I’m playing a sonata by Beethoven (op. 78. No. 24 in F# major) and there are two of these trills (see the picture I’ve attached) in the left hand. My question is, how many notes should be in this particular trill? Should it be 32nd notes so that there’s 2 notes in the left hand for every 1 in the right hand?
There should be sufficient to fill the space, using a reasonably fast but unmeasured alternation of notes. There are musical styles and occasions where a controlled, measured trill is appropriate. I suggest this is not one of them.
Keep in mind that to some extent, the trill speed is up to the performer. Some guidelines:
-- probably want the trill to be faster than the written notes in the other hand, so here at least 32nd note-speed.
-- the faster the trill, the more "urgency" or "liveliness" the mood you're portraying.
-- Decide here what the lead line (melody) is. I don't recall this piece offhand, but it may well be that the upper hand is just ornamental here, and the lower trilled chord is the actual melody (chord progression) introduction. In that case, you may well want the trill not only to be louder than the 16th notes, but to increase in speed as you lead into the grace notes which announce the next measure.