# 720edo

← 719edo | 720edo | 721edo → |

^{4}× 3^{2}× 5**720 equal divisions of the octave** (abbreviated **720edo** or **720ed2**), also called **720-tone equal temperament** (**720tet**) or **720 equal temperament** (**720et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 720 equal parts of about 1.67 ¢ each. Each step represents a frequency ratio of 2^{1/720}, or the 720th root of 2.

## Theory

720edo is only consistent to the 5-odd-limit, but it has a reasonable approximation of the full 17-limit using the patent val. It tempers out the schisma in the 5-limit. It supports octant up to the 11-limit and tetraicosic up to the 19-limit.

The patent val can also 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.

### Prime harmonics

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 (%) | +0.0 | -17.3 | +21.2 | -29.6 | +20.9 | -31.7 | +2.7 | +49.2 | +3.5 | +25.4 | -2.1 | |

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

### Subsets and supersets

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 subsets, limited modes of transposition, and fraction-octave mosses. With 720edo, it is 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 ^{[clarification needed ]}.

Since 720 = 72 × 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.

## Regular temperament properties

### Rank-2 temperaments

Periods per 8ve |
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 |

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct