26ed5
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Prime factorization
2 × 13
Step size
107.166¢
Octave
11\26ed5 (1178.83¢)
Twelfth
18\26ed5 (1928.99¢) (→9\13ed5)
Consistency limit
3
Distinct consistency limit
3
 |
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| ← 25ed5 | 26ed5 | 27ed5 → |
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 extremely obscure subgroup 5.6.41.67.97.103.151.181.193.
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 | 107.166 | 18/17 |
| 2 | 214.332 | 17/15, 25/22 |
| 3 | 321.498 | 6/5, 23/19 |
| 4 | 428.664 | 23/18 |
| 5 | 535.83 | 15/11 |
| 6 | 642.995 | |
| 7 | 750.161 | 17/11, 20/13, 23/15 |
| 8 | 857.327 | 18/11 |
| 9 | 964.493 | 7/4 |
| 10 | 1071.659 | 13/7, 24/13 |
| 11 | 1178.825 | |
| 12 | 1285.991 | 19/9, 21/10, 23/11 |
| 13 | 1393.157 | |
| 14 | 1500.323 | |
| 15 | 1607.489 | |
| 16 | 1714.655 | |
| 17 | 1821.821 | 20/7 |
| 18 | 1928.986 | |
| 19 | 2036.152 | 13/4 |
| 20 | 2143.318 | 24/7 |
| 21 | 2250.484 | 11/3 |
| 22 | 2357.65 | |
| 23 | 2464.816 | 25/6 |
| 24 | 2571.982 | 22/5 |
| 25 | 2679.148 | |
| 26 | 2786.314 | 5/1 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -21.2 | +27.0 | -42.3 | +0.0 | +5.9 | -46.7 | +43.6 | -53.1 |
| Relative (%) | -19.8 | +25.2 | -39.5 | +0.0 | +5.5 | -43.6 | +40.7 | -49.6 | |
| Steps (reduced) |
11 (11) |
18 (18) |
22 (22) |
26 (0) |
29 (3) |
31 (5) |
34 (8) |
35 (9) | |
| 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 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +13.8 | -43.9 | +8.0 | +14.8 | -33.3 | +44.2 | -41.3 | +52.2 | +10.4 | +47.6 | +13.6 |
| Relative (%) | +12.9 | -41.0 | +7.4 | +13.8 | -31.1 | +41.3 | -38.5 | +48.7 | +9.7 | +44.4 | +12.7 | |
| Steps (reduced) |
66 (14) |
66 (14) |
68 (16) |
69 (17) |
69 (17) |
71 (19) |
71 (19) |
73 (21) |
74 (22) |
75 (23) |
75 (23) | |
| Harmonic | 107 | 109 | 113 | 127 | 131 | 137 | 139 | 149 | 151 | 157 | 163 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -52.3 | +22.8 | -39.6 | -27.5 | +26.0 | -51.5 | +30.5 | +17.4 | -5.6 | +34.1 | -30.9 |
| Relative (%) | -48.8 | +21.3 | -37.0 | -25.6 | +24.3 | -48.1 | +28.5 | +16.3 | -5.3 | +31.8 | -28.8 | |
| Steps (reduced) |
75 (23) |
76 (24) |
76 (24) |
78 (0) |
79 (1) |
79 (1) |
80 (2) |
81 (3) |
81 (3) |
82 (4) |
82 (4) | |
| Harmonic | 167 | 173 | 179 | 181 | 191 | 193 | 197 | 199 | 211 | 223 | 227 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +34.3 | -26.8 | +21.4 | +2.1 | +16.2 | -1.8 | -37.4 | +52.3 | -49.1 | -37.6 | +38.7 |
| Relative (%) | +32.0 | -25.0 | +19.9 | +2.0 | +15.1 | -1.7 | -34.9 | +48.8 | -45.8 | -35.1 | +36.2 | |
| Steps (reduced) |
83 (5) |
83 (5) |
84 (6) |
84 (6) |
85 (7) |
85 (7) |
85 (7) |
86 (8) |
86 (8) |
87 (9) |
88 (10) | |