Ragismic microtemperaments

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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 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, maja, pontiac, trillium, whirrschmidt, zarvo, vishnu, and vulture.

Ennealimmal

Main article: 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 period 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

Tuning ranges:

  • 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

Vals: 27, 45, 72, 99, 171, 441, 612

Badness: 0.003610

11-limit

The ennealimmal temperament can be described as 99e&270 temperament, which tempers out 5632/5625 (vishdel comma) and 19712/19683 (symbiotic comma).

Comma list: 2401/2400, 4375/4374, 5632/5625

Mapping: [9 1 1 12 -75], 0 2 3 2 16]]

POTE generator: ~36/35 = 48.8654

Vals: 99e, 171e, 270, 909, 1179, 1449c, 1719c

Badness: 0.027332

13-limit

Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374

Mapping: [9 1 1 12 -75 93], 0 2 3 2 16 -9]]

POTE generator: ~36/35 = 48.9030

Vals: 99e, 171e, 270

Badness: 0.029404

Ennealimmia

Ennealimmal temperament has various extensions to the 11-limit. Tempering out 131072/130977 (salururu comma) leads to the ennealimmia temperament (171&270, named by Xenllium).

Comma list: 2401/2400, 4375/4374, 131072/130977

Mapping: [9 1 1 12 124], 0 2 3 2 -14]]

POTE generator: ~36/35 = 48.9244

Vals: 99, 171, 270, 711, 981, 1251, 2232e

Badness: 0.026463

13-limit

Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374

Mapping: [9 1 1 12 124 93], 0 2 3 2 -14 -9]]

POTE generator: ~36/35 = 48.9336

Vals: 99, 171, 270, 711, 981, 1692e, 2673e

Badness: 0.016607

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: 72, 171, 243

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: 72, 171, 243

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: 72, 171, 243

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: 27e, 45ef, 72

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: 27, 45, 72, 171e, 243e, 315e

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: 27, 45f, 72, 171ef, 243ef

Badness: 0.030325

Hemiennealimmal

Comma list: 2401/2400, 3025/3024, 4375/4374

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: 72, 198, 270, 342, 612, 954, 1566

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: 72, 198, 270

Badness: 0.012505

Semihemiennealimmal

Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374

Mapping: [18 0 -1 22 48 88], 0 4 6 4 2 -3]]

POTE generator: ~39/32 = 342.139

Vals: 126, 144, 270, 684, 954

Badness: 0.013104

Semiennealimmal

Comma list: 2401/2400, 4000/3993, 4375/4374

Mapping: [9 3 4 14 18], 0 6 9 6 7]]

POTE generator: ~140/121 = 250.3367

Vals: 72, 369, 441

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: 72, 297ef, 369f, 441

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: 342, 1053, 1395, 1737, 4869dd, 6606cdd

Badness: 0.021320

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: 27, 243, 270, 783, 1053, 1323

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

Vals: 26, 73, 99, 224, 323, 422, 745d

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: 26, 198, 224, 422, 646, 1068d

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: 26, 198, 224, 422, 646f, 1068df

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

Vals: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214

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: 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf

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

Vals: 19, 152, 171, 665, 836, 1007, 2185

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: 152, 342, 494, 836, 1178, 2014

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: 152, 342, 494, 836

Badness: 0.030391

Deca

Deca temperament has a period of 1/10 octave and tempers out the linus comma, [11 -10 -10 10 and [12 -3 -14 9 = 165288374272/164794921875 (satritrizo-asepbigu).

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

Vals: 80, 190, 270, 1270, 1540, 1810, 2080

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: 80, 190, 270, 1000, 1270

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: 80, 190, 270, 730, 1000

Badness: 0.016810

Mitonic

See also: Minortonic family #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

Vals: 46, 125, 171

Badness: 0.025184

Sfourth

Comma list: 4375/4374, 64827/64000

Mapping: [1 2 3 3], 0 -19 -31 -9]]

Wedgie: ⟨⟨19 31 9 5 -39 -66]]

POTE generator: ~49/48 = 26.287

Vals: 45, 46, 91, 137d

Badness: 0.123291

11-limit

Comma list: 121/120, 441/440, 4375/4374

Mapping: [1 2 3 3 4], 0 -19 -31 -9 -25]]

POTE generator: ~49/48 = 26.286

Vals: 45e, 46, 91e, 137de

Badness: 0.054098

13-limit

Comma list: 121/120, 169/168, 325/324, 441/440

Mapping: [1 2 3 3 4 4], 0 -19 -31 -9 -25 -14]]

POTE generator: ~49/48 = 26.310

Vals: 45ef, 46, 91ef, 137def

Badness: 0.033067

Sfour

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

Mapping: [1 2 3 3 3], 0 -19 -31 -9 21]]

POTE generator: ~49/48 = 26.246

Vals: 45, 46, 91, 137d

Badness: 0.076567

13-limit

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

Mapping: [1 2 3 3 3 3], 0 -19 -31 -9 21 32]]

POTE generator: ~49/48 = 26.239

Vals: 45, 46, 91, 137d

Badness: 0.051893

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

Vals: 46, 132, 178, 224, 270, 494, 764, 1034, 1798

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: 46, 132, 178, 224, 270, 494, 764

Badness: 0.012860

13-limit

Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095

Mapping: [2 7 13 -1 1 -2], 0 -11 -24 19 17 27]]

POTE generator: ~44/39 = 208.903

Vals: 46, 178, 224, 270, 494, 764, 1258

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]]

POTE generator: ~162/125 = 449.1269

Vals: 8, 163, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419

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

Vals: 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, [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

Vals: 7, 217, 224, 441, 1106, 1547

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: 7, 217, 224, 441, 665, 1771ee

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: 7, 217, 224, 441, 665, 1771eef

Badness: 0.023132

Quasithird

The quasithird temperament is featured by a major third interval which is 1600000/1594323 (amity comma) or 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 ragisma and [-60 29 0 5.

Comma: [55 -64 20

Mapping: [4 0 -11], 0 5 16]]

POTE generator: ~1594323/1280000 = 380.395

Vals: 60, 164, 224, 388, 612, 836, 1000, 1448, 1612, 2224, 2836

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

Vals: 60d, 164, 224, 388, 612, 1448, 2060

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: 60d, 164, 224, 388, 612, 836, 1448

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: 60d, 164, 224, 388, 612, 836, 1448f, 2284f

Badness: 0.029501

Semidimfourth

The semidimifourth 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.

Comma: [7 41 -31

Mapping: [1 21 28], 0 -31 -41]]

POTE generator: ~162/125 = 448.449

Vals: 8, 91, 99, 190, 289, 388, 677, 3674, 4351, 5028, 5705, 6382, 13441c, 19823bcc

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

Vals: 8d, 91, 99, 289, 388, 875, 1263d, 1651d

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: 8d, 190, 388

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: 8d, 190, 198, 388

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

Vals: 19, 251, 270

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

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: 19, 251, 270

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: 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde

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: 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf

Badness: 0.026028

Seniority

See also: Very high accuracy temperaments#Senior

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

Vals: 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d

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

Vals: 26, 244, 270, 836, 1106, 1376, 2482

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: 26, 244, 270, 566, 836, 1106

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: 26, 244, 270, 566, 836f, 1106f

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

Vals: 190, 224, 414, 638, 1052c, 1690bcc

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: 190, 224, 414, 638, 1052c

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: 190, 224, 414, 638, 1690bcc, 2328bccde

Badness: 0.022643

Octoid

The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 (ragisma) and 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.

Comma list: 4375/4374, 16875/16807

Tuning ranges:

  • 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

Vals: 8d, 72, 152, 224

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: 72, 152, 224

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: 72, 152f, 224

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: 72, 152, 224f

Badness: 0.021679

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, 39, 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

Vals: 7, 39, 46, 53, 152, 205, 463, 668, 873

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

Vals: 7, 39, 46, 53, 99, 251, 350, 601cd, 951bcdd

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: 53, 99e, 152, 555dee, 707ddee, 859bddee

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: 53, 99ef, 152f

Badness: 0.028008

Hitchcock

See also: Amity family #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: 7, 39, 46, 53, 99

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: 7, 39, 46, 53, 99

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: 14cde, 46, 106, 152, 350, 502d

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 -8 19 -102 42 -132 -222]] adding 385/384. For the 7-limit 99EDO may be preferred, but in the 11-limit it is best to stick with 118.

Comma: 124440064/1220703125

Mapping: [1 5 6], 0 -13 -14]]

POTE generator: ~6/5 = 315.240

Vals: 19, 61, 80, 99, 118, 453, 571, 689, 1496

Badness: 0.043279

7-limit

Comma list: 3136/3125, 4375/4374

Mapping: [1 5 6 12], 0 -13 -14 -35]]

Wedgie: ⟨⟨13 14 35 -8 19 42]]

POTE generator: ~6/5 = 315.181

Vals: 19, 80, 99, 217, 316, 415

Badness: 0.027431

11-limit

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

Mapping: [1 5 6 12 -6], 0 -13 -14 -35 36]]

POTE generator: ~6/5 = 315.251

Vals: 19, 99, 118

Badness: 0.049711

Paraklein

The paraklein temperament (118&217, named by Xenllium) tempers out 441/440, 5632/5625, and 19712/19683.

Comma list: 441/440, 3136/3125, 4375/4374

Mapping: [1 5 6 12 25], 0 -13 -14 -35 -82]]

POTE generator: ~6/5 = 315.220

Vals: 19e, 99e, 118, 217, 335, 552d, 887dd

Badness: 0.036027

13-limit

Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374

Mapping: [1 5 6 12 25 -16], 0 -13 -14 -35 -82 75]]

POTE generator: ~6/5 = 315.214

Vals: 99e, 118, 217, 552d, 769de

Badness: 0.044710

Parkleismic

Comma list: 176/175, 1375/1372, 2200/2187

Mapping: [1 5 6 12 20], 0 -13 -14 -35 -63]]

POTE generator: ~6/5 = 315.060

Vals: 19e, 80, 179, 259cd

Badness: 0.055884

13-limit

Comma list: 169/168, 176/175, 325/324, 1375/1372

Mapping: [1 5 6 12 20 10], 0 -13 -14 -35 -63 -24]]

POTE generator: ~6/5 = 315.075

Vals: 19e, 80, 179

Badness: 0.036559

Paradigmic

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

Mapping: [1 5 6 12 -1], 0 -13 -14 -35 17]]

POTE generator: ~6/5 = 315.096

Vals: 19, 61d, 80, 99e, 179e

Badness: 0.041720

13-limit

Comma list: 169/168, 325/324, 540/539, 832/825

Mapping: [1 5 6 12 -1 10], 0 -13 -14 -35 17 -24]]

POTE generator: ~6/5 = 315.080

Vals: 19, 61d, 80, 99e, 179e

Badness: 0.035781

Semiparakleismic

Comma list: 3025/3024, 3136/3125, 4375/4374

Mapping: [2 10 12 24 19], 0 -13 -14 -35 -23]]

POTE generator: ~6/5 = 315.181

Vals: 80, 118, 198, 316, 514c, 830c

Badness: 0.034208

13-limit

Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374

Mapping: [2 10 12 24 19 -1], 0 -13 -14 -35 -23 16]]

POTE generator: ~6/5 = 315.156

Vals: 80, 118, 198

Badness: 0.033775

Gentsemiparakleismic

Comma list: 169/168, 325/324, 364/363, 3136/3125

Mapping: [2 10 12 24 19 20], 0 -13 -14 -35 -23 -24]]

POTE generator: ~6/5 = 315.184

Vals: 80, 118f, 198f

Badness: 0.040467

Quincy

Comma list: 4375/4374, 823543/819200

Mapping: [1 2 3 3], 0 -30 -49 -14]]

Wedgie: ⟨⟨30 49 14 8 -62 -105]]

POTE generator: ~1728/1715 = 16.613

Vals: 72, 217, 289

Badness: 0.079657

11-limit

Comma list: 441/440, 4000/3993, 4375/4374

Mapping: [1 2 3 3 4], 0 -30 -49 -14 -39]]

POTE generator: ~100/99 = 16.613

Vals: 72, 217, 289

Badness: 0.030875

13-limit

Comma list: 364/363, 441/440, 676/675, 4375/4374

Mapping: [1 2 3 3 4 5], 0 -30 -49 -14 -39 -94]]

POTE generator: ~100/99 = 16.602

Vals: 72, 145, 217, 289

Badness: 0.023862

17-limit

Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155

Mapping: [1 2 3 3 4 5 5], 0 -30 -49 -14 -39 -94 -66]]

POTE generator: ~100/99 = 16.602

Vals: 72, 145, 217, 289

Badness: 0.014741

19-limit

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

Mapping: [1 2 3 3 4 5 5 4], 0 -30 -49 -14 -39 -94 -66 18]]

POTE generator: ~100/99 = 16.594

Vals: 72, 145, 217

Badness: 0.015197

Chlorine

The name of chlorine temperament comes from Chlorine, the 17th element.

Chlorine temperament has a period of 1/17 octave. It tempers out the septendecima, [-52 -17 34, by which 17 chromatic semitones (25/24) exceed an octave. This temperament can be described as 289&323 temperament, which tempers out [-49 4 22 -3 as well as the ragisma.

Comma: [-52 -17 34

Mapping: [17 26 39], 0 2 1]]

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

Vals: 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797

Badness: 0.077072

7-limit

Comma list: 4375/4374, 193119049072265625/193091834023510016

Mapping: [17 26 39 43], 0 2 1 10]]

Wedgie: ⟨⟨34 17 170 -52 174 347]]

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

Vals: 289, 323, 612, 935, 1547

Badness: 0.041658

11-limit

Comma list: 4375/4374, 41503/41472, 1879453125/1879048192

Mapping: [17 26 39 43 64], 0 2 1 10 -11]]

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

Vals: 289, 323, 612

Badness: 0.063706

Palladium

The name of palladium temperament (named by Xenllium) comes from Palladium, the 46th element.

Palladium temperament has a period of 1/46 octave. It tempers out the 46-9/5-comma, [-39 92 -46, by which 46 minortones (10/9) fall short of seven octaves. This temperament can be described as 46&414 temperament, which tempers out [-51 8 2 12 as well as the ragisma.

Comma list: 4375/4374, 2270317133144025/2251799813685248

Mapping: [46 73 107 129], 0 -1 -2 1]]

Wedgie: ⟨⟨46 92 -46 39 -202 -365]]

POTE generator: ~3/2 = 701.6074

Vals: 46, 368, 414, 460, 874d

Badness: 0.308505

11-limit

Comma list: 3025/3024, 9801/9800, 134775333/134217728

Mapping: [46 73 107 129 159], 0 -1 -2 1 1]]

POTE generator: ~3/2 = 701.5951

Vals: 46, 368, 414, 460, 874de

Badness: 0.073783

13-limit

Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364

Mapping: [46 73 107 129 159 170], 0 -1 -2 1 1 2]]

POTE generator: ~3/2 = 701.6419

Vals: 46, 368, 414, 460, 874de, 1334de

Badness: 0.040751

17-limit

Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224

Mapping: [46 73 107 129 159 170 188], 0 -1 -2 1 1 2 0]]

POTE generator: ~3/2 = 701.6425

Vals: 46, 368, 414, 460, 874de, 1334deg

Badness: 0.022441

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, 36030948116563575/36028797018963968

Mapping: [1 2 10 -25], 0 -2 -37 134]]

Wedgie: ⟨⟨2 37 -134 54 -218 -415]]

POTE generator: ~310078125/268435456 = 249.0207

Vals: 53, 559, 612, 1277, 1889

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: 53, 559, 612

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: 53, 559, 612

Badness: 0.053780