The supertemperament

For any odd prime p, there is a finite list of superparticular ratios
which belong to the p-limit. For some n, the smallest n intervals on
this list will uniquely determine a val v, which will be for an equal
temperament v(2) for which v is the standard val. This defines a
function supertemp(p) from p to the "supertemperament" for p. If I
calculated the 19-limit superparticulars correctly, we have the following:

supertemp(3) = 2
supertemp(5) = 7
supertemp(7) = 72
supertemp(11) = 72
supertemp(13) = 270
supertemp(17) = 1506
supertemp(19) = 8539

For each p, there will be a range from the first n to the first n
superparticulars which give the supertemperament. The ranges up
through 19 are as follows:

3: 1
5: 2
7: 3
11: 5-9
13: 9-12
17: 9-13
19: 14-15

>For any odd prime p, there is a finite list of superparticular ratios
>which belong to the p-limit.

Here's something I can believe but which isn't immediately obvious.
Can you prove it?

>For some n, the smallest n intervals on
>this list will uniquely determine a val v, which will be for an equal
>temperament v(2) for which v is the standard val. This defines a
>function supertemp(p) from p to the "supertemperament" for p. If I
>calculated the 19-limit superparticulars correctly, we have the following:
>
>supertemp(3) = 2
>supertemp(5) = 7
>supertemp(7) = 72
>supertemp(11) = 72
>supertemp(13) = 270
>supertemp(17) = 1506
>supertemp(19) = 8539

Cool. Howabout moving a fixed n down the list (or n's which, for each
starting point in the list, uniquely define a val)?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >For any odd prime p, there is a finite list of superparticular
ratios
> >which belong to the p-limit.
>
> Here's something I can believe but which isn't immediately obvious.
> Can you prove it?

Been there, done that. It follows from Baker's Theorem.

> Cool. Howabout moving a fixed n down the list (or n's which, for
each
> starting point in the list, uniquely define a val)?

There's a thought, but first I've got to write up the capstone
temperament (same thing but for linear temperaments.)

>>>For any odd prime p, there is a finite list of superparticular
>>>ratios which belong to the p-limit.
> >
> > Here's something I can believe but which isn't immediately
> > obvious. Can you prove it?
>
> Been there, done that. It follows from Baker's Theorem.

There are no results for "Baker's theorem" or "bakers theorem"
at mathworld, but the 2nd google result for "baker's theorem"
is this post of yours...

/tuning-math/message/1108

Hooray again for google. I wonder how much of these lists
are google-searchable?

-Carl

>
> > Cool. Howabout moving a fixed n down the list (or n's which, for
> each
> > starting point in the list, uniquely define a val)?
>
> There's a thought, but first I've got to write up the capstone
> temperament (same thing but for linear temperaments.)