# 4004edo

 ← 4003edo 4004edo 4005edo →
Prime factorization 22 × 7 × 11 × 13
Step size 0.2997¢
Fifth 2342\4004 (701.898¢) (→1171\2002)
Semitones (A1:m2) 378:302 (113.3¢ : 90.51¢)
Consistency limit 5
Distinct consistency limit 5

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

Approximation of prime harmonics in 4004edo
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 (%) +0.0 -19.0 -0.0 +35.1 +43.6 +43.9 -20.1 +29.8 -34.2 -35.6 +39.8
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.