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:

Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, ragitritonic, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:

* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]

* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]

* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]

* ''[[Modus]]'' (+64/63) → [[Tetracot family #Modus|Tetracot family]]

* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]

* ''[[Flattone]]'' (+81/80) → [[Meantone family #Flattone|Meantone family]]

* [[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]]

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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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: ''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:

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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

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

Aside from the ragisma, the seniority temperament tempers out the [[wadisma]], 201768035/201326592, and may be described as {{nowrap| 26 & 145 }}. It is so named because the senior comma ({{monzo| -17 62 -35 }}~~, quadla-sepquingu~~) is tempered out.

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.

[[Subgroup]]: 2.3.5.7

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

~~The monzismic temperament~~ tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]]. It may be described as the {{nowrap| 53 & 612 }} temperament. A notable tuning not appearing on the optimal ET sequence is [[665edo]], which is nearly equivalent to the pure-3's tuning.

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.

[[Subgroup]]: 2.3.5.7

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[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9305{{c}}, ~6/5 = 884.3923{{c}}

* [[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}}, ~6/5 = 884.4423{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.4423{{c}}

: error map: {{val| 0.000 +0.198 +0.282 -0.136 }}

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

~~The quasithird~~ temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the ragisma and {{monzo| -60 29 0 5 }}.

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

[[Subgroup]]: 2.3.5.7

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

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

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

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== Keenanose ==

Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.

Keenanose, the {{nowrap| 270 & 1889 }} temperament, was named by [[Eliora]] in 2022 for the fact that it uses [[385/384]], the keenanisma, as the generator.

[[Subgroup]]: 2.3.5.7

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

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.

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

[[Subgroup]]: 2.3.5.7

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Badness (Sintel): 1.18

== ~~Countritonic~~ ==

== Ragitritonic ==

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

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

~~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~~.

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.

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

[[Subgroup]]: 2.3.5.7

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Optimal tunings:

* WE: ~2 = 1199.7916{{c}}, ~~~768~~/~~539~~ = 611.2698{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~~~768~~/~~539~~ = 611.3754{{c}}

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

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

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

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

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

[[Optimal tuning]] ([[~~POTE~~]]): ~2278125/1605632 = 600.~~000~~{{c}}, ~~~448~~/~~405~~ = ~~176~~.~~805~~{{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| 190, 224, 414, 638, 1052c, 1690bcc }}

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

[[Badness]] (Sintel): 4.~~454~~

[[Badness]] (Sintel): 4.45

=== 11-limit ===

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

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

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

Optimal ~~tuning (POTE)~~: ~99/70 = 600.~~000~~{{c}}, ~~~448~~/~~405~~ = 176.~~806~~{{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.~~357~~

Badness (Sintel): 1.36

=== 13-limit ===

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

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

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

Optimal ~~tuning (POTE)~~: ~99/70 = 600.~~000~~{{c}}, ~~~195~~/~~176~~ = ~~176~~.~~804~~{{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| 190, 224, 414, 638~~, 1690bcc, 2328bccde~~ }}

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

Badness (Sintel): 0.936

== Moulin ==

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.

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

[[Subgroup]]: 2.3.5.7

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

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

: mapping generators: ~2, ~3796875/3211264

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1200.0000{{c}}, ~~~6422528~~/~~3796875~~ = ~~910~~.~~9323~~{{c}}

[[Optimal tuning]]s:

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

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

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

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

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

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

[[Badness]] (Sintel): 5.93

=== 11-limit ===

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

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

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

Optimal ~~tuning (CTE)~~: ~2 = 1200.0000{{c}}, ~~~1024~~/~~605~~ = ~~910~~.~~9323~~{{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.~~240~~

Badness (Sintel): 2.24

=== 13-limit ===

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

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

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

Optimal ~~tuning (CTE)~~: ~2 = 1200.0000{{c}}, ~22/13 = ~~910~~.~~9323~~{{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): 1.~~118~~

Badness (Sintel): 1.12

== Palladium ==

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

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

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

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

: mapping generators: ~83349/81920, ~3

[[Optimal tuning]] ([[~~POTE~~]]): ~83349/81920 = 26.0870{{c}}, ~3/2 = 701.~~6074~~{{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 }}

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

[[Badness]] (Sintel): 7.81

=== 11-limit ===

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

Optimal ~~tuning (POTE)~~: ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.~~5951~~{{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}}

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

Badness (Sintel): 2.44

=== 13-limit ===

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

Optimal ~~tuning (POTE)~~: ~65/64 = 26.0870{{c}}, ~3/2 = 701.~~6419~~{{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| 46, 368, 414, 460, 874de, ~~1334de~~ }}

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

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

Badness (Sintel): 1.68

=== 17-limit ===

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

Optimal ~~tuning (POTE)~~: ~65/64 = 26.0870{{c}}, ~3/2 = 701.~~6425~~{{c}}

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 ET sequence|legend=0| 46, 368, 414, 460, 874de, ~~1334deg~~ }}

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

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

Badness (Sintel): 1.14

== ~~Oviminor~~ ==

== Counterorson ==

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

~~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~~.

[[Subgroup]]: 2.3.5.7

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

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

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

: mapping generators: ~2, ~{{monzo| 66 -23 -9 -3 }}

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1200.0000{{c}}, ~~~6/5~~ = ~~315~~.~~7501~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0040{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7122{{c}}

: [[error map]]: {{val| +0.004 -0.303 -0.041 -0.015 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7113{{c}}

: error map: {{val| 0.000 +0.024 -0.051 -0.025 }}

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

[[Badness]] (Sintel): 14.~~739~~

[[Badness]] (Sintel): 7.92

== Octoid ==

== Oviminor ==

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

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

[[Subgroup]]: 2.3.5.7

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

: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0193{{c}}, ~5/3 = 884.2638{{c}}

: [[error map]]: {{val| +0.019 +0.010 -0.085 +0.032 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.2497{{c}}

: error map: {{val| 0.000 -0.011 -0.120 +0.008 }}

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

[[Badness]] (Sintel): 14.7

== Octoid ==

: {{Main| Octoid }}

: ''For the 5-limit version, see [[8th-octave temperaments #Octoid]].''

The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). 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.

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]].

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

[[Subgroup]]: 2.3.5.7

Line 1,185:

Line 1,246:

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

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

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

[[Optimal tuning]]s:

* [[WE]]: ~49/45 = 150.0003{{c}}, ~7/5 = 583.9416{{c}}

: [[error map]]: {{val| +0.002 -0.130 -0.547 +0.883 }}

* [[CWE]]: ~49/45 = 150.0000{{c}}, ~7/5 = 583.9411{{c}}

: error map: {{val| 0.000 -0.132 -0.549 +0.880 }}

[[Tuning ranges]]:

Line 1,195:

Line 1,260:

* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

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

[[Badness]] (Sintel): 1.08

~~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

Line 1,210:

Line 1,271:

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

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

Optimal tunings:

* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9477{{c}}

Tuning ranges:

Line 1,216:

Line 1,279:

* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

Badness (Sintel): 0.466

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

==== 13-limit ====

Line 1,229:

Line 1,290:

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

Optimal ~~tuning (POTE)~~: ~12/11 = 150.~~000~~{{c}}, ~7/5 = 583.~~905~~{{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): 0.631

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

~~; Music~~

* ''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 =====

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Line 1,305:

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}}

Optimal tunings:

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

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

Line 1,260:

Line 1,320:

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

Optimal ~~tuning (POTE)~~: ~12/11 = 150.~~000~~{{c}}, ~7/5 = 583.~~932~~{{c}}

Optimal tunings:

* WE: ~12/11 = 149.9785{{c}}, ~7/5 = 583.8482{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}}

Line 1,267:

Line 1,329:

==== 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{{~~cent~~}}.

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}}.

Subgroup: 2.3.5.7.11.13

Line 1,275:

Line 1,337:

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

Optimal ~~tuning (POTE)~~: ~12/11 = 150.~~000~~{{c}}, ~7/5 = 583.~~892~~{{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): 0.~~0896~~

Badness (Sintel): 0.896

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

Line 1,288:

Line 1,352:

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 tunings:

* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}

Badness (Sintel): 0.795

Line 1,301:

Line 1,367:

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 tunings:

* WE: ~12/11 = 150.0049{{c}}, ~7/5 = 584.0833{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}}

Badness (Sintel): 0.993

Line 1,321:

Line 1,389:

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

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

Optimal tunings:

* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}

* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}

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

Badness (Sintel): 1.27

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

Line 1,334:

Line 1,404:

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

Optimal ~~tuning (POTE)~~: ~117/112 = 75.~~000~~{{c}}, ~13/8 = ~~840~~.~~932~~{{c}}

Optimal tunings:

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

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

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

Badness (Sintel): 1.46

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

Line 1,345:

Line 1,417:

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

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

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

Optimal ~~tuning (POTE)~~: ~117/112 = 75.~~000~~{{c}}, ~13/8 = ~~840~~.~~896~~{{c}}

Optimal tunings:

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

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

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

Badness (Sintel): 1.44

== Parakleismic ==

Line 1,357:

Line 1,431:

: ''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.

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

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

[[Subgroup]]: 2.3.5.7

Line 1,363:

Line 1,439:

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

: mapping generators: ~2, ~6/5

: mapping generators: ~2, ~5/3

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.7820{{c}}, ~5/3 = 884.6581{{c}}

: [[error map]]: {{val| -0.218 +0.344 +0.644 -0.779 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8088{{c}}

: error map: {{val| 0.000 +0.560 +1.010 -0.516 }}

{{Optimal ET sequence|legend=1| 19, 61d, 80, 99, 217, 316, 415 }}

[[Badness]] (Sintel): 0.694

Line 1,377:

Line 1,457:

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

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

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

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

Optimal tunings:

* WE: ~2 = 1200.3296{{c}}, ~5/3 = 884.9921{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7519{{c}}

Badness (Sintel): 1.~~643~~

Badness (Sintel): 1.64

=== 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.

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

Subgroup: 2.3.5.7.11

Line 1,392:

Line 1,474:

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

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

Mapping: {{mapping| 1 -8 -8 -23 -57 | 0 13 14 35 82 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~6/5 = ~~315~~.~~220~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9940{{c}}, ~5/3 = 884.7757{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7800{{c}}

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

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

Badness (Sintel): 1.19

==== 13-limit ====

Line 1,405:

Line 1,489:

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

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

Mapping: {{mapping| 1 -8 -8 -23 -57 59 | 0 13 14 35 82 -75 }}

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

Optimal tunings:

* WE: ~2 = 1199.9218{{c}}, ~5/3 = 884.7285{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7858{{c}}

{{Optimal ET sequence|legend=0| 99e, 118, 217~~, 552d, 769de~~ }}

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

Badness (Sintel): 1.85

==== Paraklein ====

~~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]].

Paraklein (19e & 118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].

Subgroup: 2.3.5.7.11.13

Line 1,420:

Line 1,506:

Comma list: 196/195, 352/351, 625/624, 729/728

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

Mapping: {{mapping| 1 -8 -8 -23 -57 -28 | 0 13 14 35 82 43 }}

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

Optimal tunings:

* WE: ~2 = 1199.8239{{c}}, ~5/3 = 884.6449{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7709{{c}}

{{Optimal ET sequence|legend=0| 19e, 99ef, 118~~, 217ff, 335ff~~ }}

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

Badness (Sintel): 1.55

=== Parkleismic ===

Line 1,433:

Line 1,521:

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

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

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

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

Optimal tunings:

* WE: ~2 = 1199.1848{{c}}, ~5/3 = 884.3386{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9158{{c}}

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

Badness (Sintel): 1.85

==== 13-limit ====

Line 1,446:

Line 1,536:

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

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

Mapping: {{mapping| 1 -8 -8 -23 -43 -14 | 0 13 14 35 63 24 }}

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

Optimal tunings:

* WE: ~2 = 1199.5318{{c}}, ~5/3 = 884.5800{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9118{{c}}

Badness (Sintel): 1.~~511~~

Badness (Sintel): 1.51

=== Paradigmic ===

Line 1,459:

Line 1,551:

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

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

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

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

Optimal tunings:

* WE: ~2 = 1199.0616{{c}}, ~5/3 = 884.2124{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.8877{{c}}

{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e, 457bcddeeee }}

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

Badness (Sintel): 1.38

==== 13-limit ====

Line 1,472:

Line 1,566:

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

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

Mapping: {{mapping| 1 -8 -8 -23 16 -14 | 0 13 14 35 -17 24 }}

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

Optimal tunings:

* WE: ~2 = 1199.2683{{c}}, ~5/3 = 884.3805{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9061{{c}}

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

Badness (Sintel): 1.48

=== Semiparakleismic ===

Line 1,485:

Line 1,581:

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

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

Mapping: {{mapping| 2 -3 -2 -11 -4 | 0 13 14 35 23 }}

: mapping generators: ~99/70, ~33/28

Optimal ~~tuning (POTE)~~: ~99/70 = 600.~~000~~{{c}}, ~6/5 = ~~315~~.~~181~~{{c}}

Optimal tunings:

* WE: ~99/70 = 599.9270{{c}}, ~33/28 = 284.7841{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8119{{c}}

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

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

Badness (Sintel): 1.13

==== Semiparamint ====

Line 1,500:

Line 1,599:

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

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

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

Optimal ~~tuning (POTE)~~: ~99/70 = 600.~~000~~{{c}}, ~6/5 = ~~315~~.~~156~~{{c}}

Optimal tunings:

* WE: ~99/70 = 599.8253{{c}}, ~33/28 = 284.7608{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8366{{c}}

Badness (Sintel): 1.~~396~~

Badness (Sintel): 1.40

==== Semiparawolf ====

Line 1,515:

Line 1,616:

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

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

Mapping: {{mapping| 2 -3 -2 -11 -4 -4 | 0 13 14 35 23 24 }}

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

Optimal tunings:

* WE: ~99/70 = 600.0569{{c}}, ~13/11 = 284.8431{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~13/11 = 284.8216{{c}}

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

Badness (Sintel): 1.67

== Counterkleismic ==

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

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

In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses (648/625)]] 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).

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)

[[Subgroup]]: 2.3.5.7

Line 1,532:

Line 1,635:

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

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

: mapping generators: ~2, ~6/5

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1778{{c}}, ~6/5 = 316.1065{{c}}

: [[error map]]: {{val| +0.178 -0.181 -0.469 +0.388 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.0631{{c}}

: error map: {{val| 0.000 -0.377 -0.799 +0.161 }}

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

[[Badness]] (Sintel): 2.29

=== 11-limit ===

Line 1,546:

Line 1,653:

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

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

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

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

Optimal tunings:

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

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

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

Badness (Sintel): 2.35

==== 13-limit ====

Line 1,559:

Line 1,668:

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~6/5 = 316.~~070~~{{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=0| 19, 205, 224~~, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef~~ }}

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

Badness (Sintel): 1.40

=== Counterlytic ===

Line 1,572:

Line 1,683:

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~6/5 = 316.~~065~~{{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): 2.~~162~~

Badness (Sintel): 2.16

==== 13-limit ====

Line 1,585:

Line 1,698:

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~6/5 = 316.~~065~~{{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| 19e, 205e, 224, 467e, 691, 915c }}

Badness (Sintel): 1.~~231~~

Badness (Sintel): 1.23

== Quincy ==

Line 1,599:

Line 1,714:

: mapping generators: ~2, ~1728/1715

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.000{{c}}, ~1728/1715 = 16.~~613~~{{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 }}

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

[[Badness]] (Sintel): 2.02

=== 11-limit ===

Line 1,613:

Line 1,733:

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

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

Optimal tunings:

* WE: ~2 = 1200.1286{{c}}, ~100/99 = 16.6147{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6101{{c}}

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

Badness (Sintel): 1.02

=== 13-limit ===

Line 1,626:

Line 1,748:

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

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

Optimal tunings:

* WE: ~2 = 1200.0554{{c}}, ~100/99 = 16.6028{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6011{{c}}

Line 1,639:

Line 1,763:

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

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

Optimal tunings:

* WE: ~2 = 1200.0647{{c}}, ~100/99 = 16.6025{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6004{{c}}

Line 1,652:

Line 1,778:

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~100/99 = 16.~~594~~{{c}}

Optimal tunings:

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

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

Line 1,659:

Line 1,787:

== Sfourth ==

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

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

[[Subgroup]]: 2.3.5.7

Line 1,666:

Line 1,794:

: mapping generators: ~2, ~49/48

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~49/48 = 26.~~287~~{{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 }}

[[Badness]] (Sintel): 3.~~120~~

[[Badness]] (Sintel): 3.12

=== 11-limit ===

Line 1,680:

Line 1,813:

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~49/48 = 26.~~286~~{{c}}

Optimal tunings:

* WE: ~2 = 1201.1486{{c}}, ~49/48 = 26.3112{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2461{{c}}

Badness (Sintel): 1.~~788~~

Badness (Sintel): 1.78

==== 13-limit ====

Line 1,693:

Line 1,828:

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~49/48 = 26.~~310~~{{c}}

Optimal 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.~~366~~

Badness (Sintel): 1.37

=== Sfour ===

Line 1,706:

Line 1,843:

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

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

Optimal tunings:

* WE: ~2 = 1200.4402{{c}}, ~49/48 = 26.2557{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2403{{c}}

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

Badness (Sintel): 2.53

==== 13-limit ====

Line 1,719:

Line 1,858:

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

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

Optimal tunings:

* WE: ~2 = 1200.3796{{c}}, ~49/48 = 26.2473{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2372{{c}}

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

Badness (Sintel): 2.14

== Trideci ==

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

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

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

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

[[Subgroup]]: 2.3.5.7

Line 1,735:

Line 1,876:

: mapping generators: ~256/245, ~3

[[Optimal tuning]] ([[~~POTE~~]]): ~256/245 = 92.3077{{c}}, ~3/2 = 699.~~1410~~{{c}}

[[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 }}

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

[[Badness]] (Sintel): 4.67

=== 11-limit ===

Line 1,749:

Line 1,895:

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

Optimal ~~tuning (POTE)~~: ~22/21 = 92.3077{{c}}, ~3/2 = 699.~~6179~~{{c}}

Optimal tunings:

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

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

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

Badness (Sintel): 2.80

=== 13-limit ===

Line 1,762:

Line 1,910:

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

Optimal ~~tuning (POTE)~~: ~22/21 = 92.~~3077~~{{c}}, ~3/2 = 699.~~2969~~{{c}}

Optimal 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):~~ 2.~~164~~

~~== Counterorson ==~~

~~Counterorson tempers out the~~ {{~~monzo| 147 -103 7~~ }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]].

~~[[Subgroup]]: 2.3.5.7~~

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

~~{{Mapping|legend=1| 1 0 -21 85 | 0 7 103 -363 }}~~

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

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

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

Badness (Sintel): 2.16

== References ==

@@ 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:
+Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, ragitritonic, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
 * ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
 * ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
 * ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
-* ''[[Modus]]'' (+64/63) → [[Tetracot family #Modus|Tetracot family]]
+* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]
-* ''[[Flattone]]'' (+81/80) → [[Meantone family #Flattone|Meantone family]]
+* [[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]]
@@ Line 223: / Line 223: @@
 == 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 277: @@
 : ''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 409: / Line 411: @@
 : ''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 416: / Line 420: @@
 {{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:
@@ Line 444: / Line 448: @@
 == Orga ==
+Orga may be described as the {{nowrap| 26 & 270 }} temperament, and [[1106edo]] gives a strong tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 494: / Line 500: @@
 : ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]].
-Aside from the ragisma, the seniority temperament tempers out the [[wadisma]], 201768035/201326592, and may be described as {{nowrap| 26 & 145 }}. It is so named because the senior comma ({{monzo| -17 62 -35 }}, quadla-sepquingu) is tempered out.
+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.
 [[Subgroup]]: 2.3.5.7
@@ Line 563: / Line 569: @@
 : ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]].
-The monzismic temperament tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]]. It may be described as the {{nowrap| 53 & 612 }} temperament. A notable tuning not appearing on the optimal ET sequence is [[665edo]], which is nearly equivalent to the pure-3's tuning.
+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.
 [[Subgroup]]: 2.3.5.7
@@ Line 674: / Line 680: @@
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.9305{{c}}, ~6/5 = 884.3923{{c}}
+* [[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}}, ~6/5 = 884.4423{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.4423{{c}}
 : error map: {{val| 0.000 +0.198 +0.282 -0.136 }}
@@ Line 746: / Line 752: @@
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasithird]].''
-The quasithird temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the ragisma and {{monzo| -60 29 0 5 }}.
+Quasithird may be described as the {{nowrap| 224 & 388 }} temperament, featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows it to temper out the ragisma and {{monzo| -60 29 0 5 }}. Its [[ploidacot]] is tetraploid delta-pentacot.
 [[Subgroup]]: 2.3.5.7
@@ Line 798: / Line 804: @@
 : ''For 5-limit version, see [[10th-octave temperaments #Neon]].''
-Deca temperament has a period of 1/10 octave and tempers out the [[linus comma]], {{monzo| 11 -10 -10 10 }}, neon comma {{monzo| 21 60 -50 }} and {{monzo| 12 -3 -14 9 }} (165288374272/164794921875).
+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
@@ Line 863: / Line 869: @@
 == Keenanose ==
-Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.
+Keenanose, the {{nowrap| 270 & 1889 }} temperament, was named by [[Eliora]] in 2022 for the fact that it uses [[385/384]], the keenanisma, as the generator.
 [[Subgroup]]: 2.3.5.7
@@ Line 915: / Line 921: @@
 : ''For the 5-limit version, see [[13th-octave temperaments #Aluminium]].''
-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.
+Aluminium was named by [[Eliora]] in 2023 after the 13th element. It tempers out the {{monzo| 92 -39 -13 }} comma, which sets [[135/128]] interval to be equal to 1/13th of the octave.
 [[Subgroup]]: 2.3.5.7
@@ Line 964: / Line 970: @@
 Badness (Sintel): 1.18
-== Countritonic ==
+== Ragitritonic ==
-: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic (5-limit)]].''
+: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
-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.
+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.
+Ragitritonic was named by [[Flora Canou]] in 2026 as a contraction of ''ragismic'' and ''tritonic''.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,009: / Line 1,017: @@
 Optimal tunings:
-* WE: ~2 = 1199.7916{{c}}, ~768/539 = 611.2698{{c}}
+* WE: ~2 = 1199.7916{{c}}, ~91/64 = 611.2698{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~768/539 = 611.3754{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~91/64 = 611.3754{{c}}
 {{Optimal ET sequence|legend=0| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}
@@ Line 1,023: / Line 1,031: @@
 [[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
-{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }}
+{{Mapping|legend=1| 2 -6 -1 -36 | 0 13 8 59 }}
-: mapping generators: ~2278125/1605632, ~448/405
+: mapping generators: ~2278125/1605632, ~7168/5625
-[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 600.000{{c}}, ~448/405 = 176.805{{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| 190, 224, 414, 638, 1052c, 1690bcc }}
+{{Optimal ET sequence|legend=1| 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc }}
-[[Badness]] (Sintel): 4.454
+[[Badness]] (Sintel): 4.45
 === 11-limit ===
@@ Line 1,037: / Line 1,049: @@
 Comma list: 3025/3024, 4375/4374, 1265625/1261568
-Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }}
+Mapping: {{mapping| 2 -6 -1 -36 -22 | 0 13 8 59 41 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~448/405 = 176.806{{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| 190, 224, 414, 638, 1052c }}
+{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638, 1052c }}
-Badness (Sintel): 1.357
+Badness (Sintel): 1.36
 === 13-limit ===
@@ Line 1,050: / Line 1,064: @@
 Comma list: 625/624, 729/728, 1575/1573, 2200/2197
-Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }}
+Mapping: {{mapping| 2 -6 -1 -36 -22 -6 | 0 13 8 59 41 19 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~195/176 = 176.804{{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| 190, 224, 414, 638, 1690bcc, 2328bccde }}
+{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638 }}
 Badness (Sintel): 0.936
 == Moulin ==
-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.
+Moulin can be described as the {{nowrap| 494 & 1619 }} temperament. It has a generator of ~[[22/13]], and it was named by [[Eliora]] in 2022 after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". However, the functional generator is ~[[13/11]], and 73 of them octave reduced reach the [[3/2|perfect fifth]]. Since [[11/8]] is within 23 generators, the 25-tone generator chain (4L 21s) of this temperament contains the 8:11:13 triad.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,065: / Line 1,081: @@
 [[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
-{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }}
+{{Mapping|legend=1| 1 -16 -9 -75 | 0 73 47 323 }}
-: Mapping generators: ~2, ~6422528/3796875
+: mapping generators: ~2, ~3796875/3211264
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6422528/3796875 = 910.9323{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0272{{c}}, ~3796875/3211264 = 289.0675{{c}}
+: [[error map]]: {{val| +0.027 +0.007 -0.084 +0.013 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3796875/3211264 = 289.0675{{c}}
+: error map: {{val| 0.000 -0.029 -0.142 -0.029 }}
-{{Optimal ET sequence|legend=1| 494, 1125, 1619 }}
+{{Optimal ET sequence|legend=1| 494, 1125, 1619, 8589cc, 10208cc }}
-[[Badness]] (Sintel): 5.931
+[[Badness]] (Sintel): 5.93
 === 11-limit ===
@@ Line 1,079: / Line 1,099: @@
 Comma list: 4375/4374, 759375/758912, 100663296/100656875
-Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }}
+Mapping: {{mapping| 1 -16 -9 -75 9 | 0 73 47 323 -23 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~1024/605 = 910.9323{{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| 494, 1125, 1619, 2113 }}
-Badness (Sintel): 2.240
+Badness (Sintel): 2.24
 === 13-limit ===
@@ Line 1,092: / Line 1,114: @@
 Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
-Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }}
+Mapping: {{mapping| 1 -16 -9 -75 9 9 | 0 73 47 323 -23 -22 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~22/13 = 910.9323{{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| 494, 1125, 1619, 2113 }}
-Badness (Sintel): 1.118
+Badness (Sintel): 1.12
 == Palladium ==
-: ''For the 5-limit version of this temperament, see [[46th-octave temperaments #Palladium]]''.
+: ''For the 5-limit version, see [[46th-octave temperaments #Palladium]]''.
 The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minor whole tones (10/9) fall short of seven octaves. This temperament can be described as {{nowrap| 46 & 414 }} temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.
@@ Line 1,110: / Line 1,134: @@
 {{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
-: Mapping generators: ~83349/81920, ~3
+: mapping generators: ~83349/81920, ~3
-[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 26.0870{{c}}, ~3/2 = 701.6074{{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| 46, 368, 414, 460, 874d }}
+{{Optimal ET sequence|legend=1| 46, …, 368, 414, 460, 874d }}
-[[Badness]] (Sintel): 7.807
+[[Badness]] (Sintel): 7.81
 === 11-limit ===
@@ Line 1,125: / Line 1,153: @@
 Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}
-Optimal tuning (POTE): ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.5951{{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| 46, 368, 414, 460, 874de }}
+{{Optimal ET sequence|legend=0| 46, …, 368, 414, 460, 874de }}
-Badness (Sintel): 2.439
+Badness (Sintel): 2.44
 === 13-limit ===
@@ Line 1,138: / Line 1,168: @@
 Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}
-Optimal tuning (POTE): ~65/64 = 26.0870{{c}}, ~3/2 = 701.6419{{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| 46, 368, 414, 460, 874de, 1334de }}
+{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334dde }}
-Badness (Sintel): 1.684
+Badness (Sintel): 1.68
 === 17-limit ===
@@ Line 1,151: / Line 1,183: @@
 Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}
-Optimal tuning (POTE): ~65/64 = 26.0870{{c}}, ~3/2 = 701.6425{{c}}
+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 ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334deg }}
+{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334ddeg }}
-Badness (Sintel): 1.143
+Badness (Sintel): 1.14
-== Oviminor ==
+== Counterorson ==
-: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Oviminor (5-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]].
-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.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}
+[[Comma list]]: 4375/4374, {{monzo| 154 -54 -21 -7 }}
-{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }}
+{{Mapping|legend=1| 1 0 -21 85 | 0 7 103 -363 }}
-: Mapping generators: ~2, ~6/5
+: mapping generators: ~2, ~{{monzo| 66 -23 -9 -3 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 315.7501{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0040{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7122{{c}}
+: [[error map]]: {{val| +0.004 -0.303 -0.041 -0.015 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7113{{c}}
+: error map: {{val| 0.000 +0.024 -0.051 -0.025 }}
-{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}
+{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}
-[[Badness]] (Sintel): 14.739
+[[Badness]] (Sintel): 7.92
-== Octoid ==
+== Oviminor ==
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Oviminor (5-limit)]].''
+Oviminor was named by [[Eliora]] in 2022 after the facts that it takes 184 minor thirds of [[6/5]] to reach the interval class of [[4/3]], the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}
+{{Mapping|legend=1| 1 -134 -134 -401 | 0 184 185 548 }}
+: mapping generators: ~2, ~5/3
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0193{{c}}, ~5/3 = 884.2638{{c}}
+: [[error map]]: {{val| +0.019 +0.010 -0.085 +0.032 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.2497{{c}}
+: error map: {{val| 0.000 -0.011 -0.120 +0.008 }}
+{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}
+[[Badness]] (Sintel): 14.7
+== Octoid ==
+: {{Main| Octoid }}
 : ''For the 5-limit version, see [[8th-octave temperaments #Octoid]].''
-The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). 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.
+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]].
+The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimizing the average damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out [[324/323]], [[375/374]], [[495/494]], [[625/624]], [[715/714]] or [[729/728]].
 [[Subgroup]]: 2.3.5.7
@@ Line 1,185: / Line 1,246: @@
 {{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
-: Mapping generators: ~49/45, ~7/5
+: mapping generators: ~49/45, ~7/5
-[[Optimal tuning]] ([[POTE]]): ~49/45 = 150.000{{c}}, ~7/5 = 583.940{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~49/45 = 150.0003{{c}}, ~7/5 = 583.9416{{c}}
+: [[error map]]: {{val| +0.002 -0.130 -0.547 +0.883 }}
+* [[CWE]]: ~49/45 = 150.0000{{c}}, ~7/5 = 583.9411{{c}}
+: error map: {{val| 0.000 -0.132 -0.549 +0.880 }}
 [[Tuning ranges]]:
@@ Line 1,195: / Line 1,260: @@
 * 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
-{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}
+{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}
-[[Badness]] (Sintel): 1.080
+[[Badness]] (Sintel): 1.08
-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
@@ Line 1,210: / Line 1,271: @@
 Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.962{{c}}
+Optimal tunings:
+* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9477{{c}}
 Tuning ranges:
@@ Line 1,216: / Line 1,279: @@
 * 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
-{{Optimal ET sequence|legend=0| 72, 152, 224 }}
+{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224, 824d }}
 Badness (Sintel): 0.466
-Scales: [[octoid72]], [[octoid80]]
 ==== 13-limit ====
@@ Line 1,229: / Line 1,290: @@
 Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.905{{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| 72, 152f, 224 }}
 Badness (Sintel): 0.631
-Scales: [[octoid72]], [[octoid80]]
-; Music
-* ''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 =====
@@ Line 1,247: / Line 1,305: @@
 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}}
+Optimal tunings:
+* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}
 {{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}
@@ Line 1,260: / Line 1,320: @@
 Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.932{{c}}
+Optimal tunings:
+* WE: ~12/11 = 149.9785{{c}}, ~7/5 = 583.8482{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}}
 {{Optimal ET sequence|legend=0| 72, 152fg, 224 }}
@@ Line 1,267: / Line 1,329: @@
 ==== 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{{cent}}.
+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}}.
 Subgroup: 2.3.5.7.11.13
@@ Line 1,275: / Line 1,337: @@
 Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.892{{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| 72, 152, 224f }}
+{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}
-Badness (Sintel): 0.0896
+Badness (Sintel): 0.896
 ===== 17-limit =====
@@ Line 1,288: / Line 1,352: @@
 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 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=0| 72, 152, 224fg, 296ffg }}
+{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}
 Badness (Sintel): 0.795
@@ Line 1,301: / Line 1,367: @@
 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 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| 72, 152, 224fg, 376ffgh }}
+{{Optimal ET sequence|legend=0| 8d, 72, 152 }}
 Badness (Sintel): 0.993
@@ Line 1,321: / Line 1,389: @@
 : mapping generators: ~448/429, ~7/5
-Optimal tuning (POTE): ~448/429 = 75.000{{c}}, ~13/8 = 841.015{{c}}
+Optimal tunings:
+* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}
+* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}
 {{Optimal ET sequence|legend=0| 80, 144, 224 }}
-Badness (Sintel): 1.273
+Badness (Sintel): 1.27
 ===== 17-limit =====
@@ Line 1,334: / Line 1,404: @@
 Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
-Optimal tuning (POTE): ~117/112 = 75.000{{c}}, ~13/8 = 840.932{{c}}
+Optimal tunings:
+* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9626{{c}}
+* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0463{{c}}
 {{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }}
-Badness (Sintel): 1.458
+Badness (Sintel): 1.46
 ===== 19-limit =====
@@ Line 1,345: / Line 1,417: @@
 Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
-Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }}
+Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}
-Optimal tuning (POTE): ~117/112 = 75.000{{c}}, ~13/8 = 840.896{{c}}
+Optimal tunings:
+* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}
+* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}
 {{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}
-Badness (Sintel): 1.443
+Badness (Sintel): 1.44
 == Parakleismic ==
@@ Line 1,357: / Line 1,431: @@
 : ''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.
+In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat [[6/5]], 13 of which give 32/3, and 14 give 64/5. While 118 no longer has better than a cent of accuracy in the 7-limit, it is a decent temperament there nonetheless, and this allows an extension adding [[3136/3125]] and 4375/4374, for which [[99edo]], 118edo, and especially [[217edo]] are accurate tunings.
+Parakleismic does not extend easily to the 11- or 13-limit. Possible 11-limit extensions include undecimal parakleismic (99 & 118), paralytic (99e & 118), parkleismic (80 & 99), and paradigmic (80 & 99e).
 [[Subgroup]]: 2.3.5.7
@@ Line 1,363: / Line 1,439: @@
 [[Comma list]]: 3136/3125, 4375/4374
-{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }}
+{{Mapping|legend=1| 1 -8 -8 -23 | 0 13 14 35 }}
-: mapping generators: ~2, ~6/5
+: mapping generators: ~2, ~5/3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 315.181{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7820{{c}}, ~5/3 = 884.6581{{c}}
+: [[error map]]: {{val| -0.218 +0.344 +0.644 -0.779 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8088{{c}}
+: error map: {{val| 0.000 +0.560 +1.010 -0.516 }}
-{{Optimal ET sequence|legend=1| 19, 80, 99, 217, 316, 415 }}
+{{Optimal ET sequence|legend=1| 19, 61d, 80, 99, 217, 316, 415 }}
 [[Badness]] (Sintel): 0.694
@@ Line 1,377: / Line 1,457: @@
 Comma list: 385/384, 3136/3125, 4375/4374
-Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }}
+Mapping: {{mapping| 1 -8 -8 -23 30 | 0 13 14 35 -36 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.251{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3296{{c}}, ~5/3 = 884.9921{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7519{{c}}
 {{Optimal ET sequence|legend=0| 19, 99, 118 }}
-Badness (Sintel): 1.643
+Badness (Sintel): 1.64
 === 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.
+Paralytic (99e & 118) tempers out [[441/440]], [[5632/5625]], and [[19712/19683]]. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575.
 Subgroup: 2.3.5.7.11
@@ Line 1,392: / Line 1,474: @@
 Comma list: 441/440, 3136/3125, 4375/4374
-Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }}
+Mapping: {{mapping| 1 -8 -8 -23 -57 | 0 13 14 35 82 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.220{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9940{{c}}, ~5/3 = 884.7757{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7800{{c}}
-{{Optimal ET sequence|legend=0| 19e, 99e, 118, 217, 335, 552d, 887dd }}
+{{Optimal ET sequence|legend=0| 19e, …, 99e, 118, 217, 335 }}
-Badness (Sintel): 1.191
+Badness (Sintel): 1.19
 ==== 13-limit ====
@@ Line 1,405: / Line 1,489: @@
 Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
-Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }}
+Mapping: {{mapping| 1 -8 -8 -23 -57 59 | 0 13 14 35 82 -75 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.214{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9218{{c}}, ~5/3 = 884.7285{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7858{{c}}
-{{Optimal ET sequence|legend=0| 99e, 118, 217, 552d, 769de }}
+{{Optimal ET sequence|legend=0| 99e, 118, 217 }}
-Badness (Sintel): 1.847
+Badness (Sintel): 1.85
 ==== Paraklein ====
-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]].
+Paraklein (19e & 118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].
 Subgroup: 2.3.5.7.11.13
@@ Line 1,420: / Line 1,506: @@
 Comma list: 196/195, 352/351, 625/624, 729/728
-Mapping: {{mapping| 1 5 6 12 25 15 | 0 -13 -14 -35 -82 -43 }}
+Mapping: {{mapping| 1 -8 -8 -23 -57 -28 | 0 13 14 35 82 43 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.225{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.8239{{c}}, ~5/3 = 884.6449{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7709{{c}}
-{{Optimal ET sequence|legend=0| 19e, 99ef, 118, 217ff, 335ff }}
+{{Optimal ET sequence|legend=0| 19e, …, 99ef, 118 }}
-Badness (Sintel): 1.554
+Badness (Sintel): 1.55
 === Parkleismic ===
@@ Line 1,433: / Line 1,521: @@
 Comma list: 176/175, 1375/1372, 2200/2187
-Mapping: {{mapping| 1 5 6 12 20 | 0 -13 -14 -35 -63 }}
+Mapping: {{mapping| 1 -8 -8 -23 -43 | 0 13 14 35 63 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.060{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.1848{{c}}, ~5/3 = 884.3386{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9158{{c}}
-{{Optimal ET sequence|legend=0| 19e, 80, 179, 259cd }}
+{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179, 259cd }}
-Badness (Sintel): 1.848
+Badness (Sintel): 1.85
 ==== 13-limit ====
@@ Line 1,446: / Line 1,536: @@
 Comma list: 169/168, 176/175, 325/324, 1375/1372
-Mapping: {{mapping| 1 5 6 12 20 10 | 0 -13 -14 -35 -63 -24 }}
+Mapping: {{mapping| 1 -8 -8 -23 -43 -14 | 0 13 14 35 63 24 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.075{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.5318{{c}}, ~5/3 = 884.5800{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9118{{c}}
-{{Optimal ET sequence|legend=0| 19e, 80, 179 }}
+{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179 }}
-Badness (Sintel): 1.511
+Badness (Sintel): 1.51
 === Paradigmic ===
@@ Line 1,459: / Line 1,551: @@
 Comma list: 540/539, 896/891, 3136/3125
-Mapping: {{mapping| 1 5 6 12 -1 | 0 -13 -14 -35 17 }}
+Mapping: {{mapping| 1 -8 -8 -23 16 | 0 13 14 35 -17 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.096{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.0616{{c}}, ~5/3 = 884.2124{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.8877{{c}}
-{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e }}
+{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e, 457bcddeeee }}
-Badness (Sintel): 1.379
+Badness (Sintel): 1.38
 ==== 13-limit ====
@@ Line 1,472: / Line 1,566: @@
 Comma list: 169/168, 325/324, 540/539, 832/825
-Mapping: {{mapping| 1 5 6 12 -1 10 | 0 -13 -14 -35 17 -24 }}
+Mapping: {{mapping| 1 -8 -8 -23 16 -14 | 0 13 14 35 -17 24 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.080{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.2683{{c}}, ~5/3 = 884.3805{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9061{{c}}
-{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e }}
+{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e }}
-Badness (Sintel): 1.479
+Badness (Sintel): 1.48
 === Semiparakleismic ===
@@ Line 1,485: / Line 1,581: @@
 Comma list: 3025/3024, 3136/3125, 4375/4374
-Mapping: {{mapping| 2 10 12 24 19 | 0 -13 -14 -35 -23 }}
+Mapping: {{mapping| 2 -3 -2 -11 -4 | 0 13 14 35 23 }}
+: mapping generators: ~99/70, ~33/28
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~6/5 = 315.181{{c}}
+Optimal tunings:
+* WE: ~99/70 = 599.9270{{c}}, ~33/28 = 284.7841{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8119{{c}}
-{{Optimal ET sequence|legend=0| 80, 118, 198, 316, 514c, 830c }}
+{{Optimal ET sequence|legend=0| 80, 118, 198, 316, 514c }}
-Badness (Sintel): 1.131
+Badness (Sintel): 1.13
 ==== Semiparamint ====
@@ Line 1,500: / Line 1,599: @@
 Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
-Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }}
+Mapping: {{mapping| 2 -3 -2 -11 -4 15 | 0 13 14 35 23 -16 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~6/5 = 315.156{{c}}
+Optimal tunings:
+* WE: ~99/70 = 599.8253{{c}}, ~33/28 = 284.7608{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8366{{c}}
 {{Optimal ET sequence|legend=0| 80, 118, 198 }}
-Badness (Sintel): 1.396
+Badness (Sintel): 1.40
 ==== Semiparawolf ====
@@ Line 1,515: / Line 1,616: @@
 Comma list: 169/168, 325/324, 364/363, 3136/3125
-Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }}
+Mapping: {{mapping| 2 -3 -2 -11 -4 -4 | 0 13 14 35 23 24 }}
-Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~6/5 = 315.184{{c}}
+Optimal tunings:
+* WE: ~99/70 = 600.0569{{c}}, ~13/11 = 284.8431{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~13/11 = 284.8216{{c}}
 {{Optimal ET sequence|legend=0| 80, 118f, 198f }}
-Badness (Sintel): 1.672
+Badness (Sintel): 1.67
 == Counterkleismic ==
-: ''For the 5-limit temperament, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''
-In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses (648/625)]] 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).
+In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses]] ((648/625)<sup>6</sup>) fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament, tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). It was named by analogy to [[catakleismic]] and parakleismic)
 [[Subgroup]]: 2.3.5.7
@@ Line 1,532: / Line 1,635: @@
 [[Comma list]]: 4375/4374, 158203125/157351936
-{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }}
+{{Mapping|legend=1| 1 -5 -4 -18 | 0 25 24 79 }}
-: Mapping generators: ~2, ~5/3
+: mapping generators: ~2, ~6/5
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 316.060{{c}}
+[[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 }}
-{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }}
+{{Optimal ET sequence|legend=1| 19, …, 205, 224, 243, 467 }}
-[[Badness]] (Sintel): 2.292
+[[Badness]] (Sintel): 2.29
 === 11-limit ===
@@ Line 1,546: / Line 1,653: @@
 Comma list: 540/539, 4375/4374, 2097152/2096325
-Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }}
+Mapping: {{mapping| 1 -5 -4 -18 19 | 0 25 24 79 -59 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.071{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9944{{c}}, ~6/5 = 316.0690{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0705{{c}}
 {{Optimal ET sequence|legend=0| 19, 205, 224 }}
-Badness (Sintel): 2.346
+Badness (Sintel): 2.35
 ==== 13-limit ====
@@ Line 1,559: / Line 1,668: @@
 Comma list: 540/539, 625/624, 729/728, 10985/10976
-Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }}
+Mapping: {{mapping| 1 -5 -4 -18 19 -15 | 0 25 24 79 -59 71 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.070{{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=0| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}
+{{Optimal ET sequence|legend=0| 19, 205, 224 }}
-Badness (Sintel): 1.400
+Badness (Sintel): 1.40
 === Counterlytic ===
@@ Line 1,572: / Line 1,683: @@
 Comma list: 1375/1372, 4375/4374, 496125/495616
-Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }}
+Mapping: {{mapping| 1 -5 -4 -18 -40 | 0 25 24 79 165 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.065{{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 }}
+{{Optimal ET sequence|legend=1| 19e, 205e, 224, 467e, 691, 915c }}
-Badness (Sintel): 2.162
+Badness (Sintel): 2.16
 ==== 13-limit ====
@@ Line 1,585: / Line 1,698: @@
 Comma list: 625/624, 729/728, 1375/1372, 10985/10976
-Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }}
+Mapping: {{mapping| 1 -5 -4 -18 -40 -15 | 0 25 24 79 165 71 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.065{{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| 19e, 205e, 224 }}
+{{Optimal ET sequence|legend=0| 19e, 205e, 224, 467e, 691, 915c }}
-Badness (Sintel): 1.231
+Badness (Sintel): 1.23
 == Quincy ==
@@ Line 1,599: / Line 1,714: @@
 {{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
+: mapping generators: ~2, ~1728/1715
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~1728/1715 = 16.613{{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| 72, 217, 289 }}
+{{Optimal ET sequence|legend=1| 72, 217, 289, 650d, 939dd }}
-[[Badness]] (Sintel): 2.016
+[[Badness]] (Sintel): 2.02
 === 11-limit ===
@@ Line 1,613: / Line 1,733: @@
 Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.613{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1286{{c}}, ~100/99 = 16.6147{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6101{{c}}
 {{Optimal ET sequence|legend=0| 72, 217, 289 }}
-Badness (Sintel): 1.021
+Badness (Sintel): 1.02
 === 13-limit ===
@@ Line 1,626: / Line 1,748: @@
 Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.602{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0554{{c}}, ~100/99 = 16.6028{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6011{{c}}
 {{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
@@ Line 1,639: / Line 1,763: @@
 Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.602{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0647{{c}}, ~100/99 = 16.6025{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6004{{c}}
 {{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
@@ Line 1,652: / Line 1,778: @@
 Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.594{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9287{{c}}, ~100/99 = 16.5930{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.5948{{c}}
 {{Optimal ET sequence|legend=0| 72, 145, 217 }}
@@ Line 1,659: / Line 1,787: @@
 == Sfourth ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sfourth]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 1,666: / Line 1,794: @@
 {{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
+: mapping generators: ~2, ~49/48
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/48 = 26.287{{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=1| 45, 46, 91, 137d }}
-[[Badness]] (Sintel): 3.120
+[[Badness]] (Sintel): 3.12
 === 11-limit ===
@@ Line 1,680: / Line 1,813: @@
 Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.286{{c}}
+Optimal tunings:
+* WE: ~2 = 1201.1486{{c}}, ~49/48 = 26.3112{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2461{{c}}
 {{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }}
-Badness (Sintel): 1.788
+Badness (Sintel): 1.78
 ==== 13-limit ====
@@ Line 1,693: / Line 1,828: @@
 Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.310{{c}}
+Optimal 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 }}
+{{Optimal ET sequence|legend=0| 45ef, 46, 91ef, 137def, 228ddeeefff }}
-Badness (Sintel): 1.366
+Badness (Sintel): 1.37
 === Sfour ===
@@ Line 1,706: / Line 1,843: @@
 Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.246{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.4402{{c}}, ~49/48 = 26.2557{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2403{{c}}
-{{Optimal ET sequence|legend=0| 45, 46, 91, 137d }}
+{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}
-Badness (Sintel): 2.531
+Badness (Sintel): 2.53
 ==== 13-limit ====
@@ Line 1,719: / Line 1,858: @@
 Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.239{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3796{{c}}, ~49/48 = 26.2473{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2372{{c}}
-{{Optimal ET sequence|legend=0| 45, 46, 91, 137d }}
+{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}
-Badness (Sintel): 2.144
+Badness (Sintel): 2.14
 == Trideci ==
-: ''For the 5-limit version of this temperament, see [[13th-octave temperaments #Tridecatonic]].''
+: ''For the 5-limit version, see [[13th-octave temperaments #Tridecatonic]].''
-The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from "tridecim" (Latin for "[[wikipedia:13|thirteen]]").
+The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from ''tridecim'' (Latin for "thirteen").
 [[Subgroup]]: 2.3.5.7
@@ Line 1,735: / Line 1,876: @@
 {{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
+: mapping generators: ~256/245, ~3
-[[Optimal tuning]] ([[POTE]]): ~256/245 = 92.3077{{c}}, ~3/2 = 699.1410{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~256/245 = 92.4141{{c}}, ~3/2 = 699.9466{{c}}
+: [[error map]]: {{val| +1.383 -0.626 -0.210 -2.554 }}
+* [[CWE]]: ~256/245 = 92.3077{{c}}, ~3/2 = 699.4521{{c}}
+: error map: {{val| 0.000 -2.503 -2.794 -6.740 }}
-{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }}
+{{Optimal ET sequence|legend=1| 26, 65, 91 }}
-[[Badness]] (Sintel): 4.671
+[[Badness]] (Sintel): 4.67
 === 11-limit ===
@@ Line 1,749: / Line 1,895: @@
 Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}
-Optimal tuning (POTE): ~22/21 = 92.3077{{c}}, ~3/2 = 699.6179{{c}}
+Optimal tunings:
+* WE: ~22/21 = 92.3729{{c}}, ~3/2 = 700.1118{{c}}
+* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.7703{{c}}
-{{Optimal ET sequence|legend=0| 26, 65, 91, 156d, 247cdde }}
+{{Optimal ET sequence|legend=0| 26, 65, 91 }}
-Badness (Sintel): 2.796
+Badness (Sintel): 2.80
 === 13-limit ===
@@ Line 1,762: / Line 1,910: @@
 Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}
-Optimal tuning (POTE): ~22/21 = 92.3077{{c}}, ~3/2 = 699.2969{{c}}
+Optimal tunings:
+* WE: ~22/21 = 92.4003{{c}}, ~3/2 = 699.9983{{c}}
-{{Optimal ET sequence|legend=0| 26, 65f, 91f, 156dff }}
+* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.4772{{c}}
-Badness (Sintel): 2.164
-== Counterorson ==
-Counterorson tempers out the {{monzo| 147 -103 7 }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]].
-[[Subgroup]]: 2.3.5.7
+{{Optimal ET sequence|legend=0| 26, 65f, 91f }}
-[[Comma list]]: 4375/4374, {{monzo| 154 -54 -21 -7 }}
-{{Mapping|legend=1| 1 0 -21 85 | 0 7 103 -363 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7113{{c}}
-{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}
-[[Badness]] (Sintel): 7.916
+Badness (Sintel): 2.16
 == References ==