26ed5
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26 equal divisions of the 5th harmonic (abbreviated 26ed5) is a nonoctave tuning system that divides the interval of 5/1 into 26 equal parts of about 107 ¢ each. Each step represents a frequency ratio of 51/26, or the 26th root of 5.
26ed5 is a strong tuning for the obscure subgroup 5.6.41.67.97.103.
Less esoterically, it can be used as a mediocre but workable tuning for the more practical subgroup 5.6.11.17.41. Most of these harmonics are tuned sharp, so 26ed5 can be made to work better by compressing 26ed5’s equave, making 5/1 slightly flat but the other harmonics more in-tune. This can elevate 26ed5 from mediocre to pretty decent.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 107.2 | 18/17 |
| 2 | 214.3 | 17/15, 25/22 |
| 3 | 321.5 | 6/5, 23/19 |
| 4 | 428.7 | 23/18 |
| 5 | 535.8 | 15/11 |
| 6 | 643 | |
| 7 | 750.2 | 17/11, 20/13, 23/15 |
| 8 | 857.3 | 18/11 |
| 9 | 964.5 | 7/4 |
| 10 | 1071.7 | 13/7, 24/13 |
| 11 | 1178.8 | |
| 12 | 1286 | 19/9, 21/10, 23/11 |
| 13 | 1393.2 | |
| 14 | 1500.3 | |
| 15 | 1607.5 | |
| 16 | 1714.7 | |
| 17 | 1821.8 | 20/7 |
| 18 | 1929 | |
| 19 | 2036.2 | 13/4 |
| 20 | 2143.3 | 24/7 |
| 21 | 2250.5 | 11/3 |
| 22 | 2357.7 | |
| 23 | 2464.8 | 25/6 |
| 24 | 2572 | 22/5 |
| 25 | 2679.1 | |
| 26 | 2786.3 | 5/1 |
Harmonics
| Harmonic | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.9 | -46.7 | +43.6 | -53.1 | -21.2 | +28.2 | -15.3 | -46.7 | +39.3 | +27.0 | +22.5 |
| Relative (%) | +5.5 | -43.6 | +40.7 | -49.6 | -19.8 | +26.3 | -14.3 | -43.6 | +36.7 | +25.2 | +21.0 | |
| Steps (reduced) |
29 (3) |
31 (5) |
34 (8) |
35 (9) |
37 (11) |
39 (13) |
40 (14) |
41 (15) |
43 (17) |
44 (18) |
45 (19) | |
| Harmonic | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +28.2 | -46.7 | +24.7 | +46.5 | +37.2 | -42.6 | -50.9 | -35.7 | +0.9 | +25.6 | -21.2 |
| Relative (%) | +26.3 | -43.6 | +23.0 | +43.3 | +34.7 | -39.8 | -47.5 | -33.3 | +0.8 | +23.9 | -19.8 | |
| Steps (reduced) |
39 (13) |
41 (15) |
46 (20) |
48 (22) |
51 (25) |
54 (2) |
55 (3) |
58 (6) |
60 (8) |
61 (9) |
62 (10) | |
| Harmonic | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | 97 | 101 | 103 | 107 | 109 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +13.8 | -43.9 | +8.0 | +14.8 | -33.3 | +44.2 | -41.3 | +52.2 | +10.4 | +47.6 | +13.6 | -52.3 | +22.8 |
| Relative (%) | +12.9 | -41.0 | +7.4 | +13.8 | -31.1 | +41.3 | -38.5 | +48.7 | +9.7 | +44.4 | +12.7 | -48.8 | +21.3 | |
| Steps (reduced) |
66 (14) |
66 (14) |
68 (16) |
69 (17) |
69 (17) |
71 (19) |
71 (19) |
73 (21) |
74 (22) |
75 (23) |
75 (23) |
75 (23) |
76 (24) | |