The fundamental theorem of arithmetic prevents reaching the octave by perfect fifths

Evidence

Reaching C′—the octave above C—in the Pythagorean system of fifths is, profoundly, not possible. It has seismic implications for the number theory of music. C′ was already in our list to begin with as a perfect ratio of 2, before starting to build the rest of the Pythagorean system of perfect fifths. It is distressing, however, that we cannot reach the octave C′ from below by any choices of n and m in 3n/2m. Here is the proof: If there were such integers n and m, such that 3n/2m = 2, then also 3n = 2m+1. Both 2 and 3 are prime numbers, divisible only by themselves and 1. Both sides of 3n = 2m+1 are therefore factored in terms of prime numbers, and they are supposedly equal. This cannot be, since it violates the fundamental theorem of arithmetic, which says that factorization of any number in terms of prime numbers is unique. Therefore, the premise that 3n = 2m+1 is incorrect. — Eric J. Heller, Why You Hear What You Hear- An…, 2376

Categorization

• DEFINE: Invokes the fundamental theorem of arithmetic as the proof that Pythagorean fifths can never equal the octave.
• DEEPTHINKING: This is a genuinely elegant result: the very primeness of 2 and 3 mathematically forces the historical messiness of tuning. A profound connection between prime factorization and music theory.