365edo
| ← 364edo | 365edo | 366edo → |
365 equal divisions of the octave (abbreviated 365edo or 365ed2), also called 365-tone equal temperament (365tet) or 365 equal temperament (365et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 365 equal parts of about 3.29 ¢ each. Each step represents a frequency ratio of 21/365, or the 365th root of 2.
Theory
365edo is consistent to the 7-odd-limit, but both harmonics 3 and 5 are about halfway between its steps. As every other step of 730edo, it is suitable for a 2.9.15 subgroup interpretation, in which case it is identical to 730edo.
Nonetheless, it does temper out 2401/2400, 3136/3125 and 6144/6125 on the patent val in the 7-limit, with an optimal stretch of -0.52 cents, and hereby tunes the hemiwürschmidt temperament. In the 11-limit, it tempers out 3025/3024, 3388/3375, 14641/14580; in the 13-limit, 352/351, 1001/1000, and 1716/1715.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.61 | +1.63 | +1.04 | -0.07 | +1.01 | +1.12 | -0.05 | +0.25 | -1.62 | -0.64 | -0.33 |
| Relative (%) | +48.9 | +49.6 | +31.5 | -2.3 | +30.7 | +34.0 | -1.5 | +7.6 | -49.4 | -19.6 | -10.0 | |
| Steps (reduced) |
579 (214) |
848 (118) |
1025 (295) |
1157 (62) |
1263 (168) |
1351 (256) |
1426 (331) |
1492 (32) |
1550 (90) |
1603 (143) |
1651 (191) | |
Subsets and supersets
Since 365 factors into 5 × 73, 365edo contains 5edo and 73edo as subsets. A step of 365edo is exactly 2 Woolhouse units (2\730).
Miscellaneous properties
An octave stretch of -0.796 cents would compress 365edo to an interesting intepretation: the pure 2/1 would represent 365.24219edo, which is the length of solar days in a tropical year. In 23-limit, 365eeffgghiii val's octave stretch of -0.79428 cents is very close, and makes 2/1 correspond to 365.241917 days, or 365 days 5h 48m 21.7s, which is only about 20 seconds short of the tropical year in the present era. A comma basis for the 365eeffgghiii val in the 23-limit is {256/255, 300/299, 352/351, 456/455, 896/891, 1225/1224, 3136/3125, 13608/13585}.