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An example of this is the song Nobody Home by Pink Floyd. The last line of the intro is, in a C major key:
F Fm C %
I think this sounds great, and in fact I often use such passages when improvising on the piano myself, but I still don't know how exactly this can/should be used, and what it means.
Does anyone have any insights on this?
Tension and release. The point of a leading note is often to give a feeling that it needs to move, as little as possible, to the tonic. As in G chord, dominant of C, contains B, which needs to move to the tonic C. A semitone move is used in this example. The F minor chord has F, Ab and C. To move as little as possible to the tonic involves the Ab going to G, the F going to E and the C remaining static.So the A-Ab-G move (F, Fm,C) is all going in the same, inevitable direction.
The Ab works somewhat like the b9 in G7b9, as it pulls again in the same way .Another example is the augmented chord. G+ has G, B and D#. To move to tonic, the B goes a semitone to C, the D# a semitone to E, and the G remains static. It's all to do with getting as close as possible to the target. A sort of magnetic attraction, where a tone away is too far to get a pull.
The chord progression for the introduction is: Am - C+ - C - D7 - F - Fm - C
However, you can see it as Am - C - F - C (or vi - I - IV - I), and the chords in the middle are only chromatic passing notes, i.e.
• In the sequence Am - C you have (A,C,E) that goes to (C,E,G). E and C are common to both chords and A and G are one tone apart. Add a chromatic ornament in between A and G to obtain the note sequence A - Ab - G, and it may look like you passed through C+, but it is in reality just a passing note.
• Between C (C,E,G) and F(F,A,C), E goes to F through F#, and G goes to A through Ab, looking like you passed through D7.
• Finally to go form F(F,A,C) to C(C,E,G), A goes to G through Ab and F remains to go directly to E, looking like you passed through Fm.
This way you can see that you are in the key of C all the time, and all strange notes are in fact chromatic passing notes. Passing notes always create tension since they are not part of the chord and therefore create a dissonance (and chromatic passing notes create even more tension since they are not part of the key), which is released once they resolve to a note that is part of the chord, so concerning the specific part you ask about (IV-iv-I) you have a simple plagal cadence with a chromatic passing note.
Of course this is just an interpretation, but at least the way I learned analysis, the goal is to explain the harmony with as little chords and modulations as possible, as long as the remaining notes fall into the category of an embellishment note.
DP
I agree with the other answers that iv contains a chromatic passing tone that enhances the horizontal (melodic) structure.
For a vertical (harmonic) perspective, iv is closely related to the neapolitan sixth chord (in this context, Db major in first inversion), a standard 19th-century substitution for IV or ii.
Thinking this way, the substitution of interest is not Ab for A, but C for Db, which I think is easy to understand as a suspension or pedal tone carried over from the previous IV chord (and into the following I).
Zooming out, IV - I or ii - I is the plagal cadence, as @Dissidentpenguin notes.
Alternatively, the neapolitan chord is also a tritone substitution for V, so in this pop-music setting it might be appropriate to think of it as IV - V - I, using a tritone substitution for V with a suspension to boot.
You've already received several good answers, but I would like to add one thing that I consider important. The progression in your question is a well-known example of a more general concept know as modal interchange. Modal interchange is the borrowing of chords from parallel tonalities. The most common form of it, which can be found in thousands of pop songs, is borrowing natural minor chords for use in the parallel major key. In your example this means that you can use chords from C natural minor (such as Fm) in the key of C major. Some other chords commonly found in songs in C major based on this idea are G minor, Ab major, and Bb major. If you look for it, you'll start hearing the result of this concept almost everywhere. It is also a great concept for melodic improvisation.
I think what's happening is ..
So your ears pick up the sequence A Ab G as it resolves to C.
Our ears "want to" hear sequences of notes and the initial flattening fo the A implies something else is going to happen along those lines, with another flattening of the note.
It's a similar thing to playing C, C7 then F.
Your ears pick up the sequence C Bb A, which against a backdrop of chords that sound nice together anyway, seems to fit nicely.
I would say that this common progression follows one of the 'rules' (descriptions of common practice) of harmony and modulation. That is modulation to the parallel minor. I would say that the iv chord is 'borrowed' from the parallel minor key. Then, the song modulates immediately back to the original major key.
