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Nut on my Rickenbacker 4003 - too low?

Raising the nut will only affect open strings, not any fretted notes unless it is actually vibrating sympathetically behind them - I have actually had this happen with a 4003, took me ages to figure out that the buzz was below my fretted note and not above it. If it was me I would go baking soda and super glue, done properly you should not notice it and you can always rub a black sharpie over it. Again, I have done this successfully and easily on a 4003 with no issues. They come new from the factory with an incredibly high nut, so some user tweaking is to be expected.

That's true, but there is a way it also affects fretted notes, at least when playing unplugged, that can be really annoying on a low action bass with a relatively flat neck.

So yes, that nut is a tiny tiny speck too low. You might be able to raise the slots a speck with some fast drying hard glue, like superglue and a dab of water or spit in it (since super glue hardens with moisture almost instantly, hence why touching it results in permanently glued finger-to-thumb due to moisture in our skin).

Why do I say that about listening unplugged? When you fret a note, the string on EITHER side of the fretted note will vibrate and play a note just from the hammer-on effect of your left hand. Some of us fret harder than others, but yea that is a thing.

When playing unplugged, you really can hear that lower half of the string playing a really high note at the same time as the "playing" part of the string plays your desired note (hopefully LoL). On some basses the nut-to-frettted-note is louder than on others, and mostly when plugged in you don't hear it at all so it's all good. But playing unplugged or very quietly through an amp can be a little annoying due to this phenomenon.

Now.. if your nut slots are really low, not so low that they buzz when playing open strings sort of calmly, but low enough that your strings buzz when playing 1st, 2nd, or 3rd frets or so with a bit of strength in your fingering hand,.... well then the short vibrating string on the left side of your fingering hand can actually do audible fret buzz that you can hear acoustically. Again it probably isn't coming through your pickups, but if you're playing unplugged or quietly, it can be really annoying too.

So is it too short? if it's buzzing for you, it's too short. I like very low nut slots, but you can't have them so low that it buzzes for ya.
 
When I set up a nut, I do it by eye, rather than a measurement. The height of your frets are a factor, so you would have to measure that and figure out a formula.
I fret the first fret and then the 2nd and watch how much the string travels. Then I check the string travel between the open note and first fret. You want this to have just a little more travel, as open strings have just a little more excursion/movement.
 
I was wondering about back-bow too - but it only appears like that at the first few frets. Thus far I have slightly raised the strings on the nut and have had good effect with this. I have been measuring neck relief by capo-ing first fret, fretting last fret and using a feeler gauge at 8th fret - currently there's around .014" clearance at the 8th fret at both E and G strings. I have slightly adjusted the truss rods to get to this measurement. Please do let me know if there's a better way of measuring this.

That's a good poor-man's way of checking the relief if you don't have a luthiers straight edge or other tool.

But when assessing relief, what you're really looking for is the shape of the relief, not so much just a measurement at the middle which must be so-and-so. So take a look at it in "profile". Take measurements at each fret if you like. That should show any humps, high/low frets or low spots, etc. Once you see the shape of it, that will allow you to see what's causing buzzes or certain parts of the fretboard having high/firm action and so on.

That can also help you see what adjustment to make (or not make) to fix a problem. For example, if you have an upward hump near the headstock, tightening the truss rod will only make buzzing "behind" that area worse (i.e. towards the headstock). You might be able to compromise by loosening the rod, which will add some relief, but put a little more downslope on that hump which may help the buzzing in that area. But the cost will be a little higher action in the middle of the neck.

And so on. But that's the general idea. Some things the truss rod can fix but sometimes you need a compromise adjustment to make it playable overall. And if you have a bad hump or dip that you can't do anything about with the relief adjustment, then you'll have to do some repairs...

I just went through a similar process with a new jaco that I got the other day. It turned out to have a kind of "cup" depression in the fingerboard right around the 1st "fret" area. But it was bad enough that unacceptably high action and positive relief were needed to treat it. So bad I would worry about not having enough wood thickness in the board to fully fix it. Which is fatal and it's going back to Sweetwater for an exchange....
But that's how I assessed it on my bass too: I essentially just used the string method first, then examined smaller spots with a straight edge right on the fingerboard....

L
 
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If you can use this and figure it is straight, then you need some relief in the neck. Many companies don't even attempt to set a bass up properly. And that is sad.
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