This root carries two threads that share three letters. The big one is music: the song, the singer, the nightingale, the psalm, the orchestra. The quieter one is pruning a vine, an old agricultural sense from the same consonants. Treat them as two neighbours under one roof, with the musical family doing almost all the work in modern Hebrew. Every word here is verified against pealim, niqqud included.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
The two binyanim cleanly split the two senses: pa'al לִזְמֹר keeps the old "prune the vine" meaning, while pi'el לְזַמֵּר is the living verb "to sing". The other binyanim are not lexicalized; the root expresses itself mostly through its dense field of nouns.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.