3600edo: Difference between revisions

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~~The '''3600edo''' divides the octave into~~ 3600 ~~equal parts of 0.33333 (~~exactly 1/3~~) of a [[~~cent]] each~~. A cent is therefore three steps; also, the Dröbisch Angle which is 1/360 octave is ten steps~~.

{{EDO intro|3600}}, or exactly 1/3 cent each.

== Theory ==

It also has the advantage of expressing the steps of [[72edo]] in whole numbers. Aside from its relationship to cents, it is of interest as a system [[support|supporting]] [[ennealimmal temperament]], tempering out the ennealimma, {{monzo| 1 -27 18 }}, in the [[5-limit]] and (with the [[patent val]]) 2401/2400 and 4375/4374 in the [[7-limit]]. An alternative 7-limit mapping is 3600d, with the 7 slightly sharp rather than slightly flat; this no longer supports ennealimmal, but it does temper out 52734375/52706752; together with the ennealimma that leads to a sort of strange sibling to ennealimmal temperament, more accurate but also more complex. Via the [[val]] {{val| 3600 5706 8359 10106 12453 13318 }}, 3600edo also supports [[hemiennealimmal temperament]].

~~One step~~ of 3600edo is ~~also close to the~~ [[~~landscape comma~~]].

=== Number description ===

[[Category:Equal divisions of the octave|####]]

3600edo's prime factorization is

<math>3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}</math>. Its [[Number of the divisors|45 divisors]] are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 30, 36, 40, 45, 48, 50, 60, 72, 75, 80, 90, 100, 120, 144, 150, 180, 200, 225, 240, 300, 360, 400, 450, 600, 720, 900, 1200, 1800, 3600.

~~3600edo's prime factorization~~ is

A cent is therefore represented by three steps; and the Dröbisch angle, which is [[360edo|logarithmically 1/360 of the octave]], is ten steps. EDOs corresponding to other notable divisors include [[72edo]], which has found a dissemination in practice and one step of which is represented by 50 steps, and [[200edo]], which holds the continued fraction expansion record for the best perfect fifth and its step is represented by 18 steps.

~~<math>3600~~ = 2^{4} ~~\cdot 3^~~{2} ~~\cdot~~ 5^{2}~~</math>~~

=== Prime harmonics ===

Aside from its relationship to cents, it is of interest as a system [[support|supporting]] [[ennealimmal temperament]], tempering out the ennealimma, {{monzo| 1 -27 18 }}, in the [[5-limit]] and (with the [[patent val]]) 2401/2400 and 4375/4374 in the [[7-limit]]. An alternative 7-limit mapping is 3600d, with the 7 slightly sharp rather than slightly flat; this no longer supports ennealimmal, but it does temper out 52734375/52706752; together with the ennealimma that leads to a sort of strange sibling to ennealimmal temperament, more accurate but also more complex. Via the [[val]] {{val| 3600 5706 8359 10106 12453 13318 }}, 3600edo also supports [[hemiennealimmal temperament]].

~~Its~~ [[~~Number of the divisors|45 divisors~~]] ~~are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 30, 36, 40, 45, 48, 50, 60, 72, 75, 80, 90, 100, 120, 144, 150, 180, 200, 225, 240, 300, 360, 400, 450, 600, 720, 900, 1200, 1800, 3600~~.

One step of 3600edo is close to the [[landscape comma]].

[[Category:Equal divisions of the octave|####]]

[[Category:Equal divisions of the octave|####]]

[[Category:Ennealimmal]]

@@ Line 1: / Line 1: @@
-The '''3600edo''' divides the octave into 3600 equal parts of 0.33333 (exactly 1/3) of a [[cent]] each. A cent is therefore three steps; also, the Dröbisch Angle which is 1/360 octave is ten steps.
+{{EDO intro|3600}}, or exactly 1/3 cent each.
 == Theory ==
-It also has the advantage of expressing the steps of [[72edo]] in whole numbers. Aside from its relationship to cents, it is of interest as a system [[support|supporting]] [[ennealimmal temperament]], tempering out the ennealimma, {{monzo| 1 -27 18 }}, in the [[5-limit]] and (with the [[patent val]]) 2401/2400 and 4375/4374 in the [[7-limit]]. An alternative 7-limit mapping is 3600d, with the 7 slightly sharp rather than slightly flat; this no longer supports ennealimmal, but it does temper out 52734375/52706752; together with the ennealimma that leads to a sort of strange sibling to ennealimmal temperament, more accurate but also more complex. Via the [[val]] {{val| 3600 5706 8359 10106 12453 13318 }}, 3600edo also supports [[hemiennealimmal temperament]].
-One step of 3600edo is also close to the [[landscape comma]].
+=== Number description ===
+[[Category:Equal divisions of the octave|####]]
+edo's prime factorization is
+<math>3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}</math>. Its [[Number of the divisors|45 divisors]] are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 30, 36, 40, 45, 48, 50, 60, 72, 75, 80, 90, 100, 120, 144, 150, 180, 200, 225, 240, 300, 360, 400, 450, 600, 720, 900, 1200, 1800, 3600.
-edo's prime factorization is
+A cent is therefore represented by three steps; and the Dröbisch angle, which is [[360edo|logarithmically 1/360 of the octave]], is ten steps. EDOs corresponding to other notable divisors include [[72edo]], which has found a dissemination in practice and one step of which is represented by 50 steps, and [[200edo]], which holds the continued fraction expansion record for the best perfect fifth and its step is represented by 18 steps.
-<math>3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}</math>
+=== Prime harmonics ===
+{{Harmonics in equal|3600}}
+Aside from its relationship to cents, it is of interest as a system [[support|supporting]] [[ennealimmal temperament]], tempering out the ennealimma, {{monzo| 1 -27 18 }}, in the [[5-limit]] and (with the [[patent val]]) 2401/2400 and 4375/4374 in the [[7-limit]]. An alternative 7-limit mapping is 3600d, with the 7 slightly sharp rather than slightly flat; this no longer supports ennealimmal, but it does temper out 52734375/52706752; together with the ennealimma that leads to a sort of strange sibling to ennealimmal temperament, more accurate but also more complex. Via the [[val]] {{val| 3600 5706 8359 10106 12453 13318 }}, 3600edo also supports [[hemiennealimmal temperament]].
-Its [[Number of the divisors|45 divisors]] are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 30, 36, 40, 45, 48, 50, 60, 72, 75, 80, 90, 100, 120, 144, 150, 180, 200, 225, 240, 300, 360, 400, 450, 600, 720, 900, 1200, 1800, 3600.
+One step of 3600edo is close to the [[landscape comma]].
-[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
+[[Category:Equal divisions of the octave|####]]<!-- 4-digit number -->
 [[Category:Ennealimmal]]