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
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