In A minor, I play a chord with the notes E, A, F - with E being in the bass. What would you call this chord?
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Well, I hate to be vague, but it really depends on the context of the chord-it's function in what's going on. Since the "E" is on the bottom, you don't have any really obvious thirds. It seems unlikely that it's an Fmaj7/E, although it could be. It could be like an Esus(add9) too.
In A minor, I play a chord with the notes E, A, F - with E being in the bass. What would you call this chord?
Dissonant classical music, I guess. I have a tune that goes
Amin-Caug-Amin-Fdim-G#dim
Amin-(this chord we are talking about)-G#min-(B,E,C)
So to me the minor ninth sounds more like a minor ninth than an inverted major seventh, if that makes sense.
All of the other chords in his progression are from tertian harmony, functional or not.
Suspended chords are quartal and if you're finding a suspension in tertian harmony it is a nonharmonic tone called, of course, a suspension.
Fmaj7, 3rd inversion.
All of the other chords in his progression are from tertian harmony, functional or not.

It seems like all my chords here do relate to the tonal center, though, no? G# minor contains the D#, which is not a part of the key, but what is not "functional" here?
In A minor, I play a chord with the notes E, A, F - with E being in the bass. What would you call this chord?