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.