In theory, hitting the middle C on a piano should produce a sound wave with a frequency of 523.25 Hz and nothing else. However, running the resulting audio through a spectrum analyzer, it becomes obvious that there’s much more going on. This is true for all other instruments, from tubas to trumpets, basoons to flutes, contrabasses to violins. Play any note and you’ll get a package of sound waves at different frequencies rather than just one.
First of all: why is that? Let’s focus on stringed instruments. When you plug the string, it goes into its most basic vibration mode: it moves up and down as a whole at a certain frequency f. This is the so called first harmonic (or fundamental). But shortly after that, the nature of the vibration changes and the string enters a second mode: while one half of the string moves up, the other half moves down. This happens naturally and is just part of the string’s dynamics. In this mode, called the second harmonic, the vibration accelerates to a frequency of 2 * f. The story continues in this fashion as other modes of vibration appear: the third harmonic at a frequency 3 * f, the fourth harmonic at 4 * f, and so on.
A note is determined by the frequency. As already stated, the middle C on the piano should produce a sound wave with a frequency of 523.25 Hz. And indeed it does produce said sound wave, but it is only the first harmonic. As the string continues to vibrate, all the other harmonics follow, producing overtones. In the picture below you can see which notes you’ll get when playing a C (overtone series):
(The marked notes are only approximates. Taken from http://legacy.earlham.edu)
Quite the package! And note that the major chord is fully included within the first four overtones. So it’s buy a note, get a chord free. And unless you digitally produce a note, there’s no avoiding it. You might wonder why it is that we don’t seem to perceive the additional notes. Well, we do and we don’t. We don’t perceive the overtones consciously because the amplitude, and thus volume, of each harmonic is smaller then the amplitude of the previous one (however, this is a rule of thumb and exceptions are possible, any instrument will emphasize some overtones in particular). But I can assure you that when listening to a digitally produced note, you’ll feel that something’s missing. It will sound bland and cold. So unconsciously, we do perceive and desire the overtones.
If you’re not interested in mathematics, feel free to stop reading now (I hope you enjoyed the post so far). For all others: let’s get down to some mathematical business. The frequency of a note, or rather of its first harmonic, can be computed via:
(1) f(n) = 440 * 2n/12
With n = 0 being the chamber pitch and each step of n one half-tone. For example, from the chamber pitch (note A) to the middle C there are n = 3 half-tone steps (A#, B, C). So the frequency of the middle C is:
f(3) = 440 * 23/12 = 523.25 Hz
As expected. Given a fundamental frequency f = F, corresponding to a half-step-value of n = N, the freqency of the k-th harmonic is just:
(2) f(k) = k * F = k * 440 * 2N/12
Equating (1) and (2), we get a relationship that enables us to identify the musical pitch of any overtone:
440 * 2n/12 = k * 440 * 2N/12
2n/12 = k * 2N/12
n/12 * ln(2) = ln(k) + N/12 * ln(2)
n/12 = ln(k)/ln(2) + N/12
(3) n – N = 12 * ln(k) / ln(2) ≈ 17.31 * ln(k)
The equation results in this table:
n – N (rounded)
And so on. How does this tell us where the overtones are? Read it like this:
- The first harmonic (k = 1) is zero half-steps from the fundamental (n-N = 0). So far, so duh.
- The second harmonic (k = 2) is twelve half-steps, or one octave, from the fundamental (n-N = 12).
- The third harmonic (k = 3) is nineteen half-steps, or one octave and a quint, from the fundamental (n-N = 19).
- The fourth harmonic (k = 4) is twenty-four half-steps, or two octaves, from the fundamental (n-N = 24).
- The fifth harmonic (k = 5) is twenty-wight half-steps, or two octaves and a third, from the fundamental (n-N = 28).
So indeed the formula produces the correct overtone series for any note. And for any note the same is true: The second overtone is exactly one octave higher, the third harmonic one octave and a quint higher, and so on. The corresponding major chord is always contained within the first five harmonics.