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-pentaves 26ed5 is incompatible with prime limit tuning. Of all primes up to 41, 5 and 41 are the only two 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.
A good compressed tuning of 26ed5 is 46ed17, which transforms 26ed5 from a 5.41 tuning to a 3.5.11.17.23.43 tuning. The 3/1 in 46ed17 isn’t that good, with similar error to 5edo, but it’s a huge improvement on 26ed5. And the 5, 11, 17, 23 and 43 are genuinely solid approximations. Other tunings which are almost identical to 46ed17, and so provide those same benefits, are 8ed18/11 and 20ed24/7.
If one attempts to stretch 26ed5 instead of compress, one will not find any tunings that approximate primes well until reaching 11edo, so only compression is viable, not stretching.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -21.2 | +27.0 | +0.0 | -46.7 | +28.2 | -46.7 | +24.7 | +46.5 | +37.2 | -42.6 | -50.9 |
| Relative (%) | -19.8 | +25.2 | +0.0 | -43.6 | +26.3 | -43.6 | +23.0 | +43.3 | +34.7 | -39.8 | -47.5 | |
| Steps (reduced) |
11 (11) |
18 (18) |
26 (0) |
31 (5) |
39 (13) |
41 (15) |
46 (20) |
48 (22) |
51 (25) |
54 (2) |
55 (3) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -35.7 | +0.9 | +25.6 | -21.2 | -14.9 | +13.8 | -43.9 | +8.0 | +14.8 | -33.3 | +44.2 |
| Relative (%) | -33.3 | +0.8 | +23.9 | -19.8 | -13.9 | +12.9 | -41.0 | +7.4 | +13.8 | -31.1 | +41.3 | |
| Steps (reduced) |
58 (6) |
60 (8) |
61 (9) |
62 (10) |
64 (12) |
66 (14) |
66 (14) |
68 (16) |
69 (17) |
69 (17) |
71 (19) | |
Composite subgroups
If one does not restrict to primes and allows all integers, pure-pentaves 26ed5 can instead be used as a strong tuning for the giant subgroup:
5.6.12.22.32.44.49.52.56
63.81.91.98.104.117.126
Or it can be a strong tuning for any smaller subgroup that is contained within that group.
| 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
| Step | Cents | Just intonation approximation | |
|---|---|---|---|
| 5.6.12.22… subgroup described above |
5.6.12.22… subgroup (with ratios simplified) | ||
| 1 | 107.2 | 36/34, 34/32, 32/30 | 18/17, 17/16, 16/15 |
| 2 | 214.3 | 34/30, 25/22 | 17/15, 25/22 |
| 3 | 321.5 | 6/5 | 6/5 |
| 4 | 428.7 | 56/44, 63/49 | 14/11, 9/7 |
| 5 | 535.8 | (6/5)x(25/22) | 15/11 |
| 6 | 643.0 | 52/36, 32/22 | 13/9, 16/11 |
| 7 | 750.2 | 34/22 | 17/11 |
| 8 | 857.3 | (6/5)x(15/11) | 18/11 |
| 9 | 964.5 | 56/32 | 7/4 |
| 10 | 1071.7 | 104/56 | 13/7 |
| 11 | 1178.8 | 49/25 | 49/25 |
| 12 | 1286.0 | (6/5)x(56/32) | 21/10 |
| 13 | 1393.2 | 81/36 | 9/4 |
| 14 | 1500.3 | 12/5 | 12/5 |
| 15 | 1607.5 | 30/12 | 5/2 |
| 16 | 1714.7 | 32/12, 81/30 | 8/3, 27/10 |
| 17 | 1821.8 | 160/56 | 20/7 |
| 18 | 1929.0 | 110/36, 104/34 | 55/18, 52/17 |
| 19 | 2036.2 | 104/32 | 13/4 |
| 20 | 2143.3 | (32/12)x(63/49) | 24/7 |
| 21 | 2250.5 | 22/6 | 11/3 |
| 22 | 2357.7 | 117/30 | 39/10 |
| 23 | 2464.8 | 25/6 | 25/6 |
| 24 | 2572.0 | 22/5 | 22/5 |
| 25 | 2679.1 | 56/12 | 14/3 |
| 26 | 2786.3 | 5/1 | 5/1 |
Scales
13ed5plus
Inspired by the elevenplus scale of 22edo, the 13ed5plus scale is 13ed5 plus a step of 26ed5 in between two of its steps.
In other words, 13ed5plus is all of the odd-numbered steps of 26ed5, plus step 26.
The scale is useful because it includes most of 26ed5’s consonances while leaving out many of the less-used intervals. Making it practical to use on an instrument.
Properties
13ed5plus is a 14-tone scale.
As a MOS scale, it is an example of the scale 13L 1s (5/1-equivalent). The 2/1-equivalent version would be 13L 1s.
Intervals
| Step | Cents | JI approximation (5.6.12.22… subgroup; ratios simplified) |
|---|---|---|
| 1 | 107.2 | 18/17, 17/16, 16/15 |
| 3 | 321.5 | 6/5 |
| 5 | 535.8 | 15/11 |
| 7 | 750.2 | 17/11 |
| 9 | 964.5 | 7/4 |
| 11 | 1178.8 | 49/25 |
| 13 | 1393.2 | 9/4 |
| 15 | 1607.5 | 5/2 |
| 17 | 1821.8 | 20/7 |
| 19 | 2036.2 | 13/4 |
| 21 | 2250.5 | 11/3 |
| 23 | 2464.8 | 25/6 |
| 25 | 2679.1 | 14/3 |
| 26 | 2786.3 | 5/1 |