26ed5: Difference between revisions
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m Collapse harmonics tables, correct error (octave to pentave) |
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=== Prime subgroups === | === Prime subgroups === | ||
Pure- | Pure-[[pentave]]s 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' [[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 3.5.11.17.23.43 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]]. | 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 [[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. | ||
{{Harmonics in equal | |||
| steps = 26 | |||
| num = 5 | |||
| denom = 1 | |||
| intervals = prime | |||
| collapsed = 1 | |||
| start = 1 | |||
| title = Prime harmonics 2 to 31 (26ed5) | |||
}} | |||
{{Harmonics in equal | |||
| steps = 26 | |||
| num = 5 | |||
| denom = 1 | |||
| intervals = prime | |||
| collapsed = 1 | |||
| start = 12 | |||
| title = Prime harmonics 37 to 79 (26ed5) | |||
}} | |||
=== Composite subgroups === | === Composite subgroups === | ||
If one | 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''' | '''5.6.12.22.32.44.49.52.56''' | ||
| Line 19: | Line 37: | ||
Or it can be a strong tuning for any smaller subgroup that is contained within that group. | Or it can be a strong tuning for any smaller subgroup that is contained within that group. | ||
{{Harmonics in equal | {{Harmonics in equal | ||
| steps = 26 | | steps = 26 | ||
| Line 26: | Line 42: | ||
| denom = 1 | | denom = 1 | ||
| intervals = integer | | intervals = integer | ||
| collapsed = 1 | |||
| start = 1 | | start = 1 | ||
| title = | | title = Integer harmonics 2 to 12 (26ed5) | ||
}} | }} | ||
{{Harmonics in equal | {{Harmonics in equal | ||
| Line 36: | Line 53: | ||
| collapsed = 1 | | collapsed = 1 | ||
| start = 12 | | start = 12 | ||
| title = | | title = Integer harmonics 13 to 23 (26ed5) | ||
}} | }} | ||
{{Harmonics in equal | {{Harmonics in equal | ||
| Line 45: | Line 62: | ||
| collapsed = 1 | | collapsed = 1 | ||
| start = 23 | | start = 23 | ||
| title = | | title = Integer harmonics 24 to 34 (26ed5) | ||
}} | }} | ||
{{Harmonics in equal | {{Harmonics in equal | ||
| Line 54: | Line 71: | ||
| collapsed = 1 | | collapsed = 1 | ||
| start = 34 | | start = 34 | ||
| title = | | title = Integer harmonics 35 to 45 (26ed5) | ||
}} | }} | ||
{{Harmonics in equal | {{Harmonics in equal | ||
| Line 63: | Line 80: | ||
| collapsed = 1 | | collapsed = 1 | ||
| start = 45 | | start = 45 | ||
| title = | | title = Integer harmonics 46 to 56 (26ed5) | ||
}} | }} | ||
{{Harmonics in equal | {{Harmonics in equal | ||
| Line 72: | Line 89: | ||
| collapsed = 1 | | collapsed = 1 | ||
| start = 56 | | start = 56 | ||
| title = | | title = Integer harmonics 57 to 68 (26ed5) | ||
| columns = 12 | | columns = 12 | ||
}} | }} | ||