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Hemififths | Semihemi | Tertiaseptal | Hemitert | Harry | Quasiorwell | Decoid | Neominor | Emmthird | Quinmite | Unthirds | Newt | Amicable | Septidiasemi | Maviloid | Subneutral | Osiris | Gorgik | Fibo | Mintone | Catafourth

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|]
EDOs: 41, 58, 99, 239, 338
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: 42.487

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 140&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, 373ef, 544ef
Badness: 0.0369

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|]
EDOs: 31, 208, 239, 270
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

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