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The ragisma is [[4375/4374]] with a [[monzo]] of |-1 -7 4 1>, the smallest 7-limit [[superparticular]] ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the ragisma, [[4375/4374]] ({{monzo| -1 -7 4 1 }}). The ragisma is the smallest [[7-limit]] [[superparticular ratio]].  


Temperaments not discussed here include [[Jubilismic clan #Crepuscular|crepuscular]], [[Meantone family #Flattone|flattone]], [[Porcupine family #Hystrix|hystrix]], [[Starling temperaments #Sensi|sensi]], [[Gamelismic clan #Unidec|unidec]], [[Orwellismic temperaments #Quartonic|quartonic]], [[Kleismic family #Catakleismic|catakleismic]], [[Tetracot family #Modus|modus]], [[Schismatic family #Pontiac|pontiac]], [[Würschmidt family #Whirrschmidt|whirrschmidt]],  [[Gravity family #Zarvo|zarvo]], [[Vishnuzmic family #Vishnu|vishnu]], and [[Vulture family #Vulture|vulture]].  
Since {{nowrap|(10/9)<sup>4</sup> {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)<sup>2</sup> }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.


= Ennealimmal =
Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
[[Ennealimmal]] temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the [[ennealimma|ennealimmal comma]], |1 -27 18&gt;, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two 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. Its wedgie is &lt;&lt;18 27 18 1 -22 -34||.
* ''[[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]]


Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 EDOs, though its hardly likely anyone could tell the difference.
== 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. 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.


If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example.) In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS.
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 4375/4374
[[Comma list]]: 4375/4374, 52734375/52706752
 
{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082
 
{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}
 
[[Badness]]: 0.010836
 
=== Semisupermajor ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 35156250/35153041
 
Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082
 
{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}
 
Badness: 0.012773
 
== Enneadecal ==
Enneadecal temperament tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out [[703125/702464]], the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.
 
''For the 5-limit temperament, see [[19th-octave temperaments#(5-limit) enneadecal]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 703125/702464
 
{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}
 
: mapping generators: ~28/27, ~3
 
[[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)
 
{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}
 
[[Badness]]: 0.010954


[[Tuning ranges]]:
=== 11-limit ===
* valid range: [26.667, 66.667] (1\45 to 1\18)
Subgroup: 2.3.5.7.11
* nice range: [48.920, 49.179]
* strict range: [48.920, 49.179]


[[Mapping]]: [&lt;9 1 1 12|, &lt;0 2 3 2|]
Comma list: 540/539, 4375/4374, 16384/16335


[[Wedgie]]: &lt;&lt;18 27 18 1 -22 -34||
Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}


Mapping generators: ~27/25, ~5/3
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)


[[POTE tuning|POTE generators]]: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980
{{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }}


[[EDO|Vals]]: {{Val list| 27, 45, 72, 99, 171, 441, 612 }}
Badness: 0.043734


[[Badness]]: 0.003610
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


== Hemiennealimmal ==
Comma list: 540/539, 625/624, 729/728, 2205/2197
Comma list: 2401/2400, 4375/4374, 3025/3024


Tuning ranges:  
Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}
* valid range: [13.333, 22.222] (1\90 to 1\54)
* nice range: [17.304, 17.985]
* strict range:  [17.304, 17.985]


Mapping: [&lt;18 0 -1 22 48|, &lt;0 2 3 2 1|]
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)


POTE generator: ~99/98 = 17.6219
{{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }}


Vals: {{Val list| 72, 198, 270, 342, 612, 954, 1566 }}
Badness: 0.033545


Badness: 0.006283
=== Hemienneadecal ===
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 3025/3024, 4375/4374, 234375/234256
Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024


Tuning ranges:  
Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}
* valid range: [16.667, 22.222] (1\72 to 1\54)
* nice range: [17.304, 18.309]
* strict range: [17.304, 18.309]


Mapping: [&lt;18 0 -1 22 48 -19|, &lt;0 2 3 2 1 6|]
: mapping generators: ~55/54, ~3


POTE generator ~99/98 = 17.7504
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)


Vals: {{Val list| 72, 198, 270 }}
{{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}


Badness: 0.012505
Badness: 0.009985


=== Semihemiennealimmal ===
==== Hemienneadecalis ====
Comma list: 2401/2400, 4375/4374, 3025/3024, 4225/4224
Subgroup: 2.3.5.7.11.13


Mapping: [&lt;18 0 -1 22 48 88|, &lt;0 4 6 4 2 -3|]
Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256


POTE generator: ~39/32 = 342.139
Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}


Vals: {{Val list| 126, 144, 270, 684, 954 }}
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)


Badness: 0.013104
{{Optimal ET sequence|legend=1| 152f, 342f, 494 }}


== Semiennealimmal ==
Badness: 0.020782
Comma list: 2401/2400, 4375/4374, 4000/3993


Mapping: [&lt;9 3 4 14 18|, &lt;0 6 9 6 7|]
==== Hemienneadec ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~140/121 = 250.3367
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213


Vals: {{Val list| 72, 369, 441 }}
Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}


Badness: 0.034196
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444)


=== 13-limit ===
{{Optimal ET sequence|legend=1| 152, 342, 494, 1330, 1824, 2318d }}
Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374


Mapping: [&lt;9 3 4 14 18 -8|, &lt;0 6 9 6 7 22|]
Badness: 0.030391


POTE generator: ~140/121 = 250.3375
==== Semihemienneadecal ====
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 72, 297ef, 369f, 441 }}
Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078


Badness: 0.026122
Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}


== Quadraennealimmal ==
: mapping generators: ~55/54 = 1\38, ~55/54, ~429/250
Comma list: 2401/2400, 4375/4374, 234375/234256


Mapping: [&lt;9 1 1 12 -7|, &lt;0 8 12 8 23|]
Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)


POTE generator: ~77/75 = 45.595
{{Optimal ET sequence|legend=1| 190, 304d, 494, 684, 1178, 2850, 4028ce }}


Vals: {{Val list| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
Badness: 0.014694


Badness: 0.021320
=== 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.


== Ennealimnic ==
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 243/242, 441/440, 4375/4356


Tuning ranges:  
Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344
* valid range: [44.444, 53.333] (1\27 to 2\45)
* nice range: [48.920, 52.592]
* strict range: [48.920, 52.592]


Mapping: [&lt;9 1 1 12 -2|, &lt;0 2 3 2 5|]
Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}


POTE generator: ~36/35 = 49.395
Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244


Vals: {{Val list| 72, 171, 243 }}
{{Optimal ET sequence|legend=1| 855, 988, 1843 }}


Badness: 0.020347
== Semidimi ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].''


=== 13-limit ===
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, 364/363, 441/440, 625/624


Tuning ranges:
[[Subgroup]]: 2.3.5.7
* valid range: [48.485, 50.000] (4\99 to 3\72)
* nice range: [48.825, 52.592]
* strict range: [48.825, 50.000]


Mapping: [&lt;9 1 1 12 -2 -33|, &lt;0 2 3 2 5 10|]
[[Comma list]]: 4375/4374, 3955078125/3954653486


POTE generator: ~36/35 = 49.341
{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}


Vals: {{Val list| 72, 171, 243 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270


Badness: 0.023250
{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}


=== 17-limit ===
[[Badness]]: 0.015075
Comma list: 243/242, 364/363, 375/374, 441/440, 595/594


Tuning ranges:
== Brahmagupta ==
* valid range: [48.485, 50.000] (4\99 to 3\72)
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.  
* nice range: [46.363, 52.592]
* strict range: [48.485, 50.000]


Mapping: [&lt;9 1 1 12 -2 -33 -3|, &lt;0 2 3 2 5 10 6|]
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 ¢).


POTE generator: ~36/35 = 49.335
[[Subgroup]]: 2.3.5.7


Vals: {{Val list| 72, 171, 243 }}
[[Comma list]]: 4375/4374, 70368744177664/70338939985125


Badness: 0.014602
{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}


=== Ennealim ===
: mapping generators: ~1157625/1048576, ~27/20
Comma list: 169/168, 243/242, 325/324, 441/440


Mapping: [&lt;9 1 1 12 -2 20|, &lt;0 2 3 2 5 2|]
[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716


POTE generator: ~36/35 = 49.708
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }}


Vals: {{Val list| 27e, 45ef, 72, 315ff, 387cff, 459cdfff }}
[[Badness]]: 0.029122


Badness: 0.020697
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Ennealiminal ==
Comma list: 4000/3993, 4375/4374, 131072/130977
Comma list: 385/384, 1375/1372, 4375/4374


Mapping: [&lt;9 1 1 12 51|, &lt;0 2 3 2 -3|]
Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}


POTE generator: ~36/35 = 49.504
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704


Vals: {{Val list| 27, 45, 72, 171e, 243e, 315e }}
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }}


Badness: 0.031123
Badness: 0.052190


=== 13-limit ===
=== 13-limit ===
Comma list: 169/168, 325/324, 385/384, 1375/1372
Subgroup: 2.3.5.7.11.13
 
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374
 
Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}
 
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.706
 
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771eef }}
 
Badness: 0.023132
 
== Abigail ==
Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930]: "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things."</ref>
 
''For the 5-limit temperament, see [[Very high accuracy temperaments#Abigail]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 2147483648/2144153025
 
{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }}
 
: mapping generators: ~46305/32768, ~27/20
 
[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899
 
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}
 
[[Badness]]: 0.037000
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 131072/130977
 
Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }}


Mapping: [&lt;9 1 1 12 51 20|, &lt;0 2 3 2 -3 2|]
Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901


POTE generator: ~36/35 = 49.486
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }}
 
Badness: 0.012860
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 27, 45f, 72, 171ef, 243ef }}
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095


Badness: 0.030325
Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}


== Trinealimmal ==
Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903
Comma list: 2401/2400, 4375/4374, 2097152/2096325


Mapping: [&lt;27 1 0 34 177|, &lt;0 2 3 2 -4|]
{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }}


POTE generator: ~6/5 = 315.644
Badness: 0.008856


Vals: {{Val list| 27, 243, 270, 783, 1053, 1323 }}
== Gamera ==
''For the 5-limit temperament, see [[High badness temperaments#Gamera]].


Badness: 0.029812
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [&lt;1 6 10 3|, &lt;0 -23 -40 -1|]
{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }}


[[Wedgie]]: &lt;&lt;23 40 1 10 -63 -110||
: mapping generators: ~2, ~8/7


[[POTE tuning|POTE generator]] ~8/7 = 230.336
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336


[[EDO|Vals]]: {{Val list| 26, 73, 99, 224, 323, 422, 745d }}
{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}


[[Badness]]: 0.037648
[[Badness]]: 0.037648


== Hemigamera ==
=== Hemigamera ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 589824/588245
Comma list: 3025/3024, 4375/4374, 589824/588245


Mapping: [&lt;2 12 20 6 5|, &lt;0 -23 -40 -1 5|]
Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}


POTE generator: ~8/7 = 230.3370
: mapping generators: ~99/70, ~8/7


Vals: {{Val list| 26, 198, 224, 422, 646, 1068d }}
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3370
 
{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646, 1068d }}


Badness: 0.040955
Badness: 0.040955


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024


Mapping: [&lt;2 12 20 6 5 17|, &lt;0 -23 -40 -1 5 -25|]
Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}


POTE generator: ~8/7 = 230.3373
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373


Vals: {{Val list| 26, 198, 224, 422, 646f, 1068df }}
{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646f, 1068df }}


Badness: 0.020416
Badness: 0.020416


= Supermajor =
=== Semigamera ===
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.0002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of &lt;&lt;37 46 75 -13 15 45||. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 14641/14580, 15488/15435
 
Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }}
 
: mapping generators: ~2, ~77/72
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1642
 
{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }}
 
Badness: 0.078
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580
 
Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1628
 
{{Optimal ET sequence|legend=1| 73f, 125f, 198, 323, 521 }}
 
Badness: 0.044
 
== Crazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
 
Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament. [[1106edo]] is an strong tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}


[[Comma list]]: 4375/4374, 52734375/52706752
{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}


[[Mapping]]: [&lt;1 15 19 30|, &lt;0 -37 -46 -75|]
: mapping generators: ~332150625/234881024, ~1125/1024


[[Wedgie]]: &lt;&lt;37 46 75 -13 15 45||
[[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 }}


[[POTE tuning|POTE generator]]: ~9/7 = 435.082
{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}


[[EDO|Vals]]: {{Val list| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}
[[Badness]] (Smith): 0.0394


[[Badness]]: 0.010836
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Semisupermajor ==
Comma list: 3025/3024, 4375/4374, 2791309312/2790703125
Comma list: 3025/3024, 4375/4374, 35156250/35153041


Mapping: [&lt;2 30 38 60 41|, &lt;0 -37 -46 -75 -47|]
Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}


POTE generator: ~9/7 = 435.082
Optimal tunings:
* CTE: ~99/70 = 162.7485, ~1125/1024 = 162.7485
* CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481


EDOs: {{Val list| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}
{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}


Badness: 0.012773
Badness (Smith): 0.0170


= Enneadecal =
== Orga ==
Enneadecal temperament tempers out the enneadeca, |-14 -19 19&gt;, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5 or 7 limits, and [[494edo]] shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 703125/702464
[[Comma list]]: 4375/4374, 54975581388800/54936068900769


[[Mapping]]: [&lt;19 0 14 -37|, &lt;0 1 1 3|]
{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }}


[[Wedgie]]: &lt;&lt;19 19 57 -14 37 79||
: mapping generators: ~7411887/5242880, ~1310720/1058841


Mapping generators: ~28/27, ~3
[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104


[[POTE tuning|POTE generator]]: ~3/2 = 701.880
{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}


[[EDO|Vals]]: {{Val list| 19, 152, 171, 665, 836, 1007, 2185 }}
[[Badness]]: 0.040236


[[Badness]]: 0.010954
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Hemienneadecal ==
Comma list: 3025/3024, 4375/4374, 5767168/5764801
Comma list: 3025/3024, 4375/4374, 234375/234256


Mapping: [&lt;38 0 28 -74 11|, &lt;0 1 1 3 2|]
Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }}


POTE generator: ~3/2 = 701.881
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103


Vals: {{Val list| 152, 342, 494, 836, 1178, 2014 }}
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836, 1106 }}


Badness: 0.009985
Badness: 0.016188


=== 13-limit ===
=== 13-limit ===
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360
 
Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
 
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836f, 1106f }}
 
Badness: 0.021762
 
== Seniority ==
{{See also| Very high accuracy temperaments #Senior }}
 
Aside from the ragisma, the seniority temperament (26 &amp; 145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ({{monzo| -17 62 -35 }}, quadla-sepquingu) is tempered out.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 201768035/201326592
 
{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3087/2560 = 322.804
 
{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}
 
[[Badness]]: 0.044877
 
=== Senator ===
The senator temperament (26 &amp; 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It can be extended to the 13- and 17-limit immediately, by adding 364/363 and 595/594 to the comma list in this order.
 
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 4375/4374, 65536/65219
 
Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
 
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316e, 487ee }}
 
Badness: 0.092238
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 2200/2197, 4375/4374
 
Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
 
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}
 
Badness: 0.044662
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197
 
Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }}
 
Optimal tuning (POTE): ~77/64 = 322.793
 
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}


Mapping: [&lt;38 0 28 -74 11 502|, &lt;0 1 1 3 2 -6|]
Badness: 0.026562


POTE generator: ~3/2 = 701.986
== Monzismic ==
: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Monzismic]].  


Vals: {{Val list| 152, 342, 494, 836 }}
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: 0.030391
[[Subgroup]]: 2.3.5.7


= Deca =
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}
[[Comma list]]: 4375/4374, 165288374272/164794921875


[[Mapping]]: [&lt;10 4 9 2|, &lt;0 5 6 11|]
{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}


[[Wedgie]]: &lt;&lt;50 60 110 -21 34 87||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207


[[POTE tuning|POTE generator]]: ~6/5 = 315.577
{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}


[[EDO|Vals]]: {{Val list| 80, 190, 270, 1270, 1540, 1810, 2080 }}
[[Badness]]: 0.046569


[[Badness]]: 0.080637
=== Monzism ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 4375/4374, 41503/41472, 184549376/184528125
Comma list: 3025/3024, 4375/4374, 422576/421875


Mapping: [&lt;10 4 9 2 18|, &lt;0 5 6 11 7|]
Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}


POTE generator: ~6/5 = 315.582
Optimal tuning (POTE): ~231/200 = 249.0193


Vals: {{Val list| 80, 190, 270, 1000, 1270 }}
{{Optimal ET sequence|legend=1| 53, 559, 612 }}


Badness: 0.024329
Badness: 0.057083


== 13-limit ==
==== 13-limit ====
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
Subgroup: 2.3.5.7.11.13


Mapping: [&lt;10 4 9 2 18 37|, &lt;0 5 6 11 7 0|]
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625


POTE generator: ~6/5 = 315.602
Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}


Vals: {{Val list| 80, 190, 270, 730, 1000 }}
Optimal tuning (POTE): ~231/200 = 249.0199


Badness: 0.016810
{{Optimal ET sequence|legend=1| 53, 559, 612 }}


= Mitonic =
Badness: 0.053780
{{see also|Minortonic family #Mitonic}}


[[Comma list]]: 4375/4374, 2100875/2097152
== Semidimfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].''


[[Mapping]]: [&lt;1 -1 -3 6|, &lt;0 17 35 -21|]
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.


[[Wedgie]]: &lt;&lt;17 35 -21 16 -81 -147||
[[Subgroup]]: 2.3.5.7


[[POTE tuning|POTE generator]]: ~10/9 = 182.458
[[Comma list]]: 4375/4374, 235298/234375


[[EDO|Vals]]: {{Val list| 46, 125, 171 }}
[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}


[[Badness]]: 0.025184
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456


= Abigail =
{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}
[[Comma list]]: 4375/4374, 2147483648/2144153025


[[Mapping]]: [&lt;2 7 13 -1|, &lt;0 -11 -24 19|]
[[Badness]]: 0.055249


[[Wedgie]]: &lt;&lt;22 48 -38 25 -122 -223||
=== Neusec ===
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE generator]]: ~6912/6125 = 208.899
Comma list: 3025/3024, 4375/4374, 235298/234375


[[EDO|Vals]]: {{Val list| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}
Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }}


[[Badness]]: 0.037000
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547


== 11-limit ==
{{Optimal ET sequence|legend=1| 8d, 190, 388 }}
Comma list: 3025/3024, 4375/4374, 20614528/20588575


Mapping: [&lt;2 7 13 -1 1|, &lt;0 -11 -24 19 17|]
Badness: 0.059127


POTE generator: ~1155/1024 = 208.901
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 46, 132, 178, 224, 270, 494, 764 }}
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374


Badness: 0.012860
Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }}


== 13-limit ==
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095


Map: [&lt;2 7 13 -1 1 -2|, &lt;0 -11 -24 19 17 27|]
{{Optimal ET sequence|legend=1| 8d, 190, 198, 388 }}


POTE generator: ~44/39 = 208.903
Badness: 0.030941


Vals: {{Val list| 46, 178, 224, 270, 494, 764, 1258 }}
== Acrokleismic ==
[[Subgroup]]: 2.3.5.7


Badness: 0.008856
[[Comma list]]: 4375/4374, 2202927104/2197265625


= Semidimi =
{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }}
The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit |-12 -73 55&gt; and 7-limit 3955078125/3954653486, as well as 4375/4374.


[[Comma]]: |-12 -73 55&gt;
: mapping generators: ~2, ~6/5


[[Mapping]]: [&lt;1 36 48|, &lt;0 -55 -73|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557


[[Wedgie]]: &lt;&lt;55 73 -12||
{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}


[[POTE tuning|POTE generator]]: ~162/125 = 449.1269
[[Badness]]: 0.056184


[[EDO|Vals]]: {{Val list| 8, 163, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Badness]]: 0.754866
Comma list: 4375/4374, 41503/41472, 172032/171875


== 7-limit ==
Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }}
[[Comma list]]: 4375/4374, 3955078125/3954653486


[[Mapping]]: [&lt;1 36 48 61|, &lt;0 -55 -73 -93|]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558


[[Wedgie]]: &lt;&lt;55 73 93 -12 -7 11||
{{Optimal ET sequence|legend=1| 19, 251, 270, 829, 1099, 1369, 1639 }}


[[POTE tuning|POTE generator]]: ~35/27 = 449.1270
Badness: 0.036878


[[EDO|Vals]]: {{Val list| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.015075
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976


= Brahmagupta =
Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }}
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], |47 -7 -7 -7&gt; = 140737488355328 / 140710042265625.


[[Comma list]]: 4375/4374, 70368744177664/70338939985125
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557


[[Mapping]]: [&lt;7 2 -8 53|, &lt;0 3 8 -11|]
{{Optimal ET sequence|legend=1| 19, 251, 270 }}


[[Wedgie]]: &lt;&lt;21 56 -77 40 -181 -336||
Badness: 0.026818


[[POTE tuning|POTE generator]]: ~27/20 = 519.716
=== Counteracro ===
Subgroup: 2.3.5.7.11


[[EDO|Vals]]: {{Val list| 7, 217, 224, 441, 1106, 1547 }}
Comma list: 4375/4374, 5632/5625, 117649/117612


[[Badness]]: 0.029122
Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.553


== 11-limit ==
{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}
Comma list: 4000/3993, 4375/4374, 131072/130977


Mapping: [&lt;7 2 -8 53 3|, &lt;0 3 8 -11 7|]
Badness: 0.042572


POTE generator: ~27/20 = 519.704
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 7, 217, 224, 441, 665, 1771ee }}
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374


Badness: 0.052190
Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }}


== 13-limit ==
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.554
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374


Mapping: [&lt;7 2 -8 53 3 35|, &lt;0 3 8 -11 7 -3|]
{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}


POTE generator: ~27/20 = 519.706
Badness: 0.026028


Vals: {{Val list| 7, 217, 224, 441, 665, 1771eef }}
== 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}}.


Badness: 0.023132
[[Subgroup]]: 2.3.5


= Quasithird =
[[Comma list]]: {{monzo| 55 -64 20 }}
[[Comma]]: |55 -64 20&gt;


[[Mapping]]: [&lt;4 0 -11|, &lt;0 5 16|]
{{Mapping|legend=1| 4 0 -11 | 0 5 16 }}


[[Wedgie]]: &lt;&lt;20 64 55||
: mapping generators: ~51200000/43046721, ~1594323/1280000


[[POTE tuning|POTE generator]]: ~1594323/1280000 = 380.395
[[Optimal tuning]] ([[POTE]]): ~51200000/43046721, ~1594323/1280000 = 380.395


[[EDO|Vals]]: {{Val list| 60, 164, 224, 388, 612, 836, 1000, 1448, 1612, 2224, 2836 }}
{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }}


[[Badness]]: 0.099519
[[Badness]]: 0.099519


== 7-limit ==
=== 7-limit ===
[[Comma list]]: 4375/4374, 1153470752371588581/1152921504606846976
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [&lt;4 0 -11 48|, &lt;0 5 16 -29|]
[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}


[[Wedgie]]: &lt;&lt;20 64 -116 55 -240 -449||
{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}


[[POTE tuning|POTE generator]]: ~5103/4096 = 380.388  
[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 1\4, ~5103/4096 = 380.388  


[[EDO|Vals]]: {{Val list| 60d, 164, 224, 388, 612, 1448, 2060 }}
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}


[[Badness]]: 0.061813
[[Badness]]: 0.061813


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296


Mapping: [&lt;4 0 -11 48 43|, &lt;0 5 16 -29 -23|]
Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}


POTE generator: ~22/21 = 80.387 (or ~5103/4096 = 380.387)
Optimal tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387)


Vals: {{Val list| 60d, 164, 224, 388, 612, 836, 1448 }}
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448 }}


Badness: 0.021125
Badness: 0.021125


== 13-limit ==
=== 13-limit ===
Comma list: 2200/2197, 3025/3024, 4375/4374, 468512/468195
Subgroup: 2.3.5.7.11.13
 
Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374


Mapping: [&lt;4 0 -11 48 43 11|, &lt;0 5 16 -29 -23 3|]
Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}


POTE generator: ~22/21 = 80.385 (or ~5103/4096 = 380.385)
Optimal tuning (POTE): ~81/65 = 380.385 (or ~22/21 = 80.385)


Vals: {{Val list| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}


Badness: 0.029501
Badness: 0.029501


= Semidimfourth =
== Deca ==
[[Comma]]: |7 41 -31&gt;
: ''For 5-limit version of this temperament, see [[10th-octave temperaments #Neon]].''


[[Mapping]]: [&lt;1 21 28|, &lt;0 -31 -41|]
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).


[[Wedgie]]: &lt;&lt;31 41 -7||
[[Subgroup]]: 2.3.5.7


[[POTE tuning|POTE generator]]: ~162/125 = 448.449
[[Comma list]]: 4375/4374, 165288374272/164794921875


[[EDO|Vals]]: {{Val list| 8, 91, 99, 190, 289, 388, 487, 677, 875, 966 }}
{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}


[[Badness]]: 0.233376
: mapping generators: ~15/14, ~6/5


== 7-limit ==
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577
[[Comma list]]: 4375/4374, 235298/234375


[[Mapping]]: [&lt;1 21 28 36|, &lt;0 -31 -41 -53|]
{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}


[[Wedgie]]: &lt;&lt;31 41 53 -7 -3 8||
[[Badness]]: 0.080637


[[POTE tuning|POTE generator]]: ~35/27 = 448.456
Badness (Sintel): 2.041


[[EDO|Vals]]: {{Val list| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Badness]]: 0.055249
Comma list: 3025/3024, 4375/4374, 391314/390625


== Neusec ==
Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}
Comma list: 3025/3024, 4375/4374, 235298/234375


Mapping: [&lt;2 11 15 19 15|, &lt;0 -31 -41 -53 -32|]
Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.582


POTE generator: ~12/11 = 151.547
{{Optimal ET sequence|legend=1| 80, 190, 270, 1000, 1270, 1540e, 1810e }}


Vals: {{Val list| 8d, 190, 388 }}
Badness: 0.024329


Badness: 0.059127
Badness (Sintel): 0.804


=== 13-limit ===
=== 13-limit ===
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
 
Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}
 
Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.602 (~40/39 = 44.398)
 
{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}
 
Badness: 0.016810
 
Badness (Sintel): 0.695
 
=== no-17's 19-limit ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374, 1521/1520
 
Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}
 
Optimal tuning (CTE): ~15/14 = 1\10, ~6/5 = 315.581 (~39/38 = 44.419)
 
{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}


Mapping: [&lt;2 11 15 19 15 17|, &lt;0 -31 -41 -53 -32 -38|]
Badness (Sintel): 0.556


POTE generator: ~12/11 = 151.545
== Keenanose ==
Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.


Vals: {{Val list| 8d, 190, 198, 388 }}
[[Subgroup]]: 2.3.5.7


Badness: 0.030941
[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}


= Acrokleismic =
{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
[[Comma list]]: 4375/4374, 2202927104/2197265625


[[Mapping]]: [&lt;1 10 11 27|, &lt;0 -32 -33 -92|]
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}


[[Wedgie]]: &lt;&lt;32 33 92 -22 56 121||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 21 3 1 -10 }} = 4.4465


[[POTE tuning|POTE generator]]: ~6/5 = 315.557
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}


[[EDO|Vals]]: {{Val list| 19, 251, 270 }}
[[Badness]]: 0.0858


[[Badness]]: 0.056184
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 4375/4374, 117649/117612, 67110351/67108864
Comma list: 4375/4374, 41503/41472, 172032/171875


Mapping: [&lt;1 10 11 27 -16|, &lt;0 -32 -33 -92 74|]
Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}


POTE generator: ~6/5 = 315.558
Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4465


Vals: {{Val list| 19, 251, 270, 829, 1099, 1369, 1639 }}
{{Optimal ET sequence|legend=1| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}


Badness: 0.036878
Badness: 0.0308


=== 13-limit ===
=== 13-limit ===
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
Subgroup: 2.3.5.7.11.13
 
Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612
 
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4466
 
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 4048 }}
 
Badness: 0.0213
 
== Aluminium ==
Aluminium is named after the 13th element, and tempers out the {{monzo| 92 -39 -13 }} comma which sets [[135/128]] interval to be equal to 1/13th of the octave.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 92 -39 -13 }}
 
[[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}
 
: mapping generators: ~135/128, ~3
 
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897
 
{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}
 
[[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 }}


Mapping: [&lt;1 10 11 27 -16 25|, &lt;0 -32 -33 -92 74 -81|]
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 702.0024


POTE generator: ~6/5 = 315.557
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}


Vals: {{Val list| 19, 251, 270 }}
[[Badness]]: 0.126


Badness: 0.026818
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Counteracro ==
Comma list: 4375/4374, 234375/234256, 2097152/2096325
Comma list: 4375/4374, 5632/5625, 117649/117612


Mapping: [&lt;1 10 11 27 55|, &lt;0 -32 -33 -92 -196|]
Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}


POTE generator: ~6/5 = 315.553
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042


Vals: {{Val list| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 3588e, 5135e }}


Badness: 0.042572
Badness: 0.0421


=== 13-limit ===
=== 13-limit ===
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
Subgroup: 2.3.5.7.11.13
 
Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078


Mapping: [&lt;1 10 11 27 55 25|, &lt;0 -32 -33 -92 -196 -81|]
Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}


POTE generator: ~6/5 = 315.554
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099


Vals: {{Val list| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}
{{Optimal ET sequence|legend=1| 494, 1547, 2041, 4576def }}


Badness: 0.026028
Badness: 0.0286


= Seniority =
== Countritonic ==
{{see also|Very high accuracy temperaments#Senior}}
: ''For the 5-limit version of this temperament, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


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


[[POTE tuning|POTE generator]]: ~3087/2560 = 322.804
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [&lt;1 11 19 2|, &lt;0 -35 -62 3|]
[[Comma list]]: 4375/4374, 68719476736/68356598625


[[Wedgie]]: &lt;&lt;35 62 -3 17 -103 -181||
{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }}


[[EDO|Vals]]: {{Val list| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}
: mapping generators: ~2, ~45927/32768


[[Badness]]: 0.044877
[[Optimal tuning]] (CTE): ~2 = 1\1, ~45927/32768 = 588.6216


= Orga =
{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}
[[Comma list]]: 4375/4374, 54975581388800/54936068900769


[[Mapping]]: [&lt;2 21 36 5|, &lt;0 -29 -51 1|]
[[Badness]]: 0.133


[[Wedgie]]: &lt;&lt;58 102 -2 27 -166 -291||
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE generator]]: ~8/7 = 231.104
Comma list: 4375/4374, 5632/5625, 2621440/2614689


[[EDO|Vals]]: {{Val list| 26, 244, 270, 836, 1106, 1376, 2482 }}
Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 }}


[[Badness]]: 0.040236
Optimal tuning (CTE): ~2 = 1\1, ~539/384 = 588.6258


== 11-limit ==
{{Optimal ET sequence|legend=1| 53, 316e, 369, 422, 791e, 1213cde }}
Comma list: 3025/3024, 4375/4374, 5767168/5764801


Mapping: [&lt;2 21 36 5 2|, &lt;0 -29 -51 1 8|]
Badness: 0.0707


POTE generator: ~8/7 = 231.103
=== 13-limit ===
Subgroup: 2.3.5.7.11


Vals: {{Val list| 26, 244, 270, 566, 836, 1106 }}
Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625


Badness: 0.016188
Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 -74 }}


== 13-limit ==
Optimal tuning (CTE): ~2 = 1\1, ~128/91 = 588.6277
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360


Mapping: [&lt;2 21 36 5 2 24|, &lt;0 -29 -51 1 8 -27|]
{{Optimal ET sequence|legend=1| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}


POTE generator: ~8/7 = 231.103
Badness: 0.0366


Vals: {{Val list| 26, 244, 270, 566, 836f, 1106f }}
== Quatracot ==
{{See also| Stratosphere }}


Badness: 0.021762
[[Subgroup]]: 2.3.5.7


= Quatracot =
[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
[[Comma list]]: 4375/4374, 1483154296875/1473173782528


[[Mapping]]: [&lt;2 7 7 23|, &lt;0 -13 -8 -59|]
{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }}


[[Wedgie]]: &lt;&lt;26 16 118 -35 114 229||
: mapping generators: ~2278125/1605632, ~448/405


[[POTE tuning|POTE generator]]: ~448/405 = 176.805
[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 1\2, ~448/405 = 176.805


[[EDO|Vals]]: {{Val list| 190, 224, 414, 638, 1052c, 1690bcc }}
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}


[[Badness]]: 0.175982
[[Badness]]: 0.175982


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 1265625/1261568
Comma list: 3025/3024, 4375/4374, 1265625/1261568


Mapping: [&lt;2 7 7 23 19|, &lt;0 -13 -8 -59 -41|]
Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }}


POTE generator: ~448/405 = 176.806
Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 176.806


Vals: {{Val list| 190, 224, 414, 638, 1052c }}
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c }}


Badness: 0.041043
Badness: 0.041043


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 729/728, 1575/1573, 2200/2197
Comma list: 625/624, 729/728, 1575/1573, 2200/2197


Mapping: [&lt;2 7 7 23 19 13|, &lt;0 -13 -8 -59 -41 -19|]
Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }}


POTE generator: ~195/176 = 176.804
Optimal tuning (POTE): ~99/70 = 1\2, ~195/176 = 176.804


Vals: {{Val list| 190, 224, 414, 638, 1690bcc, 2328bccde }}
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1690bcc, 2328bccde }}


Badness: 0.022643
Badness: 0.022643


= Octoid =
== 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 0 -39 202 | 0 1 2 -1 }}
 
: mapping generators: ~83349/81920, ~3
 
[[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 0 -39 202 232 | 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 0 -39 202 232 316 | 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 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}
 
Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425
 
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334deg }}
 
Badness: 0.022441
 
== Oviminor ==
{{See also| Syntonic–kleismic equivalence continuum }}
 
Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}
 
{{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 ==
''For the 5-limit temperament, see [[8th-octave temperaments#Octoid (5-limit)]].''
 
The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 16875/16807
[[Comma list]]: 4375/4374, 16875/16807
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
: mapping generators: ~49/45, ~7/5
[[Optimal tuning]] ([[POTE]]): ~49/45 = 1\8, ~7/5 = 583.940


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* valid range: [578.571, 600.000] (27\56 to 4\8)
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
* nice range: [582.512, 584.359]
* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
* strict range: [582.512, 584.359]
* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]


[[Mapping]]: [&lt;8 1 3 3|, &lt;0 3 4 5|]
{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}
 
[[Badness]]: 0.042670


[[Wedgie]]: &lt;&lt;24 32 40 -5 -4 3||
Scales: [[octoid72]], [[octoid80]]


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


[[POTE tuning|POTE generator]]: ~7/5 = 583.940
Subgroup: 2.3.5.7.11


[[EDO|Vals]]: {{Val list| 8d, 72, 152, 224 }}
Comma list: 540/539, 1375/1372, 4000/3993


[[Badness]]: 0.042670
Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}


== 11-limit ==
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.962
Comma list: 540/539, 1375/1372, 4000/3993


Tuning ranges:  
Tuning ranges:  
* valid range: [581.250, 586.364] (31\64, 43\88)
* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
* nice range: [582.512, 585.084]
* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
* strict range: [582.512, 585.084]


Mapping: [&lt;8 1 3 3 16|, &lt;0 3 4 5 3|]
{{Optimal ET sequence|legend=1| 72, 152, 224 }}


POTE generator: ~7/5 = 583.962
Badness: 0.014097


Vals: {{Val list| 72, 152, 224 }}
Scales: [[octoid72]], [[octoid80]]


Badness: 0.014097
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


== 13-limit ==
Comma list: 540/539, 625/624, 729/728, 1375/1372
Comma list: 540/539, 1375/1372, 4000/3993, 625/624


Mapping: [&lt;8 1 3 3 16 -21|, &lt;0 3 4 5 3 13|]
Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}


POTE generator: ~7/5 = 583.905
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.905


Vals: {{Val list| 72, 152f, 224 }}
{{Optimal ET sequence|legend=1| 72, 152f, 224 }}


Badness: 0.015274
Badness: 0.015274


=== Music ===
Scales: [[octoid72]], [[octoid80]]
* [http://www.archive.org/details/Dreyfus http://www.archive.org/details/Dreyfus]
 
* [http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play]
; Music
* ''Dreyfus'' (archived 2010) by [[Gene Ward Smith]] – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – octoid[72] in 224edo tuning
 
===== 17-limit =====
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 ====
A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4 = 300{{cent}}.
 
Subgroup: 2.3.5.7.11.13


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


Mapping: [&lt;8 1 3 3 16 14|, &lt;0 3 4 5 3 4|]
Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}


POTE generator: ~7/5 = 583.892
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.892


Vals: {{Val list| 72, 152, 224f }}
{{Optimal ET sequence|legend=1| 72, 152, 224f }}


Badness: 0.021679
Badness: 0.021679


= Amity =
Scales: [[octoid72]], [[octoid80]]
{{main|Amity}}
 
{{see also|Amity family #Amity}}
===== 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 ====
{{ See also | 16th-octave temperaments }}
 
Hexadecoid (80 &amp; 144) has a period of 1/16 octave and tempers out 4225/4224.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
 
Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
 
: mapping generators: ~448/429, ~7/5
 
Optimal tuning (POTE): ~448/429 = 1\16, ~13/8 = 841.015
 
{{Optimal ET sequence|legend=1| 80, 144, 224 }}
 
Badness: 0.030818
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
 
Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
 
Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932
 
{{Optimal ET sequence|legend=1| 80, 144, 224, 528dg }}
 
Badness: 0.028611
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
 
Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }}
 
Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896
 
{{Optimal ET sequence|legend=1| 80, 144, 224, 304dh, 528dghh }}
 
Badness: 0.023731
 
== Parakleismic ==
{{Main| Parakleismic }}
 
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension 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
 
[[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
 
{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }}
 
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.181
 
{{Optimal ET sequence|legend=1| 19, 80, 99, 217, 316, 415 }}
 
[[Badness]]: 0.027431


The generator for [[amity]] temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&amp;53 temperament, or by its wedgie, &lt;&lt;5 13 -17 9 -41 -76||. [[99edo]] is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.
=== 11-limit ===
Subgroup: 2.3.5.7.11


In the 5-limit amity is a genuine microtemperament, with 58/205 being a possible tuning. Another good choice is (64/5)^(1/13), which gives pure major thirds.
Comma list: 385/384, 3136/3125, 4375/4374


Comma: 1600000/1594323
Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }}


POTE generator: ~243/200 = 339.519
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.251


Map: [&lt;1 3 6|, &lt;0 -5 -13|]
{{Optimal ET sequence|legend=1| 19, 99, 118 }}


EDOs: 7, 39, 46, 53, 152, 205, 463, 668, 873
Badness: 0.049711


Badness: 0.0220
=== Paralytic ===
The ''paralytic'' temperament (118&amp;217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 &amp; 217 tempers out 1001/1000, 1575/1573, and 3584/3575.


== 7-limit ==
Subgroup: 2.3.5.7.11
Commas: 4375/4374, 5120/5103


POTE generator: ~128/105 = 339.432
Comma list: 441/440, 3136/3125, 4375/4374


Map: [&lt;1 3 6 -2|, &lt;0 -5 -13 17|]
Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }}


Wedgie: &lt;&lt;5 13 -17 9 -41 -76||
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.220


EDOs: 7, 39, 46, 53, 99, 251, 350
{{Optimal ET sequence|legend=1| 19e, 99e, 118, 217, 335, 552d, 887dd }}


Badness: 0.0236
Badness: 0.036027


== 11-limit ==
==== 13-limit ====
Commas: 540/539, 4375/4374, 5120/5103
Subgroup: 2.3.5.7.11.13


POTE generator: ~128/105 = 339.464
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374


Map: [&lt;1 3 6 -2 21|, &lt;0 -5 -13 17 -62|]
Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }}


EDOs: 53, 99e, 152, 555dee, 707ddee, 859bddee
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.214


Badness: 0.0315
{{Optimal ET sequence|legend=1| 99e, 118, 217, 552d, 769de }}


=== 13-limit ===
Badness: 0.044710
Commas: 352/351, 540/539, 625/624, 847/845
 
==== Paraklein ====
The ''paraklein'' temperament (19e &amp; 118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 625/624, 729/728
 
Mapping: {{mapping| 1 5 6 12 25 15 | 0 -13 -14 -35 -82 -43 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.225
 
{{Optimal ET sequence|legend=1| 19e, 99ef, 118, 217ff, 335ff }}
 
Badness: 0.037618
 
=== Parkleismic ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 1375/1372, 2200/2187
 
Mapping: {{mapping| 1 5 6 12 20 | 0 -13 -14 -35 -63 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.060
 
{{Optimal ET sequence|legend=1| 19e, 80, 179, 259cd }}
 
Badness: 0.055884
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 176/175, 325/324, 1375/1372
 
Mapping: {{mapping| 1 5 6 12 20 10 | 0 -13 -14 -35 -63 -24 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.075
 
{{Optimal ET sequence|legend=1| 19e, 80, 179 }}
 
Badness: 0.036559
 
=== Paradigmic ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 896/891, 3136/3125
 
Mapping: {{mapping| 1 5 6 12 -1 | 0 -13 -14 -35 17 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.096
 
{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}
 
Badness: 0.041720
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 325/324, 540/539, 832/825
 
Mapping: {{mapping| 1 5 6 12 -1 10 | 0 -13 -14 -35 17 -24 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.080
 
{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}
 
Badness: 0.035781
 
=== Semiparakleismic ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 3136/3125, 4375/4374
 
Mapping: {{mapping| 2 10 12 24 19 | 0 -13 -14 -35 -23 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.181


POTE generator: ~128/105 = 339.481
{{Optimal ET sequence|legend=1| 80, 118, 198, 316, 514c, 830c }}


Map: [&lt;1 3 6 -2 21 17|, &lt;0 -5 -13 17 -62 -47|]
Badness: 0.034208


EDOS: 53, 99ef, 152f, 205
==== Semiparamint ====
This extension was named ''semiparakleismic'' in the earlier materials.


Badness: 0.0280
Subgroup: 2.3.5.7.11.13


== Hitchcock ==
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
{{see also|Amity family #Hitchcock}}


Commas: 121/120, 176/175, 2200/2187
Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }}


POTE generator: ~11/9 = 339.340
Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.156


Map: [&lt;1 3 6 -2 6|, &lt;0 -5 -13 17 -9|]
{{Optimal ET sequence|legend=1| 80, 118, 198 }}


EDOs: 7, 39, 46, 53, 99
Badness: 0.033775


Badness: 0.0352
==== Semiparawolf ====
This extension was named ''gentsemiparakleismic'' in the earlier materials.  


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Commas: 121/120, 169/168, 176/175, 325/324


POTE generator: ~11/9 = 339.419
Comma list: 169/168, 325/324, 364/363, 3136/3125


Map: [&lt;1 3 6 -2 6 2|, &lt;0 -5 -13 17 -9 6|]
Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }}


EDOs: 7, 39, 46, 53, 99
Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 315.184


Badness: 0.0224
{{Optimal ET sequence|legend=1| 80, 118f, 198f }}


== Hemiamity ==
Badness: 0.040467
Commas: 3025/3024, 4375/4374, 5120/5103


POTE generator: ~64/55 = 339.493
== Counterkleismic ==
{{See also| High badness temperaments #Counterhanson}}


Map: [&lt;2 1 -1 13 13|, &lt;0 5 13 -17 -14|]
In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as 19 &amp; 224 temperament (''counterkleismic'', named by analogy to [[catakleismic]] and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).


EDOs: 14cde, 46, 106, 152, 350
[[Subgroup]]: 2.3.5.7


Badness: 0.0313
[[Comma list]]: 4375/4374, 158203125/157351936


= Parakleismic =
{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }}
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13&gt;, 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 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being &lt;&lt;13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie &lt;&lt;13 14 35 -36 ...|| adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.


Comma: 124440064/1220703125
: mapping generators: ~2, ~5/3


POTE generator: ~6/5 = 315.240
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.060


Map: [&lt;1 5 6|, &lt;0 -13 -14|]
{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }}


EDOs: 19, 61, 80, 99, 118, 453, 571, 689, 1496
[[Badness]]: 0.090553


Badness: 0.0433
=== 11-limit ===
Subgroup: 2.3.5.7.11


==7-limit==
Comma list: 540/539, 4375/4374, 2097152/2096325
Commas: 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.181
Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }}


Map: [&lt;1 5 6 12|, &lt;0 -13 -14 -35|]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.071


EDOs: 19, 80, 99, 217, 316, 415
{{Optimal ET sequence|legend=1| 19, 205, 224 }}


Badness: 0.0274
Badness: 0.070952


==11-limit==
==== 13-limit ====
Commas: 385/384, 3136/3125, 4375/4374
Subgroup: 2.3.5.7.11.13


POTE generator: ~6/5 = 315.251
Comma list: 540/539, 625/624, 729/728, 10985/10976


Map: [&lt;1 5 6 12 -6|, &lt;0 -13 -14 -35 36|]
Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }}


EDOs: 19, 99, 118
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.070


Badness: 0.0497
{{Optimal ET sequence|legend=1| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}


==Parkleismic==
Badness: 0.033874
Commas: 176/175, 1375/1372, 2200/2187


POTE generator: ~6/5 = 315.060
=== Counterlytic ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 5 6 12 20|, &lt;0 -13 -14 -35 -63|]
Comma list: 1375/1372, 4375/4374, 496125/495616


EDOs: 80, 179, 259cd
Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }}


Badness: 0.0559
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065


===13-limit===
{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}
Commas: 169/168, 176/175, 325/324, 1375/1372


POTE generator: ~6/5 = 315.075
Badness: 0.065400


Map: [&lt;1 5 6 12 20 10|, &lt;0 -13 -14 -35 -63 -24|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 15, 19, 80, 179
Comma list: 625/624, 729/728, 1375/1372, 10985/10976


Badness: 0.0366
Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }}


==Paradigmic==
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065
Commas: 540/539, 896/891, 3136/3125


POTE generator: ~6/5 = 315.096
{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}


Map: [&lt;1 5 6 12 -1|, &lt;0 -13 -14 -35 17|]
Badness: 0.029782


EDOs: 19, 80, 99e, 179e
== Quincy ==
[[Subgroup]]: 2.3.5.7


Badness: 0.0417
[[Comma list]]: 4375/4374, 823543/819200


===13-limit===
{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
Commas: 169/168, 325/324, 540/539, 832/825


POTE generator: ~6/5 = 315.080
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1715 = 16.613


Map: [&lt;1 5 6 12 -1 10|, &lt;0 -13 -14 -35 17 -24|]
{{Optimal ET sequence|legend=1| 72, 217, 289 }}


EDOs: 19, 80, 99e, 179e
[[Badness]]: 0.079657


Badness: 0.0358
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Semiparakleismic ==
Comma list: 441/440, 4000/3993, 4375/4374
Commas: 3025/3024, 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.181
Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}


Map: [&lt;2 10 12 24 19|, &lt;0 -13 -14 -35 -23|]
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.613


EDOs: 80, 118, 198, 316, 514c, 830c
{{Optimal ET sequence|legend=1| 72, 217, 289 }}


Badness: 0.0342
Badness: 0.030875


=== 13-limit ===
=== 13-limit ===
Commas: 352/351, 1001/1000, 3025/3024, 4375/4374
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 676/675, 4375/4374
 
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602
 
{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}


POTE generator: ~6/5 = 315.1563
Badness: 0.023862


Map: [<2 10 12 24 19 -1|, <0 -13 -14 -35 -23 16|]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


EDOs: {{EDOs|80, 118, 198}}
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155


Badness: 0.0338
Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}


=== Gentsemiparakleismic ===
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602
Commas: 169/168, 325/324, 364/363, 3136/3125


POTE generator: ~6/5 = 315.1839
{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}


Map: [<2 10 12 24 19 20|, <0 -13 -14 -35 -23 -24|]
Badness: 0.014741


EDOs: {{EDOs|80, 118f, 198f}}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0405
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675


=Quincy=
Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}
Commas: 4375/4374, 823543/819200


POTE generator: ~1728/1715 = 16.613
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.594


Map: [&lt;1 2 2 3|, &lt;0 -30 -49 -14|]
{{Optimal ET sequence|legend=1| 72, 145, 217 }}


EDOs: 72, 217, 289
Badness: 0.015197


Badness: 0.0797
== Sfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 441/440, 4000/3993, 41503/41472


POTE generator: ~100/99 = 16.613
[[Comma list]]: 4375/4374, 64827/64000


Map: [&lt;1 2 2 3 4|, &lt;0 -30 -49 -14 -39|]
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}


EDOs: 72, 217, 289
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 26.287


Badness: 0.0309
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


==13-limit==
[[Badness]]: 0.123291
Commas: 364/363, 441/440, 676/675, 4375/4374


POTE generator: ~100/99 = 16.602
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 2 3 4 5|, &lt;0 -30 -49 -14 -39 -94|]
Comma list: 121/120, 441/440, 4375/4374


EDOs: 72, 145, 217, 289
Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}


Badness: 0.0239
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.286


==17-limit==
{{Optimal ET sequence|legend=1| 45e, 46, 91e, 137de }}
Commas: 364/363, 441/440, 595/594, 1001/1000, 1156/1155


POTE generator: ~100/99 = 16.602
Badness: 0.054098


Map: [&lt;1 2 2 3 4 5 5|, &lt;0 -30 -49 -14 -39 -94 -66|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 72, 145, 217, 289
Comma list: 121/120, 169/168, 325/324, 441/440


Badness: 0.0147
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}


==19-limit==
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.310
Commas: 343/342, 364/363, 441/440, 595/594, 676/675, 2601/2600


POTE generator: ~100/99 = 16.594
{{Optimal ET sequence|legend=1| 45ef, 46, 91ef, 137def }}


Map: [&lt;1 2 2 3 4 5 5 4|, &lt;0 -30 -49 -14 -39 -94 -66 18|]
Badness: 0.033067


EDOs: 72, 145, 217
=== Sfour ===
Subgroup: 2.3.5.7.11


Badness: 0.0152
Comma list: 385/384, 2401/2376, 4375/4374


= Chlorine =
Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}
The name of chlorine temperament comes from Chlorine, the 17th element.


Chlorine microtemperament has a period of 1/17 octave. It tempers out the septendecima, |-52 -17 34&gt;, by which 17 chromatic semitones (25/24) fall short of an octave. Possible tunings for chlorine are [[289edo|289]], [[323edo|323]], and [[612edo|612]] EDOs, though its hardly likely anyone could tell the difference. In the 7-limit, 289&amp;323 temperament tempers out |-49 4 22 -3&gt; as well as the ragisma.
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.246


Comma: |-52 -17 34&gt;
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2687
Badness: 0.076567


Map: [&lt;17 26 39|, &lt;0 2 1|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 34, 289, 323, 612, 901
Comma list: 196/195, 364/363, 385/384, 4375/4374


Badness: 0.0771
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}


==7-limit==
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.239
Commas: 4375/4374, 193119049072265625/193091834023510016


POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2936
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


Map: [&lt;17 26 39 43|, &lt;0 2 1 10|]
Badness: 0.051893


EDOs: 34d, 289, 323, 612, 935, 1547
== Trideci ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tridecatonic]].''


Badness: 0.0417
The trideci temperament (26 &amp; 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from "tridecim" (Latin for "[[wikipedia:13|thirteen]]").


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 4375/4374, 41503/41472, 1879453125/1879048192


POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2690
[[Comma list]]: 4375/4374, 83349/81920


Map: [&lt;17 26 39 43 64|, &lt;0 2 1 10 -11|]
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}


EDOs: 34de, 289, 323, 612, 901
[[Optimal tuning]] ([[POTE]]): ~256/245 = 1\13, ~3/2 = 699.1410


Badness: 0.0637
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }}


= Monzism =
[[Badness]]: 0.184585
The ''monzism'' temperament (53&amp;612, named by [[User:Xenllium|Xenllium]]) is a rank-two temperament which tempers out the [[monzisma]], {{monzo|54 -37 2}} and the [[nanisma]], {{monzo|109 -67 0 -1}}, as well as the ragisma, [[4375/4374]].  


[[Comma list]]: 4375/4374, {{monzo|-55 30 2 1}}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [&lt;1 2 10 -25|, &lt;0 -2 -37 134|]
Comma list: 245/242, 385/384, 4375/4374


[[POTE tuning|POTE generator]]: ~310078125/268435456 = 249.0207
Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}


[[EDO|Vals]]: {{Val list| 53, 559, 612, 1277, 1889 }}
Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.6179


[[Badness]]: 0.046569
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdde }}
 
Badness: 0.084590
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 245/242, 325/324, 385/384
 
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


== 11-limit ==
{{Optimal ET sequence|legend=1| 26, 65f, 91f, 156dff }}
Comma list: 4375/4374, 41503/41472, 184549376/184528125


Mapping: [&lt;1 2 10 -25 46|, &lt;0 -2 -37 134 -205|]
Badness: 0.052366


POTE generator: ~231/200 = 249.0193
== Counterorson ==
Counterorson tempers out the {{monzo| 147 -103 7 }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]].  


Vals: {{Val list| 53, 559, 612 }}
Subgroup: 2.3.5.7


Badness: 0.057083
Comma list: 4375/4374, {{monzo| 154 -54 -21 -7 }}


== 13-limit ==
Mapping: {{mapping| 1 0 -21 85 | 0 7 103 -363 }}
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625


Mapping: [&lt;1 2 10 -25 46 23|, &lt;0 -2 -37 134 -205 -93|]
Optimal tuning (CTE): ~2 = 1\1, ~{{monzo| 66 -23 -9 -3 }} = 271.7113


POTE generator: ~231/200 = 249.0199
{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}


Vals: {{Val list| 53, 559, 612 }}
Badness: 0.312806


Badness: 0.053780
== Notes ==


[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Ragismic microtemperaments| ]] <!-- main article -->
[[Category:Ragismic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Microtemperaments]]
[[Category:Abigail]]
[[Category:Abigail]]
[[Category:Amity]]
[[Category:Deca]]
[[Category:Deca]]
[[Category:Enneadecal]]
[[Category:Enneadecal]]
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[[Category:Octoid]]
[[Category:Octoid]]
[[Category:Parakleismic]]
[[Category:Parakleismic]]
[[Category:Quincy]]
[[Category:Supermajor]]
[[Category:Supermajor]]
[[Category:Microtemperament]]
[[Category:Ragismic]]
[[Category:Rank 2]]
[[Category:Ragismic microtemperaments| ]] <!-- main article -->
[[Category:Todo:review]]
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