720edo
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. In the 11-limit, it is a tuning for the octant temperament.
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