Breedsmic temperaments: Difference between revisions

Revision as of 11:48, 23 February 2026

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:

Decimal (+25/24, 49/48 or 50/49) → Dicot family
Beatles (+64/63 or 686/675) → Archytas clan
Squares (+81/80) → Meantone family
Myna (+126/125) → Starling temperaments
Miracle (+225/224) → Gamelismic clan
Octacot (+245/243) → Tetracot family
Greenwood (+405/392 or 1323/1280) → Greenwoodmic temperaments
Quasitemp (+875/864) → Keemic temperaments
Quadrasruta (+2048/2025) → Diaschismic family
Quadrimage (+3125/3072) → Magic family
Hemiwürschmidt (+3136/3125 or 6144/6125) → Hemimean clan
Ennealimmal (+4375/4374) → Ragismic microtemperaments
Quadritikleismic (+15625/15552) → Kleismic family
Harry (+19683/19600) → Gravity family
Sesquiquartififths (+32805/32768) → Schismatic family
Amicable (+1600000/1594323) → Amity family
Neptune (+48828125/48771072) → Gammic family
Decoid (+67108864/66976875) → Quintosec family
Tertiseptisix (+390625000/387420489) → Quartonic family
Eagle (+10485760000/10460353203) → Vulture family

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:

CTE: ~2 = 1200.0000 ¢, ~49/40 = 351.4464 ¢

error map: ⟨0.0000 +0.9379 -0.1531 -0.0224]

POTE: ~2 = 1200.0000 ¢, ~49/40 = 351.4774 ¢

error map: ⟨0.0000 +0.9999 +0.6221 +0.0307]

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: 41, 58, 99, 239, 338

Badness (Smith): 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 = 1200.0000 ¢, ~11/9 = 351.4289 ¢
POTE: ~2 = 1200.0000 ¢, ~11/9 = 351.5206 ¢

Optimal ET sequence: 17c, 41, 58, 99e

Badness (Smith): 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 = 1200.0000 ¢, ~11/9 = 351.4331 ¢
POTE: ~2 = 1200.0000 ¢, ~11/9 = 351.5734 ¢

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

Badness (Smith): 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 = 600.0000 ¢, ~49/40 = 351.4722 ¢
POTE: ~99/70 = 600.0000 ¢, ~49/40 = 351.5047 ¢

Optimal ET sequence: 58, 140, 198

Badness (Smith): 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 = 600.0000 ¢, ~49/40 = 351.4674 ¢
POTE: ~99/70 = 600.0000 ¢, ~49/40 = 351.5019 ¢

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

Badness (Smith): 0.021188

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:

CTE: ~2 = 1200.0000 ¢, ~243/220 = 175.7284 ¢
POTE: ~2 = 1200.0000 ¢, ~243/220 = 175.7378 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~72/65 = 175.7412 ¢
POTE: ~2 = 1200.0000 ¢, ~72/65 = 175.7470 ¢

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

Badness (Smith): 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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~256/245 = 77.191 ¢

Optimal ET sequence: 31, 109, 140, 171

Badness (Smith): 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 = 1200.000 ¢, ~256/245 = 77.227 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~117/112 = 77.203 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~68/65 = 77.201 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~22/21 = 77.173 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~22/21 = 77.158 ¢

Optimal ET sequence: 31, 109, 140, 311e, 451ee

Badness (Smith): 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 = 1200.000 ¢, ~22/21 = 77.162 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~256/245 = 77.169 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~117/112 = 77.168 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~68/65 = 77.169 ¢

Optimal ET sequence: 140, 171, 311

Badness (Smith): 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 = 1200.000 ¢, ~68/65 = 77.169 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~23/22 = 77.168 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~23/22 = 77.167 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~23/22 = 77.169 ¢

Optimal ET sequence: 140, 171, 311

Badness (Smith): 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 = 1200.000 ¢, ~23/22 = 77.170 ¢

Optimal ET sequence: 140, 171, 311

Badness (Smith): 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 = 1200.000 ¢, ~23/22 = 77.169 ¢

Optimal ET sequence: 140, 171, 311

Badness (Smith): 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 = 1200.000 ¢, ~45/44 = 38.596 ¢

Optimal ET sequence: 31, 280, 311, 342

Badness (Smith): 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 = 1200.000 ¢, ~45/44 = 38.588 ¢

Optimal ET sequence: 31, 280, 311, 964f, 1275f, 1586cff

Badness (Smith): 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 = 1200.000 ¢, ~45/44 = 38.589 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~256/245 = 77.193 ¢

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

Badness (Smith): 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 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 31 0 9], ⟨0 -38 3 -8]]

mapping generators: ~2, ~875/512

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~1024/875 = 271.107 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~90/77 = 271.111 ¢

Optimal ET sequence: 31, 208, 239, 270

Badness (Smith): 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 = 1200.000 ¢, ~90/77 = 271.107 ¢

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

Badness (Smith): 0.017921

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

Optimal tunings:

TE: ~2 = 1200.428 ¢, ~189/160 = 283.381 ¢
CTE: ~2 = 1200.000 ¢, ~189/160 = 283.247 ¢

Optimal ET sequence: 72, 161, 233, 305

Badness (Sintel): 2.233

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 tunings:

TE: ~2 = 1200.347 ¢, ~33/28 = 283.358 ¢
CTE: ~2 = 1200.000 ¢, ~33/28 = 283.247 ¢

Optimal ET sequence: 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 12 8 7 7], ⟨0 -6 -41 -22 -15 -14]]

Optimal tunings:

TE: ~2 = 1200.689 ¢, ~13/11 = 283.457 ¢
CTE: ~2 = 1200.000 ¢, ~13/11 = 283.233 ¢

Optimal ET sequence: 72, 161f, 233ff

Badness (Sintel): 1.113

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 12 8 7 7 14], ⟨0 -6 -41 -22 -15 -14 -42]]

Optimal tunings:

TE: ~2 = 1200.692 ¢, ~13/11 = 283.455 ¢
CTE: ~2 = 1200.000 ¢, ~13/11 = 283.229 ¢

Optimal ET sequence: 72, 161f, 233ff

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 11 42 25], ⟨0 -14 -59 -33]]

mapping generators: ~2, ~2187/1372

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~2744/2187 = 392.988 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~1372/1089 = 392.991 ¢

Optimal ET sequence: 58, 113, 171

Badness (Smith): 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 = 1200.000 ¢, ~180/143 = 392.989 ¢

Optimal ET sequence: 58, 113, 171

Badness (Smith): 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 = 1200.000 ¢, ~64/51 = 392.985 ¢

Optimal ET sequence: 58, 113, 171

Badness (Smith): 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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~25/21 = 302.997 ¢

Optimal ET sequence: 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc

Badness (Smith): 0.037322

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 29 33 25], ⟨0 -42 -47 -34]]

mapping generators: ~2, ~6125/3888

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3969/3125 = 416.717 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~14/11 = 416.718 ¢

Optimal ET sequence: 72, 167, 239, 311

Badness (Smith): 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 = 1200.000 ¢, ~14/11 = 416.716 ¢

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

Badness (Smith): 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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/40 = 351.113 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~49/40 = 351.115 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~49/40 = 351.117 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~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]]

mpping generators: ~2, ~28/15

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/14 = 119.297 ¢

Optimal ET sequence: 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd

Badness (Smith): 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 = 1200.000 ¢, ~15/14 = 119.279 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~15/14 = 119.281 ¢

Optimal ET sequence: 10, 151, 161, 171, 332, 835eeff

Badness (Smith): 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 = 1200.000 ¢, ~15/14 = 119.281 ¢

Optimal ET sequence: 10, 151, 161, 171, 332, 503ef, 835eeff

Badness (Smith): 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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~1296/875 = 678.810 ¢

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

Badness (Smith): 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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~57344/46875 = 348.301 ¢

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

Badness (Smith): 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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~2800/2187 = 428.066 ¢

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

Badness (Smith): 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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~8/7 = 227.512 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~8/7 = 227.500 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~8/7 = 227.493 ¢

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

Badness (Smith): 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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~125/96 = 454.310 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~100/77 = 454.318 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~13/10 = 454.316 ¢

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

Badness (Smith): 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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~10/9 = 186.343 ¢

Optimal ET sequence: 45, 58, 103, 161, 586b, 747bc, 908bbc

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 186.345 ¢

Optimal ET sequence: 58, 103, 161, 425b, 586b, 747bc

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 186.347 ¢

Optimal ET sequence: 58, 103, 161, 425b, 586bf

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 186.348 ¢

Optimal ET sequence: 58, 103, 161, 425b, 586bf

Badness (Smith): 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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~250/189 = 489.235 ¢

Optimal ET sequence: 27, 76, 103, 130

Badness (Smith): 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 = 1200.000 ¢, ~250/189 = 489.252 ¢

Optimal ET sequence: 103, 130, 233, 363, 493e, 856be

Badness (Smith): 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 = 1200.000 ¢, ~65/49 = 489.256 ¢

Optimal ET sequence: 103, 130, 233, 363

Badness (Smith): 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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 583.385 ¢

Optimal ET sequence: 35, 37, 72, 109, 181, 253

Badness (Smith): 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 = 1200.000 ¢, ~7/5 = 583.387 ¢

Optimal ET sequence: 35, 37, 72, 109, 181, 253

Badness (Smith): 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 = 1200.000 ¢, ~7/5 = 583.387 ¢

Optimal ET sequence: 37, 72, 109, 181f

Badness (Smith): 0.028683

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 tuning (POTE): ~2 = 1200.000 ¢, ~49/40 = 348.603 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~11/9 = 348.639 ¢

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

Badness (Smith): 0.046181

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:

CTE: ~2 = 1200.0000 ¢, ~3828125/2985984 = 431.1071 ¢

error map: ⟨0.0000 -0.0270 +0.1502 -0.1120]

CWE: ~2 = 1200.0000 ¢, ~3828125/2985984 = 431.1072 ¢

error map: ⟨0.0000 -0.0205 +0.1547 -0.1081]

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

Badness (Smith): 0.0597

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:

CTE: ~2 = 1200.0000 ¢, ~77/60 = 431.1082 ¢
CWE: ~2 = 1200.0000 ¢, ~77/60 = 431.1078 ¢

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

Badness (Smith): 0.0262

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:

CTE: ~2 = 1200.0000 ¢, ~77/60 = 431.1085 ¢
CWE: ~2 = 1200.0000 ¢, ~77/60 = 431.1069 ¢

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

Badness (Smith): 0.0160

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:

CTE: ~2 = 1200.000 ¢, ~77/60 = 431.107 ¢
CWE: ~2 = 1200.000 ¢, ~77/60 = 431.108 ¢

Optimal ET sequence: 103, 167, 270

Badness (Smith): 0.0210

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:

CTE: ~2 = 1200.000 ¢, ~41/32 = 431.107 ¢
CWE: ~2 = 1200.000 ¢, ~41/32 = 431.111 ¢

Optimal ET sequence: 103, 167, 270

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, ~7/4

Optimal tuning (CWE): ~2 = 1200.000 ¢, ~7/4 = 970.690 ¢

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 43 -74 -25], ⟨0 -70 129 47]]

mapping generators: ~2, ~675/448

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~675/448 = 709.9719 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~675/448 = 709.9720 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~98/65 = 709.9723 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~98/65 = 709.9722 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~98/65 = 709.9722 ¢

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

Badness (Smith): 0.013771

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

[1]

[2]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-This page discusses miscellaneous [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[breedsma]], {{monzo| -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.
+This page discusses miscellaneous [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[breedsma]] ({{monzo|legend=1| -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)<sup>2</sup> = 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.
@@ Line 38: / Line 38: @@
 {{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}
 : mapping generators: ~2, ~49/40
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.0000, ~49/40 = 351.4464
+* [[CTE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.4464{{c}}
 : [[error map]]: {{val| 0.0000 +0.9379 -0.1531 -0.0224 }}
-* [[POTE]]: ~2 = 1200.0000, ~49/40 = 351.4774
+* [[POTE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.4774{{c}}
 : error map: {{val| 0.0000 +0.9999 +0.6221 +0.0307 }}
@@ Line 50: / Line 49: @@
 * [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
 : {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}
-: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 [[Algebraic generator]]: (2 + sqrt(2))/2
@@ Line 66: / Line 65: @@
 Optimal tunings:
-* CTE: ~2 = 1200.0000, ~11/9 = 351.4289
+* CTE: ~2 = 1200.0000{{c}}, ~11/9 = 351.4289{{c}}
-* POTE: ~2 = 1200.0000, ~11/9 = 351.5206
+* POTE: ~2 = 1200.0000{{c}}, ~11/9 = 351.5206{{c}}
 {{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}
@@ Line 81: / Line 80: @@
 Optimal tunings:
-* CTE: ~2 = 1200.0000, ~11/9 = 351.4331
+* CTE: ~2 = 1200.0000{{c}}, ~11/9 = 351.4331{{c}}
-* POTE: ~2 = 1200.0000, ~11/9 = 351.5734
+* POTE: ~2 = 1200.0000{{c}}, ~11/9 = 351.5734{{c}}
 {{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}
@@ Line 94: / Line 93: @@
 Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
 : mapping generators: ~99/70, ~400/231
 Optimal tunings:
-* CTE: ~99/70 = 600.0000, ~49/40 = 351.4722
+* CTE: ~99/70 = 600.0000{{c}}, ~49/40 = 351.4722{{c}}
-* POTE: ~99/70 = 600.0000, ~49/40 = 351.5047
+* POTE: ~99/70 = 600.0000{{c}}, ~49/40 = 351.5047{{c}}
 {{Optimal ET sequence|legend=0| 58, 140, 198 }}
@@ Line 113: / Line 111: @@
 Optimal tunings:
-* CTE: ~99/70 = 600.0000, ~49/40 = 351.4674
+* CTE: ~99/70 = 600.0000{{c}}, ~49/40 = 351.4674{{c}}
-* POTE: ~99/70 = 600.0000, ~49/40 = 351.5019
+* POTE: ~99/70 = 600.0000{{c}}, ~49/40 = 351.5019{{c}}
 {{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}
@@ Line 128: / Line 126: @@
 Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
 : Mapping generators: ~2, ~243/220
 Optimal tunings:
-* CTE: ~2 = 1200.0000, ~243/220 = 175.7284
+* CTE: ~2 = 1200.0000{{c}}, ~243/220 = 175.7284{{c}}
-* POTE: ~2 = 1200.0000, ~243/220 = 175.7378
+* POTE: ~2 = 1200.0000{{c}}, ~243/220 = 175.7378{{c}}
 {{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}
@@ Line 147: / Line 144: @@
 Optimal tunings:
-* CTE: ~2 = 1200.0000, ~72/65 = 175.7412
+* CTE: ~2 = 1200.0000{{c}}, ~72/65 = 175.7412{{c}}
-* POTE: ~2 = 1200.0000, ~72/65 = 175.7470
+* POTE: ~2 = 1200.0000{{c}}, ~72/65 = 175.7470{{c}}
 {{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}
@@ Line 157: / Line 154: @@
 {{Main| 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 &amp; 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.
+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 {{nowrap| 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
@@ Line 164: / Line 161: @@
 {{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}
+: mapping generators: ~2, ~256/245
-: Mapping generators: ~2, ~256/245
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~256/245 = 77.191{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191
 {{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}
-[[Badness]]: 0.012995
+[[Badness]] (Smith): 0.012995
 === 11-limit ===
@@ Line 180: / Line 176: @@
 Mapping: {{mapping| 1 3 2 3 7 | 0 -22 5 -3 -55 }}
-Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~256/245 = 77.227{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171, 202 }}
+{{Optimal ET sequence|legend=0| 31, 109e, 140e, 171, 202 }}
-Badness: 0.035576
+Badness (Smith): 0.035576
 ==== 13-limit ====
@@ Line 193: / Line 189: @@
 Mapping: {{mapping| 1 3 2 3 7 1 | 0 -22 5 -3 -55 42 }}
-Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~117/112 = 77.203{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171 }}
+{{Optimal ET sequence|legend=0| 31, 109e, 140e, 171 }}
-Badness: 0.036876
+Badness (Smith): 0.036876
 ==== 17-limit ====
@@ Line 206: / Line 202: @@
 Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~68/65 = 77.201{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 31, 109eg, 140e, 171 }}
+{{Optimal ET sequence|legend=0| 31, 109eg, 140e, 171 }}
-Badness: 0.027398
+Badness (Smith): 0.027398
 === Tertia ===
@@ Line 219: / Line 215: @@
 Mapping: {{mapping| 1 3 2 3 5 | 0 -22 5 -3 -24 }}
-Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~22/21 = 77.173{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 171e, 311e }}
+{{Optimal ET sequence|legend=0| 31, 109, 140, 171e, 311e }}
-Badness: 0.030171
+Badness (Smith): 0.030171
 ==== 13-limit ====
@@ Line 232: / Line 228: @@
 Mapping: {{mapping| 1 3 2 3 5 1 | 0 -22 5 -3 -24 42 }}
-Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~22/21 = 77.158{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 311e, 451ee }}
+{{Optimal ET sequence|legend=0| 31, 109, 140, 311e, 451ee }}
-Badness: 0.028384
+Badness (Smith): 0.028384
 ==== 17-limit ====
@@ Line 245: / Line 241: @@
 Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~22/21 = 77.162{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 31, 109g, 140, 311e, 451ee }}
+{{Optimal ET sequence|legend=0| 31, 109g, 140, 311e, 451ee }}
-Badness: 0.022416
+Badness (Smith): 0.022416
 === Tertiaseptia ===
@@ Line 258: / Line 254: @@
 Mapping: {{mapping| 1 3 2 3 -4 | 0 -22 5 -3 116 }}
-Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~256/245 = 77.169{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}
+{{Optimal ET sequence|legend=0| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}
-Badness: 0.056926
+Badness (Smith): 0.056926
 ==== 13-limit ====
@@ Line 271: / Line 267: @@
 Mapping: {{mapping| 1 3 2 3 -4 1 | 0 -22 5 -3 116 42 }}
-Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~117/112 = 77.168{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}
+{{Optimal ET sequence|legend=0| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}
-Badness: 0.027474
+Badness (Smith): 0.027474
 ==== 17-limit ====
@@ Line 284: / Line 280: @@
 Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~68/65 = 77.169{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
+{{Optimal ET sequence|legend=0| 140, 171, 311 }}
-Badness: 0.018773
+Badness (Smith): 0.018773
 ==== 19-limit ====
@@ Line 297: / Line 293: @@
 Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~68/65 = 77.169{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}
+{{Optimal ET sequence|legend=0| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}
-Badness: 0.017653
+Badness (Smith): 0.017653
 ==== 23-limit ====
@@ Line 310: / Line 306: @@
 Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~23/22 = 77.168{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfgg }}
+{{Optimal ET sequence|legend=0| 140, 311, 762g, 1073g, 1384cfgg }}
-Badness: 0.015123
+Badness (Smith): 0.015123
 ==== 29-limit ====
@@ Line 323: / Line 319: @@
 Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~23/22 = 77.167{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfggj }}
+{{Optimal ET sequence|legend=0| 140, 311, 762g, 1073g, 1384cfggj }}
-Badness: 0.012181
+Badness (Smith): 0.012181
 ==== 31-limit ====
@@ Line 336: / Line 332: @@
 Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~23/22 = 77.169{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
+{{Optimal ET sequence|legend=0| 140, 171, 311 }}
-Badness: 0.012311
+Badness (Smith): 0.012311
 ==== 37-limit ====
@@ Line 349: / Line 345: @@
 Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~23/22 = 77.170{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
+{{Optimal ET sequence|legend=0| 140, 171, 311 }}
-Badness: 0.010949
+Badness (Smith): 0.010949
 ==== 41-limit ====
@@ Line 362: / Line 358: @@
 Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~23/22 = 77.169{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
+{{Optimal ET sequence|legend=0| 140, 171, 311 }}
-Badness: 0.009825
+Badness (Smith): 0.009825
 === Hemitert ===
@@ Line 374: / Line 370: @@
 Mapping: {{mapping| 1 3 2 3 6 | 0 -44 10 -6 -79 }}
+: mapping generators: ~2, ~45/44
-: Mapping generators: ~2, ~45/44
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/44 = 38.596{{c}}
-Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596
+{{Optimal ET sequence|legend=0| 31, 280, 311, 342 }}
-Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 342 }}
+Badness (Smith): 0.015633
-Badness: 0.015633
 ==== 13-limit ====
@@ Line 390: / Line 385: @@
 Mapping: {{mapping| 1 3 2 3 6 1 | 0 -44 10 -6 -79 84 }}
-Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/44 = 38.588{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 964f, 1275f, 1586cff }}
+{{Optimal ET sequence|legend=0| 31, 280, 311, 964f, 1275f, 1586cff }}
-Badness: 0.033573
+Badness (Smith): 0.033573
 ==== 17-limit ====
@@ Line 403: / Line 398: @@
 Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~45/44 = 38.589{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 653f, 964f }}
+{{Optimal ET sequence|legend=0| 31, 280, 311, 653f, 964f }}
-Badness: 0.025298
+Badness (Smith): 0.025298
 === Semitert ===
@@ Line 415: / Line 410: @@
 Mapping: {{mapping| 2 6 4 6 1 | 0 -22 5 -3 46 }}
+: mapping generators: ~99/70, ~256/245
-: Mapping generators: ~99/70, ~256/245
+Optimal tuning (POTE): ~99/70 = 1200.000{{c}}, ~256/245 = 77.193{{c}}
-Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193
+{{Optimal ET sequence|legend=0| 62e, 140, 202, 342 }}
-Optimal ET sequence: {{Optimal ET sequence| 62e, 140, 202, 342 }}
+Badness (Smith): 0.025790
-Badness: 0.025790
 == Quasiorwell ==
-In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 = {{monzo| 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 &amp; 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)<sup>1/8</sup>, giving just 7's, or 384<sup>1/38</sup>, giving pure fifths.
+In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 ({{monzo| 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)<sup>1/8</sup>, giving just 7's, or 384<sup>1/38</sup>, giving pure fifths.
 Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.
@@ Line 434: / Line 428: @@
 {{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }}
+: mapping generators: ~2, ~875/512
-: Mapping generators: ~2, ~875/512
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~1024/875 = 271.107{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107
 {{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}
-[[Badness]]: 0.035832
+[[Badness]] (Smith): 0.035832
 === 11-limit ===
@@ Line 450: / Line 443: @@
 Mapping: {{mapping| 1 31 0 9 53 | 0 -38 3 -8 -64 }}
-Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~90/77 = 271.111{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 31, 208, 239, 270 }}
+{{Optimal ET sequence|legend=0| 31, 208, 239, 270 }}
-Badness: 0.017540
+Badness (Smith): 0.017540
 === 13-limit ===
@@ Line 463: / Line 456: @@
 Mapping: {{mapping| 1 31 0 9 53 -59 | 0 -38 3 -8 -64 81 }}
-Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~90/77 = 271.107{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 31, 239, 270, 571, 841, 1111 }}
+{{Optimal ET sequence|legend=0| 31, 239, 270, 571, 841, 1111 }}
-Badness: 0.017921
+Badness (Smith): 0.017921
 == Neominor ==
@@ Line 477: / Line 470: @@
 {{Mapping|legend=1| 1 3 12 8 | 0 -6 -41 -22 }}
+: mapping generators: ~2, ~189/160
-: Mapping generators: ~2, ~189/160
 [[Optimal tuning]]s:
-* [[TE]]: ~2 = 1200.428, ~189/160 = 283.381
+* [[TE]]: ~2 = 1200.428{{c}}, ~189/160 = 283.381{{c}}
-* [[CTE]]: ~2 = 1200.000, ~189/160 = 283.247
+* [[CTE]]: ~2 = 1200.000{{c}}, ~189/160 = 283.247{{c}}
 {{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
@@ Line 496: / Line 488: @@
 Optimal tunings:
-* TE: ~2 = 1200.347, ~33/28 = 283.358
+* TE: ~2 = 1200.347{{c}}, ~33/28 = 283.358{{c}}
-* CTE: ~2 = 1200.000, ~33/28 = 283.247
+* CTE: ~2 = 1200.000{{c}}, ~33/28 = 283.247{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 72, 161, 233, 305 }}
+{{Optimal ET sequence|legend=0| 72, 161, 233, 305 }}
 Badness (Sintel): 0.924
@@ Line 511: / Line 503: @@
 Optimal tunings:
-* TE: ~2 = 1200.689, ~13/11 = 283.457
+* TE: ~2 = 1200.689{{c}}, ~13/11 = 283.457{{c}}
-* CTE: ~2 = 1200.000, ~13/11 = 283.233
+* CTE: ~2 = 1200.000{{c}}, ~13/11 = 283.233{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 72, 161f, 233ff }}
+{{Optimal ET sequence|legend=0| 72, 161f, 233ff }}
 Badness (Sintel): 1.113
@@ Line 526: / Line 518: @@
 Optimal tunings:
-* TE: ~2 = 1200.692, ~13/11 = 283.455
+* TE: ~2 = 1200.692{{c}}, ~13/11 = 283.455{{c}}
-* CTE: ~2 = 1200.000, ~13/11 = 283.229
+* CTE: ~2 = 1200.000{{c}}, ~13/11 = 283.229{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 72, 161f, 233ff }}
+{{Optimal ET sequence|legend=0| 72, 161f, 233ff }}
 Badness (Sintel): 0.918
@@ Line 541: / Line 533: @@
 {{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }}
+: mapping generators: ~2, ~2187/1372
-: Mapping generators: ~2, ~2187/1372
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~2744/2187 = 392.988{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988
 {{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
-[[Badness]]: 0.016736
+[[Badness]] (Smith): 0.016736
 === 11-limit ===
@@ Line 557: / Line 548: @@
 Mapping: {{mapping| 1 11 42 25 27 | 0 -14 -59 -33 -35 }}
-Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~1372/1089 = 392.991{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
+{{Optimal ET sequence|legend=0| 58, 113, 171 }}
-Badness: 0.052358
+Badness (Smith): 0.052358
 === 13-limit ===
@@ Line 570: / Line 561: @@
 Mapping: {{mapping| 1 11 42 25 27 38 | 0 -14 -59 -33 -35 -51 }}
-Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~180/143 = 392.989{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
+{{Optimal ET sequence|legend=0| 58, 113, 171 }}
-Badness: 0.026974
+Badness (Smith): 0.026974
 === 17-limit ===
@@ Line 583: / Line 574: @@
 Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~64/51 = 392.985{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
+{{Optimal ET sequence|legend=0| 58, 113, 171 }}
-Badness: 0.023205
+Badness (Smith): 0.023205
 == Quinmite ==
@@ Line 597: / Line 588: @@
 {{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }}
+: mapping generators: ~2, ~42/25
-: Mapping generators: ~2, ~42/25
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25/21 = 302.997{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997
 {{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}
-[[Badness]]: 0.037322
+[[Badness]] (Smith): 0.037322
 == Unthirds ==
@@ Line 616: / Line 606: @@
 {{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }}
+: mapping generators: ~2, ~6125/3888
-: Mapping generators: ~2, ~6125/3888
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3969/3125 = 416.717{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717
 {{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
-[[Badness]]: 0.075253
+[[Badness]] (Smith): 0.075253
 === 11-limit ===
@@ Line 632: / Line 621: @@
 Mapping: {{mapping| 1 29 33 25 25 | 0 -42 -47 -34 -33 }}
-Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 416.718{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 72, 167, 239, 311 }}
+{{Optimal ET sequence|legend=0| 72, 167, 239, 311 }}
-Badness: 0.022926
+Badness (Smith): 0.022926
 === 13-limit ===
@@ Line 645: / Line 634: @@
 Mapping: {{mapping| 1 29 33 25 25 99 | 0 -42 -47 -34 -33 -146 }}
-Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 416.716{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 72, 239f, 311, 694, 1005c }}
+{{Optimal ET sequence|legend=0| 72, 239f, 311, 694, 1005c }}
-Badness: 0.020888
+Badness (Smith): 0.020888
 == Newt ==
@@ Line 659: / Line 648: @@
 {{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }}
 : mapping generators: ~2, ~49/40
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/40 = 351.113{{c}}
 {{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}
-[[Badness]]: 0.041878
+[[Badness]] (Smith): 0.041878
 === 11-limit ===
@@ Line 675: / Line 663: @@
 Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}
-Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/40 = 351.115{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972 }}
+{{Optimal ET sequence|legend=0| 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972 }}
-Badness: 0.019461
+Badness (Smith): 0.019461
 === 13-limit ===
@@ Line 688: / Line 676: @@
 Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}
-Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/40 = 351.117{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b }}
+{{Optimal ET sequence|legend=0| 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b }}
-Badness: 0.013830
+Badness (Smith): 0.013830
 === 2.3.5.7.11.13.19 subgroup (neonewt) ===
@@ Line 701: / Line 689: @@
 Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~49/40 = 351.117{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb }}
+{{Optimal ET sequence|legend=0| 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb }}
 == Septidiasemi ==
@@ Line 715: / Line 703: @@
 {{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}
+: mpping generators: ~2, ~28/15
-: Mapping generators: ~2, ~28/15
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~15/14 = 119.297{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297
 {{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}
-[[Badness]]: 0.044115
+[[Badness]] (Smith): 0.044115
 === Sedia ===
-The ''sedia'' temperament (10&amp;161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.
+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
@@ Line 733: / Line 720: @@
 Mapping: {{mapping| 1 25 -31 -8 62 | 0 -26 37 12 -65 }}
-Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 119.279{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332 }}
+{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332 }}
-Badness: 0.090687
+Badness (Smith): 0.090687
 ==== 13-limit ====
@@ Line 746: / Line 733: @@
 Mapping: {{mapping| 1 25 -31 -8 62 1 | 0 -26 37 12 -65 3 }}
-Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 119.281{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 835eeff }}
+{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332, 835eeff }}
-Badness: 0.045773
+Badness (Smith): 0.045773
 ==== 17-limit ====
@@ Line 759: / Line 746: @@
 Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 119.281{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 503ef, 835eeff }}
+{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332, 503ef, 835eeff }}
-Badness: 0.027322
+Badness (Smith): 0.027322
 == Maviloid ==
@@ Line 773: / Line 760: @@
 {{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }}
+: mapping generators: ~2, ~1296/875
-: Mapping generators: ~2, ~1296/875
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~1296/875 = 678.810{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810
 {{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}
-[[Badness]]: 0.057632
+[[Badness]] (Smith): 0.057632
 == Subneutral ==
@@ Line 790: / Line 776: @@
 {{Mapping|legend=1| 1 19 0 6 | 0 -60 8 -11 }}
+: mapping generators: ~2, ~57344/46875
-: Mapping generators: ~2, ~57344/46875
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~57344/46875 = 348.301{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301
 {{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}
-[[Badness]]: 0.045792
+[[Badness]] (Smith): 0.045792
 == Osiris ==
@@ Line 807: / Line 792: @@
 {{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }}
+: mapping generators: ~2, ~2800/2187
-: Mapping generators: ~2, ~2800/2187
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~2800/2187 = 428.066{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066
 {{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}
-[[Badness]]: 0.028307
+[[Badness]] (Smith): 0.028307
 == Gorgik ==
@@ Line 822: / Line 806: @@
 {{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }}
+: mapping generators: ~2, ~8/7
-: Mapping generators: ~2, ~8/7
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~8/7 = 227.512{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512
 {{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
-[[Badness]]: 0.158384
+[[Badness]] (Smith): 0.158384
 === 11-limit ===
@@ Line 838: / Line 821: @@
 Mapping: {{mapping| 1 5 1 3 1 | 0 -18 7 -1 13 }}
-Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~8/7 = 227.500{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
+{{Optimal ET sequence|legend=0| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
-Badness: 0.059260
+Badness (Smith): 0.059260
 === 13-limit ===
@@ Line 851: / Line 834: @@
 Mapping: {{mapping| 1 5 1 3 1 2 | 0 -18 7 -1 13 9 }}
-Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~8/7 = 227.493{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bcef, 211bccdeeff }}
+{{Optimal ET sequence|legend=0| 21, 37, 58, 153bcef, 211bccdeeff }}
-Badness: 0.032205
+Badness (Smith): 0.032205
 == Fibo ==
@@ Line 863: / Line 846: @@
 {{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }}
+: mapping generators: ~2, ~125/96
-: Mapping generators: ~2, ~125/96
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~125/96 = 454.310{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310
 {{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}
-Badness: 0.100511
+Badness (Smith): 0.100511
 === 11-limit ===
@@ Line 879: / Line 861: @@
 Mapping: {{mapping| 1 19 8 10 8 | 0 -46 -15 -19 -12 }}
-Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/77 = 454.318{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}
+{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}
-Badness: 0.056514
+Badness (Smith): 0.056514
 === 13-limit ===
@@ Line 892: / Line 874: @@
 Mapping: {{mapping| 1 19 8 10 8 9 | 0 -46 -15 -19 -12 -14 }}
-Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/10 = 454.316{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}
+{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}
-Badness: 0.027429
+Badness (Smith): 0.027429
 == Mintone ==
-In addition to 2401/2400, mintone tempers out 177147/175000 = {{monzo| -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 &amp; 103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.
+In addition to 2401/2400, mintone tempers out 177147/175000 ({{monzo| -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
@@ Line 906: / Line 888: @@
 {{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }}
+: mapping generators: ~2, ~10/9
-: Mapping generators: ~2, ~10/9
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~10/9 = 186.343{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343
 {{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}
-[[Badness]]: 0.125672
+[[Badness]] (Smith): 0.125672
 === 11-limit ===
@@ Line 922: / Line 903: @@
 Mapping: {{mapping| 1 5 9 7 12 | 0 -22 -43 -27 -55 }}
-Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 186.345{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586b, 747bc }}
+{{Optimal ET sequence|legend=0| 58, 103, 161, 425b, 586b, 747bc }}
-Badness: 0.039962
+Badness (Smith): 0.039962
 === 13-limit ===
@@ Line 935: / Line 916: @@
 Mapping: {{mapping| 1 5 9 7 12 11 | 0 -22 -43 -27 -55 -47 }}
-Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 186.347{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}
+{{Optimal ET sequence|legend=0| 58, 103, 161, 425b, 586bf }}
-Badness: 0.021849
+Badness (Smith): 0.021849
 === 17-limit ===
@@ Line 948: / Line 929: @@
 Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 186.348{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}
+{{Optimal ET sequence|legend=0| 58, 103, 161, 425b, 586bf }}
-Badness: 0.020295
+Badness (Smith): 0.020295
 == Catafourth ==
@@ Line 962: / Line 943: @@
 {{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }}
+: mapping generators: ~2, ~250/189
-: Mapping generators: ~2, ~250/189
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~250/189 = 489.235{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235
 {{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
-Badness: 0.079579
+Badness (Smith): 0.079579
 === 11-limit ===
@@ Line 978: / Line 958: @@
 Mapping: {{mapping| 1 13 17 13 32 | 0 -28 -36 -25 -70 }}
-Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~250/189 = 489.252{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363, 493e, 856be }}
+{{Optimal ET sequence|legend=0| 103, 130, 233, 363, 493e, 856be }}
-Badness: 0.036785
+Badness (Smith): 0.036785
 === 13-limit ===
@@ Line 991: / Line 971: @@
 Mapping: {{mapping| 1 13 17 13 32 9 | 0 -28 -36 -25 -70 -13 }}
-Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~65/49 = 489.256{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363 }}
+{{Optimal ET sequence|legend=0| 103, 130, 233, 363 }}
-Badness: 0.021694
+Badness (Smith): 0.021694
 == Cotritone ==
@@ Line 1,003: / Line 983: @@
 {{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}
+: mappping generators: ~2, ~10/7
-: Mappping generators: ~2, ~10/7
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 583.385{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385
 {{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
-[[Badness]]: 0.098322
+[[Badness]] (Smith): 0.098322
 === 11-limit ===
@@ Line 1,019: / Line 998: @@
 Mapping: {{mapping| 1 17 9 10 5 | 0 -30 -13 -14 -3 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 583.387{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 35, 37, 72, 109, 181, 253 }}
+{{Optimal ET sequence|legend=0| 35, 37, 72, 109, 181, 253 }}
-Badness: 0.032225
+Badness (Smith): 0.032225
 === 13-limit ===
@@ Line 1,032: / Line 1,011: @@
 Mapping: {{mapping| 1 17 9 10 5 15 | 0 -30 -13 -14 -3 -22 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 583.387{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 37, 72, 109, 181f }}
+{{Optimal ET sequence|legend=0| 37, 72, 109, 181f }}
-Badness: 0.028683
+Badness (Smith): 0.028683
 == Quasimoha ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasimoha]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasimoha]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 1,046: / Line 1,025: @@
 {{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}
+: mapping generators: ~2, ~49/40
-: Mapping generators: ~2, ~49/40
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/40 = 348.603{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 348.603
 {{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}
-[[Badness]]: 0.110820
+[[Badness]] (Smith): 0.110820
 === 11-limit ===
@@ Line 1,062: / Line 1,040: @@
 Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/9 = 348.639{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 31, 86ce, 117ce, 148bce }}
+{{Optimal ET sequence|legend=0| 31, 86ce, 117ce, 148bce }}
-Badness: 0.046181
+Badness (Smith): 0.046181
 == Lockerbie ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].''
 Lockerbie can be described as the {{nowrap| 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.
@@ Line 1,081: / Line 1,060: @@
 {{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}
+: mapping generators: ~2, ~3828125/2985984
-: Mapping generators: ~2, ~3828125/2985984
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1071
+* [[CTE]]: ~2 = 1200.0000{{c}}, ~3828125/2985984 = 431.1071{{c}}
 : [[error map]]: {{val| 0.0000 -0.0270 +0.1502 -0.1120 }}
-* [[CWE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1072
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3828125/2985984 = 431.1072{{c}}
 : error map: {{val| 0.0000 -0.0205 +0.1547 -0.1081 }}
@@ Line 1,102: / Line 1,080: @@
 Optimal tunings:
-* CTE: ~2 = 1200.0000, ~77/60 = 431.1082
+* CTE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1082{{c}}
-* CWE: ~2 = 1200.0000, ~77/60 = 431.1078
+* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1078{{c}}
 {{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}
@@ Line 1,117: / Line 1,095: @@
 Optimal tunings:
-* CTE: ~2 = 1200.0000, ~77/60 = 431.1085
+* CTE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1085{{c}}
-* CWE: ~2 = 1200.0000, ~77/60 = 431.1069
+* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1069{{c}}
 {{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}
@@ Line 1,132: / Line 1,110: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~77/60 = 431.107
+* CTE: ~2 = 1200.000{{c}}, ~77/60 = 431.107{{c}}
-* CWE: ~2 = 1200.000, ~77/60 = 431.108
+* CWE: ~2 = 1200.000{{c}}, ~77/60 = 431.108{{c}}
 {{Optimal ET sequence|legend=0| 103, 167, 270 }}
@@ Line 1,147: / Line 1,125: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~41/32 = 431.107
+* CTE: ~2 = 1200.000{{c}}, ~41/32 = 431.107{{c}}
-* CWE: ~2 = 1200.000, ~41/32 = 431.111
+* CWE: ~2 = 1200.000{{c}}, ~41/32 = 431.111{{c}}
 {{Optimal ET sequence|legend=0| 103, 167, 270 }}
@@ Line 1,162: / Line 1,140: @@
 {{Mapping|legend=1| 1 21 -9 2 | 0 -24 14 1 }}
 : mapping generators: ~2, ~7/4
-[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~7/4 = 970.690
+[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000{{c}}, ~7/4 = 970.690{{c}}
 {{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}
@@ Line 1,179: / Line 1,156: @@
 {{Mapping|legend=1| 1 43 -74 -25 | 0 -70 129 47 }}
+: mapping generators: ~2, ~675/448
-: Mapping generators: ~2, ~675/448
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~675/448 = 709.9719{{c}}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~675/448 = 709.9719
 {{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 2848, 3901 }}
-[[Badness]]: 0.202249
+[[Badness]] (Smith): 0.202249
 === 11-limit ===
@@ Line 1,195: / Line 1,171: @@
 Mapping: {{mapping| 1 43 -74 -25 36 | 0 -70 129 47 -55 }}
-Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9720
+Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~675/448 = 709.9720{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795 }}
+{{Optimal ET sequence|legend=0| 120, 191, 311, 742, 1053, 1795 }}
-Badness: 0.052308
+Badness (Smith): 0.052308
 === 13-limit ===
@@ Line 1,208: / Line 1,184: @@
 Mapping: {{mapping| 1 43 -74 -25 36 25 | 0 -70 129 47 -55 -36 }}
-Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9723
+Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~98/65 = 709.9723{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795f }}
+{{Optimal ET sequence|legend=0| 120, 191, 311, 742, 1053, 1795f }}
-Badness: 0.032503
+Badness (Smith): 0.032503
 === 17-limit ===
@@ Line 1,221: / Line 1,197: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.0000{{c}}, ~98/65 = 709.9722{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}
+{{Optimal ET sequence|legend=0| 120g, 191g, 311, 431, 742, 1795f }}
-Badness: 0.020995
+Badness (Smith): 0.020995
 === 19-limit ===
@@ Line 1,234: / Line 1,210: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.0000{{c}}, ~98/65 = 709.9722{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}
+{{Optimal ET sequence|legend=0| 120g, 191g, 311, 431, 742, 1795f }}
-Badness: 0.013771
+Badness (Smith): 0.013771
-== Notes ==
+== References ==
 [[Category:Temperament collections]]