Topics in 34 : (was Re: Hey, did I find a new comma???

> From: graham@microtonal.co.uk
> Subject: Re: Hey, did I find a new comma???
>
> Bob Valentine wrote:
>
> > The interval being tempered out is
> >
> > 2^139
> > -----
> > 17^34
> >
> > which is not terribly small at 31.5C, but is probably
> > incredibly important for the many fans of 17-limit
> > JI looking for even numbered equal temperments.
>
> There's also 289:288, which is simpler and only 6 cents. Consistent with
> 58-, 72-, 80- and 94-equal.
>

Okay. While on the topic of 34, it is inconsistent in 7-limit but
if one removes 7, I think it is consistent for { 3, 5, 9, 11, 13, 17 }.

So I sat down and looked at two 3D interpretations.

3 5 11
[ -1 8 0 ]
[ 4 2 0 ]
[ -1 0 3 ]

3 5 13
[ -1 8 0 ]
[ 4 2 0 ]
[ -1 0 2 ]

I believe both of these have determinants of 34. Even the 5-limt matrix

[ -1 8 ]
[ 4 2 ]

has a determinant of 34. Is this a particular five limit identity that
has a famous name I should know by now?

I tried to come up with a good lattice for the { 3, 5, 13 } case, since
the 13's seem to be a little more accurate and it sort of seemed cool,
you know, 13, 34...

I was looking for two 17-note structures seperated by a 13/8, with more
"5"s than "3"s in the indivdual structures, since stacks of 5's remain
consistnt longer than stacks of 3's. Perhaps one of the lattice-sticians
on the list can help me out?

Bob Valentine

> Graham

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:
>
> > From: graham@m...
> > Subject: Re: Hey, did I find a new comma???
> >
> > Bob Valentine wrote:
> >
> > > The interval being tempered out is
> > >
> > > 2^139
> > > -----
> > > 17^34
> > >
> > > which is not terribly small at 31.5C, but is probably
> > > incredibly important for the many fans of 17-limit
> > > JI looking for even numbered equal temperments.
> >
> > There's also 289:288, which is simpler and only 6 cents.
Consistent with
> > 58-, 72-, 80- and 94-equal.
> >
>
> Okay. While on the topic of 34, it is inconsistent in 7-limit but
> if one removes 7, I think it is consistent for { 3, 5, 9, 11, 13,
17 }.
>
> So I sat down and looked at two 3D interpretations.
>
> 3 5 11
> [ -1 8 0 ]
> [ 4 2 0 ]
> [ -1 0 3 ]

The determinant is 102, but the implied temperament may still be 34-
tET -- have you read up on "torsion" at the tuning-math list?

> 3 5 13
> [ -1 8 0 ]
> [ 4 2 0 ]
> [ -1 0 2 ]

Determinant 68; again may be torsional 34-tET.

> Even the 5-limt matrix
>
> [ -1 8 ]
> [ 4 2 ]
>
> has a determinant of 34. Is this a particular five limit identity
that
> has a famous name I should know by now?

Which identity are you referring to? The second row is the
diaschisma; the first is 393216:390625 -- but 15552:15625, the
kleisma, and again the diaschisma provide a simpler and more common
way of definining 34-tET. The kleisma itself implies a minor-third
generator, and I believe this was most important in Larry Hanson's
investigations into 34-tET.
>
> I was looking for two 17-note structures seperated by a 13/8, with
more
> "5"s than "3"s in the indivdual structures, since stacks of 5's
remain
> consistnt longer than stacks of 3's. Perhaps one of the lattice-
sticians
> on the list can help me out?

Ask the question, as clear as you can, to tuning-
math@yahoogroups.com . . .