ETD Question

[Marc Damashek]

	Many issues have been considered in this thread -- but not 
the algorithms used by the ETD's to decide on target pitches, which 
are less than optimal when dealing with subtle imperfections in the 
strings. Here's an opinion from a physicist/tuner/computer guy -- I'm 
going to stick my neck out with a specific, testable hypothesis.

	I had a tuning epiphany about 15 years ago the first time I 
played with a friend's Conn Strobotuner at his ancient upright. I was 
thrilled to see the inharmonicity of the partials right before my 
eyes, and to see it correlate so well with what I was hearing. But 
what was really striking was that the inharmonicity of higher 
partials on various strings was not always a smoothly increasing 
function of the partial number, as one might have expected.

	If I were to plot up the number of cents each partial was 
sharp, the points would not lie on a smoothly rising curve, but would 
instead deviate randomly from that curve, with some of the deviations 
being pretty sizable. The Strobotuner showed me that the successively 
faster advance rates of the upper partials on some strings wasn't 
uniform: some partials were obviously slower or faster than they 
ought to have been, and didn't interpolate the frequencies of their 
neighbors.

	Is this higher-order effect well known? I can't recall ever 
seeing it discussed. It most certainly exists and is one reason why 
every piano has its very own personality. It results from string 
irregularities of one kind or another (density, diameter, etc., 
including nicks in the string, rust, and so on). Such imperfections 
affect the frequency of different partials differently, depending on 
the detailed physical distribution of the defects. For instance, 
altering a plain wire string right at its midpoint (for instance, by 
nicking it or loading it with some fine wire wrapped tightly around 
it) will not affect the (stretched) even harmonics, which have a node 
at that point. But it will change the frequency of the odd 
(stretched) harmonics, as well as inducing other craziness such as 
false beats between the various vibrating string segments.

	So the hierarchy of tuning subtlety goes something like this:

	perfect strings (harmonics at integer multiples of the fundamental)

	--> strings with inharmonicity (harmonics at successively 
higher frequencies than integer multiples -- stretched scales)

	--> strings with inevitable random imperfections (random 
deviations from smoothly stretched scales)

	Every piano tuning is a compromise in which we try to 
accommodate those random deviations as best we can, and that's what 
really separates the sheep from the goats.

	Now, how do existing ETD's decide what the 'best' frequency 
is for a string? Obviously not by simply targeting the equal-tempered 
fundamentals, because of inharmonicity (first-order correction). But 
do any of them explicitly optimize their targets based on accurate 
measurements of the partials on all strings (at least in octaves 2-5, 
say), which can exhibit random deviations from a smoothly stretched 
scale (second-order correction)? Probably not -- note that this is 
different from measuring just the A's up the keyboard, for instance, 
in order to find an appropriate stretching curve (which is a Good 
Thing as far as it goes).

	While it might seem awfully complicated to obtain a 
self-consistent tuning solution that minimizes the bad effects of 
random irregularities, it's not impossible, especially when you're 
toting around a fast laptop computer and can already accurately 
measure the partials. (The solution involves some straightforward 
linear algebra.) I strongly suspect that it is precisely because 
ETD's ignore random irregularities in real strings that the best 
aural tunings still outstrip the best ETD tunings, especially on the 
worst pianos. If this situation were to change -- and I claim that it 
can -- we would hear the improvement immediately.

	It would be very valuable to get more information from the 
people who have designed and done the programming on the ETD's in 
order to shed some light here. It would also be a whole lot of fun to 
produce a working prototype and set it to work on a piano-shaped 
object.


Marc Damashek
Hampstead, MD