248edo
| ← 247edo | 248edo | 249edo → |
248 equal divisions of the octave (abbreviated 248edo or 248ed2), also called 248-tone equal temperament (248tet) or 248 equal temperament (248et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 248 equal parts of about 4.84 ¢ each. Each step represents a frequency ratio of 21/248, or the 248th root of 2.
Theory
248edo shares the mapping of harmonics 5 and 7 with 31edo. It has a decent 13-limit interpretation despite not being consistent. The equal temperament tempers out 32805/32768 in the 5-limit; 3136/3125 and 420175/419904 in the 7-limit; 441/440, 8019/8000 in the 11-limit; 729/728, 847/845, 1001/1000, 1575/1573 and 2200/2197 in the 13-limit. It also notably tempers out the quartisma. 248edo, additionally, has the interesting property of its mapping for all prime harmonics 3 to 23 being a multiple of 3, and therefore derived from 131edt.
It supports the bischismic temperament, providing the optimal patent val for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for the essence temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.34 | +0.78 | -1.08 | +0.29 | +1.41 | +1.50 | -2.35 | +0.76 | +1.07 | +1.74 |
| Relative (%) | +0.0 | -7.1 | +16.2 | -22.4 | +6.1 | +29.1 | +30.9 | -48.6 | +15.7 | +22.1 | +35.9 | |
| Steps (reduced) |
248 (0) |
393 (145) |
576 (80) |
696 (200) |
858 (114) |
918 (174) |
1014 (22) |
1053 (61) |
1122 (130) |
1205 (213) |
1229 (237) | |
Subsets and supersets
Since 248 factors into 23 × 31, 248edo has subset edos 2, 4, 8, 31, 62, and 124.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [287 -181⟩ | [⟨248 393]] | +0.108 | 0.108 | 2.23 |
| 2.3.5 | 32805/32768, [12 32 -27⟩ | [⟨248 393 576]] | -0.041 | 0.228 | 4.70 |
| 2.3.5.7 | 3136/3125, 32805/32768, 420175/419904 | [⟨248 393 576 696]] | +0.066 | 0.270 | 5.58 |
| 2.3.5.7.11 | 441/440, 3136/3125, 8019/8000, 41503/41472 | [⟨248 393 576 696 858]] | +0.036 | 0.249 | 5.15 |
| 2.3.5.7.11.13 | 441/440, 729/728, 847/845, 1001/1000, 3136/3125 | [⟨248 393 576 696 858 918]] | +0.079 | 0.275 | 5.69 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 5\248 | 24.19 | 686/675 | Sengagen |
| 1 | 103\248 | 498.39 | 4/3 | Helmholtz |
| 2 | 77\248 (47\248) |
372.58 (227.42) |
26/21 (154/135) |
Essence |
| 2 | 103\248 | 498.39 | 4/3 | Bischismic |
| 8 | 117\248 (7\248) |
566.13 (33.87) |
104/75 (49/48) |
Octowerck |
| 31 | 103\248 (1\248) |
498.39 (4.84) |
4/3 (385/384) |
Birds |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct