Haydn was very inventive with his finales in op 20: three of the six are fugal which began the practice of incorporating this contrapuntal technique into classical form. From now on, composers like Mozart and Beethoven would use fugal sections and entire movements in fugue texture as vital elements. But this quartet does not have a fugal last movement. A typical last movement form was the rondo that alternates a theme with contrasting passages in this way: ABACADA. But in the opus 20 quartets, the non-fugal finales are not in rondo, but sonata form, in which both halves may be repeated, as in the performance by the Attaca Quartet. So-called 'sonata' form is really a development of Baroque dance form: two repeated halves with the first half modulating either to the dominant or the relative major (if in minor). The much-longer second half modulates back to the tonic. In the Classical development of this idea, the second half expands to include two main sections: a development, which tends to both fragment and explore the theme or themes and move through a variety of different keys, and a recapitulation in which the themes return in their original form and in the tonic.
Haydn has an odd theme, eleven measures long, that starts and stops and has a rather odd bit in the middle. The theme starts quite conventionally with four measures that go from tonic to dominant--the first line of my example. Then there are two measures of tonic, which I left out. But the last part of the theme--the last line of my example--is rather odd. It starts on the dominant, but then adds an anomalous F natural.
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As you can see that C#/F natural pair of notes is emphasized for three measures before a half cadence on the dominant. It functions pretty much like a dominant, but the F natural is neither part of the dominant, nor the tonic. What it is, is borrowed from the tonic minor, a technique known as 'mixture'. But I might speculate that Haydn is making reference here to the Hungarian recruiting music known as "verbunkos". However, Haydn never does anything superfluous and this F natural has a real function in the movement. On the next page, at first it looks like he is going to modulate to the dominant, but passes right by and has a big cadence on F major! Then works his way back to A major, the dominant, to end the first part.
The second part ranges through a number of keys, at one point obsessing over the VII diminished seven of E, which is the dominant of A, which is the dominant of D major. But works its way inevitably to A, then ending conventionally, but very quietly, in D. But on the way, there are a lot of passages that remind one of a real hoe-down!
The thing to remember when listening to these quartets, is that there were no rules yet: Haydn is experimenting with themes, harmony and form. The remarkable thing is that the things he discovered proved to be so fruitful that composers as late as Brahms were still studying what Haydn had come up with.