I am learning to play songs by ear. Based on the circle of fifths, one may harmonize C major with the chords C, F, G and Am. Similarly, will it work to harmonize A minor with the chords Am, Dm, Em, and C?
III chords can be used to harmonize most melodies in major or minor. The one modification in minor is that the
V chord will typically include a sharped scale degree 7: so, in the A minor case,
E G# B rather than
E G B, though the latter does show up sometimes. It's also not unusual to see the
IV as a major chord even when the overall key of the melody in minor: so, again in the A minor case,
D F# A might show up.
These four chords will get one through most simple melodies, even if the "correct" chord happens to be, say, the
ii chord. For the purposes of learning to play by ear, they are a great place to start, with other chords coming into play once these four are comfortable to work with.
Yes, that works. And funnily enough, all the chords mentioned are from the same key - same 'family'. I, ii, iii, IV, V, vi. in key C, for example, I, IV, V, vi are C, F, G, Am, whereas in key Am (the relative key) vi, ii, III, v are the corresponding chords. In the minor key itself, RN would be i, iv, v (V) and III. Be aware that in minor keys, v is more often V - major rather than minor. It just sounds so much more convincing in a perfect cadence.