1612edo
| ← 1611edo | 1612edo | 1613edo → |
1612 equal divisions of the octave (abbreviated 1612edo or 1612ed2), also called 1612-tone equal temperament (1612tet) or 1612 equal temperament (1612et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1612 equal parts of about 0.744 ¢ each. Each step represents a frequency ratio of 21/1612, or the 1612th root of 2.
1612edo is a strong 5-limit system, but it is only consistent to the 5-odd-limit since harmonics 7 and 11 are about halfway between its steps. Nonetheless, the patent val is a strong 2.3.5.13.17.23.29.31 subgroup tuning.
It provides a tuning for quasithird, aluminium and counterorson temperaments in the 5-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.030 | +0.039 | -0.340 | +0.295 | -0.081 | +0.007 | +0.254 | +0.013 | -0.049 | -0.122 |
| Relative (%) | +0.0 | +4.0 | +5.2 | -45.6 | +39.6 | -10.9 | +1.0 | +34.1 | +1.8 | -6.5 | -16.4 | |
| Steps (reduced) |
1612 (0) |
2555 (943) |
3743 (519) |
4525 (1301) |
5577 (741) |
5965 (1129) |
6589 (141) |
6848 (400) |
7292 (844) |
7831 (1383) |
7986 (1538) | |
Subsets and supersets
Since 1612 factors into 22 × 13 × 31, 1612edo has subset edos 2, 4, 13, 26, 31, 52, 62, 124, 403, and 806. 3224edo, which doubles 1612edo, corrects the mapping for 7 and 11.