Wedge products and the torsion mess

Perhaps wedge products are the best way of cleaning this up. If we
write 2^a 3^b 5^c 7^d as a e2 + b e3 + c e5 + d e7 we can take wedge
products by the following rule ei^ei = 0, and if i != j, then
e1^ej = - ej^ei. In the 5-limit case, the wedge product will be, in
effect, the correspodning val. In the 7-limit case, we get something
six dimensional, which if we added another interval would give us a
val. However, it still can be used to test for torsion.

50/49 = e2+2e5-2e7, 2048/2025 = 11e2+4e3-2e5. Taking the wedge
product gives us 50/49^2048/2025 =
4e2^e3 - 24e2^e5 - 8 e3^e5 - 4 e5^e7. This has a common factor of 4.
On the other hand 50/49^54/63 = -2 e2^e3 - 12 e2^e5 + 5 e2^e7
+ 4 e3^e5 + 2 e3^e7 - 2 e5^e7, with a gcd of 1 for the coefficients.
All is, therefore, not lost, I think. I'll ponder the question
further.

--- In tuning-math@y..., genewardsmith@j... wrote:

>I'll ponder the question
> further.

One way to see what is going on is this: if the wedge product has a
common factor, then whatever we pick as another basis interval in
order to compute the corresponding val will also have a common factor
when we take determinants, and hence show torsion according to our
usual test of the gcd of the coefficients of the val. Therefore the
torsion is already present in the two elements we started with.
2048/2025 and 50/49 cannot be extended in a non-torsion way to three
7-limit intervals, in other words, which would be suitable for a
block. This is the same problem as before, in a more insidious form.

--- In tuning-math@y..., genewardsmith@j... wrote:

> 2048/2025 and 50/49 cannot be extended in a non-torsion way to
three
> 7-limit intervals, in other words, which would be suitable for a
> block.

Well that's a nice clarification.

So perhaps it would have been better to focus on scales, rather than
linear temperaments, after all!

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> So perhaps it would have been better to focus on scales, rather
than
> linear temperaments, after all!

Not really--we will still get uniqueness after booting out the
torsion crud, and some of the things I am getting this way it would
not have occured to look at. I still plan on seeing if we are missing
something we shouldn't by looking at it from the other side also.

Of course this is one more piece of weirdness it probably would be a
pain to explain to a non-mathematical readership. :(