For the last movement of the Kreutzer sonata, I think questions are less necessary : the writing and form is more classical (and in fact, this movement has been written before the others, originally as the finale of the op.30 n.1 sonata), however I will only refer to one very ambiguous staccato sign at the threshold of the form : the lead-in to the recapitulation.
Here Beethoven (in the darker pencil) precised the calando to last until the very end of the gesture. Calando (‘with warmth’) indicates a loosening or unloosening of the energy, a human touch in the midst of the frantic joy of this movement (akin in that sense to a ritardando). The once underlined ‘pp’ in the violin part matches the piano’s in bar 267 (not shown) while in fact the violin must play this pp with more energy and drive (but not louder !) than the one bar 267, which is...
Keilen : Sorgen, Zagen, Weinen
Here, however, the black staccato marking in the upbeat to the ‘a tempo’ is a difficult choice for the performer : the reinforcement line of the ‘calando’ is also under this eigth note, while the keil shows the way to the main material’s return. The keil I think has been misread for its usage during the classical period : we see it through the lense of a ‘staccatissimo’ sign, an indication that each note with a keil must be separated and of equal emphasis. I have heard many many teachers describe it as such.
...of a secondary importance : but with the c sharp, a more vocal quality is required, and I would encourage a warm messa di voce over this long slur, going up for 3 bars and down in dynamic during the last bar.
How should a keil be performed ?
Let’s listen to Rudolf Kolisch’s recording of the Bagatelles op.9 by A. Webern, specifically the third one :
But its use during the classical period was in a way looser, more general. As van Oort shows it, they appear in order to emphasize not the single note, but the motif of notes (as in the Adieux piano sonata by Beethoven he uses as example). Once the motif is written once with keilen, there is no need to further emphasize it.
Two further investigations into this question of lifting the bow or not are provided by Webern and Beethoven :
In the writing of the Beethoven sonata’s last movement lead-in to the recap, we can see that the piano has a clearcut change of register : both hands go down a major seventh, while the violin in fact is the only linking point between the development and the recapitulation. This very precious eigth note in the violin part is what allows the acoustic and formal link between very opposite characters. Once again, we see that Beethoven knows how to control the open-endedness of his development through a small sign that appears at the end of an apparent dead end - how can one come back to the original tempo after a very sweet section like the 'calando' in the Kreutzer sonata? Beethoven's solution is both efficient and chaotic in that he makes a single note drive the entire form: and how to perform that note in a subtle way is the key.
For example in Mozart’s string writing, it is used rather as a rhetorical sign (discussion with Luc-Marie Aguera, second violin of the Ysaye quartet), a compromise between the shape of a motif and its contents, as well as its place and rôle within a phrase.
I think this is how it should be understood in this context.
(op.9 Bagatelle n.4, cello part bar 5) : the equality of these very soft strokes is more easily achieved through the control of the upper half rather than the ‘back-and-forth’ offstring motion of upbows and downbows that would be natural in the lower half of the bow. A last example is from the string quartet op.132 by Ludwig van :
The keilen bars 2 and 3 show the subsidiary material for the remainder of the bagatelle, and they are clearly played with resonant, broad martelé strokes. While these Webern pieces might seem far off any classical consideration, they nonetheless are Bagatelles (a genre championed by who else than Beethoven throughout his life), and I believe they capture the fleeting essence of what the viennese style of writing and performing might be ; in that the writing encapsulates the generosity and changing flow of what interpretation could be. A number of Kolisch recordings use this broad, à la corde stroke : it is very prominent in his 1966 live recording of the Schubert Fantasy D.934 with Paul Badura-Skoda, as well as, closer to Beethoven, Kolisch’s live recording of the Scherzo from the 7th violin sonata : it sounds like he is performing the opening staccato quarter notes in separate strokes in the upper half of the bow !
As Kitchen has shown, Beethoven even uses different lenghts for the ‘staccato strokes’, proving that it must follow a logic of development through the articulation of a motif’s shape. For example, due to the ‘più cresc.’ sign, the (small) keilen are not necessary anymore, because the volume and drive of the music would be hindered if the violinist were to force the bow to articulate everything.