88ed7/3

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88ed7/3

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Prime factorization	23 × 11
Step size	16.669¢
Octave	72\88ed7/3 (1200.17¢) (→9\11ed7/3)
Twelfth	114\88ed7/3 (1900.26¢) (→57\44ed7/3)
Consistency limit	18
Distinct consistency limit	10

88 equal divisions of 7/3 (abbreviated 88ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 88 equal parts of about 16.7 ¢ each. Each step represents a frequency ratio of (7/3)^1/88, or the 88th root of 7/3.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	16.669
2	33.338
3	50.007	34/33, 35/34, 36/35, 37/36
4	66.676	26/25, 27/26
5	83.345	21/20, 22/21
6	100.014	18/17, 35/33
7	116.683	15/14, 31/29
8	133.352	27/25, 40/37, 41/38
9	150.021	12/11
10	166.69	11/10
11	183.359	10/9
12	200.028	37/33
13	216.697	17/15, 42/37
14	233.366	8/7
15	250.035	15/13, 37/32
16	266.704	7/6
17	283.373	20/17, 33/28
18	300.042	19/16, 25/21
19	316.711	6/5
20	333.38	17/14, 23/19, 40/33
21	350.049	11/9, 38/31
22	366.718	21/17
23	383.387
24	400.056	29/23, 34/27
25	416.725	14/11
26	433.394	9/7
27	450.063	35/27
28	466.732	17/13, 38/29
29	483.401	33/25, 37/28, 41/31
30	500.07	4/3
31	516.739	27/20, 31/23, 35/26
32	533.408	34/25
33	550.077	11/8
34	566.746	25/18
35	583.415	7/5
36	600.084	41/29
37	616.753	10/7
38	633.422	36/25
39	650.091	16/11
40	666.76	25/17
41	683.428
42	700.097	3/2
43	716.766
44	733.435	26/17, 29/19
45	750.104	37/24
46	766.773	14/9
47	783.442	11/7
48	800.111	27/17
49	816.78
50	833.449	34/21
51	850.118	18/11, 31/19
52	866.787	33/20, 38/23
53	883.456	5/3
54	900.125	32/19, 37/22, 42/25
55	916.794	17/10
56	933.463	12/7
57	950.132	26/15
58	966.801	7/4
59	983.47	30/17
60	1000.139	41/23
61	1016.808	9/5
62	1033.477	20/11
63	1050.146	11/6
64	1066.815	37/20
65	1083.484
66	1100.153	17/9
67	1116.822	21/11, 40/21
68	1133.491	25/13
69	1150.16	33/17, 35/18
70	1166.829
71	1183.498
72	1200.167	2/1
73	1216.836
74	1233.505
75	1250.174	35/17, 37/18
76	1266.843	27/13
77	1283.512	21/10
78	1300.181	36/17
79	1316.85	15/7
80	1333.519	41/19
81	1350.188	24/11
82	1366.857	11/5
83	1383.526	20/9
84	1400.195
85	1416.864	34/15
86	1433.533	16/7
87	1450.202	30/13, 37/16
88	1466.871	7/3

Harmonics

Approximation of harmonics in 88ed7/3

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+0.17	-1.69	+0.33	-2.59	-1.52	-1.69	+0.50	-3.38	-2.43	-0.74	-1.36
	Relative (%)	+1.0	-10.1	+2.0	-15.6	-9.1	-10.1	+3.0	-20.3	-14.6	-4.4	-8.1
Steps (reduced)		72 (72)	114 (26)	144 (56)	167 (79)	186 (10)	202 (26)	216 (40)	228 (52)	239 (63)	249 (73)	258 (82)

Approximation of harmonics in 88ed7/3

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-6.58	-1.52	-4.28	+0.67	-4.27	-3.21	+3.20	-2.26	-3.38	-0.57	+5.82
	Relative (%)	-39.5	-9.1	-25.7	+4.0	-25.6	-19.3	+19.2	-13.5	-20.3	-3.4	+34.9
Steps (reduced)		266 (2)	274 (10)	281 (17)	288 (24)	294 (30)	300 (36)	306 (42)	311 (47)	316 (52)	321 (57)	326 (62)