You're definitely hearing a D4 along with this A, but are you sure about the F4?
I wonder if you're actually hearing a G3.
The reason is because of how string instruments are tuned.
Violins, for instance, are tuned in 5ths, so when they tune to an A4, they tune the D4 along with the A4 to create a perfect fifth.
Then they tune a G3 along with the D4 to create another perfect fifth.
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I have heard accompanists use a D minor chord when a violinist (or violist or cellist for that matter) is tuning many times.
One accompanist I had even followed up the D minor with a D major chord!
The rationale my cello teacher gave me was that playing a triad puts the string pitches into 'context', and that a D minor chord 'sounds purer' than D major.
I'm not sure about the latter point but putting notes into the context of a triad makes sense.
With a violin it is common to tune the middle two strings first (A and D), to avoid messing up the other strings' tunings because of the variations in tension when turning the pegs, hence the use of a D triad initially.
Cellists also go for A and D first (top two strings) because the variation of tension when tuning is not so pronounced.
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I've heard this done plenty of times, but never really thought about why before now.
It makes some sense to play the A and the D, because the piano will be (nominally) tuned to equal temperament and the piano's fifth will not be exactly pure.
Since pianos use with "stretched" tuning (the octaves are slightly bigger than a true octave), and the amount of stretching should be judged by ear depending on the design of the piano and the condition of the strings, the only way to tune the same sized fifth accurately is to hear both piano notes and copy what the piano tuner did.
Unless playing the F is just a tradition and the reason has been forgotten over the centuries, I would speculate that the F is simply to make the piano easier to hear.
The tone of a string instrument has more high-pitched harmonics than a piano, and the piano sound starts to die away as soon as the notes are played, but the string tone does not.
FWIW tuning a string instrument to a pipe organ, where the pipes that have the most stable pitches have very few harmonics, is surprisingly difficult - hence the speculation about "hearing the piano" in the previous paragraph.
Equal tempered major and minor thirds are both a long way "out of tune" compared with "pure" major and minor thirds.
A major third is sharp by about 1/7 of a semitone, and a minor third is flat by the approximately same amount.
That doesn't suggest any good reason for playing a minor chord compared with a major one.
Maybe string players just find it easier to play three white keys!
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Pure speculation here, but these are the reasons that stick out for me: When using real instruments, a chord is often easier to tune to than a unison tone
, I guess the reason is a combination of us being used to playing in tune with chordal accompaniment, and that it compensates a bit for any irregularities in the harmonic spectrum of real world resonators, and gives our ear more information to pick out an exact "centre" from.
That, and it's louder of course.
Why a D chord?
It gives a strong tonic centre to pick out a d and an a from, for the fiddles that leaves the G and the E to tune in fifths from the a and d.
For the bass, the same but in fourths.
(incidentally, for a guitar too with a B in fifths from your E).
For the violas and cellos, a bit more of a stretch, the G from the D and the C from that G, but still nice and easy.
Why a D minor?
Better than an open fifth for the reasons a fifth is better than unison.
And why is D minor better than D major?
No semitone clashes or tritones.
The aforementioned G, E, C and Bs all happily mind their own business in major seconds and perfect fourths from a D minor's chord tones, (or fifths and major sixths) making it easy those notes to to your own strings using 5ths/4ths while a D minor is still being played.
An F# would be a bit of a cat among the pigeons.
Why not G major?
Convention
I guess.
Would be worse for guitarists, but i strongly doubt that's a consideration.
5 string basses too but likewise.
Why not A minor?
Tuning to the third of a chord is never good, you're always tempted to go towards a more just interval.
Playing an A minor might mess up the Cs of the violas.
And you know what they say about viola players...
They're valuable additions to orchestras and you wouldn't want the C string out of tune.
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Cellos, Violas, and some Basses have C-strings.
The F in the D-minor chord gives them a fifth (or an octave and a fifth) to tune to.
An F# would be useless basically to every player in the orchestra.
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I've heard string players tune from a piano by playing A4, followed by D4 and F4 together. They said it makes it easier to tune. I've never gotten a good answer why this is. Is it just a strange tradition, or does something about a D minor chord on the piano make it easier to tune the violin family?
When tuning an orchestra, why do some people play A followed by D and F together?
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<tuning><orchestral-strings>
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music.stackexchange.com/users/34108/John Platter
S0: Cellos, Violas, and some Basses have C-strings. The F in the D-minor chord gives them a fifth (or an octave and a fifth) to tune to. An F# would be useless basically to every player in the orchestra. Cellos, Violas, and some Basses have C-strings. 
Cellos, Violas, and some Basses have C-strings. The F in the D-minor chord gives them a fifth to tune. An F# is useless 