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	27.586
2	55.172	31/30
3	82.759	22/21
4	110.345
5	137.931
6	165.517
7	193.103	19/17
8	220.69	25/22
9	248.276	15/13
10	275.862	27/23, 41/35
11	303.448	25/21, 31/26, 37/31
12	331.034	23/19
13	358.621
14	386.207
15	413.793
16	441.379
17	468.966
18	496.552
19	524.138	23/17
20	551.724
21	579.31
22	606.897	27/19
23	634.483	13/9
24	662.069	22/15
25	689.655
26	717.241
27	744.828
28	772.414	39/25
29	800	27/17
30	827.586	21/13
31	855.172	41/25
32	882.759	5/3
33	910.345	22/13
34	937.931
35	965.517
36	993.103	39/22
37	1020.69
38	1048.276
39	1075.862	41/22
40	1103.448	17/9
41	1131.034	25/13
42	1158.621	41/21
43	1186.207
44	1213.793
45	1241.379
46	1268.966
47	1296.552
48	1324.138
49	1351.724
50	1379.31
51	1406.897
52	1434.483
53	1462.069
54	1489.655
55	1517.241
56	1544.828	22/9
57	1572.414
58	1600
59	1627.586
60	1655.172	13/5
61	1682.759	37/14
62	1710.345
63	1737.931	41/15
64	1765.517
65	1793.103
66	1820.69
67	1848.276
68	1875.862
69	1903.448	3/1
70	1931.034
71	1958.621	31/10
72	1986.207	41/13
73	2013.793
74	2041.379
75	2068.966
76	2096.552
77	2124.138
78	2151.724
79	2179.31
80	2206.897
81	2234.483
82	2262.069
83	2289.655
84	2317.241
85	2344.828
86	2372.414
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)