This research on the analysis and simulations of historic pianos provided us with some unexpected benefits which might bring useful results applicable to piano acoustics beyond the strict framework of the study. Three examples are briefly summarized below: the energy analysis is an original method based on the available data of the piano simulations (Reference [2]). The reconstruction of hammer force is an original signal processing technique where the hammer force history is derived from measurements of the string velocity (Reference [1]). Finally, the aim of the study on the stringing scheme is to evaluate the consequences of the stringing on the timbre of the instrument. This last study is still under development.

One interesting feature of the piano simulation program is the possibility of calculating the energy evolution of each constitutive part of the instrument during a given time interval of a piano tone. Therefore, we have access to the temporal evolution of the energy for the strings, the hammer, the soundboard, and the sound pressure in the acoustic field around the instrument.

These energetic quantities are obtained by summing, at each time step, the spatial contributions of each domain. The temporal energy evolution of the strings, for example, are obtained by summing, at each time step, the energy of all discrete point along their length. For the soundboard, the energetic contributions of all surface point are summed together. For the acoustic field, the spatial integration is made over a finite volume of air around the instrument.

The benefits of these energetic quantities follow from the fact that they are independent of the observation points (by contrast with punctual measurements made with microphones or accelerometers, for example). For that reason, they can be considered as characteristic data of the instrument. Additional useful factors can be obtained by calculating the ratio between energetic quantities: the ratio between acoustic and soundboard energy, for example, tells us about the radiation efficiency of the instrument for the played note.

Measuring the force pulse imparted by the hammer on the strings of a piano is a very difficult task which is the object of a vast literature in the musical acoustics community. In summary, two groups of methods are used today. In the first group, the acceleration of the hammer is measured, close to the hammer head, and the force is obtained by multiplying this acceleration by an "effective mass" corresponding to the inertia of the system in motion. This is a rather fast and easy method yielding fair results but which, however, has several well identified drawbacks. The second emerging group of method uses optical techniques, such as high-speed imaging, for tracking the motion of hammer and strings, combined to acceleration measurements (Reference [19]).

During the present project, it turned out that none of the existing methods was appropriate for measuring the hammer force (and/or acceleration) on historic instruments, mostly for preservation reasons. Therefore, another original strategy was to reconstruct the hammer force from measurements of the string velocity. The string velocity was measured with a non-contact electromagnetic device (see Section 2. Measurements), and the hammer force was obtained through filtering of the velocity. The transfer function of the filter is derived from the theory of wave propagation along the string (Reference [1]).

Hammer Force reconstruction. Example 1. Left Figure: reconstruction of hammer force pulse for the string D3 of the Gert Hecher piano (GH05). Right figure: string velocity simulated with the reconstructed hammer force (dashed line), and comparison with the measured string velocity (solid line). The first pulse shows perfect agreement between measured and simulated velocity. The measured second pulse is more damped than the predicted one: this particular result is due to the fact that the hammer lightly slips under the string during its release, which is a typical property of the so-called Viennese action (Reference [21]). This additional damping contributes to give a soft character to the tones of pianos equipped with such an action.

Amongst the differences between modern pianos, and the pianofortes built before, say, 1870, the stringing scheme (which means the way the keyboard is strung) is one of the most relevant. In this project, we investigated, in particular, the differences between parallel and crossed stringing (see figures below).

In the parallel stringing scheme, the strings are parallel to each other, and parallel to the fibers of the soundboard. In the cross-stringing scheme, the bass strings overlap the medium strings, and the direction of the strings at not anymore parallel to the wood fibers.

As a consequence, in the parallel scheme, the bass strings only excite the left thin side of the soundboard (viewed from the keyboard), the medium strings excite the medium part, whereas the treble strings excite the right (and thicker) part of the soundboard. This means that the excited modes of the soundboard will be significantly different in the bass, medium and treble register, respectively. Therefore, each register will have a specific tone color.

By contrast, in the cross-stringed scheme, the bass and medium strings excite the same area of the soundboard. Therefore, the population of excited modes is similar, and the tone color is homogeneous. The treble register is also more similar to the other registers than in the parallel case, due to the direction of fibers and ribs which, again, contribute to homogenize the vibrating pattern of the soundboard.

In the present study, these phenomema are investigated through measurements and simulations. This work is still in progress.