16ed5

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← 15ed5 16ed5 17ed5 →
Prime factorization 24
Step size 174.145¢ 
Octave 7\16ed5 (1219.01¢)
(semiconvergent)
Twelfth 11\16ed5 (1915.59¢)
(semiconvergent)
Consistency limit 6
Distinct consistency limit 4

16 equal divisions of the 5th harmonic (abbreviated 16ed5) is a nonoctave tuning system that divides the interval of 5/1 into 16 equal parts of about 174⁠ ⁠¢ each. Each step represents a frequency ratio of 51/16, or the 16th root of 5.

Theory

16ed5 is a strong no-2s, no-13s, no-29s 41-limit tuning. Alternatively, it could include 2s, but the equave might need to be compressed to make the 2/1 and 4/1 more in tune.

This tuning tempers out 25/24 in the 5-limit; 21/20, 35/32, and 28/27 in the 7-limit; 33/32 and 35/33 in the 11-limit; 13/12 and 26/25 in the 13-limit; 17/16, 34/33, and 35/34 in the 17-limit; 19/18 and 39/38 in the 19-limit; 23/22 in the 23-limit; 29/28 and 29/27 in the 29-limit; 31/30 in the 31-limit; 37/36, 39/37, and 38/37 in the 37-limit; 41/40 and 42/41 in the 41-limit; 43/40, 43/42 and 43/41 in the 43-limit; and 47/44 in the 47-limit.

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 174.1 10/9, 11/10, 19/17, 21/19, 23/21
2 348.3 11/9, 17/14, 21/17, 23/19
3 522.4 15/11, 19/14, 23/17
4 696.6 3/2
5 870.7 5/3, 18/11, 23/14
6 1044.9 11/6, 20/11
7 1219 2/1
8 1393.2 9/4, 20/9
9 1567.3 5/2, 22/9
10 1741.4 11/4, 19/7
11 1915.6 3/1
12 2089.7 10/3
13 2263.9 11/3
14 2438
15 2612.2 9/2
16 2786.3 5/1

Harmonics

Approximation of prime harmonics in 16ed5
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +19.0 +13.6 +0.0 -60.1 +28.2 -86.9 -28.9 -47.3 -29.8 -82.8 -24.1
Relative (%) +10.9 +7.8 +0.0 -34.5 +16.2 -49.9 -16.6 -27.2 -17.1 -47.5 -13.8
Steps
(reduced)
7
(7)
11
(11)
16
(0)
19
(3)
24
(8)
25
(9)
28
(12)
29
(13)
31
(15)
33
(1)
34
(2)
Approximation of prime harmonics in 16ed5
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) +17.9 +14.3 -68.2 -48.0 -81.9 +80.8 +23.0 +34.8 -65.6 +60.4 -76.3
Relative (%) +10.3 +8.2 -39.1 -27.6 -47.0 +46.4 +13.2 +20.0 -37.7 +34.7 -43.8
Steps
(reduced)
36
(4)
37
(5)
37
(5)
38
(6)
39
(7)
41
(9)
41
(9)
42
(10)
42
(10)
43
(11)
43
(11)