idea
Primes and commas
Every whole number can be written as a product of prime numbers in only one way: 12 is 2 × 2 × 3 and nothing else. This simple fact has a musical consequence. Octaves are built from the number 2 and fifths from the number 3. However many fifths you stack, you get powers of 3, and these can never equal a power of 2. So a chain of pure fifths never lands exactly on an octave, and every tuning has a small leftover, which musicians call a comma.