Piano Acoustics - Inharmonicity and Piano Size

Inharmonicity and Piano Size

Any vibrating thing will vibrate at a number of frequencies above the fundamental, called overtones. When the overtones are integer multiples (e.g., 2×, 3× ... 6× ... ) 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 skinny strings or thin columns of air. However, for high overtones with short wavelengths approaching the diameter of the string, the string behaves more like a thick metal bar: its mechanical resistance to bending becomes an additional force to the tension, which 'raises the pitch' of the overtones. Only when the bending force is much smaller than the tension of the string, will leave it wave-speed (and the overtones pitched as harmonics) unchanged. The frequency-raised overtones (above the harmonics), called 'partials' can produce an unpleasant effect called "inharmonicity".

Basic strategies to reduce inharmonicity include decreasing the thickness of the string or increasing its length, 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, as its strength and durability allows string diameters to be minimized.

If string diameter, tension, mass, uniformity and length compromises were the only factors, then all pianos could be small, spinet-sized instruments. It has been found that 'longer wires' increase instrument power, harmonicity, reverberation and aid in producing a well-tempered tuning scale.

For longer wires, larger pianos achieve the longer wavelengths and tonal characteristics desired. 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 lower notes have the same frequency as the fundamentals of higher notes.

See also Piano wire, Piano tuning, Psychoacoustics.