Marc Damshek wrote: > 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. Marc, I don't think that all that's needed is linear algebra. Before we can even talk about a "self-consistent tuning solution that minimizes the bad effects of random irregularities" we would have to define a measure with respect to which the minimization is done. That is, we would have to define a totally unambiguous way of saying that tuning "A" is better than tuning "B" and this has not been done, even for aural tunings. There are a number of qualities that have been identified to evaluate a tuning: beatless or slightly sharp double octaves, even progression of the beat rate for thirds, tenths. Low beat rates for fours and fifths. But when the achievement of one of these goals comes at the expense of another of these goals, in the end it comes down to personal preference as far as which goal is more important and by how much. So the problem cannot be reduced to mathematics because it is not well-defined. I can't even say for sure how to tune a good unison if wound strings are involved, because the partials don't all zero-beat at the same point. Sure, in extreme cases a tuning is so bad that everyone will agree that it violates this or that criterion. But when tunings get fairly good, then you will start seeing the lack of a clear definition of the "perfect" tuning. And this as it should be in an artistic field. You also asked about the algorithms involved in current ETDs. A good start is to read Dr. Sanderson's patent #5285711 on the FAC method, which actually discloses the entire algorithm. I have read Mr. Reyburn's patents on the RCT, and although they describe a lot of the procedure, the exact formulas are not given. My TuneLab program does not contain any automatic algorithms for calculating a tuning, but instead allows an interactive input from the user to make the decisions while displaying the consequences of those decisions based on inharmonicity measurements. And if you want an automatic tuning curve calculation you can see David Porritt for his plug-in to TuneLab called Calcul8. I don't know what algorithm he uses either. Since you identify yourself as a computer-guy, I invite you to try your hand at solving the optimization problem that you identified. You can try out your ideas very easily by doing as David Porritt has done and write a plug-in for TuneLab. TuneLab will supply you with an ASCII file containing the measured inharmonicities and you can pass an ASCII file back to TuneLab containing your calculated ideal tuning. For details see the link "For Programmers Only" in my web site, http://www.wwnet.net/~rscott I have heard the assertion that if you take inharmonicity readings of every note on the piano, then, in theory, you can construct the ideal tuning. I don't support this notion for mostly practical reasons. For one thing, as others have already mentioned, it takes an inordinate amount of time to make all these readings. And no one has yet mentioned the possibility of measuring every string - not just every note. I don't think even Steve Fairchild's program does that. But if you do measure just one string of each unison, aren't you making an assumption of uniformity that is similar to the assumption that is made when only a few notes are sampled and are taken as representative of other notes not measured? The tuning optimization problem becomes more and more ill-defined the more irregular the inharmonicities. It is precisely in these cases where individual artistic judgement is most needed because of the extreme compromises required. Thus it is not likely that a computer program will be of much help in deciding what to do with one bad string. I do believe, however, that there is value to taking more inharmonicity readings rather than fewer. It is not because I want to find some irregularities. It is because I want to avoid using irregular measurements. One weakness of the FAC method is that it uses just three notes. What if one of those notes happens to be irregular and not representative of the other strings in the piano? The result is that a tuning will be generated based on a false premise, and will not match the piano as a whole. But if you start will 10 or 12 measurements, then there is an opportunity to do some filtering of bad data. The 10 or 12 notes can be analyzed for consistency and the worst (un-conforming) measurements can be discarded and the tuning can be constructed based on remaining consistent measurements. Or, at the very least, an algorithm can be devised that depends on all the measured notes equally. Thus the effect of one "bad" measurement will not be as great as it would be if it were one of only three measurements. -Robert Scott Real-Time Specialties
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