300edo
| ← 299edo | 300edo | 301edo → |
300 equal divisions of the octave (abbreviated 300edo or 300ed2), also called 300-tone equal temperament (300tet) or 300 equal temperament (300et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 300 equal parts of exactly 4 ¢ each. Each step represents a frequency ratio of 21/300, or the 300th root of 2.
300edo's step size is called a savart when used as an interval size unit.
Theory
300edo is the largest-number edo which tempers out the Pythagorean comma, 531441/524288, in the patent val.
It is inconsistent to the 5-odd-limit and higher, with three mappings possible for the 5-limit: ⟨300 475 697] (patent val), ⟨300 476 697] (300b), and ⟨300 475 696] (300c).
Using the patent val, it tempers out 531441/524288 and [47 7 -25⟩ in the 5-limit; 6144/6125, 50421/50000, and 1594323/1568000 in the 7-limit.
Using the 300b val, it tempers out 393216/390625 and [51 -38 4⟩ in the 5-limit; 153664/151875, 179200/177147, and 823543/819200 in the 7-limit. Using the 300bd val, it tempers out 10976/10935, 65536/64827, and 390625/388962 in the 7-limit.
Using the 300c val, it tempers out 531441/524288 and [-58 0 25⟩ in the 5-limit; 225/224, 250047/250000, and 69206436005/68719476736 in the 7-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.96 | +1.69 | -0.83 | +0.09 | +0.68 | -0.53 | -0.27 | -0.96 | -1.51 | +1.22 | -0.27 |
| Relative (%) | -48.9 | +42.2 | -20.6 | +2.2 | +17.1 | -13.2 | -6.7 | -23.9 | -37.8 | +30.5 | -6.9 | |
| Steps (reduced) |
475 (175) |
697 (97) |
842 (242) |
951 (51) |
1038 (138) |
1110 (210) |
1172 (272) |
1226 (26) |
1274 (74) |
1318 (118) |
1357 (157) | |
Subsets and supersets
Since 300 factors into 22 × 3 × 52, 300edo has subset edos 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, and 150. 600edo, which doubles it, gives a good correction to its approximation of the 5-limit.