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Piano acoustics is the set of physical properties of the piano that affect its sound. It is an area of study within musical acoustics.
The strings of a piano vary in diameter, and therefore in mass per length, with lower strings thicker than upper. A typical range is from .240 inches (6.1 mm) for the lowest bass strings [1] to .031 inches (0.79 mm), string size 13, for the highest treble strings. 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 will vibrate with a pitch one octave lower than the shorter. However, if one were to use this principle to design a piano, i.e. if one began with the highest notes and then doubled the length of the strings again and again for each lower octave, it would be impossible to fit the bass strings onto a frame of any reasonable size. Furthermore, when strings vibrate, the width of the vibrations is related to the string length; in such a hypothetical ultra-long piano, the lowest strings would strike one another when played. 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 shorter than the "double with each octave" rule would predict, and are much thicker than the others.
The other factor that affects pitch, other than length, density and mass, is tension. Individual string tension in a concert grand piano may average 200 pounds (91 kg), and have a cumulative tension exceeding 20 tonnes (20,000 kg) each.
Any vibrating thing produces vibrations at a number of frequencies above the fundamental pitch. These are called overtones. When the overtones are integer multiples (e.g., 2×, 3× ... 6× ... ) of the fundamental frequency (called harmonics), then - neglecting damping - the oscillation is periodic—i.e., it vibrates exactly the same way over and over. Many 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 theoretical ideal being a string of zero thickness and zero resistance to bending), 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 that approach 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, are its 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 help them minimize string diameters.
If string diameter, tension, mass, uniformity, and length compromises were the only factors—all pianos could be small, spinet-sized instruments. Piano builders, however, have found that longer strings increase instrument power, harmonicity, and reverberation, and help produce a properly tempered tuning scale.
With longer strings, 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 increases continuously as notes get further from the middle of the piano, and is one of the practical limits on the total range of the instrument. The lowest strings, which are necessarily 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 accepting greater inharmonicity.
The highest strings must be under the greatest tension, yet must also be thin enough to allow for a low mass density. The limited strength of steel (i.e. a too-thin string will break under the tension) forces the piano designer to use very short and slightly thicker 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 stretches the notes, slightly sharpening the high notes and flatting the low notes to make overtones of lower notes have the same frequency as the fundamentals of higher notes.
The Railsback curve, first measured in the 1930s by O.L. Railsback, a US college physics teacher, expresses the difference between inharmonicity-aware stretched piano tuning, and theoretically correct equal-tempered tuning 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 actual pitch of that note and its theoretical equal-tempered pitch is given in cents (hundredths of a semitone). The curve is derived empirically from actual pianos tuned to be pleasing to the ear, not from an exact mathematical equation. [2]
As the Railsback curve shows, octaves are normally stretched on a well-tuned piano. That is, the high notes are tuned higher, and the low notes tuned lower, than they are in a mathematically idealized equal-tempered scale. Railsback discovered that pianos were typically tuned in this manner not because of a lack of precision, but because of inharmonicity in the strings. For a string vibrating like an ideal harmonic oscillator, the overtone series of a single played note includes many additional, higher frequencies, each of which is an integer multiple of the fundamental frequency. But in fact, inharmonicity caused by piano strings being slightly inflexible makes the overtones actually produced successively higher than they would be if the string were perfectly harmonic.
Inharmonicity in a string is caused primarily by stiffness. That stiffness is the result of piano wire's inherent hardness and ductility, together with string tension, thickness, and length. When tuners adjust the tension of the wire during tuning, they establish pitches relative to notes that have already been tuned. Those previously tuned notes have overtones that are sharpened by inharmonicity, which causes the newly established pitch to conform to the sharpened overtone. As the tuning progresses up and down the scale, the inharmonicity, hence the stretch, accumulates.
It is a common misconception that the Railsback curve demonstrates that the middle of the piano is less inharmonic than the upper and lower regions. It only appears that way because that is where the tuning starts. "Stretch" is a comparative term: by definition, no matter what pitch the tuning begins with there can be no stretch. Further, it is often construed that the upper notes of the piano are especially inharmonic, because they appear to be stretched dramatically. In fact, their stretch is a reflection of the inharmonicity of strings in the middle of the piano. Moreover, the inharmonicity of the upper notes can have no bearing on tuning, because their upper partials are beyond the range of human hearing. [3]
As expected, the graph of the actual tuning is not a smooth curve, but a jagged line with peaks and troughs. This might be the result of imprecise tuning, inexact measurement, or the piano's innate variability in string scaling. It has also been suggested with Monte-Carlo simulation that such a shape comes from the way humans match pitch intervals. [4]
All but the lowest notes of a piano have multiple strings tuned to the same frequency. The notes with two strings are called bichords, and those with three strings are called trichords. These allow the piano to have a loud attack with a fast decay but a long sustain in the attack–decay–sustain–release (ADSR) system.
The trichords 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, strings do not all pull together, e.g., one pulls up while the other two pull down. There is a slow transfer of energy to the sounding board, generating a soft but near-constant sustain. [5]
The violoncello ( VY-ə-lən-CHEL-oh, Italian pronunciation:[vjolonˈtʃɛllo]), normally simply abbreviated as cello ( CHEL-oh), is a middle pitched bowed (sometimes plucked and occasionally hit) string instrument of the violin family. Its four strings are usually tuned in perfect fifths: from low to high, C2, G2, D3 and A3. The viola's four strings are each an octave higher. Music for the cello is generally written in the bass clef, tenor clef, alto clef and treble clef used for higher-range passages.
The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In terms of a superposition of sinusoids, the fundamental frequency is the lowest frequency sinusoidal in the sum of harmonically related frequencies, or the frequency of the difference between adjacent frequencies. In some contexts, the fundamental is usually abbreviated as f0, indicating the lowest frequency counting from zero. In other contexts, it is more common to abbreviate it as f1, the first harmonic.
The harmonic series is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a fundamental frequency.
In music, there are two common meanings for tuning:
The piano is a keyboard instrument that produces sound when its keys are depressed, activating an action mechanism where hammers strike strings. Modern pianos have a row of 88 black and white keys, tuned to a chromatic scale in equal temperament. A musician who specializes in piano is called a pianist.
The violin, sometimes referred as a fiddle, is a wooden chordophone, and is the smallest, and thus highest-pitched instrument (soprano) in regular use in the violin family. Smaller violin-type instruments exist, including the violino piccolo and the pochette, but these are virtually unused. Most violins have a hollow wooden body, and commonly have four strings, usually tuned in perfect fifths with notes G3, D4, A4, E5, and are most commonly played by drawing a bow across the strings. The violin can also be played by plucking the strings with the fingers (pizzicato) and, in specialized cases, by striking the strings with the wooden side of the bow.
In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the fundamental frequency of a periodic signal. The fundamental frequency is also called the 1st harmonic; the other harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. The set of harmonics forms a harmonic series.
An overtone is any resonant frequency above the fundamental frequency of a sound. In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental is the lowest pitch. While the fundamental is usually heard most prominently, overtones are actually present in any pitch except a true sine wave. The relative volume or amplitude of various overtone partials is one of the key identifying features of timbre, or the individual characteristic of a sound.
In musical instrument classification, string instruments, or chordophones, are musical instruments that produce sound from vibrating strings when a performer plays or sounds the strings in some manner.
In music, inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency.
A pseudo-octave, pseudooctave, or paradoxical octave in music is an interval whose frequency ratio is not exactly 2:1 = octave : tonic expected for perfectly harmonic pitches, but slightly wider or narrower in pitch – for example 1.98:1, 2.01:1, or even as large as 2.3:1 . The pseudo-octave is never-the-less perceived as if it were equivalent to the conventional 2:1 harmonic ratio, and consequently is treated the same: Pitches separated by a pseudo-octave appropriate for a given instrument are considered equivalent to each other just as with normal "pitch classes".
A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch. If the length or tension of the string is correctly adjusted, the sound produced is a musical tone. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos.
Piano tuning is the process of adjusting the tension of the strings of an acoustic piano so that the musical intervals between strings are in tune. The meaning of the term 'in tune', in the context of piano tuning, is not simply a particular fixed set of pitches. Fine piano tuning requires an assessment of the vibration interaction among notes, which is different for every piano, thus in practice requiring slightly different pitches from any theoretical standard. Pianos are usually tuned to a modified version of the system called equal temperament.
Musical acoustics or music acoustics is a multidisciplinary field that combines knowledge from physics, psychophysics, organology, physiology, music theory, ethnomusicology, signal processing and instrument building, among other disciplines. As a branch of acoustics, it is concerned with researching and describing the physics of music – how sounds are employed to make music. Examples of areas of study are the function of musical instruments, the human voice, computer analysis of melody, and in the clinical use of music in music therapy.
In music, strings are long flexible structures on string instruments that produce sound through vibration. Strings are held under tension so that they can vibrate freely, but with control. This is to make the string vibrate at the desired pitch, with looser strings producing lower pitches, and tighter strings producing higher pitches. However, a vibrating string produces very little sound in of itself. Therefore, most string instruments have a sounding board to amplify the sound.
Stretched tuning is a detail of musical tuning, applied to wire-stringed musical instruments, older, non-digital electric pianos, and some sample-based synthesizers based on these instruments, to accommodate the natural inharmonicity of their vibrating elements. In stretched tuning, two notes an octave apart, whose fundamental frequencies theoretically have an exact 2:1 ratio, are tuned slightly farther apart. "For a stretched tuning the octave is greater than a factor of 2; for a compressed tuning the octave is smaller than a factor of 2."
Acoustic resonance is a phenomenon in which an acoustic system amplifies sound waves whose frequency matches one of its own natural frequencies of vibration.
Playing the violin entails holding the instrument between the jaw and the collar bone. The strings are sounded either by drawing the bow across them (arco), or by plucking them (pizzicato). The left hand regulates the sounding length of the strings by stopping them against the fingerboard with the fingers, producing different pitches.
Violin acoustics is an area of study within musical acoustics concerned with how the sound of a violin is created as the result of interactions between its many parts. These acoustic qualities are similar to those of other members of the violin family, such as the viola.
An unpitched percussion instrument is a percussion instrument played in such a way as to produce sounds of indeterminate pitch, or an instrument normally played in this fashion.
...as the fundamental frequencies rise in pitch so the do their partials and at some point the partials of higher tones become barely audible so their effect on tuning would be negligible.
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| Audio frequency and pitch | ||
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