

So far we have taken advantage of the pitch of our notes, but not their length. We played all of our primes as quarter notes. Look for a minute at the beginning of the sequence of positive integers. Here I have made the primes red, the composites black, and the unit 1 blue: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 ... We see each prime is followed by a different length "gap" of composites. 2 is followed by none, 3 by one, 5 by one, 7 by three... We will use these gaps to assign lengths to the prime notesthe bigger the gap, the longer the note. It turns out that for the primes up to n, the average gap length is log n (natural logarithm). (This is an immediate consequence of the prime number theorem.) So let's play the prime p with a length proportional to the length of the gap after p plus one, divided by the log of p (that is, the length of the note assigned to the nth prime p_{n} is a constant times (p_{n+1}p_{n})/log p_{n}).
Again we pause for a question break, so that you can take your mind out for a walk! Questions 
Another prime page by Chris K. Caldwell <caldwell@utm.edu> 