# 720edo

← 719edo | 720edo | 721edo → |

^{4}× 3^{2}× 5**720 equal divisions of the octave** (**720edo**), or **720-tone equal temperament** (**720tet**), **720 equal temperament** (**720et**) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 720 equal parts of about 1.67 ¢ each.

## Theory

Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | absolute (¢) | +0.000 | -0.288 | +0.353 | -0.493 | +0.349 | -0.528 | +0.045 | +0.820 | +0.059 | +0.423 | -0.036 |

relative (%) | +0 | -17 | +21 | -30 | +21 | -32 | +3 | +49 | +4 | +25 | -2 | |

Steps (reduced) |
720 (0) |
1141 (421) |
1672 (232) |
2021 (581) |
2491 (331) |
2664 (504) |
2943 (63) |
3059 (179) |
3257 (377) |
3498 (618) |
3567 (687) |

720edo is the 14th superabundant EDO, and also the 6th factorial EDO (720 = 1*2*3*4*5*6 = 6!), which means it contains a massive amount of sub-EDOs, limited modes of transposition, and fraction-octave MOSses. With 720edo, it's better to use various vals mimicking smaller EDOs instead of the patent val, because it sounds as if the patent val is *creating* commas, not tempering them out.

### Simple interpretations

Nonetheless, in low-complexity tones, it is consistent in the 2.3.5.11 subgroup and provides satisfactory representation of the 17-limit.

In the 11-limit, it provides the optimal patent val for the octant temperament, period 8. This also means that 720edo tempers out the schisma.

### Highly melodic theory

Since 720 = 72 x 10, its possible to conceptualize it as a superset of 72edo and 10edo, which are interesting in their own right.

However, the patent val's 5/4 of 720edo comes from 90edo, and not 72edo.

### Other

720edo patent val can be thought of as a 2.3.17.23.31.43 subgroup-suited val, because these harmonics have error of less than 1 standard deviaiton away from step. In it, it supports the 195 & 720 temperament, period 15 with comma basis 1377/1376, 19683/19652, 67797/67712, 177147/176824.

## Rank-2 temperaments by generator

Periods
per octave |
Generator | Cents | Associated
ratio |
Temperaments |
---|---|---|---|---|

1 | 421\720 | 701.667 | 4/3 | Helmholtz |

8 | 421\720 (61\720) |
701.667 (101.667) |
4/3 (36/35) |
Octant |

80 | 421\720 (7\720) |
701.667 (11.667) |
4/3 (?) |
Octogintic |

80 | 283\720 (4\720) |
471.667 (6.667) |
130/99 (?) |
Tetraicosic |