253edo
| ← 252edo | 253edo | 254edo → |
(semiconvergent)
253 equal divisions of the octave (abbreviated 253edo or 253ed2), also called 253-tone equal temperament (253tet) or 253 equal temperament (253et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 253 equal parts of about 4.74 ¢ each. Each step represents a frequency ratio of 21/253, or the 253rd root of 2.
Theory
253edo is consistent to the 17-odd-limit, approximating the fifth by 148\253 (0.021284 cents sharper than the just 3/2), and the prime harmonics from 5 to 17 are all slightly flat. As an equal temperament, it tempers out 32805/32768 in the 5-limit; 2401/2400 in the 7-limit; 385/384, 1375/1372 and 4000/3993 in the 11-limit; 325/324, 1575/1573 and 2200/2197 in the 13-limit; 375/374 and 595/594 in the 17-limit. It provides the optimal patent val for the tertiaschis temperament, and a good tuning for the sesquiquartififths temperament in the higher limits.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.02 | -2.12 | -1.24 | -1.12 | -1.00 | -0.61 | +1.30 | -2.19 | -0.33 | -1.95 |
| Relative (%) | +0.0 | +0.4 | -44.8 | -26.1 | -23.6 | -21.1 | -12.8 | +27.4 | -46.1 | -6.9 | -41.2 | |
| Steps (reduced) |
253 (0) |
401 (148) |
587 (81) |
710 (204) |
875 (116) |
936 (177) |
1034 (22) |
1075 (63) |
1144 (132) |
1229 (217) |
1253 (241) | |
Subsets and supersets
Since 253 factors into 11 × 23, and has subset edos 11edo and 23edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [401 -253⟩ | [⟨253 401]] | −0.007 | 0.007 | 0.14 |
| 2.3.5 | 32805/32768, [-4 -37 27⟩ | [⟨253 401 587]] | +0.300 | 0.435 | 9.16 |
| 2.3.5.7 | 2401/2400, 32805/32768, 390625/387072 | [⟨253 401 587 710]] | +0.335 | 0.381 | 8.03 |
| 2.3.5.7.11 | 385/384, 1375/1372, 4000/3993, 19712/19683 | [⟨253 401 587 710 875]] | +0.333 | 0.341 | 7.19 |
| 2.3.5.7.11.13 | 325/324, 385/384, 1375/1372, 1575/1573, 2200/2197 | [⟨253 401 587 710 875 936]] | +0.323 | 0.312 | 6.58 |
| 2.3.5.7.11.13.17 | 325/324, 375/374, 385/384, 595/594, 1275/1274, 2200/2197 | [⟨253 401 587 710 875 936 1034]] | +0.298 | 0.295 | 6.22 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 35\253 | 166.01 | 11/10 | Tertiaschis |
| 1 | 37\253 | 175.49 | 448/405 | Sesquiquartififths |
| 1 | 105\253 | 498.02 | 4/3 | Helmholtz |
| 1 | 123\253 | 583.40 | 7/5 | Cotritone |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Scales
- 63 32 63 63 32: One of many pentic scales available
- 43 43 19 43 43 43 19: Helmholtz[7]
- 41 41 24 41 41 41 24: Meantone[7]
- 35 35 35 35 35 35 35 8: Porcupine[8]
- 33 33 33 11 33 33 33 33 11: "The Hendecapliqued superdiatonic of the Icositriphony"
- 31 31 31 18 31 31 31 31 18: Mavila[9]
- 26 26 15 26 26 26 15 26 26 26 15: Sensi[11]
- 20 20 20 11 20 20 20 20 11 20 20 20 20 11: Ketradektriatoh scale