26ed5: Difference between revisions
m Trying to tame the huge complex subgroups and make them more sensible: a very long way to go yet, but it’s a atart |
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!60th-basis | !60th-basis | ||
|15th-, 10th- & quarter-basis | |15th-, 10th- & quarter-basis | ||
|7/4.9/4.10/2.11/3.12/2 | |7/4.9/4.10/2.11/3.12/2 | ||
|13/4.17/15.22/5.23/15.25/6.28/15.29/20.38/15.41/30.43/15.47/10.49/30.61/20.69/20 | |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-basis | !68th-basis | ||
Revision as of 04:02, 5 January 2025
| ← 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.
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, comparable 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.
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 the 11-limit numbers: 5.6.12.22.32.44.49.56.63
- Only numbers below 40: 5.6.12.22.32.34
- 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. One can use any of the JI ratios approximated by its individual intervals as basis elements for a subgroup.
There are dozens of possible combinations, for example the 5.6.7/4.11/3.13/4 subgroup, the 5.6.7/4.9/4.9/7.11/3.13/4.13/7.13/9 subgroup, etc.
Nth-prime subgroups
These are some nth-prime subgroups[idiosyncratic term] which 26ed5 approximates well:
| Family | Most distinctive related families | Subgroup basis elements | Optional extra elements (sprinkle in any 1 or 2 of these) |
|---|---|---|---|
| 16th-prime | 8th-, quarter- & half-prime | 7/4.13/4.17/16.19/8 | |
| 18th-prime | 9th- & 6th-prime | 11/3.13/9.19/9.23/18.31/9 | |
| 30th-prime | 15th- & 10th-prime | 11/3.17/15.23/15 | 41/30.47/10 |
| 60th-prime | 15th-, 10th- & quarter-prime | 7/4.11/3.13/4.17/15 | 23/15.29/20.41/30.43/15.47/10.61/20 |
| 68th-prime | 17th- & quarter-prime | 7/4.13/4.41/34.43/17 | 67/34 |
| 88th-prime | 11th- & 8th-prime | 7/4.13/4.17/11.19/8.23/11 | 47/44.53/44.67/22 |
| 90th-prime | 15th-, 10th-, 9th- & 6th-prime | 11/3.13/9.17/15.19/9 | 23/15.23/18.31/9.41/30.43/15.47/19.49/30 |
| 140th-basis | 14th-, 10th- & quarter-basis | 7/4.13/4.23/14.29/7.31/7 | 29/20.61/20 |
Nth-basis subgroups
These are some nth-basis subgroups[idiosyncratic term] which 26ed5 approximates well.
| Family | Most distinctive related families | Subgroup basis elements | Optional extra elements (sprinkle in any 1 or 2 of these) |
|---|---|---|---|
| 11th-basis | 14/11.15/11.16/11.55/11.66/11 | 17/11.18/11.28/11.23/11 | |
| 14th-basis | 7th- & half-basis | 9/7.10/2.12/2.20/7.24/7 | 23/14.29/7.31/7.33/7 |
| 16th-basis | 8th-, quarter- & half-basis | 7/4.9/4.10/2.12/2.13/4 | 17/16.19/8 |
| 18th-basis | 9th- & 6th-basis | 10/2.12/2.11/3.25/6.35/9 | 13/9.19/9.23/18.31/9 |
| 30th-basis | 15th- & 10th-basis | 10/2.11/3.12/2.16/15.21/10.22/5 | 17/15.23/15.25/6.28/15.38/15.41/30.34/15.47/10 |
| 60th-basis | 15th-, 10th- & quarter-basis | 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-basis | 17th- & quarter-basis | 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-basis | 11th- & 8th-basis | 7/4.9/4.10/2.12/2.14/11.15/11.16/11 | 13/4.17/11.18/11.19/8.23/11.25/22.28/11.47/44.53/44.63/22.67/22 |
| 90th-basis | 15th-, 10th-, 9th- & 6th-basis | 10/2.11/3.12/2.16/15.21/10.22/5 | 13/9.17/15.19/9.23/15.23/18.25/6.28/15.31/9.35/9.38/15.41/30.43/15.47/19.49/30 |
| 140th-basis | 14th-, 10th- & quarter-basis | 7/4.9/4.9/7.10/2.12/2.20/7.24/7 | 13/4.23/14.29/7.29/20.31/7.33/7.61/20 |
Note that 5/1 = 10/2 = 55/11, & 6/1 = 12/2 = 66/11.
Note that any subset of any of these subgroup elements is still a valid nth-basis subgroup. So one can remove as many basis elements as desired to simplify the subgroup down, if they so wish.
Of all subgroup interpretations of 26ed5, be they integer or fractional, the 60th-basis subgroup interpretation might be the most useful, as it includes more simple, small-numeral 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-basis | 68th-basis | 88th-basis | 90th-basis | 140th-basis | Integer (5.6.12.22.32... as above) |
Integer (simplified) | ||
| 1 | 107.2 | 16/15 | 18/17 | 47/44 | 16/15 | 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, 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 | (6/5)x(25/22) | 15/11 | |
| 6 | 643.0 | 29/20 | 16/11 | 13/9 | 29/20 | 32/22 | 16/11 | |
| 7 | 750.2 | 23/15 | 17/11 | 23/15 | 54/35 | 34/22 | 17/11 | |
| 8 | 857.3 | 49/30 | 28/17 | 18/11 | 49/30 | 23/14 | (6/5)x(15/11) | 18/11 |
| 9 | 964.5 | 7/4 | 7/4 | 7/4 | 26/15 | 7/4 | 56/32 | 7/4 |
| 10 | 1071.7 | 28/15 | 63/34 | 28/15 | 13/7 | 63/34 | 63/34 | |
| 11 | 1178.8 | 67/34 | 49/25 | 49/25 | ||||
| 12 | 1286.0 | 21/10 | 21/10 | 23/11, 21/10 | 21/10, 19/9 | 21/10 | 46/22, (6/5)x(56/32) | 23/11, 21/10 |
| 13 | 1393.2 | 9/4 | 9/4 | 9/4 | 9/4 | |||
| 14 | 1500.3 | 19/8 | ||||||
| 15 | 1607.5 | 38/15 | 43/17 | 28/11 | 38/15 | 56/22 | 28/11 | |
| 16 | 1714.7 | 27/10 | 27/10 | 27/10 | 27/10 | 27/10 | ||
| 17 | 1821.8 | 43/15 | 63/22 | 43/15 | 20/7 | 63/22 | 63/22 | |
| 18 | 1929.0 | 61/20 | 67/22 | 61/20 | 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 | 39/10 | 39/10 | 35/9, 39/10 | 39/10 | ||
| 23 | 2464.8 | 25/6 | 25/6 | 25/6 | 25/6 | 29/7, 25/6 | 25/6 | 25/6 |
| 24 | 2572.0 | 22/5 | 75/17 | 22/5 | 31/7 | 22/5 | 22/5 | |
| 25 | 2679.1 | 47/10 | 80/17 | 47/10 | 33/7 | |||
| 26 | 2786.3 | 5/1 | 5/1 | 5/1 | 5/1 | 5/1 | 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 2
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.
Table
| Step | Cents | Just intonation approximation | ||||||
|---|---|---|---|---|---|---|---|---|
| 60th-basis | 68th-basis | 88th-basis | 90th-basis | 112th-basis | Integer (5.6.12.22.32... as above) |
Integer (simplified) | ||
| 1 | 107.2 | 16/15 | 18/17 | 47/44 | 16/15 | 36/34, 34/32 | 18/17, 17/16 | |
| 3 | 321.5 | 6/5 | 41/34 | 53/44 | 6/5 | 6/5, 41/34 | 6/5, 41/34 | |
| 5 | 535.8 | 41/30 | 15/11 | 41/30 | 19/14 | (6/5)x(25/22) | 15/11 | |
| 7 | 750.2 | 23/15 | 17/11 | 23/15 | 54/35 | 34/22 | 17/11 | |
| 9 | 964.5 | 7/4 | 7/4 | 7/4 | 26/15 | 7/4 | 56/32 | 7/4 |
| 11 | 1178.8 | 67/34 | 49/25 | 49/25 | ||||
| 13 | 1393.2 | 9/4 | 9/4 | 9/4 | 9/4 | |||
| 15 | 1607.5 | 38/15 | 43/17 | 28/11 | 38/15 | 56/22 | 28/11 | |
| 17 | 1821.8 | 43/15 | 63/22 | 43/15 | 20/7 | 63/22 | 63/22 | |
| 19 | 2036.2 | 13/4 | 13/4 | 13/4 | 13/4 | |||
| 21 | 2250.5 | 11/3 | 11/3 | 22/6 | 11/3 | |||
| 23 | 2464.8 | 25/6 | 25/6 | 25/6 | 25/6 | 29/7, 25/6 | 25/6 | 25/6 |
| 25 | 2679.1 | 47/10 | 80/17 | 47/10 | 33/7 | |||
| 26 | 2786.3 | 5/1 | 5/1 | 5/1 | 5/1 | 5/1 | 5/1 | 5/1 |