Schubert wrote two sets of four impromptus. Though some of the first movement sonata-allegro elements are well hidden, I think of both sets as four movement sonatas. The opus 142 set may be a more convincing sonata.
As in the opus 90, Opus 142 has a lengthy first movement which can be analyzed as a sonata-allegro form. The remaining impromptus include a slow one and a lively final movement (impromptu). The third movement programmed here is really not an impromptu at all. It is a theme with five variations. It has a short quiet ending coda which utilizes a fragment of the original theme. This theme, from Schubert’s Rosamunde opera, was also used in a string quartet.
Schubert’s Opus 142 variations are in the key of Bb major. In each of the variations he modulates to a related key in the second section. He hides the key motion to some extent by beginning most of the second sections of each variation on the dominant of the new key. This is a third away. For instance, in the theme, he begins in Bb major, leaps to D dominant 7th (by raising the leading tone) before continuing to g minor, the relative minor of Bb major.
This is true in the first variation and the second variation. In the third variation, for interest perhaps, he begins in Bb minor (not major) and shifts to Db major, the relative major, by using the common note Db between the two chords. He remains in Db for a short time before using the third above as the dominant of Bb before returning to Bb minor. Many variations of this period have at least one variation in the minor mode.
I know that all of that note stuff above is a bit confusing, but it is interesting to watch Schubert modulate about by shifting the use of a particular note in the chord. This becomes a solid attribute of the growing romantic period and is a basis for modulation by thirds.
You might find it interesting to note that in Beethoven’s Opus 57 middle movement, the theme and four variations are all in Db major with a modulation to the dominant. This is similar to the baroque suite in at least two ways. First it was common to hold key signatures in each of the individual pieces. Secondly, interest was created through change, in that case different dances with different moods. Beethoven’s “change” arises from a consistently rising mood as he moves from variation to variation. Like Schubert’s variations he re-states a modified version of the original theme before ending.
Schubert’s fourth variation adds a flat thus finding its center in the key of Gb. This is a third down from the original key. In the second section of this variation he moves to Eb minor the relative minor of Gb. He remains on this level until he is ready to return to Bb major. The raised leading tone in Eb minor becomes the third of the new tonic chord of Bb. Of course the two chords have Bb in common anyway. (there I go again!)
The last variation is fast with significant finger work like Beethoven’s last movement. Schubert’s return to the quiet mood of the opening aria brings his variations to a quiet ending.