Ragismic microtemperaments

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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, 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:

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, 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)

Tuning ranges:

  • 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

See also: Chords of ennealimnic

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 to 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

Main article: 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

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 ]]

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

See also: Stratosphere

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

See also: Syntonic–kleismic equivalence continuum

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

Tuning ranges:

  • 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

Scales: octoid72, octoid80

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

Scales: octoid72, octoid80

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

Scales: octoid72, octoid80

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]]

Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.842

Optimal ET sequence: 72, 152fg, 224, 296, 520g

Badness: 0.014304

Scales: octoid72, octoid80

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

Scales: octoid72, octoid80

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

Scales: octoid72, octoid80

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

Scales: Octoid72, Octoid80

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

Scales: Octoid72, Octoid80

Hexadecoid

See also: 16th-octave temperaments

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

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-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

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 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

Notes

  1. ↑ [1]: "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things."
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