207edo
| ← 206edo | 207edo | 208edo → |
207 equal divisions of the octave (abbreviated 207edo or 207ed2), also called 207-tone equal temperament (207tet) or 207 equal temperament (207et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 207 equal parts of about 5.8 ¢ each. Each step represents a frequency ratio of 21/207, or the 207th root of 2.
Theory
207et tempers out 32805/32768 (schisma) in the 5-limit, 6144/6125 and 19683/19600 in the 7-limit, 441/440 and 43923/43904 in the 11-limit, and 351/350, 676/675, 729/728, 847/845, 1716/1715 in the 13-limit. It serves as a tuning in the 11- and 13-limit for the swetneus temperament. It is significantly more accurate on the 2.3.7.11.13 subgroup, a favorite of many people, and one which contains both 729/728 and 10648/10647, which it tempers out.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.51 | +2.09 | -0.71 | -0.59 | +0.05 | -0.61 | -1.86 | -2.19 | +2.31 | +2.79 |
| Relative (%) | +0.0 | -8.7 | +36.1 | -12.2 | -10.2 | +0.9 | -10.5 | -32.1 | -37.7 | +39.8 | +48.1 | |
| Steps (reduced) |
207 (0) |
328 (121) |
481 (67) |
581 (167) |
716 (95) |
766 (145) |
846 (18) |
879 (51) |
936 (108) |
1006 (178) |
1026 (198) | |
Subsets and supersets
Since 207 factors into 32 × 23, 207edo has subset edos 3, 9, 23, and 69.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-328 207⟩ | [⟨207 328]] | +0.1595 | 0.1596 | 2.75 |
| 2.3.5 | 32805/32768, [2 31 -22⟩ | [⟨207 328 481]] | −0.1942 | 0.5166 | 8.91 |
| 2.3.5.7 | 6144/6125, 19683/19600, 50421/50000 | [⟨207 328 481 581]] | −0.0825 | 0.4874 | 8.41 |
| 2.3.5.7.11 | 441/440, 3388/3375, 6144/6125, 19683/19600 | [⟨207 328 481 581 716]] | −0.0317 | 0.4477 | 7.72 |
| 2.3.5.7.11.13 | 351/350, 441/440, 676/675, 847/845, 3584/3575 | [⟨207 328 481 581 716 766]] | −0.0287 | 0.4087 | 7.05 |
| 2.3.5.7.11.13.17 | 351/350, 441/440, 561/560, 676/675, 847/845, 1089/1088 | [⟨207 328 481 581 716 766 846]] | −0.0034 | 0.3834 | 6.61 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 25\207 | 144.93 | 49/45 | Swetneus |
| 1 | 43\207 | 249.28 | 15/13 | Hemischis |
| 1 | 86\207 | 498.55 | 4/3 | Helmholtz |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct