9ed7/6
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9 equal divisions of 7/6 (abbreviated 9ed7/6) is a nonoctave tuning system that divides the interval of 7/6 into 9 equal parts of about 29.7 ¢ each. Each step represents a frequency ratio of (7/6)1/9, or the 9th root of 7/6.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -13.9 | -4.2 | +1.8 | +1.0 | +11.5 | +11.5 | -12.1 | -8.4 | -12.9 | +0.0 | -2.4 |
| Relative (%) | -46.9 | -14.2 | +6.2 | +3.4 | +38.9 | +38.9 | -40.7 | -28.4 | -43.5 | +0.0 | -8.0 | |
| Steps (reduced) |
40 (4) |
64 (1) |
81 (0) |
94 (4) |
105 (6) |
114 (6) |
121 (4) |
128 (2) |
134 (8) |
140 (5) |
145 (1) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.3 | -2.4 | -3.2 | +3.7 | -12.3 | +7.3 | +2.7 | +2.8 | +7.3 | -13.9 | -1.9 |
| Relative (%) | +24.7 | -8.0 | -10.8 | +12.4 | -41.6 | +24.7 | +9.1 | +9.6 | +24.7 | -46.9 | -6.4 | |
| Steps (reduced) |
150 (6) |
154 (1) |
158 (5) |
162 (0) |
165 (3) |
169 (7) |
172 (1) |
175 (4) |
178 (7) |
180 (0) |
183 (3) | |
Note that this tuning is very close to 81ed4 and thus to every other step of 81edo, although by patent val mapping the double octave (~4/1) is mapped inconsistently to 80\9ed7/6, thus requiring constitution of ~4/1 as 2♭ × 2♯ to yield the direct approximation 81\9ed7/6.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 29.7 | |
| 2 | 59.3 | 20/19, 22/21 |
| 3 | 89 | 18/17 |
| 4 | 118.6 | 12/11, 14/13, 15/14, 19/18, 21/20 |
| 5 | 148.3 | |
| 6 | 177.9 | 10/9, 11/10, 21/19 |
| 7 | 207.6 | 17/15, 19/17 |
| 8 | 237.2 | 9/8, 15/13, 22/19 |
| 9 | 266.9 | 20/17 |
Instruments
A Lumatone mapping for 9ed7/6 is available.
Music
- 9ed(7/6) improv (2025)