420edo

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← 419edo 420edo 421edo →
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

Template:EDO intro

Theory

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 in terms of relative interval error, 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.

420edo is good at the 2.5.7.11.13.19.23 subgroup, and has a great potential as a near-just xenharmonic system. It also works satisfactorily with the 29-limit as a whole, though inconsistent. In the 11-limit, it notably tempers out 4000/3993, and in the 13-limit, 10648/10647.

Odd 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 (%) +31.6 -21.0 -8.9 -36.9 +3.9 -18.5 +10.6 +26.6 -13.0 +22.7 +10.4
Steps
(reduced)
666
(246)
975
(135)
1179
(339)
1331
(71)
1453
(193)
1554
(294)
1641
(381)
1717
(37)
1784
(104)
1845
(165)
1900
(220)

Subsets and supersets

420 is a largely composite number, being divisible by 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, and 210. For this reason 420edo is rich in modulation circles.

Trivia

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

Regular temperament properties

Subgroup Comma List Mapping Optimal 8ve
Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5.7.11 2401/2400, 4000/3993, 5120/5103, 15625/15552 [420 666 975 1179 1453]] -0.051 0.278 9.74
2.5.7.11.13.19.23 875/874, 5635/5632, 10241/10240, 12103/12100, 11875/11858, 10985/10976 [420 975 1179 1453 1554 1784 1900]] 0.069 0.104 3.62

Music

Mandrake