This will be a guide for people that are new machinist (like me) that wanna wrap their heads around threading custom threads in a lathe.

There is several way of getting to the correct gearing on a lathe and i will document two ways of calculating gears. This is based on the fact you have a gearbox on your lathe or that you know the pitch of your leadscrew for threading. I will work from a 3mm leadscrew or gearbox set to 3mm pitched metric thread in these two examples.

The custom thread i have been trouble with making is a 1.6666666~ thread on an old Aciera buttress collet. The diameter is 19.75×1.666 so its a bit of a strange thread. This is the first way i came up with the calculation for working this out..

Method1:

3mm leadscrew is = 3 = Pb
The goal pitch is 1.666666~ (rounded fraction) = 1.667 = Pt
Ratio of these two will become  = 1.8 = R
That is ( Pb / Pt = R ) which give us a 1:1.8 baseratio Br
You then take Br * 10  /  Pb  * 10 = 30:54  , this doesnt have to be ten, this has to be a number so that it matches one of your sets of gears.

This will give you a setup of  30 tooth gear on spindle, 120 gear idle and 54 gear on leadscrew. This is the common way of calculating threads.

There is a second method of calculating odd ratio based threads like this.

Method2:

First we need to understand that a 1.66666~ thread is not really a metric thread. It should actually be read 1mm and 2/3rds of a milimeter or 1.2/3mm which is an old way of saying 5/3 of a milimeter.. or the ratio 5:3. When simplyfying a decimal number to a ratio we will understand that the 5:3 ratio is then based on the fact that the machine has a 1:1 ratio between leadscrew pitch and lathe spindle. This is not so in our case. Remember we have set the leadscrew to a 3mm pitch by changing the gearing. So how do we calculate a 5:3 ratio thread.

First do like this 5:3 * 1:3 = 5:9. This will stepup your initial ratio from a 1:1 to a 1:3 ratio of the leadscrew multipled by the actual lead you have set your lathe to.
Then you have to arrive at a set of gears that you actually have in my case that would be 5*6 : 9*6 = 30/54 which corresponds to method 1 way of calculating a pitch.

After have done both of these.. i must say that method1 is easier.. more or less because it doesnt convert number o ratios.  But you can use this in reverse. If you know the fraction of a thread like the 1 and 2/3 of a milimeter or its stated to be a 5/3 metric thread. You know that is the pitch of 1.666 and you can use either method to arrive at the correct gearing for your lathe..

I do hope this isnt to confusing and if so please leave a comment.