What is the clearest way to signify the harmonic function of a polychord?

Question

Chords can be analyzed with roman numerals to indicate function:

C, Am, Dm, G

I, vi, ii, V (or I, VI-, II-, V in jazz theory)

And inversions may be analyzed by adding an arabic numeral indicating the note which takes the lowest voice:

C/G, Am, Dm, G/B

I/5, vi, ii, V/3

But what about polychords, and their ambiguous cousin, slash chords?

C, Am, Dm, F/G

I, vi, ii, IV/2

The "/2" seems a bit unclear. You could write "IV/V", but that's also a secondary dominant.

Answer 1

Usually, polychords are written like this: (with a fraction).

So, let's say you have Am {fraction} G7 and we are in the C major scale. You could symbolize that as VIm {fraction} V.

Notice that for the polychords, there is no slash, but a fraction. Slashes are for the slash chords or hybrid chords (inversions).

For the slash chords inversions, if the score you are written isn't really formal, you could write it as Cmin (1st inv) or II(3rd inv).

If the score is formal, the first inversion is symbolized with a 6 (actually ^6 -- power of 6) so, you could symbolize Cmin/Eb as I^6, and the second inversion with a 6 4 and the 3rd inversion with a 2.

If I'm not mistaken, a F/G slash chord, is written as F^9, or IV^9

Answer 2

Polychords, Roman numerals, and Functional Analysis

Roman numerals, in the context of functional analysis, are an analytical language, not a chord-naming symbol. As such, they don't follow the conventions of chord-naming systems. C/F is meaningful as a chord name (whether slash chord or polychord), but its functional meaning is undefined. I/4 and I/IV are not part of the language of functional analysis (except I/IV as a secondary-chord indication). In the context of functional harmony, polychords, specifically, have no meaning as such, rather they would be labeled as a single chord according to their overall harmonic function, and that would depend on context.

Using the F/G example:

F/G Polychord

Example 1

Consider a piece of music in C major. We encounter an F major chord, which is tied with a G major chord being added underneath the tied portion, followed by a C major chord.

Analytically, the chords would be labeled IV - V742 (or V974) - I. This notation is understood to mean that the seeming polychord is "just" a V7 chord with suspensions from the previous IV chord.

Example 2

But now consider a piece in which there is a sustained G chord with a variety of chord changes above it, including an F major chord. This is as close as functional harmony would come to an explicit polychord. The Roman numerals would be given according to the upper-voice chord progression with another notation indicating a G major pedal chord. (Or course the notations might change as in Example 1 should notes from the G major chord actually be functional parts of those upper chords.)

F/G Slash Chord

In this case, it is likely the Roman numeral will correspond to the F chord, with the G being interpreted either as a pedal tone or a suspension, depending on context. It also could be interpreted as an incomplete G7 chord with suspensions (as in Example 1, above). Less likely it would be considered an F chord with a non-harmonic G. For instance, consider the progression F - F/G - F. In this case, the G would probably just be seen as an ornamental upper neighbor and not as part of the harmony.

Polychords in Polytonality

One could imagine a polytonal piece in which, say, one set of voices was in F major while another set was, simultaneously, in G major. Polytonal music analysis may borrow ideas and notations from functional analysis, but the stretching of the language's meaning would be clear in that context.

Two analytical options for polytonality — provided each tonal part was well-enough behaved in functional terms:

1. Use two separate analysis, one for each part.
2. In the specific case where two simultaneous chords play the same role at the same time within their respective keys, one could perhaps "invent" a notation where a single, "big" Roman numeral is used. Something along the lines illustrated below.

In this setting, a polychord-type notation would have meaning, but at present there is no such established notation.