89edt

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89edt

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Prime factorization	89 (prime)
Step size	21.3703¢
Octave	56\89edt (1196.74¢)
Consistency limit	5
Distinct consistency limit	5

89 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 89edt or 89ed3), is a nonoctave tuning system that divides the interval of 3/1 into 89 equal parts of about 21.4 ¢ each. Each step represents a frequency ratio of 3^1/89, or the 89th root of 3.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	21.37
2	42.741	39/38, 42/41
3	64.111	27/26, 28/27
4	85.481	41/39
5	106.851	33/31
6	128.222	14/13
7	149.592	12/11
8	170.962	21/19
9	192.333	19/17
10	213.703	26/23
11	235.073	39/34
12	256.443	36/31
13	277.814	27/23, 34/29
14	299.184
15	320.554
16	341.924	28/23
17	363.295	21/17
18	384.665	5/4
19	406.035
20	427.406	32/25
21	448.776	35/27
22	470.146	38/29
23	491.516
24	512.887	35/26, 39/29
25	534.257	15/11
26	555.627
27	576.998
28	598.368	41/29
29	619.738
30	641.108	42/29
31	662.479	22/15, 41/28
32	683.849
33	705.219
34	726.59	35/23
35	747.96
36	769.33
37	790.7	41/26
38	812.071	8/5
39	833.441	34/21
40	854.811	18/11
41	876.182
42	897.552
43	918.922
44	940.292	31/18
45	961.663
46	983.033	37/21
47	1004.403
48	1025.773	38/21
49	1047.144	11/6
50	1068.514
51	1089.884	15/8
52	1111.255
53	1132.625
54	1153.995	35/18, 37/19
55	1175.365
56	1196.736
57	1218.106
58	1239.476
59	1260.847	29/14
60	1282.217
61	1303.587
62	1324.957
63	1346.328	37/17
64	1367.698	11/5
65	1389.068	29/13
66	1410.439
67	1431.809
68	1453.179
69	1474.549
70	1495.92
71	1517.29	12/5
72	1538.66	17/7
73	1560.031
74	1581.401
75	1602.771
76	1624.141	23/9
77	1645.512	31/12
78	1666.882	34/13
79	1688.252
80	1709.622
81	1730.993	19/7
82	1752.363	11/4
83	1773.733	39/14
84	1795.104	31/11
85	1816.474
86	1837.844	26/9
87	1859.214	38/13, 41/14
88	1880.585
89	1901.955	3/1

Harmonics

Approximation of harmonics in 89edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-3.26	+0.00	-6.53	-8.18	-3.26	+7.68	-9.79	+0.00	+9.93	-5.48	-6.53
	Relative (%)	-15.3	+0.0	-30.5	-38.3	-15.3	+35.9	-45.8	+0.0	+46.5	-25.7	-30.5
Steps (reduced)		56 (56)	89 (0)	112 (23)	130 (41)	145 (56)	158 (69)	168 (79)	178 (0)	187 (9)	194 (16)	201 (23)

Approximation of harmonics in 89edt

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+4.49	+4.41	-8.18	+8.31	+10.21	-3.26	+9.98	+6.66	+7.68	-8.75	-0.22
	Relative (%)	+21.0	+20.7	-38.3	+38.9	+47.8	-15.3	+46.7	+31.2	+35.9	-40.9	-1.0
Steps (reduced)		208 (30)	214 (36)	219 (41)	225 (47)	230 (52)	234 (56)	239 (61)	243 (65)	247 (69)	250 (72)	254 (76)