Piano acoustics are those physical properties of the piano which affect its acoustics.
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The strings of a piano vary in thickness, and therefore in mass per length, with bass strings thicker than treble. A typical range is from 1/30 inch (.85 mm) for the highest treble strings to 1/3 inch (8.5 mm) for the lowest bass. These differences in string thickness follow from well-understood acoustic properties of strings.
Given two strings, equally taut and heavy, one twice as long as the other, the longer would vibrate with a pitch one octave lower than the shorter. However, if one were to use this principle to design a piano it would be impossible to fit the bass strings onto a frame of any reasonable size. Furthermore, in such a hypothetical, gigantic piano, the lowest strings would travel so far in vibrating that they would strike one another. Instead, piano makers take advantage of the fact that a heavy string vibrates more slowly than a light string of identical length and tension; thus, the bass strings on the piano are much thicker than the others.
Any vibrating object will vibrate at a number of frequencies above the fundamental, called overtones. When the overtones are integer multiples (e.g., 2 x or 6 x ) of the fundamental frequency (called harmonics), then the oscillation is periodic, i.e., it vibrates in exactly the same way over and over again. Humans seem to enjoy the sound of periodic oscillations. For this reason, many musical instruments, including pianos, are designed to produce nearly periodic oscillations, that is, to have overtones as close as possible to the harmonics of the fundamental tone.
In an ideal vibrating string, when the wavelength of a wave on a stretched string is much greater than the thickness of the string, the wave velocity on the string is constant and the overtones are at the harmonics. That is why so many instruments are constructed of strings or thin columns of air. However, for high overtones with very short wavelengths, the thin string behaves more like a thick metal bar. The mechanical resistance of the string to bending becomes an additional force. Unless this bending force is much smaller than the tension of the string, it will raise the wave speed. This raises the frequency of the overtones above the harmonics of the fundamental, producing an unpleasant effect called "inharmonicity".
Basic strategies to reduce inharmonicity include decreasing the thickness or increasing the wavelength of the string, choosing a flexible material with a low bending force, and increasing the tension force so that it stays much bigger than the bending force.
Winding a string allows an effective decrease in the thickness of the string. In a wound string, only the inner core resists bending while the windings function only to increase the linear density of the string. The thickness of the inner core is limited by its strength and by its tension; stronger materials allow for thinner cores at higher tensions, reducing inharmonicity. Hence, piano designers choose high quality steel for their strings.
A larger piano allows for longer wires with longer wavelengths. Piano designers strive to fit the longest strings possible within the case; moreover, all else being equal, the sensible piano buyer tries to obtain the largest instrument compatible with budget and space.
Inharmonicity largely affects the lowest and highest notes in the piano and is one of the limits on the total range of a piano. The lowest strings, which would have to be the longest, are most limited by the size of the piano. The designer of a short piano is forced to use thick strings to increase mass density and is thus driven into inharmonicity.
The highest strings have to be under the greatest tension, yet must also be thin to allow for a low mass density. The limited strength of steel forces the piano designer to use very short strings whose short wavelengths thus generate inharmonicity.
The natural inharmonicity of a piano is used by the tuner to make slight adjustments in the tuning of a piano. The tuner will stretch the notes, slightly sharpening the high notes and lowering the low notes so that the overtones of low notes have the same frequency as the fundamentals of high notes.
The Railsback curve, first measured by O.L. Railsback, expresses the difference between normal piano tuning and an equal-tempered scale (one in which the frequencies of successive notes are related by a constant ratio, equal to the twelfth root of two). For any given note on the piano, the deviation between the normal pitch of that note and its equal-tempered pitch is given in cents (hundredths of a semitone).
As the Railsback curve shows, octaves are normally stretched on a well-tuned piano. That is, the high notes are higher, and the low notes lower, than they are in an equal-tempered scale. Not all octaves are equally stretched: the middle octaves are barely stretched at all, whereas the octaves on either end of the piano are stretched considerably.
Railsback discovered that pianos were typically tuned in this manner not because of a lack of precision, but because of inharmonicity in the strings. Ideally, the overtone series of a note consists of frequencies that are integer multiples of the note's fundamental frequency. Inharmonicity causes the successive overtones to be higher than they "should" be.
In order to tune an octave, a piano technician must reduce the speed of beating between the first overtone of a lower note and a higher note until it disappears. Because of inharmonicity, this first overtone will be sharper than a harmonic octave (which has the ratio of 2/1), making either the lower note flatter, or the higher note sharper, depending on which one is being tuned to. To produce an even tuning, the technician begins by tuning an octave in the middle of the piano first, and proceeds to tune outwards from there; notes from the upper range are not compared to notes in the lower range for the purposes of tuning.
Because string inharmonicity only causes harmonics to be sharper, the Railsback curve, which is functionally the integral of the inharmonicity at an octave, is monotonically increasing. Because the inharmonicity is lower in the middle octaves of the piano, the Railsback curve has a shallower slope in this area.
The inharmonicity in a string is caused primarily by its stiffness. Decreased length and increased thickness both contribute to inharmonicity. For the middle to high part of the piano range, string thickness remains constant as length decreases, contributing to greater inharmonicity in the higher notes. For the low range of the instrument, string thickness is drastically increased, especially in shorter pianos which cannot compensate with longer strings, producing greater inharmonicity in this range as well.
In the bass register, a second factor affecting the inharmonicity is the resonance caused by the acoustic impedance of the piano sounding board. These resonances exhibit positive feedback on the inharmonic effect: if a string vibrates at a frequency just below that of a resonance, the impedance will cause it to vibrate even lower, and if it vibrates just above a resonance, the impedance causes it to vibrate higher. The sounding board has multiple resonant frequencies which are unique to any particular piano. This contributes to the greater variance in the empirically measured Railsback curve in the lower octaves.
All but the lowest notes of a piano have multiple strings tuned to the same frequency. This allows the piano to have a loud attack with a fast decay but a long sustain in the ADSR system.
The three strings create a coupled oscillator with three normal modes (with two polarizations each). Since the strings are only weakly coupled, the normal modes have imperceptibly different frequencies. But they transfer their vibrational energy to the sounding board at significantly different rates.
The normal mode in which the three strings oscillate together is most efficient at transferring energy since all three strings pull in the same direction at the same time. It sounds loud, but decays quickly. This normal mode is responsible for the rapid staccato "Attack" part of the note.
In the other two normal modes the strings do not all pull together, e.g., one will pull up while the other two pull down. There is slow transfer of energy to the sounding board, generating a soft but near-constant "Sustain". [1]