Xenharmonic Wiki

Ragismic microtemperaments: Difference between revisions

← Older editNewer edit →
VisualWikitext
→Chlorine: normalize mappings and generators
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,036 edits
mNo edit summary
Line 34: Line 34:


Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]], [[Tritrizo clan #Undecentic|undecentic]], [[Tritrizo clan #Schisennealimmal|schisennealimmal]], and [[Tritrizo clan #Lunennealimmal|lunennealimmal]].
Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]], [[Tritrizo clan #Undecentic|undecentic]], [[Tritrizo clan #Schisennealimmal|schisennealimmal]], and [[Tritrizo clan #Lunennealimmal|lunennealimmal]].


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 69: Line 68:
POTE generator: ~5/3 = 884.4679
POTE generator: ~5/3 = 884.4679


Vals: {{Val list| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
Optimal GPV sequence: {{Val list| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}


Badness: 0.027332
Badness: 0.027332
Line 82: Line 81:
POTE generator: ~5/3 = 884.4304
POTE generator: ~5/3 = 884.4304


Vals: {{Val list| 99e, 171e, 270 }}
Optimal GPV sequence: {{Val list| 99e, 171e, 270 }}


Badness: 0.029404
Badness: 0.029404
Line 97: Line 96:
POTE generator: ~5/3 = 884.4089
POTE generator: ~5/3 = 884.4089


Vals: {{Val list| 99, 171, 270, 711, 981, 1251, 2232e }}
Optimal GPV sequence: {{Val list| 99, 171, 270, 711, 981, 1251, 2232e }}


Badness: 0.026463
Badness: 0.026463
Line 110: Line 109:
POTE generator: ~5/3 = 884.3997
POTE generator: ~5/3 = 884.3997


Vals: {{Val list| 99, 171, 270, 711, 981, 1692e, 2673e }}
Optimal GPV sequence: {{Val list| 99, 171, 270, 711, 981, 1692e, 2673e }}


Badness: 0.016607
Badness: 0.016607
Line 130: Line 129:
* 11-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 52.592]
* 11-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 52.592]


Vals: {{Val list| 72, 171, 243 }}
Optimal GPV sequence: {{Val list| 72, 171, 243 }}


Badness: 0.020347
Badness: 0.020347
Line 148: Line 147:
* 13- and 15-odd-limit diamond monotone and tradeoff: ~36/35 = [48.825, 50.000]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~36/35 = [48.825, 50.000]


Vals: {{Val list| 72, 171, 243 }}
Optimal GPV sequence: {{Val list| 72, 171, 243 }}


Badness: 0.023250
Badness: 0.023250
Line 166: Line 165:
* 17-odd-limit diamond monotone and tradeoff: ~36/35 = [48.485, 50.000]
* 17-odd-limit diamond monotone and tradeoff: ~36/35 = [48.485, 50.000]


Vals: {{Val list| 72, 171, 243 }}
Optimal GPV sequence: {{Val list| 72, 171, 243 }}


Badness: 0.014602
Badness: 0.014602
Line 179: Line 178:
POTE generator: ~5/3 = 883.6257
POTE generator: ~5/3 = 883.6257


Vals: {{Val list| 27e, 45ef, 72 }}
Optimal GPV sequence: {{Val list| 27e, 45ef, 72 }}


Badness: 0.020697
Badness: 0.020697
Line 192: Line 191:
POTE generator: ~5/3 = 883.8298
POTE generator: ~5/3 = 883.8298


Vals: {{Val list| 27, 45, 72, 171e, 243e, 315e }}
Optimal GPV sequence: {{Val list| 27, 45, 72, 171e, 243e, 315e }}


Badness: 0.031123
Badness: 0.031123
Line 205: Line 204:
POTE generator: ~5/3 = 883.8476
POTE generator: ~5/3 = 883.8476


Vals: {{Val list| 27, 45f, 72, 171ef, 243ef }}
Optimal GPV sequence: {{Val list| 27, 45f, 72, 171ef, 243ef }}


Badness: 0.030325
Badness: 0.030325
Line 227: Line 226:
* 11-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 17.985]
* 11-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 17.985]


Vals: {{Val list| 72, 198, 270, 342, 612, 954, 1566 }}
Optimal GPV sequence: {{Val list| 72, 198, 270, 342, 612, 954, 1566 }}


Badness: 0.006283
Badness: 0.006283
Line 248: Line 247:
* 15-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.926]
* 15-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.926]


Vals: {{Val list| 72, 198, 270 }}
Optimal GPV sequence: {{Val list| 72, 198, 270 }}


Badness: 0.012505
Badness: 0.012505
Line 263: Line 262:
POTE generator: ~1053/800 = 475.4727
POTE generator: ~1053/800 = 475.4727


Vals: {{Val list| 126, 144, 270, 684, 954 }}
Optimal GPV sequence: {{Val list| 126, 144, 270, 684, 954 }}


Badness: 0.013104
Badness: 0.013104
Line 280: Line 279:
POTE generator: ~140/121 = 250.3367
POTE generator: ~140/121 = 250.3367


Vals: {{Val list| 72, 369, 441 }}
Optimal GPV sequence: {{Val list| 72, 369, 441 }}


Badness: 0.034196
Badness: 0.034196
Line 293: Line 292:
POTE generator: ~140/121 = 250.3375
POTE generator: ~140/121 = 250.3375


Vals: {{Val list| 72, 297ef, 369f, 441 }}
Optimal GPV sequence: {{Val list| 72, 297ef, 369f, 441 }}


Badness: 0.026122
Badness: 0.026122
Line 308: Line 307:
POTE generator: ~25/22 = 221.0717
POTE generator: ~25/22 = 221.0717


Vals: {{Val list| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
Optimal GPV sequence: {{Val list| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}


Badness: 0.021320
Badness: 0.021320
Line 323: Line 322:
POTE generator: ~2352/1375 = 928.8000
POTE generator: ~2352/1375 = 928.8000


Vals: {{Val list| 27, 243, 270, 783, 1053, 1323 }}
Optimal GPV sequence: {{Val list| 27, 243, 270, 783, 1053, 1323 }}


Badness: 0.029812
Badness: 0.029812
Line 351: Line 350:
POTE generator: ~8/7 = 230.3370
POTE generator: ~8/7 = 230.3370


Vals: {{Val list| 26, 198, 224, 422, 646, 1068d }}
Optimal GPV sequence: {{Val list| 26, 198, 224, 422, 646, 1068d }}


Badness: 0.040955
Badness: 0.040955
Line 364: Line 363:
POTE generator: ~8/7 = 230.3373
POTE generator: ~8/7 = 230.3373


Vals: {{Val list| 26, 198, 224, 422, 646f, 1068df }}
Optimal GPV sequence: {{Val list| 26, 198, 224, 422, 646f, 1068df }}


Badness: 0.020416
Badness: 0.020416
Line 370: Line 369:
== Supermajor ==
== Supermajor ==
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of {{multival|37 46 75 -13 15 45}}. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of {{multival|37 46 75 -13 15 45}}. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 395: Line 393:
POTE generator: ~9/7 = 435.082
POTE generator: ~9/7 = 435.082


EDOs: {{Val list| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}
Optimal GPV sequence: {{Val list| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}


Badness: 0.012773
Badness: 0.012773
Line 401: Line 399:
== Enneadecal ==
== Enneadecal ==
Enneadecal temperament tempers out the enneadeca, {{monzo|-14 -19 19}}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo|19EDO]] up to just ones. [[171edo|171EDO]] is a good tuning for either the 5 or 7 limits, and [[494edo|494EDO]] shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo|665EDO]] for a tuning.
Enneadecal temperament tempers out the enneadeca, {{monzo|-14 -19 19}}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo|19EDO]] up to just ones. [[171edo|171EDO]] is a good tuning for either the 5 or 7 limits, and [[494edo|494EDO]] shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo|665EDO]] for a tuning.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 428: Line 425:
POTE generator: ~3/2 = 702.360
POTE generator: ~3/2 = 702.360


Vals: {{Val list| 19, 152, 323e, 475de, 627de }}
Optimal GPV sequence: {{Val list| 19, 152, 323e, 475de, 627de }}


Badness: 0.043734
Badness: 0.043734
Line 441: Line 438:
POTE generator: ~3/2 = 702.212
POTE generator: ~3/2 = 702.212


Vals: {{Val list| 19, 152f, 323e }}
Optimal GPV sequence: {{Val list| 19, 152f, 323e }}


Badness: 0.033545
Badness: 0.033545
Line 454: Line 451:
POTE generator: ~3/2 = 701.881
POTE generator: ~3/2 = 701.881


Vals: {{Val list| 152, 342, 494, 836, 1178, 2014 }}
Optimal GPV sequence: {{Val list| 152, 342, 494, 836, 1178, 2014 }}


Badness: 0.009985
Badness: 0.009985
Line 467: Line 464:
POTE generator: ~3/2 = 701.986
POTE generator: ~3/2 = 701.986


Vals: {{Val list| 152, 342, 494, 836 }}
Optimal GPV sequence: {{Val list| 152, 342, 494, 836 }}


Badness: 0.030391
Badness: 0.030391
Line 497: Line 494:
POTE generator: ~6/5 = 315.582
POTE generator: ~6/5 = 315.582


Vals: {{Val list| 80, 190, 270, 1000, 1270 }}
Optimal GPV sequence: {{Val list| 80, 190, 270, 1000, 1270 }}


Badness: 0.024329
Badness: 0.024329
Line 510: Line 507:
POTE generator: ~6/5 = 315.602
POTE generator: ~6/5 = 315.602


Vals: {{Val list| 80, 190, 270, 730, 1000 }}
Optimal GPV sequence: {{Val list| 80, 190, 270, 730, 1000 }}


Badness: 0.016810
Badness: 0.016810
Line 538: Line 535:
POTE generator: ~49/48 = 26.286
POTE generator: ~49/48 = 26.286


Vals: {{Val list| 45e, 46, 91e, 137de }}
Optimal GPV sequence: {{Val list| 45e, 46, 91e, 137de }}


Badness: 0.054098
Badness: 0.054098
Line 551: Line 548:
POTE generator: ~49/48 = 26.310
POTE generator: ~49/48 = 26.310


Vals: {{Val list| 45ef, 46, 91ef, 137def }}
Optimal GPV sequence: {{Val list| 45ef, 46, 91ef, 137def }}


Badness: 0.033067
Badness: 0.033067
Line 564: Line 561:
POTE generator: ~49/48 = 26.246
POTE generator: ~49/48 = 26.246


Vals: {{Val list| 45, 46, 91, 137d }}
Optimal GPV sequence: {{Val list| 45, 46, 91, 137d }}


Badness: 0.076567
Badness: 0.076567
Line 577: Line 574:
POTE generator: ~49/48 = 26.239
POTE generator: ~49/48 = 26.239


Vals: {{Val list| 45, 46, 91, 137d }}
Optimal GPV sequence: {{Val list| 45, 46, 91, 137d }}


Badness: 0.051893
Badness: 0.051893
Line 605: Line 602:
POTE generator: ~1155/1024 = 208.901
POTE generator: ~1155/1024 = 208.901


Vals: {{Val list| 46, 132, 178, 224, 270, 494, 764 }}
Optimal GPV sequence: {{Val list| 46, 132, 178, 224, 270, 494, 764 }}


Badness: 0.012860
Badness: 0.012860
Line 618: Line 615:
POTE generator: ~44/39 = 208.903
POTE generator: ~44/39 = 208.903


Vals: {{Val list| 46, 178, 224, 270, 494, 764, 1258 }}
Optimal GPV sequence: {{Val list| 46, 178, 224, 270, 494, 764, 1258 }}


Badness: 0.008856
Badness: 0.008856
Line 678: Line 675:
POTE generator: ~27/20 = 519.704
POTE generator: ~27/20 = 519.704


Vals: {{Val list| 7, 217, 224, 441, 665, 1771ee }}
Optimal GPV sequence: {{Val list| 7, 217, 224, 441, 665, 1771ee }}


Badness: 0.052190
Badness: 0.052190
Line 691: Line 688:
POTE generator: ~27/20 = 519.706
POTE generator: ~27/20 = 519.706


Vals: {{Val list| 7, 217, 224, 441, 665, 1771eef }}
Optimal GPV sequence: {{Val list| 7, 217, 224, 441, 665, 1771eef }}


Badness: 0.023132
Badness: 0.023132
Line 697: Line 694:
== Quasithird ==
== Quasithird ==
The '''quasithird''' temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}.
The '''quasithird''' temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}.


Subgroup: 2.3.5
Subgroup: 2.3.5
Line 735: Line 731:
POTE generator: ~22/21 = 80.387 (or ~5103/4096 = 380.387)
POTE generator: ~22/21 = 80.387 (or ~5103/4096 = 380.387)


Vals: {{Val list| 60d, 164, 224, 388, 612, 836, 1448 }}
Optimal GPV sequence: {{Val list| 60d, 164, 224, 388, 612, 836, 1448 }}


Badness: 0.021125
Badness: 0.021125
Line 748: Line 744:
POTE generator: ~22/21 = 80.385 (or ~5103/4096 = 380.385)
POTE generator: ~22/21 = 80.385 (or ~5103/4096 = 380.385)


Vals: {{Val list| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}
Optimal GPV sequence: {{Val list| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}


Badness: 0.029501
Badness: 0.029501
Line 791: Line 787:
POTE generator: ~12/11 = 151.547
POTE generator: ~12/11 = 151.547


Vals: {{Val list| 8d, 190, 388 }}
Optimal GPV sequence: {{Val list| 8d, 190, 388 }}


Badness: 0.059127
Badness: 0.059127
Line 804: Line 800:
POTE generator: ~12/11 = 151.545
POTE generator: ~12/11 = 151.545


Vals: {{Val list| 8d, 190, 198, 388 }}
Optimal GPV sequence: {{Val list| 8d, 190, 198, 388 }}


Badness: 0.030941
Badness: 0.030941
Line 832: Line 828:
POTE generator: ~6/5 = 315.558
POTE generator: ~6/5 = 315.558


Vals: {{Val list| 19, 251, 270, 829, 1099, 1369, 1639 }}
Optimal GPV sequence: {{Val list| 19, 251, 270, 829, 1099, 1369, 1639 }}


Badness: 0.036878
Badness: 0.036878
Line 845: Line 841:
POTE generator: ~6/5 = 315.557
POTE generator: ~6/5 = 315.557


Vals: {{Val list| 19, 251, 270 }}
Optimal GPV sequence: {{Val list| 19, 251, 270 }}


Badness: 0.026818
Badness: 0.026818
Line 858: Line 854:
POTE generator: ~6/5 = 315.553
POTE generator: ~6/5 = 315.553


Vals: {{Val list| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}
Optimal GPV sequence: {{Val list| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}


Badness: 0.042572
Badness: 0.042572
Line 871: Line 867:
POTE generator: ~6/5 = 315.554
POTE generator: ~6/5 = 315.554


Vals: {{Val list| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}
Optimal GPV sequence: {{Val list| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}


Badness: 0.026028
Badness: 0.026028
Line 905: Line 901:
POTE generator: ~77/64 = 322.793
POTE generator: ~77/64 = 322.793


Vals: {{Val list| 26, 119c, 145, 171, 316e, 487ee }}
Optimal GPV sequence: {{Val list| 26, 119c, 145, 171, 316e, 487ee }}


Badness: 0.092238
Badness: 0.092238
Line 918: Line 914:
POTE generator: ~77/64 = 322.793
POTE generator: ~77/64 = 322.793


Vals: {{Val list| 26, 119c, 145, 171, 316ef, 487eef }}
Optimal GPV sequence: {{Val list| 26, 119c, 145, 171, 316ef, 487eef }}


Badness: 0.044662
Badness: 0.044662
Line 931: Line 927:
POTE generator: ~77/64 = 322.793
POTE generator: ~77/64 = 322.793


Vals: {{Val list| 26, 119c, 145, 171, 316ef, 487eef }}
Optimal GPV sequence: {{Val list| 26, 119c, 145, 171, 316ef, 487eef }}


Badness: 0.026562
Badness: 0.026562
Line 959: Line 955:
POTE generator: ~8/7 = 231.103
POTE generator: ~8/7 = 231.103


Vals: {{Val list| 26, 244, 270, 566, 836, 1106 }}
Optimal GPV sequence: {{Val list| 26, 244, 270, 566, 836, 1106 }}


Badness: 0.016188
Badness: 0.016188
Line 972: Line 968:
POTE generator: ~8/7 = 231.103
POTE generator: ~8/7 = 231.103


Vals: {{Val list| 26, 244, 270, 566, 836f, 1106f }}
Optimal GPV sequence: {{Val list| 26, 244, 270, 566, 836f, 1106f }}


Badness: 0.021762
Badness: 0.021762
Line 1,002: Line 998:
POTE generator: ~448/405 = 176.806
POTE generator: ~448/405 = 176.806


Vals: {{Val list| 190, 224, 414, 638, 1052c }}
Optimal GPV sequence: {{Val list| 190, 224, 414, 638, 1052c }}


Badness: 0.041043
Badness: 0.041043
Line 1,015: Line 1,011:
POTE generator: ~195/176 = 176.804
POTE generator: ~195/176 = 176.804


Vals: {{Val list| 190, 224, 414, 638, 1690bcc, 2328bccde }}
Optimal GPV sequence: {{Val list| 190, 224, 414, 638, 1690bcc, 2328bccde }}


Badness: 0.022643
Badness: 0.022643
Line 1,021: Line 1,017:
== Octoid ==
== Octoid ==
The '''octoid''' temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.
The '''octoid''' temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 1,063: Line 1,058:
* 11-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 585.084]
* 11-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 585.084]


Vals: {{Val list| 72, 152, 224 }}
Optimal GPV sequence: {{Val list| 72, 152, 224 }}


Badness: 0.014097
Badness: 0.014097
Line 1,069: Line 1,064:
Scales: [[Octoid72]], [[Octoid80]]
Scales: [[Octoid72]], [[Octoid80]]


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 1,078: Line 1,073:
POTE generator: ~7/5 = 583.905
POTE generator: ~7/5 = 583.905


Vals: {{Val list| 72, 152f, 224 }}
Optimal GPV sequence: {{Val list| 72, 152f, 224 }}


Badness: 0.015274
Badness: 0.015274
Line 1,087: Line 1,082:
* [http://www.archive.org/details/Dreyfus http://www.archive.org/details/Dreyfus] [http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play]
* [http://www.archive.org/details/Dreyfus http://www.archive.org/details/Dreyfus] [http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play]


==== 17-limit ====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 1,096: Line 1,091:
POTE generator: ~7/5 = 583.842
POTE generator: ~7/5 = 583.842


Vals: {{Val list| 72, 152fg, 224, 296, 520g }}
Optimal GPV sequence: {{Val list| 72, 152fg, 224, 296, 520g }}


Badness: 0.014304
Badness: 0.014304
Line 1,102: Line 1,097:
Scales: [[Octoid72]], [[Octoid80]]
Scales: [[Octoid72]], [[Octoid80]]


==== 19-limit ====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 1,111: Line 1,106:
POTE generator: ~7/5 = 583.932
POTE generator: ~7/5 = 583.932


Vals: {{Val list| 72, 152fg, 224 }}
Optimal GPV sequence: {{Val list| 72, 152fg, 224 }}


Badness: 0.016036
Badness: 0.016036
Line 1,117: Line 1,112:
Scales: [[Octoid72]], [[Octoid80]]
Scales: [[Octoid72]], [[Octoid80]]


=== Octopus ===
==== Octopus ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 1,126: Line 1,121:
POTE generator: ~7/5 = 583.892
POTE generator: ~7/5 = 583.892


Vals: {{Val list| 72, 152, 224f }}
Optimal GPV sequence: {{Val list| 72, 152, 224f }}


Badness: 0.021679
Badness: 0.021679
Line 1,132: Line 1,127:
Scales: [[Octoid72]], [[Octoid80]]
Scales: [[Octoid72]], [[Octoid80]]


==== 17-limit ====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 1,141: Line 1,136:
POTE generator: ~7/5 = 583.811
POTE generator: ~7/5 = 583.811


Vals: {{Val list| 72, 152, 224fg, 296ffg }}
Optimal GPV sequence: {{Val list| 72, 152, 224fg, 296ffg }}


Badness: 0.015614
Badness: 0.015614
Line 1,147: Line 1,142:
Scales: [[Octoid72]], [[Octoid80]]
Scales: [[Octoid72]], [[Octoid80]]


==== 19-limit ====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 1,156: Line 1,151:
POTE generator: ~7/5 = 584.064
POTE generator: ~7/5 = 584.064


Vals: {{Val list| 72, 152, 224fg, 376ffgh }}
Optimal GPV sequence: {{Val list| 72, 152, 224fg, 376ffgh }}


Badness: 0.016321
Badness: 0.016321
Line 1,162: Line 1,157:
Scales: [[Octoid72]], [[Octoid80]]
Scales: [[Octoid72]], [[Octoid80]]


=== Hexadecoid ===
==== Hexadecoid ====
Hexadecoid (80&144) has a period of 1/16 octave and tempers out 4225/4224.
Hexadecoid (80&144) has a period of 1/16 octave and tempers out 4225/4224.


Line 1,173: Line 1,168:
POTE generator: ~13/8 = 841.015
POTE generator: ~13/8 = 841.015


Vals: {{Val list| 80, 144, 224 }}
Optimal GPV sequence: {{Val list| 80, 144, 224 }}


Badness: 0.030818
Badness: 0.030818


==== 17-limit ====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 1,186: Line 1,181:
POTE generator: ~13/8 = 840.932
POTE generator: ~13/8 = 840.932


Vals: {{Val list| 80, 144, 224, 528dg }}
Optimal GPV sequence: {{Val list| 80, 144, 224, 528dg }}


Badness: 0.028611
Badness: 0.028611


==== 19-limit ====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 1,199: Line 1,194:
POTE generator: ~13/8 = 840.896
POTE generator: ~13/8 = 840.896


Vals: {{Val list| 80, 144, 224, 304dh, 528dghh }}
Optimal GPV sequence: {{Val list| 80, 144, 224, 304dh, 528dghh }}


Badness: 0.023731
Badness: 0.023731
Line 1,210: Line 1,205:


In the 5-limit amity is a genuine microtemperament, with 58\205 being a possible tuning. Another good choice is (64/5)<sup>1/13</sup>, which gives pure major thirds.
In the 5-limit amity is a genuine microtemperament, with 58\205 being a possible tuning. Another good choice is (64/5)<sup>1/13</sup>, which gives pure major thirds.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 1,235: Line 1,229:
POTE generator: ~128/105 = 339.464
POTE generator: ~128/105 = 339.464


Vals: {{Val list| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}
Optimal GPV sequence: {{Val list| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}


Badness: 0.031506
Badness: 0.031506
Line 1,248: Line 1,242:
POTE generator: ~128/105 = 339.481
POTE generator: ~128/105 = 339.481


Vals: {{Val list| 46ef, 53, 99ef, 152f }} <nowiki>*</nowiki>
Optimal GPV sequence: {{Val list| 46ef, 53, 99ef, 152f }} <nowiki>*</nowiki>


<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
Line 1,265: Line 1,259:
POTE generator: ~11/9 = 339.390
POTE generator: ~11/9 = 339.390


Vals: {{Val list| 7, 39, 46, 53, 99 }}
Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}


Badness: 0.035187
Badness: 0.035187
Line 1,278: Line 1,272:
POTE generator: ~11/9 = 339.419
POTE generator: ~11/9 = 339.419


Vals: {{Val list| 7, 39, 46, 53, 99 }}
Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}


Badness: 0.022448
Badness: 0.022448
Line 1,291: Line 1,285:
POTE generator: ~11/9 = 339.366
POTE generator: ~11/9 = 339.366


Vals: {{Val list| 7, 39, 46, 53, 99 }}
Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}


Badness: 0.019395
Badness: 0.019395
Line 1,304: Line 1,298:
POTE generator: ~64/55 = 339.439
POTE generator: ~64/55 = 339.439


Vals: {{Val list| 14cde, 46, 106, 152, 350, 502d }}
Optimal GPV sequence: {{Val list| 14cde, 46, 106, 152, 350, 502d }}


Badness: 0.031307
Badness: 0.031307
Line 1,312: Line 1,306:


In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo|8 14 -13}}, with the [[118edo|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 {{multival|13 14 35 -8 19 42}} and adding 3136/3125 and 4375/4374, and the 11-limit wedgie {{multival|13 14 35 -36 -8 19 -102 42 -132 -222}} adding 385/384. For the 7-limit [[99edo|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, {{monzo|8 14 -13}}, with the [[118edo|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 {{multival|13 14 35 -8 19 42}} and adding 3136/3125 and 4375/4374, and the 11-limit wedgie {{multival|13 14 35 -36 -8 19 -102 42 -132 -222}} adding 385/384. For the 7-limit [[99edo|99EDO]] may be preferred, but in the 11-limit it is best to stick with 118.


Subgroup: 2.3.5
Subgroup: 2.3.5
Line 1,350: Line 1,343:
POTE generator: ~6/5 = 315.251
POTE generator: ~6/5 = 315.251


Vals: {{Val list| 19, 99, 118 }}
Optimal GPV sequence: {{Val list| 19, 99, 118 }}


Badness: 0.049711
Badness: 0.049711
Line 1,365: Line 1,358:
POTE generator: ~6/5 = 315.220
POTE generator: ~6/5 = 315.220


Vals: {{Val list| 19e, 99e, 118, 217, 335, 552d, 887dd }}
Optimal GPV sequence: {{Val list| 19e, 99e, 118, 217, 335, 552d, 887dd }}


Badness: 0.036027
Badness: 0.036027
Line 1,378: Line 1,371:
POTE generator: ~6/5 = 315.214
POTE generator: ~6/5 = 315.214


Vals: {{Val list| 99e, 118, 217, 552d, 769de }}
Optimal GPV sequence: {{Val list| 99e, 118, 217, 552d, 769de }}


Badness: 0.044710
Badness: 0.044710
Line 1,393: Line 1,386:
POTE generator: ~6/5 = 315.225
POTE generator: ~6/5 = 315.225


Vals: {{Val list| 19e, 99ef, 118, 217ff, 335ff }}
Optimal GPV sequence: {{Val list| 19e, 99ef, 118, 217ff, 335ff }}


Badness: 0.037618
Badness: 0.037618
Line 1,406: Line 1,399:
POTE generator: ~6/5 = 315.060
POTE generator: ~6/5 = 315.060


Vals: {{Val list| 19e, 80, 179, 259cd }}
Optimal GPV sequence: {{Val list| 19e, 80, 179, 259cd }}


Badness: 0.055884
Badness: 0.055884
Line 1,419: Line 1,412:
POTE generator: ~6/5 = 315.075
POTE generator: ~6/5 = 315.075


Vals: {{Val list| 19e, 80, 179 }}
Optimal GPV sequence: {{Val list| 19e, 80, 179 }}


Badness: 0.036559
Badness: 0.036559
Line 1,432: Line 1,425:
POTE generator: ~6/5 = 315.096
POTE generator: ~6/5 = 315.096


Vals: {{Val list| 19, 61d, 80, 99e, 179e }}
Optimal GPV sequence: {{Val list| 19, 61d, 80, 99e, 179e }}


Badness: 0.041720
Badness: 0.041720
Line 1,445: Line 1,438:
POTE generator: ~6/5 = 315.080
POTE generator: ~6/5 = 315.080


Vals: {{Val list| 19, 61d, 80, 99e, 179e }}
Optimal GPV sequence: {{Val list| 19, 61d, 80, 99e, 179e }}


Badness: 0.035781
Badness: 0.035781
Line 1,458: Line 1,451:
POTE generator: ~6/5 = 315.181
POTE generator: ~6/5 = 315.181


Vals: {{Val list| 80, 118, 198, 316, 514c, 830c }}
Optimal GPV sequence: {{Val list| 80, 118, 198, 316, 514c, 830c }}


Badness: 0.034208
Badness: 0.034208
Line 1,473: Line 1,466:
POTE generator: ~6/5 = 315.156
POTE generator: ~6/5 = 315.156


Vals: {{Val list| 80, 118, 198 }}
Optimal GPV sequence: {{Val list| 80, 118, 198 }}


Badness: 0.033775
Badness: 0.033775
Line 1,488: Line 1,481:
POTE generator: ~6/5 = 315.184
POTE generator: ~6/5 = 315.184


Vals: {{Val list| 80, 118f, 198f }}
Optimal GPV sequence: {{Val list| 80, 118f, 198f }}


Badness: 0.040467
Badness: 0.040467
Line 1,521: Line 1,514:
POTE generator: ~6/5 = 316.071
POTE generator: ~6/5 = 316.071


Vals: {{Val list| 19, 205, 224 }}
Optimal GPV sequence: {{Val list| 19, 205, 224 }}


Badness: 0.070952
Badness: 0.070952
Line 1,534: Line 1,527:
POTE generator: ~6/5 = 316.070
POTE generator: ~6/5 = 316.070


Vals: {{Val list| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}
Optimal GPV sequence: {{Val list| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}


Badness: 0.033874
Badness: 0.033874
Line 1,547: Line 1,540:
POTE generator: ~6/5 = 316.065
POTE generator: ~6/5 = 316.065


Vals: {{Val list| 19e, 205e, 224 }}
Optimal GPV sequence: {{Val list| 19e, 205e, 224 }}


Badness: 0.065400
Badness: 0.065400
Line 1,560: Line 1,553:
POTE generator: ~6/5 = 316.065
POTE generator: ~6/5 = 316.065


Vals: {{Val list| 19e, 205e, 224 }}
Optimal GPV sequence: {{Val list| 19e, 205e, 224 }}


Badness: 0.029782
Badness: 0.029782
Line 1,588: Line 1,581:
POTE generator: ~100/99 = 16.613
POTE generator: ~100/99 = 16.613


Vals: {{Val list| 72, 217, 289 }}
Optimal GPV sequence: {{Val list| 72, 217, 289 }}


Badness: 0.030875
Badness: 0.030875
Line 1,601: Line 1,594:
POTE generator: ~100/99 = 16.602
POTE generator: ~100/99 = 16.602


Vals: {{Val list| 72, 145, 217, 289 }}
Optimal GPV sequence: {{Val list| 72, 145, 217, 289 }}


Badness: 0.023862
Badness: 0.023862
Line 1,614: Line 1,607:
POTE generator: ~100/99 = 16.602
POTE generator: ~100/99 = 16.602


Vals: {{Val list| 72, 145, 217, 289 }}
Optimal GPV sequence: {{Val list| 72, 145, 217, 289 }}


Badness: 0.014741
Badness: 0.014741
Line 1,627: Line 1,620:
POTE generator: ~100/99 = 16.594
POTE generator: ~100/99 = 16.594


Vals: {{Val list| 72, 145, 217 }}
Optimal GPV sequence: {{Val list| 72, 145, 217 }}


Badness: 0.015197
Badness: 0.015197
Line 1,657: Line 1,650:
POTE generator: ~3/2 = 699.6179
POTE generator: ~3/2 = 699.6179


Vals: {{Val list| 26, 65, 91, 156d, 247cdde }}
Optimal GPV sequence: {{Val list| 26, 65, 91, 156d, 247cdde }}


Badness: 0.084590
Badness: 0.084590
Line 1,670: Line 1,663:
POTE generator: ~3/2 = 699.2969
POTE generator: ~3/2 = 699.2969


Vals: {{Val list| 26, 65f, 91f, 156dff }}
Optimal GPV sequence: {{Val list| 26, 65f, 91f, 156dff }}


Badness: 0.052366
Badness: 0.052366
Line 1,717: Line 1,710:
POTE generators: ~822083584/474609375 = 950.9749
POTE generators: ~822083584/474609375 = 950.9749


Vals: {{Val list| 289, 323, 612 }}
Optimal GPV sequence: {{Val list| 289, 323, 612 }}


Badness: 0.063706
Badness: 0.063706
Line 1,749: Line 1,742:
POTE generator: ~3/2 = 701.5951
POTE generator: ~3/2 = 701.5951


Vals: {{Val list| 46, 368, 414, 460, 874de }}
Optimal GPV sequence: {{Val list| 46, 368, 414, 460, 874de }}


Badness: 0.073783
Badness: 0.073783
Line 1,762: Line 1,755:
POTE generator: ~3/2 = 701.6419
POTE generator: ~3/2 = 701.6419


Vals: {{Val list| 46, 368, 414, 460, 874de, 1334de }}
Optimal GPV sequence: {{Val list| 46, 368, 414, 460, 874de, 1334de }}


Badness: 0.040751
Badness: 0.040751
Line 1,775: Line 1,768:
POTE generator: ~3/2 = 701.6425
POTE generator: ~3/2 = 701.6425


Vals: {{Val list| 46, 368, 414, 460, 874de, 1334deg }}
Optimal GPV sequence: {{Val list| 46, 368, 414, 460, 874de, 1334deg }}


Badness: 0.022441
Badness: 0.022441
Line 1,805: Line 1,798:
POTE generator: ~231/200 = 249.0193
POTE generator: ~231/200 = 249.0193


Vals: {{Val list| 53, 559, 612 }}
Optimal GPV sequence: {{Val list| 53, 559, 612 }}


Badness: 0.057083
Badness: 0.057083
Line 1,818: Line 1,811:
POTE generator: ~231/200 = 249.0199
POTE generator: ~231/200 = 249.0199


Vals: {{Val list| 53, 559, 612 }}
Optimal GPV sequence: {{Val list| 53, 559, 612 }}


Badness: 0.053780
Badness: 0.053780
Retrieved from "https://en.xen.wiki/w/Ragismic_microtemperaments"