input of another waveguide using coupling filters. The
coupling filters depend on the distance between the strings
at the bridge level: in the case of the triplet, the coupling
filters Ca of the adjacent strings are equal but different from
the coupling filters Ce, between the extreme strings.
this method, the modulus and the phase of each filter are
obtained in the neighborhood of the resonances. The
excitation is then extracted using a deconvolution process
with respect to the waveguide transfer function.
3 Design of the source model
The source (or excitation part) is modeled in the frequency
domain in three parts. These are the static spectrum, which
contains the spectral slope and the spectral modulation
caused by the hammer impact position. The static spectrum
model is linear, with the spectral modulation modeled with
the log of a sinusoidal. The spectral tilt, which models the
change in brightness when the hammer velocity is changed,
and the energy model, which models the increase of energy
when the hammer velocity is increased. All source
modelization is done (in dB) in the frequency domain on the
discrete partial frequencies or indexes.
The source model parameters are found using a nonlinear curve fit (More 1977) minimizing perceptual criteria.
It has already proven its quality (Bensa et al. 2000), both
with respect to sound quality in analysis/synthesis, but also
because it has a number of relevant performance parameters,
including in particular hammer position, velocity and felt
characteristics.
Examples of the resulting excitation spectrum for the AO
note are shown in Figure 2. It is clear that the brightness tilt
model works well for the spectral slope change between the
different velocities.
Excitation for note AO, pp, mf and ff, org(stipled), model(solid)
Figure 1. Three coupled elementary waveguide.
The input E (the excitation of the model) is a direct
consequence of the interaction between the hammer and the
strings and corresponds to the amount of energy transferred
to each partial of the resonant model. Its shape mainly
depends on the velocity and the position of the hammer
impact. The excitation is the same for each elementary
waveguide, since it is supposed that the hammer strikes the
strings in the same way. The output S corresponds to the
vibration at the bridge level.
Inverse problem. The calibration of our model has to be
made carefully since the aim of this work is to make
analysis-synthesis. The parameters are estimated using data
collected on a real piano with an experimental setup
described in part 4. The estimation method is partly similar
to the one described in (Aramaki et al. 2001). With the
analysis of only one signal (measured at the bridge level),
all the parameters of the model are estimated. The analysis
method can be summarized as follow. First, each partial of
the measured signal is isolated using band-pass filtering.
Then, a parametric method, the Steiglitz-McBride method
(Steiglitz and McBride 1965), is used in the time domain to
estimate the temporal parameters. As the sound is assumed
to be a sum of exponentially decaying sinusoids, in the case
of three strings, three amplitudes, damping coefficients and
frequencies are extracted for each partial. In a second step,
the analytical systems between those temporal parameters
and the filters' parameters are solved. For three strings,
unfortunately, only numerical solution can be found. With
M
a,
0)
-Q.
C-O
1500 2000 2500
frequency (Hz)
4000
Figure 2. mf, ff and pp excitations for the AO note.
Estimated (dashed) and modeled (solid).
4 Experiment
To collect real signals an accelerometer is positioned on
the bridge of a Disklavier piano, close to the struck strings.
The vibrations of the bridge are measured for each note of
the piano with a medium velocity of the hammer. For some
