Fauns and satyrs
To begin with:
Satyrs are Greek inventions, while Fauns are Roman. Sounds like splitting hairs, but it’s important.
Saryrs were originally portrayed as “people with certain horse-like features”. Mainly the huffed feet and the horse tail. (Not to be mistaken for centaurs.)
Satyrs were connected with uncontrolled lust, but were often rejected for being overzealous. They were also associated with the God Dionysus.
Fauns were a Roman invention, also shared with neighboring city-states. They were the goat-men that pranced around, in the woods , mostly shy, and hidden, but also connected with Bacchus, the god of feasts and wine. At some point, Roman culture appropriated the satyres as their own, giving them a faun-like appearance, but definitely not the reserved introverted charachter.
The god Pan is also a satyr, and he is often shown as an example of someone with an overactive libido.
On that note I would like to recommend Claude Debussy’s:
Prélude à l’après-midi d’un faun.
And also :
The Syrinx.
Two beautiful compositions.