6000000edo
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| ← 5999999edo | 6000000edo | 6000001edo → |
The 6 million equal divisions of the octave, or literal microtonal system, divides the octave into steps of exactly 0.0002 cent, or 200 microcents each.
While this system is absurdly large for any human application, from a technical standpoint it is one of the few that rightfully bears the name microtonal, since it divides a tone, which is 1/6th of an octave, into 1'000'000 steps, and the SI prefix "micro-" denotes division of a unit into 1 million parts. If we instead take a tone to be 9/8, we get approximately 5884949edo, which has an almost perfectly off 3/2 so that its double, 11769898edo, may be reasonable.
If the starting note is Middle C, one step of this tuning would have a beating period of 33086 seconds, or more than 9 hours. At 20 kHz, the extreme end of human range, the period of the wahwah caused by two similar frequencies is still more than 7 minutes. In order to hypothetically make any use of this system, humans would have to hear sounds as high pitched as 140 MHz. At this point, it is already just 36 air molecules' mean free path's width.
Remarkably, it has a perfect fifth that differs from just by 1 part in 231, which ultimately derives from 80000edo. In a twist of irony, it provides good approximations for the 2.3.17.19 subgroup, which is typically associated with 12edo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000000 | -0.0000009 | +0.0000861 | +0.0000935 | +0.0000576 | -0.0000618 | -0.0000095 | -0.0000161 | +0.0000527 | +0.0000058 | +0.0000275 |
| Relative (%) | +0.0 | -0.4 | +43.1 | +46.8 | +28.8 | -30.9 | -4.8 | -8.1 | +26.4 | +2.9 | +13.8 | |
| Steps (reduced) |
6000000 (0) |
9509775 (3509775) |
13931569 (1931569) |
16844130 (4844130) |
20756590 (2756590) |
22202638 (4202638) |
24524777 (524777) |
25487565 (1487565) |
27141372 (3141372) |
29147886 (5147886) |
29725178 (5725178) | |
Subsets and supersets
6'000'000 factorizes as 27 × 3 × 56. It has 112 divisors, most notable being 128 and 3125.