Sounding Number
Dr. Rachel Hall
Harmonic and Inharmonic Spectra
All the instruments we’ve studied so far—wind and stringed instruments (including the
human voice, which has vocal cords)—have harmonics that are whole-number multiples of
their fundamental frequency. That is, their spectra contain either frequencies
f0 , 2f0 , 3f0 , 4f0 , . . .
or
f0 , 3f0 , 5f0 , 7f0 , . . . .
Any instrument whose spectrum contains only whole-number multiples of its fundamental
frequency is said to have a harmonic spectrum. (For a wild ride, check out videos of Tuvan
throat singing!)
Does every instrument have a harmonic spectrum?
The answer to this question is no. For one thing, some instruments, such as most drums,
have no discernible fundamental frequency. Even drums, such as timpani, that can be tuned
don’t have a harmonic spectrum. Here’s the picture of a spectrum of the tabla, a pitched
Indian drum (pitched means that it has a fundamental frequency that we can hum along
with):
Because this sound is quite low, I used a large-sized sample (remember that low-frequency
waves are longer) and a log-frequency axis. I found peaks in the spectrum at 59 Hz, 119 Hz,
212 Hz, 326 Hz, 490 Hz, and 658 Hz. Although the irst two frequencies (59 Hz and 119 Hz)
are approximately harmonic, the others are not. We say that the tabla’s spectrum is
inharmonic.
I recorded my Tibetan singing bowl; here’s the spectrum, which is also inharmonic:
The peaks are at 264 Hz, 528 Hz, 754 Hz, and 1417 Hz. Again, the irst two frequencies form
an octave, but the others are not harmonic. Looking at the sound wave, we can see beats at
approximately 2 Hz. This means that the bowl produces two tones that are 2 Hz apart. The
spectrum plot in Audacity is not able to detect this.
The reason that drums, bells, and singing bowls (which are members of the bell family) have
inharmonic spectra goes back to the wave equation. The wave equation that we used for
stringed and wind instruments is called the one-dimensional wave equation because those
instruments are essentially “one-dimensional”—meaning that we only have one position
variable, x, in the equation. However, in a drum, you have two position variables (x and y)
because the drum head is two-dimensional, while in a bell you need three position variables
(x, y, and z). The two- and three-dimensional wave equations are needed for drums and bells.
The solutions to these equations do not result in a harmonic spectrum.
One way to understand this difference visually is to think of the standing waves that are
possible in a string (we know these result in frequencies that are multiples of the
fundamental) and then see how a drum might vibrate. There are several excellent
demonstrations on YouTube. Try searching for “drum modes vibration” and “chladni plates.”
(Chladni plates are lat square metal plates that can be vibrated, either by bowing—like this
dude does—or electronically. Different frequencies produce different standing waves.) The
Wikipedia page “Vibrations of a circular drum” has computer-generated animations.
Chladni plate experiment and Chladni patterns published by John Tyndall in 1869.
From the UCLA Physics Lab demonstrations web site.