This root and ones like it were used for “high” things for much of Tolkien’s life. It first appeared as unglossed ᴱ√TAHA in the Qenya Lexicon of the 1910s with derivatives like ᴱQ. tā “high; high above, high up”, ᴱQ. tahōra or tayóra “lofty”, and ᴱQ. tāri “queen”; it had a variant form ᴱ√TAʕA where the ʕ might be a malformed Y (QL/87). The corresponding forms in the contemporaneous Gnomish Lexicon were G. dâ “high” and G. dara “lofty” (GL/29), indicating the true form of the root was ᴱ√DAHA, since initial voiced stops were unvoiced (d- > t-) in Early Qenya (PE12/17). Primitive forms like ᴱ✶dagá > ᴱN. dâ/ᴱQ. tá “high” in Early Noldorin Word-lists of the 1920s indicates the root continued to begin with D for the following decade (PE13/141, 161).
In The Etymologies of the 1930s Tolkien gave this root as ᴹ√TĀ/TAƷ “high, lofty; noble” with derivatives like ᴹQ. tára “lofty, high”, ᴹQ. tári “queen” and N. taen “height, summit of high mountain” (Ety/TĀ). In Definitive Linguistic Notes (DLN) from 1959 Tolkien gave the root as √TAG or Tā- “high”, and in notes from around 1967 Tolkien gave √TAƷ as the explanation of the initial element of Q. Taniquetil and contrasted it with √TĂR “stand” (PE17/186). In 1970 green-ink revisions to the Outline of Phonology (OP2), Tolkien wrote a marginal note giving √TAƷ > TĀ “high”, but this note was rejected with a statement “transfer to Gen. Structure. No [ʒ] existed in Eldarin” (PE19/72-73 note #22).
This last rejection seems to be part of Tolkien’s general vacillation on the nature and phonetic evolution of velar spirants in Primitive Elvish in 1968-70. For purposes of Neo-Eldarin, I would assume the root form was √TAH or √TAƷ > √TĀ as the basis for “high” words, much like √MAH or √MAƷ > ✶mā was the basis for “hand” words.
This root meant “fix, fasten” for much of Tolkien’s life. The earliest appearance of this root was as ᴱ√TAKA “fix, fasten, stick” in the Qenya Lexicon of the 1910s, with derivatives in that document as well as in the contemporaneous Gnomish Lexicon like ᴱQ. taka-/G. tag- “fix, make firm” and ᴱQ. tanka/G. tanc “firm, steady” (QL/88; GL/68). It also had the extended root ᴱ√TAQA “fashion” = TAK + W̯Ǝ with derivative like ᴱQ. tāqa/G. tôb(a) “shape” (QL/89; GL/71). The root ᴱ√tak “stick, [stick] in, fix, firm” was mentioned in the Early Qenya Phonology of the 1920s with numerous etymological variants like ᴱ√stak, ᴱ√taku̯, and ᴱ√tu̯ak (PE14/66).
In The Etymologies of the 1930s Tolkien gave the root ᴹ√TAK “fix, make fast” with derivatives like ᴹQ. tak-/N. taetha- “fasten”, ᴹQ. tanka/N. tanc “firm” and ᴹQ. tankil/N. tachol “pin, brooch” (Ety/TAK). The root also appeared in the first version of the Tengwesta Qenderinwa (TQ1) from the 1930s with the glosses “fix” (PE18/50) and “make” (PE18/54). √TAK reappeared in the second version of Tengwesta Qenderinwa (TQ2) from around 1950 with the gloss “fasten, fix” (PE18/100), and again (unglossed) in the Outline of Phonology (OP2) from the early 1950s as part of the derivation of ✶tankla > Q. tañkal “brooch” (PE19/83). Derivatives of this root appeared regularly in Tolkien’s later writings, making this a particularly stable root in Tolkien’s conception of Elvish.