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-This is a collection of [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the ragisma, [[4375/4374]] = {{monzo| -1 -7 4 1 }}. The ragisma is the smallest [[7-limit]] [[superparticular ratio]].
+{{Technical data page}}
+This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the ragisma, [[4375/4374]] ({{monzo| -1 -7 4 1 }}). The ragisma is the smallest [[7-limit]] [[superparticular ratio]].
-Since (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 7/6 = 4375/4374 × (27/25)<sup>2</sup>, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
+Since {{nowrap|(10/9)<sup>4</sup> {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)<sup>2</sup> }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
-Microtemperaments considered below are ennealimmal, supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
+Temperaments discussed elsewhere are:
+* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
+* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]
+* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]]
 * ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
-* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
+* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]]
-* ''[[Crepuscular]]'' (+50/49) → [[Jubilismic clan #Crepuscular|Jubilismic clan]] and [[Fifive family #Crepuscular|Fifive family]]
+* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]
-* ''[[Modus]]'' (+64/63) → [[Tetracot family #Modus|Tetracot family]]
-* ''[[Flattone]]'' (+81/80) → [[Meantone family #Flattone|Meantone family]]
-* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]] and [[Sensamagic clan #Sensi|Sensamagic clan]]
-* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
 * [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
-* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]
+* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]
-* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]]
+* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]]
-* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]
 * [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]]
-* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]
+* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]
+* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
 * ''[[Zarvo]]'' (+33075/32768) → [[Gravity family #Zarvo|Gravity family]]
+* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
+* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
 * ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
-* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortonic family #Mitonic|Minortonic family]]
+* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
-* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnuzmic family #Septimal vishnu|Vishnuzmic family]]
+* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]
-* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]
+* [[Parakleismic]] (+3136/3125) → [[Parakleismic family #Septimal parakleismic|Parakleismic family]]
-* ''[[Trillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Tricot family #Trillium|Tricot family]]
+* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnu family #Septimal vishnu|Vishnu family]]
+* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Nuwell temperaments #Maja|Nuwell temperaments]]
+* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortone family #Mitonic|Minortone family]]
 * ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]]
-* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]]
+* [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
-* ''[[Chlorine]]'' (+{{monzo| -52 -17 34}}) → [[17th-octave temperaments#Chlorine|17th-octave temperaments]]
+* [[Octoid]] (+16875/16807) → [[Stearnsmic clan #Octoid|Stearnsmic clan]]
-* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]
+* ''[[Ludizemlic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments #Ludizemlic|37th-octave temperaments]]
-* ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments#Dzelic|37th-octave temperaments]]
-== Ennealimmal ==
-{{Main| Ennealimmal }}
-Ennealimmal tempers out the two smallest 7-limit [[superparticular]] commas, 2401/2400 and 4375/4374, leading to a temperament of unusual [[efficiency]]. It also tempers out the [[ennealimma]], {{monzo| 1 -27 18 }}, which leads to the identification of (27/25)<sup>9</sup> with the [[octave]], and gives ennealimmal a [[period]] of 1/9 octave. Its [[pergen]] is (P8/9, P5/2). While 27/25 is a 5-limit interval, a stack of two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit.
-Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40~60/49, all of which have their own interesting advantages. Possible tunings are 441-, 612-, or 3600edo, though its hardly likely anyone could tell the difference.
-If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example). In particular, people fond of the idea of "[[tritave]]s" as analogous to octaves might consider the 28 or 43 note [[mos]] with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave mos, which is equivalent in average step size to a 17 2/3 to the octave mos.
-Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]], [[Tritrizo clan #Undecentic|undecentic]], [[Tritrizo clan #Schisennealimmal|schisennealimmal]], and [[Tritrizo clan #Lunennealimmal|lunennealimmal]].
--limit ennealimmal's S-expression-based comma list is {[[4375/4374|S25/S27]], [[2401/2400|S49]]}. Interestingly, the [[landscape comma]] is equal to [[2401/2400|S49]]/([[4375/4374|S25/S27]]) while the [[wizma]] is equal to [[2401/2400|S49]]*[[4375/4374|S25/S27]].
-''For the 5-limit temperament, see [[Ennealimma#Ennealimmal]].''
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 2401/2400, 4375/4374
-{{Mapping|legend=1| 9 1 1 12 | 0 2 3 2 }}
-{{Multival|legend=1| 18 27 18 1 -22 -34 }}
-: mapping generators: ~27/25, ~5/3
-[[Optimal tuning]] ([[POTE]]): ~27/25 = 1\9, ~5/3 = 884.3129 (~36/35 = 49.0205)
-[[Tuning ranges]]:
-* 7-odd-limit [[diamond monotone]]: ~36/35 = [26.667, 66.667] (1\45 to 1\18)
-* 9-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
-* 7- and 9-odd-limit [[diamond tradeoff]]: ~36/35 = [48.920, 49.179]
-* 7- and 9-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 49.179]
-{{Optimal ET sequence|legend=1| 27, 45, 72, 99, 171, 441, 612 }}
-[[Badness]]: 0.003610
-=== 11-limit ===
-The ennealimmal temperament can be described as 99e &amp; 171e, which tempers out [[5632/5625]] (vishdel comma) and [[19712/19683]] (symbiotic comma).
-Subgroup: 2.3.5.7.11
-Comma list: 2401/2400, 4375/4374, 5632/5625
-Mapping: {{mapping| 9 1 1 12 -75 | 0 2 3 2 16 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4679 (~36/35 = 48.8654)
-{{Optimal ET sequence|legend=1| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
-Badness: 0.027332
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374
-Mapping: {{mapping| 9 1 1 12 -75 93 | 0 2 3 2 16 -9 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
-{{Optimal ET sequence|legend=1| 99e, 171e, 270 }}
-Badness: 0.029404
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 715/714, 1001/1000, 1716/1715, 4096/4095, 4375/4374
-Mapping: {{mapping| 9 1 1 12 -75 93 -3 | 0 2 3 2 16 -9 6 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
-{{Optimal ET sequence|legend=1| 99e, 171e, 270 }}
-===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 715/714, 1001/1000, 1216/1215, 1716/1715, 4096/4095, 4375/4374
-Mapping: {{mapping| 9 1 1 12 -75 93 -3 -48 | 0 2 3 2 16 -9 6 13 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
-{{Optimal ET sequence|legend=1| 99e, 171e, 270 }}
-==== Ennealimmalis ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 2080/2079, 2401/2400, 4375/4374, 5632/5625
-Mapping: {{mapping| 9 1 1 12 -75 -106 | 0 2 3 2 16 21 }}
-Optimal tuning (CTE): ~27/25 = 1\9, ~5/3 = 884.4560 (~36/35 = 48.8773)
-{{Optimal ET sequence|legend=1| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}
-Badness: 0.022068
-=== Ennealimmia ===
-The ennealimmia temperament is an alternative extension and can be described as 99 & 171, which tempers out [[131072/130977]] (olympia).
-Subgroup: 2.3.5.7.11
-Comma list: 2401/2400, 4375/4374, 131072/130977
-Mapping: {{mapping| 9 1 1 12 124 | 0 2 3 2 -14 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4089 (~36/35 = 48.9244)
-{{Optimal ET sequence|legend=1| 99, 171, 270, 711, 981, 1251, 2232e }}
-Badness: 0.026463
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374
-Mapping: {{mapping| 9 1 1 12 124 93 | 0 2 3 2 -14 -9 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
-{{Optimal ET sequence|legend=1| 99, 171, 270, 711, 981, 1692e, 2673e }}
-Badness: 0.016607
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 936/935, 2080/2079, 2401/2400, 4096/4095, 4375/4374
-Mapping: {{mapping| 9 1 1 12 124 93 -3 | 0 2 3 2 -14 -9 6 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
-{{Optimal ET sequence|legend=1| 99, 171, 270 }}
-===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 936/935, 1216/1215, 2080/2079, 2401/2400, 4096/4095, 4375/4374
-Mapping: {{mapping| 9 1 1 12 124 93 -3 -48 | 0 2 3 2 -14 -9 6 13 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
-{{Optimal ET sequence|legend=1| 99, 171, 270 }}
-=== Ennealimnic ===
-Ennealimnic (72 &amp; 171) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.
-Subgroup: 2.3.5.7.11
-Comma list: 243/242, 441/440, 4375/4356
-Mapping: {{mapping| 9 1 1 12 -2 | 0 2 3 2 5 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9386 (~36/35 = 49.3948)
-Tuning ranges:
-* 11-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
-* 11-odd-limit diamond tradeoff: ~36/35 = [48.920, 52.592]
-* 11-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 52.592]
-{{Optimal ET sequence|legend=1| 72, 171, 243 }}
-Badness: 0.020347
-See also: [[Chords of ennealimnic]]
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 243/242, 364/363, 441/440, 625/624
-Mapping: {{mapping| 9 1 1 12 -2 -33 | 0 2 3 2 5 10 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9920 (~36/35 = 49.3414)
-Tuning ranges:
-* 13- and 15-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)
-* 13- and 15-odd-limit diamond tradeoff: ~36/35 = [48.825, 52.592]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~36/35 = [48.825, 50.000]
-{{Optimal ET sequence|legend=1| 72, 171, 243 }}
-Badness: 0.023250
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 243/242, 364/363, 375/374, 441/440, 595/594
-Mapping: {{mapping| 9 1 1 12 -2 -33 -3 | 0 2 3 2 5 10 6 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9981 (~36/35 = 49.3353)
-Tuning ranges:
-* 17-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)
-* 17-odd-limit diamond tradeoff: ~36/35 = [46.363, 52.592]
-* 17-odd-limit diamond monotone and tradeoff: ~36/35 = [48.485, 50.000]
-{{Optimal ET sequence|legend=1| 72, 171, 243 }}
-Badness: 0.014602
-===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 243/242, 364/363, 375/374, 441/440, 513/512, 595/594
-Mapping: {{mapping| 9 1 1 12 -2 -33 -3 78  | 0 2 3 2 5 10 6 -6 }}
-{{Optimal ET sequence|legend=1| 72, 171, 243 }}
-==== Ennealim ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 243/242, 325/324, 441/440
-Mapping: {{mapping| 9 1 1 12 -2 20 | 0 2 3 2 5 2 }}
-Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
-{{Optimal ET sequence|legend=1| 27e, 45ef, 72 }}
-Badness: 0.020697
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 169/168, 221/220, 243/242, 325/324, 441/440
-Mapping: {{mapping| 9 1 1 12 -2 20 -3 | 0 2 3 2 5 2 6 }}
-Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
-{{Optimal ET sequence|legend=1| 27eg, 45efg, 72 }}
-===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 169/168, 221/220, 243/242, 325/324, 441/440
-Mapping: {{mapping| 9 1 1 12 -2 20 -3 25 | 0 2 3 2 5 2 6 2 }}
-Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
-{{Optimal ET sequence|legend=1| 27eg, 45efg, 72 }}
-=== Ennealiminal ===
-Subgroup: 2.3.5.7.11
-Comma list: 385/384, 1375/1372, 4375/4374
-Mapping: {{mapping| 9 1 1 12 51 | 0 2 3 2 -3 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.8298 (~36/35 = 49.5036)
-{{Optimal ET sequence|legend=1| 27, 45, 72, 171e, 243e, 315e }}
-Badness: 0.031123
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 325/324, 385/384, 1375/1372
-Mapping: {{mapping| 9 1 1 12 51 20 | 0 2 3 2 -3 2 }}
-Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
-{{Optimal ET sequence|legend=1| 27, 45f, 72, 171ef, 243eff }}
-Badness: 0.030325
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 169/168, 221/220, 325/324, 385/384, 1375/1372
-Mapping: {{mapping| 9 1 1 12 51 20 50 | 0 2 3 2 -3 2 -2 }}
-Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
-{{Optimal ET sequence|legend=1| 27, 45f, 72 }}
-===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 153/152, 169/168, 221/220, 325/324, 385/384, 1375/1372
-Mapping: {{mapping| 9 1 1 12 51 20 50 25 | 0 2 3 2 -3 2 -2 2 }}
-Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
-{{Optimal ET sequence|legend=1| 27, 45f, 72 }}
-=== Hemiennealimmal ===
-Hemiennealimmal (72 &amp; 198) has a period of 1/18 octave and tempers out the four smallest superparticular commas of the 11-limit JI, 2401/2400, 3025/3024, 4375/4374, and 9801/9800. Tempering out [[9801/9800]] leads an octave split into two equal parts. Notably, every one of these commas is part of one or more known infinite comma families; see directly below.
-Its S-expression-based comma list is {([[3025/3024|S22/S24 = S55 = S25/S27 * S99]],) [[4375/4374|S25/S27]], [[2401/2400|S49]], [[9801/9800|S33/S35 = S99]]}.
-Subgroup: 2.3.5.7.11
-Comma list: 2401/2400, 3025/3024, 4375/4374
-Mapping: {{mapping| 18 0 -1 22 48 | 0 2 3 2 1 }}
-: mapping generators: ~80/77, ~400/231
-Optimal tuning (POTE): ~80/77 = 1\18, ~400/231 = 950.9553
-Tuning ranges:
-* 11-odd-limit diamond monotone: ~99/98 = [13.333, 22.222] (1\90 to 1\54)
-* 11-odd-limit diamond tradeoff: ~99/98 = [17.304, 17.985]
-* 11-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 17.985]
-{{Optimal ET sequence|legend=1| 72, 198, 270, 342, 612, 954, 1566 }}
-Badness: 0.006283
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024
-Mapping: {{mapping| 18 0 -1 22 48 -19 | 0 2 3 2 1 6 }}
-Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837
-Tuning ranges:
-* 13-odd-limit diamond monotone: ~99/98 = [16.667, 22.222] (1\72 to 1\54)
-* 15-odd-limit diamond monotone: ~99/98 = [16.667, 19.048] (1\72 to 2\126)
-* 13-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.309]
-* 15-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.926]
-* 13-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.309]
-* 15-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.926]
-{{Optimal ET sequence|legend=1| 72, 198, 270 }}
-Badness: 0.012505
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 676/675, 715/714, 1001/1000, 1716/1715, 3025/3024
-Mapping: {{mapping| 18 0 -1 22 48 -19 -12 | 0 2 3 2 1 6 6 }}
+Considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, chlorine, seniority, monzismic, semidimfourth, acrokleismic, quasithird, quincy, deca, keenanose, counterkleismic, sfourth, aluminium, ragitritonic, quatracot, trideci, moulin, and palladium.
-Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837
-{{Optimal ET sequence|legend=1| 72, 198g, 270 }}
-===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 676/675, 715/714, 1001/1000, 1331/1330, 1716/1715, 3025/3024
-Mapping: {{mapping| 18 0 -1 22 48 -19 -12 48 105 | 0 2 3 2 1 6 6 -2 }}
-Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837
-{{Optimal ET sequence|legend=1| 72, 198g, 270 }}
-==== Semihemiennealimmal ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374
-Mapping: {{mapping| 18 0 -1 22 48 88 | 0 4 6 4 2 -3 }}
-: mapping generators: ~80/77, ~1053/800
-Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
-{{Optimal ET sequence|legend=1| 126, 144, 270, 684, 954 }}
-Badness: 0.013104
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 2401/2400, 2431/2430, 3025/3024, 4225/4224, 4375/4374
-Mapping: {{mapping| 18 0 -1 22 48 88 -119 | 0 4 6 4 2 -3 27 }}
-: mapping generators: ~80/77, ~1053/800
-Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
-{{Optimal ET sequence|legend=1| 270, 684, 954 }}
-Badness: 0.013104
-===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 2401/2400, 2431/2430, 2926/2925, 3025/3024, 4225/4224, 4375/4374
-Mapping: {{mapping| 18 0 -1 22 48 88 -119 -2 | 0 4 6 4 2 -3 27 11 }}
-: mapping generators: ~80/77, ~1053/800
-Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
-{{Optimal ET sequence|legend=1| 270, 684h, 954h, 1224 }}
-Badness: 0.013104
-=== Semiennealimmal ===
-Semiennealimmal tempers out [[4000/3993]], and uses a ~140/121 semifourth generator. Notably, however, two generator steps do not reach ~4/3, despite that the name may suggest so. In fact, it splits the generator of ennealimmal into three.
-Subgroup: 2.3.5.7.11
-Comma list: 2401/2400, 4000/3993, 4375/4374
-Mapping: {{mapping| 9 3 4 14 18 | 0 6 9 6 7 }}
-: mapping generators: ~27/25, ~140/121
-Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3367
-{{Optimal ET sequence|legend=1| 72, 369, 441 }}
-Badness: 0.034196
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374
-Mapping: {{mapping| 9 3 4 14 18 -8 | 0 6 9 6 7 22 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3375
-{{Optimal ET sequence|legend=1| 72, 297ef, 369f, 441 }}
-Badness: 0.026122
-=== Quadraennealimmal ===
-Subgroup: 2.3.5.7.11
-Comma list: 2401/2400, 4375/4374, 234375/234256
-Mapping: {{mapping| 9 1 1 12 -7 | 0 8 12 8 23 }}
-: mapping generators: ~27/25, ~25/22
-Optimal tuning (POTE): ~27/25 = 1\9, ~25/22 = 221.0717
-{{Optimal ET sequence|legend=1| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
-Badness: 0.021320
-=== Trinealimmal ===
-Subgroup: 2.3.5.7.11
-Comma list: 2401/2400, 4375/4374, 2097152/2096325
-Mapping: {{mapping| 27 1 0 34 177 | 0 2 3 2 -4 }}
-: mapping generators: ~2744/2673, ~2352/1375
-Optimal tuning (POTE): ~2744/2673 = 1\27, ~2352/1375 = 928.8000
-{{Optimal ET sequence|legend=1| 27, 243, 270, 783, 1053, 1323 }}
-Badness: 0.029812
-=== Rhodium ===
-{{Main| Rhodium }}
-Rhodium splits the ennealimmal period in five parts and thereby features a period of 9 × 5 = 45, thus the name is given after the 45th element.
-Subgroup: 2.3.5.7.11
-Comma list: 2401/2400, 4375/4374, 117440512/117406179
-Mapping: {{mapping| 45 1 -1 56 226 | 0 2 3 2 -2 }}
-: mapping generators: ~3072/3025, ~55/32
-Optimal tunings:
-* CTE: ~3072/3025 = 1\45, ~55/32 = 937.6658 (~385/384 = 4.3325)
-* CWE: ~3072/3025 = 1\45, ~55/32 = 937.6630 (~385/384 = 4.3397)
-Optimal ET sequence: {{Optimal ET sequence| 45, 225c, 270, 1125, 1395, 1665, 5265d }}
-Badness: 0.0381
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 2401/2400, 4225/4224, 4375/4374, 6656/6655
-Mapping: {{mapping| 45 1 -1 56 226 272 | 0 2 3 2 -2 -3 }}
-Optimal tunings:
-* CTE: ~66/65 = 1\45, ~55/32 = 937.6569 (~385/384 = 4.3236)
-* CWE: ~66/65 = 1\45, ~55/32 = 937.6515 (~385/384 = 4.3182)
-Optimal ET sequence: {{Optimal ET sequence| 45, 270, 855, 1125, 1395, 1665, 3060d, 4725df }}
-Badness: 0.0226
 == Supermajor ==
-The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2<sup>15</sup>)/3, 46 give (2<sup>19</sup>)/5, and 75 give (2<sup>30</sup>)/7, leading to a wedgie of {{multival| 37 46 75 -13 15 45 }}. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80-note mos is presumably the place to start, and if that is not enough notes for you, there is always the 171-note mos.
+The generator for supermajor temperament is a supermajor third, [[9/7]], tuned about 0.002 cents flat. Note that in the data that follow, the generator is given as its [[octave complement]]. 37 of these give 3/2<sup>22</sup>, 46 give 5/2<sup>27</sup>, and 75 give 7/2<sup>45</sup>. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: [[1106edo]] or [[1277edo]] can be used as tunings, leading to accuracy even greater than that of [[ennealimmal]]. The 80-note generator chain is presumably the place to start, and if that is not enough notes for you, there is always the 171-note generator chain.
 [[Subgroup]]: 2.3.5.7
@@ Line 543: / Line 41: @@
 [[Comma list]]: 4375/4374, 52734375/52706752
-{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }}
+{{Mapping|legend=1| 1 -22 -27 -45 | 0 37 46 75 }}
+: mapping generators: ~2, ~14/9
-{{Multival|legend=1| 37 46 75 -13 15 45 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0067{{c}}, ~14/9 = 764.9222{{c}}
+: [[error map]]: {{val| +0.007 +0.019 -0.074 +0.037 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 764.9181{{c}}
+: error map: {{val| 0.000 +0.013 -0.083 +0.029 }}
-{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}
+{{Optimal ET sequence|legend=1| 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214 }}
-[[Badness]]: 0.010836
+[[Badness]] (Sintel): 0.274
 === Semisupermajor ===
@@ Line 558: / Line 59: @@
 Comma list: 3025/3024, 4375/4374, 35156250/35153041
-Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}
+{{Mapping|legend=0| 2 -7 -8 -15 -6 | 0 37 46 75 47 }}
+: mapping generators: ~99/70, ~11/10
-Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082
+Optimal tunings:
+* WE: ~99/70 = 600.0103{{c}}, ~11/10 = 164.9205{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 164.9180{{c}}
-{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}
+{{Optimal ET sequence|legend=0| 80, 262d, 342, 764, 1106, 1448, 2554, 4002e, 6556cee }}
-Badness: 0.012773
+Badness (Sintel): 0.422
 == Enneadecal ==
-Enneadecal temperament tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out [[703125/702464]], the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Enneadecal (5-limit)]].''
-''For the 5-limit temperament, see [[19th-octave temperaments#(5-limit) enneadecal]].''
+Enneadecal tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen [[6/5|just minor thirds]] fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out [[703125/702464]], the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones.
+[[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 576: / Line 82: @@
 {{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}
-{{Multival|legend=1| 19 19 57 -14 37 79 }}
 : mapping generators: ~28/27, ~3
-[[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)
+[[Optimal tuning]]s:
+* [[WE]]: ~28/27 = 63.1599{{c}}, ~3/2 = 701.9027{{c}} (~225/224 = 7.1437{{c}})
+: [[error map]]: {{val| +0.038 -0.014 -0.134 +0.080 }}
+* [[CWE]]: ~28/27 = 63.1579{{c}}, ~3/2 = 701.9002{{c}} (~225/224 = 7.1634{{c}})
+: error map: {{val| 0.000 -0.055 -0.203 +0.033 }}
 {{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}
-[[Badness]]: 0.010954
+[[Badness]] (Sintel): 0.277
 === 11-limit ===
@@ Line 592: / Line 99: @@
 Comma list: 540/539, 4375/4374, 16384/16335
-Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}
+{{Mapping|legend=0| 19 0 14 -37 126 | 0 1 1 3 -2 }}
-Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)
+Optimal tunings:
+* WE: ~28/27 = 63.1431{{c}}, ~3/2 = 702.1956{{c}} (~225/224 = 7.6216{{c}})
+* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.3164{{c}} (~225/224 = 7.5795{{c}})
-{{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }}
+{{Optimal ET sequence|legend=0| 19, 133d, 152, 323e, 475de, 627de }}
-Badness: 0.043734
+Badness (Sintel): 1.45
 ==== 13-limit ====
@@ Line 605: / Line 114: @@
 Comma list: 540/539, 625/624, 729/728, 2205/2197
-Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}
+{{Mapping|legend=0| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}
-Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)
+Optimal tunings:
+* WE: ~28/27 = 63.1406{{c}}, ~3/2 = 702.0192{{c}} (~225/224 = 7.4730{{c}})
+* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.1539{{c}} (~225/224 = 7.4171{{c}})
-{{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }}
+{{Optimal ET sequence|legend=0| 19, 133df, 152f, 323ef }}
-Badness: 0.033545
+Badness (Sintel): 1.39
 === Hemienneadecal ===
@@ Line 618: / Line 129: @@
 Comma list: 3025/3024, 4375/4374, 234375/234256
-Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}
+{{Mapping|legend=0| 38 0 28 -74 11 | 0 1 1 3 2 }}
 : mapping generators: ~55/54, ~3
-Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)
+Optimal tunings:
+* WE: ~55/54 = 31.5800{{c}}, ~3/2 = 701.9053{{c}} (~243/242 = 7.1448{{c}})
+* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9034{{c}} (~243/242 = 7.1666{{c}})
-{{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}
+{{Optimal ET sequence|legend=0| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}
-Badness: 0.009985
+Badness (Sintel): 0.330
 ==== Hemienneadecalis ====
@@ Line 633: / Line 145: @@
 Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256
-Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}
+{{Mapping|legend=0| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}
-Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)
+Optimal tunings:
+* WE: ~55/54 = 31.5785{{c}}, ~3/2 = 701.9995{{c}} (~243/242 = 7.2727{{c}})
+* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 702.0053{{c}} (~243/242 = 7.2685{{c}})
-{{Optimal ET sequence|legend=1| 152f, 342f, 494 }}
+{{Optimal ET sequence|legend=0| 152f, 342f, 494 }}
-Badness: 0.020782
+Badness (Sintel): 0.859
 ==== Hemienneadec ====
@@ Line 646: / Line 160: @@
 Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
-Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}
+{{Mapping|legend=0| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}
-Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444)
+Optimal tunings:
+* WE: ~55/54 = 31.5784{{c}}, ~3/2 = 701.9736{{c}} (~243/242 = 7.2493{{c}})
+* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9855{{c}} (~243/242 = 7.2487{{c}})
-{{Optimal ET sequence|legend=1| 152, 342, 494, 1330, 1824, 2318d }}
+{{Optimal ET sequence|legend=0| 152, 342, 494, 1330, 1824, 2318d }}
-Badness: 0.030391
+Badness (Sintel): 1.26
 ==== Semihemienneadecal ====
@@ Line 659: / Line 175: @@
 Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078
-Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}
+{{Mapping|legend=0| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}
+: mapping generators: ~55/54, ~429/250
-: mapping generators: ~55/54 = 1\38, ~55/54, ~429/250
+Optimal tunings:
+* WE: ~55/54 = 31.5799{{c}}, ~429/250 = 935.1824{{c}} (~144/143 = 12.2152{{c}})
+* CWE: ~55/54 = 31.5789{{c}}, ~429/250 = 935.1617{{c}} (~144/143 = 12.2067{{c}})
-Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)
+{{Optimal ET sequence|legend=0| 190, 304d, 494, 684, 1178, 2850, 4028ce }}
-{{Optimal ET sequence|legend=1| 190, 304d, 494, 684, 1178, 2850, 4028ce }}
+Badness (Sintel): 0.607
-Badness: 0.014694
 === Kalium ===
@@ Line 676: / Line 193: @@
 Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344
-Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}
+{{Mapping|legend=0| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}
+Optimal tunings:
+* WE: ~28/27 = 63.1582{{c}}, ~6545/5928 = 171.2448{{c}}
+* CWE: ~28/27 = 63.1579{{c}}, ~6545/5928 = 171.2439{{c}}
-Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244
+{{Optimal ET sequence|legend=0| 855, 988, 1843 }}
-{{Optimal ET sequence|legend=1| 855, 988, 1843 }}
+Badness (Sintel): 3.15
 == Semidimi ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimi]].''
-The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374.
+The generator of semidimi is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374.
 [[Subgroup]]: 2.3.5.7
@@ Line 691: / Line 212: @@
 [[Comma list]]: 4375/4374, 3955078125/3954653486
-{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}
+{{Mapping|legend=1| 1 -19 -25 -32 | 0 55 73 93 }}
+: mapping generators: ~2, ~35/27
-{{Multival|legend=1| 55 73 93 -12 -7 11 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0018{{c}}, ~35/27 = 449.1277{{c}}
+: [[error map]]: {{val| +0.002 +0.031 -0.040 -0.012 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 449.1270{{c}}
+: error map: {{val| 0.000 +0.030 -0.043 -0.015 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270
+{{Optimal ET sequence|legend=1| 8d, …, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
-{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
+[[Badness]] (Sintel): 0.382
-[[Badness]]: 0.015075
+== Brahmagupta ==
+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.
-== Brahmagupta ==
+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}}).
-The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 70368744177664/70338939985125
+[[Comma list]]: 4375/4374, {{monzo| 46 -14 -3 -6 }}
 {{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}
 : mapping generators: ~1157625/1048576, ~27/20
-{{Multival|legend=1| 21 56 -77 40 -181 -336 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~1157625/1048576 = 171.4275{{c}}, ~27/20 = 519.7125{{c}}
-[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716
+: [[error map]]: {{val| -0.007 +0.037 -0.034 -0.004 }}
+* [[CWE]]: ~1157625/1048576 = 171.4286{{c}}, ~27/20 = 519.7156{{c}}
+: error map: {{val| 0.000 +0.049 -0.018 +0.017 }}
-{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }}
+{{Optimal ET sequence|legend=1| 7, …, 217, 224, 441, 1106, 1547 }}
-[[Badness]]: 0.029122
+[[Badness]] (Sintel): 0.737
 === 11-limit ===
@@ Line 725: / Line 252: @@
 Comma list: 4000/3993, 4375/4374, 131072/130977
-Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}
+{{Mapping|legend=0| 7 2 -8 53 3 | 0 3 8 -11 7 }}
-Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704
+Optimal tunings:
+* WE: ~243/220 = 171.4208{{c}}, ~27/20 = 519.6807{{c}}
+* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7034{{c}}
-{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }}
+{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665 }}
-Badness: 0.052190
+Badness (Sintel): 1.73
 === 13-limit ===
@@ Line 738: / Line 267: @@
 Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374
-Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}
+{{Mapping|legend=0| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}
-Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.706
+Optimal tunings:
+* WE: ~243/220 = 171.4197{{c}}, ~27/20 = 519.6789{{c}}
+* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7052{{c}}
-{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771eef }}
+{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1106e }}
-Badness: 0.023132
+Badness (Sintel): 0.956
 == Abigail ==
-Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930]: "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."</ref>
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Abigail]].''
-''For the 5-limit temperament, see [[Very high accuracy temperaments#Abigail]].''
+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 755: / Line 288: @@
 [[Comma list]]: 4375/4374, 2147483648/2144153025
-{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }}
+{{Mapping|legend=1| 2 -4 -11 18 | 0 11 24 -19 }}
+: mapping generators: ~46305/32768, ~1536/1225
-: mapping generators: ~46305/32768, ~27/20
+[[Optimal tuning]]s:
+* [[WE]]: ~46305/32768 = 599.9699{{c}}, ~1536/1225 = 391.0818{{c}}
+: [[error map]]: {{val| -0.060 +0.065 -0.021 +0.079 }}
+* [[CWE]]: ~46305/32768 = 600.0000{{c}}, ~1536/1225 = 391.1007{{c}}
+: error map: {{val| 0.000 +0.152 +0.102 +0.262 }}
-{{Multival|legend=1| 22 48 -38 25 -122 -223 }}
+{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd }}
-[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899
+[[Badness]] (Sintel): 0.936
-{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}
-[[Badness]]: 0.037000
 === 11-limit ===
@@ Line 772: / Line 306: @@
 Comma list: 3025/3024, 4375/4374, 131072/130977
-Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }}
+{{Mapping|legend=0| 2 -4 -11 18 18 | 0 11 24 -19 -17 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901
+Optimal tunings:
+* WE: ~99/70 = 599.9782{{c}}, ~1536/1225 = 391.0852{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~1536/1225 = 391.0992{{c}}
-{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }}
+{{Optimal ET sequence|legend=0| 46, 132, 178, 224, 270, 494, 764 }}
-Badness: 0.012860
+Badness (Sintel): 0.425
 === 13-limit ===
@@ Line 785: / Line 321: @@
 Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
-Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}
+{{Mapping|legend=0| 2 -4 -11 18 18 25 | 0 11 24 -19 -17 -27 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903
+Optimal tunings:
+* WE: ~99/70 = 599.9862{{c}}, ~351/280 = 391.0879{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~351/280 = 391.0969{{c}}
-{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }}
+{{Optimal ET sequence|legend=0| 46, 178, 224, 270, 494, 764, 1258 }}
-Badness: 0.008856
+Badness (Sintel): 0.366
 == Gamera ==
-''For the 5-limit temperament, see [[High badness temperaments#Gamera]].
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Gamera]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 800: / Line 338: @@
 [[Comma list]]: 4375/4374, 589824/588245
-{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }}
+{{Mapping|legend=1| 1 -17 -30 2 | 0 23 40 1 }}
+: mapping generators: ~2, ~7/4
-: mapping generators: ~2, ~8/7
-{{Multival|legend=1| 23 40 1 10 -63 -110 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8483{{c}}, ~7/4 = 969.5415{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 969.6608{{c}}
 {{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}
-[[Badness]]: 0.037648
+[[Badness]] (Sintel): 0.953
 === Hemigamera ===
@@ Line 817: / Line 354: @@
 Comma list: 3025/3024, 4375/4374, 589824/588245
-Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}
+{{Mapping|legend=0| 2 -11 -20 5 10 | 0 23 40 1 -5 }}
+: mapping generators: ~99/70, ~99/80
-: mapping generators: ~99/70, ~8/7
+Optimal tunings:
+* WE: ~99/70 = 599.9323{{c}}, ~99/80 = 369.6212{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~99/80 = 369.6610{{c}}
-Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3370
+{{Optimal ET sequence|legend=0| 26, 172c, 198, 224, 422, 646, 1068d }}
-{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646, 1068d }}
+Badness (Sintel): 1.35
-Badness: 0.040955
 ==== 13-limit ====
@@ Line 832: / Line 370: @@
 Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024
-Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}
+{{Mapping|legend=0| 2 -11 -20 5 10 -8 | 0 23 40 1 -5 25 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373
+Optimal tunings:
+* WE: ~99/70 = 599.9207{{c}}, ~26/21 = 369.6139{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~26/21 = 369.6603{{c}}
-{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646f, 1068df }}
+{{Optimal ET sequence|legend=0| 26, 172cf, 198, 224, 422, 646f, 1068df }}
-Badness: 0.020416
+Badness (Sintel): 0.844
 === Semigamera ===
@@ Line 845: / Line 385: @@
 Comma list: 4375/4374, 14641/14580, 15488/15435
-Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }}
+{{Mapping|legend=0| 1 -40 -70 1 -77 | 0 46 80 2 89 }}
+: mapping generators: ~2, ~144/77
-: mapping generators: ~2, ~77/72
+Optimal tunings:
+* WE: ~2 = 1199.8845{{c}}, ~144/77 = 1084.7314{{c}}
-Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1642
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8345{{c}}
-{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }}
+{{Optimal ET sequence|legend=0| 73, 125, 198, 323, 521 }}
-Badness: 0.078
+Badness (Sintel): 2.59
 ==== 13-limit ====
@@ Line 860: / Line 401: @@
 Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580
-Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }}
+{{Mapping|legend=0| 1 -40 -70 1 -77 -131 | 0 46 80 2 89 149 }}
-Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1628
+Optimal tunings:
+* WE: ~2 = 1199.8726{{c}}, ~144/77 = 1084.7220{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8359{{c}}
-{{Optimal ET sequence|legend=1| 73f, 125f, 198, 323, 521 }}
+{{Optimal ET sequence|legend=0| 73f, 125f, 198, 323, 521 }}
-Badness: 0.044
+Badness (Sintel): 1.82
+== Crazy ==
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
+Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament, 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.
-== Orga ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 54975581388800/54936068900769
+[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}
+{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}
+: mapping generators: ~332150625/234881024, ~1125/1024
+[[Optimal tuning]]s:
+* [[WE]]: ~332150625/234881024 = 600.0019{{c}}, ~1125/1024 = 162.7479{{c}}
+: [[error map]]: {{val| +0.004 +0.030 -0.042 -0.014 }}
+* [[CWE]]: ~332150625/234881024 = 600.0000{{c}}, ~1125/1024 = 162.7474{{c}}
+: error map: {{val| 0.000 +0.024 -0.051 -0.022 }}
+{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}
+[[Badness]] (Sintel): 0.998
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }}
+Comma list: 3025/3024, 4375/4374, 2791309312/2790703125
-: mapping generators: ~7411887/5242880, ~1310720/1058841
+{{Mapping|legend=0| 2 1 6 -15 -8 | 0 8 -5 76 55 }}
-{{Multival|legend=1| 58 102 -2 27 -166 -291 }}
+Optimal tunings:
+* WE: ~99/70 = 600.0047{{c}}, ~1125/1024 = 162.7493{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~1125/1024 = 162.7481{{c}}
-[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104
+{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}
-{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}
+Badness (Sintel): 0.562
-[[Badness]]: 0.040236
+== Orga ==
+Orga may be described as the {{nowrap| 26 & 270 }} temperament, and [[1106edo]] gives a strong tuning.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 4375/4374, {{monzo| 41 -4 2 -14 }}
+{{Mapping|legend=1| 2 -8 -15 6 | 0 29 51 -1 }}
+: mapping generators: ~7411887/5242880, ~8/7
+[[Optimal tuning]]s:
+* [[WE]]: ~7411887/5242880 = 599.9927{{c}}, ~8/7 = 231.1012{{c}}
+: [[error map]]: {{val| -0.015 +0.037 -0.045 +0.029 }}
+* [[CWE]]: ~7411887/5242880 = 600.0000{{c}}, ~8/7 = 231.1037{{c}}
+: error map: {{val| 0.000 +0.053 -0.023 +0.070 }}
+{{Optimal ET sequence|legend=1| 26, …, 244, 270, 836, 1106, 1376, 2482 }}
+[[Badness]] (Sintel): 1.02
 === 11-limit ===
@@ Line 890: / Line 475: @@
 Comma list: 3025/3024, 4375/4374, 5767168/5764801
-Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }}
+{{Mapping|legend=0| 2 -8 -15 6 10 | 0 29 51 -1 -8 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
+Optimal tunings:
+* WE: ~99/70 = 600.0025{{c}}, ~8/7 = 231.1039{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1030{{c}}
-{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836, 1106 }}
+{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836, 1106 }}
-Badness: 0.016188
+Badness (Sintel): 0.535
 === 13-limit ===
@@ Line 903: / Line 490: @@
 Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360
-Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }}
+{{Mapping|legend=0| 2 -8 -15 6 10 -3 | 0 29 51 -1 -8 27 }}
+Optimal tunings:
+* WE: ~99/70 = 600.0192{{c}}, ~8/7 = 231.1102{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1033{{c}}
+{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836f, 1106f }}
+Badness (Sintel): 0.899
+== Chlorine ==
+: ''For the 5-limit version, see [[17th-octave temperaments #Chlorine]].''
+Chlorine (named after the 17th element) tempers out the [[septendecima]] in the 5-limit, and {{monzo| -49 4 22 -3 }} as well as the ragisma in the 7-limit. It has a 1/17-octave period, and can be described as {{nowrap| 289 & 323 }} temperament. Not only the semitwelfth, but also the ~5/4 can be used as a generator.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 4375/4374, {{monzo| -49 4 22 -3 }}
+{{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~25/24 = 70.5880{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9962{{c}}
+: [[error map]]: {{val| -0.004 +0.037 -0.030 -0.019 }}
+* [[CWE]]: ~25/24 = 70.5882{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9990{{c}}
+: error map: {{val| 0.000 +0.043 -0.021 -0.013 }}
+{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547, 3706, 5253 }}
+[[Badness]] (Sintel): 1.05
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 4375/4374, 41503/41472, 1879453125/1879048192
+{{Mapping|legend=0| 17 0 26 -87 207 | 0 2 1 10 -11 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
+Optimal tunings:
+* WE: ~25/24 = 70.5905{{c}}, ~693/400 = 951.0054{{c}}
+* CWE: ~25/24 = 70.5882{{c}}, ~693/400 = 950.9754{{c}}
-{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836f, 1106f }}
+{{Optimal ET sequence|legend=0| 289, 323, 612, 3349de, 3961de, …, 5797ddee }}
-Badness: 0.021762
+Badness (Sintel): 2.11
 == Seniority ==
-{{See also| Very high accuracy temperaments #Senior }}
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]].
-Aside from the ragisma, the seniority temperament (26 &amp; 145) tempers out the wadisma, 201768035/201326592. 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 920: / Line 545: @@
 [[Comma list]]: 4375/4374, 201768035/201326592
-{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}
+{{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }}
+: mapping generators: ~2, ~5120/3087
-{{Multival|legend=1| 35 62 -3 17 -103 -181 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0745{{c}}, ~5120/3087 = 877.2500{{c}}
+: [[error map]]: {{val| +0.075 +0.008 -0.016 -0.203 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5120/3087 = 877.1965{{c}}
+: error map: {{val| 0.000 -0.077 -0.130 -0.415 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3087/2560 = 322.804
+{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d }}
-{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}
+[[Badness]] (Sintel): 1.14
-[[Badness]]: 0.044877
 === Senator ===
-The senator temperament (26 &amp; 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It can be extended to the 13- and 17-limit immediately, by adding 364/363 and 595/594 to the comma list in this order.
+Senator (26 & 145) extends seniority by tempering out [[441/440]] and [[65536/65219]], and can be extended to the 13- and 17-limit immediately by adding [[364/363]] and [[595/594]] to the comma list in this order.
 Subgroup: 2.3.5.7.11
@@ Line 937: / Line 565: @@
 Comma list: 441/440, 4375/4374, 65536/65219
-Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }}
+{{Mapping|legend=0| 1 -24 -43 5 2 | 0 35 62 -3 2 }}
-Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
+Optimal tunings:
+* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}}
-{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316e, 487ee }}
+{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e }}
-Badness: 0.092238
+Badness (Sintel): 3.05
 ==== 13-limit ====
@@ Line 950: / Line 580: @@
 Comma list: 364/363, 441/440, 2200/2197, 4375/4374
-Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }}
+{{Mapping|legend=0| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}
-Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
+Optimal tunings:
+* WE: ~2 = 1199.7136{{c}}, ~108/65 = 877.9974{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2038{{c}}
-{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}
+{{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }}
-Badness: 0.044662
+Badness (Sintel): 1.85
 ==== 17-limit ====
@@ Line 963: / Line 595: @@
 Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197
-Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }}
+{{Mapping|legend=0| 1 -24 -43 5 2 -27 -31 | 0 35 62 -3 2 42 48 }}
-Optimal tuning (POTE): ~77/64 = 322.793
+Optimal tunings:
+* WE: ~2 = 1199.7195{{c}}, ~108/65 = 877.0018{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}}
-{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}
+{{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }}
-Badness: 0.026562
+Badness (Sintel): 1.35
 == Monzismic ==
-: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Monzismic]].
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]].
-The monzismic temperament (53 &amp; 612) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]].
+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 980: / Line 614: @@
 [[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}
-{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}
+{{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }}
+: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}
-{{Multival|legend=1| 2 37 -134 54 -218 -415 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0128{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9895{{c}}
+: [[error map]]: {{val| +0.013 +0.024 -0.049 -0.019 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9793{{c}}
+: error map: {{val| 0.000 +0.004 -0.080 -0.050 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207
+{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd }}
-{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}
+[[Badness]] (Sintel): 1.18
-[[Badness]]: 0.046569
 === Monzism ===
@@ Line 995: / Line 632: @@
 Comma list: 4375/4374, 41503/41472, 184549376/184528125
-Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}
+{{Mapping|legend=0| 1 0 -27 109 -159 | 0 2 37 -134 205 }}
-Optimal tuning (POTE): ~231/200 = 249.0193
+Optimal tunings:
+* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}}
-{{Optimal ET sequence|legend=1| 53, 559, 612 }}
+{{Optimal ET sequence|legend=0| 53, 559, 612, 3619de, 4231de, …, 6067ddee }}
-Badness: 0.057083
+Badness (Sintel): 1.89
 ==== 13-limit ====
@@ Line 1,008: / Line 647: @@
 Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
-Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}
+{{Mapping|legend=0| 1 0 -27 109 -159 -70 | 0 2 37 -134 205 93 }}
-Optimal tuning (POTE): ~231/200 = 249.0199
+Optimal tunings:
+* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}}
-{{Optimal ET sequence|legend=1| 53, 559, 612 }}
+{{Optimal ET sequence|legend=0| 53, 559, 612 }}
-Badness: 0.053780
+Badness (Sintel): 2.22
 == Semidimfourth ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimfourth]].''
-The semidimfourth temperament is featured by a semi-diminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, 235298/234375.
+The semidimfourth temperament is featured by a semidiminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, [[235298/234375]].
 [[Subgroup]]: 2.3.5.7
@@ Line 1,025: / Line 666: @@
 [[Comma list]]: 4375/4374, 235298/234375
-[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}
+{{Mapping|legend=1| 1 -10 -13 -17 | 0 31 41 53 }}
+: mapping generators: ~2, ~35/27
-{{Multival|legend=1| 31 41 53 -7 -3 8 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9936{{c}}, ~35/27 = 448.4533{{c}}
+: [[error map]]: {{val| -0.007 +0.160 +0.353 -0.694 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 448.4555{{c}}
+: error map: {{val| 0.000 +0.165 +0.361 -0.685 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456
+{{Optimal ET sequence|legend=1| 8d, …, 91, 99, 289, 388, 875 }}
-{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}
+[[Badness]] (Sintel): 1.40
-[[Badness]]: 0.055249
 === Neusec ===
@@ Line 1,040: / Line 684: @@
 Comma list: 3025/3024, 4375/4374, 235298/234375
-Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }}
+{{Mapping|legend=0| 2 -20 -26 -34 -17 | 0 31 41 53 32 }}
+: mapping generators: ~99/70, ~35/27
-Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547
+Optimal tunings:
+* WE: ~99/70 = 600.0381{{c}}, ~35/27 = 448.4812{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4546{{c}}
-{{Optimal ET sequence|legend=1| 8d, 190, 388 }}
+{{Optimal ET sequence|legend=0| 8d, …, 190, 388 }}
-Badness: 0.059127
+Badness (Sintel): 1.95
 ==== 13-limit ====
@@ Line 1,053: / Line 700: @@
 Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
-Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }}
+{{Mapping|legend=0| 2 -20 -26 -34 -17 -21 | 0 31 41 53 32 38 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545
+Optimal tunings:
+* WE: ~99/70 = 600.0034{{c}}, ~35/27 = 448.4573{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4549{{c}}
-{{Optimal ET sequence|legend=1| 8d, 190, 198, 388 }}
+{{Optimal ET sequence|legend=0| 8d, …, 190, 198, 388 }}
-Badness: 0.030941
+Badness (Sintel): 1.28
 == Acrokleismic ==
@@ Line 1,066: / Line 715: @@
 [[Comma list]]: 4375/4374, 2202927104/2197265625
-{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }}
+{{Mapping|legend=1| 1 -22 -22 -65 | 0 32 33 92 }}
+: mapping generators: ~2, ~5/3
-: mapping generators: ~2, ~6/5
-{{Multival|legend=1| 32 33 92 -22 56 121 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9305{{c}}, ~5/3 = 884.3923{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557
+: [[error map]]: {{val| -0.070 +0.126 +0.160 -0.221 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.4423{{c}}
+: error map: {{val| 0.000 +0.198 +0.282 -0.136 }}
 {{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}
-[[Badness]]: 0.056184
+[[Badness]] (Sintel): 1.42
 === 11-limit ===
@@ Line 1,083: / Line 733: @@
 Comma list: 4375/4374, 41503/41472, 172032/171875
-Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }}
+{{Mapping|legend=0| 1 -22 -22 -65 58 | 0 32 33 92 -74 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558
+Optimal tunings:
+* WE: ~2 = 1199.9698{{c}}, ~5/3 = 884.4193{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4414{{c}}
-{{Optimal ET sequence|legend=1| 19, 251, 270, 829, 1099, 1369, 1639 }}
+{{Optimal ET sequence|legend=0| 19, 251, 270, 829, 1099, 1369, 1639 }}
-Badness: 0.036878
+Badness (Sintel): 1.22
 ==== 13-limit ====
@@ Line 1,096: / Line 748: @@
 Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
-Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }}
+{{Mapping|legend=0| 1 -22 -22 -65 58 -56 | 0 32 33 92 -74 81 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557
+Optimal tunings:
+* WE: ~2 = 1199.9939{{c}}, ~5/3 = 884.4384{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4429{{c}}
-{{Optimal ET sequence|legend=1| 19, 251, 270 }}
+{{Optimal ET sequence|legend=0| 19, 251, 270 }}
-Badness: 0.026818
+Badness (Sintel): 1.11
 === Counteracro ===
@@ Line 1,109: / Line 763: @@
 Comma list: 4375/4374, 5632/5625, 117649/117612
-Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }}
+{{Mapping|legend=0| 1 -22 -22 -65 -141 | 0 32 33 92 196 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.553
+Optimal tunings:
+* WE: ~2 = 1199.8877{{c}}, ~5/3 = 884.3639{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4457{{c}}
-{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}
+{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1061e, 1331c, 1601c, 1871bc }}
-Badness: 0.042572
+Badness (Sintel): 1.41
 ==== 13-limit ====
@@ Line 1,122: / Line 778: @@
 Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
-Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }}
+{{Mapping|legend=0| 1 -22 -22 -65 -141 -56 | 0 32 33 92 196 81 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.554
+Optimal tunings:
+* WE: ~2 = 1199.9285{{c}}, ~5/3 = 884.3937{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4458{{c}}
-{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}
+{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1331c }}
-Badness: 0.026028
+Badness (Sintel): 1.08
 == 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 [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}.
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasithird]].''
-[[Subgroup]]: 2.3.5
-[[Comma list]]: {{monzo| 55 -64 20 }}
-{{Mapping|legend=1| 4 0 -11 | 0 5 16 }}
-: mapping generators: ~51200000/43046721, ~1594323/1280000
-[[Optimal tuning]] ([[POTE]]): ~51200000/43046721, ~1594323/1280000 = 380.395
-{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }}
-[[Badness]]: 0.099519
+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.
-=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
@@ Line 1,153: / Line 798: @@
 {{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
+: mapping generators: ~65536/55125, ~5103/4096
-{{Multival|legend=1| 20 64 -116 55 -240 -449 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~65536/55125 = 300.0052{{c}}, ~5103/4096 = 380.3949{{c}}
-[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 1\4, ~5103/4096 = 380.388
+: [[error map]]: {{val| +0.021 +0.020 -0.052 -0.031 }}
+* [[CWE]]: ~65536/55125 = 300.0000{{c}}, ~5103/4096 = 380.3884{{c}}
+: error map: {{val| 0.000 -0.013 -0.100 -0.089 }}
 {{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}
-[[Badness]]: 0.061813
+[[Badness]] (Sintel): 1.56
 === 11-limit ===
@@ Line 1,167: / Line 815: @@
 Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
-Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}
+{{Mapping|legend=0| 4 0 -11 48 43 | 0 5 16 -29 -23 }}
-Optimal tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387)
+Optimal tunings:
+* WE: ~65536/51125 = 300.0073{{c}}, ~5103/4096 = 380.3963{{c}} (or ~22/21 = 80.3890{{c}})
+* CWE: ~65536/51125 = 300.0000{{c}}, ~5103/4096 = 380.3868{{c}} (or ~22/21 = 80.3868{{c}})
-{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448 }}
+{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee }}
-Badness: 0.021125
+Badness (Sintel): 0.698
 === 13-limit ===
@@ Line 1,180: / Line 830: @@
 Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374
-Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}
+{{Mapping|legend=0| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}
-Optimal tuning (POTE): ~81/65 = 380.385 (or ~22/21 = 80.385)
+Optimal tunings:
+* WE: ~65536/51125 = 299.9985{{c}}, ~81/65 = 380.3833{{c}} (or ~22/21 = 80.3848{{c}})
+* CWE: ~65536/51125 = 300.0000{{c}}, ~81/65 = 380.3852{{c}} (or ~22/21 = 80.3852{{c}})
-{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}
+{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836 }}
-Badness: 0.029501
+Badness (Sintel): 1.22
-== Deca ==
-: ''For 5-limit version of this temperament, 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 (satritrizo-asepbigu).
+== Quincy ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 165288374272/164794921875
+[[Comma list]]: 4375/4374, 823543/819200
-{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}
+{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
+: mapping generators: ~2, ~1728/1715
-: mapping generators: ~15/14, ~6/5
+[[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 }}
-{{Multival|legend=1| 50 60 110 -21 34 87 }}
+{{Optimal ET sequence|legend=1| 72, 217, 289, 650d, 939dd }}
-[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577
+[[Badness]] (Sintel): 2.02
-{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}
-[[Badness]]: 0.080637
-Badness (Dirichlet): 2.041
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 391314/390625
+Comma list: 441/440, 4000/3993, 4375/4374
-Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}
+{{Mapping|legend=0| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}
-Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.582
+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=1| 80, 190, 270, 1000, 1270, 1540e, 1810e }}
+{{Optimal ET sequence|legend=0| 72, 217, 289 }}
-Badness: 0.024329
+Badness (Sintel): 1.02
-Badness (Dirichlet): 0.804
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
+Comma list: 364/363, 441/440, 676/675, 4375/4374
-Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}
+{{Mapping|legend=0| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}
-Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.602 (~40/39 = 44.398)
+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=1| 80, 190, 270, 730, 1000 }}
+{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
-Badness: 0.016810
+Badness (Sintel): 0.986
-Badness (Dirichlet): 0.695
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
-=== no-17's 19-limit ===
+Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155
-Subgroup: 2.3.5.7.11.13.19
-Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374, 1521/1520
+{{Mapping|legend=0| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}
-Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}
+Optimal tunings:
+* WE: ~2 = 1200.0647{{c}}, ~100/99 = 16.6025{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6004{{c}}
-Optimal tuning (CTE): ~15/14 = 1\10, ~6/5 = 315.581 (~39/38 = 44.419)
+{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
-{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}
+Badness (Sintel): 0.751
-Badness (Dirichlet): 0.556
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
-== Keenanose ==
+Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675
-Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.
-[[Subgroup]]: 2.3.5.7
+{{Mapping|legend=0| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}
-[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}
+Optimal tunings:
+* WE: ~2 = 1199.9287{{c}}, ~100/99 = 16.5930{{c}}
-{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
+* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.5948{{c}}
-: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}
+{{Optimal ET sequence|legend=0| 72, 145, 217 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 21 3 1 -10 }} = 4.4465
+Badness (Sintel): 0.924
-{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}
+== Deca ==
+: ''For 5-limit version, see [[10th-octave temperaments#Neon]].''
-[[Badness]]: 0.0858
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 117649/117612, 67110351/67108864
-Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}
-Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4465
-{{Optimal ET sequence|legend=1| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}
-Badness: 0.0308
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612
-Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}
-Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4466
-{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 4048 }}
-Badness: 0.0213
-== 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.
-[[Subgroup]]: 2.3.5
-[[Comma list]]: {{monzo| 92 -39 -13 }}
-[[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}
-: mapping generators: ~135/128, ~3
-[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897
-{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}
+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.
-[[Badness]]: 0.123
-=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}
+[[Comma list]]: 4375/4374, 165288374272/164794921875
-[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}
+{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}
+: mapping generators: ~15/14, ~460992/390625
-[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 702.0024
+[[Optimal tuning]]s:
+* [[WE]]: ~15/14 = 119.9966{{c}}, ~460992/390625 = 284.4150{{c}} (5625/5488 = 44.4219{{c}})
+: [[error map]]: {{val| -0.034 +0.106 +0.145 -0.268 }}
+* [[CWE]]: ~15/14 = 120.0000{{c}}, ~460992/390625 = 284.4182{{c}} (5625/5488 = 44.4182{{c}})
+: error map: {{val| 0.000 +0.136 +0.195 -0.226 }}
-{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}
+{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}
-[[Badness]]: 0.126
+[[Badness]] (Sintel): 2.04
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 234375/234256, 2097152/2096325
+Comma list: 3025/3024, 4375/4374, 391314/390625
-Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}
+{{Mapping|legend=0| 10 4 9 2 18 | 0 5 6 11 7 }}
-Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042
+Optimal tunings:
+* WE: ~15/14 = 120.0004{{c}}, ~33/28 = 284.4193{{c}} (77/75 = 44.4185{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4189{{c}} (77/75 = 44.4189{{c}})
-{{Optimal ET sequence|legend=1| 494, 1053, 1547, 3588e, 5135e }}
+{{Optimal ET sequence|legend=0| 80, 190, 270, 1000, 1270, 1540e, 1810e }}
-Badness: 0.0421
+Badness (Sintel): 0.804
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078
+Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
-Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}
+{{Mapping|legend=0| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}
-Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099
+Optimal tunings:
+* WE: ~15/14 = 120.0067{{c}}, ~33/28 = 284.4139{{c}} (~40/39 = 44.4006{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4048{{c}} (~40/39 = 44.4048{{c}})
-{{Optimal ET sequence|legend=1| 494, 1547, 2041, 4576def }}
+{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}
-Badness: 0.0286
+Badness (Sintel): 0.695
-== Countritonic ==
+=== 2.3.5.7.11.13.19 subgroup ===
-: ''For the 5-limit version of this temperament, see [[Schismic–Mercator equivalence continuum #Countritonic]] and [[High badness temperaments #Countritonic]]
+Subgroup: 2.3.5.7.11.13.19
-Countritonic (''co-un-tritonic'') can be described as the 53 & 422 temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit.
+Comma list: 1001/1000, 1521/1520, 3025/3024, 4225/4224, 4375/4374
-[[Subgroup]]: 2.3.5.7
+{{Mapping|legend=0| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}
-[[Comma list]]: 4375/4374, 68719476736/68356598625
+Optimal tunings:
+* WE: ~15/14 = 120.0045{{c}}, ~33/28 = 284.4140{{c}} (~39/38 = 44.4050{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4075{{c}} (~39/38 = 44.4075{{c}})
-{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }}
+{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}
-: mapping generators: ~2, ~45927/32768
+Badness (Sintel): 0.556
-[[Optimal tuning]] (CTE): ~2 = 1\1, ~45927/32768 = 588.6216
+== Keenanose ==
+Keenanose, the {{nowrap| 270 & 1889 }} temperament, was named by [[Eliora]] in 2022 for the fact that it uses [[385/384]], the keenanisma, as the generator.
-{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}
-[[Badness]]: 0.133
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 5632/5625, 2621440/2614689
-Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 }}
-Optimal tuning (CTE): ~2 = 1\1, ~539/384 = 588.6258
-{{Optimal ET sequence|legend=1| 53, 316e, 369, 422, 791e, 1213cde }}
-Badness: 0.0707
-=== 13-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625
-Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 -74 }}
-Optimal tuning (CTE): ~2 = 1\1, ~128/91 = 588.6277
-{{Optimal ET sequence|legend=1| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}
-Badness: 0.0366
-== Quatracot ==
-{{See also| Stratosphere }}
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
+[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}
-{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }}
+{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
+: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}
-: mapping generators: ~2278125/1605632, ~448/405
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0068{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4467{{c}}
+: [[error map]]: {{val| +0.007 +0.031 -0.035 -0.032 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4466{{c}}
+: error map: {{val| 0.000 +0.025 -0.043 -0.050 }}
-{{Multival|legend=1| 26 16 118 -35 114 229 }}
+{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}
-[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 1\2, ~448/405 = 176.805
-{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}
-[[Badness]]: 0.175982
+[[Badness]] (Sintel): 2.17
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 1265625/1261568
+Comma list: 4375/4374, 117649/117612, 67110351/67108864
-Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }}
+{{Mapping|legend=0| 1 2 3 3 3 | 0 -112 -183 -52 124 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 176.806
+Optimal tunings:
+* WE: ~2 = 1199.9970{{c}}, ~385/384 = 4.4465{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4465{{c}}
-{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c }}
+{{Optimal ET sequence|legend=0| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}
-Badness: 0.041043
+Badness (Sintel): 1.02
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 729/728, 1575/1573, 2200/2197
+Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612
-Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }}
+{{Mapping|legend=0| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~195/176 = 176.804
+Optimal tunings:
+* WE: ~2 = 1200.0065{{c}}, ~385/384 = 4.4467{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4467{{c}}
-{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1690bcc, 2328bccde }}
+{{Optimal ET sequence|legend=0| 270, 1079, 1349, 1619, 1889, 4048 }}
-Badness: 0.022643
+Badness (Sintel): 0.879
-== Moulin ==
+== Counterkleismic ==
-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 494 & 1619 temperament.
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''
+In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses]] ((648/625)<sup>6</sup>) fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament, tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). It was named by analogy to [[catakleismic]] and [[parakleismic]]).
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
+[[Comma list]]: 4375/4374, 158203125/157351936
-{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }}
+{{Mapping|legend=1| 1 -5 -4 -18 | 0 25 24 79 }}
+: mapping generators: ~2, ~6/5
-: mapping generators: ~2, ~6422528/3796875
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6422528/3796875 = 910.9323
+[[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| 494, 1125, 1619 }}
+{{Optimal ET sequence|legend=1| 19, …, 205, 224, 243, 467 }}
-[[Badness]]: 0.234
+[[Badness]] (Sintel): 2.29
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 759375/758912, 100663296/100656875
+Comma list: 540/539, 4375/4374, 2097152/2096325
-Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }}
+{{Mapping|legend=0| 1 -5 -4 -18 19 | 0 25 24 79 -59 }}
-Optimal tuning (CTE): ~2 = 1\1, ~1024/605 = 910.9323
+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=1| 494, 1125, 1619, 2113 }}
+{{Optimal ET sequence|legend=0| 19, 205, 224 }}
-Badness: 0.0678
+Badness (Sintel): 2.35
-=== 13-limit ===
-Since 11/8 is within 23 generators, the 25 tone MOS (4L 21s) of this temperament contains the 8:11:13 triad.
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
+Comma list: 540/539, 625/624, 729/728, 10985/10976
-Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }}
+{{Mapping|legend=0| 1 -5 -4 -18 19 -15 | 0 25 24 79 -59 71 }}
-Optimal tuning (CTE): ~2 = 1\1, ~22/13 = 910.9323
+Optimal tunings:
+* WE: ~2 = 1199.9827{{c}}, ~6/5 = 316.0650{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0695{{c}}
-{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}
+{{Optimal ET sequence|legend=0| 19, 205, 224 }}
-Badness: 0.0271
+Badness (Sintel): 1.40
-== Palladium ==
+=== Counterlytic ===
-: ''For the 5-limit version of this temperament, see [[46th-octave temperaments]]''.
-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 minortones (10/9) fall short of seven octaves. This temperament can be described as 46 &amp; 414 temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}
-{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
-: mapping generators: ~83349/81920, ~3
-{{Multival|legend=1| 46 92 -46 39 -202 -365 }}
-[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 1\46, ~3/2 = 701.6074
-{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }}
-[[Badness]]: 0.308505
-=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 134775333/134217728
+Comma list: 1375/1372, 4375/4374, 496125/495616
-Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}
+{{Mapping|legend=0| 1 -5 -4 -18 -40 | 0 25 24 79 165 }}
-Optimal tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951
+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| 46, 368, 414, 460, 874de }}
+{{Optimal ET sequence|legend=1| 19e, 205e, 224, 467e, 691, 915c }}
-Badness: 0.073783
+Badness (Sintel): 2.16
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
+Comma list: 625/624, 729/728, 1375/1372, 10985/10976
-Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}
+{{Mapping|legend=0| 1 -5 -4 -18 -40 -15 | 0 25 24 79 165 71 }}
-Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6419
+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=1| 46, 368, 414, 460, 874de, 1334de }}
+{{Optimal ET sequence|legend=0| 19e, 205e, 224, 467e, 691, 915c }}
-Badness: 0.040751
+Badness (Sintel): 1.23
-=== 17-limit ===
+== Sfourth ==
-Subgroup: 2.3.5.7.11.13.17
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sfourth]].''
-Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
-Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}
-Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425
-{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334deg }}
-Badness: 0.022441
-== Oviminor ==
-{{See also| Syntonic-kleismic equivalence continuum }}
-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, 64827/64000
-{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }}
+{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
+: mapping generators: ~2, ~49/48
-: mapping generators: ~2, ~6/5
+[[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 tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 315.7501
+{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
-{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}
-[[Badness]]: 0.582
-== Octoid ==
-''For the 5-limit temperament, see [[8th-octave temperaments#Octoid (5-limit)]].''
-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.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 16875/16807
+[[Badness]] (Sintel): 3.12
-{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
-{{Multival|legend=1| 24 32 40 -5 -4 3 }}
-: mapping generators: ~49/45, ~7/5
-[[Optimal tuning]] ([[POTE]]): ~49/45 = 1\8, ~7/5 = 583.940
-[[Tuning ranges]]:
-* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
-* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
-* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
-* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
-{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}
-[[Badness]]: 0.042670
-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 doesn't temper 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 1375/1372, 4000/3993
+Comma list: 121/120, 441/440, 4375/4374
-Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}
+{{Mapping|legend=0| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.962
+Optimal tunings:
+* WE: ~2 = 1201.1486{{c}}, ~49/48 = 26.3112{{c}}
-Tuning ranges:
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2461{{c}}
-* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
-* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
-{{Optimal ET sequence|legend=1| 72, 152, 224 }}
-Badness: 0.014097
+{{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }}
-Scales: [[octoid72]], [[octoid80]]
+Badness (Sintel): 1.78
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 625/624, 729/728, 1375/1372
+Comma list: 121/120, 169/168, 325/324, 441/440
-Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}
+{{Mapping|legend=0| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.905
+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=1| 72, 152f, 224 }}
+{{Optimal ET sequence|legend=0| 45ef, 46, 91ef, 137def, 228ddeeefff }}
-Badness: 0.015274
+Badness (Sintel): 1.37
-Scales: [[octoid72]], [[octoid80]]
+=== Sfour ===
+Subgroup: 2.3.5.7.11
-; Music
+Comma list: 385/384, 2401/2376, 4375/4374
-* ''Dreyfus'' (archived 2010) by [[Gene Ward Smith]] – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – octoid[72] in 224edo tuning
-===== 17-limit =====
+{{Mapping|legend=0| 1 2 3 3 3 | 0 -19 -31 -9 21 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 375/374, 540/539, 625/624, 715/714, 729/728
+Optimal tunings:
+* WE: ~2 = 1200.4402{{c}}, ~49/48 = 26.2557{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2403{{c}}
-Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}
+{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.842
+Badness (Sintel): 2.53
-{{Optimal ET sequence|legend=1| 72, 152fg, 224, 296, 520g }}
-Badness: 0.014304
-Scales: [[octoid72]], [[octoid80]]
-===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
-Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.932
-{{Optimal ET sequence|legend=1| 72, 152fg, 224 }}
-Badness: 0.016036
-Scales: [[octoid72]], [[octoid80]]
-==== 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}}.
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 325/324, 364/363, 540/539
+Comma list: 196/195, 364/363, 385/384, 4375/4374
-Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
+{{Mapping|legend=0| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.892
+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=1| 72, 152, 224f }}
+{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}
-Badness: 0.021679
+Badness (Sintel): 2.14
-Scales: [[octoid72]], [[octoid80]]
+== Aluminium ==
+: ''For the 5-limit version, see [[13th-octave temperaments #Aluminium]].''
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
-Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.811
-{{Optimal ET sequence|legend=1| 72, 152, 224fg, 296ffg }}
-Badness: 0.015614
-Scales: [[Octoid72]], [[Octoid80]]
-===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
-Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 584.064
-{{Optimal ET sequence|legend=1| 72, 152, 224fg, 376ffgh }}
-Badness: 0.016321
-Scales: [[Octoid72]], [[Octoid80]]
-==== Hexadecoid ====
+Aluminium tempers out {{monzo| 92 -39 -13 }} in the 5-limit and sets [[135/128]] to 1/13 of an [[octave]]. It was named by [[Eliora]] in 2023 after the 13th element.
-{{ See also | 16th-octave temperaments }}
-Hexadecoid (80 &amp; 144) has a period of 1/16 octave and tempers out 4225/4224.
-Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
-Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
-: mapping generators: ~448/429, ~7/5
-Optimal tuning (POTE): ~448/429 = 1\16, ~13/8 = 841.015
-{{Optimal ET sequence|legend=1| 80, 144, 224 }}
-Badness: 0.030818
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
-Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
-Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932
-{{Optimal ET sequence|legend=1| 80, 144, 224, 528dg }}
-Badness: 0.028611
-===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
-Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }}
-Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896
-{{Optimal ET sequence|legend=1| 80, 144, 224, 304dh, 528dghh }}
-Badness: 0.023731
-== Parakleismic ==
-{{Main| Parakleismic }}
-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, with the 7-limit wedgie being {{multival| 13 14 35 -8 19 42 }} and adding 3136/3125 and 4375/4374, and the 11-limit wedgie {{multival| 13 14 35 -36 -8 19 -102 42 -132 -222 }} adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.
-[[Subgroup]]: 2.3.5
-[[Comma list]]: 1224440064/1220703125
-{{Mapping|legend=1| 1 5 6 | 0 -13 -14 }}
-: mapping generators: ~2, ~6/5
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.240
-{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}
-[[Badness]]: 0.043279
-=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 3136/3125, 4375/4374
+[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}
-{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }}
+[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}
+: Mapping generators: ~135/128, ~3
-{{Multival|legend=1| 13 14 35 -8 19 42 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~135/128 = 92.3072{{c}}, ~3/2 = 701.9995{{c}}
+: [[error map]]: {{val| -0.006 +0.038 -0.030 -0.013 }}
+* [[CWE]]: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0030{{c}}
+: error map: {{val| 0.000 +0.048 -0.015 +0.001 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.181
+{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}
-{{Optimal ET sequence|legend=1| 19, 80, 99, 217, 316, 415 }}
-[[Badness]]: 0.027431
+[[Badness]] (Sintel): 3.20
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 385/384, 3136/3125, 4375/4374
+Comma list: 4375/4374, 234375/234256, 2097152/2096325
-Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }}
+{{Mapping|legend=0| 13 0 92 -355 148 | 0 1 -3 19 -5 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.251
+Optimal tunings:
+* WE: ~135/128 = 92.3062{{c}}, ~3/2 = 701.9946{{c}}
+* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0056{{c}}
-{{Optimal ET sequence|legend=1| 19, 99, 118 }}
+{{Optimal ET sequence|legend=0| 494, 1053, 1547, 3588e, 5135e }}
-Badness: 0.049711
+Badness (Sintel): 1.39
-=== Paralytic ===
+=== 13-limit ===
-The ''paralytic'' temperament (118&amp;217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 &amp; 217 tempers out 1001/1000, 1575/1573, and 3584/3575.
-Subgroup: 2.3.5.7.11
-Comma list: 441/440, 3136/3125, 4375/4374
-Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.220
-{{Optimal ET sequence|legend=1| 19e, 99e, 118, 217, 335, 552d, 887dd }}
-Badness: 0.036027
-==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
+Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078
-Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }}
+{{Mapping|legend=0| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.214
+Optimal tunings:
+* WE: ~135/128 = 92.3055{{c}}, ~3/2 = 701.9928{{c}}
+* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0098{{c}}
-{{Optimal ET sequence|legend=1| 99e, 118, 217, 552d, 769de }}
+{{Optimal ET sequence|legend=0| 494, 1547, 2041, 4576def }}
-Badness: 0.044710
+Badness (Sintel): 1.18
-==== Paraklein ====
+== Ragitritonic ==
-The ''paraklein'' temperament (19e &amp; 118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].
+: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
-Subgroup: 2.3.5.7.11.13
+Ragitritonic may be described as the {{nowrap| 53 & 369 }} temperament, splitting the [[24/1|24th harmonic]] into nine tritone generators; its [[ploidacot]] is thus delta-enneacot. [[422edo]] makes for a strong tuning.
-Comma list: 196/195, 352/351, 625/624, 729/728
+Ragitritonic was named by [[Flora Canou]] in 2026 as a contraction of ''ragismic'' and ''tritonic''.
-Mapping: {{mapping| 1 5 6 12 25 15 | 0 -13 -14 -35 -82 -43 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.225
+[[Comma list]]: 4375/4374, 68719476736/68356598625
-{{Optimal ET sequence|legend=1| 19e, 99ef, 118, 217ff, 335ff }}
+{{Mapping|legend=1| 1 -3 -15 40 | 0 9 34 -73 }}
+: mapping generators: ~2, ~65536/45927
-Badness: 0.037618
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8189{{c}}, ~65536/45927 = 611.2850{{c}}
+: [[error map]]: {{val| -0.181 +0.153 +0.094 +0.123 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~65536/45927 = 611.3775{{c}}
+: error map: {{val| 0.000 +0.443 +0.522 +0.615 }}
-=== Parkleismic ===
+{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}
-Subgroup: 2.3.5.7.11
-Comma list: 176/175, 1375/1372, 2200/2187
+[[Badness]] (Sintel): 3.37
-Mapping: {{mapping| 1 5 6 12 20 | 0 -13 -14 -35 -63 }}
+=== 11-limit ===
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.060
-{{Optimal ET sequence|legend=1| 19e, 80, 179, 259cd }}
-Badness: 0.055884
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 176/175, 325/324, 1375/1372
-Mapping: {{mapping| 1 5 6 12 20 10 | 0 -13 -14 -35 -63 -24 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.075
-{{Optimal ET sequence|legend=1| 19e, 80, 179 }}
-Badness: 0.036559
-=== Paradigmic ===
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 896/891, 3136/3125
+Comma list: 4375/4374, 5632/5625, 2621440/2614689
-Mapping: {{mapping| 1 5 6 12 -1 | 0 -13 -14 -35 17 }}
+{{Mapping|legend=0| 1 -3 -15 40 -75 | 0 9 34 -73 154 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.096
+Optimal tunings:
+* WE: ~2 = 1199.8147{{c}}, ~768/539 = 611.2822{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~768/539 = 611.3762{{c}}
-{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}
+{{Optimal ET sequence|legend=0| 53, 316e, 369, 422, 791e, 1213cde }}
-Badness: 0.041720
+Badness (Sintel): 2.34
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 325/324, 540/539, 832/825
+Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625
-Mapping: {{mapping| 1 5 6 12 -1 10 | 0 -13 -14 -35 17 -24 }}
+{{Mapping|legend=0| 1 -3 -15 40 -75 -34 | 0 9 34 -73 154 74 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.080
+Optimal tunings:
+* WE: ~2 = 1199.7916{{c}}, ~91/64 = 611.2698{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~91/64 = 611.3754{{c}}
-{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}
+{{Optimal ET sequence|legend=0| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}
-Badness: 0.035781
+Badness (Sintel): 1.51
-=== Semiparakleismic ===
+== Quatracot ==
-Subgroup: 2.3.5.7.11
+{{See also| Stratosphere }}
-Comma list: 3025/3024, 3136/3125, 4375/4374
-Mapping: {{mapping| 2 10 12 24 19 | 0 -13 -14 -35 -23 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.181
-{{Optimal ET sequence|legend=1| 80, 118, 198, 316, 514c, 830c }}
-Badness: 0.034208
-==== Semiparamint ====
-This extension was named ''semiparakleismic'' in the earlier materials.
-Subgroup: 2.3.5.7.11.13
-Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
-Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.156
-{{Optimal ET sequence|legend=1| 80, 118, 198 }}
-Badness: 0.033775
-==== Semiparawolf ====
-This extension was named ''gentsemiparakleismic'' in the earlier materials.
-Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 325/324, 364/363, 3136/3125
-Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }}
-Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 315.184
-{{Optimal ET sequence|legend=1| 80, 118f, 198f }}
-Badness: 0.040467
-== Counterkleismic ==
-{{See also| High badness temperaments #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 19 &amp; 224 temperament (''counterkleismic'', named by analogy to [[catakleismic]] and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 158203125/157351936
+[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
-{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }}
+{{Mapping|legend=1| 2 -6 -1 -36 | 0 13 8 59 }}
+: mapping generators: ~2278125/1605632, ~7168/5625
-: mapping generators: ~2, ~5/3
+[[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 }}
-{{Multival|legend=1| 25 24 79 -20 55 116 }}
+{{Optimal ET sequence|legend=1| 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.060
+[[Badness]] (Sintel): 4.45
-{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }}
-[[Badness]]: 0.090553
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 4375/4374, 2097152/2096325
+Comma list: 3025/3024, 4375/4374, 1265625/1261568
-Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }}
+{{Mapping|legend=0| 2 -6 -1 -36 -22 | 0 13 8 59 41 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.071
+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=1| 19, 205, 224 }}
+{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638, 1052c }}
-Badness: 0.070952
+Badness (Sintel): 1.36
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 625/624, 729/728, 10985/10976
+Comma list: 625/624, 729/728, 1575/1573, 2200/2197
-Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }}
+{{Mapping|legend=0| 2 -6 -1 -36 -22 -6 | 0 13 8 59 41 19 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.070
+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=1| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}
+{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638 }}
-Badness: 0.033874
+Badness (Sintel): 0.936
-=== Counterlytic ===
+== Trideci ==
-Subgroup: 2.3.5.7.11
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tridecatonic]].''
-Comma list: 1375/1372, 4375/4374, 496125/495616
-Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065
-{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}
-Badness: 0.065400
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 729/728, 1375/1372, 10985/10976
-Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065
+The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic]] temperament, but with the ragisma (4375/4374) rather than the octagar comma (4000/3969) tempered out. The name ''trideci'' comes from ''tridecim'' (Latin for "thirteen").
-{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}
-Badness: 0.029782
-== Quincy ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 823543/819200
+[[Comma list]]: 4375/4374, 83349/81920
-{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
+{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
+: mapping generators: ~256/245, ~3
-{{Multival|legend=1| 30 49 14 8 -62 -105 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1715 = 16.613
+[[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| 72, 217, 289 }}
+{{Optimal ET sequence|legend=1| 26, 65, 91 }}
-[[Badness]]: 0.079657
+[[Badness]] (Sintel): 4.67
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 441/440, 4000/3993, 4375/4374
+Comma list: 245/242, 385/384, 4375/4374
-Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}
+{{Mapping|legend=0| 13 0 -11 57 45 | 0 1 2 -1 0 }}
-Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.613
+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=1| 72, 217, 289 }}
+{{Optimal ET sequence|legend=0| 26, 65, 91 }}
-Badness: 0.030875
+Badness (Sintel): 2.80
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 441/440, 676/675, 4375/4374
+Comma list: 169/168, 245/242, 325/324, 385/384
-Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}
+{{Mapping|legend=0| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}
-Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602
+Optimal tunings:
+* WE: ~22/21 = 92.4003{{c}}, ~3/2 = 699.9983{{c}}
+* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.4772{{c}}
-{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}
+{{Optimal ET sequence|legend=0| 26, 65f, 91f }}
-Badness: 0.023862
+Badness (Sintel): 2.16
-=== 17-limit ===
+== Moulin ==
-Subgroup: 2.3.5.7.11.13.17
+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.
-Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155
-Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}
-Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602
-{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}
-Badness: 0.014741
-=== 19-limit ===
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675
-Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}
-Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.594
-{{Optimal ET sequence|legend=1| 72, 145, 217 }}
-Badness: 0.015197
-== Sfourth ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 64827/64000
+[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
-{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
+{{Mapping|legend=1| 1 -16 -9 -75 | 0 73 47 323 }}
+: mapping generators: ~2, ~3796875/3211264
-{{Multival|legend=1| 19 31 9 5 -39 -66 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 26.287
+[[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| 45, 46, 91, 137d }}
+{{Optimal ET sequence|legend=1| 494, 1125, 1619, 8589cc, 10208cc }}
-[[Badness]]: 0.123291
+[[Badness]] (Sintel): 5.93
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 121/120, 441/440, 4375/4374
+Comma list: 4375/4374, 759375/758912, 100663296/100656875
-Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}
+{{Mapping|legend=0| 1 -16 -9 -75 9 | 0 73 47 323 -23 }}
-Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.286
+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=1| 45e, 46, 91e, 137de }}
+{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}
-Badness: 0.054098
+Badness (Sintel): 2.24
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 121/120, 169/168, 325/324, 441/440
+Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
-Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}
+{{Mapping|legend=0| 1 -16 -9 -75 9 9 | 0 73 47 323 -23 -22 }}
-Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.310
+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=1| 45ef, 46, 91ef, 137def }}
+{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}
-Badness: 0.033067
+Badness (Sintel): 1.12
-=== Sfour ===
+== Palladium ==
-Subgroup: 2.3.5.7.11
+: ''For the 5-limit version, see [[46th-octave temperaments #Palladium]]''.
-Comma list: 385/384, 2401/2376, 4375/4374
-Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}
+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.
-Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.246
-{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
-Badness: 0.076567
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 196/195, 364/363, 385/384, 4375/4374
-Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}
-Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.239
-{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
-Badness: 0.051893
-== Trideci ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tridecatonic]].''
-The trideci temperament (26 &amp; 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from "tridecim" (Latin for "[[wikipedia:13|thirteen]]").
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 83349/81920
+[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}
-{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
+{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
+: mapping generators: ~83349/81920, ~3
-[[Optimal tuning]] ([[POTE]]): ~256/245 = 1\13, ~3/2 = 699.1410
+[[Optimal tuning]]s:
+* [[WE]]: ~83349/81920 = 26.0910{{c}}, ~3/2 = 701.7155{{c}}
+: [[error map]]: {{val| +0.185 -0.055 -0.061 +0.349 }}
+* [[CWE]]: ~83349/81920 = 26.0870{{c}}, ~3/2 = 701.6491{{c}}
+: error map: {{val| 0.000 -0.306 -0.407 -0.910 }}
-{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }}
+{{Optimal ET sequence|legend=1| 46, …, 368, 414, 460, 874d }}
-[[Badness]]: 0.184585
+[[Badness]] (Sintel): 7.81
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 245/242, 385/384, 4375/4374
+Comma list: 3025/3024, 4375/4374, 134775333/134217728
-Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}
+{{Mapping|legend=0| 46 0 -39 202 232 | 0 1 2 -1 -1 }}
-Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.6179
+Optimal tunings:
+* WE: ~8192/8085 = 26.0912{{c}}, ~3/2 = 701.7082{{c}}
+* CWE: ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.6173{{c}}
-{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdde }}
+{{Optimal ET sequence|legend=0| 46, …, 368, 414, 460, 874de }}
-Badness: 0.084590
+Badness (Sintel): 2.44
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 245/242, 325/324, 385/384
+Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
-Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}
+{{Mapping|legend=0| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}
-Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.2969
+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=1| 26, 65f, 91f, 156dff }}
+{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334dde }}
-Badness: 0.052366
+Badness (Sintel): 1.68
-== Counterorson ==
+=== 17-limit ===
-Counterorson tempers out the {{monzo| 147 -103 7 }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]].
+Subgroup: 2.3.5.7.11.13.17
-Subgroup: 2.3.5.7
+Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
-Comma list: 4375/4374, {{monzo| 154 -54 -21 -7 }}
+{{Mapping|legend=0| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}
-Mapping: {{mapping| 1 0 -21 85 | 0 7 103 -363 }}
+Optimal tunings:
+* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7399{{c}}
+* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6464{{c}}
-Optimal tuning (CTE): ~2 = 1\1, ~{{monzo| 66 -23 -9 -3 }} = 271.7113
+{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334ddeg }}
-{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}
+Badness (Sintel): 1.14
-Badness: 0.312806
+== References ==
-== Notes ==
+[[Category:Ragismic microtemperaments| ]] <!-- main article -->
 [[Category:Temperament collections]]
-[[Category:Ragismic microtemperaments| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Ragismic| ]] <!-- key article -->
-[[Category:Rank 2]]
-[[Category:Microtemperaments]]
-[[Category:Abigail]]
-[[Category:Deca]]
-[[Category:Enneadecal]]
-[[Category:Ennealimmal]]
-[[Category:Gamera]]
-[[Category:Mitonic]]
-[[Category:Octoid]]
-[[Category:Parakleismic]]
-[[Category:Quincy]]
-[[Category:Supermajor]]

Ragismic microtemperaments: Difference between revisions