small lettering question

No, no real minimum char. per inch because there are too many variables. I go by height. I can get clean small lettering at .15-.16mm. The trick with tiny lettering is to try and use a font with equal width columns, no serifs, and hand digitize that font. Most stock fonts that u can set up in ur digitizing software aren't really good when reduced to very small. The only exception would be Wilcoms "Micro Block". It's sews really nice small letters when set up correctly. I typically use a density of .38-.40 with a .25mm pull comp. and depending on the material, I'll either use no underlay, or a straight stitch with a short stitch length of about .8. The style of font has everything to do with clean small letters. The trick is to know just by looking at if whether or not it will see clean. But ur choices go up considerably if u can set up clean vector text and digitize it yourself. Small lettering is one of the biggest challenges to digitizing. Once you've mastered it, you can do just about anything else thrown ur way. Take the time and sew out lettering as much as u can. And also watch the machine very closely. Watch how the letters are created and understand how they sew. Most of my talent as a digitizer was learned watching and running a machine. I firmly believe to be a good digitizer you have to spend a ton of time watching and studying a machine moving and forming stitches. When I digitize I am watching the design sew in my head as I'm creating the design.
I've seen great graphic artist take digitizing classes to learn the software inside and out, only to fail miserably as digitizers because they have no clue how machines actually run. And on the other hand, I've seen lowly machine operators become excellent digitizers without one ounce of artistic talent.

Going off what Sulp has written your lettering should fit within the space you have. maybe your font needs to change as he/she wrote?? Also don't forget that with smaller lettering if you sew in thinner 60wt thread then that saves a lot of headache as well.

What exactly are the 37 characters? Any option to split it up on 2 lines? There are just some things that can't be done with embroidery exactly the way it looks on screen/paper. You have to know when and what to simplify. I know you want to satisfy the customer. But sometimes you can look at something and say "this can't be done like this on a LC" (or whatever). If it's way too small to embroider the text on one line, and you know that just by looking at it, you just have to say it can't be done and look for alternatives. Let me know what the text should say and I will let you know what your possibilities are. Also let me know location and type of material youe sewing one.

@Robert, I'm a him..lol!

And yes, as Robert says, thinner thread, smaller needles all help in clean small lettering as well. But I will be honest...I rarely change my needles for small lettering and use 75/11 for almost everything.

I set it up in my system, and it is doable. I matched the dimensions exactly (3.5 x .25) and I did have to distort it a little bit (mostly stretching taller to achieve the .25 height) but it's not too bad. I could definitely get it to sew nicely at that size....And with keyboard lettering to boot.