After you complete the Face Shrine dungeon, you get a hint to go to the mountain to find the next Siren's Instrument.
Learn Frog's Song of Soul
Go to the Signpost Maze. You can find it by going one screen west of the Key Cavern entrance, then using the Pegasus Boots and Roc's Feather to jump over the gap. Then read the sign across the gap and follow the instructions. Just go directly south from that sign to find the next sign. Keep following the signs. Each sign points to a sign that is in the same line vertically or horizontally, so don't get confused and check on a sign that is not aligned with the previous one.
After you have correctly completed the maze, stairs appear. Go down.
Go up to the big frog. Agree to pay 300 to hear their unreleased song. You will learn how to play the Frog's Song of Soul on the Ocarina.
Get the Flying Rooster
Go back to Mabe Village and go to the Weathervane north of Marin and Tarin's house. Push on it from below while you have the Power Bracelet equipped. The Weathervane will move, revealing stairs. Go down.
Follow the path and you will find the remains of the Flying Rooster. Play Frog's Song of Soul. The rooster will come back to life and start following you. Go outside.
Get the Bird Key
Now you need to go to the mountains. You can get there quickly by using the warp spot east of Mabe Village. Go west from the warp spot in the Tal Tal Mountains until you find stairs leading up the mountain. Go up.
The owl will stop and talk to you. Lift a rock and go into the cave. Go through the path. Use the Pegasus Boots to dash through the crystals and go to the eastern room. Go out the door.
Go east until you find some water that you can swim in. Go up the stairs that go up from that water. Go west, then go into the cave just before the stairs up to the house.
In the cave, there is a rock puzzle. Push one of the middle rocks up into the hole, then push the one left of you into the hole. Push the one above you up, then push the left one left, then the one above you up, then the left one left. Then you can use the hookshot to go to the north exit (or just pick up the rooster and fly there).
In the next room, pick up the rooster using the Power Bracelet, then float to the west exit. Go through.
Jump over the gap and get the Bird Key from the ledge.
Reach the Next Dungeon
Go back the way you came, using the Flying Rooster to reach the south exit of the next room. Then use the east exit in the room with the rock puzzle.
Outside, go east and then down the stairs into the water. Swim east and go into the cave. Go through the cave and continue going east when you get outside. In the area with a bird statue, go up the stairs and into the cave. Go through the cave. When you get outside, go west and into the next cave. Go through it and go down the stairs. Jump over the hole, then go up the stairs and go east. Use the hookshot to go over the gap, then jump down from the ledge to reach the door to the outside.
Bomb the cracked wall to find a Fairy Fountain. Then go into the northeast cave. Follow the path through it and you will be on top of the mountain. Go west and lift the rocks, then push on the keyhole to use the Bird Key. Then you can go into Eagle's Tower.