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
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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 41/123, 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
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) |
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) |