zeta/wiki

One vote here for a zeta tuning page on the wiki... -Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> One vote here for a zeta tuning page on the wiki... -Carl
>

Noted. I had the feeling the more theoretical pages were not generating much interest.

At 10:13 AM 1/4/2011, you wrote:
>
>--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>>
>> One vote here for a zeta tuning page on the wiki... -Carl
>
>Noted. I had the feeling the more theoretical pages were not
>generating much interest.

There is a stark jump in the writing level throughout the
wiki which would be good to smooth over at some point.

One thing that comes to mind... I've seen zeta tunings, but
not zeta error. Would such a thing make sense?

-Carl

On Tue, Jan 4, 2011 at 1:13 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
> >
> > One vote here for a zeta tuning page on the wiki... -Carl
> >
>
> Noted. I had the feeling the more theoretical pages were not generating much interest.

In general, the more theoretical pages there are on that wiki, the
more I can understand what's going on. So the more the better for me.

-Mike

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> One thing that comes to mind... I've seen zeta tunings, but
> not zeta error. Would such a thing make sense?

You could certainly define something of the sort.

Gene wrote:
>> One thing that comes to mind... I've seen zeta tunings, but
>> not zeta error. Would such a thing make sense?
>
>You could certainly define something of the sort.

I would certainly interested in this error, which would remove
the need to specify prime limit. What form could it take?
Would we see error and weighting terms? -Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> I would certainly interested in this error, which would remove
> the need to specify prime limit. What form could it take?
> Would we see error and weighting terms? -Carl

It would require using something called omega theorems, which usually depend on the Riemann hypothesis, and for which I'd need to find the best available version. That doesn't quite work like logflatnes, but close.

>> I would certainly interested in this error, which would remove
>> the need to specify prime limit. What form could it take?
>> Would we see error and weighting terms? -Carl
>
>It would require using something called omega theorems, which usually
>depend on the Riemann hypothesis, and for which I'd need to find the
>best available version. That doesn't quite work like logflatnes, but close.

Sounds absolutely fascinating. I don't think assuming the
Riemann hypothesis should be any problem for the Ultimate
tuning optimization. -Carl

On Tue, Jan 4, 2011 at 5:00 PM, Carl Lumma <carl@lumma.org> wrote:
>
> >> I would certainly interested in this error, which would remove
> >> the need to specify prime limit. What form could it take?
> >> Would we see error and weighting terms? -Carl
> >
> >It would require using something called omega theorems, which usually
> >depend on the Riemann hypothesis, and for which I'd need to find the
> >best available version. That doesn't quite work like logflatnes, but close.
>
> Sounds absolutely fascinating. I don't think assuming the
> Riemann hypothesis should be any problem for the Ultimate
> tuning optimization. -Carl

OK, my interest is piqued. Can someone give me at least a rough idea
of what we're talking about? I saw the zeta integral list a while ago,
and noticed that most of the "good" equal temperaments were on there,
which I assume has something to do with the fact that the zeta
function and the primes are connected.

-Mike

On Tue, Jan 4, 2011 at 5:00 PM, Carl Lumma <carl@lumma.org> wrote:
>
> >> I would certainly interested in this error, which would remove
> >> the need to specify prime limit. What form could it take?
> >> Would we see error and weighting terms? -Carl

I'm doubly interested because, as you remember from my messages
offlist, since HE can be represented by a convolution integral, then
HE and this function are connected:

http://en.wikipedia.org/wiki/Thomae%27s_function

And since Thomae's function is connected to this...

http://en.wikipedia.org/wiki/Euclid%27s_orchard

It means that HE can be viewed as a projective view of an n-limit
interval space (I guess a "Tenney-Euclidean" space?), except where the
dots in the space are replaced with multivariate Gaussians (e.g. there
are "fuzzy" basis functions).

This would allow us to represent, more or less in a sense, how the ear
is "naturally" tempering intervals for some value s, up till the
n-limit. The dual of this space would also yield, I guess, the
offspring of harmonic entropy and tenney projective tuning space, and
that might be useful to figure out what the "goodest" temperaments
already are as far as the ear is concerned.

This would yield some kind of fuzzy group structure that I guess
wouldn't really be a free abelian fuzzy group, because every interval
will be able to be reached by more than one way via the built-in
tempering of intervals; this is the whole point.

Anyway, if this zeta tuning concept can eliminate the need to
represent things up to the n-limit, then we could perhaps extend this
concept to have an HE meets zeta tuning space, although I don't know
enough about the zeta function to really understand how that would
work...

-Mike

Carl Lumma <carl@lumma.org> wrote:

> Sounds absolutely fascinating. I don't think assuming the
> Riemann hypothesis should be any problem for the Ultimate
> tuning optimization. -Carl

What would be cool is turning it round, and proving the
Riemann hypothesis from tuning optimization ;-)

Graham

>> Sounds absolutely fascinating. I don't think assuming the
>> Riemann hypothesis should be any problem for the Ultimate
>> tuning optimization. -Carl
>
>What would be cool is turning it round, and proving the
>Riemann hypothesis from tuning optimization ;-)
>
> Graham

I daydreamed about this some years ago.

/tuning-math/message/8530
(I observed in 2 or 3 cases that that the p-limit TOP tuning
of an ET seems to oscillate around the zeta tuning until p > o,
the o-limit where the ET is no longer consistent.)

-Carl