According to elvish dot org, artistic languages can be copyrighted as they are created for artistic reasons. One could reasonably argue that Klingon was created for artistic reasons (i.e. movies, TV shows and books) and is thus copyrighted.
Since they’re “just” languages, aren’t Quenya and Sindarin in the public domain?
No. Unlike Esperanto (which was explicitly placed into the public domain by its creator), Tolkien’s languages are not in the public domain. As the artistic creations of J.R.R. Tolkien, they enjoy the same copyright protections as his literary works, including but not limited to restrictions on the amount of quotation, purpose of use, and creation of derivative works.
Klingon can provide a helpful example illustrating the point from a different but precisely parallel perspective. The creator of Klingon, Marc Okrand, has created a substantial vocabulary for his language. Not all of this vocabulary has been collected and published in a single compendium; various words and their definitions have been published in, for example, various books, in the Star Trek television shows and movies, and in the journal HolQeD. But no one would seriously suppose that it would be Fair Use to compile a complete lexicon of Klingon from these disparate sources and publish it, even for free and for purely scholarly purposes, without the permission of Marc Okrand and/or Paramount Enterprises.
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The suit against Axanar productions includes more than just using the Klingon language. So getting a pass on use of the language does not make the suit go away.
According to comments at Popehat by a member of the Language Creation Society (who are apparently supporting Axanar in their response to the suit) -
Along the same lines, can you copyright a programming language? Do any of the current languages (Python, C#, or JavaScript for example) have a copyright? I can see the language syntax diagrams and web pages of example code being copyrighted, but not the language itself. At best you can have a standards committee that would prevent you from adding features willy-nilly, but can they actually prevent you from using the language itself?
Programming languages themselves are not copyrightable. A given implementation of a programming language, like a compiler or interpreter, is copyrightable. The specification (if there is one) is also copyrightable. Standard libraries distributed with your implementation are also copyrightable. Whether the interfaces to those libraries are also copyrightable is a much thornier question.