9edt
| ← 8edt | 9edt | 10edt → |
9 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 9edt or 9ed3), is a nonoctave tuning system that divides the interval of 3/1 into 9 equal parts of about 211 ¢ each. Each step represents a frequency ratio of 31/9, or the 9th root of 3.
Theory
It has a decent seventh harmonic (7/1) which is 12.4 cents sharp, and an excellent 13/1 inherited from 3edt which is only 2.6 cents flat. However, the 5/1 is 39 cents flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth — or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845 cents, which is between 8/5 and 5/3; if this interval is also taken as an approximation to 13/8, it would temper out 40/39 — making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 subgroup, of which it tempers out 351/343 and 2197/2187.
Following 4edt, this is the next edt that supports BPS temperament. For small edts, this property is virtually the same as supporting a 3/1-equivalent 4L 5s, aka lambda, moment of symmetry scale, of which 9edt offers the "equalized" interpretation of L = s, analogous to 7edo in diatonic (5L 2s) music.
9edt is the third no-twos zeta peak edt.
Relation to edos
9edt is related to 17edo, by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to 3/1.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +68.0 | +0.0 | -75.4 | -39.0 | +68.0 | +12.4 | -7.4 | +0.0 | +28.9 | +75.2 | -75.4 |
| Relative (%) | +32.2 | +0.0 | -35.7 | -18.5 | +32.2 | +5.9 | -3.5 | +0.0 | +13.7 | +35.6 | -35.7 | |
| Steps (reduced) |
6 (6) |
9 (0) |
11 (2) |
13 (4) |
15 (6) |
16 (7) |
17 (8) |
18 (0) |
19 (1) |
20 (2) |
20 (2) | |
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +68.0 | +0.0 | -39.0 | +12.4 | +75.2 | -2.6 | -44.4 | -25.6 | +66.3 | +87.6 | -27.8 |
| Relative (%) | +32.2 | +0.0 | -18.5 | +5.9 | +35.6 | -1.2 | -21.0 | -12.1 | +31.4 | +41.5 | -13.2 | |
| Steps (reduced) |
6 (6) |
9 (0) |
13 (4) |
16 (7) |
20 (2) |
21 (3) |
23 (5) |
24 (6) |
26 (8) |
28 (1) |
28 (1) | |
| Steps | Size | Comparable intervals (in ¢) | |
|---|---|---|---|
| (in ¢) | in hekts | ||
| 0 | 1/1 | ||
| 1 | 211.328 | 144.444 | 9/8 (204) |
| 2 | 422.657 | 288.889 | 9/7 (435) |
| 3 | 633.985 | 433.333 | 13/9 (637) |
| 4 | 845.313 | 577.778 | 13/8 (841), 5/3 (884), 8/5 (814) |
| 5 | 1056.642 | 722.222 | 9/5 (1018), 11/6 (1049) |
| 6 | 1267.970 | 866.667 | 27/13 (1265) |
| 7 | 1479.298 | 1011.111 | 7/3 (1467) |
| 8 | 1690.627 | 1155.556 | 8/3 (1698) |
| 9 | 1901.955 | 1300 | 3/1 |