We want the pipes to vibrate in the first mode. The picture shows how that happens: there are two nodes, 22.4% of the way in from each end, that stay put, and the pipes vibrate around them, the ends going up when the middle goes down, and
vice versa. The pipes will sound the best when they are flexibly supported around the nodes. In the glockenspiel, the pipes will be supported by rubber bands at the nodes. If you want to try how a pipe sounds, you need to support it at the nodes and hit it in the middle.
Once you buy your pipe, the only thing you can really control is the length of the pipes. The longer the pipe, the lower the main frequency. The formula is:
where
f is the frequency,
L is the length and
A is a number that could in principle be calculated from the thickness of the pipe walls, the diameter of the pipes and the speed of sound in the material. In theory, you could measure the diameter of the pipes and the thickness of the pipe walls, and then precisely calculate
A and figure out how long your pipes should be for the desired frequencies. The problem with that is that it is very hard to measure the thickness of the pipe walls and diameter of the pipes with sufficient precision. And you can't just use my data, because every pipe will be a bit different. Instead, we will cut a test pipe section, measure its length, use a mobile phone app to measure the frequency, and then use that to calculate
A.
Once we've calculated
A, we can choose the frequencies we want for the notes and solve the equation
f=A/L^{2} to calculate the length of the pipe. The solution, of course, will be that
L is the square root of
A/
f.
My value of
A was approximately 67,600,000 mm
^{2}/s. You can use this to get an approximate idea of how long your pipes should be, if you're using type M 1/2" nominal copper pipe like I was, for planning. You can look up the frequencies for different notes
here. For instance, C6 is 1046.5 Hz, so the length
L will be approximately the square root of 67,600,000/1046.5, or 254mm. If that's your lowest note, like it was for me, this will be the length of your longest pipe.