Quick question about multivector "ranks"

The term "rank" means something different for multivectors than it
does for abelian groups. The rank of a multivector is the minimum
number of totally decomposable multivectors that you need to add
together to obtain the multivector in question. So for instance, all
of the multivectors that denote temperaments are rank 1 by definition.

The term "grade" is the thing that refers to the dimensionality of the
associated temperament; e.g. an multivector of grade n lives in the
nth exterior power of the vector space. Should I use this term then?
Or perhaps just "dimensionality"?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> The term "grade" is the thing that refers to the dimensionality of the
> associated temperament; e.g. an multivector of grade n lives in the
> nth exterior power of the vector space. Should I use this term then?

What's wrong with grade?

On Fri, Aug 3, 2012 at 11:25 AM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > The term "grade" is the thing that refers to the dimensionality of the
> > associated temperament; e.g. an multivector of grade n lives in the
> > nth exterior power of the vector space. Should I use this term then?
>
> What's wrong with grade?

That's fine, so long as everyone knows what I'm talking about.

Is there another name for the "rank" of a multivector, since it can be
particularly confusing to use that term for music theory? If I called
it the "tensor rank" would that be understandable?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Is there another name for the "rank" of a multivector, since it can be
> particularly confusing to use that term for music theory? If I called
> it the "tensor rank" would that be understandable?

I don't know of another name for rank, and "tensor rank" strikes me as a terrible name. What about "blade-sum rank" or even "wedgie-sum rank"?

On Sat, Aug 4, 2012 at 2:51 AM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > Is there another name for the "rank" of a multivector, since it can be
> > particularly confusing to use that term for music theory? If I called
> > it the "tensor rank" would that be understandable?
>
> I don't know of another name for rank, and "tensor rank" strikes me as a
> terrible name. What about "blade-sum rank" or even "wedgie-sum rank"?

Well, I don't think wedgie-sum rank is a good idea because this
applies to multivectors in general, including multimonzos. Blade-sum
rank is alright but I still think "rank" is a bit confusing.

We could just take the plunge and refer to the "decomposability" of a
vector, so that a blade is 1-decomposable, and then 2-decomposable
means they can be reduced to a sum of 2 terms, etc. We could also talk
about how simple they are, e.g. a blade is 1-simple, then there's
2-simple, etc.

If we really wanted to have fun with it, looks like there's some
physics books which call tensors of rank > 1 "entangled tensors." We
could say multivectors of rank > 1 are thus "entangled multivectors."
I kind of like it, tbh.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> We could just take the plunge and refer to the "decomposability" of a
> vector, so that a blade is 1-decomposable, and then 2-decomposable
> means they can be reduced to a sum of 2 terms, etc. We could also talk
> about how simple they are, e.g. a blade is 1-simple, then there's
> 2-simple, etc.

n-simple sounds good if you mention it's usually called rank.