First, let's assume that all the bars for an instrument are to be cut from the same thickness of metal plate, preferably from the same stock. There can and will be minor differences in metal of the same thickness from batch to batch, so best if possible to do it all from the same piece. Your local sheet metal dealer will probably have a wide selection of stock, in various widths and thicknesses. The principles of tuning are the same, regardless.
For you familiar with the instruments of the Javanese Gamelan (my chief musical milleu), I use 1/8 inch stock for slenthems and gender-like instruments, and 1/4 inch stock for instruments like the demung, saron, or peking. The rule is: the thicker the stock, the longer a key will have to be to get the right pitch.
The first step is to cut a test piece. I like to use 30 cm. as a handy length - it often comes out to be one of the keys I want anyway. For example let's say we cut a 30 cm. piece of 1/4 inch aluminum plate (width doesn't matter that much, I usually do 2.5 - 3 inches on lower instruments, 1.5 - 2 inches on the higher pitched instruments). When we measure the pitch, we get F# above middle C (your answer may be different). We convert this into Hz, or cycles per second: 370 Hz.
Next we construct a tuning table for the instrument we are building. Let's say that we are building a demung-like instrument tuned to a pentatonic scale:
1 = C 2 = D 3 = F 4 = G 5 = A
The first thing that we notice is that our test key doesn't match any of these notes exactly. But don't worry, it won't go to waste. We will be able to trim it to fit one of the other notes in our scale.
Next we need to convert the pitches in our scale to Hz. I have a program on my computer to do this for me, but if you don't, then go here and I'll show you how to do it with a scientific calculator (including the one that comes with Windows 3.1).
1 = C = 262 Hz 2 = D = 294 3 = F = 349 4 = G = 392 5 = A = 440
Now there's some math. The first thing to do take the figures for the test piece (30 cm, 370 Hz), and perform this operation on them:
L1 * L1 * F1
Where L1 is the length (in cm), and F1 is the frequency (in Hz), for this result:
Let's call this number K.
Then we need to plug it into this formula:
(L2)^2 = K/F2 Where: L2 = the length of the desired key, in cm. F2 = the desired frequency, in Hz
So, if we want to figure the length of the bar for our"1" key, we get something like this:
(L2)^2 = K/F2 (L2)^2 = 333,000/F2 substitute 333,000 for K (L2)^2 = 333,000/262 substitute 262 (from our tuning table) for F2 (L2)^2 = 1271 divide L2 = 35.7 cm take the square root of both sides
Using this procedure, we can make a table of the legths of the bars for our instrument:
|Key No.||Note||Frequency||Bar Length|
When you cut the bars, make them 2-3 millimeters too long. It's easier to shorten the keys if they're loo low in pitch, than to make them longer again if they are too high.