Breedsmic temperaments

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:

• Beatles (+64/63) → Archytas clan
• Newt (+33554432/33480783) → Garischismic clan
• Decimal (+25/24, 49/48 or 50/49) → Dicot family
• Squares (+81/80) → Meantone family
• Sesquiquartififths (+32805/32768) → Schismatic family
• Miracle (+225/224) → Gamelismic clan
• Octacot (+245/243) → Tetracot family
• Quadrasruta (+2048/2025) → Diaschismic family
• Myna (+126/125) → Starling temperaments
• Harry (+19683/19600) → Gravity family
• Quasitemp (+875/864) → Keemic temperaments
• Hemiwürschmidt (+3136/3125 or 6144/6125) → Hemimean clan
• Eagle (+10485760000/10460353203) → Vulture family
• Ennealimmal (+4375/4374) → Septiennealimmal clan
• Quadrimage (+3125/3072) → Magic family
• Amicable (+1600000/1594323) → Amity family
• Decoid (+67108864/66976875) → Quintosec family
• Quadritikleismic (+15625/15552) → Kleismic family
• Subneutral (+274877906944/274658203125) → Luna family
• Neptune (+48828125/48771072) → Gammic family
• Tertiseptisix (+390625000/387420489) → Quartonic family
• Maviloid (+1224440064/1220703125) → Parakleismic family
• Greenwood (+405/392 or 1323/1280) → Whitewood family
• Hemicountercomp (+52613349376/52301766015) → Countercomp family

Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, septidiasemi, lockerbie, unthirds, neominor, catafourth, cotritone, fibo, quasimoha, mintone, gorgik, hemigoldis, and surmarvelpyth, in the order of increasing badness.

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. The ploidacot for this temperament is 20-sheared 22-cot (or pentaseph due to a much simpler 2.5.7-subgroup structure).

171edo makes for an excellent tuning, although 140edo (= 171 - 31) also makes sense, and in very high limits 311edo (= 140 + 171) 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

This extension was considered by Gene Ward Smith as a 41-limit temperament. It can be extended as such by tempering out 875/874, 714/713, 703/702 and 697/696, and mapping 19, 31, 37 and 41 to 94, 105, -81 and +10 steps, respectively.

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

2.3.5.7.11.13.17.23 subgroup

Subgroup: 2.3.5.7.11.13.17.23

Comma list: 595/594, 625/624, 833/832, 1105/1104, 1156/1155, 2200/2197

Mapping: [⟨1 -19 7 0 112 43 49 114], ⟨0 22 -5 3 -116 -42 -48 -117]]

Optimal tunings:

• WE: ~2 = 1200.0047 ¢, ~44/23 = 1122.8363 ¢ (~23/22 = 77.1684 ¢)
• CWE: ~2 = 1200.0000 ¢, ~44/23 = 1122.8319 ¢ (~23/22 = 77.1681 ¢)

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

Badness (Sintel): 0.944

2.3.5.7.11.13.17.23.29 subgroup

Subgroup: 2.3.5.7.11.13.17.23.29

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

Mapping: [⟨1 -19 7 0 112 43 49 114 61], ⟨0 22 -5 3 -116 -42 -48 -117 -60]]

Optimal tunings:

• WE: ~2 = 1199.9945 ¢, ~44/23 = 1122.8270 ¢ (~23/22 = 77.1675 ¢)
• CWE: ~2 = 1200.0000 ¢, ~44/23 = 1122.8322 ¢ (~23/22 = 77.1678 ¢)

Optimal ET sequence: 31ei, 140, 311, 762g

Badness (Sintel): 0.858

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

Emmthird

Emmthird tempers out the scheme comma and may be described as the 58 & 171 temperament. The generator for emmthird is flatter than 81/64 by a lee comma, 177147/175616, and sharper than 5/4 by the hemimage comma, 10976/10935. The ploidacot for this temperament is delta-14-cot.

The 11-limit version, which tempers out 243/242 and 441/440, has much lower accuracy and is supported by much fewer equal temperaments.

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

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:

• 7- and 9-odd-limit minimax: ~49/40 = [1/5 0 1/25⟩

[[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 catalogued 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

Cutefourths

This extension splits the neutral third plus an octave in three, with a ploidacot signature of beta-hexacot. The generator is an acute fourth in size (but not representing 27/20), hence the name.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4000/3993, 5120/5103

Mapping: [⟨1 -1 -30 -14 -28], ⟨0 6 75 39 73]]

mapping generators: ~2, ~66/49

Optimal tunings:

• WE: ~2 = 1199.7345 ¢, ~66/49 = 517.0436 ¢
• CWE: ~2 = 1200.0000 ¢, ~66/49 = 517.1543 ¢

Optimal ET sequence: 58, 181, 239, 1014bcee

Badness (Sintel): 1.71

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 1575/1573, 2401/2400

Mapping: [⟨1 -1 -30 -14 -28 -20], ⟨0 6 75 39 73 55]]

Optimal tunings:

• WE: ~2 = 1199.6427 ¢, ~66/49 = 517.0035 ¢
• CWE: ~2 = 1200.0000 ¢, ~66/49 = 517.1524 ¢

Optimal ET sequence: 58, 181, 239f

Badness (Sintel): 1.45

Osiris

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

Semiosiris

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 1771561/1771470

Mapping: [⟨2 -6 -20 -9 -14], ⟨0 32 86 51 73]]

mapping generators: ~99/70, ~243/220

Optimal tunings:

• WE: ~99/70 = 600.0152 ¢, ~243/220 = 171.9383 ¢
• CWE: ~99/70 = 600.0000 ¢, ~243/220 = 171.9344 ¢

Optimal ET sequence: 328, 342, 1012, 1354, 1696

Badness (Sintel): 0.703

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 its ploidacot is eta-38-cot (or omega-triseph due to a much simpler 2.5.7-subgroup structure). As one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)^1/8, giving just 7's, or 384^1/38, 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

Quinmite

Quinmite may be described as the 99 & 103 temperament. The generator for quinmite is the quasi-tempered minor third 25/21, sharper than 32/27 by the marvel comma, 225/224. It is also generated by 1/5 of the 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. Its ploidacot is eta-34-cot.

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

Septidiasemi

Main article: Septidiasemi

Aside from 2401/2400, septidiasemi tempers out 2152828125/2147483648 in the 7-limit, and may be described as the 10 & 171 temperament. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of 15/14), with a ploidacot of beta-26-cot. It is an excellent temperament 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]]

mapping 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

Lockerbie

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

Lockerbie can be described as the 103 & 270 temperament. Its generator is ~77/60 from the 11-limit onwards, and 74 generator steps give the interval class of 3; its ploidacot is 26-sheared 74-cot. 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 has a join 103 & 270, hence the name. The name was proposed in 2022 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 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

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. It is generated by the interval of 14/11 (undecimal major third, hence the name) tuned less than a cent flat, 42 of which octave reduced give the perfect fifth. Its ploidacot is 14-sheared 42-cot. The 23-note mos from the generator 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.

The commas it tempers out in the 11-limit include 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

Neominor

Neominor tempers out 177147/175616 and may be described as the 72 & 89 temperament. The generator is a neogothic minor third, which represents 13/11~20/17, or its octave complement, which represents 17/10~22/13. The latter stacked six times octave reduced give the perfect fifth, and the temperament has a ploidacot of delta-hexacot. 72edo and 89edo can be used as tunings.

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

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

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

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

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 may be described as the 58 & 103 temperament. It has a generator of ~10/9, tuned to around 49/44. Note that in the data below, the generator is its octave complement, ~9/5, so that 22 of them octave reduced give the perfect fifth. Its ploidacot is 18-sheared 22-cot. 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

Gorgik

See also: Llywelynsmic clan

Gorgik may be described as the 21 & 37 temperament, with a ploidacot of 14-sheared 18-cot (or alpha-heptaseph due to a much simpler 2.5.7-subgroup restriction). 58edo makes for a strong tuning for this temperament.

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

Hemigoldis

Hemigoldis may be described as the 68 & 89 temperament. Though fairly complex in the 7-limit, it does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to higher 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 can be described as the 311 & 431 temperament, starting with the 7-limit to the 19-limit. Its ploidacot is 28-sheared 70-cot. It was named by Eliora in 2022 for the generator fifth, 675/448 being 225/224 (marvel comma) sharp of 3/2.

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