Now, this is a treat for me. It has music, mathematics and physics, all in one big bite. Let’s imagine an electric guitar (it’s easier, as its sound is amplified, so we will be hearing more clearly the tones we are to produce). Let’s pluck an open string (the first one, e -lowercase as it’s the high e, not the low E; it’s the string closest to your legs, not your chest) and produce a fundamental frequency. Now, if we touch (touch, not press) the string with out left hand, right at the end of the 12th fret and strike the string with a pick (or with our fingernails MIYAVI-style, both work just fine), we will be hearing a different tone. This tone will be an octave higher, which is twice the frequency and half the length of the string. Now, let’s move our left hand towards the 7th fret and touch the same string. The sound will change once again. This time, we’re listening to a B, which is an octave plus a perfect fifth on top of the original sound of the open e string. That’s three times the frequency of the original sound and one third of the string’s length. Similarly, if you place your hand on the 5th fret of the guitar, you will be producing another E, which will be (octave + perfect fifth + perfect fourth =) 2 octaves higher when compared to your original tone. Again, that would be four times higher in frequency and one fourth of the string’s length. And the progression goes on and on and on… Let’s go with A2 at 110 Hz, as our fundamental frequency.

*NOTE that counting is apparently hard for us musicians, so we count 1 as the fundamental frequency and not a harmonic…

2nd harmonic: A3, 220 Hz (octave)
3rd harmonic: E4, ~330 Hz (above + perfect fifth)
4th harmonic: A4, 440 Hz (above + perfect fourth)
5th harmonic: C#5, ~523 Hz (above + major third)
6th harmonic: E5, ~659 Hz (above + minor third)
7th harmonic: a tone slightly lower than G5, ~784 Hz (above + smaller minor third or better, a harmonic seventh interval)

The list goes on… But I guess you’re getting the hang of it…
Now, if you’re a pianist. Go to an acoustic piano; upright ones are fine. Press down on your right pedal, so as to sustain the sound of the strings. Look to your left. For the sake of my example above, press rather hard the same note, A2. If you listen quite closely, you’ll be able to hear higher tones, as the harmonics of the key you struck resonate with the open strings of the rest of the piano. In case it didn’t work well enough, do it again but press A1 this time; it has a thiccc sound so the harmonics will be more prominent. Cool, huh?