Re: Wedge product understood at last (was: Grassmann Algebra, by Joh

In-Reply-To: <a762uk+mcqd@eGroups.com>
dkeenanuqnetau wrote:

> If anyone is still struggling, as I was until a few minutes ago, to
> understand what the heck the exterior product is (I prefer this name
> to "wedge product"), and what the point of it is, then just read the
> first 6 pages of the Introduction. The result was a great "aha!" for
> me.

Yes, the introduction already covers a lot of ground. I'm working through
the whole thing now.

> It is really quite a beautiful conception. I still say we don't "need"
> it, but I can certainly see that some things will be described much
> more elegantly _with_ it.

Once you know the formalism, it's easier to calculate the temperaments
using exterior algebra than matrices. More people will already be
familiar with matrices, but I suppose most musicians won't be with either.
One problem is that you can think of the matrix approach as solving an
equation.

The alternative model is that the wedgie allows you to tell which
intervals are equivalent to a unison, and can be generated from a sample
set of such intervals. I'm not sure why it should also give mappings to
convert prime coordinates to tempered ones, other than by analogy with
matrices.

> Thanks guys for all your attempts. What I was missing was
> (a) Its geometrical interpretation
> "thus enabling us to bring to bear the power of geometric
> visualization and intuition into our algebraic manipulations."

Yes, it has big geometric and physical applications I didn't know about.

The thing I was calling the complement is also called the complement in
this book, so that saves me making any changes to my program. Although I
could change "Wedgable" to "ExteriorElement". From the regressive product
being like an intersection, we should have

(h31^h41)v(h19^h41) = h41

So if you set

miracle = (h31^h41).complement()
magic = (h19^h41).complement()

then miracle.complement() v magic.complement() = +/- h41

or equivalently

(miracle ^ magic).complement() = +/- h41

Hence we have an answer to the question asked some time ago about how you
find the equal temperament common to a pair of linear temperaments.
Unfortunately, it doesn't work beyond the 5-limit. Still, I can get at
the 7-limit result like this:

>>> (~(~magic^{(3,):1}) ^ ~(~miracle^{(3,):1})).invariant()
(41, 65, 95)
>>> (~(~magic^{(2,):1}) ^ ~(~miracle^{(2,):1}) ^ {(2,):1}).invariant()
(41, 65, 0, 115)

where I've experimentally redefined the ~ operator to find the complement.
The first calculation is (miracle^magic).invariant() for the 5-limit
subset. The invariant() method already gets the complement and sign
right. The second calculation is the same idea but removing the prime 5
from the reckoning. The two results between them give the correct mapping
[41 65 95 115].

Starting from the 11-limit, you'd have to knock out two primes at a time.

Graham

--- In tuning-math@y..., graham@m... wrote:

> One problem is that you can think of the matrix approach as solving an
> equation.

Why is that a problem? You can generalize Cramer's rule with wedges, for that matter.

> The thing I was calling the complement is also called the complement in
> this book, so that saves me making any changes to my program.

I didn't know you knew about compliments. This is all connected to the jabber I giving about Poincare duality, where vals map to compliments of 1-intervals and wedges of vals map to vee products of compliments.

By the way, it seems people are adopting the terminology of this book.
The wedge product is probably most often called "exterior product" (four syllables rather than one, and I didn't learn it that way, so I like wedge better) but I think "regressive product" goes all the way back to Grassmann--at least, its usually called the "intersection product", but I like "vee product" myself. There's also something called the "Grassmann product" in case this wasn't confusing enough as it is.