ETD Question

[Robert Scott]

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