9ed5

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← 8ed5 9ed5 10ed5 →
Prime factorization 32
Step size 309.59¢ 
Octave 4\9ed5 (1238.36¢)
(semiconvergent)
Twelfth 6\9ed5 (1857.54¢) (→2\3ed5)
Consistency limit 7
Distinct consistency limit 3

9 equal divisions of the 5th harmonic (abbreviated 9ed5) is a nonoctave tuning system that divides the interval of 5/1 into 9 equal parts of about 310⁠ ⁠¢ each. Each step represents a frequency ratio of 51/9, or the 9th root of 5.

Theory

This tuning tempers out 9/8 in the 3-limit; 25/24 and 27/25 in the 5-limit; 15/14, 35/32, 36/35 and 21/20 in the 7-limit; 11/10, 33/28, and 22/21 in the 11-limit; 13/12, 26/25, and 27/26 in the 13-limit; 17/16, 35/34, and 18/17 in the 17-limit; 19/17 and 19/18 in the 19-limit; 24/23, 25/23, 26/23, and 27/23 in the 23-limit; 29/28, 33/29 and 30/29 in the 29-limit; 31/29, 31/28, 33/31, and 31/30 in the 31-limit; and 37/36, 37/35, 37/34, and 37/32 in the 37-limit.

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 309.6 6/5, 11/9, 13/11, 17/14, 20/17
2 619.2 7/5, 10/7, 13/9, 17/12, 19/13
3 928.8 12/7, 17/10, 19/11, 22/13
4 1238.4 2/1
5 1548 5/2, 12/5, 17/7, 22/9
6 1857.5
7 2167.1 7/2
8 2476.7 17/4, 21/5
9 2786.3 5/1

Harmonics

Approximation of harmonics in 9ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +38 -44 +77 +0 -6 +37 +115 -89 +38 -127 +32
Relative (%) +12.4 -14.3 +24.8 +0.0 -2.0 +11.8 +37.2 -28.7 +12.4 -40.9 +10.4
Steps
(reduced)
4
(4)
6
(6)
8
(8)
9
(0)
10
(1)
11
(2)
12
(3)
12
(3)
13
(4)
13
(4)
14
(5)
Approximation of harmonics in 9ed5
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -106 +75 -44 +153 +48 -50 -144 +77 -8 -88 +144
Relative (%) -34.3 +24.2 -14.3 +49.6 +15.7 -16.3 -46.5 +24.8 -2.5 -28.5 +46.6
Steps
(reduced)
14
(5)
15
(6)
15
(6)
16
(7)
16
(7)
16
(7)
16
(7)
17
(8)
17
(8)
17
(8)
18
(0)