This root is attested only in The Etymologies of the 1930s (Ety/ÑGAN), where it was glossed “play (on stringed instrument)” and was the basis of various “harp” words, mostly from its extended form √ÑGANAD. The root ᴱ√KᴬNTᴬN from the Qenya Lexicon of the 1910s is probably the precursor of ᴹ√ÑGAN(AD), given the shift the ᴱQ. vorokantele “monotonous repetition” in the Qenya Lexicon (QL/45) to ᴹQ. vorongandele “continual repetition (harping on one tune)” in The Etymologies (Ety/ÑGAN).
The later Quenya words Q. tanta- “to harp” and tant(il)a “harp” (VT41/10) might indicate its replacement by another root (?√TANAT or ?√KYANAT). ᴱQ. kantl “large harp” (< ᴱ√KᴬNTᴬN) might have developed conceptually into the later Q. tant(il)a “harp”, which might be an indirect derivative of an unattested root ✱√KYANAT (<< ᴱ√KᴬNTᴬN) via Telerin in the same way that Q. telpë “silver” developed from √KYELEP under influence of T. telpe. We have no cognates of Q. tant(il)a in other languages, so there is no clear way to determine the actual later root.
Neo-Eldarin: For Neo-Eldarin purposes, I would suggest using ᴹ√ÑGAN(AD) and its larger set of derivatives over hypothetical √TANAT or √KYANAT, which has only Quenya derivatives. For example, nande was used in NQNT (Neo-Quenya New Testament) over tant(il)a.
A root in The Etymologies of the 1930s glossed “gate, entrance” (Ety/AD) that seems to have replaced ᴱ√BOÐO (QL/75) from the Qenya Lexicon of the 1910s. Its two main derivatives are ᴹQ. ando and N. annon “(great) gate, door”. These words in turn were transferred in 1959 to the root √ANA “to, towards”, from the original “approach, entrance” (PE17/40). Thus it seems Tolkien abandoned this root, perhaps because he wished to use it instead for √AD(A) “alongside, by the side of”, one of the etymologies Tolkien considered for the Eldarin words for “and” (PE17/71).