adjusting string height

I have an old Martin that I restored and one of the things I did was make a nut for it.I made if from buffalo horn. When I put my eletronic tuner on it and play open E and tune it, and then fret an F the reading goes up a bit. Does this mean I have to lower the action a little more?I dont want to lower it too much and have to make a new nut. I'm fairly new to doing work on my uke and really enjoy doing it, but I still have a lot to learn. Thanks.

I feel like it would have more to do with the saddle height, but I could be wrong...

This Uke has the original saddle

it could be too high off the nut, and by the time you depress the string all the way down it will go sharp...not because of the fret position, but because of the tension caused from having to stretch so far to the fret.

is the new nut higher than the old?

Does this normally happen for your ukes? You may just be pressing down too hard. Check to see if similar things happen when you play others.

There is a neat trick for making sure you don't go too low when setting up the nut where you flatten a pencil and place it flat on the frets, then draw a line on the nut.

My experience is a good set up fixes many intonation issues.
I had a uke that the intonation was way off from the second fret on.
After lowering the saddle height, it was still off. I trimmed a little of the nut and all is good.
The uke plays better and the intonation is much improved. The key to set-up is go slow.
You can always take off a little height, but it is difficult to put it back.
There are lots of articles and resources about set-up.

If you file too low on a nut slot take a take some crazy glue on the end of a toothpick and fill the slot a bit. then file it down again after it dries. I'm not a fan of shimming nuts or saddles... but what do I know.

If you use the flat pencil trick there shouldn't be a problem.

What I am doing is fine tuning my nut string height using my electronic tuner. Tuning an open string with the tuner and then fretting at the first fret and checking the reading. All readings on the four strings were a little high when fretted at the first fret.
I just spent an hour with my welding tip cleaners ( I have to buy a couple nut files ) and the uke sounds much better. When I first made the nut I used feeler gauges and the strings werent quite as low as they could be. I thought the tuner would be a good way to fine tune the nut. Has anyone tried this ? They are still a little high according to the tuner but I'll do it again later.

A trick I picked up from my days in the machine shop.. is to stick a wet cigarette paper across the first fret....and then file down the nut slot untill the string touches the paper with a heavy plucking of the string..you will see and hear where the the string just grazes the paper...then you will be about
.002" off the 1st fret..if this is too close for you then stick 2 wet papers on.