229edo
| ← 228edo | 229edo | 230edo → |
229 equal divisions of the octave (abbreviated 229edo or 229ed2), also called 229-tone equal temperament (229tet) or 229 equal temperament (229et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 229 equal parts of about 5.24 ¢ each. Each step represents a frequency ratio of 21/229, or the 229th root of 2.
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. The equal temperament 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, notably supporting hemiwürschmidt, newt, and trident.
It extends less well to the 13-limit. Using the patent val ⟨229 363 532 643 792 847], it tempers out 351/350, 1573/1568, 2080/2079, and 4096/4095. Using the alternative 229f val ⟨229 363 532 643 792 848], it tempers out 352/351, 729/728, 1001/1000, and 1716/1715.
Higher harmonics like 17, 19, and 23 are well approximated, so it shows great potential in the no-13 23-limit. It tempers out 561/560, 1089/1088, and 1701/1700 in the 17-limit; 476/475, 1216/1215, 1445/1444, and 1540/1539 in the 19-limit; and 484/483, 576/575 and 736/735 in the 23-limit.
The 229b val supports a septimal meantone close to the CTE tuning.
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 (%) | +0.0 | +4.4 | +27.8 | +11.6 | -21.0 | -40.1 | -2.9 | +22.5 | +10.4 | -47.8 | +48.9 | |
| Steps (reduced) |
229 (0) |
363 (134) |
532 (74) |
643 (185) |
792 (105) |
847 (160) |
936 (20) |
973 (57) |
1036 (120) |
1112 (196) |
1135 (219) | |
Subsets and supersets
229edo is the 50th prime edo.
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 |
| 2.3.5.7.11.17 | 561/560, 1089/1088, 1701/1700, 2401/2400, 3136/3125 | [⟨229 363 532 643 792 936]] | −0.106 | 0.288 | 5.50 |
| 2.3.5.7.11.17.19 | 476/475, 561/560, 1089/1088, 1216/1215, 1445/1444, 2401/2400 | [⟨229 363 532 643 792 936 973]] | −0.130 | 0.273 | 5.22 |
| 2.3.5.7.11.17.19.23 | 476/475, 484/483, 561/560, 576/575, 736/735, 1089/1088, 1216/1215 | [⟨229 363 532 643 792 936 973 1036]] | −0.129 | 0.256 | 4.88 |
| 2.3.5.7.11.13 | 351/350, 1573/1568, 2080/2079, 2197/2187, 3136/3125 | [⟨229 363 532 643 792 847]] (229) | −0.017 | 0.384 | 7.32 |
| 2.3.5.7.11.13 | 352/351, 729/728, 1001/1000, 1716/1715, 3025/3024 | [⟨229 363 532 643 792 848]] (229f) | −0.253 | 0.387 | 7.39 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 16\229 | 83.84 | 16807/16000 | Sextilimeans |
| 1 | 19\229 | 99.56 | 18/17 | Quintagar / quinsandra (229) / quinsandric (229) |
| 1 | 37\229 | 193.87 | 28/25 | Didacus / hemiwürschmidt |
| 1 | 67\229 | 351.09 | 49/40 | Newt (229) |
| 1 | 74\229 | 387.77 | 5/4 | Würschmidt (5-limit) |
| 1 | 95\229 | 497.82 | 4/3 | Gary |
| 1 | 75\229 | 503.06 | 147/110 | Quadrawürschmidt |
| 1 | 108\229 | 565.94 | 18/13 | Trident (229) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct