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Are the notes played here essentially random?
I was watching this video and at the linked time I was wondering if there was any technique being used because it looks almost random but sounds quite good. The only thing I could see is he was using only the white keys.
No, it isn't random, he's playing along with the bass line, which has a repeating pattern/progression. Using only the white keys just means that the music he is playing is in the key of C. Free-form playing of this style is known as "improvisation," it's done on the spot and looks and sounds very impressive but it takes years of practice to be able to play this way in any key. His improvisation is probably what gives you the feeling that it is somehow "random."
Edit: the video says it's in the key of D, but with no black notes, he may be mapping everything to white notes or something, not sure what's up with that. Might make it easier to play since the keys are so stupid tiny.
Not random - music can be improvised. In fact, most jazz solos are improvised.
For example, listen to Sonny Rollins' St. Thomas - from 55 seconds onwards, he (and the other members of the band) are just improvising, picking notes which work well with what each other are doing.
There are even some styles of jazz where almost the entire thing is improvised, with only a loose structure keeping it all together. For example, listen to the Preservation Hall Jazz Band playing New Orleans-style jazz. In the beginning, the trumpet plays the melody, but everyone else is just making shit up to fit in with that melody (and the chords the piano/bass are laying down). There's not even anyone who wrote down the pattern of the drums, or what specific notes the piano/bass should play... there are standards of the music, but the players are free to move around in them as they will.
Not sure I understand the question, doesn't seem particularly random to me. Anything in particular that makes it look random to you?
Honestly that's probably a better question for the author of the video. The term you're looking for is "Avante garde". Other good examples of this can be found in a lot of David Bowie's music.
This ted talk about writing a repetition free piece of music might be interesting to you.