90ed10

← 89ed10

90ed10

91ed10 →

Prime factorization	2 × 32 × 5
Step size	44.2924 ¢
Octave	27\90ed10 (1195.89 ¢) (→ 3\10ed10)
Twelfth	43\90ed10 (1904.57 ¢)
Consistency limit	10
Distinct consistency limit	8

90 equal divisions of the 10th harmonic (abbreviated 90ed10) is a nonoctave tuning system that divides the interval of 10/1 into 90 equal parts of about 44.3 ¢ each. Each step represents a frequency ratio of 10^1/90, or the 90th root of 10.

Theory

90ed10 is closely related to 27edo, but with the 10th harmonic rather than the octave being just, which compresses the octave by about 4.11 ¢. The local zeta peak around 27 is located at 27.086614, which has a step size of 44.3023 ¢, making 90ed10 very close to optimal for 27edo.

Harmonics

Approximation of harmonics in 90ed10

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-4.1	+2.6	-8.2	+4.1	-1.5	-2.6	-12.3	+5.2	+0.0	+12.2	-5.6
	Relative (%)	-9.3	+5.9	-18.5	+9.3	-3.4	-5.9	-27.8	+11.8	+0.0	+27.5	-12.6
Steps (reduced)		27 (27)	43 (43)	54 (54)	63 (63)	70 (70)	76 (76)	81 (81)	86 (86)	90 (0)	94 (4)	97 (7)

Approximation of harmonics in 90ed10 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	-11.3	-6.7	+6.7	-16.4	+11.5	+1.1	-3.9	-4.1	+0.0	+8.1	+19.7	-9.7
	Relative (%)	-25.5	-15.2	+15.2	-37.1	+26.0	+2.5	-8.8	-9.3	+0.0	+18.2	+44.4	-21.9
Steps (reduced)		100 (10)	103 (13)	106 (16)	108 (18)	111 (21)	113 (23)	115 (25)	117 (27)	119 (29)	121 (31)	123 (33)	124 (34)

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	44.3	38/37, 39/38, 40/39, 41/40, 42/41
2	88.6	20/19, 39/37, 41/39
3	132.9	27/25, 40/37, 41/38
4	177.2	31/28, 41/37
5	221.5	25/22, 33/29, 42/37
6	265.8	7/6
7	310
8	354.3	27/22, 38/31
9	398.6	29/23, 34/27, 39/31
10	442.9	31/24, 40/31
11	487.2
12	531.5	34/25
13	575.8	39/28
14	620.1	10/7
15	664.4	22/15
16	708.7
17	753	17/11
18	797.3	19/12
19	841.6	13/8
20	885.8	5/3
21	930.1
22	974.4
23	1018.7	9/5
24	1063	24/13, 37/20
25	1107.3	36/19
26	1151.6	35/18, 37/19
27	1195.9
28	1240.2	41/20
29	1284.5	21/10
30	1328.8	28/13, 41/19
31	1373.1	42/19
32	1417.4	34/15
33	1461.6
34	1505.9	31/13
35	1550.2
36	1594.5
37	1638.8
38	1683.1	37/14
39	1727.4	19/7
40	1771.7	39/14
41	1816	20/7
42	1860.3	41/14
43	1904.6	3/1
44	1948.9	37/12
45	1993.2	19/6
46	2037.4
47	2081.7	10/3
48	2126	41/12
49	2170.3	7/2
50	2214.6
51	2258.9
52	2303.2	34/9
53	2347.5	31/8
54	2391.8
55	2436.1
56	2480.4
57	2524.7
58	2569
59	2613.3
60	2657.5
61	2701.8
62	2746.1
63	2790.4
64	2834.7	36/7
65	2879
66	2923.3
67	2967.6
68	3011.9
69	3056.2
70	3100.5	6/1
71	3144.8
72	3189.1
73	3233.3
74	3277.6
75	3321.9
76	3366.2	7/1
77	3410.5
78	3454.8
79	3499.1
80	3543.4	31/4
81	3587.7
82	3632
83	3676.3
84	3720.6
85	3764.9
86	3809.1
87	3853.4	37/4
88	3897.7	19/2
89	3942	39/4
90	3986.3	10/1

See also

• 16edf – relative edf
• 27edo – relative edo
• 43edt – relative edt
• 70ed6 – relative ed6
• 97ed12 – relative ed12