113ed4

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113ed4

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Prime factorization	113 (prime)
Step size	21.2389¢
Octave	57\113ed4 (1210.62¢)
Twelfth	90\113ed4 (1911.5¢)
Consistency limit	1
Distinct consistency limit	1

113 equal divisions of the 4th harmonic (abbreviated 113ed4) is a nonoctave tuning system that divides the interval of 4/1 into 113 equal parts of about 21.2 ¢ each. Each step represents a frequency ratio of 4^1/113, or the 113th root of 4.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	21.239
2	42.478
3	63.717	28/27
4	84.956	21/20, 41/39
5	106.195	33/31
6	127.434
7	148.673
8	169.912	43/39
9	191.15	19/17
10	212.389	26/23, 43/38
11	233.628
12	254.867	22/19, 29/25
13	276.106
14	297.345
15	318.584
16	339.823
17	361.062
18	382.301
19	403.54
20	424.779
21	446.018	22/17
22	467.257
23	488.496
24	509.735	47/35
25	530.973
26	552.212
27	573.451
28	594.69	31/22
29	615.929	10/7
30	637.168
31	658.407	19/13
32	679.646	37/25, 40/27
33	700.885	3/2
34	722.124	47/31
35	743.363
36	764.602	14/9
37	785.841
38	807.08
39	828.319
40	849.558	31/19
41	870.796	38/23, 43/26
42	892.035
43	913.274	39/23
44	934.513
45	955.752	33/19
46	976.991
47	998.23
48	1019.469
49	1040.708	31/17
50	1061.947
51	1083.186	43/23
52	1104.425
53	1125.664
54	1146.903	33/17
55	1168.142
56	1189.381
57	1210.619
58	1231.858
59	1253.097
60	1274.336
61	1295.575
62	1316.814
63	1338.053
64	1359.292	46/21
65	1380.531	20/9
66	1401.77	9/4
67	1423.009	25/11
68	1444.248
69	1465.487	7/3
70	1486.726
71	1507.965
72	1529.204
73	1550.442
74	1571.681
75	1592.92
76	1614.159	33/13
77	1635.398
78	1656.637
79	1677.876	29/11
80	1699.115
81	1720.354
82	1741.593	41/15
83	1762.832
84	1784.071
85	1805.31
86	1826.549
87	1847.788
88	1869.027
89	1890.265
90	1911.504
91	1932.743
92	1953.982
93	1975.221	47/15
94	1996.46
95	2017.699
96	2038.938
97	2060.177	23/7
98	2081.416
99	2102.655
100	2123.894
101	2145.133
102	2166.372
103	2187.611
104	2208.85
105	2230.088
106	2251.327
107	2272.566	26/7
108	2293.805
109	2315.044
110	2336.283
111	2357.522	39/10
112	2378.761
113	2400

Harmonics

Approximation of harmonics in 113ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+10.62	+9.55	+0.00	-4.01	-1.07	+8.17	+10.62	-2.14	+6.61	-9.73	+9.55
	Relative (%)	+50.0	+45.0	+0.0	-18.9	-5.0	+38.4	+50.0	-10.1	+31.1	-45.8	+45.0
Steps (reduced)		57 (57)	90 (90)	113 (0)	131 (18)	146 (33)	159 (46)	170 (57)	179 (66)	188 (75)	195 (82)	203 (90)

Approximation of harmonics in 113ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-1.59	-2.45	+5.54	+0.00	+1.24	+8.48	-0.17	-4.01	-3.52	+0.89	+8.89
	Relative (%)	-7.5	-11.6	+26.1	+0.0	+5.8	+39.9	-0.8	-18.9	-16.6	+4.2	+41.9
Steps (reduced)		209 (96)	215 (102)	221 (108)	226 (0)	231 (5)	236 (10)	240 (14)	244 (18)	248 (22)	252 (26)	256 (30)