365edo

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← 364edo365edo366edo →
Prime factorization 5 × 73
Step size 3.28767¢ 
Fifth 214\365 (703.562¢)
Semitones (A1:m2) 38:25 (124.9¢ : 82.19¢)
Dual sharp fifth 214\365 (703.562¢)
Dual flat fifth 213\365 (700.274¢)
Dual major 2nd 62\365 (203.836¢)
Consistency limit 7
Distinct consistency limit 7

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

Approximation of odd harmonics in 365edo
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}.

See 365edo/Eliora's approach.

Interval table

see Table of 365edo intervals

Approaches