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	26.966
2	53.933	34/33
3	80.899	22/21
4	107.865
5	134.831
6	161.798
7	188.764
8	215.73	17/15
9	242.697	38/33
10	269.663
11	296.629
12	323.596	35/29
13	350.562
14	377.528
15	404.494
16	431.461
17	458.427
18	485.393	41/31
19	512.36
20	539.326	15/11
21	566.292
22	593.258
23	620.225	10/7
24	647.191
25	674.157
26	701.124	3/2
27	728.09	35/23
28	755.056	17/11
29	782.022	11/7
30	808.989
31	835.955	34/21
32	862.921
33	889.888
34	916.854	17/10
35	943.82	19/11
36	970.787
37	997.753
38	1024.719	38/21
39	1051.685
40	1078.652
41	1105.618
42	1132.584
43	1159.551
44	1186.517
45	1213.483
46	1240.449
47	1267.416
48	1294.382
49	1321.348	15/7
50	1348.315	37/17
51	1375.281
52	1402.247	9/4
53	1429.213
54	1456.18
55	1483.146	33/14
56	1510.112
57	1537.079	17/7
58	1564.045	37/15
59	1591.011
60	1617.978
61	1644.944
62	1671.91
63	1698.876
64	1725.843
65	1752.809
66	1779.775
67	1806.742
68	1833.708
69	1860.674
70	1887.64
71	1914.607
72	1941.573
73	1968.539
74	1995.506
75	2022.472
76	2049.438
77	2076.404
78	2103.371	27/8
79	2130.337
80	2157.303
81	2184.27
82	2211.236
83	2238.202
84	2265.169	37/10
85	2292.135
86	2319.101
87	2346.067
88	2373.034
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)