Fun with "e" and pi

This has probably been done, but anyway, here goes -

So it's true that the syntonic comma to the diesis is almost
exactly 11 times the grad to 21 times the grad, or 11/21. I got
to thinking, this is about pi/6.

It's true that (81/80)^6/pi =~ (128/125). Well, here's a little
more math:

((81/80)^6/pi)^(pi^2/6)= (81/80)^pi

Of course pi^2/6 is Sigma_1_oo (1/n^2) (RZF)

Now ln(81/80)^pi = pi * ln(81/80) (Useful in 1/n-comma meantone
calculations)

And log2(81/80)^pi is just pi*(81/80)/ln2 = 67.564 cents

Silly or useful?

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> This has probably been done, but anyway, here goes -
>
> So it's true that the syntonic comma to the diesis is almost
> exactly 11 times the grad to 21 times the grad, or 11/21. I got
> to thinking, this is about pi/6.
>
> It's true that (81/80)^6/pi =~ (128/125). Well, here's a little
> more math:
>
> ((81/80)^6/pi)^(pi^2/6)= (81/80)^pi
>
> Of course pi^2/6 is Sigma_1_oo (1/n^2) (RZF)
>
> Now ln(81/80)^pi = pi * ln(81/80) (Useful in 1/n-comma meantone
> calculations)
>
> And log2(81/80)^pi is just pi*(81/80)/ln2 = 67.564 cents

** Oops! pi * ln(81/80)/ln2

Forgot to add: (81/80)^pi = e^(pi*ln(81/80)) = (e^pi)^(ln(81/80))
(I meant to bring in e^pi). But why in the world would you
raise anything to pi^2/6? Which is 1 + 1/4 + 1/9 + 1/16 ....

Well, then syntonic_comma^pi = diesis^(pi^2/6)

4/1-4/3+4/5-4/7...

Also

(81/80)^22/7 =~ (128/125)^242/147
(81/80)^22/21=~ (128/125)^242/461

But

(81/80)^21/11 = (128/125) in the first place.

> Silly or useful?
>
> PGH
>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > This has probably been done, but anyway, here goes -
> >
> > So it's true that the syntonic comma to the diesis is almost
> > exactly 11 times the grad to 21 times the grad, or 11/21. I got
> > to thinking, this is about pi/6.
> >
> > It's true that (81/80)^6/pi =~ (128/125). Well, here's a little
> > more math:
> >
> > ((81/80)^6/pi)^(pi^2/6)= (81/80)^pi
> >
> > Of course pi^2/6 is Sigma_1_oo (1/n^2) (RZF)
> >
> > Now ln(81/80)^pi = pi * ln(81/80) (Useful in 1/n-comma meantone
> > calculations)
> >
> > And log2(81/80)^pi is just pi*(81/80)/ln2 = 67.564 cents
>
> ** Oops! pi * ln(81/80)/ln2
>
> Forgot to add: (81/80)^pi = e^(pi*ln(81/80)) = (e^pi)^(ln(81/80))
> (I meant to bring in e^pi). But why in the world would you
> raise anything to pi^2/6? Which is 1 + 1/4 + 1/9 + 1/16 ....
>
> Well, then syntonic_comma^pi = diesis^(pi^2/6)
>
> 4/1-4/3+4/5-4/7...
>
> Also
>
> (81/80)^22/7 =~ (128/125)^242/147
> (81/80)^22/21=~ (128/125)^242/461
>
> But
>
> (81/80)^21/11 = (128/125) in the first place.

Continuing - - - -

diesis = syntonic_comma^pi^(1/(pi^2/6))

(4/1-4/3+4/5...)^(1/(1+1/4+1/9...))

(4/(1+1/4+1/9...))
-(4/(3+3/4+3/9..))
+(4/(5+5/4+5/9..))
-(4/(7+7/4+7/9..))

Any ideas of how to compute this? It's NOT pi + pi/4 + pi/9 . . .

PGH