65edo
65 tone equal temperament
65edo divides the octave into 65 equal parts of 18.4615 cents each. It can be characterized as the temperament which tempers out the schisma, 32805/32768, the sensipent comma, 78732/78125, and the wuerschmidt comma. In the 7-limit, there are two different maps; the first is <65 103 151 182|, tempering out 126/125, 245/243 and 686/675, so that 65edo supports sensi temperament, and the second is <65 103 151 183|, tempering out 225/224, 3125/3097, 4000/3969 and 5120/5103, so that 65edo supports garibaldi temperament. In both cases, the tuning privileges the 5-limit over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit wuerschmidt temperament (wurschmidt and worschmidt) these two mappings provide.
65edo approximates the intervals 3/2, 5/4, 11/8 and 19/16 well, so that it does a good job representing the 2.3.5.11.19 just intonation subgroup. To this one may want to add 13/8 and 17/16, giving the 19-limit no-sevens subgroup 2.3.5.11.13.17.19. Also of interest is the 19-limit 2*65 subgroup 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as 130edo.
65edo contains 13edo as a subset. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see Rubble: a Xenuke Unfolded.
Intervals
| Degree | Size | ||
|---|---|---|---|
| Cents | pions | 7mus | |
| 0 | |||
| 1 | 18.4615 | 19.5692 | 23.6308 (17.A17A16) |
| 2 | 36.9231 | 39.1385 | 47.2615 (2F.42F416) |
| 3 | 55.3846 | 58.7077 | 70.8923 (46.E46E16) |
| 4 | 73.84615 | 78.2769 | 94.5231 (5E.85E816) |
| 5 | 92.3077 | 97.84615 | 118.15385 (76.276216) |
| 6 | 110.7692 | 117.4154 | 141.7846 (8D.C8DD16) |
| 7 | 129.2308 | 136.9846 | 165.4154 (A5.6A5716) |
| 8 | 147.6923 | 156.55385 | 189.04615 (BD.0BD116) |
| 9 | 166.15385 | 176.1231 | 212.6769 (D4.AD4B16 |
| 10 | 184.6154 | 195.6923 | 236.3077 (EC.4EC516) |
| 11 | 203.0769 | 215.2615 | 259.9385 (103.F03F16) |
| 12 | 221.5385 | 234.8308 | 283.5692 (11B.91B916) |
| 13 | 240 | 254.4 | 307.2 (133.333316) |
| 14 | 258.4615 | 273.9692 | 330.8308 (14A.D4AD16) |
| 15 | 276.9231 | 293.5385 | 354.4615 (162.762716) |
| 16 | 295.3846 | 313.1077 | 378.0923 (17A.17A1816) |
| 17 | 313.84615 | 332.6769 | 401.7231 (191.B91C16) |
| 18 | 332.3077 | 352.24615 | 425.35385 (1A9.5A9616) |
| 19 | 350.7692 | 371.8154 | 448.9846 (1C0.FC116) |
| 20 | 369.2308 | 391.3846 | 472.6154 (1D8.9D8A16) |
| 21 | 387.6923 | 410.95385 | 496.24615 (1F0.3F0416) |
| 22 | 406.15385 | 430.5231 | 519.8769 (207.E07E16) |
| 23 | 424.6154 | 450.0923 | 543.5077 (21F.81F816) |
| 24 | 443.0769 | 469.6615 | 567.1385 (237.237216) |
| 25 | 461.5385 | 489.2308 | 590.7692 (24E.C4ED16) |
| 26 | 480 | 508.8 | 614.4 (266.666616) |
| 27 | 498.4615 | 528.3692 | 638.0308 (27E07E0816) |
| 28 | 516.9231 | 547.9385 | 661.6615 (295.A95B16) |
| 29 | 535.3846 | 567.5077 | 685.2923 (2AD.4AD416) |
| 30 | 553.84615 | 587.0769 | 708.9231 (2C4.EC4F16) |
| 31 | 572.3077 | 606.64615 | 732.55385 (2DC.8DC916) |
| 32 | 590.7692 | 626.2154 | 756.1846 (2F4.2F4216) |
| 33 | 609.2308 | 645.7846 | 779.8154 (30B.D0BD16) |
| 34 | 627.6923 | 665.35385 | 803.44615 (323.723716) |
| 35 | 646.1538 | 684.9231 | 827.0769 (33B.13B116) |
| 36 | 664.6154 | 704.4923 | 850.7077 (352.B52B16) |
| 37 | 683.0769 | 724.0615 | 874.3385 (36A.56A516) |
| 38 | 701.5385 | 743.6308 | 897.9692 (381.F81F816) |
| 39 | 720 | 763.2 | 921.6 (399.999A16) |
| 40 | 738.4615 | 782.7692 | 945.2308 (3B1.3B1316) |
| 41 | 756.9231 | 802.3385 | 968.8615 (3C8.DC8E16) |
| 42 | 775.3846 | 821.9077 | 992.4923 (3E0.7E0816) |
| 43 | 793.84615 | 841.4769 | 1016.1231 (3F8.1F8216) |
| 44 | 812.3077 | 861.04615 | 1039.75385 (40F.C0FC16) |
| 45 | 830.7692 | 880.6154 | 1063.3846 (427.627616) |
| 46 | 849.2308 | 900.1846 | 1087.0154 (43F.03F16) |
| 47 | 867.6923 | 919.75385 | 1110.64615 (456.A56A16) |
| 48 | 886.15385 | 939.3231 | 1134.2769 (46E.46E416) |
| 49 | 904.6154 | 958.8923 | 1157.9077 (485.E85E816) |
| 50 | 923.0769 | 978.4615 | 1181.53845 (49D.89D916) |
| 51 | 941.5385 | 998.0308 | 1205.1692 (4B5.2B5316) |
| 52 | 960 | 1017.6 | 1228.8 (4CC.CCCD16) |
| 53 | 978.4615 | 1037.1692 | 1252.4308 (4E4.6E4716) |
| 54 | 996.9231 | 1056.7385 | 1276.0615 (4FC.0FC116) |
| 55 | 1015.3846 | 1076.3077 | 1299.6923 (513.B13B16) |
| 56 | 1033.84615 | 1095.8769 | 1323.3231 (52B.52B516) |
| 57 | 1052.3077 | 1115.44615 | 1346.95385 (542,F42F16) |
| 58 | 1070.7692 | 1135.0154 | 1370.5846 (55A.95A916) |
| 59 | 1089.2308 | 1154.5846 | 1394.2154 (572.372316) |
| 60 | 1107.6923 | 1174.1535 | 1417.84615 (589.D89E16) |
| 61 | 1126.15385 | 1193.7231 | 1441.4769 (5A1.7A1816) |
| 62 | 1144.6154 | 1213.2923 | 1465.1077 (5B9.1B9216) |
| 63 | 1163.0769 | 1232.8615 | 1488.7385 (5D0.BD0C16) |
| 64 | 1181.5385 | 1252.4308 | 1512.3692 (5E8.5E8616) |
| 65 | 1200 | 1272 | 1536 (60016) |