The Op.34 Variations can come across as slight (if very beautiful). In fact, they’re one of the most important sets of variations ever written, as they represent B.’s most successful attempt at large-scale organization in theme and variation form.
A bit of context on this. The major problem of theme and variation form is that - if you think about it - the form has no natural sense of order or finality (contrast this to sonata form). You can, in principle, keep varying on forever, and once the variations get going there is no natural impulse which checks the momentum you’ve built up. One standard way of solving this problem is to use a fugue to end (or almost-end) a set of variations, since a fugue represents the most thorough and internally rigorous way of examining a theme - you put it at the end to say, “And now there’s really nothing more to be said about this theme.” This is what B. does in the Diabelli and Eroica, and what Brahms does in the Haydn/St Antoni. Another method is to imply some large-scale movements within a set of variations: Rachmaninoff was particularly adept at this. And yet another is to end by building into a recapitulative structure - so in the Goldbergs the ending is heralded by the Quodlibet, which breaks the work’s structure by forcefully entering where there should instead be a canon at the 11th, which in turn clues you in to the fact that something special’s going to happen (the return of the aria).
B. uses a little of that last method here, but what is by far and away the most striking aspect of the Op.34 is how B. manages to impose a pattern of almost Schubert-like tonal organization over the set (B. was actually quite proud of this novelty of having every variation in a different key - he told his publisher to ensure that this fact was mentioned in the printed edition). The ingenious thing is not only that this structure exists, but that this structure also acts as a colouristic device: it has one foot in sonata form (structure) and another in variation form (in-the-moment texture). So a set of variations that really is a highly disparate group of character pieces (7 keys, 4 time signatures, various dance forms) sounds coherent and directed in a way no other set of variations B. produced quite manages.
On tonality: the variations move down a chain of 3rds or delayed 5ths (F-D-Bb-G-Eb-Cmin/maj-F) to land on the dominant C, which then sets up the return to the last restatement of the theme (which itself is a pretty clever thing, also serving as final variation and cadenza). It’s striking how sonata-like this scheme is - the end of Var.5 comes right out of a sonata's development section. At the same time, the largely unprepared modulations create some really nice shifts of colour between variations. The entrance of Var.1 in D has an open-ended, expansive feel characteristic of modulation down by minor 3rd (helped also by the higher tessitura), and the move from D to Bb also has a characteristic modulation-down-by-major-3rd feel - something more rounded, closed, warm.
Another structural device that’s first presented as a textural one is the vii diminished 7th over F you hear in bar 6 and which recurs in the 3rd-last bar of the theme. It’s a beautiful expressive device, functioning basically as a rootless dominant 9th floating over the tonic. But it also chains the variations together, because the progression of keys through the piece means that the same diminished 7th harmony (or something very close to it, like a borrowed iv from the parallel minor) is always heard in successive pairs of variations, but each time resolving differently. So in the theme and Var.1, you get E dim.7th (resolving to F and D), in Vars.2-3 you get A dim.7th (resolving to G min/Bb and G), and in Vars.4-5 you get D dim.7th (resolving to Eb and Cmin). There are other decorative devices that also get given structural treatment, such as the F pedal you hear in the first two bars and the dotted rhythm which closes the first phrase of the theme in bar 7: both motifs recur constantly, with the latter opening Vars.2, 4, 5, and 6.
Some other things worth noting: Var.5 anticipates the second movement of the Eroica Symphony, soon to be written after this work. The florid decoration of Var.1 mimics the sort of technique you usually see at the end of a set of variations, rather than the beginning, and Var.6 follows in the footsteps of the Goldbergs in using something festive and celebratory to anticipate the end of the piece.