How do you play the trill in measure 29 of Beethoven's "Moonlight" sonata?
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There is a general consensus that the trill should start on the principle note (A#) with finger 4 and consist of 5 total notes, including the trill ending. Beyond that, there is little agreement. There are dissenters in all aspects, especially with regard to fingering.
Here is a table reflecting the choices of various editors.
Starting Editor Note Rhythm Fingering -------------------------------------------------- Lebert/v.B. A# 3+2 or 2+3 --- Schultze A# 5 4..34 Casella A# 5 4.... Weiner A# 5 45323 Schnabel A# --- 45.24 Schenker A# --- 35... Reimann B --- --- Cooper B 6 ---
Riemann’s  editorial decision to begin his trills on the upper note upsets the sentiment that nineteenth century pianists would always start their trills on the main note. (p. 76)
Trills and turns had two primary functions during the nineteenth century. Those located at the beginning of a strong beat served as accenting ornaments. Contrariwise, those assigned to weak beats would act as connecting embellishments. The number of notes within a trill, the rhythm of the shaking notes, and the trill endings depended on the taste of individual performers. Tracing back from the eighteenth-century convention, pianists during the early nineteenth century usually began trills with the upper note on the strong beat. Later, an increasing number of pianists preferred to start on the main note. (p. 22)
1Sigmund Lebert and Hans von Bülow: https://imslp.org/wiki/Piano_Sonata_No.14%2C_Op.27_No.2_(Beethoven%2C_Ludwig_van)
2Clemens Schultze: Ibid.
3Alfredo Casella: Ibid.
4Leó Weiner: Ibid.
5Heinrich Schenker: Ibid.
6Artur Schnabel: Ibid.
7Hugo Reimann: King Yue Li, 2020, "Nineteenth-century Performance and Editorial Practice: A Study of Beethoven's Sonata in C-sharp minor, Op. 27 No. 2", Ph.D. diss, Arizona State University, https://repository.asu.edu/attachments/227327/content/Li_asu_0010E_19894.pdf
8Barry Cooper: Barry Cooper, 2007, Beethoven: The 35 Piano Sonatas Volume 2 (The Associated Board of the Royal Schools of Music).
I would play it like this (the image came out kind of big...):
You can play the lower A# quite short to avoid tension (if it suits your interpretation, of course). For me the most natural fingering is 4-5-4-3-4-5 in the upper voice, although 3-5-3-2-3 works, too, and might be easier if your hand is big and/or 4th finger weak.
There's a similar trill a couple bars later, which I find much harder. The fingering 4-5-4-3-4-5 works there, too, or you may try 5-3-5-3-5-4 or something similar. In both cases it helps to play the first note of the trill a bit longer and stronger than the others, and it works musically, too.
Beethoven always started his trills on the upper note (documented here), unlike modern practice. This would come out more fluent coming onto the G#. Although Nonpop's example could work, it's not normal to start and end on the same note. A better interpretation would be
B A# B A# G# A#. As far as fingering, without context, it is hard to say what would work best, but 4-3-4-3-2-3 should work. As far as timing, whatever goes with your interpretation is fine. I've shown two alternate timings below.
I’m inclined to lean towards playing it as whatever sounds best. The quintuplet starting on the main note feels right to me - having been attempting to play bar 32 with a convincing trill starting on the upper note for a while and not really succeeding, it feels likely to me that it is supposed to start on the main note, which is much easier to execute accurately. The other way just seems a bit messy.