410edo
| ← 409edo | 410edo | 411edo → |
The 410 equal divisions of the octave (410edo), or the 410(-tone) equal temperament (410tet, 410et) when viewed from a regular temperament perspective, is the equal division of the octave into 410 parts of about 2.93 cents each.
Theory
410edo is closely related to 205edo, but the patent val differs on the mappings for 7 and 13. It is contorted in the 5-limit, tempering out 1600000/1594323 (amity comma) and [38 -2 -15⟩ (luna/hemithirds comma), as well as [-29 -11 20⟩ (gammic comma) and [47 -15 -10⟩ (qintosec comma). It tempers out 2401/2400 (breedsma), 4802000/4782969 (canousma), and 48828125/48771072 (neptunisma) in the 7-limit; 5632/5625, 9801/9800, 14641/14580, and 117649/117612 in the 11-limit; 676/675, 1001/1000, 1716/1715, 2080/2079, 4096/4095, and 4225/4224 in the 13-limit.
410edo provides the optimal patent val for the 11- and 13-limit semiluna, hemiluna, and floral temperament, the rank-3 semicanou temperament, and the rank-4 temperament tempering out 14641/14580.
410edo works much better as a no-11 no-13 subgroup temperament, with a sharp tendency to harmonics up to 29. For example, it tempers out 1216/1215, 1225/1224, 1445/1444, and 2500/2499 in the 2.3.5.7.17.19 subgroup.
Possible usage in Georgian music
410edo's fifth is on its 240th step, which is a highly composite number. As such, it supports EDFs which are divisors of 240. In addition, it's perfect fourth is on the 170th step, which while is not highly composite, is still notable to carry a few ED4/3 scales. This can be used to play Kartvelian scales.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.48 | +0.03 | -0.05 | -1.07 | -0.53 | +0.41 | +1.02 | +0.99 | +0.67 | -0.65 |
| Relative (%) | +0.0 | +16.5 | +0.9 | -1.6 | -36.7 | -18.0 | +14.0 | +35.0 | +34.0 | +22.8 | -22.0 | |
| Steps (reduced) |
410 (0) |
650 (240) |
952 (132) |
1151 (331) |
1418 (188) |
1517 (287) |
1676 (36) |
1742 (102) |
1855 (215) |
1992 (352) |
2031 (391) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7 | 2401/2400, 1600000/1594323, 48828125/48771072 | [⟨410 650 952 1151]] | -0.0753 | 0.1332 | 4.55 |
| 2.3.5.7.17 | 1225/1224, 2401/2400, 24576/24565, 295936/295245 | [⟨410 650 952 1151 1676]] | -0.0803 | 0.1196 | 4.09 |
| 2.3.5.7.17.19 | 1216/1215, 1225/1224, 1445/1444, 2401/2400, 24576/24565 | [⟨410 650 952 1151 1676 1742]] | -0.1071 | 0.1245 | 4.25 |
Rank-2 temperaments
Note: 5-limit temperaments supported by 205et are not shown.
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 29\410 | 84.88 | 21/20 | Amicable / amical |
| 1 | 33\410 | 96.59 | 143/135 | Hemiluna |
| 1 | 118\410 | 348.29 | 57344/46875 | Subneutral |
| 1 | 199\410 | 582.44 | 7/5 | Neptune |
| 2 | 29\410 | 84.88 | 21/20 | Floral |
| 2 | 66\410 | 193.17 | 121/108 | Semiluna |
| 2 | 6\410 | 17.56 | 99/98 | Poseidon |
| 10 | 85\410 (3\410) |
248.78 (8.78) |
15/13 (176/175) |
Decoid |