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The Entropy of Music: How Many Possible Pieces of Music Are There?

notes on a page of staff paper, any more than one can count possible

English sentences by merely enumerating strings of letters or words:

this combinatorial exercise vastly overestimates the actual count

because almost all the resulting scores or texts are utter nonsense.

The patterns that humans need to remember and make sense of music impose

many constraints; beginning students often have a hard time finding

even one tolerable way to satisfy them, and some masterpieces seem

so intricately put together that it feels like one can account for

every single note, as though the composer had no choice at all.

Yet we have many thousands of good pieces of music, and many more

must be possible, if only because many are known to have been lost.

The question, then, is how much the criteria of psychological and

structural coherence actually constrain the space of viable pieces

of music. Mathematics and science (specifically statistical mechanics,

information theory, and signal processing) give a precise language for

quantifying this "how much''. Such language may indeed seem

too precise for music, but we nevertheless show how a composition that

began as a mnemonic for the digits of π also suggests a lower bound on

the information content of music.