Tuning and the Riemann Zeta Function

This is a posting by Gene Ward Smith that appeared on the sci.math
newsgroup in May 1998 regarding tuning systems and the Riemann Zeta
Function. I have also included a clarification at the bottom from
Gene that I received today. Posting this to elicit further discussion
on Gram Points, The RZF, the critical strip, and everything, related
to "good" tuning systems. I am admittedly confused by the fact that
high values exist for Re(z)>1, could this actually be Re(z)<1? Seeing
as the critical strip is 0<Re(z)<1 and this phenomenon extends "into
the critical strip." Any further commentary, Gene, or anyone? Thanks

: 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 "accidentally"
happen to be very close to simple fractions.

(Gene writes)

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 homomorphisms 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.

(Recent commentary)

The idea, very briefly, is that when Re(z)>=1 the absolute value of
the Riemann Zeta function will be high when z=s+it is near a scale
division where t = 2 pi n /ln(2), n, being the scale division; this
is because the Diophantine approximation problem for finding a good
division and finding a high value of |Zeta(z)| are essentially the
same. It turns out the relationship extrapolates into the critical
strip.