709edo
709 equal divisions of the octave (abbreviated 709edo or 709ed2), also called 709-tone equal temperament (709tet) or 709 equal temperament (709et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 709 equal parts of about 1.69 ¢ each. Each step represents a frequency ratio of 21/709, or the 709th root of 2.
Theory
709edo is only consistent to the 3-odd-limit. It can be used in the 2.3.11.13.17.19 subgroup, tempering out 2432/2431, 34816/34749, 25289/25272, 709631/708588 and 974651392/972714177.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.443 | -0.418 | -0.702 | -0.807 | +0.445 | +0.657 | +0.025 | -0.019 | +0.371 | -0.259 | -0.348 |
| Relative (%) | +26.2 | -24.7 | -41.5 | -47.7 | +26.3 | +38.8 | +1.5 | -1.1 | +21.9 | -15.3 | -20.5 | |
| Steps (reduced) |
1124 (415) |
1646 (228) |
1990 (572) |
2247 (120) |
2453 (326) |
2624 (497) |
2770 (643) |
2898 (62) |
3012 (176) |
3114 (278) |
3207 (371) | |
Subsets and supersets
709edo is the 127th prime EDO. 2127edo, which triples it, gives a good correction to the harmonics 5 and 7.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1124 -709⟩ | [⟨709 1124]] | −0.1397 | 0.1397 | 8.25 |