5ed7/4

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← 4ed7/45ed7/46ed7/4 →
Prime factorization 5 (prime)
Step size 193.765¢ 
Octave 6\5ed7/4 (1162.59¢)
(semiconvergent)
Twelfth 10\5ed7/4 (1937.65¢) (→2\1ed7/4)
Consistency limit 3
Distinct consistency limit 2

5 equal divisions of 7/4 (abbreviated 5ed7/4) is a nonoctave tuning system that divides the interval of 7/4 into 5 equal parts of about 194 ¢ each. Each step represents a frequency ratio of (7/4)1/5, or the 5th root of 7/4.

In a context with octaves, this scale is associated with didacus temperament, where one step is equated to 28/25 and two steps make 5/4, and its extension hemiwurschmidt, where sixteen steps stack to 6/1 and twenty-eight stack to 23/1, hence the extremely accurate 6th and great 23rd harmonics this tuning provides.

Intervals

Steps Cents Approximate Ratios
0 0 1/1
1 193.765 7/6, 8/7, 11/10, 12/11, 13/12, 15/13, 15/14, 17/15, 19/17, 21/19
2 387.53 5/4, 6/5, 14/11, 17/14, 19/15, 21/17
3 581.296 7/5, 10/7, 11/8, 13/9, 15/11, 16/11, 17/12, 18/13, 19/14
4 775.061 3/2, 8/5, 11/7, 17/11, 19/12, 21/13
5 968.826 7/4, 12/7, 17/10, 19/11, 20/11

Harmonics

Approximation of harmonics in 5ed7/4
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -37.4 +35.7 -74.8 -73.6 -1.7 -74.8 +81.5 +71.4 +82.8 -82.2 -39.1
Relative (%) -19.3 +18.4 -38.6 -38.0 -0.9 -38.6 +42.1 +36.8 +42.7 -42.4 -20.2
Steps
(reduced)
6
(1)
10
(0)
12
(2)
14
(4)
16
(1)
17
(2)
19
(4)
20
(0)
21
(1)
21
(1)
22
(2)
Approximation of harmonics in 5ed7/4
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +16.1 +81.5 -37.9 +44.1 -60.8 +34.0 -59.6 +45.3 -39.1 +74.1 -2.8
Relative (%) +8.3 +42.1 -19.6 +22.8 -31.4 +17.5 -30.8 +23.4 -20.2 +38.2 -1.5
Steps
(reduced)
23
(3)
24
(4)
24
(4)
25
(0)
25
(0)
26
(1)
26
(1)
27
(2)
27
(2)
28
(3)
28
(3)