--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

'
> geometry, where angles are left undefined . . . Anyhow, since both of
> these methods could be used to address a 3-limit TOP temperament, in
> 5-limit could they be still both be expressible in a single form in a
> general enough framework, say exterior algebra?

That was my suggestion. You normalize by dividing each coordinate by
log2(p) and take the wedge product up to the (normalized) multival,
and then measure complexity by taking the max of the absolute values
of the coefficients. If you start from the monzo side, you get the
same normalized coefficients up to a constant factor, but now you
might rather take the sum of the absolute values (L1 vs L infinity.)

The goal, of course, is to produce
> complexity vs. TOP error graphs for 7-limit linear temperaments,
> something I currently don't know how to do.

Why not use the formula I gave before:

|| <<w1 w2 w3 w4 w5 w6|| || = max(|w1|/p3, |w2|/p5, |w3|/p7,
|w4|/p3p5, |w5|/p3p7, |w6|/p5p7)

to measure complexity? Here p3=log2(3), etc. The corresponding
log-flat badness would be

BAD = ||TOP - JIP|| * ||Wedgie||^2

Here ||Wedgie|| is as above, and ||TOP-JIP|| is the maximum weighted
error, or distance to the JIP. If <t2 t3 t5 t7| is the top tuning for
the temperament given by Wedgie, then this is

||TOP - JIP|| = Max(|t2-1|, |t3/p3-1|, |t5/p5-1|, |t7/p7-1|)

Your choice whether to do everything in log2 or cents terms, of course.