Robert Scott <rscott@wwnet.net> wrote: > 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... <snip> > > 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. > Thanks very much for your detailed comments. I agree with everything you've said here. I hoped to point out that once optimization criteria have been chosen -- the most important and interesting part of the operation, but still feasible -- solving for actual target frequencies will involve little more than matrix algebra. The criteria themselves are where the money is, and discussion has probably been stifled by the natural assumption that taking more strings into account quickly makes the problem intractable -- which it doesn't. Another misperception seems to be that even an awful piano might somehow be made beautiful by some kind of numerical magic, if only we knew how. Of course it won't: we have only one handle on each string, namely the tension, and if the spectrum of some string is atrocious, it will always sound lousy, because you'll never come within a mile of getting its partials to simultaneously match their brethren on other strings. > Since you > identify yourself as a computer-guy, I invite you to try your hand at > solving the optimization problem that you identified. It's a labor of love (I want to play an average piano that has been made to sound exceptionally good). As time permits, I'm working on a prototype that weights the various partial matches by the spectral products of the partials involved (e.g., 4:2, 3:2, 5:4, etc.) -- there's no point worrying about a particular wild beat rate if the power's down in the dungeon. The fine details of that weighting should be decided by experimentation, and should ultimately be under the user's control. >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 > This is a very thoughtful facility. I don't own TuneLab, and I'm a Mac sort of guy, so it's not clear whether I can take advantage of it. > 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'm hoping to come back with some hard information on that claim, because I'm not sure it's really true. For the task at hand, I'd certainly commit 30-60 minutes to measuring a particular piano once and for all -- it won't change significantly from tuning to tuning. Can I not take a 1-2 second FFT for preliminary frequencies and amplitudes of partials (negligible compute time), then use those freqs and amps as initial values for a nonlinear least-squares fit of a sum of specific sinusoids (whichever partials I choose to fit) to give accurate amplitude, decay rate, frequency, and phase? Even if my computer cranks for 10 seconds on each string's data (an overestimate, I think), that still falls in my comfort zone. Probably most of the time will be spent moving a pair of mutes. I'd like to try it for octaves 2-5 or 2-6; heroic efforts at the top and bottom are probably not warranted, and I'd continue to do those by ear (subject to whatever we learn from experiment). > > 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 > This is precisely on target, and goes to the heart of experimental science and engineering. I hope it can be shown that the extra effort required is both acceptable and artistically satisfying. Do you think there would be enough interest to convene a discussion session on all these issues at the PTG Convention? At this early stage, face-to-face bandwidth has got to be more productive than all this typing. There are a lot of misconceptions to be cleared up, and a lot of worthwhile experimental work yet to be done. -- Marc Damashek Hampstead, MD ____________________________________________________________________ Get your own FREE, personal Netscape WebMail account today at http://webmail.netscape.com.
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