Piano acoustics

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Piano acoustics are those physical properties of the piano which affect its acoustics.

1 String length and thickness
2 Inharmonicity and piano size
3 The Railsback curve
- 3.1 Shape of the curve
4 References
5 External links

[edit] 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.

Assuming that two strings were equally taut and thick, a string that is twice as long as another would vibrate with a pitch one octave lower than the other. 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 thick string vibrates more slowly than a thin string of identical length and tension; thus, the bass strings on the piano are much thicker than the others.

[edit] 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 an 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)).

However, the overtones of a stiff string do not quite coincide with the simple frequency ratios of the harmonic series. To the extent that a string is thick and stiff relative to its length, its harmonics will deviate from being multiples of the fundamental; instead becoming slightly sharp. This is what is referred to by inharmonicity.

Inharmonicity is related to length: a longer, thinner string may produce the same pitch as a shorter, thicker string, but the former is less inharmonic than the latter. Thus with longer strings the piano comes closer to ideal harmonic alignment (see the Railsback curve effect below). In addition to being better tuned for participation in an ensemble, the harmonic relationship between the extreme ranges of the piano is better defined when inharmonicity is reduced.

Another effect of this harmonic alignment is on the characteristic resonance of a modern piano with the damper pedal depressed. As overtones on a low string match tones in a higher string, the strings vibrate sympathetically with one another; this effect is enhanced somewhat by reducing the inharmonicity of the piano strings.

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.); the Yamaha CFIIIS is 9 feet long (275 cm); the Bösendorfer Model 290 is 9 feet, 6 inches long (290 cm),and the longest Fazioli piano, the Model F308, is 10 feet, 2 inches (308 cm.). The shortest strings used in pianos are found in inexpensive spinet models, which are smaller than most other pianos.

Inharmonicity also explains why the lowest strings of the piano are not made of plain steel, but often steel wrapped in more dense copper. The wrapped construction adds the necessary mass to the string to compensate for its shorter length, while preventing the string from becoming inflexible, improving both durability and inharmonicity with respect to a corresponding solid steel string.

The highest notes on a keyboard suffer greatly from inharmonicity, as it is difficult to use a longer, thinner, string without it being succeptible to breakage. The highest keys on lower quality pianos often produce no note at all, only an unpleasant percussive sound.

See also Piano wire.

[edit] The Railsback curve

The Railsback curve, indicating the deviation between normal piano tuning and an equal-tempered scale.

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.

[edit] Shape of the curve

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. Increased tension, decreased length, or increased thickness all 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 soundboard. 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 soundboard 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.

[edit] References

Ortiz-Berenguer, Luis I., F. Javier Casajús-Quirós, Marisol Torres-Guijarro, J.A. Beracoechea. Piano Transcription Using Pattern Recognition: Aspects On Parameter Extraction: Proceeds of The International Conference on Digital Audio Effects, Naples, October 2004.
Railsback, O. L. (1938) Scale Temperament as Applied to Piano Tuning: The Journal of the Acoustical Society of America, Volume 9, Issue 3, p. 274
Sundberg, Johan (1991) The Science of Musical Sounds, San Diego: Academic Press. (ISBN 0-12-676948-6)