Ragismic microtemperaments: Difference between revisions

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Since {{nowrap|(10/9)4 {{=}} (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)2 }}, 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]]

* ''[[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]]

* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt 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]]

* ''[[~~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]]

* [[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]]

* ''[[~~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]]

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 ==

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== 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}}).

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: ''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

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[[Comma list]]: 4375/4374, 589824/588245

: mapping generators: ~2, ~8/7

: mapping generators: ~2, ~7/4

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~8/7 = ~~230~~.~~336~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.8483{{c}}, ~7/4 = 969.5415{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 969.6608{{c}}

{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}

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Comma list: 3025/3024, 4375/4374, 589824/588245

Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}

Mapping: {{mapping| 2 -11 -20 5 10 | 0 23 40 1 -5 }}

: mapping generators: ~99/70, ~8/7

: mapping generators: ~99/70, ~99/80

Optimal ~~tuning (POTE)~~: ~99/70 = 600.0000{{c}}, ~8/7 = ~~230~~.~~3370~~{{c}}

Optimal tunings:

* WE: ~99/70 = 599.9323{{c}}, ~99/80 = 369.6212{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~99/80 = 369.6610{{c}}

{{Optimal ET sequence|legend=0| 26, 172c, 198, 224, 422, 646, 1068d }}

Badness (Sintel): 1.~~354~~

Badness (Sintel): 1.35

==== 13-limit ====

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Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024

Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}

Mapping: {{mapping| 2 -11 -20 5 10 -8 | 0 23 40 1 -5 25 }}

Optimal ~~tuning (POTE)~~: ~99/70 = 600.0000{{c}}, ~8/7 = ~~230~~.~~3373~~{{c}}

Optimal tunings:

* WE: ~99/70 = 599.9207{{c}}, ~26/21 = 369.6139{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~26/21 = 369.6603{{c}}

{{Optimal ET sequence|legend=0| 26, 198, 224, 422, 646f, 1068df }}

{{Optimal ET sequence|legend=0| 26, 172cf, 198, 224, 422, 646f, 1068df }}

Badness (Sintel): 0.844

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Comma list: 4375/4374, 14641/14580, 15488/15435

Mapping: {{mapping| 1 ~~6 10 3 12~~ | 0 -46 -80 -2 -89 }}

Mapping: {{mapping| 1 -40 -70 1 -77 | 0 46 80 2 89 }}

: mapping generators: ~2, ~77~~/72~~

: mapping generators: ~2, ~144/77

Optimal ~~tuning (POTE)~~: ~2 = 1200.0000{{c}}, ~77~~/72~~ = ~~115~~.~~1642~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.8845{{c}}, ~144/77 = 1084.7314{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8345{{c}}

Badness (Sintel): 2.~~589~~

Badness (Sintel): 2.59

==== 13-limit ====

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Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580

Mapping: {{mapping| 1 ~~6 10 3 12 18~~ | 0 -46 -80 -2 -89 -149 }}

Mapping: {{mapping| 1 -40 -70 1 -77 -131 | 0 46 80 2 89 149 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.0000{{c}}, ~77~~/72~~ = ~~115~~.~~1628~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.8726{{c}}, ~144/77 = 1084.7220{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8359{{c}}

Badness (Sintel): 1.~~821~~

Badness (Sintel): 1.82

== Crazy ==

: ''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

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: ~~Mapping~~ generators: ~332150625/234881024, ~1125/1024

: mapping generators: ~332150625/234881024, ~1125/1024

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~332150625/234881024 = 600.~~0000~~{{c}}, ~1125/1024 = 162.~~7475~~{{c}}

* [[WE]]: ~332150625/234881024 = 600.0019{{c}}, ~1125/1024 = 162.7479{{c}}

* [[error map]]: {{val| 0.~~0000~~ +0.~~0253~~ -0.~~0514~~ -0.~~0133~~ }}

: [[error map]]: {{val| +0.004 +0.030 -0.042 -0.014 }}

* [[CWE]]: ~332150625/234881024 = 600.0000{{c}}, ~1125/1024 = 162.7474{{c}}

* error map: {{val| 0.~~0000~~ +0.~~0244~~ -0.~~0508~~ -0.~~0218~~ }}

: error map: {{val| 0.000 +0.024 -0.051 -0.022 }}

{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}

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Optimal tunings:

* ~~CTE~~: ~99/70 = ~~162~~.~~7485~~{{c}}, ~1125/1024 = 162.~~7485~~{{c}}

* WE: ~99/70 = 600.0047{{c}}, ~1125/1024 = 162.7493{{c}}

* CWE: ~99/70 = ~~162~~.~~7485~~{{c}}, ~1125/1024 = 162.7481{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~1125/1024 = 162.7481{{c}}

{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}

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== Orga ==

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, ~~54975581388800/54936068900769~~

[[Comma list]]: 4375/4374, {{monzo| 41 -4 2 -14 }}

: ~~Mapping~~ generators: ~7411887/5242880, ~~~1310720~~/~~1058841~~

: mapping generators: ~7411887/5242880, ~8/7

[[Optimal tuning]] ([[~~POTE~~]]): ~7411887/5242880 = 600.~~000~~{{c}}, ~8/7 = 231.~~104~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~7411887/5242880 = 599.9927{{c}}, ~8/7 = 231.1012{{c}}

: [[error map]]: {{val| -0.015 +0.037 -0.045 +0.029 }}

* [[CWE]]: ~7411887/5242880 = 600.0000{{c}}, ~8/7 = 231.1037{{c}}

: error map: {{val| 0.000 +0.053 -0.023 +0.070 }}

{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}

{{Optimal ET sequence|legend=1| 26, …, 244, 270, 836, 1106, 1376, 2482 }}

[[Badness]] (Sintel): 1.~~018~~

[[Badness]] (Sintel): 1.02

=== 11-limit ===

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Comma list: 3025/3024, 4375/4374, 5767168/5764801

Mapping: {{mapping| 2 ~~21 36 5 2~~ | 0 -29 -51 1 8 }}

Mapping: {{mapping| 2 -8 -15 6 10 | 0 29 51 -1 -8 }}

Optimal ~~tuning (POTE)~~: ~99/70 = 600.~~000~~{{c}}, ~8/7 = 231.~~103~~{{c}}

Optimal tunings:

* WE: ~99/70 = 600.0025{{c}}, ~8/7 = 231.1039{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1030{{c}}

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Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360

Mapping: {{mapping| 2 ~~21 36 5 2 24~~ | 0 -29 -51 1 8 -27 }}

Mapping: {{mapping| 2 -8 -15 6 10 -3 | 0 29 51 -1 -8 27 }}

Optimal ~~tuning (POTE)~~: ~99/70 = 600.~~000~~{{c}}, ~8/7 = 231.~~103~~{{c}}

Optimal tunings:

* WE: ~99/70 = 600.0192{{c}}, ~8/7 = 231.1102{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1033{{c}}

{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836f, 1106f }}

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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 (26 & 145~~) tempers out the ~~wadisma~~, ~~201768035/201326592. 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

[[Comma list]]: 4375/4374, ~~201768035/201326592~~

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

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = ~~1200~~.~~000~~{{c}}, ~~~3087/2560~~ = ~~322~~.~~804~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~25/24 = 70.5880{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9962{{c}}

: [[error map]]: {{val| -0.004 +0.037 -0.030 -0.019 }}

* [[CWE]]: ~25/24 = 70.5882{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9990{{c}}

: error map: {{val| 0.000 +0.043 -0.021 -0.013 }}

{{Optimal ET sequence|legend=1| 26, ~~145~~, ~~171~~, ~~1513d~~, ~~1684d~~, ~~1855d~~, ~~2026d, 2197d, 2368d, 2539d, 2710d~~ }}

{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547, 3706, 5253 }}

[[Badness]] (Sintel): 1.~~136~~

[[Badness]] (Sintel): 1.05

~~=== Senator ===~~

The senator temperament (26 & 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It 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

Comma list: ~~441~~/~~440~~, ~~4375~~/~~4374~~, ~~65536~~/~~65219~~

Comma list: 4375/4374, 41503/41472, 1879453125/1879048192

Mapping: {{mapping| ~~1 11 19 2 4~~ | 0 -~~35 -62 3 -2~~ }}

Mapping: {{mapping| 17 0 26 -87 207 | 0 2 1 10 -11 }}

Optimal ~~tuning (POTE)~~: ~2 = ~~1200~~.~~000~~{{c}}, ~77/64 = ~~322~~.~~793~~{{c}}

Optimal tunings:

* WE: ~25/24 = 70.5905{{c}}, ~693/400 = 951.0054{{c}}

* CWE: ~25/24 = 70.5882{{c}}, ~693/400 = 950.9754{{c}}

{{Optimal ET sequence|legend=0| 26, ~~119c~~, ~~145~~, ~~171~~, ~~316e~~, ~~487ee~~ }}

{{Optimal ET sequence|legend=0| 289, 323, 612, 3349de, 3961de, …, 5797ddee }}

Badness (Sintel): 3.~~049~~

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~~ 11 ~~19 2 4 15 | 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 tuning (POTE)~~: ~2 ~~= 1200~~.~~000{{c}}, ~77/64 = 322~~.~~793{{c}}~~

[[Subgroup]]: 2.3.5.7

~~{{Optimal ET sequence|legend=0| 26~~, ~~119c, 145, 171, 316ef, 487eef }}~~

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

~~Badness (Sintel)~~: ~~1.845~~

: 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 ~~11 19 2 4 15 17~~ | ~~0 -35 -62 3 -2 -42 -48~~ }}

~~Optimal tuning~~ (~~POTE~~): ~~~2 = 1200~~.~~000{{c}}, ~77/64 = 322.793{{c}}~~

[[Badness]] (Sintel): 1.08

~~{{Optimal ET sequence|legend~~=~~0| 26, 119c, 145, 171, 316ef, 487eef }}~~

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~Badness (Sintel)~~: ~~1.353~~

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 (53 & 612) 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]]~~. ~~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~~ }}

~~{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}~~

Badness (Sintel): 0.466

~~[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~{{monzo| -27 11 3 1 }} = 249.0207{{c}}~~

~~{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}~~

[[Badness]] (Sintel): ~~1.179~~

~~=== Monzism ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 4375/4374, 41503/41472, 184549376/184528125~~

~~Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}~~

~~Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~231/200 = 249.0193{{c}}~~

~~{{Optimal ET sequence|legend=~~0~~| 53, 559, 612 }}~~

~~Badness (Sintel): 1~~.~~887~~

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: ~~2200~~/~~2197~~, ~~4096~~/~~4095~~, ~~4375~~/~~4374~~, ~~40656~~/~~40625~~

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

Mapping: {{mapping| 1 ~~2 10~~ -~~25 46 23~~ | 0 ~~-2 -37 134 -205 -93~~ }}

Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}

Optimal ~~tuning (POTE)~~: ~2 = ~~1200~~.0000{{c}}, ~~~231~~/~~200~~ = ~~249~~.~~0199~~{{c}}

Optimal tunings:

* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}

Badness (Sintel): 2.~~222~~

Badness (Sintel): 0.631

== ~~Semidimfourth~~ ==

===== 17-limit =====

: ~~''For the~~ 5~~-limit version, see [[Miscellaneous 5-limit temperaments #Semidimfourth]]~~.''

Subgroup: 2.3.5.7.11.13.17

~~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: 375/374, 540/539, 625/624, 715/714, 729/728

~~[[Subgroup]]~~: 2.3.5.7

Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}

~~[[Comma list]]~~: ~~4375~~/~~4374~~, ~~235298~~/~~234375~~

Optimal tunings:

* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}

~~[[Mapping]]:~~ {{~~mapping~~| ~~1 21 28 36~~ | ~~0 -31 -41 -53~~ }}

~~[[Optimal tuning]]~~ (~~[[POTE]]~~): ~~~2 = 1200~~.~~000{{c}}, ~35/27 = 448.456{{c}}~~

Badness (Sintel): 0.729

~~{{Optimal ET sequence|legend~~=~~1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}~~

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

~~[[Badness]] (Sintel)~~: ~~1.398~~

Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714

~~=== Neusec ===~~

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

~~Subgroup~~: 2.3.5~~.7.11~~

~~Comma list~~: ~~3025~~/~~3024~~, ~~4375~~/~~4374~~, ~~235298~~/~~234375~~

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:~~ {{~~mapping~~| ~~2 11 15 19 15~~ | ~~0 -31 -41 -53 -32~~ }}

~~Optimal tuning~~ (~~POTE~~): ~~~99/70 = 600~~.~~000{{c}}, ~12/11 = 151.547{{c}}~~

Badness (Sintel): 0.975

==== 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): 1.955~~

~~==== 13-limit ====~~

Subgroup: 2.3.5.7.11.13

Comma list: ~~847~~/~~845~~, ~~1001~~/~~1000~~, ~~3025~~/~~3024~~, ~~4375~~/~~4374~~

Comma list: 169/168, 325/324, 364/363, 540/539

Mapping: {{mapping| ~~2 11 15 19 15 17~~ | 0 ~~-31 -41 -53 -32 -38~~ }}

Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}

Optimal ~~tuning (POTE)~~: ~99/70 = ~~1200~~.~~000~~{{c}}, ~12/11 = ~~151~~.~~545~~{{c}}

Optimal tunings:

* WE: ~12/11 = 150.0313{{c}}, ~7/5 = 584.0134{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}}

Badness (Sintel): 1.~~279~~

Badness (Sintel): 0.896

== ~~Acrokleismic~~ ==

===== 17-limit =====

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.7.11.13.17

[[Comma list]]: ~~4375~~/~~4374~~, ~~2202927104~~/~~2197265625~~

Comma list: 169/168, 221/220, 289/288, 325/324, 540/539

{{~~Mapping~~|~~legend=~~1~~| 1 10 11 27~~ | 0 ~~-32 -33 -92~~ }}

Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}

~~: mapping generators: ~2, ~6/5~~

[[Optimal ~~tuning]] ([[POTE]])~~: ~2 = ~~1200~~.~~000~~{{c}}, ~6/5 = ~~315~~.~~557~~{{c}}

Optimal tunings:

* 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=1| 19, …, ~~251~~, ~~270~~, ~~2449c~~, ~~2719c, 2989bc~~ }}

[[Badness]] (Sintel): 1.~~422~~

Badness (Sintel): 0.795

=== 11-limit ===

===== 19-limit =====

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11.13.17.19

Comma list: ~~4375~~/~~4374~~, ~~41503~~/~~41472~~, ~~172032~~/~~171875~~

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

Mapping: {{mapping| 1 ~~10 11 27 -~~16 | 0 ~~-32 -33 -92 74~~ }}

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1200~~.~~000~~{{c}}, ~6/5 = ~~315~~.~~558~~{{c}}

Optimal tunings:

* 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| 19, ~~251~~, ~~270, 829, 1099, 1369, 1639~~ }}

Badness (Sintel): 1.~~219~~

Badness (Sintel): 0.993

~~==== 13-limit ====~~

Scales: [[Octoid72]], [[Octoid80]]

~~Subgroup~~: ~~2.3.5.7.11.13~~

~~Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976~~

==== Hexadecoid ====

~~Mapping: {{mapping|~~ 1 ~~10 11 27 -~~16 ~~25 | 0 -32 -33 -92 74 -81 }}~~

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

~~Optimal tuning (POTE)~~: ~2 ~~= 1200~~.~~000{{c}}, ~6/~~5 ~~= 315~~.~~557{{c}}~~

Subgroup: 2.3.5.7.11.13

~~{{Optimal ET sequence|legend=0| 19~~, ~~251~~, ~~270 }}~~

Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224

~~Badness (Sintel)~~: 1~~.108~~

Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}

: mapping generators: ~448/429, ~7/5

==~~= Counteracro ===~~

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~~, ~~5632/5625~~, ~~117649/117612~~

~~Mapping~~: ~~{{mapping|~~ 1 ~~10 11~~ 27 ~~55 | 0 -32 -33 -92 -196 }}~~

Badness (Sintel): 1.27

~~Optimal tuning (POTE)~~: ~2 ~~= 1200~~.~~000{{c}}, ~6/~~5 ~~= 315~~.~~553{{c}}~~

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

~~{{Optimal ET sequence|legend=0| 19e~~, ~~251e~~, ~~270~~, ~~1061e~~, ~~1331c, 1601c, 1871bc, 4012bcde }}~~

Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224

~~Badness (Sintel)~~: 1~~.407~~

Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}

==~~== 13-limit ====~~

Optimal tunings:

~~Subgroup~~: ~~2.3~~.5.~~7.11.13~~

* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9626{{c}}

* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0463{{c}}

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

~~Mapping~~: ~~{{mapping|~~ 1 ~~10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }}~~

Badness (Sintel): 1.46

~~Optimal tuning (POTE)~~: ~2 ~~= 1200~~.~~000{{c}}, ~6/~~5 ~~= 315~~.~~554{{c}}~~

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

~~{{Optimal ET sequence|legend=0| 19e~~, ~~251e~~, ~~270~~, ~~1331c~~, ~~1601c~~, ~~1871bcf, 2141bcf }}~~

Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444

~~Badness (Sintel)~~: 1~~.076~~

Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}

~~== Quasithird ==~~

Optimal tunings:

~~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 [[4375~~/~~4374|ragisma]] and~~ {{~~monzo|-60 29 0 5~~}}.

* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}

* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}

~~[[Subgroup]]: 2.3.5~~

~~[[Comma list]]~~: ~~{{monzo| 55 -64 20 }}~~

Badness (Sintel): 1.44

~~{{Mapping|legend~~=~~1| 4 0 -11 | 0 5 16 }}~~

== Seniority ==

: ~~mapping generators: ~51200000/43046721~~, ~~~1594323/1280000~~

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

[[~~Optimal tuning~~]] ~~([[POTE]]): ~51200000~~/~~43046721 = 300.000~~{{c}}~~, ~1594323/1280000 = 380~~.~~395~~{{c}}

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.

~~{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }}~~

[[Subgroup]]: 2.3.5.7

[[~~Badness~~]] ~~(Sintel)~~: ~~2.335~~

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

=~~== 7~~-~~limit ===~~

~~[[Subgroup]]~~: 2~~.3.5.7~~

: mapping generators: ~2, ~5120/3087

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

[[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 }}

{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d }}

[[~~Optimal tuning~~]] (~~[[POTE]]~~): ~~~65536/55125 = 300.000{{c}}, ~5103/4096 = 380~~.~~388 {{c}}~~

[[Badness]] (Sintel): 1.14

~~{{Optimal ET sequence|legend~~=~~1| 60d, 164, 224, 388, 612, 1448~~, ~~2060 }}~~

=== 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.

~~[[Badness]] (Sintel): 1.564~~

~~=== 11-limit ===~~

Subgroup: 2.3.5.7.11

Comma list: ~~3025~~/~~3024~~, 4375/4374, ~~4296700485~~/~~4294967296~~

Comma list: 441/440, 4375/4374, 65536/65219

Mapping: {{mapping| ~~4 0~~ -~~11 48~~ 43 | 0 ~~5 16 -29~~ -23 }}

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

Optimal ~~tuning (POTE)~~: ~~~65536/51125~~ = ~~300~~.~~000~~{{c}}, ~~~5103~~/~~4096~~ = ~~380~~.~~387~~{{c}} ~~(or~~ ~22/21 = 80.~~387~~{{c}})

Optimal tunings:

* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}}

{{Optimal ET sequence|legend=0| ~~60d~~, ~~164~~, ~~224~~, ~~388~~, ~~612, 836, 1448~~ }}

Badness (Sintel): 0.~~698~~

Badness (Sintel): 3.05

=== 13-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

Mapping: {{mapping| ~~4 0~~ -~~11 48~~ 43 11 | 0 ~~5 16 -29~~ -23 3 }}

Mapping: {{mapping| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}

Optimal ~~tuning (POTE)~~: ~~~65536/51125~~ = ~~300~~.~~000~~{{c}}, ~81/65 = ~~380~~.~~385~~{{c}} ~~(or~~ ~22/21 = 80.~~385~~{{c}})

Optimal tunings:

* 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| ~~60d~~, ~~164~~, ~~224~~, ~~388~~, ~~612, 836, 1448f, 2284f~~ }}

Badness (Sintel): 1.~~219~~

Badness (Sintel): 1.85

== ~~Deca~~ ==

==== 17-limit ====

: ~~''For~~ 5~~-limit version, see [[10th-octave temperaments #Neon]]~~.''

Subgroup: 2.3.5.7.11.13.17

~~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 (satritrizo-asepbigu).~~

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~~, ~~165288374272~~/~~164794921875~~

Optimal tunings:

* WE: ~2 = 1199.7195{{c}}, ~108/65 = 877.0018{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}}

~~: mapping generators: ~15/14, ~6/5~~

~~[[Optimal tuning]]~~ (~~[[POTE]]~~): ~~~15/14 = 120~~.~~000{{c}}, ~6/5 = 315.577{{c}}~~

Badness (Sintel): 1.35

~~{{Optimal ET sequence|legend~~=~~1| 80, 190, 270, 1270, 1540, 1810, 2080 }}~~

== Monzismic ==

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

~~[[Badness]] (Sintel): 2.041~~

~~=== 11-limit~~ ===

~~Subgroup~~: ~~2.3.~~5~~.7.11~~

~~Comma list: 3025/3024~~, ~~4375/4374, 391314/390625~~

~~Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}~~

~~Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~6/5 = 315~~.~~582{{c}}~~

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

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.

~~Badness (Sintel): 0.804~~

~~=== 13~~-limit ~~===~~

~~Subgroup: 2.3.5.7.11.13~~

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

~~Mapping:~~ {{~~mapping| 10 4 9 2 18 37~~ | 0 ~~5 6 11 7 0 }}~~

~~Optimal tuning (POTE): ~15/14 = 120.000{{c~~}}, ~~~6/5 = 315.602{{c}} (~40/39 = 44.398{{c}})~~

~~{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}~~

~~Badness (Sintel): 0.695~~

~~=== 2.3.5.7.11.13.19 subgroup ===~~

~~Subgroup: 2.3.5.7.11.13.19~~

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

~~Mapping:~~ {{~~mapping~~| ~~10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}~~

~~Optimal tuning (CTE): ~15/14 = 120.000{{c~~}}, ~~~6/5 = 315~~.~~581{{c}} (~39/38 = 44.419{{c}})~~

~~{{Optimal~~ ET sequence~~|legend=0| 80, 190, 270, 730, 1000 }}~~

~~Badness (Sintel): 0.556~~

~~== Keenanose ==~~

~~Keenanose~~ is ~~named 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~~ }}

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

: mapping generators: ~2, ~{{monzo| 21 3 1 ~~-10~~ }}

: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1200.0000{{c}}, ~{{monzo| 21 3 1 ~~-10~~ }} = 4.~~4465~~{{c}}

[[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 }}

{{Optimal ET sequence|legend=1| ~~270~~, ~~1079~~, ~~1349~~, ~~1619~~, 1889, ~~2159~~, ~~4048~~, ~~18081cd~~ }}

{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd }}

[[Badness]] (Sintel): 2.~~172~~

[[Badness]] (Sintel): 1.18

=== ~~11-limit~~ ===

=== Monzism ===

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, ~~117649~~/~~117612~~, ~~67110351~~/~~67108864~~

Comma list: 4375/4374, 41503/41472, 184549376/184528125

Mapping: {{mapping| 1 ~~2 3 3 3 |~~ 0 -~~112~~ -~~183~~ -~~52 124~~ }}

Mapping: {{mapping| 1 0 -27 109 -159 | 0 2 37 -134 205 }}

Optimal ~~tuning (CTE)~~: ~2 = 1200.0000{{c}}, ~~~385~~/~~384~~ = 4.~~4465~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}}

{{Optimal ET sequence|legend=0| ~~270~~, ~~1349~~, ~~1619~~, ~~1889~~, ~~2159~~, ~~11065~~, ~~13224~~ }}

{{Optimal ET sequence|legend=0| 53, 559, 612, 3619de, 4231de, …, 6067ddee }}

Badness (Sintel): 1.~~020~~

Badness (Sintel): 1.89

=== 13-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: ~~4225~~/~~4224~~, 4375/4374, ~~6656/6655, 117649~~/~~117612~~

Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625

Mapping: {{mapping| 1 ~~2 3 3 3 3 |~~ 0 -~~112~~ -~~183~~ -~~52 124 189~~ }}

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

Optimal ~~tuning (CTE)~~: ~2 = 1200.0000{{c}}, ~~~385~~/~~384~~ = 4.~~4466~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}}

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

Badness (Sintel): 0.~~879~~

Badness (Sintel): 2.22

== ~~Aluminium~~ ==

== Semidimfourth ==

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

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

~~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~~.

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]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, ~~{{monzo| 92 -39 -13 }}~~

[[Comma list]]: 4375/4374, 235298/234375

~~[[Mapping]]:~~ {{~~mapping~~| 13 ~~0 92~~ -~~355~~ | 0 ~~1 -3 19~~ }}

: ~~Mapping~~ generators: ~~~135/128~~, ~3

: mapping generators: ~2, ~35/27

[[Optimal tuning]] ([[~~CTE~~]]): ~~~135~~/~~128~~ = 92.~~3077~~{{c}}, ~3/2 = ~~702~~.~~0024~~{{c}}

[[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 }}

{{Optimal ET sequence|legend=1| ~~494~~, ~~1053~~, ~~1547~~, ~~8788~~, ~~10335~~, ~~11882~~, ~~13429b, 14976b~~ }}

[[Badness]] (Sintel): 3.~~201~~

[[Badness]] (Sintel): 1.40

=== ~~11-limit~~ ===

=== Neusec ===

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 234375~~/234256, 2097152/2096325~~

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

Mapping: {{mapping| ~~13 0 92~~ -~~355 148~~ | 0 ~~1 -3 19 -5~~ }}

Mapping: {{mapping| 2 -20 -26 -34 -17 | 0 31 41 53 32 }}

: mapping generators: ~99/70, ~35/27

Optimal ~~tuning (CTE)~~: ~~~135~~/~~128~~ = 92.~~3077~~{{c}}, ~3/2 = ~~702~~.~~0042~~{{c}}

Optimal tunings:

* WE: ~99/70 = 600.0381{{c}}, ~35/27 = 448.4812{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4546{{c}}

{{Optimal ET sequence|legend=0| ~~494~~, ~~1053~~, ~~1547~~, ~~3588e, 5135e~~ }}

Badness (Sintel): 1.~~393~~

Badness (Sintel): 1.95

=== 13-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: ~~4096~~/~~4095~~, ~~4375~~/~~4374~~, ~~6656~~/~~6655~~, ~~78125~~/~~78078~~

Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374

Mapping: {{mapping| ~~13 0 92~~ -~~355 148 419 | 0 1~~ -~~3 19~~ -5 -18 }}

Mapping: {{mapping| 2 -20 -26 -34 -17 -21 | 0 31 41 53 32 38 }}

Optimal ~~tuning (CTE)~~: ~~~135~~/~~128~~ = 92.~~3077~~{{c}}, ~3/2 = ~~702~~.~~0099~~{{c}}

Optimal tunings:

* WE: ~99/70 = 600.0034{{c}}, ~35/27 = 448.4573{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4549{{c}}

{{Optimal ET sequence|legend=0| ~~494~~, ~~1547~~, ~~2041~~, ~~4576def~~ }}

Badness (Sintel): 1.~~180~~

Badness (Sintel): 1.28

~~== Countritonic ==~~

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

~~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.~~

== Acrokleismic ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, ~~68719476736~~/~~68356598625~~

[[Comma list]]: 4375/4374, 2202927104/2197265625

: mapping generators: ~2, ~~~45927~~/~~32768~~

: mapping generators: ~2, ~5/3

[[Optimal tuning]] ~~(CTE)~~: ~2 = 1200.0000{{c}}, ~~~45927~~/~~32768~~ = ~~588~~.~~6216~~{{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=1| 53, ~~210d~~, ~~263~~, ~~316~~, ~~369~~, ~~422~~, ~~791, 1213cd, 2004bcdd~~ }}

{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}

[[Badness]] (Sintel): 3.~~370~~

[[Badness]] (Sintel): 1.42

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, ~~5632~~/~~5625~~, ~~2621440~~/~~2614689~~

Comma list: 4375/4374, 41503/41472, 172032/171875

Mapping: {{mapping| 1 ~~6 19~~ -~~13 79~~ | 0 -~~9 -34 73 154~~ }}

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

Optimal ~~tuning (CTE)~~: ~2 = 1200.0000{{c}}, ~~~539~~/~~384~~ = ~~588~~.~~6258~~{{c}}

Optimal tunings:

* 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| 53, ~~316e~~, ~~369~~, ~~422~~, ~~791e~~, ~~1213cde~~ }}

{{Optimal ET sequence|legend=0| 19, 251, 270, 829, 1099, 1369, 1639 }}

Badness (Sintel): 2.~~336~~

Badness (Sintel): 1.22

=== 13-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11.13

Comma list: ~~2080~~/~~2079~~, ~~2200~~/~~2197~~, 4375/4374, ~~5632~~/~~5625~~

Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976

Mapping: {{mapping| 1 ~~6 19~~ -~~13 79~~ | 0 ~~-9 -34 73 154~~ -74 }}

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

Optimal ~~tuning (CTE)~~: ~2 = 1200.0000{{c}}, ~~~128~~/91 = ~~588~~.~~6277~~{{c}}

Optimal tunings:

* 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| 53, ~~316ef~~, ~~369f, 422, 1213cdeff, 1635bcdefff~~ }}

Badness (Sintel): 1.~~514~~

Badness (Sintel): 1.11

== ~~Quatracot~~ ==

=== Counteracro ===

~~{{See also| Stratosphere }}~~

~~[[Subgroup]]: 2.3.5.7~~

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

~~{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }}~~

~~: mapping generators: ~2278125/1605632, ~448/405~~

~~[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 600.000{{c}}, ~448/405 = 176.805{{c}}~~

~~{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}~~

~~[[Badness]] (Sintel): 4.454~~

~~=== 11-limit~~ ===

Subgroup: 2.3.5.7.11

Comma list: ~~3025~~/~~3024~~, ~~4375~~/~~4374~~, ~~1265625~~/~~1261568~~

Comma list: 4375/4374, 5632/5625, 117649/117612

Mapping: {{mapping| ~~2 7 7 23 19 | 0~~ -13 -8 -59 -41 }}

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

Optimal ~~tuning (POTE)~~: ~99/70 = ~~600~~.~~000~~{{c}}, ~~~448~~/~~405~~ = ~~176~~.~~806~~{{c}}

Optimal tunings:

* 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| ~~190~~, ~~224~~, ~~414~~, ~~638~~, ~~1052c~~ }}

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

Badness (Sintel): 1.~~357~~

Badness (Sintel): 1.41

=== 13-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: ~~625~~/~~624~~, ~~729~~/~~728~~, ~~1575~~/~~1573~~, ~~2200~~/~~2197~~

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

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

~~Mapping~~: {{~~mapping|~~ 2 ~~7 7 23 19 13 | 0 -13 -8 -59 -41 -19~~ }}

Optimal tunings:

* WE: ~2 = 1199.9285{{c}}, ~5/3 = 884.3937{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4458{{c}}

Optimal ~~tuning (POTE): ~99/70~~ = ~~600.000{{c}}~~, ~~~195/176 = 176.804{{c~~}}

~~{{Optimal ET sequence|legend=0| 190, 224, 414, 638, 1690bcc, 2328bccde }}~~

Badness (Sintel): 1.08

~~Badness (Sintel)~~: 0.~~936~~

== Quasithird ==

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

~~== Moulin ==~~

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.

~~Moulin 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". It can~~ be described as the {{nowrap| ~~494~~ & ~~1619~~ }} temperament~~. Since 11~~/~~8 is within 23 generators, the 25-tone mos~~ (~~4L 21s~~) of ~~this~~ temperament ~~contains~~ the ~~8:11:13 triad~~.

[[Subgroup]]: 2.3.5.7

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

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

: ~~Mapping~~ generators: ~2, ~~~6422528~~/~~3796875~~

: mapping generators: ~65536/55125, ~5103/4096

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = ~~1200~~.0000{{c}}, ~~~6422528~~/~~3796875~~ = ~~910~~.~~9323~~{{c}}

[[Optimal tuning]]s:

* [[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 }}

[[Badness]] (Sintel): 5.~~931~~

[[Badness]] (Sintel): 1.56

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, ~~759375/758912, 100663296~~/~~100656875~~

Comma list: 3025/3024, 4375/4374, 4296700485/4294967296

Mapping: {{mapping| ~~1 57 38 248~~ -14 | 0 -~~73 -47~~ -~~323~~ 23 }}

Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}

Optimal ~~tuning~~ (~~CTE~~): ~2 = ~~1200~~.0000{{c}}, ~~~1024~~/~~605~~ = ~~910~~.~~9323~~{{c}}

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}})

{{Optimal ET sequence|legend=0| ~~494~~, ~~1125~~, ~~1619~~, ~~2113~~ }}

{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee }}

Badness (Sintel): 2.~~240~~

Badness (Sintel): 0.698

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: ~~4225~~/~~4224~~, ~~4375~~/~~4374~~, ~~6656~~/~~6655~~, ~~78125~~/~~78078~~

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

Mapping: {{mapping| ~~1 57 38 248~~ -~~14 -13~~ | 0 -~~73 -47~~ -~~323~~ 23 22 }}

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

Optimal ~~tuning~~ (~~CTE~~): ~2 = ~~1200~~.0000{{c}}, ~22/13 = ~~910~~.~~9323~~{{c}}

Optimal tunings:

* 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| ~~494~~, ~~1125~~, ~~1619~~, ~~2113~~ }}

Badness (Sintel): 1.~~118~~

Badness (Sintel): 1.22

~~== Palladium ==~~

~~: ''For the 5-limit version of this temperament, 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.

== Quincy ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, ~~{{monzo| -51 8 2 12 }}~~

[[Comma list]]: 4375/4374, 823543/819200

: ~~Mapping~~ generators: ~~~83349/81920~~, ~3

: mapping generators: ~2, ~1728/1715

[[Optimal tuning]] ([[~~POTE~~]]): ~~~83349~~/~~81920~~ = 26.~~0870~~{{c}}, ~3/2 = ~~701~~.~~6074~~{{c}}

[[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| 46, ~~368~~, ~~414~~, ~~460~~, ~~874d~~ }}

[[Badness]] (Sintel): 7.~~807~~

[[Badness]] (Sintel): 2.02

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: ~~3025~~/~~3024~~, 4375/4374~~, 134775333/134217728~~

Comma list: 441/440, 4000/3993, 4375/4374

Mapping: {{mapping| 46 0 -~~39 202 232 | 0 1 2~~ -1 -1 }}

Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}

Optimal ~~tuning (POTE)~~: ~~~8192~~/~~8085~~ = 26.~~0870~~{{c}}, ~3/2 = ~~701~~.~~5951~~{{c}}

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| 46, ~~368~~, ~~414, 460, 874de~~ }}

Badness (Sintel): 2.~~439~~

Badness (Sintel): 1.02

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: ~~3025~~/~~3024~~, ~~4225~~/~~4224~~, 4375/4374~~, 26411/26364~~

Comma list: 364/363, 441/440, 676/675, 4375/4374

Mapping: {{mapping| 46 0 -~~39 202 232 316 | 0 1 2~~ -1 -1 -2 }}

Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}

Optimal ~~tuning (POTE)~~: ~65/64 = 26.~~0870~~{{c}}, ~3/2 = ~~701~~.~~6419~~{{c}}

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| 46, ~~368~~, ~~414~~, ~~460, 874de, 1334de~~ }}

Badness (Sintel): 1.~~684~~

Badness (Sintel): 0.986

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

Comma list: ~~833~~/~~832~~, ~~1089~~/~~1088~~, ~~1225~~/~~1224~~, ~~1701~~/~~1700~~, ~~4225~~/~~4224~~

Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155

Mapping: {{mapping| 46 0 -39 ~~202 232 316 188 | 0 1 2~~ -1 -~~1 -2 0~~ }}

Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}

Optimal ~~tuning (POTE)~~: ~65/64 = 26.~~0870~~{{c}}, ~3/2 = ~~701~~.~~6425~~{{c}}

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| 46, ~~368~~, ~~414~~, ~~460, 874de, 1334deg~~ }}

Badness (Sintel): 1.~~143~~

Badness (Sintel): 0.751

== ~~Oviminor~~ ==

=== 19-limit ===

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

Subgroup: 2.3.5.7.11.13.17.19

~~Oviminor is named after the facts that it takes 184 minor thirds of 6~~/~~5 to reach 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.~~

Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675

~~[[Subgroup]]~~: 2.3.5.7

Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}

~~[[Comma list]]~~: ~~4375~~/~~4374~~, {{~~monzo| -100 53 48 -34~~ }}

Optimal tunings:

* WE: ~2 = 1199.9287{{c}}, ~100/99 = 16.5930{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.5948{{c}}

~~: Mapping generators: ~2, ~6/5~~

~~[[Optimal tuning]]~~ (~~[[CTE]]~~): ~~~2 = 1200~~.~~0000{{c}}, ~6/5 = 315.7501{{c}}~~

Badness (Sintel): 0.924

~~{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}~~

== Deca ==

: ''For 5-limit version, see [[10th-octave temperaments#Neon]].''

~~[[Badness]] (Sintel): 14.739~~

== ~~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]]~~)~~. In~~ the 11-limit~~, it tempers out 540/539, 1375/1372, and 6250/6237~~. ~~In this~~ temperament, ~~one period gives both 12/11~~ and ~~49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99~~.

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

[[Comma list]]: 4375/4374, ~~16875~~/~~16807~~

[[Comma list]]: 4375/4374, 165288374272/164794921875

: ~~Mapping~~ generators: ~49/45, ~7/5

: mapping generators: ~15/14, ~460992/390625

[[Optimal tuning]] ([[~~POTE~~]]): ~49/45 = ~~150~~.~~000~~{{c}}, ~7/5 = ~~583~~.~~940~~{{c}}

[[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 }}

~~[[Tuning ranges]]:~~

{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}

* 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]~~

~~{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}~~

[[Badness]] (Sintel): 2.04

[[Badness]] (Sintel): 1.~~080~~

~~Scales: [[octoid72]], [[octoid80]]~~

=== 11-limit ===

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 minimaxing the 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, if one wants to use 80edo as the tuning, one must use octopus – not octoid – as 80edo does not temper 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.

Subgroup: 2.3.5.7.11

Comma list: ~~540~~/~~539~~, ~~1375~~/~~1372~~, ~~4000~~/~~3993~~

Comma list: 3025/3024, 4375/4374, 391314/390625

Mapping: {{mapping| ~~8 1 3 3 16~~ | 0 ~~3 4~~ 5 3 }}

Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}

Optimal ~~tuning~~ (~~POTE~~): ~12/11 = ~~150~~.~~000~~{{c}}, ~7/5 = ~~583~~.~~962~~{{c}}

Optimal tunings:

* 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}})

~~Tuning ranges:~~

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

* 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~~| 72, 152, 224 }}~~

Badness (Sintel): 0.804

~~Badness (Sintel): 0.466~~

=== 13-limit ===

~~Scales: [[octoid72]], [[octoid80]]~~

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

Mapping: {{mapping| ~~8 1 3 3 16 -21~~ | 0 ~~3 4~~ 5 ~~3 13~~ }}

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

Optimal ~~tuning~~ (~~POTE~~): ~12/11 = ~~150~~.~~000~~{{c}}, ~7/5 = ~~583~~.~~905~~{{c}}

Optimal tunings:

* 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}})

Badness (Sintel): 0.~~631~~

Badness (Sintel): 0.695

~~Scales~~: ~~[[octoid72]], [[octoid80]]~~

=== 2.3.5.7.11.13.19 subgroup ===

Subgroup: 2.3.5.7.11.13.19

~~; Music~~

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

* ''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| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}

~~Subgroup~~: 2~~.3.~~5.7~~.11.13.17~~

~~Comma list~~: ~~375~~/~~374~~, ~~540~~/~~539, 625~~/~~624~~, ~~715~~/~~714, 729~~/~~728~~

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:~~ {{~~mapping~~| ~~8 1 3 3 16 -21 -14~~ | ~~0 3 4 5 3 13 12~~ }}

~~Optimal tuning~~ (~~POTE~~): ~~~12/11 = 150~~.~~000{{c}}, ~7/5 = 583.842{{c}}~~

Badness (Sintel): 0.556

{{~~Optimal ET sequence~~|~~legend=0| 72, 152fg~~, ~~224~~, ~~296~~, ~~520g }}~~

== 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.

~~Badness (Sintel)~~: 0.~~729~~

[[Subgroup]]: 2.3.5.7

~~===== 19~~-~~limit =====~~

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

~~Subgroup: 2.3.5.7.11.13.17.19~~

~~Comma list~~: ~~324/323, 375/374, 400/399, 495/494, 540/539~~, ~~715/714~~

: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}

~~Mapping~~: {{~~mapping~~| 8 1 ~~3 3 16~~ -21 -~~14 34~~ | 0 ~~3 4 5 3 13 12~~ 0 }}

[[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 ~~tuning (POTE): ~12/11~~ = ~~150.000{{c}}~~, ~~~7/5 = 583.932{{c~~}}

{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}

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

[[Badness]] (Sintel): 2.17

~~Badness (Sintel)~~: 0.~~975~~

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~==== Octopus ====~~

Comma list: 4375/4374, 117649/117612, 67110351/67108864

~~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{{cent}}.~~

~~Subgroup~~: 2.3~~.5.7.11.13~~

Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}

~~Comma list~~: ~~169/168~~, ~~325~~/~~324, 364/363~~, ~~540~~/~~539~~

Optimal tunings:

* WE: ~2 = 1199.9970{{c}}, ~385/384 = 4.4465{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4465{{c}}

~~Mapping:~~ {{~~mapping~~| ~~8 1 3 3 16 14~~ | ~~0 3 4 5 3 4~~ }}

{{Optimal ET sequence|legend=0| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}

~~Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.892{{c}}~~

Badness (Sintel): 1.02

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

~~Badness (Sintel): 0.0896~~

~~===== 17-limit =====~~

~~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 tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.811{{c}}~~

~~{{Optimal ET sequence|legend=0| 72, 152, 224fg, 296ffg }}~~

~~Badness (Sintel): 0.795~~

~~===== 19-limit =====~~

~~Subgroup: 2.3.5.7.11.13.17.19~~

~~Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399~~

~~Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}~~

~~Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 584.064{{c}}~~

~~{{Optimal ET sequence|legend=0| 72, 152, 224fg, 376ffgh }}~~

Badness (Sintel): ~~0.993~~

~~Scales: [[Octoid72]], [[Octoid80]]~~

~~==== Hexadecoid ====~~

~~{{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

Comma list: ~~540~~/~~539~~, ~~1375~~/~~1372~~, ~~4000~~/~~3993~~, ~~4225~~/~~4224~~

Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612

Mapping: {{mapping| 16 2 ~~6 6 32 67~~ | 0 ~~3 4 5 3~~ -1 }}

Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}

~~: mapping generators: ~448/429, ~7/5~~

Optimal ~~tuning (POTE)~~: ~~~448~~/~~429~~ = 75.~~000~~{{c}}, ~13/8 = ~~841~~.~~015~~{{c}}

Optimal tunings:

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

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

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

Badness (Sintel): 1.~~273~~

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: 540~~/~~539, 715~~/~~714, 936~~/~~935, 4000~~/~~3993~~, ~~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)6) 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| 16~~ 2 ~~6 6 32 67 81 | 0~~ 3 4 5 ~~3 -1 -2 }}~~

[[Subgroup]]: 2.3.5.7

~~Optimal tuning (POTE)~~: ~~~117~~/~~112 = 75.000{{c}}~~, ~~~13~~/~~8 = 840.932{{c}}~~

[[Comma list]]: 4375/4374, 158203125/157351936

: mapping generators: ~2, ~6/5

~~Badness (Sintel)~~: 1.~~458~~

[[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 }}

=~~==== 19-limit =====~~

~~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): 2.29

~~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 tuning (POTE)~~: ~~~117~~/~~112 = 75.000{{c}}~~, ~~~13~~/~~8 = 840.896{{c}}~~

Comma list: 540/539, 4375/4374, 2097152/2096325

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

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

~~Badness (Sintel)~~: 1.~~443~~

Optimal tunings:

* WE: ~2 = 1199.9944{{c}}, ~6/5 = 316.0690{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0705{{c}}

~~== 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. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, and 11-limit adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.

Badness (Sintel): 2.35

[[Subgroup]]: 2.3.5.7

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

[[Comma list]]: ~~3136~~/~~3125~~, ~~4375~~/~~4374~~

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

Mapping: {{mapping| 1 -5 -4 -18 19 -15 | 0 25 24 79 -59 71 }}

~~: mapping generators: ~2, ~6/5~~

[[Optimal ~~tuning]] ([[POTE]])~~: ~2 = 1200.~~000~~{{c}}, ~6/5 = ~~315~~.~~181~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9827{{c}}, ~6/5 = 316.0650{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0695{{c}}

[[Badness]] (Sintel): 0.~~694~~

Badness (Sintel): 1.40

=== ~~11-limit~~ ===

=== Counterlytic ===

Subgroup: 2.3.5.7.11

Comma list: ~~385~~/~~384~~, ~~3136~~/~~3125~~, ~~4375~~/~~4374~~

Comma list: 1375/1372, 4375/4374, 496125/495616

Mapping: {{mapping| 1 5 ~~6 12~~ -6 | 0 ~~-13 -14 -35 36~~ }}

Mapping: {{mapping| 1 -5 -4 -18 -40 | 0 25 24 79 165 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~6/5 = ~~315~~.~~251~~{{c}}

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): ~~1.643~~

Badness (Sintel): 2.16

~~=== Paralytic ===~~

~~The paralytic temperament (118 & 217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575.~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 441/440, 3136/3125, 4375/4374~~

~~Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }}~~

~~Optimal tuning (POTE): ~~~2 ~~= 1200.000{{c}}, ~6/5 = 315.220{{c}}~~

~~{{Optimal ET sequence|legend=0| 19e, 99e, 118, 217, 335, 552d, 887dd }}~~

~~Badness (Sintel): 1~~.~~191~~

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: ~~441~~/~~440~~, ~~1001~~/~~1000~~, ~~3136~~/~~3125~~, ~~4375~~/~~4374~~

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

Mapping: {{mapping| 1 5 ~~6 12 25 -16 | 0~~ -13 -14 -35 -~~82 75~~ }}

Mapping: {{mapping| 1 -5 -4 -18 -40 -15 | 0 25 24 79 165 71 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~6/5 = ~~315~~.~~214~~{{c}}

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| ~~99e~~, ~~118~~, ~~217~~, ~~552d~~, ~~769de~~ }}

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

Badness (Sintel): 1.~~847~~

Badness (Sintel): 1.23

==~~== Paraklein ==~~==

== Sfourth ==

~~The paraklein temperament (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 #Sfourth]].''

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, 64827/64000

~~Mapping:~~ {{~~mapping~~| 1 ~~5 6 12 25 15~~ | 0 -13 -14 -~~35 -82 -43~~ }}

: mapping generators: ~2, ~49/48

Optimal tuning ~~(POTE)~~: ~2 = 1200.~~000~~{{c}}, ~6/5 = ~~315~~.~~225~~{{c}}

[[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=0| ~~19e~~, ~~99ef~~, ~~118~~, ~~217ff, 335ff~~ }}

Badness (Sintel): 1.~~554~~

[[Badness]] (Sintel): 3.12

=== ~~Parkleismic~~ ===

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: ~~176~~/~~175~~, ~~1375~~/~~1372~~, ~~2200~~/~~2187~~

Comma list: 121/120, 441/440, 4375/4374

Mapping: {{mapping| 1 ~~5 6 12 20~~ | 0 -13 -14 -35 -63 }}

Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~6/5 = ~~315~~.~~060~~{{c}}

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| ~~19e~~, 80, ~~179~~, ~~259cd~~ }}

Badness (Sintel): 1.~~848~~

Badness (Sintel): 1.78

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 169/168~~, 176/175~~, 325/324, ~~1375~~/~~1372~~

Comma list: 121/120, 169/168, 325/324, 441/440

Mapping: {{mapping| 1 ~~5 6 12 20 10~~ | 0 -13 -14 -35 -63 -24 }}

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~6/5 = ~~315~~.~~075~~{{c}}

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.~~511~~

Badness (Sintel): 1.37

=== ~~Paradigmic~~ ===

=== Sfour ===

Subgroup: 2.3.5.7.11

Comma list: ~~540~~/~~539~~, ~~896~~/~~891~~, ~~3136~~/~~3125~~

Comma list: 385/384, 2401/2376, 4375/4374

Mapping: {{mapping| 1 ~~5 6 12 -1~~ | 0 -13 -14 -~~35 17~~ }}

Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~6/5 = ~~315~~.~~096~~{{c}}

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| 19, ~~61d~~, 80, ~~99e~~, ~~179e~~ }}

Badness (Sintel): 1.~~379~~

Badness (Sintel): 2.53

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: ~~169~~/~~168~~, ~~325~~/~~324~~, ~~540~~/~~539~~, ~~832~~/~~825~~

Comma list: 196/195, 364/363, 385/384, 4375/4374

Mapping: {{mapping| 1 ~~5 6 12 -1 10~~ | 0 -~~13 -14~~ -~~35 17~~ -24 }}

Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~6/5 = ~~315~~.~~080~~{{c}}

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| 19, ~~61d~~, 80, ~~99e~~, ~~179e~~ }}

Badness (Sintel): 1.~~479~~

Badness (Sintel): 2.14

==~~= Semiparakleismic =~~==

== Aluminium ==

~~Subgroup~~: ~~2.3.~~5.~~7.11~~

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

~~Comma list: 3025~~/~~3024, 3136~~/~~3125, 4375/4374~~

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|~~ 2 ~~10 12 24 19 | 0 -13 -14 -35 -23 }}~~

[[Subgroup]]: 2.3.5.7

~~Optimal tuning (POTE)~~: ~~~99~~/~~70 = 600.000{{c}}~~, ~~~6/5 = 315.181~~{{c}}

[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}

{{~~Optimal ET sequence~~|~~legend=~~0| ~~80, 118, 198, 316, 514c, 830c~~ }}

[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}

: Mapping generators: ~135/128, ~3

~~Badness (Sintel)~~: 1.~~131~~

[[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 }}

=~~=== Semiparamint ====~~

{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}

~~This extension was named ''semiparakleismic'' in the earlier materials.~~

~~Subgroup~~: 2.3.~~5.7.11.13~~

[[Badness]] (Sintel): 3.20

~~Comma list~~: ~~352/351, 1001/1000, 3025/3024, 4375/4374~~

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~Mapping~~: ~~{{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }}~~

Comma list: 4375/4374, 234375/234256, 2097152/2096325

~~Optimal tuning (POTE)~~: ~~~99/70 = 600.000~~{{~~c}}, ~6/~~5 ~~= 315.156{{c~~}}

Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}

Optimal tunings:

* WE: ~135/128 = 92.3062{{c}}, ~3/2 = 701.9946{{c}}

* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0056{{c}}

~~Badness (Sintel): 1.396~~

~~==== Semiparawolf ====~~

Badness (Sintel): 1.39

~~This extension was named ''gentsemiparakleismic'' in the earlier materials~~.

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: ~~169~~/~~168~~, ~~325~~/~~324~~, ~~364~~/~~363~~, ~~3136~~/~~3125~~

Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078

Mapping: {{mapping| ~~2 10 12 24 19 20~~ | 0 -~~13 -14 -35~~ -23 -24 }}

Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}

Optimal ~~tuning (POTE)~~: ~55/39 = ~~600~~.~~000~~{{c}}, ~6/5 = ~~315~~.~~184~~{{c}}

Optimal tunings:

* WE: ~135/128 = 92.3055{{c}}, ~3/2 = 701.9928{{c}}

* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0098{{c}}

Badness (Sintel): 1.~~672~~

Badness (Sintel): 1.18

== ~~Counterkleismic~~ ==

== Ragitritonic ==

: ''For the 5-limit ~~temperament~~, see [[~~Syntonic–kleismic~~ equivalence continuum #~~Counterhanson~~]].''

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

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.

~~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)~~.

Ragitritonic was named by [[Flora Canou]] in 2026 as a contraction of ''ragismic'' and ''tritonic''.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, ~~158203125~~/~~157351936~~

[[Comma list]]: 4375/4374, 68719476736/68356598625

: ~~Mapping~~ generators: ~2, ~5/3

: mapping generators: ~2, ~65536/45927

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~6/5 = ~~316~~.~~060~~{{c}}

[[Optimal tuning]]s:

* [[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 }}

{{Optimal ET sequence|legend=1| 19, ~~205~~, ~~224~~, ~~243~~, ~~467~~ }}

{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}

[[Badness]] (Sintel): 2.~~292~~

[[Badness]] (Sintel): 3.37

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: ~~540~~/~~539~~, ~~4375~~/~~4374~~, ~~2097152~~/~~2096325~~

Comma list: 4375/4374, 5632/5625, 2621440/2614689

Mapping: {{mapping| 1 ~~20 20 61~~ -40 | 0 -~~25 -24 -79 59~~ }}

Mapping: {{mapping| 1 -3 -15 40 -75 | 0 9 34 -73 154 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~6/5 = ~~316~~.~~071~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.8147{{c}}, ~768/539 = 611.2822{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~768/539 = 611.3762{{c}}

{{Optimal ET sequence|legend=0| 53, 316e, 369, 422, 791e, 1213cde }}

Badness (Sintel): 2.~~346~~

Badness (Sintel): 2.34

==== 13-limit ====

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625

Mapping: {{mapping| 1 ~~20 20 61~~ -40 56 | 0 -~~25 -24 -79 59 -71~~ }}

Mapping: {{mapping| 1 -3 -15 40 -75 -34 | 0 9 34 -73 154 74 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~6/5 = ~~316~~.~~070~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.7916{{c}}, ~91/64 = 611.2698{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~91/64 = 611.3754{{c}}

{{Optimal ET sequence|legend=0| 19, ~~205~~, ~~224~~, ~~1587cde~~, ~~1811ccdef~~, ~~2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef~~ }}

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

Badness (Sintel): 1.~~400~~

Badness (Sintel): 1.51

=== ~~Counterlytic~~ ===

== Quatracot ==

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 1375/1372, 4375/4374, 496125/495616~~

~~Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }}~~

~~Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.065{{c}}~~

~~{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}~~

~~Badness (Sintel): 2.162~~

~~==== 13-limit ====~~

~~Subgroup: 2.3.5.7.11.13~~

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

~~Mapping:~~ {{~~mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }}~~

~~Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.065{{c}}~~

~~{{Optimal ET sequence|legend=0~~| ~~19e, 205e, 224~~ }}

~~Badness (Sintel): 1.231~~

~~== Quincy ==~~

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, ~~823543/819200~~

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

: mapping generators: ~2278125/1605632, ~7168/5625

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = ~~1200~~.~~000~~{{c}}, ~~~1728~~/~~1715~~ = 16.~~613~~{{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 }}

[[Badness]] (Sintel): 2.~~016~~

[[Badness]] (Sintel): 4.45

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: ~~441~~/~~440~~, ~~4000~~/~~3993~~, ~~4375~~/~~4374~~

Comma list: 3025/3024, 4375/4374, 1265625/1261568

Mapping: {{mapping| 1 2 ~~3 3 4 | 0~~ -30 -49 -14 -39 }}

Mapping: {{mapping| 2 -6 -1 -36 -22 | 0 13 8 59 41 }}

Optimal ~~tuning (POTE)~~: ~2 = ~~1200~~.~~000~~{{c}}, ~~~100~~/99 = 16.~~613~~{{c}}

Optimal tunings:

* WE: ~99/70 = 600.0847{{c}}, ~225/176 = 423.2536{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~225/176 = 423.1977{{c}}

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

Badness (Sintel): 1.~~021~~

Badness (Sintel): 1.36

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: ~~364~~/~~363~~, ~~441~~/~~440~~, ~~676~~/~~675~~, ~~4375~~/~~4374~~

Comma list: 625/624, 729/728, 1575/1573, 2200/2197

Mapping: {{mapping| 1 2 ~~3 3 4 5 | 0~~ -30 -49 -14 -39 -94 }}

Mapping: {{mapping| 2 -6 -1 -36 -22 -6 | 0 13 8 59 41 19 }}

Optimal ~~tuning (POTE)~~: ~2 = ~~1200~~.~~000~~{{c}}, ~~~100~~/99 = 16.~~602~~{{c}}

Optimal tunings:

* WE: ~99/70 = 600.0571{{c}}, ~143/112 = 423.2366{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~143/112 = 423.1987{{c}}

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

Badness (Sintel): 0.~~986~~

Badness (Sintel): 0.936

==~~= 17-limit =~~==

== Trideci ==

~~Subgroup~~: ~~2.3.~~5.~~7.11.13.17~~

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tridecatonic]].''

~~Comma list: 364~~/~~363, 441~~/~~440, 595~~/~~594~~, ~~676~~/~~675, 1156~~/~~1155~~

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~~ 2 3 ~~3 4 5~~ 5 ~~| 0 -30 -49 -14 -39 -94 -66 }}~~

[[Subgroup]]: 2.3.5.7

~~Optimal tuning (POTE)~~: ~~~2 = 1200.000{{c}}~~, ~~~100~~/~~99 = 16.602{{c}}~~

[[Comma list]]: 4375/4374, 83349/81920

: mapping generators: ~256/245, ~3

~~Badness (Sintel)~~: 0.~~751~~

[[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 }}

=~~== 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~~

[[Badness]] (Sintel): 4.67

~~Mapping~~: ~~{{mapping| 1~~ 2 3 ~~3 4~~ 5 ~~5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}~~

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~Optimal tuning (POTE)~~: ~~~2 = 1200.000{{c}}~~, ~~~100~~/~~99 = 16.594{{c}}~~

Comma list: 245/242, 385/384, 4375/4374

Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}

~~Badness (Sintel)~~: 0.~~924~~

Optimal tunings:

* WE: ~22/21 = 92.3729{{c}}, ~3/2 = 700.1118{{c}}

* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.7703{{c}}

=~~= Sfourth ==~~

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

~~[[Subgroup]]~~: 2.~~3.5.7~~

Badness (Sintel): 2.80

~~[[Comma list]]~~: ~~4375/4374, 64827/64000~~

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

~~{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}~~

Comma list: 169/168, 245/242, 325/324, 385/384

~~[[Optimal tuning]] ([[POTE]])~~: ~~~2 = 1200.000~~{{~~c}}, ~49/~~48 ~~= 26.287{{c~~}}

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}}

~~[[Badness]] (Sintel): 3.120~~

~~=== 11-limit ===~~

Badness (Sintel): 2.16

~~Subgroup~~: 2.~~3.5.7.11~~

~~Comma list~~: ~~121~~/~~120~~, ~~441~~/~~440~~, ~~4375/4374~~

== 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.

~~Mapping~~: ~~{{mapping| 1~~ 2 3 ~~3 4 | 0 -19 -31 -9 -25 }}~~

[[Subgroup]]: 2.3.5.7

~~Optimal tuning (POTE)~~: ~~~2 = 1200.000{{c}}~~, ~~~49/48 = 26.286~~{{c}}

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

: mapping generators: ~2, ~3796875/3211264

~~Badness (Sintel)~~: ~~1.788~~

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0272{{c}}, ~3796875/3211264 = 289.0675{{c}}

==~~== 13~~-~~limit ==~~==

: [[error map]]: {{val| +0.027 +0.007 -0.084 +0.013 }}

~~Subgroup~~: 2.3.5.~~7.11~~.13

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3796875/3211264 = 289.0675{{c}}

: error map: {{val| 0.000 -0.029 -0.142 -0.029 }}

~~Comma list: 121/120~~, ~~169/168~~, ~~325/324~~, ~~441/440~~

{{Optimal ET sequence|legend=1| 494, 1125, 1619, 8589cc, 10208cc }}

~~Mapping~~: ~~{{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}~~

[[Badness]] (Sintel): 5.93

~~Optimal tuning (POTE): ~2~~ = ~~1200.000{{c}}, ~49/48~~ = ~~26.310{{c}}~~

=== 11-limit ===

~~{{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~~

Comma list: 4375/4374, 759375/758912, 100663296/100656875

Mapping: {{mapping| 1 ~~2 3 3 3 | 0~~ -19 -31 -9 21 }}

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~49/48 = 26.~~246~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0043{{c}}, ~605/512 = 289.0687{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~605/512 = 289.0677{{c}}

Badness (Sintel): 2.~~531~~

Badness (Sintel): 2.24

==== 13-limit ====

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: ~~196~~/~~195~~, ~~364~~/~~363~~, ~~385~~/~~384~~, ~~4375~~/~~4374~~

Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078

Mapping: {{mapping| 1 ~~2 3 3 3 3~~ | 0 -~~19 -31~~ -~~9 21 32~~ }}

Mapping: {{mapping| 1 -16 -9 -75 9 9 | 0 73 47 323 -23 -22 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~49/48 = 26.~~239~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0043{{c}}, ~13/11 = 289.0687{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 289.0677{{c}}

Badness (Sintel): 2.~~144~~

Badness (Sintel): 1.12

== ~~Trideci~~ ==

== Palladium ==

: ''For the 5-limit version ~~of this temperament~~, see [[~~13th~~-octave temperaments #~~Tridecatonic~~]].''

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

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]]")~~.

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.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, ~~83349/81920~~

[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}

: mapping generators: ~83349/81920, ~3

[[Optimal tuning]] ([[~~POTE~~]]): ~~~256~~/~~245~~ = 92.~~3077~~{{c}}, ~3/2 = ~~699~~.~~1410~~{{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=1| 26, 65, 91, ~~156d~~, ~~247cdd~~ }}

[[Badness]] (Sintel): 4.~~671~~

[[Badness]] (Sintel): 7.81

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: ~~245~~/~~242~~, ~~385~~/~~384~~, ~~4375~~/~~4374~~

Comma list: 3025/3024, 4375/4374, 134775333/134217728

Mapping: {{mapping| 13 0 -~~11 57 45~~ | 0 1 2 -1 0 }}

Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}

Optimal ~~tuning (POTE)~~: ~22/21 = 92.~~3077~~{{c}}, ~3/2 = ~~699~~.~~6179~~{{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=0| 26, 65, 91, ~~156d~~, ~~247cdde~~ }}

Badness (Sintel): 2.~~796~~

Badness (Sintel): 2.44

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: ~~169~~/~~168~~, ~~245~~/~~242~~, ~~325~~/~~324~~, ~~385~~/~~384~~

Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364

Mapping: {{mapping| 13 0 -~~11 57 45 48~~ | 0 1 2 -1 ~~0 0~~ }}

Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}

Optimal ~~tuning (POTE)~~: ~22/21 = 92.~~3077~~{{c}}, ~3/2 = ~~699~~.~~2969~~{{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| 26, ~~65f~~, ~~91f~~, ~~156dff~~ }}

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

Badness (Sintel): 2.~~164~~

Badness (Sintel): 1.68

== ~~Counterorson~~ ==

=== 17-limit ===

~~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: 2.3.5.7.11.13.17

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

Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224

~~[[Comma list]]~~: ~~4375/4374,~~ {{~~monzo~~| ~~154~~ -54 -21 -7 }}

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

Optimal tunings:

* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7399{{c}}

* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6464{{c}}

~~[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000~~{{~~c}}, ~{{monzo~~| ~~66 -23 -9 -3 }}~~ = ~~271.7113{{c~~}}

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

~~{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}~~

Badness (Sintel): 1.14

[[Badness]] (Sintel): 7.~~916~~

== References ==

[[Category:Ragismic microtemperaments| ]]

[[Category:Temperament collections]]

[[Category:~~Ragismic microtemperaments| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

@@ Line 4: / Line 4: @@
 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]]
+* ''[[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]]
+* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt 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]]
-* ''[[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]]
-* [[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]]
-* ''[[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]]
+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 ==
@@ Line 223: / Line 225: @@
 == 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}}).
@@ Line 277: / Line 279: @@
 : ''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
@@ Line 333: / Line 337: @@
 [[Comma list]]: 4375/4374, 589824/588245
-{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }}
+{{Mapping|legend=1| 1 -17 -30 2 | 0 23 40 1 }}
-: mapping generators: ~2, ~8/7
+: mapping generators: ~2, ~7/4
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~8/7 = 230.336{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8483{{c}}, ~7/4 = 969.5415{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 969.6608{{c}}
 {{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}
@@ Line 347: / Line 353: @@
 Comma list: 3025/3024, 4375/4374, 589824/588245
-Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}
+Mapping: {{mapping| 2 -11 -20 5 10 | 0 23 40 1 -5 }}
-: mapping generators: ~99/70, ~8/7
+: mapping generators: ~99/70, ~99/80
-Optimal tuning (POTE): ~99/70 = 600.0000{{c}}, ~8/7 = 230.3370{{c}}
+Optimal tunings:
+* WE: ~99/70 = 599.9323{{c}}, ~99/80 = 369.6212{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~99/80 = 369.6610{{c}}
-{{Optimal ET sequence|legend=0| 26, 198, 224, 422, 646, 1068d }}
+{{Optimal ET sequence|legend=0| 26, 172c, 198, 224, 422, 646, 1068d }}
-Badness (Sintel): 1.354
+Badness (Sintel): 1.35
 ==== 13-limit ====
@@ Line 361: / Line 369: @@
 Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024
-Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}
+Mapping: {{mapping| 2 -11 -20 5 10 -8 | 0 23 40 1 -5 25 }}
-Optimal tuning (POTE): ~99/70 = 600.0000{{c}}, ~8/7 = 230.3373{{c}}
+Optimal tunings:
+* WE: ~99/70 = 599.9207{{c}}, ~26/21 = 369.6139{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~26/21 = 369.6603{{c}}
-{{Optimal ET sequence|legend=0| 26, 198, 224, 422, 646f, 1068df }}
+{{Optimal ET sequence|legend=0| 26, 172cf, 198, 224, 422, 646f, 1068df }}
 Badness (Sintel): 0.844
@@ Line 374: / Line 384: @@
 Comma list: 4375/4374, 14641/14580, 15488/15435
-Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }}
+Mapping: {{mapping| 1 -40 -70 1 -77 | 0 46 80 2 89 }}
-: mapping generators: ~2, ~77/72
+: mapping generators: ~2, ~144/77
-Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.1642{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.8845{{c}}, ~144/77 = 1084.7314{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8345{{c}}
-{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }}
+{{Optimal ET sequence|legend=0| 73, 125, 198, 323, 521 }}
-Badness (Sintel): 2.589
+Badness (Sintel): 2.59
 ==== 13-limit ====
@@ Line 388: / Line 400: @@
 Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580
-Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }}
+Mapping: {{mapping| 1 -40 -70 1 -77 -131 | 0 46 80 2 89 149 }}
-Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.1628{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.8726{{c}}, ~144/77 = 1084.7220{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8359{{c}}
 {{Optimal ET sequence|legend=0| 73f, 125f, 198, 323, 521 }}
-Badness (Sintel): 1.821
+Badness (Sintel): 1.82
 == Crazy ==
 : ''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
@@ Line 406: / Line 422: @@
 {{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:
-* [[CTE]]: ~332150625/234881024 = 600.0000{{c}}, ~1125/1024 = 162.7475{{c}}
+* [[WE]]: ~332150625/234881024 = 600.0019{{c}}, ~1125/1024 = 162.7479{{c}}
-* [[error map]]: {{val| 0.0000 +0.0253 -0.0514 -0.0133 }}
+: [[error map]]: {{val| +0.004 +0.030 -0.042 -0.014 }}
 * [[CWE]]: ~332150625/234881024 = 600.0000{{c}}, ~1125/1024 = 162.7474{{c}}
-* error map: {{val| 0.0000 +0.0244 -0.0508 -0.0218 }}
+: error map: {{val| 0.000 +0.024 -0.051 -0.022 }}
 {{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}
@@ Line 426: / Line 442: @@
 Optimal tunings:
-* CTE: ~99/70 = 162.7485{{c}}, ~1125/1024 = 162.7485{{c}}
+* WE: ~99/70 = 600.0047{{c}}, ~1125/1024 = 162.7493{{c}}
-* CWE: ~99/70 = 162.7485{{c}}, ~1125/1024 = 162.7481{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~1125/1024 = 162.7481{{c}}
 {{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}
@@ Line 434: / Line 450: @@
 == Orga ==
+Orga may be described as the {{nowrap| 26 & 270 }} temperament, and [[1106edo]] gives a strong tuning.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 54975581388800/54936068900769
+[[Comma list]]: 4375/4374, {{monzo| 41 -4 2 -14 }}
-{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }}
+{{Mapping|legend=1| 2 -8 -15 6 | 0 29 51 -1 }}
-: Mapping generators: ~7411887/5242880, ~1310720/1058841
+: mapping generators: ~7411887/5242880, ~8/7
-[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 600.000{{c}}, ~8/7 = 231.104{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~7411887/5242880 = 599.9927{{c}}, ~8/7 = 231.1012{{c}}
+: [[error map]]: {{val| -0.015 +0.037 -0.045 +0.029 }}
+* [[CWE]]: ~7411887/5242880 = 600.0000{{c}}, ~8/7 = 231.1037{{c}}
+: error map: {{val| 0.000 +0.053 -0.023 +0.070 }}
-{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}
+{{Optimal ET sequence|legend=1| 26, …, 244, 270, 836, 1106, 1376, 2482 }}
-[[Badness]] (Sintel): 1.018
+[[Badness]] (Sintel): 1.02
 === 11-limit ===
@@ Line 452: / Line 474: @@
 Comma list: 3025/3024, 4375/4374, 5767168/5764801
-Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }}
+Mapping: {{mapping| 2 -8 -15 6 10 | 0 29 51 -1 -8 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~8/7 = 231.103{{c}}
+Optimal tunings:
+* WE: ~99/70 = 600.0025{{c}}, ~8/7 = 231.1039{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1030{{c}}
 {{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836, 1106 }}
@@ Line 465: / Line 489: @@
 Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360
-Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }}
+Mapping: {{mapping| 2 -8 -15 6 10 -3 | 0 29 51 -1 -8 27 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~8/7 = 231.103{{c}}
+Optimal tunings:
+* WE: ~99/70 = 600.0192{{c}}, ~8/7 = 231.1102{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1033{{c}}
 {{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836f, 1106f }}
@@ Line 473: / Line 499: @@
 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 (26 & 145) tempers out the wadisma, 201768035/201326592. 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
-[[Comma list]]: 4375/4374, 201768035/201326592
+[[Comma list]]: 4375/4374, {{monzo| -49 4 22 -3 }}
-{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}
+{{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3087/2560 = 322.804{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~25/24 = 70.5880{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9962{{c}}
+: [[error map]]: {{val| -0.004 +0.037 -0.030 -0.019 }}
+* [[CWE]]: ~25/24 = 70.5882{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9990{{c}}
+: error map: {{val| 0.000 +0.043 -0.021 -0.013 }}
-{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}
+{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547, 3706, 5253 }}
-[[Badness]] (Sintel): 1.136
+[[Badness]] (Sintel): 1.05
-=== Senator ===
-The senator temperament (26 & 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It 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
-Comma list: 441/440, 4375/4374, 65536/65219
+Comma list: 4375/4374, 41503/41472, 1879453125/1879048192
-Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }}
+Mapping: {{mapping| 17 0 26 -87 207 | 0 2 1 10 -11 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~77/64 = 322.793{{c}}
+Optimal tunings:
+* WE: ~25/24 = 70.5905{{c}}, ~693/400 = 951.0054{{c}}
+* CWE: ~25/24 = 70.5882{{c}}, ~693/400 = 950.9754{{c}}
-{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e, 487ee }}
+{{Optimal ET sequence|legend=0| 289, 323, 612, 3349de, 3961de, …, 5797ddee }}
-Badness (Sintel): 3.049
+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 11 19 2 4 15 | 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 tuning (POTE): ~2 = 1200.000{{c}}, ~77/64 = 322.793{{c}}
+[[Subgroup]]: 2.3.5.7
-{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316ef, 487eef }}
+[[Comma list]]: 4375/4374, 16875/16807
-Badness (Sintel): 1.845
+{{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 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }}
+{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~77/64 = 322.793{{c}}
+[[Badness]] (Sintel): 1.08
-{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316ef, 487eef }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Badness (Sintel): 1.353
+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 (53 & 612) 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]]. 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 2 10 -25 | 0 -2 -37 134 }}
+Badness (Sintel): 0.466
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~{{monzo| -27 11 3 1 }} = 249.0207{{c}}
-{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}
-[[Badness]] (Sintel): 1.179
-=== Monzism ===
-Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 41503/41472, 184549376/184528125
-Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}
-Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~231/200 = 249.0193{{c}}
-{{Optimal ET sequence|legend=0| 53, 559, 612 }}
-Badness (Sintel): 1.887
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
+Comma list: 540/539, 625/624, 729/728, 1375/1372
-Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}
+Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}
-Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~231/200 = 249.0199{{c}}
+Optimal tunings:
+* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}
-{{Optimal ET sequence|legend=0| 53, 559, 612 }}
+{{Optimal ET sequence|legend=0| 72, 152f, 224 }}
-Badness (Sintel): 2.222
+Badness (Sintel): 0.631
-== Semidimfourth ==
+===== 17-limit =====
-: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimfourth]].''
+Subgroup: 2.3.5.7.11.13.17
-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: 375/374, 540/539, 625/624, 715/714, 729/728
-[[Subgroup]]: 2.3.5.7
+Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}
-[[Comma list]]: 4375/4374, 235298/234375
+Optimal tunings:
+* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}
-[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}
+{{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~35/27 = 448.456{{c}}
+Badness (Sintel): 0.729
-{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-[[Badness]] (Sintel): 1.398
+Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
-=== Neusec ===
+Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}
-Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 235298/234375
+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: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }}
+{{Optimal ET sequence|legend=0| 72, 152fg, 224 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~12/11 = 151.547{{c}}
+Badness (Sintel): 0.975
-{{Optimal ET sequence|legend=0| 8d, 190, 388 }}
+==== 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): 1.955
-==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
+Comma list: 169/168, 325/324, 364/363, 540/539
-Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }}
+Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
-Optimal tuning (POTE): ~99/70 = 1200.000{{c}}, ~12/11 = 151.545{{c}}
+Optimal tunings:
+* WE: ~12/11 = 150.0313{{c}}, ~7/5 = 584.0134{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}}
-{{Optimal ET sequence|legend=0| 8d, 190, 198, 388 }}
+{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}
-Badness (Sintel): 1.279
+Badness (Sintel): 0.896
-== Acrokleismic ==
+===== 17-limit =====
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11.13.17
-[[Comma list]]: 4375/4374, 2202927104/2197265625
+Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
-{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }}
+Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}
-: mapping generators: ~2, ~6/5
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 315.557{{c}}
+Optimal tunings:
+* 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=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}
+{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}
-[[Badness]] (Sintel): 1.422
+Badness (Sintel): 0.795
-=== 11-limit ===
+===== 19-limit =====
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 4375/4374, 41503/41472, 172032/171875
+Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
-Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }}
+Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.558{{c}}
+Optimal tunings:
+* 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| 19, 251, 270, 829, 1099, 1369, 1639 }}
+{{Optimal ET sequence|legend=0| 8d, 72, 152 }}
-Badness (Sintel): 1.219
+Badness (Sintel): 0.993
-==== 13-limit ====
+Scales: [[Octoid72]], [[Octoid80]]
-Subgroup: 2.3.5.7.11.13
-Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
+==== Hexadecoid ====
+{{See also| 16th-octave temperaments }}
-Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }}
+Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.557{{c}}
+Subgroup: 2.3.5.7.11.13
-{{Optimal ET sequence|legend=0| 19, 251, 270 }}
+Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
-Badness (Sintel): 1.108
+Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
+: mapping generators: ~448/429, ~7/5
-=== Counteracro ===
+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, 5632/5625, 117649/117612
+{{Optimal ET sequence|legend=0| 80, 144, 224 }}
-Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }}
+Badness (Sintel): 1.27
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.553{{c}}
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-{{Optimal ET sequence|legend=0| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}
+Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
-Badness (Sintel): 1.407
+Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
-==== 13-limit ====
+Optimal tunings:
-Subgroup: 2.3.5.7.11.13
+* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9626{{c}}
+* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0463{{c}}
-Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
+{{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }}
-Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }}
+Badness (Sintel): 1.46
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.554{{c}}
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-{{Optimal ET sequence|legend=0| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}
+Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
-Badness (Sintel): 1.076
+Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}
-== Quasithird ==
+Optimal tunings:
-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 [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}.
+* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}
+* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}
-[[Subgroup]]: 2.3.5
+{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}
-[[Comma list]]: {{monzo| 55 -64 20 }}
+Badness (Sintel): 1.44
-{{Mapping|legend=1| 4 0 -11 | 0 5 16 }}
+== Seniority ==
-: mapping generators: ~51200000/43046721, ~1594323/1280000
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]].
-[[Optimal tuning]] ([[POTE]]): ~51200000/43046721 = 300.000{{c}}, ~1594323/1280000 = 380.395{{c}}
+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.
-{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }}
+[[Subgroup]]: 2.3.5.7
-[[Badness]] (Sintel): 2.335
+[[Comma list]]: 4375/4374, 201768035/201326592
-=== 7-limit ===
+{{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }}
-[[Subgroup]]: 2.3.5.7
+: mapping generators: ~2, ~5120/3087
-[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}
+[[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 }}
-{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
+{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d }}
-[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 300.000{{c}}, ~5103/4096 = 380.388 {{c}}
+[[Badness]] (Sintel): 1.14
-{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}
+=== 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.
-[[Badness]] (Sintel): 1.564
-=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
+Comma list: 441/440, 4375/4374, 65536/65219
-Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}
+Mapping: {{mapping| 1 -24 -43 5 2 | 0 35 62 -3 2 }}
-Optimal tuning (POTE): ~65536/51125 = 300.000{{c}}, ~5103/4096 = 380.387{{c}} (or ~22/21 = 80.387{{c}})
+Optimal tunings:
+* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}}
-{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448 }}
+{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e }}
-Badness (Sintel): 0.698
+Badness (Sintel): 3.05
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374
+Comma list: 364/363, 441/440, 2200/2197, 4375/4374
-Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}
+Mapping: {{mapping| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}
-Optimal tuning (POTE): ~65536/51125 = 300.000{{c}}, ~81/65 = 380.385{{c}} (or ~22/21 = 80.385{{c}})
+Optimal tunings:
+* 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| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}
+{{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }}
-Badness (Sintel): 1.219
+Badness (Sintel): 1.85
-== Deca ==
+==== 17-limit ====
-: ''For 5-limit version, see [[10th-octave temperaments #Neon]].''
+Subgroup: 2.3.5.7.11.13.17
-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 (satritrizo-asepbigu).
+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, 165288374272/164794921875
+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| 10 4 9 2 | 0 5 6 11 }}
+{{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }}
-: mapping generators: ~15/14, ~6/5
-[[Optimal tuning]] ([[POTE]]): ~15/14 = 120.000{{c}}, ~6/5 = 315.577{{c}}
+Badness (Sintel): 1.35
-{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}
+== Monzismic ==
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]].
-[[Badness]] (Sintel): 2.041
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 391314/390625
-Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}
-Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~6/5 = 315.582{{c}}
-{{Optimal ET sequence|legend=0| 80, 190, 270, 1000, 1270, 1540e, 1810e }}
+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.
-Badness (Sintel): 0.804
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
-Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}
-Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~6/5 = 315.602{{c}} (~40/39 = 44.398{{c}})
-{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}
-Badness (Sintel): 0.695
-=== 2.3.5.7.11.13.19 subgroup ===
-Subgroup: 2.3.5.7.11.13.19
-Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374, 1521/1520
-Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}
-Optimal tuning (CTE): ~15/14 = 120.000{{c}}, ~6/5 = 315.581{{c}} (~39/38 = 44.419{{c}})
-{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}
-Badness (Sintel): 0.556
-== Keenanose ==
-Keenanose is named 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 }}
+[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}
-{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
+{{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }}
-: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}
+: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4465{{c}}
+[[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 }}
-{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}
+{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd }}
-[[Badness]] (Sintel): 2.172
+[[Badness]] (Sintel): 1.18
-=== 11-limit ===
+=== Monzism ===
 Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 117649/117612, 67110351/67108864
+Comma list: 4375/4374, 41503/41472, 184549376/184528125
-Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}
+Mapping: {{mapping| 1 0 -27 109 -159 | 0 2 37 -134 205 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~385/384 = 4.4465{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}}
-{{Optimal ET sequence|legend=0| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}
+{{Optimal ET sequence|legend=0| 53, 559, 612, 3619de, 4231de, …, 6067ddee }}
-Badness (Sintel): 1.020
+Badness (Sintel): 1.89
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612
+Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
-Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}
+Mapping: {{mapping| 1 0 -27 109 -159 -70 | 0 2 37 -134 205 93 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~385/384 = 4.4466{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}}
-{{Optimal ET sequence|legend=0| 270, 1079, 1349, 1619, 1889, 4048 }}
+{{Optimal ET sequence|legend=0| 53, 559, 612 }}
-Badness (Sintel): 0.879
+Badness (Sintel): 2.22
-== Aluminium ==
+== Semidimfourth ==
-: ''For the 5-limit version, see [[13th-octave temperaments #Aluminium]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimfourth]].''
-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.
+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]].
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}
+[[Comma list]]: 4375/4374, 235298/234375
-[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}
+{{Mapping|legend=1| 1 -10 -13 -17 | 0 31 41 53 }}
-: Mapping generators: ~135/128, ~3
+: mapping generators: ~2, ~35/27
-[[Optimal tuning]] ([[CTE]]): ~135/128 = 92.3077{{c}}, ~3/2 = 702.0024{{c}}
+[[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 }}
-{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}
+{{Optimal ET sequence|legend=1| 8d, …, 91, 99, 289, 388, 875 }}
-[[Badness]] (Sintel): 3.201
+[[Badness]] (Sintel): 1.40
-=== 11-limit ===
+=== Neusec ===
 Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 234375/234256, 2097152/2096325
+Comma list: 3025/3024, 4375/4374, 235298/234375
-Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}
+Mapping: {{mapping| 2 -20 -26 -34 -17 | 0 31 41 53 32 }}
+: mapping generators: ~99/70, ~35/27
-Optimal tuning (CTE): ~135/128 = 92.3077{{c}}, ~3/2 = 702.0042{{c}}
+Optimal tunings:
+* WE: ~99/70 = 600.0381{{c}}, ~35/27 = 448.4812{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4546{{c}}
-{{Optimal ET sequence|legend=0| 494, 1053, 1547, 3588e, 5135e }}
+{{Optimal ET sequence|legend=0| 8d, …, 190, 388 }}
-Badness (Sintel): 1.393
+Badness (Sintel): 1.95
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078
+Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
-Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}
+Mapping: {{mapping| 2 -20 -26 -34 -17 -21 | 0 31 41 53 32 38 }}
-Optimal tuning (CTE): ~135/128 = 92.3077{{c}}, ~3/2 = 702.0099{{c}}
+Optimal tunings:
+* WE: ~99/70 = 600.0034{{c}}, ~35/27 = 448.4573{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4549{{c}}
-{{Optimal ET sequence|legend=0| 494, 1547, 2041, 4576def }}
+{{Optimal ET sequence|legend=0| 8d, …, 190, 198, 388 }}
-Badness (Sintel): 1.180
+Badness (Sintel): 1.28
-== Countritonic ==
-: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic (5-limit)]].''
-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.
+== Acrokleismic ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 68719476736/68356598625
+[[Comma list]]: 4375/4374, 2202927104/2197265625
-{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }}
+{{Mapping|legend=1| 1 -22 -22 -65 | 0 32 33 92 }}
-: mapping generators: ~2, ~45927/32768
+: mapping generators: ~2, ~5/3
-[[Optimal tuning]] (CTE): ~2 = 1200.0000{{c}}, ~45927/32768 = 588.6216{{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=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}
+{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}
-[[Badness]] (Sintel): 3.370
+[[Badness]] (Sintel): 1.42
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 5632/5625, 2621440/2614689
+Comma list: 4375/4374, 41503/41472, 172032/171875
-Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 }}
+Mapping: {{mapping| 1 -22 -22 -65 58 | 0 32 33 92 -74 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~539/384 = 588.6258{{c}}
+Optimal tunings:
+* 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| 53, 316e, 369, 422, 791e, 1213cde }}
+{{Optimal ET sequence|legend=0| 19, 251, 270, 829, 1099, 1369, 1639 }}
-Badness (Sintel): 2.336
+Badness (Sintel): 1.22
-=== 13-limit ===
+==== 13-limit ====
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13
-Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625
+Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
-Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 -74 }}
+Mapping: {{mapping| 1 -22 -22 -65 58 -56 | 0 32 33 92 -74 81 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~128/91 = 588.6277{{c}}
+Optimal tunings:
+* 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| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}
+{{Optimal ET sequence|legend=0| 19, 251, 270 }}
-Badness (Sintel): 1.514
+Badness (Sintel): 1.11
-== Quatracot ==
+=== Counteracro ===
-{{See also| Stratosphere }}
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
-{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }}
-: mapping generators: ~2278125/1605632, ~448/405
-[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 600.000{{c}}, ~448/405 = 176.805{{c}}
-{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}
-[[Badness]] (Sintel): 4.454
-=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 1265625/1261568
+Comma list: 4375/4374, 5632/5625, 117649/117612
-Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }}
+Mapping: {{mapping| 1 -22 -22 -65 -141 | 0 32 33 92 196 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~448/405 = 176.806{{c}}
+Optimal tunings:
+* 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| 190, 224, 414, 638, 1052c }}
+{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1061e, 1331c, 1601c, 1871bc }}
-Badness (Sintel): 1.357
+Badness (Sintel): 1.41
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 729/728, 1575/1573, 2200/2197
+Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
+Mapping: {{mapping| 1 -22 -22 -65 -141 -56 | 0 32 33 92 196 81 }}
-Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }}
+Optimal tunings:
+* WE: ~2 = 1199.9285{{c}}, ~5/3 = 884.3937{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4458{{c}}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~195/176 = 176.804{{c}}
+{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1331c }}
-{{Optimal ET sequence|legend=0| 190, 224, 414, 638, 1690bcc, 2328bccde }}
+Badness (Sintel): 1.08
-Badness (Sintel): 0.936
+== Quasithird ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasithird]].''
-== Moulin ==
+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.
-Moulin 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". It can be described as the {{nowrap| 494 & 1619 }} temperament. Since 11/8 is within 23 generators, the 25-tone mos (4L 21s) of this temperament contains the 8:11:13 triad.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
+[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}
-{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }}
+{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
-: Mapping generators: ~2, ~6422528/3796875
+: mapping generators: ~65536/55125, ~5103/4096
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6422528/3796875 = 910.9323{{c}}
+[[Optimal tuning]]s:
+* [[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| 494, 1125, 1619 }}
+{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}
-[[Badness]] (Sintel): 5.931
+[[Badness]] (Sintel): 1.56
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 759375/758912, 100663296/100656875
+Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
-Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }}
+Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~1024/605 = 910.9323{{c}}
+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}})
-{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}
+{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee }}
-Badness (Sintel): 2.240
+Badness (Sintel): 0.698
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
+Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374
-Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }}
+Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~22/13 = 910.9323{{c}}
+Optimal tunings:
+* 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| 494, 1125, 1619, 2113 }}
+{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836 }}
-Badness (Sintel): 1.118
+Badness (Sintel): 1.22
-== Palladium ==
-: ''For the 5-limit version of this temperament, 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.
+== Quincy ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}
+[[Comma list]]: 4375/4374, 823543/819200
-{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
+{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
-: Mapping generators: ~83349/81920, ~3
+: mapping generators: ~2, ~1728/1715
-[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 26.0870{{c}}, ~3/2 = 701.6074{{c}}
+[[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| 46, 368, 414, 460, 874d }}
+{{Optimal ET sequence|legend=1| 72, 217, 289, 650d, 939dd }}
-[[Badness]] (Sintel): 7.807
+[[Badness]] (Sintel): 2.02
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 134775333/134217728
+Comma list: 441/440, 4000/3993, 4375/4374
-Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}
+Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}
-Optimal tuning (POTE): ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.5951{{c}}
+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| 46, 368, 414, 460, 874de }}
+{{Optimal ET sequence|legend=0| 72, 217, 289 }}
-Badness (Sintel): 2.439
+Badness (Sintel): 1.02
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
+Comma list: 364/363, 441/440, 676/675, 4375/4374
-Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}
+Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}
-Optimal tuning (POTE): ~65/64 = 26.0870{{c}}, ~3/2 = 701.6419{{c}}
+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| 46, 368, 414, 460, 874de, 1334de }}
+{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
-Badness (Sintel): 1.684
+Badness (Sintel): 0.986
 === 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
+Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155
-Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}
+Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}
-Optimal tuning (POTE): ~65/64 = 26.0870{{c}}, ~3/2 = 701.6425{{c}}
+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| 46, 368, 414, 460, 874de, 1334deg }}
+{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
-Badness (Sintel): 1.143
+Badness (Sintel): 0.751
-== Oviminor ==
+=== 19-limit ===
-: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Oviminor (5-limit)]].''
+Subgroup: 2.3.5.7.11.13.17.19
-Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 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.
+Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675
-[[Subgroup]]: 2.3.5.7
+Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}
-[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}
+Optimal tunings:
+* WE: ~2 = 1199.9287{{c}}, ~100/99 = 16.5930{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.5948{{c}}
-{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }}
+{{Optimal ET sequence|legend=0| 72, 145, 217 }}
-: Mapping generators: ~2, ~6/5
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 315.7501{{c}}
+Badness (Sintel): 0.924
-{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}
+== Deca ==
+: ''For 5-limit version, see [[10th-octave temperaments#Neon]].''
-[[Badness]] (Sintel): 14.739
-== 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]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.
+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
-[[Comma list]]: 4375/4374, 16875/16807
+[[Comma list]]: 4375/4374, 165288374272/164794921875
-{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
+{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}
-: Mapping generators: ~49/45, ~7/5
+: mapping generators: ~15/14, ~460992/390625
-[[Optimal tuning]] ([[POTE]]): ~49/45 = 150.000{{c}}, ~7/5 = 583.940{{c}}
+[[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 }}
-[[Tuning ranges]]:
+{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}
-* 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]
-{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}
+[[Badness]] (Sintel): 2.04
-[[Badness]] (Sintel): 1.080
-Scales: [[octoid72]], [[octoid80]]
 === 11-limit ===
-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 minimaxing the 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, if one wants to use 80edo as the tuning, one must use octopus – not octoid – as 80edo does not temper 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 1375/1372, 4000/3993
+Comma list: 3025/3024, 4375/4374, 391314/390625
-Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}
+Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.962{{c}}
+Optimal tunings:
+* 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}})
-Tuning ranges:
+{{Optimal ET sequence|legend=0| 80, 190, 270, 1000, 1270, 1540e, 1810e }}
-* 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| 72, 152, 224 }}
+Badness (Sintel): 0.804
-Badness (Sintel): 0.466
+=== 13-limit ===
-Scales: [[octoid72]], [[octoid80]]
-==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 625/624, 729/728, 1375/1372
+Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
-Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}
+Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.905{{c}}
+Optimal tunings:
+* 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| 72, 152f, 224 }}
+{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}
-Badness (Sintel): 0.631
+Badness (Sintel): 0.695
-Scales: [[octoid72]], [[octoid80]]
+=== 2.3.5.7.11.13.19 subgroup ===
+Subgroup: 2.3.5.7.11.13.19
-; Music
+Comma list: 1001/1000, 1521/1520, 3025/3024, 4225/4224, 4375/4374
-* ''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| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 375/374, 540/539, 625/624, 715/714, 729/728
+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: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}
+{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.842{{c}}
+Badness (Sintel): 0.556
-{{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}
+== 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.
-Badness (Sintel): 0.729
+[[Subgroup]]: 2.3.5.7
-===== 19-limit =====
+[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
+{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
+: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}
-Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}
+[[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 tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.932{{c}}
+{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}
-{{Optimal ET sequence|legend=0| 72, 152fg, 224 }}
+[[Badness]] (Sintel): 2.17
-Badness (Sintel): 0.975
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-==== Octopus ====
+Comma list: 4375/4374, 117649/117612, 67110351/67108864
-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{{cent}}.
-Subgroup: 2.3.5.7.11.13
+Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}
-Comma list: 169/168, 325/324, 364/363, 540/539
+Optimal tunings:
+* WE: ~2 = 1199.9970{{c}}, ~385/384 = 4.4465{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4465{{c}}
-Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
+{{Optimal ET sequence|legend=0| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.892{{c}}
+Badness (Sintel): 1.02
-{{Optimal ET sequence|legend=0| 72, 152, 224f }}
-Badness (Sintel): 0.0896
-===== 17-limit =====
-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 tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.811{{c}}
-{{Optimal ET sequence|legend=0| 72, 152, 224fg, 296ffg }}
-Badness (Sintel): 0.795
-===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
-Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 584.064{{c}}
-{{Optimal ET sequence|legend=0| 72, 152, 224fg, 376ffgh }}
-Badness (Sintel): 0.993
-Scales: [[Octoid72]], [[Octoid80]]
-==== Hexadecoid ====
-{{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
-Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
+Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612
-Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
+Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}
-: mapping generators: ~448/429, ~7/5
-Optimal tuning (POTE): ~448/429 = 75.000{{c}}, ~13/8 = 841.015{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0065{{c}}, ~385/384 = 4.4467{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4467{{c}}
-{{Optimal ET sequence|legend=0| 80, 144, 224 }}
+{{Optimal ET sequence|legend=0| 270, 1079, 1349, 1619, 1889, 4048 }}
-Badness (Sintel): 1.273
+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: 540/539, 715/714, 936/935, 4000/3993, 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| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~117/112 = 75.000{{c}}, ~13/8 = 840.932{{c}}
+[[Comma list]]: 4375/4374, 158203125/157351936
-{{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }}
+{{Mapping|legend=1| 1 -5 -4 -18 | 0 25 24 79 }}
+: mapping generators: ~2, ~6/5
-Badness (Sintel): 1.458
+[[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 }}
-===== 19-limit =====
+{{Optimal ET sequence|legend=1| 19, …, 205, 224, 243, 467 }}
-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): 2.29
-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 tuning (POTE): ~117/112 = 75.000{{c}}, ~13/8 = 840.896{{c}}
+Comma list: 540/539, 4375/4374, 2097152/2096325
-{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}
+Mapping: {{mapping| 1 -5 -4 -18 19 | 0 25 24 79 -59 }}
-Badness (Sintel): 1.443
+Optimal tunings:
+* WE: ~2 = 1199.9944{{c}}, ~6/5 = 316.0690{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0705{{c}}
-== Parakleismic ==
+{{Optimal ET sequence|legend=0| 19, 205, 224 }}
-{{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. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, and 11-limit adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.
+Badness (Sintel): 2.35
-[[Subgroup]]: 2.3.5.7
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-[[Comma list]]: 3136/3125, 4375/4374
+Comma list: 540/539, 625/624, 729/728, 10985/10976
-{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }}
+Mapping: {{mapping| 1 -5 -4 -18 19 -15 | 0 25 24 79 -59 71 }}
-: mapping generators: ~2, ~6/5
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 315.181{{c}}
+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=1| 19, 80, 99, 217, 316, 415 }}
+{{Optimal ET sequence|legend=0| 19, 205, 224 }}
-[[Badness]] (Sintel): 0.694
+Badness (Sintel): 1.40
-=== 11-limit ===
+=== Counterlytic ===
 Subgroup: 2.3.5.7.11
-Comma list: 385/384, 3136/3125, 4375/4374
+Comma list: 1375/1372, 4375/4374, 496125/495616
-Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }}
+Mapping: {{mapping| 1 -5 -4 -18 -40 | 0 25 24 79 165 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.251{{c}}
+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=0| 19, 99, 118 }}
+{{Optimal ET sequence|legend=1| 19e, 205e, 224, 467e, 691, 915c }}
-Badness (Sintel): 1.643
+Badness (Sintel): 2.16
-=== Paralytic ===
-The paralytic temperament (118 & 217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 &amp; 217 tempers out 1001/1000, 1575/1573, and 3584/3575.
-Subgroup: 2.3.5.7.11
-Comma list: 441/440, 3136/3125, 4375/4374
-Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.220{{c}}
-{{Optimal ET sequence|legend=0| 19e, 99e, 118, 217, 335, 552d, 887dd }}
-Badness (Sintel): 1.191
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
+Comma list: 625/624, 729/728, 1375/1372, 10985/10976
-Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }}
+Mapping: {{mapping| 1 -5 -4 -18 -40 -15 | 0 25 24 79 165 71 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.214{{c}}
+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| 99e, 118, 217, 552d, 769de }}
+{{Optimal ET sequence|legend=0| 19e, 205e, 224, 467e, 691, 915c }}
-Badness (Sintel): 1.847
+Badness (Sintel): 1.23
-==== Paraklein ====
+== Sfourth ==
-The paraklein temperament (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 #Sfourth]].''
-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, 64827/64000
-Mapping: {{mapping| 1 5 6 12 25 15 | 0 -13 -14 -35 -82 -43 }}
+{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
+: mapping generators: ~2, ~49/48
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.225{{c}}
+[[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=0| 19e, 99ef, 118, 217ff, 335ff }}
+{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
-Badness (Sintel): 1.554
+[[Badness]] (Sintel): 3.12
-=== Parkleismic ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 176/175, 1375/1372, 2200/2187
+Comma list: 121/120, 441/440, 4375/4374
-Mapping: {{mapping| 1 5 6 12 20 | 0 -13 -14 -35 -63 }}
+Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.060{{c}}
+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| 19e, 80, 179, 259cd }}
+{{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }}
-Badness (Sintel): 1.848
+Badness (Sintel): 1.78
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 176/175, 325/324, 1375/1372
+Comma list: 121/120, 169/168, 325/324, 441/440
-Mapping: {{mapping| 1 5 6 12 20 10 | 0 -13 -14 -35 -63 -24 }}
+Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.075{{c}}
+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| 19e, 80, 179 }}
+{{Optimal ET sequence|legend=0| 45ef, 46, 91ef, 137def, 228ddeeefff }}
-Badness (Sintel): 1.511
+Badness (Sintel): 1.37
-=== Paradigmic ===
+=== Sfour ===
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 896/891, 3136/3125
+Comma list: 385/384, 2401/2376, 4375/4374
-Mapping: {{mapping| 1 5 6 12 -1 | 0 -13 -14 -35 17 }}
+Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.096{{c}}
+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| 19, 61d, 80, 99e, 179e }}
+{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}
-Badness (Sintel): 1.379
+Badness (Sintel): 2.53
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 325/324, 540/539, 832/825
+Comma list: 196/195, 364/363, 385/384, 4375/4374
-Mapping: {{mapping| 1 5 6 12 -1 10 | 0 -13 -14 -35 17 -24 }}
+Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.080{{c}}
+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| 19, 61d, 80, 99e, 179e }}
+{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}
-Badness (Sintel): 1.479
+Badness (Sintel): 2.14
-=== Semiparakleismic ===
+== Aluminium ==
-Subgroup: 2.3.5.7.11
+: ''For the 5-limit version, see [[13th-octave temperaments #Aluminium]].''
-Comma list: 3025/3024, 3136/3125, 4375/4374
+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| 2 10 12 24 19 | 0 -13 -14 -35 -23 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~6/5 = 315.181{{c}}
+[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}
-{{Optimal ET sequence|legend=0| 80, 118, 198, 316, 514c, 830c }}
+[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}
+: Mapping generators: ~135/128, ~3
-Badness (Sintel): 1.131
+[[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 }}
-==== Semiparamint ====
+{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}
-This extension was named ''semiparakleismic'' in the earlier materials.
-Subgroup: 2.3.5.7.11.13
+[[Badness]] (Sintel): 3.20
-Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }}
+Comma list: 4375/4374, 234375/234256, 2097152/2096325
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~6/5 = 315.156{{c}}
+Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}
-{{Optimal ET sequence|legend=0| 80, 118, 198 }}
+Optimal tunings:
+* WE: ~135/128 = 92.3062{{c}}, ~3/2 = 701.9946{{c}}
+* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0056{{c}}
-Badness (Sintel): 1.396
+{{Optimal ET sequence|legend=0| 494, 1053, 1547, 3588e, 5135e }}
-==== Semiparawolf ====
+Badness (Sintel): 1.39
-This extension was named ''gentsemiparakleismic'' in the earlier materials.
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 325/324, 364/363, 3136/3125
+Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078
-Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }}
+Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}
-Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~6/5 = 315.184{{c}}
+Optimal tunings:
+* WE: ~135/128 = 92.3055{{c}}, ~3/2 = 701.9928{{c}}
+* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0098{{c}}
-{{Optimal ET sequence|legend=0| 80, 118f, 198f }}
+{{Optimal ET sequence|legend=0| 494, 1547, 2041, 4576def }}
-Badness (Sintel): 1.672
+Badness (Sintel): 1.18
-== Counterkleismic ==
+== Ragitritonic ==
-: ''For the 5-limit temperament, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''
+: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
+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.
-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).
+Ragitritonic was named by [[Flora Canou]] in 2026 as a contraction of ''ragismic'' and ''tritonic''.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 158203125/157351936
+[[Comma list]]: 4375/4374, 68719476736/68356598625
-{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }}
+{{Mapping|legend=1| 1 -3 -15 40 | 0 9 34 -73 }}
-: Mapping generators: ~2, ~5/3
+: mapping generators: ~2, ~65536/45927
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 316.060{{c}}
+[[Optimal tuning]]s:
+* [[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 }}
-{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }}
+{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}
-[[Badness]] (Sintel): 2.292
+[[Badness]] (Sintel): 3.37
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 4375/4374, 2097152/2096325
+Comma list: 4375/4374, 5632/5625, 2621440/2614689
-Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }}
+Mapping: {{mapping| 1 -3 -15 40 -75 | 0 9 34 -73 154 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.071{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.8147{{c}}, ~768/539 = 611.2822{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~768/539 = 611.3762{{c}}
-{{Optimal ET sequence|legend=0| 19, 205, 224 }}
+{{Optimal ET sequence|legend=0| 53, 316e, 369, 422, 791e, 1213cde }}
-Badness (Sintel): 2.346
+Badness (Sintel): 2.34
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 625/624, 729/728, 10985/10976
+Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625
-Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }}
+Mapping: {{mapping| 1 -3 -15 40 -75 -34 | 0 9 34 -73 154 74 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.070{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7916{{c}}, ~91/64 = 611.2698{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~91/64 = 611.3754{{c}}
-{{Optimal ET sequence|legend=0| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}
+{{Optimal ET sequence|legend=0| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}
-Badness (Sintel): 1.400
+Badness (Sintel): 1.51
-=== Counterlytic ===
+== Quatracot ==
-Subgroup: 2.3.5.7.11
+{{See also| Stratosphere }}
-Comma list: 1375/1372, 4375/4374, 496125/495616
-Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.065{{c}}
-{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}
-Badness (Sintel): 2.162
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 729/728, 1375/1372, 10985/10976
-Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.065{{c}}
-{{Optimal ET sequence|legend=0| 19e, 205e, 224 }}
-Badness (Sintel): 1.231
-== Quincy ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 823543/819200
+[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
-{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
+{{Mapping|legend=1| 2 -6 -1 -36 | 0 13 8 59 }}
+: mapping generators: ~2278125/1605632, ~7168/5625
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~1728/1715 = 16.613{{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| 72, 217, 289 }}
+{{Optimal ET sequence|legend=1| 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc }}
-[[Badness]] (Sintel): 2.016
+[[Badness]] (Sintel): 4.45
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 441/440, 4000/3993, 4375/4374
+Comma list: 3025/3024, 4375/4374, 1265625/1261568
-Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}
+Mapping: {{mapping| 2 -6 -1 -36 -22 | 0 13 8 59 41 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.613{{c}}
+Optimal tunings:
+* WE: ~99/70 = 600.0847{{c}}, ~225/176 = 423.2536{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~225/176 = 423.1977{{c}}
-{{Optimal ET sequence|legend=0| 72, 217, 289 }}
+{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638, 1052c }}
-Badness (Sintel): 1.021
+Badness (Sintel): 1.36
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 441/440, 676/675, 4375/4374
+Comma list: 625/624, 729/728, 1575/1573, 2200/2197
-Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}
+Mapping: {{mapping| 2 -6 -1 -36 -22 -6 | 0 13 8 59 41 19 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.602{{c}}
+Optimal tunings:
+* WE: ~99/70 = 600.0571{{c}}, ~143/112 = 423.2366{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~143/112 = 423.1987{{c}}
-{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
+{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638 }}
-Badness (Sintel): 0.986
+Badness (Sintel): 0.936
-=== 17-limit ===
+== Trideci ==
-Subgroup: 2.3.5.7.11.13.17
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tridecatonic]].''
-Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155
+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 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.602{{c}}
+[[Comma list]]: 4375/4374, 83349/81920
-{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
+{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
+: mapping generators: ~256/245, ~3
-Badness (Sintel): 0.751
+[[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 }}
-=== 19-limit ===
+{{Optimal ET sequence|legend=1| 26, 65, 91 }}
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675
+[[Badness]] (Sintel): 4.67
-Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.594{{c}}
+Comma list: 245/242, 385/384, 4375/4374
-{{Optimal ET sequence|legend=0| 72, 145, 217 }}
+Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}
-Badness (Sintel): 0.924
+Optimal tunings:
+* WE: ~22/21 = 92.3729{{c}}, ~3/2 = 700.1118{{c}}
+* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.7703{{c}}
-== Sfourth ==
+{{Optimal ET sequence|legend=0| 26, 65, 91 }}
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''
-[[Subgroup]]: 2.3.5.7
+Badness (Sintel): 2.80
-[[Comma list]]: 4375/4374, 64827/64000
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
+Comma list: 169/168, 245/242, 325/324, 385/384
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/48 = 26.287{{c}}
+Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}
-{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
+Optimal tunings:
+* WE: ~22/21 = 92.4003{{c}}, ~3/2 = 699.9983{{c}}
+* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.4772{{c}}
-[[Badness]] (Sintel): 3.120
+{{Optimal ET sequence|legend=0| 26, 65f, 91f }}
-=== 11-limit ===
+Badness (Sintel): 2.16
-Subgroup: 2.3.5.7.11
-Comma list: 121/120, 441/440, 4375/4374
+== 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.
-Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.286{{c}}
+[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
-{{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }}
+{{Mapping|legend=1| 1 -16 -9 -75 | 0 73 47 323 }}
+: mapping generators: ~2, ~3796875/3211264
-Badness (Sintel): 1.788
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0272{{c}}, ~3796875/3211264 = 289.0675{{c}}
-==== 13-limit ====
+: [[error map]]: {{val| +0.027 +0.007 -0.084 +0.013 }}
-Subgroup: 2.3.5.7.11.13
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3796875/3211264 = 289.0675{{c}}
+: error map: {{val| 0.000 -0.029 -0.142 -0.029 }}
-Comma list: 121/120, 169/168, 325/324, 441/440
+{{Optimal ET sequence|legend=1| 494, 1125, 1619, 8589cc, 10208cc }}
-Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}
+[[Badness]] (Sintel): 5.93
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.310{{c}}
+=== 11-limit ===
-{{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
+Comma list: 4375/4374, 759375/758912, 100663296/100656875
-Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}
+Mapping: {{mapping| 1 -16 -9 -75 9 | 0 73 47 323 -23 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.246{{c}}
+Optimal tunings:
+* 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| 45, 46, 91, 137d }}
+{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}
-Badness (Sintel): 2.531
+Badness (Sintel): 2.24
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 196/195, 364/363, 385/384, 4375/4374
+Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
-Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}
+Mapping: {{mapping| 1 -16 -9 -75 9 9 | 0 73 47 323 -23 -22 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.239{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0043{{c}}, ~13/11 = 289.0687{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 289.0677{{c}}
-{{Optimal ET sequence|legend=0| 45, 46, 91, 137d }}
+{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}
-Badness (Sintel): 2.144
+Badness (Sintel): 1.12
-== Trideci ==
+== Palladium ==
-: ''For the 5-limit version of this temperament, see [[13th-octave temperaments #Tridecatonic]].''
+: ''For the 5-limit version, see [[46th-octave temperaments #Palladium]]''.
-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]]").
+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.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 83349/81920
+[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}
-{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
+{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
+: mapping generators: ~83349/81920, ~3
-[[Optimal tuning]] ([[POTE]]): ~256/245 = 92.3077{{c}}, ~3/2 = 699.1410{{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=1| 26, 65, 91, 156d, 247cdd }}
+{{Optimal ET sequence|legend=1| 46, …, 368, 414, 460, 874d }}
-[[Badness]] (Sintel): 4.671
+[[Badness]] (Sintel): 7.81
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 245/242, 385/384, 4375/4374
+Comma list: 3025/3024, 4375/4374, 134775333/134217728
-Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}
+Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}
-Optimal tuning (POTE): ~22/21 = 92.3077{{c}}, ~3/2 = 699.6179{{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=0| 26, 65, 91, 156d, 247cdde }}
+{{Optimal ET sequence|legend=0| 46, …, 368, 414, 460, 874de }}
-Badness (Sintel): 2.796
+Badness (Sintel): 2.44
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 245/242, 325/324, 385/384
+Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
-Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}
+Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}
-Optimal tuning (POTE): ~22/21 = 92.3077{{c}}, ~3/2 = 699.2969{{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| 26, 65f, 91f, 156dff }}
+{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334dde }}
-Badness (Sintel): 2.164
+Badness (Sintel): 1.68
-== Counterorson ==
+=== 17-limit ===
-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: 2.3.5.7.11.13.17
-[[Subgroup]]: 2.3.5.7
+Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
-[[Comma list]]: 4375/4374, {{monzo| 154 -54 -21 -7 }}
+Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}
-{{Mapping|legend=1| 1 0 -21 85 | 0 7 103 -363 }}
+Optimal tunings:
+* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7399{{c}}
+* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6464{{c}}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7113{{c}}
+{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334ddeg }}
-{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}
+Badness (Sintel): 1.14
-[[Badness]] (Sintel): 7.916
 == References ==
+[[Category:Ragismic microtemperaments| ]] <!-- main article -->
 [[Category:Temperament collections]]
-[[Category:Ragismic microtemperaments| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]