Sorry to buck the trend of everyone saying it's a P, but it sounds like a J to me - definitely two single coil pickups.
That bass has a lot of what I call 'wood' in the tone. If it's not there acoustically, no tweaking of pickups or electronics or strings will make it come out, but there are ways to help it along.
- flatwound strings
- foam mutes
If you're looking to buy a bass with more of that tone, there are some other thing will help it along.
- shorter scale
- fully hollow body (archtop, not semi hollow)
- lighter wood (more resonant)
Check out the
Classic 4. Check out
Invalid Link Removed. I don't know what he's playing but, again it approaches the tone you're looking for. Looks short scale, not sure if it's hollow or not.
This Hofner gets a similar tone, but not as well as some others I've heard, obviously you'd need more bass than is on this recording.
A good P bass (not all P basses are created equal) will get a similar tone. Check out Jamerson's solo'd bass track from
What's Going On. Warmer & woolier, but if you turned up the treble you'd get that papery warm wood tone you're after.
Invalid Link Removed gets a killer P tone too, I can almost feel it thump my chest through the earphones! The emphasis is in a different place than the track you showed me - the P pickup location puts the punch in a different place than a Jazz. But the woody tone of the bass coming through is the same.
The Roadworn Fender Jazz I played recently approaches that sort of tone, though you'd need some EQ... the Sansamp VT bass wouldn't hurt either. I'd wager a Road Worn through a VT bass pedal could get you there.
The bottom line is - unplug the bass. How close does it get to the tone you want without the electronics? I'd almost be willing to buy a bass based on its unplugged tone alone, provided the eletronics configuration allowed me to swap out whatever it had for something better if I had to.
Never buy a bass you haven't played (unless it's a music man, one music man is the same as any other with the same configuration - the modern ones anyway), especially if that's the tone you're going after. That tone is ALL ABOUT the wood.