The Well-Tempered Tune

Devoted to perfecting the sound of pianos and harpsichords
7D2_3536-4The Well-Tempered Tune
Partials and Overtones

It is helpful to have a look at the number and range of the partials for various piano notes (strings). Recordings of the single (central) strings of the 8 A's of a Steinway B were made with a digital recorder. That of A0 is shown here, with an acquisition time of 11 sec and a sampling frequency of 44.1 kHz (485,100 data points).

A0 transient
Stacks Image 40
Fourier transformation of the recording provides a distribution of the partials for each A (see methods below). Over 40 overtones can be observed for A0, while only 2 (the fundamental and the first overtone) is observable for A6, and only the fundamental can be reliably defined for A7 (frequencies are in Hz). The variation of intensities of the overtones is what gives each piano its characteristic color.

A0 overtones (0.1 Hz resolution)
Stacks Image 51

A1 overtones
Stacks Image 62

A2 overtones
Stacks Image 73

A3 overtones
Stacks Image 84

A4 overtones
Stacks Image 95

A5 overtones
Stacks Image 106

A6 overtones
Stacks Image 117

A7 overtones
Stacks Image 128
To make the figures above, digital recordings were made of the transient ringing of single strings for selected notes following a mf to f 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 to remove silent sections before the transient, 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.)