109ed4

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109ed4

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Prime factorization	109 (prime)
Step size	22.0183 ¢
Octave	55\109ed4 (1211.01 ¢)
Twelfth	86\109ed4 (1893.58 ¢)
Consistency limit	1
Distinct consistency limit	1

109 equal divisions of the 4th harmonic (abbreviated 109ed4) is a nonoctave tuning system that divides the interval of 4/1 into 109 equal parts of about 22 ¢ each. Each step represents a frequency ratio of 4^1/109, or the 109th root of 4.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	22
2	44
3	66.1	26/25
4	88.1	41/39
5	110.1	33/31
6	132.1
7	154.1	47/43
8	176.1	41/37
9	198.2	28/25, 37/33
10	220.2	25/22, 42/37
11	242.2
12	264.2
13	286.2
14	308.3	37/31, 43/36
15	330.3	23/19
16	352.3
17	374.3	36/29, 41/33
18	396.3	39/31
19	418.3	14/11
20	440.4
21	462.4	17/13, 47/36
22	484.4	41/31
23	506.4
24	528.4	19/14
25	550.5
26	572.5
27	594.5
28	616.5
29	638.5
30	660.6
31	682.6	43/29
32	704.6
33	726.6	35/23, 38/25
34	748.6
35	770.6
36	792.7	30/19
37	814.7
38	836.7	47/29
39	858.7	23/14
40	880.7
41	902.8
42	924.8	29/17
43	946.8	19/11
44	968.8
45	990.8
46	1012.8
47	1034.9
48	1056.9	35/19, 46/25
49	1078.9
50	1100.9
51	1122.9
52	1145
53	1167
54	1189
55	1211
56	1233
57	1255	31/15
58	1277.1	23/11
59	1299.1	36/17
60	1321.1	15/7
61	1343.1
62	1365.1	11/5
63	1387.2	29/13
64	1409.2
65	1431.2
66	1453.2
67	1475.2
68	1497.2
69	1519.3
70	1541.3
71	1563.3	37/15
72	1585.3	5/2
73	1607.3	43/17
74	1629.4
75	1651.4
76	1673.4
77	1695.4
78	1717.4
79	1739.4	41/15
80	1761.5	36/13, 47/17
81	1783.5	14/5
82	1805.5
83	1827.5
84	1849.5
85	1871.6
86	1893.6
87	1915.6
88	1937.6
89	1959.6
90	1981.7
91	2003.7	35/11
92	2025.7
93	2047.7
94	2069.7	43/13
95	2091.7
96	2113.8
97	2135.8
98	2157.8
99	2179.8
100	2201.8
101	2223.9	47/13
102	2245.9
103	2267.9
104	2289.9
105	2311.9	19/5
106	2333.9
107	2356
108	2378
109	2400

Harmonics

Approximation of harmonics in 109ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+11.01	-8.38	+0.00	+10.02	+2.63	-0.02	+11.01	+5.26	-0.99	+10.15	-8.38
	Relative (%)	+50.0	-38.0	+0.0	+45.5	+12.0	-0.1	+50.0	+23.9	-4.5	+46.1	-38.0
Steps (reduced)		55 (55)	86 (86)	109 (0)	127 (18)	141 (32)	153 (44)	164 (55)	173 (64)	181 (72)	189 (80)	195 (86)

Approximation of harmonics in 109ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+7.18	+10.99	+1.64	+0.00	+5.14	-5.74	+10.74	+10.02	-8.40	-0.86	+10.26
	Relative (%)	+32.6	+49.9	+7.4	+0.0	+23.3	-26.1	+48.8	+45.5	-38.1	-3.9	+46.6
Steps (reduced)		202 (93)	208 (99)	213 (104)	218 (0)	223 (5)	227 (9)	232 (14)	236 (18)	239 (21)	243 (25)	247 (29)