9ed9/8
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Prime factorization
32
Step size
22.6567¢
Octave
53\9ed9/8 (1200.8¢)
(convergent)
Twelfth
84\9ed9/8 (1903.16¢) (→28\3ed9/8)
Consistency limit
10
Distinct consistency limit
4
← 8ed9/8 | 9ed9/8 | 10ed9/8 → |
(convergent)
9 equal divisions of 9/8 (abbreviated 9ed9/8) is a nonoctave tuning system that divides the interval of 9/8 into 9 equal parts of about 22.7 ¢ each. Each step represents a frequency ratio of (9/8)1/9, or the 9th root of 9/8. It corresponds to 52.9645 edo, which is closely related to 53edo but with the whole tone instead of the octave tuned pure.
Harmonics
Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +0.80 | +1.21 | +1.61 | +0.46 | +2.01 | +7.02 | +2.41 | +2.41 | +1.26 | -5.15 | +2.81 |
Relative (%) | +3.5 | +5.3 | +7.1 | +2.0 | +8.9 | +31.0 | +10.6 | +10.6 | +5.6 | -22.7 | +12.4 | |
Steps (reduced) |
53 (8) |
84 (3) |
106 (7) |
123 (6) |
137 (2) |
149 (5) |
159 (6) |
168 (6) |
176 (5) |
183 (3) |
190 (1) |
Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +0.18 | +7.82 | +1.66 | +3.21 | -11.12 | +3.21 | +0.24 | +2.06 | +8.22 | -4.34 | +9.33 |
Relative (%) | +0.8 | +34.5 | +7.3 | +14.2 | -49.1 | +14.2 | +1.0 | +9.1 | +36.3 | -19.2 | +41.2 | |
Steps (reduced) |
196 (7) |
202 (4) |
207 (0) |
212 (5) |
216 (0) |
221 (5) |
225 (0) |
229 (4) |
233 (8) |
236 (2) |
240 (6) |
Intervals
# | Cents Value | Ratio |
---|---|---|
0 | 0.0000 | 1/1 |
1 | 22.6567 | (9/8)1/9 |
2 | 45.3133 | (9/8)2/9 |
3 | 67.9700 | (9/8)1/3 |
4 | 90.6267 | (9/8)4/9 |
5 | 113.2833 | (9/8)5/9 |
6 | 135.9400 | (9/8)2/3 |
7 | 158.5967 | (9/8)7/9 |
8 | 181.2533 | (9/8)8/9 |
9 | 203.9100 | 9/8 |
10 | 226.5667 | (9/8)10/9 |
11 | 249.2233 | (9/8)11/9 |
12 | 271.8800 | (9/8)4/3 |
13 | 294.5367 | (9/8)13/9 |
14 | 317.1933 | (9/8)14/9 |
15 | 339.8500 | (9/8)5/3 |
16 | 362.5067 | (9/8)16/9 |
17 | 385.1633 | (9/8)17/9 |
18 | 407.8200 | (9/8)2 = 81/64 |
19 | 430.4767 | (9/8)19/9 |
20 | 453.1333 | (9/8)20/9 |
21 | 475.7900 | (9/8)7/3 |
22 | 498.4467 | (9/8)22/9 |
23 | 521.1033 | (9/8)23/9 |
24 | 543.7600 | (9/8)8/3 |
25 | 566.4167 | (9/8)25/9 |
26 | 589.0733 | (9/8)26/9 |
27 | 611.7300 | (9/8)3 = 729/512 |
28 | 634.3867 | (9/8)28/9 |
29 | 657.0433 | (9/8)29/9 |
30 | 679.7000 | (9/8)10/3 |
31 | 702.3567 | (9/8)31/9 |
32 | 725.0133 | (9/8)32/9 |
33 | 747.6700 | (9/8)11/3 |
34 | 770.3267 | (9/8)34/9 |
35 | 792.9833 | (9/8)35/9 |
36 | 815.6400 | (9/8)4 = 6561/4096 |
37 | 838.2967 | (9/8)37/9 |
38 | 860.9533 | (9/8)38/9 |
39 | 883.6100 | (9/8)13/3 |
40 | 906.2667 | (9/8)40/9 |
41 | 928.9233 | (9/8)41/9 |
42 | 951.5800 | (9/8)14/3 |
43 | 974.2367 | (9/8)43/9 |
44 | 996.8933 | (9/8)44/9 |
45 | 1019.5500 | (9/8)5 = 59049/32768 |
46 | 1042.2067 | (9/8)46/9 |
47 | 1064.8633 | (9/8)47/9 |
48 | 1087.5200 | (9/8)16/3 |
49 | 1110.1767 | (9/8)49/9 |
50 | 1132.8333 | (9/8)50/9 |
51 | 1155.4900 | (9/8)17/3 |
52 | 1178.1467 | (9/8)52/9 |
53 | 1200.8033 | (9/8)53/9 |
54 | 1223.4600 | (9/8)6 = 531441/262144 |
Approximation to JI
15-odd-limit mappings
The following table shows how 15-odd-limit intervals are represented in 9ed9/8 (ordered by absolute error).
Interval(s) | Error (abs, ¢) |
---|---|
9/8 | 0.000 |
3/2, 4/3 | 0.402 |
26/15 | 0.679 |
15/8, 5/3 | 0.749 |
16/9 | 0.803 |
13/10 | 1.081 |
5/4, 10/9 | 1.150 |
15/13 | 1.482 |
6/5, 16/15 | 1.552 |
20/13 | 1.884 |
9/5, 8/5 | 1.954 |
13/8, 13/9 | 2.231 |
13/12 | 2.633 |
16/13, 18/13 | 3.034 |
24/13 | 3.436 |
12/7 | 4.206 |
22/13 | 4.524 |
9/7, 8/7 | 4.607 |
7/6 | 5.009 |
13/11 | 5.327 |
7/4, 14/9 | 5.411 |
10/7 | 5.758 |
22/15 | 6.006 |
15/14 | 6.159 |
11/10 | 6.408 |
7/5 | 6.561 |
15/11 | 6.809 |
13/7 | 6.838 |
28/15 | 6.963 |
11/6 | 7.156 |
20/11 | 7.211 |
11/9, 11/8 | 7.558 |
14/13 | 7.642 |
12/11 | 7.960 |
18/11, 16/11 | 8.361 |
14/11 | 9.688 |
11/7 | 10.491 |