This is one of the pieces currently in my repertoire, so I thought I'd analyse it to get a better understanding of it.
The piece is, as the name would suggest, written for four pianos (though originally of course it would have been written for harpsichords, however Vivaldi composed the original piece for four violins, and Bach merely arranged it for keyboards-"cembali".) Sometimes there are strings involved too, because that is how Vivaldi composed it and it was the Baroque period! The piece is in A minor, and is in 4/4. The four keyboards are constantly in interaction with each other, whether it be imitation or playing in harmony.
The structure of the piece is not clear, though at the end the first sections return. The piece begins on the dominant, with only the 1st cembalo playing single notes, then the 2nd cembalo comes in on the same notes, so that they are playing in unison, although their parts are different. They play together for a bit, then in bar 5, the other two parts come in and all the parts play chords, the tonic chord. For much of the rest of the piece, the parts all play different melodic ideas at different points, all of the ideas interchanging between parts. This creates the idea of imitation. Each part gets a solo, the 1st cembalo having the first one, then the 4th cembalo comes in, it's left hand playing in rhythmic imitation of the 1st cembalo (three semiquavers). A few bars later the other parts come back in. Then later on the 2nd cembalo has a solo, and the 3rd cembalo comes in half way through, just like the 1st and 4th cembali. However then the 3rd cembalo goes away again and the 2nd cembalo keeps playing.
I used this version: http://www.youtube.com/watch?v=4_pbxdfwi7Y The movement ends at 4:50.
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