3600edo: Difference between revisions
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{{Infobox ET}} | |||
{{ED intro}} | |||
== | == 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. | ||
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: | |||
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|####]] | |||
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|####]]<!-- 4-digit number --> | |||
[[Category:Ennealimmal]] | |||
Latest revision as of 14:47, 20 February 2025
| ← 3599edo | 3600edo | 3601edo → |
3600 equal divisions of the octave (abbreviated 3600edo or 3600ed2), also called 3600-tone equal temperament (3600tet) or 3600 equal temperament (3600et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3600 equal parts of about 0.333 ¢ each. Each step represents a frequency ratio of 21/3600, or the 3600th root of 2.
Theory
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 supports the ennealimmal temperament, tempering out the ennealimma, [1 -27 18⟩, and (with the patent val) 2401/2400 and 4375/4374 in the 7-limit. Via the 3600e 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.
One step of 3600edo is close to the landscape comma.
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) | |
Subsets and supersets
3600edo factors as 24 × 32 × 52, and has subset 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 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.