EGs: I I I6 V V V V7 I vs. I I I Imaj7 IV IV IV IV6 I
Sixth chords move up a fifth, seventh chords move down a fifth. (the seventh of a chord falls, the sixth rises - one can also view the IV7 chord in blues music that moves to the I to be a IV+6)
The fundamental tetrad concept introduces the idea of "homophonic chords" - for example, a major sixth chord and a minor seventh chord have the exact same pattern, and the designation depends on context. Often, two designations exist in a superposition until it is settled later on (or never).
Also interesting is chordal metamorphosis, where for example a subtle shift in notes can modulate you into a new mode -for example, V morphing into III7 by chromatically altering a note, taking you from major mode to relative minor. (and this particular V III7 change is a nice way to make a strong chord change when stuck on an already strong chord) EG: Hallelujah - |I |IV V |vi |IV |V |III7 |vi |
Music theory is a very interesting topic, and I've arrived at models that do a pretty good job of explaining the underworkings, allowing me to appreciate the beauty of songs analytically as well as emotionally.