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Piano tones, like many other musical tones, contain together slowly and rapidly varying fluctuations. Slowly-varying fluctuations (typically less than 16 Hz) can be followed in time by the ears: amplitude modulations, beats and decay are examples of such slow temporal variations. Slowly-varying features of are obtained through low-pass filtering of the tones. The results are displayed in the form of time envelopes (see the figures below).
Piano tones are characterized by a short rise time and a long decay time. Measuring the slope in the decaying part yields an estimate of the global decay factor. This decay factor increases with the energy dissipated in the different components of the instruments (hammers, strings, soundboard) and on the radiated acoustic energy. As a result, the decay time decreases as the dissipated energy increases.
Most envelopes of piano tones show a so-called "double decay" composed of a two different slopes: an initially rapidly decreasing part followed by a more slowly decaying part. The origin of this phenomenon is complex: it depends together on the detuning of the strings inside a triplet of a given note, and on the exchange of energy between the two transverse components of the string motion (vertical and horizontal). It has been recently shown that this energy exchange critically depends on the string-bridge coupling.
The temporal envelope is an attribute of the tone quality (or timbre) of piano tones. Other time-domain descriptors can be used, such as the temporal centroid which accounts for the more or less "percussive" quality of the tone.
This figure shows an example of sound pressure envelope for the string C4 of of J.B. Streicher piano (1851). The dash-dotted is an exponential fit which yields an estimate for the mean decay factor of this tone.
This figure shows a comparison between of temporal envelopes for the note C2 played on four different pianos. In this example the pianos JBS36 and JBS50 are characterized by a single decay, whereas the two others show a distinct decay behavior before and after 1s of sound, with clear amplitude fluctuations.
This figure shows the temporal evolution of the note A3 for the two pianos J. B. Streicher 1836 and 1873. It is interesting to notice that, despite these two pianos were built 37 years apart, with significant different building schemes (see Section 1. Description), their temporal evolution are very similar. In fact, the two pianos show differences in tone qualities which are visible in their spectral properties (see subsection B. Frequency domain).
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