A simplified diagram of the piano (from ``Five Lectures on the Acoustics of the Piano'', edited by Anders Askenfelt).
A concert grand piano has 8 single wrapped strings, 5 pairs of wrapped strings, 7 sets of three wrapped strings, and 68 sets of three unwrapped strings.
To achieve a greater loudness, the piano strings are held at very high tensions. To offset these high forces, a sturdy cast iron frame is used.
The high string tensions demand high-strength wires.
Higher-strength wires are generally stiffer (or have a higher Young's modulus). This bending stiffness provides an additional restoring force (besides tension) that slightly raises the frequency of all the modes.
String bending is greater for the higher modes, resulting in greater frequency stretching at higher frequencies..
The resulting inharmonicity of strings is approximately given by
where fn is the frequency of the nth harmonic and f0 is the frequency of the fundamental. For a solid wire without wrapping,
where r is the radius of the string, Y is the Young's modulus, T is the tension, and L is the length of the string.
To minimize inharmonicities due to string stiffness, the smallest string diameter possible should be used (since B scales by r4).
In order to maintain string mass but minimize stiffness, the lower strings are wrapped.
Because of the inherent inharmonicity of its strings, the piano is ``stretch-tuned''.
Over most of its playing range, the piano has three strings per note.
In order to maximize decay time, these strings are slightly mistuned by about one to two cents from each other.
The initial ``in-phase'' excitation of all three strings produces a rapid initial decay of string energy into the sound board. Because of mistunings, however, the vibrations soon grow out of phase and result in a much longer secondary decay.
Modern grand piano action (from ``From touch to string vibration'' by Anders Askenfelt & Erik Jansson).
The hammer action of the piano produces a ``striking'' excitation.
The hammer is typically ``thrown'' away from the string by the first reflected pulse on the string. Depending on its weight, however, the hammer may remain in contact with the string for longer or shorter periods of time.
The soundboard is nearly always made of spruce (of approximately 1 cm thickness) and is the main source of radiated sound.
Strips of spruce are glued together and then ribs are added at right angles to the grain so that cross-grain stiffness is roughly equal to the natural stiffness along the grain.
Modern pianos have two bridges.
The lowest mode of a soundboard is typically around 50 Hz.
The lossless one-dimension wave equation was previously derived for a string by assuming the restoring force due to string stiffness was negligible. However, it turns out that string stiffness cannot be ignored in many musical contexts.
In general, strings of greater diameter have larger restoring forces due to bending.
This restoring force can be accounted for in the wave equation with a term proportional to the fourth spatial derivative of the string displacement:
where T is tension,
is the moment constant for a cylindrical string of radius r and Young's modulus Y.
In a digital waveguide model, the effects of dispersion can be accounted for by considering the relationship between temporal and spatial sampling intervals:
where Ts0 is the unit delay time without dispersion.
As a result, unit delays (z-1) must be replaced by
We can interpret this new delay unit as an allpass filter that approximates the corresponding frequency-dependent delay.
In a digital waveguide model, dispersive wave propagation can thus be simulated by replacing unit delays with allpass filters, as illustrated in Fig. 3.
Discrete-time simulation of dispersive wave propagation, where digital allpass filters Ha(z) replace unit delay elements.
Because allpass filters are linear and time invariant, they can be commuted and implemented at discrete spatial locations in the model, in exactly the same way as previously discussed for other linear, time invariant gain factors or filters.
In general, good approximations to dispersion require fairly high-order allpass filters. Contrary to the case with commuted frequency-dependent loss filters, the consolidation of allpass units does not necessarily lead to computational savings by way of lower-order filters.
In general, digital waveguide models of string instruments make use of allpass delay-line interpolation techniques because FIR filter techniques result in too much high-frequency attenuation.
The delay-line lengths necessary to produce very high fundamental frequencies may sometimes become too short for a given implementation. This problem could especially occur in conjunction with vectorized computational structures.
The fundamental frequency of the highest piano note (C8) is 4186 Hz. At a sample rate of 44100 Hz, only four (unstretched) partials will possibly fall within the range of human hearing.
Thus, for such high notes, one can simply model each partial with a separate second-order digital resonance filter. The model then consists of several resonance filters combined in parallel.
When the piano hammer strikes the string(s), a pulse travels to the agraffe and back. As long as the hammer remains in contact with the string, there will be subsequent pulses and reflections.
The piano hammer is typically ``thrown away'' from the string by one of the reflected pulses.
A plot depicting a theoretical hammer-string interaction force is shown in Fig. 8. This plot corresponds to three pulse reflections.
A theoretical hammer force characteristic.
The actual number of pulse reflections that occur is dependent on the string being hit and the initial velocity of the hammer.
The hammer-string interaction is inherently non-linear, in large part because of the compression characteristics of the hammer felt.
However, because the pulse-hammer interactions occur at discrete times, it is possible to consider them as distinct events that overlap to produce a complete hammer-string characteristic.
Each single pulse can be modeled as a lowpass filtered impulse signal. This suggests a hammer-excited piano synthesis system as diagrammed in Fig. 9.
A hammer-excited piano synthesis block diagram implementation.
In the system of Fig. 9, an input velocity control triggers an impulse signal. The tapped delay-line is used to produce three subsequent time-delayed impulses, each of which drive a particular lowpass filter. The resultant linear summation of time-delayed lowpass filter impulse responses then drives the string model and its soundboard response filter.
Hammer-string characteristics can be evaluated for various strings and input velocities and appropriate lowpass filter responses designed. The input velocity can then be used as a control parameter to select the appropriate filter characteristics.
Finally, because the implementation described by Fig. 9 is linear, the piano soundboard impulse response can be ``commuted'' through the system and used to trigger the hammer-string filters as shown in Fig. 10.
A commuted piano hammer block-diagram implementation.