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The fugue subject comes from a placement test example for the Peabody Institute...

First exposition...

Audio

Second exposition...

Audio

My question are about the interplay of harmony and counterpoint.

It seems to me the counterpoint of many fugue expositions weaves harmonically between the tonic and dominant or sub-dominant chords.

One might expect that when a subject (which is often an elaboration of tonic chord) is transposed up a fifth for the answer that the harmony of the answer should be the dominant chord. But this subject ends on the mediant scale degree, not part of the dominant. A quick check of some Bach fugues shows that in such cases the final mediant in the subject and initial dominant degree of the answer will form a I6 first inversion tonic chord. So in some cases the initial harmony of the answer may be tonic (ex. WTC I, no. 2), but other times the harmony could be dominant (ex. WTC I, no. 6), depending on the final tone of the subject. Are there occasions where the answer might start with another harmony, like the sub-dominant chord?

The second part of my question is about bar 2 of the subject/answer. In the subject bar 2 moves around the subdominant and that transposes to the tonic in the answer. However, in the answer there is a question of whether should be heard as returning to the tonic (D and F in D minor) or should it be heard as being of the dominant (D and F in A minor?)

I split the difference and wrote two alternate expositions.

In the first I treated the D and F as being of the dominant key and harmonized it with a G# to form a viio7 diminished seventh chord in A minor.

In the second I treated the D and F as being of the tonic key and harmonized it with a voice exchange F and D.

Are both options viable by 18th century fugue conventions?

Both sound OK to me, but I'm teaching myself so my confidence isn't too high.

BTW, if anyone recognizes the theme as one from a famous composer, please let me know. I've always wondered with there was a real finished product to study.

I suppose the Peabody Institute might hope that candidates realized that Bach has written an extended textbook on this exact situation, namely "Das Kunst der Fuge." Bach's fugue subject does have a C sharp to define a dominant harmony before ending on the mediant, but he often replaces it with a C natural to avoid the "obvious" cadential harmonic progression.

The rest of this answer is entirely about the first answer in the fugue, since two-part writing is inherently more ambiguous harmonically than three or more parts, and in that sense more difficult.

In both your versions, you chose to use a real answer. That has the consequence that you can't use C sharp as the leading note of D minor, and implies the harmony will tend to be in the Dorian mode instead of the D minor. There is nothing "wrong" with that decision, but it has consequences.

Without a clear leading note, you can easily end up with the audible tonal center ambiguously meandering between any pair of chords that happen to sound like a dominant and tonic - i.e. to F major or B flat major instead of D Dorian where you want it to be.

I think bar 3 of your first version suffers from that problem. Beats 2 3 and 4 sound like three C major chords which define a "tonic" by repetition, and that takes you off track relative to what a fugue is expected to be doing. Bar 4 recovers the situation by an unambiguous progression of E major - A minor, but the damage has been done before that.

The second version solves the problem by staying firmly in D Dorian mode for the answer, except that to my ears the B flat on the last beat of bar 3 sounds too much like a B flat major harmony which again is wandering away from a "dominant" of "A Aeolian." Try replacing beat 4 in the bass with a quarter note E or A and hear the difference. (To my ears E is harmonically OK but a bit insipid, but I don't know if the "Peabody placement marking scheme" would lose you marks for an unprepared 4th A-D.)

An alternative idea is to cut through all of this modal ambiguity and use a tonal answer with unambiguous A major - D minor harmony, as below. If the final F in the question was in parentheses, you are at liberty to replace it with anything you like in the answer, or subsequent entries of the subject.