Common-practice functional harmony is predicated on the progression from predominant to dominant to tonic. In the below example of Mozart's Piano Sonata No. 13, those functions are abbreviated by their initial.
Like classical harmony, pop harmony is also functional in the sense that certain chords lead to other chords, usually culminating in a cadence. However, unlike classical music, pop chords are subject to far less strict rules of progression. In other words, the progressions found in pop music are more flexible than those found in classical, with the same chords (or at least the same Roman Numerals) functioning in multiple ways. And that means any theory of popular music harmony must account for this multifunctional flexibility.
Enter Christopher Doll. In his book Hearing Harmony (University of Michigan Press, 2017), Doll implements Greek letters to indicate distance from tonic:
Doll goes into some depth in an appendix, specifying all possible chords in each of these functions and providing names like "hypo pre-subdominant (mediant of a subdominant)" and "medial pre-dominant (mediant of a dominant)". While I appreciate the thoroughness of his theorizing, I find the exhaustive details cumbersome to the point of being unusable in practice. But I find his notion of Greek letter functions quite compelling and practical.
Applying Doll's Greek letters to the same Mozart example shown above yields the following:
In this case, ii (c) functions as the predominant (γ), V7 (F7) as dominant (β), and I (Bb) as tonic (α). It's a straight-forward, textbook example. But what about a pop song, such as Alanis Morissette's 'You Oughta Know'? Here's the chorus from that song, which clearly employs functional chord progressions but in a rather different way from Mozart:
In this case, IV (B) is the pretonic chord (β). It resolves to I (α) on the subsequent downbeat. bIII (A), then, is the pre-pretonic (γ); and bVII (E) the pre-prepretonic (δ).
But that is just one example. What makes these Greek letter functions so compelling and useful is how flexible they are. The same chords can function in different ways depending on their syntactical order. Here is the chorus from Metallica's 'The Unforgiven', which incorporates similar chords but with different functions:
This time i (a) is α, V (E) is β, bVII (G) is γ, and bIII (C) is δ.
To avoid constantly scrolling up and down to compare the two songs, here are the Greek letter functions of both 'You Oughta Know' and 'The Unforgiven' side-by-side:
Tonic, whether major (I) or minor (i), will always be α - that will not change from one progression to another. But all other Greek functions are liable to change. Notice how β is IV in one song but V in the other. And how both bIII and bVII are γ in one but δ in the other. This is what I mean when I say pop harmony is multifunctional - the same chords can function in different ways. This is contrast to classical harmony, where chords typically function one way, and only one way. The value of Christopher Doll's Greek letter functional nomenclature, then, is that it addresses pop music's more flexible rules of progression - it illustrates the linear harmonic motion without the restrictive and loaded terminology of classical analysis.
Aaron Krerowicz, pop music scholar
An informal but highly analytic study of popular music.