226edo
| ← 225edo | 226edo | 227edo → |
226 equal divisions of the octave (abbreviated 226edo or 226ed2), also called 226-tone equal temperament (226tet) or 226 equal temperament (226et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 226 equal parts of about 5.31 ¢ each. Each step represents a frequency ratio of 21/226, or the 226th root of 2.
Theory
226edo is closely related to 113edo, but its mapping of harmonic 5 is sharp instead of flat. Unlike 113, 226 is only consistent to the 5-odd-limit. Using the patent val, the equal temperament tempers out 1029/1024 and 19683/19600 in the 7-limit; 243/242, 9801/9800 and notably the quartisma in the 11-limit; and 364/363 and 729/728 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.07 | +1.30 | -2.45 | -2.14 | +0.89 | -1.59 | +0.23 | +1.24 | -0.17 | +1.79 | -1.73 |
| Relative (%) | -20.2 | +24.4 | -46.2 | -40.3 | +16.8 | -29.9 | +4.3 | +23.3 | -3.2 | +33.6 | -32.5 | |
| Steps (reduced) |
358 (132) |
525 (73) |
634 (182) |
716 (38) |
782 (104) |
836 (158) |
883 (205) |
924 (20) |
960 (56) |
993 (89) |
1022 (118) | |
Subsets and supersets
226 factors into 2 × 113, with 2edo and 113edo as its subset edos. 904edo, which quadruples it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [17 1 -8⟩, [-32 29 -6⟩ | [⟨226 358 525]] | +0.0386 | 0.5044 | 9.50 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 73\226 | 387.61 | 5/4 | Würschmidt (5-limit) |
| 1 | 91\226 | 483.19 | 320/243 | Hemiseven (7-limit) |
| 2 | 23\226 | 122.12 | 15/14 | Lagaca |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct