Stormer's theorem

I got Haluska's book from the library, and already have learned one
thing of great interest. I gave a proof here that for any p-limit,
only a finite number m/n exist with m-n below a fixed bound. This, it
turns out, is a theorem of Stormer dating to 1898, and is
*constructive*. Rather than use Baker's theorem, which was unknown
back then, Stormer showed how to constuct the entire list using Pell's
equation. Lehmer has computed these up to the 41-limit; I have a huge
box of old reprints of Lehmer's I got after he died, and I wonder if
this was in it. Anyway, Lehmer's paper is "On a problem of Stormer",
Illinois Journal of Mathematics 8(1964), 57-79, and I plan to dig it
out if I can.

In the preface we find:

The author thanks composers of contemporary music and computer music
theorists for their information-sharing and feedback. My thanks go to
D. Benson, D. Canright, P. Erlich, K. Grady, M. Haluska, Y.
Hellegouarch, B. Hero, D. Keenan, V. A. Lefebvre, H. L. Mittdendorf,
J. L. Monzo, G. Morrison, G. Mazzola, E. Neuwirth, M. Op de Coul, R.
Ruzicka, M. Schulter, J. Starrett, Ch. Stoddard, and W.Sethares.

This should be communicated to John Chalmers post haste; in the most
recent (6 years ago) issue of Xenharmonikon, he wrote an article
called _The number of 23-Prime-Limit Superparticular Ratios Less Than
10,000,000_ -- since that's as high as his system would go. I'm sure
he would be most interested to know that at least since 1964, the
answer was in some sense known with the 10,000,000 relaxed to
infinity.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I got Haluska's book from the library, and already have learned one
> thing of great interest. I gave a proof here that for any p-limit,
> only a finite number m/n exist with m-n below a fixed bound. This,
it
> turns out, is a theorem of Stormer dating to 1898, and is
> *constructive*. Rather than use Baker's theorem, which was unknown
> back then, Stormer showed how to constuct the entire list using
Pell's
> equation. Lehmer has computed these up to the 41-limit;

For what fixed bound for m-n?

> I have a huge
> box of old reprints of Lehmer's I got after he died, and I wonder if
> this was in it. Anyway, Lehmer's paper is "On a problem of Stormer",
> Illinois Journal of Mathematics 8(1964), 57-79, and I plan to dig it
> out if I can.
>
> In the preface we find:
>
> The author thanks composers of contemporary music and computer music
> theorists for their information-sharing and feedback. My thanks go
to
> D. Benson, D. Canright, P. Erlich, K. Grady, M. Haluska, Y.
> Hellegouarch, B. Hero, D. Keenan, V. A. Lefebvre, H. L. Mittdendorf,
> J. L. Monzo, G. Morrison, G. Mazzola, E. Neuwirth, M. Op de Coul, R.
> Ruzicka, M. Schulter, J. Starrett, Ch. Stoddard, and W.Sethares.