8edt
| ← 7edt | 8edt | 9edt → |
(convergent)
8 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 8edt or 8ed3), is a nonoctave tuning system that divides the interval of 3/1 into 8 equal parts of about 238 ¢ each. Each step represents a frequency ratio of 31/8, or the 8th root of 3.
Theory
As the double of 4edt, it is the analog of 10edo being the double of 5edo. However, the full 3:5:7 triad is already present in 4edt which is unlike the situation in 10edo where 4:5:6 gains a new better approximation than the sus4 triad in 5edo. More precisely, 8edt is enfactored in the 3.5.7 subgroup.
What it does introduce are flat 2/1 pseudo-octaves and sharp 3/2 perfect fifths, making it related to 5edo melodically. It is equivalent to 5edo with the 3/1 made just, by compressing the octave by 11.3 cents.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -11 | +0 | -23 | +67 | -11 | -40 | -34 | +0 | +55 | -110 | -23 |
| relative (%) | -5 | +0 | -9 | +28 | -5 | -17 | -14 | +0 | +23 | -46 | -9 | |
| Steps (reduced) |
5 (5) |
8 (0) |
10 (2) |
12 (4) |
13 (5) |
14 (6) |
15 (7) |
16 (0) |
17 (1) |
17 (1) |
18 (2) | |
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -11 | +0 | +67 | -40 | -110 | +77 | +88 | -105 | +40 | +114 | -1 |
| relative (%) | -5 | +0 | +28 | -17 | -46 | +32 | +37 | -44 | +17 | +48 | -1 | |
| Steps (reduced) |
5 (5) |
8 (0) |
12 (4) |
14 (6) |
17 (1) |
19 (3) |
21 (5) |
21 (5) |
23 (7) |
25 (1) |
25 (1) | |
Interval table
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1, 33/32, 36/35, 40/39, 49/48, 55/54, 56/55, 64/63, 81/80 |
| 1 | 237.744 | 7/6, 8/7, 9/8, 10/9, 15/13, 32/27, 33/28, 39/35, 49/44, 52/45, 55/48, 55/49, 63/55, 64/55, 81/70 |
| 2 | 475.489 | 4/3, 9/7, 13/10, 14/11, 21/16, 27/20, 35/26, 35/27, 49/36, 50/39, 55/42, 64/49, 66/49, 72/55 |
| 3 | 713.233 | 3/2, 14/9, 20/13, 32/21, 39/25, 40/27, 49/32, 49/33, 52/35, 54/35, 55/36, 72/49, 81/52, 81/55 |
| 4 | 950.978 | 7/4, 12/7, 16/9, 26/15, 27/16, 45/26, 55/32, 56/33 |
| 5 | 1188.72 | 2/1, 25/13, 27/14, 35/18, 39/20, 49/24, 52/27, 55/27, 55/28, 63/32, 64/33, 80/39, 81/40 |
| 6 | 1426.47 | 7/3, 9/4, 16/7, 20/9, 30/13, 33/14, 49/22, 55/24, 78/35, 81/35 |
| 7 | 1664.21 | 8/3, 13/5, 18/7, 21/8, 27/10, 28/11, 35/13, 55/21, 70/27, 81/32 |
| 8 | 1901.96 | 3/1, 32/11, 35/12, 40/13, 49/16, 55/18, 64/21, 80/27 |