6000000edo

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 ← 5999999edo 6000000edo 6000001edo →
Prime factorization 27 × 3 × 56
Step size 0.0002¢
Fifth 3509775\6000000 (701.955¢) (→46797\80000)
Semitones (A1:m2) 568425:451125 (113.7¢ : 90.23¢)
Consistency limit 11
Distinct consistency limit 11

The 6 million divisions of the octave, or literal microtonal system, divides the 2/1 into steps of exactly 0.0002 cent, or 200 microcent each.

While this system is absurdly large for any human application, from a technical standpoint it is the only one 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.

This title can be contested by 1000000ed9/8, approx. 5884949edo, if a "tone" is defined to be a just 9/8 interval instead of an octave fraction.

Theory

Approximation of prime harmonics in 6000000edo
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 +43 +47 +29 -31 -5 -8 +26 +3 +14
Steps
(reduced)
6000000
(0)
9509775
(3509775)
13931569
(1931569)
16844130
(4844130)
20756590
(2756590)
22202638
(4202638)
24524777
(524777)
25487565
(1487565)
27141372
(3141372)
29147886
(5147886)
29725178
(5725178)

6'000'000 factorizes as $2^7 \cdot 3 \cdot 5^6$. It has 112 divisors, most notable being 128 and 3125.

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 proveds good approximations for the 2.3.17.19 subgroup, which is typically associated with 12edo.