...The odd number of blades will break those types of waves up, while the even number can add to them.
I take exception with this. There is no "breaking up" effect. Sound waves are merely superimposed - in all cases. When peaks coincide in the same direction they are additive in amplitude, when they are opposed they cancel out, and they all dissipate according to the medium and environmental conditions.
The rotating blades don't really work as separate entities they create a bulk pressure a difference and the air moves to equalize the difference. If the blades move at a constant speed and the sound that is made is very complicated. If the air simply moved constantly with no loss, there would be no sound at all; but instead you have edge effects, the center velocity is slower than the outer - until they blend, there are vortices shed, etc., each effect creating its own characteristic noise and they all bend together. There isn't even a rotational motion any distance from the fan as we easily intuit.
The momentum imparted by the blades has no rotational effect its direction only according the resulting vector of the air before it is effected and the angle and instantaneous direction when the blade interacts with the air - pushes the air. An unducted fan spills most of this of the ends of the blades in a radial direction, the air that has angular velocity as a bulk property in a ducted fan quickly loses the rotation when clear of the duct.
We hear a lot of high frequency noise from some fans where the edges are blunt, there are fancy things that can be done to reduce the sound they make. There is a Skystream wind turbine outside my lab that has very fancy profiles to reduce noise. The quiet fan at my feet has a similar, but much shorter shape, the loud box fan at my side has blades roughly analogous to rectangles with rounded corners - simple and loud. If you reduce the noise they will likely be more efficient, but they may not move the largest amount of air - there is a trade off.
I imagine that odd numbers of blades is easier on the motor, and mounts. No two blades will dysfunctionally add to a greater radial magnitude (resonate), so vibration of the motor and structure is reduced, and the overall vibration will have a higher frequency for a given rotational speed.
If you have opposed blades (an even number of evenly spaced blades) where one is a little lighter than average and the other a little heavier, they will couple creating a greater vibration - but not particularly in the air, the system will have vibrations in the structure and mounting. IF the blades are matched well they won't cause vibration. In manufacture it is easier to just have odd numbers, rather than even and a balancing act.
None of this is very pertinent to specing a livestock fan. Buy from a supplier you trust, odd numbers of blades may be marginally better - you don't care much about noise, but they may be more durable with odd numbers of blades. Then buy capacity to move air - you need to turn the air in the building over some number of times an hour. A Walmart store turns over ALL the air inside on the order of 5 times an hour - lest it be come stale and discomfiting. It takes a lot of energy and the fan runs 24/7, so an efficient fan IS important, where cooling one person in a room it is not such a big deal. There should be effective ducting about the fan. You want all the air to stay in the fan and be accelerated, rather than spilling away from the blades easily.
The fan should be easy to repair. The motor should be commonly available - including how it mounts.
