Any vibrating object will create a sound wave, even if we cannot hear it. The frequency at which an object vibrates is its natural frequency. Some instruments, such as the flute, vibrate at only its natural frequency. This is said to be a pure tone.
Most other instruments vibrate at more than one frequency in a mathematical relationship that can be represented in whole numbers. A tuba or a string vibrates with a simple mathematical ratio. Other instruments have more complex ratios, which determines their unique timbre or sound. A French mathematician named Fourier showed how waves of different frequencies, when they have a simple mathematical relationship, are perceived as one sound by the human ear.
Musicians must change that fundamental frequency for each pitch they play. For example, a stringed instrument player stops the string against a fretboard to get a higher pitch, and a trombonist would lengthen the slide to get a lower pitch. When that fundamental frequency is changed, the harmonics above it will change as well, but will preserve the distinctive mathematical ratio of that instrument. Once altered in this way, the fundamental of the pitch or note we hear is no longer the natural frequency of the instrument.
A 19th-century acoustician named Helmholtz showed that some musical intervals, or the sound of two pitches played together, had differing amounts of consonance and dissonance. Two pitches with similar frequency ratios would sound consonant to our ear because the sound waves or harmonics are synchronized. Two pitches with different frequency ratios would sound dissonant because the waves or harmonics do not match up and therefore create what Helmholtz called "beats."
Harmonics are measured by how many nodes and antinodes they have. A node is the frequency at rest (picture a dot on a line), and an antinode is the crest (highest point) or trough (lowest point) above or below that line. The first harmonic in a series has two nodes and one antinode, the second harmonic has three nodes and two antinodes (high and low), and so on. When put together with the frequency of the specific vibrating object (instrument), this is known as the harmonic series.