John, you have posted on your web site the formula most people
contributed: (S + L) x 44 = T
[ http://www.playercare.com/tubing.html#trackerbar ]
Maybe I'm overanxious and overcritical, but what you did is in
my opinion quite risky and "wrong": you posted a formula based on
a mathematical model without verifying it. This is the classical
error which leads to the prejudice, "See, mathematics does not work
in the real world."
I have never repaired a stack myself up to now, but just reading the
postings reveals the following:
The data you yourself gave, and data posted by others, indicate that
the formula does not work: you measured 10" and 24" for shortest and
longest tube respectively, and told us you would guess that 150' should
be enough. Now, (10"+24")x44 = 124.66'  quite a lot less. In other
posts we heard about lengths of 200', 180' (and that 300' are too
much). Also, your measurement of length differences (1/2") indicates
that something is wrong.
The formula above is based on the following assumptions (or
"mathematical model"):
* Lengths increase linearly.
* The two extreme tubes measured are wellchosen representatives of
the whole tube bundle.
Both assumptions are, as I see it, not at all proven. So, if we want
to be good engineers, we should do what all engineers have done in the
last 200 years in such cases:
(a) Try to improve the mathematical model. This is hard (or at least
much) work, e.g., measuring 20 or 30 tube bundles tube by tube and
finding a better model (e.g., a length increase with the square root
of length, or whatever). I would do this only if the methods below are
not sufficient.
(b) Add practical correction factors. My suggestion of putting in
a factor of 1.2 is such a suggestion; however, because it is based on
a single measurement (your indication that 150' should be needed,
instead of the 124.66' from the formula), it is also open to critics.
At least, it should be tried out a few times.
(c) Add more _stability_ to the formula: therefore, I made the suggestion
of measuring 8 tubes instead of 2 (which leads to the, in my opinion,
nice algorithm following:
1. Measure, in _inches_, the 2 longest tubes on each side and the 4
shortest tubes, altogether 8 tubes.
2. Add up all the lengths.
3. Add 10%.
4. The sum is the length of _feet_ needed.
All this is not to say that the suggested formula (S+L)x44 is "bad"
for the intended purpose of telling onetime customers/users how much
to order; this I cannot prove. But I try to argue that a practical
formula must be more than an invention based on some abstract theory:
Unfortunately, the world is not so simple, in most experiences.
On the other hand, I would never want to agree with the people saying
that there's no need for such a formula. One reader asked something
like, "Who would ever need/want to know more?" (More than that you
need 200', if I understood it correctly; the rest will be used "later".)
First, we all want to know more! There have been postings here of
mathematical models of sound production in organ pipes. Of course,
neither organ builders, nor organ players nor composers _need_ to
know this. Still, being human is being curious. To get a little
philosophical: Yes, we (some) want to know it, even if we do not need
it (right now, I'd add).
Second, as John pointed out, there _is_ a need for knowing more "even"
in practical settings, like the people who repair only one stack.
Actually, such computations (or estimates) do "pay off"...
Hopefully, the theoretical formula is okay, and your customers are happy.
I would not (yet) bet on it.
Regards
Harald M. Mueller
