ï~~4. Pedals
There are, on most grand pianos, three pedals, each with
a slightly different function. The left-most pedal is called
the "soft" pedal or una corda pedal. It shifts the action,
including the hammers, slightly to the right so that each
hammer hits fewer strings or hits its string with a less
centered blow of the hammer. The chief result of this is not
so much to reduce the overall amplitude of the sound as to
alter the timbre and, possibly, the decay characteristics.
The middle pedal simply serves as an extra hand to hold
down any notes that are sounding when it is pressed. It is
not necessary to go into this here.
The right-most pedal is probably the most important and
the most often used. It raises the dampers on all the strings,
whether the pianist has played them or not. The effect of
this is to add a whole bunch of sympathetic resonances
to the sound. Whatever string is struck by the keys has
its energy slowly "picked-up" by the other strings that are
now free to vibrate. Quantitive studies of this effect are
currently underway at CCRMA.
IV. The basie model
Each of the aforementioned structures has a corresponding module in the synthesis model.
Basically, the piano structure is approximated by a complex network of resonators fed from a nearly impulsive
source. The main sections of the digital model are then
(see Figure 9):
* The hammer blow to the string (initial impulse)
* The resonance of the strings (primary resonator)
* The reflection, transmission and absorption of the
bridge
* The resonance and radiation of the soundboard (secondary resonator)
* The sympathetic vibrations of the pedal system
Each of the resonator systems is modeled by one or more
waveguide digital filter sections [17) that are then coupled to
the other resonating systems and the input signal. A more
detailed discussion of each of these component systems is
presented next.
1. Hammer
My model uses as its excitation a hanning function as
previously described. Though this has been shown to be
accurate only to a first approximation (see, particularly,
[4]), it will be shown in the next section that my model for
the string accounts for a number of the secondary effects of
the hammer-string interaction as well.
smokeP"m
Elm!1
Figure 9. The basic waveguide piano.
The hanning pulse is applied directly to the string resonators (marked a and b in Figure 9) through a three-way junction. th is made to vary proportionally with key velocity so
the faster the hammer is moving, the narrower the pulse will
be and the wider the bandwidth of the resulting spectrum.
See [2], [4], [5], [6) and [21) for further discussion of this.
2. Strings
Each string is treated as one-dimensional and is therefore modeled by a bi-directional waveguide that is initially
split into two parts at the point of contact with the hammer. The first section, from the capo d'astro bar (near the
keyboard) to the hammer strike position (in most pianos
the hammer strikes the string 1/7 to 1/8 of its length from
the capo d'astro); second, from the hammer strike position
to the bridge. The blow of the hammer can be thought
of as driving a smoothed pulse away from the strike position in both directions simultaneously. In some cases the
pulse traveling toward the capo has time to reflect from
the capo d'aatro and return to the hammer strike position
before the hammer has left the string and this has several
consequences.
* The original hammer pulse is deformed by the returning wave (see [21] and [4],[5],[6]).
* The hammer is thrown off of the string sooner than
it would be otherwise.
* Some partials of the fundamental frequency of the
string are damped by the action of the hammer (see
[2]).
My model accounts for this very simply, the point of
contact between the hammer and the string is explicitly
modeled with a time-varying waveguide junction. From the
moment the hammer strikes the string there is a gradual
1987 ICMC Proceedings
93
