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	21.4
2	42.7	39/38, 42/41
3	64.1	27/26, 28/27
4	85.5	41/39
5	106.9	33/31
6	128.2	14/13
7	149.6	12/11
8	171	21/19
9	192.3	19/17
10	213.7	26/23
11	235.1	39/34
12	256.4	36/31
13	277.8	27/23, 34/29
14	299.2
15	320.6
16	341.9	28/23
17	363.3	21/17
18	384.7	5/4
19	406
20	427.4	32/25
21	448.8	35/27
22	470.1	38/29
23	491.5
24	512.9	35/26, 39/29
25	534.3	15/11
26	555.6
27	577
28	598.4	41/29
29	619.7
30	641.1	42/29
31	662.5	22/15, 41/28
32	683.8
33	705.2
34	726.6	35/23
35	748
36	769.3
37	790.7	41/26
38	812.1	8/5
39	833.4	34/21
40	854.8	18/11
41	876.2
42	897.6
43	918.9
44	940.3	31/18
45	961.7
46	983	37/21
47	1004.4
48	1025.8	38/21
49	1047.1	11/6
50	1068.5
51	1089.9	15/8
52	1111.3
53	1132.6
54	1154	35/18, 37/19
55	1175.4
56	1196.7
57	1218.1
58	1239.5
59	1260.8	29/14
60	1282.2
61	1303.6
62	1325
63	1346.3	37/17
64	1367.7	11/5
65	1389.1	29/13
66	1410.4
67	1431.8
68	1453.2
69	1474.5
70	1495.9
71	1517.3	12/5
72	1538.7	17/7
73	1560
74	1581.4
75	1602.8
76	1624.1	23/9
77	1645.5	31/12
78	1666.9	34/13
79	1688.3
80	1709.6
81	1731	19/7
82	1752.4	11/4
83	1773.7	39/14
84	1795.1	31/11
85	1816.5
86	1837.8	26/9
87	1859.2	38/13, 41/14
88	1880.6
89	1902	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)