3600edo: Difference between revisions

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The '''3600odo''' 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. 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 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]].~~

== ~~Divisors~~ ==

== Theory ==

~~The prime factorization~~ is

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

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

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

~~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~~.

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.

[[Category:Equal divisions of the octave]]

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:~~ennealimmal~~]]

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

=== Prime harmonics ===

=== 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}}.

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]]

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-The '''3600odo''' 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. 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 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]].
+{{Infobox ET}}
+{{ED intro}}
-== Divisors ==
+== Theory ==
-The prime factorization is
+[[Category:Equal divisions of the octave|####]]
-<math>3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}</math>
+edo is consistent in the 5-limit and it is a good 2.3.5.11.17.23.31.37.41 subgroup tuning.
-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.
+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.
-[[Category:Equal divisions of the octave]]
+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:ennealimmal]]
+One step of 3600edo is close to the [[landscape comma]].
+=== Prime harmonics ===
+{{Harmonics in equal|3600}}
+=== 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}}.
+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]]