Re: [tuning] Re: Ums

>This is because eg a Tenney space is *not* Euclidean.

Maybe Tenney *space* isn't, but the Tenney *lattice*
sure seems like it fits in Euclidean space. Because
it's rectangular, the axes can be scaled however with
no worries. But maybe you can give us examples of
how Tenney space differs from R. Maybe even use some
math. But please be gentle.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >This is because eg a Tenney space is *not* Euclidean.
>
> Maybe Tenney *space* isn't, but the Tenney *lattice*
> sure seems like it fits in Euclidean space.

That is not possible; the distances along diagonals do not correspond
to the distances along sides. If you take a box with corners 1, 2, 3,
6, 1/5, 2/5, 3/5 and 6/5 you get 6/5 in the opposite corner from 1.
The Euclidean distance is sqrt(1+log2(3)^2+log2(5)^2) = 2.98, but the
taxicab distance is log2(30) = 4.91. Obviously, this isn't a normal
kind of box.