3125edo

The 3125 equal divisions of the octave (3125edo), or the 3125-tone equal temperament (3125tet), 3125 equal temperament (3125et) when viewed from a regular temperament perspective, divides the octave into 5⁵ = 3125 equal parts of exactly 384 millicents each.

Theory

It is notable for being an extremely strong 7-limit system, being the first equal division past 171edo with a lower relative error. It is also distinctly consistent through the 15-odd-limit. A basis for its 7-limit commas is 78125000/78121827, 645700815/645657712 and 281484423828125/281474976710656. In the 11-limit, 151263/151250, 820125/819896, 21437500/21434787 and 117440512/117406179 are tempered out – it should be noted this edo is so far the only one known to have been confirmed as tempering out 117440512/117406179 prior to the independent discovery of this comma's significance as the difference between a stack of five 33/32 quartertones and one 7/6 subminor third. In the 13-limit, 6656/6655, 123201/123200, 140625/140608, 151263/151250 and 1399680/1399489 are all tempered out.

The fact that 3125 = 5⁵ makes curious notations possible based on the symmetric base 5 positional number system, by converting the number to base 5 with digits {-2, -1, 0, 1, 2}. 3125 has subset edos 5, 25, 125, and 625.