9ed9/8
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| ← 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.
Theory
9ed9/8 corresponds to 52.9645…edo, which is closely related to 53edo but with the whole tone instead of the octave tuned pure. Like 53edo, 9ed9/8 is consistent to the 10-integer-limit, but it has a sharp tendency, with all the harmonics within 1 to 16 but 11 tuned sharp.
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 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.18 | +7.82 | +1.66 | +3.21 | -11.12 | +3.21 | +0.24 | +2.06 | +8.22 | -4.34 | +9.33 | +3.62 |
| Relative (%) | +0.8 | +34.5 | +7.3 | +14.2 | -49.1 | +14.2 | +1.0 | +9.1 | +36.3 | -19.2 | +41.2 | +16.0 | |
| Steps (reduced) |
196 (7) |
202 (4) |
207 (0) |
212 (5) |
216 (0) |
221 (5) |
225 (0) |
229 (4) |
233 (8) |
236 (2) |
240 (6) |
243 (0) | |
Subsets and supersets
9ed9/8 is the first odd composite ed9/8, containing 3ed9/8 as a subset.
Intervals
| # | Cents | Ratio |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 22.7 | (9/8)1/9 |
| 2 | 45.3 | (9/8)2/9 |
| 3 | 68.0 | (9/8)1/3 |
| 4 | 90.6 | (9/8)4/9 |
| 5 | 113.3 | (9/8)5/9 |
| 6 | 135.9 | (9/8)2/3 |
| 7 | 158.6 | (9/8)7/9 |
| 8 | 181.3 | (9/8)8/9 |
| 9 | 203.9 | 9/8 |
| 10 | 226.6 | (9/8)10/9 |
| 11 | 249.2 | (9/8)11/9 |
| 12 | 271.9 | (9/8)4/3 |
| 13 | 294.5 | (9/8)13/9 |
| 14 | 317.2 | (9/8)14/9 |
| 15 | 339.9 | (9/8)5/3 |
| 16 | 362.5 | (9/8)16/9 |
| 17 | 385.2 | (9/8)17/9 |
| 18 | 407.8 | (9/8)2 = 81/64 |
| 19 | 430.5 | (9/8)19/9 |
| 20 | 453.1 | (9/8)20/9 |
| 21 | 475.8 | (9/8)7/3 |
| 22 | 498.4 | (9/8)22/9 |
| 23 | 521.1 | (9/8)23/9 |
| 24 | 543.8 | (9/8)8/3 |
| 25 | 566.4 | (9/8)25/9 |
| 26 | 589.1 | (9/8)26/9 |
| 27 | 611.7 | (9/8)3 = 729/512 |
| 28 | 634.4 | (9/8)28/9 |
| 29 | 657.0 | (9/8)29/9 |
| 30 | 679.7 | (9/8)10/3 |
| 31 | 702.4 | (9/8)31/9 |
| 32 | 725.0 | (9/8)32/9 |
| 33 | 747.7 | (9/8)11/3 |
| 34 | 770.3 | (9/8)34/9 |
| 35 | 792.0 | (9/8)35/9 |
| 36 | 815.6 | (9/8)4 = 6561/4096 |
| 37 | 838.3 | (9/8)37/9 |
| 38 | 861.0 | (9/8)38/9 |
| 39 | 883.6 | (9/8)13/3 |
| 40 | 906.3 | (9/8)40/9 |
| 41 | 928.9 | (9/8)41/9 |
| 42 | 951.6 | (9/8)14/3 |
| 43 | 974.2 | (9/8)43/9 |
| 44 | 996.9 | (9/8)44/9 |
| 45 | 1019.6 | (9/8)5 = 59049/32768 |
| 46 | 1042.2 | (9/8)46/9 |
| 47 | 1064.9 | (9/8)47/9 |
| 48 | 1087.5 | (9/8)16/3 |
| 49 | 1110.2 | (9/8)49/9 |
| 50 | 1132.8 | (9/8)50/9 |
| 51 | 1155.5 | (9/8)17/3 |
| 52 | 1178.1 | (9/8)52/9 |
| 53 | 1200.8 | (9/8)53/9 |
| 54 | 1223.5 | (9/8)6 = 531441/262144 |