Chopin goes unstable

[Some while ago, I wrote:]
>>Oh, and, at Robert Walker's request, I've allowed negative melodic
>>spring constants. As predicted, this can cause the matrix to go
>>unstable beyond some critical value, but smaller values cause the
>>melodic steps to spread (though why Robert wants this, I'll let _him_
>>explain!).

But, when I went back to run these negative melodic springs, I found
unstable matrices at every turn, even when the coefficients were only
_very_ weakly negative!

My early optimism was at a time when I'd hooked up only a few melodic
springs. Later work, with looser timing requirements for the sensation
of melodic motion, and thus many more melodic springs, was blowing up
more or less constantly!

This surprised me: my relaxation process had what I thought was a clever
instability-defeating feature: in relaxing each node (the tuning of one
scale degree over a short period of time in which notes sound
consistently), if the total spring constant was negative, I'd put the
tuning at the _top_ of the energy curve rather than the bottom. Zero
force was the calculation.

When this failed to ward off instability, I tried another "clever" idea:
each node would evaluate its sum spring strength; if negative, all the
negative springs would be modified so that the sum spring strength was
positive. This would achieve "local stability", and I thought, global
stability as well.

What an idiot! There are paths of instability which do not depend upon
a lack of local stability. All this is intensely interesting to me. I
had a sudden flash about the way there are so many, uhhh, inappropriate
romances in life: they happen to people who are seemingly stable, but
when confronted with "positive feedback loops" with some other person,
both succumb to destructive behavior. A mathematical and moral lesson
all in one!

I added several mechanisms to try to recover stability. All of them
dynamically adjust negative spring constants on the fly when it is
detected that unstable things are happening. This allows negative
springs in other locations to exist unmodified as long as their overall
effect is not unstable.

This was moderately effective, and I can at least produce a
semi-reasonable negative melodic 5-limit Chopin, which I have posted on
my web page (see tuning list for details, or just go to adaptune.com).

What does it sound like? Very interesting!! So far melodic spring
strengths are modelled in such a way that they are relatively more
powerful when motion is fast. The frenetic parts of the Chopin get
heavily distorted in an appealingly frenetic kind of way, while the slow
lyrical parts get tuned nicely.

JdL