Dual space example

I had thought of this business of subspaces and the JIP as a way of
formulating what we were already doing, but didn't realize it led to
anything new. I wish I'd seen the possible link to Paul's heuristic,
but another interesting aspect I didn't consider is this bounded
relative error business. Here's a basic example of what I mean.

As I've pointed out from time to time, the norm which gives the
equilateral triangular lattice for 5-limit octave classes is

||3^a 5^b|| = sqrt(a^2 + ab + b^2)

where the class is represented by 3^a 5^b. The norm on the dual space
is then

||(x3, x5)|| = sqrt(x3^2 - x3 x5 + x5^2)

where x2 is the tuning of 3 (in log2, cents or whatever your favorite
log is terms) and x5 is the tuning of 5. The nearest point to [log(3),
log(5)] on a subspace corresponding to a temperament is the 5-limit
rms tuning. If we look on the line x5=4x3 - 4, for instance, we get
the Woolhouse tuning.

If we now take the error in cents for the Woolhouse tuning of any
5-limit interval q and divide it by the norm of q, it will be bounded
by the worst case, namely 81/80, for which we get 5.965. Other values
are 5.790 for 3/2, 1.654 for 5/4, 4.136 for 5/3, 5.940 for 27/20 and
so forth--all bounded by 5.965.

>I had thought of this business of subspaces and the JIP as a way of
>formulating what we were already doing, but didn't realize it led to
>anything new. I wish I'd seen the possible link to Paul's heuristic,
>but another interesting aspect I didn't consider is this bounded
>relative error business. Here's a basic example of what I mean.
>
>As I've pointed out from time to time, the norm which gives the
>equilateral triangular lattice for 5-limit octave classes is
>
>||3^a 5^b|| = sqrt(a^2 + ab + b^2)
>
>where the class is represented by 3^a 5^b.

I can believe that.

>The norm on the dual space is then
>
>||(x3, x5)|| = sqrt(x3^2 - x3 x5 + x5^2)
>
>where x2 is the tuning of 3 (in log2, cents or whatever your favorite
>log is terms) and x5 is the tuning of 5.

I'll take your word on that.

>The nearest point to [log(3), log(5)] on a subspace corresponding
>to a temperament is the 5-limit rms tuning. If we look on the line
>x5=4x3 - 4, for instance, we get the Woolhouse tuning.

Yes!!

Is there anything fundamentally keeping the 2s out of this?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Yes!!
>
> Is there anything fundamentally keeping the 2s out of this?

No; you get something similar in each even limit, for instance.