Can anyone explain to me a conundrum?

The note "E" is a part of a number of common chords we play at the nut, e.g., C, A, Am. So why doesn't it "work" to play the low E string as part of the chords, i.e., when one plays a C chord, why shouldn't playing the low E string sound good with the rest of the chord? Thanks!

B

I'll take a stab at this. If you look at the 1st position C chord as triads starting from the 5th string:
6-5-4-3-2-1
Root Position: C-E-G
1st Inversion: E-G-C
2nd Inversion: G-C-E

So if you want to play the 6th string the triad would be the 2nd inversion:

G-C-E-G-C-E

and that sounds right. E is a chord tone but not the right tone to complete the triad. Maybe that's why it doesn't always sound right. I wonder if the right way to refer to it is C/E, not sure.

Oops. I meant to get the string number and triad to line up:

6-5-4-3-2-1
x-C-E-G-x-x root position
x-x-E-G-C-x 1st inversion
x-x-x-G-C-E 2nd inversion

G-C-E-x-x-x 2nd inversion

It works; just doesn't sound that played by itself. Try playing a progression of first inversion triads, maybe go up the scale. First inversions sound great for that.