Breedsmic temperaments: Difference between revisions

'''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.

Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with [[99edo|99EDO]] and [[140edo|140EDO]] providing good tunings, and [[239edo|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.

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.

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: [[31edo|31]], [[177edo|177]], [[208edo|208]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]

==11-limit==

Commas: 2401/2400, 3025/3024, 5632/5625

Map: [<1 31 0 9 53|, <0 -38 3 -8 -64|]

EDOs: [[31edo|31]], [[208edo|208]], [[239edo|239]], [[270edo|270]]

Badness: 0.0175

Map: [<1 31 0 9 53 -59|, <0 -38 3 -8 -64 81|]

EDOs: [[31edo|31]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]

{{see also|Qintosec family #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 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 family|qintosec temperament]].

[[POTE tuning|POTE generator]]: ~8/7 = 231.099

[[Map]]: [<10 0 47 36|, <0 2 -3 -1|]

Wedgie: <<20 -30 -10 -94 -72 61||

{{Vals|legend=1| 10, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}

==11-limit==

Map: [<10 0 47 36 98|, <0 2 -3 -1 -8|]

Vals: {{Vals| 10e, 130, 270, 670, 940, 1210, 2150c }}

Badness: 0.018735

Map: [<10 0 47 36 98 37|, <0 2 -3 -1 -8 0|]

Vals: {{Vals| 10e, 130, 270, 940, 1210f, 1480cf }}

Badness: 0.013475

=Neominor=

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

Commas: 169/168, 243/242, 364/363, 441/440

Map: [<1 3 12 8 7 7|, <0 -6 -41 -22 -15 -14|]

EDOs: 72, 161f, 233f

Badness: 0.0269

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

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=

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

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

Map: [<1 29 33 25 25 99|, <0 -42 -47 -34 -33 -146|]

EDOs: 72, 311, 694, 1005c, 1699cd

 

=Newt=

==11-limit==

==13-limit==

=Amicable=

=Septidiasemi=

Map: [<1 25 -31 -8|, <0 -26 37 12|]

=Cotritone=

Commas: 2401/2400, 390625/387072

POTE generator: ~7/5 = 583.3848

Map: [<1 -13 -4 -4|, <0 30 13 14|]

EDOs: 35, 37, 72, 109, 181, 253

==11-limit==

Commas: 385/384, 1375/1372, 4000/3993

POTE generator: ~7/5 = 583.3872

Map: [<1 -13 -4 -4 2|, <0 30 13 14 3|]

EDOs: 35, 37, 72, 109, 181, 253

==13-limit==

Commas: 169/168, 364/363, 385/384, 625/624

POTE generator: ~7/5 = 583.3866

Map: [<1 -13 -4 -4 2 -7|, <0 30 13 14 3 22|]

EDOs: 37, 72, 109, 181f