10edf
| ← 9edf | 10edf | 11edf → |
(semiconvergent)
(semiconvergent)
10 equal divisions of the perfect fifth (abbreviated 10edf or 10ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 10 equal parts of about 70.2 ¢ each. Each step represents a frequency ratio of (3/2)1/10, or the 10th root of 3/2.
Theory
10edf is related to 17edo, but with the perfect fifth rather than the octave being just. The octave is compressed by about 6.68 ¢, a small but significant deviation. 10edf is consistent to the 7-integer-limit, but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the 4-integer-limit. This makes 10edf a suitable tuning perhaps in the 5-limit, but overcompressed in any other limits, as well as the no-5 13-limit, where 17edo is best at.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -6.7 | -6.7 | -13.4 | +21.5 | -13.4 | +0.6 | -20.0 | -13.4 | +14.8 | -9.8 | -20.0 |
| Relative (%) | -9.5 | -9.5 | -19.0 | +30.6 | -19.0 | +0.8 | -28.5 | -19.0 | +21.1 | -13.9 | -28.5 | |
| Steps (reduced) |
17 (7) |
27 (7) |
34 (4) |
40 (0) |
44 (4) |
48 (8) |
51 (1) |
54 (4) |
57 (7) |
59 (9) |
61 (1) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -18.2 | -6.1 | +14.8 | -26.7 | +8.7 | -20.0 | +26.8 | +8.2 | -6.1 | -16.5 | -23.2 | -26.7 |
| Relative (%) | -25.9 | -8.7 | +21.1 | -38.0 | +12.4 | -28.5 | +38.1 | +11.6 | -8.7 | -23.4 | -33.1 | -38.0 | |
| Steps (reduced) |
63 (3) |
65 (5) |
67 (7) |
68 (8) |
70 (0) |
71 (1) |
73 (3) |
74 (4) |
75 (5) |
76 (6) |
77 (7) |
78 (8) | |
Subsets and supersets
Since 10 factors into primes as 2 × 5, 10edf contains 2edf and 5edf as subset edfs.
Intervals
| # | Cents | Neptunian notation using 8\10edf |
Neapolitan notation using 3/10edf |
|---|---|---|---|
| 0 | 0.0 | C | F |
| 1 | 70.2 | ^C, vDb | F^, Gb |
| 2 | 140.4 | C#, Db | F#, Gd |
| 3 | 210.6 | vD | G |
| 4 | 280.8 | D | G^, Ab |
| 5 | 351.0 | ^D, vE | G#, Ad |
| 6 | 421.2 | E | A |
| 7 | 491.4 | ^E, vF | A^, Hb |
| 8 | 561.6 | F | A#, Hd |
| 9 | 631.8 | ^F, vC | H |
| 10 | 702.0 | C | B |
| 11 | 772.2 | ^C, vDb | B^, Cb |
| 12 | 842.3 | C#, Db | B#, Cd |
| 13 | 912.5 | vD | C |
| 14 | 982.7 | D | C^, Db |
| 15 | 1052.9 | ^D, vE | C#, Dd |
| 16 | 1123.1 | E | D |
| 17 | 1193.3 | ^E, vF | D^, Eb |
| 18 | 1263.5 | F | D#, Eb |
| 19 | 1333.7 | ^F, vC | E |
| 20 | 1403.9 | C | F |
Music
- 10 edf (archived 2011)