403edo
| ← 402edo | 403edo | 404edo → |
403 equal divisions of the octave (abbreviated 403edo or 403ed2), also called 403-tone equal temperament (403tet) or 403 equal temperament (403et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 403 equal parts of about 2.98 ¢ each. Each step represents a frequency ratio of 21/403, or the 403rd root of 2.
Theory
403edo is only consistent to the 5-odd-limit, since the error of harmonic 7 is quite large. To start with, the 403def val ⟨403 639 936 1132 1395 1492], the 403df val ⟨403 639 936 1132 1394 1492], and the patent val ⟨403 639 936 1131 1394 1491] are worth considering.
The equal temperament tempers out 1600000/1594323 (amity comma) and [70 0 -31⟩ (31-5, or birds comma) in the 5-limit.
Using the 403d val, it tempers out 4375/4374, 5120/5103, and 6144/6125 in the 7-limit, so that it supports septimal amity, the 152 & 251 temperament. Extending it by the 403def val, it tempers out 540/539, 5632/5625, 6250/6237, and 19712/19683 in the 11-limit, supporting 11-limit amity; and 1575/1573, 1716/1715, 2200/2197, 3584/3575 in the 13-limit. Extending it by the alternative 403df val, 1375/1372, 14641/14580 in the 11-limit; 352/351, 847/845, and 2080/2079 in the 13-limit.
Using the patent val, it tempers out 3136/3125, 2100875/2097152, and 78125000/78121827 in the 7-limit; 3025/3024, 3388/3375, 12005/11979, 14641/14580, 42875/42768, and 131072/130977 in the 11-limit; 2080/2079, 4096/4095, 4225/4224, 6656/6655, and 10648/10647 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.77 | +0.78 | -1.08 | -1.43 | -0.45 | -0.83 | -1.42 | -0.74 | +0.25 | -0.31 | +0.01 |
| Relative (%) | +26.0 | +26.3 | -36.4 | -48.0 | -15.1 | -27.7 | -47.7 | -24.8 | +8.5 | -10.4 | +0.5 | |
| Steps (reduced) |
639 (236) |
936 (130) |
1131 (325) |
1277 (68) |
1394 (185) |
1491 (282) |
1574 (365) |
1647 (35) |
1712 (100) |
1770 (158) |
1823 (211) | |
Subsets and supersets
Since 403 factors into 13 × 31, 403edo contains 13edo and 31edo as subsets.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [639 -403⟩ | [⟨403 639]] | -0.2443 | 0.2443 | 8.20 |
| 2.3.5 | 1600000/1594323, [81 -13 -26⟩ | [⟨403 639 936]] | -0.2753 | 0.2042 | 6.86 |
| 2.3.5.7 | 3136/3125, 1600000/1594323, 2100875/2097152 | [⟨403 639 936 1131]] (403) | -0.3751 | 0.2473 | 8.31 |
| 2.3.5.7.11 | 3025/3024, 3136/3125, 12005/11979, 131072/130977 | [⟨403 639 936 1131 1394]] (403) | -0.0621 | 0.3160 | 10.61 |
| 2.3.5.7.11.13 | 2080/2079, 3025/3024, 3136/3125, 4096/4095, 12005/11979 | [⟨403 639 936 1131 1394 1491]] (403) | -0.0146 | 0.3074 | 10.32 |
| 2.3.5.7.11.13.17 | 595/594, 833/832, 1225/1224, 3025/3024, 3136/3125, 4096/4095 | [⟨403 639 936 1131 1394 1491 1647]] (403) | +0.0133 | 0.2927 | 9.83 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 53\403 | 157.82 | 36756909/33554432 | Hemiegads |
| 1 | 106\403 | 315.63 | 6/5 | Egads |
| 1 | 114\403 | 339.45 | 243/200 | Amity (403defff) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct