87ed4

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87ed4

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Prime factorization	3 × 29
Step size	27.5862¢
Octave	44\87ed4 (1213.79¢)
Twelfth	69\87ed4 (1903.45¢) (→23\29ed4)
Consistency limit	1
Distinct consistency limit	1

87 equal divisions of the 4th harmonic (abbreviated 87ed4) is a nonoctave tuning system that divides the interval of 4/1 into 87 equal parts of about 27.6 ¢ each. Each step represents a frequency ratio of 4^1/87, or the 87th root of 4.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	27.6
2	55.2	31/30
3	82.8	22/21
4	110.3
5	137.9
6	165.5
7	193.1	19/17
8	220.7	25/22
9	248.3	15/13
10	275.9	27/23, 41/35
11	303.4	25/21, 31/26, 37/31
12	331	23/19
13	358.6
14	386.2
15	413.8
16	441.4
17	469
18	496.6
19	524.1	23/17
20	551.7
21	579.3
22	606.9	27/19
23	634.5	13/9
24	662.1	22/15
25	689.7
26	717.2
27	744.8
28	772.4	39/25
29	800	27/17
30	827.6	21/13
31	855.2	41/25
32	882.8	5/3
33	910.3	22/13
34	937.9
35	965.5
36	993.1	39/22
37	1020.7
38	1048.3
39	1075.9	41/22
40	1103.4	17/9
41	1131	25/13
42	1158.6	41/21
43	1186.2
44	1213.8
45	1241.4
46	1269
47	1296.6
48	1324.1
49	1351.7
50	1379.3
51	1406.9
52	1434.5
53	1462.1
54	1489.7
55	1517.2
56	1544.8	22/9
57	1572.4
58	1600
59	1627.6
60	1655.2	13/5
61	1682.8	37/14
62	1710.3
63	1737.9	41/15
64	1765.5
65	1793.1
66	1820.7
67	1848.3
68	1875.9
69	1903.4	3/1
70	1931
71	1958.6	31/10
72	1986.2	41/13
73	2013.8
74	2041.4
75	2069
76	2096.6
77	2124.1
78	2151.7
79	2179.3
80	2206.9
81	2234.5
82	2262.1
83	2289.7
84	2317.2
85	2344.8
86	2372.4
87	2400

Harmonics

Approximation of harmonics in 87ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+13.8	+1.5	+0.0	-0.1	-12.3	-3.3	+13.8	+3.0	+13.7	-13.4	+1.5
	Relative (%)	+50.0	+5.4	+0.0	-0.4	-44.6	-12.0	+50.0	+10.8	+49.6	-48.5	+5.4
Steps (reduced)		44 (44)	69 (69)	87 (0)	101 (14)	112 (25)	122 (35)	131 (44)	138 (51)	145 (58)	150 (63)	156 (69)

Approximation of harmonics in 87ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+0.9	+10.5	+1.4	+0.0	+5.4	-10.8	+5.9	-0.1	-1.8	+0.4	+6.2
	Relative (%)	+3.1	+38.0	+5.0	+0.0	+19.5	-39.2	+21.5	-0.4	-6.6	+1.5	+22.5
Steps (reduced)		161 (74)	166 (79)	170 (83)	174 (0)	178 (4)	181 (7)	185 (11)	188 (14)	191 (17)	194 (20)	197 (23)