57edo: Difference between revisions
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{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | [[Degree|Degree]] | ! rowspan="2"| [[Degree|Degree]] | ||
! | Size | ! colspan="3"| Size | ||
|- | |||
![[cent|Cents]] | |||
!pions | |||
!7mus | |||
|- | |- | ||
| style="text-align:center;" | 0 | | style="text-align:center;" | 0 | ||
| style="text-align:right;" | | | colspan="3" style="text-align:right;" | 0 | ||
|- | |- | ||
| style="text-align:center;" | 1 | | style="text-align:center;" | 1 | ||
| style="text-align:right;" | 21.0526 | | style="text-align:right;" | 21.0526 | ||
|22.3158 | |||
|26.9474 (1A.F287<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 2 | | style="text-align:center;" | 2 | ||
| style="text-align:right;" | 42.1053 | | style="text-align:right;" | 42.1053 | ||
|44.6316 | |||
|53.8947 (35.E50E<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 3 | | style="text-align:center;" | 3 | ||
| style="text-align:right;" | 63.1579 | | style="text-align:right;" | 63.1579 | ||
|66.9474 | |||
|80.8421 (50.D794<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 4 | | style="text-align:center;" | 4 | ||
| style="text-align:right;" | 84.2105 | | style="text-align:right;" | 84.2105 | ||
|89.2632 | |||
|107.7895 (6B.CA1B<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 5 | | style="text-align:center;" | 5 | ||
| style="text-align:right;" | 105.2632 | | style="text-align:right;" | 105.2632 | ||
|111.57895 | |||
|134.7368 (86.BCA2<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 6 | | style="text-align:center;" | 6 | ||
| style="text-align:right;" | 126.3158 | | style="text-align:right;" | 126.3158 | ||
|133.8947 | |||
|161.6842 (A1.AF29<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 7 | | style="text-align:center;" | 7 | ||
| style="text-align:right;" | 147.3684 | | style="text-align:right;" | 147.3684 | ||
|156.2105 | |||
|188.6316 (BC.A1AF<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 8 | | style="text-align:center;" | 8 | ||
| style="text-align:right;" | 168. | | style="text-align:right;" | 168.42105 | ||
|178.5263 | |||
|215.57895 (D7.9436<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 9 | | style="text-align:center;" | 9 | ||
| style="text-align:right;" | 189.4737 | | style="text-align:right;" | 189.4737 | ||
|200.8421 | |||
|242.5263 (F2.86BD<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 10 | | style="text-align:center;" | 10 | ||
| style="text-align:right;" | 210.5263 | | style="text-align:right;" | 210.5263 | ||
|223.1579 | |||
|269.4737 (10D.7943<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 11 | | style="text-align:center;" | 11 | ||
| style="text-align:right;" | 231. | | style="text-align:right;" | 231.57895 | ||
|245.4737 | |||
|296.42105 (128.6BCA<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 12 | | style="text-align:center;" | 12 | ||
| style="text-align:right;" | 252.6316 | | style="text-align:right;" | 252.6316 | ||
|267.7895 | |||
|323.3684 (143.5E51<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 13 | | style="text-align:center;" | 13 | ||
| style="text-align:right;" | 273.6842 | | style="text-align:right;" | 273.6842 | ||
|290.1053 | |||
|350.3158 (15E.50D7<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 14 | | style="text-align:center;" | 14 | ||
| style="text-align:right;" | 294.7368 | | style="text-align:right;" | 294.7368 | ||
|312.42105 | |||
|377.2632 (17B.435E<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 15 | | style="text-align:center;" | 15 | ||
| style="text-align:right;" | 315.7895 | | style="text-align:right;" | 315.7895 | ||
|334.7368 | |||
|404.2105 (194.35E5<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 16 | | style="text-align:center;" | 16 | ||
| style="text-align:right;" | 336.8421 | | style="text-align:right;" | 336.8421 | ||
|357.0526 | |||
|431.1579 (1AF.286C<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 17 | | style="text-align:center;" | 17 | ||
| style="text-align:right;" | 357.8947 | | style="text-align:right;" | 357.8947 | ||
|379.3684 | |||
|458.1053 (1CA.1AF2<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 18 | | style="text-align:center;" | 18 | ||
| style="text-align:right;" | 378.9474 | | style="text-align:right;" | 378.9474 | ||
|401.6842 | |||
|485.0526 (1E5.0D79<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 19 | | style="text-align:center;" | 19 | ||
| style="text-align:right;" | 400 | | style="text-align:right;" | 400 | ||
|424 | |||
|512 (200<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 20 | | style="text-align:center;" | 20 | ||
| style="text-align:right;" | 421.0526 | | style="text-align:right;" | 421.0526 | ||
|446.3158 | |||
|538.9474 (21A.F287<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 21 | | style="text-align:center;" | 21 | ||
| style="text-align:right;" | 442.1053 | | style="text-align:right;" | 442.1053 | ||
|468.6316 | |||
|565.8947 (235.E50E<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 22 | | style="text-align:center;" | 22 | ||
| style="text-align:right;" | 463.1579 | | style="text-align:right;" | 463.1579 | ||
|490.9474 | |||
|592.8421 (250.D794<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 23 | | style="text-align:center;" | 23 | ||
| style="text-align:right;" | 484.2105 | | style="text-align:right;" | 484.2105 | ||
|513.2632 | |||
|619.7895 (26B.CA1B<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 24 | | style="text-align:center;" | 24 | ||
| style="text-align:right;" | 505.2632 | | style="text-align:right;" | 505.2632 | ||
|535.57895 | |||
|646.7368 (286.BCA2<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 25 | | style="text-align:center;" | 25 | ||
| style="text-align:right;" | 526.3158 | | style="text-align:right;" | 526.3158 | ||
|557.8947 | |||
|673.6842 (2A1.AF29<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 26 | | style="text-align:center;" | 26 | ||
| style="text-align:right;" | 547.3684 | | style="text-align:right;" | 547.3684 | ||
|580.2105 | |||
|700.6316 (2BC.A1AF<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 27 | | style="text-align:center;" | 27 | ||
| style="text-align:right;" | 568. | | style="text-align:right;" | 568.42105 | ||
|602.5263 | |||
|727.57895 (2D7.9436<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 28 | | style="text-align:center;" | 28 | ||
| style="text-align:right;" | 589.4737 | | style="text-align:right;" | 589.4737 | ||
|624.8421 | |||
|754.5263 (2F2.86BD<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 29 | | style="text-align:center;" | 29 | ||
| style="text-align:right;" | 610.5263 | | style="text-align:right;" | 610.5263 | ||
|647.1579 | |||
|781.4737 (30D.7943<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 30 | | style="text-align:center;" | 30 | ||
| style="text-align:right;" | 631. | | style="text-align:right;" | 631.57895 | ||
|669.4737 | |||
|808.42105 (328.6BCA<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 31 | | style="text-align:center;" | 31 | ||
| style="text-align:right;" | 652.6316 | | style="text-align:right;" | 652.6316 | ||
|691.7895 | |||
|835.3684 (343.5E51<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 32 | | style="text-align:center;" | 32 | ||
| style="text-align:right;" | 673.6842 | | style="text-align:right;" | 673.6842 | ||
|714.1053 | |||
|862.3158 (35E.50D7<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 33 | | style="text-align:center;" | 33 | ||
| style="text-align:right;" | 694.7368 | | style="text-align:right;" | 694.7368 | ||
|736.42105 | |||
|889.2632 (37B.435E<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 34 | | style="text-align:center;" | 34 | ||
| style="text-align:right;" | 715.7895 | | style="text-align:right;" | 715.7895 | ||
|758.7368 | |||
|916.2105 (394.35E5<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 35 | | style="text-align:center;" | 35 | ||
| style="text-align:right;" | 736.8421 | | style="text-align:right;" | 736.8421 | ||
|781.0526 | |||
|943.1579 (3AF.286C<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 36 | | style="text-align:center;" | 36 | ||
| style="text-align:right;" | 757.8947 | | style="text-align:right;" | 757.8947 | ||
|803.3684 | |||
|970.1053 (3CA.1AF2<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 37 | | style="text-align:center;" | 37 | ||
| style="text-align:right;" | 778.9474 | | style="text-align:right;" | 778.9474 | ||
|825.6842 | |||
|997.0526 (3E5.0D79<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 38 | | style="text-align:center;" | 38 | ||
| style="text-align:right;" | 800 | | style="text-align:right;" | 800 | ||
|848 | |||
|1024 (400<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 39 | | style="text-align:center;" | 39 | ||
| style="text-align:right;" | 821.0526 | | style="text-align:right;" | 821.0526 | ||
|870.3158 | |||
|1050.9474 (41A.F287<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 40 | | style="text-align:center;" | 40 | ||
| style="text-align:right;" | 842.1053 | | style="text-align:right;" | 842.1053 | ||
|892.6316 | |||
|1077.8947 (435.E50E<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 41 | | style="text-align:center;" | 41 | ||
| style="text-align:right;" | 863.1579 | | style="text-align:right;" | 863.1579 | ||
|914.9474 | |||
|1104.8421 (450.D794<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 42 | | style="text-align:center;" | 42 | ||
| style="text-align:right;" | 884.2105 | | style="text-align:right;" | 884.2105 | ||
|937.2632 | |||
|1131.7895 (46B.CA1B<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 43 | | style="text-align:center;" | 43 | ||
| style="text-align:right;" | 905.2632 | | style="text-align:right;" | 905.2632 | ||
|959.57895 | |||
|1158.7368 (486.BCA2<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 44 | | style="text-align:center;" | 44 | ||
| style="text-align:right;" | 926.3158 | | style="text-align:right;" | 926.3158 | ||
|981.8947 | |||
|1175.6842 (4A1.AF29<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 45 | | style="text-align:center;" | 45 | ||
| style="text-align:right;" | 947.3684 | | style="text-align:right;" | 947.3684 | ||
|1004.2105 | |||
|1212.6316 (4BC.A1AF<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 46 | | style="text-align:center;" | 46 | ||
| style="text-align:right;" | 968. | | style="text-align:right;" | 968.42105 | ||
|1026.5263 | |||
|1239.57895 (4D7.9436<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 47 | | style="text-align:center;" | 47 | ||
| style="text-align:right;" | 989.4737 | | style="text-align:right;" | 989.4737 | ||
|1048.8421 | |||
|1266.5263 (4F2.86BD<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 48 | | style="text-align:center;" | 48 | ||
| style="text-align:right;" | 1010.5263 | | style="text-align:right;" | 1010.5263 | ||
|1071.1579 | |||
|1293.4737 (50D.7943<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 49 | | style="text-align:center;" | 49 | ||
| style="text-align:right;" | 1031. | | style="text-align:right;" | 1031.57895 | ||
|1093.4737 | |||
|1320.42105 (528.6BCA<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 50 | | style="text-align:center;" | 50 | ||
| style="text-align:right;" | 1052.6316 | | style="text-align:right;" | 1052.6316 | ||
|1115.7895 | |||
|1347.3684 (543.5E51<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 51 | | style="text-align:center;" | 51 | ||
| style="text-align:right;" | 1073.6842 | | style="text-align:right;" | 1073.6842 | ||
|1138.1053 | |||
|1374.3158 (55E.50D7<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 52 | | style="text-align:center;" | 52 | ||
| style="text-align:right;" | 1094.7368 | | style="text-align:right;" | 1094.7368 | ||
|1160.42105 | |||
|1401.2632 (57B.435E<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 53 | | style="text-align:center;" | 53 | ||
| style="text-align:right;" | 1115.7895 | | style="text-align:right;" | 1115.7895 | ||
|1182.7368 | |||
|1428.2105 (594.35E5<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 54 | | style="text-align:center;" | 54 | ||
| style="text-align:right;" | 1136.8421 | | style="text-align:right;" | 1136.8421 | ||
|1205.0526 | |||
|1455.1579 (5AF.286C<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 55 | | style="text-align:center;" | 55 | ||
| style="text-align:right;" | 1157.8947 | | style="text-align:right;" | 1157.8947 | ||
|1227.3684 | |||
|1482.1053 (5CA.1AF2<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 56 | | style="text-align:center;" | 56 | ||
| style="text-align:right;" | 1178.9474 | | style="text-align:right;" | 1178.9474 | ||
|1249.6842 | |||
|1509.0526 (5E5.0D79<sub>16</sub>) | |||
|- | |- | ||
| style="text-align:center;" | 57 | | style="text-align:center;" | 57 | ||
| style="text-align:right;" | 1200 | | style="text-align:right;" | 1200 | ||
|1272 | |||
|1536 (600<sub>16</sub>) | |||
|} | |} | ||
Revision as of 23:55, 24 April 2019
57 tone equal temperament
57edo divides the octave into 57 parts of size 21.053. It can be used to tune mothra temperament, and is an excellent tuning for the 2.5/3.7.11.13.17.19 just intonation subgroup. One way to describe 57 is that it has a 5-limit part consisting of three versions of 19, plus a no-threes no-fives part which is much more accurate. A good generator to exploit the 2.5/3.7.11.13.17.19 aspect of 57 is the approximate 11/8, which is 26\57. This gives the 19-limit 46&57 temperament Heinz.
5-limit commas: 81/80, 3125/3072
7-limit commas: 81/80, 3125/3072, 1029/1024
11-limit commas: 99/98, 385/384, 441/440, 625/616
Intervals
| Degree | Size | ||
|---|---|---|---|
| Cents | pions | 7mus | |
| 0 | 0 | ||
| 1 | 21.0526 | 22.3158 | 26.9474 (1A.F28716) |
| 2 | 42.1053 | 44.6316 | 53.8947 (35.E50E16) |
| 3 | 63.1579 | 66.9474 | 80.8421 (50.D79416) |
| 4 | 84.2105 | 89.2632 | 107.7895 (6B.CA1B16) |
| 5 | 105.2632 | 111.57895 | 134.7368 (86.BCA216) |
| 6 | 126.3158 | 133.8947 | 161.6842 (A1.AF2916) |
| 7 | 147.3684 | 156.2105 | 188.6316 (BC.A1AF16) |
| 8 | 168.42105 | 178.5263 | 215.57895 (D7.943616) |
| 9 | 189.4737 | 200.8421 | 242.5263 (F2.86BD16) |
| 10 | 210.5263 | 223.1579 | 269.4737 (10D.794316) |
| 11 | 231.57895 | 245.4737 | 296.42105 (128.6BCA16) |
| 12 | 252.6316 | 267.7895 | 323.3684 (143.5E5116) |
| 13 | 273.6842 | 290.1053 | 350.3158 (15E.50D716) |
| 14 | 294.7368 | 312.42105 | 377.2632 (17B.435E16) |
| 15 | 315.7895 | 334.7368 | 404.2105 (194.35E516) |
| 16 | 336.8421 | 357.0526 | 431.1579 (1AF.286C16) |
| 17 | 357.8947 | 379.3684 | 458.1053 (1CA.1AF216) |
| 18 | 378.9474 | 401.6842 | 485.0526 (1E5.0D7916) |
| 19 | 400 | 424 | 512 (20016) |
| 20 | 421.0526 | 446.3158 | 538.9474 (21A.F28716) |
| 21 | 442.1053 | 468.6316 | 565.8947 (235.E50E16) |
| 22 | 463.1579 | 490.9474 | 592.8421 (250.D79416) |
| 23 | 484.2105 | 513.2632 | 619.7895 (26B.CA1B16) |
| 24 | 505.2632 | 535.57895 | 646.7368 (286.BCA216) |
| 25 | 526.3158 | 557.8947 | 673.6842 (2A1.AF2916) |
| 26 | 547.3684 | 580.2105 | 700.6316 (2BC.A1AF16) |
| 27 | 568.42105 | 602.5263 | 727.57895 (2D7.943616) |
| 28 | 589.4737 | 624.8421 | 754.5263 (2F2.86BD16) |
| 29 | 610.5263 | 647.1579 | 781.4737 (30D.794316) |
| 30 | 631.57895 | 669.4737 | 808.42105 (328.6BCA16) |
| 31 | 652.6316 | 691.7895 | 835.3684 (343.5E5116) |
| 32 | 673.6842 | 714.1053 | 862.3158 (35E.50D716) |
| 33 | 694.7368 | 736.42105 | 889.2632 (37B.435E16) |
| 34 | 715.7895 | 758.7368 | 916.2105 (394.35E516) |
| 35 | 736.8421 | 781.0526 | 943.1579 (3AF.286C16) |
| 36 | 757.8947 | 803.3684 | 970.1053 (3CA.1AF216) |
| 37 | 778.9474 | 825.6842 | 997.0526 (3E5.0D7916) |
| 38 | 800 | 848 | 1024 (40016) |
| 39 | 821.0526 | 870.3158 | 1050.9474 (41A.F28716) |
| 40 | 842.1053 | 892.6316 | 1077.8947 (435.E50E16) |
| 41 | 863.1579 | 914.9474 | 1104.8421 (450.D79416) |
| 42 | 884.2105 | 937.2632 | 1131.7895 (46B.CA1B16) |
| 43 | 905.2632 | 959.57895 | 1158.7368 (486.BCA216) |
| 44 | 926.3158 | 981.8947 | 1175.6842 (4A1.AF2916) |
| 45 | 947.3684 | 1004.2105 | 1212.6316 (4BC.A1AF16) |
| 46 | 968.42105 | 1026.5263 | 1239.57895 (4D7.943616) |
| 47 | 989.4737 | 1048.8421 | 1266.5263 (4F2.86BD16) |
| 48 | 1010.5263 | 1071.1579 | 1293.4737 (50D.794316) |
| 49 | 1031.57895 | 1093.4737 | 1320.42105 (528.6BCA16) |
| 50 | 1052.6316 | 1115.7895 | 1347.3684 (543.5E5116) |
| 51 | 1073.6842 | 1138.1053 | 1374.3158 (55E.50D716) |
| 52 | 1094.7368 | 1160.42105 | 1401.2632 (57B.435E16) |
| 53 | 1115.7895 | 1182.7368 | 1428.2105 (594.35E516) |
| 54 | 1136.8421 | 1205.0526 | 1455.1579 (5AF.286C16) |
| 55 | 1157.8947 | 1227.3684 | 1482.1053 (5CA.1AF216) |
| 56 | 1178.9474 | 1249.6842 | 1509.0526 (5E5.0D7916) |
| 57 | 1200 | 1272 | 1536 (60016) |
Modes of 57edo
2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 1 - 3MOS of type 18L 21s (augene)