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Breedsmic temperaments

Revision as of 20:06, 26 February 2026 by FloraC (talk | contribs) (- redundant category)
This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This page discusses miscellaneous rank-2 temperaments tempering out the breedsma (monzo: [-5 -1 -2 4⟩, ratio: 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 may be described as the 41 & 58 temperament, tempering out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator; its ploidacot is dicot. 99edo and 140edo provides 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 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

Optimal tunings:

  • WE: ~2 = 1199.7412 ¢, ~49/40 = 351.4016 ¢
error map: ⟨-0.259 +0.590 +0.021 -0.346]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.4671 ¢
error map: ⟨0.000 +0.979 +0.364 +0.246]

Minimax tuning:

[[1 0 0 0⟩, [7/5 0 2/25 0⟩, [0 0 1 0⟩, [8/5 0 13/25 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

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

Optimal ET sequence: 17c, 41, 58, 99, 239, 338

Badness (Sintel): 0.563

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:

  • WE: ~2 = 1199.2845 ¢, ~11/9 = 351.3110 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.4956 ¢

Optimal ET sequence: 17c, 41, 58, 99e

Badness (Sintel): 0.777

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:

  • WE: ~2 = 1198.8875 ¢, ~11/9 = 351.2475 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.5438 ¢

Optimal ET sequence: 17c, 41, 58, 99ef, 157eff

Badness (Sintel): 0.789

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:

  • WE: ~99/70 = 599.8556 ¢, ~400/231 = 951.2757 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~400/231 = 951.4939 ¢

Optimal ET sequence: 58, 140, 198

Badness (Sintel): 1.40

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:

  • WE: ~99/70 = 599.8513 ¢, ~26/15 = 951.2662 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~26/15 = 951.4905 ¢

Optimal ET sequence: 58, 140, 198, 536f

Badness (Sintel): 0.876

Quadrafifths

This has been logged as semihemififths in Graham Breed's temperament finder, but quadrafifths arguably makes more sense because it straight-up splits the fifth in four.

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:

  • WE: ~2 = 1199.7520 ¢, ~243/220 = 175.7015 ¢
  • CWE: ~2 = 1200.0000 ¢, ~243/220 = 175.7360 ¢

Optimal ET sequence: 41, 157, 198, 239, 676b, 915be

Badness (Sintel): 1.33

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:

  • WE: ~2 = 1199.6502 ¢, ~72/65 = 175.6957 ¢
  • CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.7461 ¢

Optimal ET sequence: 41, 157, 198, 437f, 635bcff

Badness (Sintel): 1.29

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 -19 7 0], ⟨0 22 -5 3]]

mapping generators: ~2, ~245/128

Optimal tunings:

  • WE: ~2 = 1200.1004 ¢, ~245/128 = 1122.9024 ¢ (~256/245 = 77.1979)
error map: ⟨+0.100 -0.008 -0.123 -0.119]
  • CWE: ~2 = 1200.0000 ¢, ~245/128 = 1122.8101 ¢ (~256/245 = 77.1899)
error map: ⟨0.000 -0.133 -0.364 -0.396]

Optimal ET sequence: 31, 109, 140, 171

Badness (Sintel): 0.329

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -19 7 0 -48], ⟨0 22 -5 3 55]]

Optimal tunings:

  • WE: ~2 = 1200.1034 ¢, ~245/128 = 1122.8694 ¢ (~256/245 = 77.2340 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~245/128 = 1122.7743 ¢ (~256/245 = 77.2257 ¢)

Optimal ET sequence: 31, 109e, 140e, 171, 202

Badness (Sintel): 1.18

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -19 7 0 -48 43], ⟨0 22 -5 3 55 -42]]

Optimal tunings:

  • WE: ~2 = 1199.8783 ¢, ~224/117 = 1122.6835 ¢ (~117/112 = 77.1948 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~224/117 = 1122.7968 ¢ (~117/112 = 77.2032 ¢)

Optimal ET sequence: 31, 140e, 171, 373ef

Badness (Sintel): 1.52

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -19 7 0 -48 43 49], ⟨0 22 -5 3 55 -42 -48]]

Optimal tunings:

  • WE: ~2 = 1199.8677 ¢, ~65/34 = 1122.6748 ¢ (~68/65 = 77.1929 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~65/34 = 1122.7985 ¢ (~68/65 = 77.2015 ¢)

Optimal ET sequence: 31, 140e, 171

Badness (Sintel): 1.40

Tertia

Subgroup:2.3.5.7.11

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

Mapping: [⟨1 -19 7 0 -19], ⟨0 22 -5 3 24]]

Optimal tunings:

  • WE: ~2 = 1200.2336 ¢, ~21/11 = 1123.0454 ¢ (~22/21 = 77.1882 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~21/11 = 1122.8311 ¢ (~22/21 = 77.1689 ¢)

Optimal ET sequence: 31, 109, 140, 171e, 311e

Badness (Sintel): 0.997

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -19 7 0 -19 43], ⟨0 22 -5 3 24 -42]]

Optimal tunings:

  • WE: ~2 = 1200.1395 ¢, ~21/11 = 1122.9727 ¢ (~22/21 = 77.1669 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~21/11 = 1122.8426 ¢ (~22/21 = 77.1574 ¢)

Optimal ET sequence: 31, 78f, 109, 140

Badness (Sintel): 1.17

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -19 7 0 -19 43 49], ⟨0 22 -5 3 24 -42 -48]]

Optimal tunings:

  • WE: ~2 = 1200.1655 ¢, ~21/11 = 1122.9926 ¢ (~22/21 = 77.1729 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~21/11 = 1122.8376 ¢ (~22/21 = 77.1624 ¢)

Optimal ET sequence: 31, 78fg, 109g, 140

Badness (Sintel): 1.14

Tertiaseptia

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -19 7 0 112], ⟨0 22 -5 3 -116]]

Optimal tunings:

  • WE: ~2 = 1200.0053 ¢, ~245/128 = 1122.8357 ¢ (~256/245 = 77.1696 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~245/128 = 1122.8308 ¢ (~256/245 = 77.1692 ¢)

Optimal ET sequence: 31e, 140, 171, 311

Badness (Sintel): 1.88

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -19 7 0 112 43], ⟨0 22 -5 3 -116 -42]]

Optimal tunings:

  • WE: ~2 = 1199.9823 ¢, ~224/117 = 1122.8150 ¢ (~117/112 = 77.1673 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~224/117 = 1122.8316 ¢ (~117/112 = 77.1684 ¢)

Optimal ET sequence: 31e, 140, 171, 311, 1073

Badness (Sintel): 1.14

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -19 7 0 112 43 49], ⟨0 22 -5 3 -116 -42 -48]]

Optimal tunings:

  • WE: ~2 = 1200.0092 ¢, ~65/34 = 1122.8392 ¢ (~68/65 = 77.1700 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~65/34 = 1122.8305 ¢ (~68/65 = 77.1695 ¢)

Optimal ET sequence: 31e, 140, 171, 311

Badness (Sintel): 0.956

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 -19 7 0 112 43 49 -94], ⟨0 22 -5 3 -116 -42 -48 105]]

Optimal tunings:

  • WE: ~2 = 1200.0187 ¢, ~65/34 = 1122.8489 ¢ (~68/65 = 77.1698 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~65/34 = 1122.8313 ¢ (~68/65 = 77.1687 ¢)

Optimal ET sequence: 140, 171, 311

Badness (Sintel): 1.07

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 -19 7 0 112 43 49 -94 114], ⟨0 22 -5 3 -116 -42 -48 105 -117]]

Optimal tunings:

  • WE: ~2 = 1200.0101 ¢, ~44/23 = 1122.8418 ¢ (~23/22 = 77.1683 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~44/23 = 1122.8323 ¢ (~23/22 = 77.1677 ¢)

Optimal ET sequence: 140, 311, 762g

Badness (Sintel): 1.08

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 -19 7 0 112 43 49 -94 114 61], ⟨0 22 -5 3 -116 -42 -48 105 -117 -60]]

Optimal tunings:

  • WE: ~2 = 1200.0007 ¢, ~44/23 = 1122.8332 ¢ (~23/22 = 77.1675 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~44/23 = 1122.8326 ¢ (~23/22 = 77.1674 ¢)

Optimal ET sequence: 140, 311, 762g

Badness (Sintel): 1.02

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 -19 7 0 112 43 49 -94 114 61 -83], ⟨0 22 -5 3 -116 -42 -48 105 -117 -60 94]]

Optimal tunings:

  • WE: ~2 = 1199.9721 ¢, ~44/23 = 1122.8047 ¢ (~23/22 = 77.1673 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~44/23 = 1122.8309 ¢ (~23/22 = 77.1691 ¢)

Optimal ET sequence: 140, 171, 311

Badness (Sintel): 1.18

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 -19 7 0 112 43 49 -94 114 61 -83 81], ⟨0 22 -5 3 -116 -42 -48 105 -117 -60 94 -81]]

Optimal tunings:

  • WE: ~2 = 1199.9824 ¢, ~44/23 = 1122.8139 ¢ (~23/22 = 77.1685 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~44/23 = 1122.8304 ¢ (~23/22 = 77.1696 ¢)

Optimal ET sequence: 140, 171, 311

Badness (Sintel): 1.19

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 -19 7 0 112 43 49 -94 114 61 -83 81 -4], ⟨0 22 -5 3 -116 -42 -48 105 -117 -60 94 -81 10]]

Optimal tunings:

  • WE: ~2 = 1199.9957 ¢, ~44/23 = 1122.8266 ¢ (~23/22 = 77.1691 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~44/23 = 1122.8306 ¢ (~23/22 = 77.1694 ¢)

Optimal ET sequence: 140, 171, 311

Badness (Sintel): 1.20

Hemitert

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -41 12 -3 -73], ⟨0 44 -10 6 79]]

mapping generators: ~2, ~88/45

Optimal tunings:

  • WE: ~2 = 1200.1008 ¢, ~88/45 = 1161.5020 ¢ (~45/44 = 38.5988 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~88/45 = 1161.4053 ¢ (~45/44 = 38.5947 ¢)

Optimal ET sequence: 31, …, 280, 311, 342, 2021cde, 2363cde, …, 3389ccddee, 3731ccddee

Badness (Sintel): 0.517

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -41 12 -3 -73 85], ⟨0 44 -10 6 79 -84]]

Optimal tunings:

  • WE: ~2 = 1199.9822 ¢, ~88/45 = 1161.3952 ¢ (~45/44 = 38.5871 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~88/45 = 1161.4123 ¢ (~45/44 = 38.5877 ¢)

Optimal ET sequence: 31, 280, 311

Badness (Sintel): 1.39

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -41 12 -3 -73 85 97], ⟨0 44 -10 6 79 -84 -96]]

Optimal tunings:

  • WE: ~2 = 1200.0042 ¢, ~88/45 = 1161.4149 ¢ (~45/44 = 38.5893 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~88/45 = 1161.4109 ¢ (~45/44 = 38.5891 ¢)

Optimal ET sequence: 31, 280, 311, 653f

Badness (Sintel): 1.29

Semitert

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 -16 9 3 47], ⟨0 22 -5 3 -46]]

mapping generators: ~99/70, ~693/512

Optimal tunings:

  • WE: ~99/70 = 600.0548 ¢, ~693/512 = 522.8547 ¢ (~256/245 = 77.2002 ¢)
  • CWE: ~99/70 = 600.0000 ¢, ~693/512 = 522.8069 ¢ (~256/245 = 77.1931 ¢)

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

Badness (Sintel): 0.853

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 as expected, 270 remains an excellent tuning.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 29360128/29296875

Mapping: [⟨1 -7 3 1], ⟨0 38 -3 8]]

mapping generators: ~2, ~1024/875

Optimal tunings:

  • WE: ~2 = 1199.9403 ¢, ~1024/875 = 271.0935 ¢
error map: ⟨-0.060 +0.018 +0.226 -0.137]
  • CWE: ~2 = 1200.0000 ¢, ~1024/875 = 271.1064 ¢
error map: ⟨0.000 +0.087 +0.367 +0.025]

Optimal ET sequence: 31, …, 177, 208, 239, 270, 571, 841, 1111

Badness (Sintel): 0.907

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -7 3 1 -11], ⟨0 38 -3 8 64]]

Optimal tunings:

  • WE: ~2 = 1199.9484 ¢, ~90/77 = 271.0989 ¢
  • CWE: ~2 = 1200.0000 ¢, ~90/77 = 271.1099 ¢

Optimal ET sequence: 31, …, 177e, 208, 239, 270

Badness (Sintel): 0.580

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -7 3 1 -11 22], ⟨0 38 -3 8 64 -81]]

Optimal tunings:

  • WE: ~2 = 1199.9916 ¢, ~90/77 = 271.1051 ¢
  • CWE: ~2 = 1200.0000 ¢, ~90/77 = 271.1070 ¢

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

Badness (Sintel): 0.741

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 -29 -14], ⟨0 6 41 22]]

mapping generators: ~2, ~320/189

Optimal tunings:

  • WE: ~2 = 1200.4276 ¢, ~320/189 = 917.0471 ¢
error map: ⟨+0.428 -0.955 +0.216 +0.224]
  • CWE: ~2 = 1200.0000 ¢, ~320/189 = 916.7320 ¢
error map: ⟨0.000 -1.563 -0.301 -0.722]

Optimal ET sequence: 17c, 55c, 72, 161, 233, 305

Badness (Sintel): 2.23

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -3 -29 -14 -8], ⟨0 6 41 22 15]]

Optimal tunings:

  • WE: ~2 = 1200.3466 ¢, ~56/33 = 916.9889 ¢
  • CWE: ~2 = 1200.0000 ¢, ~56/33 = 916.7330 ¢

Optimal ET sequence: 17c, 55c, 72, 161, 233, 305

Badness (Sintel): 0.924

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -3 -29 -14 -8 -7], ⟨0 6 41 22 15 14]]

Optimal tunings:

  • WE: ~2 = 1200.6874 ¢, ~22/13 = 917.2313 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/13 = 916.7228 ¢

Optimal ET sequence: 17c, 55cf, 72

Badness (Sintel): 1.11

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 243/242, 273/272, 364/363

Mapping: [⟨1 -3 -29 -14 -8 -7 -28], ⟨0 6 41 22 15 14 42]]

Optimal tunings:

  • WE: ~2 = 1200.6905 ¢, ~17/10 = 917.2356 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/10 = 916.7252 ¢

Optimal ET sequence: 17cg, 55cfg, 72

Badness (Sintel): 0.918

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 -3 -17 -8], ⟨0 14 59 33]]

mapping generators: ~2, ~2744/2187

Optimal tunings:

  • WE: ~2 = 1200.0435 ¢, ~2744/2187 = 393.0021 ¢
error map: ⟨+0.043 -0.057 +0.069 -0.106]
  • CWE: ~2 = 1200.0000 ¢, ~2744/2187 = 392.9887 ¢
error map: ⟨0.000 -0.113 +0.022 -0.197]

Optimal ET sequence: 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d

Badness (Sintel): 0.424

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]]

Optimal tunings:

  • WE: ~2 = 1199.8090 ¢, ~1372/1089 = 392.9286 ¢
  • CWE: ~2 = 1200.0000 ¢, ~1372/1089 = 392.9870 ¢

Optimal ET sequence: 58, 113, 171

Badness (Sintel): 1.73

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]]

Optimal tunings:

  • WE: ~2 = 1199.7756 ¢, ~180/143 = 392.9154 ¢
  • CWE: ~2 = 1200.0000 ¢, ~180/143 = 392.9840 ¢

Optimal ET sequence: 58, 113, 171

Badness (Sintel): 1.11

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 tunings:

  • WE: ~2 = 1199.8396 ¢, ~64/51 = 392.9322 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/51 = 392.9826 ¢

Optimal ET sequence: 58, 113, 171

Badness (Sintel): 1.18

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 -7 -5 -3], ⟨0 34 29 23]]

mapping generators: ~2, ~25/21

Optimal tunings:

  • WE: ~2 = 1199.9361 ¢, ~25/21 = 302.9808 ¢
error map: ⟨-0.064 -0.162 +0.448 -0.077]
  • CWE: ~2 = 1200.0000 ¢, ~25/21 = 302.9953 ¢
error map: ⟨0.000 -0.116 +0.549 +0.065]

Optimal ET sequence: 99, 202, 301, 400, 701, 1101c, 1802c

Badness (Sintel): 0.945

Unthirds

Despite the complexity of its mapping, unthirds is an important temperament to the structure of the 11-limit; this is hinted at by unthirds' representation as the 72 & 311 temperament, the join of two tuning systems well-known for their high accuracy in the 11-limit and 41-limit respectively. It is generated by the interval of 14/11 (undecimal major third, hence the name) tuned less than a cent flat, and the 23-note MOS this interval generates serves as a well temperament of, of all things, 23edo. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.

The commas it tempers out include the breedsma (2401/2400), the lehmerisma (3025/3024), the pine comma (4000/3993), the unisquary comma (12005/11979), the argyria (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a restriction of the temperament to the 2.5/3.7/3.11/3 fractional subgroup that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with sqrtphi (whose generator is tuned flat of 72edo's).

Subgroup: 2.3.5.7

Comma list: 2401/2400, 68359375/68024448

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

mapping generators: ~2, ~3969/3125

Optimal tunings:

  • WE: ~2 = 1200.0859 ¢, ~3969/3125 = 416.7465 ¢
error map: ⟨+0.086 +0.281 -0.431 -0.218]
  • CWE: ~2 = 1200.0000 ¢, ~3969/3125 = 416.7184 ¢
error map: ⟨0.000 +0.220 -0.547 -0.399]

Optimal ET sequence: 72, 167, 239, 311, 694, 1005c

Badness (Sintel): 1.90

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]]

Optimal tunings:

  • WE: ~2 = 1200.0246 ¢, ~14/11 = 416.7270 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/11 = 416.7190 ¢

Optimal ET sequence: 72, 167, 239, 311

Badness (Sintel): 0.758

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -13 -14 -9 -8 -47], ⟨0 42 47 34 33 146]]

Optimal tunings:

  • WE: ~2 = 1200.0536 ¢, ~14/11 = 416.7343 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/11 = 416.7164 ¢

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

Badness (Sintel): 0.863

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

Optimal tunings:

  • WE: ~2 = 1199.9315 ¢, ~49/40 = 351.0932 ¢
error map: ⟨-0.068 +0.163 +0.075 -0.188]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1141 ¢
error map: ⟨0.000 +0.273 +0.180 -0.022]

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

Badness (Sintel): 1.06

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 tunings:

  • WE: ~2 = 1199.9603 ¢, ~49/40 = 351.1038 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1155 ¢

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

Badness (Sintel): 0.643

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 tunings:

  • WE: ~2 = 1199.9747 ¢, ~49/40 = 351.1094 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1168 ¢

Optimal ET sequence: 41, 229, 270, 581, 851, 2283b

Badness (Sintel): 0.571

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 tunings:

  • WE: ~2 = 1199.9782 ¢, ~49/40 = 351.1102 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1166 ¢

Optimal ET sequence: 41, 229, 270, 581, 851

Badness (Sintel): 0.438

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]]

mpping generators: ~2, ~15/14

Optimal tunings:

  • WE: ~2 = 1200.1043 ¢, ~15/14 = 119.3076 ¢
error map: ⟨+0.104 -0.061 -0.070 -0.100]
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 119.2971 ¢
error map: ⟨0.000 -0.230 -0.307 -0.391]

Optimal ET sequence: 10, 151, 161, 171, 3581bcdd, 3752bcdd, …, 5633bbccddd, 5804bbccddd

Badness (Sintel): 1.12

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]]

Optimal tunings:

  • WE: ~2 = 1199.9635 ¢, ~15/14 = 119.2755 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 119.2791 ¢

Optimal ET sequence: 10, 151, 161, 171, 332

Badness (Sintel): 3.00

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]]

Optimal tunings:

  • WE: ~2 = 1199.8922 ¢, ~15/14 = 119.2700 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 119.2804 ¢

Optimal ET sequence: 10, 151, 161, 171, 332

Badness (Sintel): 1.89

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]]

Optimal tunings:

  • WE: ~2 = 1199.9088 ¢, ~15/14 = 119.2719 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 119.2808 ¢

Optimal ET sequence: 10, 151, 161, 171, 332, 503ef

Badness (Sintel): 1.39

Maviloid

See also: Ragismic microtemperaments § Parakleismic

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1224440064/1220703125

Mapping: [⟨1 -21 -22 -15], ⟨0 52 56 41]]

mapping generators: ~2, ~875/648

Optimal tunings:

  • WE: ~2 = 1199.9863 ¢, ~875/648 = 521.1837 ¢
error map: ⟨-0.014 -0.115 +0.274 -0.089]
  • CWE: ~2 = 1200.0000 ¢, ~875/648 = 521.1894 ¢
error map: ⟨0.000 -0.106 +0.293 -0.060]

Optimal ET sequence: 76, 99, 274, 373, 472, 571, 1043, 1614

Badness (Sintel): 1.46

Subneutral

See also: Luna family

Subgroup: 2.3.5.7

Comma list: 2401/2400, 274877906944/274658203125

Mapping: [⟨1 -41 8 -5], ⟨0 60 -8 11]]

mapping generators: ~2, ~46875/28672

Optimal tunings:

  • WE: ~2 = 1199.9998 ¢, ~46875/28672 = 851.6994 (~57344/46875 = 348.3005 ¢)
error map: ⟨-0.000 +0.013 +0.090 -0.132]
  • CWE: ~2 = 1200.0000 ¢, ~46875/28672 = 851.6995 ¢ (~57344/46875 = 348.3005 ¢)
error map: ⟨0.000 +0.014 +0.090 -0.132]

Optimal ET sequence: 31, …, 348, 379, 410, 441, 1354, 1795, 2236

Badness (Sintel): 1.16

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, ~2187/1400

Optimal tunings:

  • WE: ~2 = 1200.0285 ¢, ~2187/1400 = 771.9522 ¢
error map: ⟨+0.028 -0.025 +0.068 -0.117]
  • CWE: ~2 = 1200.0000 ¢, ~2187/1400 = 771.9343 ¢
error map: ⟨0.000 -0.056 +0.039 -0.175]

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

Badness (Sintel): 0.716

Gorgik

Subgroup: 2.3.5.7

Comma list: 2401/2400, 28672/28125

Mapping: [⟨1 -13 8 2], ⟨0 18 -7 1]]

mapping generators: ~2, ~7/4

Optimal tunings:

  • WE: ~2 = 1198.5503 ¢, ~7/4 = 971.3132 ¢ (~8/7 = 227.2371 ¢)
error map: ⟨-1.450 +0.528 +2.896 -0.412]
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 972.4675 ¢ (~8/7 = 227.5325 ¢)
error map: ⟨0.000 +2.460 +6.414 +3.642]

Optimal ET sequence: 21, 37, 58, 153bc, 211bccd, 269bccd

Badness (Sintel): 4.01

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -13 8 2 14], ⟨0 18 -7 1 -13]]

Optimal tunings:

  • WE: ~2 = 1198.4615 ¢, ~7/4 = 971.2535 ¢ (~8/7 = 227.2079 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 972.4918 ¢ (~8/7 = 227.5082 ¢)

Optimal ET sequence: 21, 37, 58, 153bce, 211bccdee, 269bccdee

Badness (Sintel): 1.96

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -13 8 2 14 11], ⟨0 18 -7 1 -13 -9]]

Optimal tunings:

  • WE: ~2 = 1198.4012 ¢, ~7/4 = 971.2110 ¢ (~8/7 = 227.1903 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 972.5030 ¢ (~8/7 = 227.4970 ¢)

Optimal ET sequence: 21, 37, 58, 153bcef, 211bccdeeff

Badness (Sintel): 1.33

Fibo

Subgroup: 2.3.5.7

Comma list: 2401/2400, 341796875/339738624

Mapping: [⟨1 -27 -7 -9], ⟨0 46 15 19]]

mapping generators: ~2, ~192/125

Optimal tunings:

  • WE: ~2 = 1200.2050 ¢, ~192/125 = 745.8170 ¢
error map: ⟨+0.205 +0.094 -0.493 -0.147]
  • CWE: ~2 = 1200.0000 ¢, ~192/125 = 745.6927 ¢
error map: ⟨0.000 -0.092 -0.924 -0.665]

Optimal ET sequence: 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd

Badness (Sintel): 2.54

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -27 -7 -9 -4], ⟨0 46 15 19 12]]

Optimal tunings:

  • WE: ~2 = 1200.4064 ¢, ~77/50 = 745.9349 ¢
  • CWE: ~2 = 1200.0000 ¢, ~77/50 = 745.6876 ¢

Optimal ET sequence: 37, 66b, 103, 140, 243e

Badness (Sintel): 1.87

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -27 -7 -9 -4 -5], ⟨0 46 15 19 12 14]]

Optimal tunings:

  • WE: ~2 = 1200.3728 ¢, ~20/13 = 745.9152 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 745.6879 ¢

Optimal ET sequence: 37, 66b, 103, 140, 243e

Badness (Sintel): 1.13

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 -17 -34 -20], ⟨0 22 43 27]]

mapping generators: ~2, ~9/5

Optimal tunings:

  • WE: ~2 = 1200.1458 ¢, ~9/5 = 1013.7798 ¢
error map: ⟨+0.146 -1.277 +1.263 +0.314]
  • CWE: ~2 = 1200.0000 ¢, ~9/5 = 1013.6611 ¢
error map: ⟨0.000 -1.410 +1.116 +0.025]

Optimal ET sequence: 45, 58, 103, 161

Badness (Sintel): 3.18

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -17 -34 -20 -43], ⟨0 22 43 27 55]]

Optimal tunings:

  • WE: ~2 = 1200.1491 ¢, ~9/5 = 1013.7809 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/5 = 1013.6593 ¢

Optimal ET sequence: 45e, 58, 103, 161, 425b

Badness (Sintel): 1.32

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -17 -34 -20 -43 -36], ⟨0 22 43 27 55 47]]

Optimal tunings:

  • WE: ~2 = 1200.0928 ¢, ~9/5 = 1013.7311 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/5 = 1013.6556 ¢

Optimal ET sequence: 45ef, 58, 103, 161

Badness (Sintel): 0.903

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -17 -34 -20 -43 -36 10], ⟨0 22 43 27 55 47 -7]]

Optimal tunings:

  • WE: ~2 = 1200.1085 ¢, ~9/5 = 1013.7433 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/5 = 1013.6537 ¢

Optimal ET sequence: 45ef, 58, 103, 161

Badness (Sintel): 1.03

Catafourth

See also: Sensipent family

Subgroup: 2.3.5.7

Comma list: 2401/2400, 78732/78125

Mapping: [⟨1 -15 -19 -12], ⟨0 28 36 25]]

mapping generators: ~2, ~189/125

Optimal tunings:

  • WE: ~2 = 1199.9278 ¢, ~189/125 = 710.7220 ¢
error map: ⟨-0.072 -0.656 +1.050 +0.091]
  • CWE: ~2 = 1200.0000 ¢, ~189/125 = 710.7626 ¢
error map: ⟨0.000 -0.603 +1.139 +0.238]

Optimal ET sequence: 27, 76, 103, 130

Badness (Sintel): 2.01

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -15 -19 -12 -38], ⟨0 28 36 25 70]]

Optimal tunings:

  • WE: ~2 = 1200.0219 ¢, ~189/125 = 710.7610 ¢
  • CWE: ~2 = 1200.0000 ¢, ~189/125 = 710.7487 ¢

Optimal ET sequence: 27e, 76e, 103, 130, 233, 363, 493e

Badness (Sintel): 1.22

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -15 -19 -12 -38 -4], ⟨0 28 36 25 70 13]]

Optimal tunings:

  • WE: ~2 = 1200.1023 ¢, ~98/65 = 710.8043 ¢
  • CWE: ~2 = 1200.0000 ¢, ~98/65 = 710.7459 ¢

Optimal ET sequence: 27e, 76e, 103, 130, 233, 363

Badness (Sintel): 0.896

Cotritone

Subgroup: 2.3.5.7

Comma list: 2401/2400, 390625/387072

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

mappping generators: ~2, ~7/5

Optimal tunings:

  • WE: ~2 = 1199.9278 ¢, ~7/5 = 583.5994 ¢
error map: ⟨+0.441 +0.289 -1.287 -0.200]
  • CWE: ~2 = 1200.0000 ¢, ~7/5 = 583.3956 ¢
error map: ⟨0.000 -0.086 -2.170 -1.287]

Optimal ET sequence: 35, 37, 72, 181, 253, 325c

Badness (Sintel): 2.49

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]]

Optimal tunings:

  • WE: ~2 = 1200.4058 ¢, ~7/5 = 583.5845 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/5 = 583.3950 ¢

Optimal ET sequence: 35, 37, 72, 181, 253, 325c

Badness (Sintel): 1.07

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]]

Optimal tunings:

  • WE: ~2 = 1200.6111 ¢, ~7/5 = 583.6837 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/5 = 583.3987 ¢

Optimal ET sequence: 35f, 37, 72, 181f, 253ff

Badness (Sintel): 1.19

Quasimoha

For the 5-limit version, see Miscellaneous 5-limit 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 tunings:

  • WE: ~2 = 1201.5059 ¢, ~49/40 = 348.0409 ¢
error map: ⟨+1.506 -2.367 -0.702 +0.759]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 348.5582 ¢
error map: ⟨0.000 -4.839 -3.152 -2.966]

Optimal ET sequence: 24c, 31, 117c, 148bc, 179bcd

Badness (Sintel): 2.80

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 tunings:

  • WE: ~2 = 1201.7630 ¢, ~11/9 = 349.1510 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.6050 ¢

Optimal ET sequence: 24c, 31, 86ce, 117ce, 148bce

Badness (Sintel): 1.53

Lockerbie

For the 5-limit version, see Miscellaneous 5-limit temperaments #Lockerbie.

Lockerbie can be described as the 103 & 270 temperament. Its generator is 120/77 or 77/60. An obvious tuning is given by 270edo, but 373edo and especially 643edo work as well.

The temperament derives its name from the Scottish town, where a flight numbered 103 crashed with 270 casualties, and the temperament is defined as 103 & 270, hence the name. The name is proposed by Eliora, who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.

Lockerbie also has a unique extension that adds the 41st harmonic such that the generator below 600 cents is also on the same step in 103 or 270 as 41/32, which means that 616/615 is tempered out.

Subgroup: 2.3.5.7

Comma list: 2401/2400, [24 13 -18 -1⟩

Mapping: [⟨1 -25 -16 -13], ⟨0 74 51 44]]

mapping generators: ~2, ~3828125/2985984

Optimal tunings:

  • WE: ~2 = 1199.9950 ¢, ~3828125/2985984 = 431.1055 ¢
error map: ⟨-0.005 -0.024 +0.146 -0.120]
  • CWE: ~2 = 1200.0000 ¢, ~3828125/2985984 = 431.1072 ¢
error map: ⟨0.0000 -0.020 +0.155 -0.108]

Optimal ET sequence: 103, 167, 270, 643, 913, 1183

Badness (Sintel): 1.51

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 766656/765625

Mapping: [⟨1 -25 -16 -13 -26], ⟨0 74 51 44 82]]

Optimal tunings:

  • WE: ~2 = 1200.0199 ¢, ~77/60 = 431.1147 ¢
  • CWE: ~2 = 1200.0000 ¢, ~77/60 = 431.1078 ¢

Optimal ET sequence: 103, 167, 270, 643, 913, 1183e

Badness (Sintel): 0.865

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224

Mapping: [⟨1 -25 -16 -13 -26 -6], ⟨0 74 51 44 82 27]]

Optimal tunings:

  • WE: ~2 = 1200.0707 ¢, ~77/60 = 431.1316 ¢
  • CWE: ~2 = 1200.0000 ¢, ~77/60 = 431.1069 ¢

Optimal ET sequence: 103, 167, 270, 643, 913f

Badness (Sintel): 0.662

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224

Mapping: [⟨1 -25 -16 -13 -26 -6 -11], ⟨0 74 51 44 82 27 42]]

Optimal tunings:

  • WE: ~2 = 1199.9639 ¢, ~77/60 = 431.0957 ¢
  • CWE: ~2 = 1200.0000 ¢, ~77/60 = 431.1083 ¢

Optimal ET sequence: 103, 167, 270

Badness (Sintel): 1.07

2.3.5.7.11.13.17.41 subgroup

Subgroup: 2.3.5.7.11.13.17.41

Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224

Mapping: [⟨1 -25 -16 -13 -26 -6 -11 5], ⟨0 74 51 44 82 27 42 1]]

Optimal tunings:

  • WE: ~2 = 1199.8693 ¢, ~41/32 = 431.0650 ¢
  • CWE: ~2 = 1200.000 ¢, ~41/32 = 431.1109 ¢

Optimal ET sequence: 103, 167, 270

Badness (Sintel): 1.25

Hemigoldis

For the 5-limit version, see Diaschismic–gothmic equivalence continuum #Goldis.

Though fairly complex in the 7-limit, hemigoldis does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to other primes. For example, two periods minus six generators yields a "tetracot second" which can be interpreted as ~21/19 to add prime 19 or perhaps more accurately ~31/28 to add prime 7, or even simply as ~32/29 to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again 89edo is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 549755813888/533935546875

Mapping: [⟨1 21 -9 2], ⟨0 24 -14 -1]]

mapping generators: ~2, ~8/7

Optimal tunings:

  • WE: ~2 = 1199.2264 ¢, ~8/7 = 229.1679 ¢
error map: ⟨-0.774 +0.394 +1.468 -0.314]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 229.3103 ¢
error map: ⟨0.000 +1.491 +3.343 +1.864]

Optimal ET sequence: 21, 47b, 68, 157, 382bccd, 529bccd

Badness (Sintel): 4.40

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 -27 55 22], ⟨0 70 -129 -47]]

mapping generators: ~2, ~896/675

Optimal tunings:

  • WE: ~2 = 1200.0051 ¢, ~896/675 = 490.0303 ¢
error map: ⟨+0.005 +0.025 +0.063 -0.136]
  • CWE: ~2 = 1200.0000 ¢, ~896/675 = 490.0282 ¢
error map: ⟨0.000 +0.017 +0.052 -0.150]

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

Badness (Sintel): 5.12

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -27 55 22 -19], ⟨0 70 -129 -47 55]]

Optimal tunings:

  • WE: ~2 = 1199.9901 ¢, ~896/675 = 490.0239 ¢
  • CWE: ~2 = 1200.000 ¢, ~896/675 = 490.0279 ¢

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

Badness (Sintel): 1.73

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -27 55 22 -19 -11], ⟨0 70 -129 -47 55 36]]

Optimal tunings:

  • WE: ~2 = 1199.9701 ¢, ~65/49 = 490.0155 ¢
  • CWE: ~2 = 1200.0000 ¢, ~65/49 = 490.0277 ¢

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

Badness (Sintel): 1.34

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -27 55 22 -19 -11 78], ⟨0 70 -129 -47 55 36 -181]]

Optimal tunings:

  • WE: ~2 = 1199.9726 ¢, ~65/49 = 490.0164 ¢
  • CWE: ~2 = 1200.0000 ¢, ~65/49 = 490.0276 ¢

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

Badness (Sintel): 1.07

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 -27 55 22 -19 -11 78 41], ⟨0 70 -129 -47 55 36 -181 -90]]

Optimal tunings:

  • WE: ~2 = 1199.9756 ¢, ~65/49 = 490.0176 ¢
  • CWE: ~2 = 1200.0000 ¢, ~65/49 = 490.0276 ¢

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

Badness (Sintel): 0.838

References

  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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