Ragismic microtemperaments
This is a collection of rank-2 temperaments tempering out the ragisma, 4375/4374 = [-1 -7 4 1⟩. The ragisma is 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.
Microtemperaments considered below are ennealimmal, supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, orga, chlorine, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
- Hystrix (+36/35) → Porcupine family
- Rhinoceros (+49/48) → Unicorn family
- Crepuscular (+50/49) → Jubilismic clan and Fifive family
- Modus (+64/63) → Tetracot family
- Flattone (+81/80) → Meantone family
- Sensi (+126/125 or 245/243) → Sensipent family and Sensamagic clan
- Catakleismic (+225/224) → Kleismic family
- Unidec (+1029/1024) → Gamelismic clan
- Quartonic (+1728/1715 or 4000/3969) → Quartonic family
- Srutal (+2048/2025) → Diaschismic family
- Maja (+2430/2401 or 3125/3087) → Maja family
- Amity (+5120/5103) → Amity family
- Pontiac (+32805/32768) → Schismatic family
- Zarvo (+33075/32768) → Gravity family
- Whirrschmidt (+393216/390625) → Würschmidt family
- Mitonic (+2100875/2097152) → Minortonic family
- Vishnu (+29360128/29296875) → Vishnuzmic family
- Vulture (+33554432/33480783) → Vulture family
- Trillium (+[40 -22 -1 -1⟩) → Tricot family
- Vacuum (+[-68 18 17⟩) → Vavoom family
- Unlit (+[41 -20 -4⟩) → Undim family
- Quindro (+[56 -28 -5⟩) → Quindromeda family
- Dzelic (+[-223 47 -11 62⟩) → 37th-octave temperaments
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, a stack of 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.
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.
7-limit ennealimmal's S-expression-based comma list is {S25/S27, S49}. Interestingly, the landscape comma is equal to S49/(S25/S27) while the wizma is equal to S49*S25/S27.
For the 5-limit temperament, see Ennealimma#Ennealimmal.
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
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3129 (~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 ET sequence: 27, 45, 72, 99, 171, 441, 612
Badness: 0.003610
11-limit
The ennealimmal temperament can be described as 99e & 171e, 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]]
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4679 (~36/35 = 48.8654)
Optimal ET 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]]
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
Optimal ET sequence: 99e, 171e, 270
Badness: 0.029404
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 715/714, 1001/1000, 1716/1715, 4096/4095, 4375/4374
Mapping: [⟨9 1 1 12 -75 93 -3], ⟨0 2 3 2 16 -9 6]]
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
Optimal ET sequence: 99e, 171e, 270
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 715/714, 1001/1000, 1216/1215, 1716/1715, 4096/4095, 4375/4374
Mapping: [⟨9 1 1 12 -75 93 -3 -48], ⟨0 2 3 2 16 -9 6 13]]
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
Optimal ET sequence: 99e, 171e, 270
Ennealimmalis
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2401/2400, 4375/4374, 5632/5625
Mapping: [⟨9 1 1 12 -75 -106], ⟨0 2 3 2 16 21]]
Optimal tuning (CTE): ~27/25 = 1\9, ~5/3 = 884.4560 (~36/35 = 48.8773)
Optimal ET sequence: 99ef, 171ef, 270, 639, 909, 1179, 2088bce
Badness: 0.022068
Ennealimmia
The ennealimmia temperament is an alternative extension and can be described as 99 & 171, which tempers out 131072/130977 (olympia).
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]]
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4089 (~36/35 = 48.9244)
Optimal ET 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]]
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
Optimal ET sequence: 99, 171, 270, 711, 981, 1692e, 2673e
Badness: 0.016607
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 936/935, 2080/2079, 2401/2400, 4096/4095, 4375/4374
Mapping: [⟨9 1 1 12 124 93 -3], ⟨0 2 3 2 -14 -9 6]]
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
Optimal ET sequence: 99, 171, 270
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 936/935, 1216/1215, 2080/2079, 2401/2400, 4096/4095, 4375/4374
Mapping: [⟨9 1 1 12 124 93 -3 -48], ⟨0 2 3 2 -14 -9 6 13]]
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
Optimal ET sequence: 99, 171, 270
Ennealimnic
Ennealimnic (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]]
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9386 (~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 ET 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]]
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9920 (~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 ET 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]]
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9981 (~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 ET sequence: 72, 171, 243
Badness: 0.014602
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 243/242, 364/363, 375/374, 441/440, 513/512, 595/594
Mapping: [⟨9 1 1 12 -2 -33 -3 78], ⟨0 2 3 2 5 10 6 -6]]
Optimal ET sequence: 72, 171, 243
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]]
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
Optimal ET sequence: 27e, 45ef, 72
Badness: 0.020697
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 169/168, 221/220, 243/242, 325/324, 441/440
Mapping: [⟨9 1 1 12 -2 20 -3], ⟨0 2 3 2 5 2 6]]
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
Optimal ET sequence: 27eg, 45efg, 72
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 169/168, 221/220, 243/242, 325/324, 441/440
Mapping: [⟨9 1 1 12 -2 20 -3 25], ⟨0 2 3 2 5 2 6 2]]
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
Optimal ET sequence: 27eg, 45efg, 72
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]]
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.8298 (~36/35 = 49.5036)
Optimal ET 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]]
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
Optimal ET sequence: 27, 45f, 72, 171ef, 243eff
Badness: 0.030325
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 169/168, 221/220, 325/324, 385/384, 1375/1372
Mapping: [⟨9 1 1 12 51 20 50], ⟨0 2 3 2 -3 2 -2]]
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
Optimal ET sequence: 27, 45f, 72
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 153/152, 169/168, 221/220, 325/324, 385/384, 1375/1372
Mapping: [⟨9 1 1 12 51 20 50 25], ⟨0 2 3 2 -3 2 -2 2]]
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
Optimal ET sequence: 27, 45f, 72
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. Notably, every one of these commas is part of one or more known infinite comma families; see directly below.
Its S-expression-based comma list is {(S22/S24 = S55 = S25/S27 * S99,) S25/S27, S49, S33/S35 = S99}.
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
Optimal tuning (POTE): ~80/77 = 1\18, ~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 ET 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]]
Optimal tuning (POTE): ~27/26 = 1\18, ~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 ET sequence: 72, 198, 270
Badness: 0.012505
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 676/675, 715/714, 1001/1000, 1716/1715, 3025/3024
Mapping: [⟨18 0 -1 22 48 -19 -12], ⟨0 2 3 2 1 6 6]]
Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837
Optimal ET sequence: 72, 198g, 270
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 676/675, 715/714, 1001/1000, 1331/1330, 1716/1715, 3025/3024
Mapping: [⟨18 0 -1 22 48 -19 -12 48 105], ⟨0 2 3 2 1 6 6 -2]]
Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837
Optimal ET sequence: 72, 198g, 270
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
Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
Optimal ET sequence: 126, 144, 270, 684, 954
Badness: 0.013104
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 2401/2400, 2431/2430, 3025/3024, 4225/4224, 4375/4374
Mapping: [⟨18 0 -1 22 48 88 -119], ⟨0 4 6 4 2 -3 27]]
- mapping generators: ~80/77, ~1053/800
Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
Optimal ET sequence: 270, 684, 954
Badness: 0.013104
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 2401/2400, 2431/2430, 2926/2925, 3025/3024, 4225/4224, 4375/4374
Mapping: [⟨18 0 -1 22 48 88 -119 -2], ⟨0 4 6 4 2 -3 27 11]]
- mapping generators: ~80/77, ~1053/800
Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
Optimal ET sequence: 270, 684h, 954h, 1224
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
Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3367
Optimal ET 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]]
Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3375
Optimal ET 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
Optimal tuning (POTE): ~27/25 = 1\9, ~25/22 = 221.0717
Optimal ET 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
Optimal tuning (POTE): ~2744/2673 = 1\27, ~2352/1375 = 928.8000
Optimal ET sequence: 27, 243, 270, 783, 1053, 1323
Badness: 0.029812
Rhodium
Rhodium splits the ennealimmal period in five parts and thereby features a period of 9 × 5 = 45, thus the name is given after the 45th element.
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 4375/4374, 117440512/117406179
Mapping: [⟨45 1 -1 56 226], ⟨0 2 3 2 -2]]
- mapping generators: ~3072/3025, ~55/32
Optimal tunings:
- CTE: ~3072/3025 = 1\45, ~55/32 = 937.6658 (~385/384 = 4.3325)
- CWE: ~3072/3025 = 1\45, ~55/32 = 937.6630 (~385/384 = 4.3397)
Optimal ET sequence: 45, 225c, 270, 1125, 1395, 1665, 5265d
Badness: 0.0381
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2401/2400, 4225/4224, 4375/4374, 6656/6655
Mapping: [⟨45 1 -1 56 226 272], ⟨0 2 3 2 -2 -3]]
Optimal tunings:
- CTE: ~66/65 = 1\45, ~55/32 = 937.6569 (~385/384 = 4.3236)
- CWE: ~66/65 = 1\45, ~55/32 = 937.6515 (~385/384 = 4.3182)
Optimal ET sequence: 45, 270, 855, 1125, 1395, 1665, 3060d, 4725df
Badness: 0.0226
Supermajor
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (215)/3, 46 give (219)/5, and 75 give (230)/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 is not enough notes for you, there is 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 ]]
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 435.082
Optimal ET 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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082
Optimal ET 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-limit, and 494edo shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use 665edo for a tuning.
For the 5-limit temperament, see 19th-octave temperaments#(5-limit) enneadecal.
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
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)
Optimal ET sequence: 19, …, 152, 171, 665, 836, 1007, 2185, 3192c
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]]
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)
Optimal ET sequence: 19, 133d, 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]]
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)
Optimal ET sequence: 19, 133df, 152f, 323ef
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
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)
Optimal ET sequence: 152, 342, 836, 1178, 2014, 3192ce, 5206ce
Badness: 0.009985
Hemienneadecalis
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256
Mapping: [⟨38 0 28 -74 11 -281], ⟨0 1 1 3 2 7]]
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)
Optimal ET sequence: 152f, 342f, 494
Badness: 0.020782
Hemienneadec
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
Mapping: [⟨38 0 28 -74 11 502], ⟨0 1 1 3 2 -6]]
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444)
Optimal ET sequence: 152, 342, 494, 1330, 1824, 2318d
Badness: 0.030391
Semihemienneadecal
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078
Mapping: [⟨38 1 29 -71 13 111], ⟨0 2 2 6 4 1]]
- mapping generators: ~55/54 = 1\38, ~55/54, ~429/250
Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)
Optimal ET sequence: 190, 304d, 494, 684, 1178, 2850, 4028ce
Badness: 0.014694
Kalium
Named after the 19th element, potassium, and after an archaic variant of the element's name to resolve a name conflict. 19/16 can be used as a generator. Since it is enfactored in the 17-limit and lower, it makes no sense to name it for the lower subgroups.
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344
Mapping: [⟨19 3 17 -28 82 92 159 78], ⟨0 10 10 30 -6 -8 -30 1]]
Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244
Optimal ET sequence: 855, 988, 1843
Semidimi
- For the 5-limit version of this temperament, see High badness temperaments #Semidimi.
The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit [-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 ]]
Optimal tuning (POTE): ~2 = 1\1, ~35/27 = 449.1270
Optimal ET 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]]
- mapping generators: ~1157625/1048576, ~27/20
Wedgie: ⟨⟨ 21 56 -77 40 -181 -336 ]]
Optimal tuning (POTE): ~1157625/1048576 = 1\7, ~27/20 = 519.716
Optimal ET 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]]
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704
Optimal ET 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]]
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.706
Optimal ET sequence: 7, 217, 224, 441, 665, 1771eef
Badness: 0.023132
Abigail
Abigail temperament tempers out the pessoalisma in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.[1]
For the 5-limit temperament, see Very high accuracy temperaments#Abigail.
Subgroup: 2.3.5.7
Comma list: 4375/4374, 2147483648/2144153025
Mapping: [⟨2 7 13 -1], ⟨0 -11 -24 19]]
- mapping generators: ~46305/32768, ~27/20
Wedgie: ⟨⟨ 22 48 -38 25 -122 -223 ]]
Optimal tuning (POTE): ~46305/32768 = 1\2, ~6912/6125 = 208.899
Optimal ET 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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901
Optimal ET 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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903
Optimal ET sequence: 46, 178, 224, 270, 494, 764, 1258
Badness: 0.008856
Gamera
For the 5-limit temperament, see High badness temperaments#Gamera.
Subgroup: 2.3.5.7
Comma list: 4375/4374, 589824/588245
Mapping: [⟨1 6 10 3], ⟨0 -23 -40 -1]]
- mapping generators: ~2, ~8/7
Wedgie: ⟨⟨ 23 40 1 10 -63 -110 ]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 230.336
Optimal ET 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
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3370
Optimal ET 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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373
Optimal ET 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
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1642
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1628
Optimal ET sequence: 73f, 125f, 198, 323, 521
Badness: 0.044
Orga
Subgroup: 2.3.5.7
Comma list: 4375/4374, 54975581388800/54936068900769
Mapping: [⟨2 21 36 5], ⟨0 -29 -51 1]]
- mapping generators: ~7411887/5242880, ~1310720/1058841
Wedgie: ⟨⟨ 58 102 -2 27 -166 -291 ]]
Optimal tuning (POTE): ~7411887/5242880 = 1\2, ~8/7 = 231.104
Optimal ET 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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
Optimal ET 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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
Optimal ET sequence: 26, 244, 270, 566, 836f, 1106f
Badness: 0.021762
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 list: [-52 -17 34⟩
Mapping: [⟨17 0 26], ⟨0 2 1]]
- mapping generators: ~25/24, ~[26 9 -17⟩
Optimal tuning (POTE): ~[26 9 -17⟩ = 950.9746
Optimal ET 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, [-49 4 22 -3⟩
Mapping: [⟨17 0 26 -87], ⟨0 2 1 10]]
Wedgie: ⟨⟨ 34 17 170 -52 174 347 ]]
Optimal tuning (POTE): ~[24 -5 -9 2⟩ = 950.9995
Optimal ET 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]]
Optimal tuning (POTE): ~[24 -5 -9 2⟩ = 950.9749
Optimal ET sequence: 289, 323, 612
Badness: 0.063706
Seniority
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 ]]
Optimal tuning (POTE): ~2 = 1\1, ~3087/2560 = 322.804
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
Optimal ET 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]]
Optimal tuning (POTE): ~77/64 = 322.793
Optimal ET sequence: 26, 119c, 145, 171, 316ef, 487eef
Badness: 0.026562
Monzismic
- For the 5-limit version of this temperament, see Very high accuracy temperaments #Monzismic.
The monzismic temperament (53 & 612) tempers out the monzisma, [54 -37 2⟩, and in the 7-limit, the nanisma, [109 -67 0 -1⟩, as well as the ragisma, 4375/4374.
Subgroup: 2.3.5.7
Comma list: 4375/4374, [-55 30 2 1⟩
Mapping: [⟨1 2 10 -25], ⟨0 -2 -37 134]]
Wedgie: ⟨⟨ 2 37 -134 54 -218 -415 ]]
Optimal tuning (POTE): ~2 = 1\1, ~[-27 11 3 1⟩ = 249.0207
Optimal ET sequence: 53, …, 559, 612, 1277, 1889
Badness: 0.046569
Monzism
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]]
Optimal tuning (POTE): ~231/200 = 249.0193
Optimal ET 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]]
Optimal tuning (POTE): ~231/200 = 249.0199
Optimal ET sequence: 53, 559, 612
Badness: 0.053780
Semidimfourth
- For the 5-limit version of this temperament, see High badness temperaments #Semidimfourth.
The semidimfourth temperament is featured by a semi-diminished fourth inverval which is 128/125 above the pythagorean major third 81/64. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, 235298/234375.
Subgroup: 2.3.5.7
Comma list: 4375/4374, 235298/234375
Mapping: [⟨1 21 28 36], ⟨0 -31 -41 -53]]
Wedgie: ⟨⟨ 31 41 53 -7 -3 8 ]]
Optimal tuning (POTE): ~2 = 1\1, ~35/27 = 448.456
Optimal ET 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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547
Optimal ET 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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545
Optimal ET 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]]
- mapping generators: ~2, ~6/5
Wedgie: ⟨⟨ 32 33 92 -22 56 121 ]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557
Optimal ET sequence: 19, …, 251, 270, 2449c, 2719c, 2989bc
Badness: 0.056184
11-limit
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 41503/41472, 172032/171875
Mapping: [⟨1 10 11 27 -16], ⟨0 -32 -33 -92 74]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.553
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.554
Optimal ET sequence: 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf
Badness: 0.026028
Quasithird
The quasithird temperament is featured by a major third interval which is 1600000/1594323 (amity comma) or 5120/5103 (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 list: [55 -64 20⟩
Mapping: [⟨4 0 -11], ⟨0 5 16]]
- mapping generators: ~51200000/43046721, ~1594323/1280000
Optimal tuning (POTE): ~51200000/43046721, ~1594323/1280000 = 380.395
Optimal ET 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, [-60 29 0 5⟩
Mapping: [⟨4 0 -11 48], ⟨0 5 16 -29]]
Wedgie: ⟨⟨ 20 64 -116 55 -240 -449 ]]
Optimal tuning (POTE): ~65536/55125 = 1\4, ~5103/4096 = 380.388
Optimal ET 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]]
Optimal tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387)
Optimal ET 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]]
Optimal tuning (POTE): ~81/65 = 380.385 (or ~22/21 = 80.385)
Optimal ET sequence: 60d, 164, 224, 388, 612, 836, 1448f, 2284f
Badness: 0.029501
Deca
- For 5-limit version of this temperament, see 10th-octave temperaments #Neon.
Deca temperament has a period of 1/10 octave and tempers out the linus comma, [11 -10 -10 10⟩, neon comma [21 60 -50⟩ 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]]
- mapping generators: ~15/14, ~6/5
Wedgie: ⟨⟨ 50 60 110 -21 34 87 ]]
Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.577
Optimal ET sequence: 80, 190, 270, 1270, 1540, 1810, 2080
Badness: 0.080637
Badness (Dirichlet): 2.041
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 391314/390625
Mapping: [⟨10 4 9 2 18], ⟨0 5 6 11 7]]
Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.582
Optimal ET sequence: 80, 190, 270, 1000, 1270, 1540e, 1810e
Badness: 0.024329
Badness (Dirichlet): 0.804
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]]
Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.602 (~40/39 = 44.398)
Optimal ET sequence: 80, 190, 270, 730, 1000
Badness: 0.016810
Badness (Dirichlet): 0.695
no-17's 19-limit
Subgroup: 2.3.5.7.11.13.19
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374, 1521/1520
Mapping: [⟨10 4 9 2 18 37 33], ⟨0 5 6 11 7 0 4]]
Optimal tuning (CTE): ~15/14 = 1\10, ~6/5 = 315.581 (~39/38 = 44.419)
Optimal ET sequence: 80, 190, 270, 730, 1000
Badness (Dirichlet): 0.556
Keenanose
Keenanose is named for the fact that it uses 385/384, the keenanisma, as the generator.
Subgroup: 2.3.5.7
Comma list: 4375/4374, [-56 1 -8 26⟩
Mapping: [⟨1 2 3 3], ⟨0 -112 -183 -52]]
- mapping generators: ~2, ~[21 3 1 -10⟩
Optimal tuning (CTE): ~2 = 1\1, ~[21 3 1 -10⟩ = 4.4465
Optimal ET sequence: 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd
Badness: 0.0858
11-limit
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 117649/117612, 67110351/67108864
Mapping: [⟨1 2 3 3 3], ⟨0 -112 -183 -52 124]]
Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4465
Optimal ET sequence: 270, 1349, 1619, 1889, 2159, 11065, 13224
Badness: 0.0308
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612
Mapping: [⟨1 2 3 3 3 3], ⟨0 -112 -183 -52 124 189]]
Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4466
Optimal ET sequence: 270, 1079, 1349, 1619, 1889, 4048
Badness: 0.0213
Aluminium
Aluminium is named after the 13th element, and tempers out the [92 -39 -13⟩ comma which sets 135/128 interval to be equal to 1/13th of the octave.
Subgroup: 2.3.5
Comma list: [92 -39 -13⟩
Mapping: [⟨13 0 92], ⟨0 1 -3]]
- mapping generators: ~135/128, ~3
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 701.9897
Optimal ET sequence: 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc
Badness: 0.123
7-limit
Subgroup: 2.3.5.7
Comma list: 4375/4374, [92 -39 -13⟩
Mapping: [⟨13 0 92 -355], ⟨0 1 -3 19]]
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0024
Optimal ET sequence: 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b
Badness: 0.126
11-limit
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 234375/234256, 2097152/2096325
Mapping: [⟨13 0 92 -355 148], ⟨0 1 -3 19 -5]]
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042
Optimal ET sequence: 494, 1053, 1547, 3588e, 5135e
Badness: 0.0421
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078
Mapping: [⟨13 0 92 -355 148 419], ⟨0 1 -3 19 -5 -18]]
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099
Optimal ET sequence: 494, 1547, 2041, 4576def
Badness: 0.0286
Countritonic
- For the 5-limit version of this temperament, see Schismic-Mercator equivalence continuum #Countritonic and High badness temperaments #Countritonic
Countritonic (co-un-tritonic) can be described as the 53 & 422 temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit.
Subgroup: 2.3.5.7
Comma list: 4375/4374, 68719476736/68356598625
Mapping: [⟨1 6 19 -33], ⟨0 -9 -34 73]]
- mapping generators: ~2, ~45927/32768
Optimal tuning (CTE): ~2 = 1\1, ~45927/32768 = 588.6216
Optimal ET sequence: 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd
Badness: 0.133
11-limit
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 5632/5625, 2621440/2614689
Mapping: [⟨1 6 19 -13 79], ⟨0 -9 -34 73 154]]
Optimal tuning (CTE): ~2 = 1\1, ~539/384 = 588.6258
Optimal ET sequence: 53, 316e, 369, 422, 791e, 1213cde
Badness: 0.0707
13-limit
Subgroup: 2.3.5.7.11
Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625
Mapping: [⟨1 6 19 -13 79], ⟨0 -9 -34 73 154 -74]]
Optimal tuning (CTE): ~2 = 1\1, ~128/91 = 588.6277
Optimal ET sequence: 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff
Badness: 0.0366
Quatracot
Subgroup: 2.3.5.7
Comma list: 4375/4374, [-32 5 14 -3⟩
Mapping: [⟨2 7 7 23], ⟨0 -13 -8 -59]]
- mapping generators: ~2278125/1605632, ~448/405
Wedgie: ⟨⟨ 26 16 118 -35 114 229 ]]
Optimal tuning (POTE): ~2278125/1605632 = 1\2, ~448/405 = 176.805
Optimal ET 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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 176.806
Optimal ET 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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~195/176 = 176.804
Optimal ET sequence: 190, 224, 414, 638, 1690bcc, 2328bccde
Badness: 0.022643
Moulin
Moulin has a generator of 22/13, and it is named after the Law & Order: Special Victims Unit episode Season 22, Episode 13. "Trick-Rolled At The Moulin". It can be described as the 494 & 1619 temperament.
Subgroup: 2.3.5.7
Comma list: 4375/4374, [-88 2 45 -7⟩
Mapping: [⟨1 57 38 248], ⟨0 -73 -47 -323]]
- mapping generators: ~2, ~6422528/3796875
Optimal tuning (CTE): ~2 = 1\1, ~6422528/3796875 = 910.9323
Optimal ET sequence: 494, 1125, 1619
Badness: 0.234
11-limit
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 759375/758912, 100663296/100656875
Mapping: [⟨1 57 38 248 -14], ⟨0 -73 -47 -323 23]]
Optimal tuning (CTE): ~2 = 1\1, ~1024/605 = 910.9323
Optimal ET sequence: 494, 1125, 1619, 2113
Badness: 0.0678
13-limit
Since 11/8 is within 23 generators, the 25 tone MOS (4L 21s) of this temperament contains the 8:11:13 triad.
Subgroup: 2.3.5.7.11.13
Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
Mapping: [⟨1 57 38 248 -14 -13], ⟨0 -73 -47 -323 23 22]]
Optimal tuning (CTE): ~2 = 1\1, ~22/13 = 910.9323
Optimal ET sequence: 494, 1125, 1619, 2113
Badness: 0.0271
Palladium
- For the 5-limit version of this temperament, see 46th-octave temperaments.
The name of the palladium temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, [-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, [-51 8 2 12⟩
Mapping: [⟨46 0 -39 202], ⟨0 1 2 -1]]
- mapping generators: ~83349/81920, ~3
Wedgie: ⟨⟨ 46 92 -46 39 -202 -365 ]]
Optimal tuning (POTE): ~83349/81920 = 1\46, ~3/2 = 701.6074
Optimal ET sequence: 46, 368, 414, 460, 874d
Badness: 0.308505
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 134775333/134217728
Mapping: [⟨46 0 -39 202 232], ⟨0 1 2 -1 -1]]
Optimal tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951
Optimal ET 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 0 -39 202 232 316], ⟨0 1 2 -1 -1 -2]]
Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6419
Optimal ET 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 0 -39 202 232 316 188], ⟨0 1 2 -1 -1 -2 0]]
Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425
Optimal ET sequence: 46, 368, 414, 460, 874de, 1334deg
Badness: 0.022441
Oviminor
Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past egads, though it is less accurate.
Subgroup: 2.3.5.7
Comma list: 4375/4374, [-100 53 48 -34⟩
Mapping: [⟨1 50 51 147], ⟨0 -184 -185 -548]]
- mapping generators: ~2, ~6/5
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 315.7501
Optimal ET sequence: 19, …, 1600, 1619, 4838, 6457c
Badness: 0.582
Octoid
For the 5-limit temperament, see 8th-octave temperaments#Octoid (5-limit).
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
Optimal tuning (POTE): ~49/45 = 1\8, ~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]
Optimal ET sequence: 8d, 72, 152, 224
Badness: 0.042670
11-limit
The 11-limit is the last place where all the extensions of octoid shown here agree in the mappings of primes. 80edo is an alternative tuning for octoid in the 11-limit; though 72edo does better for minimaxing the damage on the 11-odd-limit, 80edo damages prime 7 in favor of practically-just 17/16's, 11/10's and 9/7's. In higher limits, if one wants to use 80edo as the tuning, one must use octopus — not octoid — as 80edo doesn't temper 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.
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]]
Optimal tuning (POTE): ~12/11 = 1\8, ~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]
Optimal ET 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]]
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.905
Optimal ET sequence: 72, 152f, 224
Badness: 0.015274
- Music
- Dreyfus (archived 2010) by Gene Ward Smith – SoundCloud | details | play – octoid[72] in 224edo tuning
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]]
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.842
Optimal ET 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]]
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.932
Optimal ET sequence: 72, 152fg, 224
Badness: 0.016036
Octopus
A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is 80edo, which has a strong sharp tendency that can be thought of as matching the sharpness of mapping 19/16 to 1\4 = 300 ¢.
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]]
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.892
Optimal ET 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]]
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.811
Optimal ET 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]]
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 584.064
Optimal ET 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 2 6 6 32 67], ⟨0 3 4 5 3 -1]]
- mapping generators: ~448/429, ~7/5
Optimal tuning (POTE): ~448/429 = 1\16, ~13/8 = 841.015
Optimal ET 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 2 6 6 32 67 81], ⟨0 3 4 5 3 -1 -2]]
Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932
Optimal ET 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 2 6 6 32 67 81 68], ⟨0 -3 -4 -5 -3 1 2 0]]
Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896
Optimal ET sequence: 80, 144, 224, 304dh, 528dghh
Badness: 0.023731
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-limit, 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]]
- mapping generators: ~2, ~6/5
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.240
Optimal ET 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 ]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.181
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.251
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.220
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.214
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.225
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.060
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.075
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.096
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.080
Optimal ET 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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.181
Optimal ET 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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.156
Optimal ET 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]]
Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 315.184
Optimal ET sequence: 80, 118f, 198f
Badness: 0.040467
Counterkleismic
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 20 20 61], ⟨0 -25 -24 -79]]
- mapping generators: ~2, ~5/3
Wedgie: ⟨⟨ 25 24 79 -20 55 116 ]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.060
Optimal ET 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 20 20 61 -40], ⟨0 -25 -24 -79 59]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.071
Optimal ET 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 20 20 61 -40 56], ⟨0 -25 -24 -79 59 -71]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.070
Optimal ET 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 20 20 61 125], ⟨0 -25 -24 -79 -165]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065
Optimal ET 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 20 20 61 125 56], ⟨0 -25 -24 -79 -165 -71]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065
Optimal ET 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 ]]
Optimal tuning (POTE): ~2 = 1\1, ~1728/1715 = 16.613
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.613
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.594
Optimal ET sequence: 72, 145, 217
Badness: 0.015197
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 ]]
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.287
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.286
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.310
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.246
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.239
Optimal ET sequence: 45, 46, 91, 137d
Badness: 0.051893
Trideci
- For the 5-limit version of this temperament, see 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 0 -11 57], ⟨0 1 2 -1]]
Optimal tuning (POTE): ~256/245 = 1\13, ~3/2 = 699.1410
Optimal ET 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 0 -11 57 45], ⟨0 1 2 -1 0]]
Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.6179
Optimal ET 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 0 -11 57 45 48], ⟨0 1 2 -1 0 0]]
Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.2969
Optimal ET sequence: 26, 65f, 91f, 156dff
Badness: 0.052366
Counterorson
Counterorson tempers out the [147 -103 7⟩ comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the semicomma family, 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on 53edo.
Subgroup: 2.3.5.7
Comma list: 4375/4374, [154 -54 -21 -7⟩
Mapping: [⟨1 0 -21 85], ⟨0 7 103 -363]]
Optimal tuning (CTE): ~2 = 1\1, ~[66 -23 -9 -3⟩ = 271.7113
Optimal ET sequence: 53, …, 1612, 1665, 1718
Badness: 0.312806