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Ragismic microtemperaments: Difference between revisions

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Since {{nowrap|(10/9)<sup>4</sup> {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)<sup>2</sup> }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
Since {{nowrap|(10/9)<sup>4</sup> {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)<sup>2</sup> }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.


Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, ragitritonic, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
Temperaments discussed elsewhere are:
* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]
* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]]
* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]]
* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]
* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]
* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]
* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]
* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]]
* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]]
* [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]
* [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]]
* [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]]
* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]
* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]
* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Zarvo]]'' (+33075/32768) → [[Gravity family #Zarvo|Gravity family]]
* ''[[Zarvo]]'' (+33075/32768) → [[Gravity family #Zarvo|Gravity family]]
* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortonic family #Mitonic|Minortonic family]]
* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnuzmic family #Septimal vishnu|Vishnuzmic family]]
* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]
* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]
* [[Parakleismic]] (+3136/3125) → [[Parakleismic family #Septimal parakleismic|Parakleismic family]]
* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]]
* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnu family #Septimal vishnu|Vishnu family]]
* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]
* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortone family #Mitonic|Minortone family]]
* ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]]
* ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]]
* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]]
* [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Chlorine]]'' (+{{monzo| -52 -17 34}}) → [[17th-octave temperaments #Chlorine|17th-octave temperaments]]
* ''[[Chlorine]]'' (+{{monzo| -52 -17 34}}) → [[17th-octave temperaments #Chlorine|17th-octave temperaments]]
* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]
* ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments #Dzelic|37th-octave temperaments]]
* ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments #Dzelic|37th-octave temperaments]]
Considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, octoid, seniority, monzismic, semidimfourth, acrokleismic, quasithird, quincy, deca, keenanose, counterkleismic, sfourth, aluminium, ragitritonic, quatracot, trideci, moulin, and palladium.


== Supermajor ==
== Supermajor ==
Line 497: Line 500:
Badness (Sintel): 0.899
Badness (Sintel): 0.899


== Seniority ==
== Octoid ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]].  
: {{Main| Octoid }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Octoid]].''
 
The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai comma]]). In the 11-limit, it tempers out [[540/539]], [[1375/1372]], and [[6250/6237]]. In this temperament, one period gives ~[[12/11]], two give ~[[25/21]], three give ~[[35/27]], and four give [[99/70]]~[[140/99]].  


Aside from the ragisma, the seniority temperament tempers out the [[wadisma]], 201768035/201326592, and may be described as {{nowrap| 26 & 145 }}. It is so named because the [[senior comma]] ({{monzo| -17 62 -35 }}) is tempered out.
The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimizing the average damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out [[324/323]], [[375/374]], [[495/494]], [[625/624]], [[715/714]] or [[729/728]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 201768035/201326592
[[Comma list]]: 4375/4374, 16875/16807


{{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }}
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
: mapping generators: ~2, ~5120/3087
: mapping generators: ~49/45, ~7/5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0745{{c}}, ~5120/3087 = 877.2500{{c}}
* [[WE]]: ~49/45 = 150.0003{{c}}, ~7/5 = 583.9416{{c}}
: [[error map]]: {{val| +0.075 +0.008 -0.016 -0.203 }}
: [[error map]]: {{val| +0.002 -0.130 -0.547 +0.883 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5120/3087 = 877.1965{{c}}
* [[CWE]]: ~49/45 = 150.0000{{c}}, ~7/5 = 583.9411{{c}}
: error map: {{val| 0.000 -0.077 -0.130 -0.415 }}
: error map: {{val| 0.000 -0.132 -0.549 +0.880 }}


{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d }}
[[Tuning ranges]]:
 
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
[[Badness]] (Sintel): 1.14
* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
 
{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}


=== Senator ===
[[Badness]] (Sintel): 1.08
Senator (26 & 145) extends seniority by tempering out [[441/440]] and [[65536/65219]], and can be extended to the 13- and 17-limit immediately by adding [[364/363]] and [[595/594]] to the comma list in this order.


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


Comma list: 441/440, 4375/4374, 65536/65219
Comma list: 540/539, 1375/1372, 4000/3993


Mapping: {{mapping| 1 -24 -43 5 2 | 0 35 62 -3 2 }}
Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}}
* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9477{{c}}
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]


{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e }}
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224, 824d }}


Badness (Sintel): 3.05
Badness (Sintel): 0.466


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


Comma list: 364/363, 441/440, 2200/2197, 4375/4374
Comma list: 540/539, 625/624, 729/728, 1375/1372


Mapping: {{mapping| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}
Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7136{{c}}, ~108/65 = 877.9974{{c}}
* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2038{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}


{{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }}
{{Optimal ET sequence|legend=0| 72, 152f, 224 }}


Badness (Sintel): 1.85
Badness (Sintel): 0.631


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


Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197
Comma list: 375/374, 540/539, 625/624, 715/714, 729/728


Mapping: {{mapping| 1 -24 -43 5 2 -27 -31 | 0 35 62 -3 2 42 48 }}
Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7195{{c}}, ~108/65 = 877.0018{{c}}
* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}


{{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }}
{{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}


Badness (Sintel): 1.35
Badness (Sintel): 0.729


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


Monzismic tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]]. It may be described as the {{nowrap| 53 & 612 }} temperament, with a [[ploidacot]] signature of alpha-dicot. A notable tuning not appearing on the optimal ET sequence is [[665edo]], which is nearly equivalent to the pure-3's tuning.
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}


[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}
Optimal tunings:  
* WE: ~12/11 = 149.9785{{c}}, ~7/5 = 583.8482{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}}


{{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }}
{{Optimal ET sequence|legend=0| 72, 152fg, 224 }}
: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}


[[Optimal tuning]]s:  
Badness (Sintel): 0.975
* [[WE]]: ~2 = 1200.0128{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9895{{c}}
: [[error map]]: {{val| +0.013 +0.024 -0.049 -0.019 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9793{{c}}
: error map: {{val| 0.000 +0.004 -0.080 -0.050 }}


{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd }}
==== Octopus ====
A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4 = 300{{c}}.


[[Badness]] (Sintel): 1.18
Subgroup: 2.3.5.7.11.13


=== Monzism ===
Comma list: 169/168, 325/324, 364/363, 540/539
Subgroup: 2.3.5.7.11


Comma list: 4375/4374, 41503/41472, 184549376/184528125
Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
 
Mapping: {{mapping| 1 0 -27 109 -159 | 0 2 37 -134 205 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}}
* WE: ~12/11 = 150.0313{{c}}, ~7/5 = 584.0134{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}}


{{Optimal ET sequence|legend=0| 53, 559, 612, 3619de, 4231de, …, 6067ddee }}
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}


Badness (Sintel): 1.89
Badness (Sintel): 0.896


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


Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
Comma list: 169/168, 221/220, 289/288, 325/324, 540/539


Mapping: {{mapping| 1 0 -27 109 -159 -70 | 0 2 37 -134 205 93 }}
Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}}
* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}


{{Optimal ET sequence|legend=0| 53, 559, 612 }}
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}


Badness (Sintel): 2.22
Badness (Sintel): 0.795


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


The semidimfourth temperament is featured by a semidiminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, [[235298/234375]].
Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}


[[Comma list]]: 4375/4374, 235298/234375
Optimal tunings:  
* WE: ~12/11 = 150.0049{{c}}, ~7/5 = 584.0833{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}}


{{Mapping|legend=1| 1 -10 -13 -17 | 0 31 41 53 }}
{{Optimal ET sequence|legend=0| 8d, 72, 152 }}
: mapping generators: ~2, ~35/27


[[Optimal tuning]]s:  
Badness (Sintel): 0.993
* [[WE]]: ~2 = 1199.9936{{c}}, ~35/27 = 448.4533{{c}}
: [[error map]]: {{val| -0.007 +0.160 +0.353 -0.694 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 448.4555{{c}}
: error map: {{val| 0.000 +0.165 +0.361 -0.685 }}


{{Optimal ET sequence|legend=1| 8d, …, 91, 99, 289, 388, 875 }}
Scales: [[Octoid72]], [[Octoid80]]


[[Badness]] (Sintel): 1.40
==== Hexadecoid ====
{{See also| 16th-octave temperaments }}


=== Neusec ===
Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4375/4374, 235298/234375
Subgroup: 2.3.5.7.11.13


Mapping: {{mapping| 2 -20 -26 -34 -17 | 0 31 41 53 32 }}
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
: mapping generators: ~99/70, ~35/27
 
Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
: mapping generators: ~448/429, ~7/5


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.0381{{c}}, ~35/27 = 448.4812{{c}}
* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4546{{c}}
* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}


{{Optimal ET sequence|legend=0| 8d, …, 190, 388 }}
{{Optimal ET sequence|legend=0| 80, 144, 224 }}


Badness (Sintel): 1.95
Badness (Sintel): 1.27


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


Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224


Mapping: {{mapping| 2 -20 -26 -34 -17 -21 | 0 31 41 53 32 38 }}
Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.0034{{c}}, ~35/27 = 448.4573{{c}}
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9626{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4549{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0463{{c}}


{{Optimal ET sequence|legend=0| 8d, …, 190, 198, 388 }}
{{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }}


Badness (Sintel): 1.28
Badness (Sintel): 1.46


== Acrokleismic ==
===== 19-limit =====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17.19


[[Comma list]]: 4375/4374, 2202927104/2197265625
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444


{{Mapping|legend=1| 1 -22 -22 -65 | 0 32 33 92 }}
Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}
: mapping generators: ~2, ~5/3


[[Optimal tuning]]s:  
Optimal tunings:  
* [[WE]]: ~2 = 1199.9305{{c}}, ~5/3 = 884.3923{{c}}
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}
: [[error map]]: {{val| -0.070 +0.126 +0.160 -0.221 }}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.4423{{c}}
: error map: {{val| 0.000 +0.198 +0.282 -0.136 }}


{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}
{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}


[[Badness]] (Sintel): 1.42
Badness (Sintel): 1.44


=== 11-limit ===
== Seniority ==
Subgroup: 2.3.5.7.11
: ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]].  


Comma list: 4375/4374, 41503/41472, 172032/171875
Aside from the ragisma, the seniority temperament tempers out the [[wadisma]], 201768035/201326592, and may be described as {{nowrap| 26 & 145 }}. It is so named because the [[senior comma]] ({{monzo| -17 62 -35 }}) is tempered out.


Mapping: {{mapping| 1 -22 -22 -65 58 | 0 32 33 92 -74 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 4375/4374, 201768035/201326592
* WE: ~2 = 1199.9698{{c}}, ~5/3 = 884.4193{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4414{{c}}


{{Optimal ET sequence|legend=0| 19, 251, 270, 829, 1099, 1369, 1639 }}
{{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }}
: mapping generators: ~2, ~5120/3087


Badness (Sintel): 1.22
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0745{{c}}, ~5120/3087 = 877.2500{{c}}
: [[error map]]: {{val| +0.075 +0.008 -0.016 -0.203 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5120/3087 = 877.1965{{c}}
: error map: {{val| 0.000 -0.077 -0.130 -0.415 }}


==== 13-limit ====
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d }}
Subgroup: 2.3.5.7.11.13


Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
[[Badness]] (Sintel): 1.14


Mapping: {{mapping| 1 -22 -22 -65 58 -56 | 0 32 33 92 -74 81 }}
=== Senator ===
Senator (26 & 145) extends seniority by tempering out [[441/440]] and [[65536/65219]], and can be extended to the 13- and 17-limit immediately by adding [[364/363]] and [[595/594]] to the comma list in this order.


Optimal tunings:  
Subgroup: 2.3.5.7.11
* WE: ~2 = 1199.9939{{c}}, ~5/3 = 884.4384{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4429{{c}}


{{Optimal ET sequence|legend=0| 19, 251, 270 }}
Comma list: 441/440, 4375/4374, 65536/65219


Badness (Sintel): 1.11
Mapping: {{mapping| 1 -24 -43 5 2 | 0 35 62 -3 2 }}
 
=== Counteracro ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 5632/5625, 117649/117612
 
Mapping: {{mapping| 1 -22 -22 -65 -141 | 0 32 33 92 196 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8877{{c}}, ~5/3 = 884.3639{{c}}
* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4457{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}}


{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1061e, 1331c, 1601c, 1871bc }}
{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e }}


Badness (Sintel): 1.41
Badness (Sintel): 3.05


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


Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
Comma list: 364/363, 441/440, 2200/2197, 4375/4374


Mapping: {{mapping| 1 -22 -22 -65 -141 -56 | 0 32 33 92 196 81 }}
Mapping: {{mapping| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9285{{c}}, ~5/3 = 884.3937{{c}}
* WE: ~2 = 1199.7136{{c}}, ~108/65 = 877.9974{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4458{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2038{{c}}


{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1331c }}
{{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }}


Badness (Sintel): 1.08
Badness (Sintel): 1.85


== Quasithird ==
==== 17-limit ====
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasithird]].''
Subgroup: 2.3.5.7.11.13.17


Quasithird may be described as the {{nowrap| 224 & 388 }} temperament, featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows it to temper out the ragisma and {{monzo| -60 29 0 5 }}. Its [[ploidacot]] is tetraploid delta-pentacot.
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 1 -24 -43 5 2 -27 -31 | 0 35 62 -3 2 42 48 }}


[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}
Optimal tunings:  
* WE: ~2 = 1199.7195{{c}}, ~108/65 = 877.0018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}}


{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
{{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }}
: mapping generators: ~65536/55125, ~5103/4096


[[Optimal tuning]]s:  
Badness (Sintel): 1.35
* [[WE]]: ~65536/55125 = 300.0052{{c}}, ~5103/4096 = 380.3949{{c}}
: [[error map]]: {{val| +0.021 +0.020 -0.052 -0.031 }}
* [[CWE]]: ~65536/55125 = 300.0000{{c}}, ~5103/4096 = 380.3884{{c}}
: error map: {{val| 0.000 -0.013 -0.100 -0.089 }}


{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}
== Monzismic ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]].


[[Badness]] (Sintel): 1.56
Monzismic tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]]. It may be described as the {{nowrap| 53 & 612 }} temperament, with a [[ploidacot]] signature of alpha-dicot. A notable tuning not appearing on the optimal ET sequence is [[665edo]], which is nearly equivalent to the pure-3's tuning.


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


Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}


Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}
{{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }}
: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}


Optimal tunings:  
[[Optimal tuning]]s:  
* WE: ~65536/51125 = 300.0073{{c}}, ~5103/4096 = 380.3963{{c}} (or ~22/21 = 80.3890{{c}})
* [[WE]]: ~2 = 1200.0128{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9895{{c}}
* CWE: ~65536/51125 = 300.0000{{c}}, ~5103/4096 = 380.3868{{c}} (or ~22/21 = 80.3868{{c}})
: [[error map]]: {{val| +0.013 +0.024 -0.049 -0.019 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9793{{c}}
: error map: {{val| 0.000 +0.004 -0.080 -0.050 }}


{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee }}
{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd }}


Badness (Sintel): 0.698
[[Badness]] (Sintel): 1.18


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


Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374
Comma list: 4375/4374, 41503/41472, 184549376/184528125


Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}
Mapping: {{mapping| 1 0 -27 109 -159 | 0 2 37 -134 205 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~65536/51125 = 299.9985{{c}}, ~81/65 = 380.3833{{c}} (or ~22/21 = 80.3848{{c}})
* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}}
* CWE: ~65536/51125 = 300.0000{{c}}, ~81/65 = 380.3852{{c}} (or ~22/21 = 80.3852{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}}


{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836 }}
{{Optimal ET sequence|legend=0| 53, 559, 612, 3619de, 4231de, …, 6067ddee }}


Badness (Sintel): 1.22
Badness (Sintel): 1.89


== Deca ==
==== 13-limit ====
: ''For 5-limit version, see [[10th-octave temperaments #Neon]].''
Subgroup: 2.3.5.7.11.13


Deca has a period of 1/10 octave and tempers out the neon comma {{monzo| 21 60 -50 }} in the 5-limit, the [[linus comma]]{{monzo| 11 -10 -10 10 }} and {{monzo| 12 -3 -14 9 }} (165288374272/164794921875) in the 7-limit. It may be described as the {{nowrap| 80 & 190 }} temperament, and has a [[ploidacot]] of decaploid wau-pentacot.
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 1 0 -27 109 -159 -70 | 0 2 37 -134 205 93 }}


[[Comma list]]: 4375/4374, 165288374272/164794921875
Optimal tunings:  
* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}}


{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}
{{Optimal ET sequence|legend=0| 53, 559, 612 }}
: mapping generators: ~15/14, ~460992/390625


[[Optimal tuning]]s:
Badness (Sintel): 2.22
* [[WE]]: ~15/14 = 119.9966{{c}}, ~460992/390625 = 284.4150{{c}} (5625/5488 = 44.4219{{c}})
: [[error map]]: {{val| -0.034 +0.106 +0.145 -0.268 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~460992/390625 = 284.4182{{c}} (5625/5488 = 44.4182{{c}})
: error map: {{val| 0.000 +0.136 +0.195 -0.226 }}


{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}
== Semidimfourth ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimfourth]].''


[[Badness]] (Sintel): 2.04
The semidimfourth temperament is featured by a semidiminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, [[235298/234375]].


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


Comma list: 3025/3024, 4375/4374, 391314/390625
[[Comma list]]: 4375/4374, 235298/234375


Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}
{{Mapping|legend=1| 1 -10 -13 -17 | 0 31 41 53 }}
: mapping generators: ~2, ~35/27


Optimal tunings:  
[[Optimal tuning]]s:  
* WE: ~15/14 = 120.0004{{c}}, ~33/28 = 284.4193{{c}} (77/75 = 44.4185{{c}})
* [[WE]]: ~2 = 1199.9936{{c}}, ~35/27 = 448.4533{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4189{{c}} (77/75 = 44.4189{{c}})
: [[error map]]: {{val| -0.007 +0.160 +0.353 -0.694 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 448.4555{{c}}
: error map: {{val| 0.000 +0.165 +0.361 -0.685 }}


{{Optimal ET sequence|legend=0| 80, 190, 270, 1000, 1270, 1540e, 1810e }}
{{Optimal ET sequence|legend=1| 8d, …, 91, 99, 289, 388, 875 }}


Badness (Sintel): 0.804
[[Badness]] (Sintel): 1.40


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


Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
Comma list: 3025/3024, 4375/4374, 235298/234375


Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}
Mapping: {{mapping| 2 -20 -26 -34 -17 | 0 31 41 53 32 }}
: mapping generators: ~99/70, ~35/27


Optimal tunings:  
Optimal tunings:  
* WE: ~15/14 = 120.0067{{c}}, ~33/28 = 284.4139{{c}} (~40/39 = 44.4006{{c}})
* WE: ~99/70 = 600.0381{{c}}, ~35/27 = 448.4812{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4048{{c}} (~40/39 = 44.4048{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4546{{c}}


{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}
{{Optimal ET sequence|legend=0| 8d, …, 190, 388 }}


Badness (Sintel): 0.695
Badness (Sintel): 1.95


=== 2.3.5.7.11.13.19 subgroup ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.19
Subgroup: 2.3.5.7.11.13


Comma list: 1001/1000, 1521/1520, 3025/3024, 4225/4224, 4375/4374
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374


Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}
Mapping: {{mapping| 2 -20 -26 -34 -17 -21 | 0 31 41 53 32 38 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~15/14 = 120.0045{{c}}, ~33/28 = 284.4140{{c}} (~39/38 = 44.4050{{c}})
* WE: ~99/70 = 600.0034{{c}}, ~35/27 = 448.4573{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4075{{c}} (~39/38 = 44.4075{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4549{{c}}


{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}
{{Optimal ET sequence|legend=0| 8d, …, 190, 198, 388 }}


Badness (Sintel): 0.556
Badness (Sintel): 1.28
 
== Keenanose ==
Keenanose, the {{nowrap| 270 & 1889 }} temperament, was named by [[Eliora]] in 2022 for the fact that it uses [[385/384]], the keenanisma, as the generator.


== Acrokleismic ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}
[[Comma list]]: 4375/4374, 2202927104/2197265625


{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
{{Mapping|legend=1| 1 -22 -22 -65 | 0 32 33 92 }}
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}
: mapping generators: ~2, ~5/3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0068{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4467{{c}}
* [[WE]]: ~2 = 1199.9305{{c}}, ~5/3 = 884.3923{{c}}
: [[error map]]: {{val| +0.007 +0.031 -0.035 -0.032 }}
: [[error map]]: {{val| -0.070 +0.126 +0.160 -0.221 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4466{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.4423{{c}}
: error map: {{val| 0.000 +0.025 -0.043 -0.050 }}
: error map: {{val| 0.000 +0.198 +0.282 -0.136 }}


{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}
{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}


[[Badness]] (Sintel): 2.17
[[Badness]] (Sintel): 1.42


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


Comma list: 4375/4374, 117649/117612, 67110351/67108864
Comma list: 4375/4374, 41503/41472, 172032/171875


Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}
Mapping: {{mapping| 1 -22 -22 -65 58 | 0 32 33 92 -74 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9970{{c}}, ~385/384 = 4.4465{{c}}
* WE: ~2 = 1199.9698{{c}}, ~5/3 = 884.4193{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4414{{c}}


{{Optimal ET sequence|legend=0| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}
{{Optimal ET sequence|legend=0| 19, 251, 270, 829, 1099, 1369, 1639 }}


Badness (Sintel): 1.02
Badness (Sintel): 1.22


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


Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976


Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}
Mapping: {{mapping| 1 -22 -22 -65 58 -56 | 0 32 33 92 -74 81 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0065{{c}}, ~385/384 = 4.4467{{c}}
* WE: ~2 = 1199.9939{{c}}, ~5/3 = 884.4384{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4467{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4429{{c}}


{{Optimal ET sequence|legend=0| 270, 1079, 1349, 1619, 1889, 4048 }}
{{Optimal ET sequence|legend=0| 19, 251, 270 }}


Badness (Sintel): 0.879
Badness (Sintel): 1.11


== Aluminium ==
=== Counteracro ===
: ''For the 5-limit version, see [[13th-octave temperaments #Aluminium]].''
Subgroup: 2.3.5.7.11


Aluminium was named by [[Eliora]] in 2023 after the 13th element. It tempers out the {{monzo| 92 -39 -13 }} comma, which sets [[135/128]] interval to be equal to 1/13th of the octave.
Comma list: 4375/4374, 5632/5625, 117649/117612


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 1 -22 -22 -65 -141 | 0 32 33 92 196 }}


[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}
Optimal tunings:  
* WE: ~2 = 1199.8877{{c}}, ~5/3 = 884.3639{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4457{{c}}


[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}
{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1061e, 1331c, 1601c, 1871bc }}
: Mapping generators: ~135/128, ~3


[[Optimal tuning]]s:  
Badness (Sintel): 1.41
* [[WE]]: ~135/128 = 92.3072{{c}}, ~3/2 = 701.9995{{c}}
: [[error map]]: {{val| -0.006 +0.038 -0.030 -0.013 }}
* [[CWE]]: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0030{{c}}
: error map: {{val| 0.000 +0.048 -0.015 +0.001 }}


{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Badness]] (Sintel): 3.20
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374


=== 11-limit ===
Mapping: {{mapping| 1 -22 -22 -65 -141 -56 | 0 32 33 92 196 81 }}
Subgroup: 2.3.5.7.11


Comma list: 4375/4374, 234375/234256, 2097152/2096325
Optimal tunings:  
* WE: ~2 = 1199.9285{{c}}, ~5/3 = 884.3937{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4458{{c}}


Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}
{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1331c }}


Optimal tunings:  
Badness (Sintel): 1.08
* WE: ~135/128 = 92.3062{{c}}, ~3/2 = 701.9946{{c}}
* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0056{{c}}


{{Optimal ET sequence|legend=0| 494, 1053, 1547, 3588e, 5135e }}
== Quasithird ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasithird]].''


Badness (Sintel): 1.39
Quasithird may be described as the {{nowrap| 224 & 388 }} temperament, featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows it to temper out the ragisma and {{monzo| -60 29 0 5 }}. Its [[ploidacot]] is tetraploid delta-pentacot.  


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


Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078
[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}


Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}
{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
: mapping generators: ~65536/55125, ~5103/4096


Optimal tunings:  
[[Optimal tuning]]s:  
* WE: ~135/128 = 92.3055{{c}}, ~3/2 = 701.9928{{c}}
* [[WE]]: ~65536/55125 = 300.0052{{c}}, ~5103/4096 = 380.3949{{c}}
* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0098{{c}}
: [[error map]]: {{val| +0.021 +0.020 -0.052 -0.031 }}
* [[CWE]]: ~65536/55125 = 300.0000{{c}}, ~5103/4096 = 380.3884{{c}}
: error map: {{val| 0.000 -0.013 -0.100 -0.089 }}


{{Optimal ET sequence|legend=0| 494, 1547, 2041, 4576def }}
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}


Badness (Sintel): 1.18
[[Badness]] (Sintel): 1.56


== Ragitritonic ==
=== 11-limit ===
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
Subgroup: 2.3.5.7.11


Ragitritonic may be described as the {{nowrap| 53 & 369 }} temperament, splitting the [[24/1|24th harmonic]] into nine tritone generators; its [[ploidacot]] is thus delta-enneacot. [[422edo]] makes for a strong tuning.
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296


Ragitritonic was named by [[Flora Canou]] in 2026 as a contraction of ''ragismic'' and ''tritonic''.
Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}


[[Subgroup]]: 2.3.5.7
Optimal tunings:  
* WE: ~65536/51125 = 300.0073{{c}}, ~5103/4096 = 380.3963{{c}} (or ~22/21 = 80.3890{{c}})
* CWE: ~65536/51125 = 300.0000{{c}}, ~5103/4096 = 380.3868{{c}} (or ~22/21 = 80.3868{{c}})


[[Comma list]]: 4375/4374, 68719476736/68356598625
{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee }}


{{Mapping|legend=1| 1 -3 -15 40 | 0 9 34 -73 }}
Badness (Sintel): 0.698
: mapping generators: ~2, ~65536/45927


[[Optimal tuning]]s:
=== 13-limit ===
* [[WE]]: ~2 = 1199.8189{{c}}, ~65536/45927 = 611.2850{{c}}
Subgroup: 2.3.5.7.11.13
: [[error map]]: {{val| -0.181 +0.153 +0.094 +0.123 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~65536/45927 = 611.3775{{c}}
: error map: {{val| 0.000 +0.443 +0.522 +0.615 }}


{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}
Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374


[[Badness]] (Sintel): 3.37
Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}


=== 11-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~65536/51125 = 299.9985{{c}}, ~81/65 = 380.3833{{c}} (or ~22/21 = 80.3848{{c}})
* CWE: ~65536/51125 = 300.0000{{c}}, ~81/65 = 380.3852{{c}} (or ~22/21 = 80.3852{{c}})


Comma list: 4375/4374, 5632/5625, 2621440/2614689
{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836 }}


Mapping: {{mapping| 1 -3 -15 40 -75 | 0 9 34 -73 154 }}
Badness (Sintel): 1.22


Optimal tunings:
== Quincy ==
* WE: ~2 = 1199.8147{{c}}, ~768/539 = 611.2822{{c}}
[[Subgroup]]: 2.3.5.7
* CWE: ~2 = 1200.0000{{c}}, ~768/539 = 611.3762{{c}}


{{Optimal ET sequence|legend=0| 53, 316e, 369, 422, 791e, 1213cde }}
[[Comma list]]: 4375/4374, 823543/819200


Badness (Sintel): 2.34
{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
: mapping generators: ~2, ~1728/1715


=== 13-limit ===
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13
* [[WE]]: ~2 = 1200.2169{{c}}, ~1728/1715 = 16.6160{{c}}
: [[error map]]: {{val| +0.217 +0.000 +0.155 -0.799 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1728/1715 = 16.6083{{c}}
: error map: {{val| 0.000 -0.205 -0.122 -1.343 }}


Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625
{{Optimal ET sequence|legend=1| 72, 217, 289, 650d, 939dd }}


Mapping: {{mapping| 1 -3 -15 40 -75 -34 | 0 9 34 -73 154 74 }}
[[Badness]] (Sintel): 2.02
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 4000/3993, 4375/4374
 
Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7916{{c}}, ~91/64 = 611.2698{{c}}
* WE: ~2 = 1200.1286{{c}}, ~100/99 = 16.6147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~91/64 = 611.3754{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6101{{c}}


{{Optimal ET sequence|legend=0| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}
{{Optimal ET sequence|legend=0| 72, 217, 289 }}


Badness (Sintel): 1.51
Badness (Sintel): 1.02


== Quatracot ==
=== 13-limit ===
{{See also| Stratosphere }}
Subgroup: 2.3.5.7.11.13


[[Subgroup]]: 2.3.5.7
Comma list: 364/363, 441/440, 676/675, 4375/4374


[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}


{{Mapping|legend=1| 2 -6 -1 -36 | 0 13 8 59 }}
Optimal tunings:  
: mapping generators: ~2278125/1605632, ~7168/5625
* WE: ~2 = 1200.0554{{c}}, ~100/99 = 16.6028{{c}}
 
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6011{{c}}
[[Optimal tuning]]s:  
* [[WE]]: ~2278125/1605632 = 600.0888{{c}}, ~7168/5625 = 423.2574{{c}}
: [[error map]]: {{val| +0.178 -0.141 -0.343 +0.165 }}
* [[CWE]]: ~2278125/1605632 = 600.0000{{c}}, ~7168/5625 = 423.1986{{c}}
: error map: {{val| 0.000 -0.374 -0.725 -0.111 }}


{{Optimal ET sequence|legend=1| 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc }}
{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}


[[Badness]] (Sintel): 4.45
Badness (Sintel): 0.986


=== 11-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17


Comma list: 3025/3024, 4375/4374, 1265625/1261568
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155


Mapping: {{mapping| 2 -6 -1 -36 -22 | 0 13 8 59 41 }}
Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.0847{{c}}, ~225/176 = 423.2536{{c}}
* WE: ~2 = 1200.0647{{c}}, ~100/99 = 16.6025{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~225/176 = 423.1977{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6004{{c}}


{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638, 1052c }}
{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}


Badness (Sintel): 1.36
Badness (Sintel): 0.751


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


Comma list: 625/624, 729/728, 1575/1573, 2200/2197
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675


Mapping: {{mapping| 2 -6 -1 -36 -22 -6 | 0 13 8 59 41 19 }}
Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.0571{{c}}, ~143/112 = 423.2366{{c}}
* WE: ~2 = 1199.9287{{c}}, ~100/99 = 16.5930{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~143/112 = 423.1987{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.5948{{c}}
 
{{Optimal ET sequence|legend=0| 72, 145, 217 }}


{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638 }}
Badness (Sintel): 0.924


Badness (Sintel): 0.936
== Deca ==
: ''For 5-limit version, see [[10th-octave temperaments#Neon]].''


== Moulin ==
Deca has a period of 1/10 octave and tempers out the [[neon comma]] ({{monzo| 21 60 -50 }}) in the 5-limit, the [[linus comma]] ({{monzo| 11 -10 -10 10 }}) and {{monzo| 12 -3 -14 9 }} (165288374272/164794921875) in the 7-limit. It may be described as the {{nowrap| 80 & 190 }} temperament, and has a [[ploidacot]] of decaploid wau-pentacot.  
Moulin can be described as the {{nowrap| 494 & 1619 }} temperament. It has a generator of ~[[22/13]], and it was named by [[Eliora]] in 2022 after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". However, the functional generator is ~[[13/11]], and 73 of them octave reduced reach the [[3/2|perfect fifth]]. Since [[11/8]] is within 23 generators, the 25-tone generator chain (4L 21s) of this temperament contains the 8:11:13 triad.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
[[Comma list]]: 4375/4374, 165288374272/164794921875


{{Mapping|legend=1| 1 -16 -9 -75 | 0 73 47 323 }}
{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}
: mapping generators: ~2, ~3796875/3211264
: mapping generators: ~15/14, ~460992/390625


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0272{{c}}, ~3796875/3211264 = 289.0675{{c}}
* [[WE]]: ~15/14 = 119.9966{{c}}, ~460992/390625 = 284.4150{{c}} (5625/5488 = 44.4219{{c}})
: [[error map]]: {{val| +0.027 +0.007 -0.084 +0.013 }}
: [[error map]]: {{val| -0.034 +0.106 +0.145 -0.268 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3796875/3211264 = 289.0675{{c}}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~460992/390625 = 284.4182{{c}} (5625/5488 = 44.4182{{c}})
: error map: {{val| 0.000 -0.029 -0.142 -0.029 }}
: error map: {{val| 0.000 +0.136 +0.195 -0.226 }}


{{Optimal ET sequence|legend=1| 494, 1125, 1619, 8589cc, 10208cc }}
{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}


[[Badness]] (Sintel): 5.93
[[Badness]] (Sintel): 2.04


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


Comma list: 4375/4374, 759375/758912, 100663296/100656875
Comma list: 3025/3024, 4375/4374, 391314/390625


Mapping: {{mapping| 1 -16 -9 -75 9 | 0 73 47 323 -23 }}
Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0043{{c}}, ~605/512 = 289.0687{{c}}
* WE: ~15/14 = 120.0004{{c}}, ~33/28 = 284.4193{{c}} (77/75 = 44.4185{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~605/512 = 289.0677{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4189{{c}} (77/75 = 44.4189{{c}})


{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}
{{Optimal ET sequence|legend=0| 80, 190, 270, 1000, 1270, 1540e, 1810e }}


Badness (Sintel): 2.24
Badness (Sintel): 0.804


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


Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374


Mapping: {{mapping| 1 -16 -9 -75 9 9 | 0 73 47 323 -23 -22 }}
Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0043{{c}}, ~13/11 = 289.0687{{c}}
* WE: ~15/14 = 120.0067{{c}}, ~33/28 = 284.4139{{c}} (~40/39 = 44.4006{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 289.0677{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4048{{c}} (~40/39 = 44.4048{{c}})


{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}
{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}


Badness (Sintel): 1.12
Badness (Sintel): 0.695


== Palladium ==
=== 2.3.5.7.11.13.19 subgroup ===
: ''For the 5-limit version, see [[46th-octave temperaments #Palladium]]''.
Subgroup: 2.3.5.7.11.13.19


The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minor whole tones (10/9) fall short of seven octaves. This temperament can be described as {{nowrap| 46 & 414 }} temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.
Comma list: 1001/1000, 1521/1520, 3025/3024, 4225/4224, 4375/4374


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}


[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}
Optimal tunings:  
* WE: ~15/14 = 120.0045{{c}}, ~33/28 = 284.4140{{c}} (~39/38 = 44.4050{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4075{{c}} (~39/38 = 44.4075{{c}})


{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}
: mapping generators: ~83349/81920, ~3


[[Optimal tuning]]s:  
Badness (Sintel): 0.556
* [[WE]]: ~83349/81920 = 26.0910{{c}}, ~3/2 = 701.7155{{c}}
: [[error map]]: {{val| +0.185 -0.055 -0.061 +0.349 }}
* [[CWE]]: ~83349/81920 = 26.0870{{c}}, ~3/2 = 701.6491{{c}}
: error map: {{val| 0.000 -0.306 -0.407 -0.910 }}


{{Optimal ET sequence|legend=1| 46, …, 368, 414, 460, 874d }}
== Keenanose ==
Keenanose, the {{nowrap| 270 & 1889 }} temperament, was named by [[Eliora]] in 2022 for the fact that it uses [[385/384]], the keenanisma, as the generator.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}
 
{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0068{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4467{{c}}
: [[error map]]: {{val| +0.007 +0.031 -0.035 -0.032 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4466{{c}}
: error map: {{val| 0.000 +0.025 -0.043 -0.050 }}
 
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}


[[Badness]] (Sintel): 7.81
[[Badness]] (Sintel): 2.17


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


Comma list: 3025/3024, 4375/4374, 134775333/134217728
Comma list: 4375/4374, 117649/117612, 67110351/67108864


Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}
Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~8192/8085 = 26.0912{{c}}, ~3/2 = 701.7082{{c}}
* WE: ~2 = 1199.9970{{c}}, ~385/384 = 4.4465{{c}}
* CWE: ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.6173{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4465{{c}}


{{Optimal ET sequence|legend=0| 46, …, 368, 414, 460, 874de }}
{{Optimal ET sequence|legend=0| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}


Badness (Sintel): 2.44
Badness (Sintel): 1.02


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


Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612


Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7411{{c}}
* WE: ~2 = 1200.0065{{c}}, ~385/384 = 4.4467{{c}}
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4467{{c}}


{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334dde }}
{{Optimal ET sequence|legend=0| 270, 1079, 1349, 1619, 1889, 4048 }}


Badness (Sintel): 1.68
Badness (Sintel): 0.879


=== 17-limit ===
== Counterkleismic ==
Subgroup: 2.3.5.7.11.13.17
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''


Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses]] ((648/625)<sup>6</sup>) fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament, tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). It was named by analogy to [[catakleismic]] and [[parakleismic]]).


Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 4375/4374, 158203125/157351936
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7399{{c}}
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6464{{c}}


{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334ddeg }}
{{Mapping|legend=1| 1 -5 -4 -18 | 0 25 24 79 }}
: mapping generators: ~2, ~6/5


Badness (Sintel): 1.14
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1778{{c}}, ~6/5 = 316.1065{{c}}
: [[error map]]: {{val| +0.178 -0.181 -0.469 +0.388 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.0631{{c}}
: error map: {{val| 0.000 -0.377 -0.799 +0.161 }}


== Counterorson ==
{{Optimal ET sequence|legend=1| 19, …, 205, 224, 243, 467 }}
Counterorson tempers out the {{monzo| 147 -103 7 }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]].


[[Subgroup]]: 2.3.5.7
[[Badness]] (Sintel): 2.29


[[Comma list]]: 4375/4374, {{monzo| 154 -54 -21 -7 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Mapping|legend=1| 1 0 -21 85 | 0 7 103 -363 }}
Comma list: 540/539, 4375/4374, 2097152/2096325
: mapping generators: ~2, ~{{monzo| 66 -23 -9 -3 }}


[[Optimal tuning]]s:  
Mapping: {{mapping| 1 -5 -4 -18 19 | 0 25 24 79 -59 }}
* [[WE]]: ~2 = 1200.0040{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7122{{c}}
: [[error map]]: {{val| +0.004 -0.303 -0.041 -0.015 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7113{{c}}
: error map: {{val| 0.000 +0.024 -0.051 -0.025 }}


{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}
Optimal tunings:
* WE: ~2 = 1199.9944{{c}}, ~6/5 = 316.0690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0705{{c}}


[[Badness]] (Sintel): 7.92
{{Optimal ET sequence|legend=0| 19, 205, 224 }}


== Oviminor ==
Badness (Sintel): 2.35
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Oviminor (5-limit)]].''


Oviminor was named by [[Eliora]] in 2022 after the facts that it takes 184 minor thirds of [[6/5]] to reach the interval class of [[4/3]], the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.  
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Subgroup]]: 2.3.5.7
Comma list: 540/539, 625/624, 729/728, 10985/10976


[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}
Mapping: {{mapping| 1 -5 -4 -18 19 -15 | 0 25 24 79 -59 71 }}


{{Mapping|legend=1| 1 -134 -134 -401 | 0 184 185 548 }}
Optimal tunings:
: mapping generators: ~2, ~5/3
* WE: ~2 = 1199.9827{{c}}, ~6/5 = 316.0650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0695{{c}}
 
{{Optimal ET sequence|legend=0| 19, 205, 224 }}


[[Optimal tuning]]s:  
Badness (Sintel): 1.40
* [[WE]]: ~2 = 1200.0193{{c}}, ~5/3 = 884.2638{{c}}
: [[error map]]: {{val| +0.019 +0.010 -0.085 +0.032 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.2497{{c}}
: error map: {{val| 0.000 -0.011 -0.120 +0.008 }}


{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}
=== Counterlytic ===
Subgroup: 2.3.5.7.11


[[Badness]] (Sintel): 14.7
Comma list: 1375/1372, 4375/4374, 496125/495616


== Octoid ==
Mapping: {{mapping| 1 -5 -4 -18 -40 | 0 25 24 79 165 }}
: {{Main| Octoid }}
: ''For the 5-limit version, see [[8th-octave temperaments #Octoid]].''


The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai comma]]). In the 11-limit, it tempers out [[540/539]], [[1375/1372]], and [[6250/6237]]. In this temperament, one period gives ~[[12/11]], two give ~[[25/21]], three give ~[[35/27]], and four give [[99/70]]~[[140/99]].  
Optimal tunings:
* WE: ~2 = 1200.1247{{c}}, ~6/5 = 316.0976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0660{{c}}


The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimizing the average damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out [[324/323]], [[375/374]], [[495/494]], [[625/624]], [[715/714]] or [[729/728]].
{{Optimal ET sequence|legend=1| 19e, 205e, 224, 467e, 691, 915c }}


[[Subgroup]]: 2.3.5.7
Badness (Sintel): 2.16


[[Comma list]]: 4375/4374, 16875/16807
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
Comma list: 625/624, 729/728, 1375/1372, 10985/10976
: mapping generators: ~49/45, ~7/5


[[Optimal tuning]]s:  
Mapping: {{mapping| 1 -5 -4 -18 -40 -15 | 0 25 24 79 165 71 }}
* [[WE]]: ~49/45 = 150.0003{{c}}, ~7/5 = 583.9416{{c}}
: [[error map]]: {{val| +0.002 -0.130 -0.547 +0.883 }}
* [[CWE]]: ~49/45 = 150.0000{{c}}, ~7/5 = 583.9411{{c}}
: error map: {{val| 0.000 -0.132 -0.549 +0.880 }}


[[Tuning ranges]]:  
Optimal tunings:  
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
* WE: ~2 = 1200.0987{{c}}, ~6/5 = 316.0908{{c}}
* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0658{{c}}
* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]


{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}
{{Optimal ET sequence|legend=0| 19e, 205e, 224, 467e, 691, 915c }}


[[Badness]] (Sintel): 1.08
Badness (Sintel): 1.23


=== 11-limit ===
== Sfourth ==
Subgroup: 2.3.5.7.11
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sfourth]].''


Comma list: 540/539, 1375/1372, 4000/3993
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}
[[Comma list]]: 4375/4374, 64827/64000


Optimal tunings:
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}}
: mapping generators: ~2, ~49/48
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9477{{c}}


Tuning ranges:  
[[Optimal tuning]]s:  
* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
* [[WE]]: ~2 = 1200.8332{{c}}, ~49/48 = 26.3053{{c}}
* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
: [[error map]]: {{val| +0.833 -0.090 +0.721 -3.074 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 26.2590{{c}}
: error map: {{val| 0.000 -0.876 -0.343 -5.157 }}


{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224, 824d }}
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


Badness (Sintel): 0.466
[[Badness]] (Sintel): 3.12


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


Comma list: 540/539, 625/624, 729/728, 1375/1372
Comma list: 121/120, 441/440, 4375/4374


Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}
* WE: ~2 = 1201.1486{{c}}, ~49/48 = 26.3112{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2461{{c}}


{{Optimal ET sequence|legend=0| 72, 152f, 224 }}
{{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }}


Badness (Sintel): 0.631
Badness (Sintel): 1.78


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


Comma list: 375/374, 540/539, 625/624, 715/714, 729/728
Comma list: 121/120, 169/168, 325/324, 441/440


Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}
* WE: ~2 = 1201.4956{{c}}, ~49/48 = 26.3423{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2614{{c}}


{{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}
{{Optimal ET sequence|legend=0| 45ef, 46, 91ef, 137def, 228ddeeefff }}


Badness (Sintel): 0.729
Badness (Sintel): 1.37


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


Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
Comma list: 385/384, 2401/2376, 4375/4374


Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}
Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~12/11 = 149.9785{{c}}, ~7/5 = 583.8482{{c}}
* WE: ~2 = 1200.4402{{c}}, ~49/48 = 26.2557{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2403{{c}}


{{Optimal ET sequence|legend=0| 72, 152fg, 224 }}
{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}


Badness (Sintel): 0.975
Badness (Sintel): 2.53
 
==== Octopus ====
A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4 = 300{{c}}.


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


Comma list: 169/168, 325/324, 364/363, 540/539
Comma list: 196/195, 364/363, 385/384, 4375/4374


Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~12/11 = 150.0313{{c}}, ~7/5 = 584.0134{{c}}
* WE: ~2 = 1200.3796{{c}}, ~49/48 = 26.2473{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2372{{c}}


{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}
{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}


Badness (Sintel): 0.896
Badness (Sintel): 2.14


===== 17-limit =====
== Aluminium ==
Subgroup: 2.3.5.7.11.13.17
: ''For the 5-limit version, see [[13th-octave temperaments #Aluminium]].''


Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
Aluminium tempers out {{monzo| 92 -39 -13 }} in the 5-limit and sets [[135/128]] to 1/13 of an [[octave]]. It was named by [[Eliora]] in 2023 after the 13th element.


Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}
* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}


{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}
[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}
: Mapping generators: ~135/128, ~3


Badness (Sintel): 0.795
[[Optimal tuning]]s:  
* [[WE]]: ~135/128 = 92.3072{{c}}, ~3/2 = 701.9995{{c}}
: [[error map]]: {{val| -0.006 +0.038 -0.030 -0.013 }}
* [[CWE]]: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0030{{c}}
: error map: {{val| 0.000 +0.048 -0.015 +0.001 }}


===== 19-limit =====
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
[[Badness]] (Sintel): 3.20


Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Optimal tunings:  
Comma list: 4375/4374, 234375/234256, 2097152/2096325
* WE: ~12/11 = 150.0049{{c}}, ~7/5 = 584.0833{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}}


{{Optimal ET sequence|legend=0| 8d, 72, 152 }}
Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}


Badness (Sintel): 0.993
Optimal tunings:  
* WE: ~135/128 = 92.3062{{c}}, ~3/2 = 701.9946{{c}}
* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0056{{c}}


Scales: [[Octoid72]], [[Octoid80]]
{{Optimal ET sequence|legend=0| 494, 1053, 1547, 3588e, 5135e }}


==== Hexadecoid ====
Badness (Sintel): 1.39
{{See also| 16th-octave temperaments }}
 
Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.


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


Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078


Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}
: mapping generators: ~448/429, ~7/5


Optimal tunings:  
Optimal tunings:  
* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}
* WE: ~135/128 = 92.3055{{c}}, ~3/2 = 701.9928{{c}}
* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}
* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0098{{c}}


{{Optimal ET sequence|legend=0| 80, 144, 224 }}
{{Optimal ET sequence|legend=0| 494, 1547, 2041, 4576def }}


Badness (Sintel): 1.27
Badness (Sintel): 1.18


===== 17-limit =====
== Ragitritonic ==
Subgroup: 2.3.5.7.11.13.17
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
Ragitritonic may be described as the {{nowrap| 53 & 369 }} temperament, splitting the [[24/1|24th harmonic]] into nine tritone generators; its [[ploidacot]] is thus delta-enneacot. [[422edo]] makes for a strong tuning.


Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
Ragitritonic was named by [[Flora Canou]] in 2026 as a contraction of ''ragismic'' and ''tritonic''.


Optimal tunings:  
[[Subgroup]]: 2.3.5.7
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9626{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0463{{c}}


{{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }}
[[Comma list]]: 4375/4374, 68719476736/68356598625


Badness (Sintel): 1.46
{{Mapping|legend=1| 1 -3 -15 40 | 0 9 34 -73 }}
: mapping generators: ~2, ~65536/45927


===== 19-limit =====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13.17.19
* [[WE]]: ~2 = 1199.8189{{c}}, ~65536/45927 = 611.2850{{c}}
: [[error map]]: {{val| -0.181 +0.153 +0.094 +0.123 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~65536/45927 = 611.3775{{c}}
: error map: {{val| 0.000 +0.443 +0.522 +0.615 }}


Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}


Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}
[[Badness]] (Sintel): 3.37


Optimal tunings:
=== 11-limit ===
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}
Subgroup: 2.3.5.7.11
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}


{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}
Comma list: 4375/4374, 5632/5625, 2621440/2614689


Badness (Sintel): 1.44
Mapping: {{mapping| 1 -3 -15 40 -75 | 0 9 34 -73 154 }}


== Parakleismic ==
Optimal tunings:
{{Main| Parakleismic }}
* WE: ~2 = 1199.8147{{c}}, ~768/539 = 611.2822{{c}}
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic (5-limit)]].''
* CWE: ~2 = 1200.0000{{c}}, ~768/539 = 611.3762{{c}}


In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat [[6/5]], 13 of which give 32/3, and 14 give 64/5. While 118 no longer has better than a cent of accuracy in the 7-limit, it is a decent temperament there nonetheless, and this allows an extension adding [[3136/3125]] and 4375/4374, for which [[99edo]], 118edo, and especially [[217edo]] are accurate tunings.
{{Optimal ET sequence|legend=0| 53, 316e, 369, 422, 791e, 1213cde }}


Parakleismic does not extend easily to the 11- or 13-limit. Possible 11-limit extensions include undecimal parakleismic (99 & 118), paralytic (99e & 118), parkleismic (80 & 99), and paradigmic (80 & 99e).  
Badness (Sintel): 2.34


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


[[Comma list]]: 3136/3125, 4375/4374
Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625


{{Mapping|legend=1| 1 -8 -8 -23 | 0 13 14 35 }}
Mapping: {{mapping| 1 -3 -15 40 -75 -34 | 0 9 34 -73 154 74 }}
: mapping generators: ~2, ~5/3


[[Optimal tuning]]s:  
Optimal tunings:  
* [[WE]]: ~2 = 1199.7820{{c}}, ~5/3 = 884.6581{{c}}
* WE: ~2 = 1199.7916{{c}}, ~91/64 = 611.2698{{c}}
: [[error map]]: {{val| -0.218 +0.344 +0.644 -0.779 }}
* CWE: ~2 = 1200.0000{{c}}, ~91/64 = 611.3754{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8088{{c}}
: error map: {{val| 0.000 +0.560 +1.010 -0.516 }}


{{Optimal ET sequence|legend=1| 19, 61d, 80, 99, 217, 316, 415 }}
{{Optimal ET sequence|legend=0| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}


[[Badness]] (Sintel): 0.694
Badness (Sintel): 1.51


=== 11-limit ===
== Quatracot ==
Subgroup: 2.3.5.7.11
{{See also| Stratosphere }}


Comma list: 385/384, 3136/3125, 4375/4374
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 -8 -8 -23 30 | 0 13 14 35 -36 }}
[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}


Optimal tunings:
{{Mapping|legend=1| 2 -6 -1 -36 | 0 13 8 59 }}
* WE: ~2 = 1200.3296{{c}}, ~5/3 = 884.9921{{c}}
: mapping generators: ~2278125/1605632, ~7168/5625
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7519{{c}}


{{Optimal ET sequence|legend=0| 19, 99, 118 }}
[[Optimal tuning]]s:
* [[WE]]: ~2278125/1605632 = 600.0888{{c}}, ~7168/5625 = 423.2574{{c}}
: [[error map]]: {{val| +0.178 -0.141 -0.343 +0.165 }}
* [[CWE]]: ~2278125/1605632 = 600.0000{{c}}, ~7168/5625 = 423.1986{{c}}
: error map: {{val| 0.000 -0.374 -0.725 -0.111 }}


Badness (Sintel): 1.64
{{Optimal ET sequence|legend=1| 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc }}


=== Paralytic ===
[[Badness]] (Sintel): 4.45
Paralytic (99e & 118) tempers out [[441/440]], [[5632/5625]], and [[19712/19683]]. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575.


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


Comma list: 441/440, 3136/3125, 4375/4374
Comma list: 3025/3024, 4375/4374, 1265625/1261568


Mapping: {{mapping| 1 -8 -8 -23 -57 | 0 13 14 35 82 }}
Mapping: {{mapping| 2 -6 -1 -36 -22 | 0 13 8 59 41 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9940{{c}}, ~5/3 = 884.7757{{c}}
* WE: ~99/70 = 600.0847{{c}}, ~225/176 = 423.2536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7800{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~225/176 = 423.1977{{c}}


{{Optimal ET sequence|legend=0| 19e, …, 99e, 118, 217, 335 }}
{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638, 1052c }}


Badness (Sintel): 1.19
Badness (Sintel): 1.36


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


Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
Comma list: 625/624, 729/728, 1575/1573, 2200/2197


Mapping: {{mapping| 1 -8 -8 -23 -57 59 | 0 13 14 35 82 -75 }}
Mapping: {{mapping| 2 -6 -1 -36 -22 -6 | 0 13 8 59 41 19 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9218{{c}}, ~5/3 = 884.7285{{c}}
* WE: ~99/70 = 600.0571{{c}}, ~143/112 = 423.2366{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7858{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~143/112 = 423.1987{{c}}


{{Optimal ET sequence|legend=0| 99e, 118, 217 }}
{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638 }}


Badness (Sintel): 1.85
Badness (Sintel): 0.936


==== Paraklein ====
== Trideci ==
Paraklein (19e & 118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tridecatonic]].''


Subgroup: 2.3.5.7.11.13
The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic]] temperament, but with the ragisma (4375/4374) rather than the octagar comma (4000/3969) tempered out. The name ''trideci'' comes from ''tridecim'' (Latin for "thirteen").


Comma list: 196/195, 352/351, 625/624, 729/728
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 -8 -8 -23 -57 -28 | 0 13 14 35 82 43 }}
[[Comma list]]: 4375/4374, 83349/81920
 
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
: mapping generators: ~256/245, ~3


Optimal tunings:  
[[Optimal tuning]]s:  
* WE: ~2 = 1199.8239{{c}}, ~5/3 = 884.6449{{c}}
* [[WE]]: ~256/245 = 92.4141{{c}}, ~3/2 = 699.9466{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7709{{c}}
: [[error map]]: {{val| +1.383 -0.626 -0.210 -2.554 }}
* [[CWE]]: ~256/245 = 92.3077{{c}}, ~3/2 = 699.4521{{c}}
: error map: {{val| 0.000 -2.503 -2.794 -6.740 }}


{{Optimal ET sequence|legend=0| 19e, …, 99ef, 118 }}
{{Optimal ET sequence|legend=1| 26, 65, 91 }}


Badness (Sintel): 1.55
[[Badness]] (Sintel): 4.67


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


Comma list: 176/175, 1375/1372, 2200/2187
Comma list: 245/242, 385/384, 4375/4374


Mapping: {{mapping| 1 -8 -8 -23 -43 | 0 13 14 35 63 }}
Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.1848{{c}}, ~5/3 = 884.3386{{c}}
* WE: ~22/21 = 92.3729{{c}}, ~3/2 = 700.1118{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9158{{c}}
* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.7703{{c}}


{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179, 259cd }}
{{Optimal ET sequence|legend=0| 26, 65, 91 }}


Badness (Sintel): 1.85
Badness (Sintel): 2.80


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


Comma list: 169/168, 176/175, 325/324, 1375/1372
Comma list: 169/168, 245/242, 325/324, 385/384


Mapping: {{mapping| 1 -8 -8 -23 -43 -14 | 0 13 14 35 63 24 }}
Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.5318{{c}}, ~5/3 = 884.5800{{c}}
* WE: ~22/21 = 92.4003{{c}}, ~3/2 = 699.9983{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9118{{c}}
* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.4772{{c}}


{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179 }}
{{Optimal ET sequence|legend=0| 26, 65f, 91f }}


Badness (Sintel): 1.51
Badness (Sintel): 2.16
 
== Moulin ==
Moulin can be described as the {{nowrap| 494 & 1619 }} temperament. It has a generator of ~[[22/13]], and it was named by [[Eliora]] in 2022 after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". However, the functional generator is ~[[13/11]], and 73 of them octave reduced reach the [[3/2|perfect fifth]]. Since [[11/8]] is within 23 generators, the 25-tone generator chain (4L 21s) of this temperament contains the 8:11:13 triad.


=== Paradigmic ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11


Comma list: 540/539, 896/891, 3136/3125
[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}


Mapping: {{mapping| 1 -8 -8 -23 16 | 0 13 14 35 -17 }}
{{Mapping|legend=1| 1 -16 -9 -75 | 0 73 47 323 }}
: mapping generators: ~2, ~3796875/3211264


Optimal tunings:  
[[Optimal tuning]]s:  
* WE: ~2 = 1199.0616{{c}}, ~5/3 = 884.2124{{c}}
* [[WE]]: ~2 = 1200.0272{{c}}, ~3796875/3211264 = 289.0675{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.8877{{c}}
: [[error map]]: {{val| +0.027 +0.007 -0.084 +0.013 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3796875/3211264 = 289.0675{{c}}
: error map: {{val| 0.000 -0.029 -0.142 -0.029 }}


{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e, 457bcddeeee }}
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 8589cc, 10208cc }}


Badness (Sintel): 1.38
[[Badness]] (Sintel): 5.93


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


Comma list: 169/168, 325/324, 540/539, 832/825
Comma list: 4375/4374, 759375/758912, 100663296/100656875


Mapping: {{mapping| 1 -8 -8 -23 16 -14 | 0 13 14 35 -17 24 }}
Mapping: {{mapping| 1 -16 -9 -75 9 | 0 73 47 323 -23 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.2683{{c}}, ~5/3 = 884.3805{{c}}
* WE: ~2 = 1200.0043{{c}}, ~605/512 = 289.0687{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9061{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~605/512 = 289.0677{{c}}


{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e }}
{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}


Badness (Sintel): 1.48
Badness (Sintel): 2.24


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


Comma list: 3025/3024, 3136/3125, 4375/4374
Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078


Mapping: {{mapping| 2 -3 -2 -11 -4 | 0 13 14 35 23 }}
Mapping: {{mapping| 1 -16 -9 -75 9 9 | 0 73 47 323 -23 -22 }}
: mapping generators: ~99/70, ~33/28


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 599.9270{{c}}, ~33/28 = 284.7841{{c}}
* WE: ~2 = 1200.0043{{c}}, ~13/11 = 289.0687{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8119{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 289.0677{{c}}


{{Optimal ET sequence|legend=0| 80, 118, 198, 316, 514c }}
{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}


Badness (Sintel): 1.13
Badness (Sintel): 1.12


==== Semiparamint ====
== Palladium ==
This extension was named ''semiparakleismic'' in the earlier materials.  
: ''For the 5-limit version, see [[46th-octave temperaments #Palladium]]''.


Subgroup: 2.3.5.7.11.13
The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minor whole tones (10/9) fall short of seven octaves. This temperament can be described as {{nowrap| 46 & 414 }} temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.


Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 2 -3 -2 -11 -4 15 | 0 13 14 35 23 -16 }}
[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}


Optimal tunings:
{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
* WE: ~99/70 = 599.8253{{c}}, ~33/28 = 284.7608{{c}}
: mapping generators: ~83349/81920, ~3
* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8366{{c}}


{{Optimal ET sequence|legend=0| 80, 118, 198 }}
[[Optimal tuning]]s:
* [[WE]]: ~83349/81920 = 26.0910{{c}}, ~3/2 = 701.7155{{c}}
: [[error map]]: {{val| +0.185 -0.055 -0.061 +0.349 }}
* [[CWE]]: ~83349/81920 = 26.0870{{c}}, ~3/2 = 701.6491{{c}}
: error map: {{val| 0.000 -0.306 -0.407 -0.910 }}


Badness (Sintel): 1.40
{{Optimal ET sequence|legend=1| 46, …, 368, 414, 460, 874d }}


==== Semiparawolf ====
[[Badness]] (Sintel): 7.81
This extension was named ''gentsemiparakleismic'' in the earlier materials.  


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


Comma list: 169/168, 325/324, 364/363, 3136/3125
Comma list: 3025/3024, 4375/4374, 134775333/134217728


Mapping: {{mapping| 2 -3 -2 -11 -4 -4 | 0 13 14 35 23 24 }}
Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.0569{{c}}, ~13/11 = 284.8431{{c}}
* WE: ~8192/8085 = 26.0912{{c}}, ~3/2 = 701.7082{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~13/11 = 284.8216{{c}}
* CWE: ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.6173{{c}}


{{Optimal ET sequence|legend=0| 80, 118f, 198f }}
{{Optimal ET sequence|legend=0| 46, …, 368, 414, 460, 874de }}


Badness (Sintel): 1.67
Badness (Sintel): 2.44


== Counterkleismic ==
=== 13-limit ===
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''
Subgroup: 2.3.5.7.11.13


In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses]] ((648/625)<sup>6</sup>) fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament, tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). It was named by analogy to [[catakleismic]] and parakleismic)
Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}


[[Comma list]]: 4375/4374, 158203125/157351936
Optimal tunings:  
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7411{{c}}
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6465{{c}}


{{Mapping|legend=1| 1 -5 -4 -18 | 0 25 24 79 }}
{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334dde }}
: mapping generators: ~2, ~6/5


[[Optimal tuning]]s:  
Badness (Sintel): 1.68
* [[WE]]: ~2 = 1200.1778{{c}}, ~6/5 = 316.1065{{c}}
: [[error map]]: {{val| +0.178 -0.181 -0.469 +0.388 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.0631{{c}}
: error map: {{val| 0.000 -0.377 -0.799 +0.161 }}


{{Optimal ET sequence|legend=1| 19, …, 205, 224, 243, 467 }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Badness]] (Sintel): 2.29
Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224


=== 11-limit ===
Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4375/4374, 2097152/2096325
 
Mapping: {{mapping| 1 -5 -4 -18 19 | 0 25 24 79 -59 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9944{{c}}, ~6/5 = 316.0690{{c}}
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7399{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0705{{c}}
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6464{{c}}


{{Optimal ET sequence|legend=0| 19, 205, 224 }}
{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334ddeg }}


Badness (Sintel): 2.35
Badness (Sintel): 1.14


==== 13-limit ====
== References ==
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 625/624, 729/728, 10985/10976
 
Mapping: {{mapping| 1 -5 -4 -18 19 -15 | 0 25 24 79 -59 71 }}
 
Optimal tunings:
* WE: ~2 = 1199.9827{{c}}, ~6/5 = 316.0650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0695{{c}}
 
{{Optimal ET sequence|legend=0| 19, 205, 224 }}
 
Badness (Sintel): 1.40
 
=== Counterlytic ===
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 4375/4374, 496125/495616
 
Mapping: {{mapping| 1 -5 -4 -18 -40 | 0 25 24 79 165 }}
 
Optimal tunings:
* WE: ~2 = 1200.1247{{c}}, ~6/5 = 316.0976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0660{{c}}
 
{{Optimal ET sequence|legend=1| 19e, 205e, 224, 467e, 691, 915c }}
 
Badness (Sintel): 2.16
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 729/728, 1375/1372, 10985/10976
 
Mapping: {{mapping| 1 -5 -4 -18 -40 -15 | 0 25 24 79 165 71 }}
 
Optimal tunings:
* WE: ~2 = 1200.0987{{c}}, ~6/5 = 316.0908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0658{{c}}
 
{{Optimal ET sequence|legend=0| 19e, 205e, 224, 467e, 691, 915c }}
 
Badness (Sintel): 1.23
 
== Quincy ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 823543/819200
 
{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
: mapping generators: ~2, ~1728/1715
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2169{{c}}, ~1728/1715 = 16.6160{{c}}
: [[error map]]: {{val| +0.217 +0.000 +0.155 -0.799 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1728/1715 = 16.6083{{c}}
: error map: {{val| 0.000 -0.205 -0.122 -1.343 }}
 
{{Optimal ET sequence|legend=1| 72, 217, 289, 650d, 939dd }}
 
[[Badness]] (Sintel): 2.02
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 4000/3993, 4375/4374
 
Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}
 
Optimal tunings:
* WE: ~2 = 1200.1286{{c}}, ~100/99 = 16.6147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6101{{c}}
 
{{Optimal ET sequence|legend=0| 72, 217, 289 }}
 
Badness (Sintel): 1.02
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 676/675, 4375/4374
 
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}
 
Optimal tunings:
* WE: ~2 = 1200.0554{{c}}, ~100/99 = 16.6028{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6011{{c}}
 
{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
 
Badness (Sintel): 0.986
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155
 
Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}
 
Optimal tunings:
* WE: ~2 = 1200.0647{{c}}, ~100/99 = 16.6025{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6004{{c}}
 
{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
 
Badness (Sintel): 0.751
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675
 
Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}
 
Optimal tunings:
* WE: ~2 = 1199.9287{{c}}, ~100/99 = 16.5930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.5948{{c}}
 
{{Optimal ET sequence|legend=0| 72, 145, 217 }}
 
Badness (Sintel): 0.924
 
== Sfourth ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sfourth]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 64827/64000
 
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
: mapping generators: ~2, ~49/48
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8332{{c}}, ~49/48 = 26.3053{{c}}
: [[error map]]: {{val| +0.833 -0.090 +0.721 -3.074 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 26.2590{{c}}
: error map: {{val| 0.000 -0.876 -0.343 -5.157 }}
 
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
 
[[Badness]] (Sintel): 3.12
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 441/440, 4375/4374
 
Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}
 
Optimal tunings:
* WE: ~2 = 1201.1486{{c}}, ~49/48 = 26.3112{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2461{{c}}
 
{{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }}
 
Badness (Sintel): 1.78
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 169/168, 325/324, 441/440
 
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}
 
Optimal tunings:
* WE: ~2 = 1201.4956{{c}}, ~49/48 = 26.3423{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2614{{c}}
 
{{Optimal ET sequence|legend=0| 45ef, 46, 91ef, 137def, 228ddeeefff }}
 
Badness (Sintel): 1.37
 
=== Sfour ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 2401/2376, 4375/4374
 
Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}
 
Optimal tunings:
* WE: ~2 = 1200.4402{{c}}, ~49/48 = 26.2557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2403{{c}}
 
{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}
 
Badness (Sintel): 2.53
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 364/363, 385/384, 4375/4374
 
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}
 
Optimal tunings:
* WE: ~2 = 1200.3796{{c}}, ~49/48 = 26.2473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2372{{c}}
 
{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}
 
Badness (Sintel): 2.14
 
== Trideci ==
: ''For the 5-limit version, see [[13th-octave temperaments #Tridecatonic]].''
 
The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from ''tridecim'' (Latin for "thirteen").
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 83349/81920
 
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
: mapping generators: ~256/245, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~256/245 = 92.4141{{c}}, ~3/2 = 699.9466{{c}}
: [[error map]]: {{val| +1.383 -0.626 -0.210 -2.554 }}
* [[CWE]]: ~256/245 = 92.3077{{c}}, ~3/2 = 699.4521{{c}}
: error map: {{val| 0.000 -2.503 -2.794 -6.740 }}
 
{{Optimal ET sequence|legend=1| 26, 65, 91 }}
 
[[Badness]] (Sintel): 4.67
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/242, 385/384, 4375/4374
 
Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}
 
Optimal tunings:
* WE: ~22/21 = 92.3729{{c}}, ~3/2 = 700.1118{{c}}
* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.7703{{c}}
 
{{Optimal ET sequence|legend=0| 26, 65, 91 }}
 
Badness (Sintel): 2.80
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 245/242, 325/324, 385/384
 
Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}
 
Optimal tunings:
* WE: ~22/21 = 92.4003{{c}}, ~3/2 = 699.9983{{c}}
* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.4772{{c}}
 
{{Optimal ET sequence|legend=0| 26, 65f, 91f }}
 
Badness (Sintel): 2.16
 
== References ==


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