Ok, this is a fun one, and it's easy to solve once you know :) as someone else mentioned, get the Gary Katz DVD's on wainscot, it's a huge time saver.
The one thing that you will need here is a construction master pro calculator, or the buildcalc app on your iphone/android device. The calculator is (in Canada) between $80-120. The "app" is $20 bucks. I use the app all the time in my business for this type of thing.
The small wall, you want four frames. The "hard" non-variable measurement is the 4" spacing that you would like to use, which means that in order for everything to work out perfectly your frames might be a little bit different size that what you would like.
We need to treat the frame and the 4" spacing as one unit. So in order to get it so that we have 4" at the end, deduct 4" from your overall measurement. 116-4=112"
Now using our calculator divide 112 by 4. this gives you 28.
So now measure on your wall (or story stick) from the (start) corner to the end of the first frame: 28"
Now on your calculator push the "+" button and the "=" button. the next measurement will pop up, and it will be 56". Remember this is a 4" space plus your frame.
Here's the tricky part: do NOT touch the "+" button again!!! only push the "=" button. this gives you your third measurement which is 84"
The last measurement is the same, push the "=" button and your measurement will be 112"
This leaves you exactly 4" at the end.
Subtract 4" from a measurement to get your frame size, in this case it will be 28-4=24".
The larger wall is similar. I would recommend keeping the 4" spacing as I think that is more noticeable than the frame size.
A few things to consider here. How many frames/spaces do you want? as close to the smaller wall as possible I think
Now lets use our long wall measurement, which is 157". We want a 4" space at the end, so we deduct 4" so 157-4=153".
Here is the estimating part, we know that we want roughly a 24" frame, so how many frames do we want on this wall?
Ok, lets assume we want 5 frames, well divide 153/5 that gives us 30 5/8", and if we subtract 4" from that it means that our actual frame size will be 26 5/8" Seems a bit big, so lets try another amount.
Assume now that we want 6 frames, so Divide 153/6 = 25 1/2". Subtract 4" from that and your panel size will be 21 1/2". Now this seems a bit small. What needs to happen here is you using your judgement, what would look best to you? slightly larger frames? or slightly smaller?
I am going to assume slightly larger as the wall is longer and you don't want it to look too busy.
So that means that we are going to have 5 frames.
Now to the fun part:
153/5= 30 5/8" this is the first measurement from your corner/wall. Remember this is a 4" space plus our frame size. Don't worry about the frame size at this point, we can find it when we are done. Actually we already did this but anyway.
Next measurement hit your "+" button and "=" button, and the result will be 61 3/16". Remember this is one space(4") and one frame(26 5/8")
Ok, now hit the "=" button again and we get 91 13/16" (hands off of the "+" button!)
"="next one is 122 3/8"
"="next one is 153"
Right dead on!!! :) and exactly 4" left over at the end.
Figure out your frame size by subtracting 4" from 30 5/8", which we already know is 26 5/8".
Your spacing will be accurate within 1/16`1/32 either way depending upon what setting you have used in your calculator.
The calculator program (and app) figures out these measurements to something like 9 decimal places, then rounds up or down to the nearest 16th or 32nd. Each measurement is INDEPENDANT of the previous measurement, so you do not have "cumulative" error.
We've all tried to measure out a space, then using a spacer block marked it off- usually with about an inch error at the end. This calculator option solves all those problems.
There's a lot more to wainscoting, I highly recommend that you purchase Gary Katz dvd's on Wainscotting, he explains this in a very easy to understand way.
Show us the results when you are done :)