# 8edt

 ← 7edt 8edt 9edt →
Prime factorization 23
Step size 237.744¢
Octave 5\8edt (1188.72¢)
(convergent)
Consistency limit 10
Distinct consistency limit 4

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

Approximation of harmonics in 8edt
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)
Approximation of prime harmonics in 8edt
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 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