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.
Comma list: 2401/2400, 4375/4374
- valid range: [26.667, 66.667] (1\45 to 1\18)
- nice range: [48.920, 49.179]
- strict range: [48.920, 49.179]
Mapping: [<9 1 1 12|, <0 2 3 2|]
Wedgie: <<18 27 18 1 -22 -34||
Mapping generators: ~27/25, ~5/3
POTE generators: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980
Badness: 0.003610
Hemiennealimmal
Comma list: 2401/2400, 4375/4374, 3025/3024
Tuning ranges:
- valid range: [13.333, 22.222] (1\90 to 1\54)
- nice range: [17.304, 17.985]
- strict range: [17.304, 17.985]
Mapping: [<18 0 -1 22 48|, <0 2 3 2 1|]
POTE generator: ~99/98 = 17.6219
Vals: Template:Val list
Badness: 0.006283
13-limit
Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024
Tuning ranges:
- valid range: [16.667, 22.222] (1\72 to 1\54)
- nice range: [17.304, 18.309]
- strict range: [17.304, 18.309]
Mapping: [<18 0 -1 22 48 -19|, <0 2 3 2 1 6|]
POTE generator ~99/98 = 17.7504
Vals: Template:Val list
Badness: 0.012505
Semihemiennealimmal
Comma list: 2401/2400, 4375/4374, 3025/3024, 4225/4224
Mapping: [<18 0 -1 22 48 88|, <0 4 6 4 2 -3|]
POTE generator: ~39/32 = 342.139
Vals: Template:Val list
Badness: 0.013104
Semiennealimmal
Comma list: 2401/2400, 4375/4374, 4000/3993
Mapping: [<9 3 4 14 18|, <0 6 9 6 7|]
POTE generator: ~140/121 = 250.3367
Vals: Template:Val list
Badness: 0.034196
13-limit
Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374
Mapping: [<9 3 4 14 18 -8|, <0 6 9 6 7 22|]
POTE generator: ~140/121 = 250.3375
Vals: Template:Val list
Badness: 0.026122
Quadraennealimmal
Comma list: 2401/2400, 4375/4374, 234375/234256
Mapping: [<9 1 1 12 -7|, <0 8 12 8 23|]
POTE generator: ~77/75 = 45.595
Vals: Template:Val list
Badness: 0.021320
Ennealimnic
Comma list: 243/242, 441/440, 4375/4356
Tuning ranges:
- valid range: [44.444, 53.333] (1\27 to 2\45)
- nice range: [48.920, 52.592]
- strict range: [48.920, 52.592]
Mapping: [<9 1 1 12 -2|, <0 2 3 2 5|]
POTE generator: ~36/35 = 49.395
Vals: Template:Val list
Badness: 0.020347
13-limit
Comma list: 243/242, 364/363, 441/440, 625/624
Tuning ranges:
- valid range: [48.485, 50.000] (4\99 to 3\72)
- nice range: [48.825, 52.592]
- strict range: [48.825, 50.000]
Mapping: [<9 1 1 12 -2 -33|, <0 2 3 2 5 10|]
POTE generator: ~36/35 = 49.341
Vals: Template:Val list
Badness: 0.023250
17-limit
Comma list: 243/242, 364/363, 375/374, 441/440, 595/594
Tuning ranges:
- valid range: [48.485, 50.000] (4\99 to 3\72)
- nice range: [46.363, 52.592]
- strict range: [48.485, 50.000]
Mapping: [<9 1 1 12 -2 -33 -3|, <0 2 3 2 5 10 6|]
POTE generator: ~36/35 = 49.335
Vals: Template:Val list
Badness: 0.014602
Ennealim
Comma list: 169/168, 243/242, 325/324, 441/440
Mapping: [<9 1 1 12 -2 20|, <0 2 3 2 5 2|]
POTE generator: ~36/35 = 49.708
Vals: Template:Val list
Badness: 0.020697
Ennealiminal
Comma list: 385/384, 1375/1372, 4375/4374
Mapping: [<9 1 1 12 51|, <0 2 3 2 -3|]
POTE generator: ~36/35 = 49.504
Vals: Template:Val list
Badness: 0.031123
13-limit
Comma list: 169/168, 325/324, 385/384, 1375/1372
Mapping: [<9 1 1 12 51 20|, <0 2 3 2 -3 2|]
POTE generator: ~36/35 = 49.486
Vals: Template:Val list
Badness: 0.030325
Trinealimmal
Comma list: 2401/2400, 4375/4374, 2097152/2096325
Mapping: [<27 1 0 34 177|, <0 2 3 2 -4|]
POTE generator: ~6/5 = 315.644
Vals: Template:Val list
Badness: 0.029812
Gamera
Comma list: 4375/4374, 589824/588245
Mapping: [<1 6 10 3|, <0 -23 -40 -1|]
Wedgie: <<23 40 1 10 -63 -110||
POTE generator ~8/7 = 230.336
Badness: 0.037648
Hemigamera
Comma list: 3025/3024, 4375/4374, 589824/588245
Mapping: [<2 12 20 6 5|, <0 -23 -40 -1 5|]
POTE generator: ~8/7 = 230.3370
Vals: Template:Val list
Badness: 0.040955
13-limit
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024
Mapping: [<2 12 20 6 5 17|, <0 -23 -40 -1 5 -25|]
POTE generator: ~8/7 = 230.3373
Vals: Template:Val list
Badness: 0.020416
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.
Comma list: 4375/4374, 52734375/52706752
Mapping: [<1 15 19 30|, <0 -37 -46 -75|]
Wedgie: <<37 46 75 -13 15 45||
POTE generator: ~9/7 = 435.082
Badness: 0.010836
Semisupermajor
Comma list: 3025/3024, 4375/4374, 35156250/35153041
Mapping: [<2 30 38 60 41|, <0 -37 -46 -75 -47|]
POTE generator: ~9/7 = 435.082
EDOs: Template:Val list
Badness: 0.012773
Enneadecal
Enneadecal 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.
Comma list: 4375/4374, 703125/702464
Mapping: [<19 0 14 -37|, <0 1 1 3|]
Wedgie: <<19 19 57 -14 37 79||
Mapping generators: ~28/27, ~3
POTE generator: ~3/2 = 701.880
Badness: 0.010954
Hemienneadecal
Comma list: 3025/3024, 4375/4374, 234375/234256
Mapping: [<38 0 28 -74 11|, <0 1 1 3 2|]
POTE generator: ~3/2 = 701.881
Vals: Template:Val list
Badness: 0.009985
13-limit
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
Mapping: [<38 0 28 -74 11 502|, <0 1 1 3 2 -6|]
POTE generator: ~3/2 = 701.986
Vals: Template:Val list
Badness: 0.030391
Deca
Comma list: 4375/4374, 165288374272/164794921875
Mapping: [<10 4 9 2|, <0 5 6 11|]
Wedgie: <<50 60 110 -21 34 87||
POTE generator: ~6/5 = 315.577
Badness: 0.080637
11-limit
Comma list: 3025/3024, 4375/4374, 422576/421875
Mapping: [<10 4 9 2 18|, <0 5 6 11 7|]
POTE generator: ~6/5 = 315.582
Vals: Template:Val list
Badness: 0.024329
13-limit
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
Mapping: [<10 4 9 2 18 37|, <0 5 6 11 7 0|]
POTE generator: ~6/5 = 315.602
Vals: Template:Val list
Badness: 0.016810
Mitonic
Comma list: 4375/4374, 2100875/2097152
Mapping: [<1 -1 -3 6|, <0 17 35 -21|]
Wedgie: <<17 35 -21 16 -81 -147||
POTE generator: ~10/9 = 182.458
Badness: 0.025184
Abigail
Comma list: 4375/4374, 2147483648/2144153025
Mapping: [<2 7 13 -1|, <0 -11 -24 19|]
Wedgie: <<22 48 -38 25 -122 -223||
POTE generator: ~6912/6125 = 208.899
Badness: 0.037000
11-limit
Comma list: 3025/3024, 4375/4374, 20614528/20588575
Mapping: [<2 7 13 -1 1|, <0 -11 -24 19 17|]
POTE generator: ~1155/1024 = 208.901
Vals: Template:Val list
Badness: 0.012860
13-limit
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
Map: [<2 7 13 -1 1 -2|, <0 -11 -24 19 17 27|]
POTE generator: ~44/39 = 208.903
Vals: Template:Val list
Badness: 0.008856
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>
Mapping: [<1 36 48|, <0 -55 -73|]
Wedgie: <<55 73 -12||
POTE generator: ~162/125 = 449.1269
Badness: 0.754866
7-limit
Comma list: 4375/4374, 3955078125/3954653486
Mapping: [<1 36 48 61|, <0 -55 -73 -93|]
Wedgie: <<55 73 93 -12 -7 11||
POTE generator: ~35/27 = 449.1270
Badness: 0.015075
Brahmagupta
The brahmagupta temperament has a period of 1/7 octave, tempering out the akjaysma, |47 -7 -7 -7> = 140737488355328 / 140710042265625.
Comma list: 4375/4374, 70368744177664/70338939985125
Mapping: [<7 2 -8 53|, <0 3 8 -11|]
Wedgie: <<21 56 -77 40 -181 -336||
POTE generator: ~27/20 = 519.716
Badness: 0.029122
11-limit
Comma list: 4000/3993, 4375/4374, 131072/130977
Mapping: [<7 2 -8 53 3|, <0 3 8 -11 7|]
POTE generator: ~27/20 = 519.704
Vals: Template:Val list
Badness: 0.052190
13-limit
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374
Mapping: [<7 2 -8 53 3 35|, <0 3 8 -11 7 -3|]
POTE generator: ~27/20 = 519.706
Vals: Template:Val list
Badness: 0.023132
Quasithird
Comma: |55 -64 20>
Mapping: [<4 0 -11|, <0 5 16|]
Wedgie: <<20 64 55||
POTE generator: ~1594323/1280000 = 380.395
Badness: 0.099519
7-limit
Comma list: 4375/4374, 1153470752371588581/1152921504606846976
Mapping: [<4 0 -11 48|, <0 5 16 -29|]
Wedgie: <<20 64 -116 55 -240 -449||
POTE generator: ~5103/4096 = 380.388
Badness: 0.061813
11-limit
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
Mapping: [<4 0 -11 48 43|, <0 5 16 -29 -23|]
POTE generator: ~22/21 = 80.387 (or ~5103/4096 = 380.387)
Vals: Template:Val list
Badness: 0.021125
13-limit
Comma list: 2200/2197, 3025/3024, 4375/4374, 468512/468195
Mapping: [<4 0 -11 48 43 11|, <0 5 16 -29 -23 3|]
POTE generator: ~22/21 = 80.385 (or ~5103/4096 = 380.385)
Vals: Template:Val list
Badness: 0.029501
Semidimfourth
Comma: |7 41 -31>
Mapping: [<1 21 28|, <0 -31 -41|]
Wedgie: <<31 41 -7||
POTE generator: ~162/125 = 448.449
Badness: 0.233376
7-limit
Comma list: 4375/4374, 235298/234375
Mapping: [<1 21 28 36|, <0 -31 -41 -53|]
Wedgie: <<31 41 53 -7 -3 8||
POTE generator: ~35/27 = 448.456
Badness: 0.055249
Neusec
Comma list: 3025/3024, 4375/4374, 235298/234375
Mapping: [<2 11 15 19 15|, <0 -31 -41 -53 -32|]
POTE generator: ~12/11 = 151.547
Vals: Template:Val list
Badness: 0.059127
13-limit
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
Mapping: [<2 11 15 19 15 17|, <0 -31 -41 -53 -32 -38|]
POTE generator: ~12/11 = 151.545
Vals: Template:Val list
Badness: 0.030941
Acrokleismic
Comma list: 4375/4374, 2202927104/2197265625
Mapping: [<1 10 11 27|, <0 -32 -33 -92|]
Wedgie: <<32 33 92 -22 56 121||
POTE generator: ~6/5 = 315.557
Badness: 0.056184
11-limit
Comma list: 4375/4374, 41503/41472, 172032/171875
Mapping: [<1 10 11 27 -16|, <0 -32 -33 -92 74|]
POTE generator: ~6/5 = 315.558
Vals: Template:Val list
Badness: 0.036878
13-limit
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
Mapping: [<1 10 11 27 -16 25|, <0 -32 -33 -92 74 -81|]
POTE generator: ~6/5 = 315.557
Vals: Template:Val list
Badness: 0.026818
Counteracro
Comma list: 4375/4374, 5632/5625, 117649/117612
Mapping: [<1 10 11 27 55|, <0 -32 -33 -92 -196|]
POTE generator: ~6/5 = 315.553
Vals: Template:Val list
Badness: 0.042572
13-limit
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
Mapping: [<1 10 11 27 55 25|, <0 -32 -33 -92 -196 -81|]
POTE generator: ~6/5 = 315.554
Vals: Template:Val list
Badness: 0.026028
Seniority
Comma list: 4375/4374, 201768035/201326592
Mapping: [<1 11 19 2|, <0 -35 -62 3|]
Wedgie: <<35 62 -3 17 -103 -181||
POTE generator: ~3087/2560 = 322.804
Badness: 0.044877
Orga
Comma list: 4375/4374, 54975581388800/54936068900769
Mapping: [<2 21 36 5|, <0 -29 -51 1|]
Wedgie: <<58 102 -2 27 -166 -291||
POTE generator: ~8/7 = 231.104
Badness: 0.040236
11-limit
Comma list: 3025/3024, 4375/4374, 5767168/5764801
Mapping: [<2 21 36 5 2|, <0 -29 -51 1 8|]
POTE generator: ~8/7 = 231.103
Vals: Template:Val list
Badness: 0.016188
13-limit
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360
Mapping: [<2 21 36 5 2 24|, <0 -29 -51 1 8 -27|]
POTE generator: ~8/7 = 231.103
Vals: Template:Val list
Badness: 0.021762
Quatracot
Comma list: 4375/4374, 1483154296875/1473173782528
Mapping: [<2 7 7 23|, <0 -13 -8 -59|]
Wedgie: <<26 16 118 -35 114 229||
POTE generator: ~448/405 = 176.805
Badness: 0.175982
11-limit
Comma list: 3025/3024, 4375/4374, 1265625/1261568
Mapping: [<2 7 7 23 19|, <0 -13 -8 -59 -41|]
POTE generator: ~448/405 = 176.806
Vals: Template:Val list
Badness: 0.041043
13-limit
Comma list: 625/624, 729/728, 1575/1573, 2200/2197
Mapping: [<2 7 7 23 19 13|, <0 -13 -8 -59 -41 -19|]
POTE generator: ~195/176 = 176.804
Vals: Template:Val list
Badness: 0.022643
Octoid
Comma list: 4375/4374, 16875/16807
- valid range: [578.571, 600.000] (27\56 to 4\8)
- nice range: [582.512, 584.359]
- strict range: [582.512, 584.359]
Mapping: [<8 1 3 3|, <0 3 4 5|]
Wedgie: <<24 32 40 -5 -4 3||
Mapping generators: ~49/45, ~7/5
POTE generator: ~7/5 = 583.940
Badness: 0.042670
11-limit
Comma list: 540/539, 1375/1372, 4000/3993
Tuning ranges:
- valid range: [581.250, 586.364] (31\64, 43\88)
- nice range: [582.512, 585.084]
- strict range: [582.512, 585.084]
Mapping: [<8 1 3 3 16|, <0 3 4 5 3|]
POTE generator: ~7/5 = 583.962
Vals: Template:Val list
Badness: 0.014097
13-limit
Comma list: 540/539, 1375/1372, 4000/3993, 625/624
Mapping: [<8 1 3 3 16 -21|, <0 3 4 5 3 13|]
POTE generator: ~7/5 = 583.905
Vals: Template:Val list
Badness: 0.015274
Music
Octopus
Comma list: 169/168, 325/324, 364/363, 540/539
Mapping: [<8 1 3 3 16 14|, <0 3 4 5 3 4|]
POTE generator: ~7/5 = 583.892
Vals: Template:Val list
Badness: 0.021679
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
Mapping: [<1 3 6|, <0 -5 -13|]
POTE generator: ~243/200 = 339.519
Badness: 0.021960
7-limit
Comma list: 4375/4374, 5120/5103
Mapping: [<1 3 6 -2|, <0 -5 -13 17|]
Wedgie: <<5 13 -17 9 -41 -76||
POTE generator: ~128/105 = 339.432
Badness: 0.023649
11-limit
Comma list: 540/539, 4375/4374, 5120/5103
Mapping: [<1 3 6 -2 21|, <0 -5 -13 17 -62|]
POTE generator: ~128/105 = 339.464
Vals: Template:Val list
Badness: 0.031506
13-limit
Comma list: 352/351, 540/539, 625/624, 847/845
Mapping: [<1 3 6 -2 21 17|, <0 -5 -13 17 -62 -47|]
POTE generator: ~128/105 = 339.481
Vals: Template:Val list
Badness: 0.028008
Hitchcock
Comma list: 121/120, 176/175, 2200/2187
Mapping: [<1 3 6 -2 6|, <0 -5 -13 17 -9|]
POTE generator: ~11/9 = 339.390
Vals: Template:Val list
Badness: 0.035187
13-limit
Comma list: 121/120, 169/168, 176/175, 325/324
Mapping: [<1 3 6 -2 6 2|, <0 -5 -13 17 -9 6|]
POTE generator: ~11/9 = 339.419
Vals: Template:Val list
Badness: 0.022448
Hemiamity
Comma list: 3025/3024, 4375/4374, 5120/5103
Mapping: [<2 1 -1 13 13|, <0 5 13 -17 -14|]
POTE generator: ~64/55 = 339.439
Vals: Template:Val list
Badness: 0.031307
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
Monzism
The monzism temperament (53&612, named by Xenllium) is a rank-two temperament which tempers out the monzisma, [54 -37 2⟩ and the nanisma, [109 -67 0 -1⟩, as well as the ragisma, 4375/4374.
Comma list: 4375/4374, [-55 30 2 1⟩
Mapping: [<1 2 10 -25|, <0 -2 -37 134|]
POTE generator: ~310078125/268435456 = 249.0207
Badness: 0.046569
11-limit
Comma list: 4375/4374, 41503/41472, 184549376/184528125
Mapping: [<1 2 10 -25 46|, <0 -2 -37 134 -205|]
POTE generator: ~231/200 = 249.0193
Vals: Template:Val list
Badness: 0.057083
13-limit
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
Mapping: [<1 2 10 -25 46 23|, <0 -2 -37 134 -205 -93|]
POTE generator: ~231/200 = 249.0199
Vals: Template:Val list
Badness: 0.053780