422edo
| ← 421edo | 422edo | 423edo → |
422 equal divisions of the octave (abbreviated 422edo or 422ed2), also called 422-tone equal temperament (422tet) or 422 equal temperament (422et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 422 equal parts of about 2.84 ¢ each. Each step represents a frequency ratio of 21/422, or the 422nd root of 2.
Theory
422edo is a zeta peak edo, though not zeta integral nor zeta gap. It is distinctly consistent through the 27-odd-limit, with harmonics of 3 through 23 all tuned sharp. The equal temperament tempers out the vishnuzma, [23 6 -14⟩ and the countritonic comma, [33 -34 9⟩, in the 5-limit; 4375/4374 and 589824/588245 in the 7-limit; 3025/3024, 5632/5625, and 9801/9800 in the 11-limit; 1716/1715, 2080/2079, and 2200/2197 in the 13-limit; 1156/1155, 1275/1274, and 2431/2430 in the 17-limit; 1216/1215, 1331/1330, 1445/1444, and 2432/2431 in the 19-limit; and 736/735, 1496/1495, and 1863/1862 in the 23-limit. It supports provides the optimal patent vals for gamera in the 7-limit and hemigamera in the 13-limit. Other notable temperaments it supports are vishnu, semisupermajor, and countritonic.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.41 | +0.42 | +0.84 | +0.34 | +1.18 | +0.26 | +1.07 | +0.16 | -0.19 | +0.94 |
| Relative (%) | +0.0 | +14.6 | +14.6 | +29.6 | +12.0 | +41.4 | +9.1 | +37.5 | +5.7 | -6.8 | +32.9 | |
| Steps (reduced) |
422 (0) |
669 (247) |
980 (136) |
1185 (341) |
1460 (194) |
1562 (296) |
1725 (37) |
1793 (105) |
1909 (221) |
2050 (362) |
2091 (403) | |
Subsets and supersets
Since 422 factors into 2 × 211, 422edo has subset edos 2edo and 211edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [669 -422⟩ | [⟨422 669]] | −0.1308 | 0.1308 | 4.60 |
| 2.3.5 | [23 6 -14⟩, [33 -34 9⟩ | [⟨422 669 980]] | −0.1469 | 0.1092 | 3.84 |
| 2.3.5.7 | 4375/4374, 589824/588245, 29360128/29296875 | [⟨422 669 980 1185]] | −0.1852 | 0.1155 | 4.06 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 5632/5625, 589824/588245 | [⟨422 669 980 1185 1460]] | −0.1679 | 0.1090 | 3.83 |
| 2.3.5.7.11.13 | 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625 | [⟨422 669 980 1185 1460 1562]] | −0.1930 | 0.1142 | 4.02 |
| 2.3.5.7.11.13.17 | 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430 | [⟨422 669 980 1185 1460 1562 1725]] | −0.1744 | 0.1151 | 4.05 |
| 2.3.5.7.11.13.17.19 | 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197 | [⟨422 669 980 1185 1460 1562 1725 1793]] | −0.1839 | 0.1106 | 3.89 |
| 2.3.5.7.11.13.17.19.23 | 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715 | [⟨422 669 980 1185 1460 1562 1725 1793 1909]] | −0.1675 | 0.1142 | 4.02 |
- 422et has lower absolute errors than any previous equal temperaments in the 17-, 19-, and 23-limit. In the 17- and 19-limit it beats 400 and is bettered by 460. In the 23-limit it beats 373g and is bettered by 525.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 81\422 | 230.33 | 8/7 | Gamera |
| 1 | 111\422 | 315.64 | 6/5 | Egads |
| 1 | 153\422 | 435.07 | 9/7 | Supermajor |
| 1 | 207\422 | 588.63 | 128/91 | Countritonic |
| 2 | 25\422 | 71.09 | 25/24 | Vishnu / acyuta |
| 2 | 81\422 | 230.33 | 8/7 | Hemigamera |
| 2 | 153\422 (58\422) |
435.07 (164.93) |
9/7 (11/10) |
Semisupermajor |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct