What's the theory behind and E7-F chord progression in the key of C? Shouldn't E major and E Dominant 7th chords be followed by an A minor chord?
'Should' isn't really the appropriate word. I guess it's sprung from some theory.
True, many many times the chord after E7 will be Am (or A major), so theory has jumped on this fact and given (some of) us the impression that that's what has to be.
The 'theory' behind E7>? is that music sometimes wants to be unpredictable, or have tensions introduced which can then be released. Also, notes/chords like to resolve by as little movement as possible. Think leading note to tonic; sub-dominant to median, both from V>I.
Here, the expectation is that following E7 will be A/Am. Another diatonic chord appears instead - F. It's nearly right! At least two of its notes are the same - A and C! It also fulfils the 'let's move as little as possible 'theory'. And it's unexpected. Ticks a few boxes, that one.
As stated many times - it's theory, not law. What sounds good usually is. That's the real bottom line. Just like this...
Rosie's answer is very good, but I wanted to flesh it out a bit and add one thing:
Analysts often restrict applied (or "secondary") chords to just one chord; in the key of C, we say that E(7) to A is "V/vi" moving to vi. But we can use that notation for multiple chords in a row!
Imagine you're in the key of A minor; E(7) to F is just V(7) moving to VI. In the US we call this a "deceptive cadence," or "deceptive motion" if it's not at a cadential point. We call it "deceptive" because the urge is so strong for V to move to I, and our ears are deceived when this resolution is taken from us. In other countries they call this an "interrupted cadence."
So when we explain these chords, we can explain both of them as they are used in the key of A minor. Some would say that this is V/vi (E) moving to VI/vi (F). I would personally recommend the notation of an "extended tonicization," where we bracket multiple chords, provide a temporary tonality, and give Roman numerals in that key:
In the above example, we say that the E and F chords are V and VI, respectively, in the key of vi (which is A in C major).
It isn't clear to me from the OP whether the chord of F is at the end of a phrase (which would indicate that it's at a cadence point) or the phrase continues on after it. The rest of this answer supposes that it is at the end of a phrase.
As Tim says, after E7, you expect A or Am (that would be a perfect cadence), but you get some unexpected chord instead. When that happens, it's called an interrupted cadence.
With apologies for repeating some points from Tim's excellent answer, there are a number of reasons why you might consider F a good choice to follow E7 in a C major context:
- If the listener's expectation is that it will move to Am, F has two tones in common with Am (A and C), so it's a good substitution for Am.
- The smoothness of the bassline is maintained by moving one semitone from E to F
- the whole chord motion from E to F (ignoring the 7th tone), is a semitone side-step into the F chord
- F is solidly in C major
As others have said here, music theory has no over-arching rules you have to follow; however, certain styles of music may imply such rules.
(I would also add to the above list: If you consider chords from the point of view of their harmonic 'quality', rather than their diatonic function, the major quality of F is consistent with E major (and the C major chord that I presume would also be present somewhere in the progression), maintaining consistency in the progression. However, I don't know if there's a proper 'theory' word for this, question here!)
In the key of C, an E7 is V of VI. The motion from E7 to F in this case would be called "deceptive motion" as in deceptive cadence.
One way to notate it in roman numeral form would be
V of ( VI )
In my opinion, it depends on the chord that precedes the E7. The case where the progression is something like C-E7-F is covered in other answers.
Another case is if the progression is like F E7 F, which is also common. I know that in the progression fx Fdim to F, the Fdim is called an auxiliary diminished chord. If I remember correctly, the E7-F can then be regarded as a small modification of the Fdim-F progression, only the bass note note is changed. Unfortunately I can't find a reference to support this.
Maybe try substituting your E7 with Fdim in the progression you are analysing and see if the function stays roughly the same.