Alan, the answer takes you a bit into prop theory.

A prop's immediate function is to speed up a column of air. The thrust you get from it is actually a byproduct of this speed-up -- but obviously a handy by-product.

The amount (# of pounds) of thrust is a function of two things: (1) the volume of air you speed up and (2) the amount by which you speed it up.

"Efficiency" of props is a measure of how many pounds of thrust you can get from a given amount of engine horsepower. The more efficient your prop, the more bang you get for the buck -- or more specifically, the more pounds of thrust you can get from your 46 hp 503SC Rotax.

It turns out that it's more efficient to give a LARGE column of air a SMALL change in speed than to give a small column of air a large change in speed. You just get more pounds of thrust with the first arrangement than with the second. That's why rotors are large in diameter: it's more efficient that way.

Therefore, we want the prop to be as large in diameter as possible. One limit on diameter, however, is prop tip speed. If you use your 2.24 ratio and run a 60" prop that allows the engine to rev to 6500 RPM full throttle (as it should), you'll have a prop RPM of 2900 and tip speed of 759 ft./sec. That's fast enough to make for a very noisy prop. It's also fast enough to create extra drag losses as the blade tips start approaching the speed of sound.

A 60" prop with 2.58 redrive and 6500 engine RPM gives you 2520 prop RPM and a tip speed of 659 ft./sec. That's enough slower that tip drag will be significantly less, and you'll waste less power just making noise.

There's another factor involved, too. You're going to have to de-pitch the prop a fair amount to get the engine to rev properly. As you de-pitch, you make the change-in-speed that you give to the air smaller and smaller. This becomes a factor in higher-speed flight. With too little pitch, you'll climb well, but have no added speed available for cruising once you get to altitude.

The happy compromise with 447's and 503's on open-frame gyros occurs with a 60" prop, a 2.54 ratio and 34-36" pitch (or 8-9 degrees at the tip if you're using a protractor on the flat side of the blades). Much less pitch than that and you'll have problems getting to an appropriate cruise speed without running the engine wide open.

(It's possible to have so little prop pitch that the gyro will levitate off the ground at 10 mph, but will be unable to speed up even to its best climb speed, never mind cruise. The gyro will be stuck permanently behind the power curve. A student once brought me his 503DC-powered gyro to fly that (unbeknownst to me) had a prop set with way too little pitch. It made for one VERY exciting 20 mph pattern flight...a mile behind the power curve the whole time, and sluggish as a loaded dump truck. Once we cranked up the pitch, it flew beautifully.)