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).