Breedsmic temperaments
This page discusses miscellaneous rank-2 temperaments tempering out the breedsma, [-5 -1 -2 4⟩ = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.
The breedsma is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, 25/24. We might note also that (49/40)(10/7) = 7/4 and (49/40)(10/7)2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.
Temperaments discussed elsewhere include:
- 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
- Tertiseptisix (+390625000/387420489) → Quartonic family
- Eagle (+10485760000/10460353203) → Vulture family
Hemififths
Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with 99edo and 140edo providing good tunings, and 239edo an even better one; and other possible tunings are 160(1/25), giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14(1/13), giving just 7's. It may be called the 41 & 58 temperament. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17- and 24-note mos are suited. The full force of this highly accurate temperament can be found using the 41-note mos or even the 34-note 2mos [clarification needed ].
By adding 243/242 (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99edo is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding 144/143 brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be 16/13. 99 remains a good tuning choice.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 5120/5103
Mapping: [⟨1 1 -5 -1], ⟨0 2 25 13]]
- mapping generators: ~2, ~49/40
Wedgie: ⟨⟨ 2 25 13 35 15 -40 ]]
- 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⟩]
- eigenmonzo (unchanged-interval) basis: 2.5
Algebraic generator: (2 + sqrt(2))/2
Optimal ET sequence: 41, 58, 99, 239, 338
Badness: 0.022243
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 896/891
Mapping: [⟨1 1 -5 -1 2], ⟨0 2 25 13 5]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 351.4289
- POTE: ~2 = 1\1, ~11/9 = 351.521
Optimal ET sequence: 17c, 41, 58, 99e
Badness: 0.023498
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 144/143, 196/195, 243/242, 364/363
Mapping: [⟨1 1 -5 -1 2 4], ⟨0 2 25 13 5 -1]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 351.4331
- POTE: ~2 = 1\1, ~11/9 = 351.573
Optimal ET sequence: 17c, 41, 58, 99ef, 157eff
Badness: 0.019090
Semihemi
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3388/3375, 5120/5103
Mapping: [⟨2 0 -35 -15 -47], ⟨0 2 25 13 34]]
- mapping generators: ~99/70, ~400/231
Optimal tunings:
- CTE: ~99/70 = 1\2, ~49/40 = 351.4722
- POTE: ~99/70 = 1\2, ~49/40 = 351.505
Optimal ET sequence: 58, 140, 198
Badness: 0.042487
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 676/675, 847/845, 1716/1715
Mapping: [⟨2 0 -35 -15 -47 -37], ⟨0 2 25 13 34 28]]
Optimal tunings:
- CTE: ~99/70 = 1\2, ~49/40 = 351.4674
- POTE: ~99/70 = 1\2, ~49/40 = 351.502
Optimal ET sequence: 58, 140, 198, 536f
Badness: 0.021188
Quadrafifths
This has been logged as semihemififths in Graham Breed's temperament finder, but quadrafifths arguably makes more sense.
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 5120/5103
Mapping: [⟨1 1 -5 -1 8], ⟨0 4 50 26 -31]]
- Mapping generators: ~2, ~243/220
Optimal tunings:
- CTE: ~2 = 1\1, ~243/220 = 175.7284
- POTE: ~2 = 1\1, ~243/220 = 175.7378
Optimal ET sequence: 41, 157, 198, 239, 676b, 915be
Badness: 0.040170
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 847/845, 2401/2400, 3025/3024
Mapping: [⟨1 1 -5 -1 8 10], ⟨0 4 50 26 -31 -43]]
Optimal tunings:
- CTE: ~2 = 1\1, ~72/65 = 175.7412
- POTE: ~2 = 1\1, ~72/65 = 175.7470
Optimal ET sequence: 41, 157, 198, 437f, 635bcff
Badness: 0.031144
Tertiaseptal
Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152, the rainy comma. It can be described as the 31 & 171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. 171edo makes for an excellent tuning, although 171edo - 31edo = 140edo also makes sense, and in very high limits 140edo + 171edo = 311edo is especially notable. The 15- or 16-note mos can be used to explore no-threes harmony, and the 31-note mos gives plenty of room for those as well.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 65625/65536
Mapping: [⟨1 3 2 3], ⟨0 -22 5 -3]]
- Mapping generators: ~2, ~256/245
Wedgie: ⟨⟨ 22 -5 3 -59 -57 21 ]]
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.191
Optimal ET sequence: 31, 109, 140, 171
Badness: 0.012995
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 65625/65536
Mapping: [⟨1 3 2 3 7], ⟨0 -22 5 -3 -55]]
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227
Optimal ET sequence: 31, 109e, 140e, 171, 202
Badness: 0.035576
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 441/440, 625/624, 3584/3575
Mapping: [⟨1 3 2 3 7 1], ⟨0 -22 5 -3 -55 42]]
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203
Optimal ET sequence: 31, 109e, 140e, 171
Badness: 0.036876
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
Mapping: [⟨1 3 2 3 7 1 1], ⟨0 -22 5 -3 -55 42 48]]
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201
Optimal ET sequence: 31, 109eg, 140e, 171
Badness: 0.027398
Tertia
Subgroup:2.3.5.7.11
Comma list: 385/384, 1331/1323, 1375/1372
Mapping: [⟨1 3 2 3 5], ⟨0 -22 5 -3 -24]]
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173
Optimal ET sequence: 31, 109, 140, 171e, 311e
Badness: 0.030171
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 385/384, 625/624, 1331/1323
Mapping: [⟨1 3 2 3 5 1], ⟨0 -22 5 -3 -24 42]]
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158
Optimal ET sequence: 31, 109, 140, 311e, 451ee
Badness: 0.028384
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
Mapping: [⟨1 3 2 3 5 1 1], ⟨0 -22 5 -3 -24 42 48]]
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162
Optimal ET sequence: 31, 109g, 140, 311e, 451ee
Badness: 0.022416
Tertiaseptia
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 6250/6237, 65625/65536
Mapping: [⟨1 3 2 3 -4], ⟨0 -22 5 -3 116]]
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169
Optimal ET sequence: 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde
Badness: 0.056926
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400
Mapping: [⟨1 3 2 3 -4 1], ⟨0 -22 5 -3 116 42]]
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168
Optimal ET sequence: 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf
Badness: 0.027474
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197
Mapping: [⟨1 3 2 3 -4 1 1], ⟨0 -22 5 -3 116 42 48]]
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169
Optimal ET sequence: 140, 171, 311
Badness: 0.018773
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197
Mapping: [⟨1 3 2 3 -4 1 1 11], ⟨0 -22 5 -3 116 42 48 -105]]
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169
Optimal ET sequence: 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg
Badness: 0.017653
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215
Mapping: [⟨1 3 2 3 -4 1 1 11 -3], ⟨0 -22 5 -3 116 42 48 -105 117]]
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168
Optimal ET sequence: 140, 311, 762g, 1073g, 1384cfgg
Badness: 0.015123
29-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29
Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155
Mapping: [⟨1 3 2 3 -4 1 1 11 -3 1], ⟨0 -22 5 -3 116 42 48 -105 117 60]]
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167
Optimal ET sequence: 140, 311, 762g, 1073g, 1384cfggj
Badness: 0.012181
31-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
Mapping: [⟨1 3 2 3 -4 1 1 11 -3 1 11], ⟨0 -22 5 -3 116 42 48 -105 117 60 -94]]
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
Optimal ET sequence: 140, 171, 311
Badness: 0.012311
37-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
Comma list: 595/594, 625/624, 703/702, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
Mapping: [⟨1 3 2 3 -4 1 1 11 -3 1 11 0], ⟨0 -22 5 -3 116 42 48 -105 117 60 -94 81]]
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.170
Optimal ET sequence: 140, 171, 311
Badness: 0.010949
41-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
Comma list: 595/594, 625/624, 697/696, 703/702, 714/713, 784/783, 820/819, 833/832, 875/874, 900/899, 931/930
Mapping: [⟨1 3 2 3 -4 1 1 11 -3 1 11 0 6], ⟨0 -22 5 -3 116 42 48 -105 117 60 -94 81 -10]]
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
Optimal ET sequence: 140, 171, 311
Badness: 0.009825
Hemitert
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 65625/65536
Mapping: [⟨1 3 2 3 6], ⟨0 -44 10 -6 -79]]
- Mapping generators: ~2, ~45/44
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596
Optimal ET sequence: 31, 280, 311, 342
Badness: 0.015633
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095
Mapping: [⟨1 3 2 3 6 1], ⟨0 -44 10 -6 -79 84]]
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588
Optimal ET sequence: 31, 280, 311, 964f, 1275f, 1586cff
Badness: 0.033573
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095
Mapping: [⟨1 3 2 3 6 1 1], ⟨0 -44 10 -6 -79 84 96]]
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589
Optimal ET sequence: 31, 280, 311, 653f, 964f
Badness: 0.025298
Semitert
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 9801/9800, 65625/65536
Mapping: [⟨2 6 4 6 1], ⟨0 -22 5 -3 46]]
- Mapping generators: ~99/70, ~256/245
Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193
Optimal ET sequence: 62e, 140, 202, 342
Badness: 0.025790
Quasiorwell
In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 = [22 -1 -10 1⟩. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31 & 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)1/8, giving just 7's, or 3841/38, giving pure fifths.
Adding 3025/3024 extends to the 11-limit and gives ⟨⟨ 38 -3 8 64 … ]] for the initial wedgie, and as expected, 270 remains an excellent tuning.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 29360128/29296875
Mapping: [⟨1 31 0 9], ⟨0 -38 3 -8]]
- Mapping generators: ~2, ~875/512
Wedgie: ⟨⟨ 38 -3 8 -93 -94 27 ]]
Optimal tuning (POTE): ~2 = 1\1, ~1024/875 = 271.107
Optimal ET sequence: 31, 177, 208, 239, 270, 571, 841, 1111
Badness: 0.035832
11-limit
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 5632/5625
Mapping: [⟨1 31 0 9 53], ⟨0 -38 3 -8 -64]]
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111
Optimal ET sequence: 31, 208, 239, 270
Badness: 0.017540
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095
Mapping: [⟨1 31 0 9 53 -59], ⟨0 -38 3 -8 -64 81]]
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107
Optimal ET sequence: 31, 239, 270, 571, 841, 1111
Badness: 0.017921
Decoid
Decoid tempers out 2401/2400 and 67108864/66976875, as well as the linus comma, [11 -10 -10 10⟩. Either 8/7 or 16/15 can be used as its generator. It may be described as the 130 & 270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the quintosec temperament.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 67108864/66976875
Mapping: [⟨10 0 47 36], ⟨0 2 -3 -1]]
- mapping generators: ~15/14, ~8192/4725
Wedgie: ⟨⟨ 20 -30 -10 -94 -72 61 ]]
- CTE: ~15/14 = 1\10, ~8192/4725 = 951.1086 (~16/15 = 111.1086, or ~225/224 = 8.8914)
- POTE: ~15/14 = 1\10, ~8192/4725 = 951.099 (~16/15 = 111.099, or ~225/224 = 8.901)
Optimal ET sequence: 10, …, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc
Badness: 0.033902
11-limit
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 5632/5625, 9801/9800
Mapping: [⟨10 0 47 36 98], ⟨0 2 -3 -1 -8]]
Optimal tunings:
- CTE: ~15/14 = 1\10, ~400/231 = 951.0943 (~16/15 = 111.0943, or ~225/224 = 8.9057)
- POTE: ~15/14 = 1\10, ~400/231 = 951.070 (~16/15 = 111.070, or ~225/224 = 8.930)
Optimal ET sequence: 10e, …, 130, 270, 670, 940, 1210, 2150c
Badness: 0.018735
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095
Mapping: [⟨10 0 47 36 98 37], ⟨0 2 -3 -1 -8 0]]
Optimal tunings:
- CTE: ~15/14 = 1\10, ~26/15 = 951.0943 (~16/15 = 111.0943, or ~196/195 = 8.9057)
- POTE: ~15/14 = 1\10, ~26/15 = 951.083 (~16/15 = 111.083, or ~196/195 = 8.917)
Optimal ET sequence: 10e, …, 130, 270, 940, 1210f, 1480cf
Badness: 0.013475
Neominor
The generator for neominor temperament is tridecimal minor third 13/11, also known as Neo-gothic minor third.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 177147/175616
Mapping: [⟨1 3 12 8], ⟨0 -6 -41 -22]]
- Mapping generators: ~2, ~189/160
Wedgie: ⟨⟨ 6 41 22 51 18 -64 ]]
Optimal tuning (POTE): ~2 = 1\1, ~189/160 = 283.280
Optimal ET sequence: 72, 161, 233, 305
Badness: 0.088221
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 35937/35840
Mapping: [⟨1 3 12 8 7], ⟨0 -6 -41 -22 -15]]
Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 283.276
Optimal ET sequence: 72, 161, 233, 305
Badness: 0.027959
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 243/242, 364/363, 441/440
Mapping: [⟨1 3 12 8 7 7], ⟨0 -6 -41 -22 -15 -14]]
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 283.294
Optimal ET sequence: 72, 161f, 233f
Badness: 0.026942
Emmthird
The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 14348907/14336000
Mapping: [⟨1 11 42 25], ⟨0 -14 -59 -33]]
- Mapping generators: ~2, ~2187/1372
Wedgie: ⟨⟨ 14 59 33 61 13 -8 9 ]]
Optimal tuning (POTE): ~2 = 1\1, ~2744/2187 = 392.988
Optimal ET sequence: 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d
Badness: 0.016736
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 1792000/1771561
Mapping: [⟨1 11 42 25 27], ⟨0 -14 -59 -33 -35]]
Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991
Optimal ET sequence: 58, 113, 171
Badness: 0.052358
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 364/363, 441/440, 2200/2197
Mapping: [⟨1 11 42 25 27 38], ⟨0 -14 -59 -33 -35 -51]]
Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989
Optimal ET sequence: 58, 113, 171
Badness: 0.026974
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197
Mapping: [⟨1 -3 -17 -8 -8 -13 9], ⟨0 14 59 33 35 51 -15]]
Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985
Optimal ET sequence: 58, 113, 171
Badness: 0.023205
Quinmite
The generator for quinmite is quasi-tempered minor third 25/21, flatter than 6/5 by the starling comma, 126/125. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth", given by Petr Pařízek in 2011[1][2].
Subgroup: 2.3.5.7
Comma list: 2401/2400, 1959552/1953125
Mapping: [⟨1 27 24 20], ⟨0 -34 -29 -23]]
- Mapping generators: ~2, ~42/25
Wedgie: ⟨⟨ 34 29 23 -33 -59 -28 ]]
Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 302.997
Optimal ET sequence: 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc
Badness: 0.037322
Unthirds
The generator for unthirds temperament is undecimal major third, 14/11.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 68359375/68024448
Mapping: [⟨1 29 33 25], ⟨0 -42 -47 -34]]
- Mapping generators: ~2, ~6125/3888
Wedgie: ⟨⟨ 42 47 34 -23 -64 -53 ]]
Optimal tuning (POTE): ~2 = 1\1, ~3969/3125 = 416.717
Optimal ET sequence: 72, 167, 239, 311, 694, 1005c
Badness: 0.075253
11-limit
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 4000/3993
Mapping: [⟨1 29 33 25 25], ⟨0 -42 -47 -34 -33]]
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718
Optimal ET sequence: 72, 167, 239, 311
Badness: 0.022926
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
Mapping: [⟨1 29 33 25 25 99], ⟨0 -42 -47 -34 -33 -146]]
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716
Optimal ET sequence: 72, 239f, 311, 694, 1005c
Badness: 0.020888
Newt
Newt has a generator of a neutral third (0.2 cents flat of 49/40) and tempers out the garischisma. It can be described as the 41 & 270 temperament, and extends naturally to the no-17 19-limit, a.k.a. neonewt. 270edo and 311edo are obvious tuning choices, but 581edo and especially 851edo work much better.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 33554432/33480783
Mapping: [⟨1 1 19 11], ⟨0 2 -57 -28]]
- mapping generators: ~2, ~49/40
Wedgie: ⟨⟨ 2 -57 -28 -95 -50 95 ]]
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.113
Optimal ET sequence: 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201
Badness: 0.041878
11-limit
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 19712/19683
Mapping: [⟨1 1 19 11 -10], ⟨0 2 -57 -28 46]]
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115
Optimal ET sequence: 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972
Badness: 0.019461
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095
Mapping: [⟨1 1 19 11 -10 -20], ⟨0 2 -57 -28 46 81]]
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
Optimal ET sequence: 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b
Badness: 0.013830
2.3.5.7.11.13.19 subgroup (neonewt)
Subgroup: 2.3.5.7.11.13.19
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400
Mapping: [⟨1 1 19 11 -10 -20 18], ⟨0 2 -57 -28 46 81 -47]]
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
Optimal ET sequence: 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb
Septidiasemi
Aside from 2401/2400, septidiasemi tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of 15/14). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 2152828125/2147483648
Mapping: [⟨1 25 -31 -8], ⟨0 -26 37 12]]
- Mapping generators: ~2, ~28/15
Wedgie: ⟨⟨ 26 -37 -12 -119 -92 76 ]]
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.297
Optimal ET sequence: 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd
Badness: 0.044115
Sedia
The sedia temperament (10&161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 939524096/935859375
Mapping: [⟨1 25 -31 -8 62], ⟨0 -26 37 12 -65]]
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279
Optimal ET sequence: 10, 151, 161, 171, 332
Badness: 0.090687
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 441/440, 2200/2197, 3584/3575
Mapping: [⟨1 25 -31 -8 62 1], ⟨0 -26 37 12 -65 3]]
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281
Optimal ET sequence: 10, 151, 161, 171, 332, 835eeff
Badness: 0.045773
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575
Mapping: [⟨1 25 -31 -8 62 1 23], ⟨0 -26 37 12 -65 3 -21]]
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281
Optimal ET sequence: l 10, 151, 161, 171, 332, 503ef, 835eeff
Badness: 0.027322
Maviloid
Subgroup: 2.3.5.7
Comma list: 2401/2400, 1224440064/1220703125
Mapping: [⟨1 31 34 26], ⟨0 -52 -56 -41]]
- Mapping generators: ~2, ~1296/875
Wedgie: ⟨⟨ 52 56 41 -32 -81 -62 ]]
Optimal tuning (POTE): ~2 = 1\1, ~1296/875 = 678.810
Optimal ET sequence: 76, 99, 274, 373, 472, 571, 1043, 1614
Badness: 0.057632
Subneutral
Subgroup: 2.3.5.7
Comma list: 2401/2400, 274877906944/274658203125
Mapping: [⟨1 19 0 6], ⟨0 -60 8 -11]]
- Mapping generators: ~2, ~57344/46875
Wedgie: ⟨⟨ 60 -8 11 -152 -151 48 ]]
Optimal tuning (POTE): ~2 = 1\1, ~57344/46875 = 348.301
Optimal ET sequence: 31, …, 348, 379, 410, 441, 1354, 1795, 2236
Badness: 0.045792
Osiris
Subgroup: 2.3.5.7
Comma list: 2401/2400, 31381059609/31360000000
Mapping: [⟨1 13 33 21], ⟨0 -32 -86 -51]]
- Mapping generators: ~2, ~2800/2187
Wedgie: ⟨⟨ 32 86 51 62 -9 -123 ]]
Optimal tuning (POTE): ~2 = 1\1, ~2800/2187 = 428.066
Optimal ET sequence: 157, 171, 1012, 1183, 1354, 1525, 1696
Badness: 0.028307
Gorgik
Subgroup: 2.3.5.7
Comma list: 2401/2400, 28672/28125
Mapping: [⟨1 5 1 3], ⟨0 -18 7 -1]]
- Mapping generators: ~2, ~8/7
Wedgie: ⟨⟨ 18 -7 1 -53 -49 22 ]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.512
Optimal ET sequence: 21, 37, 58, 153bc, 211bccd, 269bccd
Badness: 0.158384
11-limit
Subgroup: 2.3.5.7.11
Comma list: 176/175, 2401/2400, 2560/2541
Mapping: [⟨1 5 1 3 1], ⟨0 -18 7 -1 13]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500
Optimal ET sequence: 21, 37, 58, 153bce, 211bccdee, 269bccdee
Badness: 0.059260
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 196/195, 364/363, 512/507
Mapping: [⟨1 5 1 3 1 2], ⟨0 -18 7 -1 13 9]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493
Optimal ET sequence: 21, 37, 58, 153bcef, 211bccdeeff
Badness: 0.032205
Fibo
Subgroup: 2.3.5.7
Comma list: 2401/2400, 341796875/339738624
Mapping: [⟨1 19 8 10], ⟨0 -46 -15 -19]]
- Mapping generators: ~2, ~125/96
Wedgie: ⟨⟨ 46 15 19 -83 -99 2 ]]
Optimal tuning (POTE): ~2 = 1\1, ~125/96 = 454.310
Optimal ET sequence: 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd
Badness: 0.100511
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 43923/43750
Mapping: [⟨1 19 8 10 8], ⟨0 -46 -15 -19 -12]]
Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318
Optimal ET sequence: 37, 66b, 103, 140, 243e
Badness: 0.056514
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 385/384, 625/624, 847/845, 1375/1372
Mapping: [⟨1 19 8 10 8 9], ⟨0 -46 -15 -19 -12 -14]]
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316
Optimal ET sequence: 37, 66b, 103, 140, 243e
Badness: 0.027429
Mintone
In addition to 2401/2400, mintone tempers out 177147/175000 = [-3 11 -5 -1⟩ in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58 & 103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 177147/175000
Mapping: [⟨1 5 9 7], ⟨0 -22 -43 -27]]
- Mapping generators: ~2, ~10/9
Wedgie: ⟨⟨ 22 43 27 17 -19 -58 ]]
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.343
Optimal ET sequence: 45, 58, 103, 161, 586b, 747bc, 908bbc
Badness: 0.125672
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 43923/43750
Mapping: [⟨1 5 9 7 12], ⟨0 -22 -43 -27 -55]]
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345
Optimal ET sequence: 58, 103, 161, 425b, 586b, 747bc
Badness: 0.039962
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 351/350, 441/440, 847/845
Mapping: [⟨1 5 9 7 12 11], ⟨0 -22 -43 -27 -55 -47]]
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347
Optimal ET sequence: 58, 103, 161, 425b, 586bf
Badness: 0.021849
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
Mapping: [⟨1 5 9 7 12 11 3], ⟨0 -22 -43 -27 -55 -47 7]]
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348
Optimal ET sequence: 58, 103, 161, 425b, 586bf
Badness: 0.020295
Catafourth
Subgroup: 2.3.5.7
Comma list: 2401/2400, 78732/78125
Mapping: [⟨1 13 17 13], ⟨0 -28 -36 -25]]
- Mapping generators: ~2, ~250/189
Wedgie: ⟨⟨ 28 36 25 -8 -39 -43 ]]
Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.235
Optimal ET sequence: 27, 76, 103, 130
Badness: 0.079579
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 78408/78125
Mapping: [⟨1 13 17 13 32], ⟨0 -28 -36 -25 -70]]
Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252
Optimal ET sequence: 103, 130, 233, 363, 493e, 856be
Badness: 0.036785
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 351/350, 441/440, 10985/10976
Mapping: [⟨1 13 17 13 32 9], ⟨0 -28 -36 -25 -70 -13]]
Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256
Optimal ET sequence: 103, 130, 233, 363
Badness: 0.021694
Cotritone
Subgroup: 2.3.5.7
Comma list: 2401/2400, 390625/387072
Mapping: [⟨1 17 9 10], ⟨0 -30 -13 -14]]
- Mappping generators: ~2, ~10/7
Wedgie: ⟨⟨ 30 13 14 -49 -62 -4 ]]
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.385
Optimal ET sequence: 35, 37, 72, 109, 181, 253
Badness: 0.098322
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 4000/3993
Mapping: [⟨1 17 9 10 5], ⟨0 -30 -13 -14 -3]]
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
Optimal ET sequence: 35, 37, 72, 109, 181, 253
Badness: 0.032225
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 364/363, 385/384, 625/624
Mapping: [⟨1 17 9 10 5 15], ⟨0 -30 -13 -14 -3 -22]]
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
Optimal ET sequence: 37, 72, 109, 181f
Badness: 0.028683
Quasimoha
- For the 5-limit version of this temperament, see High badness temperaments #Quasimoha.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 3645/3584
Mapping: [⟨1 1 9 6], ⟨0 2 -23 -11]]
- Mapping generators: ~2, ~49/40
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 348.603
Optimal ET sequence: 31, 117c, 148bc, 179bc
Badness: 0.110820
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 1815/1792
Mapping: [⟨1 1 9 6 2], ⟨0 2 -23 -11 5]]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639
Optimal ET sequence: 31, 86ce, 117ce, 148bce
Badness: 0.046181
Surmarvelpyth
Surmarvelpyth is named for the generator fifth, 675/448 being 225/224 (marvel comma) sharp of 3/2. It can be described as the 311 & 431 temperament, starting with the 7-limit to the 19-limit.
Subgroup: 2.3.5.7
Comma list: 2401/2400, [93 -32 -17 -1⟩
Mapping: [⟨1 43 -74 -25], ⟨0 -70 129 47]]
- Mapping generators: ~2, ~675/448
Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9719
Optimal ET sequence: 120, 191, 311, 742, 1053, 2848, 3901
Badness: 0.202249
11-limit
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 820125/819896, 2097152/2096325
Mapping: [⟨1 43 -74 -25 36], ⟨0 -70 129 47 -55]]
Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9720
Optimal ET sequence: 120, 191, 311, 742, 1053, 1795
Badness: 0.052308
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167
Mapping: [⟨1 43 -74 -25 36 25], ⟨0 -70 129 47 -55 -36]]
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9723
Optimal ET sequence: 120, 191, 311, 742, 1053, 1795f
Badness: 0.032503
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619
Mapping: [⟨1 43 -74 -25 36 25 -103], ⟨0 -70 129 47 -55 -36 181]]
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722
Optimal ET sequence: 120g, 191g, 311, 431, 742, 1795f
Badness: 0.020995
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 2401/2400, 2601/2600, 2926/2925, 3136/3135, 3213/3211, 5985/5984
Mapping: [⟨1 43 -74 -25 36 25 -103 -49], ⟨0 -70 129 47 -55 -36 181 90]]
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722
Optimal ET sequence: 120g, 191g, 311, 431, 742, 1795f
Badness: 0.013771