[music] So, we’ve arrived at the finale – the main body of the finale, post-introduction – and it is a straight-ahead sonata form. Here is the exposition. [music] So, this music provided the resolution that the entire piece has cried out for. It does so, first of all, on a literal level: in two and a half movements, we never got a 5-1 cadence in A major, and that’s what the first two notes provide us with. [music] But more significantly, they bring a joy, which to this point had never been fully allowed to spring forth. In the opening movement, it is compromised by this great vulnerability; in the second movement it is more apparent, but it’s a very specific kind of joy – a gruff, elbows out, rough-and-tumble kind. Only after the piece reaches its lowest moment in the third movement introduction, a point of true darkness, do we then move fully into the light. And then, when we do, we do. All of the material in the exposition is, in one way or another, ecstatic. The first theme is exultant. [music] The second theme is more one of quiet contentment, but it, too, moves towards effusiveness. [music] By the way, the one formal quirk in this otherwise straight-forward sonata movement is that this second theme is on the tonic, not the dominant. [music] The move to the dominant comes only as the theme is coming to a close. But anyway, throughout, we have the feeling that something that has been held in for the whole piece, thus far, is now being released. The other way in which this last movement represents a fulfillment is in its obsession with counterpoint. In the development, 15 minutes into a work that has been more-or-less constantly interested in the interplay of the voices, we finally get what seems inevitable – a fugue, a dazzling fugue, really. Now, Beethoven was very big on fugues in his late period, and nearly all of the late sonatas contain one. But this one, along with the one in Opus 110, is the most fully fleshed-out – it has the least “license,” which is the word Beethoven used to describe the fugue of the Hammerklavier Sonata. License or no license, the Hammerklavier fugue is dizzyingly intricate, but Opus 101 is really not far behind. Like most fugues, this one modulates frequently – almost every phrase, at a certain point – but it both begins and ends not in A major, the key of the movement, or E major, the key the exposition ended in, but A minor – a callback to the introduction. The first half of the fugue's subject is very similar to the movement’s first theme, but the second half of the subject is new. [music] Although this second half makes oblique reference to a passage from the exposition [music], it is essentially new material, which gives this fugue the feeling of questing – of heading into uncharted territory. [music] So, as you can hear, each phrase introduces a new voice, with the fugue subject, until we have 4 voices. In the previous lecture, I spoke about the “fake fugue” of Opus 10, Number 2’s finale: it introduces three voices, and then more-or-less abandons the idea of them functioning independently. In this fugue, however, once the fourth voice makes its first appearance, all four of them remain present and active. There are a minimum of three independently moving voices going at any given time, and often there are four. This makes this fugue VERY complex, because the subject has a lot of motion in it. It’s one thing to keep track of four voices when they're slow moving – you know, quarter and eighth notes – but when there are sixteenths involved [music], things get very dense very quickly. [music] Given that the motion is constant – and that one instrument, and therefore one timbre, is responsible for all of the voices -- it takes a lot of effort to distinguish them from one another. The Hammerklavier fugue has an even faster-moving subject, but it only has three voices, and there are many moments when only two of them are in action. So, as discussed, the fugue eventually works its way back to A minor, and once there, it moves towards an absolutely ENORMOUS 5-1 cadence, which resolves back to the major. [music] The final phrase unfolds over this E pedal tone, and its first half is really just an embellishment of the equivalent passage in the slow introduction. [music] But whereas that version got…lost… and had to cycle through first a cadenza [music], etcetera, and a reference to the opening [music] before finally getting to 5-1 [music] here, the asked-for cadence comes immediately, and thrillingly. [music] Beethoven is making up for lost time in this movement: he is establishing the A major with increasing force and exhilaration, having shrouded it in mist for the first two and a half movements.