Riemann Zeta Function and Tuning Systems...

Hello,

Would like to stimulate further discussion on this posting from the
sci.math newsgroup. I am especially interested in the part that talks
about "streching or shrinking the octave to the nearest Gram point".
What does this mean exactly? I have also included, at the bottom, a
short explanation that I received from Gene a couple days ago.

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 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.

(Gene wrote to me, recently, this clarification)

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.

--- In tuning-math@y..., "paulhjelmstad" <paul.hjelmstad@u...> wrote:
> Hello,
>
> Would like to stimulate further discussion on this posting from the
> sci.math newsgroup. I am especially interested in the part that
talks
> about "streching or shrinking the octave to the nearest Gram
point".
> What does this mean exactly? I have also included, at the bottom, a
> short explanation that I received from Gene a couple days ago.

Hi, Paul; I'm some minor computer problems so I'm afraid I'm a little
tardy in my reply. Probably the best thing to do would be to take a
look at my tuning-math postings on this first, and then go from there.
The Gram point business arises because the Gram points are easily
computed and close to the critical values of Z(t) in question.

Here is something to start the discussion off with:

/tuning-math/message/879

/tuning-math/message/894

/tuning-math/message/946

As you can see, there is much, much more going on on this list than
the Riemann Zeta function discussion, which I commend to your
attention if you are interested in the musical aspect of all this.