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Muting problem?

I'm having a problem with my playing/technique. I find that no matter what I do, I always hear "extra" noises, usually a string will be slightly vibrating even when not playing the string. My fingers will slightly touch it when I'm playing the string above it. What can I do/practice to prevent this? I have been using the floating thumb technique but only to a certain degree. Depending on what I'm playing, my thumb will rest on either the pickup, low B or E string. When I'm playing higher up, it stays on the E string.

Also, I notice that whenever I play a note, for example, on the C string, it will cause all strings to ring slightly, even when all I'm doing is playing that one note at a low volume. One string will usually ring more than the others. I apologize for the vague/newbie question.
 
I took the term "floating thumb" quite literally when I started and for the most part my muting is pretty solid. This is what I do, and I'll describe it to you:

Put your index and middle fingers on the A string, and lean your thumb against the E string. Don't rest it on top of the string, place it against the string. The point of where the thumb and string touch on your thumb should be just above where your thumb starts to curve at the tip (not on any part of the tip, but where the thumb becomes flat). Keep your fingers and thumb this distance apart wherever you are. So if you're playing the G string, your thumb will be on the same spot on the D string as it was on the E. I play a 4-string, so when I play the E, I keep my thumb floating in the air. This will keep it free and ready for quick movements. I never rest my thumb on anything, it just leans against the lower string and as I slide up a string it will slide up a string too. Also, I try to keep my thumb flat so when I'm playing the G string, the thumb is leaning against all three lower strings. It takes a little practice to transfer strings without the thumb making a noise, but it's much easier than placing it on a string. One important note about this tip though, is to NOT rest your wrist on the bass itself. Sharp angles are bad for the wrist. Try to keep your elbow sticking out at your side, and basically just raise and lower your elbow as you go higher and lower on the strings. Again, keep your wrists straight!

Also, when I'm moving down the strings (say from G to D), then I use any finger (could be a free one or I commonly use the finger I'm fretting with) on my fretting hand to mute the ringing string (ie. G). And I try to keep them relatively close to the strings. If you play a G on the E string, you will see that the G string will actually vibrate as well (something to do with the vibrations from playing a G causes this). So keep that in mind too.

As to why all of your strings ring slightly, I'm not sure. Just practice on staying tight. In terms of muting, I find it usually takes more work than fretting itself. It will seem hard at first, but keep to it, eventually you'll do it naturally and may not even realize you're doing it.
 
"I have been using the floating thumb technique but only to a certain degree."

What exactly do you mean by this? The floating thumb is really the most efficient way to play, in my opinion. This technique takes a little time to get used to, but is a much more fluent way to play compared to resting your thumb on the pickup or low string. Speed, fluency, and versatility all come from the floating thumb technique. The closer you angle your fingers perpendicular to the strings the easier it is to play. Stick with it.
 
Almost all stringed instruments have what is called "sympathetic" vibrations. Sitars are made deliberately with extra drone and sympathetic strings so they vibrate when the playing strings are plucked. Guitars do it and basses do it.

I don't have a solution other than what everyone else is saying about muting the string with another finger, just saying it's not the player, but an effect of the instrument.