I've got a '60 P, and after decades of Steinberger, Sadowsky, Alleva-Coppolo, and modded Fenders, I'm back with it as my main bass, now with la Bella flats and the tone knob all the way down. Who knows what happened to the ashtray, so I made a mute, out of a piece of sponge wrapped in felt, that I just stick under the strings at the bridge. I made half a dozen before I found the right amount of muting, which I have to say is quite a bit. It's pretty big and it does affect the intonation as depicted on an electronic tuner -- I tune at the third fret so the lower-fretted notes are a touch flat and it goes a little sharp as you go up the neck -- But I've gotten lots of positive responses, both live and in the studio, and not a single complaint about intonation. I do keep the G string a little less muted, but it's still damped down noticeably.
For reference it's interesting to hear Jamerson's isolated bass track for "What's Going On" (). Note the high C#'s at ) 0:32 and 2:18. They might be on the D string but I think it's more likely he's on the G string, just given his general near-the-nut, open-string-centric approach. They're quite heavily muted -- the first one in particular barely makes it to the middle of the bar before fading away -- and to me they sound pretty sharp, but only in isolation. In the mix, they sound just great to me. Once again, a solo'd bass track is a world away from the same track embedded in the music.
-Steve