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.