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Two-Cylinder John Deere Engines
Well, it's just a quirk of math and physics really.
It started out as fluke, based on the "hit & miss" engines of the time, and became the signature sound for a whole reutation based around the "Poppin' Johnnies".
Basically the theory then was not to have a steady, almost continous series of detonations feeding power to the flyweel which smoothed the power out and kept it constant and not just in spurts which is hard on both man & machine.
Bear in mind, the only thing they had to base their designs on at the time was horses or mules or oxen, and anyone who has ever worked a draft animal will tell you, under sever load, or when starting a heavy load, a horse will break stride and kick out. That is to say, they don't push with one leg at a time, they lean forward into the traces against the load, squat slightly and give a big mighty push with both hind legs at once.
So, when 'horsepower' got mechanised, they had only this example. The early engines, especially the hit & miss ones, had an enormous flywheel and a relatively few 'kicks' from the power strokes, spun it up and lept it spinning, and it was just the stored energy in that flywheel that did the work, not each power stroke against the crankshaft like todays engines do.
The math & physics enter into it because when John Deere designed the 2 cylinder engines, the did something unusual, they didn't spead out the strokes over the entire circle. Various historians think this was because they either, designed the engine as a 4 cyl. and cut it in half, or designed it such that it had 'room for expansion' later on. Nobody knows for sure.
We do know, that the two power strokes are at 0° and at 180°, and then the engine coasts for 540° and then the whole process starts over again. It is this 'pop, pop, coast' that gives them their distinctive sound.
Why they didn't have them fire at 0° and 360° to even things out? Well, we do know that there was one really good reason, shake!! Having them fire at 0°& 360° on a 2 cylinder motor would have meant that both pistons hooked to the crank on the same side, the crank would have been a U-shape instead of a Z-shape. This would have caused a huge shake in the machine.
Of course 3 or 4 cylinders would have solved this problem too, but that's not what they did.
Regardless of the reasoning, I'm glad they did it the way they did. I've got a JD AR that has been in the family since new, it still looks that way too. Every year it appears in countless parades, and you can see the people stop and stare, especially the old-timers when it comes pop, pop.....poppin by them. 
Best of luck.
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Two-Cylinder John Deere Engines
Yup, they even get fussy about them up here in Cannuckistan too Art.
I had to 'buy' the tractor from the farm corp. then write up a storage contract and pay the corp. a few bucks a year for 'storage' to even be allowed to keep it on the farm.
But there's still something magical about them.
Best of luck.
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