--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Some evidence you've actually considered it would be nice. A plot
> would be grand. Some attempt to theoretically justify what you two are
> doing would be appreciated.

I'm not sure what "it" is that you think we haven't considered. If
it's log-flat badness then that seems to have been the only such
measure being considered on this list for the past several years,
despite the objections (from psychology) that I thought I spelled out
in great detail when it was first mooted.

And by "theoretically justify" do you mean justify purely from
mathematical considerations? I believe that to be futile. It
eventually needs to be grounded in human psychology, both perceptual
and cognitive.

I understand you're still in favour of log-flat cutoffs which can be
written in the form

log(err) + k * log(complexity) < x

Paul and I have been considering those of the form

err^p + k * comp^p < x

which can be made to look a lot like the previous one when 0<p<0.5.

Paul and I have not so much been trying to theoretically justify, but
rather empirically determine, appropriate values for p, admittedly
based on some pretty sketchy and anecdotal evidence. But that's all we
have.

By far the greatest body of evidence, about which temperaments people
consider musically interesting or useful, relates to equal
temperaments, particularly at the 5-limit.

And we find that what works best is a value of p that's slightly less
than one, i.e. the cutoff functions that we construct based on our
knowledge of which ETs have been popular historically, are somewhere
between log and linear, but much closer to linear.

Since you and Paul seem to have done a marvelous job of giving us
error and complexity measures that generalise from equal temps to
linear temps and beyond, then it seems likely that the general shape
of equal-interest contours we find for equal temps will be repeated
for higher dimensions. I suppose you could say this is the theoretical
part of the justification.

But rather than trying to come up with precise values for p and the
scaling constants for cutoffs, we are looking for what we call
"moats". These are places where moderate changes in these constants
will make no difference to which temperaments are included. They would
ideally look like a band of whitespace on the graph shaped like a pair
of back-to-back horns (something I hadn't realised before). In other
words it doesn't matter so much if a moat has a narrow waist. What is
most important is that it is wide near the axes.

But we can't just use any old moat. There are bound to be some very
wide moats that are unusable because they bear no resemblance to an
equal-interest cutoff.

The idea is that they should agree with the subjective cutoff
functions (implicit or otherwise) of as many different people as possible.