3600edo: Difference between revisions

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

== Theory ==

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

3600edo is consistent in the 5-odd-limit and it is a good 2.3.5.11.17.23.31.37.41 subgroup tuning.

3600edo is consistent in the 5-limit and it is a good 2.3.5.11.17.23.31.37.41 subgroup tuning.

In the 5-limit, 3600edo [[support~~|supports~~]] the [[ennealimmal ~~temperament~~]], tempering out the ennealimma, {{monzo| 1 -27 18 }}, and (with the [[patent val]]) 2401/2400 and 4375/4374 in the [[7-limit]]. Via the 3600e [[val]] {{val| 3600 5706 8359 10106 12453}}, 3600edo also supports the [[hemiennealimmal ~~temperament~~]] in the 11-limit.

In the 5-limit, 3600edo [[support]]s the [[ennealimmal]] temperament, tempering out the ennealimma, {{monzo| 1 -27 18 }}, and (with the [[patent val]]) 2401/2400 and 4375/4374 in the [[7-limit]]. Via the 3600e [[val]] {{val| 3600 5706 8359 10106 12453}}, 3600edo also supports the [[hemiennealimmal]] temperament in the 11-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.

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=== Subsets and supersets ===

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

3600edo factors as

3600edo factors as {{Factorization|3600}}, and has subset edos {{EDOs|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}}.

~~<math>3600 = 2^~~{~~4} \cdot 3^~~{2} ~~\cdot 5^{2~~}~~</math>~~, and has subset edos {{EDOs|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}}.

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.

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|3600}}, or exactly 1/3 cent each.
+{{ED intro}}
 == Theory ==
-[[Category:Equal divisions of the octave|####]]
+edo is consistent in the 5-odd-limit and it is a good 2.3.5.11.17.23.31.37.41 subgroup tuning.
-edo is consistent in the 5-limit and it is a good 2.3.5.11.17.23.31.37.41 subgroup tuning.
-In the 5-limit, 3600edo [[support|supports]] the [[ennealimmal temperament]], tempering out the ennealimma, {{monzo| 1 -27 18 }}, and (with the [[patent val]]) 2401/2400 and 4375/4374 in the [[7-limit]]. Via the 3600e [[val]] {{val| 3600 5706 8359 10106 12453}}, 3600edo also supports the [[hemiennealimmal temperament]] in the 11-limit.
+In the 5-limit, 3600edo [[support]]s the [[ennealimmal]] temperament, tempering out the ennealimma, {{monzo| 1 -27 18 }}, and (with the [[patent val]]) 2401/2400 and 4375/4374 in the [[7-limit]]. Via the 3600e [[val]] {{val| 3600 5706 8359 10106 12453}}, 3600edo also supports the [[hemiennealimmal]] temperament in the 11-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.
@@ Line 16: / Line 15: @@
 === Subsets and supersets ===
 [[Category:Equal divisions of the octave|####]]
-edo factors as
+edo factors as {{Factorization|3600}}, and has subset edos {{EDOs|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}}.
-<math>3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}</math>, and has subset edos {{EDOs|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}}.
 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.