Kjinnovatn wrote:

Here's a little puzzler that I've been wondering about: If you investigate the number-theoretic basis of music theory, it all hinges on the fact that certain simple fractional powers of 2 "accidentally" happen to be very close to simple fractions. I noticed a quarter century ago (but never published) that this Diophantine approximation problem is closely connected to the Riemann Zeta function, in that good values correspond to high values along lines whose real part is fixed. This relationship extends into the critical strip, and along the line Re(z) = 1/2, which allows some amusing formulas to come into play. One can distinguish different microtonal systems by the argument of zeta, and adjust them by slightly stretching or shrinking the octave to the nearest Gram point, with an eye to slight improvements of the approximations involved on average, in some sense of average. You may now double your fun by bringing in group theory, and noting that a microtonal system is also closely related to homomorphisms from finitely generated subgroups of the group of positive rational numbers under multiplication to the free group of rank one. The kernels of these homomorphims determine the relations between such systems - a system with 81/80 in the kernel behaves very differently in terms of harmonic theory than one without 81/80 in the kernel.