1429edo
| ← 1428edo | 1429edo | 1430edo → |
1429 equal divisions of the octave (abbreviated 1429edo or 1429ed2), also called 1429-tone equal temperament (1429tet) or 1429 equal temperament (1429et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1429 equal parts of about 0.84 ¢ each. Each step represents a frequency ratio of 21/1429, or the 1429th root of 2.
Theory
1429edo has a reasonable approximation of the full 17-limit. It is consistent to the 9-odd-limit with only 11/10 barely missing the line. The 11-limit optimal tuning of the equal temperament is consistent to the 18-integer-limit; however, the 13- and 17-limit optimal tunings, which have less of octave compression, are not, so one might want to keep the compression tight.
The equal temperament tempers out 4375/4374 in the 7-limit; 131072/130977, 759375/758912, 1953125/1951488, 2359296/2358125, 2657205/2656192, and 3294225/3294172 in the 11-limit; 2080/2079, 4096/4095, 4225/4224, 78125/78078, and 123201/123200 in the 13-limit; 2500/2499, 5832/5831, 11016/11011, and 12376/12375 in the 17-limit. It supports the gross temperament and provides the optimal patent val for the 11- and 13-limit trillium temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.074 | -0.030 | +0.243 | +0.397 | +0.060 | +0.013 | -0.242 | -0.143 | -0.046 | +0.381 |
| Relative (%) | +0.0 | +8.9 | -3.5 | +29.0 | +47.2 | +7.2 | +1.6 | -28.8 | -17.0 | -5.5 | +45.3 | |
| Steps (reduced) |
1429 (0) |
2265 (836) |
3318 (460) |
4012 (1154) |
4944 (657) |
5288 (1001) |
5841 (125) |
6070 (354) |
6464 (748) |
6942 (1226) |
7080 (1364) | |
Subsets and supersets
1429edo is the 226th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [2265 -1429⟩ | [⟨1429 2265]] | −0.0235 | 0.0234 | 2.80 |
| 2.3.5 | [39 -29 3⟩, [-66 -36 53⟩ | [⟨1429 2265 3318]] | −0.0114 | 0.0257 | 3.06 |
| 2.3.5.7 | 4375/4374, [26 4 -3 -14⟩, [40 -22 -1 -1⟩ | [⟨1429 2265 3318 4012]] | −0.0302 | 0.0395 | 4.70 |
| 2.3.5.7.11 | 4375/4374, 131072/130977, 759375/758912, 3294225/3294172 | [⟨1429 2265 3318 4012 4944]] | −0.0471 | 0.0488 | 5.81 |
| 2.3.5.7.11.13 | 2080/2079, 4096/4095, 4375/4374, 78125/78078, 3294225/3294172 | [⟨1429 2265 3318 4012 4944 5288]] | −0.0420 | 0.0460 | 5.48 |
| 2.3.5.7.11.13.17 | 2080/2079, 2500/2499, 4096/4095, 4375/4374, 11016/11011, 108086/108045 | [⟨1429 2265 3318 4012 4944 5288 5841]] | −0.0364 | 0.0447 | 5.32 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 109\1429 | 91.533 | [144 -22 -47⟩ | Gross |
| 1 | 674\1429 | 565.990 | 25/18 | Trillium |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct