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-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.

~~=== Number description ===~~

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.

~~[[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~~.

~~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.

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

=== 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]]~~.

=== Subsets and supersets ===

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

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}}.

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

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.

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

[[Category:Ennealimmal]]

@@ 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-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.
-=== Number description ===
+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.
-[[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.
-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.
+One step of 3600edo is close to the [[landscape comma]].
 === 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]].
+=== Subsets and supersets ===
+[[Category:Equal divisions of the octave|####]]
+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}}.
-One step of 3600edo is close to the [[landscape comma]].
+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.
 [[Category:Equal divisions of the octave|####]]<!-- 4-digit number -->
 [[Category:Ennealimmal]]