Standing waves are waves that are confined to a certain distance, much like strings on a violin in which the strings are fixed at both ends. These fixed points are called "nodes," which have no motion. The question is which wavelengths are there such that each end of the wavelength meets at a node, or in other words, which wavelengths would "fit" on the given fixed distance? The largest possible wavelength is one that covers the entire distance. We call this the fundamental wavelength.

This wavelength, when vibrating, would alternate back and forth from the top to the bottom of the string. The second possible wavelength would oscillate one more than the fundamental wavelength before hitting the node. This is often called the 2nd harmonic, since it is the second possible resonance that can happen (harmonic is often used when referring to resonances in musical instruments).

You can continue this process infinitely to the 3rd, 4th, 5th harmonic and so on. What is important to note is that the points where the wave intersects the horizontal line is also a node. This node is where destructive interference happens or where there is no sound. Conversely, the points where the perpendicular projections of the peaks of the waves intersects the horizontal line are where constructive interference happens, and those points are called anti-nodes.

The harmonic series is essentially the set of all the possible wavelengths on a fixed distance, which is exactly what was illustrated in the previous section of standing waves. The first harmonic or the fundamental wavelength has a frequency of 400 Hz and the second harmonic is, using musical terms, an octave above the fundamental wavelength, which means that it has double the frequency or vibrates with 800 Hz. The third harmonic is a perfect fifth above the second harmonic with a frequency of 1200 Hz and so on. The following is an illustration of even-numbered string harmonics from 2nd up to the 64th (five octaves).

The twelfth root of two represents the frequency ratio (another word for musical interval) of a semitone (another word for half-step) in an equal temperament, which is a system in music that divides the octave into 12 equal steps/intervals. A semitone itself is divided into 100 cents, which is a logarithmic unit of measure for musical intervals (1 cent = 1200th root of 2). Applying this value to the tones of a chromatic scale, it produces the following sequence of pitches:

The Fourier series is essentially the expansion of a periodic function in terms of an infinite sum of sines and cosines. Fourier initially intended the series for solving heat equations, but his idea was applied to other areas in mathematics. The Fourier series soon became synonymous with breaking down functions and patterns into combinations of simple oscillations, and one of its many applications is in music. The Fourier series can be used to model sound, and it can be broken into trigonometric functions with various frequencies and amplitudes (or the fundamental and its harmonics). These functions would represent the fundamental, the first harmonic, and so on until the nth harmonic:

You can read more about the connection between the Fourier series and music here: