A lyre is crafted by a player during The Fremennik Trials quest. The lyre is needed in the quest to perform in the longhall to get Olaf the Bard's vote to become a Fremennik.
The lyre can then be enchanted by Fossegrimen to make an enchanted lyre and progress with the quest. While equipped, the colour of the lyre's middle section is the same as the player's shoe colour.
The lyre can be made by using an axe to cut a branch from the swaying tree east of Rellekka, and then using a knife on the branch to obtain an unstrung lyre, then attaching golden wool. Doing so requires 25 Fletching.
Golden wool can be obtained from buying golden fleece from Lalli (1,000gp), then spinning it on a spinning wheel (nearest spinning wheel is in Seers' Village, or Rellekka after completing The Fremennik Trials).
Alternatively, players can kill Lanzig, Borrokar, Lensa or Freidir, who can drop the lyre. Lanzig, who is commonly killed by players for the lyre, respawns in approximately 18 seconds and can be killed quicker by world hopping. For mage and range users, you can use the fire in the middle of the room (as a blockade between you and Lanzig) to safespot. You can also safespot Lanzig by opening the door east, attacking Lanzig, then going two steps east of the door to kill him from a distance.
For an exhaustive list of all known sources for this item, see here