Bass is picking up hum - did I damage the pickup?
I have this cheap copy of a 50s Precision Bass, which came with an ugly pickguard. So I made a new pickguard, and in the process of fitting it to the bass, I scraped off the protection tape on the side of the pickup, and the wires became visible.
When I was done, I noticed that the bass is now picking up hum when I'm not touching the strings, which I'm fairly sure it didn't do before. The hum goes away when I touch the strings, and I can turn it off by turning off the volume pot on the bass. It doesn't seem to be the cable, or the amp - plugging it into my guitar doesn't result in hum.
So I have a couple of related questions:
What is wrong with the bass? Did I accidentally damage the pickup, or tear off a ground wire? How do I diagnose this?
What can I do to fix it? A brand replacement pickup would cost roughly as much as the bass...
Edit: As requested, here are some images, and the results of some tests with the multimeter.
As far as I can tell, all points that should be grounded have a good connection (1 Ohm or less) to the outside of the output jack. What worries me is that I cannot measure the resistance of the pickup - from what I understand, there should be some 5-20 kiloohm between the two poles of the pickup, depending on the construction, but my multimeter shows infinite resistance.
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Please add some pictures. You have an earthing fault. The most likely explanation is that you accidentally tugged on the wires at the jack socket, or pulled an earth from one of the knobs. Unscrew the control plate (where the knobs are) and inspect carefully for loose wires (probably black).
If you have a multimeter you can test for earth continuity to the jack socket's outer ring where the black wire connects.
I disagree that this could be an earthing fault. You need to go right back to the basic design of this bass. The P bass from 1951 to 1957 was built with a single coil pickup, which inherently picks up electrostatic interference. Conversely the human body is a good antenna for attracting AC electromagnetic interference. Your bass gets quieter when you touch the strings, or other grounded metal hardware, as this is all connected to ground, forming a pathway for this interference to go to ground. Your bass doesn't have any shielding, so will have always been likely to have picked up background interference from any number of sources.
From the picture you posted, the control cavity and pickup cavity in your bass would benefit from shielding (either copper tape or shielding paint) to cut down on electromagnetic interference. This will minimise the difference in noise levels from your bass, regardless of whether you are touching any grounded part of the bass or not.
However the pickup in your bass will always pick up electrostatic interference, and there isn't much you can do about that. The damage to the pickup coil might be superficial or serious. The best test is to see if you still get a DC resistance reading from the pickup coil. If it is between ~ 5 and 7 k ohms then the pickup is ok. If it reads as an open circuit then you cut into the coil and broke it. A pickup with a broken coil still produces output (Geddy Lee's famous black Jazz bass, which he purchased in a pawn shop, has an open coil in the bridge pickup. Likewise Roy Buchanan's famously bright Telecaster tone came, in part, from a broken pickup coil).
A broken pickup coil still passes high frequencies, but the tone control acts like a second volume control when this happens. In the case of your bass I think I can see a broken copper wire within the pickup coil, but I can't be sure.
If the pickup is broken, there are cheap Wilkinson and other unbranded single coil P bass pickups on Ebay for a lot less than you paid for the bass (assuming that it is 1) the Harley Henton '51 P bass copy and 2) you bought it new).
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