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 discussed elsewhere include:
- Hystrix, {36/35, 160/147} → Porcupine family
- Rhinoceros, {49/48, 4375/4374} → Unicorn family
- Crepuscular, {50/49, 4375/4374} → Jubilismic clan and Fifive family
- Modus, {64/63, 4375/4374} → Tetracot family
- Flattone, {81/80, 525/512} → Meantone family
- Sensi, {126/125, 245/243} → Sensipent family and Sensamagic clan
- Catakleismic, {225/224, 4375/4374} → Kleismic family
- Unidec, {1029/1024, 4375/4374} → Gamelismic clan
- Quartonic, {1728/1715, 4000/3969} → Orwellismic temperaments
- Srutal, {2048/2025, 4375/4374} → Diaschismic family
- Maja, {2430/2401, 3125/3087} → Maja family
- Pontiac, {4375/4374, 32805/32768} → Schismatic family
- Zarvo, {4375/4374, 33075/32768} → Gravity family
- Whirrschmidt, {4375/4374, 393216/390625} → Würschmidt family
- Mitonic, {4375/4374, 2100875/2097152} → Minortonic family
- Vishnu, {4375/4374, 29360128/29296875} → Vishnuzmic family
- Vulture, {4375/4374, 33554432/33480783} → Vulture family
- Trillium, {4375/4374, [40 -22 -1 -1⟩} → Tricot family
- Unlit, {4375/4374, [41 -20 -4⟩} → Undim family
- Quindro, {4375/4374, [56 -28 -5⟩} → Quindromeda family
Considered below are ennealimmal, gamera, supermajor, enneadecal, decal, sfourth, abigail, semidimi, brahmagupta, quasithird, semidimfourth, acrokleismic, seniority, orga, quatracot, octoid, amity, parakleismic, counterkleismic, quincy, trideci, chlorine, palladium, and monzism.
Ennealimmal
- Main article: Ennealimmal
Ennealimmal 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 ennealimma, [1 -27 18⟩, which leads to the identification of (27/25)9 with the octave, and gives ennealimmal a period of 1/9 octave. Its pergen is (P8/9, P5/2). 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.
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 49/40~60/49, all of which have their own interesting advantages. Possible tunings are 441-, 612-, or 3600edo, 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.
Ennealimmal extensions discussed elsewhere include omicronbeta, undecentic, schisennealimmal, and lunennealimmal.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 4375/4374
Mapping: [⟨9 1 1 12], ⟨0 2 3 2]]
Wedgie: ⟨⟨18 27 18 1 -22 -34]]
Mapping generators: ~27/25, ~5/3
POTE generators: ~5/3 = 884.3129 or ~36/35 = 49.0205
- 7-odd-limit diamond monotone: ~36/35 = [26.667, 66.667] (1\45 to 1\18)
- 9-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
- 7- and 9-odd-limit diamond tradeoff: ~36/35 = [48.920, 49.179]
- 7- and 9-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 49.179]
Optimal GPV sequence: 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).
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 4375/4374, 5632/5625
Mapping: [⟨9 1 1 12 -75], ⟨0 2 3 2 16]]
POTE generator: ~5/3 = 884.4679 or ~36/35 = 48.8654
Optimal GPV sequence: 99e, 171e, 270, 909, 1179, 1449c, 1719c
Badness: 0.027332
13-limit
Subgroup: 2.3.5.7.11.13
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: ~5/3 = 884.4304 or ~36/35 = 48.9030
Optimal GPV sequence: 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).
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 4375/4374, 131072/130977
Mapping: [⟨9 1 1 12 124], ⟨0 2 3 2 -14]]
POTE generator: ~5/3 = 884.4089 or ~36/35 = 48.9244
Optimal GPV sequence: 99, 171, 270, 711, 981, 1251, 2232e
Badness: 0.026463
13-limit
Subgroup: 2.3.5.7.11.13
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: ~5/3 = 884.3997 or ~36/35 = 48.9336
Optimal GPV sequence: 99, 171, 270, 711, 981, 1692e, 2673e
Badness: 0.016607
Ennealimnic
Ennealimnic temperament (72&171) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 4375/4356
Mapping: [⟨9 1 1 12 -2], ⟨0 2 3 2 5]]
POTE generator: ~5/3 = 883.9386 or ~36/35 = 49.3948
Tuning ranges:
- 11-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
- 11-odd-limit diamond tradeoff: ~36/35 = [48.920, 52.592]
- 11-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 52.592]
Optimal GPV sequence: 72, 171, 243
Badness: 0.020347
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 364/363, 441/440, 625/624
Mapping: [⟨9 1 1 12 -2 -33], ⟨0 2 3 2 5 10]]
POTE generator: ~5/3 = 883.9920 or ~36/35 = 49.3414
Tuning ranges:
- 13- and 15-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)
- 13- and 15-odd-limit diamond tradeoff: ~36/35 = [48.825, 52.592]
- 13- and 15-odd-limit diamond monotone and tradeoff: ~36/35 = [48.825, 50.000]
Optimal GPV sequence: 72, 171, 243
Badness: 0.023250
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 364/363, 375/374, 441/440, 595/594
Mapping: [⟨9 1 1 12 -2 -33 -3], ⟨0 2 3 2 5 10 6]]
POTE generator: ~5/3 = 883.9981 or ~36/35 = 49.3353
Tuning ranges:
- 17-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)
- 17-odd-limit diamond tradeoff: ~36/35 = [46.363, 52.592]
- 17-odd-limit diamond monotone and tradeoff: ~36/35 = [48.485, 50.000]
Optimal GPV sequence: 72, 171, 243
Badness: 0.014602
Ennealim
Subgroup: 2.3.5.7.11.13
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: ~5/3 = 883.6257 or ~36/35 = 49.7076
Optimal GPV sequence: 27e, 45ef, 72
Badness: 0.020697
Ennealiminal
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 4375/4374
Mapping: [⟨9 1 1 12 51], ⟨0 2 3 2 -3]]
POTE generator: ~5/3 = 883.8298 or ~36/35 = 49.5036
Optimal GPV sequence: 27, 45, 72, 171e, 243e, 315e
Badness: 0.031123
13-limit
Subgroup: 2.3.5.7.11.13
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: ~5/3 = 883.8476 or ~36/35 = 49.4857
Optimal GPV sequence: 27, 45f, 72, 171ef, 243ef
Badness: 0.030325
Hemiennealimmal
Hemiennealimmal (72&198) has a period of 1/18 octave and tempers out the four smallest superparticular commas of the 11-limit JI, 2401/2400, 3025/3024, 4375/4374, and 9801/9800. Tempering out 9801/9800 leads an octave split into two equal parts.
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 4375/4374
Mapping: [⟨18 0 -1 22 48], ⟨0 2 3 2 1]]
Mapping generators: ~80/77, ~400/231
POTE generator: ~400/231 = 950.9553
Tuning ranges:
- 11-odd-limit diamond monotone: ~99/98 = [13.333, 22.222] (1\90 to 1\54)
- 11-odd-limit diamond tradeoff: ~99/98 = [17.304, 17.985]
- 11-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 17.985]
Optimal GPV sequence: 72, 198, 270, 342, 612, 954, 1566
Badness: 0.006283
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024
Mapping: [⟨18 0 -1 22 48 -19], ⟨0 2 3 2 1 6]]
POTE generator ~26/15 = 951.0837
Tuning ranges:
- 13-odd-limit diamond monotone: ~99/98 = [16.667, 22.222] (1\72 to 1\54)
- 15-odd-limit diamond monotone: ~99/98 = [16.667, 19.048] (1\72 to 2\126)
- 13-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.309]
- 15-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.926]
- 13-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.309]
- 15-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.926]
Optimal GPV sequence: 72, 198, 270
Badness: 0.012505
Semihemiennealimmal
Subgroup: 2.3.5.7.11.13
Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374
Mapping: [⟨18 0 -1 22 48 88], ⟨0 4 6 4 2 -3]]
Mapping generators: ~80/77, ~1053/800
POTE generator: ~1053/800 = 475.4727
Optimal GPV sequence: 126, 144, 270, 684, 954
Badness: 0.013104
Semiennealimmal
Semiennealimmal tempers out 4000/3993, and uses a ~140/121 semifourth generator. Notably, however, two generator steps do not reach ~4/3, despite that the name may suggest so. In fact, it splits the generator of ennealimmal into three.
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 4000/3993, 4375/4374
Mapping: [⟨9 3 4 14 18], ⟨0 6 9 6 7]]
Mapping generators: ~27/25, ~140/121
POTE generator: ~140/121 = 250.3367
Optimal GPV sequence: 72, 369, 441
Badness: 0.034196
13-limit
Subgroup: 2.3.5.7.11.13
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
Optimal GPV sequence: 72, 297ef, 369f, 441
Badness: 0.026122
Quadraennealimmal
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 4375/4374, 234375/234256
Mapping: [⟨9 1 1 12 -7], ⟨0 8 12 8 23]]
Mapping generators: ~27/25, ~25/22
POTE generator: ~25/22 = 221.0717
Optimal GPV sequence: 342, 1053, 1395, 1737, 4869dd, 6606cdd
Badness: 0.021320
Trinealimmal
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 4375/4374, 2097152/2096325
Mapping: [⟨27 1 0 34 177], ⟨0 2 3 2 -4]]
Mapping generators: ~2744/2673, ~2352/1375
POTE generator: ~2352/1375 = 928.8000
Optimal GPV sequence: 27, 243, 270, 783, 1053, 1323
Badness: 0.029812
Gamera
Subgroup: 2.3.5.7
Comma list: 4375/4374, 589824/588245
Mapping: [⟨1 6 10 3], ⟨0 -23 -40 -1]]
Mapping generators: ~2, ~8/7
Wedgie: ⟨⟨23 40 1 10 -63 -110]]
POTE generator ~8/7 = 230.336
Optimal GPV sequence: 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: [⟨2 12 20 6 5], ⟨0 -23 -40 -1 5]]
Mapping generators: ~99/70, ~8/7
POTE generator: ~8/7 = 230.3370
Optimal GPV sequence: 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: [⟨2 12 20 6 5 17], ⟨0 -23 -40 -1 5 -25]]
POTE generator: ~8/7 = 230.3373
Optimal GPV sequence: 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: [⟨1 6 10 3 12], ⟨0 -46 -80 -2 -89]]
Mapping generators: ~2, ~77/72
POTE generator: ~77/72 = 115.1642
Optimal GPV sequence: 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: [⟨1 6 10 3 12 18], ⟨0 -46 -80 -2 -89 -149]]
POTE generator: ~77/72 = 115.1628
Optimal GPV sequence: 73f, 125f, 198, 323, 521
Badness: 0.044
Supermajor
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 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.
Subgroup: 2.3.5.7
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
Optimal GPV sequence: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214
Badness: 0.010836
Semisupermajor
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 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.
Subgroup: 2.3.5.7
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
Optimal GPV sequence: 19, 152, 171, 665, 836, 1007, 2185
Badness: 0.010954
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4375/4374, 16384/16335
Mapping: [⟨19 0 14 -37 126], ⟨0 1 1 3 -2]]
POTE generator: ~3/2 = 702.360
Optimal GPV sequence: 19, 152, 323e, 475de, 627de
Badness: 0.043734
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 625/624, 729/728, 2205/2197
Mapping: [⟨19 0 14 -37 126 -20], ⟨0 1 1 3 -2 3]]
POTE generator: ~3/2 = 702.212
Optimal GPV sequence: 19, 152f, 323e
Badness: 0.033545
Hemienneadecal
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 234375/234256
Mapping: [⟨38 0 28 -74 11], ⟨0 1 1 3 2]]
Mapping generators: ~55/54, ~3
POTE generator: ~3/2 = 701.881
Optimal GPV sequence: 152, 342, 494, 836, 1178, 2014
Badness: 0.009985
13-limit
Subgroup: 2.3.5.7.11.13
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
Optimal GPV sequence: 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).
Subgroup: 2.3.5.7
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
Optimal GPV sequence: 80, 190, 270, 1270, 1540, 1810, 2080
Badness: 0.080637
11-limit
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 80, 190, 270, 1000, 1270
Badness: 0.024329
13-limit
Subgroup: 2.3.5.7.11.13
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
Optimal GPV sequence: 80, 190, 270, 730, 1000
Badness: 0.016810
Sfourth
- For the 5-limit version of this temperament, see High badness temperaments #Sfourth.
Subgroup: 2.3.5.7
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
Optimal GPV sequence: 45, 46, 91, 137d
Badness: 0.123291
11-limit
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 45e, 46, 91e, 137de
Badness: 0.054098
13-limit
Subgroup: 2.3.5.7.11.13
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
Optimal GPV sequence: 45ef, 46, 91ef, 137def
Badness: 0.033067
Sfour
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 45, 46, 91, 137d
Badness: 0.076567
13-limit
Subgroup: 2.3.5.7.11.13
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
Optimal GPV sequence: 45, 46, 91, 137d
Badness: 0.051893
Abigail
Subgroup: 2.3.5.7
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
Optimal GPV sequence: 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: [⟨2 7 13 -1 1], ⟨0 -11 -24 19 17]]
POTE generator: ~1155/1024 = 208.901
Optimal GPV sequence: 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: [⟨2 7 13 -1 1 -2], ⟨0 -11 -24 19 17 27]]
POTE generator: ~44/39 = 208.903
Optimal GPV sequence: 46, 178, 224, 270, 494, 764, 1258
Badness: 0.008856
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 [-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: [⟨1 36 48 61], ⟨0 -55 -73 -93]]
Wedgie: ⟨⟨55 73 93 -12 -7 11]]
POTE generator: ~35/27 = 449.1270
Optimal GPV sequence: 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.
Subgroup: 2.3.5.7
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
Optimal GPV sequence: 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: [⟨7 2 -8 53 3], ⟨0 3 8 -11 7]]
POTE generator: ~27/20 = 519.704
Optimal GPV sequence: 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: [⟨7 2 -8 53 3 35], ⟨0 3 8 -11 7 -3]]
POTE generator: ~27/20 = 519.706
Optimal GPV sequence: 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⟩.
Subgroup: 2.3.5
Comma: [55 -64 20⟩
Mapping: [⟨4 0 -11], ⟨0 5 16]]
Mapping generators: ~51200000/43046721, ~1594323/1280000
POTE generator: ~1594323/1280000 = 380.395
Optimal GPV sequence: 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, 1153470752371588581/1152921504606846976
Mapping: [⟨4 0 -11 48], ⟨0 5 16 -29]]
Wedgie: ⟨⟨20 64 -116 55 -240 -449]]
POTE generator: ~5103/4096 = 380.388
Optimal GPV sequence: 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: [⟨4 0 -11 48 43], ⟨0 5 16 -29 -23]]
POTE generator: ~5103/4096 = 380.387 (or ~22/21 = 80.387)
Optimal GPV sequence: 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: [⟨4 0 -11 48 43 11], ⟨0 5 16 -29 -23 3]]
POTE generator: ~81/65 = 380.385 (or ~22/21 = 80.385)
Optimal GPV sequence: 60d, 164, 224, 388, 612, 836, 1448f, 2284f
Badness: 0.029501
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: [⟨1 21 28 36], ⟨0 -31 -41 -53]]
Wedgie: ⟨⟨31 41 53 -7 -3 8]]
POTE generator: ~35/27 = 448.456
Optimal GPV sequence: 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: [⟨2 11 15 19 15], ⟨0 -31 -41 -53 -32]]
POTE generator: ~12/11 = 151.547
Optimal GPV sequence: 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: [⟨2 11 15 19 15 17], ⟨0 -31 -41 -53 -32 -38]]
POTE generator: ~12/11 = 151.545
Optimal GPV sequence: 8d, 190, 198, 388
Badness: 0.030941
Acrokleismic
Subgroup: 2.3.5.7
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
Optimal GPV sequence: 19, 251, 270
Badness: 0.056184
11-limit
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 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: [⟨1 10 11 27 -16 25], ⟨0 -32 -33 -92 74 -81]]
POTE generator: ~6/5 = 315.557
Optimal GPV sequence: 19, 251, 270
Badness: 0.026818
Counteracro
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 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: [⟨1 10 11 27 55 25], ⟨0 -32 -33 -92 -196 -81]]
POTE generator: ~6/5 = 315.554
Optimal GPV sequence: 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf
Badness: 0.026028
Seniority
- See also: Very high accuracy temperaments #Senior
Aside from the ragisma, the seniority temperament (26&145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ([-17 62 -35⟩, quadla-sepquingu) is tempered out.
Subgroup: 2.3.5.7
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
Optimal GPV sequence: 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d
Badness: 0.044877
Senator
The senator temperament (26&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: [⟨1 11 19 2 4], ⟨0 -35 -62 3 -2]]
POTE generator: ~77/64 = 322.793
Optimal GPV sequence: 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: [⟨1 11 19 2 4 15], ⟨0 -35 -62 3 -2 -42]]
POTE generator: ~77/64 = 322.793
Optimal GPV sequence: 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: [⟨1 11 19 2 4 15 17], ⟨0 -35 -62 3 -2 -42 -48]]
POTE generator: ~77/64 = 322.793
Optimal GPV sequence: 26, 119c, 145, 171, 316ef, 487eef
Badness: 0.026562
Orga
Subgroup: 2.3.5.7
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
Optimal GPV sequence: 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: [⟨2 21 36 5 2], ⟨0 -29 -51 1 8]]
POTE generator: ~8/7 = 231.103
Optimal GPV sequence: 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: [⟨2 21 36 5 2 24], ⟨0 -29 -51 1 8 -27]]
POTE generator: ~8/7 = 231.103
Optimal GPV sequence: 26, 244, 270, 566, 836f, 1106f
Badness: 0.021762
Quatracot
- See also: Stratosphere
Subgroup: 2.3.5.7
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
Optimal GPV sequence: 190, 224, 414, 638, 1052c, 1690bcc
Badness: 0.175982
11-limit
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 190, 224, 414, 638, 1052c
Badness: 0.041043
13-limit
Subgroup: 2.3.5.7.11.13
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
Optimal GPV sequence: 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.
Subgroup: 2.3.5.7
Comma list: 4375/4374, 16875/16807
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
- 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]
- 7-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 584.359]
- 9-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 585.084]
Optimal GPV sequence: 8d, 72, 152, 224
Badness: 0.042670
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 4000/3993
Mapping: [⟨8 1 3 3 16], ⟨0 3 4 5 3]]
POTE generator: ~7/5 = 583.962
Tuning ranges:
- 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
- 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
- 11-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 585.084]
Optimal GPV sequence: 72, 152, 224
Badness: 0.014097
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 625/624, 729/728, 1375/1372
Mapping: [⟨8 1 3 3 16 -21], ⟨0 3 4 5 3 13]]
POTE generator: ~7/5 = 583.905
Optimal GPV sequence: 72, 152f, 224
Badness: 0.015274
- Music
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 375/374, 540/539, 625/624, 715/714, 729/728
Mapping: [⟨8 1 3 3 16 -21 -14], ⟨0 3 4 5 3 13 12]]
POTE generator: ~7/5 = 583.842
Optimal GPV sequence: 72, 152fg, 224, 296, 520g
Badness: 0.014304
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
Mapping: [⟨8 1 3 3 16 -21 -14 34], ⟨0 3 4 5 3 13 12 0]]
POTE generator: ~7/5 = 583.932
Optimal GPV sequence: 72, 152fg, 224
Badness: 0.016036
Octopus
Subgroup: 2.3.5.7.11.13
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
Optimal GPV sequence: 72, 152, 224f
Badness: 0.021679
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
Mapping: [⟨8 1 3 3 16 14 21], ⟨0 3 4 5 3 4 3]]
POTE generator: ~7/5 = 583.811
Optimal GPV sequence: 72, 152, 224fg, 296ffg
Badness: 0.015614
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
Mapping: [⟨8 1 3 3 16 14 21 34], ⟨0 3 4 5 3 4 3 0]]
POTE generator: ~7/5 = 584.064
Optimal GPV sequence: 72, 152, 224fg, 376ffgh
Badness: 0.016321
Hexadecoid
Hexadecoid (80&144) has a period of 1/16 octave and tempers out 4225/4224.
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
Mapping: [⟨16 26 38 46 56 59], ⟨0 -3 -4 -5 -3 1]]
POTE generator: ~13/8 = 841.015
Optimal GPV sequence: 80, 144, 224
Badness: 0.030818
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
Mapping: [⟨16 26 38 46 56 59 65], ⟨0 -3 -4 -5 -3 1 2]]
POTE generator: ~13/8 = 840.932
Optimal GPV sequence: 80, 144, 224, 528dg
Badness: 0.028611
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
Mapping: [⟨16 26 38 46 56 59 65 68], ⟨0 -3 -4 -5 -3 1 2 0]]
POTE generator: ~13/8 = 840.896
Optimal GPV sequence: 80, 144, 224, 304dh, 528dghh
Badness: 0.023731
Amity
- Main article: Amity
- See also: Amity family #Amity
The generator for amity temperament is the acute minor third, which means the 6/5 just 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. 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.
Subgroup: 2.3.5.7
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
Optimal GPV sequence: 7, 32c, 39, 46, 53, 99, 251, 350, 601cd, 951bcdd
Badness: 0.023649
11-limit
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee
Badness: 0.031506
13-limit
Subgroup: 2.3.5.7.11.13
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
Optimal GPV sequence: 46ef, 53, 99ef, 152f *
* optimal patent val: 205
Badness: 0.028008
Hitchcock
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 7, 39, 46, 53, 99
Badness: 0.035187
13-limit
Subgroup: 2.3.5.7.11.13
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
Optimal GPV sequence: 7, 39, 46, 53, 99
Badness: 0.022448
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 121/120, 154/153, 169/168, 176/175, 273/272
Mapping: [⟨1 3 6 -2 6 2 -1], ⟨0 -5 -13 17 -9 6 18]]
POTE generator: ~11/9 = 339.366
Optimal GPV sequence: 7, 39, 46, 53, 99
Badness: 0.019395
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189
Mapping: [⟨1 3 6 -2 6 2 -1 0], ⟨0 -5 -13 17 -9 6 18 15]]
POTE generator: ~11/9 = 339.407
Optimal GPV sequence: 7, 39h, 46, 53, 99h
Badness: 0.017513
Catamite
Subgroup: 2.3.5.7.11
Comma list: 441/440, 896/891, 4375/4374
Mapping: [⟨1 3 6 -2 -7], ⟨0 -5 -13 17 37]]
POTE generator: ~128/105 = 339.340
Optimal GPV sequence: 46, 99e, 145, 244e
Badness: 0.040976
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 364/363, 4375/4374
Mapping: [⟨1 3 6 -2 -7 -11], ⟨0 -5 -13 17 37 52]]
POTE generator: ~128/105 = 339.313
Optimal GPV sequence: 46, 99ef, 145
Badness: 0.034215
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155
Mapping: [⟨1 3 6 -2 -7 -11 -1], ⟨0 -5 -13 17 37 52 18]]
POTE generator: ~17/14 = 339.313
Optimal GPV sequence: 46, 99ef, 145
Badness: 0.021193
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475
Mapping: [⟨1 3 6 -2 -7 -11 -1 -13], ⟨0 -5 -13 17 37 52 18 61]]
POTE generator: ~17/14 = 339.325
Optimal GPV sequence: 46, 99ef, 145
Badness: 0.018864
Hemiamity
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 5120/5103
Mapping: [⟨2 1 -1 13 13], ⟨0 5 13 -17 -14]]
Mapping generators: ~99/70, ~64/55
POTE generator: ~64/55 = 260.561
Optimal GPV sequence: 14cde, 46, 106, 152, 350, 502d
Badness: 0.031307
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 847/845, 1716/1715, 3025/3024
Mapping: [⟨2 1 -1 13 13 20], ⟨0 5 13 -17 -14 -29]]
POTE generator: ~64/55 = 260.583
Optimal GPV sequence: 46, 106f, 152f, 198, 350f, 548cdff
Badness: 0.025784
Parakleismic
- Main article: 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.
Subgroup: 2.3.5
Comma list: 1224440064/1220703125
Mapping: [⟨1 5 6], ⟨0 -13 -14]]
POTE generator: ~6/5 = 315.240
Optimal GPV sequence: 19, 61, 80, 99, 118, 453, 571, 689, 1496
Badness: 0.043279
7-limit
Subgroup: 2.3.5.7
Comma list: 3136/3125, 4375/4374
Mapping: [⟨1 5 6 12], ⟨0 -13 -14 -35]]
Wedgie: ⟨⟨13 14 35 -8 19 42]]
POTE generator: ~6/5 = 315.181
Optimal GPV sequence: 19, 80, 99, 217, 316, 415
Badness: 0.027431
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 3136/3125, 4375/4374
Mapping: [⟨1 5 6 12 -6], ⟨0 -13 -14 -35 36]]
POTE generator: ~6/5 = 315.251
Optimal GPV sequence: 19, 99, 118
Badness: 0.049711
Paralytic
The paralytic temperament (118&217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118&217 tempers out 1001/1000, 1575/1573, and 3584/3575.
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 4375/4374
Mapping: [⟨1 5 6 12 25], ⟨0 -13 -14 -35 -82]]
POTE generator: ~6/5 = 315.220
Optimal GPV sequence: 19e, 99e, 118, 217, 335, 552d, 887dd
Badness: 0.036027
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
Mapping: [⟨1 5 6 12 25 -16], ⟨0 -13 -14 -35 -82 75]]
POTE generator: ~6/5 = 315.214
Optimal GPV sequence: 99e, 118, 217, 552d, 769de
Badness: 0.044710
Paraklein
The paraklein temperament (19e&118) is another 13-limit extension of paralytic, which equates 13/11 with 32/27, 14/13 with 15/14, 25/24 with 26/25, and 27/26 with 28/27.
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 625/624, 729/728
Mapping: [⟨1 5 6 12 25 15], ⟨0 -13 -14 -35 -82 -43]]
POTE generator: ~6/5 = 315.225
Optimal GPV sequence: 19e, 99ef, 118, 217ff, 335ff
Badness: 0.037618
Parkleismic
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 19e, 80, 179, 259cd
Badness: 0.055884
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 176/175, 325/324, 1375/1372
Mapping: [⟨1 5 6 12 20 10], ⟨0 -13 -14 -35 -63 -24]]
POTE generator: ~6/5 = 315.075
Optimal GPV sequence: 19e, 80, 179
Badness: 0.036559
Paradigmic
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 19, 61d, 80, 99e, 179e
Badness: 0.041720
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 325/324, 540/539, 832/825
Mapping: [⟨1 5 6 12 -1 10], ⟨0 -13 -14 -35 17 -24]]
POTE generator: ~6/5 = 315.080
Optimal GPV sequence: 19, 61d, 80, 99e, 179e
Badness: 0.035781
Semiparakleismic
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 80, 118, 198, 316, 514c, 830c
Badness: 0.034208
Semiparamint
This extension was named semiparakleismic in the earlier materials.
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
Mapping: [⟨2 10 12 24 19 -1], ⟨0 -13 -14 -35 -23 16]]
POTE generator: ~6/5 = 315.156
Optimal GPV sequence: 80, 118, 198
Badness: 0.033775
Semiparawolf
This extension was named gentsemiparakleismic in the earlier materials.
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 325/324, 364/363, 3136/3125
Mapping: [⟨2 10 12 24 19 20], ⟨0 -13 -14 -35 -23 -24]]
POTE generator: ~6/5 = 315.184
Optimal GPV sequence: 80, 118f, 198f
Badness: 0.040467
Counterkleismic
- See also: High badness temperaments #Counterhanson
In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, [-20 -24 25⟩, the amount by which six major dieses (648/625) fall short of the classic major third (5/4). It can be described as 19&224 temperament (counterkleismic, named by analogy to catakleismic and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).
Subgroup: 2.3.5.7
Comma list: 4375/4374, 158203125/157351936
Mapping: [⟨1 -5 -4 -18], ⟨0 25 24 79]]
Wedgie: ⟨⟨25 24 79 -20 55 116]]
POTE generator: ~6/5 = 316.060
Optimal GPV sequence: 19, 205, 224, 243, 467
Badness: 0.090553
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4375/4374, 2097152/2096325
Mapping: [⟨1 -5 -4 -18 19], ⟨0 25 24 79 -59]]
POTE generator: ~6/5 = 316.071
Optimal GPV sequence: 19, 205, 224
Badness: 0.070952
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 625/624, 729/728, 10985/10976
Mapping: [⟨1 -5 -4 -18 19 -15], ⟨0 25 24 79 -59 71]]
POTE generator: ~6/5 = 316.070
Optimal GPV sequence: 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef
Badness: 0.033874
Counterlytic
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 4375/4374, 496125/495616
Mapping: [⟨1 -5 -4 -18 -40], ⟨0 25 24 79 165]]
POTE generator: ~6/5 = 316.065
Optimal GPV sequence: 19e, 205e, 224
Badness: 0.065400
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 729/728, 1375/1372, 10985/10976
Mapping: [⟨1 -5 -4 -18 -40 -15], ⟨0 25 24 79 165 71]]
POTE generator: ~6/5 = 316.065
Optimal GPV sequence: 19e, 205e, 224
Badness: 0.029782
Quincy
Subgroup: 2.3.5.7
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
Optimal GPV sequence: 72, 217, 289
Badness: 0.079657
11-limit
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 72, 217, 289
Badness: 0.030875
13-limit
Subgroup: 2.3.5.7.11.13
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
Optimal GPV sequence: 72, 145, 217, 289
Badness: 0.023862
17-limit
Subgroup: 2.3.5.7.11.13.17
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
Optimal GPV sequence: 72, 145, 217, 289
Badness: 0.014741
19-limit
Subgroup: 2.3.5.7.11.13.17.19
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
Optimal GPV sequence: 72, 145, 217
Badness: 0.015197
Trideci
- See also: High badness temperaments #Tridecatonic
The trideci temperament (26&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 tridecatonic temperament, but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name trideci comes from "tridecim" (Latin for "thirteen").
Subgroup: 2.3.5.7
Comma list: 4375/4374, 83349/81920
Mapping: [⟨13 21 31 36], ⟨0 -1 -2 1]]
POTE generator: ~3/2 = 699.1410
Optimal GPV sequence: 26, 65, 91, 156d, 247cdd
Badness: 0.184585
11-limit
Subgroup: 2.3.5.7.11
Comma list: 245/242, 385/384, 4375/4374
Mapping: [⟨13 21 31 36 45], ⟨0 -1 -2 1 0]]
POTE generator: ~3/2 = 699.6179
Optimal GPV sequence: 26, 65, 91, 156d, 247cdde
Badness: 0.084590
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 245/242, 325/324, 385/384
Mapping: [⟨13 21 31 36 45 48], ⟨0 -1 -2 1 0 0]]
POTE generator: ~3/2 = 699.2969
Optimal GPV sequence: 26, 65f, 91f, 156dff
Badness: 0.052366
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. Not only the semitwelfth, but also the ~5/4 can be used as a generator.
Subgroup: 2.3.5
Comma: [-52 -17 34⟩
Mapping: [⟨17 0 26], ⟨0 2 1]]
Mapping generators: ~25/24, ~[26 9 -17⟩
POTE generator: ~[26 9 -17⟩ = 950.9746
Optimal GPV sequence: 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797
Badness: 0.077072
7-limit
Subgroup: 2.3.5.7
Comma list: 4375/4374, 193119049072265625/193091834023510016
Mapping: [⟨17 0 26 -87], ⟨0 2 1 10]]
Wedgie: ⟨⟨34 17 170 -52 174 347]]
POTE generator: ~822083584/474609375 = 950.9995
Optimal GPV sequence: 289, 323, 612, 935, 1547
Badness: 0.041658
11-limit
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 41503/41472, 1879453125/1879048192
Mapping: [⟨17 0 26 -87 207], ⟨0 2 1 10 -11]]
POTE generators: ~822083584/474609375 = 950.9749
Optimal GPV sequence: 289, 323, 612
Badness: 0.063706
Palladium
The name of palladium temperament 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.
Subgroup: 2.3.5.7
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
Optimal GPV sequence: 46, 368, 414, 460, 874d
Badness: 0.308505
11-limit
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 46, 368, 414, 460, 874de
Badness: 0.073783
13-limit
Subgroup: 2.3.5.7.11.13
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
Optimal GPV sequence: 46, 368, 414, 460, 874de, 1334de
Badness: 0.040751
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
Mapping: [⟨46 73 107 129 159 170 188], ⟨0 -1 -2 1 1 2 0]]
POTE generator: ~3/2 = 701.6425
Optimal GPV sequence: 46, 368, 414, 460, 874de, 1334deg
Badness: 0.022441
Monzism
The monzism temperament (53&612) 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.
Subgroup: 2.3.5.7
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
Optimal GPV sequence: 53, 559, 612, 1277, 1889
Badness: 0.046569
11-limit
Subgroup: 2.3.5.7.11
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
Optimal GPV sequence: 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: [⟨1 2 10 -25 46 23], ⟨0 -2 -37 134 -205 -93]]
POTE generator: ~231/200 = 249.0199
Optimal GPV sequence: 53, 559, 612
Badness: 0.053780