Why does the 7th of a dominant 7th chord have to resolve down in voice leading?
Because it is a tendency tone. The really strong tendency tones are
TI resolving respectively to
You can either accept they are handled this way by convention, or you can explain it as they must resolve this way, because these two movements are the half step movements in
V7 I. (If you want a rationale of
ME in minor, consider it to be following the half step voice leading model of major with the minor third of tonic
i being a modal modification after the fact.)
But that isn't your real question. The real question is how to get fully voiced chords in
V7 I with good voice leading, or turned around, why strict voice leading for
V7 I results in incomplete chords.
It may help to first look at the voice leading for triads only and then the implications of adding the seventh to the
The way to voice lead
V I is ideally hold one voice move the other two by step, or possibly you could move all voices in the same direction. If we add basses in contrary motion we have:
V chord the
E3 in the bass is a given so that means the
E4 in the tenor is the duplicate. If we move one voice to get the seventh of a fully voiced
V7, we will move the duplicate
E of the tenor.
If we make that change, and strictly resolve the tendency tones, we get doubled tones and incomplete tonic chords:
If one of the tendency tones is not strictly resolved, we could move
FA up to
TI down to
I'm not really sure how "acceptable" the first one is, the second is more normal. In either case, burying the unconventional, undesirable movement in the inner voices is the normal way to disguise it. You would not want to put, for example,
SOL in the soprano.
Another way to approach the problem is work backwards from where you want to end.
Start at a fully voiced tonic chord, then build the
V7 chord, first add the two tendency tones, then add the two remaining voices as step-wise motion:
Notice that none of the voices move
DO. That means that no matter how you try to invert those voices you will not be able to get both chords in root position! That's a detail you didn't add in the question, but is an essential aspect of the voice leading "problem." There really isn't a voice leading problem until both chords must be root position.
A particular issue is the held tone from the ideal, triadic voice leading, which is
SOL held to
SOL. If you put that in the bass, you will get a
I6/4 tonic chord:
If we "fix" the bass by moving it to
...we have two root position chords, and good voice leading, but arrive at the incomplete chord problem through a different process. The "solution" to making that
I complete will be the same as above, move one of the tendency tones against its tendency and bury it in the inner voices.