# Functional analysis, huh

I was reading this and it seemed to make way too much sense—I feel like it describes the framework for what I do when I improvise chords on the piano. Here’s some stuff that I thought about. (I apologize in advance for mixing roman numeral notation with the functional analysis notation.)

The idea is that having just I, IV, and V is boring, so you can add variation (and do other stuff) by using “secondary” chords. You do this by taking whatever chord you have—tonic (T), predominant (P), or dominant (D), in this notation—and substituting another chord that has two notes in common with the original chord. For example, the C major chord has C, E, and G. So if you want to keep C and E, then you play A minor, the relative minor. If you want to keep E and G, then you play E minor. We’re calling these two chords “tonic relative” (Tr) and “tonic variant” (Tv) respectively. (Lowercase and uppercase show whether the chord is major or minor.)

Here’s the secondary chords (r, relative, and v, variant) written out for I (T, tonic), IV (P, predominant), and V (D, dominant), given that we’re in C:

Tr = A minor; Tv = E minor.
Pr = D minor; Pv = A minor.
Dr = E minor; Dv = B diminished.

Notice that Tr is the same chord as Pv, and Dr is the same chord as Tv. I did a little experimenting, and it seems that the variant function tends to trump the relative function, if just slightly. (This also may just be the result of the voicings that I chose when testing it.) So if I’m in C major and I play an A minor, it’s more of a Pv than a Tr—so it acts as a substitute for the IV better than for the I. From this I conclude tentatively that relative chords used as secondary chords are weaker substitutes than variant chords.

Then I did an experiment.

I took the idea of “substitution chords that work because they share multiple notes with the main chord” and applied it to seventh chords.

First, the seventh chords sharing three notes with IM7. Since variant trumps relative (or at least I’m assuming this), I’m not going to worry about the relative substitutions. I’ll pay attention only to variants. And since iiim7 is the variant, I can conclude it is a reasonable substitution chord for IM7; a similar conclusion can be made for vim7 substituted for IVM7. This is the equivalent conclusion as the one that was reached earlier when not worrying about seventh chords.
1, 3, 5: vim7 ([I relative], IV variant)
1, 3, 7: none
1, 5, 7: none
3, 5, 7: iiim7 (I variant, [V relative])

So, no surprises there.

If I take this a step further and look for seventh chords that share only two notes with IM7, then I get a host of secondary seventh chords that can all act as the variant of I, and a bunch of secondary seventh chords that can act as the variant of IV (or the relative of I—same thing).

Thus I can split this into two groups: predominant variant seventh chords (Pv7: chords stemming from the variant of IV) and tonic variant seventh chords (Tv7: chords stemming from the variant of I).

Now, the seventh chords that share two notes with IM7
Pv7 Chords: (similar to vi chord, which is the main P variant)
1, 3: IVM7, #IVhalfdim7
1, 5: Im7, ♭viM7, vihalfdim7
Tv7 Chords: (similar to iii chord, which is the main T variant)
3, 5: VIMm7, iiidim7
3, 7: IIIM7, IIIMm7, ♭iim7
5, 7: VM7, VMm7

If my theory is correct, then I should be able to take any chord sequence in a major key and substitute a tonic with one of the Tv7 chords and substitute a predominant with any of the Pv7 chords and it would still sound decent. So I tested that, and it worked!

As a little test, I’m going to take the waltz I wrote last week and write the chords in this T, P, D kind of notation. The notation is explained in more detail over here.

The Last Waltz

Normal chord notation:
| A Bm | C#m D | A F#m | E E |
| A Bm | C#m D | A E | A A |
| D Dm | A F#m | D E | A F#m |
| D Dm | A F#m | D E | A A |

Letter notation:
A: | T Pr | Tv P | T Pv | D D |
| T Pr | Tv P | T D | T T |
| P p | T Tr | P D | T Tr |
| P p | T Tr | P D | T T |