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	Hekts	Approximate ratios
0	0	0	1/1
1	21.4	14.6
2	42.7	29.2	39/38, 42/41
3	64.1	43.8	27/26, 28/27
4	85.5	58.4	41/39
5	106.9	73	33/31
6	128.2	87.6	14/13
7	149.6	102.2	12/11
8	171	116.9	21/19
9	192.3	131.5	19/17
10	213.7	146.1	26/23
11	235.1	160.7	39/34
12	256.4	175.3	36/31
13	277.8	189.9	27/23, 34/29
14	299.2	204.5
15	320.6	219.1
16	341.9	233.7	28/23
17	363.3	248.3	21/17
18	384.7	262.9	5/4
19	406	277.5
20	427.4	292.1	32/25
21	448.8	306.7	35/27
22	470.1	321.3	38/29
23	491.5	336
24	512.9	350.6	35/26, 39/29
25	534.3	365.2	15/11
26	555.6	379.8
27	577	394.4
28	598.4	409	41/29
29	619.7	423.6
30	641.1	438.2	42/29
31	662.5	452.8	22/15, 41/28
32	683.8	467.4
33	705.2	482
34	726.6	496.6	35/23
35	748	511.2
36	769.3	525.8
37	790.7	540.4	41/26
38	812.1	555.1	8/5
39	833.4	569.7	34/21
40	854.8	584.3	18/11
41	876.2	598.9
42	897.6	613.5
43	918.9	628.1
44	940.3	642.7	31/18
45	961.7	657.3
46	983	671.9	37/21
47	1004.4	686.5
48	1025.8	701.1	38/21
49	1047.1	715.7	11/6
50	1068.5	730.3
51	1089.9	744.9	15/8
52	1111.3	759.6
53	1132.6	774.2
54	1154	788.8	35/18, 37/19
55	1175.4	803.4
56	1196.7	818
57	1218.1	832.6
58	1239.5	847.2
59	1260.8	861.8	29/14
60	1282.2	876.4
61	1303.6	891
62	1325	905.6
63	1346.3	920.2	37/17
64	1367.7	934.8	11/5
65	1389.1	949.4	29/13
66	1410.4	964
67	1431.8	978.7
68	1453.2	993.3
69	1474.5	1007.9
70	1495.9	1022.5
71	1517.3	1037.1	12/5
72	1538.7	1051.7	17/7
73	1560	1066.3
74	1581.4	1080.9
75	1602.8	1095.5
76	1624.1	1110.1	23/9
77	1645.5	1124.7	31/12
78	1666.9	1139.3	34/13
79	1688.3	1153.9
80	1709.6	1168.5
81	1731	1183.1	19/7
82	1752.4	1197.8	11/4
83	1773.7	1212.4	39/14
84	1795.1	1227	31/11
85	1816.5	1241.6
86	1837.8	1256.2	26/9
87	1859.2	1270.8	38/13, 41/14
88	1880.6	1285.4
89	1902	1300	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)