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Breedsmic temperaments: Difference between revisions

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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]].  
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=
= Hemififths =
{{main|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]] 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.
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.
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==
Subgroup: 2.3.5
Comma: 858993459200/847288609443


POTE generator: ~655360/531441 = 351.476
[[Comma]]: 858993459200/847288609443


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


EDOs: 41, 58, 99, 239, 338, 915b, 1253bc
[[POTE generator]]: ~655360/531441 = 351.476


Badness: 0.3728
{{Val list|legend=1| 41, 58, 99, 239, 338, 915b, 1253bc }}


==7-limit==
[[Badness]]: 0.372848
Commas: 2401/2400, 5120/5103


7 and 9-limit minimax
== 7-limit ==
Subgroup: 2.3.5.7


[|1 0 0 0>, |7/5, 0, 2/25, 0>, |0 0 1 0>, |8/5 0 13/25 0>]
[[Comma list]]: 2401/2400, 5120/5103


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


Algebraic generator: (2 + sqrt(2))/2
{{Multival|legend=1| 2 25 13 35 15 -40 }}


Map: [<1 1 -5 -1|, <0 2 25 13|]
[[POTE generator]]: ~49/40 = 351.477


EDOs: [[41edo|41]], [[58edo|58]], [[99edo|99]], [[239edo|239]], [[338edo|338]]
[[Minimax tuning]]:
* 7 and 9-limit minimax
: [{{monzo|1 0 0 0}}, {{monzo|7/5, 0, 2/25, 0}}, {{monzo|0 0 1 0}}, {{monzo|8/5 0 13/25 0}}]
: Eigenvalues: 2, 5


Badness: 0.0222
[[Algebraic generator]]: (2 + sqrt(2))/2


==11-limit==
{{Val list|legend=1| 41, 58, 99, 239, 338, 1253bbc, 1591bbc }}
Commas: 243/242, 441/440, 896/891
 
[[Badness]]: 0.022243
 
== 11-limit ==
Subgroup: 2.3.5.7.11
 
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
POTE generator: ~11/9 = 351.521


Map: [<1 1 -5 -1 2|, <0 2 25 13 5|]
Vals: {{Val list| 17c, 41, 58, 99e }}
 
Badness: 0.023498


EDOs: 7ccd, 17c, 41, 58, 99e
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0235
Comma list: 144/143, 196/195, 243/242, 364/363


=== 13-limit ===
Mapping: [{{val| 1 1 -5 -1 2 4 }}, {{val| 0 2 25 13 5 -1 }}]
Commas: 144/143, 196/195, 243/242, 364/363


POTE generator: ~11/9 = 351.573
POTE generator: ~11/9 = 351.573


Map: [<1 1 -5 -1 2 4|, <0 2 25 13 5 -1|]
Vals: {{Val list| 17c, 41, 58, 99ef }}
 
Badness: 0.019090


EDOs: 7ccd, 17c, 41, 58, 99ef
== Semihemi ==
Subgroup: 2.3.5.7.11


Badness: 0.0191
Comma list: 2401/2400, 3388/3375, 9801/9800


== Semihemi ==
Mapping: [{{val| 2 0 -35 -15 -47 }}, {{val| 0 2 25 13 34 }}]
Commas: 2401/2400, 3388/3375, 9801/9800


POTE generator: ~49/40 = 351.505
POTE generator: ~49/40 = 351.505


Map: [<2 0 -35 -15 -47|, <0 2 25 13 34|]
Vals: {{Val list| 58, 140, 198, 734bc, 932bcd, 1130bcd }}
 
EDOs: 58, 140, 198, 734bc, 932bcd, 1130bcd


Badness: 0.042487
Badness: 0.042487


=== 13-limit ===
=== 13-limit ===
Commas: 352/351, 676/675, 847/845, 1716/1715
Subgroup: 2.3.5.7.11.13
 
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
POTE generator: ~49/40 = 351.502


Map: [<2 0 -35 -15 -47 -37|, <0 2 25 13 34 28|]
Vals: {{Val list| 58, 140, 198, 536f, 734bcf, 932bcdf }}
 
Badness: 0.021188
 
= 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.


EDOs: 58, 140, 198, 536f, 734bcf, 932bcdf
Subgroup: 2.3.5.7


Badness: 0.0212
[[Comma list]]: 2401/2400, 65625/65536


=Tertiaseptal=
[[Mapping]]: [{{val| 1 3 2 3 }}, {{val| 0 -22 5 -3 }}]
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.
 
{{Multival|legend=1| 22 -5 3 -59 -57 21 }}


Commas: 2401/2400, 65625/65536
[[POTE generator]]: ~256/245 = 77.191


POTE generator: ~256/245 = 77.191
{{Val list|legend=1| 31, 109, 140, 171 }}


Map: [<1 3 2 3|, <0 -22 5 -3|]
[[Badness]]: 0.012995


EDOs: 15, 16, 31, 109, 140, 171
== 11-limit ==
Subgroup: 2.3.5.7.11


Badness: 0.0130
Comma list: 243/242, 441/440, 65625/65536


==11-limit==
Mapping: [{{val| 1 3 2 3 7 }}, {{val| 0 -22 5 -3 -55 }}]
Commas: 243/242, 441/440, 65625/65536


POTE generator: ~256/245 = 77.227
POTE generator: ~256/245 = 77.227


Map: [<1 3 2 3 7|, <0 -22 5 -3 -55|]
Vals: {{Val list| 31, 109e, 140e, 171, 202 }}


EDOs: 15, 16, 31, 171, 202
Badness: 0.035576


Badness: 0.0356
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
Comma list: 243/242, 441/440, 625/624, 3584/3575
Commas: 243/242, 441/440, 625/624, 3584/3575
 
Mapping: [{{val| 1 3 2 3 7 1 }}, {{val| 0 -22 5 -3 -55 42 }}]


POTE generator: ~117/112 = 77.203
POTE generator: ~117/112 = 77.203


Map: [<1 3 2 3 7 1|, <0 -22 5 -3 -55 42|]
Vals: {{Val list| 31, 109e, 140e, 171 }}


EDOs: 31, 140e, 171
Badness: 0.036876
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0369
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575


=== 17-limit ===
Mapping: [{{val| 1 3 2 3 7 1 1 }}, {{val| 0 -22 5 -3 -55 42 48 }}]
Commas: 243/242, 375/374, 441/440, 625/624, 3584/3575


POTE generator: ~68/65 = 77.201
POTE generator: ~68/65 = 77.201


Map: [<1 3 2 3 7 1 1|, <0 -22 5 -3 -55 42 48|]
Vals: {{Val list| 31, 109eg, 140e, 171 }}


EDOs: 31, 140e, 171
Badness: 0.027398


Badness: 0.0274
== Tertia ==
Subgroup:2.3.5.7.11
 
Comma list: 385/384, 1331/1323, 1375/1372


==Tertia==
Mapping: [{{val| 1 3 2 3 5 }}, {{val| 0 -22 5 -3 -24 }}]
Commas: 385/384, 1331/1323, 1375/1372


POTE generator: ~22/21 = 77.173
POTE generator: ~22/21 = 77.173


Map: [<1 3 2 3 5|, <0 -22 5 -3 -24|]
Vals: {{Val list| 31, 109, 140, 171e, 311e }}
 
Badness: 0.030171
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 385/384, 625/624, 1331/1323
 
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 }}
 
Badness: 0.028384
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


EDOs: 31, 109, 140, 171e, 311e
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714


Badness: 0.0302
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, 171e, 311e, 451ee }}
 
Badness: 0.022416


== Hemitert ==
== Hemitert ==
Commas: 2401/2400, 3025/3024, 65625/65536
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 65625/65536
 
Mapping: [{{val| 1 3 2 3 6 }}, {{val| 0 -44 10 -6 -79 }}]


POTE generator: ~45/44 = 38.596
POTE generator: ~45/44 = 38.596


Map: [<1 3 2 3 6|, <0 -44 10 -6 -79|]
Vals: {{Val list| 31, 280, 311, 342 }}
 
EDOs: 31, 280, 311, 342, 2021cde, 3731cde


Badness: 0.0156
Badness: 0.015633


=Harry=
= Harry =
{{main|Harry}}
{{main|Harry}}
{{see also|Gravity family #Harry}}
{{see also|Gravity family #Harry}}
Line 164: Line 218:
Commas: 2401/2400, 19683/19600
Commas: 2401/2400, 19683/19600


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


Map: [<2 4 7 7|, <0 -6 -17 -10|]
Map: [<2 4 7 7|, <0 -6 -17 -10|]
Line 558: Line 612:
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.
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.


==7-limit==
Commas: 2401/2400, 177147/175000
Commas: 2401/2400, 177147/175000


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