Building a Xylophone, Part 2: Xylo-brations
Once I decided to build a xylophone, the first question I had to answer was how I would tune the bars. I didn’t expect a simple answer, but initial searches bombarded me with an avalanche of terms like ‘antinodes’ and ‘Fast Fourier Transforms.’
Here’s a snowball to ease us in: the variable that affects frequency most for a bar or tube is length. If you brush a set of wind chimes from short bars to long, it clearly descends in pitch. The same tenet holds for organ pipes, glockenspiels, and xylophones, along with many other percussion instruments. For my xylophone, then, could tuning simply be a matter of cutting the bars to the right length?
Of course it couldn’t be that easy. It turns out that when an object resonates, it does so at multiple frequencies. The lowest, and generally loudest, frequency is the fundamental frequency, which is the note that we primarily hear. The other resonant frequencies are called modes or overtones, named the second mode/first overtone, third mode/second overtone, etc. in order of increasing frequency. The fundamental frequency is the first mode. If those overtones happen to be integer multiples of the fundamental, they are called harmonics, and sound particularly… harmonious.
For any sound, we can identify the modes, or resonant frequencies, by plotting all frequencies present in that sound against their power level in decibels (by using software that performs the above-mentioned Fast Fourier Transform). Such a plot is called a power spectrum, and the peaks in the spectrum are the different modes within the sound.
One interesting aspect of power spectra is they can give us clues as to the tone or quality of the sound. For example, a spectrum with a very dominant fundamental frequency might have a crystal clear, pure quality, and might have been produced by a flute or violin. Conversely, a spectrum with many modes vying for top billing might have a fuzzy, woody, or even cacophonous quality, and might have been made by a bassoon or bagpipes. The plot above has a particularly high number of peaks because there are many instruments playing different notes simultaneously.
Each mode has a waveform associated with it, which describes the oscillating vibration of the air molecules that propagate the sound. When the waveforms from each mode are added together, they create a single complex waveform that reconstructs the original sound. In some cases, we can easily visualize the waves being created in the sound-producing instrument, such as a guitar. When you pluck a guitar string, it vibrates strongly in the middle, which creates the fundamental frequency, and also activates other modes.
The antinodes of the wave are the points along the string that move the most, and the nodes of the wave don’t move at all. In a string instrument like a guitar, the two ends of the string are fixed, which forces the wave to have nodes at those points. If you were to pluck the guitar string directly in the center, that would favor the modes that have an antinode there (odd modes) and dampen those with a node (even modes). This is likely why it sounds better to strike the string off center—that way all modes can be excited.
In a xylophone, the bar itself acts like the string — it physically vibrates, albeit less dramatically. However, the ends of a xylophone are not fixed, which means the natural nodes of the fundamental frequency are located about a quarter of the way into the bar.
Because of the free nature of the vibration, the second and third modes of a xylohone bar produce frequencies that are 2.76 and 5.4 times the fundamental. You may have noticed those are not integers— if you didn’t, then take a moment to observe so. This means the sound of a struck block of wood is naturally discordant, which begs the question, how do we make the block of wood sound more pleasant, and perhaps more importantly, do we care?
It turns out we do, though interestingly, people didn’t care about tuning the higher modes of xylophones and marimbas until the 1920s. A marimba is like a xylophone but with larger bars and a wider range. Another key difference, I discovered, is that the second and third modes of a xylophone are tuned to multiples of 3 and 5 times the fundamental, whereas a marimba is tuned to 4 and 10 times the fundamental, which creates a deeper, richer tone.
The trick to tuning higher modes is to carve out the underside of the bar, which affects the shape of the waveforms when struck. In commercial instruments, this tuning is done first by machine, then by xylophone tuning prodigies who meticulously guess and check. Being fluent in computer and not a xylophone tuning prodigy, my next question to research was, how can I get a computer to do the work for me?
Next Up: Xylo-speriments
