365edo: Difference between revisions
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== Theory == | == Theory == | ||
365edo is [[consistent]] to the [[7-odd-limit]], but both [[harmonic]]s [[3/1|3]] and [[5/1|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 === | |||
{{Harmonics in equal|365}} | {{Harmonics in equal|365}} | ||
=== Subsets and supersets === | |||
Since 365 factors into {{factorization|365}}, 365edo contains [[5edo]] and [[73edo]] as subsets. A step of 365edo is exactly 2 [[Woolhouse unit]]s (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 == | == Approaches == | ||
* [[365edo/Eliora's approach|Eliora's approach]] | * [[365edo/Eliora's approach|Eliora's approach]] | ||
Latest revision as of 23:00, 20 February 2025
| ← 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}.