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} → Dicot family #Decimal
- Beatles, {64/63, 686/675} → Archytas clan #Beatles
- Squares, {81/80, 2401/2400} → Meantone family #Squares
- Myna, {126/125, 1728/1715} → Starling temperaments #Myna
- Miracle, {225/224, 1029/1024} → Gamelismic clan #Miracle
- Octacot, {245/243, 2401/2400} → Tetracot family #Octacot
- Greenwood, {405/392, 1323/1280} → Greenwoodmic temperaments #Greenwood
- Quasitemp, {875/864, 2401/2400} → Keemic temperaments #Quasitemp
- Quadrasruta, {2048/2025, 2401/2400} → Diaschismic family #Quadrasruta
- Quadrimage, {2401/2400, 3125/3072} → Magic family #Quadrimage
- Hemiwürschmidt, {2401/2400, 3136/3125} → Würschmidt family #Hemiwürschmidt
- Ennealimmal, {2401/2400, 4375/4374} → Ragismic microtemperaments #Ennealimmal
- Quadritikleismic, {2401/2400, 15625/15552} → Kleismic family #Quadritikleismic
- Sesquiquartififths, {2401/2400, 32805/32768} → Schismatic family #Sesquiquartififths
- Neptune, {2401/2400, 48828125/48771072} → Gammic family #Neptune
- Eagle, {2401/2400, 10485760000/10460353203} → Vulture family #Eagle
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-odd-limit minimax tuning, or 14(1/13), giving just 7s. 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]]
Wedgie: ⟨⟨ 2 25 13 35 15 -40 ]]
POTE generator: ~49/40 = 351.477
- 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⟩]
- Eigenmonzos: 2, 5
Algebraic generator: (2 + sqrt(2))/2
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]]
POTE generator: ~11/9 = 351.521
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~11/9 = 351.573
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~49/40 = 351.505
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~49/40 = 351.502
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~243/220 = 175.7378
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~72/65 = 175.7470
Optimal GPV sequence: Template:Val list
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. 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]]
Wedgie: ⟨⟨ 22 -5 3 -59 -57 21 ]]
POTE generator: ~256/245 = 77.191
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]]
POTE generator: ~256/245 = 77.227
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~117/112 = 77.203
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~68/65 = 77.201
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~22/21 = 77.173
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~22/21 = 77.158
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~22/21 = 77.162
Optimal GPV sequence: Template:Val list
Badness: 0.022416
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]]
POTE generator: ~45/44 = 38.596
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~45/44 = 38.588
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~45/44 = 38.589
Optimal GPV sequence: Template:Val list
Badness: 0.025298
Harry
Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&72 temperament. 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.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 19683/19600
Mapping: [⟨2 4 7 7], ⟨0 -6 -17 -10]]
Wedgie: ⟨⟨ 12 34 20 26 -2 -49 ]]
POTE generator: ~21/20 = 83.156
Badness: 0.034077
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 4000/3993
Mapping: [⟨2 4 7 7 9], ⟨0 -6 -17 -10 -15]]
POTE generator: ~21/20 = 83.167
Optimal GPV sequence: Template:Val list
Badness: 0.015867
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 351/350, 364/363, 441/440
Mapping: [⟨2 4 7 7 9 11], ⟨0 -6 -17 -10 -15 -26]]
POTE generator: ~21/20 = 83.116
Optimal GPV sequence: Template:Val list
Badness: 0.013046
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 221/220, 243/242, 289/288, 351/350, 441/440
Mapping: [⟨2 4 7 7 9 11 9], ⟨0 -6 -17 -10 -15 -26 -6]]
POTE generator: ~21/20 = 83.168
Optimal GPV sequence: Template:Val list
Badness: 0.012657
Quasiorwell
In addition to 2401/2400, quasiorwell tempers out 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 7s, 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]]
Wedgie: ⟨⟨ 38 -3 8 -93 -94 27 ]]
POTE generator: ~1024/875 = 271.107
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]]
POTE generator: ~90/77 = 271.111
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~90/77 = 271.107
Optimal GPV sequence: Template:Val list
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 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 qintosec 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 ]]
POTE generator: ~16/15 = 111.099
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]]
POTE generator: ~16/15 = 111.070
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~16/15 = 111.083
Optimal GPV sequence: Template:Val list
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]]
Wedgie: ⟨⟨ 6 41 22 51 18 -64 ]]
POTE generator: ~189/160 = 283.280
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]]
POTE generator: ~33/28 = 283.276
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~13/11 = 283.294
Optimal GPV sequence: Template:Val list
Badness: 0.026942
Emmthird
The generator for emmthird temperament 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 -3 -17 -8], ⟨0 14 59 33]]
Wedgie: ⟨⟨ 14 59 33 61 13 -89 ]]
POTE generator: ~2744/2187 = 392.988
Badness: 0.016736
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 1792000/1771561
Mapping: [⟨1 -3 -17 -8 -8], ⟨0 14 59 33 35]]
POTE generator: ~1372/1089 = 392.991
Optimal GPV sequence: Template:Val list
Badness: 0.052358
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 364/363, 441/440, 2200/2197
Mapping: [⟨1 -3 -17 -8 -8 -13], ⟨0 14 59 33 35 51]]
POTE generator: ~180/143 = 392.989
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~64/51 = 392.985
Optimal GPV sequence: Template:Val list
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".
Subgroup: 2.3.5.7
Comma list: 2401/2400, 1959552/1953125
Mapping: [⟨1 -7 -5 -3], ⟨0 34 29 23]]
Wedgie: ⟨⟨ 34 29 23 -33 -59 -28 ]]
POTE generator: ~25/21 = 302.997
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 -13 -14 -9], ⟨0 42 47 34]]
Wedgie: ⟨⟨ 42 47 34 -23 -64 -53 ]]
POTE generator: ~3969/3125 = 416.717
Badness: 0.075253
11-limit
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 4000/3993
Mapping: [⟨1 -13 -14 -9 -8], ⟨0 42 47 34 33]]
POTE generator: ~14/11 = 416.718
Optimal GPV sequence: Template:Val list
Badness: 0.022926
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
Mapping: [⟨1 -13 -14 -9 -9 -47], ⟨0 42 47 34 33 146]]
POTE generator: ~14/11 = 416.716
Optimal GPV sequence: Template:Val list
Badness: 0.020888
Newt
This temperament has a generator of neutral third (0.2 cents flat of 49/40) and tempers out the garischisma.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 33554432/33480783
Mapping: [⟨1 1 19 11], ⟨0 2 -57 -28]]
Wedgie: ⟨⟨ 2 -57 -28 -95 -50 95 ]]
POTE generator: ~49/40 = 351.113
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]]
POTE generator: ~49/40 = 351.115
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~49/40 = 351.117
Optimal GPV sequence: Template:Val list
Badness: 0.013830
Amicable
The amicable temperament tempers out the amity comma and the canousma in addition to the breedsma, and is closely associated with the canou temperament.
While it extends well into 2.3.5.7.13/11, there are multiple reasonable places for the prime 11 and 13 in the interval chain. Amical (311 & 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 & 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 & 113) and pseudoamorous (14cf & 99ef) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semioctave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 1600000/1594323
Mapping: [⟨1 3 6 5], ⟨0 -20 -52 -31]]
Wedgie: ⟨⟨ 20 52 31 36 -7 -74 ]]
POTE generator: ~21/20 = 84.880
Badness: 0.045473
Amical
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 131072/130977, 1600000/1594323
Mapping: [⟨1 3 6 5 -8], ⟨0 -20 -52 -31 162]]
POTE generator: ~21/20 = 84.8843
Optimal GPV sequence: Template:Val list
Badness: 0.100668
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206
Mapping: [⟨1 3 6 5 -8 -5], ⟨0 -20 -52 -31 162 123]]
POTE generator: ~21/20 = 84.8838
Optimal GPV sequence: Template:Val list
Badness: 0.049893
Amorous
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 6250/6237, 19712/19683
Mapping: [⟨1 3 6 5 14], ⟨0 -20 -52 -31 -149]]
POTE generator: ~21/20 = 84.8896
Optimal GPV sequence: Template:Val list
Badness: 0.048924
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647
Mapping: [⟨1 3 6 5 14 17], ⟨0 -20 -52 -31 -149 -188]]
POTE generator: ~21/20 = 84.8910
Optimal GPV sequence: Template:Val list
Badness: 0.034681
Pseudoamical
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 1600000/1594323
Mapping: [⟨1 3 6 5 -1], ⟨0 -20 -52 -31 63]]
POTE generator: ~21/20 = 84.9091
Optimal GPV sequence: Template:Val list
Badness: 0.085837
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 1375/1372, 19773/19712
Mapping: [⟨1 3 6 5 -1 2], ⟨0 -20 -52 -31 63 24]]
POTE generator: ~21/20 = 84.9127
Optimal GPV sequence: Template:Val list
Badness: 0.047025
Pseudoamorous
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 980000/970299
Mapping: [⟨1 3 6 5 7], ⟨0 -20 -52 -31 -50]]
POTE generator: ~21/20 = 84.8917
Optimal GPV sequence: Template:Val list
Badness: 0.056583
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 364/363, 441/440, 1875/1859
Mapping: [⟨1 3 6 5 7 10], ⟨0 -20 -52 -31 -50 -89]]
POTE generator: ~21/20 = 84.9164
Optimal GPV sequence: Template:Val list
Badness: 0.042826
Floral
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 9801/9800, 14641/14580
Mapping: [⟨2 6 12 10 13], ⟨0 -20 -52 -31 -43]]
POTE generator: ~21/20 = 84.8788
Optimal GPV sequence: Template:Val list
Badness: 0.065110
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580
Mapping: [⟨2 6 12 10 13 19], ⟨0 -20 -52 -31 -43 -82]]
POTE generator: ~21/20 = 84.8750
Optimal GPV sequence: Template:Val list
Badness: 0.037013
Humorous
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 1600000/1594323
Mapping: [⟨1 3 6 5 3], ⟨0 -40 -104 -62 13]]
POTE generator: ~4096/3993 = 42.4391
Optimal GPV sequence: Template:Val list
Badness: 0.058249
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024
Mapping: [⟨1 3 6 5 3 6], ⟨0 -40 -104 -62 13 -65]]
POTE generator: ~40/39 = 42.4391
Optimal GPV sequence: Template:Val list
Badness: 0.028267
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 -1 6 4], ⟨0 26 -37 -12]]
Wedgie: ⟨⟨ 26 -37 -12 -119 -92 76 ]]
POTE generator: ~15/14 = 119.297
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 -1 6 4 -3], ⟨0 26 -37 -12 65]]
POTE generator: ~15/14 = 119.279
Optimal GPV sequence: Template:Val list
Badness: 0.090687
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 441/440, 2200/2197, 3584/3575
Mapping: [⟨1 -1 6 4 -3 4], ⟨0 26 -37 -12 65 -3]]
POTE generator: ~15/14 = 119.281
Optimal GPV sequence: Template:Val list
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 -1 6 4 -3 4 2], ⟨0 26 -37 -12 65 -3 21]]
POTE generator: ~15/14 = 119.281
Optimal GPV sequence: Template:Val list
Badness: 0.027322
Maviloid
Subgroup: 2.3.5.7
Comma list: 2401/2400, 1224440064/1220703125
Mapping: [⟨1 31 34 26], ⟨0 -52 -56 -41]]
Wedgie: ⟨⟨ 52 56 41 -32 -81 -62 ]]
POTE generator: ~1296/875 = 678.810
Badness: 0.057632
Subneutral
Subgroup: 2.3.5.7
Comma list: 2401/2400, 274877906944/274658203125
Mapping: [⟨1 19 0 6], ⟨0 -60 8 -11]]
Wedgie: ⟨⟨ 60 -8 11 -152 -151 48 ]]
POTE generator: ~57344/46875 = 348.301
Badness: 0.045792
Osiris
Subgroup: 2.3.5.7
Comma list: 2401/2400, 31381059609/31360000000
Mapping: [⟨1 13 33 21], ⟨0 -32 -86 -51]]
Wedgie: ⟨⟨ 32 86 51 62 -9 -123 ]]
POTE generator: ~2800/2187 = 428.066
Badness: 0.028307
Gorgik
Subgroup: 2.3.5.7
Comma list: 2401/2400, 28672/28125
Mapping: [⟨1 5 1 3], ⟨0 -18 7 -1]]
Wedgie: ⟨⟨ 18 -7 1 -53 -49 22 ]]
POTE generator: ~8/7 = 227.512
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]]
POTE generator: ~8/7 = 227.500
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~8/7 = 227.493
Optimal GPV sequence: Template:Val list
Badness: 0.032205
Fibo
Subgroup: 2.3.5.7
Comma list: 2401/2400, 341796875/339738624
Mapping: [⟨1 19 8 10], ⟨0 -46 -15 -19]]
Wedgie: ⟨⟨ 46 15 19 -83 -99 2 ]]
POTE generator: ~125/96 = 454.310
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]]
POTE generator: ~100/77 = 454.318
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~13/10 = 454.316
Optimal GPV sequence: Template:Val list
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]]
Wedgie: ⟨⟨ 22 43 27 17 -19 -58 ]]
POTE generator: ~10/9 = 186.343
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]]
POTE generator: ~10/9 = 186.345
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~10/9 = 186.347
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~10/9 = 186.348
Optimal GPV sequence: Template:Val list
Badness: 0.020295
Catafourth
Subgroup: 2.3.5.7
Comma list: 2401/2400, 78732/78125
Mapping: [⟨1 13 17 13], ⟨0 -28 -36 -25]]
Wedgie: ⟨⟨ 28 36 25 -8 -39 -43 ]]
POTE generator: ~250/189 = 489.235
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]]
POTE generator: ~250/189 = 489.252
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~65/49 = 489.256
Optimal GPV sequence: Template:Val list
Badness: 0.021694
Cotritone
Subgroup: 2.3.5.7
Comma list: 2401/2400, 390625/387072
Mapping: [⟨1 -13 -4 -4], ⟨0 30 13 14]]
Wedgie: ⟨⟨ 30 13 14 -49 -62 -4 ]]
POTE generator: ~7/5 = 583.385
Badness: 0.098322
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 4000/3993
Mapping: [⟨1 -13 -4 -4 2], ⟨0 30 13 14 3]]
POTE generator: ~7/5 = 583.387
Optimal GPV sequence: Template:Val list
Badness: 0.032225
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 364/363, 385/384, 625/624
Mapping: [⟨1 -13 -4 -4 2 -7], ⟨0 30 13 14 3 22]]
POTE generator: ~7/5 = 583.387
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~49/40 = 348.603
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]]
POTE generator: ~11/9 = 348.639
Optimal GPV sequence: Template:Val list
Badness: 0.046181