poincare duality

I think Mathworld could use your help with this one, Gene:

http://mathworld.wolfram.com/PoincareDuality.html

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> I think Mathworld could use your help with this one, Gene:
>
> http://mathworld.wolfram.com/PoincareDuality.html

It's a little succinct, isn't it? Actually, if I did write something
it would be for Wikipedia. I started out thinking about adding
something on the Riemann-Siegel zeta function, just so I could refer
to it, and ended up deciding that some more important special
functions were missing. So I added the Jacobi theta function and
defined Jacobi and Weierstrass elliptic functions in terms of it,
just as I always thought was the best method, added a Dedekind eta
function, a j invariant, and a Dirichlet eta function. In all of
which I managed to lose sight of my original goal of relevance to
music, but the encyclopedia is much improved in the area of special
functions relevant to number theory.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> I think Mathworld could use your help with this one, Gene:
>
> http://mathworld.wolfram.com/PoincareDuality.html

By the way, there's a discussion of this page

http://en.wikipedia.org/wiki/Wedge_product =

http://en.wikipedia.org/wiki/Exterior_power

on the corresponding talk page ("discuss this page.") I was thinking
of writing something on the wedge product which could actually be
understandable by non-mathematicians, and Charles was not too keen on
the idea. However, I see that immediately afterward he moved material
from the Grassmann algebra page to this one, giving it an entire new
introductory section, and thereby making it more concrete, just as we
had discussed. I'd be interested if people here could tell me if it
makes any sense to them, and how much.

Gene Ward Smith wrote:

> By the way, there's a discussion of this page
> > http://en.wikipedia.org/wiki/Wedge_product =
> > http://en.wikipedia.org/wiki/Exterior_power
> > on the corresponding talk page ("discuss this page.") I was thinking > of writing something on the wedge product which could actually be > understandable by non-mathematicians, and Charles was not too keen on > the idea. However, I see that immediately afterward he moved material > from the Grassmann algebra page to this one, giving it an entire new > introductory section, and thereby making it more concrete, just as we > had discussed. I'd be interested if people here could tell me if it > makes any sense to them, and how much.

The definition there seems easier to understand than the one from Mathworld that was posted way back, but that could just be that I've spent enough time looking at actual wedgies used in temperaments and seeing how they relate to commas and maps that the concept in general makes more sense now. The problem is that this page describes in an abstract sense what a wedge product is rather than explaining how to calculate it. What would be more useful to musicians is a page describing wedge products strictly as applied to vals and monzos, rather than the more general definition that mathematicians need, with clear algorithms and examples of how to wedge vals to get linear temperaments, how to find commas and maps from wedgies, and anything else that might be of interest in making music.

I'd also start with the 7-limit examples, since it's in the 7-limit that wedgies start being useful in representing temperaments. One problem with the http://66.98.148.43/~xenharmo/wedge.html page is that the 7-limit examples are the last thing on the page. I think that any page intended for musicians should start with these 7-limit definitions of monzo product and val product, with examples of familiar temperaments like meantone, and progress from there to explain how to generalize these definitions and use them for different kinds of temperaments. And technical terms such as "abelian group", which aren't immediately relevant to the musical use of wedge products, should not be in an introductory page. The Wiki page starts right out with references to tensor products and concepts from Grassman algebra theory that aren't of much interest to someone wanting to figure out the mathematics of temperaments, so it's even less useful than the Xenharmony page in that respect.