227edo
| ← 226edo | 227edo | 228edo → |
227 equal divisions of the octave (abbreviated 227edo or 227ed2), also called 227-tone equal temperament (227tet) or 227 equal temperament (227et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 227 equal parts of about 5.286 ¢ each. Each step represents a frequency ratio of 21/227, or the 227th root of 2.
Theory
The equal temperament tempers out 15625/15552 (kleisma) and [61 -37 -1⟩ in the 5-limit; 5120/5103, 65625/65536, and 117649/116640 in the 7-limit, so that it supports countercata. In the 11-limit, it tempers out 385/384, 2200/2187, 3388/3375, and 12005/11979, so that it provides the optimal patent val for 11-limit countercata. In the 13-limit, it tempers out 325/324, 352/351, 625/624, 676/675, and 847/845, and again supplies a good tuning for 13-limit countercata, although 140edo tunes it better in this case.
227edo is accurate for the 13th harmonic, as the denominator of a convergent to log213, after 10 and before 5231.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | +1.13 | -0.41 | -1.43 | -1.54 | +0.00 | +0.77 | -1.48 | +0.80 | +1.26 | +2.10 |
| relative (%) | +0 | +21 | -8 | -27 | -29 | +0 | +15 | -28 | +15 | +24 | +40 | |
| Steps (reduced) |
227 (0) |
360 (133) |
527 (73) |
637 (183) |
785 (104) |
840 (159) |
928 (20) |
964 (56) |
1027 (119) |
1103 (195) |
1125 (217) | |
Subsets and supersets
227edo is the 49th prime edo.
Intervals
- Main article: Table of 227edo intervals
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [360 -227⟩ | [⟨227 360]] | -0.3561 | 0.3560 | 6.73 |
| 2.3.5 | 15625/15552, [61 -37 -1⟩ | [⟨227 360 527]] | -0.1785 | 0.3842 | 7.27 |
| 2.3.5.7 | 5120/5103, 15625/15552, 117649/116640 | [⟨227 360 527 637]] | -0.0071 | 0.4461 | 8.44 |
| 2.3.5.7.11 | 385/384, 2200/2187, 3388/3375, 12005/11979 | [⟨227 360 527 637 785]] | +0.0832 | 0.4380 | 8.29 |
| 2.3.5.7.11.13 | 325/324, 352/351, 385/384, 625/624, 12005/11979 | [⟨227 360 527 637 785 840]] | +0.0693 | 0.4010 | 7.59 |
| 2.3.5.7.11.13.17 | 325/324, 352/351, 385/384, 595/594, 625/624, 3185/3179 | [⟨227 360 527 637 785 840 928]] | +0.0324 | 0.3821 | 7.23 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 25\227 | 132.16 | 121/112 | Kastro |
| 1 | 60\227 | 317.18 | 6/5 | Countercata |
| 1 | 94\227 | 496.92 | 4/3 | Undecental |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct