optimizer

Paul,

you wrote:
>I think I showed that one can take a smooth ride through these
>well-temperaments "downhill" to 12-tET -- its just that the optimizer got a
>little lost.

I'm not sure I'm clear here: if the optimizer only returns rel. min's then
there's no smooth ride which is all downhill, right?

>Well I could always use various different starting points in order to find
>that out, and as I mentioned, I tried quite a few. Any suggestions?

Ok, I have one suggestion: to imput the optimizer with random 12-t scales,
asking it to discard any results which contain unisons, and see if you get
any strongly differing from the well/equal temperments. But as I say this
I'm thinking, couldn't we reason that we won't get any results differing
widely from 12-tET since 12-tET is the highest ET to give results w/o
unisons. Higher numbered scales tend to settle into unisons because the
entropy in the adjacent notes is too high and close enough to the other
side of the largest hump in the entropy curve. So if any of the notes are
much closer together than 100 cents, we should expect the same results.
But this is very loose reasoning. Maybe the random inputs would be useful
for lower cardinalities, if you can do them without memory overloads. I'm
interested, especially in the 12, 7, and 5 tone cases, to see if we find
any results which aren't meantonish.
I have another idea. Could you qualify the entropy calculation so that,
instead of decreasing to minimum entropy at the unison, the entropy
increases asymptotically approaching unison? This would prevent pairs from
settling into unisons, and the optimizer could then give good results for
9, 11, 13 and higher cardinalities.

jason

--- In tuning@egroups.com, Jason_Yust <jason_yust@b...> wrote:
> Paul,
>
> you wrote:
> >I think I showed that one can take a smooth ride through these
> >well-temperaments "downhill" to 12-tET -- its just that the
optimizer got a
> >little lost.
>
> I'm not sure I'm clear here: if the optimizer only returns
rel. min's then
> there's no smooth ride which is all downhill, right?

I'm almost positive I could construct such a smooth ride for any of
the well-temperaments it found, which means that the points it found
were not really local minima but only looked that way to the
optimizer's algorithm (which, based on numerical methods, is not
foolproof).
>
> Ok, I have one suggestion: to imput the optimizer with random
12-t scales,
> asking it to discard any results which contain unisons, and see if
you get
> any strongly differing from the well/equal temperments. But as I
say this
> I'm thinking, couldn't we reason that we won't get any results
differing
> widely from 12-tET since 12-tET is the highest ET to give results
w/o
> unisons. Higher numbered scales tend to settle into unisons
because the
> entropy in the adjacent notes is too high and close enough to the
other
> side of the largest hump in the entropy curve. So if any of the
notes are
> much closer together than 100 cents, we should expect the same
results.
> But this is very loose reasoning. Maybe the random inputs would be
useful
> for lower cardinalities, if you can do them without memory
overloads. I'm
> interested, especially in the 12, 7, and 5 tone cases, to see if we
find
> any results which aren't meantonish.

I agree with your reasoning and I'll try all these things.

> I have another idea. Could you qualify the entropy
calculation so that,
> instead of decreasing to minimum entropy at the unison, the entropy
> increases asymptotically approaching unison? This would prevent
pairs from
> settling into unisons, and the optimizer could then give good
results for
> 9, 11, 13 and higher cardinalities.

As I replied to John, I don't think any "new" local minima I found
this way would be meaningful, but would be an artifact of the
artificial modification of the curve.