105ed4

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105ed4

107ed4 →

Prime factorization	3 × 5 × 7
Step size	22.8571 ¢
Octave	53\105ed4 (1211.43 ¢)
Twelfth	83\105ed4 (1897.14 ¢)
Consistency limit	1
Distinct consistency limit	1

105 equal divisions of the 4th harmonic (abbreviated 105ed4) is a nonoctave tuning system that divides the interval of 4/1 into 105 equal parts of about 22.9 ¢ each. Each step represents a frequency ratio of 4^1/105, or the 105th root of 4.

Theory

This tuning tempers out 135/128 in the 5-limit; 126/125 in the 7-limit; 45/44, 33/32, and 121/120 in the 11-limit; 91/90 and 91/88 in the 13-limit; 57/56 in the 19-limit; 115/114 in the 23-limit; 93/92, 125/124, and 63/62 in the 31-limit; 123/121 and 41/40 in the 41-limit; 87/86 in the 43-limit; 95/94 and 141/140 in the 47-limit; 119/118 in the 59-limit; 122/121, 61/60, and 123/122 in the 61-limit; 67/66, 67/64, and 135/134 in the 67-limit; 143/142 in the 71-limit; and 98/97 in the 97-limit.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	22.9
2	45.7
3	68.6	26/25
4	91.4	19/18, 39/37
5	114.3	31/29
6	137.1
7	160	45/41
8	182.9
9	205.7
10	228.6
11	251.4
12	274.3	41/35
13	297.1
14	320
15	342.9
16	365.7
17	388.6
18	411.4	19/15, 33/26
19	434.3	9/7
20	457.1	43/33
21	480	33/25
22	502.9
23	525.7	42/31
24	548.6
25	571.4
26	594.3
27	617.1
28	640	42/29
29	662.9
30	685.7
31	708.6
32	731.4	29/19
33	754.3	17/11
34	777.1
35	800	46/29
36	822.9	37/23
37	845.7	31/19
38	868.6
39	891.4
40	914.3	39/23
41	937.1	43/25
42	960
43	982.9	30/17, 37/21
44	1005.7	25/14
45	1028.6
46	1051.4	11/6
47	1074.3
48	1097.1
49	1120
50	1142.9	29/15
51	1165.7
52	1188.6
53	1211.4
54	1234.3
55	1257.1	31/15
56	1280
57	1302.9
58	1325.7
59	1348.6
60	1371.4	42/19
61	1394.3	38/17
62	1417.1
63	1440
64	1462.9
65	1485.7	33/14
66	1508.6	43/18
67	1531.4	46/19
68	1554.3
69	1577.1
70	1600
71	1622.9	23/9
72	1645.7
73	1668.6
74	1691.4
75	1714.3	35/13
76	1737.1	30/11
77	1760
78	1782.9	14/5
79	1805.7
80	1828.6
81	1851.4
82	1874.3
83	1897.1
84	1920
85	1942.9	43/14
86	1965.7
87	1988.6	41/13
88	2011.4
89	2034.3
90	2057.1	23/7
91	2080
92	2102.9
93	2125.7
94	2148.6	45/13
95	2171.4
96	2194.3
97	2217.1	18/5
98	2240
99	2262.9
100	2285.7
101	2308.6
102	2331.4
103	2354.3
104	2377.1
105	2400

Harmonics

Approximation of harmonics in 105ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+11.4	-4.8	+0.0	+2.3	+6.6	-8.8	+11.4	-9.6	-9.2	+8.7	-4.8
	Relative (%)	+50.0	-21.1	+0.0	+9.9	+28.9	-38.6	+50.0	-42.1	-40.1	+38.0	-21.1
Steps (reduced)		53 (53)	83 (83)	105 (0)	122 (17)	136 (31)	147 (42)	158 (53)	166 (61)	174 (69)	182 (77)	188 (83)

Approximation of harmonics in 105ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-6.2	+2.6	-2.6	+0.0	+9.3	+1.8	-0.4	+2.3	+9.2	-2.7	-11.1
	Relative (%)	-27.3	+11.4	-11.2	+0.0	+40.8	+7.9	-1.6	+9.9	+40.3	-12.0	-48.7
Steps (reduced)		194 (89)	200 (95)	205 (100)	210 (0)	215 (5)	219 (9)	223 (13)	227 (17)	231 (21)	234 (24)	237 (27)