365edo: Difference between revisions

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The '''365 equal divisions of the octave''' ('''365edo'''), or the '''365(-tone) equal temperament''' ('''365tet''', '''365et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 365 [[equal]] parts of about 3.29 [[cent]]s each.

== 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 ===

~~365edo does not have good harmonics of 3, 5, 7 and as such could benefit from octave stretching.~~

~~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. In the 11 limit~~, 365edo tempers out [[3025~~/~~3024]]~~, ~~3388~~/~~3375~~, ~~14641/14580. In 13-limit it tempers out~~ 352/351, ~~1001~~/~~1000~~, ~~and 1716~~/~~1715. In the 23-limit it tempers out 595~~/~~594~~, ~~1496~~/~~1495~~, ~~1729~~/~~1725, 3136~~/~~3135 in the 23-limit~~.

=== 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]]''

~~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. Such a temperament eliminates 300/299, 875/874, 1729/1725, 3060/3059, 4235/4232.

== Approaches ==

* [[365edo/Eliora's approach|Eliora's approach]]

@@ Line 1: / Line 1: @@
-The '''365 equal divisions of the octave''' ('''365edo'''), or the '''365(-tone) equal temperament''' ('''365tet''', '''365et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 365 [[equal]] parts of about 3.29 [[cent]]s each.
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
+edo 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}}
-edo does not have good harmonics of 3, 5, 7 and as such could benefit from octave stretching.
-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. In the 11 limit, 365edo tempers out [[3025/3024]], 3388/3375, 14641/14580. In 13-limit it tempers out 352/351, 1001/1000, and 1716/1715. In the 23-limit it tempers out 595/594, 1496/1495, 1729/1725, 3136/3135 in the 23-limit.
+=== 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]]''
-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. Such a temperament eliminates 300/299, 875/874, 1729/1725, 3060/3059, 4235/4232.
+== Approaches ==
+* [[365edo/Eliora's approach|Eliora's approach]]