Yahoo Tuning Groups Ultimate Backup tuning-math Wedgie dual

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I want to make sure my dual routine works correctly. Am I working out
> the dual of 11-limit meantone right?
>
> 11-limit meantone: <<1 4 10 18 4 13 25 12 28 16||
> Dual: ||16 -28 12 25 -13 4 -18 10 -4 1>>
>
> Do I have all of the signs right and everything? I note that for the
> 11-limit, it's not just that the sign of every other entry flips like
> it does for the 5- and 7-limits.

I get ||16 -28 12 -25 13 -4 18 10 -4 1>>

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > I want to make sure my dual routine works correctly. Am I working out
> > the dual of 11-limit meantone right?
> >
> > 11-limit meantone: <<1 4 10 18 4 13 25 12 28 16||
> > Dual: ||16 -28 12 25 -13 4 -18 10 -4 1>>
> >
> > Do I have all of the signs right and everything? I note that for the
> > 11-limit, it's not just that the sign of every other entry flips like
> > it does for the 5- and 7-limits.
>
> I get ||16 -28 12 -25 13 -4 18 10 -4 1>>

Sorry, that should be |||16 -28 12 -25 13 -4 18 10 -4 1>>>. I think I misled you by an error in the first sentence of "The dual", obviously a bad place for an error.

🔗Mike Battaglia <battaglia01@gmail.com>

Er, yeah, should be a 3-monzo. I actually knew that, that was just a typo
on my part. But I still don't get why the signs aren't exactly as I said.

I wrote this

> > 11-limit meantone: <<1 4 10 18 4 13 25 12 28 16||
> > Dual: |||16 -28 12 25 -13 4 -18 10 -4 1>>>

I corrected it above so that the multimonzo is of rank 3 instead of rank 2.
However, the signs I wrote above seem to be in agreement with the wiki
article, whereas yours seem to be reversed with respect to the article. For
instance, the article says this:

"Hence, we may take the (n-i)th element of the k-vector, and this becomes
the ith element of the dual if the permutation of the k basis elements in
order, concatenated with the remaining (n-k) elements in order, is an even
permutation. If it is an odd permutation, then minus the (n-i)th element
becomes the ith element of the dual."

The 4th coefficient in the original multival for meantone, which has the
value 18, represents e2^e11. If I follow your instructions above, and
concatenate (e2, e11) with (e3, e5, e7), I get (e2, e11, e3, e5, e7), which
is an odd permutation. Proof:

(2,11,3,5,7) - 0 reversals (starting point)
(2,3,11,5,7) - 1 swap
(2,3,5,11,7) - 2 swaps
(2,3,5,7,11) - 3 swaps

Since the 4th coefficient of the bival in this case leads to an odd
permutation, the 4th from last coefficient of the dual trimonzo should be
-1 * 18 = 18. However, the result you gave has positive 18 in the 4th from
last coefficient, indicating a discrepancy between the result you gave and
the wiki article.

-Mike

On Apr 23, 2012, at 4:42 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...>
wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > I want to make sure my dual routine works correctly. Am I working out
> > the dual of 11-limit meantone right?
> >
> > 11-limit meantone: <<1 4 10 18 4 13 25 12 28 16||
> > Dual: ||16 -28 12 25 -13 4 -18 10 -4 1>>
> >
> > Do I have all of the signs right and everything? I note that for the
> > 11-limit, it's not just that the sign of every other entry flips like
> > it does for the 5- and 7-limits.
>
> I get ||16 -28 12 -25 13 -4 18 10 -4 1>>

Sorry, that should be |||16 -28 12 -25 13 -4 18 10 -4 1>>>. I think I
misled you by an error in the first sentence of "The dual", obviously a bad
place for an error.