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Ragismic microtemperaments: Difference between revisions

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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]].  


=Ennealimmal=
Since {{nowrap|(10/9)<sup>4</sup> {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)<sup>2</sup> }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
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||.


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.
Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
* ''[[Modus]]'' (+64/63) → [[Tetracot family #Modus|Tetracot family]]
* ''[[Flattone]]'' (+81/80) → [[Meantone family #Flattone|Meantone family]]
* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]
* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]]
* [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]
* [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]]
* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]
* ''[[Zarvo]]'' (+33075/32768) → [[Gravity family #Zarvo|Gravity family]]
* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortonic family #Mitonic|Minortonic family]]
* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnuzmic family #Septimal vishnu|Vishnuzmic family]]
* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]
* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]]
* ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]]
* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]]
* ''[[Chlorine]]'' (+{{monzo| -52 -17 34}}) → [[17th-octave temperaments #Chlorine|17th-octave temperaments]]
* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]
* ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments#Dzelic|37th-octave temperaments]]


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.
== 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.


[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [26.667, 66.667] (45bcd to 18bcd)
[[Subgroup]]: 2.3.5.7


nice range: [48.920, 49.179]
[[Comma list]]: 4375/4374, 52734375/52706752


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


Commas: 2401/2400, 4375/4374
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082


POTE generators: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980
{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}


Map: [&lt;9 1 1 2|, &lt;0 2 3 2|]
[[Badness]]: 0.010836


Wedgie: &lt;&lt;18 27 18 1 -22 -34||
=== Semisupermajor ===
Subgroup: 2.3.5.7.11


EDOs: [[27edo|27]], [[45edo|45]], [[72edo|72]], [[99edo|99]], [[171edo|171]], [[270edo|270]], [[441edo|441]], [[612edo|612]], [[3600edo|3600]]
Comma list: 3025/3024, 4375/4374, 35156250/35153041


Badness: 0.00361
Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}


==Hemiennealimmal==
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082
Commas: 2401/2400, 4375/4374, 3025/3024


valid range: [13.333, 22.222] (90bcd, 54c)
{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}


nice range: [17.304, 17.985]
Badness: 0.012773


strict 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.


POTE generator: ~99/98 = 17.6219
''For the 5-limit temperament, see [[19th-octave temperaments#(5-limit) enneadecal]].''


Map: [&lt;18 0 -1 22 48|, &lt;0 2 3 2 1|]
[[Subgroup]]: 2.3.5.7


EDOs: 72, 198, 270, 342, 612, 954, 1566
[[Comma list]]: 4375/4374, 703125/702464


Badness: 0.00628
{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}


===13-limit===
: mapping generators: ~28/27, ~3
Commas: 676/675, 1001/1000, 1716/1715, 3025/3024


valid range: [16.667, 22.222] (72 to 54cf)
[[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)


nice range: [17.304, 18.309]
{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}


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


POTE generator ~99/98 = 17.7504
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;18 0 -1 22 48 -19|, &lt;0 2 3 2 1 6|]
Comma list: 540/539, 4375/4374, 16384/16335


EDOs: 72, 198, 270
Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}


Badness: 0.0125
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)


==Semiennealimmal==
{{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }}
Commas: 2401/2400, 4375/4374, 4000/3993


POTE generator: ~140/121 = 250.3367
Badness: 0.043734


Map: [&lt;9 3 4 14 18|, &lt;0 6 9 6 7|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 72, 369, 441
Comma list: 540/539, 625/624, 729/728, 2205/2197


Badness: 0.0342
Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}


===13-limit===
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)
Commas: 1575/1573, 2080/2079, 2401/2400, 4375/4374


POTE generator: ~140/121 = 250.3375
{{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }}


Map: [&lt;9 3 4 14 18 -8|, &lt;0 6 9 6 7 22|]
Badness: 0.033545


EDOs: 72, 441
=== Hemienneadecal ===
Subgroup: 2.3.5.7.11


Badness: 0.0261
Comma list: 3025/3024, 4375/4374, 234375/234256


==Quadraennealimmal==
Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}
Commas: 2401/2400, 4375/4374, 234375/234256


POTE generator: ~77/75 = 45.595
: mapping generators: ~55/54, ~3


Map: [&lt;9 1 1 12 -7|, &lt;0 8 12 8 23|]
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)


EDOs: 342, 1053, 1395, 1737, 4869d, 6606cd
{{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}
 
Badness: 0.009985
 
==== Hemienneadecalis ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256
 
Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}
 
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)
 
{{Optimal ET sequence|legend=1| 152f, 342f, 494 }}
 
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
 
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
 
Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903
 
{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }}
 
Badness: 0.008856
 
== Gamera ==
''For the 5-limit temperament, see [[High badness temperaments#Gamera]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 589824/588245
 
{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }}
 
: mapping generators: ~2, ~8/7
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336
 
{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}
 
[[Badness]]: 0.037648
 
=== Hemigamera ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 589824/588245
 
Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}
 
: mapping generators: ~99/70, ~8/7
 
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
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024
 
Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373
 
{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646f, 1068df }}
 
Badness: 0.020416
 
=== Semigamera ===
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 }}
 
{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}
 
: mapping generators: ~332150625/234881024, ~1125/1024
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}
 
[[Badness]] (Smith): 0.0394
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 2791309312/2790703125
 
Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}
 
Optimal tunings:
* CTE: ~99/70 = 162.7485, ~1125/1024 = 162.7485
* CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481
 
{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}
 
Badness (Smith): 0.0170
 
== Orga ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 54975581388800/54936068900769
 
{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }}
 
: mapping generators: ~7411887/5242880, ~1310720/1058841
 
[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104
 
{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}
 
[[Badness]]: 0.040236
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 5767168/5764801
 
Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
 
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836, 1106 }}
 
Badness: 0.016188
 
=== 13-limit ===
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 }}
 
Badness: 0.026562
 
== Monzismic ==
: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Monzismic]].
 
The monzismic temperament (53 &amp; 612) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}
 
{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207
 
{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}
 
[[Badness]]: 0.046569
 
=== Monzism ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 41503/41472, 184549376/184528125
 
Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}
 
Optimal tuning (POTE): ~231/200 = 249.0193
 
{{Optimal ET sequence|legend=1| 53, 559, 612 }}
 
Badness: 0.057083
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
 
Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}
 
Optimal tuning (POTE): ~231/200 = 249.0199
 
{{Optimal ET sequence|legend=1| 53, 559, 612 }}
 
Badness: 0.053780
 
== Semidimfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].''
 
The semidimfourth temperament is featured by a semi-diminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, 235298/234375.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 235298/234375
 
[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456
 
{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}
 
[[Badness]]: 0.055249
 
=== Neusec ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 235298/234375
 
Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547
 
{{Optimal ET sequence|legend=1| 8d, 190, 388 }}
 
Badness: 0.059127
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
 
Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545
 
{{Optimal ET sequence|legend=1| 8d, 190, 198, 388 }}
 
Badness: 0.030941
 
== Acrokleismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 2202927104/2197265625
 
{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }}
 
: mapping generators: ~2, ~6/5
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557
 
{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}
 
[[Badness]]: 0.056184
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 41503/41472, 172032/171875
 
Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558
 
{{Optimal ET sequence|legend=1| 19, 251, 270, 829, 1099, 1369, 1639 }}
 
Badness: 0.036878
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
 
Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557
 
{{Optimal ET sequence|legend=1| 19, 251, 270 }}
 
Badness: 0.026818
 
=== Counteracro ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 5632/5625, 117649/117612
 
Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.553
 
{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}
 
Badness: 0.042572
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
 
Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.554
 
{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}
 
Badness: 0.026028
 
== 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}}.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 55 -64 20 }}
 
{{Mapping|legend=1| 4 0 -11 | 0 5 16 }}
 
: mapping generators: ~51200000/43046721, ~1594323/1280000
 
[[Optimal tuning]] ([[POTE]]): ~51200000/43046721, ~1594323/1280000 = 380.395
 
{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }}
 
[[Badness]]: 0.099519
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}
 
{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
 
[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 1\4, ~5103/4096 = 380.388
 
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}
 
[[Badness]]: 0.061813
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
 
Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}
 
Optimal tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387)
 
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448 }}
 
Badness: 0.021125
 
=== 13-limit ===
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]].''
 
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).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 165288374272/164794921875
 
{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}
 
: mapping generators: ~15/14, ~6/5
 
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577
 
{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}
 
[[Badness]]: 0.080637
 
Badness (Sintel): 2.041
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 391314/390625
 
Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}
 
Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.582
 
{{Optimal ET sequence|legend=1| 80, 190, 270, 1000, 1270, 1540e, 1810e }}
 
Badness: 0.024329
 
Badness (Sintel): 0.804
 
=== 13-limit ===
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 }}
 
Badness (Sintel): 0.556
 
== Keenanose ==
Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}
 
{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
 
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 21 3 1 -10 }} = 4.4465
 
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}
 
[[Badness]]: 0.0858
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 117649/117612, 67110351/67108864
 
Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4465
 
{{Optimal ET sequence|legend=1| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}
 
Badness: 0.0308
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612
 
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4466
 
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 4048 }}


Badness: 0.0213
Badness: 0.0213


==Ennealimnic==
== Aluminium ==
Commas: 243/242, 441/440, 4375/4356
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.


valid range: [44.444, 53.333] (27e to 45e)
[[Subgroup]]: 2.3.5


nice range: [48.920, 52.592]
[[Comma list]]: {{monzo| 92 -39 -13 }}


strict range: [48.920, 52.592]
[[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}


POTE generator: ~36/35 = 49.395
: mapping generators: ~135/128, ~3


Map: [&lt;9 1 1 12 -2|, &lt;0 2 3 2 5|]
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897


EDOs: 72, 171, 243
{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}


Badness: 0.0203
[[Badness]]: 0.123


===13-limit===
=== 7-limit ===
Commas: 243/242, 364/363, 441/440, 625/624
[[Subgroup]]: 2.3.5.7


valid range: [48.485, 50.000] (99ef to 72)
[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}


nice range: [48.825, 52.592]
[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}


strict range: [48.825, 50.000]
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 702.0024


POTE generator: ~36/35 = 49.341
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}


Map: [&lt;9 1 1 12 -2 -33|, &lt;0 2 3 2 5 10|]
[[Badness]]: 0.126


EDOs: 72, 171, 243
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0233
Comma list: 4375/4374, 234375/234256, 2097152/2096325


===17-limit===
Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}
Commas: 243/242, 364/363, 375/374, 441/440, 595/594


valid range: [48.485, 50.000] (99ef to 72)
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042


nice range: [46.363, 52.592]
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 3588e, 5135e }}


strict range: [48.485, 50.000]
Badness: 0.0421


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: 4096/4095, 4375/4374, 6656/6655, 78125/78078


EDOs: 72, 171, 243
Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}


Badness: 0.0146
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099


==Ennealim==
{{Optimal ET sequence|legend=1| 494, 1547, 2041, 4576def }}
Commas: 169/168, 243/242, 325/324, 441/440


POTE generator: ~36/35 = 49.708
Badness: 0.0286


Map: [&lt;9 1 1 12 -2 20|, &lt;0 2 3 2 5 2|]
== Countritonic ==
: ''For the 5-limit version of this temperament, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


EDOs: 27e, 45ef, 72, 315ff, 387cff, 459cdfff
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.


Badness: 0.0207
[[Subgroup]]: 2.3.5.7


==Ennealiminal==
[[Comma list]]: 4375/4374, 68719476736/68356598625
Commas: 385/384, 1375/1372, 4375/4374


POTE generator: ~36/35 = 49.504
{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }}


Map: [&lt;9 1 1 12 51|, &lt;0 2 3 2 -3|]
: mapping generators: ~2, ~45927/32768


EDOs: 27, 45, 72, 171e, 243e, 315e
[[Optimal tuning]] (CTE): ~2 = 1\1, ~45927/32768 = 588.6216


Badness: 0.0311
{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}


===13-limit===
[[Badness]]: 0.133
Commas: 169/168, 325/324, 385/384, 1375/1372


POTE generator: ~36/35 = 49.486
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;9 1 1 12 51 20|, &lt;0 2 3 2 -3 2|]
Comma list: 4375/4374, 5632/5625, 2621440/2614689


EDOs: 27, 45f, 72, 171ef, 243ef
Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 }}


Badness: 0.0303
Optimal tuning (CTE): ~2 = 1\1, ~539/384 = 588.6258


==Trinealimmal==
{{Optimal ET sequence|legend=1| 53, 316e, 369, 422, 791e, 1213cde }}
Commas: 2401/2400, 4375/4374, 2097152/2096325


POTE generator: ~6/5 = 315.644
Badness: 0.0707


Map: [&lt;27 1 0 34 177|, &lt;0 2 3 2 -4|]
=== 13-limit ===
Subgroup: 2.3.5.7.11


EDOs: 27, 243, 270, 783, 1053, 1323, 10854bcde
Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625


Badness: 0.0298
Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 -74 }}


==Semihemiennealimmal==
Optimal tuning (CTE): ~2 = 1\1, ~128/91 = 588.6277
Commas: 2401/2400, 4375/4374, 3025/3024, 4225/4224


POTE generator: ~39/32 = 342.139
{{Optimal ET sequence|legend=1| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}


Map: [&lt;18 0 -1 22 48 88|, &lt;0 4 6 4 2 -3|]
Badness: 0.0366


EDOs: 126, 144, 270, 684, 954
== Quatracot ==
{{See also| Stratosphere }}


Badness: 0.0131
[[Subgroup]]: 2.3.5.7


=Gamera=
[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
Commas: 4375/4374, 589824/588245


POTE generator ~8/7 = 230.336
{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }}


Map: [&lt;1 6 10 3|, &lt;0 -23 -40 -1|]
: mapping generators: ~2278125/1605632, ~448/405


EDOs: 26, 73, 99, 224, 323, 422, 735
[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 1\2, ~448/405 = 176.805


Badness: 0.0376
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}


==Hemigamera==
[[Badness]]: 0.175982
Commas: 3025/3024, 4375/4374, 202397184/201768035


POTE generator: ~8/7 = 230.337
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 12 20 6 5|, &lt;0 -23 -40 -1 5|]
Comma list: 3025/3024, 4375/4374, 1265625/1261568


EDOs: 26, 198, 224, 422, 646, 1068d
Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }}


Badness: 0.0410
Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 176.806


===13-limit===
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c }}
Commas: 1716/1715 2080/2079 2200/2197 3025/3024


Map: [&lt;2 12 20 6 5 17|, &lt;0 -23 -40 -1 5 -25|]
Badness: 0.041043


EDOs: 26, 198, 224, 422, 646f, 1068df
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0204
Comma list: 625/624, 729/728, 1575/1573, 2200/2197


=Supermajor=
Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }}
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 tuning (POTE): ~99/70 = 1\2, ~195/176 = 176.804


POTE generator: ~9/7 = 435.082
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1690bcc, 2328bccde }}


Map: [&lt;1 15 19 30|, &lt;0 -37 -46 -75|]
Badness: 0.022643


EDOs: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214
== 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.


Badness: 0.0108
[[Subgroup]]: 2.3.5.7


==Semisupermajor==
[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
Commas: 3025/3024, 4375/4374, 35156250/35153041


POTE generator: ~9/7 = 435.082
{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }}


Map: [&lt;2 30 38 60 41|, &lt;0 -37 -46 -75 -47|]
: mapping generators: ~2, ~6422528/3796875


EDOs: 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6422528/3796875 = 910.9323


Badness: 0.0128
{{Optimal ET sequence|legend=1| 494, 1125, 1619 }}


=Enneadecal=
[[Badness]]: 0.234
Enndedecal 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
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 701.880
Comma list: 4375/4374, 759375/758912, 100663296/100656875


Map: [&lt;19 0 14 -37|, &lt;0 1 1 3|]
Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }}


Generators: 28/27, 3
Optimal tuning (CTE): ~2 = 1\1, ~1024/605 = 910.9323


EDOs: 19, 152, 171, 665, 836, 1007, 2185
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}


Badness: 0.0110
Badness: 0.0678


==Hemienneadecal==
=== 13-limit ===
Commas: 3025/3024, 4375/4374, 234375/234256
Since 11/8 is within 23 generators, the 25 tone MOS (4L 21s) of this temperament contains the 8:11:13 triad.


POTE generator: ~3/2 = 701.881
Subgroup: 2.3.5.7.11.13


Map: [&lt;38 0 28 -74 11|, &lt;0 1 1 3 2|]
Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078


EDOs: 152, 342, 494, 836, 1178, 2014
Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }}


Badness: 0.00999
Optimal tuning (CTE): ~2 = 1\1, ~22/13 = 910.9323


===13-limit===
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}
Commas: 3025/3024, 4096/4095, 4375/4374, 31250/31213


POTE generator: ~3/2 = 701.986
Badness: 0.0271


Map: [&lt;38 0 28 -74 11 502|, &lt;0 1 1 3 2 -6|]
== Palladium ==
: ''For the 5-limit version of this temperament, see [[46th-octave temperaments]]''.


EDOs: 152, 342, 494, 836
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.


Badness: 0.0304
[[Subgroup]]: 2.3.5.7


=Deca=
[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}
Commas: 4375/4374, 165288374272/164794921875


POTE generator: ~460992/390625 = 284.423
{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}


Map: [&lt;10 4 2 9|, &lt;0 5 6 11|]
: mapping generators: ~83349/81920, ~3


EDOs: 80, 190, 270, 1270, 1540, 1810, 2080
[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 1\46, ~3/2 = 701.6074


Badness: 0.0806
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }}


==11-limit==
[[Badness]]: 0.308505
Commas: 3025/3024, 4375/4374, 422576/421875


POTE generator: ~33/28 = 284.418
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;10 4 2 9 18|, &lt;0 5 6 11 7|]
Comma list: 3025/3024, 4375/4374, 134775333/134217728


EDOs: 80, 190, 270, 1000, 1270
Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}


Badness: 0.0243
Optimal tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951


==13-limit==
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de }}
Commas: 1001/1000, 3025/3024, 4225/4224, 4375/4374


POTE generator: ~33/28 = 284.398
Badness: 0.073783


Map: [&lt;10 4 2 9 18 37|, &lt;0 5 6 11 7 0|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 80, 190, 270, 730, 1000
Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364


Badness: 0.0168
Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}


= Mitonic =
Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6419
{{see also|Minortonic family #Mitonic}}


Commas: 4375/4374, 2100875/2097152
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334de }}


POTE generator: ~10/9 = 182.458
Badness: 0.040751


Map: [&lt;1 16 32 -15|, &lt;0 -17 -35 21|]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


EDOs: 46, 125, 171
Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224


Badness: 0.0252
Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}


=Abigail=
Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425
Commas: 4375/4374, 2147483648/2144153025


[[POTE_tuning|POTE generator]]: 208.899
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334deg }}


Map: [&lt;2 7 13 -1|, &lt;0 -11 -24 19|]
Badness: 0.022441


Wedgie: &lt;&lt;22 48 -38 25 -122 -223||
== Oviminor ==
{{See also| Syntonic–kleismic equivalence continuum }}


EDOs: 46, 132, 178, 224, 270, 494, 764, 1034, 1798
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.


Badness: 0.0370
[[Subgroup]]: 2.3.5.7


==11-limit==
[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}
Comma: 3025/3024, 4375/4374, 20614528/20588575


[[POTE_tuning|POTE generator]]: 208.901
{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }}


Map: [&lt;2 7 13 -1 1|, &lt;0 -11 -24 19 17|]
: mapping generators: ~2, ~6/5


EDOs: 46, 132, 178, 224, 270, 494, 764
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 315.7501


Badness: 0.0129
{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}


==13-limit==
[[Badness]]: 0.582
Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095


[[POTE_tuning|POTE generator]]: 208.903
== Octoid ==
''For the 5-limit temperament, see [[8th-octave temperaments#Octoid (5-limit)]].''


Map: [&lt;2 7 13 -1 1 -2|, &lt;0 -11 -24 19 17 27|]
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.


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


Badness: 0.00886
[[Comma list]]: 4375/4374, 16875/16807


=Semidimi=
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
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: ~49/45, ~7/5


POTE generator: ~162/125 = 449.127
[[Optimal tuning]] ([[POTE]]): ~49/45 = 1\8, ~7/5 = 583.940


Map: [&lt;1 36 48|, &lt;0 -55 -73|]
[[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]


Wedgie: &lt;&lt;55 73 -12||
{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}


EDOs: 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419
[[Badness]]: 0.042670


Badness: 0.7549
Scales: [[octoid72]], [[octoid80]]


==7-limit==
=== 11-limit ===
Commas: 4375/4374, 3955078125/3954653486
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 generator: ~35/27 = 449.127
Subgroup: 2.3.5.7.11


Map: [&lt;1 36 48 61|, &lt;0 -55 -73 -93|]
Comma list: 540/539, 1375/1372, 4000/3993


Wedgie: &lt;&lt;55 73 93 -12 -7 11||
Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}


EDOs: 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.962


Badness: 0.0151
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]


=Brahmagupta=
{{Optimal ET sequence|legend=1| 72, 152, 224 }}
Commas: 4375/4374, 70368744177664/70338939985125


POTE generator: ~27/20 = 519.716
Badness: 0.014097


Map: [&lt;7 2 -8 53|, &lt;0 3 8 -11|]
Scales: [[octoid72]], [[octoid80]]


Wedgie: &lt;&lt;21 56 -77 40 -181 -336||
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 217, 224, 441, 1106, 1547
Comma list: 540/539, 625/624, 729/728, 1375/1372


Badness: 0.0291
Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}


==11-limit==
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.905
Commas: 4000/3993, 4375/4374, 131072/130977


POTE generator: ~27/20 = 519.704
{{Optimal ET sequence|legend=1| 72, 152f, 224 }}


Map: [&lt;7 2 -8 53 3|, &lt;0 3 8 -11 7|]
Badness: 0.015274


EDOs: 217, 224, 441, 665, 1771ee
Scales: [[octoid72]], [[octoid80]]


Badness: 0.0522
; 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


==13-limit==
===== 17-limit =====
Commas: 1575/1573, 2080/2079, 4096/4095, 4375/4374
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~27/20 = 519.706
Comma list: 375/374, 540/539, 625/624, 715/714, 729/728


Map: [&lt;7 2 -8 53 3 35|, &lt;0 3 8 -11 7 -3|]
Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}


EDOs: 217, 224, 441, 665, 1771eef
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.842


Badness: 0.0231
{{Optimal ET sequence|legend=1| 72, 152fg, 224, 296, 520g }}


=Quasithird=
Badness: 0.014304
Comma: |55 -64 20&gt;


POTE generator: ~1594323/1280000 = 380.395
Scales: [[octoid72]], [[octoid80]]


Map: [&lt;4 0 -11|, &lt;0 5 16|]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Wedgie: &lt;&lt;20 64 55||
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714


EDOs: 164, 224, 388, 612, 836, 1000, 1448, 1612, 2224, 2836
Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}


Badness: 0.0995
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.932


==7-limit==
{{Optimal ET sequence|legend=1| 72, 152fg, 224 }}
Commas: 4375/4374, 1153470752371588581/1152921504606846976


POTE generator: ~5103/4096 = 380.388
Badness: 0.016036


Map: [&lt;4 0 -11 48|, &lt;0 5 16 -29|]
Scales: [[octoid72]], [[octoid80]]


Wedgie: &lt;&lt;20 64 -116 55 -240 -449||
==== 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}}.


EDOs: 164, 224, 388, 612, 1448, 2060
Subgroup: 2.3.5.7.11.13


Badness: 0.0618
Comma list: 169/168, 325/324, 364/363, 540/539


==11-limit==
Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
Commas: 3025/3024, 4375/4374, 4296700485/4294967296


POTE generator: ~5103/4096 = 380.387
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.892


Map: [&lt;4 0 -11 48 43|, &lt;0 5 16 -29 -23|]
{{Optimal ET sequence|legend=1| 72, 152, 224f }}


EDOs: 164, 224, 388, 612, 836, 1448
Badness: 0.021679


Badness: 0.0211
Scales: [[octoid72]], [[octoid80]]


==13-limit==
===== 17-limit =====
Commas: 2200/2197, 3025/3024, 4375/4374, 468512/468195
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~5103/4096 = 380.385
Comma list: 169/168, 221/220, 289/288, 325/324, 540/539


Map: [&lt;4 0 -11 48 43 11|, &lt;0 5 16 -29 -23 3|]
Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}


EDOs: 164, 224, 388, 612, 836, 1448f, 2284f
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.811


Badness: 0.0295
{{Optimal ET sequence|legend=1| 72, 152, 224fg, 296ffg }}


=Semidimfourth=
Badness: 0.015614
Comma: |7 41 -31&gt;


POTE generator: ~162/125 = 448.449
Scales: [[Octoid72]], [[Octoid80]]


Map: [&lt;1 21 28|, &lt;0 -31 -41|]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Wedgie: &lt;&lt;31 41 -7||
Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399


EDOs: 91, 99, 190, 289, 388, 487, 677, 875, 966
Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}


Badness: 0.1930
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 584.064


==7-limit==
{{Optimal ET sequence|legend=1| 72, 152, 224fg, 376ffgh }}
Commas: 4375/4374, 235298/234375


POTE generator: ~35/27 = 448.457
Badness: 0.016321


Map: [&lt;1 21 28 36|, &lt;0 -31 -41 -53|]
Scales: [[Octoid72]], [[Octoid80]]


Wedgie: &lt;&lt;31 41 53 -7 -3 8||
==== Hexadecoid ====
{{ See also | 16th-octave temperaments }}


EDOs: 91, 99, 289, 388, 875, 1263d, 1651d
Hexadecoid (80 &amp; 144) has a period of 1/16 octave and tempers out 4225/4224.


Badness: 0.0552
Subgroup: 2.3.5.7.11.13


=Neusec=
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
Commas: 3025/3024, 4375/4374, 235298/234375


POTE generator: ~12/11 = 151.547
Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}


Map: [&lt;2 11 15 19 15|, &lt;0 -31 -41 -53 -32|]
: mapping generators: ~448/429, ~7/5


EDOs: 190, 388
Optimal tuning (POTE): ~448/429 = 1\16, ~13/8 = 841.015


Badness: 0.0591
{{Optimal ET sequence|legend=1| 80, 144, 224 }}


==13-limit==
Badness: 0.030818
Commas: 847/845, 1001/1000, 3025/3024, 4375/4374


POTE generator: ~12/11 = 151.545
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 11 15 19 15 17|, &lt;0 -31 -41 -53 -32 -38|]
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224


EDOs: 190, 198, 388
Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}


Badness: 0.0309
Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932


=Acrokleismic=
{{Optimal ET sequence|legend=1| 80, 144, 224, 528dg }}
Commas: 4375/4374, 2202927104/2197265625


POTE generator: ~6/5 = 315.557
Badness: 0.028611


Map: [&lt;1 10 11 27|, &lt;0 -32 -33 -92|]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Wedgie: &lt;&lt;32 33 92 -22 56 121||
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444


EDOs: 19, 251, 270
Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }}


Badness: 0.0562
Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896


==11-limit==
{{Optimal ET sequence|legend=1| 80, 144, 224, 304dh, 528dghh }}
Commas: 4375/4374, 41503/41472, 172032/171875


POTE generator: ~6/5 = 315.558
Badness: 0.023731


Map: [&lt;1 10 11 27 -16|, &lt;0 -32 -33 -92 74|]
== Parakleismic ==
{{Main| Parakleismic }}


EDOs: 19, 251, 270, 829, 1099, 1369, 1639
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.


Badness: 0.0369
[[Subgroup]]: 2.3.5


==13-limit==
[[Comma list]]: 1224440064/1220703125
Commas: 676/675, 1001/1000, 4375/4374, 10985/10976


POTE generator: ~6/5 = 315.557
{{Mapping|legend=1| 1 5 6 | 0 -13 -14 }}


Map: [&lt;1 10 11 27 -16 25|, &lt;0 -32 -33 -92 74 -81|]
: mapping generators: ~2, ~6/5


EDOs: 19, 251, 270
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.240


Badness: 0.0268
{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}


==Counteracro==
[[Badness]]: 0.043279
Commas: 4375/4374, 5632/5625, 117649/117612


POTE generator: ~6/5 = 315.553
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 10 11 27 55|, &lt;0 -32 -33 -92 -196|]
[[Comma list]]: 3136/3125, 4375/4374


EDOs: 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde
{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }}


Badness: 0.0426


===13-limit===
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.181
Commas: 676/675, 1716/1715, 4225/4224, 4375/4374


POTE generator: ~6/5 = 315.554
{{Optimal ET sequence|legend=1| 19, 80, 99, 217, 316, 415 }}


Map: [&lt;1 10 11 27 55 25|, &lt;0 -32 -33 -92 -196 -81|]
[[Badness]]: 0.027431


EDOs: 270, 1331c, 1601c, 1871bcf, 2141bcf
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0260
Comma list: 385/384, 3136/3125, 4375/4374


=Seniority=
Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }}
Commas: 4375/4374, 201768035/201326592


POTE generator: ~3087/2560 = 322.804
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.251


Map: [&lt;1 11 19 2|, &lt;0 -35 -62 3|]
{{Optimal ET sequence|legend=1| 19, 99, 118 }}


Wedgie: &lt;&lt;35 62 -3 17 -103 -181||
Badness: 0.049711


EDOs: 26, 145, 171, 2710d
=== 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.


Badness: 0.0449
Subgroup: 2.3.5.7.11


=Orga=
Comma list: 441/440, 3136/3125, 4375/4374
Commas: 4375/4374, 54975581388800/54936068900769


POTE generator: ~8/7 = 231.104
Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }}


Map: [&lt;2 21 36 5|, &lt;0 -29 -51 1|]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.220


Wedgie: &lt;&lt;58 102 -2 27 -166 -291||
{{Optimal ET sequence|legend=1| 19e, 99e, 118, 217, 335, 552d, 887dd }}


EDOs: 26, 244, 270, 836, 1106, 1376, 2482
Badness: 0.036027


Badness: 0.0402
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
Commas: 3025/3024, 4375/4374, 5767168/5764801


POTE generator: ~8/7 = 231.103
Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }}


Map: [&lt;2 21 36 5 2|, &lt;0 -29 -51 1 8|]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.214


EDOs: 26, 244, 270, 566, 836, 1106
{{Optimal ET sequence|legend=1| 99e, 118, 217, 552d, 769de }}


Badness: 0.0162
Badness: 0.044710


==13-limit==
==== Paraklein ====
Commas: 1716/1715, 2080/2079, 3025/3024, 15379/15360
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]].


POTE generator: ~8/7 = 231.103
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 21 36 5 2 24|, &lt;0 -29 -51 1 8 -27|]
Comma list: 196/195, 352/351, 625/624, 729/728


EDOs: 26, 244, 270, 566, 836f, 1106f
Mapping: {{mapping| 1 5 6 12 25 15 | 0 -13 -14 -35 -82 -43 }}


Badness: 0.0218
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.225


=Quatracot=
{{Optimal ET sequence|legend=1| 19e, 99ef, 118, 217ff, 335ff }}
Commas: 4375/4374, 1483154296875/1473173782528


POTE generator: ~448/405 = 176.805
Badness: 0.037618


Map: [&lt;2 7 7 23|, &lt;0 -13 -8 -59|]
=== Parkleismic ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;26 16 118 -35 114 229||
Comma list: 176/175, 1375/1372, 2200/2187


EDOs: 190, 224, 414, 638, 1052c, 1690bc
Mapping: {{mapping| 1 5 6 12 20 | 0 -13 -14 -35 -63 }}


Badness: 0.1760
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.060


==11-limit==
{{Optimal ET sequence|legend=1| 19e, 80, 179, 259cd }}
Commas: 3025/3024, 4375/4374, 1265625/1261568


POTE generator: ~448/405 = 176.806
Badness: 0.055884


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


EDOs: 190, 224, 414, 638, 1052c
Comma list: 169/168, 176/175, 325/324, 1375/1372


Badness: 0.0410
Mapping: {{mapping| 1 5 6 12 20 10 | 0 -13 -14 -35 -63 -24 }}


==13-limit==
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.075
Commas: 625/624, 729/728, 1575/1573, 2200/2197


POTE generator: ~448/405 = 176.804
{{Optimal ET sequence|legend=1| 19e, 80, 179 }}


Map: [&lt;2 7 7 23 19 13|, &lt;0 -13 -8 -59 -41 -19|]
Badness: 0.036559


EDOs: 190, 224, 414, 638, 1690bc, 2328bcde
=== Paradigmic ===
Subgroup: 2.3.5.7.11


Badness: 0.0226
Comma list: 540/539, 896/891, 3136/3125


=Octoid=
Mapping: {{mapping| 1 5 6 12 -1 | 0 -13 -14 -35 17 }}
Commas: 4375/4374, 16875/16807


valid range: [578.571, 600.000] (56bcd to 8d)
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.096


nice range: [582.512, 584.359]
{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}


strict range: [582.512, 584.359]
Badness: 0.041720


POTE generator: ~7/5 = 583.940
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;8 1 3 3|, &lt;0 3 4 5|]
Comma list: 169/168, 325/324, 540/539, 832/825


Generators: 49/45, 7/5
Mapping: {{mapping| 1 5 6 12 -1 10 | 0 -13 -14 -35 17 -24 }}


EDOs: 72, 152, 224
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.080


Badness: 0.0427
{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}


==11-limit==
Badness: 0.035781
Commas: 540/539, 1375/1372, 4000/3993


valid range: [581.250, 586.364] (64cd, 88bcde)
=== Semiparakleismic ===
Subgroup: 2.3.5.7.11


nice range: [582.512, 585.084]
Comma list: 3025/3024, 3136/3125, 4375/4374


strict range: [582.512, 585.084]
Mapping: {{mapping| 2 10 12 24 19 | 0 -13 -14 -35 -23 }}


POTE generator: ~7/5 = 583.692
Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.181


Map: [&lt;8 1 3 3 16|, &lt;0 3 4 5 3|]
{{Optimal ET sequence|legend=1| 80, 118, 198, 316, 514c, 830c }}


EDOs: 72, 152, 224
Badness: 0.034208


Badness: 0.0141
==== Semiparamint ====
This extension was named ''semiparakleismic'' in the earlier materials.  


==13-limit==
Subgroup: 2.3.5.7.11.13
Commas: 540/539, 1375/1372, 4000/3993, 625/624


POTE generator: ~7/5 = 583.905
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374


Map: [&lt;8 1 3 3 16 -21|, &lt;0 3 4 5 3 13|]
Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }}


EDOs: 72, 224
Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.156


Badness: 0.0153
{{Optimal ET sequence|legend=1| 80, 118, 198 }}


==Music==
Badness: 0.033775
[http://www.archive.org/details/Dreyfus http://www.archive.org/details/Dreyfus]


[http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play]
==== Semiparawolf ====
This extension was named ''gentsemiparakleismic'' in the earlier materials.  


==Octopus==
Subgroup: 2.3.5.7.11.13
Commas: 169/168, 325/324, 364/363, 540/539


POTE generator: ~7/5 = 583.892
Comma list: 169/168, 325/324, 364/363, 3136/3125


Map: [&lt;8 1 3 3 16 14|, &lt;0 3 4 5 3 4|]
Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }}


EDOs: 72, 152, 224f
Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 315.184


Badness: 0.0217
{{Optimal ET sequence|legend=1| 80, 118f, 198f }}


= Amity =
Badness: 0.040467
{{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.
== Counterkleismic ==
{{See also| High badness temperaments #Counterhanson}}


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.
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).


Comma: 1600000/1594323
[[Subgroup]]: 2.3.5.7


POTE generator: ~243/200 = 339.519
[[Comma list]]: 4375/4374, 158203125/157351936


Map: [&lt;1 3 6|, &lt;0 -5 -13|]
{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }}


EDOs: 7, 39, 46, 53, 152, 205, 463, 668, 873
: mapping generators: ~2, ~5/3


Badness: 0.0220
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.060


==7-limit==
{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }}
Commas: 4375/4374, 5120/5103


POTE generator: ~128/105 = 339.432
[[Badness]]: 0.090553


Map: [&lt;1 3 6 -2|, &lt;0 -5 -13 17|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;5 13 -17 9 -41 -76||
Comma list: 540/539, 4375/4374, 2097152/2096325


EDOs: 7, 39, 46, 53, 99, 251, 350
Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }}


Badness: 0.0236
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.071


==11-limit==
{{Optimal ET sequence|legend=1| 19, 205, 224 }}
Commas: 540/539, 4375/4374, 5120/5103


POTE generator: ~128/105 = 339.464
Badness: 0.070952


Map: [&lt;1 3 6 -2 21|, &lt;0 -5 -13 17 -62|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 53, 99e, 152, 555de, 707de, 859bde
Comma list: 540/539, 625/624, 729/728, 10985/10976


Badness: 0.0315
Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }}


==13-limit==
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.070
Commas: 352/351, 540/539, 625/624, 847/845


POTE generator: ~128/105 = 339.481
{{Optimal ET sequence|legend=1| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}


Map: [&lt;1 3 6 -2 21 17|, &lt;0 -5 -13 17 -62 -47|]
Badness: 0.033874


EDOS: 53, 99ef, 152f, 205
=== Counterlytic ===
Subgroup: 2.3.5.7.11


Badness: 0.0280
Comma list: 1375/1372, 4375/4374, 496125/495616


==Hitchcock==
Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }}
Commas: 121/120, 176/175, 2200/2187


POTE generator: ~11/9 = 339.340
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065


Map: [&lt;1 3 6 -2 6|, &lt;0 -5 -13 17 -9|]
{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}


EDOs: 7, 39, 46, 53, 99
Badness: 0.065400


Badness: 0.0352
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


===13-limit===
Comma list: 625/624, 729/728, 1375/1372, 10985/10976
Commas: 121/120, 169/168, 176/175, 325/324


POTE generator: ~11/9 = 339.419
Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }}


Map: [&lt;1 3 6 -2 6 2|, &lt;0 -5 -13 17 -9 6|]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065


EDOs: 7, 39, 46, 53, 99
{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}


Badness: 0.0224
Badness: 0.029782


==Hemiamity==
== Quincy ==
Commas: 3025/3024, 4375/4374, 5120/5103
[[Subgroup]]: 2.3.5.7


POTE generator: ~64/55 = 339.493
[[Comma list]]: 4375/4374, 823543/819200


Map: [&lt;2 1 -1 13 13|, &lt;0 5 13 -17 -14|]
{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}


EDOs: 14cde, 46, 106, 152, 198, 350
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1715 = 16.613


Badness: 0.0313
{{Optimal ET sequence|legend=1| 72, 217, 289 }}


=Parakleismic=
[[Badness]]: 0.079657
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
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~6/5 = 315.240
Comma list: 441/440, 4000/3993, 4375/4374


Map: [&lt;1 5 6|, &lt;0 -13 -14|]
Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}


EDOs: 19, 61, 80, 99, 118, 453, 571, 689, 1496
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.613


Badness: 0.0433
{{Optimal ET sequence|legend=1| 72, 217, 289 }}


==7-limit==
Badness: 0.030875
Commas: 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.181
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 5 6 12|, &lt;0 -13 -14 -35|]
Comma list: 364/363, 441/440, 676/675, 4375/4374


EDOs: 19, 80, 99, 217, 316, 415
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}


Badness: 0.0274
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602


==11-limit==
{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}
Commas: 385/384, 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.251
Badness: 0.023862


Map: [&lt;1 5 6 12 -6|, &lt;0 -13 -14 -35 36|]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


EDOs: 19, 99, 118
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155


Badness: 0.0497
Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}


==Parkleismic==
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602
Commas: 176/175, 1375/1372, 2200/2187


POTE generator: ~6/5 = 315.060
{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}


Map: [&lt;1 5 6 12 20|, &lt;0 -13 -14 -35 -63|]
Badness: 0.014741


EDOs: 80, 179, 259cd
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0559
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675


===13-limit===
Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}
Commas: 169/168, 176/175, 325/324, 1375/1372


POTE generator: ~6/5 = 315.075
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.594


Map: [&lt;1 5 6 12 20 10|, &lt;0 -13 -14 -35 -63 -24|]
{{Optimal ET sequence|legend=1| 72, 145, 217 }}


EDOs: 15, 19, 80, 179
Badness: 0.015197


Badness: 0.0366
== Sfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''
 
[[Subgroup]]: 2.3.5.7


==Paradigmic==
[[Comma list]]: 4375/4374, 64827/64000
Commas: 540/539, 896/891, 3136/3125


POTE generator: ~6/5 = 315.096
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}


Map: [&lt;1 5 6 12 -1|, &lt;0 -13 -14 -35 17|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 26.287


EDOs: 19, 80, 99e, 179e
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


Badness: 0.0417
[[Badness]]: 0.123291


===13-limit===
=== 11-limit ===
Commas: 169/168, 325/324, 540/539, 832/825
Subgroup: 2.3.5.7.11


POTE generator: ~6/5 = 315.080
Comma list: 121/120, 441/440, 4375/4374


Map: [&lt;1 5 6 12 -1 10|, &lt;0 -13 -14 -35 17 -24|]
Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}


EDOs: 19, 80, 99e, 179e
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.286


Badness: 0.0358
{{Optimal ET sequence|legend=1| 45e, 46, 91e, 137de }}


=Semiparakleismic=
Badness: 0.054098
Commas: 3025/3024, 3136/3125, 4375/4374


POTE generator: 315.181
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 10 12 24 19|, &lt;0 -13 -14 -35 -23|]
Comma list: 121/120, 169/168, 325/324, 441/440


EDOs: 80, 118, 198, 316, 514c, 830c
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}


Badness: 0.0342
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.310


=Quincy=
{{Optimal ET sequence|legend=1| 45ef, 46, 91ef, 137def }}
Commas: 4375/4374, 823543/819200


POTE generator: ~1728/1715 = 16.613
Badness: 0.033067


Map: [&lt;1 2 2 3|, &lt;0 -30 -49 -14|]
=== Sfour ===
Subgroup: 2.3.5.7.11


EDOs: 72, 217, 289
Comma list: 385/384, 2401/2376, 4375/4374


Badness: 0.0797
Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}


==11-limit==
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.246
Commas: 441/440, 4000/3993, 41503/41472


POTE generator: ~100/99 = 16.613
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


Map: [&lt;1 2 2 3 4|, &lt;0 -30 -49 -14 -39|]
Badness: 0.076567


EDOs: 72, 217, 289
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0309
Comma list: 196/195, 364/363, 385/384, 4375/4374


==13-limit==
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}
Commas: 364/363, 441/440, 676/675, 4375/4374


POTE generator: ~100/99 = 16.602
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.239


Map: [&lt;1 2 2 3 4 5|, &lt;0 -30 -49 -14 -39 -94|]
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


EDOs: 72, 145, 217, 289
Badness: 0.051893


Badness: 0.0239
== Trideci ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tridecatonic]].''


==17-limit==
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]]").
Commas: 364/363, 441/440, 595/594, 1001/1000, 1156/1155


POTE generator: ~100/99 = 16.602
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 2 3 4 5 5|, &lt;0 -30 -49 -14 -39 -94 -66|]
[[Comma list]]: 4375/4374, 83349/81920


EDOs: 72, 145, 217, 289
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}


Badness: 0.0147
[[Optimal tuning]] ([[POTE]]): ~256/245 = 1\13, ~3/2 = 699.1410


==19-limit==
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }}
Commas: 343/342, 364/363, 441/440, 595/594, 676/675, 2601/2600


POTE generator: ~100/99 = 16.594
[[Badness]]: 0.184585


Map: [&lt;1 2 2 3 4 5 5 4|, &lt;0 -30 -49 -14 -39 -94 -66 18|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 72, 145, 217
Comma list: 245/242, 385/384, 4375/4374


Badness: 0.0152
Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}


=Chlorine=
Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.6179
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 ET sequence|legend=1| 26, 65, 91, 156d, 247cdde }}


Comma: |-52 -17 34&gt;
Badness: 0.084590


POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2687
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;17 26 39|, &lt;0 2 1|]
Comma list: 169/168, 245/242, 325/324, 385/384


EDOs: 34, 289, 323, 612, 901
Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}


Badness: 0.0771
Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.2969


==7-limit==
{{Optimal ET sequence|legend=1| 26, 65f, 91f, 156dff }}
Commas: 4375/4374, 193119049072265625/193091834023510016


POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2936
Badness: 0.052366


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


EDOs: 34d, 289, 323, 612, 935, 1547
Subgroup: 2.3.5.7


Badness: 0.0417
Comma list: 4375/4374, {{monzo| 154 -54 -21 -7 }}


==11-limit==
Mapping: {{mapping| 1 0 -21 85 | 0 7 103 -363 }}
Commas: 4375/4374, 41503/41472, 1879453125/1879048192


POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2690
Optimal tuning (CTE): ~2 = 1\1, ~{{monzo| 66 -23 -9 -3 }} = 271.7113


Map: [&lt;17 26 39 43 64|, &lt;0 2 1 10 -11|]
{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}


EDOs: 34de, 289, 323, 612, 901
Badness: 0.312806


Badness: 0.0637
== 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]]
Line 1,011: Line 1,721:
[[Category:Octoid]]
[[Category:Octoid]]
[[Category:Parakleismic]]
[[Category:Parakleismic]]
[[Category:Quincy]]
[[Category:Supermajor]]
[[Category:Supermajor]]
[[Category:Microtemperament]]
[[Category:Ragismic]]
Retrieved from "https://en.xen.wiki/w/Ragismic_microtemperaments"