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]]
* [[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]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Fog]]'', {225/224, 156250/151263} → [[Misty family #Fog]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* ''[[Compton]]'', {225/224, 250047/250000} → [[Compton family #Compton]]
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Houborizic]]'' {225/224, 1250000/1240029} → [[Amity family #Houborizic]]
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Qintosec]]'', {225/224, 2560000/2470629} → [[Qintosec family #Qintosec]]
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* ''[[Maquila]]'', {225/224, 30233088/28824005} → [[Maquila family #Maquila]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Marvo]]'', {225/224, 78125000/78121827} → [[Gravity family #Marvo]]
* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore 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, gracecordial, and quintapole.  
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]]: 2.3.5.7


Subgroup: 2.3.5
[[Comma list]]: 225/224, 118098/117649


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


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}
: error map: {{val| 0.000 -1.169 -3.111 -0.849 }}


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


=== 11-limit ===
Badness (Sintel): 0.741
Subgroup: 2.3.5.7.11
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


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


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


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


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


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


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


Badness: 0.021559
Badness (Sintel): 0.950
 
==== 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.  
{{Main|Wizard}}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #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
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
Comma list: 225/224, 275/273, 385/384, 1331/1323


Mapping generators: ~77/54, ~55/48
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


POTE generator: ~5/4 = 383.124
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Vals: {{Val list| 22, 50e, 72ee, 94ee }}
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Badness: 0.057799
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]].''


Subgroup: 2.3.5.7
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


[[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 404: Line 509:
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|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


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


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


[[POTE generator]]: ~7/5 = 580.286
[[Badness]] (Sintel): 2.11
 
{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
 
[[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
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


POTE generator: ~7/5 = 580.108
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Vals: {{Val list| 29, 31, 60e, 151cde }}
Badness (Sintel): 0.760


Badness: 0.022993
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


=== Tritoni ===
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 27783/27500
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


POTE generator: ~7/5 = 580.389
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Vals: {{Val list| 31, 91, 122, 153d }}
Badness (Sintel): 0.895


Badness: 0.045456
== Decic ==
{{Main| Decic }}


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


Subgroup: 2.3.5.7
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.


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


[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
[[Comma list]]: 225/224, 16807/16384


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


[[POTE generator]]: ~7/5 = 585.585
[[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 }}


{{Val list|legend=1| 41, 84, 125 }}
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


[[Badness]]: 0.055078
[[Badness]] (Sintel): 2.26


=== 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, 3087/3025


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}


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


Vals: {{Val list| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Badness: 0.036383
Badness (Sintel): 2.11


=== 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: 105/104, 144/143, 196/195, 2200/2197
 
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 10, 40, 50 }}
 
Badness (Sintel): 1.52
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220
 
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 10, 40, 50 }}
 
Badness (Sintel): 1.28
 
=== 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''.
 
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 1617/1600, 2401/2376
 
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}
 
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 ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}
 
Badness (Sintel): 1.98
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


POTE generator: ~7/5 = 585.657
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


Vals: {{Val list| 41, 84, 125e, 209ef, 293ef }}
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}})


Badness: 0.027544
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


== Untriton ==
Badness (Sintel): 1.57
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Untriton]].''


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


[[Comma list]]: 225/224, 125000000/121060821
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374


[[Mapping]]: [{{Val|1 6 -7 -7}}, {{Val|0 -9 19 20}}]
Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


{{Multival|legend=1|9 -19 -20 -51 -57 7}}
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]]: ~7/5 = 588.641
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


{{Val list|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}
Badness (Sintel): 1.33


[[Badness]]: 0.143976
=== 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''.


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


Comma list: 121/120, 225/224, 22000/21609
Comma list: 225/224, 441/440, 5929/5832


Mapping: [{{Val|1 6 -7 -7 1}}, {{Val|0 -9 19 20 5}}]
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}


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


Vals: {{Val list| 51, 53 }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness: 0.074295
Badness (Sintel): 2.20


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


Comma list: 121/120, 225/224, 275/273, 1040/1029
Comma list: 105/104, 196/195, 275/273, 5929/5832
 
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Mapping: [{{Val|1 6 -7 -7 1 -12}}, {{Val|0 -9 19 20 5 32}}]
Badness (Sintel): 1.73


POTE generator: ~7/5 = 588.654
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Vals: {{Val list| 51f, 53 }}
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


Badness: 0.047441
Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


== Septimin ==
Optimal tunings:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
* 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 ET sequence|legend=0| 10, 50e, 60e }}


Subgroup: 2.3.5.7
Badness (Sintel): 1.41


[[Comma list]]: 225/224, 84035/82944
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
[[Comma list]]: 225/224, 40353607/40310784


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


[[POTE generator]]: ~7/6 = 263.632
[[Optimal tuning]]s:  
* [[WE]]: ~2592/2401 = 133.4174{{c}}, ~12005/6912 = 950.7667{{c}} (~1728/1715 = 16.8452{{c}})
: [[error map]]: {{val| +0.756 -0.422 -1.395 +0.298 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}})
: error map: {{val| 0.000 -1.361 -3.277 -1.565 }}


{{Val list|legend=1| 41, 91, 132d }}
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


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


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


Comma list: 225/224, 385/384, 2401/2376
Comma list: 225/224, 385/384, 12005/11979


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


POTE generator: ~7/6 = 263.634
Optimal tunings:  
* WE: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}})
* CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}})


Vals: {{Val list| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.031309
Badness (Sintel): 1.01


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


Comma list: 105/104, 144/143, 196/195, 245/242
Comma list: 169/168, 225/224, 364/363, 1716/1715


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


POTE generator: ~7/6 = 263.700
Optimal tunings:  
* WE: ~14/13 = 133.4245{{c}}, ~26/15 = 950.9362{{c}} (~105/104 = 16.9650{{c}})
* CWE: ~14/13 = 133.3333{{c}}, ~26/15 = 950.4364{{c}} (~99/98 = 17.1031{{c}})


Vals: {{Val list| 41, 91 }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.023117
Badness (Sintel): 0.922


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 4194304/4117715
[[Comma list]]: 225/224, 781250000/771895089


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


{{Multival|legend=1|15 -14 2 -57 -39 44}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


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


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


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


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


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


POTE generator: ~16/15 = 113.323
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Vals: {{Val list| 21, 32e, 53, 127, 180de }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Badness: 0.085390
Badness (Sintel): 2.96


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


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


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


POTE generator: ~16/15 = 113.323
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


Vals: {{Val list| 21, 32e, 53, 127, 180de }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Badness: 0.045569
Badness (Sintel): 2.16


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


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


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


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


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


Badness: 0.087333
Badness (Sintel): 1.96


== Slender ==
==== 19-limit ====
Slender (31&amp;32) tempers out the [[Hewuermera temperaments|hewuermera comma]] in addition to the marvel comma, this has a generator [[49/48]], 3 of which equal marvel's 16/15 - 15/14, and 10 generators is 5/4.
Subgroup: 2.3.5.7.11.13.17.19


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


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


[[Comma list]]: 225/224, 589824/588245
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
Badness (Sintel): 1.71


[[POTE generator]]: ~49/48 = 38.413
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


{{Val list|legend=1| 31, 94, 125 }}
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


[[Badness]]: 0.056934
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


=== 11-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Comma list: 225/224, 385/384, 1331/1323
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
Badness (Sintel): 1.57


POTE generator: ~49/48 = 38.387
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Vals: {{Val list| 31, 63, 94, 125 }}
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Badness: 0.025342
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


=== 13-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11.13
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Comma list: 225/224, 275/273, 385/384, 1331/1323
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
Badness (Sintel): 1.50


POTE generator: ~49/48 = 38.314
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Vals: {{Val list| 31, 63, 94 }}
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


Badness: 0.025913
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


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


[[Comma list]]: 225/224, 156250000/155649627
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
Badness (Sintel): 1.53


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
=== Gracecord ===
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~21/20 = 83.348
Comma list: 225/224, 441/440, 109375/107811


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


[[Badness]]: 0.083338
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


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


Comma list: 225/224, 441/440, 4000/3993
Badness (Sintel): 2.21


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~21/20 = 83.340
Comma list: 225/224, 364/363, 441/440, 6125/6084


Vals: {{Val list| 29, 43, 72 }}
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Badness: 0.028965
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


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


Comma list: 169/168, 225/224, 364/363, 441/440
Badness (Sintel): 1.83


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~21/20 = 83.330
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Vals: {{Val list| 29, 43, 72, 115f }}
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Badness: 0.021470
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


== Amavil ==
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Mabila]].''


Subgroup: 2.3.5.7
Badness (Sintel): 1.87


[[Comma list]]: 225/224, 17496/16807
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


[[POTE generator]]: ~48/35 = 529.979
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


[[Badness]]: 0.109625
Badness (Sintel): 1.68


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


Comma list: 99/98, 176/175, 864/847
[[Comma list]]: 225/224, 5764801/5668704


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


POTE generator: ~15/11 = 529.974
[[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 }}


Vals: {{Val list| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Badness: 0.042649
[[Badness]] (Sintel): 3.27


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


Comma list: 78/77, 99/98, 144/143, 176/175
Comma list: 225/224, 385/384, 12250/11979


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


POTE generator: ~15/11 = 529.951
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Vals: {{Val list| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Badness: 0.025791
Badness (Sintel): 2.43


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


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


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


{{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]]: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
Optimal tunings:
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


[[POTE generator]]: ~48/35 = 556.221
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
Badness (Sintel): 1.43


[[Badness]]: 0.129258
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


=== 11-limit ===
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 12250/11979
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


POTE generator: ~11/8 = 556.144
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Vals: {{Val list| 28d, 41, 315cde }}
Badness (Sintel): 1.26


Badness: 0.073459
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.  


== Tertiosec ==
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5


[[Comma list]]: {{monzo| -89 21 24 }}
[[Comma list]]: 225/224, 9805926501/9765625000


[[Mapping]]: [{{val| 3 7 5 }}, {{val| 0 -8 7 }}]
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


[[POTE generator]]: ~16/15 = 112.281
[[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, 171 }}
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


[[Badness]]: 0.851003
[[Badness]] (Sintel): 3.81


=== 7-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 225/224, 14495514624/13841287201
Comma list: 225/224, 441/440, 43923/43750


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


{{Multival|legend=1| 24 -21 6 -89 -58 73 }}
Optimal tunings:
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


[[POTE generator]]: ~15/14 = 112.283
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


{{Val list|legend=1| 21, 54, 75, 96, 171d }}
Badness (Sintel): 1.74


[[Badness]]: 0.431636
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 11-limit ===
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
Subgroup: 2.3.5.7.11


Comma list: 225/224, 3840/3773, 12005/11979
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


Mapping: [{{val| 3 7 5 9 9 }}, {{val| 0 -8 7 -2 5 }}]
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


POTE generator: ~15/14 = 112.171
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Vals: {{Val list| 21, 54, 75e }}
Badness (Sintel): 1.35


Badness: 0.173485
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== Gwazy ==
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
{{see also|Very high accuracy temperaments #Kwazy}}


Subgroup: 2.3.5.7
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


[[Comma list]]: 225/224, 5971968/5764801
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 }}, {{val| 0 8 -5 6 }}]
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


{{Multival|legend=1| 16 -10 12 -53 -26 56 }}
Badness (Sintel): 1.17


[[POTE generator]]: ~35/32 = 162.658
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


{{Val list|legend=1| 22, 74, 96, 118d }}
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.


[[Badness]]: 0.178826
[[Subgroup]]: 2.3.5.7


=== 11-limit ===
[[Comma list]]: 225/224, 122880/117649
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 65536/65219
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


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


POTE generator: ~11/10 = 162.592
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}


Vals: {{Val list| 22, 74, 96, 118d }}
[[Badness]] (Sintel): 4.37


Badness: 0.068410
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Gracecordial ==
Comma list: 121/120, 225/224, 352/343
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 781250000/771895089
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}


[[Mapping]]: [{{val| 1 0 34 63 }}, {{val| 0 1 -20 -38 }}]
Optimal tunings:  
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


{{Multival|legend=1| 1 -20 -38 -34 -63 -32 }}
{{Optimal ET sequence|legend=0| 22, 55d }}


[[POTE generator]]: ~3/2 = 700.824
Badness (Sintel): 2.46


{{Val list|legend=1| 12, 113, 125, 238c, 363c }}
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


[[Badness]]: 0.096279
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.  


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


Comma list: 225/224, 385/384, 236328125/234365481
[[Comma list]]: 225/224, 5971968/5764801


Mapping: [{{val| 1 0 34 63 -90 }}, {{val| 0 1 -20 -38 59 }}]
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


POTE generator: ~3/2 = 700.834
[[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 }}


Vals: {{Val list| 12e, 113, 125, 238c }}
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


Badness: 0.089588
[[Badness]] (Sintel): 4.53


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


Comma list: 225/224, 325/324, 385/384, 831875/830466
Comma list: 99/98, 176/175, 65536/65219


Mapping: [{{val| 1 0 34 63 -90 -66 }}, {{val| 0 1 -20 -38 59 44 }}]
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


POTE generator: ~3/2 = 700.841
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, 125f, 238cf }}
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


Badness: 0.052235
Badness (Sintel): 2.26


== Quintapole ==
== Tertiosec ==
: ''See also: [[16ed5/2 #16ed5/2 as a generator]] or [[42ed11 #42ed11 as a generator]]''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


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).''
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, 7812500/7411887


[[Mapping]]: [{{val|1 2 1 1}}, {{val|0 -5 16 22}}]
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


[[POTE generator]]: ~21/20 = 98.994
[[Optimal tuning]]s:  
* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
: [[error map]]: {{val| -0.523 -1.044 -1.549 +5.138 }}
* [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}}
: error map: {{val| 0.000 -0.284 -0.276 +6.592 }}


{{Val list|legend=1| 12, 73c, 85, 97d }}
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


[[Badness]]: 0.192498
[[Badness]] (Sintel): 10.9


=== 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]]

Latest revision as of 11:09, 5 July 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This page discusses miscellaneous rank-2 temperaments tempering out 225/224, the marvel comma or septimal kleisma.

Temperaments considered in families and clans are:

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)2 = (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.

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.)

Wizard

Main article: 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 22 & 72. Its ploidacot is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the perfect twelfth. 72edo, 94edo, and especially 166edo are good tunings for it.

Subgroup: 2.3.5.7

Comma list: 225/224, 118098/117649

Mapping: [⟨2 1 5 2], ⟨0 6 -1 10]]

mapping generators: ~1225/864, ~245/216

Optimal tunings:

  • WE: ~1225/864 = 600.3438 ¢, ~245/216 = 216.8680 ¢
error map: ⟨+0.688 -0.403 -1.463 +0.541]
  • CWE: ~1225/864 = 600.0000 ¢, ~245/216 = 216.7977 ¢
error map: ⟨0.000 -1.169 -3.111 -0.849]

Optimal ET sequence: 22, 50, 72, 238c, 310c, 382c, 454bccd

Badness (Sintel): 1.03

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 1 5 2 8], ⟨0 6 -1 10 -3]]

Optimal tunings:

  • WE: ~99/70 = 600.3051 ¢, ~25/22 = 216.8782 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~25/22 = 216.7961 ¢

Optimal ET sequence: 22, 50, 72, 166, 238c, 310c

Badness (Sintel): 0.613

Lizard

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 364/363, 385/384

Mapping: [⟨2 1 5 2 8 11], ⟨0 6 -1 10 -3 -10]]

Optimal tunings:

  • WE: ~55/39 = 600.4824 ¢, ~25/22 = 216.7852 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~25/22 = 216.6247 ¢

Optimal ET sequence: 22, 50, 72

Badness (Sintel): 0.900

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 273/272, 289/288, 351/350, 375/374

Mapping: [⟨2 1 5 2 8 11 6], ⟨0 6 -1 10 -3 -10 6]]

Optimal tunings:

  • WE: ~17/12 = 600.5032 ¢, ~17/15 = 216.8002 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.6361 ¢

Optimal ET sequence: 22, 50, 72

Badness (Sintel): 0.741

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342

Mapping: [⟨2 1 5 2 8 11 6 2], ⟨0 6 -1 10 -3 -10 6 18]]

Optimal tunings:

  • WE: ~17/12 = 600.4698 ¢, ~17/15 = 216.6925 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.5434 ¢

Optimal ET sequence: 22h, 50, 72, 122g, 194dfg

Badness (Sintel): 0.955

Gizzard

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 1573/1568

Mapping: [⟨2 1 5 2 8 -2], ⟨0 6 -1 10 -3 26]]

Optimal tunings:

  • WE: ~99/70 = 600.2896 ¢, ~25/22 = 216.9343 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~25/22 = 216.8501 ¢

Optimal ET sequence: 22f, 72, 166, 238cf

Badness (Sintel): 0.837

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨2 1 5 2 8 -2 6], ⟨0 6 -1 10 -3 26 6]]

Optimal tunings:

  • WE: ~17/12 = 600.3227 ¢, ~17/15 = 216.9414 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.8469 ¢

Optimal ET sequence: 22f, 72, 166g, 238cfg

Badness (Sintel): 0.694

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨2 1 5 2 8 -2 6 15], ⟨0 6 -1 10 -3 26 6 -18]]

Optimal tunings:

  • WE: ~17/12 = 600.2637 ¢, ~17/15 = 216.9570 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.8687 ¢

Optimal ET sequence: 72, 94, 166g

Badness (Sintel): 0.901

Mage

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1331/1296

Mapping: [⟨2 1 5 2 4], ⟨0 6 -1 10 8]]

Optimal tunings:

  • WE: ~77/54 = 600.6486 ¢, ~55/48 = 217.1099 ¢
  • CWE: ~77/54 = 600.0000 ¢, ~55/48 = 216.9841 ¢

Optimal ET sequence: 22, 50e, 72ee

Badness (Sintel): 1.91

Tritonic

For the 5-limit version, see Miscellaneous 5-limit temperaments #Tritonic.

Tritonic tempers out 50421/50000 and may be described as the 29 & 31 temperament. It splits the 6th harmonic into five generators of ~10/7 tritones, 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: 225/224, 50421/50000

Mapping: [⟨1 -1 8 9], ⟨0 5 -11 -12]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1201.3539 ¢, ~10/7 = 620.4131 ¢
error map: ⟨+1.354 -1.243 -0.027 -1.598]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6778 ¢
error map: ⟨0.000 -3.566 -2.769 -4.959]

Optimal ET sequence: 29, 31, 60, 91, 122, 213bcd

Badness (Sintel): 1.20

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 8 9 5], ⟨0 5 -11 -12 -3]]

Optimal tunings:

  • WE: ~2 = 1201.7116 ¢, ~10/7 = 620.6166 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6890 ¢

Optimal ET sequence: 29, 31, 60e, 91e, 213bcdeee

Badness (Sintel): 0.782

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 196/195, 275/273

Mapping: [⟨1 -1 8 9 5 13], ⟨0 5 -11 -12 -3 -18]]

Optimal tunings:

  • WE: ~2 = 1201.5355 ¢, ~10/7 = 620.6855 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.8469 ¢

Optimal ET sequence: 29, 31, 60e

Badness (Sintel): 0.950

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 121/120, 154/153, 196/195, 273/272

Mapping: [⟨1 -1 8 9 5 13 17], ⟨0 5 -11 -12 -3 -18 -25]]

Optimal tunings:

  • WE: ~2 = 1201.5260 ¢, ~10/7 = 620.7330 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.8986 ¢

Optimal ET sequence: 29g, 31, 60e

Badness (Sintel): 0.973

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272

Mapping: [⟨1 -1 8 9 5 13 17 12], ⟨0 5 -11 -12 -3 -18 -25 -15]]

Optimal tunings:

  • WE: ~2 = 1201.3100 ¢, ~10/7 = 620.6509 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.9328 ¢

Optimal ET sequence: 29g, 31, 60e

Badness (Sintel): 1.03

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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

Mapping: [⟨1 -1 8 9 5 13 17 12 4], ⟨0 5 -11 -12 -3 -18 -25 -15 1]]

Optimal tunings:

  • WE: ~2 = 1201.4074 ¢, ~10/7 = 620.7185 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.9548 ¢

Optimal ET sequence: 29g, 31, 60e

Badness (Sintel): 1.04

Tritoni

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 27783/27500

Mapping: [⟨1 -1 8 9 -11], ⟨0 5 -11 -12 28]]

Optimal tunings:

  • WE: ~2 = 1201.0888 ¢, ~10/7 = 620.1733 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6146 ¢

Optimal ET sequence: 31, 91, 122, 153d

Badness (Sintel): 1.50

Septimin

For the 5-limit version, see Miscellaneous 5-limit temperaments #Septimin.

Septimin may be described as the 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 reduced give the perfect fifth; its ploidacot is thus eta-hendecacot. 91edo may be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 225/224, 84035/82944

Mapping: [⟨1 -7 7 -5], ⟨0 11 -6 10]]

mapping generators: ~2, ~12/7

Optimal tunings:

  • WE: ~2 = 1201.2452 ¢, ~12/7 = 937.3394 ¢
error map: ⟨+1.245 +0.062 -1.633 -1.658]
  • CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.4036 ¢
error map: ⟨0.000 -1.516 -4.735 -4.790]

Optimal ET sequence: 41, 91, 132d

Badness (Sintel): 1.38

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -7 7 -5 -2], ⟨0 11 -6 10 7]]

Optimal tunings:

  • WE: ~2 = 1200.8059 ¢, ~12/7 = 936.9952 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.3906 ¢

Optimal ET sequence: 41, 91, 223cdef

Badness (Sintel): 1.04

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195, 245/242

Mapping: [⟨1 -7 7 -5 -2 -8], ⟨0 11 -6 10 7 15]]

Optimal tunings:

  • WE: ~2 = 1200.5990 ¢, ~12/7 = 936.7670 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.3196 ¢

Optimal ET sequence: 41, 91

Badness (Sintel): 0.955

Merman

For the 5-limit version, see Miscellaneous 5-limit temperaments #Merman.

Merman may be described as the 41 & 43 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, seven generator steps give the interval class of 3. The ploidacot for this temperament is gamma-heptacot.

The name was likely derived from Triton, which was in turn derived from tritonic.

Subgroup: 2.3.5.7

Comma list: 225/224, 2500000/2470629

Mapping: [⟨1 -2 10 11], ⟨0 7 -15 -16]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1200.3898 ¢, ~10/7 = 614.6413 ¢
error map: ⟨+0.390 -0.435 -1.630 +1.634]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4073 ¢
error map: ⟨0.000 -1.104 -2.423 +0.657]

Optimal ET sequence: 41, 84, 125

Badness (Sintel): 1.39

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 1344/1331

Mapping: [⟨1 -2 10 11 5], ⟨0 7 -15 -16 -3]]

Optimal tunings:

  • WE: ~2 = 1199.9578 ¢, ~10/7 = 614.3720 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.3943 ¢

Optimal ET sequence: 41, 84, 125e

Badness (Sintel): 1.20

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 225/224, 364/363, 441/440

Mapping: [⟨1 -2 10 11 5 -5], ⟨0 7 -15 -16 -3 17]]

Optimal tunings:

  • WE: ~2 = 1199.7422 ¢, ~10/7 = 614.2110 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.3442 ¢

Optimal ET sequence: 41, 84, 125e, 209ef, 293ef

Badness (Sintel): 1.14

Mermaid

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 532400/531441

Mapping: [⟨1 -2 10 11 -16], ⟨0 7 -15 -16 38]]

Optimal tunings:

  • WE: ~2 = 1199.4973 ¢, ~10/7 = 614.7004 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4470 ¢

Optimal ET sequence: 41, 84e, 125, 166

Badness (Sintel): 1.46

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -2 10 11 22 32], ⟨0 7 -15 -16 38 58]]

Optimal tunings:

  • WE: ~2 = 1200.5126 ¢, ~10/7 = 614.7152 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4562 ¢

Optimal ET sequence: 41, 84ef, 125f, 166

Badness (Sintel): 1.47

Slender

Slender tempers out the hewuermera comma in addition to the marvel comma, and may be described as the 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.

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

Subgroup: 2.3.5.7

Comma list: 225/224, 589824/588245

Mapping: [⟨1 2 2 3], ⟨0 -13 10 -6]]

mapping generators: ~2, ~49/48

Optimal tunings:

  • WE: ~2 = 1200.3816 ¢, ~49/48 = 38.4256 ¢
error map: ⟨+0.382 -0.725 -1.295 +1.765]
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.4079 ¢
error map: ⟨0.000 -1.257 -2.235 +0.727]

Optimal ET sequence: 31, 94, 125, 406c

Badness (Sintel): 1.44

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 2 3 4], ⟨0 -13 10 -6 -17]]

Optimal tunings:

  • WE: ~2 = 1199.4983 ¢, ~49/48 = 38.4030 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.3775 ¢

Optimal ET sequence: 31, 63, 94, 125

Badness (Sintel): 0.838

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 2 3 4 3], ⟨0 -13 10 -6 -17 22]]

Optimal tunings:

  • WE: ~2 = 1200.1728 ¢, ~49/48 = 38.3192 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.3129 ¢

Optimal ET sequence: 31, 63, 94

Badness (Sintel): 1.07

Triton

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Stump.

Triton may be described as the 19 & 21 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, three generator steps give the interval class of 3. The ploidacot for this temperament is alpha-tricot.

Subgroup: 2.3.5.7

Comma list: 225/224, 1029/1000

Mapping: [⟨1 0 6 7], ⟨0 3 -7 -8]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1203.3828 ¢, ~10/7 = 632.9137 ¢
error map: ⟨+3.383 -3.214 +3.587 -8.457]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 630.9827 ¢
error map: ⟨0.000 -9.007 -3.192 -16.687]

Optimal ET sequence: 2, 17d, 19, 78bd, 97bd

Badness (Sintel): 1.50

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55, 1029/1000

Mapping: [⟨1 0 6 7 4], ⟨0 3 -7 -8 -1]]

Optimal tunings:

  • WE: ~2 = 1201.3875 ¢, ~10/7 = 631.5852 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 630.8007 ¢

Optimal ET sequence: 2, 17d, 19

Badness (Sintel): 1.51

Marvolo

Subgroup: 2.3.5.7

Comma list: 225/224, 156250000/155649627

Mapping: [⟨1 2 1 1], ⟨0 -6 19 26]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • WE: ~2 = 1200.7714 ¢, ~21/20 = 83.4014 ¢
error map: ⟨+0.772 -0.820 -0.916 +0.381]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3640 ¢
error map: ⟨0.000 -2.139 -2.398 -1.362]

Optimal ET sequence: 29, 43, 72, 619bbccd, 691bbccd

Badness (Sintel): 2.11

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 1 1 2], ⟨0 -6 19 26 21]]

Optimal tunings:

  • WE: ~2 = 1200.7075 ¢, ~21/20 = 83.3888 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3564 ¢

Optimal ET sequence: 29, 43, 72

Badness (Sintel): 0.958

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 1 1 2 3], ⟨0 -6 19 26 21 10]]

Optimal tunings:

  • WE: ~2 = 1200.9467 ¢, ~21/20 = 83.3956 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3516 ¢

Optimal ET sequence: 29, 43, 72

Badness (Sintel): 0.887

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 2 1 1 2 3 2], ⟨0 -6 19 26 21 10 30]]

Optimal tunings:

  • WE: ~2 = 1200.9606 ¢, ~21/20 = 83.4030 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3594 ¢

Optimal ET sequence: 29g, 43, 72

Badness (Sintel): 0.760

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 2 1 1 2 3 2 3], ⟨0 -6 19 26 21 10 30 18]]

Optimal tunings:

  • WE: ~2 = 1200.7625 ¢, ~21/20 = 83.3895 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3551 ¢

Optimal ET sequence: 29g, 43, 72

Badness (Sintel): 0.895

Decic

Main article: Decic

Named by Xenllium in 2021, decic tempers out 16807/16384, the cloudy comma, and [11 -10 -10 10⟩, the linus comma, in addition to the marvel comma. It may be described as the 50 & 60 temperament, with a period of 1/10 octave and a ploidacot signature of decaploid monocot. It is supported by 10-, 50-, and 60edo.

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.

Subgroup: 2.3.5.7

Comma list: 225/224, 16807/16384

Mapping: [⟨10 0 39 28], ⟨0 1 -1 0]]

mapping generators: ~15/14, ~3

Optimal tunings:

  • WE: ~15/14 = 120.1837 ¢, ~3/2 = 699.7654 ¢ (~49/48 = 21.3366 ¢)
error map: ⟨+1.837 -0.353 -0.753 -3.683]
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 698.8236 ¢ (~49/48 = 21.1764 ¢)
error map: ⟨0.000 -3.131 -5.137 -8.826]

Optimal ET sequence: 10, 30b, 40, 50, 60, 110d, 170cdd

Badness (Sintel): 2.26

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 3087/3025

Mapping: [⟨10 0 39 28 3], ⟨0 1 -1 0 2]]

Optimal tunings:

  • WE: ~15/14 = 120.1406 ¢, ~3/2 = 697.6075 ¢ (~56/55 = 23.2360 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 697.0142 ¢ (~56/55 = 22.9858 ¢)

Optimal ET sequence: 10, 40, 50

Badness (Sintel): 2.11

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195, 2200/2197

Mapping: [⟨10 0 39 28 3 37], ⟨0 1 -1 0 2 0]]

Optimal tunings:

  • WE: ~15/14 = 120.1166 ¢, ~3/2 = 697.6705 ¢ (~78/77 = 23.0289 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 697.1492 ¢ (~78/77 = 22.8508 ¢)

Optimal ET sequence: 10, 40, 50

Badness (Sintel): 1.52

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 144/143, 170/169, 196/195, 221/220

Mapping: [⟨10 0 39 28 3 37 25], ⟨0 1 -1 0 2 0 1]]

Optimal tunings:

  • WE: ~15/14 = 120.1262 ¢, ~3/2 = 697.8185 ¢ (~78/77 = 22.9388 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 697.2757 ¢ (~78/77 = 22.7243 ¢)

Optimal ET sequence: 10, 40, 50

Badness (Sintel): 1.28

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.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 1617/1600, 2401/2376

Mapping: [⟨10 0 39 28 82], ⟨0 1 -1 0 -3]]

Optimal tunings:

  • WE: ~15/14 = 120.1874 ¢, ~3/2 = 699.6085 ¢ (~99/98 = 21.5156 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 698.3531 ¢ (~99/98 = 21.6469 ¢)

Optimal ET sequence: 10e, 40e, 50, 60e, 110de, 170cddee

Badness (Sintel): 1.98

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 1001/1000, 1188/1183

Mapping: [⟨10 0 39 28 82 37], ⟨0 1 -1 0 -3 0]]

Optimal tunings:

  • WE: ~15/14 = 120.1565 ¢, ~3/2 = 699.2756 ¢ (~91/90 = 21.6631 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 698.2480 ¢ (~91/90 = 21.7520 ¢)

Optimal ET sequence: 10e, 40e, 50, 60e, 110de

Badness (Sintel): 1.57

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 170/169, 196/195, 289/288, 375/374

Mapping: [⟨10 0 39 28 82 37 25], ⟨0 1 -1 0 -3 0 1]]

Optimal tunings:

  • WE: ~15/14 = 120.1571 ¢, ~3/2 = 699.2892 ¢ (~91/90 = 21.6536 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 698.3144 ¢ (~91/90 = 21.6856 ¢)

Optimal ET sequence: 10e, 50, 60e, 110deg

Badness (Sintel): 1.33

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.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 5929/5832

Mapping: [⟨10 0 39 28 -13], ⟨0 1 -1 0 3]]

Optimal tunings:

  • WE: ~15/14 = 120.2024 ¢, ~3/2 = 701.3908 ¢ (~55/54 = 19.8237 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 700.5235 ¢ (~55/54 = 19.4765 ¢)

Optimal ET sequence: 10, 50e, 60e

Badness (Sintel): 2.20

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 275/273, 5929/5832

Mapping: [⟨10 0 39 28 -13 37], ⟨0 1 -1 0 3 0]]

Optimal tunings:

  • WE: ~15/14 = 120.1654 ¢, ~3/2 = 701.4683 ¢ (~91/90 = 19.5242 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 700.7175 ¢ (~91/90 = 19.2825 ¢)

Optimal ET sequence: 10, 50e, 60e

Badness (Sintel): 1.73

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 154/153, 170/169, 196/195, 289/288

Mapping: [⟨10 0 39 28 -13 37 25], ⟨0 1 -1 0 3 0 1]]

Optimal tunings:

  • WE: ~15/14 = 120.1577 ¢, ~3/2 = 701.3950 ¢ (~91/90 = 19.5514 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 700.6932 ¢ (~91/90 = 19.3068 ¢)

Optimal ET sequence: 10, 50e, 60e

Badness (Sintel): 1.41

Enneaportent

Subgroup: 2.3.5.7

Comma list: 225/224, 40353607/40310784

Mapping: [⟨9 0 28 11], ⟨0 2 -1 2]]

mapping generators: ~2592/2401, ~12005/6912

Optimal tunings:

  • WE: ~2592/2401 = 133.4174 ¢, ~12005/6912 = 950.7667 ¢ (~1728/1715 = 16.8452 ¢)
error map: ⟨+0.756 -0.422 -1.395 +0.298]
  • CWE: ~2592/2401 = 133.3333 ¢, ~12005/6912 = 950.2969 ¢ (~1728/1715 = 16.9636 ¢)
error map: ⟨0.000 -1.361 -3.277 -1.565]

Optimal ET sequence: 9, 54, 63, 72, 495bccd, 567bcccd

Badness (Sintel): 2.37

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 12005/11979

Mapping: [⟨9 0 28 11 24], ⟨0 2 -1 2 1]]

Optimal tunings:

  • WE: ~121/112 = 133.4071 ¢, ~210/121 = 950.7131 ¢ (~99/98 = 16.8633 ¢)
  • CWE: ~121/112 = 133.3333 ¢, ~210/121 = 950.2994 ¢ (~99/98 = 16.9661 ¢)

Optimal ET sequence: 9, 54, 63, 72

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 364/363, 1716/1715

Mapping: [⟨9 0 28 11 24 19], ⟨0 2 -1 2 1 2]]

Optimal tunings:

  • WE: ~14/13 = 133.4245 ¢, ~26/15 = 950.9362 ¢ (~105/104 = 16.9650 ¢)
  • CWE: ~14/13 = 133.3333 ¢, ~26/15 = 950.4364 ¢ (~99/98 = 17.1031 ¢)

Optimal ET sequence: 9, 54, 63, 72

Badness (Sintel): 0.922

Gracecordial

For the 5-limit version, see Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit).

Subgroup: 2.3.5.7

Comma list: 225/224, 781250000/771895089

Mapping: [⟨1 0 34 63], ⟨0 1 -20 -38]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1200.4904 ¢, ~3/2 = 701.1103 ¢
error map: ⟨+0.490 -0.354 -1.655 +1.241]
  • CWE: ~2 = 1200.3333 ¢, ~3/2 = 700.8112 ¢
error map: ⟨0.000 -1.144 -2.537 +0.349]

Optimal ET sequence: 12, …, 113, 125, 238c, 363c

Badness (Sintel): 2.44

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 34 63 -90], ⟨0 1 -20 -38 59]]

Optimal tunings:

  • WE: ~2 = 1200.5571 ¢, ~3/2 = 701.1589 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8328 ¢

Optimal ET sequence: 12e, 113, 125, 238c

Badness (Sintel): 2.96

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 831875/830466

Mapping: [⟨1 0 34 63 -90 -66], ⟨0 1 -20 -38 59 44]]

Optimal tunings:

  • WE: ~2 = 1200.6282 ¢, ~3/2 = 701.2080 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8421 ¢

Optimal ET sequence: 12e, 113, 125f, 238cf

Badness (Sintel): 2.16

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 34 63 -90 -66 -7], ⟨0 1 -20 -38 59 44 7]]

Optimal tunings:

  • WE: ~2 = 1200.5058 ¢, ~3/2 = 701.1360 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8414 ¢

Optimal ET sequence: 12e, 113, 125f, 238cf

Badness (Sintel): 1.96

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 0 34 63 -90 -66 -7 9], ⟨0 1 -20 -38 59 44 7 -3]]

Optimal tunings:

  • WE: ~2 = 1200.4418 ¢, ~3/2 = 701.0999 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8425 ¢

Optimal ET sequence: 12e, 113, 125f, 238cf

Badness (Sintel): 1.71

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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

Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43], ⟨0 1 -20 -38 59 44 7 -3 30]]

Optimal tunings:

  • WE: ~2 = 1200.4641 ¢, ~3/2 = 701.1145 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8444 ¢

Optimal ET sequence: 12e, 113, 238cfi

Badness (Sintel): 1.57

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

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

Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43 -49], ⟨0 1 -20 -38 59 44 7 -3 30 34]]

Optimal tunings:

  • WE: ~2 = 1200.4400 ¢, ~3/2 = 701.0986 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8428 ¢

Optimal ET sequence: 12e, 113, 125f, 238cfi

Badness (Sintel): 1.50

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

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

Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43 -49 -79], ⟨0 1 -20 -38 59 44 7 -3 30 34 53]]

Optimal tunings:

  • WE: ~2 = 1200.4178 ¢, ~3/2 = 701.0822 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8396 ¢

Optimal ET sequence: 12e, 113, 125f, 238cfi

Badness (Sintel): 1.53

Gracecord

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 34 63 89], ⟨0 1 -20 -38 -54]]

Optimal tunings:

  • WE: ~2 = 1200.6064 ¢, ~3/2 = 701.2398 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8718 ¢

Optimal ET sequence: 12, …, 101cd, 113

Badness (Sintel): 2.21

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 34 63 89 113], ⟨0 1 -20 -38 -54 -69]]

Optimal tunings:

  • WE: ~2 = 1200.6225 ¢, ~3/2 = 701.2539 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8781 ¢

Optimal ET sequence: 12f, …, 101cdf, 113

Badness (Sintel): 1.83

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 34 63 89 113 -7], ⟨0 1 -20 -38 -54 -69 7]]

Optimal tunings:

  • WE: ~2 = 1200.3308 ¢, ~3/2 = 701.0632 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8654 ¢

Optimal ET sequence: 12f, 101cdf, 113

Badness (Sintel): 1.87

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 0 34 63 89 113 -7 9], ⟨0 1 -20 -38 -54 -69 7 -3]]

Optimal tunings:

  • WE: ~2 = 1200.2658 ¢, ~3/2 = 701.0213 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8629 ¢

Optimal ET sequence: 12f, 101cdf, 113

Badness (Sintel): 1.68

Alphorn

Subgroup: 2.3.5.7

Comma list: 225/224, 5764801/5668704

Mapping: [⟨1 -7 5 -9], ⟨0 16 -5 22]]

mapping generators: ~2, ~35/24

Optimal tunings:

  • WE: ~2 = 1201.3004 ¢, ~35/24 = 644.4767 ¢
error map: ⟨+1.300 +0.569 -2.195 -2.043]
  • CWE: ~2 = 1200.3333 ¢, ~35/24 = 643.8137 ¢
error map: ⟨0.000 -0.936 -5.382 -4.924]

Optimal ET sequence: 13d, 28d, 41, 151cd, 192cdd, 233ccdd

Badness (Sintel): 3.27

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 12250/11979

Mapping: [⟨1 -7 5 -9 4], ⟨0 16 -5 22 -1]]

Optimal tunings:

  • WE: ~2 = 1200.5123 ¢, ~16/11 = 644.1307 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 643.8662 ¢

Optimal ET sequence: 13d, 28d, 41

Badness (Sintel): 2.43

Misneb

For the 5-limit version, see Miscellaneous 5-limit temperaments #Misneb.

Subgroup: 2.3.5.7

Comma list: 225/224, 4194304/4117715

Mapping: [⟨1 -12 15 1], ⟨0 15 -14 2]]

mapping generators: ~2, ~15/8

Optimal tunings:

  • WE: ~2 = 1199.7642 ¢, ~15/8 = 1086.5513 ¢
error map: ⟨-0.236 -0.856 -1.569 +4.041]
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.7633 ¢
error map: ⟨0.000 -0.506 -0.999 +4.701]

Optimal ET sequence: 21, 32, 53

Badness (Sintel): 3.57

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1310720/1294139

Mapping: [⟨1 -12 15 1 27], ⟨0 15 -14 2 -26]]

Optimal tunings:

  • WE: ~2 = 1200.1654 ¢, ~15/8 = 1086.8269 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.6766 ¢

Optimal ET sequence: 21, 32e, 53, 127

Badness (Sintel): 2.82

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 640/637, 847/845

Mapping: [⟨1 -12 15 1 27 20], ⟨0 15 -14 2 -26 -18]]

Optimal tunings:

  • WE: ~2 = 1200.1687 ¢, ~15/8 = 1086.8295 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.6757 ¢

Optimal ET sequence: 21, 32e, 53, 127

Badness (Sintel): 1.88

Musneb

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 1 3 6], ⟨0 -15 14 -2 -27]]

Optimal tunings:

  • WE: ~2 = 1200.0839 ¢, ~15/8 = 1086.9343 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.8593 ¢

Optimal ET sequence: 21e, 32, 53

Badness (Sintel): 2.89

Untriton

For the 5-limit version, see Miscellaneous 5-limit temperaments #Untriton.

Named by Petr Pařízek in 2011[1], untriton may be described as the 51 & 53 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, nine generator steps give the interval class of 3. The ploidacot for this temperament is delta-enneacot.

Subgroup: 2.3.5.7

Comma list: 225/224, 125000000/121060821

Mapping: [⟨1 -3 12 13], ⟨0 9 -19 -20]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1199.8275 ¢, ~10/7 = 611.2710 ¢
error map: ⟨-0.172 +0.002 -2.533 +3.511]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3614 ¢
error map: ⟨0.000 +0.298 -2.181 +3.946]

Optimal ET sequence: 51, 53

Badness (Sintel): 3.64

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 22000/21609

Mapping: [⟨1 -3 12 13 6], ⟨0 9 -19 -20 -5]]

Optimal tunings:

  • WE: ~2 = 1200.3591 ¢, ~10/7 = 611.5569 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3690 ¢

Optimal ET sequence: 51, 53

Badness (Sintel): 2.46

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 225/224, 275/273, 1040/1029

Mapping: [⟨1 -3 12 13 6 20], ⟨0 9 -19 -20 -5 -32]]

Optimal tunings:

  • WE: ~2 = 1200.4078 ¢, ~10/7 = 611.5536 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3392 ¢

Optimal ET sequence: 51f, 53

Badness (Sintel): 1.96

Naiadical

Named by Xenllium in 2026, naiadical may be described as the 21 & 29 temperament.

Subgroup: 2.3.5.7

Comma list: 225/224, 823543/800000

Mapping: [⟨1 -4 11 9], ⟨0 9 -14 -10]]

mapping generators: ~2, ~32/21

Optimal tunings:

  • WE: ~2 = 1202.1198 ¢, ~32/21 = 745.4675 ¢
error map: ⟨+2.120 -1.227 +0.459 -4.423]
  • CWE: ~2 = 1200.0000 ¢, ~32/21 = 744.1318 ¢
error map: ⟨0.000 -4.769 -4.159 -10.144]

Optimal ET sequence: 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: [⟨1 -4 11 9 14], ⟨0 9 -14 -10 -17]]

Optimal tunings:

  • WE: ~2 = 1201.9008 ¢, ~21/16 = 745.3867 ¢
  • CWE: ~2 = 1200.0000 ¢, ~32/21 = 744.1777 ¢

Optimal ET sequence: 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: [⟨1 -4 11 9 14 13], ⟨0 9 -14 -10 -17 -15]]

Optimal tunings:

  • WE: ~2 = 1201.7863 ¢, ~20/13 = 745.3344 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.1931 ¢

Optimal ET sequence: 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: [⟨1 -4 11 9 14 13 14], ⟨0 9 -14 -10 -17 -15 -16]]

Optimal tunings:

  • WE: ~2 = 1201.9208 ¢, ~20/13 = 745.3976 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.1669 ¢

Optimal ET sequence: 21, 29g, 50, 79dg

Badness (Sintel): 1.26

Quintannic

Named by Scott Dakota, quintannic may be described as the 43 & 60 temperament.

Subgroup: 2.3.5.7

Comma list: 225/224, 9805926501/9765625000

Mapping: [⟨1 1 5 7], ⟨0 5 -23 -36]]

mapping generators: ~2, ~10000/9261

Optimal tunings:

  • WE: ~2 = 1200.9803 ¢, ~10000/9261 = 139.9522 ¢
error map: ⟨+0.980 -1.214 -0.313 -0.243]
  • CWE: ~2 = 1200.0000 ¢, ~10000/9261 = 139.8184 ¢
error map: ⟨0.000 -2.863 -2.136 -2.287]

Optimal ET sequence: 43, 60, 103, 266bcd, 369bcd

Badness (Sintel): 3.81

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 43923/43750

Mapping: [⟨1 1 5 7 8], ⟨0 5 -23 -36 -39]]

Optimal tunings:

  • WE: ~2 = 1201.0031 ¢, ~320/297 = 139.9435 ¢
  • CWE: ~2 = 1200.0000 ¢, ~320/297 = 139.8053 ¢

Optimal ET sequence: 43, 60e, 103, 369bcdeee, 472bbcddeee

Badness (Sintel): 1.74

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 5 7 8 3], ⟨0 5 -23 -36 -39 6]]

Optimal tunings:

  • WE: ~2 = 1200.8354 ¢, ~13/12 = 139.9095 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.7997 ¢

Optimal ET sequence: 43, 60e, 103

Badness (Sintel): 1.35

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 1 5 7 8 3 7], ⟨0 5 -23 -36 -39 6 -25]]

Optimal tunings:

  • WE: ~2 = 1200.7402 ¢, ~13/12 = 139.9015 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.8038 ¢

Optimal ET sequence: 43, 60e, 103

Badness (Sintel): 1.17

Hendeca

Not to be confused with Hendec.
For the 5-limit version, see Miscellaneous 5-limit temperaments #Hendecatonic.

Hendeca tempers out the same 5-limit comma as hendecatonic, and has a period of 1/11 octave. However, in this temperament, nine periods represent 7/4, the same as 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: [⟨11 0 43 31], ⟨0 1 -1 0]]

mapping generators: ~16/15, ~3

Optimal tunings:

  • WE: ~16/15 = 108.9526 ¢, ~3/2 = 702.4215 ¢
error map: ⟨-1.521 -1.055 -2.252 +8.705]
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.2071 ¢
error map: ⟨0.000 +1.252 +1.388 +12.992]

Optimal ET sequence: 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: [⟨11 0 43 31 38], ⟨0 1 -1 0 0]]

Optimal tunings:

  • WE: ~16/15 = 109.0109 ¢, ~3/2 = 702.5746 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.0395 ¢

Optimal ET sequence: 22, 55d

Badness (Sintel): 2.46

Gwazy

For the 5-limit version, see Very high accuracy temperaments #Kwazy.

Named by Petr Pařízek in 2011[1], gwazy may be described as the 22 & 74 temperament.

Subgroup: 2.3.5.7

Comma list: 225/224, 5971968/5764801

Mapping: [⟨2 1 6 4], ⟨0 8 -5 6]]

mapping generators: ~2401/1728, ~35/32

Optimal tunings:

  • WE: ~2401/1728 = 599.7132 ¢, ~35/32 = 162.5806 ¢
error map: ⟨-0.574 -1.597 -0.937 +5.510]
  • CWE: ~2401/1728 = 600.0000 ¢, ~35/32 = 162.6388 ¢
error map: ⟨0.000 -0.844 +0.492 +7.007]

Optimal ET sequence: 22, 74, 96, 118d

Badness (Sintel): 4.53

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 1 6 4 8], ⟨0 8 -5 6 -4]]

Optimal tunings:

  • WE: ~363/256 = 599.8517 ¢, ~11/10 = 162.5518 ¢
  • CWE: ~363/256 = 600.0000 ¢, ~11/10 = 162.5863 ¢

Optimal ET sequence: 22, 74, 96

Badness (Sintel): 2.26

Tertiosec

For the 5-limit version, see Miscellaneous 5-limit temperaments #Tertiosec.

Tertiosec may be described as the 21 & 75 temperament. It was initially named tertiomar by Petr Pařízek in 2011[1], but was changed to tertiosec in 2012[2].

Subgroup: 2.3.5.7

Comma list: 225/224, 14495514624/13841287201

Mapping: [⟨3 -1 12 7], ⟨0 8 -7 2]]

mapping generators: ~3072/2401, ~2048/1715

Optimal tunings:

  • WE: ~3072/2401 = 399.8257 ¢, ~2048/1715 = 287.5920 ¢
error map: ⟨-0.523 -1.044 -1.549 +5.138]
  • CWE: ~3072/2401 = 400.0000 ¢, ~2048/1715 = 287.7088 ¢
error map: ⟨0.000 -0.284 -0.276 +6.592]

Optimal ET sequence: 21, 54, 75, 96, 171d

Badness (Sintel): 10.9

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 3840/3773, 12005/11979

Mapping: [⟨3 -1 12 7 14], ⟨0 8 -7 2 -5]]

Optimal tunings:

  • WE: ~44/35 = 399.6550 ¢, ~33/28 = 287.5803 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~33/28 = 287.8224 ¢

Optimal ET sequence: 21, 54, 75e

Badness (Sintel): 5.74

References

  1. ↑ 1.0 1.1 1.2 Yahoo! Tuning Group | Suggested names for the unclasified temperaments
  2. ↑ Yahoo! Tuning Group | 2D temperament names, part I -- reclassified temperaments from message #101780
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