56ed5

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56ed5

57ed5 →

Prime factorization	23 × 7
Step size	49.7556 ¢
Octave	24\56ed5 (1194.13 ¢) (→ 3\7ed5)
Twelfth	38\56ed5 (1890.71 ¢) (→ 19\28ed5)
Consistency limit	6
Distinct consistency limit	6

56 equal divisions of the 5th harmonic (abbreviated 56ed5) is a nonoctave tuning system that divides the interval of 5/1 into 56 equal parts of about 49.8 ¢ each. Each step represents a frequency ratio of 5^1/56, or the 56th root of 5.

Theory

56ed5 is related to 24edo, but with the 5th harmonic rather than the octave being just. The octave is compressed by about 5.8 ¢, a deviation that despite being small is enough to be significant. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning is also a hyperpyth, tempering out 135/133, 171/169, 225/221, and 1521/1445 in the patent val.

Harmonics

Approximation of harmonics in 56ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-5.9	-11.2	-11.7	+0.0	-17.1	+14.6	-17.6	-22.5	-5.9	-21.6	-23.0
	Relative (%)	-11.8	-22.6	-23.6	+0.0	-34.4	+29.3	-35.4	-45.2	-11.8	-43.4	-46.2
Steps (reduced)		24 (24)	38 (38)	48 (48)	56 (0)	62 (6)	68 (12)	72 (16)	76 (20)	80 (24)	83 (27)	86 (30)

Approximation of harmonics in 56ed5 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	-12.3	+8.7	-11.2	-23.5	+20.8	+21.4	-22.4	-11.7	+3.3	+22.3	-4.9	+20.9
	Relative (%)	-24.7	+17.5	-22.6	-47.2	+41.9	+43.0	-45.1	-23.6	+6.7	+44.8	-9.9	+42.0
Steps (reduced)		89 (33)	92 (36)	94 (38)	96 (40)	99 (43)	101 (45)	102 (46)	104 (48)	106 (50)	108 (52)	109 (53)	111 (55)

Subsets and supersets

Since 56 factors into primes as 2³ × 7, 56ed5 contains subset ed5's 2, 4, 7, 8, 14, and 28.

Intervals

#	Cents	Approximated ratios
0	0.0	1/1
1	49.8	35/34, 36/35
2	99.5	18/17
3	149.3	12/11
4	199.0	55/49
5	248.8	15/13
6	298.5	19/16
7	348.3	11/9
8	398.0	5/4
9	447.8	35/27
10	497.6	4/3
11	547.3	70/51
12	597.1	24/17
13	646.8
14	696.6	3/2
15	746.3	20/13
16	796.1	19/12
17	845.8	44/27, 75/46
18	895.6	5/3
19	945.4	19/11
20	995.1	9/5, 16/9
21	1044.9	64/35
22	1094.6	32/17
23	1144.4
24	1194.1	2/1
25	1243.9	39/19, 80/39
26	1293.6	19/9
27	1343.4	50/23
28	1393.2	38/17, 85/38
29	1442.9	23/10
30	1492.7	45/19
31	1542.4	39/16
32	1592.2	5/2
33	1641.9	13/5
34	1691.7	85/32
35	1741.4	175/64
36	1791.2	45/16
37	1841.0	55/19
38	1890.7	3/1
39	1940.5	46/15, 135/44
40	1990.2	60/19
41	2040.0	13/4
42	2089.7	10/3
43	2139.5	17/5
44	2189.2	85/24
45	2239.0	51/14
46	2288.8	15/4, 19/5
47	2338.5	27/7
48	2388.3	4/1
49	2438.0	45/11
50	2487.8	21/5
51	2537.5	13/3
52	2587.3	49/11
53	2637.0	55/12
54	2686.8	85/18
55	2736.6	34/7
56	2786.3	5/1

See also

• 14edf – relative edf
• 24edo – relative edo
• 38edt – relative edt
• 62ed6 – relative ed6
• 86ed12 – relative ed12
• 198ed304 – close to the zeta-optimized tuning for 24edo