 # Cutting and Tuning Robotic Glockenspiel Chimes

After a few weeks of experimentation, I think I can now write sensible notes on how to cut and tune the chimes for a glockenspiel (metal xylophone) out of metal conduit. This is the first step of my Robotic Glockenspiel project, which I hope to end with a network-connected, Arduino-controlled set of chimes that can play Christmas carols.

Parts:

A few bits of background, which you can ignore if you like:

• I’m talking about pipes what will each be suspended near each end and struck in the middle.
• Physics of such pipes says that the suspension points – the Nodes – for a pipe of length L should be at L * 0.224 from each end of the pipe.
• A chromatic tuner usually tunes to what’s called an Even-Tempered scale. In such a scale, a note that sounds one octave above another has twice the frequency of the first note. Further, that octave is divided into 12 steps, called semitones, each of which is equally spaced (in frequency ratios) from the previous one.
• Modern even-tempered tuning is based on a standard frequency of 440Hz – cycles per second – as an “A” in the scale.
• Mathematically, to divide a ratio of 2 into 12 equal ratios, each ratio is the 12th root of 2, or about 1.0594630. So the frequency of each semitone is 1.0594630 times the frequency of the next lower one. Similarly, each semitone’s frequency is 1 / 1.0594630 times the frequency of the next higher one.

First cut an arbitrary length of pipe that, when tuned, will be the reference for all your pipe lengths. For 1/2″ conduit, I’d suggest cutting a piece that’s something like 400mm long – you don’t have to be exact at this point.

Once the pipe is cut, calculate the suspension points – the Nodes.
N = L * 0.224, so for a 400mm pipe, the nodes are at 400mm * 0.224 from each end. That’s 89.6mm. Since the nodes don’t have to be exact, we can use 90mm instead.

Place the 400mm pipe on the two thin supports, each support at 90mm from an end of the pipe.

Measure the frequency of the pipe by turning on the tuner then striking the pipe with your mallet (or pen).

Now you’ll need to tune the pipe by repeatedly measuring the frequency and trimming off pieces of the pipe:

• If the tuner says that the pipe is tuned sharp- that is, above the frequency of the nearest semitone – cut a few millimeters from the pipe length and measure the pipe frequency again. Repeat this process until the tuner says the pipe is flat – that is below the frequency of the nearest semitone.
• If the tuner says that the pipe is tuned quite flat, cut 2 or 3mm from an end of the pipe and measure the pipe frequency again. Repeat this process until the tuner says that the pipe is either slightly flat or is in tune.
• If the tuner says that the pipe is tuned only slightly flat, use the file to file off a very little bit from the pipe, then measure the frequency again. Repeat this process until the tuner says that the pipe is in tune.  If you happen to file off too much and the pipe is tuned sharp, cut off a few mm of pipe as in the step above.

Now you have one properly tuned pipe.  Next you’ll get to cut and tune all the other pipes, based on the length of this standard pipe.

Suppose that the tuner says that your tuned pipe is an F and that measuring the length with a meter stick, you find the pipe is 396mm long.

A little more math comes in here.  Given a pipe of length L1 that rings at frequency F1, you can calculate the length, L2 of a second pipe that will ring at a desired frequency, F2.  Put mathematically, with “sqrt” standing for square root, L2 = L1 / sqrt(F2 / F1).

A little music theory comes in here as well. One way of noting the order of semitones in an octave is with sharps: C, C#, D, D#, E, F, F#, G, G#, A, A#, B, C. Also, each octave (for example, a C to the C above it) is 2 times the frequency of the lower note. You can use this list and (and the fact that an octave is 12 semitones) to count the number of semitones between your first pipe’s frequency and the frequency you want.

Suppose that you want to find the length of a pipe that is the A above your F.  From our list above, we find that A is 4 semitones above F.  So our desired frequency, F2, is F1 * 1.0594630^4 (where ^ means “raised to the power”), or F1 * 1.0594630 * 1.0594630 * 1.0594630 * 1.0594630.

Substituting our calculation of F2 into the pipe length equation, we have
L2 = L1 / sqrt(F1 * 1.0594630^4 / F1), or
L2  = L1 / sqrt(1.0594630^4)

So, to calculate the length of a pipe that sounds N semitones above your first pipe (where semitones below are negative numbers), the math works out to
L2 = L1 / sqrt(1.0594630^N)

Plugging in all the stuff above, L1 = 396mm and N (of the A) is 4:
L2 = 396 / sqrt(1.0594630^4)

Wow, that was boring! At any rate, now you can cut and tune all the pipes in your glockenspiel, using these two formulae:

1. L2 = L1 / sqrt(1.0594630^N)
2. Nodes at =L2 * 0.224
3. Strike the chime at the center = L2 / 2.0

In my next post, I describe a build fail of the wooden frame. Making is not always as linear as it seems.

References: