In Search of the Missing Fundamental: by Richard K. Jones
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Harmonic Series

A Harmonic Series Written as Notes on the Grand Staff and Treble Clef


Below are the first 24 partials of a harmonic series for the fundamental pitch C2 that vibrates at about sixty-four and one half cycles per second (ca. 65.4 Hz.) It is expressed in traditional musical staff notation with frequency multiples (a.k.a. partial numbers) indicated between the staves. The – and + symbols indicate that the notated pitch is significantly lower or higher, respectively, than the same pitch in the equal tempered tuning system. The modern piano is tuned using the equal temperament system. If you played through each example on a well tuned piano and then listened to each audio clip, you would notice that even the some of the lower intervals in the audio clips are out of tune with the piano as well. This is because the intervals in the harmonic series are pure or just and the intervals in the equal tempered system are not. Equally tempered intervals have  been equally adjusted or equally tempered to facilitate key changes on fixed pitched devices i.e. the modern piano.  The adjustments are subtle enough that they generally don’t disturb the ear.

Partials 1-12

Partials 8-20

Partials 16-24

Note that some of the partials are significantly “out of tune”with the Equal Tempered tuning system. These notes are indicated with a – or + .

Listen for the sharper and flatter notes as you progress through the entire series.

The sound clip below is an example of what the fundamental frequency  of C2 65.4 Hz.  with all harmonics present would sound like. All of the harmonics blend together evenly creating a composite harmonic wave and your ear perceives just one pitch. This electronically generated sample is a specific  type of waveform called a sawtooth wave. A sawtooth wave’s sound contains both even and odd harmonics of the fundamental frequency. Because it contains all the integer harmonics, it is very similar to the sound of bowed string instruments like violins and cellos.