Probably a thread for the technique forum.
Diagnosing fret buzz:
1.) does it buzz when you are amplified and playing at regular volume?
unamplified fret buzz is a non-issue
My expensive bass with a pro setup will still buzz when unplugged. But not when I'm amplified...
2.)Is it my technique or is it my instrument setup?
Turn up the volume, cleanly fret a note (right on top or just behind the fret), and pluck lightly.
Keep your fretting hand in place after plucking.
If if buzzes while let ring the cleanly fretted note, it is likely to be a instrument setup issue.
Which could be string height OR neck bow/truss rod.
(since it only happens on your E string, I'd suggest raising the E string up first)
If it only buzzes when you are playing phrases and riffs, it is likely to be a technique issue, generally lifting your fingers off the string too slowly.
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try this exercise
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Put all 4 fingers on the E string, 1 finger per fret, starting wherever that 4 fret stretch is comfortable.
press them all against the string enough to fret a note ( avoid the 'death grip')
One at a time, starting with the pinky, lift a finger off the E and move it onto the A string.
keep the remaining fingers in place
Conciously move each finger in three steps : UP, ACROSS, DOWN.
After you move your pinky, move the ring finger
keeping all the other fingers in place
Then move your middle finger, then your index finger
keeping all the other fingers in place.
after that, Starting with your pinky, reverse the movenment form A string to E string.
(If you are feeling ambitious, you can climb accross all the strings this way -its a great exercise.)
Do this slowly , concentrating on a clean three step motion: UP, ACROSS, DOWN.
It may even help to say it aloud to youself "up", "accross" ," Down" in time with your finger movement.
Keep your left hand relaxed with the
minimum pressure needed to fret, move one finger at atome and keep the rest in place.
Once your left hand starts to get used to this motion, start plucking the notes you are fretting.
Chances are you'll get less buzz at this point, but if you still do, renew your concentration on UP, ACROSS, DOWN as you pluck.
Do not worry at all about speed:
the real key to clean technique is practing slowly and steadily until your muscles begin to remember the movements.
If playign a riff at speed causes your technique to slip, simply slow it down until your hands get used to the motion.