720edo

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← 719edo720edo721edo →
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

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

Approximation of prime harmonics in 720edo
Harmonic 2 3 5 7 11 13 17 19 23 29
Error absolute (¢) +0.000 -0.288 +0.353 -0.493 +0.349 -0.528 +0.045 +0.820 +0.059 +0.423
relative (%) +0 -17 +21 -30 +21 -32 +3 +49 +4 +25
Steps
(reduced)
720
(0)
1141
(421)
1672
(232)
2021
(581)
2491
(331)
2664
(504)
2943
(63)
3059
(179)
3257
(377)
3498
(618)

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 3/2 Helmholtz
8 421\720
(61\720)
701.667
(101.667)
3/2
(?)
Octant
80 421\720
(7\720)
701.667
(11.667)
3/2
(?)
Octogintic