In order to better understand the reasons for this, we should turn to Tolkien himself: in addition to being a gifted student of languages, he also had the ability to feel them. He favored Latin and Greek; he detested French; and he loved Finnish (he described the finding of a Finnish grammar book as "entering a complete wine-cellar filled with bottles of an amazing wine of a kind and flavour never tasted before", Letters, 214).
Language-making had been Tolkien's hobby for much of his conscious life. He is known to have constructed his first languages (Animalic and Nevbosh) at a little over 13; he continued to ponder upon his creations up until his death more than sixty-five years later. This work had always been tightly connected to the mythology that Tolkien was developing, since he believed that a language is incomplete without the story of the people who spoke it - exactly as these people could never be fully realistic if he chose to describe them in English only. One could see in that the reason why Tolkien took the stance of a translator, rather than the original author of his works.
Although the Elvish languages Sindarin and Quenya are the most famous and the most mature languages of all those that Tolkien invented for his mythology, they are by no means the only ones. They belong to a whole family of Elvish dialects, that originate in a single Proto-Elvish language (see Common Eldarin). In addition to that, there is a separate language family that is spoken by Men, the most prominent member of which was Westron (derived from the Númenorean speech Adûnaic) or the "Common speech" of the peoples of the Lord of the Rings. Most Mannish tongues showed influences by Elvish, as well as some Dwarvish influences. Several independent languages were drafted as well, for example the Khuzdul language of the Dwarves. Other languages are Valarin (the tongue of the Valar), and the Black Speech created by Sauron during the Second Age. The language families are also seen evolving (both because of languages' natural tendency to change and due to borrowing of features from each other).
Sindarin and Quenya are often written in the flowing and beautiful Tengwar script, which Tolkien especially devised for this puspose. Another rune-like script exists, which is known as Cirth. When, for purposes of convenience, Middle Earth languages are set with the Latin alphabet, diaereses (ä, ë, ï) mark the short vowels wherever English pronunciation rules differ (for example the last syllable of Ainulindalë is supposed to sound like "dah-lay" and not "dale"), and acute accents (á, é, í, ó ú) mark long vowels.
Finnish morphology (particularly its advanced system of inflection) gave rise to Quenya. Another of Tolkien's favorites was Welsh - and features of Welsh phonology found their way to Sindarin. Numerous words were loaned from existing languages (la for "no" in Quenya is obviously borrowed from Arabic), others were invented by Tolkien because they "sounded right" - such is the root "lint" for swift (you may find it in Galadriel's lament at the end of the Fellowship of the Ring).
Because Tolkien replaced Westron everywhere by English, the Mannish languages are the least known languages. He took this replacing even further, by also changing all languages related to Westron: Rohirric, related to an older form of Westron, was replaced by Anglo-Saxon, and the tongue of Dale in the north of Rhovanion was replaced by Old Norse, a language related to Anglo-Saxon just like Dalish was related to Rohirric.
See also the Lhammas.
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