87ed7

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87ed7

88ed7 →

Prime factorization	3 × 29
Step size	38.7221 ¢
Octave	31\87ed7 (1200.39 ¢) (semiconvergent)
Twelfth	49\87ed7 (1897.38 ¢)
Consistency limit	12
Distinct consistency limit	9

87 equal divisions of the 7th harmonic (abbreviated 87ed7) is a nonoctave tuning system that divides the interval of 7/1 into 87 equal parts of about 38.7 ¢ each. Each step represents a frequency ratio of 7^1/87, or the 87th root of 7.

Theory

87ed7 is related to 31edo, but with the 7/1 rather than the 2/1 being just. The octave is slightly stretched (about 0.3862 cents).

Harmonics

Approximation of harmonics in 87ed7

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13
Error	Absolute (¢)	+0.4	-4.6	+0.8	+1.7	-4.2	+0.0	+1.2	-9.1	+2.1	-8.0	-3.8	+12.5
	Relative (%)	+1.0	-11.8	+2.0	+4.3	-10.8	+0.0	+3.0	-23.6	+5.3	-20.8	-9.8	+32.3
Steps (reduced)		31 (31)	49 (49)	62 (62)	72 (72)	80 (80)	87 (0)	93 (6)	98 (11)	103 (16)	107 (20)	111 (24)	115 (28)

Approximation of harmonics in 87ed7 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+12.5	+0.4	-2.9	+1.5	+12.8	-8.8	+13.8	+2.5	-4.6	-7.7	-7.2	-3.4
	Relative (%)	+32.3	+1.0	-7.5	+4.0	+32.9	-22.6	+35.7	+6.3	-11.8	-19.8	-18.5	-8.8
Steps (reduced)		115 (28)	118 (31)	121 (34)	124 (37)	127 (40)	129 (42)	132 (45)	134 (47)	136 (49)	138 (51)	140 (53)	142 (55)

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	38.7
2	77.4	23/22
3	116.2	31/29
4	154.9	23/21, 35/32
5	193.6	19/17, 28/25
6	232.3	8/7
7	271.1	41/35
8	309.8
9	348.5	11/9
10	387.2	5/4
11	425.9	23/18, 32/25
12	464.7	17/13
13	503.4
14	542.1	26/19, 41/30
15	580.8	7/5
16	619.6	10/7
17	658.3	19/13, 41/28
18	697
19	735.7	26/17
20	774.4	25/16, 36/23
21	813.2	8/5
22	851.9	18/11
23	890.6
24	929.3	41/24
25	968.1	7/4
26	1006.8	34/19
27	1045.5
28	1084.2
29	1122.9
30	1161.7
31	1200.4	2/1
32	1239.1
33	1277.8	23/11
34	1316.6	15/7
35	1355.3	35/16
36	1394	38/17
37	1432.7	16/7
38	1471.4
39	1510.2
40	1548.9	22/9
41	1587.6	5/2
42	1626.3	23/9, 41/16
43	1665.1	34/13
44	1703.8
45	1742.5	41/15
46	1781.2	14/5
47	1819.9	20/7
48	1858.7	38/13, 41/14
49	1897.4
50	1936.1
51	1974.8	25/8
52	2013.6	16/5
53	2052.3	36/11
54	2091
55	2129.7	41/12
56	2168.4	7/2
57	2207.2
58	2245.9
59	2284.6
60	2323.3
61	2362.1
62	2400.8	4/1
63	2439.5
64	2478.2
65	2516.9	30/7
66	2555.7	35/8
67	2594.4
68	2633.1	32/7
69	2671.8
70	2710.5
71	2749.3
72	2788	5/1
73	2826.7	41/8
74	2865.4
75	2904.2
76	2942.9
77	2981.6	28/5
78	3020.3	40/7
79	3059	41/7
80	3097.8
81	3136.5
82	3175.2	25/4
83	3213.9	32/5
84	3252.7
85	3291.4
86	3330.1
87	3368.8	7/1

See also

• 18edf – relative edf
• 31edo – relative edo
• 49edt – relative edt
• 72ed5 – relative ed5
• 80ed6 – relative ed6
• 111ed12 – relative ed12
• 39cET