@@ Line 1: / Line 1: @@
-The ragisma is [[4375/4374]] with a [[monzo]] of {{monzo| -1 -7 4 1 }}, the smallest 7-limit [[superparticular]] ratio. Since (10/9)<sup>4</sup> = 4375/4374 × 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 × (27/25)<sup>2</sup>, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
+{{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 discussed elsewhere include:
+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.
-* ''[[Hystrix]]'', {36/35, 160/147} → [[Porcupine family #Hystrix|Porcupine family]]
-* ''[[Rhinoceros]]'', {49/48, 4375/4374} → [[Unicorn family #Rhinoceros|Unicorn family]]
-* ''[[Crepuscular]]'', {50/49, 4375/4374} → [[Jubilismic clan #Crepuscular|Jubilismic clan]] and [[Fifive family #Crepuscular|Fifive family]]
-* ''[[Modus]]'', {64/63, 4375/4374} → [[Tetracot family #Modus|Tetracot family]]
-* ''[[Flattone]]'', {81/80, 525/512} → [[Meantone family #Flattone|Meantone family]]
-* [[Sensi]], {126/125, 245/243} → [[Sensipent family #Sensi|Sensipent family]] and [[Sensamagic clan #Sensi|Sensamagic clan]]
-* [[Catakleismic]], {225/224, 4375/4374} → [[Kleismic family #Catakleismic|Kleismic family]]
-* [[Unidec]], {1029/1024, 4375/4374} → [[Gamelismic clan #Unidec|Gamelismic clan]]
-* ''[[Quartonic]]'', {1728/1715, 4000/3969} → [[Orwellismic temperaments #Quartonic|Orwellismic temperaments]]
-* ''[[Srutal]]'', {2048/2025, 4375/4374} → [[Diaschismic family #Srutal|Diaschismic family]]
-* ''[[Maja]]'', {2430/2401, 3125/3087} → [[Maja family #Septimal maja|Maja family]]
-* [[Amity]], {4375/4374, 5120/5103} → [[Amity family #Septimal amity|Amity family]]
-* [[Pontiac]], {4375/4374, 32805/32768} → [[Schismatic family #Pontiac|Schismatic family]]
-* ''[[Zarvo]]'', {4375/4374, 33075/32768} → [[Gravity family #Zarvo|Gravity family]]
-* ''[[Whirrschmidt]]'', {4375/4374, 393216/390625} → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
-* ''[[Mitonic]]'', {4375/4374, 2100875/2097152} → [[Minortonic family #Mitonic|Minortonic family]]
-* ''[[Vishnu]]'', {4375/4374, 29360128/29296875} → [[Vishnuzmic family #Septimal vishnu|Vishnuzmic family]]
-* ''[[Vulture]]'', {4375/4374, 33554432/33480783} → [[Vulture family #Septimal vulture|Vulture family]]
-* ''[[Trillium]]'', {4375/4374, {{monzo| 40 -22 -1 -1 }}} → [[Tricot family #Trillium|Tricot family]]
-* ''[[Unlit]]'', {4375/4374, {{monzo| 41 -20 -4 }}} → [[Undim family #Unlit|Undim family]]
-* ''[[Quindro]]'', {4375/4374, {{monzo| 56 -28 -5 }}} → [[Quindromeda family #Quindro|Quindromeda family]]
-* ''[[Dzelic]]'', {4375/4374, {{monzo|-223 47 -11 62}}} → [[37th-octave temperaments#Dzelic|37th-octave temperaments]]
-Microtemperaments considered below are ennealimmal, supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, orga, chlorine, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium.
-Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci.
+Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
+* ''[[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 ==
+== Supermajor ==
-{{Main| Ennealimmal }}
+The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2<sup>15</sup>)/3, 46 give (2<sup>19</sup>)/5, and 75 give (2<sup>30</sup>)/7. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80-note mos is presumably the place to start, and if that is not enough notes for you, there is always the 171-note mos.
-Ennealimmal tempers out the two smallest 7-limit [[superparticular]] commas, 2401/2400 and 4375/4374, leading to a temperament of unusual [[efficiency]]. It also tempers out the [[ennealimma]], {{monzo| 1 -27 18 }}, which leads to the identification of (27/25)<sup>9</sup> with the [[octave]], and gives ennealimmal a [[period]] of 1/9 octave. Its [[pergen]] is (P8/9, P5/2). While 27/25 is a 5-limit interval, a stack of two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit.
+[[Subgroup]]: 2.3.5.7
-Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40~60/49, all of which have their own interesting advantages. Possible tunings are 441-, 612-, or 3600edo, though its hardly likely anyone could tell the difference.
+[[Comma list]]: 4375/4374, 52734375/52706752
-If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example). In particular, people fond of the idea of "[[tritave]]s" as analogous to octaves might consider the 28 or 43 note [[mos]] with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave mos, which is equivalent in average step size to a 17 2/3 to the octave mos.
+{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }}
-Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]], [[Tritrizo clan #Undecentic|undecentic]], [[Tritrizo clan #Schisennealimmal|schisennealimmal]], and [[Tritrizo clan #Lunennealimmal|lunennealimmal]].
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082
-[[Subgroup]]: 2.3.5.7
+{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}
-[[Comma list]]: 2401/2400, 4375/4374
+[[Badness]]: 0.010836
-{{Mapping|legend=1| 9 1 1 12 | 0 2 3 2 }}
+=== Semisupermajor ===
+Subgroup: 2.3.5.7.11
-{{Multival|legend=1| 18 27 18 1 -22 -34 }}
+Comma list: 3025/3024, 4375/4374, 35156250/35153041
-: mapping generators: ~27/25, ~5/3
+Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}
-[[Optimal tuning]] ([[POTE]]): ~27/25 = 1\9, ~5/3 = 884.3129 (~36/35 = 49.0205)
+Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082
-[[Tuning ranges]]:
+{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}
-* 7-odd-limit [[diamond monotone]]: ~36/35 = [26.667, 66.667] (1\45 to 1\18)
-* 9-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
-* 7- and 9-odd-limit [[diamond tradeoff]]: ~36/35 = [48.920, 49.179]
-* 7- and 9-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 49.179]
-{{Optimal ET sequence|legend=1| 27, 45, 72, 99, 171, 441, 612 }}
+Badness: 0.012773
-[[Badness]]: 0.003610
+== Enneadecal ==
+Enneadecal temperament tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out [[703125/702464]], the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.
-=== 11-limit ===
+''For the 5-limit temperament, see [[19th-octave temperaments#(5-limit) enneadecal]].''
-The ennealimmal temperament can be described as 99e &amp; 171e, which tempers out [[5632/5625]] (vishdel comma) and [[19712/19683]] (symbiotic comma).
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7
-Comma list: 2401/2400, 4375/4374, 5632/5625
+[[Comma list]]: 4375/4374, 703125/702464
-Mapping: {{mapping| 9 1 1 12 -75 | 0 2 3 2 16 }}
+{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4679 (~36/35 = 48.8654)
+: mapping generators: ~28/27, ~3
-{{Optimal ET sequence|legend=1| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
+[[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)
-Badness: 0.027332
+{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}
-==== 13-limit ====
+[[Badness]]: 0.010954
-Subgroup: 2.3.5.7.11.13
-Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 9 1 1 12 -75 93 | 0 2 3 2 16 -9 }}
+Comma list: 540/539, 4375/4374, 16384/16335
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
+Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}
-{{Optimal ET sequence|legend=1| 99e, 171e, 270 }}
+Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)
-Badness: 0.029404
+{{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }}
-===== 17-limit =====
+Badness: 0.043734
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 715/714, 1001/1000, 1716/1715, 4096/4095, 4375/4374
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Mapping: {{mapping| 9 1 1 12 -75 93 -3 | 0 2 3 2 16 -9 6 }}
+Comma list: 540/539, 625/624, 729/728, 2205/2197
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
+Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}
-{{Optimal ET sequence|legend=1| 99e, 171e, 270 }}
+Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)
-===== 19-limit =====
+{{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }}
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 715/714, 1001/1000, 1216/1215, 1716/1715, 4096/4095, 4375/4374
+Badness: 0.033545
-Mapping: {{mapping| 9 1 1 12 -75 93 -3 -48 | 0 2 3 2 16 -9 6 13 }}
+=== Hemienneadecal ===
+Subgroup: 2.3.5.7.11
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
+Comma list: 3025/3024, 4375/4374, 234375/234256
-{{Optimal ET sequence|legend=1| 99e, 171e, 270 }}
+Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}
-==== Ennealimmalis ====
+: mapping generators: ~55/54, ~3
-Subgroup: 2.3.5.7.11.13
-Comma list: 2080/2079, 2401/2400, 4375/4374, 5632/5625
+Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)
-Mapping: {{mapping| 9 1 1 12 -75 -106 | 0 2 3 2 16 21 }}
+{{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}
-Optimal tuning (CTE): ~27/25 = 1\9, ~5/3 = 884.4560 (~36/35 = 48.8773)
+Badness: 0.009985
-{{Optimal ET sequence|legend=1| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}
+==== Hemienneadecalis ====
+Subgroup: 2.3.5.7.11.13
-Badness: 0.022068
+Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256
-=== Ennealimmia ===
+Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}
-The ennealimmia temperament is an alternative extension and can be described as 99 & 171, which tempers out [[131072/130977]] (olympia).
-Subgroup: 2.3.5.7.11
+Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)
-Comma list: 2401/2400, 4375/4374, 131072/130977
+{{Optimal ET sequence|legend=1| 152f, 342f, 494 }}
-Mapping: {{mapping| 9 1 1 12 124 | 0 2 3 2 -14 }}
+Badness: 0.020782
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4089 (~36/35 = 48.9244)
+==== Hemienneadec ====
+Subgroup: 2.3.5.7.11.13
-{{Optimal ET sequence|legend=1| 99, 171, 270, 711, 981, 1251, 2232e }}
+Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
-Badness: 0.026463
+Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}
-==== 13-limit ====
+Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444)
-Subgroup: 2.3.5.7.11.13
-Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374
+{{Optimal ET sequence|legend=1| 152, 342, 494, 1330, 1824, 2318d }}
-Mapping: {{mapping| 9 1 1 12 124 93 | 0 2 3 2 -14 -9 }}
+Badness: 0.030391
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
+==== Semihemienneadecal ====
+Subgroup: 2.3.5.7.11.13
-{{Optimal ET sequence|legend=1| 99, 171, 270, 711, 981, 1692e, 2673e }}
+Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078
-Badness: 0.016607
+Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}
-===== 17-limit =====
+: mapping generators: ~55/54 = 1\38, ~55/54, ~429/250
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 936/935, 2080/2079, 2401/2400, 4096/4095, 4375/4374
+Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)
-Mapping: {{mapping| 9 1 1 12 124 93 -3 | 0 2 3 2 -14 -9 6 }}
+{{Optimal ET sequence|legend=1| 190, 304d, 494, 684, 1178, 2850, 4028ce }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
+Badness: 0.014694
-{{Optimal ET sequence|legend=1| 99, 171, 270 }}
+=== Kalium ===
+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.
-===== 19-limit =====
 Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 936/935, 1216/1215, 2080/2079, 2401/2400, 4096/4095, 4375/4374
+Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344
-Mapping: {{mapping| 9 1 1 12 124 93 -3 -48 | 0 2 3 2 -14 -9 6 13 }}
+Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
+Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244
-{{Optimal ET sequence|legend=1| 99, 171, 270 }}
+{{Optimal ET sequence|legend=1| 855, 988, 1843 }}
-=== Ennealimnic ===
+== Semidimi ==
-Ennealimnic (72 &amp; 171) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.
+: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].''
-Subgroup: 2.3.5.7.11
+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 list: 243/242, 441/440, 4375/4356
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 9 1 1 12 -2 | 0 2 3 2 5 }}
+[[Comma list]]: 4375/4374, 3955078125/3954653486
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9386 (~36/35 = 49.3948)
+{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}
-Tuning ranges:
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270
-* 11-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
-* 11-odd-limit diamond tradeoff: ~36/35 = [48.920, 52.592]
-* 11-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 52.592]
-{{Optimal ET sequence|legend=1| 72, 171, 243 }}
+{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
-Badness: 0.020347
+[[Badness]]: 0.015075
-==== 13-limit ====
+== Brahmagupta ==
-Subgroup: 2.3.5.7.11.13
+The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.
-Comma list: 243/242, 364/363, 441/440, 625/624
+Early in the design of the [[Sagittal]] notation system, Secor and 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 ¢ many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of Brahmagupta temperament that has pure octaves and pure fifths, which can also be described as a 17-limit extension having 1/7th octave period (171.4286 ¢) and 1/21st apotome generator (5.4136 ¢).
-Mapping: {{mapping| 9 1 1 12 -2 -33 | 0 2 3 2 5 10 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9920 (~36/35 = 49.3414)
+[[Comma list]]: 4375/4374, 70368744177664/70338939985125
-Tuning ranges:
+{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}
-* 13- and 15-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)
-* 13- and 15-odd-limit diamond tradeoff: ~36/35 = [48.825, 52.592]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~36/35 = [48.825, 50.000]
-{{Optimal ET sequence|legend=1| 72, 171, 243 }}
+: mapping generators: ~1157625/1048576, ~27/20
-Badness: 0.023250
+[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716
-===== 17-limit =====
+{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 243/242, 364/363, 375/374, 441/440, 595/594
+[[Badness]]: 0.029122
-Mapping: {{mapping| 9 1 1 12 -2 -33 -3 | 0 2 3 2 5 10 6 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9981 (~36/35 = 49.3353)
+Comma list: 4000/3993, 4375/4374, 131072/130977
-Tuning ranges:
+Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}
-* 17-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)
-* 17-odd-limit diamond tradeoff: ~36/35 = [46.363, 52.592]
-* 17-odd-limit diamond monotone and tradeoff: ~36/35 = [48.485, 50.000]
-{{Optimal ET sequence|legend=1| 72, 171, 243 }}
+Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704
-Badness: 0.014602
+{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }}
-===== 19-limit =====
+Badness: 0.052190
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 243/242, 364/363, 375/374, 441/440, 513/512, 595/594
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Mapping: {{mapping| 9 1 1 12 -2 -33 -3 78  | 0 2 3 2 5 10 6 -6 }}
+Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374
-{{Optimal ET sequence|legend=1| 72, 171, 243 }}
+Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}
-==== Ennealim ====
+Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.706
-Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 243/242, 325/324, 441/440
+{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771eef }}
-Mapping: {{mapping| 9 1 1 12 -2 20 | 0 2 3 2 5 2 }}
+Badness: 0.023132
-Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
+== Abigail ==
+Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930]: "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things."</ref>
-{{Optimal ET sequence|legend=1| 27e, 45ef, 72 }}
+''For the 5-limit temperament, see [[Very high accuracy temperaments#Abigail]].''
-Badness: 0.020697
+[[Subgroup]]: 2.3.5.7
-===== 17-limit =====
+[[Comma list]]: 4375/4374, 2147483648/2144153025
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 169/168, 221/220, 243/242, 325/324, 441/440
+{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }}
-Mapping: {{mapping| 9 1 1 12 -2 20 -3 | 0 2 3 2 5 2 6 }}
+: mapping generators: ~46305/32768, ~27/20
-Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
+[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899
-{{Optimal ET sequence|legend=1| 27eg, 45efg, 72 }}
+{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}
-===== 19-limit =====
+[[Badness]]: 0.037000
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 169/168, 221/220, 243/242, 325/324, 441/440
+=== 11-limit ===
-Mapping: {{mapping| 9 1 1 12 -2 20 -3 25 | 0 2 3 2 5 2 6 2 }}
-Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
-{{Optimal ET sequence|legend=1| 27eg, 45efg, 72 }}
-=== Ennealiminal ===
 Subgroup: 2.3.5.7.11
-Comma list: 385/384, 1375/1372, 4375/4374
+Comma list: 3025/3024, 4375/4374, 131072/130977
-Mapping: {{mapping| 9 1 1 12 51 | 0 2 3 2 -3 }}
+Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.8298 (~36/35 = 49.5036)
+Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901
-{{Optimal ET sequence|legend=1| 27, 45, 72, 171e, 243e, 315e }}
+{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }}
-Badness: 0.031123
+Badness: 0.012860
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 325/324, 385/384, 1375/1372
+Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
-Mapping: {{mapping| 9 1 1 12 51 20 | 0 2 3 2 -3 2 }}
+Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}
-Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
+Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903
-{{Optimal ET sequence|legend=1| 27, 45f, 72, 171ef, 243eff }}
+{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }}
-Badness: 0.030325
+Badness: 0.008856
-===== 17-limit =====
+== Gamera ==
-Subgroup: 2.3.5.7.11.13.17
+''For the 5-limit temperament, see [[High badness temperaments#Gamera]].
-Comma list: 169/168, 221/220, 325/324, 385/384, 1375/1372
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 9 1 1 12 51 20 50 | 0 2 3 2 -3 2 -2 }}
+[[Comma list]]: 4375/4374, 589824/588245
-Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
+{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }}
-{{Optimal ET sequence|legend=1| 27, 45f, 72 }}
+: mapping generators: ~2, ~8/7
-===== 19-limit =====
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 153/152, 169/168, 221/220, 325/324, 385/384, 1375/1372
+{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}
-Mapping: {{mapping| 9 1 1 12 51 20 50 25 | 0 2 3 2 -3 2 -2 2 }}
+[[Badness]]: 0.037648
-Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
-{{Optimal ET sequence|legend=1| 27, 45f, 72 }}
-=== Hemiennealimmal ===
-Hemiennealimmal (72 &amp; 198) has a period of 1/18 octave and tempers out the four smallest superparticular commas of the 11-limit JI, 2401/2400, 3025/3024, 4375/4374, and 9801/9800. Tempering out [[9801/9800]] leads an octave split into two equal parts.
+=== Hemigamera ===
 Subgroup: 2.3.5.7.11
-Comma list: 2401/2400, 3025/3024, 4375/4374
+Comma list: 3025/3024, 4375/4374, 589824/588245
-Mapping: {{mapping| 18 0 -1 22 48 | 0 2 3 2 1 }}
+Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}
-: mapping generators: ~80/77, ~400/231
+: mapping generators: ~99/70, ~8/7
-Optimal tuning (POTE): ~80/77 = 1\18, ~400/231 = 950.9553
+Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3370
-Tuning ranges:
+{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646, 1068d }}
-* 11-odd-limit diamond monotone: ~99/98 = [13.333, 22.222] (1\90 to 1\54)
-* 11-odd-limit diamond tradeoff: ~99/98 = [17.304, 17.985]
-* 11-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 17.985]
-{{Optimal ET sequence|legend=1| 72, 198, 270, 342, 612, 954, 1566 }}
+Badness: 0.040955
-Badness: 0.006283
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024
+Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024
-Mapping: {{mapping| 18 0 -1 22 48 -19 | 0 2 3 2 1 6 }}
+Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}
-Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837
+Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373
-Tuning ranges:
+{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646f, 1068df }}
-* 13-odd-limit diamond monotone: ~99/98 = [16.667, 22.222] (1\72 to 1\54)
-* 15-odd-limit diamond monotone: ~99/98 = [16.667, 19.048] (1\72 to 2\126)
-* 13-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.309]
-* 15-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.926]
-* 13-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.309]
-* 15-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.926]
-{{Optimal ET sequence|legend=1| 72, 198, 270 }}
+Badness: 0.020416
-Badness: 0.012505
+=== Semigamera ===
+Subgroup: 2.3.5.7.11
-===== 17-limit =====
+Comma list: 4375/4374, 14641/14580, 15488/15435
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 676/675, 715/714, 1001/1000, 1716/1715, 3025/3024
+Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }}
-Mapping: {{mapping| 18 0 -1 22 48 -19 -12 | 0 2 3 2 1 6 6 }}
+: mapping generators: ~2, ~77/72
-Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837
+Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1642
-{{Optimal ET sequence|legend=1| 72, 198g, 270 }}
+{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }}
-===== 19-limit =====
+Badness: 0.078
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 676/675, 715/714, 1001/1000, 1331/1330, 1716/1715, 3025/3024
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Mapping: {{mapping| 18 0 -1 22 48 -19 -12 48 105 | 0 2 3 2 1 6 6 -2 }}
+Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580
-Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837
+Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }}
-{{Optimal ET sequence|legend=1| 72, 198g, 270 }}
+Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1628
-==== Semihemiennealimmal ====
+{{Optimal ET sequence|legend=1| 73f, 125f, 198, 323, 521 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374
+Badness: 0.044
-Mapping: {{mapping| 18 0 -1 22 48 88 | 0 4 6 4 2 -3 }}
+== Crazy ==
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
-: mapping generators: ~80/77, ~1053/800
+Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament. [[1106edo]] is an strong tuning.
-Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
+[[Subgroup]]: 2.3.5.7
-{{Optimal ET sequence|legend=1| 126, 144, 270, 684, 954 }}
+[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}
-Badness: 0.013104
+{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}
-===== 17-limit =====
+: mapping generators: ~332150625/234881024, ~1125/1024
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 2401/2400, 2431/2430, 3025/3024, 4225/4224, 4375/4374
+[[Optimal tuning]]s:
+* [[CTE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7475
+* [[error map]]: {{val| 0.0000 +0.0253 -0.0514 -0.0133 }}
+* [[CWE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7474
+* error map: {{val| 0.0000 +0.0244 -0.0508 -0.0218 }}
-Mapping: {{mapping| 18 0 -1 22 48 88 -119 | 0 4 6 4 2 -3 27 }}
+{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}
-: mapping generators: ~80/77, ~1053/800
+[[Badness]] (Smith): 0.0394
-Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-{{Optimal ET sequence|legend=1| 270, 684, 954 }}
+Comma list: 3025/3024, 4375/4374, 2791309312/2790703125
-Badness: 0.013104
+Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}
-===== 19-limit =====
+Optimal tunings:
-Subgroup: 2.3.5.7.11.13.17.19
+* CTE: ~99/70 = 162.7485, ~1125/1024 = 162.7485
+* CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481
-Comma list: 2401/2400, 2431/2430, 2926/2925, 3025/3024, 4225/4224, 4375/4374
+{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}
-Mapping: {{mapping| 18 0 -1 22 48 88 -119 -2 | 0 4 6 4 2 -3 27 11 }}
+Badness (Smith): 0.0170
-: mapping generators: ~80/77, ~1053/800
+== Orga ==
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
+[[Comma list]]: 4375/4374, 54975581388800/54936068900769
-{{Optimal ET sequence|legend=1| 270, 684h, 954h, 1224 }}
+{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }}
-Badness: 0.013104
+: mapping generators: ~7411887/5242880, ~1310720/1058841
-=== Semiennealimmal ===
+[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104
-Semiennealimmal tempers out [[4000/3993]], and uses a ~140/121 semifourth generator. Notably, however, two generator steps do not reach ~4/3, despite that the name may suggest so. In fact, it splits the generator of ennealimmal into three.
+{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}
+[[Badness]]: 0.040236
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 2401/2400, 4000/3993, 4375/4374
+Comma list: 3025/3024, 4375/4374, 5767168/5764801
-Mapping: {{mapping| 9 3 4 14 18 | 0 6 9 6 7 }}
+Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }}
-: mapping generators: ~27/25, ~140/121
+Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
-Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3367
+{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836, 1106 }}
-{{Optimal ET sequence|legend=1| 72, 369, 441 }}
+Badness: 0.016188
-Badness: 0.034196
+=== 13-limit ===
-==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374
+Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360
-Mapping: {{mapping| 9 3 4 14 18 -8 | 0 6 9 6 7 22 }}
+Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }}
-Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3375
+Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
-{{Optimal ET sequence|legend=1| 72, 297ef, 369f, 441 }}
+{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836f, 1106f }}
-Badness: 0.026122
+Badness: 0.021762
-=== Quadraennealimmal ===
+== Seniority ==
-Subgroup: 2.3.5.7.11
+{{See also| Very high accuracy temperaments #Senior }}
-Comma list: 2401/2400, 4375/4374, 234375/234256
+Aside from the ragisma, the seniority temperament (26 &amp; 145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ({{monzo| -17 62 -35 }}, quadla-sepquingu) is tempered out.
-Mapping: {{mapping| 9 1 1 12 -7 | 0 8 12 8 23 }}
+[[Subgroup]]: 2.3.5.7
-: mapping generators: ~27/25, ~25/22
+[[Comma list]]: 4375/4374, 201768035/201326592
-Optimal tuning (POTE): ~27/25 = 1\9, ~25/22 = 221.0717
+{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}
-{{Optimal ET sequence|legend=1| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3087/2560 = 322.804
-Badness: 0.021320
+{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}
-=== Trinealimmal ===
+[[Badness]]: 0.044877
-Subgroup: 2.3.5.7.11
-Comma list: 2401/2400, 4375/4374, 2097152/2096325
+=== Senator ===
+The senator temperament (26 &amp; 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It can be extended to the 13- and 17-limit immediately, by adding 364/363 and 595/594 to the comma list in this order.
-Mapping: {{mapping| 27 1 0 34 177 | 0 2 3 2 -4 }}
+Subgroup: 2.3.5.7.11
-: mapping generators: ~2744/2673, ~2352/1375
+Comma list: 441/440, 4375/4374, 65536/65219
-Optimal tuning (POTE): ~2744/2673 = 1\27, ~2352/1375 = 928.8000
+Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }}
-{{Optimal ET sequence|legend=1| 27, 243, 270, 783, 1053, 1323 }}
+Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
-Badness: 0.029812
+{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316e, 487ee }}
-=== Rhodium ===
+Badness: 0.092238
-{{Main| Rhodium }}
-Rhodium splits the ennealimmal period in five parts and thereby features a period of 9 × 5 = 45, thus the name is given after the 45th element.
-Subgroup: 2.3.5.7.11
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Comma list: 2401/2400, 4375/4374, 117440512/117406179
+Comma list: 364/363, 441/440, 2200/2197, 4375/4374
-Mapping: {{mapping| 45 1 -1 56 226 | 0 2 3 2 -2 }}
+Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }}
-: mapping generators: ~3072/3025, ~55/32
+Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
-Optimal tuning (CTE): ~3072/3025 = 1\45, ~55/32 = 937.6658
+{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}
-{{Optimal ET sequence|legend=1| 45, 225c, 270, 1125, 1395, 1665, 5265d }}
+Badness: 0.044662
-Badness: 0.0381
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-==== 13-limit ====
+Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197
-Subgroup: 2.3.5.7.11.13
-Comma list: 2401/2400, 4225/4224, 4375/4374, 6656/6655
+Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }}
-Mapping: {{mapping| 45 1 -1 56 226 272 | 0 2 3 2 -2 -3 }}
+Optimal tuning (POTE): ~77/64 = 322.793
-Optimal tuning (CTE): ~66/65 = 1\45, ~55/32 = 937.657
+{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}
-{{Optimal ET sequence|legend=1| 45, 270, 855, 1125, 1395, 1665, 3060d, 4725df }}
+Badness: 0.026562
-Badness: 0.0226
+== Monzismic ==
+: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Monzismic]].
-== Supermajor ==
-The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2<sup>15</sup>)/3, 46 give (2<sup>19</sup>)/5, and 75 give (2<sup>30</sup>)/7, leading to a wedgie of {{multival| 37 46 75 -13 15 45 }}. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80-note mos is presumably the place to start, and if that is not enough notes for you, there is always the 171-note mos.
+The monzismic temperament (53 &amp; 612) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]].
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 52734375/52706752
+[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}
-{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }}
+{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}
-{{Multival|legend=1| 37 46 75 -13 15 45 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082
+{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}
-{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}
+[[Badness]]: 0.046569
-[[Badness]]: 0.010836
+=== Monzism ===
-=== Semisupermajor ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 35156250/35153041
+Comma list: 4375/4374, 41503/41472, 184549376/184528125
-Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}
+Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082
+Optimal tuning (POTE): ~231/200 = 249.0193
-{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}
+{{Optimal ET sequence|legend=1| 53, 559, 612 }}
-Badness: 0.012773
+Badness: 0.057083
-== Enneadecal ==
+==== 13-limit ====
-Enneadecal temperament tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out [[703125/702464]], the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.
+Subgroup: 2.3.5.7.11.13
-[[Subgroup]]: 2.3.5.7
+Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
-[[Comma list]]: 4375/4374, 703125/702464
+Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}
-{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}
+Optimal tuning (POTE): ~231/200 = 249.0199
-{{Multival|legend=1| 19 19 57 -14 37 79 }}
+{{Optimal ET sequence|legend=1| 53, 559, 612 }}
-: mapping generators: ~28/27, ~3
+Badness: 0.053780
-[[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)
+== Semidimfourth ==
+: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].''
-{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}
+The semidimfourth temperament is featured by a semi-diminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, 235298/234375.
-[[Badness]]: 0.010954
+[[Subgroup]]: 2.3.5.7
-=== 11-limit ===
+[[Comma list]]: 4375/4374, 235298/234375
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 4375/4374, 16384/16335
+[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}
-Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456
-Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)
+{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}
-{{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }}
+[[Badness]]: 0.055249
-Badness: 0.043734
+=== Neusec ===
+Subgroup: 2.3.5.7.11
-==== 13-limit ====
+Comma list: 3025/3024, 4375/4374, 235298/234375
-Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 625/624, 729/728, 2205/2197
+Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }}
-Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}
+Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547
-Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)
+{{Optimal ET sequence|legend=1| 8d, 190, 388 }}
-{{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }}
+Badness: 0.059127
-Badness: 0.033545
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-=== Hemienneadecal ===
+Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
-Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 234375/234256
+Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }}
-Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}
+Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545
-: mapping generators: ~55/54, ~3
+{{Optimal ET sequence|legend=1| 8d, 190, 198, 388 }}
-Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)
+Badness: 0.030941
-{{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}
+== Acrokleismic ==
+[[Subgroup]]: 2.3.5.7
-Badness: 0.009985
+[[Comma list]]: 4375/4374, 2202927104/2197265625
-==== Hemienneadecalis ====
+{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256
+: mapping generators: ~2, ~6/5
-Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557
-Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)
+{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}
-{{Optimal ET sequence|legend=1| 152f, 342f, 494 }}
+[[Badness]]: 0.056184
-Badness: 0.020782
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-==== Hemienneadec ====
+Comma list: 4375/4374, 41503/41472, 172032/171875
-Subgroup: 2.3.5.7.11.13
-Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
+Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }}
-Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}
+Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558
-Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444)
+{{Optimal ET sequence|legend=1| 19, 251, 270, 829, 1099, 1369, 1639 }}
-{{Optimal ET sequence|legend=1| 152, 342, 494, 1330, 1824, 2318d }}
+Badness: 0.036878
-Badness: 0.030391
+==== 13-limit ====
-==== Semihemienneadecal ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078
+Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
-Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}
+Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }}
-: mapping generators: ~55/54 = 1\38, ~55/54, ~429/250
+Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557
-Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)
+{{Optimal ET sequence|legend=1| 19, 251, 270 }}
-{{Optimal ET sequence|legend=1| 190, 304d, 494, 684, 1178, 2850, 4028ce }}
+Badness: 0.026818
-Badness: 0.014694
+=== Counteracro ===
+Subgroup: 2.3.5.7.11
-=== Kalium ===
+Comma list: 4375/4374, 5632/5625, 117649/117612
-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.
-Subgroup: 2.3.5.7.11.13.17.19
+Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }}
-Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344
+Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.553
-Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}
+{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}
-Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244
+Badness: 0.042572
-{{Optimal ET sequence|legend=1| 855, 988, 1843 }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-== Semidimi ==
+Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].''
-The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374.
+Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }}
-[[Subgroup]]: 2.3.5.7
+Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.554
-[[Comma list]]: 4375/4374, 3955078125/3954653486
+{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}
-{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}
+Badness: 0.026028
-{{Multival|legend=1| 55 73 93 -12 -7 11 }}
+== Quasithird ==
+The quasithird temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}.
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270
+[[Subgroup]]: 2.3.5
-{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
+[[Comma list]]: {{monzo| 55 -64 20 }}
-[[Badness]]: 0.015075
+{{Mapping|legend=1| 4 0 -11 | 0 5 16 }}
-== Brahmagupta ==
+: mapping generators: ~51200000/43046721, ~1594323/1280000
-The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.
+[[Optimal tuning]] ([[POTE]]): ~51200000/43046721, ~1594323/1280000 = 380.395
-[[Subgroup]]: 2.3.5.7
+{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }}
-[[Comma list]]: 4375/4374, 70368744177664/70338939985125
+[[Badness]]: 0.099519
-{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}
+=== 7-limit ===
+[[Subgroup]]: 2.3.5.7
-: mapping generators: ~1157625/1048576, ~27/20
+[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}
-{{Multival|legend=1| 21 56 -77 40 -181 -336 }}
+{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
-[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716
+[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 1\4, ~5103/4096 = 380.388
-{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }}
+{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}
-[[Badness]]: 0.029122
+[[Badness]]: 0.061813
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 4000/3993, 4375/4374, 131072/130977
+Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
-Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}
+Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}
-Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704
+Optimal tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387)
-{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }}
+{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448 }}
-Badness: 0.052190
+Badness: 0.021125
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374
+Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374
+Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}
-Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}
+Optimal tuning (POTE): ~81/65 = 380.385 (or ~22/21 = 80.385)
-Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.706
+{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}
-{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771eef }}
+Badness: 0.029501
-Badness: 0.023132
+== Deca ==
+: ''For 5-limit version of this temperament, see [[10th-octave temperaments #Neon]].''
-== Abigail ==
+Deca temperament has a period of 1/10 octave and tempers out the [[linus comma]], {{monzo| 11 -10 -10 10 }}, neon comma {{monzo| 21 60 -50 }} and {{monzo| 12 -3 -14 9 }} = 165288374272/164794921875 (satritrizo-asepbigu).
-Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930]: "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things."</ref>
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 2147483648/2144153025
+[[Comma list]]: 4375/4374, 165288374272/164794921875
-{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }}
+{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}
-: mapping generators: ~46305/32768, ~27/20
+: mapping generators: ~15/14, ~6/5
-{{Multival|legend=1| 22 48 -38 25 -122 -223 }}
+[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577
-[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899
+{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}
-{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}
+[[Badness]]: 0.080637
-[[Badness]]: 0.037000
+Badness (Sintel): 2.041
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 131072/130977
+Comma list: 3025/3024, 4375/4374, 391314/390625
+Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}
-Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }}
+Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.582
-Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901
+{{Optimal ET sequence|legend=1| 80, 190, 270, 1000, 1270, 1540e, 1810e }}
-{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }}
+Badness: 0.024329
-Badness: 0.012860
+Badness (Sintel): 0.804
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
+Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
-Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}
+Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903
+Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.602 (~40/39 = 44.398)
-{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }}
+{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}
-Badness: 0.008856
+Badness: 0.016810
-== Gamera ==
+Badness (Sintel): 0.695
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 589824/588245
+=== no-17's 19-limit ===
+Subgroup: 2.3.5.7.11.13.19
-{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }}
+Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374, 1521/1520
-: mapping generators: ~2, ~8/7
+Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}
-{{Multival|legend=1| 23 40 1 10 -63 -110 }}
+Optimal tuning (CTE): ~15/14 = 1\10, ~6/5 = 315.581 (~39/38 = 44.419)
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336
+{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}
-{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}
+Badness (Sintel): 0.556
-[[Badness]]: 0.037648
+== Keenanose ==
+Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.
-=== Hemigamera ===
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 589824/588245
+[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}
-Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}
+{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
-: mapping generators: ~99/70, ~8/7
+: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3370
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 21 3 1 -10 }} = 4.4465
-{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646, 1068d }}
+{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}
-Badness: 0.040955
+[[Badness]]: 0.0858
-==== 13-limit ====
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11
-Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024
+Comma list: 4375/4374, 117649/117612, 67110351/67108864
-Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}
+Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373
+Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4465
-{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646f, 1068df }}
+{{Optimal ET sequence|legend=1| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}
-Badness: 0.020416
+Badness: 0.0308
-=== Semigamera ===
+=== 13-limit ===
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13
-Comma list: 4375/4374, 14641/14580, 15488/15435
+Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612
-Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }}
+Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}
-: mapping generators: ~2, ~77/72
+Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4466
-Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1642
+{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 4048 }}
-{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }}
+Badness: 0.0213
-Badness: 0.078
+== Aluminium ==
+Aluminium is named after the 13th element, and tempers out the {{monzo| 92 -39 -13 }} comma which sets [[135/128]] interval to be equal to 1/13th of the octave.
-==== 13-limit ====
+[[Subgroup]]: 2.3.5
-Subgroup: 2.3.5.7.11.13
-Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580
+[[Comma list]]: {{monzo| 92 -39 -13 }}
-Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }}
+[[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}
+: mapping generators: ~135/128, ~3
-Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1628
+[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897
-{{Optimal ET sequence|legend=1| 73f, 125f, 198, 323, 521 }}
+{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}
-Badness: 0.044
+[[Badness]]: 0.123
-== Orga ==
+=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 54975581388800/54936068900769
+[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}
-{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }}
+[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}
-: mapping generators: ~7411887/5242880, ~1310720/1058841
+[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 702.0024
-{{Multival|legend=1| 58 102 -2 27 -166 -291 }}
+{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}
-[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104
+[[Badness]]: 0.126
-{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}
-[[Badness]]: 0.040236
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 5767168/5764801
+Comma list: 4375/4374, 234375/234256, 2097152/2096325
-Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }}
+Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
+Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042
-{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836, 1106 }}
+{{Optimal ET sequence|legend=1| 494, 1053, 1547, 3588e, 5135e }}
-Badness: 0.016188
+Badness: 0.0421
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360
+Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078
-Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }}
+Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
+Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099
-{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836f, 1106f }}
+{{Optimal ET sequence|legend=1| 494, 1547, 2041, 4576def }}
-Badness: 0.021762
+Badness: 0.0286
-== Chlorine ==
+== Countritonic ==
-The name of chlorine temperament comes from Chlorine, the 17th element.
+: ''For the 5-limit version of this temperament, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
-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. Not only the semitwelfth, but also the ~5/4 can be used as a generator.
+Countritonic (''co-un-tritonic'') can be described as the 53 & 422 temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit.
-[[Subgroup]]: 2.3.5
+[[Subgroup]]: 2.3.5.7
-[[Comma list]]: {{monzo| -52 -17 34 }}
+[[Comma list]]: 4375/4374, 68719476736/68356598625
-{{Mapping|legend=1| 17 0 26 | 0 2 1 }}
+{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }}
-: mapping generators: ~25/24, ~{{monzo| 26 9 -17 }}
+: mapping generators: ~2, ~45927/32768
-[[Optimal tuning]] ([[POTE]]): ~{{monzo| 26 9 -17 }} = 950.9746
+[[Optimal tuning]] (CTE): ~2 = 1\1, ~45927/32768 = 588.6216
-{{Optimal ET sequence|legend=1| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}
+{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}
-[[Badness]]: 0.077072
+[[Badness]]: 0.133
-=== 7-limit ===
+=== 11-limit ===
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11
-[[Comma list]]: 4375/4374, {{monzo| -49 4 22 -3 }}
+Comma list: 4375/4374, 5632/5625, 2621440/2614689
-{{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }}
+Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 }}
-{{Multival|legend=1| 34 17 170 -52 174 347 }}
+Optimal tuning (CTE): ~2 = 1\1, ~539/384 = 588.6258
-[[Optimal tuning]] ([[POTE]]): ~{{monzo| 24 -5 -9 2 }} = 950.9995
+{{Optimal ET sequence|legend=1| 53, 316e, 369, 422, 791e, 1213cde }}
-{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547 }}
+Badness: 0.0707
-[[Badness]]: 0.041658
+=== 13-limit ===
-=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 41503/41472, 1879453125/1879048192
+Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625
-Mapping: {{mapping|  17 0 26 -87 207 | 0 2 1 10 -11  }}
+Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 -74 }}
-Optimal tuning (POTE): ~{{monzo| 24 -5 -9 2 }} = 950.9749
+Optimal tuning (CTE): ~2 = 1\1, ~128/91 = 588.6277
-{{Optimal ET sequence|legend=1| 289, 323, 612 }}
+{{Optimal ET sequence|legend=1| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}
-Badness: 0.063706
+Badness: 0.0366
-== Seniority ==
+== Quatracot ==
-{{See also| Very high accuracy temperaments #Senior }}
+{{See also| Stratosphere }}
-Aside from the ragisma, the seniority temperament (26 &amp; 145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ({{monzo| -17 62 -35 }}, quadla-sepquingu) is tempered out.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 201768035/201326592
+[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
-{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}
+{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }}
-{{Multival|legend=1| 35 62 -3 17 -103 -181 }}
+: mapping generators: ~2278125/1605632, ~448/405
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3087/2560 = 322.804
+[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 1\2, ~448/405 = 176.805
-{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}
+{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}
-[[Badness]]: 0.044877
+[[Badness]]: 0.175982
-=== Senator ===
-The senator temperament (26 &amp; 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It can be extended to the 13- and 17-limit immediately, by adding 364/363 and 595/594 to the comma list in this order.
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 441/440, 4375/4374, 65536/65219
+Comma list: 3025/3024, 4375/4374, 1265625/1261568
-Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }}
+Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }}
-Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
+Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 176.806
-{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316e, 487ee }}
+{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c }}
-Badness: 0.092238
+Badness: 0.041043
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 441/440, 2200/2197, 4375/4374
+Comma list: 625/624, 729/728, 1575/1573, 2200/2197
-Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }}
+Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }}
-Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
+Optimal tuning (POTE): ~99/70 = 1\2, ~195/176 = 176.804
-{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}
+{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1690bcc, 2328bccde }}
-Badness: 0.044662
+Badness: 0.022643
-==== 17-limit ====
+== Moulin ==
-Subgroup: 2.3.5.7.11.13.17
+Moulin has a generator of 22/13, and it is named after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". It can be described as the 494 & 1619 temperament.
-Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }}
+[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
-Optimal tuning (POTE): ~77/64 = 322.793
+{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }}
-{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}
+: mapping generators: ~2, ~6422528/3796875
-Badness: 0.026562
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6422528/3796875 = 910.9323
-== Monzismic ==
+{{Optimal ET sequence|legend=1| 494, 1125, 1619 }}
-: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Monzismic]].
-The monzismic temperament (53 &amp; 612) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]].
+[[Badness]]: 0.234
-[[Subgroup]]: 2.3.5.7
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}
+Comma list: 4375/4374, 759375/758912, 100663296/100656875
-{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}
+Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }}
-{{Multival|legend=1| 2 37 -134 54 -218 -415 }}
+Optimal tuning (CTE): ~2 = 1\1, ~1024/605 = 910.9323
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207
+{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}
-{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}
+Badness: 0.0678
-[[Badness]]: 0.046569
+=== 13-limit ===
+Since 11/8 is within 23 generators, the 25 tone MOS (4L 21s) of this temperament contains the 8:11:13 triad.
-=== Monzism ===
+Subgroup: 2.3.5.7.11.13
-Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 41503/41472, 184549376/184528125
+Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
-Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}
+Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }}
-Optimal tuning (POTE): ~231/200 = 249.0193
+Optimal tuning (CTE): ~2 = 1\1, ~22/13 = 910.9323
-{{Optimal ET sequence|legend=1| 53, 559, 612 }}
+{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}
-Badness: 0.057083
+Badness: 0.0271
-==== 13-limit ====
+== Palladium ==
-Subgroup: 2.3.5.7.11.13
+: ''For the 5-limit version of this temperament, see [[46th-octave temperaments]]''.
-Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
+The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minortones (10/9) fall short of seven octaves. This temperament can be described as 46 &amp; 414 temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.
-Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~231/200 = 249.0199
+[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}
-{{Optimal ET sequence|legend=1| 53, 559, 612 }}
+{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
-Badness: 0.053780
+: mapping generators: ~83349/81920, ~3
-== Semidimfourth ==
+[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 1\46, ~3/2 = 701.6074
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].''
-The semidimfourth temperament is featured by a semi-diminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, 235298/234375.
+{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }}
-[[Subgroup]]: 2.3.5.7
+[[Badness]]: 0.308505
-[[Comma list]]: 4375/4374, 235298/234375
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}
+Comma list: 3025/3024, 4375/4374, 134775333/134217728
-{{Multival|legend=1| 31 41 53 -7 -3 8 }}
+Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456
+Optimal tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951
-{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}
+{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de }}
-[[Badness]]: 0.055249
+Badness: 0.073783
-=== Neusec ===
+=== 13-limit ===
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13
-Comma list: 3025/3024, 4375/4374, 235298/234375
+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 tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6419
-Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }}
+{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334de }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547
+Badness: 0.040751
-{{Optimal ET sequence|legend=1| 8d, 190, 388 }}
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
-Badness: 0.059127
+Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
-==== 13-limit ====
+Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
+Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425
-Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }}
+{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334deg }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545
+Badness: 0.022441
-{{Optimal ET sequence|legend=1| 8d, 190, 198, 388 }}
+== Oviminor ==
+{{See also| Syntonic–kleismic equivalence continuum }}
-Badness: 0.030941
+Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.
-== Acrokleismic ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 2202927104/2197265625
+[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}
-{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }}
+{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }}
 : mapping generators: ~2, ~6/5
-{{Multival|legend=1| 32 33 92 -22 56 121 }}
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 315.7501
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557
+{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}
-{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}
+[[Badness]]: 0.582
-[[Badness]]: 0.056184
+== Octoid ==
+''For the 5-limit temperament, see [[8th-octave temperaments#Octoid (5-limit)]].''
-=== 11-limit ===
+The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.
-Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 41503/41472, 172032/171875
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }}
+[[Comma list]]: 4375/4374, 16875/16807
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558
+{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
-{{Optimal ET sequence|legend=1| 19, 251, 270, 829, 1099, 1369, 1639 }}
+: mapping generators: ~49/45, ~7/5
-Badness: 0.036878
+[[Optimal tuning]] ([[POTE]]): ~49/45 = 1\8, ~7/5 = 583.940
-==== 13-limit ====
+[[Tuning ranges]]:
-Subgroup: 2.3.5.7.11.13
+* 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]
-Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
+{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}
-Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }}
+[[Badness]]: 0.042670
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557
+Scales: [[octoid72]], [[octoid80]]
-{{Optimal ET sequence|legend=1| 19, 251, 270 }}
+=== 11-limit ===
+The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimaxing the damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, if one wants to use 80edo as the tuning, one must use octopus — not octoid — as 80edo doesn't temper 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.
-Badness: 0.026818
-=== Counteracro ===
 Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 5632/5625, 117649/117612
+Comma list: 540/539, 1375/1372, 4000/3993
-Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }}
+Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.553
+Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.962
-{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}
+Tuning ranges:
+* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
+* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
-Badness: 0.042572
+{{Optimal ET sequence|legend=1| 72, 152, 224 }}
-==== 13-limit ====
+Badness: 0.014097
-Subgroup: 2.3.5.7.11.13
-Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
+Scales: [[octoid72]], [[octoid80]]
-Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.554
+Comma list: 540/539, 625/624, 729/728, 1375/1372
-{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}
+Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}
-Badness: 0.026028
+Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.905
-== Quasithird ==
+{{Optimal ET sequence|legend=1| 72, 152f, 224 }}
-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}}.
-[[Subgroup]]: 2.3.5
+Badness: 0.015274
-[[Comma list]]: {{monzo| 55 -64 20 }}
+Scales: [[octoid72]], [[octoid80]]
-{{Mapping|legend=1| 4 0 -11 | 0 5 16 }}
+; Music
+* ''Dreyfus'' (archived 2010) by [[Gene Ward Smith]] – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – octoid[72] in 224edo tuning
-: mapping generators: ~51200000/43046721, ~1594323/1280000
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-[[Optimal tuning]] ([[POTE]]): ~51200000/43046721, ~1594323/1280000 = 380.395
+Comma list: 375/374, 540/539, 625/624, 715/714, 729/728
-{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }}
+Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}
-[[Badness]]: 0.099519
+Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.842
-=== 7-limit ===
+{{Optimal ET sequence|legend=1| 72, 152fg, 224, 296, 520g }}
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}
+Badness: 0.014304
-{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
+Scales: [[octoid72]], [[octoid80]]
-{{Multival|legend=1| 20 64 -116 55 -240 -449 }}
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 1\4, ~5103/4096 = 380.388
+Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
-{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}
+Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}
-[[Badness]]: 0.061813
+Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.932
-=== 11-limit ===
+{{Optimal ET sequence|legend=1| 72, 152fg, 224 }}
-Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
+Badness: 0.016036
-Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}
+Scales: [[octoid72]], [[octoid80]]
-Optimal tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387)
+==== Octopus ====
+A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4 = 300{{cent}}.
-{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448 }}
+Subgroup: 2.3.5.7.11.13
-Badness: 0.021125
+Comma list: 169/168, 325/324, 364/363, 540/539
-=== 13-limit ===
+Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374
+Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.892
-Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}
+{{Optimal ET sequence|legend=1| 72, 152, 224f }}
-Optimal tuning (POTE): ~81/65 = 380.385 (or ~22/21 = 80.385)
+Badness: 0.021679
-{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}
+Scales: [[octoid72]], [[octoid80]]
-Badness: 0.029501
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-== Deca ==
+Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
-: ''For 5-limit version of this temperament, see [[10th-octave temperaments #Neon]].''
-Deca temperament has a period of 1/10 octave and tempers out the [[linus comma]], {{monzo| 11 -10 -10 10 }}, neon comma {{monzo| 21 60 -50 }} and {{monzo| 12 -3 -14 9 }} = 165288374272/164794921875 (satritrizo-asepbigu).
+Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}
-[[Subgroup]]: 2.3.5.7
+Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.811
-[[Comma list]]: 4375/4374, 165288374272/164794921875
+{{Optimal ET sequence|legend=1| 72, 152, 224fg, 296ffg }}
-{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}
+Badness: 0.015614
-: mapping generators: ~15/14, ~6/5
+Scales: [[Octoid72]], [[Octoid80]]
-{{Multival|legend=1| 50 60 110 -21 34 87 }}
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577
+Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
-{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}
+Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}
-[[Badness]]: 0.080637
+Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 584.064
-=== 11-limit ===
+{{Optimal ET sequence|legend=1| 72, 152, 224fg, 376ffgh }}
-Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 391314/390625
+Badness: 0.016321
-Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}
+Scales: [[Octoid72]], [[Octoid80]]
-Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.582
+==== Hexadecoid ====
+{{ See also | 16th-octave temperaments }}
-{{Optimal ET sequence|legend=1| 80, 190, 270, 1000, 1270, 1540e, 1810e }}
+Hexadecoid (80 &amp; 144) has a period of 1/16 octave and tempers out 4225/4224.
-Badness: 0.024329
-=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
+Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
-Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}
+Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
-Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.602
+: mapping generators: ~448/429, ~7/5
-{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}
+Optimal tuning (POTE): ~448/429 = 1\16, ~13/8 = 841.015
-Badness: 0.016810
+{{Optimal ET sequence|legend=1| 80, 144, 224 }}
-== Keenanose ==
+Badness: 0.030818
-Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.
-[[Subgroup]]: 2.3.5.7
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}
+Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
-{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
+Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
-: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}
+Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 21 3 1 -10 }} = 4.4465
+{{Optimal ET sequence|legend=1| 80, 144, 224, 528dg }}
-{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}
+Badness: 0.028611
-[[Badness]]: 0.0858
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-=== 11-limit ===
+Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
-Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 117649/117612, 67110351/67108864
+Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }}
-Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}
+Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896
-Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4465
+{{Optimal ET sequence|legend=1| 80, 144, 224, 304dh, 528dghh }}
-{{Optimal ET sequence|legend=1| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}
+Badness: 0.023731
-Badness: 0.0308
+== Parakleismic ==
+{{Main| Parakleismic }}
-=== 13-limit ===
+In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, and 11-limit adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.
-Subgroup: 2.3.5.7.11.13
-Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612
+[[Subgroup]]: 2.3.5
-Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}
+[[Comma list]]: 1224440064/1220703125
-Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4466
+{{Mapping|legend=1| 1 5 6 | 0 -13 -14 }}
-{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 4048 }}
+: mapping generators: ~2, ~6/5
-Badness: 0.0213
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.240
-== Aluminium ==
+{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}
-Aluminium is named after the 13th element, and tempers out the {{monzo| 92 -39 -13 }} comma which sets [[135/128]] interval to be equal to 1/13th of the octave.
-[[Subgroup]]: 2.3.5
+[[Badness]]: 0.043279
-[[Comma list]]: {{monzo| 92 -39 -13 }}
+=== 7-limit ===
-[[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}
-: mapping generators: ~135/128, ~3
-[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897
-{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}
-[[Badness]]: 0.123
-=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}
-[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}
-[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 702.0024
-{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}
-[[Badness]]: 0.126
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 234375/234256, 2097152/2096325
-Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}
-Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042
-{{Optimal ET sequence|legend=1| 494, 1053, 1547, 3588e, 5135e }}
-Badness: 0.0421
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078
-Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}
-Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099
-{{Optimal ET sequence|legend=1| 494, 1547, 2041, 4576def }}
-Badness: 0.0286
-== Countritonic ==
-: ''For the 5-limit version of this temperament, see [[Schismic-Mercator equivalence continuum #Countritonic]] and [[High badness temperaments #Countritonic]]
-Countritonic (''co-un-tritonic'') can be described as the 53 & 422 temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 68719476736/68356598625
-{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }}
-: mapping generators: ~2, ~45927/32768
-[[Optimal tuning]] (CTE): ~2 = 1\1, ~45927/32768 = 588.6216
-{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}
-[[Badness]]: 0.133
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 5632/5625, 2621440/2614689
-Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 }}
-Optimal tuning (CTE): ~2 = 1\1, ~539/384 = 588.6258
-{{Optimal ET sequence|legend=1| 53, 316e, 369, 422, 791e, 1213cde }}
-Badness: 0.0707
-=== 13-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625
-Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 -74 }}
-Optimal tuning (CTE): ~2 = 1\1, ~128/91 = 588.6277
-{{Optimal ET sequence|legend=1| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}
-Badness: 0.0366
-== Quatracot ==
-{{See also| Stratosphere }}
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
-{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }}
-: mapping generators: ~2278125/1605632, ~448/405
-{{Multival|legend=1| 26 16 118 -35 114 229 }}
-[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 1\2, ~448/405 = 176.805
-{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}
-[[Badness]]: 0.175982
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 1265625/1261568
-Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 176.806
-{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c }}
-Badness: 0.041043
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 729/728, 1575/1573, 2200/2197
-Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~195/176 = 176.804
-{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1690bcc, 2328bccde }}
-Badness: 0.022643
-== Moulin ==
-Moulin has a generator of 22/13, and it is named after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". It can be described as the 494 & 1619 temperament.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
-{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }}
-: mapping generators: ~2, ~6422528/3796875
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6422528/3796875 = 910.9323
-{{Optimal ET sequence|legend=1| 494, 1125, 1619 }}
-[[Badness]]: 0.234
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 4375/4374, 759375/758912, 100663296/100656875
-Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }}
-Optimal tuning (CTE): ~2 = 1\1, ~1024/605 = 910.9323
-{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}
-Badness: 0.0678
-=== 13-limit ===
-Since 11/8 is within 23 generators, the 25 tone MOS (4L 21s) of this temperament contains the 8:11:13 triad.
-Subgroup: 2.3.5.7.11.13
-Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
-Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }}
-Optimal tuning (CTE): ~2 = 1\1, ~22/13 = 910.9323
-{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}
-Badness: 0.0271
-== Palladium ==
-: ''For the 5-limit version of this temperament, see [[46th-octave temperaments]]''.
-The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minortones (10/9) fall short of seven octaves. This temperament can be described as 46 &amp; 414 temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}
-{{Mapping|legend=1| 46 73 107 129 | 0 -1 -2 1 }}
-: mapping generators: ~83349/81920, ~3
-{{Multival|legend=1| 46 92 -46 39 -202 -365 }}
-[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 1\46, ~3/2 = 701.6074
-{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }}
-[[Badness]]: 0.308505
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4375/4374, 134775333/134217728
-Mapping: {{mapping|  46 73 107 129 159 | 0 -1 -2 1 1  }}
-Optimal tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951
-{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de }}
-Badness: 0.073783
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
-Mapping: {{mapping| 46 73 107 129 159 170 | 0 -1 -2 1 1 2 }}
-Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6419
-{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334de }}
-Badness: 0.040751
-=== 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 73 107 129 159 170 188 | 0 -1 -2 1 1 2 0 }}
-Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425
-{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334deg }}
-Badness: 0.022441
-== Oviminor ==
-{{See also| Syntonic-kleismic equivalence continuum }}
-Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}
-{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }}
-: mapping generators: ~2, ~6/5
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 315.7501
-{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}
-[[Badness]]: 0.582
-== Octoid ==
-The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, 16875/16807
-{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
-{{Multival|legend=1| 24 32 40 -5 -4 3 }}
-: mapping generators: ~49/45, ~7/5
-[[Optimal tuning]] ([[POTE]]): ~49/45 = 1\8, ~7/5 = 583.940
-[[Tuning ranges]]:
-* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
-* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
-* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
-* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
-* 7-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 584.359]
-* 9-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 585.084]
-{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}
-[[Badness]]: 0.042670
-Scales: [[octoid72]], [[octoid80]]
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 1375/1372, 4000/3993
-Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.962
-Tuning ranges:
-* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
-* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
-* 11-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 585.084]
-{{Optimal ET sequence|legend=1| 72, 152, 224 }}
-Badness: 0.014097
-Scales: [[octoid72]], [[octoid80]]
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 625/624, 729/728, 1375/1372
-Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.905
-{{Optimal ET sequence|legend=1| 72, 152f, 224 }}
-Badness: 0.015274
-Scales: [[octoid72]], [[octoid80]]
-; Music
-* [https://www.archive.org/details/Dreyfus ''Dreyfus''] [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] by [[Gene Ward Smith]]
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 375/374, 540/539, 625/624, 715/714, 729/728
-Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.842
-{{Optimal ET sequence|legend=1| 72, 152fg, 224, 296, 520g }}
-Badness: 0.014304
-Scales: [[octoid72]], [[octoid80]]
-===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
-Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.932
-{{Optimal ET sequence|legend=1| 72, 152fg, 224 }}
-Badness: 0.016036
-Scales: [[octoid72]], [[octoid80]]
-==== Octopus ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 325/324, 364/363, 540/539
-Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.892
-{{Optimal ET sequence|legend=1| 72, 152, 224f }}
-Badness: 0.021679
-Scales: [[octoid72]], [[octoid80]]
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
-Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.811
-{{Optimal ET sequence|legend=1| 72, 152, 224fg, 296ffg }}
-Badness: 0.015614
-Scales: [[Octoid72]], [[Octoid80]]
-===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
-Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 584.064
-{{Optimal ET sequence|legend=1| 72, 152, 224fg, 376ffgh }}
-Badness: 0.016321
-Scales: [[Octoid72]], [[Octoid80]]
-==== Hexadecoid ====
-Hexadecoid (80 &amp; 144) has a period of 1/16 octave and tempers out 4225/4224.
-Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
-Mapping: {{mapping| 16 26 38 46 56 59 | 0 -3 -4 -5 -3 1 }}
-: mapping generators: ~448/429, ~7/5
-Optimal tuning (POTE): ~448/429 = 1\16, ~13/8 = 841.015
-{{Optimal ET sequence|legend=1| 80, 144, 224 }}
-Badness: 0.030818
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
-Mapping: {{mapping| 16 26 38 46 56 59 65 | 0 -3 -4 -5 -3 1 2 }}
-Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932
-{{Optimal ET sequence|legend=1| 80, 144, 224, 528dg }}
-Badness: 0.028611
-===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
-Mapping: {{mapping| 16 26 38 46 56 59 65 68 | 0 -3 -4 -5 -3 1 2 0 }}
-Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896
-{{Optimal ET sequence|legend=1| 80, 144, 224, 304dh, 528dghh }}
-Badness: 0.023731
-== Parakleismic ==
-{{Main| Parakleismic }}
-In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being {{multival| 13 14 35 -8 19 42 }} and adding 3136/3125 and 4375/4374, and the 11-limit wedgie {{multival| 13 14 35 -36 -8 19 -102 42 -132 -222 }} adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.
-[[Subgroup]]: 2.3.5
-[[Comma list]]: 1224440064/1220703125
-{{Mapping|legend=1| 1 5 6 | 0 -13 -14 }}
-: mapping generators: ~2, ~6/5
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.240
-{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}
-[[Badness]]: 0.043279
-=== 7-limit ===
-Subgroup: 2.3.5.7
 [[Comma list]]: 3136/3125, 4375/4374
@@ Line 1,787: / Line 1,286: @@
 {{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }}
-{{Multival|legend=1| 13 14 35 -8 19 42 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.181
@@ Line 1,958: / Line 1,456: @@
 : mapping generators: ~2, ~5/3
-{{Multival|legend=1| 25 24 79 -20 55 116 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.060
@@ Line 2,025: / Line 1,521: @@
 {{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
-{{Multival|legend=1| 30 49 14 8 -62 -105 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1715 = 16.613
@@ Line 2,094: / Line 1,588: @@
 {{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
-{{Multival|legend=1| 19 31 9 5 -39 -66 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 26.287
@@ Line 2,164: / Line 1,656: @@
 [[Comma list]]: 4375/4374, 83349/81920
-{{Mapping|legend=1| 13 21 31 36 | 0 -1 -2 1 }}
+{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
 [[Optimal tuning]] ([[POTE]]): ~256/245 = 1\13, ~3/2 = 699.1410
@@ Line 2,177: / Line 1,669: @@
 Comma list: 245/242, 385/384, 4375/4374
-Mapping: {{mapping| 13 21 31 36 45 | 0 -1 -2 1 0 }}
+Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}
 Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.6179
@@ Line 2,190: / Line 1,682: @@
 Comma list: 169/168, 245/242, 325/324, 385/384
-Mapping: {{mapping| 13 21 31 36 45 48 | 0 -1 -2 1 0 0 }}
+Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}
 Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.2969
@@ Line 2,211: / Line 1,703: @@
 {{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}
-Badness: 0.313
+Badness: 0.312806
 == Notes ==
 [[Category:Temperament collections]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Ragismic microtemperaments| ]] <!-- main article -->
 [[Category:Ragismic| ]] <!-- key article -->

Ragismic microtemperaments: Difference between revisions