Stringed and Piped Instruments

All sound consists of pressure waves that travel through the air from their source to your ear. Some of these pressure waves are created in turn by standing waves—anything from a vibrating violin string to an oscillating pipe organ air column. When you pluck a guitar string, the string itself wobbles back and forth, pinched at each end, and this movement is what makes the sound wave that travels to your ear.

However, most instrument standing waves do not just oscillate at one frequency. Instead, they produce many different frequencies at once. These different frequencies are integer multiples of the base frequency, called harmonics.

From left to right: base wave with frequency f_o, harmonic at 2f_o, harmonic at 4f_o.

For example, if a guitar string was tuned to play a middle A (a note with a frequency of exactly 440 Hz), it could also create sounds an octave above at 880 Hz or two octaves above at 1760 Hz. ⁹.

The same thing happens in a pipe organ, though for pipes the position of the wave's nodes and antinodes are reversed:

(Source)

These harmonics combine together to create what human ears hear as a single note, and different combinations of harmonics can change the timbre of the sound, or even what note is heard.

When a guitar maker goes to make an instrument, the sound the strings make is very much a science. We can find the resonant frequency of a string based on its mass m, length L, and the tension T it's held under.¹⁰ These three variables give us the velocity of the waves that interfere to form our standing wave:

The resonant frequency of a string is given by dividing this value by twice the length of the string:

So the frequency that the string vibrates at is only dependent on its length, mass, and tension. Of these three, mass and length are difficult to change on the fly—but tension is very easy to increase and decrease. That's why, whenever your guitar or violin or cello or ukelele gets out of tune, you can adjust the note each string makes by twisting the tuning knobs and either pulling the string more taute or adding slack. This changes T, which in turn changes f_o and the sound of the string.