700edo
| ← 699edo | 700edo | 701edo → |
700 equal divisions of the octave (abbreviated 700edo or 700ed2), also called 700-tone equal temperament (700tet) or 700 equal temperament (700et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 700 equal parts of about 1.71 ¢ each. Each step represents a frequency ratio of 21/700, or the 700th root of 2.
Theory
700edo is consistent to the 7-odd-limit, but its harmonic 3 is about halfway between its steps. It is strong in the 2.9.15.7.31 subgroup, tempering out 3969/3968, 420175/419904, 10255177611/10240000000 and 2202927104/2197265625. The equal temperament can also be used in the 2.9.5.7.13.17.23 subgroup, tempering out 1225/1224, 46000/45927, 8281/8280, 426496/426465, 31250/31213 and 262395/262144.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.812 | -0.599 | -0.254 | +0.090 | +0.682 | -0.528 | +0.303 | -0.384 | +0.773 | +0.648 | -0.846 |
| Relative (%) | -47.4 | -35.0 | -14.8 | +5.2 | +39.8 | -30.8 | +17.7 | -22.4 | +45.1 | +37.8 | -49.3 | |
| Steps (reduced) |
1109 (409) |
1625 (225) |
1965 (565) |
2219 (119) |
2422 (322) |
2590 (490) |
2735 (635) |
2861 (61) |
2974 (174) |
3075 (275) |
3166 (366) | |
Subsets and supersets
Since 700 factors into 22 × 52 × 7, 700edo has subset edos 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, and 350. 1400edo, which doubles it, gives a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [317 -100⟩ | [⟨700 2219]] | -0.0142 | 0.0142 | 0.83 |
| 2.9.5 | [-65 0 28⟩, [63 -25 7⟩ | [⟨700 2219 1625]] | +0.0766 | 0.1289 | 7.52 |
| 2.9.5.7 | 2100875/2097152, 184528125/184473632, 3500000000/3486784401 | [⟨700 2219 1625 1965]] | +0.0801 | 0.1118 | 6.52 |