Acoustic Background

1. Aspects of Construction:
   • Most pianos have 88 keys.

     Figure 1: 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
     $$f_{n} = n f_{0} \sqrt{1 + B n^2},$$
     where $f_{n}$ is the frequency of the $n$th harmonic and $f_0$ is the frequency of the fundamental. For a solid wire without wrapping,
     $$B = \frac{\pi^3 r^4 Y}{16 T L^2},$$
     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 $r^4$).

   • 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”.

2. Coupled Strings:
   • 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.

3. Hammer-String Interaction:

   Figure 2: 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.

4. The Soundboard:
   • 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.