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Ragismic microtemperaments: Difference between revisions

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The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit |-12 -73 55> and 7-limit 3955078125/3954653486, as well as 4375/4374.
The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit |-12 -73 55> and 7-limit 3955078125/3954653486, as well as 4375/4374.


Comma: |-12 -73 55>
[[Comma]]: |-12 -73 55>


POTE generator: ~162/125 = 449.127
[[Mapping]]: [<1 36 48|, <0 -55 -73|]


Map: [<1 36 48|, <0 -55 -73|]
[[Wedgie]]: <<55 73 -12||


Wedgie: <<55 73 -12||
[[POTE tuning|POTE generator]]: ~162/125 = 449.1269


EDOs: 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419
[[EDO|Vals]]: {{Val list| 8, 163, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}


Badness: 0.7549
[[Badness]]: 0.754866


==7-limit==
== 7-limit ==
Commas: 4375/4374, 3955078125/3954653486
[[Comma list]]: 4375/4374, 3955078125/3954653486
 
[[Mapping]]: [<1 36 48 61|, <0 -55 -73 -93|]
 
[[Wedgie]]: <<55 73 93 -12 -7 11||


POTE generator: ~35/27 = 449.127
[[POTE tuning|POTE generator]]: ~35/27 = 449.1270


Map: [<1 36 48 61|, <0 -55 -73 -93|]
[[EDO|Vals]]: {{Val list| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}


Wedgie: <<55 73 93 -12 -7 11||
[[Badness]]: 0.015075


EDOs: 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419
= Brahmagupta =
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], |47 -7 -7 -7> = 140737488355328 / 140710042265625.


Badness: 0.0151
[[Comma list]]: 4375/4374, 70368744177664/70338939985125


=Brahmagupta=
[[Mapping]]: [<7 2 -8 53|, <0 3 8 -11|]
Commas: 4375/4374, 70368744177664/70338939985125


POTE generator: ~27/20 = 519.716
[[Wedgie]]: <<21 56 -77 40 -181 -336||


Map: [<7 2 -8 53|, <0 3 8 -11|]
[[POTE tuning|POTE generator]]: ~27/20 = 519.716


Wedgie: <<21 56 -77 40 -181 -336||
[[EDO|Vals]]: {{Val list| 7, 217, 224, 441, 1106, 1547 }}


EDOs: 217, 224, 441, 1106, 1547
[[Badness]]: 0.029122


Badness: 0.0291
== 11-limit ==
Comma list: 4000/3993, 4375/4374, 131072/130977


==11-limit==
Mapping: [<7 2 -8 53 3|, <0 3 8 -11 7|]
Commas: 4000/3993, 4375/4374, 131072/130977


POTE generator: ~27/20 = 519.704
POTE generator: ~27/20 = 519.704


Map: [<7 2 -8 53 3|, <0 3 8 -11 7|]
Vals: {{Val list| 7, 217, 224, 441, 665, 1771ee }}


EDOs: 217, 224, 441, 665, 1771ee
Badness: 0.052190


Badness: 0.0522
== 13-limit ==
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374


==13-limit==
Mapping: [<7 2 -8 53 3 35|, <0 3 8 -11 7 -3|]
Commas: 1575/1573, 2080/2079, 4096/4095, 4375/4374


POTE generator: ~27/20 = 519.706
POTE generator: ~27/20 = 519.706


Map: [<7 2 -8 53 3 35|, <0 3 8 -11 7 -3|]
Vals: {{Val list| 7, 217, 224, 441, 665, 1771eef }}


EDOs: 217, 224, 441, 665, 1771eef
Badness: 0.023132


Badness: 0.0231
= Quasithird =
[[Comma]]: |55 -64 20>


=Quasithird=
[[Mapping]]: [<4 0 -11|, <0 5 16|]
Comma: |55 -64 20>


POTE generator: ~1594323/1280000 = 380.395
[[Wedgie]]: <<20 64 55||


Map: [<4 0 -11|, <0 5 16|]
[[POTE tuning|POTE generator]]: ~1594323/1280000 = 380.395


Wedgie: <<20 64 55||
[[EDO|Vals]]: {{Val list| 60, 164, 224, 388, 612, 836, 1000, 1448, 1612, 2224, 2836 }}


EDOs: 164, 224, 388, 612, 836, 1000, 1448, 1612, 2224, 2836
[[Badness]]: 0.099519


Badness: 0.0995
== 7-limit ==
[[Comma list]]: 4375/4374, 1153470752371588581/1152921504606846976


==7-limit==
[[Mapping]]: [<4 0 -11 48|, <0 5 16 -29|]
Commas: 4375/4374, 1153470752371588581/1152921504606846976


POTE generator: ~5103/4096 = 380.388
[[Wedgie]]: <<20 64 -116 55 -240 -449||


Map: [<4 0 -11 48|, <0 5 16 -29|]
[[POTE tuning|POTE generator]]: ~5103/4096 = 380.388


Wedgie: <<20 64 -116 55 -240 -449||
[[EDO|Vals]]: {{Val list| 60d, 164, 224, 388, 612, 1448, 2060 }}


EDOs: 164, 224, 388, 612, 1448, 2060
[[Badness]]: 0.061813


Badness: 0.0618
== 11-limit ==
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296


==11-limit==
Mapping: [<4 0 -11 48 43|, <0 5 16 -29 -23|]
Commas: 3025/3024, 4375/4374, 4296700485/4294967296


POTE generator: ~5103/4096 = 380.387
POTE generator: ~22/21 = 80.387 (or ~5103/4096 = 380.387)


Map: [<4 0 -11 48 43|, <0 5 16 -29 -23|]
Vals: {{Val list| 60d, 164, 224, 388, 612, 836, 1448 }}


EDOs: 164, 224, 388, 612, 836, 1448
Badness: 0.021125


Badness: 0.0211
== 13-limit ==
Comma list: 2200/2197, 3025/3024, 4375/4374, 468512/468195


==13-limit==
Mapping: [<4 0 -11 48 43 11|, <0 5 16 -29 -23 3|]
Commas: 2200/2197, 3025/3024, 4375/4374, 468512/468195


POTE generator: ~5103/4096 = 380.385
POTE generator: ~22/21 = 80.385 (or ~5103/4096 = 380.385)


Map: [<4 0 -11 48 43 11|, <0 5 16 -29 -23 3|]
Vals: {{Val list| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}


EDOs: 164, 224, 388, 612, 836, 1448f, 2284f
Badness: 0.029501


Badness: 0.0295
= Semidimfourth =
[[Comma]]: |7 41 -31>


=Semidimfourth=
[[Mapping]]: [<1 21 28|, <0 -31 -41|]
Comma: |7 41 -31>


POTE generator: ~162/125 = 448.449
[[Wedgie]]: <<31 41 -7||


Map: [<1 21 28|, <0 -31 -41|]
[[POTE tuning|POTE generator]]: ~162/125 = 448.449


Wedgie: <<31 41 -7||
[[EDO|Vals]]: {{Val list| 8, 91, 99, 190, 289, 388, 487, 677, 875, 966 }}


EDOs: 91, 99, 190, 289, 388, 487, 677, 875, 966
[[Badness]]: 0.233376


Badness: 0.1930
== 7-limit ==
[[Comma list]]: 4375/4374, 235298/234375


==7-limit==
[[Mapping]]: [<1 21 28 36|, <0 -31 -41 -53|]
Commas: 4375/4374, 235298/234375


POTE generator: ~35/27 = 448.457
[[Wedgie]]: <<31 41 53 -7 -3 8||


Map: [<1 21 28 36|, <0 -31 -41 -53|]
[[POTE tuning|POTE generator]]: ~35/27 = 448.456


Wedgie: <<31 41 53 -7 -3 8||
[[EDO|Vals]]: {{Val list| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}


EDOs: 91, 99, 289, 388, 875, 1263d, 1651d
[[Badness]]: 0.055249


Badness: 0.0552
== Neusec ==
Comma list: 3025/3024, 4375/4374, 235298/234375


== Neusec ==
Mapping: [<2 11 15 19 15|, <0 -31 -41 -53 -32|]
Commas: 3025/3024, 4375/4374, 235298/234375


POTE generator: ~12/11 = 151.547
POTE generator: ~12/11 = 151.547


Map: [<2 11 15 19 15|, <0 -31 -41 -53 -32|]
Vals: {{Val list| 8d, 190, 388 }}


EDOs: 190, 388
Badness: 0.059127


Badness: 0.0591
=== 13-limit ===
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374


=== 13-limit ===
Mapping: [<2 11 15 19 15 17|, <0 -31 -41 -53 -32 -38|]
Commas: 847/845, 1001/1000, 3025/3024, 4375/4374


POTE generator: ~12/11 = 151.545
POTE generator: ~12/11 = 151.545


Map: [<2 11 15 19 15 17|, <0 -31 -41 -53 -32 -38|]
Vals: {{Val list| 8d, 190, 198, 388 }}


EDOs: 190, 198, 388
Badness: 0.030941


Badness: 0.0309
= Acrokleismic =
[[Comma list]]: 4375/4374, 2202927104/2197265625


=Acrokleismic=
[[Mapping]]: [<1 10 11 27|, <0 -32 -33 -92|]
Commas: 4375/4374, 2202927104/2197265625


POTE generator: ~6/5 = 315.557
[[Wedgie]]: <<32 33 92 -22 56 121||


Map: [<1 10 11 27|, <0 -32 -33 -92|]
[[POTE tuning|POTE generator]]: ~6/5 = 315.557


Wedgie: <<32 33 92 -22 56 121||
[[EDO|Vals]]: {{Val list| 19, 251, 270 }}


EDOs: 19, 251, 270
[[Badness]]: 0.056184


Badness: 0.0562
== 11-limit ==
Comma list: 4375/4374, 41503/41472, 172032/171875


==11-limit==
Mapping: [<1 10 11 27 -16|, <0 -32 -33 -92 74|]
Commas: 4375/4374, 41503/41472, 172032/171875


POTE generator: ~6/5 = 315.558
POTE generator: ~6/5 = 315.558


Map: [<1 10 11 27 -16|, <0 -32 -33 -92 74|]
Vals: {{Val list| 19, 251, 270, 829, 1099, 1369, 1639 }}


EDOs: 19, 251, 270, 829, 1099, 1369, 1639
Badness: 0.036878


Badness: 0.0369
=== 13-limit ===
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976


=== 13-limit ===
Mapping: [<1 10 11 27 -16 25|, <0 -32 -33 -92 74 -81|]
Commas: 676/675, 1001/1000, 4375/4374, 10985/10976


POTE generator: ~6/5 = 315.557
POTE generator: ~6/5 = 315.557


Map: [<1 10 11 27 -16 25|, <0 -32 -33 -92 74 -81|]
Vals: {{Val list| 19, 251, 270 }}


EDOs: 19, 251, 270
Badness: 0.026818


Badness: 0.0268
== Counteracro ==
Comma list: 4375/4374, 5632/5625, 117649/117612


==Counteracro==
Mapping: [<1 10 11 27 55|, <0 -32 -33 -92 -196|]
Commas: 4375/4374, 5632/5625, 117649/117612


POTE generator: ~6/5 = 315.553
POTE generator: ~6/5 = 315.553


Map: [<1 10 11 27 55|, <0 -32 -33 -92 -196|]
Vals: {{Val list| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}


EDOs: 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde
Badness: 0.042572


Badness: 0.0426
=== 13-limit ===
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374


===13-limit===
Mapping: [<1 10 11 27 55 25|, <0 -32 -33 -92 -196 -81|]
Commas: 676/675, 1716/1715, 4225/4224, 4375/4374


POTE generator: ~6/5 = 315.554
POTE generator: ~6/5 = 315.554


Map: [<1 10 11 27 55 25|, <0 -32 -33 -92 -196 -81|]
Vals: {{Val list| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}


EDOs: 270, 1331c, 1601c, 1871bcf, 2141bcf
Badness: 0.026028


Badness: 0.0260
= Seniority =
{{see also|Very high accuracy temperaments#Senior}}


=Seniority=
[[Comma list]]: 4375/4374, 201768035/201326592
Commas: 4375/4374, 201768035/201326592


POTE generator: ~3087/2560 = 322.804
[[POTE tuning|POTE generator]]: ~3087/2560 = 322.804


Map: [<1 11 19 2|, <0 -35 -62 3|]
[[Mapping]]: [<1 11 19 2|, <0 -35 -62 3|]


Wedgie: <<35 62 -3 17 -103 -181||
[[Wedgie]]: <<35 62 -3 17 -103 -181||


EDOs: 26, 145, 171, 2710d
[[EDO|Vals]]: {{Val list| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}


Badness: 0.0449
[[Badness]]: 0.044877


=Orga=
= Orga =
Commas: 4375/4374, 54975581388800/54936068900769
Commas: 4375/4374, 54975581388800/54936068900769


Line 622: Line 626:
Badness: 0.0402
Badness: 0.0402


==11-limit==
== 11-limit ==
Commas: 3025/3024, 4375/4374, 5767168/5764801
Commas: 3025/3024, 4375/4374, 5767168/5764801


Line 633: Line 637:
Badness: 0.0162
Badness: 0.0162


==13-limit==
== 13-limit ==
Commas: 1716/1715, 2080/2079, 3025/3024, 15379/15360
Commas: 1716/1715, 2080/2079, 3025/3024, 15379/15360


Line 644: Line 648:
Badness: 0.0218
Badness: 0.0218


=Quatracot=
= Quatracot =
Commas: 4375/4374, 1483154296875/1473173782528
Commas: 4375/4374, 1483154296875/1473173782528


Line 657: Line 661:
Badness: 0.1760
Badness: 0.1760


==11-limit==
== 11-limit ==
Commas: 3025/3024, 4375/4374, 1265625/1261568
Commas: 3025/3024, 4375/4374, 1265625/1261568


Line 668: Line 672:
Badness: 0.0410
Badness: 0.0410


==13-limit==
== 13-limit ==
Commas: 625/624, 729/728, 1575/1573, 2200/2197
Commas: 625/624, 729/728, 1575/1573, 2200/2197


Line 679: Line 683:
Badness: 0.0226
Badness: 0.0226


=Octoid=
= Octoid =
Commas: 4375/4374, 16875/16807
[[Comma list]]: 4375/4374, 16875/16807


valid range: [578.571, 600.000] (56bcd to 8d)
[[Tuning ranges]]:
* valid range: [578.571, 600.000] (27\56 to 4\8)
* nice range: [582.512, 584.359]
* strict range:  [582.512, 584.359]


nice range: [582.512, 584.359]
[[Mapping]]: [<8 1 3 3|, <0 3 4 5|]


strict range:  [582.512, 584.359]
[[Wedgie]]: <<24 32 40 -5 -4 3||


POTE generator: ~7/5 = 583.940
Mapping generators: ~49/45, ~7/5


Map: [<8 1 3 3|, <0 3 4 5|]
[[POTE tuning|POTE generator]]: ~7/5 = 583.940


Generators: 49/45, 7/5
[[EDO|Vals]]: {{ 8d, 72, 152, 224 }}


EDOs: 72, 152, 224
[[Badness]]: 0.042670


Badness: 0.0427
== 11-limit ==
 
Comma list: 540/539, 1375/1372, 4000/3993
==11-limit==
Commas: 540/539, 1375/1372, 4000/3993


valid range: [581.250, 586.364] (64cd, 88bcde)
Tuning ranges:
* valid range: [581.250, 586.364] (31\64, 43\88)
* nice range: [582.512, 585.084]
* strict range: [582.512, 585.084]


nice range: [582.512, 585.084]
Mapping: [<8 1 3 3 16|, <0 3 4 5 3|]


strict range: [582.512, 585.084]
POTE generator: ~7/5 = 583.962


POTE generator: ~7/5 = 583.692
Vals: {{Val list| 72, 152, 224 }}


Map: [<8 1 3 3 16|, <0 3 4 5 3|]
Badness: 0.014097


EDOs: 72, 152, 224
== 13-limit ==
Comma list: 540/539, 1375/1372, 4000/3993, 625/624


Badness: 0.0141
Mapping: [<8 1 3 3 16 -21|, <0 3 4 5 3 13|]
 
=== 13-limit ===
Commas: 540/539, 1375/1372, 4000/3993, 625/624


POTE generator: ~7/5 = 583.905
POTE generator: ~7/5 = 583.905


Map: [<8 1 3 3 16 -21|, <0 3 4 5 3 13|]
Vals: {{Val list| 72, 152f, 224 }}
 
EDOs: 72, 224


Badness: 0.0153
Badness: 0.015274


=== Music ===
=== Music ===
Line 730: Line 734:
* [http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play]
* [http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play]


=== Octopus ===
== Octopus ==
Commas: 169/168, 325/324, 364/363, 540/539
Comma list: 169/168, 325/324, 364/363, 540/539
 
Mapping: [<8 1 3 3 16 14|, <0 3 4 5 3 4|]


POTE generator: ~7/5 = 583.892
POTE generator: ~7/5 = 583.892


Map: [<8 1 3 3 16 14|, <0 3 4 5 3 4|]
Vals: {{Val list| 72, 152, 224f }}
 
EDOs: 72, 152, 224f


Badness: 0.0217
Badness: 0.021679


= Amity =
= Amity =
Line 829: Line 833:
Badness: 0.0313
Badness: 0.0313


=Parakleismic=
= Parakleismic =
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13>, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being <<13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie <<13 14 35 -36 ...|| adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13>, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being <<13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie <<13 14 35 -36 ...|| adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.


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