8edt: Difference between revisions
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{{ED intro}} | |||
As the double of [[ | == 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 [[enfactoring|enfactored]] in the 3.5.7 subgroup. | |||
What it does introduce are flat | 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, which has the side effect of bringing the step size slightly closer to [[8/7]]. | ||
=== Harmonics === | |||
{{Harmonics in equal|8|3|1|}} | |||
{{Harmonics in equal|8|3|1|intervals=prime}} | |||
== Interval table == | |||
{{Interval table}} | |||
[[Category:Macrotonal]] | |||
[[Category: | |||
Latest revision as of 18:42, 8 March 2025
| ← 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, which has the side effect of bringing the step size slightly closer to 8/7.
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 (%) | -4.7 | +0.0 | -9.5 | +28.0 | -4.7 | -17.0 | -14.2 | +0.0 | +23.3 | -46.1 | -9.5 | |
| 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 (%) | -4.7 | +0.0 | +28.0 | -17.0 | -46.1 | +32.2 | +36.9 | -44.1 | +16.8 | +48.0 | -0.6 | |
| 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 | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 237.7 | 162.5 | 7/6, 8/7, 9/8, 15/13, 17/15, 20/17, 22/19 |
| 2 | 475.5 | 325 | 4/3, 9/7, 13/10, 17/13, 21/16 |
| 3 | 713.2 | 487.5 | 3/2, 20/13 |
| 4 | 951 | 650 | 7/4, 12/7, 17/10, 19/11 |
| 5 | 1188.7 | 812.5 | 2/1 |
| 6 | 1426.5 | 975 | 7/3, 9/4, 16/7, 20/9 |
| 7 | 1664.2 | 1137.5 | 8/3, 13/5, 18/7, 21/8 |
| 8 | 1902 | 1300 | 3/1 |