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 6 cents, but 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 |