tuning bars

About scaling steel bars for tuning...

Heres one way to go about it, though of course even the best laid
calculations will require fine tuning once youre done, but this should
get you close. A hint: always cut the bars a tiny bit too long, since its
much easier to file off the extra than to try and reintroduce it
afterwards...

Say the pitch (fundamental) of the chime is at frequency f1, its got
length L1 and radius r1. Then

f1 = P r1/ (L1^2)

(the denominator here is L1 squared)
and P is a constant that depends on Youngs modulus, the density of
the material and a bunch of other things that will turn out to be
irrelevant.

Similarly, the fundamental of the second bar is at

f2= P r2/ (L2^2)

where r2 is the radius and L2 the length.

Now divide these two equations to get

f2/f1 = (r2 L1^2 ) / (r1 L2^2 )

where the constant P has disappeared.
You can use this formula to approximate the tuning for uniform bars
of different lengths and radii. For instance, if you use the same
radius material, then both r1 and r2 drop out. Let the ratio f2/f1 be
how you want the two tuned in relation to each other, and then solve
for L2.

Hope this helps

Bill Sethares