26ed5
| ← 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.
Theory
Subgroup interpretation
26ed5 is a weak tuning for prime limit tuning. It can instead be used as a strong tuning for the obscure subgroup 5.6.12.22.32.34.41.44.46.49.53.56.59.63.67.
One can also use any subset of that subgroup for example:
- Only numbers below 40: 5.6.12.22.32.34
- Only numbers below 50: 5.6.12.22.32.34.44.46.49
- Only 5 and the composite numbers: 5.6.12.22.32.34.44.46.49.53.56.63
- Only 6 and the primes: 5.6.41.59.67
Fractional subgroups might also be an option for 26ed5.
Equave stretch
Many of 26ed5’s 'near-miss' primes are tuned sharp, so 26ed5 can be made to work more normally by compressing 26ed5’s equave, making 5/1 slightly flat but still okay and the other primes more in-tune.
29ed6 is a compressed version of 26ed5, compressing 5/1 by roughly 5 cents, but even it is not enough to bring many primes into line. Further compression than that is required.
Stretching rather than compressing the equave is also an option. It will change a lot of vals, so the tuning may not longer be fully recognisable as 26ed5, however the right amount of stretching will improve primes.
Tables of harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -21.2 | +27.0 | -42.3 | +0.0 | +5.9 | -46.7 | +43.6 | -53.1 | -21.2 | +28.2 | -15.3 |
| Relative (%) | -19.8 | +25.2 | -39.5 | +0.0 | +5.5 | -43.6 | +40.7 | -49.6 | -19.8 | +26.3 | -14.3 | |
| Steps (reduced) |
11 (11) |
18 (18) |
22 (22) |
26 (0) |
29 (3) |
31 (5) |
34 (8) |
35 (9) |
37 (11) |
39 (13) |
40 (14) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -46.7 | +39.3 | +27.0 | +22.5 | +24.7 | +32.9 | +46.5 | -42.3 | -19.7 | +7.0 | +37.2 |
| Relative (%) | -43.6 | +36.7 | +25.2 | +21.0 | +23.0 | +30.7 | +43.3 | -39.5 | -18.3 | +6.5 | +34.7 | |
| Steps (reduced) |
41 (15) |
43 (17) |
44 (18) |
45 (19) |
46 (20) |
47 (21) |
48 (22) |
48 (22) |
49 (23) |
50 (24) |
51 (25) | |
| Harmonic | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -36.5 | +0.0 | +39.3 | -26.1 | +18.1 | -42.6 | +5.9 | -50.9 | +1.3 | -52.0 | +3.5 |
| Relative (%) | -34.1 | +0.0 | +36.6 | -24.3 | +16.9 | -39.8 | +5.5 | -47.5 | +1.2 | -48.5 | +3.3 | |
| Steps (reduced) |
51 (25) |
52 (0) |
53 (1) |
53 (1) |
54 (2) |
54 (2) |
55 (3) |
55 (3) |
56 (4) |
56 (4) |
57 (5) | |
| Harmonic | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -46.7 | +11.7 | -35.7 | +25.3 | -19.7 | +43.6 | +0.9 | -40.8 | +25.6 | -14.2 | -53.1 |
| Relative (%) | -43.6 | +10.9 | -33.3 | +23.6 | -18.4 | +40.7 | +0.8 | -38.1 | +23.9 | -13.2 | -49.6 | |
| Steps (reduced) |
57 (5) |
58 (6) |
58 (6) |
59 (7) |
59 (7) |
60 (8) |
60 (8) |
60 (8) |
61 (9) |
61 (9) |
61 (9) | |
| Harmonic | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +16.0 | -21.2 | +49.5 | +13.8 | -21.2 | +51.7 | +18.1 | -14.9 | -47.2 | +28.2 | -3.0 |
| Relative (%) | +14.9 | -19.8 | +46.2 | +12.9 | -19.8 | +48.3 | +16.9 | -13.9 | -44.1 | +26.3 | -2.8 | |
| Steps (reduced) |
62 (10) |
62 (10) |
63 (11) |
63 (11) |
63 (11) |
64 (12) |
64 (12) |
64 (12) |
64 (12) |
65 (13) |
65 (13) | |
| Harmonic | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -33.7 | +43.4 | +13.8 | -15.3 | -43.9 | +35.1 | +7.4 | -19.9 | -46.7 | +34.0 | +8.0 | -17.7 |
| Relative (%) | -31.4 | +40.5 | +12.9 | -14.3 | -41.0 | +32.7 | +6.9 | -18.6 | -43.6 | +31.7 | +7.4 | -16.5 | |
| Steps (reduced) |
65 (13) |
66 (14) |
66 (14) |
66 (14) |
66 (14) |
67 (15) |
67 (15) |
67 (15) |
67 (15) |
68 (16) |
68 (16) |
68 (16) | |
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 |