# 720edo

 ← 719edo 720edo 721edo →
Prime factorization 24 × 32 × 5
Step size 1.66667¢
Fifth 421\720 (701.667¢)
Semitones (A1:m2) 67:55 (111.7¢ : 91.67¢)
Consistency limit 5
Distinct consistency limit 5
Special properties

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 21/720, 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

Approximation of prime harmonics in 720edo
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

Table of rank-2 temperaments by generator
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