420edo

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← 419edo420edo421edo →
Prime factorization 22 × 3 × 5 × 7
Step size 2.85714¢
Fifth 246\420 (702.857¢) (→41\70)
Semitones (A1:m2) 42:30 (120¢ : 85.71¢)
Consistency limit 3
Distinct consistency limit 3

420 equal divisions of the octave (420edo), or 420-tone equal temperament (420tet), 420 equal temperament (420et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 420 equal parts of about 2.86 ¢ each.

Theory

420 is a largely composite number, being divisible by all numbers inclusively from 2 to 7. It's other divisors are 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, and 210.

Remarkably, approximation to the third harmonic, which it derives from 70edo, constitutes 666 steps of 420edo. Nice.

Largely composite number theory

Being a largely composite number of steps, 420edo is rich in modulation circles. 420edo is enfactored in the 7-limit, with the same tuning of 3, 5, and 7 as 140edo. The 13th harmonic is also present in 140edo, and ultimately derives from 10edo. The 29th harmonic, while having significantly drifted, has retained its step position from 7edo.

In addition, in the 29-limit, only 11 and 17 have step correspondences coprime with 420. This means that all other approximations are preserved from smaller edos, thus enabling EDO mergers and mashups.

Regular temperament theory

420edo can be adapted for use with 2.5.7.11.13.19.23 subgroup, and it works satisfactorily with the 29-limit as a whole, although due to over 25% error on some harmonics, it's inconsistent. In the 11-limit, it notably tempers out 4000/3993, and in the 13-limit, 10648/10647.

Harmonics

Approximation of odd harmonics in 420edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) +0.90 -0.60 -0.25 -1.05 +0.11 -0.53 +0.30 +0.76 -0.37 +0.65 +0.30
relative (%) +32 -21 -9 -37 +4 -18 +11 +27 -13 +23 +10
Steps
(reduced)
666
(246)
975
(135)
1179
(339)
1331
(71)
1453
(193)
1554
(294)
1641
(381)
1717
(37)
1784
(104)
1845
(165)
1900
(220)