Breedsmic temperaments
Breedsmic temperaments are rank two 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.
It 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 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.
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 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&58 temperament and has wedgie <<2 25 13 35 15 -40||, which tells us that it requires 25 generator steps to get to the class for major thirds, 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.
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.
5-limit
Comma: 858993459200/847288609443
POTE generator: ~655360/531441 = 351.476
Map: [<1 1 -5|, <0 2 25|]
EDOs: 41, 58, 99, 239, 338, 915b, 1253bc
Badness: 0.3728
7-limit
Commas: 2401/2400, 5120/5103
7 and 9-limit minimax
[|1 0 0 0>, |7/5, 0, 2/25, 0>, |0 0 1 0>, |8/5 0 13/25 0>]
Eigenvalues: 2, 5
Algebraic generator: (2 + sqrt(2))/2
Map: [<1 1 -5 -1|, <0 2 25 13|]
Badness: 0.0222
11-limit
Commas: 243/242, 441/440, 896/891
POTE generator: ~11/9 = 351.521
Map: [<1 1 -5 -1 2|, <0 2 25 13 5|]
EDOs: 7, 17, 41, 58, 99
Badness: 0.0235
13-limit
Commas: 144/143, 196/195, 243/242, 364/363
POTE generator: ~11/9 = 351.573
Map: [<1 1 -5 -1 2 4|, <0 2 25 13 5 -1|]
EDOs: 7, 17, 41, 58, 99
Badness: 0.0191
Semihemi
Commas: 2401/2400, 3388/3375, 9801/9800
POTE generator: ~49/40 = 351.505
Map: [<2 0 -35 -15 -47|, <0 2 25 13 34|]
EDOs: 58, 140, 198, 734bc, 932bcd, 1130bcd
Badness: 0.042487
13-limit
Commas: 352/351, 676/675, 847/845, 1716/1715
POTE generator: ~49/40 = 351.502
Map: [<2 0 -35 -15 -47 -37|, <0 2 25 13 34 28|]
EDOs: 58, 140, 198, 536f, 734bcf, 932bcdf
Badness: 0.0212
Tertiaseptal
Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. 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. 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.
Commas: 2401/2400, 65625/65536
POTE generator: ~256/245 = 77.191
Map: [<1 3 2 3|, <0 -22 5 -3|]
EDOs: 15, 16, 31, 109, 140, 171
Badness: 0.0130
11-limit
Commas: 243/242, 441/440, 65625/65536
POTE generator: ~256/245 = 77.227
Map: [<1 3 2 3 7|, <0 -22 5 -3 -55|]
EDOs: 15, 16, 31, 171, 202
Badness: 0.0356
13-limit
Commas: 243/242, 441/440, 625/624, 3584/3575
POTE generator: ~117/112 = 77.203
Map: [<1 3 2 3 7 1|, <0 -22 5 -3 -55 42|]
EDOs: 31, 140e, 171
Badness: 0.0369
17-limit
Commas: 243/242, 375/374, 441/440, 625/624, 3584/3575
POTE generator: ~68/65 = 77.201
Map: [<1 3 2 3 7 1 1|, <0 -22 5 -3 -55 42 48|]
EDOs: 31, 140e, 171
Badness: 0.0274
Tertia
Commas: 385/384, 1331/1323, 1375/1372
POTE generator: ~22/21 = 77.173
Map: [<1 3 2 3 5|, <0 -22 5 -3 -24|]
EDOs: 31, 109, 140, 171e, 311e
Badness: 0.0302
Hemitert
Commas: 2401/2400 3025/3024 65625/65536
POTE generator: ~45/44 = 38.596
Map: [<1 3 2 3 6|, <0 -44 10 -6 -79|]
EDOs: 31, 280, 311, 342, 2021cde, 3731cde
Badness: 0.0156
Harry
Commas: 2401/2400, 19683/19600
Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&72 temperament, with wedgie <<12 34 20 26 -2 -49||. The period is half an octave, and the generator 21/20, with generator tunings of 9\130 or 14\202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices.
Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is <<12 34 20 30 ...||.
Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with <<12 34 20 30 52 ...|| as the octave wedgie. 130edo is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.
POTE generator: ~21/20 = 83.156
Map: [<2 4 7 7|, <0 -6 -17 -10|]
Wedgie: <<12 34 20 26 -2 -49||
EDOs: 14, 58, 72, 130, 202, 534, 938
Badness: 0.0341
11-limit
Commas: 243/242, 441/440, 4000/3993
POTE generator: ~21/20 = 83.167
Map: [<2 4 7 7 9|, <0 -6 -17 -10 -15|]
EDOs: 14, 58, 72, 130, 202
Badness: 0.0159
13-limit
Commas: 243/242, 351/350, 441/440, 676/675
POTE generator: ~21/20 = 83.116
Map: [<2 4 7 7 9 11|, <0 -6 -17 -10 -15 -26|]
EDOs: 14, 58, 72, 130, 462
Badness: 0.0130
Quasiorwell
In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1>. It has a wedgie <<38 -3 8 -93 -94 27||. 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 7s, or 384^(1/38), 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.
Commas: 2401/2400, 29360128/29296875
POTE generator: ~1024/875 = 271.107
Map: [<1 31 0 9|, <0 -38 3 -8|]
EDOs: 31, 177, 208, 239, 270, 571, 841, 1111
Badness: 0.0358
11-limit
Commas: 2401/2400, 3025/3024, 5632/5625
POTE generator: ~90/77 = 271.111
Map: [<1 31 0 9 53|, <0 -38 3 -8 -64|]
Badness: 0.0175
13-limit
Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095
POTE generator: ~90/77 = 271.107
Map: [<1 31 0 9 53 -59|, <0 -38 3 -8 -64 81|]
EDOs: 31, 239, 270, 571, 841, 1111
Badness: 0.0179
Decoid
In addition to 2401/2400, decoid tempers out 67108864/66976875 to temper 15/14 into one degree of 10edo. Either 8/7 or 16/15 can be used its generator. It may be described as the 10&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 qintosec temperament.
Commas: 2401/2400, 67108864/66976875
POTE generator: ~8/7 = 231.099
Map: [<10 0 47 36|, <0 2 -3 -1|]
Wedgie: <<20 -30 -10 -94 -72 61||
EDOs: 10, 120, 130, 270
Badness: 0.0339
11-limit
Commas: 2401/2400, 5832/5825, 9801/9800
POTE generator: ~8/7 = 231.070
Map: [<10 0 47 36 98|, <0 2 -3 -1 -8|]
EDOs: 130, 270, 670, 940, 1210
Badness: 0.0187
13-limit
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224
POTE generator: ~8/7 = 231.083
Map: [<10 0 47 36 98 37|, <0 2 -3 -1 -8 0|]
EDOs: 130, 270, 940, 1480
Badness: 0.0135
Neominor
Commas: 2401/2400, 177147/175616
POTE generator: ~189/160 = 283.280
Map: [<1 3 12 8|, <0 -6 -41 -22|]
Weggie: <<6 41 22 51 18 -64||
EDOs: 72, 161, 233, 305
Badness: 0.0882
11-limit
Commas: 243/242, 441/440, 35937/35840
POTE: ~33/28 = 283.276
Map: [<1 3 12 8 7|, <0 -6 -41 -22 -15|]
EDOs: 72, 161, 233, 305
Badness: 0.0280
13-limit
Commas: 169/168, 243/242, 364/363, 441/440
POTE generator: ~13/11 = 283.294
Map: [<1 3 12 8 7 7|, <0 -6 -41 -22 -15 -14|]
EDOs: 72, 161f, 233f
Badness: 0.0269
Emmthird
The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.
Commas: 2401/2400, 14348907/14336000
POTE generator: ~2744/2187 = 392.988
Map: [<1 11 42 25|, <0 -14 -59 -33|]
Wedgie: <<14 59 33 61 13 -89||
EDOs: 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d
Badness: 0.0167
Quinmite
Commas: 2401/2400, 1959552/1953125
POTE generator: ~25/21 = 302.997
Map: [<1 27 24 20|, <0 -34 -29 -23|]
Wedgie: <<34 29 23 -33 -59 -28||
EDOs: 95, 99, 202, 301, 400, 701, 1001c, 1802c, 2903c
Badness: 0.0373
Unthirds
Commas: 2401/2400, 68359375/68024448
POTE generator: ~3969/3125 = 416.717
Map: [<1 29 33 25|, <0 -42 -47 -34|]
Wedgie: <<42 47 34 -23 -64 -53||
EDOs: 72, 167, 239, 311, 694, 1005c
Badness: 0.0753
11-limit
Commas: 2401/2400, 3025/3024, 4000/3993
POTE generator: ~14/11 = 416.718
Map: [<1 29 33 25 25|, <0 -42 -47 -34 -33|]
EDOs: 72, 167, 239, 311, 1316c
Badness: 0.0229
13-limit
Commas: 625/624, 1575/1573, 2080/2079, 2401/2400
POTE generator: ~14/11 = 416.716
Map: [<1 29 33 25 25 99|, <0 -42 -47 -34 -33 -146|]
EDOs: 72, 311, 694, 1005c, 1699cd
Badness: 0.0209
Newt
Commas: 2401/2400, 33554432/33480783
POTE generator: ~49/40 = 351.113
Map: [<1 1 19 11|, <0 2 -57 -28|]
Wedgie: <<2 -57 -28 -95 -50 95||
EDOs: 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bc
Badness: 0.0419
11-limit
Commas: 2401/2400, 3025/3024, 19712/19683
POTE generator: ~49/40 = 351.115
Map: [<1 1 19 11 -10|, <0 2 -57 -28 46|]
EDOs: 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b
Badness: 0.0195
13-limit
Commas: 2080/2079, 2401/2400, 3025/3024, 4096/4095
POTE genertaor: ~49/40 = 351.117
Map: [<1 1 19 11 -10 -20|, <0 2 -57 -28 46 81|]
EDOs: 41, 229, 270, 581, 851, 2283b, 3134b
Badness: 0.0138
Amicable
Commas: 2401/2400, 1600000/1594323
POTE generator: ~21/20 = 84.880
Map: [<1 3 6 5|, <0 -20 -52 -31|]
Wedgie: <<20 52 31 36 -7 -74||
EDOs: 99, 212, 311, 410, 1131, 1541b
Badness: 0.0455
Septidiasemi
Commas: 2401/2400, 2152828125/2147483648
POTE generator: ~15/14 = 119.297
Map: [<1 25 -31 -8|, <0 -26 37 12|]
Wedgie: <<26 -37 -12 -119 -92 76||
EDOs: 10, 151, 161, 171, 3581bcd, 3752bcd, 3923bcd, 4094bcd, 4265bcd, 4436bcd, 4607bcd
Badness: 0.0441
Maviloid
Commas: 2401/2400, 1224440064/1220703125
POTE generator: ~1296/875 = 678.810
Map: [<1 31 34 26|, <0 -52 -56 -41|]
Wedgie: <<52 56 41 -32 -81 -62||
EDOs: 76, 99, 274, 373, 472, 571, 1043, 1614
Badness: 0.0576
Subneutral
Commas: 2401/2400, 274877906944/274658203125
POTE generator: ~57344/46875 = 348.301
Map: [<1 19 0 6}, <0 -60 8 -11|]
Wedgie: <<60 -8 11 -152 -151 48||
EDOs: 31, 348, 379, 410, 441, 1354, 1795, 2236
Badness: 0.0458
Osiris
Commas: 2401/2400, 31381059609/31360000000
POTE generator: ~2800/2187 = 428.066
Map: [<1 13 33 21|, <0 -32 -86 -51|]
Wedgie: <<32 86 51 62 -9 -123||
EDOs: 157, 171, 1012, 1183, 1354, 1525, 1696, 6955d
Badness: 0.0283
Gorgik
Commas: 2401/2400, 28672/28125
POTE generator: ~8/7 = 227.512
Map: [<1 5 1 3|, <0 -18 7 -1|]
Wedgie: <<18 -7 1 -53 -49 22||
EDOs: 21, 37, 58, 153bc, 211bcd, 269bcd
Badness: 0.1584
11-limit
Commas: 176/175, 2401/2400, 2560/2541
POTE generator: ~8/7 = 227.500
Map: [<1 5 1 3 1|, <0 -18 7 -1 13|]
EDOs: 21, 37, 58, 153bce, 211bcde, 269bcde
Badness: 0.059
13-limit
Commas: 176/175, 196/195, 364/363, 512/507
POTE generator: ~8/7 = 227.493
Map: [<1 5 1 3 1 2|, <0 -18 7 -1 13 9|]
EDOs: 21, 37, 58, 153bcef, 211bcdef
Badness: 0.0322
Fibo
Commas: 2401/2400, 341796875/339738624
POTE generator: ~125/96 = 454.310
Map: [<1 19 8 10|, <0 -46 -15 -19|]
Wedgie: <<46 15 19 -83 -99 2||
EDOs: 37, 103, 140, 243, 383, 1009cd, 1392cd
Badness: 0.1005
11-limit
Commas: 385/384, 1375/1372, 43923/43750
POTE generator: ~100/77 = 454.318
Map: [<1 19 8 10 8|, <0 -46 -15 -19 -12|]
EDOs: 37, 103, 140, 243e
Badness: 0.0565
13-limit
Commas: 385/384, 625/624, 847/845, 1375/1372
POTE generator: ~13/10 = 454.316
Map: [<1 19 8 10 8 9|, <0 -46 -15 -19 -12 -14|]
EDOs: 37, 103, 140, 243e
Badness: 0.0274
Mintone
Commas: 2401/2400, 177147/175000
POTE generator: ~10/9 = 186.343
Map: [<1 5 9 7|, <0 -22 -43 -27|]
EDOs: 45, 58, 103, 161, 586b, 747bc, 908bc
Badness: 0.12567
11-limit
Commas: 243/242, 441/440, 43923/43750
POTE generator: ~10/9 = 186.345
Map: [<1 5 9 7 12|, <0 -22 -43 -27 -55|]
EDOs: 58, 103, 161, 425b, 586b, 747bc
Badness: 0.0400
13-limit
Commas: 243/242, 351/350, 441/440, 847/845
POTE generator: ~10/9 = 186.347
Map: [<1 5 9 7 12 11|, <0 -22 -43 -27 -55 -47|]
EDOs: 58, 103, 161
Badness: 0.0218
Catafourth
Commas: 2400/2401, 78732/78125
POTE generator: ~250/189 = 489.235
Map: [<1 13 17 13|, <0 -28 -36 -25|]
Wedgie: <<[28 36 25 -8 -39 -43||
EDOs: 27, 76, 103, 130
Badness: 0.0796
11-limit
Commas: 243/242, 441/440, 78408/78125
POTE generator: ~250/189 = 489.252
Map: [<1 13 17 13 32|, <0 -28 -36 -25 -70|]
EDOs: 103, 130, 233, 363, 493e, 856be
Badness: 0.0368
13-limit
Commas: 243/242, 351/350, 441/440, 10985/10976
POTE generator: ~65/49 = 489.256
Map: [<1 13 17 13 32 9|, <0 -28 -36 -25 -70 -13|]
EDOs: 103, 130, 233, 363
Badness: 0.0217