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 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.
In the 2.5.11.13.19.23.29.31 subgroup, it supports a temperament called estates general and defined as 1789 & 3125.
Prime harmonics
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Regular temperament properties
It is notable for being an extremely strong 7-limit system, being the first equal division past 171edo with a lower relative error.
Rank-2 temperaments by generator
| Periods
per octave |
Generator
(reduced) |
Cents
(reduced) |
Associated
ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 1359\3125 | 249.00 | 80275/59392 | Estates general |