Ragismic microtemperaments
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 microtemperaments 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.
Temperaments not discussed here include crepuscular, flattone, hystrix, sensi, unidec, quartonic, catakleismic, modus, pontiac, whirrschmidt, zarvo, vishnu, and vulture.
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>, 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 <<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.
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.
valid range: [26.667, 66.667] (45bcd to 18bcd)
nice range: [48.920, 49.179]
strict range: [48.920, 49.179]
Commas: 2401/2400, 4375/4374
POTE generators: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980
Map: [<9 1 1 2|, <0 2 3 2|]
Wedgie: <<18 27 18 1 -22 -34||
EDOs: 27, 45, 72, 99, 171, 270, 441, 612, 3600
Badness: 0.00361
Hemiennealimmal
Commas: 2401/2400, 4375/4374, 3025/3024
valid range: [13.333, 22.222] (90bcd, 54c)
nice range: [17.304, 17.985]
strict range: [17.304, 17.985]
POTE generator: ~99/98 = 17.6219
Map: [<18 0 -1 22 48|, <0 2 3 2 1|]
EDOs: 72, 198, 270, 342, 612, 954, 1566
Badness: 0.00628
13-limit
Commas: 676/675, 1001/1000, 1716/1715, 3025/3024
valid range: [16.667, 22.222] (72 to 54cf)
nice range: [17.304, 18.309]
strict range: [17.304, 18.309]
POTE generator ~99/98 = 17.7504
Map: [<18 0 -1 22 48 -19|, <0 2 3 2 1 6|]
EDOs: 72, 198, 270
Badness: 0.0125
Semihemiennealimmal
Commas: 2401/2400, 4375/4374, 3025/3024, 4225/4224
POTE generator: ~39/32 = 342.139
Map: [<18 0 -1 22 48 88|, <0 4 6 4 2 -3|]
EDOs: 126, 144, 270, 684, 954
Badness: 0.0131
Semiennealimmal
Commas: 2401/2400, 4375/4374, 4000/3993
POTE generator: ~140/121 = 250.3367
Map: [<9 3 4 14 18|, <0 6 9 6 7|]
EDOs: 72, 369, 441
Badness: 0.0342
13-limit
Commas: 1575/1573, 2080/2079, 2401/2400, 4375/4374
POTE generator: ~140/121 = 250.3375
Map: [<9 3 4 14 18 -8|, <0 6 9 6 7 22|]
EDOs: 72, 441
Badness: 0.0261
Quadraennealimmal
Commas: 2401/2400, 4375/4374, 234375/234256
POTE generator: ~77/75 = 45.595
Map: [<9 1 1 12 -7|, <0 8 12 8 23|]
EDOs: 342, 1053, 1395, 1737, 4869d, 6606cd
Badness: 0.0213
Ennealimnic
Commas: 243/242, 441/440, 4375/4356
valid range: [44.444, 53.333] (27e to 45e)
nice range: [48.920, 52.592]
strict range: [48.920, 52.592]
POTE generator: ~36/35 = 49.395
Map: [<9 1 1 12 -2|, <0 2 3 2 5|]
EDOs: 72, 171, 243
Badness: 0.0203
13-limit
Commas: 243/242, 364/363, 441/440, 625/624
valid range: [48.485, 50.000] (99ef to 72)
nice range: [48.825, 52.592]
strict range: [48.825, 50.000]
POTE generator: ~36/35 = 49.341
Map: [<9 1 1 12 -2 -33|, <0 2 3 2 5 10|]
EDOs: 72, 171, 243
Badness: 0.0233
17-limit
Commas: 243/242, 364/363, 375/374, 441/440, 595/594
valid range: [48.485, 50.000] (99ef to 72)
nice range: [46.363, 52.592]
strict range: [48.485, 50.000]
POTE generator: ~36/35 = 49.335
Map: [<9 1 1 12 -2 -33 -3|, <0 2 3 2 5 10 6|]
EDOs: 72, 171, 243
Badness: 0.0146
Ennealim
Commas: 169/168, 243/242, 325/324, 441/440
POTE generator: ~36/35 = 49.708
Map: [<9 1 1 12 -2 20|, <0 2 3 2 5 2|]
EDOs: 27e, 45ef, 72, 315ff, 387cff, 459cdfff
Badness: 0.0207
Ennealiminal
Commas: 385/384, 1375/1372, 4375/4374
POTE generator: ~36/35 = 49.504
Map: [<9 1 1 12 51|, <0 2 3 2 -3|]
EDOs: 27, 45, 72, 171e, 243e, 315e
Badness: 0.0311
13-limit
Commas: 169/168, 325/324, 385/384, 1375/1372
POTE generator: ~36/35 = 49.486
Map: [<9 1 1 12 51 20|, <0 2 3 2 -3 2|]
EDOs: 27, 45f, 72, 171ef, 243ef
Badness: 0.0303
Trinealimmal
Commas: 2401/2400, 4375/4374, 2097152/2096325
POTE generator: ~6/5 = 315.644
Map: [<27 1 0 34 177|, <0 2 3 2 -4|]
EDOs: 27, 243, 270, 783, 1053, 1323, 10854bcde
Badness: 0.0298
Gamera
Commas: 4375/4374, 589824/588245
POTE generator ~8/7 = 230.336
Map: [<1 6 10 3|, <0 -23 -40 -1|]
EDOs: 26, 73, 99, 224, 323, 422, 745d
Badness: 0.0376
Hemigamera
Commas: 3025/3024, 4375/4374, 589824/588245
POTE generator: ~8/7 = 230.337
Map: [<2 12 20 6 5|, <0 -23 -40 -1 5|]
EDOs: 26, 198, 224, 422, 646, 1068d
Badness: 0.0410
13-limit
Commas: 1716/1715, 2080/2079, 2200/2197, 3025/3024
Map: [<2 12 20 6 5 17|, <0 -23 -40 -1 5 -25|]
EDOs: 26, 198, 224, 422, 646f, 1068df
Badness: 0.0204
Supermajor
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 <<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
POTE generator: ~9/7 = 435.082
Map: [<1 15 19 30|, <0 -37 -46 -75|]
EDOs: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214
Badness: 0.0108
Semisupermajor
Commas: 3025/3024, 4375/4374, 35156250/35153041
POTE generator: ~9/7 = 435.082
Map: [<2 30 38 60 41|, <0 -37 -46 -75 -47|]
EDOs: 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf
Badness: 0.0128
Enneadecal
Enndedecal temperament tempers out the enneadeca, |-14 -19 19>, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of 19edo up to just ones. 171edo is a good tuning for either the 5 or 7 limits, and 494edo shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use 665edo for a tuning.
Commas: 4375/4374, 703125/702464
POTE generator: ~3/2 = 701.880
Map: [<19 0 14 -37|, <0 1 1 3|]
Generators: 28/27, 3
EDOs: 19, 152, 171, 665, 836, 1007, 2185
Badness: 0.0110
Hemienneadecal
Commas: 3025/3024, 4375/4374, 234375/234256
POTE generator: ~3/2 = 701.881
Map: [<38 0 28 -74 11|, <0 1 1 3 2|]
EDOs: 152, 342, 494, 836, 1178, 2014
Badness: 0.00999
13-limit
Commas: 3025/3024, 4096/4095, 4375/4374, 31250/31213
POTE generator: ~3/2 = 701.986
Map: [<38 0 28 -74 11 502|, <0 1 1 3 2 -6|]
EDOs: 152, 342, 494, 836
Badness: 0.0304
Deca
Commas: 4375/4374, 165288374272/164794921875
POTE generator: ~460992/390625 = 284.423
Map: [<10 4 2 9|, <0 5 6 11|]
EDOs: 80, 190, 270, 1270, 1540, 1810, 2080
Badness: 0.0806
11-limit
Commas: 3025/3024, 4375/4374, 422576/421875
POTE generator: ~33/28 = 284.418
Map: [<10 4 2 9 18|, <0 5 6 11 7|]
EDOs: 80, 190, 270, 1000, 1270
Badness: 0.0243
13-limit
Commas: 1001/1000, 3025/3024, 4225/4224, 4375/4374
POTE generator: ~33/28 = 284.398
Map: [<10 4 2 9 18 37|, <0 5 6 11 7 0|]
EDOs: 80, 190, 270, 730, 1000
Badness: 0.0168
Mitonic
- See also: Minortonic family #Mitonic
Commas: 4375/4374, 2100875/2097152
POTE generator: ~10/9 = 182.458
Map: [<1 16 32 -15|, <0 -17 -35 21|]
EDOs: 46, 125, 171
Badness: 0.0252
Abigail
Commas: 4375/4374, 2147483648/2144153025
POTE generator: 208.899
Map: [<2 7 13 -1|, <0 -11 -24 19|]
Wedgie: <<22 48 -38 25 -122 -223||
EDOs: 46, 132, 178, 224, 270, 494, 764, 1034, 1798
Badness: 0.0370
11-limit
Comma: 3025/3024, 4375/4374, 20614528/20588575
POTE generator: 208.901
Map: [<2 7 13 -1 1|, <0 -11 -24 19 17|]
EDOs: 46, 132, 178, 224, 270, 494, 764
Badness: 0.0129
13-limit
Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095
POTE generator: 208.903
Map: [<2 7 13 -1 1 -2|, <0 -11 -24 19 17 27|]
EDOs: 46, 178, 224, 270, 494, 764, 1258
Badness: 0.00886
Semidimi
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> and 7-limit 3955078125/3954653486, as well as 4375/4374.
Comma: |-12 -73 55>
POTE generator: ~162/125 = 449.127
Map: [<1 36 48|, <0 -55 -73|]
Wedgie: <<55 73 -12||
EDOs: 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419
Badness: 0.7549
7-limit
Commas: 4375/4374, 3955078125/3954653486
POTE generator: ~35/27 = 449.127
Map: [<1 36 48 61|, <0 -55 -73 -93|]
Wedgie: <<55 73 93 -12 -7 11||
EDOs: 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419
Badness: 0.0151
Brahmagupta
Commas: 4375/4374, 70368744177664/70338939985125
POTE generator: ~27/20 = 519.716
Map: [<7 2 -8 53|, <0 3 8 -11|]
Wedgie: <<21 56 -77 40 -181 -336||
EDOs: 217, 224, 441, 1106, 1547
Badness: 0.0291
11-limit
Commas: 4000/3993, 4375/4374, 131072/130977
POTE generator: ~27/20 = 519.704
Map: [<7 2 -8 53 3|, <0 3 8 -11 7|]
EDOs: 217, 224, 441, 665, 1771ee
Badness: 0.0522
13-limit
Commas: 1575/1573, 2080/2079, 4096/4095, 4375/4374
POTE generator: ~27/20 = 519.706
Map: [<7 2 -8 53 3 35|, <0 3 8 -11 7 -3|]
EDOs: 217, 224, 441, 665, 1771eef
Badness: 0.0231
Quasithird
Comma: |55 -64 20>
POTE generator: ~1594323/1280000 = 380.395
Map: [<4 0 -11|, <0 5 16|]
Wedgie: <<20 64 55||
EDOs: 164, 224, 388, 612, 836, 1000, 1448, 1612, 2224, 2836
Badness: 0.0995
7-limit
Commas: 4375/4374, 1153470752371588581/1152921504606846976
POTE generator: ~5103/4096 = 380.388
Map: [<4 0 -11 48|, <0 5 16 -29|]
Wedgie: <<20 64 -116 55 -240 -449||
EDOs: 164, 224, 388, 612, 1448, 2060
Badness: 0.0618
11-limit
Commas: 3025/3024, 4375/4374, 4296700485/4294967296
POTE generator: ~5103/4096 = 380.387
Map: [<4 0 -11 48 43|, <0 5 16 -29 -23|]
EDOs: 164, 224, 388, 612, 836, 1448
Badness: 0.0211
13-limit
Commas: 2200/2197, 3025/3024, 4375/4374, 468512/468195
POTE generator: ~5103/4096 = 380.385
Map: [<4 0 -11 48 43 11|, <0 5 16 -29 -23 3|]
EDOs: 164, 224, 388, 612, 836, 1448f, 2284f
Badness: 0.0295
Semidimfourth
Comma: |7 41 -31>
POTE generator: ~162/125 = 448.449
Map: [<1 21 28|, <0 -31 -41|]
Wedgie: <<31 41 -7||
EDOs: 91, 99, 190, 289, 388, 487, 677, 875, 966
Badness: 0.1930
7-limit
Commas: 4375/4374, 235298/234375
POTE generator: ~35/27 = 448.457
Map: [<1 21 28 36|, <0 -31 -41 -53|]
Wedgie: <<31 41 53 -7 -3 8||
EDOs: 91, 99, 289, 388, 875, 1263d, 1651d
Badness: 0.0552
Neusec
Commas: 3025/3024, 4375/4374, 235298/234375
POTE generator: ~12/11 = 151.547
Map: [<2 11 15 19 15|, <0 -31 -41 -53 -32|]
EDOs: 190, 388
Badness: 0.0591
13-limit
Commas: 847/845, 1001/1000, 3025/3024, 4375/4374
POTE generator: ~12/11 = 151.545
Map: [<2 11 15 19 15 17|, <0 -31 -41 -53 -32 -38|]
EDOs: 190, 198, 388
Badness: 0.0309
Acrokleismic
Commas: 4375/4374, 2202927104/2197265625
POTE generator: ~6/5 = 315.557
Map: [<1 10 11 27|, <0 -32 -33 -92|]
Wedgie: <<32 33 92 -22 56 121||
EDOs: 19, 251, 270
Badness: 0.0562
11-limit
Commas: 4375/4374, 41503/41472, 172032/171875
POTE generator: ~6/5 = 315.558
Map: [<1 10 11 27 -16|, <0 -32 -33 -92 74|]
EDOs: 19, 251, 270, 829, 1099, 1369, 1639
Badness: 0.0369
13-limit
Commas: 676/675, 1001/1000, 4375/4374, 10985/10976
POTE generator: ~6/5 = 315.557
Map: [<1 10 11 27 -16 25|, <0 -32 -33 -92 74 -81|]
EDOs: 19, 251, 270
Badness: 0.0268
Counteracro
Commas: 4375/4374, 5632/5625, 117649/117612
POTE generator: ~6/5 = 315.553
Map: [<1 10 11 27 55|, <0 -32 -33 -92 -196|]
EDOs: 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde
Badness: 0.0426
13-limit
Commas: 676/675, 1716/1715, 4225/4224, 4375/4374
POTE generator: ~6/5 = 315.554
Map: [<1 10 11 27 55 25|, <0 -32 -33 -92 -196 -81|]
EDOs: 270, 1331c, 1601c, 1871bcf, 2141bcf
Badness: 0.0260
Seniority
Commas: 4375/4374, 201768035/201326592
POTE generator: ~3087/2560 = 322.804
Map: [<1 11 19 2|, <0 -35 -62 3|]
Wedgie: <<35 62 -3 17 -103 -181||
EDOs: 26, 145, 171, 2710d
Badness: 0.0449
Orga
Commas: 4375/4374, 54975581388800/54936068900769
POTE generator: ~8/7 = 231.104
Map: [<2 21 36 5|, <0 -29 -51 1|]
Wedgie: <<58 102 -2 27 -166 -291||
EDOs: 26, 244, 270, 836, 1106, 1376, 2482
Badness: 0.0402
11-limit
Commas: 3025/3024, 4375/4374, 5767168/5764801
POTE generator: ~8/7 = 231.103
Map: [<2 21 36 5 2|, <0 -29 -51 1 8|]
EDOs: 26, 244, 270, 566, 836, 1106
Badness: 0.0162
13-limit
Commas: 1716/1715, 2080/2079, 3025/3024, 15379/15360
POTE generator: ~8/7 = 231.103
Map: [<2 21 36 5 2 24|, <0 -29 -51 1 8 -27|]
EDOs: 26, 244, 270, 566, 836f, 1106f
Badness: 0.0218
Quatracot
Commas: 4375/4374, 1483154296875/1473173782528
POTE generator: ~448/405 = 176.805
Map: [<2 7 7 23|, <0 -13 -8 -59|]
Wedgie: <<26 16 118 -35 114 229||
EDOs: 190, 224, 414, 638, 1052c, 1690bc
Badness: 0.1760
11-limit
Commas: 3025/3024, 4375/4374, 1265625/1261568
POTE generator: ~448/405 = 176.806
Map: [<2 7 7 23 19|, <0 -13 -8 -59 -41|]
EDOs: 190, 224, 414, 638, 1052c
Badness: 0.0410
13-limit
Commas: 625/624, 729/728, 1575/1573, 2200/2197
POTE generator: ~448/405 = 176.804
Map: [<2 7 7 23 19 13|, <0 -13 -8 -59 -41 -19|]
EDOs: 190, 224, 414, 638, 1690bc, 2328bcde
Badness: 0.0226
Octoid
Commas: 4375/4374, 16875/16807
valid range: [578.571, 600.000] (56bcd to 8d)
nice range: [582.512, 584.359]
strict range: [582.512, 584.359]
POTE generator: ~7/5 = 583.940
Map: [<8 1 3 3|, <0 3 4 5|]
Generators: 49/45, 7/5
EDOs: 72, 152, 224
Badness: 0.0427
11-limit
Commas: 540/539, 1375/1372, 4000/3993
valid range: [581.250, 586.364] (64cd, 88bcde)
nice range: [582.512, 585.084]
strict range: [582.512, 585.084]
POTE generator: ~7/5 = 583.692
Map: [<8 1 3 3 16|, <0 3 4 5 3|]
EDOs: 72, 152, 224
Badness: 0.0141
13-limit
Commas: 540/539, 1375/1372, 4000/3993, 625/624
POTE generator: ~7/5 = 583.905
Map: [<8 1 3 3 16 -21|, <0 3 4 5 3 13|]
EDOs: 72, 224
Badness: 0.0153
Music
Octopus
Commas: 169/168, 325/324, 364/363, 540/539
POTE generator: ~7/5 = 583.892
Map: [<8 1 3 3 16 14|, <0 3 4 5 3 4|]
EDOs: 72, 152, 224f
Badness: 0.0217
Amity
- Main article: 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&53 temperament, or by its wedgie, <<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.
In the 5-limit amity is a genuine microtemperament, with 58/205 being a possible tuning. Another good choice is (64/5)^(1/13), which gives pure major thirds.
Comma: 1600000/1594323
POTE generator: ~243/200 = 339.519
Map: [<1 3 6|, <0 -5 -13|]
EDOs: 7, 39, 46, 53, 152, 205, 463, 668, 873
Badness: 0.0220
7-limit
Commas: 4375/4374, 5120/5103
POTE generator: ~128/105 = 339.432
Map: [<1 3 6 -2|, <0 -5 -13 17|]
Wedgie: <<5 13 -17 9 -41 -76||
EDOs: 7, 39, 46, 53, 99, 251, 350
Badness: 0.0236
11-limit
Commas: 540/539, 4375/4374, 5120/5103
POTE generator: ~128/105 = 339.464
Map: [<1 3 6 -2 21|, <0 -5 -13 17 -62|]
EDOs: 53, 99e, 152, 555dee, 707ddee, 859bddee
Badness: 0.0315
13-limit
Commas: 352/351, 540/539, 625/624, 847/845
POTE generator: ~128/105 = 339.481
Map: [<1 3 6 -2 21 17|, <0 -5 -13 17 -62 -47|]
EDOS: 53, 99ef, 152f, 205
Badness: 0.0280
Hitchcock
- See also: Amity family #Hitchcock
Commas: 121/120, 176/175, 2200/2187
POTE generator: ~11/9 = 339.340
Map: [<1 3 6 -2 6|, <0 -5 -13 17 -9|]
EDOs: 7, 39, 46, 53, 99
Badness: 0.0352
13-limit
Commas: 121/120, 169/168, 176/175, 325/324
POTE generator: ~11/9 = 339.419
Map: [<1 3 6 -2 6 2|, <0 -5 -13 17 -9 6|]
EDOs: 7, 39, 46, 53, 99
Badness: 0.0224
Hemiamity
Commas: 3025/3024, 4375/4374, 5120/5103
POTE generator: ~64/55 = 339.493
Map: [<2 1 -1 13 13|, <0 5 13 -17 -14|]
EDOs: 14cde, 46, 106, 152, 350
Badness: 0.0313
Parakleismic
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |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 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being <<13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie <<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
POTE generator: ~6/5 = 315.240
Map: [<1 5 6|, <0 -13 -14|]
EDOs: 19, 61, 80, 99, 118, 453, 571, 689, 1496
Badness: 0.0433
7-limit
Commas: 3136/3125, 4375/4374
POTE generator: ~6/5 = 315.181
Map: [<1 5 6 12|, <0 -13 -14 -35|]
EDOs: 19, 80, 99, 217, 316, 415
Badness: 0.0274
11-limit
Commas: 385/384, 3136/3125, 4375/4374
POTE generator: ~6/5 = 315.251
Map: [<1 5 6 12 -6|, <0 -13 -14 -35 36|]
EDOs: 19, 99, 118
Badness: 0.0497
Parkleismic
Commas: 176/175, 1375/1372, 2200/2187
POTE generator: ~6/5 = 315.060
Map: [<1 5 6 12 20|, <0 -13 -14 -35 -63|]
EDOs: 80, 179, 259cd
Badness: 0.0559
13-limit
Commas: 169/168, 176/175, 325/324, 1375/1372
POTE generator: ~6/5 = 315.075
Map: [<1 5 6 12 20 10|, <0 -13 -14 -35 -63 -24|]
EDOs: 15, 19, 80, 179
Badness: 0.0366
Paradigmic
Commas: 540/539, 896/891, 3136/3125
POTE generator: ~6/5 = 315.096
Map: [<1 5 6 12 -1|, <0 -13 -14 -35 17|]
EDOs: 19, 80, 99e, 179e
Badness: 0.0417
13-limit
Commas: 169/168, 325/324, 540/539, 832/825
POTE generator: ~6/5 = 315.080
Map: [<1 5 6 12 -1 10|, <0 -13 -14 -35 17 -24|]
EDOs: 19, 80, 99e, 179e
Badness: 0.0358
Semiparakleismic
Commas: 3025/3024, 3136/3125, 4375/4374
POTE generator: ~6/5 = 315.181
Map: [<2 10 12 24 19|, <0 -13 -14 -35 -23|]
EDOs: 80, 118, 198, 316, 514c, 830c
Badness: 0.0342
13-limit
Commas: 352/351, 1001/1000, 3025/3024, 4375/4374
POTE generator: ~6/5 = 315.1563
Map: [<2 10 12 24 19 -1|, <0 -13 -14 -35 -23 16|]
Badness: 0.0338
Gentsemiparakleismic
Commas: 169/168, 325/324, 364/363, 3136/3125
POTE generator: ~6/5 = 315.1839
Map: [<2 10 12 24 19 20|, <0 -13 -14 -35 -23 -24|]
Badness: 0.0405
Quincy
Commas: 4375/4374, 823543/819200
POTE generator: ~1728/1715 = 16.613
Map: [<1 2 2 3|, <0 -30 -49 -14|]
EDOs: 72, 217, 289
Badness: 0.0797
11-limit
Commas: 441/440, 4000/3993, 41503/41472
POTE generator: ~100/99 = 16.613
Map: [<1 2 2 3 4|, <0 -30 -49 -14 -39|]
EDOs: 72, 217, 289
Badness: 0.0309
13-limit
Commas: 364/363, 441/440, 676/675, 4375/4374
POTE generator: ~100/99 = 16.602
Map: [<1 2 2 3 4 5|, <0 -30 -49 -14 -39 -94|]
EDOs: 72, 145, 217, 289
Badness: 0.0239
17-limit
Commas: 364/363, 441/440, 595/594, 1001/1000, 1156/1155
POTE generator: ~100/99 = 16.602
Map: [<1 2 2 3 4 5 5|, <0 -30 -49 -14 -39 -94 -66|]
EDOs: 72, 145, 217, 289
Badness: 0.0147
19-limit
Commas: 343/342, 364/363, 441/440, 595/594, 676/675, 2601/2600
POTE generator: ~100/99 = 16.594
Map: [<1 2 2 3 4 5 5 4|, <0 -30 -49 -14 -39 -94 -66 18|]
EDOs: 72, 145, 217
Badness: 0.0152
Chlorine
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>, by which 17 chromatic semitones (25/24) fall short of an octave. Possible tunings for chlorine are 289, 323, and 612 EDOs, though its hardly likely anyone could tell the difference. In the 7-limit, 289&323 temperament tempers out |-49 4 22 -3> as well as the ragisma.
Comma: |-52 -17 34>
POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2687
Map: [<17 26 39|, <0 2 1|]
EDOs: 34, 289, 323, 612, 901
Badness: 0.0771
7-limit
Commas: 4375/4374, 193119049072265625/193091834023510016
POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2936
Map: [<17 26 39 43|, <0 2 1 10|]
EDOs: 34d, 289, 323, 612, 935, 1547
Badness: 0.0417
11-limit
Commas: 4375/4374, 41503/41472, 1879453125/1879048192
POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2690
Map: [<17 26 39 43 64|, <0 2 1 10 -11|]
EDOs: 34de, 289, 323, 612, 901
Badness: 0.0637