40ed5

← 39ed5

40ed5

41ed5 →

Prime factorization	23 × 5
Step size	69.6578 ¢
Octave	17\40ed5 (1184.18 ¢)
Twelfth	27\40ed5 (1880.76 ¢)
Consistency limit	5
Distinct consistency limit	5

40 equal divisions of the 5th harmonic (abbreviated 40ed5) is a nonoctave tuning system that divides the interval of 5/1 into 40 equal parts of about 69.7 ¢ each. Each step represents a frequency ratio of 5^1/40, or the 40th root of 5.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	69.7	26/25, 28/27
2	139.3	25/23
3	209	9/8, 26/23
4	278.6	20/17
5	348.3
6	417.9
7	487.6
8	557.3
9	626.9
10	696.6	3/2
11	766.2	14/9
12	835.9
13	905.6	22/13, 27/16
14	975.2	7/4
15	1044.9
16	1114.5	19/10
17	1184.2
18	1253.8
19	1323.5	15/7
20	1393.2	29/13
21	1462.8	7/3
22	1532.5	17/7
23	1602.1
24	1671.8	21/8, 29/11
25	1741.4
26	1811.1	20/7
27	1880.8
28	1950.4
29	2020.1
30	2089.7	10/3
31	2159.4
32	2229.1
33	2298.7
34	2368.4
35	2438
36	2507.7	17/4
37	2577.3
38	2647	23/5
39	2716.7
40	2786.3	5/1

Harmonics

Approximation of prime harmonics in 40ed5

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-15.8	-21.2	+0.0	-25.2	+28.2	+17.6	-28.9	-12.5	+5.0	+21.7	-24.1
	Relative (%)	-22.7	-30.4	+0.0	-36.2	+40.4	+25.2	-41.5	-17.9	+7.2	+31.1	-34.6
Steps (reduced)		17 (17)	27 (27)	40 (0)	48 (8)	60 (20)	64 (24)	70 (30)	73 (33)	78 (38)	84 (4)	85 (5)

(17edo for comparison)

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0	+3.9	-33.4	+19.4	+13.4	+6.5	-34.4	-15.2	+7.0	+29.2	-15.6
	Relative (%)	+0.0	+5.6	-47.3	+27.5	+19.0	+9.3	-48.7	-21.5	+9.9	+41.4	-22.1
Steps (reduced)		17 (0)	27 (10)	39 (5)	48 (14)	59 (8)	63 (12)	69 (1)	72 (4)	77 (9)	83 (15)	84 (16)

Comparison with 17edo

40ed5 has the same scale shape as 17edo but with a very compressed octave about 16 ¢ flat. This dramatically changes the nature of the tuning:

• 17edo's strongest simple primes are 2, 3, 13 and 23
• 17edo's weakest are 5, 17 and 29
• 40ed5's strongest simple primes are 5, 19 and 23
• 40ed5's weakest are 7, 11 and 17
• The mappings of prime 5, 11, 13, 17, 19, 23, 29 and 31 are all changed from 17edo

17edo can be thought of as a tuning for either:

• the no-5s 13-limit (high accuracy for its size)
• the no-5s no-17s 23-limit (high accuracy for its size)

40ed5 can be thought of as a tuning for either:

• the full 13-limit (low accuracy for its size)
• the no-7s no-11s 41-limit (moderate accuracy for its size)

Music

• escape yourself - Merct (2025)

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