89ed4

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89ed4

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Prime factorization	89 (prime)
Step size	26.9663 ¢
Octave	45\89ed4 (1213.48 ¢)
Twelfth	71\89ed4 (1914.61 ¢)
Consistency limit	1
Distinct consistency limit	1

89 equal divisions of the 4th harmonic (abbreviated 89ed4) is a nonoctave tuning system that divides the interval of 4/1 into 89 equal parts of about 27 ¢ each. Each step represents a frequency ratio of 4^1/89, or the 89th root of 4.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	27
2	53.9	34/33
3	80.9	22/21
4	107.9
5	134.8
6	161.8
7	188.8
8	215.7	17/15
9	242.7	38/33
10	269.7
11	296.6
12	323.6	35/29
13	350.6
14	377.5
15	404.5
16	431.5
17	458.4
18	485.4	41/31
19	512.4
20	539.3	15/11
21	566.3
22	593.3
23	620.2	10/7
24	647.2
25	674.2
26	701.1	3/2
27	728.1	35/23
28	755.1	17/11
29	782	11/7
30	809
31	836	34/21
32	862.9
33	889.9
34	916.9	17/10
35	943.8	19/11
36	970.8
37	997.8
38	1024.7	38/21
39	1051.7
40	1078.7
41	1105.6
42	1132.6
43	1159.6
44	1186.5
45	1213.5
46	1240.4
47	1267.4
48	1294.4
49	1321.3	15/7
50	1348.3	37/17
51	1375.3
52	1402.2	9/4
53	1429.2
54	1456.2
55	1483.1	33/14
56	1510.1
57	1537.1	17/7
58	1564	37/15
59	1591
60	1618
61	1644.9
62	1671.9
63	1698.9
64	1725.8
65	1752.8
66	1779.8
67	1806.7
68	1833.7
69	1860.7
70	1887.6
71	1914.6
72	1941.6
73	1968.5
74	1995.5
75	2022.5
76	2049.4
77	2076.4
78	2103.4	27/8
79	2130.3
80	2157.3
81	2184.3
82	2211.2
83	2238.2
84	2265.2	37/10
85	2292.1
86	2319.1
87	2346.1
88	2373
89	2400

Harmonics

Approximation of harmonics in 89ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+13.5	+12.7	+0.0	-8.8	-0.8	+2.0	+13.5	-1.7	+4.7	+1.5	+12.7
	Relative (%)	+50.0	+46.9	+0.0	-32.6	-3.1	+7.3	+50.0	-6.2	+17.4	+5.5	+46.9
Steps (reduced)		45 (45)	71 (71)	89 (0)	103 (14)	115 (26)	125 (36)	134 (45)	141 (52)	148 (59)	154 (65)	160 (71)

Approximation of harmonics in 89ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+8.9	-11.5	+3.9	+0.0	+2.9	+11.8	-0.9	-8.8	-12.4	-12.0	-8.0
	Relative (%)	+33.0	-42.7	+14.3	+0.0	+10.8	+43.8	-3.3	-32.6	-45.8	-44.5	-29.9
Steps (reduced)		165 (76)	169 (80)	174 (85)	178 (0)	182 (4)	186 (8)	189 (11)	192 (14)	195 (17)	198 (20)	201 (23)