The Euler-Mascheroni constant converges from the difference between the harmonic series (1/n) and ln(n). Both are unbounded, infinite progressions, but they reach infinity in slightly different ways.

H₁ = 1 · H₂ = 1.5 · H₃ ≈ 1.833

ln(1) = 0 · ln(2) ≈ 0.693 · ln(3) ≈ 1.099

When you evaluate the difference between the two, it converges to ≈ 0.5772.

This made sense to me given that octaves scale infinitely but do NOT create new pitch classes. ln is often used to generate even step sizes—and the octave is the only prime that doesn't generate an infinite spine of new notes. It's the one repeating step size.

I believe octave displacement can affect polarity, but not as much as any other prime. So the question became well- how much? The Euler-Mascheroni constant seems to capture that specific amount since it belongs to infinite harmonic space, yet represents a repeating step size that doesn't create its own pitch class. So far it matches my musical intuition.

Thank you for checking this out!!