Music theory question about finding keys

Question

If I have chords Dm, F Maj, Am and no other melodic context is provided, I just have these three chords as an information, how do I know what key is it?

From my understanding , it can be either Am or Dm (or its relative F Maj) , because the two keys/scales are kind of similar (Am scale has an B but doesn't have A#, Dm scale has A# but doesn't have B), and I have no sharps or flats, or any other melodic context to distinguish between the scales, so what is the process behind figuring this out since it's so ambiguous, at least for my level of knowledge?

The only way I could try is to translate the progression to each of the possible keys/scales and looking at the chord functions for each of them? idk

In the key of Dm: i - III - v

In the key of F Maj: vi - I - ii

In the key of Am: iv - VI - I

Answer 1

You actually cannot know what key it is with just those 3 chords as information. You CAN figure out which keys it can possibly be and which keys it cannot be. The key is determined by the context in which those chords are used. Of course we are speaking strictly diatonically in this scenario.

You’ve done a good job at figuring out what keys it could be, F major or its relative minor, Dm or Am. You can probably eliminate the relative major of A minor, C major because there is no C chord.

One correction, to your statement: “Dm scale has A# but doesn't have B”; Dm has a single flat, Bb, not A#, Dm has an A in the scale and you cannot have two of the same letter in a single key.

You can also look at the chord tones of the 3 chords: D,F,A,C,E. This eliminates all sharp keys (all have F#) and all flat keys starting with Bb, 2 flats or more (all have Eb). That leaves C/Am (no sharps or flats) and F/Dm (one flat), the same result as with your chord function analysis.

Since you’re playing detective, upon further inspection it is more likely to be F or Am over Dm because V or V7 is more common than v in minor keys (A or A7 instead of Am in Dm).

Answer 2

A list of backing chords can give you probable candidates, but not the key. I made two example tunes with those backing chords.

Example 1:

Example 2:

In my opinion, in the first tune, Dm is a good ending chord and Am in the second one, but you can try it out yourself.

When you see chord progressions, they're just backing chords, instructions for accompaniment. They're not results of some kind of statistical analysis of all frequencies on a recording, and how people on average perceive the whole.