The fastest possible trill

For the trilly bits in Mozart, starting at the pickup to 51 and again at 120, I long ago decided I wanted to do 32nd notes, something like this:

Three turns, starting from the bottom. The advantage of straight 32nd notes over something like a 7-tuplet is that it's easier to practice slowly with a subdivided beat. When I crank up the metronome, I don't always manage to get all three turns in, or maybe there will be two clean ones plus a little bounce near the end, but this is the plan anyway.

There are lots of ways to play these, of course. My teacher I think told me he played 5 turns, which seems like a lot to me. And of course, when he demonstrated, they went too fast for me to count. With recordings, we can slow things down. Here's that section from Klaus Thunemann's recording, which has pretty fast trills:

  Kt-normal by TFox17

It's too fast for me to count, unless I slow it down:
  Kt slowed by TFox17

From that, it's pretty easy to write out what he is doing:

I count four turns, starting from the top, for straight 32nd notes all the way through. So not, on paper, any faster than my plan, although faster in practice, since he both takes a faster tempo, and is able to execute the whole thing beautifully.

Now, how fast can a trill go? It turns out that there are limits in principle, not just practice. First of all, there are limits as to how fast fingers can move. Kochevitsky's book on piano technique goes through some of the studies.   Apparently the 2nd and 3rd fingers can make 5-6 movements per second, 4-5 with the other fingers. Training doesn't help peak speed: great pianists and members of the general public were about the same, with some untrained participants able to make 7 per second, which some pianists were only able to do 5. For a trill on piano, you can alternate two fingers, so 6 movements per second can give you 12 notes per second. On a wind instrument, you can only move one finger, but each full movement cycle, up and down, gives you two notes. So it's more or less the same, 12 notes per second. It turns out that there's a limit to what the ear will perceive of as distinct notes. Passages played faster than this will blur together. This makes sense: after all, every note is just a cyclical progression of pressures, and A0, reachable by a contra with an extension, has a frequency of about 22 Hz. The limit of hearing notes depends on several things: the pitch of the notes (low notes get muddier faster), the complexity of the passage (scales, trills and tremolos are easier to perceive than more complex passages), and the listener. This limit is typically around 12 notes per second, sextuplets at 120 BPM, which interestingly corresponds to about the limit of what's possible to produce. For a trill, the limit quoted is more like 15 notes per second, which is around what Klaus Thunemann is performing his trill. We can test this ourselves, though, by shifting the tempo on the recording, and seeing what happens.

Here's the tempo shifted to about 150 bpm, which gives us about 20 notes per second:

  Kt 151 by TFox17

And again, to about 180, which gives us 24 notes per second:
  Kt 182 by TFox17

For me, at 150 it still sounds like a trill, though I certainly can't count the notes. At 180, it's degenerated into a sort of fluttery effect on a sustained tone. At 240, the 16ths become indistinct.

From this I think we can conclude that Klaus Thunemann's trills are, not just fast, but essentially at or near the limit of being the fastest possible, either to perform or to hear.