Chords and Their Giant Extended Families

Most of us who have done musical training will have heard the term relative used to describe the relationship between two keys - a major and its relative minor. Two relatives is a pretty small family, so lets invite everyone to the family reunion!
Rather than looking at the relationships between keys, we're going to look at the way we can build up a collection of chords that belong to the same family as one original chord by using extensions.

Let's start with a simple chord, in the key of C, the first chord we ever learned, C major (C E G)
I'm going to let every chord that shares at least two of those three notes in C major join our family. I'll be underlining the notes in each chord that relate back to our original C major chord.

In three note chords (major/minor) we have:
A minor (A C E); and
E minor (E G B).

In four note chords (7ths):
Cmaj7 (C E G B)
Am7 (A C E G); 
Em7 (E G B D); and
Fmaj7 (F A C E);

In five note chords (9ths):
Cmaj9 (C E G B D)
Am9 (A C E G B);
Fmaj9 (F A C E G); and
​Dm9 (D F A C E)

All of the above chords stay strictly within the key of C major (none of the chords use any sharps or flats whatsoever). As an exercise, try playing the chord sequence Fmaj7, Dm9, C, Am7. Notice how adding in our extensions, our common notes from C major, bring the chords closer together. Now try it without the extensions: F, Dm, C, Am - does it still sound like they're as closely related?

(For those music theory buffs - notice that we haven't gone to chord V in our key - G major? Using the above chords and avoiding chord V is a great way to build a narrative or a simple melodic idea without the jarring tension that chord V introduces.)

See if you can work out the same family groups in other keys - start with a new major chord and build those extensions!
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To me it doesn't make sense that traditional music theory only introduces (or never introduces) these chord extensions to our musical awareness. I believe they are a natural extension that is worth exploring early in your musical education.