26ed5: Difference between revisions
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Many of 26ed5’s 'near-miss' [[prime]]s are tuned sharp, so 26ed5 can be made to work more normally by [[Octave shrinking|compressing]] 26ed5’s [[equave]], making [[5/1]] slightly flat but still okay and the other primes more in-tune. | Many of 26ed5’s 'near-miss' [[prime]]s are tuned sharp, so 26ed5 can be made to work more normally by [[Octave shrinking|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. 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 and 23 are genuinely solid approximations. | A good compressed tuning of 26ed5 is [[46ed17]], which transforms 26ed5 from a 5.41 tuning to a 3.5.11.17.23 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 and 23 are genuinely solid approximations. | ||
If one attempts to [[Octave stretch|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. | If one attempts to [[Octave stretch|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. | ||
Revision as of 05:21, 22 December 2024
| ← 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 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 and 23 are genuinely solid approximations.
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 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. 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, 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-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-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 |