# Piano Acoustics

Piano acoustics is an exploration of how physical science, particularly acoustics, can help to explain a number of important properties of the piano.

String length and thickness

The strings of a piano vary in thickness, with bass strings thicker than treble. A typical range is from 1/30 inch for the highest treble strings to 1/3 inch for the lowest bass. These differences in string thickness follow from well-understood acoustic properties of strings.

The relationship of string length to pitch was worked out by Pythagoras ca. 500 B.C. Assuming that all strings are equally taut and thick, a string that is twice as long as another vibrates one octave lower. However, if one were to use this principle to design a piano, it would be impossible to fit the bass strings into a case of any reasonable size; moreover, in such a hypothetical huge "Pythagorean 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 thick string vibrates more slowly than a thin string of identical length and tension, and thus make the strings of the lower notes progressively thicker.

Inharmonicity and piano size

In pianos, long strings are considered desirable. Piano design strives to fit the longest possible strings within a given case size; moreover, all else being equal, the sensible piano buyer tries to obtain the largest instrument compatible with budget and space. The desirability of long strings is the result of a phenomenon called inharmonicity.

Every piano string, when struck, vibrates both at its own natural pitch (called the fundamental frequency), and many overtones, each--as a rough approximation--at a pitch which is a multiple of the fundamental. The lowest overtone is one octave above the fundamental (twice the pitch of the fundamental), the next overtone an octave and a fifth (3 times the fundamental), the next two octaves (4), the next two octaves and a third (5), and so on (see Harmonic series (music)). Since the overtones match other notes on the piano--closely related notes in the theory of musical harmony--the strings vibrate sympathetically with one another whenever they are not covered by their dampers. This creates the characteristic rich tone of a piano.

Unfortunately, in practice, the overtones do not quite coincide with harmonically related musical notes. To the extent that a string is thick and stiff relative to its length, its harmonics will deviate from being multiples of the fundamental; thus they will be in a sense "unmusical". This phenomenon is referred to as inharmonicity.

Because inharmonicity depends on string length, the longer the strings are, the more they approximate ideal theoretical strings, and the more they will vibrate sympathetically with other, musically related notes. Thus, the most prized pianos are (all else being equal) those with the longest strings. The flagship model of Steinway, the Model D, is 8 feet, 11 3/4 inches long (274 cm.); and the longest Fazioli piano is 10 feet, 2 inches (308 cm.). The shortest strings used in pianos are found in cheap spinet models. In these, the highest keys often produce no note at all, only an unpleasant percussive sound.

Inharmonicity also explains why the lowest strings of the piano are not made of plain steel, but rather of steel wrapped in copper. The wrapped construction adds the necessary mass to the string, while minimizing the addition of stiffness (and thus of inharmonicity).

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