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	21.622
2	43.243
3	64.865	27/26
4	86.486	41/39
5	108.108	33/31
6	129.73
7	151.351
8	172.973	21/19
9	194.595	19/17
10	216.216	17/15
11	237.838	31/27, 47/41
12	259.459	43/37
13	281.081
14	302.703	25/21, 31/26
15	324.324	35/29, 47/39
16	345.946	11/9
17	367.568	21/17, 26/21
18	389.189
19	410.811	19/15, 33/26
20	432.432
21	454.054
22	475.676	25/19
23	497.297
24	518.919	31/23
25	540.541
26	562.162
27	583.784	7/5
28	605.405
29	627.027	33/23
30	648.649
31	670.27
32	691.892
33	713.514
34	735.135	26/17
35	756.757
36	778.378
37	800	27/17, 46/29
38	821.622	37/23
39	843.243
40	864.865
41	886.486	5/3
42	908.108
43	929.73
44	951.351	26/15, 45/26
45	972.973
46	994.595
47	1016.216	9/5
48	1037.838
49	1059.459	35/19
50	1081.081	43/23
51	1102.703	17/9
52	1124.324
53	1145.946
54	1167.568
55	1189.189
56	1210.811
57	1232.432
58	1254.054
59	1275.676	23/11
60	1297.297
61	1318.919	15/7
62	1340.541
63	1362.162
64	1383.784
65	1405.405
66	1427.027
67	1448.649
68	1470.27
69	1491.892	45/19
70	1513.514
71	1535.135	17/7
72	1556.757
73	1578.378
74	1600
75	1621.622
76	1643.243
77	1664.865
78	1686.486	45/17
79	1708.108
80	1729.73	19/7
81	1751.351
82	1772.973
83	1794.595	31/11
84	1816.216
85	1837.838	26/9
86	1859.459
87	1881.081
88	1902.703	3/1
89	1924.324
90	1945.946
91	1967.568
92	1989.189	41/13
93	2010.811
94	2032.432
95	2054.054
96	2075.676
97	2097.297
98	2118.919	17/5
99	2140.541	31/9
100	2162.162
101	2183.784
102	2205.405	25/7
103	2227.027	47/13
104	2248.649	11/3
105	2270.27	26/7
106	2291.892
107	2313.514
108	2335.135	27/7
109	2356.757
110	2378.378
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)