A question on wedgies

Hi there.
I've been away from the list for a few weeks recently so I haven't carefully followed the topics for some time. But I hope someone can give me a clue on this.
I always thought that a 2D temperament could be uniquely defined by the wedgie. To my surprise, this doesn't seem to be the case if there's contorsion -- or I must be doing something wrong. For example, if I try to pair up 24d and 14c and then I try to pair up 24d and 26, I get identical wedgies. However, according to Graham's Temperament Finder, each of the resulting 2D temperaments uses a different generator and therefore a different mapping. May I know what on earth is going on here?
Thanks.
Petr

On Wed, Jan 25, 2012 at 5:01 AM, petrparizek2000
<petrparizek2000@...> wrote:
>
> Hi there.
> I've been away from the list for a few weeks recently so I haven't carefully followed the topics for some time. But I hope someone can give me a clue on this.
> I always thought that a 2D temperament could be uniquely defined by the wedgie. To my surprise, this doesn't seem to be the case if there's contorsion -- or I must be doing something wrong. For example, if I try to pair up 24d and 14c and then I try to pair up 24d and 26, I get identical wedgies. However, according to Graham's Temperament Finder, each of the resulting 2D temperaments uses a different generator and therefore a different mapping. May I know what on earth is going on here?
> Thanks.
> Petr

Hi Petr - what limit are you in?

The term "wedgie" refers to the thing which has contorsion eliminated.
The term "multival" has it still left in. So in this case, the
multival for both of these temperaments is <<4 16 16 16 14 -8||, but
the wedgie is <<2 8 8 8 7 -4||.

Multivals don't distinguish between different forms of contorsion. For
example, in the 5-limit, 7&24, 5&19, and 12&14c are all differently
contorted forms of meantone, but they have the same multival (and
hence, also the same wedgie).

You can use the Hermite normal form of the resultant mapping matrices
to uniquely identify every contorted temperament. I've been devoting
some thought recently to coming up with an elegant way to do it that
mimics the look and feel of something like the exterior product, but
I'm not sure how to do it really.

-Mike

Mike wrote:

> Hi Petr - what limit are you in?

I thought 7, you got it right.

> The term "wedgie" refers to the thing which has contorsion eliminated.
> The term "multival" has it still left in. So in this case, the
> multival for both of these temperaments is <<4 16 16 16 14 -8||, but

Yes, that's what I meant.

> the wedgie is <<2 8 8 8 7 -4||.

Understood.

> Multivals don't distinguish between different forms of contorsion. For
> example, in the 5-limit, 7&24, 5&19, and 12&14c are all differently
> contorted forms of meantone, but they have the same multival (and
> hence, also the same wedgie).

Hohohoh, I knew why I'd been avoiding contorted temperaments for so long. :-))) And I see I'll probably keep on doing so.

> You can use the Hermite normal form of the resultant mapping matrices
> to uniquely identify every contorted temperament. I've been devoting
> some thought recently to coming up with an elegant way to do it that
> mimics the look and feel of something like the exterior product, but
> I'm not sure how to do it really.

Wow, looks like I'll have to read something about that stuff on the web; I'm pretty new to those hermite reductions and all that.

Petr

I wrote:

> Wow, looks like I'll have to read something about that stuff on the web; I'm pretty new to those hermite reductions and all that.

Or rather "pretty unfamiliar with them" -- to make it sound more like English. :-D

Petr