From: gwsmith@gwi.net (Gene Ward Smith)
Newsgroups: sci.math
Subject: Re: number theoretic (or statistical?) basis of music theory and harmony
Date: 8 May 1998 16:59:27 GMT
: 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.