1205edo
1205 equal divisions of the octave (abbreviated 1205edo or 1205ed2), also called 1205-tone equal temperament (1205tet) or 1205 equal temperament (1205et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1205 equal parts of about 0.996 ¢ each. Each step represents a frequency ratio of 21/1205, or the 1205th root of 2.
Theory
1205edo shares the same perfect fifth with 241edo, but the approximation to the higher harmonics are very much improved. It is a strong no-17 higher-limit system, fully consistent to the no-17 31-odd-limit.
As an equal temperament, it tempers out [38 -2 -15⟩ (luna comma) in the 5-limit, 4375/4374 (ragisma) in the 7-limit, supporting and providing a great tuning for lunatic. It further tempers out 117649/117612, 131072/130977, 151263/151250, 234375/234256 in the 11-limit; 2080/2079, 4096/4095, 4225/4224 in the 13-limit; and 3136/3135, 3250/3249, 5776/5775, 6175/6174 in the 2.3.5.7.11.13.19 subgroup.
If intervals of 17 are desired, the sharp-tending 1205g val blends best with the rest, and in fact lends itself as a tuning with a record-low 31-limit TE error. This maps 17/16, 17/10, 17/13, 23/17 and their octave complements inconsistently, and tempers out 2058/2057 and 2500/2499. The less accurate patent val tempers out 1156/1155, 2431/2430 and 2601/2600 instead.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.120 | +0.076 | +0.137 | +0.383 | -0.030 | -0.391 | +0.246 | +0.107 | +0.132 | +0.193 |
| Relative (%) | +0.0 | +12.0 | +7.7 | +13.7 | +38.5 | -3.0 | -39.3 | +24.7 | +10.8 | +13.3 | +19.3 | |
| Steps (reduced) |
1205 (0) |
1910 (705) |
2798 (388) |
3383 (973) |
4169 (554) |
4459 (844) |
4925 (105) |
5119 (299) |
5451 (631) |
5854 (1034) |
5970 (1150) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.390 | +0.149 | +0.350 | -0.278 | -0.144 | +0.413 | +0.460 | +0.361 | -0.443 | +0.260 | -0.056 |
| Relative (%) | -39.1 | +15.0 | +35.1 | -28.0 | -14.4 | +41.5 | +46.2 | +36.3 | -44.5 | +26.1 | -5.6 | |
| Steps (reduced) |
6277 (252) |
6456 (431) |
6539 (514) |
6693 (668) |
6902 (877) |
7089 (1064) |
7147 (1122) |
7310 (80) |
7410 (180) |
7459 (229) |
7596 (366) | |
Subsets and supersets
Since 1205 factors into primes as 5 × 241, 1205edo contains 5edo and 241edo as subset edos.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [38 -2 -15⟩, [26 -75 40⟩ | [⟨1205 1910 2798]] | −0.0361 | 0.0309 | 3.10 |
| 2.3.5.7 | 4375/4374, [-1 4 11 -11⟩, [36 -5 0 -10⟩ | [⟨1205 1910 2798 3383]] | −0.0393 | 0.0273 | 2.74 |
| 2.3.5.7.11 | 4375/4374, 117649/117612, 131072/130977, 234375/234256 | [⟨1205 1910 2798 3383 4169]] | −0.0536 | 0.0376 | 3.78 |
| 2.3.5.7.11.13 | 2080/2079, 4096/4095, 4375/4374, 117649/117612, 225000/224939 | [⟨1205 1910 2798 3383 4169 4459]] | −0.0433 | 0.0413 | 4.15 |
| 2.3.5.7.11.13.19 | 2080/2079, 3136/3135, 3250/3249, 4096/4095, 4375/4374, 225000/224939 | [⟨1205 1910 2798 3383 4169 4459 5119]] | −0.0454 | 0.0386 | 3.88 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 194\1205 | 193.195 | [-38 5 13⟩ | Lunatic |