# 420edo

← 419edo | 420edo | 421edo → |

^{2}× 3 × 5 × 7**420 equal divisions of the octave** (abbreviated **420edo** or **420ed2**), also called **420-tone equal temperament** (**420tet**) or **420 equal temperament** (**420et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 420 equal parts of about 2.86 ¢ each. Each step represents a frequency ratio of 2^{1/420}, or the 420th root of 2.

## Theory

420edo is enfactored in the 7-limit, with the same tuning of harmonics 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 the harmonics 11 and 17 have step numbers 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

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, its nontrivial subset edos being 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.

## 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

*Follow In Is*(2022) – a superset of 12edo, 5edo, and 7edo, least common multiple of which is 420edo.