I'm using an IGH with semi-horizontal dropouts. The ratio I have now is not working well. As it is, I'm at the very back of the dropouts and can't really maintain adequate chain tension. If I take a link out, the non-turn washers stick out the front of the dropouts.

So, question is: the axle-crank distance will change by how much for each incremental increase in chainring size? If it matters, I'm using a ~38 right now.

The chain wraps 1/2 of the way around the ring so each 2 teeth of chainring need 1 more link of chain. The chain is split equally between the upper and lower loops, so each link of chain (1/2") would move the wheel 1/4"

Answer the wheel moves 1/8" per added chainring tooth.

Outstanding answer. Thanks.

I'm a little lost here. The pitch of the gears is the same as the chain, 1/2". So the wheel should move 1/4".

Follow the reasoning in my first post. I could be wrong (it's not like that never happened) but I don't think so.

The chain wraps 1/2 way around so total chain length change is 1/4" per tooth, but that's is split between the upper and lower loops, moving the axle 1/8". Did I miss something?

Looking at it another way, the chainring is roughly circular. Increasing the circumference by 1" (2 teeth) increases the diameter by .318, or the radius by .159" or roughly 5/32". This is a slightly more than 1/8" and I think that's because a chain on chainring doesn't form a circle, but a polygon of X sides of 1/2".

Either way we're still looking at about 1/8 - 5/32" or so per tooth.