410edo
| ← 409edo | 410edo | 411edo → |
410 equal divisions of the octave (abbreviated 410edo or 410ed2), also called 410-tone equal temperament (410tet) or 410 equal temperament (410et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 410 equal parts of about 2.93 ¢ each. Each step represents a frequency ratio of 21/410, or the 410th root of 2.
Theory
410edo is enfactored in the 5-limit, with the same tuning as 205edo characterized by 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⟩ (quintosec comma), but the approximations to harmonics 7 and 13 are much improved. The equal temperament 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 temperaments, 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.
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) | |
Subsets and supersets
Since 410 factors into 2 × 5 × 41, 410edo has subset edos 2, 5, 10, 41, 82, and 205. Meanwhile, as every sixth step of 2460edo, a step of 410edo is exactly 6 minas.
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 8ve |
Generator* | Cents* | 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 |
| 41 | 61\410 (1\410) |
178.54 (2.93) |
567/512 (352/351) |
Hemicountercomp |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Scales
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, its 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.
- Kartvelian Tetratonic: 120 120 85 85 (simplifies to 82edo)
- Kartvelian Decatonic: 48 48 48 48 48 34 34 34 34 34 (simplifies to 205edo)
- Kartvelian 22-tonic: 20 20 20 20 20 20 20 20 20 20 20 20 17 17 17 17 17 17 17 17 17
Music
- Mercury Amalgam (2023)
- All Time Best – decoid[40], cover of Phlub