123ed4

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123ed4

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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	19.5
2	39	43/42, 47/46
3	58.5	30/29, 31/30
4	78
5	97.6	18/17
6	117.1	31/29, 46/43
7	136.6
8	156.1	23/21
9	175.6
10	195.1	47/42
11	214.6	43/38
12	234.1
13	253.7
14	273.2
15	292.7
16	312.2
17	331.7	23/19
18	351.2	38/31
19	370.7	31/25
20	390.2
21	409.8	19/15
22	429.3
23	448.8
24	468.3	38/29
25	487.8
26	507.3
27	526.8	42/31
28	546.3
29	565.9	43/31
30	585.4
31	604.9
32	624.4	33/23, 43/30
33	643.9
34	663.4
35	682.9	43/29, 46/31
36	702.4
37	722	47/31
38	741.5	23/15
39	761
40	780.5
41	800	46/29
42	819.5
43	839
44	858.5
45	878
46	897.6	42/25
47	917.1
48	936.6
49	956.1	33/19
50	975.6
51	995.1
52	1014.6
53	1034.1
54	1053.7
55	1073.2	13/7
56	1092.7	47/25
57	1112.2
58	1131.7	25/13
59	1151.2
60	1170.7
61	1190.2
62	1209.8
63	1229.3
64	1248.8	37/18
65	1268.3
66	1287.8
67	1307.3
68	1326.8
69	1346.3	37/17
70	1365.9	11/5
71	1385.4	49/22
72	1404.9
73	1424.4	41/18
74	1443.9
75	1463.4
76	1482.9
77	1502.4
78	1522
79	1541.5
80	1561
81	1580.5
82	1600
83	1619.5
84	1639
85	1658.5
86	1678	29/11
87	1697.6
88	1717.1
89	1736.6	30/11
90	1756.1
91	1775.6
92	1795.1	31/11
93	1814.6
94	1834.1
95	1853.7	35/12
96	1873.2
97	1892.7
98	1912.2
99	1931.7
100	1951.2
101	1970.7
102	1990.2
103	2009.8
104	2029.3	42/13
105	2048.8
106	2068.3
107	2087.8
108	2107.3
109	2126.8
110	2146.3	38/11
111	2165.9
112	2185.4
113	2204.9	25/7
114	2224.4	47/13
115	2243.9
116	2263.4
117	2282.9
118	2302.4
119	2322
120	2341.5
121	2361	43/11
122	2380.5
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)