I am, of course, always glad to have support for the rising line as background, although it's hardly needed any more, given the 1000+ examples of ascending cadence gestures I have found (so far) in the repertoire of musics of all kinds, but I am obliged to disagree with Rothstein here, if we are talking in Schenkerian terms. Beethoven is not "fitting a figure in" to an existing system here--he is using that distinctive figure to transcend the system altogether. (David Lewin discusses this idea of transcending the system in terms of patriarchy and women's voices -- see the reference at the bottom of this post. In Beethoven's case, it is almost certainly a philosophical-religious-pantheistic transcendence of the kind one finds elsewhere in his music.)
Here is the title page for the first edition, with the publisher's hopeful marketing note "faciles et agréables."
The bagatelle is an odd little bricolage of musical bits that resembles a cut-and-paste job more than a coherent composition. I have exaggerated the point by "cutting up" the score, separating it into its three components: first, a more or less normal opening phrase of six bars;
. . . then an eight-bar "continuation" whose only connections to the preceding are staccato notes (cf. bar 6) and simple presentations of invertible counterpoint and stretto;
. . . and finally what looks rhetorically like a structural cadence, but (a) offers only a second inversion ii chord; then (b) subverts the dominant by providing the proper bass (eighth note G2 in the second bar) but with Bb, not B-natural. The persistent subdominant -- it's been there since bar 11 -- and the very extended tonic pedal point are both familiar features of Baroque preludes and so are not strange here, given the display of old-fashioned devices that preceded. Nor, even, is the wandering into the instrument's highest register -- recall BWV 924 & 924a and Niedt's recommendation (link) -- but, still, the long ascent combined with an equally extended crescendo does seem a bit much in context. (Yet again, though, as many writers have noted, there is an obvious connection between this little bagatelle and the attention to registral extremes in the variation movement of Op. 111, which must have been written around the same time as this bagatelle.)
In Schenkerian terms, the turn to the subdominant subverts a cadence to the final C in the bass. We are therefore obliged to read an Ursatz that concludes in ten bars with a by no means hidden Urlinie from ^3:
This obligatory reading is clumsy, of course, but given that the music heads off to the subdominant immediately thereafter, it makes sense. Note, of course, that there is no ascending Urlinie -- much as it bludgeons our ears, the ascent over the pedal point in the second half of this bagatelle-prelude is a foreground feature at best.
I am not overly inclined to defend this bagatelle, as you may have guessed, but I am willing to suggest that it is at least possible to draw the final ascending figure into an effective reading based on register, tonal frames, and invertible counterpoint. In the example below, the upper voice pair ^3/^5 in bar 1 is flipped to the sixth ^5/^10 in bar 2 (invertible counterpoint, remember). By bar 5 the ^5/^10 has become ^5/^9 -- or ^5/^2. In the sequence of bars 8 and following, ^2 becomes ^1 and ^5 becomes ^4. In bars 15 and following, ^1 (or ^8) returns by step to ^5, and ^4 drops to ^3, thus recovering, in its original position, the third-pair from bar 1. It's that interval that is looped and threaded through overlappings until it finally makes a direct (if chromatic) ascent to ^8 (as C7).
Lewin, David. [1992] 2006. "Women's Voices and the Fundamental Bass." In his Studies in Music with Text. New York: Oxford University Press, 267-81.
Rothstein, William. 2006. "Transformations of Cadential Formulae in the Music of Corelli and His Successors." In Essays from the Third International Schenker Symposium, edited by Allen Cadwallader, 245-278.
