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

Intervals of 87ed7

degree	cents value	corresponding JI intervals	comments
0	0.0000	exact 1/1
1	38.7221
2	77.4443
3	116.1664	77/72, 15/14
4	154.8885
5	193.6107	19/17, 85/76
6	232.3328	8/7
7	271.0550
8	309.7771
9	348.4992	11/9, 49/40
10	387.2214	5/4
11	425.9435
12	464.6656	98/75
13	503.3878	91/68, 75/56
14	542.1099
15	580.8321	7/5
16	619.5542	10/7
17	658.2763
18	696.9985	112/75, 121/81, 136/91, 187/125
19	735.7206
20	774.4427
21	813.1649	8/5
22	851.8870
23	890.6091
24	929.3313	65/38
25	968.0534
26	1006.7756
27	1045.4977	64/35
28	1084.2198
29	1122.9420
30	1161.6641	88/45, 96/49, 49/25
31	1200.3862	2/1
32	1239.1084
33	1277.8305	44/21
34	1316.5527
35	1355.2748
36	1393.9969	38/17, 85/38
37	1432.7191
38	1471.4412
39	1510.1633
40	1548.8855
41	1587.6076
42	1626.3297	64/25
43	1665.0519
44	1703.7740
45	1742.4962	52/19
46	1781.2183
47	1819.9404
48	1858.6626	38/13
49	1897.3847
50	1936.1068
51	1974.8290	25/8
52	2013.5511
53	2052.2733
54	2090.9954
55	2129.7175
56	2168.4397
57	2207.1618	68/19
58	2245.8839
59	2284.6061
60	2323.3282	65/17
61	2362.0503
62	2400.7725
63	2439.4946	45/11
64	2478.2168
65	2516.9389
66	2555.6610	35/8
67	2594.3832
68	2633.1053
69	2671.8274
70	2710.5496
71	2749.2717
72	2787.9939	5/1
73	2826.7160
74	2865.4381
75	2904.1603	75/14
76	2942.8824
77	2981.6045
78	3020.3267	40/7, 63/11
79	3059.0488
80	3097.7709
81	3136.4931	49/8
82	3175.2152
83	3213.9374
84	3252.6595	98/15, 72/11
85	3291.3816
86	3330.1038
87	3368.8259	exact 7/1	harmonic seventh plus two octaves

See also

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