There are multiple regal rings but if your talking about the one for Linebeck to fix the bridge: Play the song of light next to the beacon then look at the direction it is shining swing your boomerang in that direction then procceed then go to top left corner of all the square tiles and play the song of discovery there you will find... A BLUE RUPPEE oh wait i mean THE REGAL RING
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Well they're mostly spread all over the game, but if you're talking about the one you need to pay for fixing the bridge, go through the door next to the trading post, defeat the monsters in the cave (make sure they don't eat you, or your shield will get eaten too), then, go through the back of the cave, use your fan (or whirlwind) to blow away all the piles of leaves in the area (they're orange, pretty easy to find), now go to the stone rocket, tap it, you'll see little colorful lights come out, play those colors in same order on your spirit Flute. That is the song of light. Make sure you play it close to the rocket, or it won't work. Once you've played the song, the rocket will fly away and a big crystal will come out the ground. It will create a straight blue light pointing to a purple orb switch. Switch to your boomerang and hit the orb (make sure to be close to the river, but not too close or you'll fall in and lose a life) so you can hit it. Once you've hit it, a bridge will open. Go across the bridge and you'll see a cave opening. Go in. When you're in, you'll find two stone crystals, play the song of light near the crystals. When they are activated, their lights will come together. Just to save some time, you can find the ring at the top-left corner of the screen. Sorry if this answer was kinda long.
Hope I helped!
im not exactly sure but im sure that you have to follow some istructions or someting then you play the song of discovery in a certain place, i think.
wrong first you go through the cave at the west side of the map and pass the shield eating monsters. go up the steps and you will come out at the north part of the map. go to the right and stand on the edge. grab your boomarang and make a path east where you will see a switch hit it and a bridge will extend go over it and onpen the chest.
in the phantom hourglass it may or may not be in chests containing treasures and the same for spirit tracks
If you're talking about the one Linebeck tells you to get, go into the cave, go to the top-left corner of the stone path, and play the Song of Discovery.