1783edo
| ← 1782edo | 1783edo | 1784edo → |
1783 equal divisions of the octave (abbreviated 1783edo or 1783ed2), also called 1783-tone equal temperament (1783tet) or 1783 equal temperament (1783et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1783 equal parts of about 0.673 ¢ each. Each step represents a frequency ratio of 21/1783, or the 1783rd root of 2.
1783edo is a very strong 5-limit system, with a lower 5-limit relative error than anything until 2513. It is consistent to the 9-odd-limit, but there is a large relative delta for the harmonic 7. Together with decent approximations to 11, 13, 17, and 19, it makes for a good no-7 19-limit tuning.
In the 5-limit the equal temperament tempers out the monzisma, [54 -37 2⟩; egads, [-36 -52 51⟩; gross, [144 -22 -47⟩; and pirate, [-90 -15 49⟩.
Using the patent val, it tempers out 2460375/2458624 in the 7-limit; 3025/3024 and 180224/180075 in the 11-limit; 1716/1715 and 4096/4095 in the 13-limit. The alternative 1783d val tempers out 4375/4374 in the 7-limit; 41503/41472 and 160083/160000 in the 11-limit; 10648/10647 in the 13-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0080 | +0.0015 | +0.3272 | -0.1121 | +0.0781 | +0.0362 | -0.0369 | +0.3291 | +0.1480 | -0.2235 |
| Relative (%) | +0.0 | +1.2 | +0.2 | +48.6 | -16.7 | +11.6 | +5.4 | -5.5 | +48.9 | +22.0 | -33.2 | |
| Steps (reduced) |
1783 (0) |
2826 (1043) |
4140 (574) |
5006 (1440) |
6168 (819) |
6598 (1249) |
7288 (156) |
7574 (442) |
8066 (934) |
8662 (1530) |
8833 (1701) | |
Subsets and supersets
1783edo is the 276th prime edo. 3566edo, which doubles it, provides a good correction to the approximation of harmonic 7.