Breedsmic temperaments

From Xenharmonic Wiki
(Redirected from Subneutral)
Jump to navigation Jump to search

This page discusses miscellaneous rank-2 temperaments tempering out the breedsma, [-5 -1 -2 4 = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12EDO, for example) which does not possess a neutral third cannot be tempering out the breedsma.

The breedsma is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, 25/24. We might note also that 49/40 × 10/7 = 7/4 and 49/40 × (10/7)2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.

Temperaments discussed elsewhere include:

Hemififths

Main article: Hemififths

Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with 99EDO and 140EDO providing good tunings, and 239EDO an even better one; and other possible tunings are 160(1/25), giving just 5s, the 7- and 9-odd-limit minimax tuning, or 14(1/13), giving just 7s. It may be called the 41&58 temperament. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS[clarification needed].

By adding 243/242 (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99EDO is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding 144/143 brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be 16/13. 99 remains a good tuning choice.


Subgroup: 2.3.5.7

Comma list: 2401/2400, 5120/5103

Mapping: [1 1 -5 -1], 0 2 25 13]]

Wedgie⟨⟨2 25 13 35 15 -40]]

POTE generator: ~49/40 = 351.477

Minimax tuning:

[[1 0 0 0, [7/5 0 2/25 0, [0 0 1 0, [8/5 0 13/25 0]
Eigenmonzos: 2, 5

Algebraic generator: (2 + sqrt(2))/2

Vals41, 58, 99, 239, 338

Badness: 0.022243

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 896/891

Mapping: [1 1 -5 -1 2], 0 2 25 13 5]]

POTE generator: ~11/9 = 351.521

Vals: 17c, 41, 58, 99e

Badness: 0.023498

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 243/242, 364/363

Mapping: [1 1 -5 -1 2 4], 0 2 25 13 5 -1]]

POTE generator: ~11/9 = 351.573

Vals: 17c, 41, 58, 99ef, 157eff

Badness: 0.019090

Semihemi

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3388/3375, 5120/5103

Mapping: [2 0 -35 -15 -47], 0 2 25 13 34]]

POTE generator: ~49/40 = 351.505

Vals: 58, 140, 198

Badness: 0.042487

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845, 1716/1715

Mapping: [2 0 -35 -15 -47 -37], 0 2 25 13 34 28]]

POTE generator: ~49/40 = 351.502

Vals: 58, 140, 198, 536f

Badness: 0.021188

Quadrafifths

This has been logged as semihemififths in Graham Breed's temperament finder, but quadrafifths arguably makes more sense.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 5120/5103

Mapping: [1 1 -5 -1 8], 0 4 50 26 -31]]

POTE generator: ~243/220 = 175.7378

Vals: 41, 157, 198, 239, 676b, 915be

Badness: 0.040170

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 2401/2400, 3025/3024

Mapping: [1 1 -5 -1 8 10], 0 4 50 26 -31 -43]]

POTE generator: ~72/65 = 175.7470

Vals: 41, 157, 198, 437f, 635bcff

Badness: 0.031144

Tertiaseptal

Main article: Tertiaseptal

Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152, the rainy comma. It can be described as the 31&171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. 171EDO makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.


Subgroup: 2.3.5.7

Comma list: 2401/2400, 65625/65536

Mapping: [1 3 2 3], 0 -22 5 -3]]

Wedgie⟨⟨22 -5 3 -59 -57 21]]

POTE generator: ~256/245 = 77.191

Vals31, 109, 140, 171

Badness: 0.012995

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 65625/65536

Mapping: [1 3 2 3 7], 0 -22 5 -3 -55]]

POTE generator: ~256/245 = 77.227

Vals: 31, 109e, 140e, 171, 202

Badness: 0.035576

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 441/440, 625/624, 3584/3575

Mapping: [1 3 2 3 7 1], 0 -22 5 -3 -55 42]]

POTE generator: ~117/112 = 77.203

Vals: 31, 109e, 140e, 171

Badness: 0.036876

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575

Mapping: [1 3 2 3 7 1 1], 0 -22 5 -3 -55 42 48]]

POTE generator: ~68/65 = 77.201

Vals: 31, 109eg, 140e, 171

Badness: 0.027398

Tertia

Subgroup:2.3.5.7.11

Comma list: 385/384, 1331/1323, 1375/1372

Mapping: [1 3 2 3 5], 0 -22 5 -3 -24]]

POTE generator: ~22/21 = 77.173

Vals: 31, 109, 140, 171e, 311e

Badness: 0.030171

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 625/624, 1331/1323

Mapping: [1 3 2 3 5 1], 0 -22 5 -3 -24 42]]

POTE generator: ~22/21 = 77.158

Vals: 31, 109, 140, 311e, 451ee

Badness: 0.028384

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 385/384, 561/560, 625/624, 715/714

Mapping: [1 3 2 3 5 1 1], 0 -22 5 -3 -24 42 48]]

POTE generator: ~22/21 = 77.162

Vals: 31, 109g, 140, 311e, 451ee

Badness: 0.022416

Hemitert

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 65625/65536

Mapping: [1 3 2 3 6], 0 -44 10 -6 -79]]

POTE generator: ~45/44 = 38.596

Vals: 31, 280, 311, 342

Badness: 0.015633

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095

Mapping: [1 3 2 3 6 1], 0 -44 10 -6 -79 84]]

POTE generator: ~45/44 = 38.588

Vals: 31, 280, 311, 964f, 1275f, 1586cff

Badness: 0.033573

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095

Mapping: [1 3 2 3 6 1 1], 0 -44 10 -6 -79 84 96]]

POTE generator: ~45/44 = 38.589

Vals: 31, 280, 311, 653f, 964f

Badness: 0.025298

Harry

Main article: Harry
See also: Gravity family #Harry

Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&72 temperament. The period is half an octave, and the generator 21/20, with generator tunings of 9\130 or 14\202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices.

Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is ⟨⟨12 34 20 30 …]].

Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with ⟨⟨12 34 20 30 52 …]] as the octave wedgie. 130EDO is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.


Subgroup: 2.3.5.7

Comma list: 2401/2400, 19683/19600

Mapping: [2 4 7 7], 0 -6 -17 -10]]

Wedgie⟨⟨12 34 20 26 -2 -49]]

POTE generator: ~21/20 = 83.156

Vals14c, 58, 72, 130, 202, 534, 736b, 938b

Badness: 0.034077

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 4000/3993

Mapping: [2 4 7 7 9], 0 -6 -17 -10 -15]]

POTE generator: ~21/20 = 83.167

Vals: 14c, 58, 72, 130, 202

Badness: 0.015867

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 364/363, 441/440

Mapping: [2 4 7 7 9 11], 0 -6 -17 -10 -15 -26]]

POTE generator: ~21/20 = 83.116

Vals: 14cf, 58, 72, 130, 332f, 462ef

Badness: 0.013046

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 243/242, 289/288, 351/350, 441/440

Mapping: [2 4 7 7 9 11 9], 0 -6 -17 -10 -15 -26 -6]]

POTE generator: ~21/20 = 83.168

Vals: 14cf, 58, 72, 130, 202g

Badness: 0.012657

Quasiorwell

In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = [22 -1 -10 1. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)1/8, giving just 7s, or 3841/38, giving pure fifths.

Adding 3025/3024 extends to the 11-limit and gives ⟨⟨38 -3 8 64 …]] for the initial wedgie, and as expected, 270 remains an excellent tuning.


Subgroup: 2.3.5.7

Comma list: 2401/2400, 29360128/29296875

Mapping: [1 31 0 9], 0 -38 3 -8]]

Wedgie⟨⟨38 -3 8 -93 -94 27]]

POTE generator: ~1024/875 = 271.107

Vals31, 177, 208, 239, 270, 571, 841, 1111

Badness: 0.035832

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 5632/5625

Mapping: [1 31 0 9 53], 0 -38 3 -8 -64]]

POTE generator: ~90/77 = 271.111

Vals: 31, 208, 239, 270

Badness: 0.017540

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095

Mapping: [1 31 0 9 53 -59], 0 -38 3 -8 -64 81]]

POTE generator: ~90/77 = 271.107

Vals: 31, 239, 270, 571, 841, 1111

Badness: 0.017921

Decoid

See also: Qintosec family #Decoid

Decoid tempers out 2401/2400 and 67108864/66976875, as well as the linus comma, [11 -10 -10 10. Either 8/7 or 16/15 can be used its generator. It may be described as the 130&270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the qintosec temperament.


Subgroup: 2.3.5.7

Comma list: 2401/2400, 67108864/66976875

Mapping: [10 0 47 36], 0 2 -3 -1]]

Mapping generators: ~15/14, ~8192/4725

Wedgie⟨⟨20 -30 -10 -94 -72 61]]

POTE generator: ~16/15 = 111.099

Vals10, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc

Badness: 0.033902

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 5632/5625, 9801/9800

Mapping: [10 0 47 36 98], 0 2 -3 -1 -8]]

POTE generator: ~16/15 = 111.070

Vals: 10e, 130, 270, 670, 940, 1210, 2150c

Badness: 0.018735

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095

Mapping: [10 0 47 36 98 37], 0 2 -3 -1 -8 0]]

POTE generator: ~16/15 = 111.083

Vals: 10e, 130, 270, 940, 1210f, 1480cf

Badness: 0.013475

Neominor

The generator for neominor temperament is tridecimal minor third 13/11, also known as Neo-gothic minor third.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 177147/175616

Mapping: [1 3 12 8], 0 -6 -41 -22]]

Wedgie⟨⟨6 41 22 51 18 -64]]

POTE generator: ~189/160 = 283.280

Vals72, 161, 233, 305

Badness: 0.088221

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 35937/35840

Mapping: [1 3 12 8 7], 0 -6 -41 -22 -15]]

POTE generator: ~33/28 = 283.276

Vals: 72, 161, 233, 305

Badness: 0.027959

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 243/242, 364/363, 441/440

Mapping: [1 3 12 8 7 7], 0 -6 -41 -22 -15 -14]]

POTE generator: ~13/11 = 283.294

Vals: 72, 161f, 233f

Badness: 0.026942

Emmthird

The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 14348907/14336000

Mapping: [1 -3 -17 -8], 0 14 59 33]]

Wedgie⟨⟨14 59 33 61 13 -89]]

POTE generator: ~2744/2187 = 392.988

Vals58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d

Badness: 0.016736

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 1792000/1771561

Mapping: [1 -3 -17 -8 -8], 0 14 59 33 35]]

POTE generator: ~1372/1089 = 392.991

Vals: 58, 113, 171

Badness: 0.052358

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 2200/2197

Mapping: [1 -3 -17 -8 -8 -13], 0 14 59 33 35 51]]

POTE generator: ~180/143 = 392.989

Vals: 58, 113, 171

Badness: 0.026974

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197

Mapping: [1 -3 -17 -8 -8 -13 9], 0 14 59 33 35 51 -15]]

POTE generator: ~64/51 = 392.985

Vals: 58, 113, 171

Badness: 0.023205

Quinmite

The generator for quinmite is quasi-tempered minor third 25/21, flatter than 6/5 by the starling comma, 126/125. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth".

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1959552/1953125

Mapping: [1 -7 -5 -3], 0 34 29 23]]

Wedgie⟨⟨34 29 23 -33 -59 -28]]

POTE generator: ~25/21 = 302.997

Vals95, 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc

Badness: 0.037322

Unthirds

The generator for unthirds temperament is undecimal major third, 14/11.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 68359375/68024448

Mapping: [1 -13 -14 -9], 0 42 47 34]]

Wedgie⟨⟨42 47 34 -23 -64 -53]]

POTE generator: ~3969/3125 = 416.717

Vals72, 167, 239, 311, 694, 1005c

Badness: 0.075253

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 4000/3993

Mapping: [1 -13 -14 -9 -8], 0 42 47 34 33]]

POTE generator: ~14/11 = 416.718

Vals: 72, 167, 239, 311, 1316c

Badness: 0.022926

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400

Mapping: [1 -13 -14 -9 -9 -47], 0 42 47 34 33 146]]

POTE generator: ~14/11 = 416.716

Vals: 72, 311, 694, 1005c, 1699cd

Badness: 0.020888

Newt

This temperament has a generator of neutral third (0.2 cents flat of 49/40) and tempers out the garischisma.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 33554432/33480783

Mapping: [1 1 19 11], 0 2 -57 -28]]

Wedgie⟨⟨2 -57 -28 -95 -50 95]]

POTE generator: ~49/40 = 351.113

Vals41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bbcc

Badness: 0.041878

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 19712/19683

Mapping: [1 1 19 11 -10], 0 2 -57 -28 46]]

POTE generator: ~49/40 = 351.115

Vals: 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b

Badness: 0.019461

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095

Mapping: [1 1 19 11 -10 -20], 0 2 -57 -28 46 81]]

POTE generator: ~49/40 = 351.117

Vals: 41, 229, 270, 581, 851, 2283b, 3134b

Badness: 0.013830

Amicable

See also: Amity family

The amicable temperament tempers out the amity comma and the canousma in addition to the breedsma, and is closely associated with the canou temperament.

While it extends well into 2.3.5.7.13/11, there are multiple reasonable places for the prime 11 and 13 in the interval chain. Amical (311 & 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 & 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 & 113) and pseudoamorous (14cf & 99ef) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semioctave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two.


Subgroup: 2.3.5.7

Comma list: 2401/2400, 1600000/1594323

Mapping: [1 3 6 5], 0 -20 -52 -31]]

Wedgie⟨⟨20 52 31 36 -7 -74]]

POTE generator: ~21/20 = 84.880

Vals99, 212, 311, 410, 1131, 1541b

Badness: 0.045473

Amical

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 131072/130977, 1600000/1594323

Mapping: [1 3 6 5 -8], 0 -20 -52 -31 162]]

POTE generator: ~21/20 = 84.8843

Vals: 99, 212e, 311, 410, 721, 1032, 1343

Badness: 0.101

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206

Mapping: [1 3 6 5 -8 -5], 0 -20 -52 -31 162 123]]

POTE generator: ~21/20 = 84.8838

Vals: 99, 212ef, 311, 410, 721, 1032

Badness: 0.0499

Amorous

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 6250/6237, 19712/19683

Mapping: [1 3 6 5 14], 0 -20 -52 -31 -149]]

POTE generator: ~21/20 = 84.8896

Vals: 99e, 212, 311, 1145c, 1456cd

Badness: 0.0489

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647

Mapping: [1 3 6 5 14 17], 0 -20 -52 -31 -149 -188]]

POTE generator: ~21/20 = 84.8910

Vals: 99ef, 212, 311, 834, 1145c

Badness: 0.0347

Pseudoamical

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 1600000/1594323

Mapping: [1 3 6 5 -1], 0 -20 -52 -31 63]]

POTE generator: ~21/20 = 84.9091

Vals: 99, 113, 212, 961ccdeee

Badness: 0.0858

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1375/1372, 19773/19712

Mapping: [1 3 6 5 -1 2], 0 -20 -52 -31 63 24]]

POTE generator: ~21/20 = 84.9127

Vals: 99, 113, 212, 537cdeff, 749ccdeefff

Badness: 0.0470

Pseudoamorous

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 980000/970299

Mapping: [1 3 6 5 7], 0 -20 -52 -31 -50]]

POTE generator: ~21/20 = 84.8917

Vals: 99e, 212e

Badness: 0.0566

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 1875/1859

Mapping: [1 3 6 5 7 10], 0 -20 -52 -31 -50 -89]]

POTE generator: ~21/20 = 84.9164

Vals: 99ef, 113, 212ef

Badness: 0.0428

Floral

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 14641/14580

Mapping: [2 6 12 10 13], 0 -20 -52 -31 -43]]

POTE generator: ~21/20 = 84.8788

Vals: 198, 212, 410

Badness: 0.0651

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580

Mapping: [2 6 12 10 13 19], 0 -20 -52 -31 -43 -82]]

POTE generator: ~21/20 = 84.8750

Vals: 198, 410

Badness: 0.0370

Humorous

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 1600000/1594323

Mapping: [1 3 6 5 3], 0 -40 -104 -62 13]]

POTE generator: ~4096/3993 = 42.4391

Vals: 85c, 113, 198, 311, 509, 820

Badness: 0.0582

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024

Mapping: [1 3 6 5 3 6], 0 -40 -104 -62 13 -65]]

POTE generator: ~40/39 = 42.4391

Vals: 85c, 113, 198, 311, 509, 820f

Badness: 0.0283

Septidiasemi

Main article: Septidiasemi

Aside from 2401/2400, septidiasemi tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of 15/14). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.


Subgroup: 2.3.5.7

Comma list: 2401/2400, 2152828125/2147483648

Mapping: [1 -1 6 4], 0 26 -37 -12]]

Wedgie⟨⟨26 -37 -12 -119 -92 76]]

POTE generator: ~15/14 = 119.297

Vals10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd

Badness: 0.044115

Sedia

The sedia temperament (10&161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 939524096/935859375

Mapping: [1 -1 6 4 -3], 0 26 -37 -12 65]]

POTE generator: ~15/14 = 119.279

Vals: 10, 151, 161, 171, 332

Badness: 0.090687

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 441/440, 2200/2197, 3584/3575

Mapping: [1 -1 6 4 -3 4], 0 26 -37 -12 65 -3]]

POTE generator: ~15/14 = 119.281

Vals: 10, 151, 161, 171, 332, 835eeff

Badness: 0.045773

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575

Mapping: [1 -1 6 4 -3 4 2], 0 26 -37 -12 65 -3 21]]

POTE generator: ~15/14 = 119.281

Vals: 10, 151, 161, 171, 332, 503ef, 835eeff

Badness: 0.027322

Maviloid

See also: Ragismic microtemperaments #Parakleismic

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1224440064/1220703125

Mapping: [1 31 34 26], 0 -52 -56 -41]]

Wedgie⟨⟨52 56 41 -32 -81 -62]]

POTE generator: ~1296/875 = 678.810

Vals76, 99, 274, 373, 472, 571, 1043, 1614

Badness: 0.057632

Subneutral

See also: Luna family

Subgroup: 2.3.5.7

Comma list: 2401/2400, 274877906944/274658203125

Mapping: [1 19 0 6], 0 -60 8 -11]]

Wedgie⟨⟨60 -8 11 -152 -151 48]]

POTE generator: ~57344/46875 = 348.301

Vals31, 348, 379, 410, 441, 1354, 1795, 2236

Badness: 0.045792

Osiris

See also: Metric microtemperaments #Geb

Subgroup: 2.3.5.7

Comma list: 2401/2400, 31381059609/31360000000

Mapping: [1 13 33 21], 0 -32 -86 -51]]

Wedgie⟨⟨32 86 51 62 -9 -123]]

POTE generator: ~2800/2187 = 428.066

Vals157, 171, 1012, 1183, 1354, 1525, 1696, 6955dd

Badness: 0.028307

Gorgik

Subgroup: 2.3.5.7

Comma list: 2401/2400, 28672/28125

Mapping: [1 5 1 3], 0 -18 7 -1]]

Wedgie⟨⟨18 -7 1 -53 -49 22]]

POTE generator: ~8/7 = 227.512

Vals21, 37, 58, 153bc, 211bccd, 269bccd

Badness: 0.158384

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 2401/2400, 2560/2541

Mapping: [1 5 1 3 1], 0 -18 7 -1 13]]

POTE generator: ~8/7 = 227.500

Vals: 21, 37, 58, 153bce, 211bccdee, 269bccdee

Badness: 0.059260

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 196/195, 364/363, 512/507

Mapping: [1 5 1 3 1 2], 0 -18 7 -1 13 9]]

POTE generator: ~8/7 = 227.493

Vals: 21, 37, 58, 153bcef, 211bccdeeff

Badness: 0.032205

Fibo

Subgroup: 2.3.5.7

Comma list: 2401/2400, 341796875/339738624

Mapping: [1 19 8 10], 0 -46 -15 -19]]

Wedgie⟨⟨46 15 19 -83 -99 2]]

POTE generator: ~125/96 = 454.310

Vals37, 103, 140, 243, 383, 1009cd, 1392ccd

Badness: 0.100511

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 43923/43750

Mapping: [1 19 8 10 8], 0 -46 -15 -19 -12]]

POTE generator: ~100/77 = 454.318

Vals: 37, 103, 140, 243e

Badness: 0.056514

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624, 847/845, 1375/1372

Mapping: [1 19 8 10 8 9], 0 -46 -15 -19 -12 -14]]

POTE generator: ~13/10 = 454.316

Vals: 37, 103, 140, 243e

Badness: 0.027429

Mintone

In addition to 2401/2400, mintone tempers out 177147/175000 = [-3 11 -5 -1 in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58&103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.


Subgroup: 2.3.5.7

Comma list: 2401/2400, 177147/175000

Mapping: [1 5 9 7], 0 -22 -43 -27]]

Wedgie⟨⟨22 43 27 17 -19 -58]]

POTE generator: ~10/9 = 186.343

Vals45, 58, 103, 161, 586b, 747bc, 908bbc

Badness: 0.125672

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 43923/43750

Mapping: [1 5 9 7 12], 0 -22 -43 -27 -55]]

POTE generator: ~10/9 = 186.345

Vals: 58, 103, 161, 425b, 586b, 747bc

Badness: 0.039962

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 441/440, 847/845

Mapping: [1 5 9 7 12 11], 0 -22 -43 -27 -55 -47]]

POTE generator: ~10/9 = 186.347

Vals: 58, 103, 161, 425b, 586bf

Badness: 0.021849

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 351/350, 441/440, 561/560, 847/845

Mapping: [1 5 9 7 12 11 3], 0 -22 -43 -27 -55 -47 7]]

POTE generator: ~10/9 = 186.348

Vals: 58, 103, 161, 425b, 586bf

Badness: 0.020295

Catafourth

See also: Sensipent family

Subgroup: 2.3.5.7

Comma list: 2401/2400, 78732/78125

Mapping: [1 13 17 13], 0 -28 -36 -25]]

Wedgie⟨⟨28 36 25 -8 -39 -43]]

POTE generator: ~250/189 = 489.235

Vals27, 76, 103, 130

Badness: 0.079579

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 78408/78125

Mapping: [1 13 17 13 32], 0 -28 -36 -25 -70]]

POTE generator: ~250/189 = 489.252

Vals: 103, 130, 233, 363, 493e, 856be

Badness: 0.036785

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 441/440, 10985/10976

Mapping: [1 13 17 13 32 9], 0 -28 -36 -25 -70 -13]]

POTE generator: ~65/49 = 489.256

Vals: 103, 130, 233, 363

Badness: 0.021694

Cotritone

Subgroup: 2.3.5.7

Comma list: 2401/2400, 390625/387072

Mapping: [1 -13 -4 -4], 0 30 13 14]]

Wedgie⟨⟨30 13 14 -49 -62 -4]]

POTE generator: ~7/5 = 583.385

Vals35, 37, 72, 109, 181, 253

Badness: 0.098322

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 4000/3993

Mapping: [1 -13 -4 -4 2], 0 30 13 14 3]]

POTE generator: ~7/5 = 583.387

Vals: 35, 37, 72, 109, 181, 253

Badness: 0.032225

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 364/363, 385/384, 625/624

Mapping: [1 -13 -4 -4 2 -7], 0 30 13 14 3 22]]

POTE generator: ~7/5 = 583.387

Vals: 37, 72, 109, 181f

Badness: 0.028683

Quasimoha

For the 5-limit version of this temperament, see High badness temperaments #Quasimoha.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3645/3584

Mapping: [1 1 9 6], 0 2 -23 -11]]

POTE generator: ~49/40 = 348.603

Vals31, 117c, 148bc, 179bc

Badness: 0.110820

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 1815/1792

Mapping: [1 1 9 6 2], 0 2 -23 -11 5]]

POTE generator: ~11/9 = 348.639

Vals: 31, 86ce, 117ce, 148bce

Badness: 0.046181