In response to John Tuttle's search for a simple mathematical formula
to determine the total amount of tubing needed for a player piano,
if the shortest and longest lengths are known, I offer the following
simple solution.

If you have a numeric series where the increment between any two
numbers is the same (called a geometric series), then the total of all
the numbers is the sum of the first number plus the last number all
multiplied by half the number of values in the series.  Take the
numbers 5 through 10.  Add the 5 and 10 to get 25.

There are a total of 6 numbers, so half of that is 3.  If you now
multiply 25 by 3, you get a total of 75 for the sum of the numbers in
the series.  This can be checked by adding 10 + 11 + 12 + 13 + 14 + 15
to get 75.  One way to convince yourself that this works is to look at
the series I just wrote and notice that the first number plus the last
is 25, as is the second number plus the second to last and so on.
Also, there does not need to be an even number of values for this to
work.

Now, for the tubing problem, we know that for half a stack, we have 44
pieces of tubing.  Since we have two identical half stacks, we have two
identical series.  Therefore, we can multiply the sum of the shortest
and longest by 44 to get the total length needed for the entire piano
(44 times half from the above formula times two half-stacks is 44).

So, in the example John gave, we add the shortest tube (10") to the
longest tube (24") for 34" and multiply it by 44 to get a total length
of 1496".  Divide this by 12 to get 124 feet 9 inches.  To write it as
a formula, we have:

  (S + L) * 44 = T, where S is the shortest length, L is the longest
length and T is the total length.

I hope this answers the question and that my explanation was simple
enough to be followed by all those interested in it.

Jack Breen
Southborough, MA

P.S. [later]:  I realized after sending the exact solution to the
problem that John posed that there was a much easier way to estimate
the amount of tubing required if the purpose of the estimate was to
purchase tubing and you wanted to be guaranteed that you had enough.

Using the earlier explanation, if you take the shortest length in
inches and add it to the longest length in inches, and multiply the
total by 4 you will get approximately the number of feet needed for the
job.  This answer will only be high by about 8% (1 - 44/48).

So, to apply this to John's problem, (10 + 24) * 4 is 136 feet.  The
exact answer given previously was just under 125 feet, so the job would
have 11 feet left over, which could easily be eaten up by making each
tube only 1.5" longer.

Jack Breen