How should I consider this part ?
Is there something different that we should hear, or that should appear when I play apart from the regular scale ?
Why aren't the notes organized regularly 4-4-4 or 8-8 ?
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Guessing the time sig. is 4/8, it is written regularly. the 1st bar shown has two lots of four semidemis (7+rest) probably played with l.h., then eight beamed together, played r.h. No problems there.
Next bar is 8 semidemis beamed together, then four, but the last 5 are played in the time of 4, so no other way to write it.
Bar 3 - they're all triplets, admittedly not written perfectly clearly, but you'll play 6 in the time of 4, and again to finish the bar off.So, the notes are structured in 8+8. They couldn't be in just 4+4+4 without another +4.
The scale seems to be an Eb Mixolydian mode run.Regular Eb scale, but with a flattened 7.
Hope that all clears up the confusion!
The notes would be grouped 8 - 8, except that the notation in the first bar shows (by the stem directions and the rests) that the left hand plays the first 7 notes (written with stems down) and the right hand the remainder (with stems up).
Beethoven wrote this concerto to perform himself, so the notation that shows how the notes are split between the hands is probably by Beethoven himself, not a later editorial addition.
Notation conventions say that you shouldn't write dotted rests, so it makes sense to beam the first 7 notes in groups that match up with the three rests.
In the second bar, the beaming of 4 - 5 is easier to read than a single beam over 9 notes.
This has nothing to do with "Mixolydian mode". It's just a scale that happens to start on G in the key of Ab major, over an Eb7 chord. The scale doesn't start on Eb for the rather prosaic reason that when Beethoven wrote it, the lowest note on his piano was F - the compass had not yet increased beyond 5 octaves F to F.
The beaming in the third bar may be following a notation convention that was common in Haydn and Mozart, where the length of the beams tended to correspond to what would now be marked by short slurs. If Beethoven was following that convention, beaming the triplets in 4 groups of 3 would have implied something different from 2 groups of 6.
Contemporary evidence suggests that Beethoven was somewhat "mathematically challenged", so perhaps one shouldn't read too much into the notation of triplets rather than sextuplets!