Originally Posted by FBinNY

I doubt it's related to the smaller diameter (more flex when coming on or off), because the difference in ring size isn't enough for any difference to be noticeable.

The biggest factors affecting chain noise - not counting lubrication - are chain or sprocket wear, cut of the sprocket teeth, and alignment. This assumes we're talking about chain/sprocket noise and the chain isn't actually rubbing anywhere.

Normal chain noise - sort of a gentle whirring or buzz - happens because the sprockets aren't actually round pulleys but ace as polygons causing the chain to rise and fall slightly as the teeth go by - think of it as the chain being wrapped around a 39 or 53 sided polygon. This sets up a vibration with a pitch equal to the rpm times the number of teeth. There's also a slight shift in the speed of the chain or sprocket as high and low diameter zones go by, which also contributes.

That's the normal or minimum noise of chain drives, but differences in the cuts of the sprocket teeth, or alignment of the system can increase the noise. For example when the chain comes from an angle, the teeth will hold the chain straight, and when they dis-engage the chain will shift sideways under tension. This repeats with every tooth, and can be a more or less sudden movement depending on how the teeth are beveled to a point. For an extreme example, ride big/big for a minute and listen carefully, and you'll hear it.

Chain or sprocket wear also add to noise especially when the chain is "stretched" to more than the original 1/2" pitch, by pin wear. When new each engaging tooth perfectly meets the space between the rollers, slipping in smoothly. When the chain is worn, the roller is slightly out of position and the front of the engaging tooth bumps into it as it slides in. If you've every ridden a ferry it's akin the the difference between a pilot who enters the slip perfectly centered vs.one who bumps the guides on either side as he enters. Again, this happens with every tooth.

As to why your two rings sound different, it could be the wear condition or the original cut of the teeth. It could also be that the pitch change brings it into a better hearing range for you. There's also more tension on the chain on smaller rings, both because of the greater crank leverage, and because you tend to use the inner for climbing or headwinds.

Those are the various possibilities, but other than intellectual curiosity, you should simply ignore it as a normal characteristic of chain drives. If it's loud enough to bug you, changing chain oils might quiet the drivetrain enough that you no longer notice it either way.