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
Prime subgroups
Pure-octaves 26ed5 is incompatible with prime limit tuning. Of all primes up to 37, 5 is the only one it approximates well.
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.
Composite subgroups
If one ignores primes and focuses on integers in general, 26ed5 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
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) | |
Fractional subgroups
Fractional subgroups are another approach to taming 26ed5. Once can use any of the JI ratios approximated by its individual intervals as basis elements for a subgroup.
There are dozens of possible combinations, here is a small sampling of possible ones:
- 5.6.7/4.11/3.13/4 subgroup
- 5.6.7/4.9/4.9/7.11/3.13/4.13/7.13/9 subgroup
- 5.6.7/4.11/3.13/4.17/11.19/8.23/11.29/7.31/7 subgroup
Nth-prime subgroups
These are some nth-prime subgroups which 26ed5 approximates well:
| Family | Most distinctive related families | Subgroup basis elements |
|---|---|---|
| 11th-prime | 14/11.15/11.16/11.17/11.18/11.23/11.28/11.55/11.66/11 | |
| 14th-prime | 7th- & half-prime | 9/7.10/2.12/2.20/7.23/14.24/7.29/7.31/7.33/7 |
| 16th-prime | 8th-, quarter- & half-prime | 7/4.9/4.10/2.12/2.13/4.17/16.19/8.49/16 |
| 18th-prime | 9th- & 6th-prime | 10/2.12/2.11/3.13/9.19/9.23/18.25/6.31/9.35/9 |
| 30th-prime | 15th- & 10th-prime | 10/2.11/3.12/2.16/15.17/15.21/10.22/5.23/15.25/6.28/15.38/15.41/30.34/15.47/10 |
| 60th-prime | 15th-, 10th- & quarter-prime | 7/4.9/4.10/2.11/3.12/2.13/4.16/15.17/15.21/10.22/5.23/15.25/6.28/15.29/20.38/15.41/30.43/15.47/10.49/30.61/20.69/20 |
| 68th-prime | 17th- & quarter-prime | 7/4.9/4.10/2.12/2.13/4.18/17.28/17.41/34.43/17.63/34.67/34.75/17.80/17 |
| 88th-prime | 11th- & eighth-prime | 7/4.9/4.10/2.12/2.13/4.14/11.15/11.16/11.17/11.18/11.19/8.23/11.25/22.28/11.47/44.53/44.63/22.67/22 |
| 90th-prime | 15th-, 10th-, 9th- & 6th-prime | 10/2.11/3.12/2.13/9.16/15.17/15.19/9.21/10.22/5.23/15.23/18.25/6.28/15.31/9.35/9.38/15.41/30.43/15.47/19.49/30 |
| 112th-prime | 16th- & 14th-prime | 7/4.9/4.9/7.10/2.12/2.13/4.17/16.19/8.20/7.23/14.24/7.29/7.31/7.33/7.49/16 |
| 130th-prime | 13th- & 10th-prime | 6/5.10/2.12/2.21/10.22/5.24/13.27/10.29/13.35/13.39/10.47/10.54/13 |
Note that 5/1 = 10/2 = 55/11, and 6/1 = 12/2 = 66/11.
Of all subgroup interpretations of 26ed5, be they integer or fractional, the 60th-prime subgroup interpretation might be the most useful, as it includes more simple, small-integer consonances than any other interpretation. It includes a 6/5, 7/4, 9/4, 13/4, 11/3 and of course 5/1.
Intervals
| Step | Cents | Just intonation approximation | |||||||
|---|---|---|---|---|---|---|---|---|---|
| 60th-prime | 68th-prime | 88th-prime | 90th-prime | 112th-prime | 130th-prime | Integer (5.6.12.22.32... as above) | Integer (simplified) | ||
| 1 | 107.2 | 16/15 | 18/17 | 47/44 | 16/15 | 17/16 | 36/34, 34/32 | 18/17, 17/16 | |
| 2 | 214.3 | 17/15 | 25/22 | 17/15 | 25/22 | 25/22 | |||
| 3 | 321.5 | 6/5 | 41/34 | 53/44 | 6/5 | 6/5 | 6/5, 41/34 | 6/5, 41/34 | |
| 4 | 428.7 | 14/11 | 23/18 | 9/7 | 63/49 | 9/7 | |||
| 5 | 535.8 | 41/30 | 15/11 | 41/30 | 19/14 | ||||
| 6 | 643.0 | 29/20 | 16/11 | 13/9 | 32/22 | 16/11 | |||
| 7 | 750.2 | 23/15 | 17/11 | 23/15 | 34/22 | 17/11 | |||
| 8 | 857.3 | 49/30 | 28/17 | 18/11 | 49/30 | 23/14 | |||
| 9 | 964.5 | 7/4 | 7/4 | 7/4 | 7/4 | 56/32 | 7/4 | ||
| 10 | 1071.7 | 28/15 | 63/34 | 28/15 | 13/7 | 24/13 | 63/34 | 63/34 | |
| 11 | 1178.8 | 67/34 | 49/25 | 49/25 | |||||
| 12 | 1286.0 | 21/10 | 23/11 | 21/10, 19/9 | 21/10 | 46/22 | 23/11 | ||
| 13 | 1393.2 | 9/4 | 9/4 | 9/4 | 9/4 | 29/13 | |||
| 14 | 1500.3 | 19/8 | 19/8 | ||||||
| 15 | 1607.5 | 38/15 | 43/17 | 28/11 | 38/15 | 56/22 | 28/11 | ||
| 16 | 1714.7 | 27/10 | 27/10 | 35/13, 27/10 | |||||
| 17 | 1821.8 | 43/15 | 63/22 | 43/15 | 20/7 | 63/22 | 63/22 | ||
| 18 | 1929.0 | 61/20 | 67/22 | 49/16 | 67/22 | 67/22 | |||
| 19 | 2036.2 | 13/4 | 13/4 | 13/4 | 13/4 | ||||
| 20 | 2143.3 | 69/20 | 31/9 | 24/7 | |||||
| 21 | 2250.5 | 11/3 | 11/3 | 22/6 | 11/3 | ||||
| 22 | 2357.7 | 39/10 | 35/9, 39/10 | 39/10 | |||||
| 23 | 2464.8 | 25/6 | 25/6 | 29/7 | 54/13 | 25/6 | 25/6 | ||
| 24 | 2572.0 | 22/5 | 75/17 | 22/5 | 31/7 | 22/5 | 22/5 | 22/5 | |
| 25 | 2679.1 | 47/10 | 80/17 | 47/10 | 33/7 | 47/10 | |||
| 26 | 2786.3 | 5/1 | 5/1 | 5/1 | 5/1 | 5/1 | 5/1 | 5/1 | 5/1 |