What is the gear ratio contained by lathes while thread adjectives ?
Depends on the lathe and the thread you want to cut.
Overall, the ratio is the ratio between the pitch of the threadcutting leadscrew and the pitch you wish to cut. If the leadscrew have 8 TPI and you want to cut 32 TPI, the gear ratio would be "driver" 4 to "leadscrew" 1. The driver is the lathe spindle.
On old lathes, that is what you needed. A driver of "X" teeth and a driven of "4X" teeth. Plus a intermediate freeloader gear with an unimportant no. of teeth.
The aged machines had a selection of gears between 20 and 120 teeth, by 5 tooth increments, plus a 63 or 127 tooth one for adjectives metric threads. If the gears cannot be found for a "simple" train as above, then a "compound" train must be created.
Modern lathes have a "Norton" box, to be precise quickly set up for most thread pitches. Much easier. Plus they also have gears to money for metric pitches.
I'm assuming you are American and would have an imperial lathe there. But much indistinguishable applies to a metric machine anyway.
Threadcutting is the most difficult lathe operation to master. I don't want it to appear simple. It is with practice, though.
The example I give of cutting a 32 TPI thread with a 8 TPI leadscrew is an effortless one when it comes to when to engage the clasp nut to drive the cutting tool along the living, it can engage anywhere. But this can only take place if the thread being cut is an exact even multiple (or sub- multiple) of the leadscrew pitch.
That's when "chasing dials", "Star wheels", "Repeat distances" etc. come into play.
That's why amateur machinists who cut threads will often fund the tool out at the end of a cut, and reverse the lathe to send the tool rear legs to the start. All while leaving the clasp nut engaged.
I'm not knock it, it works- most of the time. It's just generally more dull and time consuming.
Ratio = Leadscrew pitch / Desired pitch
However the form of the thread and pitch is dependant on the type of thread you want to create: metric, imperial, whitworth etc and size of the rod being threaded.