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.

Theory

This tuning tempers out 49/48 in the 7-limit; 99/98 and 33/32 in the 11-limit; 91/90 and 105/104 in the 13-limit; 57/56 in the 19-limit; 115/114 and 115/112 in the 23-limit; 93/92 in the 31-limit; 37/36 in the 37-limit; 82/81 in the 41-limit; 87/86 in the 43-limit; 95/94 in the 47-limit; and 118/117 in the 59-limit.

Harmonics

Approximation of harmonics in 113ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Error	Absolute (¢)	+10.62	+9.55	+0.00	-4.01	-1.07	+8.17	+10.62	-2.14	+6.61	-9.73	+9.55	-1.59	-2.45	+5.54	+0.00
	Relative (%)	+50.0	+45.0	+0.0	-18.9	-5.0	+38.4	+50.0	-10.1	+31.1	-45.8	+45.0	-7.5	-11.6	+26.1	+0.0
Steps (reduced)		57 (57)	90 (90)	113 (0)	131 (18)	146 (33)	159 (46)	170 (57)	179 (66)	188 (75)	195 (82)	203 (90)	209 (96)	215 (102)	221 (108)	226 (0)

Approximation of harmonics in 113ed4

Harmonic		17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
Error	Absolute (¢)	+1.24	+8.48	-0.17	-4.01	-3.52	+0.89	+8.89	-1.07	-8.03	+9.03	+7.41	+8.17	-10.11	-5.08	+1.87
	Relative (%)	+5.8	+39.9	-0.8	-18.9	-16.6	+4.2	+41.9	-5.0	-37.8	+42.5	+34.9	+38.4	-47.6	-23.9	+8.8
Steps (reduced)		231 (5)	236 (10)	240 (14)	244 (18)	248 (22)	252 (26)	256 (30)	259 (33)	262 (36)	266 (40)	269 (43)	272 (46)	274 (48)	277 (51)	280 (54)

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	21.2
2	42.5
3	63.7	28/27
4	85	21/20, 41/39
5	106.2	33/31
6	127.4
7	148.7
8	169.9	43/39
9	191.2	19/17
10	212.4	26/23, 43/38
11	233.6
12	254.9	22/19, 29/25
13	276.1
14	297.3
15	318.6
16	339.8
17	361.1
18	382.3
19	403.5
20	424.8
21	446	22/17
22	467.3
23	488.5
24	509.7	47/35
25	531
26	552.2
27	573.5
28	594.7	31/22
29	615.9	10/7
30	637.2
31	658.4	19/13
32	679.6	37/25, 40/27
33	700.9	3/2
34	722.1	47/31
35	743.4
36	764.6	14/9
37	785.8
38	807.1
39	828.3
40	849.6	31/19
41	870.8	38/23, 43/26
42	892
43	913.3	39/23
44	934.5
45	955.8	33/19
46	977
47	998.2
48	1019.5
49	1040.7	31/17
50	1061.9
51	1083.2	43/23
52	1104.4
53	1125.7
54	1146.9	33/17
55	1168.1
56	1189.4
57	1210.6
58	1231.9
59	1253.1
60	1274.3
61	1295.6
62	1316.8
63	1338.1
64	1359.3	46/21
65	1380.5	20/9
66	1401.8	9/4
67	1423	25/11
68	1444.2
69	1465.5	7/3
70	1486.7
71	1508
72	1529.2
73	1550.4
74	1571.7
75	1592.9
76	1614.2	33/13
77	1635.4
78	1656.6
79	1677.9	29/11
80	1699.1
81	1720.4
82	1741.6	41/15
83	1762.8
84	1784.1
85	1805.3
86	1826.5
87	1847.8
88	1869
89	1890.3
90	1911.5
91	1932.7
92	1954
93	1975.2	47/15
94	1996.5
95	2017.7
96	2038.9
97	2060.2	23/7
98	2081.4
99	2102.7
100	2123.9
101	2145.1
102	2166.4
103	2187.6
104	2208.8
105	2230.1
106	2251.3
107	2272.6	26/7
108	2293.8
109	2315
110	2336.3
111	2357.5	39/10
112	2378.8
113	2400