Tertian arithmetic

The idea

This has been stewing around in my head for a little while now, and I think there’s something significant in it, but I have not yet found the pow! way to explain it. So, please forgive me if this discussion still seems a bit formative–it is.

The topic is looking at extended complex chords (ninths, elevenths, thirteenths) as polychords. The intent here is to come up with a deliberate device to understand these complex chords better, by viewing them as the products of multiple simpler chords (triads, sevenths) that one may already have in the hands. What I like is that there is a very simple logic at work that makes this daunting subject rather easy to grasp–at least in a practical way.

And that is my goal. Many of us find ourselves in a group context in which we are one voice among several, and so this concept of reducing a complex chord into two simpler chords, each to be played by a separate person, has a great deal of appeal. And for the improvising musician, the ability to add harmonic complexity spontaneously (without having to know where, say, a G13 or an Am11 is) is a real bonus.

You’ve probably heard the phrase “chords are built in thirds”, and the term “tertian chords”. It turns out that there’s a lot more to this than you might think. Most of our chords are indeed built from stacked thirds, but the stacking can be applied further to whole chords a diatonic third apart. Somewhere in there is a sort of “tertian arithmetic” that can be described, that just unlocks the whole mess like a master key. It is the description of this arithmetic that I continue to struggle with; I can see it by permutation, but the teacher in me wants to be able to distill out the best way to ‘splain it. (I’ll get there.)

The basics

Okay, so think in the box of a seven-note scale and its diatonic, tertian chords. I’ll stick with the Western major scale for simplicity–the example will be good old C major.

Our givens:

Diatonic triads are built by stacking two thirds on top of one another. The sequence and nature of the component thirds determines the type of triad: a diminished triad is two minor thirds (BDF), a major triad is a minor third on top of a major third (CEG), a minor triad is a major third on top of a minor third (DFA), and an augmented triad is two major thirds (unavailable in the Western major scale, this would be CEG#). In C major, then, the diatonic triads are C major (CEG), D minor (DFA), E minor (EGB), F major (FAC), G major (GBD), A minor (ACE), B diminished (BDF).

Diatonic sevenths take these triads and add another diatonic third onto each one. The combinations become much more complex, and C major only yields four of the possible variants. The diatonic seventh chords for C major, then, are Cmaj7 (CEGB), Dm7 (DFAC), Em7 (EGBD), Fmaj7 (FACE), G7 (GBDF), Am7 (ACEG), and Bm7b5 (BDFA), also known as the “half-diminished seventh”.

Ninths add another diatonic third onto the seventh, making them a stacking of four thirds. Elevenths add a fifth third, and thirteenths are six thirds stacked together. (At this point, you are playing every note in the scale simultaneously, so there is no such thing as a fifteenth or beyond.) The possible permutations quickly get ridiculous here, and so does the harmonic density. After a few too many of these chords, even the most diehard jazz lover can appreciate the clarity and resolution of a triad or power chord.

Great, so where is all this going?

Consider C major: C-E-G. Now, consider E minor: E-G-B. What happens if you get your buddy to play a C major at the same time you play an E minor? Together, you’re playing Cmaj7, C-E-G-B, and are overlapping on the E and the G. Huh.

You may also have heard that Em is a good “substitute chord” for C major. The logic goes that the bass player is probably covering the root note, so if you play an Em somewhere in the upper register, it should sound “right”, and a little fuller. What you’re doing is extending the basic chord by another diatonic third, making a more complex chord in the process – Cmaj7.

Now, what makes Em a suitable substitute chord for C? Here is the simple, master-key answer: it’s a diatonic third up from C.

From here, everything starts to fall into place. If you understand diatonic triads in your scale, you can easily create sevenths by playing two triads a third apart. If you understand diatonic sevenths in your scale, you can easily create diatonic ninths by playing a “root” seventh chord and the diatonic seventh one-third above it. (Three tones will overlap.) That is, you can easily create Cmaj9 by playing Cmaj7 and having your buddy play Em7 over it. C-E-G-B plus E-G-B-D equals C-E-G-B-D, or Cmaj9. Work up the scale: Dm7 + Fmaj7 = Dm9, Em7 + G7 = Em7b9, etc.

Now, here’s the real kicker: consider those overlapping tones. In the case of Cmaj9, the notes are C-E-G-B-D. If you start to break apart the ninth into constituent chords, do you notice that you could also play the G triad over the C triad, and come up with Cmaj9? C-E-G plus G-B-D equals C-E-G-B-D. The G triad is two thirds up from C, and now you’re playing a ninth from two triads with one note still overlapping. Why not go a step further? Three thirds up from C is the B diminished triad, B-D-F. C major (C-E-G) plus B diminished (B-D-F) yields C-E-G-B-D-F, or Cmaj11. Imagine playing eleventh chords simply from knowing your triads, and knowing how to go up three diatonic thirds up from a root chord. Not bad!

Now, combine a seventh with a triad. Hold down your Cmaj7 and have your buddy play a D minor (D-F-A). Poof! Cmaj13 (C-E-G-B-D-F-A), no overlapping notes. You could do the same playing a Bm7b5 over a C major triad.

There will be practical limits to this, of course, but this is a powerful concept, and once you have mastered triads and are comfortable with sevenths, these extended chords are almost instantly available to you. Commit these ideas to immediate recall:

Look at complex chords as a series of thirds stacked together in the scale. That is, for Cmaj13, look at the chord as C-E-G-B-D-F-A. Do you suddenly see all the triads in there? Just break them up, assign them and play them!

To extend a chord by one “factor” (triad to seventh, seventh to ninth, ninth to eleventh, eleventh to thirteenth), you need to add a third. You can do that by creating a polychord from a root chord of appropriate complexity and a chord a third above of appropriate complexity. (Here, we start to get limited by terminology. I’m working on this!)

You can extend a chord by more than one “factor” at a time (thereby keeping constituent chords simpler and fewer notes overlapping) by going up two or three thirds for your “upper” component of the polychord, and/or by extending the upper chord by one or more “factors”.

In general, you can cover any diatonic ninth, eleventh or thirteenth chord with one seventh and one triad.

Hopefully I will get clearer with this each time I explain it. For now, here are some practical examples that illustrate the explanation. Try these things!

Cmaj9 (CEGBD) = C (CEG) + Em7 (EGBD) (also w/Cmaj7)
Cmaj9 (CEGBD) = C (CEG) + G (GBD) (also w/Cmaj7)
Cmaj11 (CEGBDF) = C (CEG) + G7 (GBDF) (also w/Cmaj7)
Cmaj11 (CEGBDF) = C (CEG) + Bdim (BDF) (also w/Cmaj7)
Cmaj13 (CEGBDFA) = Cmaj7 (CEGB) + Bm7b5 (BDFA)
Cmaj13 (CEGBDFA) = Cmaj7 (CEGB) + Dm (DFA)

Note that in each of those pairs of examples, the first suggestion goes up the minimum number of thirds and uses a seventh form; the second suggestion goes up an additional third from the root, but simplifies the upper chord form.

There is a little arithmetic involved here, sure, but it is very logical, and I imagine that mastering it would make one a very powerful improviser. I can envision a situation in which I’m playing mandolin and reading a chart which vamps on a dominant chord for a few bars. The band has the G7 covered, and I want to add tension on top of the mix. For the first bar I play G7, then I play Dm (making a G9), then F (making a G11), then cap it off with Am (G13)–just by knowing to go up in thirds.

I know I’m a geek, but that’s cool.

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