9ed7/3
9 equal divisions of 7/3 (abbreviated 9ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 9 equal parts of about 163 ¢ each. Each step represents a frequency ratio of (7/3)1/9, or the 9th root of 7/3.
Theory
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -59.1 | +53.9 | +44.8 | -15.6 | -5.2 | +53.9 | -14.3 | -55.2 | -74.7 | -76.7 | -64.3 |
| Relative (%) | -36.3 | +33.1 | +27.5 | -9.5 | -3.2 | +33.1 | -8.8 | -33.9 | -45.8 | -47.0 | -39.5 | |
| Steps (reduced) |
7 (7) |
12 (3) |
15 (6) |
17 (8) |
19 (1) |
21 (3) |
22 (4) |
23 (5) |
24 (6) |
25 (7) |
26 (8) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -39.9 | -5.2 | +38.3 | -73.4 | -15.4 | +48.6 | -45.0 | +29.2 | -55.2 | +27.2 | -49.7 | +39.6 |
| Relative (%) | -24.5 | -3.2 | +23.5 | -45.0 | -9.4 | +29.8 | -27.6 | +17.9 | -33.9 | +16.7 | -30.5 | +24.3 | |
| Steps (reduced) |
27 (0) |
28 (1) |
29 (2) |
29 (2) |
30 (3) |
31 (4) |
31 (4) |
32 (5) |
32 (5) |
33 (6) |
33 (6) |
34 (7) | |
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 163 | 11/10, 12/11, 13/12, 14/13, 19/17 |
| 2 | 326 | 6/5, 13/11, 17/14 |
| 3 | 489 | 13/10, 17/13 |
| 4 | 651.9 | 19/13 |
| 5 | 814.9 | 19/12 |
| 6 | 977.9 | 19/11 |
| 7 | 1140.9 | 19/10 |
| 8 | 1303.9 | 13/6, 15/7 |
| 9 | 1466.9 | 7/3 |
Lumatone mappings
On the Xenharmonic Alliance Discord in September 2025, Maeve Gutierrez noted that the notes of 3ed7/3 make for a nice chord when played simultaneously. Budjarn Lambeth noted that 9ed7/3 is a good tuning for using said chord.
Lambeth suggested the following mappings for using 9ed7/3 on an isomorphic keyboard like the Lumatone. He suggested using 1\9<7/3> for the x-steps and 3\9<7/3>\49 for the y-steps or vice versa.
This gives easy access to Gutierrez's 3ed7/3 chord above any note, while also allowing satisfying bends using the neutral second interval of ~11/10.
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