Like Bach's Sonata in E major, BWV 1035, this is a technically challenging work dedicated to Potsdam flutist Michael Gabriel Fredersdorff, an employee of the flute-loving King Frederick the Great. Or so says a very late copy of the manuscript, not prepared by Bach. There's also an early copy by organist Johann Peter Kellner made around 1725 or 1726, nearly 20 years before BWV 1035 was written. Either the copy with the Fredersdorff dedication is incorrect, or Bach dusted off an old sonata for Fredersdorff's use.
In any event, BWV 1034, like BWV 1035, is in the four-movement, slow-fast-slow-fast sonata di chiesa, or "church sonata," format. The first movement, Adagio ma non tanto, is usually performed at a fairly deliberate pace despite Bach's ma non tanto admonition. There's something implacable about the music's steady trudge and something obsessive about the melody, which tends to break into repeated two-note units. Next comes an Allegro, based on a burbling flute melody over a descending bass figure; the bass line soon levels out, but it has already started to pull the melody in a downward slide. The flute indulges in rapid passagework as Bach provides a short series of variations on this material. The ensuing Andante begins with an extended introduction by the continuo instruments (usually harpsichord and gamba or cello). Once the flute enters with its spacious theme, the bass line repeats fairly steadily, as if for a chaconne or passacaglia, with the flute singing freely above. The B section delivers a variation on the theme, and the final section essentially repeats the movement's opening measures. The sonata's concluding Allegro is dark and quick, something of a Bourrée, extremely busy and energetic while engaging the continuo in full-fledged counterpoint. It's a flashy finale for an extroverted -- and highly skilled -- soloist.