The last movement echoes classical form in that it is like a rondo and like a dance, both qualities suitable for a classical finale. Here is how it begins:
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Even though the key signature is E major, the F naturals give the melody an Eastern European folk inflection. There are moments when the melody sounds a bit like one that Bartók might have collected. The quality of the melodies is usually connected with Jewish folk music, which Shostakovich was interested in at the time and also connects with the dedicatee, the Jewish Ivan Sollertinsky. The bare, almost rudimentary opening slowly gains weight and complexity:
Here the piano is playing a new theme, an expansion of the first theme, in Lydian mode on C, while the strings accompany with C minor harmony for a very exotic effect. Later a new, contrasting theme in 5/8 is introduced in the cello:
Another contrasting section with piano arpeggios is presented in E minor:
This leads to a restatement of the main themes from the movement:
Finally, on the last page of the piece, the texture liquidates (a term from Schoenberg, meaning that the texture is thinned down to almost nothing) and, under harmonics that recall the first movement opening, the piano states the chord progression of the passacaglia. Then, after one last statement of the initial theme, the movement ends with simple E major chords.
Honestly, you don't have to beef up the piece with extra-musical references: there is plenty of musical beef here already. Now, let's listen to a performance of the movement. The Argerich, Kremer, Maisky version of the last movement is broken up into two parts, so let's listen to some other Russians. Here are Dmitri Vinnik (piano), Sviatoslav Moroz (violin), and Natalia Gutman (cello), with all of the last movement of the Piano Trio, Op. 67, by Shostakovich: