135ed4
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Prime factorization
33 × 5
Step size
17.7778¢
Octave
68\135ed4 (1208.89¢)
Twelfth
107\135ed4 (1902.22¢)
(semiconvergent)
Consistency limit
1
Distinct consistency limit
1
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← 133ed4 | 135ed4 | 137ed4 → |
(semiconvergent)
135 equal divisions of the 4th harmonic (abbreviated 135ed4) is a nonoctave tuning system that divides the interval of 4/1 into 135 equal parts of about 17.8 ¢ each. Each step represents a frequency ratio of 41/135, or the 135th root of 4.
Intervals
Steps | Cents | Approximate ratios |
---|---|---|
0 | 0 | 1/1 |
1 | 17.8 | |
2 | 35.6 | 47/46 |
3 | 53.3 | 34/33 |
4 | 71.1 | |
5 | 88.9 | |
6 | 106.7 | |
7 | 124.4 | 29/27 |
8 | 142.2 | 51/47 |
9 | 160 | 45/41 |
10 | 177.8 | 41/37, 51/46 |
11 | 195.6 | 28/25, 47/42 |
12 | 213.3 | |
13 | 231.1 | |
14 | 248.9 | 15/13 |
15 | 266.7 | |
16 | 284.4 | 33/28, 46/39 |
17 | 302.2 | |
18 | 320 | |
19 | 337.8 | 17/14, 45/37 |
20 | 355.6 | 43/35 |
21 | 373.3 | |
22 | 391.1 | |
23 | 408.9 | 19/15 |
24 | 426.7 | |
25 | 444.4 | |
26 | 462.2 | |
27 | 480 | 33/25 |
28 | 497.8 | |
29 | 515.6 | 31/23 |
30 | 533.3 | 34/25 |
31 | 551.1 | |
32 | 568.9 | 25/18 |
33 | 586.7 | |
34 | 604.4 | |
35 | 622.2 | |
36 | 640 | 42/29 |
37 | 657.8 | 19/13 |
38 | 675.6 | 31/21 |
39 | 693.3 | |
40 | 711.1 | |
41 | 728.9 | |
42 | 746.7 | |
43 | 764.4 | 14/9 |
44 | 782.2 | |
45 | 800 | 27/17, 46/29 |
46 | 817.8 | |
47 | 835.6 | 47/29 |
48 | 853.3 | 18/11 |
49 | 871.1 | |
50 | 888.9 | |
51 | 906.7 | |
52 | 924.4 | 29/17 |
53 | 942.2 | |
54 | 960 | 47/27 |
55 | 977.8 | 51/29 |
56 | 995.6 | |
57 | 1013.3 | |
58 | 1031.1 | |
59 | 1048.9 | 11/6 |
60 | 1066.7 | |
61 | 1084.4 | 43/23 |
62 | 1102.2 | 17/9 |
63 | 1120 | |
64 | 1137.8 | 27/14 |
65 | 1155.6 | 37/19 |
66 | 1173.3 | |
67 | 1191.1 | |
68 | 1208.9 | |
69 | 1226.7 | |
70 | 1244.4 | 39/19 |
71 | 1262.2 | 29/14 |
72 | 1280 | |
73 | 1297.8 | |
74 | 1315.6 | |
75 | 1333.3 | |
76 | 1351.1 | |
77 | 1368.9 | |
78 | 1386.7 | |
79 | 1404.4 | |
80 | 1422.2 | 25/11 |
81 | 1440 | |
82 | 1457.8 | |
83 | 1475.6 | |
84 | 1493.3 | 45/19 |
85 | 1511.1 | |
86 | 1528.9 | |
87 | 1546.7 | |
88 | 1564.4 | 37/15, 42/17 |
89 | 1582.2 | |
90 | 1600 | |
91 | 1617.8 | 28/11 |
92 | 1635.6 | |
93 | 1653.3 | 13/5 |
94 | 1671.1 | |
95 | 1688.9 | |
96 | 1706.7 | |
97 | 1724.4 | 46/17 |
98 | 1742.2 | 41/15 |
99 | 1760 | 47/17 |
100 | 1777.8 | |
101 | 1795.6 | |
102 | 1813.3 | |
103 | 1831.1 | |
104 | 1848.9 | |
105 | 1866.7 | |
106 | 1884.4 | |
107 | 1902.2 | 3/1 |
108 | 1920 | |
109 | 1937.8 | |
110 | 1955.6 | |
111 | 1973.3 | |
112 | 1991.1 | |
113 | 2008.9 | |
114 | 2026.7 | 29/9 |
115 | 2044.4 | |
116 | 2062.2 | |
117 | 2080 | |
118 | 2097.8 | 47/14 |
119 | 2115.6 | |
120 | 2133.3 | |
121 | 2151.1 | 45/13 |
122 | 2168.9 | |
123 | 2186.7 | 46/13 |
124 | 2204.4 | |
125 | 2222.2 | |
126 | 2240 | |
127 | 2257.8 | |
128 | 2275.6 | |
129 | 2293.3 | |
130 | 2311.1 | 19/5 |
131 | 2328.9 | |
132 | 2346.7 | |
133 | 2364.4 | |
134 | 2382.2 | |
135 | 2400 |
Harmonics
Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +8.89 | +0.27 | +0.00 | +4.80 | -8.62 | -8.83 | +8.89 | +0.53 | -4.09 | +8.68 | +0.27 |
Relative (%) | +50.0 | +1.5 | +0.0 | +27.0 | -48.5 | -49.6 | +50.0 | +3.0 | -23.0 | +48.8 | +1.5 | |
Steps (reduced) |
68 (68) |
107 (107) |
135 (0) |
157 (22) |
174 (39) |
189 (54) |
203 (68) |
214 (79) |
224 (89) |
234 (99) |
242 (107) |
Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +3.92 | +0.06 | +5.06 | +0.00 | +1.71 | -8.35 | +4.71 | +4.80 | -8.56 | -0.21 | -6.05 |
Relative (%) | +22.0 | +0.4 | +28.5 | +0.0 | +9.6 | -47.0 | +26.5 | +27.0 | -48.1 | -1.2 | -34.0 | |
Steps (reduced) |
250 (115) |
257 (122) |
264 (129) |
270 (0) |
276 (6) |
281 (11) |
287 (17) |
292 (22) |
296 (26) |
301 (31) |
305 (35) |