56ed5

← 55ed5

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

Division of the 5th harmonic into 56 equal parts (56ed5) is related to 24 edo, but with the 5/1 rather than the 2/1 being just. The octave is about 5.8656 cents compressed and the step size is about 49.7556 cents. 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.

Intervals

degree	cents value	corresponding JI intervals	comments
0	0.0000	exact 1/1
1	49.7556	36/35, 35/34
2	99.5112	18/17
3	149.2668	12/11
4	199.0224	55/49
5	248.7780	15/13
6	298.5336	19/16
7	348.2892	11/9
8	398.0448	34/27	pseudo-5/4
9	447.8004	35/27
10	497.5560	4/3
11	547.3116	70/51
12	597.0672	24/17
13	646.8228
14	696.5784		meantone fifth (pseudo-3/2)
15	746.3340	20/13
16	796.0896	19/12
17	845.8452	44/27, 75/46
18	895.6008	57/34	pseudo-5/3
19	945.3564	19/11
20	995.1120	16/9	pseudo-9/5
21	1044.8676	64/35
22	1094.6232	32/17
23	1144.3788
24	1194.1344	255/128	pseudo-octave
25	1243.8901	80/39, 39/19
26	1293.6457	19/9
27	1343.4013	50/23
28	1393.1569	38/17, 85/38	meantone major second plus an octave
29	1442.9125	23/10
30	1492.6681	45/19
31	1542.4237	39/16
32	1592.1793	128/51	pseudo-5/2
33	1641.9349		pseudo-13/5
34	1691.6905	85/32
35	1741.4461	175/64
36	1791.2017	45/16
37	1840.9573	55/19
38	1890.7129	170/57	pseudo-3/1
39	1940.4685	46/15, 135/44
40	1990.2241	60/19
41	2039.9797	13/4
42	2089.7353		meantone major sixth plus an octave (pseudo-10/3)
43	2139.4909		pseudo-17/5
44	2189.2465	85/24
45	2239.0021	51/14
46	2288.7577	15/4	pseudo-19/5
47	2338.5133	27/7
48	2388.2689	135/34	pseudo-4/1
49	2438.0245	45/11
50	2487.7801	80/19	pseudo-21/5
51	2537.5357	13/3
52	2587.2913	49/11
53	2637.0469	55/12
54	2686.8025	85/18
55	2736.5581	34/7
56	2786.3137	exact 5/1	just major third plus two octaves

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

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