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Silvan Elvish was a name for the tongue spoken by the Silvan Elves.
Silvan Elvish descended from Nandorin, the language of the Nandor.
After the First Age, the surviving Elves of Beleriand carried their speech back into the east. Sindarin-speakers settled in both the main Silvan realms, and in Lórien, at least, Sindarin quickly overtook Silvan Elvish in common use. The history of the tongue in Mirkwood is more difficult to determine - some sources suggest that it, too, was quickly overtaken by Sindarin, but other sources dispute this.