3125edo
| ← 3124edo | 3125edo | 3126edo → |
3125 equal divisions of the octave (abbreviated 3125edo or 3125ed2), also called 3125-tone equal temperament (3125tet) or 3125 equal temperament (3125et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3125 equal parts of exactly 0.384 ¢ each. Each step represents a frequency ratio of 21/3125, or the 3125th root of 2.
Theory
3125edo is 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.
In the 2.5.11.13.19.23.29.31 subgroup, it supports a temperament called estates general, described as 1789 & 3125.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.003 | -0.010 | +0.006 | +0.106 | +0.048 | -0.123 | +0.087 | -0.050 | -0.073 | +0.052 |
| Relative (%) | +0.0 | -0.8 | -2.5 | +1.6 | +27.6 | +12.6 | -32.1 | +22.7 | -13.1 | -19.1 | +13.7 | |
| Steps (reduced) |
3125 (0) |
4953 (1828) |
7256 (1006) |
8773 (2523) |
10811 (1436) |
11564 (2189) |
12773 (273) |
13275 (775) |
14136 (1636) |
15181 (2681) |
15482 (2982) | |
Subsets and supersets
3125 = 55, and as such it is the 5th edo of the form nn. It has subset edos 5, 25, 125, and 625.
Regular temperament properties
3125et 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
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 139\3125 | 53.376 | 33/32 | Prequartismic |
| 1 | 411\3125 | 157.824 | 36756909/33554432 | Hemiegads |
| 1 | 577\3125 | 221.568 | 8388608/7381125 | Fortune |
| 1 | 822\3125 | 315.648 | 6/5 | Egads |
| 1 | 894\3125 | 343.296 | 8000/6561 | Raider |
| 1 | 1359\3125 | 521.856 | 80275/59392 | Estates general |
| 1 | 1412\3125 | 542.208 | 16807/12288 | Revopent |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct