With regard to radial piston engines: Wasn't the odd number of cylinders (per row) so you would have an even number of articulating rods connected to the master rod for engine balance?

Even number cylinder (per row) radials have been tried.

From : Don Stackhouse

The typical radial piston engines are four-strokes. Because of this, each cylinder has a compression stroke and power stroke every other rotation of the crankshaft, with an exhaust stroke and intake stroke on the other rotations of the crankshaft in between. If you had an even number of cylinders, you would end up with every cylinder having a power stroke on one revolution of the crank, then no power strokes on the next crank rotation, then power strokes on all cylinders again, and so on. The torsional vibration from that would be horrendous!

With an odd number of cylinders, they can have power strokes on every other rotation of the crank, so that all the power strokes are equally spaced, and all cylinders get a power stroke every other rotation of the crank. This gives the most power per revolution, with the smoothest torque output.

There have been radial piston engines with an even number of cylinders, but they were all two-stroke engines. Each cylinder on a two-stroke engine has a power stroke on each rotation of the crankshaft, so it isn't an issue for them.

As far as vibration goes, radials in theory seem like they should be almost perfectly balanced, but in practice tend to be anything but!. The problem is the master rod concept. Most radials have a "master cylinder" (typically the top one) with its "master" connecting rod running directly to the crankshaft. The other cylinders have their connecting rods running to lobes on the master rod, not to the crankshaft itself. This takes care of keeping the connecting rods from trying to wind themselves around the throw on the crank. However, as the crank rotates, the master rod swings back and forth on it, causing the big ends of all the slave rods to swing back and forth with it. That means that the slave rods are now changing the distance from the crankshaft to their individual pistons in concert with what the master rod is doing.

This causes all sorts of higher order harmonics. For example, depending on the arrangement, a nine cylinder radial could have a nine-per rev harmonic, or a four-and-a-half-per rev. They often use "pendulum dampers", counterweights mounted on special pivots on the crankshaft that swing back and forth like a pendulum, creating a torsional vibration that tends to cancel out the ones caused by the master rod's shenanigans.

Of course those have to be tuned to exactly the right vibrational mode to work effectively. The results of getting it wrong can be catastrophic. I recall hearing about a Beech 18 that went in for an engine overhaul many years ago. The mechanic reassembling the engine mistakenly put ninth order dampers in the engine instead of the required four-and-a-half order dampers. The undamped torsional vibrations that resulted were not immediately clear to the people on board the plane, but the effects on the prop were another matter. When the flight crew tried to take off with the freshly overhauled airplane, the bending stresses in the blade roots from the improperly damped vibrations were bad enough to cause a low-cycle fatigue failure of one of the blade roots just at liftoff on that first takeoff. The 50,000 pounds of unbalanced, rotating centrifugal force created by the loss of the blade yanked the entire engine from its mounts and left it dangling from the firewall by just its plumbing and control cables! Fortunately they were on a very long runway, and were able to get the plane back down and stopped, but it was an extremely close call.

Vibrations can be an insidious thing, and engines and props can be terribly unforgiving.

Don Stackhouse
DJ Aerotech