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-The ragisma is [[4375/4374]] with a [[monzo]] of {{monzo|-1 -7 4 1}}, the smallest 7-limit [[superparticular]] ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
+{{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]].
-Temperaments not discussed here include [[Jubilismic clan #Crepuscular|crepuscular]], [[Meantone family #Flattone|flattone]], [[Porcupine family #Hystrix|hystrix]], [[Starling temperaments #Sensi|sensi]], [[Gamelismic clan #Unidec|unidec]], [[Orwellismic temperaments #Quartonic|quartonic]], [[Kleismic family #Catakleismic|catakleismic]], [[Tetracot family #Modus|modus]], [[Maja family|maja]], [[Schismatic family #Pontiac|pontiac]], [[Tricot family|trillium]], [[Würschmidt family #Whirrschmidt|whirrschmidt]],  [[Gravity family #Zarvo|zarvo]], [[Vishnuzmic family #Vishnu|vishnu]], and [[Vulture family #Vulture|vulture]].
+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.
-= Ennealimmal =
+Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, ragitritonic, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
-{{main|Ennealimmal}}
+* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
+* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
+* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
+* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]
+* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
+* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
+* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
+* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
+* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]
+* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]]
+* [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
+* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]
+* [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]]
+* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]
+* ''[[Zarvo]]'' (+33075/32768) → [[Gravity family #Zarvo|Gravity family]]
+* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
+* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortonic family #Mitonic|Minortonic family]]
+* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnuzmic family #Septimal vishnu|Vishnuzmic family]]
+* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]
+* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]]
+* ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]]
+* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]]
+* ''[[Chlorine]]'' (+{{monzo| -52 -17 34}}) → [[17th-octave temperaments #Chlorine|17th-octave temperaments]]
+* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]
+* ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments #Dzelic|37th-octave temperaments]]
-[[Ennealimmal]] temperament 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|ennealimmal comma]], {{monzo|1 -27 18}}, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two period 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. Its wedgie is {{multival|18 27 18 1 -22 -34}}.
+== Supermajor ==
+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.
-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 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 EDOs, though its hardly likely anyone could tell the difference.
+[[Subgroup]]: 2.3.5.7
-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 "tritaves" 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.
+[[Comma list]]: 4375/4374, 52734375/52706752
+{{Mapping|legend=1| 1 -22 -27 -45 | 0 37 46 75 }}
+: mapping generators: ~2, ~14/9
+[[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| 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214 }}
+[[Badness]] (Sintel): 0.274
+=== Semisupermajor ===
+Subgroup: 2.3.5.7.11
+Comma list: 3025/3024, 4375/4374, 35156250/35153041
+Mapping: {{mapping| 2 -7 -8 -15 -6 | 0 37 46 75 47 }}
+: mapping generators: ~99/70, ~11/10
+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=0| 80, 262d, 342, 764, 1106, 1448, 2554, 4002e, 6556cee }}
+Badness (Sintel): 0.422
+== Enneadecal ==
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Enneadecal (5-limit)]].''
+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.
-[[Comma list]]: 2401/2400, 4375/4374
+[[Subgroup]]: 2.3.5.7
-[[Tuning ranges]]:
+[[Comma list]]: 4375/4374, 703125/702464
-* [[diamond monotone]] range: [26.667, 66.667] (1\45 to 1\18)
-* [[diamond purer]] range: [48.920, 49.179]
-* [[diamond nice]] range: [48.920, 49.179]
-[[Mapping]]: [{{val|9 1 1 12}}, {{val|0 2 3 2}}]
+{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}
+: mapping generators: ~28/27, ~3
-{{Multival|legend=1|18 27 18 1 -22 -34}}
+[[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 }}
-Mapping generators: ~27/25, ~5/3
+{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}
-[[POTE generator]]s: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980
+[[Badness]] (Sintel): 0.277
-{{Val list|legend=1| 27, 45, 72, 99, 171, 441, 612 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-[[Badness]]: 0.003610
+Comma list: 540/539, 4375/4374, 16384/16335
-== 11-limit ==
+Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}
-The ennealimmal temperament can be described as 99e&amp;270 temperament, which tempers out 5632/5625 (vishdel comma) and 19712/19683 (symbiotic comma).
-Comma list: 2401/2400, 4375/4374, 5632/5625
+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}})
-Mapping: [{{val|9 1 1 12 -75}}, {{val|0 2 3 2 16}}]
+{{Optimal ET sequence|legend=0| 19, 133d, 152, 323e, 475de, 627de }}
-POTE generator: ~36/35 = 48.8654
+Badness (Sintel): 1.45
-Vals: {{Val list| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Badness: 0.027332
+Comma list: 540/539, 625/624, 729/728, 2205/2197
-=== 13-limit ===
+Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}
-Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374
-Mapping: [{{val|9 1 1 12 -75 93}}, {{val|0 2 3 2 16 -9}}]
+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}})
-POTE generator: ~36/35 = 48.9030
+{{Optimal ET sequence|legend=0| 19, 133df, 152f, 323ef }}
-Vals: {{Val list| 99e, 171e, 270 }}
+Badness (Sintel): 1.39
-Badness: 0.029404
+=== Hemienneadecal ===
+Subgroup: 2.3.5.7.11
-== Ennealimmia ==
+Comma list: 3025/3024, 4375/4374, 234375/234256
-Ennealimmal temperament has various extensions to the 11-limit. Tempering out 131072/130977 (salururu comma) leads to the ''ennealimmia'' temperament (171&amp;270, named by [[User:Xenllium|Xenllium]]).
-Comma list: 2401/2400, 4375/4374, 131072/130977
+Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}
+: mapping generators: ~55/54, ~3
-Mapping: [{{val|9 1 1 12 124}}, {{val|0 2 3 2 -14}}]
+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}})
-POTE generator: ~36/35 = 48.9244
+{{Optimal ET sequence|legend=0| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}
-Vals: {{Val list| 99, 171, 270, 711, 981, 1251, 2232e }}
+Badness (Sintel): 0.330
-Badness: 0.026463
+==== Hemienneadecalis ====
+Subgroup: 2.3.5.7.11.13
-=== 13-limit ===
+Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256
-Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374
-Mapping: [{{val|9 1 1 12 124 93}}, {{val|0 2 3 2 -14 -9}}]
+Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}
-POTE generator: ~36/35 = 48.9336
+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}})
-Vals: {{Val list| 99, 171, 270, 711, 981, 1692e, 2673e }}
+{{Optimal ET sequence|legend=0| 152f, 342f, 494 }}
-Badness: 0.016607
+Badness (Sintel): 0.859
-== Ennealimnic ==
+==== Hemienneadec ====
-Comma list: 243/242, 441/440, 4375/4356
+Subgroup: 2.3.5.7.11.13
-Tuning ranges:
+Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
-* [[diamond monotone]] range: [44.444, 53.333] (1\27 to 2\45)
-* [[diamond purer]] range: [48.920, 52.592]
-* [[diamond nice]] range: [48.920, 52.592]
-Mapping: [{{val|9 1 1 12 -2}}, {{val|0 2 3 2 5}}]
+Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}
-POTE generator: ~36/35 = 49.395
+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}})
-Vals: {{Val list| 72, 171, 243 }}
+{{Optimal ET sequence|legend=0| 152, 342, 494, 1330, 1824, 2318d }}
-Badness: 0.020347
+Badness (Sintel): 1.26
-=== 13-limit ===
+==== Semihemienneadecal ====
-Comma list: 243/242, 364/363, 441/440, 625/624
+Subgroup: 2.3.5.7.11.13
-Tuning ranges:
+Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078
-* [[diamond monotone]] range: [48.485, 50.000] (4\99 to 3\72)
-* [[diamond purer]] range: [48.825, 52.592]
-* [[diamond nice]] range: [48.825, 50.000]
-Mapping: [{{val|9 1 1 12 -2 -33}}, {{val|0 2 3 2 5 10}}]
+Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}
+: mapping generators: ~55/54, ~429/250
-POTE generator: ~36/35 = 49.341
+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}})
-Vals: {{Val list| 72, 171, 243 }}
+{{Optimal ET sequence|legend=0| 190, 304d, 494, 684, 1178, 2850, 4028ce }}
-Badness: 0.023250
+Badness (Sintel): 0.607
-==== 17-limit ====
+=== Kalium ===
-Comma list: 243/242, 364/363, 375/374, 441/440, 595/594
+Named after the 19th element, potassium, and after an archaic variant of the element's name to resolve a name conflict. [[19/16]] can be used as a generator. Since it is enfactored in the 17-limit and lower, it makes no sense to name it for the lower subgroups.
-Tuning ranges:
+Subgroup: 2.3.5.7.11.13.17.19
-* [[diamond monotone]] range: [48.485, 50.000] (4\99 to 3\72)
-* [[diamond purer]] range: [46.363, 52.592]
-* [[diamond nice]] range: [48.485, 50.000]
-Mapping: [{{val|9 1 1 12 -2 -33 -3}}, {{val|0 2 3 2 5 10 6}}]
+Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344
-POTE generator: ~36/35 = 49.335
+Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}
-Vals: {{Val list| 72, 171, 243 }}
+Optimal tunings:
+* WE: ~28/27 = 63.1582{{c}}, ~6545/5928 = 171.2448{{c}}
+* CWE: ~28/27 = 63.1579{{c}}, ~6545/5928 = 171.2439{{c}}
-Badness: 0.014602
+{{Optimal ET sequence|legend=0| 855, 988, 1843 }}
-=== Ennealim ===
+Badness (Sintel): 3.15
-Comma list: 169/168, 243/242, 325/324, 441/440
-Mapping: [{{val|9 1 1 12 -2 20}}, {{val|0 2 3 2 5 2}}]
+== Semidimi ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimi]].''
-POTE generator: ~36/35 = 49.708
+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.
-Vals: {{Val list| 27e, 45ef, 72 }}
+[[Subgroup]]: 2.3.5.7
-Badness: 0.020697
+[[Comma list]]: 4375/4374, 3955078125/3954653486
-== Ennealiminal ==
+{{Mapping|legend=1| 1 -19 -25 -32 | 0 55 73 93 }}
-Comma list: 385/384, 1375/1372, 4375/4374
+: mapping generators: ~2, ~35/27
-Mapping: [{{val|9 1 1 12 51}}, {{val|0 2 3 2 -3}}]
+[[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 }}
-POTE generator: ~36/35 = 49.504
+{{Optimal ET sequence|legend=1| 8d, …, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
-Vals: {{Val list| 27, 45, 72, 171e, 243e, 315e }}
+[[Badness]] (Sintel): 0.382
-Badness: 0.031123
+== 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.
-=== 13-limit ===
+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}}).
-Comma list: 169/168, 325/324, 385/384, 1375/1372
-Mapping: [{{val|9 1 1 12 51 20}}, {{val|0 2 3 2 -3 2}}]
+[[Subgroup]]: 2.3.5.7
-POTE generator: ~36/35 = 49.486
+[[Comma list]]: 4375/4374, {{monzo| 46 -14 -3 -6 }}
-Vals: {{Val list| 27, 45f, 72, 171ef, 243ef }}
+{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}
+: mapping generators: ~1157625/1048576, ~27/20
-Badness: 0.030325
+[[Optimal tuning]]s:
+* [[WE]]: ~1157625/1048576 = 171.4275{{c}}, ~27/20 = 519.7125{{c}}
+: [[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 }}
-== Hemiennealimmal ==
+{{Optimal ET sequence|legend=1| 7, …, 217, 224, 441, 1106, 1547 }}
-Comma list: 2401/2400, 3025/3024, 4375/4374
-Tuning ranges:
+[[Badness]] (Sintel): 0.737
-* [[diamond monotone]] range: [13.333, 22.222] (1\90 to 1\54)
-* [[diamond purer]] range: [17.304, 17.985]
-* [[diamond nice]] range:  [17.304, 17.985]
-Mapping: [{{val|18 0 -1 22 48}}, {{val|0 2 3 2 1}}]
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-POTE generator: ~99/98 = 17.6219
+Comma list: 4000/3993, 4375/4374, 131072/130977
-Vals: {{Val list| 72, 198, 270, 342, 612, 954, 1566 }}
+Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}
-Badness: 0.006283
+Optimal tunings:
+* WE: ~243/220 = 171.4208{{c}}, ~27/20 = 519.6807{{c}}
+* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7034{{c}}
-=== 13-limit ===
+{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665 }}
-Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024
-Tuning ranges:
+Badness (Sintel): 1.73
-* [[diamond monotone]] range: [16.667, 22.222] (1\72 to 1\54)
-* [[diamond purer]] range: [17.304, 18.309]
-* [[diamond nice]] range: [17.304, 18.309]
-Mapping: [{{val|18 0 -1 22 48 -19}}, {{val|0 2 3 2 1 6}}]
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-POTE generator ~99/98 = 17.7504
+Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374
-Vals: {{Val list| 72, 198, 270 }}
+Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}
-Badness: 0.012505
+Optimal tunings:
+* WE: ~243/220 = 171.4197{{c}}, ~27/20 = 519.6789{{c}}
+* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7052{{c}}
-=== Semihemiennealimmal ===
+{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1106e }}
-Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374
-Mapping: [{{val|18 0 -1 22 48 88}}, {{val|0 4 6 4 2 -3}}]
+Badness (Sintel): 0.956
-POTE generator: ~39/32 = 342.139
+== Abigail ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Abigail]].''
-Vals: {{Val list| 126, 144, 270, 684, 954 }}
+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.
-Badness: 0.013104
+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>
-== Semiennealimmal ==
+[[Subgroup]]: 2.3.5.7
-Comma list: 2401/2400, 4000/3993, 4375/4374
-Mapping: [{{val|9 3 4 14 18}}, {{val|0 6 9 6 7}}]
+[[Comma list]]: 4375/4374, 2147483648/2144153025
-POTE generator: ~140/121 = 250.3367
+{{Mapping|legend=1| 2 -4 -11 18 | 0 11 24 -19 }}
+: mapping generators: ~46305/32768, ~1536/1225
-Vals: {{Val list| 72, 369, 441 }}
+[[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 }}
-Badness: 0.034196
+{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd }}
-=== 13-limit ===
+[[Badness]] (Sintel): 0.936
-Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374
-Mapping: [{{val|9 3 4 14 18 -8}}, {{val|0 6 9 6 7 22}}]
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-POTE generator: ~140/121 = 250.3375
+Comma list: 3025/3024, 4375/4374, 131072/130977
-Vals: {{Val list| 72, 297ef, 369f, 441 }}
+Mapping: {{mapping| 2 -4 -11 18 18 | 0 11 24 -19 -17 }}
-Badness: 0.026122
+Optimal tunings:
+* WE: ~99/70 = 599.9782{{c}}, ~1536/1225 = 391.0852{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~1536/1225 = 391.0992{{c}}
-== Quadraennealimmal ==
+{{Optimal ET sequence|legend=0| 46, 132, 178, 224, 270, 494, 764 }}
-Comma list: 2401/2400, 4375/4374, 234375/234256
-Mapping: [{{val|9 1 1 12 -7}}, {{val|0 8 12 8 23}}]
+Badness (Sintel): 0.425
-POTE generator: ~77/75 = 45.595
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Vals: {{Val list| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
+Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
-Badness: 0.021320
+Mapping: {{mapping| 2 -4 -11 18 18 25 | 0 11 24 -19 -17 -27 }}
-== Trinealimmal ==
+Optimal tunings:
-Comma list: 2401/2400, 4375/4374, 2097152/2096325
+* WE: ~99/70 = 599.9862{{c}}, ~351/280 = 391.0879{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~351/280 = 391.0969{{c}}
-Mapping: [{{val|27 1 0 34 177}}, {{val|0 2 3 2 -4}}]
+{{Optimal ET sequence|legend=0| 46, 178, 224, 270, 494, 764, 1258 }}
-POTE generator: ~6/5 = 315.644
+Badness (Sintel): 0.366
-Vals: {{Val list| 27, 243, 270, 783, 1053, 1323 }}
+== Gamera ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Gamera]].''
-Badness: 0.029812
+[[Subgroup]]: 2.3.5.7
-= Gamera =
 [[Comma list]]: 4375/4374, 589824/588245
-[[Mapping]]: [{{val|1 6 10 3}}, {{val|0 -23 -40 -1}}]
+{{Mapping|legend=1| 1 -17 -30 2 | 0 23 40 1 }}
+: mapping generators: ~2, ~7/4
-{{Multival|legend=1|23 40 1 10 -63 -110}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8483{{c}}, ~7/4 = 969.5415{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 969.6608{{c}}
-[[POTE generator]] ~8/7 = 230.336
+{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}
-{{Val list|legend=1| 26, 73, 99, 224, 323, 422, 745d }}
+[[Badness]] (Sintel): 0.953
-[[Badness]]: 0.037648
+=== Hemigamera ===
+Subgroup: 2.3.5.7.11
-== Hemigamera ==
 Comma list: 3025/3024, 4375/4374, 589824/588245
-Mapping: [{{val|2 12 20 6 5}}, {{val|0 -23 -40 -1 5}}]
+Mapping: {{mapping| 2 -11 -20 5 10 | 0 23 40 1 -5 }}
+: mapping generators: ~99/70, ~99/80
-POTE generator: ~8/7 = 230.3370
+Optimal tunings:
+* WE: ~99/70 = 599.9323{{c}}, ~99/80 = 369.6212{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~99/80 = 369.6610{{c}}
-Vals: {{Val list| 26, 198, 224, 422, 646, 1068d }}
+{{Optimal ET sequence|legend=0| 26, 172c, 198, 224, 422, 646, 1068d }}
-Badness: 0.040955
+Badness (Sintel): 1.35
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-=== 13-limit ===
 Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024
-Mapping: [{{val|2 12 20 6 5 17}}, {{val|0 -23 -40 -1 5 -25}}]
+Mapping: {{mapping| 2 -11 -20 5 10 -8 | 0 23 40 1 -5 25 }}
+Optimal tunings:
+* WE: ~99/70 = 599.9207{{c}}, ~26/21 = 369.6139{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~26/21 = 369.6603{{c}}
+{{Optimal ET sequence|legend=0| 26, 172cf, 198, 224, 422, 646f, 1068df }}
+Badness (Sintel): 0.844
+=== Semigamera ===
+Subgroup: 2.3.5.7.11
+Comma list: 4375/4374, 14641/14580, 15488/15435
+Mapping: {{mapping| 1 -40 -70 1 -77 | 0 46 80 2 89 }}
+: mapping generators: ~2, ~144/77
+Optimal tunings:
+* WE: ~2 = 1199.8845{{c}}, ~144/77 = 1084.7314{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8345{{c}}
+{{Optimal ET sequence|legend=0| 73, 125, 198, 323, 521 }}
+Badness (Sintel): 2.59
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580
+Mapping: {{mapping| 1 -40 -70 1 -77 -131 | 0 46 80 2 89 149 }}
-POTE generator: ~8/7 = 230.3373
+Optimal tunings:
+* WE: ~2 = 1199.8726{{c}}, ~144/77 = 1084.7220{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8359{{c}}
-Vals: {{Val list| 26, 198, 224, 422, 646f, 1068df }}
+{{Optimal ET sequence|legend=0| 73f, 125f, 198, 323, 521 }}
-Badness: 0.020416
+Badness (Sintel): 1.82
-= Supermajor =
+== Crazy ==
-The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/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 isn't enough notes for you, there's always the 171 note MOS.
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
-[[Comma list]]: 4375/4374, 52734375/52706752
+Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament, with a [[ploidacot]] of diploid alpha-octacot. [[1106edo]] gives a strong tuning.
+Crazy was named by [[Flora Canou]] in 2025 by removing the mutation from ''kwazy'', the name for the 5-limit microtemperament.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}
-[[Mapping]]: [{{val|1 15 19 30}}, {{val|0 -37 -46 -75}}]
+{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}
+: mapping generators: ~332150625/234881024, ~1125/1024
-{{Multival|legend=1|37 46 75 -13 15 45}}
+[[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 }}
-[[POTE generator]]: ~9/7 = 435.082
+{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}
-{{Val list|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}
+[[Badness]] (Sintel): 0.998
-[[Badness]]: 0.010836
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== Semisupermajor ==
+Comma list: 3025/3024, 4375/4374, 2791309312/2790703125
-Comma list: 3025/3024, 4375/4374, 35156250/35153041
-Mapping: [{{val|2 30 38 60 41}}, {{val|0 -37 -46 -75 -47}}]
+Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}
-POTE generator: ~9/7 = 435.082
+Optimal tunings:
+* WE: ~99/70 = 600.0047{{c}}, ~1125/1024 = 162.7493{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~1125/1024 = 162.7481{{c}}
-EDOs: {{Val list| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}
+{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}
-Badness: 0.012773
+Badness (Sintel): 0.562
-= Enneadecal =
+== Orga ==
-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)^(1/3). 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 limits, 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.
+Orga may be described as the {{nowrap| 26 & 270 }} temperament, and [[1106edo]] gives a strong tuning.
-[[Comma list]]: 4375/4374, 703125/702464
+[[Subgroup]]: 2.3.5.7
-[[Mapping]]: [{{val|19 0 14 -37}}, {{val|0 1 1 3}}]
+[[Comma list]]: 4375/4374, {{monzo| 41 -4 2 -14 }}
-{{Multival|legend=1|19 19 57 -14 37 79}}
+{{Mapping|legend=1| 2 -8 -15 6 | 0 29 51 -1 }}
+: mapping generators: ~7411887/5242880, ~8/7
-Mapping generators: ~28/27, ~3
+[[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 }}
-[[POTE generator]]: ~3/2 = 701.880
+{{Optimal ET sequence|legend=1| 26, …, 244, 270, 836, 1106, 1376, 2482 }}
-{{Val list|legend=1| 19, 152, 171, 665, 836, 1007, 2185 }}
+[[Badness]] (Sintel): 1.02
-[[Badness]]: 0.010954
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== Hemienneadecal ==
+Comma list: 3025/3024, 4375/4374, 5767168/5764801
-Comma list: 3025/3024, 4375/4374, 234375/234256
-Mapping: [{{val|38 0 28 -74 11}}, {{val|0 1 1 3 2}}]
+Mapping: {{mapping| 2 -8 -15 6 10 | 0 29 51 -1 -8 }}
-POTE generator: ~3/2 = 701.881
+Optimal tunings:
+* WE: ~99/70 = 600.0025{{c}}, ~8/7 = 231.1039{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1030{{c}}
-Vals: {{Val list| 152, 342, 494, 836, 1178, 2014 }}
+{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836, 1106 }}
-Badness: 0.009985
+Badness (Sintel): 0.535
 === 13-limit ===
-Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
+Subgroup: 2.3.5.7.11.13
-Mapping: [{{val|38 0 28 -74 11 502}}, {{val|0 1 1 3 2 -6}}]
+Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360
-POTE generator: ~3/2 = 701.986
+Mapping: {{mapping| 2 -8 -15 6 10 -3 | 0 29 51 -1 -8 27 }}
-Vals: {{Val list| 152, 342, 494, 836 }}
+Optimal tunings:
+* WE: ~99/70 = 600.0192{{c}}, ~8/7 = 231.1102{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1033{{c}}
-Badness: 0.030391
+{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836f, 1106f }}
-= Deca =
+Badness (Sintel): 0.899
-Deca temperament has a period of 1/10 octave and tempers out the [[15/14ths equal temperament #Linus temperaments|linus comma]], {{monzo|11 -10 -10 10}} and {{monzo|12 -3 -14 9}} = 165288374272/164794921875 (satritrizo-asepbigu).
-[[Comma list]]: 4375/4374, 165288374272/164794921875
+== Seniority ==
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]].
-[[Mapping]]: [{{val|10 4 9 2}}, {{val|0 5 6 11}}]
+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.
-{{Multival|legend=1|50 60 110 -21 34 87}}
+[[Subgroup]]: 2.3.5.7
-[[POTE generator]]: ~6/5 = 315.577
+[[Comma list]]: 4375/4374, 201768035/201326592
-{{Val list|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}
+{{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }}
+: mapping generators: ~2, ~5120/3087
-[[Badness]]: 0.080637
+[[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 }}
-== 11-limit ==
+{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d }}
-Comma list: 3025/3024, 4375/4374, 422576/421875
-Mapping: [{{val|10 4 9 2 18}}, {{val|0 5 6 11 7}}]
+[[Badness]] (Sintel): 1.14
-POTE generator: ~6/5 = 315.582
+=== Senator ===
+Senator (26 & 145) extends seniority by tempering out [[441/440]] and [[65536/65219]], and can be extended to the 13- and 17-limit immediately by adding [[364/363]] and [[595/594]] to the comma list in this order.
-Vals: {{Val list| 80, 190, 270, 1000, 1270 }}
+Subgroup: 2.3.5.7.11
-Badness: 0.024329
+Comma list: 441/440, 4375/4374, 65536/65219
-== 13-limit ==
+Mapping: {{mapping| 1 -24 -43 5 2 | 0 35 62 -3 2 }}
-Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
-Mapping: [{{val|10 4 9 2 18 37}}, {{val|0 5 6 11 7 0}}]
+Optimal tunings:
+* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}}
-POTE generator: ~6/5 = 315.602
+{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e }}
-Vals: {{Val list| 80, 190, 270, 730, 1000 }}
+Badness (Sintel): 3.05
-Badness: 0.016810
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-= Mitonic =
+Comma list: 364/363, 441/440, 2200/2197, 4375/4374
-{{see also|Minortonic family #Mitonic}}
-[[Comma list]]: 4375/4374, 2100875/2097152
+Mapping: {{mapping| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}
-[[Mapping]]: [{{val|1 -1 -3 6}}, {{val|0 17 35 -21}}]
+Optimal tunings:
+* WE: ~2 = 1199.7136{{c}}, ~108/65 = 877.9974{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2038{{c}}
-{{Multival|legend=1|17 35 -21 16 -81 -147}}
+{{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }}
-[[POTE generator]]: ~10/9 = 182.458
+Badness (Sintel): 1.85
-{{Val list|legend=1| 46, 125, 171 }}
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-[[Badness]]: 0.025184
+Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197
-= Sfourth =
+Mapping: {{mapping| 1 -24 -43 5 2 -27 -31 | 0 35 62 -3 2 42 48 }}
-[[Comma list]]: 4375/4374, 64827/64000
-[[Mapping]]: [{{val|1 2 3 3}}, {{val|0 -19 -31 -9}}]
+Optimal tunings:
+* WE: ~2 = 1199.7195{{c}}, ~108/65 = 877.0018{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}}
-{{Multival|legend=1|19 31 9 5 -39 -66}}
+{{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }}
-[[POTE generator]]: ~49/48 = 26.287
+Badness (Sintel): 1.35
-{{Val list|legend=1| 45, 46, 91, 137d }}
+== Monzismic ==
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]].
-[[Badness]]: 0.123291
+Monzismic tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]]. It may be described as the {{nowrap| 53 & 612 }} temperament, with a [[ploidacot]] signature of alpha-dicot. A notable tuning not appearing on the optimal ET sequence is [[665edo]], which is nearly equivalent to the pure-3's tuning.
-== 11-limit ==
+[[Subgroup]]: 2.3.5.7
-Comma list: 121/120, 441/440, 4375/4374
-Mapping: [{{val|1 2 3 3 4}}, {{val|0 -19 -31 -9 -25}}]
+[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}
-POTE generator: ~49/48 = 26.286
+{{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }}
+: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}
-Vals: {{Val list| 45e, 46, 91e, 137de }}
+[[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 }}
-Badness: 0.054098
+{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd }}
-=== 13-limit ===
+[[Badness]] (Sintel): 1.18
-Comma list: 121/120, 169/168, 325/324, 441/440
-Mapping: [{{val|1 2 3 3 4 4}}, {{val|0 -19 -31 -9 -25 -14}}]
+=== Monzism ===
+Subgroup: 2.3.5.7.11
-POTE generator: ~49/48 = 26.310
+Comma list: 4375/4374, 41503/41472, 184549376/184528125
-Vals: {{Val list| 45ef, 46, 91ef, 137def }}
+Mapping: {{mapping| 1 0 -27 109 -159 | 0 2 37 -134 205 }}
-Badness: 0.033067
+Optimal tunings:
+* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}}
-== Sfour ==
+{{Optimal ET sequence|legend=0| 53, 559, 612, 3619de, 4231de, …, 6067ddee }}
-Comma list: 385/384, 2401/2376, 4375/4374
-Mapping: [{{val|1 2 3 3 3}}, {{val|0 -19 -31 -9 21}}]
+Badness (Sintel): 1.89
-POTE generator: ~49/48 = 26.246
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Vals: {{Val list| 45, 46, 91, 137d }}
+Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
-Badness: 0.076567
+Mapping: {{mapping| 1 0 -27 109 -159 -70 | 0 2 37 -134 205 93 }}
-=== 13-limit ===
+Optimal tunings:
-Comma list: 196/195, 364/363, 385/384, 4375/4374
+* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}}
-Mapping: [{{val|1 2 3 3 3 3}}, {{val|0 -19 -31 -9 21 32}}]
+{{Optimal ET sequence|legend=0| 53, 559, 612 }}
-POTE generator: ~49/48 = 26.239
+Badness (Sintel): 2.22
-Vals: {{Val list| 45, 46, 91, 137d }}
+== Semidimfourth ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimfourth]].''
-Badness: 0.051893
+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]].
-= Abigail =
+[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 2147483648/2144153025
-[[Mapping]]: [{{val|2 7 13 -1}}, {{val|0 -11 -24 19}}]
+[[Comma list]]: 4375/4374, 235298/234375
-{{Multival|legend=1|22 48 -38 25 -122 -223}}
+{{Mapping|legend=1| 1 -10 -13 -17 | 0 31 41 53 }}
+: mapping generators: ~2, ~35/27
-[[POTE generator]]: ~6912/6125 = 208.899
+[[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 }}
-{{Val list|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}
+{{Optimal ET sequence|legend=1| 8d, …, 91, 99, 289, 388, 875 }}
-[[Badness]]: 0.037000
+[[Badness]] (Sintel): 1.40
-== 11-limit ==
+=== Neusec ===
-Comma list: 3025/3024, 4375/4374, 20614528/20588575
+Subgroup: 2.3.5.7.11
-Mapping: [{{val|2 7 13 -1 1}}, {{val|0 -11 -24 19 17}}]
+Comma list: 3025/3024, 4375/4374, 235298/234375
-POTE generator: ~1155/1024 = 208.901
+Mapping: {{mapping| 2 -20 -26 -34 -17 | 0 31 41 53 32 }}
+: mapping generators: ~99/70, ~35/27
-Vals: {{Val list| 46, 132, 178, 224, 270, 494, 764 }}
+Optimal tunings:
+* WE: ~99/70 = 600.0381{{c}}, ~35/27 = 448.4812{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4546{{c}}
-Badness: 0.012860
+{{Optimal ET sequence|legend=0| 8d, …, 190, 388 }}
-== 13-limit ==
+Badness (Sintel): 1.95
-Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
-Mapping: [{{val|2 7 13 -1 1 -2}}, {{val|0 -11 -24 19 17 27}}]
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-POTE generator: ~44/39 = 208.903
+Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
-Vals: {{Val list| 46, 178, 224, 270, 494, 764, 1258 }}
+Mapping: {{mapping| 2 -20 -26 -34 -17 -21 | 0 31 41 53 32 38 }}
-Badness: 0.008856
+Optimal tunings:
+* WE: ~99/70 = 600.0034{{c}}, ~35/27 = 448.4573{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4549{{c}}
-= Semidimi =
+{{Optimal ET sequence|legend=0| 8d, …, 190, 198, 388 }}
-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.
-[[Comma]]: {{monzo|-12 -73 55}}
+Badness (Sintel): 1.28
-[[Mapping]]: [{{val|1 36 48}}, {{val|0 -55 -73}}]
+== Acrokleismic ==
+[[Subgroup]]: 2.3.5.7
-[[POTE generator]]: ~162/125 = 449.1269
+[[Comma list]]: 4375/4374, 2202927104/2197265625
-{{Val list|legend=1| 8, 163, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
+{{Mapping|legend=1| 1 -22 -22 -65 | 0 32 33 92 }}
+: mapping generators: ~2, ~5/3
-[[Badness]]: 0.754866
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9305{{c}}, ~5/3 = 884.3923{{c}}
+: [[error map]]: {{val| -0.070 +0.126 +0.160 -0.221 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.4423{{c}}
+: error map: {{val| 0.000 +0.198 +0.282 -0.136 }}
-== 7-limit ==
+{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}
-[[Comma list]]: 4375/4374, 3955078125/3954653486
-[[Mapping]]: [{{val|1 36 48 61}}, {{val|0 -55 -73 -93}}]
+[[Badness]] (Sintel): 1.42
-{{Multival|legend=1|55 73 93 -12 -7 11}}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-[[POTE generator]]: ~35/27 = 449.1270
+Comma list: 4375/4374, 41503/41472, 172032/171875
-{{Val list|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
+Mapping: {{mapping| 1 -22 -22 -65 58 | 0 32 33 92 -74 }}
-[[Badness]]: 0.015075
+Optimal tunings:
+* WE: ~2 = 1199.9698{{c}}, ~5/3 = 884.4193{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4414{{c}}
-= Brahmagupta =
+{{Optimal ET sequence|legend=0| 19, 251, 270, 829, 1099, 1369, 1639 }}
-The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo|47 -7 -7 -7}} = 140737488355328 / 140710042265625.
-[[Comma list]]: 4375/4374, 70368744177664/70338939985125
+Badness (Sintel): 1.22
-[[Mapping]]: [{{val|7 2 -8 53}}, {{val|0 3 8 -11}}]
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-{{Multival|legend=1|21 56 -77 40 -181 -336}}
+Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
-[[POTE generator]]: ~27/20 = 519.716
+Mapping: {{mapping| 1 -22 -22 -65 58 -56 | 0 32 33 92 -74 81 }}
-{{Val list|legend=1| 7, 217, 224, 441, 1106, 1547 }}
+Optimal tunings:
+* WE: ~2 = 1199.9939{{c}}, ~5/3 = 884.4384{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4429{{c}}
-[[Badness]]: 0.029122
+{{Optimal ET sequence|legend=0| 19, 251, 270 }}
-== 11-limit ==
+Badness (Sintel): 1.11
-Comma list: 4000/3993, 4375/4374, 131072/130977
-Mapping: [{{val|7 2 -8 53 3}}, {{val|0 3 8 -11 7}}]
+=== Counteracro ===
+Subgroup: 2.3.5.7.11
-POTE generator: ~27/20 = 519.704
+Comma list: 4375/4374, 5632/5625, 117649/117612
-Vals: {{Val list| 7, 217, 224, 441, 665, 1771ee }}
+Mapping: {{mapping| 1 -22 -22 -65 -141 | 0 32 33 92 196 }}
-Badness: 0.052190
+Optimal tunings:
+* WE: ~2 = 1199.8877{{c}}, ~5/3 = 884.3639{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4457{{c}}
-== 13-limit ==
+{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1061e, 1331c, 1601c, 1871bc }}
-Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374
-Mapping: [{{val|7 2 -8 53 3 35}}, {{val|0 3 8 -11 7 -3}}]
+Badness (Sintel): 1.41
-POTE generator: ~27/20 = 519.706
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Vals: {{Val list| 7, 217, 224, 441, 665, 1771eef }}
+Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
-Badness: 0.023132
+Mapping: {{mapping| 1 -22 -22 -65 -141 -56 | 0 32 33 92 196 81 }}
-= Quasithird =
+Optimal tunings:
-The '''quasithird''' temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}.
+* WE: ~2 = 1199.9285{{c}}, ~5/3 = 884.3937{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4458{{c}}
-[[Comma]]: {{monzo|55 -64 20}}
+{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1331c }}
-[[Mapping]]: [{{val|4 0 -11}}, {{val|0 5 16}}]
+Badness (Sintel): 1.08
-[[POTE generator]]: ~1594323/1280000 = 380.395
+== Quasithird ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasithird]].''
-[[Vals]]: {{Val list| 60, 164, 224, 388, 612, 836, 1000, 1448, 1612, 2224, 2836 }}
+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.
-[[Badness]]: 0.099519
+[[Subgroup]]: 2.3.5.7
-== 7-limit ==
+[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}
-[[Comma list]]: 4375/4374, 1153470752371588581/1152921504606846976
-[[Mapping]]: [{{val|4 0 -11 48}}, {{val|0 5 16 -29}}]
+{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
+: mapping generators: ~65536/55125, ~5103/4096
-[[Wedgie]]: {{multival|20 64 -116 55 -240 -449}}
+[[Optimal tuning]]s:
+* [[WE]]: ~65536/55125 = 300.0052{{c}}, ~5103/4096 = 380.3949{{c}}
+: [[error map]]: {{val| +0.021 +0.020 -0.052 -0.031 }}
+* [[CWE]]: ~65536/55125 = 300.0000{{c}}, ~5103/4096 = 380.3884{{c}}
+: error map: {{val| 0.000 -0.013 -0.100 -0.089 }}
-[[POTE generator]]: ~5103/4096 = 380.388
+{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}
-[[Vals]]: {{Val list| 60d, 164, 224, 388, 612, 1448, 2060 }}
+[[Badness]] (Sintel): 1.56
-[[Badness]]: 0.061813
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== 11-limit ==
 Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
-Mapping: [{{val|4 0 -11 48 43}}, {{val|0 5 16 -29 -23}}]
+Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}
-POTE generator: ~22/21 = 80.387 (or ~5103/4096 = 380.387)
+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}})
-Vals: {{Val list| 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 ==
+=== 13-limit ===
-Comma list: 2200/2197, 3025/3024, 4375/4374, 468512/468195
+Subgroup: 2.3.5.7.11.13
-Mapping: [{{val|4 0 -11 48 43 11}}, {{val|0 5 16 -29 -23 3}}]
+Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374
-POTE generator: ~22/21 = 80.385 (or ~5103/4096 = 380.385)
+Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}
-Vals: {{Val list| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}
+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}})
-Badness: 0.029501
+{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836 }}
-= Semidimfourth =
+Badness (Sintel): 1.22
-The '''semidimifourth''' 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.
-[[Comma]]: {{monzo|7 41 -31}}
+== Deca ==
+: ''For 5-limit version, see [[Miscellaneous 5-limit temperaments #Neon]].''
-[[Mapping]]: [{{val|1 21 28}}, {{val|0 -31 -41}}]
+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.
-[[POTE generator]]: ~162/125 = 448.449
+[[Subgroup]]: 2.3.5.7
-[[Vals]]: {{Val list| 8, 91, 99, 190, 289, 388, 677, 3674, 4351, 5028, 5705, 6382, 13441c, 19823bcc }}
+[[Comma list]]: 4375/4374, 165288374272/164794921875
-[[Badness]]: 0.233376
+{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}
+: mapping generators: ~15/14, ~460992/390625
-== 7-limit ==
+[[Optimal tuning]]s:
-[[Comma list]]: 4375/4374, 235298/234375
+* [[WE]]: ~15/14 = 119.9966{{c}}, ~460992/390625 = 284.4150{{c}} (5625/5488 = 44.4219{{c}})
+: [[error map]]: {{val| -0.034 +0.106 +0.145 -0.268 }}
+* [[CWE]]: ~15/14 = 120.0000{{c}}, ~460992/390625 = 284.4182{{c}} (5625/5488 = 44.4182{{c}})
+: error map: {{val| 0.000 +0.136 +0.195 -0.226 }}
+{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}
+[[Badness]] (Sintel): 2.04
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 3025/3024, 4375/4374, 391314/390625
+Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}
-[[Mapping]]: [{{val|1 21 28 36}}, {{val|0 -31 -41 -53}}]
+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}})
-[[Wedgie]]: {{multival|31 41 53 -7 -3 8}}
+{{Optimal ET sequence|legend=0| 80, 190, 270, 1000, 1270, 1540e, 1810e }}
-[[POTE generator]]: ~35/27 = 448.456
+Badness (Sintel): 0.804
-[[Vals]]: {{Val list| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-[[Badness]]: 0.055249
+Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
-== Neusec ==
+Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}
-Comma list: 3025/3024, 4375/4374, 235298/234375
+Optimal tunings:
+* WE: ~15/14 = 120.0067{{c}}, ~33/28 = 284.4139{{c}} (~40/39 = 44.4006{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4048{{c}} (~40/39 = 44.4048{{c}})
+{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}
-Mapping: [{{val|2 11 15 19 15}}, {{val|0 -31 -41 -53 -32}}]
+Badness (Sintel): 0.695
-POTE generator: ~12/11 = 151.547
+=== 2.3.5.7.11.13.19 subgroup ===
+Subgroup: 2.3.5.7.11.13.19
-Vals: {{Val list| 8d, 190, 388 }}
+Comma list: 1001/1000, 1521/1520, 3025/3024, 4225/4224, 4375/4374
-Badness: 0.059127
+Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}
-=== 13-limit ===
+Optimal tunings:
-Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
+* 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: [{{val|2 11 15 19 15 17}}, {{val|0 -31 -41 -53 -32 -38}}]
+{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}
-POTE generator: ~12/11 = 151.545
+Badness (Sintel): 0.556
-Vals: {{Val list| 8d, 190, 198, 388 }}
+== Keenanose ==
+Keenanose, the {{nowrap| 270 & 1889 }} temperament, was named by [[Eliora]] in 2022 for the fact that it uses [[385/384]], the keenanisma, as the generator.
-Badness: 0.030941
+[[Subgroup]]: 2.3.5.7
-= Acrokleismic =
+[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}
-[[Comma list]]: 4375/4374, 2202927104/2197265625
-[[Mapping]]: [{{val|1 10 11 27}}, {{val|0 -32 -33 -92}}]
+{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
+: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}
-[[Wedgie]]: {{multival|32 33 92 -22 56 121}}
+[[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 }}
-[[POTE generator]]: ~6/5 = 315.557
+{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}
-[[Vals]]: {{Val list| 19, 251, 270 }}
+[[Badness]] (Sintel): 2.17
-[[Badness]]: 0.056184
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== 11-limit ==
+Comma list: 4375/4374, 117649/117612, 67110351/67108864
-Comma list: 4375/4374, 41503/41472, 172032/171875
-Mapping: [{{val|1 10 11 27 -16}}, {{val|0 -32 -33 -92 74}}]
+Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}
-POTE generator: ~6/5 = 315.558
+Optimal tunings:
+* WE: ~2 = 1199.9970{{c}}, ~385/384 = 4.4465{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4465{{c}}
-Vals: {{Val list| 19, 251, 270, 829, 1099, 1369, 1639 }}
+{{Optimal ET sequence|legend=0| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}
-Badness: 0.036878
+Badness (Sintel): 1.02
 === 13-limit ===
-Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
+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 tunings:
+* WE: ~2 = 1200.0065{{c}}, ~385/384 = 4.4467{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4467{{c}}
+{{Optimal ET sequence|legend=0| 270, 1079, 1349, 1619, 1889, 4048 }}
+Badness (Sintel): 0.879
+== Aluminium ==
+: ''For the 5-limit version, see [[13th-octave temperaments #Aluminium]].''
+Aluminium was named by [[Eliora]] in 2023 after the 13th element. It tempers out the {{monzo| 92 -39 -13 }} comma, which sets [[135/128]] interval to be equal to 1/13th of the octave.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}
+[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}
+: Mapping generators: ~135/128, ~3
-Mapping: [{{val|1 10 11 27 -16 25}}, {{val|0 -32 -33 -92 74 -81}}]
+[[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 }}
-POTE generator: ~6/5 = 315.557
+{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}
-Vals: {{Val list| 19, 251, 270 }}
+[[Badness]] (Sintel): 3.20
-Badness: 0.026818
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== Counteracro ==
+Comma list: 4375/4374, 234375/234256, 2097152/2096325
-Comma list: 4375/4374, 5632/5625, 117649/117612
-Mapping: [{{val|1 10 11 27 55}}, {{val|0 -32 -33 -92 -196}}]
+Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}
-POTE generator: ~6/5 = 315.553
+Optimal tunings:
+* WE: ~135/128 = 92.3062{{c}}, ~3/2 = 701.9946{{c}}
+* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0056{{c}}
-Vals: {{Val list| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}
+{{Optimal ET sequence|legend=0| 494, 1053, 1547, 3588e, 5135e }}
-Badness: 0.042572
+Badness (Sintel): 1.39
 === 13-limit ===
-Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
+Subgroup: 2.3.5.7.11.13
+Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078
-Mapping: [{{val|1 10 11 27 55 25}}, {{val|0 -32 -33 -92 -196 -81}}]
+Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}
-POTE generator: ~6/5 = 315.554
+Optimal tunings:
+* WE: ~135/128 = 92.3055{{c}}, ~3/2 = 701.9928{{c}}
+* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0098{{c}}
-Vals: {{Val list| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}
+{{Optimal ET sequence|legend=0| 494, 1547, 2041, 4576def }}
-Badness: 0.026028
+Badness (Sintel): 1.18
-= Seniority =
+== Ragitritonic ==
-{{see also|Very high accuracy temperaments#Senior}}
+: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
-[[Comma list]]: 4375/4374, 201768035/201326592
+Ragitritonic may be described as the {{nowrap| 53 & 369 }} temperament, splitting the [[24/1|24th harmonic]] into nine tritone generators; its [[ploidacot]] is thus delta-enneacot. [[422edo]] makes for a strong tuning.
-[[Mapping]]: [{{val|1 11 19 2}}, {{val|0 -35 -62 3}}]
+Ragitritonic was named by [[Flora Canou]] in 2026 as a contraction of ''ragismic'' and ''tritonic''.
-[[Wedgie]]: {{multival|35 62 -3 17 -103 -181}}
+[[Subgroup]]: 2.3.5.7
-[[POTE generator]]: ~3087/2560 = 322.804
+[[Comma list]]: 4375/4374, 68719476736/68356598625
-[[Vals]]: {{Val list| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}
+{{Mapping|legend=1| 1 -3 -15 40 | 0 9 34 -73 }}
+: mapping generators: ~2, ~65536/45927
-[[Badness]]: 0.044877
+[[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 }}
-= Orga =
+{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}
-[[Comma list]]: 4375/4374, 54975581388800/54936068900769
-[[Mapping]]: [{{val|2 21 36 5}}, {{val|0 -29 -51 1}}]
+[[Badness]] (Sintel): 3.37
-[[Wedgie]]: {{multival|58 102 -2 27 -166 -291}}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-[[POTE generator]]: ~8/7 = 231.104
+Comma list: 4375/4374, 5632/5625, 2621440/2614689
-[[Vals]]: {{Val list| 26, 244, 270, 836, 1106, 1376, 2482 }}
+Mapping: {{mapping| 1 -3 -15 40 -75 | 0 9 34 -73 154 }}
-[[Badness]]: 0.040236
+Optimal tunings:
+* WE: ~2 = 1199.8147{{c}}, ~768/539 = 611.2822{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~768/539 = 611.3762{{c}}
-== 11-limit ==
+{{Optimal ET sequence|legend=0| 53, 316e, 369, 422, 791e, 1213cde }}
-Comma list: 3025/3024, 4375/4374, 5767168/5764801
-Mapping: [{{val|2 21 36 5 2}}, {{val|0 -29 -51 1 8}}]
+Badness (Sintel): 2.34
-POTE generator: ~8/7 = 231.103
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Vals: {{Val list| 26, 244, 270, 566, 836, 1106 }}
+Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625
-Badness: 0.016188
+Mapping: {{mapping| 1 -3 -15 40 -75 -34 | 0 9 34 -73 154 74 }}
-== 13-limit ==
+Optimal tunings:
-Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360
+* WE: ~2 = 1199.7916{{c}}, ~91/64 = 611.2698{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~91/64 = 611.3754{{c}}
-Mapping: [{{val|2 21 36 5 2 24}}, {{val|0 -29 -51 1 8 -27}}]
+{{Optimal ET sequence|legend=0| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}
-POTE generator: ~8/7 = 231.103
+Badness (Sintel): 1.51
-Vals: {{Val list| 26, 244, 270, 566, 836f, 1106f }}
+== Quatracot ==
+{{See also| Stratosphere }}
-Badness: 0.021762
+[[Subgroup]]: 2.3.5.7
-= Quatracot =
+[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
-[[Comma list]]: 4375/4374, 1483154296875/1473173782528
-[[Mapping]]: [{{val|2 7 7 23}}, {{val|0 -13 -8 -59}}]
+{{Mapping|legend=1| 2 -6 -1 -36 | 0 13 8 59 }}
+: mapping generators: ~2278125/1605632, ~7168/5625
-[[Wedgie]]: {{multival|26 16 118 -35 114 229}}
+[[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 }}
-[[POTE generator]]: ~448/405 = 176.805
+{{Optimal ET sequence|legend=1| 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc }}
-[[Vals]]: {{Val list| 190, 224, 414, 638, 1052c, 1690bcc }}
+[[Badness]] (Sintel): 4.45
-[[Badness]]: 0.175982
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== 11-limit ==
 Comma list: 3025/3024, 4375/4374, 1265625/1261568
-Mapping: [{{val|2 7 7 23 19}}, {{val|0 -13 -8 -59 -41}}]
+Mapping: {{mapping| 2 -6 -1 -36 -22 | 0 13 8 59 41 }}
+Optimal tunings:
+* WE: ~99/70 = 600.0847{{c}}, ~225/176 = 423.2536{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~225/176 = 423.1977{{c}}
-POTE generator: ~448/405 = 176.806
+{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638, 1052c }}
-Vals: {{Val list| 190, 224, 414, 638, 1052c }}
+Badness (Sintel): 1.36
-Badness: 0.041043
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-== 13-limit ==
 Comma list: 625/624, 729/728, 1575/1573, 2200/2197
-Mapping: [{{val|2 7 7 23 19 13}}, {{val|0 -13 -8 -59 -41 -19}}]
+Mapping: {{mapping| 2 -6 -1 -36 -22 -6 | 0 13 8 59 41 19 }}
+Optimal tunings:
+* WE: ~99/70 = 600.0571{{c}}, ~143/112 = 423.2366{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~143/112 = 423.1987{{c}}
+{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638 }}
+Badness (Sintel): 0.936
+== Moulin ==
+Moulin can be described as the {{nowrap| 494 & 1619 }} temperament. It has a generator of ~[[22/13]], and it was named by [[Eliora]] in 2022 after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". However, the functional generator is ~[[13/11]], and 73 of them octave reduced reach the [[3/2|perfect fifth]]. Since [[11/8]] is within 23 generators, the 25-tone generator chain (4L 21s) of this temperament contains the 8:11:13 triad.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
+{{Mapping|legend=1| 1 -16 -9 -75 | 0 73 47 323 }}
+: mapping generators: ~2, ~3796875/3211264
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0272{{c}}, ~3796875/3211264 = 289.0675{{c}}
+: [[error map]]: {{val| +0.027 +0.007 -0.084 +0.013 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3796875/3211264 = 289.0675{{c}}
+: error map: {{val| 0.000 -0.029 -0.142 -0.029 }}
+{{Optimal ET sequence|legend=1| 494, 1125, 1619, 8589cc, 10208cc }}
+[[Badness]] (Sintel): 5.93
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 4375/4374, 759375/758912, 100663296/100656875
+Mapping: {{mapping| 1 -16 -9 -75 9 | 0 73 47 323 -23 }}
+Optimal tunings:
+* WE: ~2 = 1200.0043{{c}}, ~605/512 = 289.0687{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~605/512 = 289.0677{{c}}
+{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}
+Badness (Sintel): 2.24
-POTE generator: ~195/176 = 176.804
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Vals: {{Val list| 190, 224, 414, 638, 1690bcc, 2328bccde }}
+Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
-Badness: 0.022643
+Mapping: {{mapping| 1 -16 -9 -75 9 9 | 0 73 47 323 -23 -22 }}
-= Octoid =
+Optimal tunings:
-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.
+* WE: ~2 = 1200.0043{{c}}, ~13/11 = 289.0687{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 289.0677{{c}}
+{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}
+Badness (Sintel): 1.12
+== Palladium ==
+: ''For the 5-limit version, see [[46th-octave temperaments #Palladium]]''.
+The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minor whole tones (10/9) fall short of seven octaves. This temperament can be described as {{nowrap| 46 & 414 }} temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.
+[[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
+[[Optimal tuning]]s:
+* [[WE]]: ~83349/81920 = 26.0910{{c}}, ~3/2 = 701.7155{{c}}
+: [[error map]]: {{val| +0.185 -0.055 -0.061 +0.349 }}
+* [[CWE]]: ~83349/81920 = 26.0870{{c}}, ~3/2 = 701.6491{{c}}
+: error map: {{val| 0.000 -0.306 -0.407 -0.910 }}
+{{Optimal ET sequence|legend=1| 46, …, 368, 414, 460, 874d }}
+[[Badness]] (Sintel): 7.81
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 3025/3024, 4375/4374, 134775333/134217728
+Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}
+Optimal tunings:
+* WE: ~8192/8085 = 26.0912{{c}}, ~3/2 = 701.7082{{c}}
+* CWE: ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.6173{{c}}
+{{Optimal ET sequence|legend=0| 46, …, 368, 414, 460, 874de }}
+Badness (Sintel): 2.44
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
+Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}
+Optimal tunings:
+* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7411{{c}}
+* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6465{{c}}
+{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334dde }}
+Badness (Sintel): 1.68
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+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 tunings:
+* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7399{{c}}
+* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6464{{c}}
+{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334ddeg }}
+Badness (Sintel): 1.14
+== Octoid ==
+: {{Main| Octoid }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Octoid]].''
+The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai comma]]). In the 11-limit, it tempers out [[540/539]], [[1375/1372]], and [[6250/6237]]. In this temperament, one period gives ~[[12/11]], two give ~[[25/21]], three give ~[[35/27]], and four give [[99/70]]~[[140/99]].
+The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimizing the average damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out [[324/323]], [[375/374]], [[495/494]], [[625/624]], [[715/714]] or [[729/728]].
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 4375/4374, 16875/16807
-[[Tuning ranges]]:
+{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
-* [[diamond monotone]] range: [578.571, 600.000] (27\56 to 4\8)
+: mapping generators: ~49/45, ~7/5
-* [[diamond purer]] range: [582.512, 584.359]
-* [[diamond nice]] range:  [582.512, 584.359]
-[[Mapping]]: [{{val|8 1 3 3}}, {{val|0 3 4 5}}]
+[[Optimal tuning]]s:
+* [[WE]]: ~49/45 = 150.0003{{c}}, ~7/5 = 583.9416{{c}}
+: [[error map]]: {{val| +0.002 -0.130 -0.547 +0.883 }}
+* [[CWE]]: ~49/45 = 150.0000{{c}}, ~7/5 = 583.9411{{c}}
+: error map: {{val| 0.000 -0.132 -0.549 +0.880 }}
-[[Wedgie]]: {{multival|24 32 40 -5 -4 3}}
+[[Tuning ranges]]:
+* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
+* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
+* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
+* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
-Mapping generators: ~49/45, ~7/5
+{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}
-[[POTE generator]]: ~7/5 = 583.940
+[[Badness]] (Sintel): 1.08
-[[Vals]]: {{Val list| 8d, 72, 152, 224 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-[[Badness]]: 0.042670
+Comma list: 540/539, 1375/1372, 4000/3993
-Scales: [[Octoid72]], [[Octoid80]]
+Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}
-== 11-limit ==
+Optimal tunings:
-Comma list: 540/539, 1375/1372, 4000/3993
+* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9477{{c}}
 Tuning ranges:
-* [[diamond monotone]] range: [581.250, 586.364] (31\64, 43\88)
+* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
-* [[diamond purer]] range: [582.512, 585.084]
+* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
-* [[diamond nice]] range: [582.512, 585.084]
-Mapping: [{{val|8 1 3 3 16}}, {{val|0 3 4 5 3}}]
+{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224, 824d }}
-POTE generator: ~7/5 = 583.962
+Badness (Sintel): 0.466
-Vals: {{Val list| 72, 152, 224 }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Badness: 0.014097
+Comma list: 540/539, 625/624, 729/728, 1375/1372
-Scales: [[Octoid72]], [[Octoid80]]
+Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}
-== 13-limit ==
+Optimal tunings:
-Comma list: 540/539, 1375/1372, 4000/3993, 625/624
+* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}
-Mapping: [{{val|8 1 3 3 16 -21}}, {{val|0 3 4 5 3 13}}]
+{{Optimal ET sequence|legend=0| 72, 152f, 224 }}
-POTE generator: ~7/5 = 583.905
+Badness (Sintel): 0.631
-Vals: {{Val list| 72, 152f, 224 }}
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-Badness: 0.015274
+Comma list: 375/374, 540/539, 625/624, 715/714, 729/728
-Scales: [[Octoid72]], [[Octoid80]]
+Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}
-=== Music ===
+Optimal tunings:
-* [http://www.archive.org/details/Dreyfus http://www.archive.org/details/Dreyfus] [http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play]
+* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}
-== Octopus ==
+{{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}
-Comma list: 169/168, 325/324, 364/363, 540/539
-Mapping: [{{val|8 1 3 3 16 14}}, {{val|0 3 4 5 3 4}}]
+Badness (Sintel): 0.729
-POTE generator: ~7/5 = 583.892
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-Vals: {{Val list| 72, 152, 224f }}
+Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
-Badness: 0.021679
+Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}
-Scales: [[Octoid72]], [[Octoid80]]
+Optimal tunings:
+* WE: ~12/11 = 149.9785{{c}}, ~7/5 = 583.8482{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}}
-= Amity =
+{{Optimal ET sequence|legend=0| 72, 152fg, 224 }}
-{{main|Amity}}
-{{see also|Amity family #Amity}}
-The generator for [[amity]] temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&amp;53 temperament, or by its wedgie, {{multival|5 13 -17 9 -41 -76}}. [[99edo|99EDO]] is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 39, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.
+Badness (Sintel): 0.975
-In the 5-limit amity is a genuine microtemperament, with 58/205 being a possible tuning. Another good choice is (64/5)^(1/13), which gives pure major thirds.
+==== Octopus ====
+A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4 = 300{{c}}.
-[[Comma]]: 1600000/1594323
+Subgroup: 2.3.5.7.11.13
-[[Mapping]]: [{{val|1 3 6}}, {{val|0 -5 -13}}]
+Comma list: 169/168, 325/324, 364/363, 540/539
-[[POTE generator]]: ~243/200 = 339.519
+Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
-[[Vals]]: {{Val list| 7, 39, 46, 53, 152, 205, 463, 668, 873 }}
+Optimal tunings:
+* WE: ~12/11 = 150.0313{{c}}, ~7/5 = 584.0134{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}}
-[[Badness]]: 0.021960
+{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}
-== 7-limit ==
+Badness (Sintel): 0.896
-[[Comma list]]: 4375/4374, 5120/5103
-[[Mapping]]: [{{val|1 3 6 -2}}, {{val|0 -5 -13 17}}]
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-[[Wedgie]]: {{multival|5 13 -17 9 -41 -76}}
+Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
-[[POTE generator]]: ~128/105 = 339.432
+Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}
-[[Vals]]: {{Val list| 7, 39, 46, 53, 99, 251, 350, 601cd, 951bcdd }}
+Optimal tunings:
+* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}
-[[Badness]]: 0.023649
+{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}
-== 11-limit ==
+Badness (Sintel): 0.795
-Comma list: 540/539, 4375/4374, 5120/5103
-Mapping: [{{val|1 3 6 -2 21}}, {{val|0 -5 -13 17 -62}}]
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-POTE generator: ~128/105 = 339.464
+Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
-Vals: {{Val list| 53, 99e, 152, 555dee, 707ddee, 859bddee }}
+Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}
-Badness: 0.031506
+Optimal tunings:
+* WE: ~12/11 = 150.0049{{c}}, ~7/5 = 584.0833{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}}
-=== 13-limit ===
+{{Optimal ET sequence|legend=0| 8d, 72, 152 }}
-Comma list: 352/351, 540/539, 625/624, 847/845
-Mapping: [{{val|1 3 6 -2 21 17}}, {{val|0 -5 -13 17 -62 -47}}]
+Badness (Sintel): 0.993
-POTE generator: ~128/105 = 339.481
+Scales: [[Octoid72]], [[Octoid80]]
-Vals: {{Val list| 53, 99ef, 152f }}
+==== Hexadecoid ====
+{{See also| 16th-octave temperaments }}
-Badness: 0.028008
+Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.
-== Hitchcock ==
+Subgroup: 2.3.5.7.11.13
-{{see also|Amity family #Hitchcock}}
-Comma list: 121/120, 176/175, 2200/2187
+Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
-Mapping: [{{val|1 3 6 -2 6}}, {{val|0 -5 -13 17 -9}}]
+Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
+: mapping generators: ~448/429, ~7/5
-POTE generator: ~11/9 = 339.390
+Optimal tunings:
+* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}
+* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}
-Vals: {{Val list| 7, 39, 46, 53, 99 }}
+{{Optimal ET sequence|legend=0| 80, 144, 224 }}
-Badness: 0.035187
+Badness (Sintel): 1.27
-=== 13-limit ===
+===== 17-limit =====
-Comma list: 121/120, 169/168, 176/175, 325/324
+Subgroup: 2.3.5.7.11.13.17
-Mapping: [{{val|1 3 6 -2 6 2}}, {{val|0 -5 -13 17 -9 6}}]
+Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
-POTE generator: ~11/9 = 339.419
+Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
-Vals: {{Val list| 7, 39, 46, 53, 99 }}
+Optimal tunings:
+* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9626{{c}}
+* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0463{{c}}
-Badness: 0.022448
+{{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }}
-== Hemiamity ==
+Badness (Sintel): 1.46
-Comma list: 3025/3024, 4375/4374, 5120/5103
-Mapping: [{{val|2 1 -1 13 13}}, {{val|0 5 13 -17 -14}}]
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-POTE generator: ~64/55 = 339.439
+Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
-Vals: {{Val list| 14cde, 46, 106, 152, 350, 502d }}
+Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}
-Badness: 0.031307
+Optimal tunings:
+* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}
+* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}
-= Parakleismic =
+{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}
-{{main| Parakleismic }}
-In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo|8 14 -13}}, with the [[118edo|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 limits, 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|99EDO]] may be preferred, but in the 11-limit it is best to stick with 118.
+Badness (Sintel): 1.44
-[[Comma list]]: 1224440064/1220703125
+== Parakleismic ==
+{{Main| Parakleismic }}
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic (5-limit)]].''
-[[Mapping]]: [{{val|1 5 6}}, {{val|0 -13 -14}}]
+In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat [[6/5]], 13 of which give 32/3, and 14 give 64/5. While 118 no longer has better than a cent of accuracy in the 7-limit, it is a decent temperament there nonetheless, and this allows an extension adding [[3136/3125]] and 4375/4374, for which [[99edo]], 118edo, and especially [[217edo]] are accurate tunings.
-[[POTE generator]]: ~6/5 = 315.240
+Parakleismic does not extend easily to the 11- or 13-limit. Possible 11-limit extensions include undecimal parakleismic (99 & 118), paralytic (99e & 118), parkleismic (80 & 99), and paradigmic (80 & 99e).
-[[Vals]]: {{Val list| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}
+[[Subgroup]]: 2.3.5.7
-[[Badness]]: 0.043279
-== 7-limit ==
 [[Comma list]]: 3136/3125, 4375/4374
-[[Mapping]]: [{{val|1 5 6 12}}, {{val|0 -13 -14 -35}}]
+{{Mapping|legend=1| 1 -8 -8 -23 | 0 13 14 35 }}
+: mapping generators: ~2, ~5/3
-[[Wedgie]]: {{multival|13 14 35 -8 19 42}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7820{{c}}, ~5/3 = 884.6581{{c}}
+: [[error map]]: {{val| -0.218 +0.344 +0.644 -0.779 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8088{{c}}
+: error map: {{val| 0.000 +0.560 +1.010 -0.516 }}
-[[POTE generator]]: ~6/5 = 315.181
+{{Optimal ET sequence|legend=1| 19, 61d, 80, 99, 217, 316, 415 }}
-[[Vals]]: {{Val list| 19, 80, 99, 217, 316, 415 }}
+[[Badness]] (Sintel): 0.694
-[[Badness]]: 0.027431
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== 11-limit ==
 Comma list: 385/384, 3136/3125, 4375/4374
-Mapping: [{{val|1 5 6 12 -6}}, {{val|0 -13 -14 -35 36}}]
+Mapping: {{mapping| 1 -8 -8 -23 30 | 0 13 14 35 -36 }}
-POTE generator: ~6/5 = 315.251
+Optimal tunings:
+* WE: ~2 = 1200.3296{{c}}, ~5/3 = 884.9921{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7519{{c}}
-Vals: {{Val list| 19, 99, 118 }}
+{{Optimal ET sequence|legend=0| 19, 99, 118 }}
-Badness: 0.049711
+Badness (Sintel): 1.64
-== Paralytic ==
+=== Paralytic ===
-The ''paralytic'' temperament (118&amp;217, named by [[User:FloraC|Flora Canou]]) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118&amp;217 tempers out 1001/1000, 1575/1573, and 3584/3575.
+Paralytic (99e & 118) tempers out [[441/440]], [[5632/5625]], and [[19712/19683]]. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575.
+Subgroup: 2.3.5.7.11
 Comma list: 441/440, 3136/3125, 4375/4374
-Mapping: [{{val|1 5 6 12 25}}, {{val|0 -13 -14 -35 -82}}]
+Mapping: {{mapping| 1 -8 -8 -23 -57 | 0 13 14 35 82 }}
+Optimal tunings:
+* WE: ~2 = 1199.9940{{c}}, ~5/3 = 884.7757{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7800{{c}}
-POTE generator: ~6/5 = 315.220
+{{Optimal ET sequence|legend=0| 19e, …, 99e, 118, 217, 335 }}
-Vals: {{Val list| 19e, 99e, 118, 217, 335, 552d, 887dd }}
+Badness (Sintel): 1.19
-Badness: 0.036027
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-=== 13-limit ===
 Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
-Mapping: [{{val|1 5 6 12 25 -16}}, {{val|0 -13 -14 -35 -82 75}}]
+Mapping: {{mapping| 1 -8 -8 -23 -57 59 | 0 13 14 35 82 -75 }}
+Optimal tunings:
+* WE: ~2 = 1199.9218{{c}}, ~5/3 = 884.7285{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7858{{c}}
-POTE generator: ~6/5 = 315.214
+{{Optimal ET sequence|legend=0| 99e, 118, 217 }}
-Vals: {{Val list| 99e, 118, 217, 552d, 769de }}
+Badness (Sintel): 1.85
-Badness: 0.044710
+==== Paraklein ====
+Paraklein (19e & 118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].
-=== Paraklein ===
+Subgroup: 2.3.5.7.11.13
-The ''paraklein'' temperament (19e&amp;118, named by [[User:Xenllium|Xenllium]]) 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]].
 Comma list: 196/195, 352/351, 625/624, 729/728
-Mapping: [{{val|1 5 6 12 25 15}}, {{val|0 -13 -14 -35 -82 -43}}]
+Mapping: {{mapping| 1 -8 -8 -23 -57 -28 | 0 13 14 35 82 43 }}
+Optimal tunings:
+* WE: ~2 = 1199.8239{{c}}, ~5/3 = 884.6449{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7709{{c}}
-POTE generator: ~6/5 = 315.225
+{{Optimal ET sequence|legend=0| 19e, …, 99ef, 118 }}
-Vals: {{Val list| 19e, 99ef, 118, 217ff, 335ff }}
+Badness (Sintel): 1.55
-Badness: 0.037618
+=== Parkleismic ===
+Subgroup: 2.3.5.7.11
-== Parkleismic ==
 Comma list: 176/175, 1375/1372, 2200/2187
-Mapping: [{{val|1 5 6 12 20}}, {{val|0 -13 -14 -35 -63}}]
+Mapping: {{mapping| 1 -8 -8 -23 -43 | 0 13 14 35 63 }}
+Optimal tunings:
+* WE: ~2 = 1199.1848{{c}}, ~5/3 = 884.3386{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9158{{c}}
-POTE generator: ~6/5 = 315.060
+{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179, 259cd }}
-Vals: {{Val list| 19e, 80, 179, 259cd }}
+Badness (Sintel): 1.85
-Badness: 0.055884
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-=== 13-limit ===
 Comma list: 169/168, 176/175, 325/324, 1375/1372
-Mapping: [{{val|1 5 6 12 20 10}}, {{val|0 -13 -14 -35 -63 -24}}]
+Mapping: {{mapping| 1 -8 -8 -23 -43 -14 | 0 13 14 35 63 24 }}
+Optimal tunings:
+* WE: ~2 = 1199.5318{{c}}, ~5/3 = 884.5800{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9118{{c}}
-POTE generator: ~6/5 = 315.075
+{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179 }}
-Vals: {{Val list| 19e, 80, 179 }}
+Badness (Sintel): 1.51
-Badness: 0.036559
+=== Paradigmic ===
+Subgroup: 2.3.5.7.11
-== Paradigmic ==
 Comma list: 540/539, 896/891, 3136/3125
-Mapping: [{{val|1 5 6 12 -1}}, {{val|0 -13 -14 -35 17}}]
+Mapping: {{mapping| 1 -8 -8 -23 16 | 0 13 14 35 -17 }}
+Optimal tunings:
+* WE: ~2 = 1199.0616{{c}}, ~5/3 = 884.2124{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.8877{{c}}
-POTE generator: ~6/5 = 315.096
+{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e, 457bcddeeee }}
-Vals: {{Val list| 19, 61d, 80, 99e, 179e }}
+Badness (Sintel): 1.38
-Badness: 0.041720
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-=== 13-limit ===
 Comma list: 169/168, 325/324, 540/539, 832/825
-Mapping: [{{val|1 5 6 12 -1 10}}, {{val|0 -13 -14 -35 17 -24}}]
+Mapping: {{mapping| 1 -8 -8 -23 16 -14 | 0 13 14 35 -17 24 }}
-POTE generator: ~6/5 = 315.080
+Optimal tunings:
+* WE: ~2 = 1199.2683{{c}}, ~5/3 = 884.3805{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9061{{c}}
-Vals: {{Val list| 19, 61d, 80, 99e, 179e }}
+{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e }}
-Badness: 0.035781
+Badness (Sintel): 1.48
+=== Semiparakleismic ===
+Subgroup: 2.3.5.7.11
-== Semiparakleismic ==
 Comma list: 3025/3024, 3136/3125, 4375/4374
-Mapping: [{{val|2 10 12 24 19}}, {{val|0 -13 -14 -35 -23}}]
+Mapping: {{mapping| 2 -3 -2 -11 -4 | 0 13 14 35 23 }}
+: mapping generators: ~99/70, ~33/28
+Optimal tunings:
+* WE: ~99/70 = 599.9270{{c}}, ~33/28 = 284.7841{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8119{{c}}
+{{Optimal ET sequence|legend=0| 80, 118, 198, 316, 514c }}
-POTE generator: ~6/5 = 315.181
+Badness (Sintel): 1.13
-Vals: {{Val list| 80, 118, 198, 316, 514c, 830c }}
+==== Semiparamint ====
+This extension was named ''semiparakleismic'' in the earlier materials.
-Badness: 0.034208
+Subgroup: 2.3.5.7.11.13
-=== 13-limit ===
 Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
-Mapping: [{{val|2 10 12 24 19 -1}}, {{val|0 -13 -14 -35 -23 16}}]
+Mapping: {{mapping| 2 -3 -2 -11 -4 15 | 0 13 14 35 23 -16 }}
-POTE generator: ~6/5 = 315.156
+Optimal tunings:
+* WE: ~99/70 = 599.8253{{c}}, ~33/28 = 284.7608{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8366{{c}}
-Vals: {{Val list| 80, 118, 198 }}
+{{Optimal ET sequence|legend=0| 80, 118, 198 }}
-Badness: 0.033775
+Badness (Sintel): 1.40
+==== Semiparawolf ====
+This extension was named ''gentsemiparakleismic'' in the earlier materials.
+Subgroup: 2.3.5.7.11.13
-=== Gentsemiparakleismic ===
 Comma list: 169/168, 325/324, 364/363, 3136/3125
-Mapping: [{{val|2 10 12 24 19 20}}, {{val|0 -13 -14 -35 -23 -24}}]
+Mapping: {{mapping| 2 -3 -2 -11 -4 -4 | 0 13 14 35 23 24 }}
-POTE generator: ~6/5 = 315.184
+Optimal tunings:
+* WE: ~99/70 = 600.0569{{c}}, ~13/11 = 284.8431{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~13/11 = 284.8216{{c}}
-Vals: {{Val list| 80, 118f, 198f }}
+{{Optimal ET sequence|legend=0| 80, 118f, 198f }}
-Badness: 0.040467
+Badness (Sintel): 1.67
-= Counterkleismic =
+== Counterkleismic ==
-{{see also|Syntonic-enneadecal equivalence continuum #Counterhanson}}
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''
-In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo|-20 -24 25}}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as 19&amp;224 temperament (''counterkleismic'', named by [[User:Xenllium|Xenllium]]), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).
+In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses]] ((648/625)<sup>6</sup>) fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament, tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). It was named by analogy to [[catakleismic]] and parakleismic)
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 4375/4374, 158203125/157351936
-[[Mapping]]: [{{val|1 -5 -4 -18}}, {{val|0 25 24 79}}]
+{{Mapping|legend=1| 1 -5 -4 -18 | 0 25 24 79 }}
+: mapping generators: ~2, ~6/5
-[[Wedgie]]: {{multival|25 24 79 -20 55 116}}
+[[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 }}
-[[POTE generator]]: ~6/5 = 316.060
+{{Optimal ET sequence|legend=1| 19, …, 205, 224, 243, 467 }}
-[[Vals]]: {{Val list| 19, 205, 224, 243, 467 }}
+[[Badness]] (Sintel): 2.29
-[[Badness]]: 0.090553
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== 11-limit ==
 Comma list: 540/539, 4375/4374, 2097152/2096325
-Mapping: [{{val|1 -5 -4 -18 19}}, {{val|0 25 24 79 -59}}]
+Mapping: {{mapping| 1 -5 -4 -18 19 | 0 25 24 79 -59 }}
-POTE generator: ~6/5 = 316.071
+Optimal tunings:
+* WE: ~2 = 1199.9944{{c}}, ~6/5 = 316.0690{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0705{{c}}
-Vals: {{Val list| 19, 205, 224 }}
+{{Optimal ET sequence|legend=0| 19, 205, 224 }}
-Badness: 0.070952
+Badness (Sintel): 2.35
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-=== 13-limit ===
 Comma list: 540/539, 625/624, 729/728, 10985/10976
-Mapping: [{{val|1 -5 -4 -18 19 -15}}, {{val|0 25 24 79 -59 71}}]
+Mapping: {{mapping| 1 -5 -4 -18 19 -15 | 0 25 24 79 -59 71 }}
+Optimal tunings:
+* WE: ~2 = 1199.9827{{c}}, ~6/5 = 316.0650{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0695{{c}}
-POTE generator: ~6/5 = 316.070
+{{Optimal ET sequence|legend=0| 19, 205, 224 }}
-Vals: {{Val list| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}
+Badness (Sintel): 1.40
-Badness: 0.033874
+=== Counterlytic ===
+Subgroup: 2.3.5.7.11
-== Counterlytic ==
 Comma list: 1375/1372, 4375/4374, 496125/495616
-Mapping: [{{val|1 -5 -4 -18 -40}}, {{val|0 25 24 79 165}}]
+Mapping: {{mapping| 1 -5 -4 -18 -40 | 0 25 24 79 165 }}
+Optimal tunings:
+* WE: ~2 = 1200.1247{{c}}, ~6/5 = 316.0976{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0660{{c}}
-POTE generator: ~6/5 = 316.065
+{{Optimal ET sequence|legend=1| 19e, 205e, 224, 467e, 691, 915c }}
-Vals: {{Val list| 19e, 205e, 224 }}
+Badness (Sintel): 2.16
-Badness: 0.065400
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-=== 13-limit ===
 Comma list: 625/624, 729/728, 1375/1372, 10985/10976
-Mapping: [{{val|1 -5 -4 -18 -40 -15}}, {{val|0 25 24 79 165 71}}]
+Mapping: {{mapping| 1 -5 -4 -18 -40 -15 | 0 25 24 79 165 71 }}
+Optimal tunings:
+* WE: ~2 = 1200.0987{{c}}, ~6/5 = 316.0908{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0658{{c}}
-POTE generator: ~6/5 = 316.065
+{{Optimal ET sequence|legend=0| 19e, 205e, 224, 467e, 691, 915c }}
-Vals: {{Val list| 19e, 205e, 224 }}
+Badness (Sintel): 1.23
-Badness: 0.029782
+== Quincy ==
+[[Subgroup]]: 2.3.5.7
-= Quincy =
 [[Comma list]]: 4375/4374, 823543/819200
-[[Mapping]]: [{{val|1 2 3 3}}, {{val|0 -30 -49 -14}}]
+{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
+: mapping generators: ~2, ~1728/1715
-[[Wedgie]]: {{multival|30 49 14 8 -62 -105}}
+[[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 }}
-[[POTE generator]]: ~1728/1715 = 16.613
+{{Optimal ET sequence|legend=1| 72, 217, 289, 650d, 939dd }}
-[[Vals]]: {{Val list| 72, 217, 289 }}
+[[Badness]] (Sintel): 2.02
-[[Badness]]: 0.079657
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== 11-limit ==
 Comma list: 441/440, 4000/3993, 4375/4374
-Mapping: [{{val|1 2 3 3 4}}, {{val|0 -30 -49 -14 -39}}]
+Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}
+Optimal tunings:
+* WE: ~2 = 1200.1286{{c}}, ~100/99 = 16.6147{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6101{{c}}
-POTE generator: ~100/99 = 16.613
+{{Optimal ET sequence|legend=0| 72, 217, 289 }}
-Vals: {{Val list| 72, 217, 289 }}
+Badness (Sintel): 1.02
-Badness: 0.030875
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-== 13-limit ==
 Comma list: 364/363, 441/440, 676/675, 4375/4374
-Mapping: [{{val|1 2 3 3 4 5}}, {{val|0 -30 -49 -14 -39 -94}}]
+Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}
-POTE generator: ~100/99 = 16.602
+Optimal tunings:
+* WE: ~2 = 1200.0554{{c}}, ~100/99 = 16.6028{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6011{{c}}
-Vals: {{Val list| 72, 145, 217, 289 }}
+{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
-Badness: 0.023862
+Badness (Sintel): 0.986
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
-== 17-limit ==
 Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155
-Mapping: [{{val|1 2 3 3 4 5 5}}, {{val|0 -30 -49 -14 -39 -94 -66}}]
+Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}
+Optimal tunings:
+* WE: ~2 = 1200.0647{{c}}, ~100/99 = 16.6025{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6004{{c}}
-POTE generator: ~100/99 = 16.602
+{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
-Vals: {{Val list| 72, 145, 217, 289 }}
+Badness (Sintel): 0.751
-Badness: 0.014741
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
-== 19-limit ==
 Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675
-Mapping: [{{val|1 2 3 3 4 5 5 4}}, {{val|0 -30 -49 -14 -39 -94 -66 18}}]
+Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}
-POTE generator: ~100/99 = 16.594
+Optimal tunings:
+* WE: ~2 = 1199.9287{{c}}, ~100/99 = 16.5930{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.5948{{c}}
-Vals: {{Val list| 72, 145, 217 }}
+{{Optimal ET sequence|legend=0| 72, 145, 217 }}
-Badness: 0.015197
+Badness (Sintel): 0.924
-= Chlorine =
+== Sfourth ==
-The name of chlorine temperament comes from Chlorine, the 17th element.
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sfourth]].''
-Chlorine temperament has a period of 1/17 octave. It tempers out the septendecima, {{monzo|-52 -17 34}}, by which 17 chromatic semitones (25/24) exceed an octave. This temperament can be described as 289&amp;323 temperament, which tempers out {{monzo|-49 4 22 -3}} as well as the ragisma.
+[[Subgroup]]: 2.3.5.7
-[[Comma]]: {{monzo|-52 -17 34}}
+[[Comma list]]: 4375/4374, 64827/64000
-[[Mapping]]: [{{val|17 26 39}}, {{val|0 2 1}}]
+{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
+: mapping generators: ~2, ~49/48
-[[POTE tuning|POTE generators]]: ~25/24 = 70.5882, ~5/4 = 386.2687
+[[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 }}
-[[Vals]]: {{Val list| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}
+{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
-[[Badness]]: 0.077072
+[[Badness]] (Sintel): 3.12
-== 7-limit ==
+=== 11-limit ===
-[[Comma list]]: 4375/4374, 193119049072265625/193091834023510016
+Subgroup: 2.3.5.7.11
-[[Mapping]]: [{{val|17 26 39 43}}, {{val|0 2 1 10}}]
+Comma list: 121/120, 441/440, 4375/4374
-[[Wedgie]]: {{multival|34 17 170 -52 174 347}}
+Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}
-[[POTE tuning|POTE generators]]: ~25/24 = 70.5882, ~5/4 = 386.2936
+Optimal tunings:
+* WE: ~2 = 1201.1486{{c}}, ~49/48 = 26.3112{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2461{{c}}
-[[Vals]]: {{Val list| 289, 323, 612, 935, 1547 }}
+{{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }}
-[[Badness]]: 0.041658
+Badness (Sintel): 1.78
-== 11-limit ==
+==== 13-limit ====
-Comma list: 4375/4374, 41503/41472, 1879453125/1879048192
+Subgroup: 2.3.5.7.11.13
-Mapping: [{{val|17 26 39 43 64}}, {{val|0 2 1 10 -11}}]
+Comma list: 121/120, 169/168, 325/324, 441/440
-POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2690
+Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}
-Vals: {{Val list| 289, 323, 612 }}
+Optimal tunings:
+* WE: ~2 = 1201.4956{{c}}, ~49/48 = 26.3423{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2614{{c}}
-Badness: 0.063706
+{{Optimal ET sequence|legend=0| 45ef, 46, 91ef, 137def, 228ddeeefff }}
-= Palladium =
+Badness (Sintel): 1.37
-The name of ''palladium temperament'' (named by [[User:Xenllium|Xenllium]]) comes from Palladium, the 46th element.
-Palladium temperament 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.
+=== Sfour ===
+Subgroup: 2.3.5.7.11
-[[Comma list]]: 4375/4374, 2270317133144025/2251799813685248
+Comma list: 385/384, 2401/2376, 4375/4374
-[[Mapping]]: [{{val|46 73 107 129}}, {{val|0 -1 -2 1}}]
+Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}
-[[Wedgie]]: {{multival|46 92 -46 39 -202 -365}}
+Optimal tunings:
+* WE: ~2 = 1200.4402{{c}}, ~49/48 = 26.2557{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2403{{c}}
-[[POTE generator]]: ~3/2 = 701.6074
+{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}
-[[Vals]]: {{Val list| 46, 368, 414, 460, 874d }}
+Badness (Sintel): 2.53
-[[Badness]]: 0.308505
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-== 11-limit ==
+Comma list: 196/195, 364/363, 385/384, 4375/4374
-Comma list: 3025/3024, 9801/9800, 134775333/134217728
-Mapping: [{{val|46 73 107 129 159}}, {{val|0 -1 -2 1 1}}]
+Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}
-POTE generator: ~3/2 = 701.5951
+Optimal tunings:
+* WE: ~2 = 1200.3796{{c}}, ~49/48 = 26.2473{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2372{{c}}
-Vals: {{Val list| 46, 368, 414, 460, 874de }}
+{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}
-Badness: 0.073783
+Badness (Sintel): 2.14
-== 13-limit ==
+== Trideci ==
-Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
+: ''For the 5-limit version, see [[13th-octave temperaments #Tridecatonic]].''
-Mapping: [{{val|46 73 107 129 159 170}}, {{val|0 -1 -2 1 1 2}}]
+The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from ''tridecim'' (Latin for "thirteen").
-POTE generator: ~3/2 = 701.6419
+[[Subgroup]]: 2.3.5.7
-Vals: {{Val list| 46, 368, 414, 460, 874de, 1334de }}
+[[Comma list]]: 4375/4374, 83349/81920
-Badness: 0.040751
+{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
+: mapping generators: ~256/245, ~3
-== 17-limit ==
+[[Optimal tuning]]s:
-Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
+* [[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 }}
-Mapping: [{{val|46 73 107 129 159 170 188}}, {{val|0 -1 -2 1 1 2 0}}]
+{{Optimal ET sequence|legend=1| 26, 65, 91 }}
-POTE generator: ~3/2 = 701.6425
+[[Badness]] (Sintel): 4.67
-Vals: {{Val list| 46, 368, 414, 460, 874de, 1334deg }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Badness: 0.022441
+Comma list: 245/242, 385/384, 4375/4374
-= Monzism =
+Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}
-The ''monzism'' temperament (53&amp;612, named by [[User:Xenllium|Xenllium]]) is a rank-two temperament which tempers out the [[monzisma]], {{monzo|54 -37 2}} and the [[nanisma]], {{monzo|109 -67 0 -1}}, as well as the ragisma, [[4375/4374]].
-[[Comma list]]: 4375/4374, 36030948116563575/36028797018963968
+Optimal tunings:
+* WE: ~22/21 = 92.3729{{c}}, ~3/2 = 700.1118{{c}}
+* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.7703{{c}}
-[[Mapping]]: [{{val|1 2 10 -25}}, {{val|0 -2 -37 134}}]
+{{Optimal ET sequence|legend=0| 26, 65, 91 }}
-[[Wedgie]]: {{multival|2 37 -134 54 -218 -415}}
+Badness (Sintel): 2.80
-[[POTE generator]]: ~310078125/268435456 = 249.0207
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-[[Vals]]: {{Val list| 53, 559, 612, 1277, 1889 }}
+Comma list: 169/168, 245/242, 325/324, 385/384
-[[Badness]]: 0.046569
+Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}
-== 11-limit ==
+Optimal tunings:
-Comma list: 4375/4374, 41503/41472, 184549376/184528125
+* WE: ~22/21 = 92.4003{{c}}, ~3/2 = 699.9983{{c}}
+* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.4772{{c}}
-Mapping: [{{val|1 2 10 -25 46}}, {{val|0 -2 -37 134 -205}}]
+{{Optimal ET sequence|legend=0| 26, 65f, 91f }}
-POTE generator: ~231/200 = 249.0193
+Badness (Sintel): 2.16
-Vals: {{Val list| 53, 559, 612 }}
+== References ==
-Badness: 0.057083
+[[Category:Temperament collections]]
+[[Category:Ragismic microtemperaments| ]] <!-- main article -->
-== 13-limit ==
-Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
-Mapping: [{{val|1 2 10 -25 46 23}}, {{val|0 -2 -37 134 -205 -93}}]
-POTE generator: ~231/200 = 249.0199
-Vals: {{Val list| 53, 559, 612 }}
-Badness: 0.053780
-[[Category:Abigail]]
-[[Category:Amity]]
-[[Category:Deca]]
-[[Category:Enneadecal]]
-[[Category:Ennealimmal]]
-[[Category:Gamera]]
-[[Category:Mitonic]]
-[[Category:Octoid]]
-[[Category:Parakleismic]]
-[[Category:Supermajor]]
-[[Category:Microtemperament]]
-[[Category:Ragismic]]
 [[Category:Rank 2]]
-[[Category:Ragismic microtemperaments| ]] <!-- main article -->
-[[Category:Todo:review]]

Ragismic microtemperaments: Difference between revisions