89edf

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89edf

90edf →

Prime factorization	89 (prime)
Step size	7.88713 ¢
Octave	152\89edf (1198.84 ¢)
Twelfth	241\89edf (1900.8 ¢)
Consistency limit	15
Distinct consistency limit	15

89edf is the equal division of the just perfect fifth into eighty nine parts of 7.887 cents each, corresponding to 152.1465 edo.

Theory

89edf is similar to Carlos Gamma in that it includes a just perfect fifth and an incredibly precise approximation of the major and minor thirds, and is next in line in the sequence {1, 2, 5, 7, 9, 11, 20, 89, 109, 129, 149} of divisions of the perfect fifth to approximate 5/4 with increasing accuracy.

Harmonics

Approximation of prime harmonics in 89edf

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-1.16	-1.16	-2.16	-1.02	-2.69	-0.07	+0.84	-2.42	-1.93	-0.98	+1.86
	Relative (%)	-14.7	-14.7	-27.3	-12.9	-34.0	-0.9	+10.7	-30.7	-24.4	-12.5	+23.6
Steps (reduced)		152 (63)	241 (63)	353 (86)	427 (71)	526 (81)	563 (29)	622 (88)	646 (23)	688 (65)	739 (27)	754 (42)

Approximation of prime harmonics in 89edf

Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	+3.15	-1.05	+3.26	-0.88	-3.81	-0.19	-2.69	+0.52	+2.66	+1.89	-0.77
	Relative (%)	+40.0	-13.3	+41.3	-11.1	-48.3	-2.4	-34.1	+6.6	+33.7	+24.0	-9.8
Steps (reduced)		793 (81)	815 (14)	826 (25)	845 (44)	871 (70)	895 (5)	902 (12)	923 (33)	936 (46)	942 (52)	959 (69)

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	7.9
2	15.8
3	23.7
4	31.5
5	39.4	42/41
6	47.3	36/35, 39/38
7	55.2	33/32
8	63.1	27/26, 28/27, 29/28
9	71	24/23, 25/24
10	78.9	22/21, 23/22
11	86.8	20/19, 21/20, 41/39
12	94.6	19/18
13	102.5	35/33
14	110.4	16/15
15	118.3	15/14
16	126.2	14/13, 29/27
17	134.1	27/25, 41/38
18	142	25/23, 38/35
19	149.9	12/11
20	157.7	23/21, 34/31, 35/32
21	165.6	11/10
22	173.5	21/19
23	181.4	10/9
24	189.3	29/26, 39/35
25	197.2	28/25
26	205.1	9/8
27	213	26/23
28	220.8	25/22
29	228.7	8/7
30	236.6	39/34
31	244.5	23/20, 38/33
32	252.4	22/19
33	260.3
34	268.2	7/6
35	276	27/23, 34/29, 41/35
36	283.9	33/28
37	291.8	13/11, 32/27
38	299.7	19/16, 25/21
39	307.6	37/31
40	315.5	6/5
41	323.4	35/29, 41/34
42	331.3	23/19, 40/33
43	339.1	28/23
44	347	11/9
45	354.9	27/22
46	362.8
47	370.7	26/21
48	378.6	41/33
49	386.5	5/4
50	394.4
51	402.2	24/19, 29/23
52	410.1	19/15, 33/26
53	418	14/11
54	425.9	23/18, 32/25
55	433.8	9/7
56	441.7
57	449.6	35/27
58	457.5	30/23
59	465.3	17/13, 38/29
60	473.2	21/16, 25/19
61	481.1	33/25
62	489
63	496.9	4/3
64	504.8
65	512.7	35/26, 39/29
66	520.6	23/17, 27/20
67	528.4	19/14
68	536.3	15/11
69	544.2	26/19
70	552.1	11/8
71	560	29/21
72	567.9	25/18
73	575.8	39/28
74	583.6	7/5
75	591.5	38/27
76	599.4	41/29
77	607.3	27/19
78	615.2	10/7
79	623.1	33/23
80	631	23/16, 36/25
81	638.9	13/9, 42/29
82	646.7	16/11
83	654.6	19/13, 35/24
84	662.5	22/15, 41/28
85	670.4	28/19
86	678.3	34/23, 40/27
87	686.2
88	694.1
89	702	3/2

Music

Bryan Deister

• 89ed(3/2) groove (2026)