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:

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.