| The bridge cover had the mute glued to it, so when you put on the cover it damped the sustain of the strings. The PUP cover functioned to support the wrist. I can't find it, but there's a great picture of Leo Fender holding an early P bass, and he's playing it by strumming it with his thumb. His hand is resting on the PUP cover, and his fingers are wrapped around the finger-rest (that's why it's under the strings). Plus, the instrument was designed in the '50s, the time of chrome on everything. I'm sure part of the reason for those big covers was to dress up the appearance.
Lots of players found that the covers got in the way of playing so they strated coming off. But if you wanted the muted sound, then they started stuffing foam of some sort under the strings just in front of the bridge. Try it and see if it works for you.
I used to keep a chunk of foam in my case, and would stuff it in for specific songs- but I've not done that in decades. I did put some under the bridge of my P bass for a while, but decided it's not the sound I really want all the time.
jte
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JTE Spelling, grammar, and punctuation do matter, despite the threats of death by grease fire!
"Without space, music is just noise piling up on itself." TRK
Lakland Owners' Club # 248
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