In the early 1800's, Joseph Fourier, a French mathematician and physicist studying the propagation of heat in solids and mechanical vibrations described, in mathematical terms, the idea that a periodic waveform is composed of a summation of periodic sine and cosine waveforms of different frequencies and is referred to mathematically as the Fourier Series. The Fourier Series is the basis of the branch of mathematics referred to as Fourier Analysis and is an important branch of mathematics that has given rise to digital audio, among other applications. In musical terms, the Fourier Series describes the standing waves that develop on a vibrating musical string. The page Fourier Series formula has a simplified graphic of the harmonic divisions of a vibrating musical string. Note that the divisions are integer divisions of the entire length of the string; 1, 1/2, 1/3, 1/4, 1/5, 1/6, and 1/7 in the graphic. What this means in physical terms is that when the note C is played on a piano, the string simultaneously vibrates in integer divisions yielding harmonics that sound at an C octave higher at 1/2, and at G at 1/3 an octave and a 5th higher, C at 1/4 two octaves higher. You can hear these harmonics by setting a single harmonic value to maximum and comparing it to other harmonics.

Piano wires have an attribute referred to as “inharmonicity”. Inharmonicity describes the fact that the harmonics of a piano wire are all slightly sharp compared to the fundamental pitch of the wire, that is, its entire vibrating length. Inharmonicity arises from the fact that the high tensile strength piano wire that is at a tension of approximately 185 lbs. of tension does not exhibit mathematically perfect harmonic lengths because of the stiffness of the piano wire and this makes the harmonics slightly short of the mathematical pure integer division causing them to be sharp. Note that the shorter the length a music wire, the sharper the pitch it produces. It’s a fact that the octave tunings of a piano are actually not a tuning of the pitch of some tonic note to an octave higher note, but as close a tuning to the coincidental harmonics of the two wires. This said, the bass notes of a piano that have copper windings on them to increase their mass produce harmonics that are often very sharp to the fundamental pitch, and also are sort of noisy for lack of a better word. When I’m tuning bass notes on a piano I hold the bass note I’m tuning an octave lower than the note I’m tuning it with, I’ll hold the damper up with the sostenuto pedal, play the notes of the harmonic series above and release them. If the bass string is well in tune, it will vibrate the harmonic series on the single bass string due to sympathetic vibrations – you can pick out which harmonics the bass string is best in tune with.

So, the Fourier Series description is a very useful tool for acoustic piano tuning along with giving rise to modern digital signal processing and other applications like sonogram machines!