Please help me understand this. Thank you in advance. Screenshot attached. From Mozart's "Hafner" Serenade (K 250 ), Andante (2). In measure 8, the harmony is A7b9 / Bb. My current understanding is that A7b9 "should" function as a pre-dominant to D minor. Yet the next measure is D Major. And it sounds magnificent. I have changed the F# to F (D minor) in measure 9, and it does not have the melodic quality inherent in that passage. Should I understand that a V7alt. -when in a pre-dominant function in a classical piece- is not always bound by post-modern music theory? And that the "chromatic" movement (A7b9 / Bb [as a V7/V] to D MAJOR) is actually not uncommon? Hopefully this will help me analyze other Mozart pieces, where his chromatics are daunting.
It's not unusual for a dominant chord to move to either the major or minor (relative) tonic chord, even in the presence of the chordal b9, which can be justified through modal mixture. Consider that even the presence of the leading tone in minor is the result of an alteration.
In this case, since D major is itself the dominant, that's where Mozart wants to be. D minor would serve a different function.
Perhaps the more interesting question is how Mozart works B♭ into the eighth bar of a movement in G major. That the phrase ends with a V-I cadence in the tonic, therefore using a D major chord, is not at all surprising.
First, I disagree with the analysis of A7♭9. The A appears only in the second beat of the measure as a passing tone. It isn't structural. A more natural analysis is C♯º7/B♭, so it's the third inversion of viiº7/V.
This is also a pre-dominant harmony, of course; the main difference with this analysis is that it relegates A to a less important role. It's also expressed in terms more favored by classical theory, but that is not particularly critical here.
It also doesn't answer the question. To do so, let's analyze the whole phrase. In the first five measures, the harmonic rhythm is slow. The harmony changes every two measures. We have six beats of G major, six beats of D7, and a cadence on G major. In the sixth bar, the harmonic rhythm changes: there is a new harmony on each beat:
G - C/E - G/D - A7/C♯ - F♯ø7/C♮ - G/B - C♯º7/B♭ - A - C♯º7/G - D/F♯ - D7/F♯
In Roman numerals, that's
I - IV6 - I(6/4) - V(6/5)/V - vii(4/3) - I6 - vii(4/2)/V - V/V - vii(4/3)/V - V6 - V(6/5)
There are two things of note. The first is easier to see in the chord/bass analysis, which is the descending bass line G-E-D-C♯-C♮-B-B♭-A-G-F♯. This helps to explain how Mozart introduces the B♭. The second is increased instability and ambiguity of the harmonies in the passage where the harmonic rhythm is faster. You could argue a bit about it; for example, you could analyze the progression from the third or fourth chord relative to the dominant, in which case the fifth and sixth chords are not vii(4/3) and I6 but vii(4/3)/IV (of V) and IV6 (of V). This gives us a progression of
[G maj:] I - IV6 - I(6/4) (= D maj: IV(6/4)) [D maj:] V(6/5) - vii(4/3)/IV - IV6 - vii(4/2) - V - vii(4/3) - I6 (= G maj: V6) [G maj:] V(6/5)
When you look at it this way, you see that the progression has the standard functional pattern of movement to subdominant to dominant to tonic.
In any event, the C♯º7 chord has a secondary dominant function as you correctly note. I would suggest, however, that thinking of it as the dominant of D minor is slightly off the mark; it's better to continue to think of it as a pre-dominant. Since it has a B♭, you can think of it as a pre-dominant "borrowed" from the parallel minor key of G minor. After all, this passage still makes musical sense and still sounds like Mozart if we use B♮s instead of B♭s.
Looking at it this way, we can say that the B♭ is a feint toward G minor, not D minor. But since we have to go through the dominant of D major (especially D7), we're not fully committed to G minor, and in fact when we leave D for G, we go to G major instead, back to the home tonality.
As a postscript, this use of the fully diminished seventh chord to introduce the dissonance and tension of minor tonality into major-key music is fairly common. The first thing I thought of was the Recordare from Mozart's Requiem, but of course Bach also loved his diminished seventh chords, and, sure enough, there are several in the first prelude from the Well-Tempered Clavier. Those in measures 12 and 14 are, in particular, part of a similar chromatic descent.