111ed4

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111ed4

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Prime factorization	3 × 37
Step size	21.6216 ¢
Octave	56\111ed4 (1210.81 ¢)
Twelfth	88\111ed4 (1902.7 ¢)
Consistency limit	1
Distinct consistency limit	1

111 equal divisions of the 4th harmonic (abbreviated 111ed4) is a nonoctave tuning system that divides the interval of 4/1 into 111 equal parts of about 21.6 ¢ each. Each step represents a frequency ratio of 4^1/111, or the 111th root of 4.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	21.6
2	43.2
3	64.9	27/26
4	86.5	41/39
5	108.1	33/31
6	129.7
7	151.4
8	173	21/19
9	194.6	19/17
10	216.2	17/15
11	237.8	31/27, 47/41
12	259.5	43/37
13	281.1
14	302.7	25/21, 31/26
15	324.3	35/29, 47/39
16	345.9	11/9
17	367.6	21/17, 26/21
18	389.2
19	410.8	19/15, 33/26
20	432.4
21	454.1
22	475.7	25/19
23	497.3
24	518.9	31/23
25	540.5
26	562.2
27	583.8	7/5
28	605.4
29	627	33/23
30	648.6
31	670.3
32	691.9
33	713.5
34	735.1	26/17
35	756.8
36	778.4
37	800	27/17, 46/29
38	821.6	37/23
39	843.2
40	864.9
41	886.5	5/3
42	908.1
43	929.7
44	951.4	26/15, 45/26
45	973
46	994.6
47	1016.2	9/5
48	1037.8
49	1059.5	35/19
50	1081.1	43/23
51	1102.7	17/9
52	1124.3
53	1145.9
54	1167.6
55	1189.2
56	1210.8
57	1232.4
58	1254.1
59	1275.7	23/11
60	1297.3
61	1318.9	15/7
62	1340.5
63	1362.2
64	1383.8
65	1405.4
66	1427
67	1448.6
68	1470.3
69	1491.9	45/19
70	1513.5
71	1535.1	17/7
72	1556.8
73	1578.4
74	1600
75	1621.6
76	1643.2
77	1664.9
78	1686.5	45/17
79	1708.1
80	1729.7	19/7
81	1751.4
82	1773
83	1794.6	31/11
84	1816.2
85	1837.8	26/9
86	1859.5
87	1881.1
88	1902.7	3/1
89	1924.3
90	1945.9
91	1967.6
92	1989.2	41/13
93	2010.8
94	2032.4
95	2054.1
96	2075.7
97	2097.3
98	2118.9	17/5
99	2140.5	31/9
100	2162.2
101	2183.8
102	2205.4	25/7
103	2227	47/13
104	2248.6	11/3
105	2270.3	26/7
106	2291.9
107	2313.5
108	2335.1	27/7
109	2356.8
110	2378.4
111	2400

Harmonics

Approximation of harmonics in 111ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+10.81	+0.75	+0.00	+2.88	-10.06	+4.15	+10.81	+1.50	-7.94	+0.03	+0.75
	Relative (%)	+50.0	+3.5	+0.0	+13.3	-46.5	+19.2	+50.0	+6.9	-36.7	+0.2	+3.5
Steps (reduced)		56 (56)	88 (88)	111 (0)	129 (18)	143 (32)	156 (45)	167 (56)	176 (65)	184 (73)	192 (81)	199 (88)

Approximation of harmonics in 111ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-8.10	-6.66	+3.62	+0.00	+3.15	-9.32	+5.19	+2.88	+4.89	-10.78	-1.25
	Relative (%)	-37.4	-30.8	+16.8	+0.0	+14.6	-43.1	+24.0	+13.3	+22.6	-49.8	-5.8
Steps (reduced)		205 (94)	211 (100)	217 (106)	222 (0)	227 (5)	231 (9)	236 (14)	240 (18)	244 (22)	247 (25)	251 (29)