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.
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)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.
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)
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.