Why a fugue, let alone a 'great' fugue? Why would Beethoven go back to what was, at this point, an archaic form? Don't the rules of music history state that composers are always progressive? Always developing new techniques and never returning to old ones? Well, those are certainly the rules of 20th century modernism, but I doubt they apply generally. Beethoven, along with Haydn and Mozart, was fascinated with the fugue as he was fascinated by Bach. It is not so widely known that one of the ways Beethoven made a name for himself as a salon pianist in his early days in Vienna was by playing preludes and fugues from Bach's Well-Tempered Clavier. He must have owned a manuscript copy because the music, at that point, had not even been published in a printed form. For a long time after Bach's death, his works circulated amongst the cognoscenti in manuscript form.
Beethoven's obsession with fugue only grew as he developed as a composer. One of his great achievements as a composer was to fuse the forms of the sonata and the fugue. He produced examples such as the finale of the Piano Sonata in A, op 101, which contains 100 measures of fugue as its development section. The Piano Sonata op 106, the "Hammerklavier" contains an earth-shaking fugal finale and the Piano Sonata in A flat, op 110, has a more lyric fugue in its finale. There are many more examples in the quartets, the Missa Solemnis and the Ninth Symphony.
One of the attractions of fugue for Beethoven was as a solution to the problem of the finale. How to achieve a powerful and appropriate final movement was a problem that only grew keener as his ability to write convincing prior movements grew. The 'classic' solution was the rondo, a movement with a single theme and contrasting episodes. This already reminds one of fugue with its single theme and developing episodes. Beethoven may have thought that fugue would enable a different sort of finale that might avoid the harmonic clichés typical of the rondo. As Kerman says, "a fugue subject is at once more pliable and interesting than a rondo tune, easier to bring in frequently and easier to vary."
Beethoven also saw fugue as a challenge. I have remarked before that the two greatest composers of all are usually counted as Bach and Beethoven. Beethoven was very aware of Bach and, just as his "Diabelli" Variations are a response to Bach's Goldberg Variations, so too, the Great Fugue is Beethoven's response to Bach's The Art of Fugue. In his own way, Beethoven is trying to exhaust every possibility. In The Art of Fugue, Bach showed all the resources of contrapuntal combination and some of thematic transformation. In the Great Fugue, Beethoven will explore more than Bach the possibilities of thematic transformation. The idea was in the air--Berlioz' Symphonie fantastique was only five years in the future!
As Kerman notes, Beethoven opens the piece by hurling various transformations of the theme at the listener's head like a handful of rocks:
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One remarkable aspect of the notation is found in the third line, last two measures. The theme is given in simple off-beat quarter notes, but Beethoven writes them as eighth-notes tied! Why? Any string player is going to play these notes more intensely and hold them out more fully than if they were written simply as quarters. Let us go ahead and listen to the music:
This performance, by the Talich Quartet, is not my favourite, but it does have the score, which is pretty important. Anyway, after you listen to that, I suggest listening to a more full-blooded version such as this with the Alban Berg Quartet:
The Great Fugue is several fugues on several transformations of the theme in several different keys. It is a piece well worth your study. I could write several posts just on the rhythmic ingenuity of the writing, or, heck, a doctoral dissertation! But I think I will end here by encouraging you to listen to the fugue several times. It grows with each hearing...