoni ===
== Triton ==
Subgroup: 2.3.5.7.11
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


Comma list: 225/224, 385/384, 27783/27500
Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot.


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~7/5 = 580.389
[[Comma list]]: 225/224, 1029/1000
 
Optimal GPV sequence: {{Val list| 31, 91, 122, 153d }}
 
Badness: 0.045456
 
== 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| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


[[POTE generator]]: ~7/6 = 263.632
[[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| 41, 91, 132d }}
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


[[Badness]]: 0.054502
[[Badness]] (Sintel): 1.50


=== 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: 45/44, 56/55, 1029/1000


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


POTE generator: ~7/6 = 263.634
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| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


Badness: 0.031309
Badness (Sintel): 1.51


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


Comma list: 105/104, 144/143, 196/195, 245/242
[[Comma list]]: 225/224, 156250000/155649627


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


POTE generator: ~7/6 = 263.700
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7714{{c}}, ~21/20 = 83.4014{{c}}
: [[error map]]: {{val| +0.772 -0.820 -0.916 +0.381 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}}
: error map: {{val| 0.000 -2.139 -2.398 -1.362 }}


Optimal GPV sequence: {{Val list| 41, 91 }}
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


Badness: 0.023117
[[Badness]] (Sintel): 2.11
 
== Merman ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 225/224, 2500000/2470629
 
[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
 
{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
 
[[POTE generator]]: ~7/5 = 585.585
 
{{Val list|legend=1| 41, 84, 125 }}
 
[[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


== Slender ==
=== 17-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


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


[[Comma list]]: 225/224, 589824/588245
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
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| 13 -10 6 -46 -27 42 }}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


[[POTE generator]]: ~49/48 = 38.413
Badness (Sintel): 0.760


{{Val list|legend=1| 31, 94, 125 }}
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
[[Badness]]: 0.056934
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 1331/1323
 
Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]


POTE generator: ~49/48 = 38.387
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Optimal GPV sequence: {{Val list| 31, 63, 94, 125 }}
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Badness: 0.025342
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 275/273, 385/384, 1331/1323
Badness (Sintel): 0.895


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


POTE generator: ~49/48 = 38.314
Named by [[Xenllium]] in 2021, decic tempers out 16807/16384, the [[cloudy comma]], and {{monzo| 11 -10 -10 10 }}, the [[linus comma]], in addition to the marvel comma. It may be described as the {{nowrap| 50 & 60 }} temperament, with a period of 1/10 octave and a [[ploidacot]] signature of decaploid monocot. It is [[support]]ed by [[10edo|10-]], [[50edo|50-]], and [[60edo]].  


Optimal GPV sequence: {{Val list| 31, 63, 94 }}
It can be extended to the 11-, 13-, and 17-limit by adding [[385/384]], [[105/104]], and [[170/169]] to the comma list in this order.


Badness: 0.025913
[[Subgroup]]: 2.3.5.7


== Triton ==
[[Comma list]]: 225/224, 16807/16384
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''


Subgroup: 2.3.5.7
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


[[Comma list]]: 225/224, 1029/1000
[[Optimal tuning]]s:  
* [[WE]]: ~15/14 = 120.1837{{c}}, ~3/2 = 699.7654{{c}} (~49/48 = 21.3366{{c}})
: [[error map]]: {{val| +1.837 -0.353 -0.753 -3.683 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 698.8236{{c}} (~49/48 = 21.1764{{c}})
: error map: {{val| 0.000 -3.131 -5.137 -8.826 }}


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
[[Badness]] (Sintel): 2.26
 
[[POTE generator]]: ~7/5 = 568.865
 
{{Val list|legend=1| 2, 17d, 19, 78bd, 97bd }}
 
[[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, 3087/3025


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}


POTE generator: ~7/5 = 569.144
Optimal tunings:  
* WE: ~15/14 = 120.1406{{c}}, ~3/2 = 697.6075{{c}} (~56/55 = 23.2360{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.0142{{c}} (~56/55 = 22.9858{{c}})


Optimal GPV sequence: {{Val list| 2, 17d, 19, 59bde, 78bde, 97bde }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Badness: 0.045675
Badness (Sintel): 2.11


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


[[Comma list]]: 69198046875/68719476736
Comma list: 105/104, 144/143, 196/195, 2200/2197


[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


[[POTE generator]]: ~10125/8192 = 362.321
Optimal tunings:  
* WE: ~15/14 = 120.1166{{c}}, ~3/2 = 697.6705{{c}} (~78/77 = 23.0289{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.1492{{c}} (~78/77 = 22.8508{{c}})


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


[[Badness]]: 0.130236
Badness (Sintel): 1.52


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


[[Comma list]]: 225/224, 51200/50421
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


[[Mapping]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
Optimal tunings:
* WE: ~15/14 = 120.1262{{c}}, ~3/2 = 697.8185{{c}} (~78/77 = 22.9388{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.2757{{c}} (~78/77 = 22.7243{{c}})


[[POTE generator]]: ~49/40 = 362.255
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


{{Val list|legend=1| 10, 33, 43, 53 }}
Badness (Sintel): 1.28


[[Badness]]: 0.060533
=== Splendecic ===
Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[fantastic]] rank-3 temperament, its name is a portmanteau of ''splendid'' and ''decic''.


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


Comma list: 225/224, 385/384, 6655/6561
Comma list: 225/224, 1617/1600, 2401/2376


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}


POTE generator: ~27/22 = 362.101
Optimal tunings:  
* WE: ~15/14 = 120.1874{{c}}, ~3/2 = 699.6085{{c}} (~99/98 = 21.5156{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3531{{c}} (~99/98 = 21.6469{{c}})


Optimal GPV sequence: {{Val list| 10, 43e, 53, 116, 169de, 285cde }}
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


Badness: 0.050582
Badness (Sintel): 1.98


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


Comma list: 169/168, 225/224, 275/273, 385/384
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


POTE generator: ~16/13 = 362.105
Optimal tunings:  
* WE: ~15/14 = 120.1565{{c}}, ~3/2 = 699.2756{{c}} (~91/90 = 21.6631{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.2480{{c}} (~91/90 = 21.7520{{c}})


Optimal GPV sequence: {{Val list| 10, 43e, 53, 116, 169de, 285cdef }}
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


Badness: 0.027689
Badness (Sintel): 1.57


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


Comma list: 99/98, 176/175, 51200/50421
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374


POTE generator: ~49/40 = 362.418
Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
Optimal tunings:  
* WE: ~15/14 = 120.1571{{c}}, ~3/2 = 699.2892{{c}} (~91/90 = 21.6536{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3144{{c}} (~91/90 = 21.6856{{c}})


POTE generator: ~49/40 = 362.418
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


Optimal GPV sequence: {{Val list| 43, 53, 96, 149d }}
Badness (Sintel): 1.33


Badness: 0.051806
=== Prodecic ===
Prodecic (50e & 60e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the [[prodigy]] rank-3 temperament, its name is a portmanteau of ''prodigy'' and ''decic''.


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


Comma list: 99/98, 169/168, 176/175, 640/637
Comma list: 225/224, 441/440, 5929/5832


POTE generator: ~16/13 = 362.402
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
Optimal tunings:  
* WE: ~15/14 = 120.2024{{c}}, ~3/2 = 701.3908{{c}} (~55/54 = 19.8237{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.5235{{c}} (~55/54 = 19.4765{{c}})


POTE generator: ~16/13 = 362.402
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Optimal GPV sequence: {{Val list| 43, 53, 96, 149d }}
Badness (Sintel): 2.20


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


== Marvolo ==
Comma list: 105/104, 196/195, 275/273, 5929/5832
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 156250000/155649627
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
Optimal tunings:  
* WE: ~15/14 = 120.1654{{c}}, ~3/2 = 701.4683{{c}} (~91/90 = 19.5242{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.7175{{c}} (~91/90 = 19.2825{{c}})


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


[[POTE generator]]: ~21/20 = 83.348
Badness (Sintel): 1.73


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
==== 17-limit ====
 
Subgroup: 2.3.5.7.11.13.17
[[Badness]]: 0.083338
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 4000/3993
 
Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
 
POTE generator: ~21/20 = 83.340
 
Optimal GPV sequence: {{Val list| 29, 43, 72 }}
 
Badness: 0.028965
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 225/224, 364/363, 441/440
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


POTE generator: ~21/20 = 83.330
Optimal tunings:  
* WE: ~15/14 = 120.1577{{c}}, ~3/2 = 701.3950{{c}} (~91/90 = 19.5514{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.6932{{c}} (~91/90 = 19.3068{{c}})


Optimal GPV sequence: {{Val list| 29, 43, 72, 115f }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness: 0.021470
Badness (Sintel): 1.41


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


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


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


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


[[POTE generator]]: ~5/4 = 383.165
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


{{Val list|legend=1| 9, 63, 72, 495bcd }}
[[Badness]] (Sintel): 2.37
 
[[Badness]]: 0.093679


=== 11-limit ===
=== 11-limit ===
Line 746: Line 783:
Comma list: 225/224, 385/384, 12005/11979
Comma list: 225/224, 385/384, 12005/11979


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


POTE generator: ~5/4 = 383.146
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| 9, 63, 72, 423cd, 495bcd }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.030426
Badness (Sintel): 1.01


=== 13-limit ===
=== 13-limit ===
Line 759: Line 798:
Comma list: 169/168, 225/224, 364/363, 1716/1715
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 }}]
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


POTE generator: ~5/4 = 383.047
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| 9, 63, 72, 279cf }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.022322
Badness (Sintel): 0.922


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


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


{{Multival|legend=1| 1 -20 -38 -34 -63 -32 }}
[[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 = 1200.0000{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


[[POTE generator]]: ~3/2 = 700.824
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


{{Val list|legend=1| 12, 113, 125, 238c, 363c }}
[[Badness]] (Sintel): 2.44
 
[[Badness]]: 0.096279


=== 11-limit ===
=== 11-limit ===
Line 789: Line 833:
Comma list: 225/224, 385/384, 236328125/234365481
Comma list: 225/224, 385/384, 236328125/234365481


Mapping: [{{val| 1 0 34 63 -90 }}, {{val| 0 1 -20 -38 59 }}]
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


POTE generator: ~3/2 = 700.834
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| 12e, 101cde, 113, 125, 238c }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Badness: 0.089588
Badness (Sintel): 2.96


==== 13-limit ====
==== 13-limit ====
Line 802: Line 848:
Comma list: 225/224, 325/324, 385/384, 831875/830466
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 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


POTE generator: ~3/2 = 700.841
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| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.052235
Badness (Sintel): 2.16


==== 17-limit ====
==== 17-limit ====
Line 815: Line 863:
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


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


POTE generator: ~3/2 = 700.841
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| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.038565
Badness (Sintel): 1.96


==== 19-limit ====
==== 19-limit ====
Line 828: Line 878:
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


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


POTE generator: ~3/2 = 700.842
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


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


Badness: 0.028165
Badness (Sintel): 1.71


==== 23-limit ====
==== 23-limit ====
Line 841: Line 893:
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 -43 }}, {{val| 0 1 -20 -38 59 44 7 -3 30 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


POTE generator: ~3/2 = 700.843
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 238cfi }}
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


Badness: 0.021879
Badness (Sintel): 1.57


==== 29-limit ====
==== 29-limit ====
Line 854: Line 908:
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 -43 -49 }}, {{val| 0 1 -20 -38 59 44 7 -3 30 34 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


POTE generator: ~3/2 = 700.842
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


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


Badness: 0.018011
Badness (Sintel): 1.50


==== 31-limit ====
==== 31-limit ====
Line 867: Line 923:
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


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: {{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: ~3/2 = 700.838
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


Optimal GPV sequence: {{Val list| 12e, 101cdek, 113, 125f, 238cfi }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Badness: 0.016007
Badness (Sintel): 1.53


=== Gracecord ===
=== Gracecord ===
Line 880: Line 938:
Comma list: 225/224, 441/440, 109375/107811
Comma list: 225/224, 441/440, 109375/107811


Mapping: [{{val| 1 0 34 63 89 }}, {{val| 0 1 -20 -38 -54 }}]
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


POTE generator: ~3/2 = 700.885
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| 12, 101cd, 113 }}
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Badness: 0.066964
Badness (Sintel): 2.21


==== 13-limit ====
==== 13-limit ====
Line 893: Line 953:
Comma list: 225/224, 364/363, 441/440, 6125/6084
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 }}]
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


POTE generator: ~3/2 = 700.890
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| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Badness: 0.044196
Badness (Sintel): 1.83


==== 17-limit ====
==== 17-limit ====
Line 906: Line 968:
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
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 }}]
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


POTE generator: ~3/2 = 700.870
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


Optimal GPV sequence: {{Val list| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Badness: 0.036637
Badness (Sintel): 1.87


==== 19-limit ====
==== 19-limit ====
Line 919: Line 983:
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663
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 }}]
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


POTE generator: ~3/2 = 700.866
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


Optimal GPV sequence: {{Val list| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Badness: 0.027559
Badness (Sintel): 1.68


== Alphorn ==
== Alphorn ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
[[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]]: ~48/35 = 556.221
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
[[Badness]] (Sintel): 3.27
 
[[Badness]]: 0.129258


=== 11-limit ===
=== 11-limit ===
Line 947: Line 1,016:
Comma list: 225/224, 385/384, 12250/11979
Comma list: 225/224, 385/384, 12250/11979


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


POTE generator: ~11/8 = 556.144
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| 28d, 41, 315cde }}
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Badness: 0.073459
Badness (Sintel): 2.43


== Misneb ==
== Misneb ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Misneb]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 4194304/4117715  
[[Comma list]]: 225/224, 4194304/4117715  


[[Mapping]]: [{{Val|1 3 1 3}}, {{Val|0 -15 14 -2}}]
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
 
: mapping generators: ~2, ~15/8
{{Multival|legend=1|15 -14 2 -57 -39 44}}


[[POTE generator]]: ~16/15 = 113.235
[[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| 21, 32, 53 }}
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


[[Badness]]: 0.140970
[[Badness]] (Sintel): 3.57


=== 11-limit ===
=== 11-limit ===
Line 977: Line 1,051:
Comma list: 99/98, 176/175, 1310720/1294139
Comma list: 99/98, 176/175, 1310720/1294139


Mapping: [{{Val|1 3 1 3 1}}, {{Val|0 -15 14 -2 26}}]
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


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


Badness: 0.085390
Badness (Sintel): 2.82


==== 13-limit ====
==== 13-limit ====
Line 990: Line 1,066:
Comma list: 99/98, 176/175, 640/637, 847/845
Comma list: 99/98, 176/175, 640/637, 847/845


Mapping: [{{Val|1 3 1 3 1 2}}, {{Val|0 -15 14 -2 26 18}}]
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


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


Badness: 0.045569
Badness (Sintel): 1.88


=== Musneb ===
=== Musneb ===
Line 1,003: Line 1,081:
Comma list: 225/224, 385/384, 66550/64827
Comma list: 225/224, 385/384, 66550/64827


Mapping: [{{Val|1 3 1 3 6}}, {{Val|0 -15 14 -2 -27}}]
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


POTE generator: ~16/15 = 113.142
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


Optimal GPV sequence: {{Val list| 32, 53, 191de, 244cddee, 297cddee }}
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Badness: 0.087333
Badness (Sintel): 2.89


== Untriton ==
== Untriton ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Untriton]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''
 
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


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


{{Multival|legend=1|9 -19 -20 -51 -57 7}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
: [[error map]]: {{val| -0.172 +0.002 -2.533 +3.511 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}}
: error map: {{val| 0.000 +0.298 -2.181 +3.946 }}


[[POTE generator]]: ~7/5 = 588.641
{{Optimal ET sequence|legend=1| 51, 53 }}


{{Val list|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}
[[Badness]] (Sintel): 3.64
 
[[Badness]]: 0.143976


=== 11-limit ===
=== 11-limit ===
Line 1,033: Line 1,118:
Comma list: 121/120, 225/224, 22000/21609
Comma list: 121/120, 225/224, 22000/21609


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


POTE generator: ~7/5 = 588.626
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| 51, 53 }}
{{Optimal ET sequence|legend=0| 51, 53 }}


Badness: 0.074295
Badness (Sintel): 2.46


=== 13-limit ===
=== 13-limit ===
Line 1,046: Line 1,133:
Comma list: 121/120, 225/224, 275/273, 1040/1029
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}}]
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


POTE generator: ~7/5 = 588.654
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


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


Badness: 0.047441
Badness (Sintel): 1.96
 
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 823543/800000
 
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}
 
[[Badness]] (Sintel): 3.67
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 245/242, 1617/1600
 
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}
 
Optimal tunings:
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}
 
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
 
Badness (Sintel): 2.00
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 196/195, 245/242, 1001/1000
 
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}
 
Optimal tunings:
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}
 
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
 
Badness (Sintel): 1.43
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242
 
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}
 
Optimal tunings:
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}
 
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}
 
Badness (Sintel): 1.26


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


[[Comma list]]: 225/224, 9805926501/9765625000
[[Comma list]]: 225/224, 9805926501/9765625000


[[Mapping]]: [{{val| 1 1 5 7 }}, {{val| 0 5 -23 -36 }}]
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
 
: mapping generators: ~2, ~10000/9261
{{Multival|legend=1| 5 -23 -36 -48 -71 -19 }}


[[POTE generator]]: ~10000/9261 = 139.838
[[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| 43, 60, 103, 266bcd, 369bcd }}
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


[[Badness]]: 0.150565
[[Badness]] (Sintel): 3.81


=== 11-limit ===
=== 11-limit ===
Line 1,074: Line 1,233:
Comma list: 225/224, 441/440, 43923/43750
Comma list: 225/224, 441/440, 43923/43750


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


POTE generator: ~320/297 = 139.827
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| 43, 60e, 103 }}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.052590
Badness (Sintel): 1.74


=== 13-limit ===
=== 13-limit ===
Line 1,087: Line 1,248:
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


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


POTE generator: ~13/12 = 139.812
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| 43, 60e, 103 }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 0.032730
Badness (Sintel): 1.35


=== 17-limit ===
=== 17-limit ===
Line 1,100: Line 1,263:
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Mapping: [{{val| 1 1 5 7 8 3 7 }}, {{val| 0 5 -23 -36 -39 6 -25 }}]
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
 
Optimal tunings:
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}
 
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
 
Badness (Sintel): 1.17
 
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''
 
Hendeca tempers out the same 5-limit comma as [[porwell temperaments #Hendecatonic|hendecatonic]], and has a period of 1/11 octave. However, in this temperament, nine periods represent [[7/4]], the same as [[keemic temperaments #Undeka|undeka]]. It can be tuned to [[22edo]] or [[33edo]] using the [[patent val]], or [[55edo]] using the 55d val. It was named by [[Xenllium]] in 2025 as a low-accuracy variant of hendecatonic.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 122880/117649
 
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~16/15 = 108.9526{{c}}, ~3/2 = 702.4215{{c}}
: [[error map]]: {{val| -1.521 -1.055 -2.252 +8.705 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 703.2071{{c}}
: error map: {{val| 0.000 +1.252 +1.388 +12.992 }}
 
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}
 
[[Badness]] (Sintel): 4.37
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 225/224, 352/343
 
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}


POTE generator: ~13/12 = 139.815
Optimal tunings:  
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


Optimal GPV sequence: {{Val list| 43, 60e, 103 }}
{{Optimal ET sequence|legend=0| 22, 55d }}


Badness: 0.023038
Badness (Sintel): 2.46


== Gwazy ==
== Gwazy ==
{{see also|Very high accuracy temperaments #Kwazy}}
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
 
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 2 1 6 4 }}, {{val| 0 8 -5 6 }}]
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
 
: mapping generators: ~2401/1728, ~35/32
{{Multival|legend=1| 16 -10 12 -53 -26 56 }}


[[POTE generator]]: ~35/32 = 162.658
[[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 }}


{{Val list|legend=1| 22, 74, 96, 118d }}
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


[[Badness]]: 0.178826
[[Badness]] (Sintel): 4.53


=== 11-limit ===
=== 11-limit ===
Line 1,130: Line 1,338:
Comma list: 99/98, 176/175, 65536/65219
Comma list: 99/98, 176/175, 65536/65219


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


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


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


Badness: 0.068410
Badness (Sintel): 2.26


== Tertiosec ==
== Tertiosec ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tertiosec]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''
 
Tertiosec may be described as the {{nowrap| 21 & 75 }} temperament. It was initially named ''tertiomar'' by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, but was changed to ''tertiosec'' in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 3 7 5 9 }}, {{val| 0 -8 7 -2 }}]
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
 
: mapping generators: ~3072/2401, ~2048/1715
{{Multival|legend=1| 24 -21 6 -89 -58 73 }}


[[POTE generator]]: ~15/14 = 112.283
[[Optimal tuning]]s:  
* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
: [[error map]]: {{val| -0.523 -1.044 -1.549 +5.138 }}
* [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}}
: error map: {{val| 0.000 -0.284 -0.276 +6.592 }}


{{Val list|legend=1| 21, 54, 75, 96, 171d }}
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


[[Badness]]: 0.431636
[[Badness]] (Sintel): 10.9


=== 11-limit ===
=== 11-limit ===
Line 1,160: Line 1,375:
Comma list: 225/224, 3840/3773, 12005/11979
Comma list: 225/224, 3840/3773, 12005/11979


Mapping: [{{val| 3 7 5 9 9 }}, {{val| 0 -8 7 -2 5 }}]
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
 
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: ~15/14 = 112.171
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


Optimal GPV sequence: {{Val list| 21, 54, 75e }}
Badness (Sintel): 5.74


Badness: 0.173485
== References ==


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