Tenney integral tuning

I've adapted Gene's Kees integral tuning, as far as I understand it, to Tenney weighting for arbitrary regular temperaments. The result is that it gives the TOP-RMS tuning. Because formulae don't come across well in plain text, I put the proof in the files section as a PDF instead.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> I've adapted Gene's Kees integral tuning, as far as I
> understand it, to Tenney weighting for arbitrary regular
> temperaments. The result is that it gives the TOP-RMS
> tuning. Because formulae don't come across well in plain
> text, I put the proof in the files section as a PDF instead.

Thanks, Graham. I was going to try to see if TOP-RMS is
what it did (which seemed as if it ought to be the case.)

Note that this makes SDT the Kees and TOP-RMS the
Tenney version of the same tuning idea, and provides
another justification for both.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >>I've adapted Gene's Kees integral tuning, as far as I >>understand it, to Tenney weighting for arbitrary regular >>temperaments. The result is that it gives the TOP-RMS >>tuning. Because formulae don't come across well in plain >>text, I put the proof in the files section as a PDF instead.
> > > Thanks, Graham. I was going to try to see if TOP-RMS is
> what it did (which seemed as if it ought to be the case.)

Right. I think the proof's actually wrong but the result's still correct. I'll fix it sometime.

> Note that this makes SDT the Kees and TOP-RMS the
> Tenney version of the same tuning idea, and provides
> another justification for both.

We don't know it always works for the STD: only in the 5-limit. The integral's harder because integrating over one variable brings in another. I'm still hoping for a clever solution to that one.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> Right. I think the proof's actually wrong but the result's
> still correct. I'll fix it sometime.

It didn't make sense to me, since it seemed you were using
measure polytopes = rectangular hypersolids and not
cross-polytopes (n-dimensional octahedrons) to integrate
over. Bit I checked a little and the result does seem to
be true, as it seems it should be.

> > Note that this makes SDT the Kees and TOP-RMS the
> > Tenney version of the same tuning idea, and provides
> > another justification for both.
>
> We don't know it always works for the STD: only in the
> 5-limit. The integral's harder because integrating over one
> variable brings in another. I'm still hoping for a clever
> solution to that one.

Note if it *isn't* true there should be, in higher prime
limits, something like the Kees metric for which it *is*
true that we would want to know about.

If you do some work you can get it up to the 7-limit and
check that, but we would really need an idea for the general
p-limit.
>
> Graham
>

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > >>Right. I think the proof's actually wrong but the result's >>still correct. I'll fix it sometime.
> > It didn't make sense to me, since it seemed you were using
> measure polytopes = rectangular hypersolids and not
> cross-polytopes (n-dimensional octahedrons) to integrate
> over. Bit I checked a little and the result does seem to
> be true, as it seems it should be.

Octahedrons? I can see replacing hypercubes with hyperspheres would be more valid, but also more difficult. For now you get weighted hypercubes. What I really want is a general formula for a function integrated over a hypersolid, which is why I've been looking at geometric algebra, advanced calculus and the like. So far to no avail.

As the result is true for any size of solid, it should also be true for any surface of a solid. So perhaps solving a hypersurface integral would help. That's the cleverest thing I can think of so far.

>>>Note that this makes SDT the Kees and TOP-RMS the
>>>Tenney version of the same tuning idea, and provides
>>>another justification for both.
>>
>>We don't know it always works for the STD: only in the >>5-limit. The integral's harder because integrating over one >>variable brings in another. I'm still hoping for a clever >>solution to that one.
> > Note if it *isn't* true there should be, in higher prime
> limits, something like the Kees metric for which it *is*
> true that we would want to know about.

There should certainly be something interesting in the higher limits.

> If you do some work you can get it up to the 7-limit and
> check that, but we would really need an idea for the general
> p-limit.

So what is the 7-limit problem and why can't Maple solve it?

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> So what is the 7-limit problem and why can't Maple solve it?

Tenney wouldn't be a problem, but for 7-limit Kees
you are integrating over the Kees unit ball. You'd
need to decompose that into pieces you could integrate
over, which since it isn't so simple a thing as
a cross-polytope becomes a problem. Obviously, it
is a problem which could be solved with enough
work.