810edo
| ← 809edo | 810edo | 811edo → |
810 equal divisions of the octave (abbreviated 810edo or 810ed2), also called 810-tone equal temperament (810tet) or 810 equal temperament (810et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 810 equal parts of about 1.48 ¢ each. Each step represents a frequency ratio of 21/810, or the 810th root of 2.
Theory
810 = 270 × 3, and 810edo has three copies of 270edo in the 13-limit (and the 2.3.5.7.11.13.19 subgroup). It makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, only consistent to the 9-odd-limit. 11/9, 13/12, 13/9, 13/10, and their octave complements are all mapped inconsistently in this edo.
As an equal temperament, it tempers out 4914/4913 in the 17-limit; and 2024/2023, 2737/2736, and 3520/3519 in the 23-limit. Although it does quite well in these limits, it is way less efficient than 270edo's or 540edo's mappings, as it has greater relative errors (→ #Regular temperament properties). It is therefore a question of whether one thinks these tuning improvements and differently supplied essentially tempered chords are worth the load of all the extra notes.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.267 | +0.353 | +0.063 | -0.207 | -0.528 | +0.230 | +0.265 | -0.126 | +0.052 | +0.150 | +0.508 | +0.567 | -0.407 | -0.321 |
| Relative (%) | +0.0 | +18.0 | +23.8 | +4.3 | -14.0 | -35.6 | +15.5 | +17.9 | -8.5 | +3.5 | +10.1 | +34.3 | +38.3 | -27.4 | -21.7 | |
| Steps (reduced) |
810 (0) |
1284 (474) |
1881 (261) |
2274 (654) |
2802 (372) |
2997 (567) |
3311 (71) |
3441 (201) |
3664 (424) |
3935 (695) |
4013 (773) |
4220 (170) |
4340 (290) |
4395 (345) |
4499 (449) | |
Subsets and supersets
Since 810 factors into 2 × 34 × 5, 810edo has subset edos 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7.11.13.17 | 676/675, 1001/1000, 1716/1715, 3025/3024, 4096/4095, 4914/4913 | [⟨810 1284 1881 2274 2802 2997 3311]] | -0.0281 | 0.1025 | 6.92 |
| 2.3.5.7.11.13.17.19 | 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728, 4914/4913 | [⟨810 1284 1881 2274 2802 2997 3311 3441]] | -0.0324 | 0.0966 | 6.52 |
| 2.3.5.7.11.13.17.19.23 | 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728, 2024/2023, 2737/2736 | [⟨810 1284 1881 2274 2802 2997 3311 3441 3664]] | -0.0257 | 0.0930 | 6.28 |