Partials and Overtones

A vibrating string vibrates not only at its fundamental frequency (f1, the first partial) but also at multiples of the fundamental (the overtones, or partials f2 and above). The second partial for an ideal string would be twice the fundamental, or an octave above the fundamental, the third would be three times the fundamental or an octave plus a fifth, the fourth 4 times the fundamental or two octaves above f1, etc. The overtones of an ideal vibrating string are the harmonics. The overtones of a real piano string do not occur exactly at the harmonics. The deviation between the harmonics and the real overtones is characterized by the inharmonicity.

The number, range, and intensities of the overtones (partials) for piano notes (strings) vary signiticantly. I have made recordings of the single (central) strings of each note of a Steinway B with a digital recorder. That of A0 (the lowest A on a piano) is shown here, with a very long deep bass resonance and sustain (11 sec here, with and a sampling frequency of 44.1 kHz (485,100 data points)). This is decaying oscillation composed of a number of frequencies that are not obvious in this view.

Fourier transformation of the decaying transient tells us the frequencies of the multiple overtones, and their relative importance (see methods below). Over 40 overtones can be observed for the A0 decay shown above, while only 8 are clear for A4 (the A above middle C), and only one for A6, two octaves above.. The variation of intensities of the overtones is what gives each piano (or any instrument) its characteristic color.

For any note, the first overtone is an octave above the fundamental (first partial). The second overtone is a fifth above that, and the third is two octaves above the fundamental.

A0 overtones (0.1 Hz resolution). The fundamental is almost imperceptible, and the largest overtone is the second at around 82 Hz.

A4 Overtones. The fundamental is at 440 Hz, and it is the largest.

A6 Overtone. Only the fundamental can be seen at 1770.8 Hz.

Recording Information:
Digital recordings were made of the transient ringing of single strings for selected notes following a key strike (only the center string of three-string notes, and the upper string for two-string notes were utilized, the other strings were muted with rubber mutes). Recordings were usually made with a portable handheld Tascam DR-1 recorder with either the internal mic (20-20,000 Hz) or an external Audio Technica AT822 mic (30-20,000 Hz response).

Signal processing:
Digital (wav) files were imported into GoldWave or Audacity and zero filled to 22 seconds if needed (giving 1 million points per file). The two channels were matched and normalized, then the file was saved as a monaural file in text format. The files were imported into Mathematica for Fourier transformation and analysis. Digital resolution of the frequency files were defined by 1/AT where AT is the length of the transient (e.g. 0.045 Hz for a 22 sec transient). (Thanks to my son Christopher for writing a very efficient routine in Mathematica for extracting the overtone frequencies.)