In my last post, I was looking at how the same chord can have different jobs depending on what key you're in. Am is a good example: it occurs naturally in the keys of G, C, and F major.
G major: Am is the ii chord
C major: Am is the vi chord
F major: Am is the iii chord
When I posted that elsewhere, Brian Kelly pointed out something I hadn't considered: those three positions also tell us which mode we're playing if we start on A.
In G, A is the 2nd degree, so we get A Dorian.
In C, A is the 6th degree, so we get A Aeolian.
In F, A is the 3rd degree, so we get A Phrygian.
What I thought was really interesting was seeing why those three modes are different. (My apologies, I didn't realize this forum would mangle my carefully crafted table. I've attached an image of what the table is supposed to look like, and it should be much clearer than the below mess.)
Mode Notes Compared with A Major
A Major A B C♯ D E F♯ G♯ 1 2 3 4 5 6 7
A Dorian A B C D E F♯ G 1 2 ♭3 4 5 6 ♭7
A Aeolian A B C D E F G 1 2 ♭3 4 5 ♭6 ♭7
A Phrygian A B♭ C D E F G 1 ♭2 ♭3 4 5 ♭6 ♭7
Notice that A, C, and E are present in all three modes—those notes form the Am chord. The Am chord itself hasn't changed. What changes are the other notes surrounding it, determined by the key we're in. Those different notes are what give us A Dorian, A Aeolian, or A Phrygian.