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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?

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    Should they? Says who? This will depend on the style. If it's a pop song you can do what you like. If it's a chorale in the style of J. S. Bach then voice leading might pose problems. Please edit your question with the context. – user48353 Apr 25 '18 at 6:55
  • Well, E7 should really point you to an A major chord, changing key pretty severely, but we're still drawn to the A minor, just less strongly than to that key change. Now, that A minor chord contains the notes A, C, E... The F major contains F, A, C... Two notes overlapping. As for the probable reason this is used: it sounds nice. – AJFaraday Apr 25 '18 at 11:00
  • @AJFaraday - I don't even think it's a key change - it's still in C, just moved to IV ( F) for a bar or two. – Tim Apr 25 '18 at 11:10
  • @Tim That's correct, this sequence does not change key into A major. – AJFaraday Apr 25 '18 at 11:11
  • @AJFaraday - or even A minor, as hinted from the E7. – Tim Apr 25 '18 at 11:13
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'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...

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    Adding to this, the "theory" could still be correct, and the entire earlier passage could be in A minor. The next passage could be in C major though. There's no reason that you can't modulate from A minor to C major and still finish your A minor section on an E chord - indeed this is an entirely standard thing to do. For an example of this, listen to the verse of Sultans of Swing. The first two lines follow a descending Dm - C - Bb - A progression, but the third line goes to F (the relative major) before going back to Dm at the end of the fourth line. All very normal, and not jarring. – Graham Apr 25 '18 at 12:25
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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:

 V   VI
|------|
   vi

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).

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    But what's the point of analysing it as being in A minor if it's demonstrably in C major? That's theology, not theory! Tweaking the evidence so that it doesn't challenge a preconception. – Laurence Payne Apr 25 '18 at 18:55
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    Respectfully, if either of us is guilty of "tweaking the evidence" it's you claiming I tweaked any evidence. The OP asked how E7 can move to F, and I showed a harmonic environment in which that happens very often. If you think that's tweaking evidence, then that's on you. – Richard Apr 25 '18 at 19:03
  • @LaurencePayne I don't know about "tweaking the evidence", but the secondary functional analysis can be argued against sometimes. – user45266 Mar 14 at 3:52
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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.

  • Temporary -1 from me, because as it stands this answer may mislead beginners into thinking that E7 followed by A or Am is a perfect cadence, categorically. It is only a perfect cadence if it is a cadence at all. It should not be called this if it is only part of a continuing progression. Please let me know in a comment if you revise the answer to clarify, and I'll gladly remove the downvote. – user48353 Apr 25 '18 at 9:03
  • @replete - not even sure that it would be a perfect cadence in this case, in key C. However, it could very well be an interrupted cadence in key C. – Tim Apr 25 '18 at 9:18
  • @Tim, if it is a cadence then there may have been a modulation to A. That is not the point, the point is that the question doesn't say that the progression is at a cadence, and the OP might just be picking out a couple of chords from a longer sequence. The intent is just to make sure people don't start calling every "this chord followed by that" a cadence. – user48353 Apr 25 '18 at 9:21
  • @replete - point taken about any old E7>A/Am. It's only a cadence at a cadence point. That's the point of cadences! As such, this answer is misleading. However, in key C, there's no particular need for a modulation to have happened at a cadence point, where E7>Am would be called interrupted, as hinted at, in the answer – Tim Apr 25 '18 at 9:29
  • @Tim, you said "there's no particular need for a modulation to have happened", when I had said "there may have been a modulation" (emphasis). We are therefore not in disagreement and this irrelevant tangent need not be pursued. – user48353 Apr 25 '18 at 9:34
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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!)

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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 )
       IV
0

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.

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    What a COMPLICATED justification for a simple, commonplace chord sequence! – Laurence Payne Apr 25 '18 at 9:40
  • I do not agree. IF the progression is F E7 F then I would say it feels the same as F Fdim F. Theres nothing complicated about diminished chords, they are both lovely and interesting. – hlynbech Apr 26 '18 at 10:50
  • I disagree with your disagreement! There are many ways of getting from C to F. C, C/E, F. C, C7/E, F. C, C+, F. C, E, F. C, E7, F. They all feel different. That's why we choose one instead of another. There's function, and there's feel. – Laurence Payne Apr 26 '18 at 12:57
  • Yes but you are talking about progressions that starts on C and ends on F, which is not included in the original question. I will update my answer to illustrate that it depends on where you are coming from, regardless if you talk feel or function. – hlynbech Apr 27 '18 at 9:19
  • For me the trouble with this is that E7 belongs to A minor, and nearly belongs to C major. But doesn't belong to either. Here it would make more sense to me to view as a substitution for E7 than the other way around. That is the sense in which this explanation seems too complicated. – David Bowling Apr 27 '18 at 9:39

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