123ed4

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123ed4

125ed4 →

Prime factorization	3 × 41
Step size	19.5122¢
Octave	62\123ed4 (1209.76¢)
Twelfth	97\123ed4 (1892.68¢)
Consistency limit	1
Distinct consistency limit	1

123 equal divisions of the 4th harmonic (abbreviated 123ed4) is a nonoctave tuning system that divides the interval of 4/1 into 123 equal parts of about 19.5 ¢ each. Each step represents a frequency ratio of 4^1/123, or the 123rd root of 4.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	19.512
2	39.024	43/42, 47/46
3	58.537	30/29, 31/30
4	78.049
5	97.561	18/17
6	117.073	31/29, 46/43
7	136.585
8	156.098	23/21
9	175.61
10	195.122	47/42
11	214.634	43/38
12	234.146
13	253.659
14	273.171
15	292.683
16	312.195
17	331.707	23/19
18	351.22	38/31
19	370.732	31/25
20	390.244
21	409.756	19/15
22	429.268
23	448.78
24	468.293	38/29
25	487.805
26	507.317
27	526.829	42/31
28	546.341
29	565.854	43/31
30	585.366
31	604.878
32	624.39	33/23, 43/30
33	643.902
34	663.415
35	682.927	43/29, 46/31
36	702.439
37	721.951	47/31
38	741.463	23/15
39	760.976
40	780.488
41	800	46/29
42	819.512
43	839.024
44	858.537
45	878.049
46	897.561	42/25
47	917.073
48	936.585
49	956.098	33/19
50	975.61
51	995.122
52	1014.634
53	1034.146
54	1053.659
55	1073.171	13/7
56	1092.683	47/25
57	1112.195
58	1131.707	25/13
59	1151.22
60	1170.732
61	1190.244
62	1209.756
63	1229.268
64	1248.78	37/18
65	1268.293
66	1287.805
67	1307.317
68	1326.829
69	1346.341	37/17
70	1365.854	11/5
71	1385.366	49/22
72	1404.878
73	1424.39	41/18
74	1443.902
75	1463.415
76	1482.927
77	1502.439
78	1521.951
79	1541.463
80	1560.976
81	1580.488
82	1600
83	1619.512
84	1639.024
85	1658.537
86	1678.049	29/11
87	1697.561
88	1717.073
89	1736.585	30/11
90	1756.098
91	1775.61
92	1795.122	31/11
93	1814.634
94	1834.146
95	1853.659	35/12
96	1873.171
97	1892.683
98	1912.195
99	1931.707
100	1951.22
101	1970.732
102	1990.244
103	2009.756
104	2029.268	42/13
105	2048.78
106	2068.293
107	2087.805
108	2107.317
109	2126.829
110	2146.341	38/11
111	2165.854
112	2185.366
113	2204.878	25/7
114	2224.39	47/13
115	2243.902
116	2263.415
117	2282.927
118	2302.439
119	2321.951
120	2341.463
121	2360.976	43/11
122	2380.488
123	2400

Harmonics

Approximation of harmonics in 123ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+9.76	-9.27	+0.00	+3.93	+0.48	+6.78	+9.76	+0.97	-5.83	+4.78	-9.27
	Relative (%)	+50.0	-47.5	+0.0	+20.1	+2.5	+34.8	+50.0	+5.0	-29.9	+24.5	-47.5
Steps (reduced)		62 (62)	97 (97)	123 (0)	143 (20)	159 (36)	173 (50)	185 (62)	195 (72)	204 (81)	213 (90)	220 (97)

Approximation of harmonics in 123ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+8.25	-2.97	-5.34	+0.00	-7.39	-8.79	-4.83	+3.93	-2.49	-4.98	-3.88
	Relative (%)	+42.3	-15.2	-27.4	+0.0	-37.9	-45.0	-24.8	+20.1	-12.8	-25.5	-19.9
Steps (reduced)		228 (105)	234 (111)	240 (117)	246 (0)	251 (5)	256 (10)	261 (15)	266 (20)	270 (24)	274 (28)	278 (32)