45ed4
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Prime factorization
32 × 5
Step size
53.3333¢
Octave
23\45ed4 (1226.67¢)
Twelfth
36\45ed4 (1920¢) (→4\5ed4)
Consistency limit
1
Distinct consistency limit
1
← 43ed4 | 45ed4 | 47ed4 → |
45 equal divisions of the 4th harmonic (abbreviated 45ed4) is a nonoctave tuning system that divides the interval of 4/1 into 45 equal parts of about 53.3 ¢ each. Each step represents a frequency ratio of 41/45, or the 45th root of 4. It is a tuning system equvalent to 22.5edo, representing every second step of 45edo.
Its step size is very close to the Alpharabian quarter tone 33/32. In addition, it tempers out the tritrizo comma in the 4.7/6 subgroup, and along with it the quartisma in the 33/32.7/6 subgroup.
Intervals
Steps | Cents | Approximate ratios |
---|---|---|
0 | 0 | 1/1 |
1 | 53.3 | |
2 | 106.7 | |
3 | 160 | 11/10, 23/21 |
4 | 213.3 | 17/15, 26/23 |
5 | 266.7 | |
6 | 320 | |
7 | 373.3 | 26/21, 31/25 |
8 | 426.7 | |
9 | 480 | |
10 | 533.3 | 15/11, 19/14 |
11 | 586.7 | 7/5 |
12 | 640 | |
13 | 693.3 | |
14 | 746.7 | |
15 | 800 | |
16 | 853.3 | 23/14 |
17 | 906.7 | |
18 | 960 | |
19 | 1013.3 | |
20 | 1066.7 | 13/7 |
21 | 1120 | 21/11 |
22 | 1173.3 | |
23 | 1226.7 | |
24 | 1280 | 21/10, 23/11 |
25 | 1333.3 | |
26 | 1386.7 | 20/9, 29/13 |
27 | 1440 | 23/10 |
28 | 1493.3 | 26/11 |
29 | 1546.7 | 22/9 |
30 | 1600 | |
31 | 1653.3 | 13/5 |
32 | 1706.7 | |
33 | 1760 | |
34 | 1813.3 | |
35 | 1866.7 | |
36 | 1920 | |
37 | 1973.3 | |
38 | 2026.7 | |
39 | 2080 | 10/3 |
40 | 2133.3 | |
41 | 2186.7 | |
42 | 2240 | |
43 | 2293.3 | |
44 | 2346.7 | |
45 | 2400 |
Harmonics
Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +26.7 | +18.0 | +0.0 | -13.0 | -8.6 | -8.8 | +26.7 | -17.2 | +13.7 | +8.7 | +18.0 |
Relative (%) | +50.0 | +33.8 | +0.0 | -24.3 | -16.2 | -16.5 | +50.0 | -32.3 | +25.7 | +16.3 | +33.8 | |
Steps (reduced) |
23 (23) |
36 (36) |
45 (0) |
52 (7) |
58 (13) |
63 (18) |
68 (23) |
71 (26) |
75 (30) |
78 (33) |
81 (36) |
Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | -13.9 | +17.8 | +5.1 | +0.0 | +1.7 | +9.4 | +22.5 | -13.0 | +9.2 | -18.0 | +11.7 |
Relative (%) | -26.0 | +33.5 | +9.5 | +0.0 | +3.2 | +17.7 | +42.2 | -24.3 | +17.3 | -33.7 | +22.0 | |
Steps (reduced) |
83 (38) |
86 (41) |
88 (43) |
90 (0) |
92 (2) |
94 (4) |
96 (6) |
97 (7) |
99 (9) |
100 (10) |
102 (12) |