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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 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


[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [26.667, 66.667] (45bcd to 18bcd)
[[Comma list]]: 4375/4374, 52734375/52706752


nice range: [48.920, 49.179]
{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }}


strict range: [48.920, 49.179]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082


Commas: 2401/2400, 4375/4374
{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}


POTE generators: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980
[[Badness]]: 0.010836


Map: [&lt;9 1 1 2|, &lt;0 2 3 2|]
=== Semisupermajor ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;18 27 18 1 -22 -34||
Comma list: 3025/3024, 4375/4374, 35156250/35153041


EDOs: [[27edo|27]], [[45edo|45]], [[72edo|72]], [[99edo|99]], [[171edo|171]], [[270edo|270]], [[441edo|441]], [[612edo|612]], [[3600edo|3600]]
Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}


Badness: 0.00361
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082


==Hemiennealimmal==
{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}
Commas: 2401/2400, 4375/4374, 3025/3024


valid range: [13.333, 22.222] (90bcd, 54c)
Badness: 0.012773


nice range: [17.304, 17.985]
== 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.


strict range:  [17.304, 17.985]
''For the 5-limit temperament, see [[19th-octave temperaments#(5-limit) enneadecal]].''


POTE generator: ~99/98 = 17.6219
[[Subgroup]]: 2.3.5.7


Map: [&lt;18 0 -1 22 48|, &lt;0 2 3 2 1|]
[[Comma list]]: 4375/4374, 703125/702464


EDOs: 72, 198, 270, 342, 612, 954, 1566
{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}


Badness: 0.00628
: mapping generators: ~28/27, ~3


===13-limit===
[[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)
Commas: 676/675, 1001/1000, 1716/1715, 3025/3024


valid range: [16.667, 22.222] (72 to 54cf)
{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}


nice range: [17.304, 18.309]
[[Badness]]: 0.010954


strict range: [17.304, 18.309]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator ~99/98 = 17.7504
Comma list: 540/539, 4375/4374, 16384/16335


Map: [&lt;18 0 -1 22 48 -19|, &lt;0 2 3 2 1 6|]
Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}


EDOs: 72, 198, 270
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)


Badness: 0.0125
{{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }}


=== Semihemiennealimmal ===
Badness: 0.043734
Commas: 2401/2400, 4375/4374, 3025/3024, 4225/4224


POTE generator: ~39/32 = 342.139
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;18 0 -1 22 48 88|, &lt;0 4 6 4 2 -3|]
Comma list: 540/539, 625/624, 729/728, 2205/2197


EDOs: 126, 144, 270, 684, 954
Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}


Badness: 0.0131
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)


==Semiennealimmal==
{{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }}
Commas: 2401/2400, 4375/4374, 4000/3993


POTE generator: ~140/121 = 250.3367
Badness: 0.033545


Map: [&lt;9 3 4 14 18|, &lt;0 6 9 6 7|]
=== Hemienneadecal ===
Subgroup: 2.3.5.7.11


EDOs: 72, 369, 441
Comma list: 3025/3024, 4375/4374, 234375/234256


Badness: 0.0342
Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}


===13-limit===
: mapping generators: ~55/54, ~3
Commas: 1575/1573, 2080/2079, 2401/2400, 4375/4374


POTE generator: ~140/121 = 250.3375
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)


Map: [&lt;9 3 4 14 18 -8|, &lt;0 6 9 6 7 22|]
{{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}


EDOs: 72, 441
Badness: 0.009985


Badness: 0.0261
==== Hemienneadecalis ====
Subgroup: 2.3.5.7.11.13


==Quadraennealimmal==
Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256
Commas: 2401/2400, 4375/4374, 234375/234256


POTE generator: ~77/75 = 45.595
Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}


Map: [&lt;9 1 1 12 -7|, &lt;0 8 12 8 23|]
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)


EDOs: 342, 1053, 1395, 1737, 4869d, 6606cd
{{Optimal ET sequence|legend=1| 152f, 342f, 494 }}


Badness: 0.0213
Badness: 0.020782
 
==== Hemienneadec ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
 
Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}
 
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444)
 
{{Optimal ET sequence|legend=1| 152, 342, 494, 1330, 1824, 2318d }}
 
Badness: 0.030391
 
==== Semihemienneadecal ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078
 
Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}
 
: mapping generators: ~55/54 = 1\38, ~55/54, ~429/250
 
Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)
 
{{Optimal ET sequence|legend=1| 190, 304d, 494, 684, 1178, 2850, 4028ce }}
 
Badness: 0.014694
 
=== 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.
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344
 
Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}
 
Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244
 
{{Optimal ET sequence|legend=1| 855, 988, 1843 }}
 
== Semidimi ==
: ''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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 3955078125/3954653486
 
{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270
 
{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
 
[[Badness]]: 0.015075
 
== Brahmagupta ==
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.
 
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 ¢).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 70368744177664/70338939985125
 
{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}
 
: mapping generators: ~1157625/1048576, ~27/20
 
[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716
 
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }}
 
[[Badness]]: 0.029122
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4000/3993, 4375/4374, 131072/130977
 
Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}
 
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704
 
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }}
 
Badness: 0.052190
 
=== 13-limit ===
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 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901
 
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }}
 
Badness: 0.012860
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==Ennealimnic==
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
Commas: 243/242, 441/440, 4375/4356


valid range: [44.444, 53.333] (27e to 45e)
Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}


nice range: [48.920, 52.592]
Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903


strict range: [48.920, 52.592]
{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }}


POTE generator: ~36/35 = 49.395
Badness: 0.008856


Map: [&lt;9 1 1 12 -2|, &lt;0 2 3 2 5|]
== Gamera ==
''For the 5-limit temperament, see [[High badness temperaments#Gamera]].


EDOs: 72, 171, 243
[[Subgroup]]: 2.3.5.7


Badness: 0.0203
[[Comma list]]: 4375/4374, 589824/588245


===13-limit===
{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }}
Commas: 243/242, 364/363, 441/440, 625/624


valid range: [48.485, 50.000] (99ef to 72)
: mapping generators: ~2, ~8/7


nice range: [48.825, 52.592]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336


strict range: [48.825, 50.000]
{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}


POTE generator: ~36/35 = 49.341
[[Badness]]: 0.037648


Map: [&lt;9 1 1 12 -2 -33|, &lt;0 2 3 2 5 10|]
=== Hemigamera ===
Subgroup: 2.3.5.7.11


EDOs: 72, 171, 243
Comma list: 3025/3024, 4375/4374, 589824/588245


Badness: 0.0233
Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}


==== 17-limit ====
: mapping generators: ~99/70, ~8/7
Commas: 243/242, 364/363, 375/374, 441/440, 595/594


valid range: [48.485, 50.000] (99ef to 72)
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3370


nice range: [46.363, 52.592]
{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646, 1068d }}


strict range: [48.485, 50.000]
Badness: 0.040955


POTE generator: ~36/35 = 49.335
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;9 1 1 12 -2 -33 -3|, &lt;0 2 3 2 5 10 6|]
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024


EDOs: 72, 171, 243
Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}


Badness: 0.0146
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373


=== Ennealim ===
{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646f, 1068df }}
Commas: 169/168, 243/242, 325/324, 441/440


POTE generator: ~36/35 = 49.708
Badness: 0.020416


Map: [&lt;9 1 1 12 -2 20|, &lt;0 2 3 2 5 2|]
=== Semigamera ===
Subgroup: 2.3.5.7.11


EDOs: 27e, 45ef, 72, 315ff, 387cff, 459cdfff
Comma list: 4375/4374, 14641/14580, 15488/15435


Badness: 0.0207
Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }}


==Ennealiminal==
: mapping generators: ~2, ~77/72
Commas: 385/384, 1375/1372, 4375/4374


POTE generator: ~36/35 = 49.504
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1642


Map: [&lt;9 1 1 12 51|, &lt;0 2 3 2 -3|]
{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }}


EDOs: 27, 45, 72, 171e, 243e, 315e
Badness: 0.078


Badness: 0.0311
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


===13-limit===
Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580
Commas: 169/168, 325/324, 385/384, 1375/1372


POTE generator: ~36/35 = 49.486
Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }}


Map: [&lt;9 1 1 12 51 20|, &lt;0 2 3 2 -3 2|]
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1628


EDOs: 27, 45f, 72, 171ef, 243ef
{{Optimal ET sequence|legend=1| 73f, 125f, 198, 323, 521 }}


Badness: 0.0303
Badness: 0.044


==Trinealimmal==
== Crazy ==
Commas: 2401/2400, 4375/4374, 2097152/2096325
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


POTE generator: ~6/5 = 315.644
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.  


Map: [&lt;27 1 0 34 177|, &lt;0 2 3 2 -4|]
[[Subgroup]]: 2.3.5.7


EDOs: 27, 243, 270, 783, 1053, 1323, 10854bcde
[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}


Badness: 0.0298
{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}


=Gamera=
: mapping generators: ~332150625/234881024, ~1125/1024
Commas: 4375/4374, 589824/588245


POTE generator ~8/7 = 230.336
[[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 }}


Map: [&lt;1 6 10 3|, &lt;0 -23 -40 -1|]
{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}


EDOs: 26, 73, 99, 224, 323, 422, 745d
[[Badness]] (Smith): 0.0394


Badness: 0.0376
=== 11-limit ===
Subgroup: 2.3.5.7.11


==Hemigamera==
Comma list: 3025/3024, 4375/4374, 2791309312/2790703125
Commas: 3025/3024, 4375/4374, 589824/588245


POTE generator: ~8/7 = 230.337
Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}


Map: [&lt;2 12 20 6 5|, &lt;0 -23 -40 -1 5|]
Optimal tunings:
* CTE: ~99/70 = 162.7485, ~1125/1024 = 162.7485
* CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481


EDOs: 26, 198, 224, 422, 646, 1068d
{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}


Badness: 0.0410
Badness (Smith): 0.0170


===13-limit===
== Orga ==
Commas: 1716/1715, 2080/2079, 2200/2197, 3025/3024
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 12 20 6 5 17|, &lt;0 -23 -40 -1 5 -25|]
[[Comma list]]: 4375/4374, 54975581388800/54936068900769


EDOs: 26, 198, 224, 422, 646f, 1068df
{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }}


Badness: 0.0204
: mapping generators: ~7411887/5242880, ~1310720/1058841


=Supermajor=
[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104
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.


Commas: 4375/4374, 52734375/52706752
{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}


POTE generator: ~9/7 = 435.082
[[Badness]]: 0.040236


Map: [&lt;1 15 19 30|, &lt;0 -37 -46 -75|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214
Comma list: 3025/3024, 4375/4374, 5767168/5764801


Badness: 0.0108
Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }}


==Semisupermajor==
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
Commas: 3025/3024, 4375/4374, 35156250/35153041


POTE generator: ~9/7 = 435.082
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836, 1106 }}


Map: [&lt;2 30 38 60 41|, &lt;0 -37 -46 -75 -47|]
Badness: 0.016188


EDOs: 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0128
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360


=Enneadecal=
Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }}
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.


Commas: 4375/4374, 703125/702464
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103


POTE generator: ~3/2 = 701.880
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836f, 1106f }}


Map: [&lt;19 0 14 -37|, &lt;0 1 1 3|]
Badness: 0.021762


Generators: 28/27, 3
== Seniority ==
{{See also| Very high accuracy temperaments #Senior }}


EDOs: 19, 152, 171, 665, 836, 1007, 2185
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.


Badness: 0.0110
[[Subgroup]]: 2.3.5.7


==Hemienneadecal==
[[Comma list]]: 4375/4374, 201768035/201326592
Commas: 3025/3024, 4375/4374, 234375/234256


POTE generator: ~3/2 = 701.881
{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}


Map: [&lt;38 0 28 -74 11|, &lt;0 1 1 3 2|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3087/2560 = 322.804


EDOs: 152, 342, 494, 836, 1178, 2014
{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}


Badness: 0.00999
[[Badness]]: 0.044877


===13-limit===
=== Senator ===
Commas: 3025/3024, 4096/4095, 4375/4374, 31250/31213
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.


POTE generator: ~3/2 = 701.986
Subgroup: 2.3.5.7.11


Map: [&lt;38 0 28 -74 11 502|, &lt;0 1 1 3 2 -6|]
Comma list: 441/440, 4375/4374, 65536/65219


EDOs: 152, 342, 494, 836
Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }}


Badness: 0.0304
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793


=Deca=
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316e, 487ee }}
Commas: 4375/4374, 165288374272/164794921875


POTE generator: ~460992/390625 = 284.423
Badness: 0.092238


Map: [&lt;10 4 2 9|, &lt;0 5 6 11|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 80, 190, 270, 1270, 1540, 1810, 2080
Comma list: 364/363, 441/440, 2200/2197, 4375/4374


Badness: 0.0806
Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }}


==11-limit==
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
Commas: 3025/3024, 4375/4374, 422576/421875


POTE generator: ~33/28 = 284.418
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}


Map: [&lt;10 4 2 9 18|, &lt;0 5 6 11 7|]
Badness: 0.044662


EDOs: 80, 190, 270, 1000, 1270
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0243
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197


==13-limit==
Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }}
Commas: 1001/1000, 3025/3024, 4225/4224, 4375/4374


POTE generator: ~33/28 = 284.398
Optimal tuning (POTE): ~77/64 = 322.793


Map: [&lt;10 4 2 9 18 37|, &lt;0 5 6 11 7 0|]
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}


EDOs: 80, 190, 270, 730, 1000
Badness: 0.026562


Badness: 0.0168
== Monzismic ==
: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Monzismic]].  


= Mitonic =
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]].
{{see also|Minortonic family #Mitonic}}


Commas: 4375/4374, 2100875/2097152
[[Subgroup]]: 2.3.5.7


POTE generator: ~10/9 = 182.458
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}


Map: [&lt;1 16 32 -15|, &lt;0 -17 -35 21|]
{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}


EDOs: 46, 125, 171
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207


Badness: 0.0252
{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}


=Abigail=
[[Badness]]: 0.046569
Commas: 4375/4374, 2147483648/2144153025


[[POTE_tuning|POTE generator]]: 208.899
=== Monzism ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 7 13 -1|, &lt;0 -11 -24 19|]
Comma list: 4375/4374, 41503/41472, 184549376/184528125


Wedgie: &lt;&lt;22 48 -38 25 -122 -223||
Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}


EDOs: 46, 132, 178, 224, 270, 494, 764, 1034, 1798
Optimal tuning (POTE): ~231/200 = 249.0193


Badness: 0.0370
{{Optimal ET sequence|legend=1| 53, 559, 612 }}


==11-limit==
Badness: 0.057083
Comma: 3025/3024, 4375/4374, 20614528/20588575


[[POTE_tuning|POTE generator]]: 208.901
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 7 13 -1 1|, &lt;0 -11 -24 19 17|]
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625


EDOs: 46, 132, 178, 224, 270, 494, 764
Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}


Badness: 0.0129
Optimal tuning (POTE): ~231/200 = 249.0199


==13-limit==
{{Optimal ET sequence|legend=1| 53, 559, 612 }}
Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095


[[POTE_tuning|POTE generator]]: 208.903
Badness: 0.053780


Map: [&lt;2 7 13 -1 1 -2|, &lt;0 -11 -24 19 17 27|]
== Semidimfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].''


EDOs: 46, 178, 224, 270, 494, 764, 1258
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: 0.00886
[[Subgroup]]: 2.3.5.7


=Semidimi=
[[Comma list]]: 4375/4374, 235298/234375
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]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}


POTE generator: ~162/125 = 449.127
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456


Map: [&lt;1 36 48|, &lt;0 -55 -73|]
{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}


Wedgie: &lt;&lt;55 73 -12||
[[Badness]]: 0.055249


EDOs: 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419
=== Neusec ===
Subgroup: 2.3.5.7.11


Badness: 0.7549
Comma list: 3025/3024, 4375/4374, 235298/234375


==7-limit==
Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }}
Commas: 4375/4374, 3955078125/3954653486


POTE generator: ~35/27 = 449.127
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547


Map: [&lt;1 36 48 61|, &lt;0 -55 -73 -93|]
{{Optimal ET sequence|legend=1| 8d, 190, 388 }}


Wedgie: &lt;&lt;55 73 93 -12 -7 11||
Badness: 0.059127


EDOs: 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0151
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374


=Brahmagupta=
Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }}
Commas: 4375/4374, 70368744177664/70338939985125


POTE generator: ~27/20 = 519.716
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545


Map: [&lt;7 2 -8 53|, &lt;0 3 8 -11|]
{{Optimal ET sequence|legend=1| 8d, 190, 198, 388 }}


Wedgie: &lt;&lt;21 56 -77 40 -181 -336||
Badness: 0.030941


EDOs: 217, 224, 441, 1106, 1547
== Acrokleismic ==
[[Subgroup]]: 2.3.5.7


Badness: 0.0291
[[Comma list]]: 4375/4374, 2202927104/2197265625


==11-limit==
{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }}
Commas: 4000/3993, 4375/4374, 131072/130977


POTE generator: ~27/20 = 519.704
: mapping generators: ~2, ~6/5


Map: [&lt;7 2 -8 53 3|, &lt;0 3 8 -11 7|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557


EDOs: 217, 224, 441, 665, 1771ee
{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}


Badness: 0.0522
[[Badness]]: 0.056184


==13-limit==
=== 11-limit ===
Commas: 1575/1573, 2080/2079, 4096/4095, 4375/4374
Subgroup: 2.3.5.7.11


POTE generator: ~27/20 = 519.706
Comma list: 4375/4374, 41503/41472, 172032/171875


Map: [&lt;7 2 -8 53 3 35|, &lt;0 3 8 -11 7 -3|]
Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }}


EDOs: 217, 224, 441, 665, 1771eef
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558


Badness: 0.0231
{{Optimal ET sequence|legend=1| 19, 251, 270, 829, 1099, 1369, 1639 }}


=Quasithird=
Badness: 0.036878
Comma: |55 -64 20&gt;


POTE generator: ~1594323/1280000 = 380.395
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;4 0 -11|, &lt;0 5 16|]
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976


Wedgie: &lt;&lt;20 64 55||
Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }}


EDOs: 164, 224, 388, 612, 836, 1000, 1448, 1612, 2224, 2836
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557


Badness: 0.0995
{{Optimal ET sequence|legend=1| 19, 251, 270 }}


==7-limit==
Badness: 0.026818
Commas: 4375/4374, 1153470752371588581/1152921504606846976


POTE generator: ~5103/4096 = 380.388
=== Counteracro ===
Subgroup: 2.3.5.7.11


Map: [&lt;4 0 -11 48|, &lt;0 5 16 -29|]
Comma list: 4375/4374, 5632/5625, 117649/117612


Wedgie: &lt;&lt;20 64 -116 55 -240 -449||
Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }}


EDOs: 164, 224, 388, 612, 1448, 2060
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.553


Badness: 0.0618
{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}


==11-limit==
Badness: 0.042572
Commas: 3025/3024, 4375/4374, 4296700485/4294967296


POTE generator: ~5103/4096 = 380.387
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;4 0 -11 48 43|, &lt;0 5 16 -29 -23|]
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374


EDOs: 164, 224, 388, 612, 836, 1448
Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }}


Badness: 0.0211
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.554


==13-limit==
{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}
Commas: 2200/2197, 3025/3024, 4375/4374, 468512/468195


POTE generator: ~5103/4096 = 380.385
Badness: 0.026028


Map: [&lt;4 0 -11 48 43 11|, &lt;0 5 16 -29 -23 3|]
== 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}}.


EDOs: 164, 224, 388, 612, 836, 1448f, 2284f
[[Subgroup]]: 2.3.5


Badness: 0.0295
[[Comma list]]: {{monzo| 55 -64 20 }}


=Semidimfourth=
{{Mapping|legend=1| 4 0 -11 | 0 5 16 }}
Comma: |7 41 -31&gt;


POTE generator: ~162/125 = 448.449
: mapping generators: ~51200000/43046721, ~1594323/1280000


Map: [&lt;1 21 28|, &lt;0 -31 -41|]
[[Optimal tuning]] ([[POTE]]): ~51200000/43046721, ~1594323/1280000 = 380.395


Wedgie: &lt;&lt;31 41 -7||
{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }}


EDOs: 91, 99, 190, 289, 388, 487, 677, 875, 966
[[Badness]]: 0.099519


Badness: 0.1930
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


==7-limit==
[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}
Commas: 4375/4374, 235298/234375


POTE generator: ~35/27 = 448.457
{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}


Map: [&lt;1 21 28 36|, &lt;0 -31 -41 -53|]
[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 1\4, ~5103/4096 = 380.388


Wedgie: &lt;&lt;31 41 53 -7 -3 8||
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}


EDOs: 91, 99, 289, 388, 875, 1263d, 1651d
[[Badness]]: 0.061813


Badness: 0.0552
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Neusec ==
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
Commas: 3025/3024, 4375/4374, 235298/234375


POTE generator: ~12/11 = 151.547
Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}


Map: [&lt;2 11 15 19 15|, &lt;0 -31 -41 -53 -32|]
Optimal tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387)


EDOs: 190, 388
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448 }}


Badness: 0.0591
Badness: 0.021125


=== 13-limit ===
=== 13-limit ===
Commas: 847/845, 1001/1000, 3025/3024, 4375/4374
Subgroup: 2.3.5.7.11.13
 
Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374
 
Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}
 
Optimal tuning (POTE): ~81/65 = 380.385 (or ~22/21 = 80.385)
 
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}
 
Badness: 0.029501
 
== Deca ==
: ''For 5-limit version of this temperament, see [[10th-octave temperaments #Neon]].''


POTE generator: ~12/11 = 151.545
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).


Map: [&lt;2 11 15 19 15 17|, &lt;0 -31 -41 -53 -32 -38|]
[[Subgroup]]: 2.3.5.7


EDOs: 190, 198, 388
[[Comma list]]: 4375/4374, 165288374272/164794921875


Badness: 0.0309
{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}


=Acrokleismic=
: mapping generators: ~15/14, ~6/5
Commas: 4375/4374, 2202927104/2197265625


POTE generator: ~6/5 = 315.557
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577


Map: [&lt;1 10 11 27|, &lt;0 -32 -33 -92|]
{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}


Wedgie: &lt;&lt;32 33 92 -22 56 121||
[[Badness]]: 0.080637


EDOs: 19, 251, 270
Badness (Sintel): 2.041


Badness: 0.0562
=== 11-limit ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 3025/3024, 4375/4374, 391314/390625
Commas: 4375/4374, 41503/41472, 172032/171875


POTE generator: ~6/5 = 315.558
Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}


Map: [&lt;1 10 11 27 -16|, &lt;0 -32 -33 -92 74|]
Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.582


EDOs: 19, 251, 270, 829, 1099, 1369, 1639
{{Optimal ET sequence|legend=1| 80, 190, 270, 1000, 1270, 1540e, 1810e }}


Badness: 0.0369
Badness: 0.024329
 
Badness (Sintel): 0.804


=== 13-limit ===
=== 13-limit ===
Commas: 676/675, 1001/1000, 4375/4374, 10985/10976
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)


POTE generator: ~6/5 = 315.557
{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}


Map: [&lt;1 10 11 27 -16 25|, &lt;0 -32 -33 -92 74 -81|]
Badness: 0.016810


EDOs: 19, 251, 270
Badness (Sintel): 0.695


Badness: 0.0268
=== no-17's 19-limit ===
Subgroup: 2.3.5.7.11.13.19


==Counteracro==
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374, 1521/1520
Commas: 4375/4374, 5632/5625, 117649/117612


POTE generator: ~6/5 = 315.553
Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}


Map: [&lt;1 10 11 27 55|, &lt;0 -32 -33 -92 -196|]
Optimal tuning (CTE): ~15/14 = 1\10, ~6/5 = 315.581 (~39/38 = 44.419)


EDOs: 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde
{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}


Badness: 0.0426
Badness (Sintel): 0.556


===13-limit===
== Keenanose ==
Commas: 676/675, 1716/1715, 4225/4224, 4375/4374
Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.


POTE generator: ~6/5 = 315.554
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 10 11 27 55 25|, &lt;0 -32 -33 -92 -196 -81|]
[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}


EDOs: 270, 1331c, 1601c, 1871bcf, 2141bcf
{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}


Badness: 0.0260
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}


=Seniority=
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 21 3 1 -10 }} = 4.4465
Commas: 4375/4374, 201768035/201326592


POTE generator: ~3087/2560 = 322.804
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}


Map: [&lt;1 11 19 2|, &lt;0 -35 -62 3|]
[[Badness]]: 0.0858


Wedgie: &lt;&lt;35 62 -3 17 -103 -181||
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 26, 145, 171, 2710d
Comma list: 4375/4374, 117649/117612, 67110351/67108864


Badness: 0.0449
Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}


=Orga=
Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4465
Commas: 4375/4374, 54975581388800/54936068900769


POTE generator: ~8/7 = 231.104
{{Optimal ET sequence|legend=1| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}


Map: [&lt;2 21 36 5|, &lt;0 -29 -51 1|]
Badness: 0.0308


Wedgie: &lt;&lt;58 102 -2 27 -166 -291||
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 26, 244, 270, 836, 1106, 1376, 2482
Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612


Badness: 0.0402
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}


==11-limit==
Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4466
Commas: 3025/3024, 4375/4374, 5767168/5764801


POTE generator: ~8/7 = 231.103
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 4048 }}


Map: [&lt;2 21 36 5 2|, &lt;0 -29 -51 1 8|]
Badness: 0.0213


EDOs: 26, 244, 270, 566, 836, 1106
== 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.


Badness: 0.0162
[[Subgroup]]: 2.3.5


==13-limit==
[[Comma list]]: {{monzo| 92 -39 -13 }}
Commas: 1716/1715, 2080/2079, 3025/3024, 15379/15360


POTE generator: ~8/7 = 231.103
[[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}


Map: [&lt;2 21 36 5 2 24|, &lt;0 -29 -51 1 8 -27|]
: mapping generators: ~135/128, ~3


EDOs: 26, 244, 270, 566, 836f, 1106f
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897


Badness: 0.0218
{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}


=Quatracot=
[[Badness]]: 0.123
Commas: 4375/4374, 1483154296875/1473173782528


POTE generator: ~448/405 = 176.805
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 7 7 23|, &lt;0 -13 -8 -59|]
[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}


Wedgie: &lt;&lt;26 16 118 -35 114 229||
[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}


EDOs: 190, 224, 414, 638, 1052c, 1690bc
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 702.0024


Badness: 0.1760
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}


==11-limit==
[[Badness]]: 0.126
Commas: 3025/3024, 4375/4374, 1265625/1261568


POTE generator: ~448/405 = 176.806
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 7 7 23 19|, &lt;0 -13 -8 -59 -41|]
Comma list: 4375/4374, 234375/234256, 2097152/2096325


EDOs: 190, 224, 414, 638, 1052c
Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}


Badness: 0.0410
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042


==13-limit==
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 3588e, 5135e }}
Commas: 625/624, 729/728, 1575/1573, 2200/2197


POTE generator: ~448/405 = 176.804
Badness: 0.0421


Map: [&lt;2 7 7 23 19 13|, &lt;0 -13 -8 -59 -41 -19|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 190, 224, 414, 638, 1690bc, 2328bcde
Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078


Badness: 0.0226
Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}


=Octoid=
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099
Commas: 4375/4374, 16875/16807


valid range: [578.571, 600.000] (56bcd to 8d)
{{Optimal ET sequence|legend=1| 494, 1547, 2041, 4576def }}


nice range: [582.512, 584.359]
Badness: 0.0286


strict range: [582.512, 584.359]
== Countritonic ==
: ''For the 5-limit version of this temperament, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


POTE generator: ~7/5 = 583.940
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.  


Map: [&lt;8 1 3 3|, &lt;0 3 4 5|]
[[Subgroup]]: 2.3.5.7


Generators: 49/45, 7/5
[[Comma list]]: 4375/4374, 68719476736/68356598625


EDOs: 72, 152, 224
{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }}


Badness: 0.0427
: mapping generators: ~2, ~45927/32768


==11-limit==
[[Optimal tuning]] (CTE): ~2 = 1\1, ~45927/32768 = 588.6216
Commas: 540/539, 1375/1372, 4000/3993


valid range: [581.250, 586.364] (64cd, 88bcde)
{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}


nice range: [582.512, 585.084]
[[Badness]]: 0.133


strict range: [582.512, 585.084]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~7/5 = 583.692
Comma list: 4375/4374, 5632/5625, 2621440/2614689


Map: [&lt;8 1 3 3 16|, &lt;0 3 4 5 3|]
Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 }}


EDOs: 72, 152, 224
Optimal tuning (CTE): ~2 = 1\1, ~539/384 = 588.6258


Badness: 0.0141
{{Optimal ET sequence|legend=1| 53, 316e, 369, 422, 791e, 1213cde }}
 
Badness: 0.0707


=== 13-limit ===
=== 13-limit ===
Commas: 540/539, 1375/1372, 4000/3993, 625/624
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


POTE generator: ~7/5 = 583.905
[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 1\2, ~448/405 = 176.805


Map: [&lt;8 1 3 3 16 -21|, &lt;0 3 4 5 3 13|]
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}


EDOs: 72, 224
[[Badness]]: 0.175982


Badness: 0.0153
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== Music ===
Comma list: 3025/3024, 4375/4374, 1265625/1261568
* [http://www.archive.org/details/Dreyfus http://www.archive.org/details/Dreyfus]
* [http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play]


=== Octopus ===
Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }}
Commas: 169/168, 325/324, 364/363, 540/539


POTE generator: ~7/5 = 583.892
Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 176.806


Map: [&lt;8 1 3 3 16 14|, &lt;0 3 4 5 3 4|]
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c }}


EDOs: 72, 152, 224f
Badness: 0.041043


Badness: 0.0217
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


= Amity =
Comma list: 625/624, 729/728, 1575/1573, 2200/2197
{{main|Amity}}
{{see also|Amity family #Amity}}


The generator for [[amity]] temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&amp;53 temperament, or by its wedgie, &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.
Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }}


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


Comma: 1600000/1594323
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1690bcc, 2328bccde }}


POTE generator: ~243/200 = 339.519
Badness: 0.022643


Map: [&lt;1 3 6|, &lt;0 -5 -13|]
== 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.


EDOs: 7, 39, 46, 53, 152, 205, 463, 668, 873
[[Subgroup]]: 2.3.5.7


Badness: 0.0220
[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}


== 7-limit ==
{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }}
Commas: 4375/4374, 5120/5103


POTE generator: ~128/105 = 339.432
: mapping generators: ~2, ~6422528/3796875


Map: [&lt;1 3 6 -2|, &lt;0 -5 -13 17|]
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6422528/3796875 = 910.9323


Wedgie: &lt;&lt;5 13 -17 9 -41 -76||
{{Optimal ET sequence|legend=1| 494, 1125, 1619 }}


EDOs: 7, 39, 46, 53, 99, 251, 350
[[Badness]]: 0.234


Badness: 0.0236
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 4375/4374, 759375/758912, 100663296/100656875
Commas: 540/539, 4375/4374, 5120/5103


POTE generator: ~128/105 = 339.464
Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }}


Map: [&lt;1 3 6 -2 21|, &lt;0 -5 -13 17 -62|]
Optimal tuning (CTE): ~2 = 1\1, ~1024/605 = 910.9323


EDOs: 53, 99e, 152, 555dee, 707ddee, 859bddee
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}


Badness: 0.0315
Badness: 0.0678


=== 13-limit ===
=== 13-limit ===
Commas: 352/351, 540/539, 625/624, 847/845
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 }}


POTE generator: ~128/105 = 339.481
{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}


Map: [&lt;1 3 6 -2 21 17|, &lt;0 -5 -13 17 -62 -47|]
: mapping generators: ~83349/81920, ~3


EDOS: 53, 99ef, 152f, 205
[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 1\46, ~3/2 = 701.6074


Badness: 0.0280
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }}


== Hitchcock ==
[[Badness]]: 0.308505
{{see also|Amity family #Hitchcock}}


Commas: 121/120, 176/175, 2200/2187
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~11/9 = 339.340
Comma list: 3025/3024, 4375/4374, 134775333/134217728


Map: [&lt;1 3 6 -2 6|, &lt;0 -5 -13 17 -9|]
Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}


EDOs: 7, 39, 46, 53, 99
Optimal tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951


Badness: 0.0352
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de }}
 
Badness: 0.073783


=== 13-limit ===
=== 13-limit ===
Commas: 121/120, 169/168, 176/175, 325/324
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
 
{{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]]:
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
 
{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}
 
[[Badness]]: 0.042670
 
Scales: [[octoid72]], [[octoid80]]
 
=== 11-limit ===
The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimaxing the damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, if one wants to use 80edo as the tuning, one must use octopus — not octoid — as 80edo doesn't temper 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.
 
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 4000/3993
 
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]
 
{{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


POTE generator: ~11/9 = 339.419
{{Optimal ET sequence|legend=1| 72, 152f, 224 }}


Map: [&lt;1 3 6 -2 6 2|, &lt;0 -5 -13 17 -9 6|]
Badness: 0.015274


EDOs: 7, 39, 46, 53, 99
Scales: [[octoid72]], [[octoid80]]


Badness: 0.0224
; 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


== Hemiamity ==
===== 17-limit =====
Commas: 3025/3024, 4375/4374, 5120/5103
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~64/55 = 339.493
Comma list: 375/374, 540/539, 625/624, 715/714, 729/728


Map: [&lt;2 1 -1 13 13|, &lt;0 5 13 -17 -14|]
Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}


EDOs: 14cde, 46, 106, 152, 350
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.842


Badness: 0.0313
{{Optimal ET sequence|legend=1| 72, 152fg, 224, 296, 520g }}


=Parakleismic=
Badness: 0.014304
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
Scales: [[octoid72]], [[octoid80]]


POTE generator: ~6/5 = 315.240
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 5 6|, &lt;0 -13 -14|]
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714


EDOs: 19, 61, 80, 99, 118, 453, 571, 689, 1496
Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}


Badness: 0.0433
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.932


==7-limit==
{{Optimal ET sequence|legend=1| 72, 152fg, 224 }}
Commas: 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.181
Badness: 0.016036


Map: [&lt;1 5 6 12|, &lt;0 -13 -14 -35|]
Scales: [[octoid72]], [[octoid80]]


EDOs: 19, 80, 99, 217, 316, 415
==== 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}}.


Badness: 0.0274
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 169/168, 325/324, 364/363, 540/539
Commas: 385/384, 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.251
Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}


Map: [&lt;1 5 6 12 -6|, &lt;0 -13 -14 -35 36|]
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.892


EDOs: 19, 99, 118
{{Optimal ET sequence|legend=1| 72, 152, 224f }}


Badness: 0.0497
Badness: 0.021679


==Parkleismic==
Scales: [[octoid72]], [[octoid80]]
Commas: 176/175, 1375/1372, 2200/2187


POTE generator: ~6/5 = 315.060
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 5 6 12 20|, &lt;0 -13 -14 -35 -63|]
Comma list: 169/168, 221/220, 289/288, 325/324, 540/539


EDOs: 80, 179, 259cd
Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}


Badness: 0.0559
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.811


===13-limit===
{{Optimal ET sequence|legend=1| 72, 152, 224fg, 296ffg }}
Commas: 169/168, 176/175, 325/324, 1375/1372


POTE generator: ~6/5 = 315.075
Badness: 0.015614


Map: [&lt;1 5 6 12 20 10|, &lt;0 -13 -14 -35 -63 -24|]
Scales: [[Octoid72]], [[Octoid80]]


EDOs: 15, 19, 80, 179
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0366
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
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 3136/3125, 4375/4374
 
Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.251
 
{{Optimal ET sequence|legend=1| 19, 99, 118 }}
 
Badness: 0.049711
 
=== 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.
 
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 3136/3125, 4375/4374
 
Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.220
 
{{Optimal ET sequence|legend=1| 19e, 99e, 118, 217, 335, 552d, 887dd }}
 
Badness: 0.036027
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
 
Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.214
 
{{Optimal ET sequence|legend=1| 99e, 118, 217, 552d, 769de }}
 
Badness: 0.044710
 
==== 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
 
{{Optimal ET sequence|legend=1| 80, 118, 198, 316, 514c, 830c }}
 
Badness: 0.034208
 
==== Semiparamint ====
This extension was named ''semiparakleismic'' in the earlier materials.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
 
Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.156
 
{{Optimal ET sequence|legend=1| 80, 118, 198 }}
 
Badness: 0.033775
 
==== Semiparawolf ====
This extension was named ''gentsemiparakleismic'' in the earlier materials.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 325/324, 364/363, 3136/3125
 
Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }}
 
Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 315.184
 
{{Optimal ET sequence|legend=1| 80, 118f, 198f }}
 
Badness: 0.040467
 
== Counterkleismic ==
{{See also| High badness temperaments #Counterhanson}}
 
In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as 19 &amp; 224 temperament (''counterkleismic'', named by analogy to [[catakleismic]] and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 158203125/157351936
 
{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }}
 
: mapping generators: ~2, ~5/3
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.060
 
{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }}
 
[[Badness]]: 0.090553
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4375/4374, 2097152/2096325
 
Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.071
 
{{Optimal ET sequence|legend=1| 19, 205, 224 }}
 
Badness: 0.070952
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 625/624, 729/728, 10985/10976
 
Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.070
 
{{Optimal ET sequence|legend=1| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}
 
Badness: 0.033874
 
=== Counterlytic ===
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 4375/4374, 496125/495616
 
Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065
 
{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}
 
Badness: 0.065400
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 729/728, 1375/1372, 10985/10976
 
Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }}


==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


POTE generator: ~6/5 = 315.1563
Comma list: 364/363, 441/440, 676/675, 4375/4374


Map: [<2 10 12 24 19 -1|, <0 -13 -14 -35 -23 16|]
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}


EDOs: {{EDOs|80, 118, 198}}
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602


Badness: 0.0338
{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}


=== Gentsemiparakleismic ===
Badness: 0.023862
Commas: 169/168, 325/324, 364/363, 3136/3125


POTE generator: ~6/5 = 315.1839
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [<2 10 12 24 19 20|, <0 -13 -14 -35 -23 -24|]
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155


EDOs: {{EDOs|80, 118f, 198f}}
Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}


Badness: 0.0405
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602


=Quincy=
{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}
Commas: 4375/4374, 823543/819200


POTE generator: ~1728/1715 = 16.613
Badness: 0.014741


Map: [&lt;1 2 2 3|, &lt;0 -30 -49 -14|]
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


EDOs: 72, 217, 289
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675


Badness: 0.0797
Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}


==11-limit==
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.594
Commas: 441/440, 4000/3993, 41503/41472


POTE generator: ~100/99 = 16.613
{{Optimal ET sequence|legend=1| 72, 145, 217 }}


Map: [&lt;1 2 2 3 4|, &lt;0 -30 -49 -14 -39|]
Badness: 0.015197


EDOs: 72, 217, 289
== Sfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''


Badness: 0.0309
[[Subgroup]]: 2.3.5.7


==13-limit==
[[Comma list]]: 4375/4374, 64827/64000
Commas: 364/363, 441/440, 676/675, 4375/4374


POTE generator: ~100/99 = 16.602
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}


Map: [&lt;1 2 2 3 4 5|, &lt;0 -30 -49 -14 -39 -94|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 26.287


EDOs: 72, 145, 217, 289
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


Badness: 0.0239
[[Badness]]: 0.123291


==17-limit==
=== 11-limit ===
Commas: 364/363, 441/440, 595/594, 1001/1000, 1156/1155
Subgroup: 2.3.5.7.11


POTE generator: ~100/99 = 16.602
Comma list: 121/120, 441/440, 4375/4374


Map: [&lt;1 2 2 3 4 5 5|, &lt;0 -30 -49 -14 -39 -94 -66|]
Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}


EDOs: 72, 145, 217, 289
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.286


Badness: 0.0147
{{Optimal ET sequence|legend=1| 45e, 46, 91e, 137de }}


==19-limit==
Badness: 0.054098
Commas: 343/342, 364/363, 441/440, 595/594, 676/675, 2601/2600


POTE generator: ~100/99 = 16.594
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 2 3 4 5 5 4|, &lt;0 -30 -49 -14 -39 -94 -66 18|]
Comma list: 121/120, 169/168, 325/324, 441/440


EDOs: 72, 145, 217
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}


Badness: 0.0152
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.310


= Chlorine =
{{Optimal ET sequence|legend=1| 45ef, 46, 91ef, 137def }}
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.
Badness: 0.033067


Comma: |-52 -17 34&gt;
=== Sfour ===
Subgroup: 2.3.5.7.11


POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2687
Comma list: 385/384, 2401/2376, 4375/4374


Map: [&lt;17 26 39|, &lt;0 2 1|]
Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}


EDOs: 34, 289, 323, 612, 901
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.246


Badness: 0.0771
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


==7-limit==
Badness: 0.076567
Commas: 4375/4374, 193119049072265625/193091834023510016


POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2936
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;17 26 39 43|, &lt;0 2 1 10|]
Comma list: 196/195, 364/363, 385/384, 4375/4374


EDOs: 34d, 289, 323, 612, 935, 1547
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}


Badness: 0.0417
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.239


==11-limit==
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
Commas: 4375/4374, 41503/41472, 1879453125/1879048192


POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2690
Badness: 0.051893


Map: [&lt;17 26 39 43 64|, &lt;0 2 1 10 -11|]
== Trideci ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tridecatonic]].''


EDOs: 34de, 289, 323, 612, 901
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]]").


Badness: 0.0637
[[Subgroup]]: 2.3.5.7


= Monzism =
[[Comma list]]: 4375/4374, 83349/81920
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}}
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}


[[Mapping]]: [&lt;1 2 10 -25|, &lt;0 -2 -37 134|]
[[Optimal tuning]] ([[POTE]]): ~256/245 = 1\13, ~3/2 = 699.1410


[[POTE tuning|POTE generator]]: ~310078125/268435456 = 249.0207
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }}


[[EDO|Vals]]: {{Val list| 53, 559, 612, 1277, 1889 }}
[[Badness]]: 0.184585


[[Badness]]: 0.046569
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/242, 385/384, 4375/4374
 
Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}
 
Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.6179
 
{{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]]
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[[Category:Enneadecal]]
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[[Category:Octoid]]
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[[Category:Parakleismic]]
[[Category:Parakleismic]]
[[Category:Quincy]]
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