4004edo

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← 4003edo4004edo4005edo →
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 4004-tone equal temperament (4004tet), 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 of 4004edo 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 -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.