I Wrestling with the demon.

In common with many or most members of the list, I'm not exactly rolling
in dough. Also in common with many or most, I'd love to make a living,
or even a tiny token dollar, from the pursuit of music. So there is a
part of me that wants to hoard any bit of knowledge I might have.

On the other hand, the heck with that attitude; this is MUSIC.


II Methods.

Springs. Thousands of springs. Hundreds of thousands, if necessary.
Springs are a way that competing interests can communicate with each
other for the purpose of overall reduction of energy, or "pain".

At least one list member has sneered at the idea of using the word
"pain" in a meaningful model of adaptive tuning. To this I would say,
if your quibble is with the choice of words, suggest others. If
your thesis is that there is no meaningful way to measure musical
tradeoffs, well, I'd say that history is in the process of proving you
wrong.

We all know that tuning, whether adaptive or fixed, is a matter of
tradeoffs. If anyone is in doubt, then consider the simple sequence
C to A to D to G to C. This is the "comma pump", and there is no way to
avoid some "pain". Either the ending C is shy of the beginning by
80:81, or someone along the way has to give.

How much do the competing needs hurt when they're not fully met? In
general, my belief is that pain is proportional to the square of
deviation. Deviation can be: mistuned intervals, motion in the tuning
of a note continuously sounding or remembered, or an overall drift of
the center of tuning away from what is expected. And this is only a
partial list, to be sure, but it includes perhaps the three most
important factors.

When a fifth, say, is mistuned by 1 cent, most ears cannot tell. Bump
it to 2 cents, as in 12-tET, and the tuning is still very good. But
double it again, to be near 1/4 comma meantone reduction, and the pain
is real. Double it again and the interval really starts to hurt. I
would measure the relative pain as 1, 4, 16, and 64, the square of
deviation.

Again, when a continuously sounding note is retuned by 1 cent, even the
most sensitive ear grasps for a clue. But with each doubling of motion,
pain jumps, I would say by the square of motion.

It happens that this relationship is in close correspondence to physical
reality. In a former life, I worked in the design of nuclear power
plants (don't all hiss at once!), and in that life I learned a lot about
springs. An ideal spring has some point of rest, and its resisting
force to deviation is linearly proportional to that deviation. If
deflecting 1 inch causes 10 pounds of resisting force, then deflecting 2
inches will cause 20 pounds of back pressure.

But didn't I say squared, not linear??? Ah, but the energy held by a
spring is proportional to the square of deviation, and in this model,
energy and pain are equivalent.

In the physical model, it can be shown mathematically that minimum
total system energy is represented by a summation of spring force at
each node of zero. To put it another way, any change of the state of
deflection which is supported by net spring force also reduces the
total energy, or "pain", of the system.

So, my tuning model in principle is extremely simple. I load the
sequence and start wiring up springs. Across every simultaneous
("vertical") interval, springs pull toward ideal JI tuning, while at
the same time allowing deviation therefrom at measured cost. Each
note's tuning is sprung to "ground", the center expectation for tuning
of that note. And each note is sprung horizontally in time to
previous and successive tunings of itself.

The strength of each spring is dependent upon both the particulars of
the moment (largely the loudness of the notes) and upon chosen
coefficients of importance that partially reflect individual taste.
I tolerate a lot of motion for the sake of good tuning; other list
members have ears that easily cringe at motion and tolerate greater
mistuning.

Once the springs are wired, I simply move the tunings of each node to
achieve zero total spring force everywhere, this being in one-to-one
correspondence with minimum pain, to the extent that the model is
correct.

One way of solving a spring matrix is by inverting the matrix. But
a large musical sequence is too large for this, I believe. I just use
successive approximation, in conjunction with "monte carlo" pseudo-
random motion through the piece.

Note that the springs need not be linear. What is important is that the
spring force represents the derivative of pain with respect to tuning
motion, whether linear or not. Once non-linearities are possible,
matrix inversion is not an option; only successive approximation can
do the job.

Many important factors are still not represented by this method. But
at the same time, the results are impressive, I believe. I'll soon
be posting more examples to illustrate. All feedback is welcome,
ESPECIALLY (constructive) negative feedback.

JdL