327edo
| ← 326edo | 327edo | 328edo → |
327 equal divisions of the octave (abbreviated 327edo or 327ed2), also called 327-tone equal temperament (327tet) or 327 equal temperament (327et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 327 equal parts of about 3.67 ¢ each. Each step represents a frequency ratio of 21/327, or the 327th root of 2.
Theory
327edo is consistent to the 7-odd-limit, though it has a reasonable approximation to the full 13-limit in its patent val. The equal temperament tempers out the semicomma in the 5-limit; 16875/16807, 19683/19600, 250047/250000, and 2100875/2097152 in the 7-limit; 540/539, 1375/1372, 3025/3024, 8019/8000, 35937/35840, 46656/46585, 102487/102400, 137781/137500, and 160083/160000 in the 11-limit; and 625/624, 1575/1573, 1716/1715, 2200/2197, 4225/4224, and 10648/10647 in the 13-limit. It supports mirkat, pnict, and the subgroup temperament petrtri.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.04 | -0.99 | -0.02 | +1.59 | -0.86 | -0.16 | +1.64 | +1.47 | -0.27 | -1.06 | -0.75 |
| Relative (%) | -28.3 | -27.0 | -0.5 | +43.5 | -23.4 | -4.4 | +44.7 | +40.0 | -7.2 | -28.8 | -20.5 | |
| Steps (reduced) |
518 (191) |
759 (105) |
918 (264) |
1037 (56) |
1131 (150) |
1210 (229) |
1278 (297) |
1337 (29) |
1389 (81) |
1436 (128) |
1479 (171) | |
Subsets and supersets
Since 327 factors into 3 × 109, 327edo has 3edo and 109edo as its subsets.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-518 327⟩ | [⟨327 518]] | 0.3273 | 0.3274 | 8.92 |
| 2.3.5 | 2109375/2097152, [-20 39 -18⟩ | [⟨327 518 759]] | 0.3608 | 0.2715 | 7.40 |
| 2.3.5.7 | 16875/16807, 19683/19600, 2100875/2097152 | [⟨327 518 759 918]] | 0.2722 | 0.2807 | 7.65 |
| 2.3.5.7.11 | 540/539, 1375/1372, 8019/8000, 2100875/2097152 | [⟨327 518 759 918 1131]] | 0.2674 | 0.2512 | 6.85 |
| 2.3.5.7.11.13 | 540/539, 625/624, 1575/1573, 2200/2197, 8019/8000 | [⟨327 518 759 918 1131 1210]] | 0.2301 | 0.2441 | 6.65 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 74\327 | 271.56 | 75/64 | Orson |
| 3 | 44\327 | 161.47 | 192/175 | Pnict |
| 3 | 50\327 | 183.49 | 10/9 | Mirkat |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct