Time to go transcribe some Jamerson.
Start with What Becomes Of The Broken Hearted. The original arrangement is an excellent example of not playing roots.
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Time to go transcribe some Jamerson.
Play whatever note you want. Just know that you're changing he chord by not playing the root. If the chord is A minor and you play a B the chord is now a Bsusb9 which will sound out of place in a country song. If you play an F you'll change the chord to an Fmaj7. Play a D and you end up with a Dsus chord. Whether it's appropriate to reharmonize he tune from the bass chair depends on the gig. Here's Ron Carter playing damn near any note but the root:
Actually,the bassline is not doing just the root E.Recently, I've re-listened to a lot of music from "Mezzoforte."
Here is one of those "boring" bass-lines with mostly root notes.
(Just a rhetorical question. When I hear a bass player playing and properly dynamically/rhythmically articulating that simple (just the roots) bass line, do I think
A. Thank bass player really sucks,
or
B. Wow, what a cool bass-line?
In Herbie Hancocks composition Maiden Voyage, the bass does not emphasize the root during the head, it is only used as a passing tone by the bass.
The 1st chord is Am7, the bass plays D
chord changes to Cm7, bass plays F
above is twice thru
next change is
Bbm7 with Eb bass to C#m9 with an F# bass
Herbie Hancock - Maiden Voyage

By the way, I have played a lot of improvised music (jazz, jam bands, Grateful Dead covers, etc.) and it's always painfully obvious which musicians have done the work of learning the song, and which are just running up and down the scale.
It's like saying, "there are only 26 letters in the English language, can't you just type those letters in any order and write a great novel?"