To my way of thinking, there is nothing very mysterious about inharmonicity readings that fail to follow the theoretical formula (proportional to the square of the partial number). This theory is based on assumptions about an ideal string - for example, that the string be perfectly uniform along its length (both dimensionally and metalurgically), and that the termination points be absolutely rigid and without lead-in bends. If anything is surprising it is that, given all the imperfections of real strings in pianos, the inharmonicity conforms to theory as well as it does. One facet of this problem that does not get much attention is the process of measuring inharmonicity itself. The assumption is implicitly made that the cents offset for each partial is a well-defined quantity that can be measured as accurately as desired. This is far from true. Let's look at the ways in which inharmonicity can be measured. If you use an Accu-Tuner, then the measurement of each partial is dependent on the user's ability to judge when the lights are stopped. Let's be honest. Those lights never stop. All strings are wild to some extent. What do you do if the lights seem to drift very slowly to the right, then, just before the sustain dies out, the pattern makes a sudden shift to the position it was in at the start of the note? This is exactly what can happen with a slightly wild string. You could say that since the pattern started and stopped at the same place, the overall movement was zero, so the lights are effectively "stopped". But is this interpretation the only one? What if the sustain was slightly shorter and the display goes out just before you can see the jump back to the original postion? Then you would be justified in saying that the SAT needs to be raised in pitch to match the string. But does it really make sense that there are two incompatible interpretations of the same string? You could also use a computer program like Dean's or mine to measure inharmonicity. I can't speak for Dean, but my measurement algorithm is based on criteria that are just a bit arbitrary. (If you really want to discuss peak-finding in the Fourier transform, e-mail me privately.) In 1993 I wrote an article for the Journal that described what I thought was the ultimate in accuracy for measuring inharmonicity. I would eliminate the sustain as a limiting factor in the measurement process. Instead of striking the string and analyzing the sound as it dies out, I would inject a small continuous excitation in the form of a pulsating electro-magnet held near the string. Then I would measure the strength of the response of the string optically. The pulsing frequency of the exciting electro-magnet was very slowly swept across the region for each partial. At each new frequency I waited until the response had stabilized before I measured it (sometimes taking as long as 30 seconds at each micro-increment of frequency). The whole process was automated so I could leave the room. It might take 30 minutes to measure one string. But I would get measurements as accurate as I wanted, right? Wrong! Although I could control the exciting frequency to any precision needed, and although I could measure response at each frequency as accurately as I wanted, the graph of response as a function of frequency did not always have one nice neat peak. On strings with audible false beats, the response curves had at least two completely separate peaks. (These peaks were separated by a frequency difference which was, not coincidently, the same as the false beat rate.) Even on "good" strings, the response curve did not define the peak frequency any less ambiguously than what you can get with the traditional (percusive) methods. So perhaps the problem is not with the measurement method, but is in the definition of the quantity being measured. The ambiguity is built-in. When you get different FAC readings for the same piano, don't assume that something in the piano changed. Maybe nothing changed. I would also like to comment on Ken's explanation of how taking into account para-inharmonicity can supposedly make a better tuning. Although I agree that the more inharmonicity readings the better, I don't think the deviations from the theory can in themselves be used constructively. Let me explain: Unless you take inharmonicity readings of all 88 notes, there are going to be some notes that you do not measure, but you still have to tune them. Somehow, the measurements that you do make are assumed to correlate with measurements that you did not make. Thus, a little predictive theory is built up whenever a tuning curve is generated. Maybe it's not identical to the general theory, but it is a theory just the same. But do the "deviant" measurements apply? That is, if I measure one deviant inharmonicity on one string in one piano, what are the chances that all the other strings of that piano are also deviant in a similar manner? Well, that depends on the deviation. If it was caused by an accident of stringing, then I would not expect to gain much useful information about the rest of that piano from that one string. In fact, I would be tempted to throw away that reading and measure a different, more representative string. The best use that I think can be made of more inharmonicity readings is to stick with a theory that represents most of the strings and use the multiple inharmonicity readings to average out the measurement errors on individual readings. This is what I see as the main benefit of ETDs that use more inharmonicity samples. The SAT is vulnerable to a single bad reading because so much depends on each of the FAC numbers. I think what experienced SAT users must do is filter all FAC numbers through their own judgement of what is reasonable for the particular piano. That way they can recognize a bad reading and take corrective action, sometimes manually adjusting the FAC numbers from their original measured values. RCT and TuneLab rely more on redundant measurements to achieve the higher level of confidence in the readings. Robert Scott Real-Time Specialties
This PTG archive page provided courtesy of Moy Piano Service, LLC