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North Sindarin is an extinct dialect of Sindarin.
The Sindar of Beleriand were divided in several groups, and their language had developed some dialects. North Sindarin, the flavour of Sindarin spoken by the Mithrim, the northernmost group of the Sindar, differed from the Sindarin of Beleriand proper in many aspects. It was this language which was adopted by the exiled Noldor after their return to Middle-earth, and by their mortal allies. During this time North Sindarin was changed much, partially due to the adoption of Quenya features, and partially due to the love of the Noldor for making linguistic changes. Beren's heritage was clear to Thingol of Doriath as he spoke the North Sindarin of his homeland.
North Sindarin retained many features of Archaic Sindarin which had been lost in the Sindarin of Beleriand proper, but also went through several changes of its own: lenition occurred far less in this dialect than in the other dialects.
After the end of the First Age, the survivors of Beleriand's realms generally adopted the more southern variants of Sindarin, but several proper names which are uninterpretable in normal Sindarin which remained in use during the Third Age show North Sindarin influence.
Other versions of the legendarium
In Tolkien's earliest writings, there was a language called Ilkorin, the language of the Ilkorindi (the Dark Elves who stayed in Great Lands). After Tolkien evolved his mythology, the background and setting of the stories changed, and Ilkorin was replaced by Sindarin, the language of the Sindar.
Edward Kloczko observed (Tyalië Tyelelliéva #9, October 1996)that the few words mentioned as "North Sindarin" fit to the phonological changes and principles seen in Ilkorin and he theorized that Tolkien recycled parts of Ilkorin as North Sindarin.
However Sindarin and North Sindarin descented from Common Telerin, where the primitive sound kw was simplified in p (cf. Quenya quàr and Telerin pár and Sindarin paur).
This setting is part of the later conception and Ilkorin does not reflect this: primitive kw stays cw or c in some Ilkorin words such as alch (ALÁKwÂ), côm (KwAM), cwess (KwESS), salch (SALÁKwÊ) something that would be impossible in a Sindarin dialect.