Exterior algebra question

Let's say that I take the wedge product of the monzos |-4 4 -1> and |7
0 -3>. This leads to the multimonzo ||28 -19 12>>.

Now, say that I want to "factor" this multimonzo by "dividing" by |7 0
-3>, hence leading me back to |-4 4 -1> again. Is there a name for
this sort of factoring operation?

I thought it might be the "interior product," but all of our
discussions of the interior product have involved taking the interior
product of "multicovectors" and "multivectors." This would be like the
interior product of "multivectors" and "other multivectors."

-Mike

I just realized my question makes no sense. What happens if I factor
|-4 4 -1> out of ||28 -19 12>>? What's left over?

Is it 128/125? Is it 648/625? etc.

-Mike

On Wed, Mar 7, 2012 at 1:26 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> Let's say that I take the wedge product of the monzos |-4 4 -1> and |7
> 0 -3>. This leads to the multimonzo ||28 -19 12>>.
>
> Now, say that I want to "factor" this multimonzo by "dividing" by |7 0
> -3>, hence leading me back to |-4 4 -1> again. Is there a name for
> this sort of factoring operation?
>
> I thought it might be the "interior product," but all of our
> discussions of the interior product have involved taking the interior
> product of "multicovectors" and "multivectors." This would be like the
> interior product of "multivectors" and "other multivectors."
>
>
> -Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I just realized my question makes no sense. What happens if I factor
> |-4 4 -1> out of ||28 -19 12>>? What's left over?
>
> Is it 128/125? Is it 648/625? etc.

It's "diminished second", that is, that equivalence class of intervals.

...524288/531441, 32768/32805, 2048/2025, 128/125, 648/625, 6561/6250...

Keenan

On Wed, Mar 7, 2012 at 1:45 PM, Keenan Pepper <keenanpepper@gmail.com>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
> >
> > I just realized my question makes no sense. What happens if I factor
> > |-4 4 -1> out of ||28 -19 12>>? What's left over?
> >
> > Is it 128/125? Is it 648/625? etc.
>
> It's "diminished second", that is, that equivalence class of intervals.
>
> ...524288/531441, 32768/32805, 2048/2025, 128/125, 648/625, 6561/6250...

So if I divide the multivector ||28 -19 12>> by the vector |-4 4 -1>,
the quotient is an affine subspace of interval space? Wow.

-Mike

Mike, in general you cannot 'divide' with respect to wedge product, since you have zero divisors, i.e., you can have nonzero elements A & B such that A /\ B =0
Bests,
Claudi

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I just realized my question makes no sense. What happens if I factor
> |-4 4 -1> out of ||28 -19 12>>? What's left over?
>
> Is it 128/125? Is it 648/625? etc.
>
> -Mike
>
>
>
> On Wed, Mar 7, 2012 at 1:26 AM, Mike Battaglia <battaglia01@...> wrote:
> > Let's say that I take the wedge product of the monzos |-4 4 -1> and |7
> > 0 -3>. This leads to the multimonzo ||28 -19 12>>.
> >
> > Now, say that I want to "factor" this multimonzo by "dividing" by |7 0
> > -3>, hence leading me back to |-4 4 -1> again. Is there a name for
> > this sort of factoring operation?
> >
> > I thought it might be the "interior product," but all of our
> > discussions of the interior product have involved taking the interior
> > product of "multicovectors" and "multivectors." This would be like the
> > interior product of "multivectors" and "other multivectors."
> >
> >
> > -Mike
>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> So if I divide the multivector ||28 -19 12>> by the vector |-4 4 -1>,
> the quotient is an affine subspace of interval space? Wow.

Or you can just say it's a meantone interval.

Keenan

On Thu, Mar 8, 2012 at 2:21 AM, Keenan Pepper <keenanpepper@gmail.com>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
> > So if I divide the multivector ||28 -19 12>> by the vector |-4 4 -1>,
> > the quotient is an affine subspace of interval space? Wow.
>
> Or you can just say it's a meantone interval.
>
> Keenan

But it's also that meantone interval squared.

-Mike