# 420edo

← 419edo | 420edo | 421edo → |

^{2}× 3 × 5 × 7**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 highly 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 (perfect fifth plus an octave, or tritave) constitutes 666 steps of 420edo. Nice.

Being a highly composite number of steps, 420edo is rich in modulation circles. In addition, of the first 10 prime harmonics, 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, and showing the vibrant and highly composite nature of 420.

420edo can be adapted for use with 2.7.11.13.19.23 subgroup.

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 the 11-limit, it notably tempers out 4000/3993, and in the 13-limit, 10648/10647.

### Prime harmonics

Prime number | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | |
---|---|---|---|---|---|---|---|---|---|---|---|

Error | absolute (¢) | +0.00 | +0.90 | -0.60 | -0.25 | +0.11 | -0.53 | +0.76 | -0.37 | +0.30 | -1.01 |

relative (%) | +0 | +32 | -21 | -9 | +4 | -18 | +27 | -13 | +10 | -35 | |

Steps (reduced) | 420 (0) | 666 (246) | 975 (135) | 1179 (339) | 1453 (193) | 1554 (294) | 1717 (37) | 1784 (104) | 1900 (220) | 2040 (360) |