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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, 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]]
* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]]
* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]
* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]]
* [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity 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]]
* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive 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]]
* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]
* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]
* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]]
* [[Parakleismic]] (+3136/3125) → [[Parakleismic family #Septimal parakleismic|Parakleismic family]]
* [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnu family #Septimal vishnu|Vishnu family]]
* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]
* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]
* [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]]
* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortone family #Mitonic|Minortone family]]
* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]
* ''[[Zarvo]]'' (+33075/32768) → [[Gravity family #Zarvo|Gravity family]]
* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortonic family #Mitonic|Minortonic family]]
* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnuzmic family #Septimal vishnu|Vishnuzmic family]]
* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]
* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot 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]]
* ''[[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, chlorine, octoid, seniority, monzismic, semidimfourth, acrokleismic, quasithird, quincy, deca, keenanose, counterkleismic, sfourth, aluminium, ragitritonic, quatracot, trideci, moulin, and palladium.


== Supermajor ==
== Supermajor ==
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== Brahmagupta ==
== Brahmagupta ==
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}).  
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}), and may be described as the {{nowrap| 217 & 224 }} temperament.  


Early in the design of the [[Sagittal]] notation system, [[George Secor|Secor]] and [[Dave Keenan|Keenan]] found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4{{c}} many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of brahmagupta that has pure octaves and pure fifths, which can also be described as a 17-limit extension having a 1/7-octave period (171.4286{{c}}) and 1/21-apotome generator (5.4136{{c}}).
Early in the design of the [[Sagittal]] notation system, [[George Secor|Secor]] and [[Dave Keenan|Keenan]] found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4{{c}} many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of brahmagupta that has pure octaves and pure fifths, which can also be described as a 17-limit extension having a 1/7-octave period (171.4286{{c}}) and 1/21-apotome generator (5.4136{{c}}).
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: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Abigail]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Abigail]].''


Abigail tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by [[Gene Ward Smith]] in 2010 after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930 Yahoo! Tuning Group | ''11-limit rank 2 using only wedgies''] "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things." —Gene Ward Smith</ref>
Abigail tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit, and may be described as the {{nowrap| 46 & 224 }} temperament, with a [[ploidacot]] signature of diploid wau-hendecacot. It extends into a very strong 11- and 13-limit temperament. [[494edo]], [[764edo]] and [[1258edo]] are among the possible tunings.
 
Abigail was named by [[Gene Ward Smith]] in 2010 after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930 Yahoo! Tuning Group | ''11-limit rank 2 using only wedgies''] "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things." —Gene Ward Smith</ref>


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament. [[1106edo]] is a strong tuning.  
Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament, with a [[ploidacot]] of diploid alpha-octacot. [[1106edo]] gives a strong tuning.
 
Crazy was named by [[Flora Canou]] in 2025 by removing the mutation from ''kwazy'', the name for the 5-limit microtemperament.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}
{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}
: Mapping generators: ~332150625/234881024, ~1125/1024
: mapping generators: ~332150625/234881024, ~1125/1024


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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== Orga ==
== Orga ==
Orga may be described as the {{nowrap| 26 & 270 }} temperament, and [[1106edo]] gives a strong tuning.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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Badness (Sintel): 0.899
Badness (Sintel): 0.899


== Seniority ==
== Chlorine ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]].  
: ''For the 5-limit version, see [[17th-octave temperaments #Chlorine]].''


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 }}, quadla-sepquingu) is tempered out.
Chlorine (named after the 17th element) tempers out the [[septendecima]] in the 5-limit, and {{monzo| -49 4 22 -3 }} as well as the ragisma in the 7-limit. It has a 1/17-octave period, and can be described as {{nowrap| 289 & 323 }} temperament. Not only the semitwelfth, but also the ~5/4 can be used as a generator.  


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


[[Comma list]]: 4375/4374, 201768035/201326592
[[Comma list]]: 4375/4374, {{monzo| -49 4 22 -3 }}


{{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }}
{{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }}
: mapping generators: ~2, ~5120/3087


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0745{{c}}, ~5120/3087 = 877.2500{{c}}
* [[WE]]: ~25/24 = 70.5880{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9962{{c}}
: [[error map]]: {{val| +0.075 +0.008 -0.016 -0.203 }}
: [[error map]]: {{val| -0.004 +0.037 -0.030 -0.019 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5120/3087 = 877.1965{{c}}
* [[CWE]]: ~25/24 = 70.5882{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9990{{c}}
: error map: {{val| 0.000 -0.077 -0.130 -0.415 }}
: error map: {{val| 0.000 +0.043 -0.021 -0.013 }}


{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d }}
{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547, 3706, 5253 }}


[[Badness]] (Sintel): 1.14
[[Badness]] (Sintel): 1.05
 
=== 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.


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


Comma list: 441/440, 4375/4374, 65536/65219
Comma list: 4375/4374, 41503/41472, 1879453125/1879048192


Mapping: {{mapping| 1 -24 -43 5 2 | 0 35 62 -3 2 }}
Mapping: {{mapping| 17 0 26 -87 207 | 0 2 1 10 -11 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}}
* WE: ~25/24 = 70.5905{{c}}, ~693/400 = 951.0054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}}
* CWE: ~25/24 = 70.5882{{c}}, ~693/400 = 950.9754{{c}}


{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e }}
{{Optimal ET sequence|legend=0| 289, 323, 612, 3349de, 3961de, …, 5797ddee }}


Badness (Sintel): 3.05
Badness (Sintel): 2.11


==== 13-limit ====
== Octoid ==
Subgroup: 2.3.5.7.11.13
: {{Main| Octoid }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Octoid]].''


Comma list: 364/363, 441/440, 2200/2197, 4375/4374
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]].


Mapping: {{mapping| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}
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 tunings:  
[[Subgroup]]: 2.3.5.7
* WE: ~2 = 1199.7136{{c}}, ~108/65 = 877.9974{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2038{{c}}


{{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }}
[[Comma list]]: 4375/4374, 16875/16807


Badness (Sintel): 1.85
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
: mapping generators: ~49/45, ~7/5


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


Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197
[[Tuning ranges]]:  
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
* 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]


Mapping: {{mapping| 1 -24 -43 5 2 -27 -31 | 0 35 62 -3 2 42 48 }}
{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}


Optimal tunings:  
[[Badness]] (Sintel): 1.08
* WE: ~2 = 1199.7195{{c}}, ~108/65 = 877.0018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}}


{{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness (Sintel): 1.35
Comma list: 540/539, 1375/1372, 4000/3993


== Monzismic ==
Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}
: ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]].


The monzismic temperament 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. A notable tuning not appearing on the optimal ET sequence is [[665edo]], which is nearly equivalent to the pure-3's tuning.
Optimal tunings:
* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9477{{c}}


[[Subgroup]]: 2.3.5.7
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]


[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224, 824d }}


{{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }}
Badness (Sintel): 0.466
: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}


[[Optimal tuning]]s:
==== 13-limit ====
* [[WE]]: ~2 = 1200.0128{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9895{{c}}
Subgroup: 2.3.5.7.11.13
: [[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 }}
Comma list: 540/539, 625/624, 729/728, 1375/1372


[[Badness]] (Sintel): 1.18
Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}


=== Monzism ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}


Comma list: 4375/4374, 41503/41472, 184549376/184528125
{{Optimal ET sequence|legend=0| 72, 152f, 224 }}


Mapping: {{mapping| 1 0 -27 109 -159 | 0 2 37 -134 205 }}
Badness (Sintel): 0.631


Optimal tunings:
===== 17-limit =====
* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}}
Subgroup: 2.3.5.7.11.13.17
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}}


{{Optimal ET sequence|legend=0| 53, 559, 612, 3619de, 4231de, …, 6067ddee }}
Comma list: 375/374, 540/539, 625/624, 715/714, 729/728


Badness (Sintel): 1.89
Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}


==== 13-limit ====
Optimal tunings:
Subgroup: 2.3.5.7.11.13
* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}


Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
{{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}


Mapping: {{mapping| 1 0 -27 109 -159 -70 | 0 2 37 -134 205 93 }}
Badness (Sintel): 0.729


Optimal tunings:
===== 19-limit =====
* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}}
Subgroup: 2.3.5.7.11.13.17.19
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}}


{{Optimal ET sequence|legend=0| 53, 559, 612 }}
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714


Badness (Sintel): 2.22
Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}


== Semidimfourth ==
Optimal tunings:
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimfourth]].''
* WE: ~12/11 = 149.9785{{c}}, ~7/5 = 583.8482{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}}


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]].
{{Optimal ET sequence|legend=0| 72, 152fg, 224 }}


[[Subgroup]]: 2.3.5.7
Badness (Sintel): 0.975


[[Comma list]]: 4375/4374, 235298/234375
==== 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}}.


{{Mapping|legend=1| 1 -10 -13 -17 | 0 31 41 53 }}
Subgroup: 2.3.5.7.11.13
: mapping generators: ~2, ~35/27


[[Optimal tuning]]s:  
Comma list: 169/168, 325/324, 364/363, 540/539
* [[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 }}
Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}


[[Badness]] (Sintel): 1.40
Optimal tunings:  
* WE: ~12/11 = 150.0313{{c}}, ~7/5 = 584.0134{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}}


=== Neusec ===
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4375/4374, 235298/234375
Badness (Sintel): 0.896


Mapping: {{mapping| 2 -20 -26 -34 -17 | 0 31 41 53 32 }}
===== 17-limit =====
: mapping generators: ~99/70, ~35/27
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
 
Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.0381{{c}}, ~35/27 = 448.4812{{c}}
* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4546{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}


{{Optimal ET sequence|legend=0| 8d, …, 190, 388 }}
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}


Badness (Sintel): 1.95
Badness (Sintel): 0.795


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


Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399


Mapping: {{mapping| 2 -20 -26 -34 -17 -21 | 0 31 41 53 32 38 }}
Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.0034{{c}}, ~35/27 = 448.4573{{c}}
* WE: ~12/11 = 150.0049{{c}}, ~7/5 = 584.0833{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4549{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}}


{{Optimal ET sequence|legend=0| 8d, …, 190, 198, 388 }}
{{Optimal ET sequence|legend=0| 8d, 72, 152 }}


Badness (Sintel): 1.28
Badness (Sintel): 0.993


== Acrokleismic ==
Scales: [[Octoid72]], [[Octoid80]]
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 2202927104/2197265625
==== Hexadecoid ====
{{See also| 16th-octave temperaments }}


{{Mapping|legend=1| 1 -22 -22 -65 | 0 32 33 92 }}
Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.
: mapping generators: ~2, ~5/3


[[Optimal tuning]]s:  
Subgroup: 2.3.5.7.11.13
* [[WE]]: ~2 = 1199.9305{{c}}, ~6/5 = 884.3923{{c}}
: [[error map]]: {{val| -0.070 +0.126 +0.160 -0.221 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 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 }}
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224


[[Badness]] (Sintel): 1.42
Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
: mapping generators: ~448/429, ~7/5


=== 11-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}
* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}


Comma list: 4375/4374, 41503/41472, 172032/171875
{{Optimal ET sequence|legend=0| 80, 144, 224 }}


Mapping: {{mapping| 1 -22 -22 -65 58 | 0 32 33 92 -74 }}
Badness (Sintel): 1.27
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
 
Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9698{{c}}, ~5/3 = 884.4193{{c}}
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9626{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4414{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0463{{c}}


{{Optimal ET sequence|legend=0| 19, 251, 270, 829, 1099, 1369, 1639 }}
{{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }}


Badness (Sintel): 1.22
Badness (Sintel): 1.46


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


Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444


Mapping: {{mapping| 1 -22 -22 -65 58 -56 | 0 32 33 92 -74 81 }}
Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9939{{c}}, ~5/3 = 884.4384{{c}}
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4429{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}


{{Optimal ET sequence|legend=0| 19, 251, 270 }}
{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}


Badness (Sintel): 1.11
Badness (Sintel): 1.44


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


Comma list: 4375/4374, 5632/5625, 117649/117612
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 -141 | 0 32 33 92 196 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 4375/4374, 201768035/201326592
* WE: ~2 = 1199.8877{{c}}, ~5/3 = 884.3639{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4457{{c}}


{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1061e, 1331c, 1601c, 1871bc }}
{{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }}
: mapping generators: ~2, ~5120/3087


Badness (Sintel): 1.41
[[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
 
[[Badness]] (Sintel): 1.14


Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
=== 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.


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


Optimal tunings:  
Comma list: 441/440, 4375/4374, 65536/65219
* WE: ~2 = 1199.9285{{c}}, ~5/3 = 884.3937{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4458{{c}}


{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1331c }}
Mapping: {{mapping| 1 -24 -43 5 2 | 0 35 62 -3 2 }}


Badness (Sintel): 1.08
Optimal tunings:  
* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}}


== Quasithird ==
{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasithird]].''


The quasithird temperament is 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 to temper out the ragisma and {{monzo| -60 29 0 5 }}.
Badness (Sintel): 3.05


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


[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}
Comma list: 364/363, 441/440, 2200/2197, 4375/4374


{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
Mapping: {{mapping| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}
: mapping generators: ~65536/55125, ~5103/4096


[[Optimal tuning]]s:  
Optimal tunings:  
* [[WE]]: ~65536/55125 = 300.0052{{c}}, ~5103/4096 = 380.3949{{c}}
* WE: ~2 = 1199.7136{{c}}, ~108/65 = 877.9974{{c}}
: [[error map]]: {{val| +0.021 +0.020 -0.052 -0.031 }}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2038{{c}}
* [[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 }}
{{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }}


[[Badness]] (Sintel): 1.56
Badness (Sintel): 1.85


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


Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197


Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}
Mapping: {{mapping| 1 -24 -43 5 2 -27 -31 | 0 35 62 -3 2 42 48 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~65536/51125 = 300.0073{{c}}, ~5103/4096 = 380.3963{{c}} (or ~22/21 = 80.3890{{c}})
* WE: ~2 = 1199.7195{{c}}, ~108/65 = 877.0018{{c}}
* CWE: ~65536/51125 = 300.0000{{c}}, ~5103/4096 = 380.3868{{c}} (or ~22/21 = 80.3868{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}}


{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee }}
{{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }}


Badness (Sintel): 0.698
Badness (Sintel): 1.35


=== 13-limit ===
== Monzismic ==
Subgroup: 2.3.5.7.11.13
: ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]].  


Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374
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.


Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}
* 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}})


{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836 }}
{{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }}
: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}


Badness (Sintel): 1.22
[[Optimal tuning]]s:  
* [[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 }}


== Deca ==
{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd }}
: ''For 5-limit version, see [[10th-octave temperaments #Neon]].''


Deca temperament has a period of 1/10 octave and tempers out the [[linus comma]], {{monzo| 11 -10 -10 10 }}, neon comma {{monzo| 21 60 -50 }} and {{monzo| 12 -3 -14 9 }} (165288374272/164794921875).
[[Badness]] (Sintel): 1.18


[[Subgroup]]: 2.3.5.7
=== Monzism ===
Subgroup: 2.3.5.7.11


[[Comma list]]: 4375/4374, 165288374272/164794921875
Comma list: 4375/4374, 41503/41472, 184549376/184528125


{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}
Mapping: {{mapping| 1 0 -27 109 -159 | 0 2 37 -134 205 }}
: mapping generators: ~15/14, ~460992/390625


[[Optimal tuning]]s:  
Optimal tunings:  
* [[WE]]: ~15/14 = 119.9966{{c}}, ~460992/390625 = 284.4150{{c}} (5625/5488 = 44.4219{{c}})
* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}}
: [[error map]]: {{val| -0.034 +0.106 +0.145 -0.268 }}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}}
* [[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 }}
{{Optimal ET sequence|legend=0| 53, 559, 612, 3619de, 4231de, …, 6067ddee }}


[[Badness]] (Sintel): 2.04
Badness (Sintel): 1.89


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


Comma list: 3025/3024, 4375/4374, 391314/390625
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625


Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}
Mapping: {{mapping| 1 0 -27 109 -159 -70 | 0 2 37 -134 205 93 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~15/14 = 120.0004{{c}}, ~33/28 = 284.4193{{c}} (77/75 = 44.4185{{c}})
* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4189{{c}} (77/75 = 44.4189{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}}


{{Optimal ET sequence|legend=0| 80, 190, 270, 1000, 1270, 1540e, 1810e }}
{{Optimal ET sequence|legend=0| 53, 559, 612 }}


Badness (Sintel): 0.804
Badness (Sintel): 2.22


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


Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
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]].


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


Optimal tunings:  
[[Comma list]]: 4375/4374, 235298/234375
* WE: ~15/14 = 120.0067{{c}}, ~33/28 = 284.4139{{c}} (~40/39 = 44.4006{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4048{{c}} (~40/39 = 44.4048{{c}})


{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}
{{Mapping|legend=1| 1 -10 -13 -17 | 0 31 41 53 }}
: mapping generators: ~2, ~35/27


Badness (Sintel): 0.695
[[Optimal tuning]]s:  
* [[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 }}


=== 2.3.5.7.11.13.19 subgroup ===
{{Optimal ET sequence|legend=1| 8d, …, 91, 99, 289, 388, 875 }}
Subgroup: 2.3.5.7.11.13.19


Comma list: 1001/1000, 1521/1520, 3025/3024, 4225/4224, 4375/4374
[[Badness]] (Sintel): 1.40


Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}
=== Neusec ===
Subgroup: 2.3.5.7.11


Optimal tunings:  
Comma list: 3025/3024, 4375/4374, 235298/234375
* 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}})


{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}
Mapping: {{mapping| 2 -20 -26 -34 -17 | 0 31 41 53 32 }}
: mapping generators: ~99/70, ~35/27


Badness (Sintel): 0.556
Optimal tunings:  
* WE: ~99/70 = 600.0381{{c}}, ~35/27 = 448.4812{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4546{{c}}


== Keenanose ==
{{Optimal ET sequence|legend=0| 8d, …, 190, 388 }}
Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.


[[Subgroup]]: 2.3.5.7
Badness (Sintel): 1.95


[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}


[[Optimal tuning]]s:  
Mapping: {{mapping| 2 -20 -26 -34 -17 -21 | 0 31 41 53 32 38 }}
* [[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 }}
Optimal tunings:
* WE: ~99/70 = 600.0034{{c}}, ~35/27 = 448.4573{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4549{{c}}


[[Badness]] (Sintel): 2.17
{{Optimal ET sequence|legend=0| 8d, …, 190, 198, 388 }}


=== 11-limit ===
Badness (Sintel): 1.28
Subgroup: 2.3.5.7.11


Comma list: 4375/4374, 117649/117612, 67110351/67108864
== Acrokleismic ==
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}
[[Comma list]]: 4375/4374, 2202927104/2197265625


Optimal tunings:  
{{Mapping|legend=1| 1 -22 -22 -65 | 0 32 33 92 }}
* WE: ~2 = 1199.9970{{c}}, ~385/384 = 4.4465{{c}}
: mapping generators: ~2, ~5/3
* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4465{{c}}
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9305{{c}}, ~5/3 = 884.3923{{c}}
: [[error map]]: {{val| -0.070 +0.126 +0.160 -0.221 }}
* [[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=0| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}
{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}


Badness (Sintel): 1.02
[[Badness]] (Sintel): 1.42


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


Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612
Comma list: 4375/4374, 41503/41472, 172032/171875


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0065{{c}}, ~385/384 = 4.4467{{c}}
* WE: ~2 = 1199.9698{{c}}, ~5/3 = 884.4193{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4467{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4414{{c}}


{{Optimal ET sequence|legend=0| 270, 1079, 1349, 1619, 1889, 4048 }}
{{Optimal ET sequence|legend=0| 19, 251, 270, 829, 1099, 1369, 1639 }}


Badness (Sintel): 0.879
Badness (Sintel): 1.22


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


Aluminium is named after the 13th element, and tempers out the {{monzo| 92 -39 -13 }} comma which sets [[135/128]] interval to be equal to 1/13th of the octave.
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 1 -22 -22 -65 58 -56 | 0 32 33 92 -74 81 }}


[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}
Optimal tunings:  
* WE: ~2 = 1199.9939{{c}}, ~5/3 = 884.4384{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4429{{c}}


[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}
{{Optimal ET sequence|legend=0| 19, 251, 270 }}
: Mapping generators: ~135/128, ~3


[[Optimal tuning]]s:  
Badness (Sintel): 1.11
* [[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 }}
=== Counteracro ===
Subgroup: 2.3.5.7.11


[[Badness]] (Sintel): 3.20
Comma list: 4375/4374, 5632/5625, 117649/117612


=== 11-limit ===
Mapping: {{mapping| 1 -22 -22 -65 -141 | 0 32 33 92 196 }}
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 234375/234256, 2097152/2096325
 
Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~135/128 = 92.3062{{c}}, ~3/2 = 701.9946{{c}}
* WE: ~2 = 1199.8877{{c}}, ~5/3 = 884.3639{{c}}
* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4457{{c}}


{{Optimal ET sequence|legend=0| 494, 1053, 1547, 3588e, 5135e }}
{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1061e, 1331c, 1601c, 1871bc }}


Badness (Sintel): 1.39
Badness (Sintel): 1.41


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


Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374


Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}
Mapping: {{mapping| 1 -22 -22 -65 -141 -56 | 0 32 33 92 196 81 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~135/128 = 92.3055{{c}}, ~3/2 = 701.9928{{c}}
* WE: ~2 = 1199.9285{{c}}, ~5/3 = 884.3937{{c}}
* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0098{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4458{{c}}


{{Optimal ET sequence|legend=0| 494, 1547, 2041, 4576def }}
{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1331c }}


Badness (Sintel): 1.18
Badness (Sintel): 1.08


== Countritonic ==
== Quasithird ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic (5-limit)]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasithird]].''


Countritonic (''co-un-tritonic'') can be described as the {{nowrap| 53 & 422 }} temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit.  
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.  


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


[[Comma list]]: 4375/4374, 68719476736/68356598625
[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}


{{Mapping|legend=1| 1 -3 -15 40 | 0 9 34 -73 }}
{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
: mapping generators: ~2, ~65536/45927
: mapping generators: ~65536/55125, ~5103/4096


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.8189{{c}}, ~65536/45927 = 611.2850{{c}}
* [[WE]]: ~65536/55125 = 300.0052{{c}}, ~5103/4096 = 380.3949{{c}}
: [[error map]]: {{val| -0.181 +0.153 +0.094 +0.123 }}
: [[error map]]: {{val| +0.021 +0.020 -0.052 -0.031 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~65536/45927 = 611.3775{{c}}
* [[CWE]]: ~65536/55125 = 300.0000{{c}}, ~5103/4096 = 380.3884{{c}}
: error map: {{val| 0.000 +0.443 +0.522 +0.615 }}
: error map: {{val| 0.000 -0.013 -0.100 -0.089 }}


{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}


[[Badness]] (Sintel): 3.37
[[Badness]] (Sintel): 1.56


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


Comma list: 4375/4374, 5632/5625, 2621440/2614689
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296


Mapping: {{mapping| 1 -3 -15 40 -75 | 0 9 34 -73 154 }}
Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8147{{c}}, ~768/539 = 611.2822{{c}}
* WE: ~65536/51125 = 300.0073{{c}}, ~5103/4096 = 380.3963{{c}} (or ~22/21 = 80.3890{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~768/539 = 611.3762{{c}}
* CWE: ~65536/51125 = 300.0000{{c}}, ~5103/4096 = 380.3868{{c}} (or ~22/21 = 80.3868{{c}})


{{Optimal ET sequence|legend=0| 53, 316e, 369, 422, 791e, 1213cde }}
{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee }}


Badness (Sintel): 2.34
Badness (Sintel): 0.698


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


Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625
Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374


Mapping: {{mapping| 1 -3 -15 40 -75 -34 | 0 9 34 -73 154 74 }}
Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7916{{c}}, ~768/539 = 611.2698{{c}}
* WE: ~65536/51125 = 299.9985{{c}}, ~81/65 = 380.3833{{c}} (or ~22/21 = 80.3848{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~768/539 = 611.3754{{c}}
* CWE: ~65536/51125 = 300.0000{{c}}, ~81/65 = 380.3852{{c}} (or ~22/21 = 80.3852{{c}})


{{Optimal ET sequence|legend=0| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}
{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836 }}


Badness (Sintel): 1.51
Badness (Sintel): 1.22
 
== Quatracot ==
{{See also| Stratosphere }}


== Quincy ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
[[Comma list]]: 4375/4374, 823543/819200


{{Mapping|legend=1| 2 -6 -1 -36 | 0 13 8 59 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
: mapping generators: ~2278125/1605632, ~7168/5625
: mapping generators: ~2, ~1728/1715


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2278125/1605632 = 600.0888{{c}}, ~7168/5625 = 423.2574{{c}}
* [[WE]]: ~2 = 1200.2169{{c}}, ~1728/1715 = 16.6160{{c}}
: [[error map]]: {{val| +0.178 -0.141 -0.343 +0.165 }}
: [[error map]]: {{val| +0.217 +0.000 +0.155 -0.799 }}
* [[CWE]]: ~2278125/1605632 = 600.0000{{c}}, ~7168/5625 = 423.1986{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1728/1715 = 16.6083{{c}}
: error map: {{val| 0.000 -0.374 -0.725 -0.111 }}
: error map: {{val| 0.000 -0.205 -0.122 -1.343 }}
 
 
{{Optimal ET sequence|legend=1| 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc }}
{{Optimal ET sequence|legend=1| 72, 217, 289, 650d, 939dd }}
 
 
[[Badness]] (Sintel): 4.45
[[Badness]] (Sintel): 2.02


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


Comma list: 3025/3024, 4375/4374, 1265625/1261568
Comma list: 441/440, 4000/3993, 4375/4374


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


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.0847{{c}}, ~225/176 = 423.2536{{c}}
* WE: ~2 = 1200.1286{{c}}, ~100/99 = 16.6147{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~225/176 = 423.1977{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6101{{c}}


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


Badness (Sintel): 1.36
Badness (Sintel): 1.02


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


Comma list: 625/624, 729/728, 1575/1573, 2200/2197
Comma list: 364/363, 441/440, 676/675, 4375/4374


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


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.0571{{c}}, ~143/112 = 423.2366{{c}}
* WE: ~2 = 1200.0554{{c}}, ~100/99 = 16.6028{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~143/112 = 423.1987{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6011{{c}}


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


Badness (Sintel): 0.936
Badness (Sintel): 0.986


== Moulin ==
=== 17-limit ===
Moulin can be described as the {{nowrap| 494 & 1619 }} temperament. It has a generator of ~[[22/13]], and it is named 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: 2.3.5.7.11.13.17


[[Subgroup]]: 2.3.5.7
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155


[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}


{{Mapping|legend=1| 1 -16 -9 -75 | 0 73 47 323 }}
Optimal tunings:
: mapping generators: ~2, ~3796875/3211264
* WE: ~2 = 1200.0647{{c}}, ~100/99 = 16.6025{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6004{{c}}


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
* [[WE]]: ~2 = 1200.0272{{c}}, ~3796875/3211264 = 289.0675{{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=1| 494, 1125, 1619, 8589cc, 10208cc }}
Badness (Sintel): 0.751


[[Badness]] (Sintel): 5.93
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


=== 11-limit ===
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675
Subgroup: 2.3.5.7.11


Comma list: 4375/4374, 759375/758912, 100663296/100656875
Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}


Mapping: {{mapping| 1 -16 -9 -75 9 | 0 73 47 323 -23 }}
Optimal tunings:  
* WE: ~2 = 1199.9287{{c}}, ~100/99 = 16.5930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.5948{{c}}


Optimal tunings:
{{Optimal ET sequence|legend=0| 72, 145, 217 }}
* WE: ~2 = 1200.0043{{c}}, ~605/512 = 289.0687{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~605/512 = 289.0677{{c}}


{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}
Badness (Sintel): 0.924


Badness (Sintel): 2.24
== Deca ==
: ''For 5-limit version, see [[10th-octave temperaments#Neon]].''


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


Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 -16 -9 -75 9 9 | 0 73 47 323 -23 -22 }}
[[Comma list]]: 4375/4374, 165288374272/164794921875


Optimal tunings:
{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}
* WE: ~2 = 1200.0043{{c}}, ~13/11 = 289.0687{{c}}
: mapping generators: ~15/14, ~460992/390625
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 289.0677{{c}}


{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}
[[Optimal tuning]]s:
* [[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 }}


Badness (Sintel): 1.12
{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}


== Palladium ==
[[Badness]] (Sintel): 2.04
: ''For the 5-limit version, see [[46th-octave temperaments #Palladium]]''.


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


[[Subgroup]]: 2.3.5.7
Comma list: 3025/3024, 4375/4374, 391314/390625


[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}
Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}


{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
Optimal tunings:
: mapping generators: ~83349/81920, ~3
* WE: ~15/14 = 120.0004{{c}}, ~33/28 = 284.4193{{c}} (77/75 = 44.4185{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4189{{c}} (77/75 = 44.4189{{c}})


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=0| 80, 190, 270, 1000, 1270, 1540e, 1810e }}
* [[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 }}
Badness (Sintel): 0.804


[[Badness]] (Sintel): 7.81
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 11-limit ===
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4375/4374, 134775333/134217728
Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}
 
Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~8192/8085 = 26.0912{{c}}, ~3/2 = 701.7082{{c}}
* WE: ~15/14 = 120.0067{{c}}, ~33/28 = 284.4139{{c}} (~40/39 = 44.4006{{c}})
* CWE: ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.6173{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4048{{c}} (~40/39 = 44.4048{{c}})


{{Optimal ET sequence|legend=0| 46, …, 368, 414, 460, 874de }}
{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}


Badness (Sintel): 2.44
Badness (Sintel): 0.695


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


Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
Comma list: 1001/1000, 1521/1520, 3025/3024, 4225/4224, 4375/4374


Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}
Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7411{{c}}
* WE: ~15/14 = 120.0045{{c}}, ~33/28 = 284.4140{{c}} (~39/38 = 44.4050{{c}})
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6465{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4075{{c}} (~39/38 = 44.4075{{c}})


{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334dde }}
{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}


Badness (Sintel): 1.68
Badness (Sintel): 0.556


=== 17-limit ===
== Keenanose ==
Subgroup: 2.3.5.7.11.13.17
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.


Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}
[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}


Optimal tunings:
{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7399{{c}}
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6464{{c}}


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


Badness (Sintel): 1.14
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}


== Oviminor ==
[[Badness]] (Sintel): 2.17
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Oviminor (5-limit)]].''


Oviminor is named 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.  
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Subgroup]]: 2.3.5.7
Comma list: 4375/4374, 117649/117612, 67110351/67108864


[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}
Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}


{{Mapping|legend=1| 1 -134 -134 -401 | 0 184 185 548 }}
Optimal tunings:
: mapping generators: ~2, ~5/3
* WE: ~2 = 1199.9970{{c}}, ~385/384 = 4.4465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4465{{c}}


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=0| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}
* [[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 }}
Badness (Sintel): 1.02


[[Badness]] (Sintel): 14.7
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== Octoid ==
Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612
: ''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]].
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}


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 tunings:
* WE: ~2 = 1200.0065{{c}}, ~385/384 = 4.4467{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4467{{c}}


[[Subgroup]]: 2.3.5.7
{{Optimal ET sequence|legend=0| 270, 1079, 1349, 1619, 1889, 4048 }}


[[Comma list]]: 4375/4374, 16875/16807
Badness (Sintel): 0.879


{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
== Counterkleismic ==
: mapping generators: ~49/45, ~7/5
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''
 
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]]).
 
[[Subgroup]]: 2.3.5.7


[[Optimal tuning]]s:  
[[Comma list]]: 4375/4374, 158203125/157351936
* [[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]]:
{{Mapping|legend=1| 1 -5 -4 -18 | 0 25 24 79 }}
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
: mapping generators: ~2, ~6/5
* 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 }}
[[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 }}


[[Badness]] (Sintel): 1.08
{{Optimal ET sequence|legend=1| 19, …, 205, 224, 243, 467 }}


Scales: [[octoid72]], [[octoid80]]
[[Badness]] (Sintel): 2.29


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


Comma list: 540/539, 1375/1372, 4000/3993
Comma list: 540/539, 4375/4374, 2097152/2096325


Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}
Mapping: {{mapping| 1 -5 -4 -18 19 | 0 25 24 79 -59 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}}
* WE: ~2 = 1199.9944{{c}}, ~6/5 = 316.0690{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9477{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0705{{c}}


Tuning ranges:
{{Optimal ET sequence|legend=0| 19, 205, 224 }}
* 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| 8d, …, 72, 152, 224, 824d }}
Badness (Sintel): 2.35
 
Badness (Sintel): 0.466
 
Scales: [[octoid72]], [[octoid80]]


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


Comma list: 540/539, 625/624, 729/728, 1375/1372
Comma list: 540/539, 625/624, 729/728, 10985/10976


Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}
Mapping: {{mapping| 1 -5 -4 -18 19 -15 | 0 25 24 79 -59 71 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}
* WE: ~2 = 1199.9827{{c}}, ~6/5 = 316.0650{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0695{{c}}


{{Optimal ET sequence|legend=0| 72, 152f, 224 }}
{{Optimal ET sequence|legend=0| 19, 205, 224 }}


Badness (Sintel): 0.631
Badness (Sintel): 1.40


Scales: [[octoid72]], [[octoid80]]
=== Counterlytic ===
Subgroup: 2.3.5.7.11


; Music
Comma list: 1375/1372, 4375/4374, 496125/495616
* ''Dreyfus'' (archived 2010) by [[Gene Ward Smith]] – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – Octoid[72] in 224edo tuning


===== 17-limit =====
Mapping: {{mapping| 1 -5 -4 -18 -40 | 0 25 24 79 165 }}
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 375/374, 540/539, 625/624, 715/714, 729/728
 
Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}
* WE: ~2 = 1200.1247{{c}}, ~6/5 = 316.0976{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0660{{c}}


{{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}
{{Optimal ET sequence|legend=1| 19e, 205e, 224, 467e, 691, 915c }}


Badness (Sintel): 0.729
Badness (Sintel): 2.16


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


Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
Comma list: 625/624, 729/728, 1375/1372, 10985/10976


Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}
Mapping: {{mapping| 1 -5 -4 -18 -40 -15 | 0 25 24 79 165 71 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~12/11 = 149.9785{{c}}, ~7/5 = 583.8482{{c}}
* WE: ~2 = 1200.0987{{c}}, ~6/5 = 316.0908{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0658{{c}}


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


Badness (Sintel): 0.975
Badness (Sintel): 1.23


==== Octopus ====
== Sfourth ==
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}}.
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sfourth]].''


Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7


Comma list: 169/168, 325/324, 364/363, 540/539
[[Comma list]]: 4375/4374, 64827/64000


Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
: mapping generators: ~2, ~49/48


Optimal tunings:  
[[Optimal tuning]]s:  
* WE: ~12/11 = 150.0313{{c}}, ~7/5 = 584.0134{{c}}
* [[WE]]: ~2 = 1200.8332{{c}}, ~49/48 = 26.3053{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{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=0| 8d, …, 72, 152, 224f }}
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


Badness (Sintel): 0.896
[[Badness]] (Sintel): 3.12


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


Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
Comma list: 121/120, 441/440, 4375/4374


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


Optimal tunings:  
Optimal tunings:  
* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}
* WE: ~2 = 1201.1486{{c}}, ~49/48 = 26.3112{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2461{{c}}


{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}
{{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }}


Badness (Sintel): 0.795
Badness (Sintel): 1.78


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


Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
Comma list: 121/120, 169/168, 325/324, 441/440


Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~12/11 = 150.0049{{c}}, ~7/5 = 584.0833{{c}}
* WE: ~2 = 1201.4956{{c}}, ~49/48 = 26.3423{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2614{{c}}


{{Optimal ET sequence|legend=0| 8d, 72, 152 }}
{{Optimal ET sequence|legend=0| 45ef, 46, 91ef, 137def, 228ddeeefff }}


Badness (Sintel): 0.993
Badness (Sintel): 1.37


Scales: [[Octoid72]], [[Octoid80]]
=== Sfour ===
Subgroup: 2.3.5.7.11


==== Hexadecoid ====
Comma list: 385/384, 2401/2376, 4375/4374
{{See also| 16th-octave temperaments }}
 
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 }}


Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.
Badness (Sintel): 2.53


==== 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: 196/195, 364/363, 385/384, 4375/4374


Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}
: mapping generators: ~448/429, ~7/5


Optimal tunings:  
Optimal tunings:  
* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}
* WE: ~2 = 1200.3796{{c}}, ~49/48 = 26.2473{{c}}
* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2372{{c}}


{{Optimal ET sequence|legend=0| 80, 144, 224 }}
{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}


Badness (Sintel): 1.27
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: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
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| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}
* 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 }}
[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}
: Mapping generators: ~135/128, ~3


Badness (Sintel): 1.46
[[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: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
[[Badness]] (Sintel): 3.20


Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Optimal tunings:  
Comma list: 4375/4374, 234375/234256, 2097152/2096325
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}


{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}
Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}


Badness (Sintel): 1.44
Optimal tunings:  
* WE: ~135/128 = 92.3062{{c}}, ~3/2 = 701.9946{{c}}
* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0056{{c}}


== Parakleismic ==
{{Optimal ET sequence|legend=0| 494, 1053, 1547, 3588e, 5135e }}
{{Main| Parakleismic }}
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic (5-limit)]].''


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.  
Badness (Sintel): 1.39


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).  
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Subgroup]]: 2.3.5.7
Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078


[[Comma list]]: 3136/3125, 4375/4374
Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}


{{Mapping|legend=1| 1 -8 -8 -23 | 0 13 14 35 }}
Optimal tunings:
: mapping generators: ~2, ~5/3
* WE: ~135/128 = 92.3055{{c}}, ~3/2 = 701.9928{{c}}
* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0098{{c}}


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=0| 494, 1547, 2041, 4576def }}
* [[WE]]: ~2 = 1199.7820{{c}}, ~5/3 = 884.6581{{c}}
: [[error map]]: {{val| -0.218 +0.344 +0.644 -0.779 }}
* [[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 }}
Badness (Sintel): 1.18


[[Badness]] (Sintel): 0.694
== Ragitritonic ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


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


Comma list: 385/384, 3136/3125, 4375/4374
Ragitritonic was named by [[Flora Canou]] in 2026 as a contraction of ''ragismic'' and ''tritonic''.


Mapping: {{mapping| 1 -8 -8 -23 30 | 0 13 14 35 -36 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 4375/4374, 68719476736/68356598625
* WE: ~2 = 1200.3296{{c}}, ~5/3 = 884.9921{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7519{{c}}


{{Optimal ET sequence|legend=0| 19, 99, 118 }}
{{Mapping|legend=1| 1 -3 -15 40 | 0 9 34 -73 }}
: mapping generators: ~2, ~65536/45927


Badness (Sintel): 1.64
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.8189{{c}}, ~65536/45927 = 611.2850{{c}}
=== Paralytic ===
: [[error map]]: {{val| -0.181 +0.153 +0.094 +0.123 }}
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.
* [[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 }}
 
[[Badness]] (Sintel): 3.37


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


Comma list: 441/440, 3136/3125, 4375/4374
Comma list: 4375/4374, 5632/5625, 2621440/2614689


Mapping: {{mapping| 1 -8 -8 -23 -57 | 0 13 14 35 82 }}
Mapping: {{mapping| 1 -3 -15 40 -75 | 0 9 34 -73 154 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9940{{c}}, ~5/3 = 884.7757{{c}}
* WE: ~2 = 1199.8147{{c}}, ~768/539 = 611.2822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7800{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~768/539 = 611.3762{{c}}


{{Optimal ET sequence|legend=0| 19e, …, 99e, 118, 217, 335 }}
{{Optimal ET sequence|legend=0| 53, 316e, 369, 422, 791e, 1213cde }}


Badness (Sintel): 1.19
Badness (Sintel): 2.34


==== 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: 2080/2079, 2200/2197, 4375/4374, 5632/5625


Mapping: {{mapping| 1 -8 -8 -23 -57 59 | 0 13 14 35 82 -75 }}
Mapping: {{mapping| 1 -3 -15 40 -75 -34 | 0 9 34 -73 154 74 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9218{{c}}, ~5/3 = 884.7285{{c}}
* WE: ~2 = 1199.7916{{c}}, ~91/64 = 611.2698{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7858{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~91/64 = 611.3754{{c}}


{{Optimal ET sequence|legend=0| 99e, 118, 217 }}
{{Optimal ET sequence|legend=0| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}


Badness (Sintel): 1.85
Badness (Sintel): 1.51


==== Paraklein ====
== Quatracot ==
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]].
{{See also| Stratosphere }}


Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7


Comma list: 196/195, 352/351, 625/624, 729/728
[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}


Mapping: {{mapping| 1 -8 -8 -23 -57 -28 | 0 13 14 35 82 43 }}
{{Mapping|legend=1| 2 -6 -1 -36 | 0 13 8 59 }}
: mapping generators: ~2278125/1605632, ~7168/5625


Optimal tunings:  
[[Optimal tuning]]s:  
* WE: ~2 = 1199.8239{{c}}, ~5/3 = 884.6449{{c}}
* [[WE]]: ~2278125/1605632 = 600.0888{{c}}, ~7168/5625 = 423.2574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7709{{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=0| 19e, …, 99ef, 118 }}
{{Optimal ET sequence|legend=1| 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc }}


Badness (Sintel): 1.55
[[Badness]] (Sintel): 4.45


=== 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: 3025/3024, 4375/4374, 1265625/1261568


Mapping: {{mapping| 1 -8 -8 -23 -43 | 0 13 14 35 63 }}
Mapping: {{mapping| 2 -6 -1 -36 -22 | 0 13 8 59 41 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.1848{{c}}, ~5/3 = 884.3386{{c}}
* WE: ~99/70 = 600.0847{{c}}, ~225/176 = 423.2536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9158{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~225/176 = 423.1977{{c}}


{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179, 259cd }}
{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638, 1052c }}


Badness (Sintel): 1.85
Badness (Sintel): 1.36


==== 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: 625/624, 729/728, 1575/1573, 2200/2197


Mapping: {{mapping| 1 -8 -8 -23 -43 -14 | 0 13 14 35 63 24 }}
Mapping: {{mapping| 2 -6 -1 -36 -22 -6 | 0 13 8 59 41 19 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.5318{{c}}, ~5/3 = 884.5800{{c}}
* WE: ~99/70 = 600.0571{{c}}, ~143/112 = 423.2366{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9118{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~143/112 = 423.1987{{c}}


{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179 }}
{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638 }}


Badness (Sintel): 1.51
Badness (Sintel): 0.936


=== Paradigmic ===
== Trideci ==
Subgroup: 2.3.5.7.11
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tridecatonic]].''


Comma list: 540/539, 896/891, 3136/3125
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").


Mapping: {{mapping| 1 -8 -8 -23 16 | 0 13 14 35 -17 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 4375/4374, 83349/81920
* WE: ~2 = 1199.0616{{c}}, ~5/3 = 884.2124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.8877{{c}}


{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e, 457bcddeeee }}
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
: mapping generators: ~256/245, ~3


Badness (Sintel): 1.38
[[Optimal tuning]]s:  
 
* [[WE]]: ~256/245 = 92.4141{{c}}, ~3/2 = 699.9466{{c}}
==== 13-limit ====
: [[error map]]: {{val| +1.383 -0.626 -0.210 -2.554 }}
Subgroup: 2.3.5.7.11.13
* [[CWE]]: ~256/245 = 92.3077{{c}}, ~3/2 = 699.4521{{c}}
: error map: {{val| 0.000 -2.503 -2.794 -6.740 }}


Comma list: 169/168, 325/324, 540/539, 832/825
{{Optimal ET sequence|legend=1| 26, 65, 91 }}


Mapping: {{mapping| 1 -8 -8 -23 16 -14 | 0 13 14 35 -17 24 }}
[[Badness]] (Sintel): 4.67


Optimal tunings:
=== 11-limit ===
* WE: ~2 = 1199.2683{{c}}, ~5/3 = 884.3805{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9061{{c}}
 
{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e }}
 
Badness (Sintel): 1.48
 
=== Semiparakleismic ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 3136/3125, 4375/4374
Comma list: 245/242, 385/384, 4375/4374


Mapping: {{mapping| 2 -3 -2 -11 -4 | 0 13 14 35 23 }}
Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}
: mapping generators: ~99/70, ~33/28


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 599.9270{{c}}, ~33/28 = 284.7841{{c}}
* WE: ~22/21 = 92.3729{{c}}, ~3/2 = 700.1118{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8119{{c}}
* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.7703{{c}}


{{Optimal ET sequence|legend=0| 80, 118, 198, 316, 514c }}
{{Optimal ET sequence|legend=0| 26, 65, 91 }}


Badness (Sintel): 1.13
Badness (Sintel): 2.80
 
==== Semiparamint ====
This extension was named ''semiparakleismic'' in the earlier materials.  


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


Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
Comma list: 169/168, 245/242, 325/324, 385/384


Mapping: {{mapping| 2 -3 -2 -11 -4 15 | 0 13 14 35 23 -16 }}
Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 599.8253{{c}}, ~33/28 = 284.7608{{c}}
* WE: ~22/21 = 92.4003{{c}}, ~3/2 = 699.9983{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8366{{c}}
* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.4772{{c}}


{{Optimal ET sequence|legend=0| 80, 118, 198 }}
{{Optimal ET sequence|legend=0| 26, 65f, 91f }}


Badness (Sintel): 1.40
Badness (Sintel): 2.16


==== Semiparawolf ====
== Moulin ==
This extension was named ''gentsemiparakleismic'' in the earlier materials.  
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: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7


Comma list: 169/168, 325/324, 364/363, 3136/3125
[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}


Mapping: {{mapping| 2 -3 -2 -11 -4 -4 | 0 13 14 35 23 24 }}
{{Mapping|legend=1| 1 -16 -9 -75 | 0 73 47 323 }}
: mapping generators: ~2, ~3796875/3211264


Optimal tunings:  
[[Optimal tuning]]s:  
* WE: ~99/70 = 600.0569{{c}}, ~13/11 = 284.8431{{c}}
* [[WE]]: ~2 = 1200.0272{{c}}, ~3796875/3211264 = 289.0675{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~13/11 = 284.8216{{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| 80, 118f, 198f }}
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 8589cc, 10208cc }}


Badness (Sintel): 1.67
[[Badness]] (Sintel): 5.93


== Counterkleismic ==
=== 11-limit ===
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''
Subgroup: 2.3.5.7.11


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)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament (''counterkleismic'', named by analogy to [[catakleismic]] and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).
Comma list: 4375/4374, 759375/758912, 100663296/100656875


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 1 -16 -9 -75 9 | 0 73 47 323 -23 }}


[[Comma list]]: 4375/4374, 158203125/157351936
Optimal tunings:  
* WE: ~2 = 1200.0043{{c}}, ~605/512 = 289.0687{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~605/512 = 289.0677{{c}}


{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }}
{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}
: Mapping generators: ~2, ~5/3


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 316.060{{c}}
Badness (Sintel): 2.24


{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Badness]] (Sintel): 2.292
Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078


=== 11-limit ===
Mapping: {{mapping| 1 -16 -9 -75 9 9 | 0 73 47 323 -23 -22 }}
Subgroup: 2.3.5.7.11


Comma list: 540/539, 4375/4374, 2097152/2096325
Optimal tunings:  
* WE: ~2 = 1200.0043{{c}}, ~13/11 = 289.0687{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 289.0677{{c}}


Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }}
{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.071{{c}}
Badness (Sintel): 1.12


{{Optimal ET sequence|legend=0| 19, 205, 224 }}
== Palladium ==
: ''For the 5-limit version, see [[46th-octave temperaments #Palladium]]''.


Badness (Sintel): 2.346
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.


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


Comma list: 540/539, 625/624, 729/728, 10985/10976
[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}


Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }}
{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
: mapping generators: ~83349/81920, ~3


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.070{{c}}
[[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 }}


{{Optimal ET sequence|legend=0| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}
{{Optimal ET sequence|legend=1| 46, …, 368, 414, 460, 874d }}


Badness (Sintel): 1.400
[[Badness]] (Sintel): 7.81


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


Comma list: 1375/1372, 4375/4374, 496125/495616
Comma list: 3025/3024, 4375/4374, 134775333/134217728


Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }}
Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.065{{c}}
Optimal tunings:  
* WE: ~8192/8085 = 26.0912{{c}}, ~3/2 = 701.7082{{c}}
* CWE: ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.6173{{c}}


{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}
{{Optimal ET sequence|legend=0| 46, …, 368, 414, 460, 874de }}


Badness (Sintel): 2.162
Badness (Sintel): 2.44


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


Comma list: 625/624, 729/728, 1375/1372, 10985/10976
Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364


Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }}
Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.065{{c}}
Optimal tunings:  
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7411{{c}}
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6465{{c}}


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


Badness (Sintel): 1.231
Badness (Sintel): 1.68


== Quincy ==
=== 17-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 4375/4374, 823543/819200
Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224


{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~1728/1715 = 16.613{{c}}
Optimal tunings:  
 
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7399{{c}}
{{Optimal ET sequence|legend=1| 72, 217, 289 }}
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6464{{c}}
 
[[Badness]] (Sintel): 2.016
 
=== 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 tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.613{{c}}
 
{{Optimal ET sequence|legend=0| 72, 217, 289 }}
 
Badness (Sintel): 1.021
 
=== 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 tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.602{{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 tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.602{{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 tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.594{{c}}
 
{{Optimal ET sequence|legend=0| 72, 145, 217 }}
 
Badness (Sintel): 0.924
 
== Sfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 64827/64000
 
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/48 = 26.287{{c}}
 
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
 
[[Badness]] (Sintel): 3.120
 
=== 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 tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.286{{c}}
 
{{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }}
 
Badness (Sintel): 1.788
 
==== 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 tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.310{{c}}
 
{{Optimal ET sequence|legend=0| 45ef, 46, 91ef, 137def }}
 
Badness (Sintel): 1.366
 
=== 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 tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.246{{c}}
 
{{Optimal ET sequence|legend=0| 45, 46, 91, 137d }}
 
Badness (Sintel): 2.531
 
==== 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 tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.239{{c}}
 
{{Optimal ET sequence|legend=0| 45, 46, 91, 137d }}
 
Badness (Sintel): 2.144
 
== Trideci ==
: ''For the 5-limit version of this temperament, 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 "[[wikipedia:13|thirteen]]").
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 83349/81920
 
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
 
[[Optimal tuning]] ([[POTE]]): ~256/245 = 92.3077{{c}}, ~3/2 = 699.1410{{c}}
 
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }}
 
[[Badness]] (Sintel): 4.671
 
=== 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 ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334ddeg }}


Optimal tuning (POTE): ~22/21 = 92.3077{{c}}, ~3/2 = 699.6179{{c}}
Badness (Sintel): 1.14
 
{{Optimal ET sequence|legend=0| 26, 65, 91, 156d, 247cdde }}
 
Badness (Sintel): 2.796
 
=== 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 tuning (POTE): ~22/21 = 92.3077{{c}}, ~3/2 = 699.2969{{c}}
 
{{Optimal ET sequence|legend=0| 26, 65f, 91f, 156dff }}
 
Badness (Sintel): 2.164
 
== Counterorson ==
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
 
[[Comma list]]: 4375/4374, {{monzo| 154 -54 -21 -7 }}
 
{{Mapping|legend=1| 1 0 -21 85 | 0 7 103 -363 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7113{{c}}
 
{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}
 
[[Badness]] (Sintel): 7.916


== References ==
== 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"