Tuesday, July 14, 2009

Music state space exploration

Obviously the state space of music is extremely large and humans will never really thoroughly explore it. At least no one particular human will since listening to all possible music sounds like one of those endeavors which would take a tad longer than the age of the universe. Of course in order to make the possible musical selections finite you need to simultaneously discretize the signal and limit its duration.

Even if we confine ourselves to sampling at a rate of say that of cd audio which is 44.1 khz or 44,100 samples a second and consider only a single minute of music that makes us consider a vector space of 2,646,000 dimensions. so even if we allow only say 100 different intensities at each time step that allows for 100^2646000 different sound bytes.

Every 1 minute sound clip (with the discretized intensity restriction kept in mind) is some point in this vector space. Now although it would be crazy to think that any human being might really thoroughly explore this space (as in listen to most of or even a tiny fraction of all possible sound clips in the space) we can still ask how well we have explored this space.

Obviously if you look at say only gregorian chant you are exploring a smaller region of the music state space than if you include also soft rock and heavy metal classical music etc.

So I propose a project that I probably will never do but intrigues me nonetheless. Why not use as a measure of the level of exploration of the music state space the convex hull of pieces of music. So we say pick a representative sample of rock music and we take the convex hull of these points in our music space and take the ratio of the volume of the convex hull to the total volume of the space as a measure of the level of exploration.

But say we took as "music" the basis vectors of the space. Then the convex hull of these "music" points would be the simplex for that dimension and while that might actually have a relatively low volume for the space as a whole it is still a volume which we can't really realistically expect our music to much out achieve and obviously the vector basis (namely a vector of one 1 and all zeroes else) is not something that really explores the music state space. So what we really want probably is to do something like take a spectrograph of the music and do our state space analysis with that.

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