4004edo
| ← 4003edo | 4004edo | 4005edo → |
4004 equal divisions of the octave (abbreviated 4004edo or 4004ed2), also called 4004-tone equal temperament (4004tet) or 4004 equal temperament (4004et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 4004 equal parts of about 0.3 ¢ each. Each step represents a frequency ratio of 21/4004, or the 4004th root of 2.
4004edo has an extremely accurate 5/4, as it is a convergent to the approximation of log25. Unfortunately it is consistent only this far, in the 5-odd-limit.
In higher limits, there is a number of representations to be considered. The 2.5.13/11 subgroup is very precise, where 4004edo tempers out 2.5.13/11 [17 -4 -32⟩. Alternately, 2.7.11.13.19 can be considered as an all-sharp system, and 2.3.17.19.23 as an all-flat system. In the 2.3.17, it tempers out [0 49 -19⟩, and in 2.3.5.17, 531441/531250.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.000 | -0.057 | -0.000 | +0.105 | +0.131 | +0.132 | -0.060 | +0.089 | -0.103 | -0.107 | +0.119 |
| relative (%) | +0 | -19 | -0 | +35 | +44 | +44 | -20 | +30 | -34 | -36 | +40 | |
| Steps (reduced) |
4004 (0) |
6346 (2342) |
9297 (1289) |
11241 (3233) |
13852 (1840) |
14817 (2805) |
16366 (350) |
17009 (993) |
18112 (2096) |
19451 (3435) |
19837 (3821) | |
Subsets and supersets
Since 4004edo factors as 22 × 7 × 11 × 13, it has subset edos 1, 2, 4, 7, 11, 13, 14, 22, 26, 28, 44, 52, 77, 91, 143, 154, 182, 286, 308, 364, 572, 1001, 2002.