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 5's, the 7- and 9-odd-limit minimax tuning, or 14(1/13), giving just 7's. 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]]

mapping generators: ~2, ~49/40

Wedgie: ⟨⟨ 2 25 13 35 15 -40 ]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~49/40 = 351.4464
  • POTE: ~2 = 1\1, ~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⟩]
eigenmonzo (unchanged-interval) basis: 2.5

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

Optimal ET sequence: 41, 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]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 351.4289
  • POTE: ~2 = 1\1, ~11/9 = 351.521

Optimal ET 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]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 351.4331
  • POTE: ~2 = 1\1, ~11/9 = 351.573

Optimal ET 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]]

mapping generators: ~99/70, ~400/231

Optimal tunings:

  • CTE: ~99/70 = 1\2, ~49/40 = 351.4722
  • POTE: ~99/70 = 1\2, ~49/40 = 351.505

Optimal ET 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]]

Optimal tunings:

  • CTE: ~99/70 = 1\2, ~49/40 = 351.4674
  • POTE: ~99/70 = 1\2, ~49/40 = 351.502

Optimal ET 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]]

Mapping generators: ~2, ~243/220

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/220 = 175.7284
  • POTE: ~2 = 1\1, ~243/220 = 175.7378

Optimal ET 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]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~72/65 = 175.7412
  • POTE: ~2 = 1\1, ~72/65 = 175.7470

Optimal ET 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, although 171edo - 31edo = 140edo also makes sense, and in very high limits 140edo + 171edo = 311edo is especially notable. 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]]

Mapping generators: ~2, ~256/245

Wedgie: ⟨⟨ 22 -5 3 -59 -57 21 ]]

Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.191

Optimal ET sequence: 31, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.170

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169

Optimal ET 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]]

Mapping generators: ~2, ~45/44

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589

Optimal ET 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]]

Mapping generators: ~99/70, ~256/245

Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193

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

Badness: 0.025790

Quasiorwell

In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 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 7's, 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]]

Mapping generators: ~2, ~875/512

Wedgie: ⟨⟨ 38 -3 8 -93 -94 27 ]]

Optimal tuning (POTE): ~2 = 1\1, ~1024/875 = 271.107

Optimal ET sequence: 31, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107

Optimal ET sequence: 31, 239, 270, 571, 841, 1111

Badness: 0.017921

Decoid

See also: Quintosec 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 as 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 quintosec 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 ]]

Optimal tunings:

  • CTE: ~15/14 = 1\10, ~8192/4725 = 951.1086 (~16/15 = 111.1086, or ~225/224 = 8.8914)
  • POTE: ~15/14 = 1\10, ~8192/4725 = 951.099 (~16/15 = 111.099, or ~225/224 = 8.901)

Optimal ET sequence: 10, …, 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]]

Optimal tunings:

  • CTE: ~15/14 = 1\10, ~400/231 = 951.0943 (~16/15 = 111.0943, or ~225/224 = 8.9057)
  • POTE: ~15/14 = 1\10, ~400/231 = 951.070 (~16/15 = 111.070, or ~225/224 = 8.930)

Optimal ET 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]]

Optimal tunings:

  • CTE: ~15/14 = 1\10, ~26/15 = 951.0943 (~16/15 = 111.0943, or ~196/195 = 8.9057)
  • POTE: ~15/14 = 1\10, ~26/15 = 951.083 (~16/15 = 111.083, or ~196/195 = 8.917)

Optimal ET 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]]

Mapping generators: ~2, ~189/160

Wedgie: ⟨⟨ 6 41 22 51 18 -64 ]]

Optimal tuning (POTE): ~2 = 1\1, ~189/160 = 283.280

Optimal ET sequence: 72, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 283.276

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 283.294

Optimal ET sequence: 72, 161f, 233f

Badness: 0.026942

Emmthird

The generator for emmthird 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 11 42 25], ⟨0 -14 -59 -33]]

Mapping generators: ~2, ~2187/1372

Wedgie: ⟨⟨ 14 59 33 61 13 -8 9 ]]

Optimal tuning (POTE): ~2 = 1\1, ~2744/2187 = 392.988

Optimal ET sequence: 58, 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 11 42 25 27], ⟨0 -14 -59 -33 -35]]

Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991

Optimal ET 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 11 42 25 27 38], ⟨0 -14 -59 -33 -35 -51]]

Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985

Optimal ET 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", given by Petr Pařízek in 2011[1][2].

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1959552/1953125

Mapping: [⟨1 27 24 20], ⟨0 -34 -29 -23]]

Mapping generators: ~2, ~42/25

Wedgie: ⟨⟨ 34 29 23 -33 -59 -28 ]]

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 302.997

Optimal ET sequence: 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 29 33 25], ⟨0 -42 -47 -34]]

Mapping generators: ~2, ~6125/3888

Wedgie: ⟨⟨ 42 47 34 -23 -64 -53 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3969/3125 = 416.717

Optimal ET sequence: 72, 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 29 33 25 25], ⟨0 -42 -47 -34 -33]]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718

Optimal ET sequence: 72, 167, 239, 311

Badness: 0.022926

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 29 33 25 25 99], ⟨0 -42 -47 -34 -33 -146]]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716

Optimal ET sequence: 72, 239f, 311, 694, 1005c

Badness: 0.020888

Newt

Newt has a generator of a neutral third (0.2 cents flat of 49/40) and tempers out the garischisma. It can be described as the 41 & 270 temperament, and extends naturally to the no-17 19-limit, a.k.a. neonewt. 270edo and 311edo are obvious tuning choices, but 581edo and especially 851edo work much better.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 33554432/33480783

Mapping: [⟨1 1 19 11], ⟨0 2 -57 -28]]

mapping generators: ~2, ~49/40

Wedgie: ⟨⟨ 2 -57 -28 -95 -50 95 ]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.113

Optimal ET sequence: 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201

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]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115

Optimal ET sequence: 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972

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]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117

Optimal ET sequence: 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b

Badness: 0.013830

2.3.5.7.11.13.19 subgroup (neonewt)

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400

Mapping: [⟨1 1 19 11 -10 -20 18], ⟨0 2 -57 -28 46 81 -47]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117

Optimal ET sequence: 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb

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 25 -31 -8], ⟨0 -26 37 12]]

Mapping generators: ~2, ~28/15

Wedgie: ⟨⟨ 26 -37 -12 -119 -92 76 ]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.297

Optimal ET sequence: 10, 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 25 -31 -8 62], ⟨0 -26 37 12 -65]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279

Optimal ET 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 25 -31 -8 62 1], ⟨0 -26 37 12 -65 3]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281

Optimal ET 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 25 -31 -8 62 1 23], ⟨0 -26 37 12 -65 3 -21]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281

Optimal ET sequence: l 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]]

Mapping generators: ~2, ~1296/875

Wedgie: ⟨⟨ 52 56 41 -32 -81 -62 ]]

Optimal tuning (POTE): ~2 = 1\1, ~1296/875 = 678.810

Optimal ET sequence: 76, 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]]

Mapping generators: ~2, ~57344/46875

Wedgie: ⟨⟨ 60 -8 11 -152 -151 48 ]]

Optimal tuning (POTE): ~2 = 1\1, ~57344/46875 = 348.301

Optimal ET sequence: 31, …, 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]]

Mapping generators: ~2, ~2800/2187

Wedgie: ⟨⟨ 32 86 51 62 -9 -123 ]]

Optimal tuning (POTE): ~2 = 1\1, ~2800/2187 = 428.066

Optimal ET sequence: 157, 171, 1012, 1183, 1354, 1525, 1696

Badness: 0.028307

Gorgik

Subgroup: 2.3.5.7

Comma list: 2401/2400, 28672/28125

Mapping: [⟨1 5 1 3], ⟨0 -18 7 -1]]

Mapping generators: ~2, ~8/7

Wedgie: ⟨⟨ 18 -7 1 -53 -49 22 ]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.512

Optimal ET sequence: 21, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493

Optimal ET 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]]

Mapping generators: ~2, ~125/96

Wedgie: ⟨⟨ 46 15 19 -83 -99 2 ]]

Optimal tuning (POTE): ~2 = 1\1, ~125/96 = 454.310

Optimal ET sequence: 37, 66b, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318

Optimal ET sequence: 37, 66b, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316

Optimal ET sequence: 37, 66b, 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]]

Mapping generators: ~2, ~10/9

Wedgie: ⟨⟨ 22 43 27 17 -19 -58 ]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.343

Optimal ET sequence: 45, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348

Optimal ET 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]]

Mapping generators: ~2, ~250/189

Wedgie: ⟨⟨ 28 36 25 -8 -39 -43 ]]

Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.235

Optimal ET sequence: 27, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256

Optimal ET sequence: 103, 130, 233, 363

Badness: 0.021694

Cotritone

Subgroup: 2.3.5.7

Comma list: 2401/2400, 390625/387072

Mapping: [⟨1 17 9 10], ⟨0 -30 -13 -14]]

Mappping generators: ~2, ~10/7

Wedgie: ⟨⟨ 30 13 14 -49 -62 -4 ]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.385

Optimal ET sequence: 35, 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 17 9 10 5], ⟨0 -30 -13 -14 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387

Optimal ET 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 17 9 10 5 15], ⟨0 -30 -13 -14 -3 -22]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387

Optimal ET 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]]

Mapping generators: ~2, ~49/40

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 348.603

Optimal ET sequence: 31, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639

Optimal ET 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): ~2 = 1\1, ~675/448 = 709.9719

Optimal ET sequence: 120, 191, 311, 742, 1053, 2848, 3901

Badness: 0.202249

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): ~2 = 1\1, ~675/448 = 709.9720

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

Badness: 0.052308

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): ~2 = 1\1, ~98/65 = 709.9723

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

Badness: 0.032503

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): ~2 = 1\1, ~98/65 = 709.9722

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

Badness: 0.020995

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): ~2 = 1\1, ~98/65 = 709.9722

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

Badness: 0.013771

  1. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
  2. ↑ Yahoo! Tuning Group | 2D temperament names, part I -- reclassified temperaments from message #101780
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