229edo
| ← 228edo | 229edo | 230edo → |
The 229 equal divisions of the octave (229edo), or the 229(-tone) equal temperament (229tet, 229et), is the equal division of the octave into 229 parts of about 5.24 cents each.
Theory
While not highly accurate for its size, 229edo is the point where a few important temperaments meet, and is distinctly consistent in the 11-odd-limit. It tempers out 393216/390625 (würschmidt comma) and [39 -29 3⟩ (tricot comma) in the 5-limit; 2401/2400, 3136/3125, 6144/6125, and 14348907/14336000 in the 7-limit; 3025/3024, 3388/3375, 8019/8000, 14641/14580 and 15488/15435 in the 11-limit, and using the patent val, 351/350, 1573/1568, 2080/2079, and 4096/4095 in the 13-limit, notably supporting hemiwürschmidt, newt, and trident.
The 229b val supports a septimal meantone close to the CTE tuning.
229edo is the 50th prime EDO.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | +0.23 | +1.46 | +0.61 | -1.10 | -2.10 | -0.15 | +1.18 | +0.55 | -2.50 | +2.56 |
| relative (%) | +0 | +4 | +28 | +12 | -21 | -40 | -3 | +22 | +10 | -48 | +49 | |
| Steps (reduced) |
229 (0) |
363 (134) |
532 (74) |
643 (185) |
792 (105) |
847 (160) |
936 (20) |
973 (57) |
1036 (120) |
1112 (196) |
1135 (219) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [363 -229⟩ | [⟨229 363]] | -0.072 | 0.072 | 1.38 |
| 2.3.5 | 393216/390625, [39 -29 3⟩ | [⟨229 363 532]] | -0.258 | 0.269 | 5.13 |
| 2.3.5.7 | 2401/2400, 3136/3125, 14348907/14336000 | [⟨229 363 532 643]] | -0.247 | 0.233 | 4.46 |
| 2.3.5.7.11 | 2401/2400, 3025/3024, 3136/3125, 8019/8000 | [⟨229 363 532 643 792]] | -0.134 | 0.308 | 5.87 |
| 2.3.5.7.11.13 | 351/350, 1573/1568, 2080/2079, 3136/3125, 4096/4095 | [⟨229 363 532 643 792 847]] | -0.017 | 0.384 | 7.32 |
| 2.3.5.7.11.13.17 | 351/350, 442/441, 561/560, 715/714, 3136/3125, 4096/4095 | [⟨229 363 532 643 792 847 936]] | -0.009 | 0.356 | 6.79 |
| 2.3.5.7.11.13.17.19 | 286/285, 351/350, 442/441, 476/475, 561/560, 1216/1215, 1729/1728 | [⟨229 363 532 643 792 847 936 973]] | -0.043 | 0.344 | 6.57 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 16\229 | 83.84 | 16807/16000 | Sextilimeans |
| 1 | 19\229 | 99.56 | 18/17 | Quintagar / quinsandra / quinsandric |
| 1 | 37\229 | 193.87 | 28/25 | Didacus / hemiwürschmidt |
| 1 | 67\229 | 351.09 | 49/40 | Newt |
| 1 | 74\229 | 387.77 | 5/4 | Würschmidt |
| 1 | 95\229 | 497.82 | 4/3 | Gary |
| 1 | 75\229 | 503.06 | 147/110 | Quadrawürschmidt |
| 1 | 108\229 | 565.94 | 18/13 | Tricot / trident |