Adding Tempo Page 4 from the Prime Listening Guide
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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 notes--the 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 pn is a constant times (pn+1-pn)/log pn).

 Click the button on the right to hear the first 300 primes modulo 41 in this manner. (Played on "Pad 6 (metallic).") Here are the first 300 primes after 1,000,000 played in the same way (modulo 41, with note lengths proportional to the gaps, played on acoustic gand).

Again we pause for a question break, so that you can take your mind out for a walk!

### Questions

1. * The prime 2 is followed by a gap of zero composites. Are there any other primes followed by a gap of length zero?

yes, no.

2. ** The primes 3, 5, 11, 17, and 29 are each followed by a gap of length one. Are there infinitely many primes p followed by a gap of length one?

yes, no, probably yes.

3. *** Using this method to play all of the primes, can the notes get arbitrarily short? (That is, given any e>0, are there notes with length less than e?)

yes, no. probably yes.

4. *** Using this method to play all of the primes, can the notes get arbitrarily long? (That is, for every real number b, are there notes whose lengths are greater than b?)

yes, no, probably yes.

 Another prime page by Chris K. Caldwell