This root was conceptually intermingled with √LIR “sing”, both of which had to do with music. The earliest iteration of this root was ᴱ√LINI “gentle” from the Qenya Lexicon of the 1910s with derivatives like ᴱQ. linda “gentle, kind; soft” and ᴱQ. linta- “soothe” (QL/54). According to Tolkien this early root was confused with ᴱ√LIŘI “sing” (PME/54), which itself was the earliest precursor to √LIR; this early root √LIŘI [LIÐI] had derivatives like ᴱQ. liri- “to sing” and ᴱQ. lindele “song, music” (QL/54). The picture in the contemporaneous Gnomish Lexicon is more muddled, with words like G. lin- “sound” (as well as lintha- “ring bell, play an instrument”) and G. lir- “sing” hinting at two distinct musical roots ✱ᴱ√LINI and ✱ᴱ√LIRI.
Indeed, in The Etymologies of the 1930s Tolkien reorganized the two roots into ᴹ√LIN “sing” and ᴹ√LIR “sing, trill”, the former taking on music words beginning with lind- and the latter musical words beginning with lir- (Ety/LIN², LIR¹). Both these had strengthened forms ᴹ√GLIN and ᴹ√GLIR used in Noldorin words like N. glinn “song, poem, lay” and N. glaer “long lay, narrative poem”, but entry for the root ᴹ√GLIN was struck through and its Noldorin words adapted to unstrengthened ᴹ√LIN, as in N. lhinn “air, tune” (Ety/GLIN, GLIR). The Etymologies also had another strengthened root ᴹ√LINDĀ “fair (especially of voice)”, with a line indicating it was derived from ᴹ√LIN (Ety/LIND; EtyAC/LIND); this strengthened root in turn was blended with ᴹ√SLIN, unglossed but apparently meaning something like “✱fine, delicate” (Ety/SLIN).
Both root √LIN “sing” (PE17/27, UT/253) and √LIR “sing, warble” (PE17/27, 67) continued to appear in Tolkien’s later writings, along with derivatives like Q. lindalë “music” and Q. lírë “song”. Tolkien discussed the root √LIN at length in the Quendi and Eldar essay from 1959-60, where he said:
> The name ✱Lindā is therefore clearly a derivative of the primitive stem ✱LIN (showing reinforcement of the medial N and adjectival -ā). This stem was possibly one of the contributions of the Nelyar [Teleri] to Primitive Quendian, for it reflects their predilections and associations, and produces more derivatives in Lindarin [Telerin] tongues than in others. Its primary reference was to melodious or pleasing sound, but it also refers (especially in Lindarin) to water, the motions of which were always by the Lindar associated with vocal (Elvish) sound. The reinforcements, either medial lind- or initial glin-, glind-, were however almost solely used of musical, especially vocal, sounds produced with intent to please (WJ/382).
Tolkien’s statement that it “also refers (especially in Lindarin) to water” is probably an allusion to √LIN “pool, mere, lake” (Ety/LIN¹; PE17/160). In a footnote in Quendi and Eldar essay Tolkien added: “Though this clan-name [S. Glinnel] has ✱glind- in Sindarin, the g- does not appear in Amanya Telerin, nor in Nandorin, so that in this case it may be an addition in Sindarin, which favoured and much increased initial groups of this kind” (WJ/411 note #13). Despite this statement, Sindarin had several derivatives from the base root √LIN(D)-, such as S. linna- “sing, chant” (LotR/238; RGEO/64; PE17/27). The sense “gentle” from the 1910s root ᴱ√LINI also seems to have survived in Tolkien’s later writings, since the adjective Q. linda “soft, gentle, light” appears in notes associated with the 1955 version of the poem Nieninquë (PE16/96).
Neo-Eldarin: For purposes of Neo-Eldarin, I think it is best to assume √LIN referred to melodious sounds, as well as pools of water (√LIN) by way of the pleasant sounds that water makes, and gentleness (Q. linda) by way of the affect such sounds have on one’s mood. However, I think √LIR more directly referred to vocal music (song) and other rhythmic vocal sounds (chanting) such as poetry, as in Q. [ᴹQ] laire/S. glaer “poem”.
A root for “pool, mere, lake” appearing in etymological notes from 1957 (PE17/145, 160), and also appearing as ᴹ√LIN “pool” in The Etymologies of the 1930s (Ety/LIN¹). In both instances it was the second element in Q. ailin “(large) lake”, and so connected to S. ael “lake” (N. oel) as suggested by Christopher Tolkien (SA/lin¹). In the 1957 notes Tolkien said the root √LIN had a “Sindarin differentiation > glin-” (PE17/160), but I can find no indication of this in any attested words.
In The Etymologies of the 1930s it was connected to ᴹ√LINKWI with derivatives ᴹQ. linqe/N. lhimp “wet” and N. lhimmid “moisten” (Ety/LINKWI; EtyAC/LINKWI). This is turn was probably a later iteration of the early root ᴱ√LIQI “flow, water; clear, transparent” with derivatives like ᴱQ. linqe “water”, ᴱQ. liqin(a) “wet” and ᴱQ. liqis(tea) “transparence (transparent)” (QL/54). For purposes of Neo-Eldarin, I think it is best to assume the “wetness” senses were transferred to ᴹ√LINKWI, but I think it is worth positing a Neo-Eldarin root ✱ᴺ√LIKWIS “clear, transparent” to preserve words associated with tranparency.
This root may be associated with √LIN “make a musical sound”; see that entry for details.