I dreamt about that this morning =D. While viewing in 3rd person, an orchestra playing a familiar piece, that then morphed into just rhythms created by footsteps of people wearing different types of shoes on the wooden stage, and then balls falling down an obstacle course designed to create certain sounds as they fell. It got quite chaotic after a while, and I was saying, "Hey, when did it stop being music and started being noise?"
I wondered much about that when I woke up, and the first thing that popped into my mind was subjectivity. Yet, I hate to use that because it offers no definite conclusion. I searched around for memories tagged "noise" and "music", and especially those that were supposed to be music that were noise, and vice versa. I found out one thing.... patterns.
The difference between a pure tone vs white noise, is that the pure tone has only one frequency, i.e. a pattern for the soundwave, and that white noise is totally random with no pattern at all.
For a piece to be music, it generally has to have patterns. Not only small ones like the frequency it produces, but large ones as well. A pop piece typically has a verse, chorus, verse, maybe bridge, chorus, etc. Each verse has patterns too since each phrase may follow the next with the same notes or, in music, we call motifs. The phrasing is also rather even, like 4 bars + 4 bars, one question, another answer, and all these are patterns.
If a piece has no tones, which what I would consider, the most basic pattern, then I think you could quite safely say it won't be music. Even drums have their tones; it may not be pure, but there are still tones. Then like some of John Cage's pieces, they have tones, but no order, no overall pattern, and seems more random than anything else, I wouldn't consider it to be music either.
It seems like it's actually possible to form an objective way to judge whether a piece is music or not. But like I said, since this has subjectivity in it, the objectivity part of it would not be 100% correct. Nevertheless, it should be somewhat close. (higher than 50% would be good enough for me)