Breedsmic temperaments

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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

Optimal GPV sequence41, 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence31, 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 31, 109g, 140, 311e, 451ee

Badness: 0.022416

Tertiaseptia

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 6250/6237, 65625/65536

Mapping: [1 3 2 3 -4], 0 -22 5 -3 116]]

POTE generator: ~256/245 = 77.169

Optimal GPV sequence: 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde

Badness: 0.056926

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400

Mapping: [1 3 2 3 -4 1], 0 -22 5 -3 116 42]]

POTE generator: ~117/112 = 77.168

Optimal GPV sequence: 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf

Badness: 0.027474

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197

Mapping: [1 3 2 3 -4 1 1], 0 -22 5 -3 116 42 48]]

POTE generator: ~68/65 = 77.169

Optimal GPV sequence: 140, 171, 311

Badness: 0.018773

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197

Mapping: [1 3 2 3 -4 1 1 11], 0 -22 5 -3 116 42 48 -105]]

POTE generator: ~68/65 = 77.169

Optimal GPV sequence: 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg

Badness: 0.017653

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215

Mapping: [1 3 2 3 -4 1 1 11 -3], 0 -22 5 -3 116 42 48 -105 117]]

POTE generator: ~23/22 = 77.168

Optimal GPV sequence: 140, 311, 762g, 1073g, 1384cfgg

Badness: 0.015123

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155

Mapping: [1 3 2 3 -4 1 1 11 -3 1], 0 -22 5 -3 116 42 48 -105 117 60]]

POTE generator: ~23/22 = 77.167

Optimal GPV sequence: 140, 311, 762g, 1073g, 1384cfggj

Badness: 0.012181

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014

Mapping: [1 3 2 3 -4 1 1 11 -3 1 11], 0 -22 5 -3 116 42 48 -105 117 60 -94]]

POTE generator: ~23/22 = 77.169

Optimal GPV sequence: 140, 171, 311

Badness: 0.012311

37-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37

Comma list: 595/594, 625/624, 703/702, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014

Mapping: [1 3 2 3 -4 1 1 11 -3 1 11 0], 0 -22 5 -3 116 42 48 -105 117 60 -94 81]]

POTE generator: ~23/22 = 77.170

Optimal GPV sequence: 140, 171, 311

Badness: 0.010949

41-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41

Comma list: 595/594, 625/624, 697/696, 703/702, 714/713, 784/783, 820/819, 833/832, 875/874, 900/899, 931/930

Mapping: [1 3 2 3 -4 1 1 11 -3 1 11 0 6], 0 -22 5 -3 116 42 48 -105 117 60 -94 81 -10]]

POTE generator: ~23/22 = 77.169

Optimal GPV sequence: 140, 171, 311

Badness: 0.009825

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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 31, 280, 311, 653f, 964f

Badness: 0.025298

Semitert

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 65625/65536

Mapping: [2 6 4 6 1], 0 -22 5 -3 46]]

POTE generator: ~256/245 = 77.193

Optimal GPV sequence: 62e, 140, 202, 342

Badness: 0.025790

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

Optimal GPV sequence31, 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence10, 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence72, 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence58, 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence95, 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

Optimal GPV sequence72, 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence41, 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

Optimal GPV sequence: 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

Optimal GPV sequence: 41, 229, 270, 581, 851, 2283b, 3134b

Badness: 0.013830

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

Optimal GPV sequence10, 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence76, 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

Optimal GPV sequence31, 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

Optimal GPV sequence157, 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

Optimal GPV sequence21, 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence37, 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence45, 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence27, 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence35, 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence31, 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

Optimal GPV sequence: 31, 86ce, 117ce, 148bce

Badness: 0.046181

Surmarvelpyth

Surmarvelpyth is named for the generator fifth, 675/448 being 225/224 (marvel comma) sharp of 3/2. It can be described as the 311 & 431 temperament, starting with the 7-limit to the 19-limit.

Subgroup: 2.3.5.7

Comma list: 2401/2400, [93 -32 -17 -1

Mapping: 1 43 -74 -25], 0 -70 129 47]

Mapping generators: ~2, ~675/448

Optimal tuning (CTE): ~675/448 = 709.9719

Optimal GPV sequence120, 191, 311, 742, 1053, 2848, 3901

Badness: 0.202

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 820125/819896, 2097152/2096325

Mapping: 1 43 -74 -25 36], 0 -70 129 47 -55]

Optimal tuning (CTE): ~675/448 = 709.9720

Optimal GPV sequence: 120, 191, 311, 742, 1053, 1795

Badness: 0.0523

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167

Mapping: 1 43 -74 -25 36 25], 0 -70 129 47 -55 -36]

Optimal tuning (CTE): ~98/65 = 709.9723

Optimal GPV sequence: 120, 191, 311, 742, 1053, 1795f

Badness: 0.0325

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619

Mapping: 1 43 -74 -25 36 25 -103], 0 -70 129 47 -55 -36 181]

Optimal tuning (CTE): ~98/65 = 709.9722

Optimal GPV sequence: 120g, 191g, 311, 431, 742, 1795f

Badness: 0.0325

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2401/2400, 2601/2600, 2926/2925, 3136/3135, 3213/3211, 5985/5984

Mapping: 1 43 -74 -25 36 25 -103 -49], 0 -70 129 47 -55 -36 181 90]

Optimal tuning (CTE): ~98/65 = 709.9722

Optimal GPV sequence: 120g, 191g, 311, 431, 742, 1795f

Badness: 0.0138