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.

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.

Temperaments not discussed here include [[Dicot family #Decimal|decimal]], [[Archytas clan #Beatles|beatles]], [[Meantone family #Squares|squares]], [[Starling temperaments #Myna|myna]], [[Keemic temperaments #Quasitemp|quasitemp]], [[Gamelismic clan #Miracle|miracle]], [[Magic family #Quadrimage|quadrimage]], [[Ragismic microtemperaments #Ennealimmal|ennealimmal]], [[Tetracot family #Octacot|octacot]], [[Kleismic family #Quadritikleismic|quadritikleismic]], [[Schismatic family #Sesquiquartififths|sesquiquartififths]], [[Würschmidt family #Hemiwürschmidt|hemiwürschmidt]], and [[Gammic family #Neptune|neptune]].

= Hemififths =

{{main|Hemififths}}

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.

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

[[Comma]]: 858993459200/847288609443

[[Mapping]]: [{{val| 1 1 -5 }}, {{val| 0 2 25 }}]

[[POTE generator]]: ~655360/531441 = 351.476

{{Val list|legend=1| 41, 58, 99, 239, 338, 915b, 1253bc }}

[[Badness]]: 0.372848

Subgroup: 2.3.5.7

[[Comma list]]: 2401/2400, 5120/5103

[[Mapping]]: [{{val| 1 1 -5 -1 }}, {{val| 0 2 25 13 }}]

{{Multival|legend=1| 2 25 13 35 15 -40 }}

[[POTE generator]]: ~49/40 = 351.477

[[Minimax tuning]]:

: [{{monzo|1 0 0 0}}, {{monzo|7/5, 0, 2/25, 0}}, {{monzo|0 0 1 0}}, {{monzo|8/5 0 13/25 0}}]

[[Algebraic generator]]: (2 + sqrt(2))/2

{{Val list|legend=1| 41, 58, 99, 239, 338, 1253bbc, 1591bbc }}

[[Badness]]: 0.022243

== 11-limit ==

Comma list: 243/242, 441/440, 896/891

Mapping: [{{val| 1 1 -5 -1 2 }}, {{val| 0 2 25 13 5 }}]

POTE generator: ~11/9 = 351.521

Vals: {{Val list| 17c, 41, 58, 99e }}

Badness: 0.023498

=== 13-limit ===

Comma list: 144/143, 196/195, 243/242, 364/363

Mapping: [{{val| 1 1 -5 -1 2 4 }}, {{val| 0 2 25 13 5 -1 }}]

POTE generator: ~11/9 = 351.573

Vals: {{Val list| 17c, 41, 58, 99ef }}

Badness: 0.019090

Comma list: 2401/2400, 3388/3375, 9801/9800

Mapping: [{{val| 2 0 -35 -15 -47 }}, {{val| 0 2 25 13 34 }}]

POTE generator: ~49/40 = 351.505

Vals: {{Val list| 58, 140, 198, 734bc, 932bcd, 1130bcd }}

Badness: 0.042487

=== 13-limit ===

Comma list: 352/351, 676/675, 847/845, 1716/1715

Mapping: [{{val| 2 0 -35 -15 -47 -37 }}, {{val| 0 2 25 13 34 28 }}]

POTE generator: ~49/40 = 351.502

Vals: {{Val list| 58, 140, 198, 536f, 734bcf, 932bcdf }}

Badness: 0.021188

= Tertiaseptal =

{{main|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

[[Mapping]]: [{{val| 1 3 2 3 }}, {{val| 0 -22 5 -3 }}]

[[POTE generator]]: ~256/245 = 77.191

{{Val list|legend=1| 31, 109, 140, 171 }}

== 11-limit ==

Mapping: [{{val| 1 3 2 3 7 }}, {{val| 0 -22 5 -3 -55 }}]

POTE generator: ~256/245 = 77.227

Vals: {{Val list| 31, 109e, 140e, 171, 202 }}

=== 13-limit ===

Mapping: [{{val| 1 3 2 3 7 1 }}, {{val| 0 -22 5 -3 -55 42 }}]

POTE generator: ~117/112 = 77.203

Vals: {{Val list| 31, 109e, 140e, 171 }}

=== 17-limit ===

Mapping: [{{val| 1 3 2 3 7 1 1 }}, {{val| 0 -22 5 -3 -55 42 48 }}]

POTE generator: ~68/65 = 77.201

Vals: {{Val list| 31, 109eg, 140e, 171 }}

== Tertia ==

Mapping: [{{val| 1 3 2 3 5 }}, {{val| 0 -22 5 -3 -24 }}]

POTE generator: ~22/21 = 77.173

Vals: {{Val list| 31, 109, 140, 171e, 311e }}

=== 13-limit ===

Mapping: [{{val| 1 3 2 3 5 1 }}, {{val| 0 -22 5 -3 -24 42 }}]

POTE generator: ~22/21 = 77.158

Vals: {{Val list| 31, 109, 140, 311e, 451ee }}

=== 17-limit ===

Mapping: [{{val| 1 3 2 3 5 1 1 }}, {{val| 0 -22 5 -3 -24 42 48 }}]

POTE generator: ~22/21 = 77.162

Vals: {{Val list| 31, 109g, 140, 311e, 451ee }}

== Hemitert ==

Mapping: [{{val| 1 3 2 3 6 }}, {{val| 0 -44 10 -6 -79 }}]

POTE generator: ~45/44 = 38.596

Vals: {{Val list| 31, 280, 311, 342 }}

=== 13-limit ===

Mapping: [{{val| 1 3 2 3 6 1 }}, {{val| 0 -44 10 -6 -79 84 }}]

POTE generator: ~45/44 = 38.588

Vals: {{Val list| 31, 280, 311, 964f, 1275f, 1586cff }}

=== 17-limit ===

Mapping: [{{val| 1 3 2 3 6 1 1 }}, {{val| 0 -44 10 -6 -79 84 96 }}]

POTE generator: ~45/44 = 38.589

Vals: {{Val list| 31, 280, 311, 653f, 964f }}

= Harry =

Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&72 temperament, with wedgie {{multival|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 {{multival|12 34 20 30 ...}}.

Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with {{multival|12 34 20 30 52 ...}} as the octave wedgie. [[130edo|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]]: [{{val| 2 4 7 7 }}, {{val| 0 -6 -17 -10 }}]

{{Multival|legend=1| 12 34 20 26 -2 -49 }}

[[POTE generator]]: ~21/20 = 83.156

[[Badness]]: 0.034077

== 11-limit ==

Comma list: 243/242, 441/440, 4000/3993

Mapping: [{{val| 2 4 7 7 9 }}, {{val| 0 -6 -17 -10 -15 }}]

POTE generator: ~21/20 = 83.167

Vals: {{Val list| 14c, 58, 72, 130, 202 }}

Badness: 0.015867

== 13-limit ==

Comma list: 243/242, 351/350, 441/440, 676/675

Mapping: [{{val| 2 4 7 7 9 11 }}, {{val| 0 -6 -17 -10 -15 -26 }}]

POTE generator: ~21/20 = 83.116

Vals: {{Val list| 58, 72, 130, 332f, 462ef }}

Badness: 0.013046

== 17-limit ==

Comma list: 221/220, 243/242, 289/288, 351/350, 441/440

Mapping: [{{val| 2 4 7 7 9 11 9 }}, {{val| 0 -6 -17 -10 -15 -26 -6 }}]

POTE generator: ~21/20 = 83.168

Vals: {{Val list| 58, 72, 130, 202g }}

Badness: 0.012657

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

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

Badness: 0.0175

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

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||

[[Badness]]: 0.033902

Comma list: 2401/2400, 5832/5825, 9801/9800

POTE generator: ~8/7 = 231.070

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

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

Comma list: 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|]

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

Badness: 0.013475

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

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

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

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

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

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

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

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

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==

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

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

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