One root for the whole world of instrumental music. To play an instrument, the player who does it, the melody that results, and the very act of playing all live in נ-ג-ן. Notice there is no plain pa'al here: the action runs entirely through the intensive pi'el לְנַגֵּן, which fits a deliberate, sustained activity like making music. 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.
This root skips the plain pa'al entirely: "to play" is the intensive pi'el לְנַגֵּן, because making music is a deliberate, repeated act. Its reflexive partner hitpa'el לְהִתְנַגֵּן means the music "plays itself", what comes out of the speakers. The remaining binyanim are not lexicalized for this root.
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.