I'm an adult who just saw this for the first time. (I must be from Mars.) As a Kodaly trained musician, this musical part of the film felt both personal and expressive. It made me happy to see Kodaly mentioned and to hear sol fa tones used as a means of communication, to communicate the notes with the hand signals, and to refer to the melody in terms of the intervals between the notes. (The actor didn't do his signals exactly correctly, nor were they coordinated well with the pitches, but I could hardly make a fuss as it was such a lovely touch anyway. ) What's written above about Curwen is correct. I use the hand signals to teach rhythm and melody with my students every day. I hear every piece of music in these terms. Just as sign language can be extremely expressive of emotion and nuance, even more so than verbally spoken language, this film showed that in some cases non verbal communication reaches across the gulf better than all the pretty words in the world.
Indeed, this film is largely a metaphor about the quality of communication and what that can yield for distinct entities and beings. Between the aliens and the humans, the ability to communicate was established by the initial series of pentatonic tones. Communication became more and more friendly and enthusiastic after the basic "Re Mi Doh (Doh) Sol" tune got established and shared, and then the conversation grew much more like a conversation between friends; richer sounds, nuanced, rapid back and forth parley, expressive, animated, colorful.
But as between humans? We got a bad grade for effective communication! In the first scene the wind and flying sand precluded both speaking and hearing, leading to degraded communication. Talking past each other in Mexico, in Indiana, in Siberia, in India, in the flight control room, in the government control room, by a government that took extreme steps to block all communication with the public. In the Dreyfuss' character's dysfunctional family there was absolutely no communication! no listening, no understanding, no guidance from the parents, no thoughts being passed, no empathy. Only those who'd experienced an "encounter" were communicative with each other, the people from the past were able to communicate with the present, and people felt passionate to tell the world about their experience.