3600edo: Difference between revisions
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{{EDO intro|3600}}, or exactly 1/3 cent each. | {{EDO intro|3600}}, or exactly 1/3 cent each. | ||
Revision as of 22:17, 4 October 2022
| ← 3599edo | 3600edo | 3601edo → |
Template:EDO intro, or exactly 1/3 cent each.
Theory
Number description
3600edo's prime factorization is [math]\displaystyle{ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} }[/math]. Its 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 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.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.045 | +0.020 | -0.159 | +0.015 | +0.139 | +0.045 | +0.154 | +0.059 | +0.089 | -0.036 |
| Relative (%) | +0.0 | +13.5 | +5.9 | -47.8 | +4.6 | +41.7 | +13.4 | +46.1 | +17.7 | +26.8 | -10.7 | |
| Steps (reduced) |
3600 (0) |
5706 (2106) |
8359 (1159) |
10106 (2906) |
12454 (1654) |
13322 (2522) |
14715 (315) |
15293 (893) |
16285 (1885) |
17489 (3089) |
17835 (3435) | |
Aside from its relationship to cents, it is of interest as a system supporting ennealimmal temperament, tempering out the ennealimma, [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 ⟨3600 5706 8359 10106 12453 13318], 3600edo also supports hemiennealimmal temperament.
One step of 3600edo is close to the landscape comma.