I love to listen to period-appropriate music when I'm reading (and writing). In fact, my fascination with 13th century France began with love of early music. I have a growing library of these recordings.
Most of them are probably not authentic in detail. Instrumental arrangements and vocal style may differ greatly from performances at the time. In some cases only lyrics survived, and music was created to go with them.
The best way to understand their music is to hear it. Here are examples from contemporary performers. If you like the sample, I encourage you to buy their albums!
My favorite performer of early music is the late Owain Phyfe and the New World Renaissance Band. Some of his performances are on Youtube. This piece, Ja Nus Hons Pris, was composed by Richard I of England, the Lionheart; it was a lament about his imprisonment on his way home from the Crusades.
You'll find information about Owain Phyfe's recordings at Nightwatch Recording; his albums are available through online stores. Many of the pieces he performs are from periods later than the 13th century or locations other than France.
Other favorite performers include:
(more will be added)