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.[ citation needed ] Pianos are usually tuned to a modified version of the system called equal temperament. (See Piano key frequencies for the theoretical piano tuning.)
In all systems of tuning, every pitch may be derived from its relationship to a chosen fixed pitch, which is usually A440 (440 Hz), the note A above middle C. For a classical piano and musical theory, the middle C is usually labelled as C4 (as in scientific pitch notation); However, in the MIDI standard definition this middle C (261.626 Hz) is labelled C3. In practice, a MIDI software can label middle C as C3-C5, which can cause confusion, especially for beginners.
Piano tuning is done by a wide range of independent piano technicians, piano rebuilders, piano-store technical personnel, and hobbyists. Professional training and certification is available from organizations or guilds, such as the Piano Technicians Guild. Many piano manufacturers recommend that pianos be tuned twice a year.
Many factors cause pianos to go out of tune, particularly atmospheric changes. For instance, changes in humidity will affect the pitch of a piano; high humidity causes the sound board to swell, stretching the strings and causing the pitch to go sharp, while low humidity has the opposite effect. [1] Changes in temperature can also affect the overall pitch of a piano. In newer pianos the strings gradually stretch and wooden parts compress, causing the piano to go flat, while in older pianos the tuning pins (that hold the strings in tune) can become loose and not hold the piano in tune as well. [2] Frequent and hard playing can also cause a piano to go out of tune. [2] For these reasons, many piano manufacturers recommend that new pianos be tuned four times during the first year and twice a year thereafter. [3]
An out-of-tune piano can often be identified by the characteristic "honky tonk" or beating sound it produces. This fluctuation in the sound intensity is a result of two (or more) tones of similar frequencies being played together. For example, if a piano string tuned to 440 Hz (vibrations per second) is played together with a piano string tuned to 442 Hz, the resulting tone beats at a frequency of 2 Hz, due to the constructive and destructive interference between the two sound waves. Likewise, if a string tuned to 220 Hz (with a harmonic at 440 Hz) is played together with a string tuned at 442 Hz, the same 2 Hz beat is heard. [4] Because pianos typically have multiple strings for each piano key, these strings must be tuned to the same frequency to eliminate beats.
The pitch of a note is determined by the frequency of vibrations. For a vibrating string, the frequency is determined by the string's length, mass, and tension. [5] Piano strings are wrapped around tuning pins, which are turned to adjust the tension of the strings.
Piano tuning became a profession around the beginning of the 1800s, as the "pianoforte" became mainstream. [6] Previously musicians owned harpsichords, which were much easier to tune, and which the musicians generally tuned themselves. Early piano tuners were trained and employed in piano factories, and often underwent an apprenticeship of about 5–7 years. Early tuners faced challenges related to a large variety of new and changing pianos and non-standardized pitches.
Historically, keyboard instruments were tuned using just intonation, pythagorean tuning and meantone temperament meaning that such instruments could sound "in tune" in one key, or some keys, but would then have more dissonance in other keys. [7] The development of well temperament allowed fixed-pitch instruments to play reasonably well in all of the keys. The famous "Well-Tempered Clavier" by Johann Sebastian Bach took advantage of this breakthrough, with preludes and fugues written for all 24 major and minor keys. [8] However, while unpleasant intervals (such as the wolf interval) were avoided, the sizes of intervals were still not consistent between keys, and so each key still had its own distinctive character. During the 1800s this variation led to an increase in the use of quasi- equal temperament, in which the frequency ratio between each pair of adjacent notes on the keyboard were nearly equal, allowing music to be transposed between keys without changing the relationship between notes. [9]
Pianos are generally tuned to an A440 pitch standard that was adopted during the early 20th century in response to widely varying standards. [10] Previously the pitch standards had gradually risen from about A415 during the late 18th century and early 19th century to A435 during the late 19th century. Though A440 is generally the standard, some orchestras, particularly in Europe, use a higher pitch standard, such as A442. [11]
A stretched string can vibrate in different modes, or harmonics, and when a piano hammer strikes a string it excites multiple harmonics at the same time. The first harmonic (or fundamental frequency) is usually the loudest, and determines the pitch that is perceived. [12] In theory the higher harmonics (also called overtones or partials) vibrate at integer multiples of the fundamental frequency. (e.g. a string with a fundamental frequency of 100 Hz would have overtones at 200 Hz, 300 Hz, 400 Hz, etc.) In reality, the frequencies of the overtones are shifted up slightly, due to inharmonicity caused by the stiffness of the strings. [13]
The relationship between two pitches, called an interval, is the ratio of their absolute frequencies. The easiest intervals to identify and tune are those where the note frequencies have a simple whole-number ratio (e.g. octave with a 2:1 ratio, perfect fifth with 3:2, etc.) because the harmonics of these intervals coincide and beat when they are out of tune. (For a perfect fifth, the 3rd harmonic of the lower note coincides with the 2nd harmonic of the top note.)
The term temperament refers to a tuning system that allows intervals to beat instead of tuning pure or "just intervals". In equal temperament, for instance, a fifth would be tempered by narrowing it slightly, achieved by flattening its upper pitch slightly, or raising its lower pitch slightly.
Tempering an interval causes it to beat. Because the actual tone of a vibrating piano string is not just one pitch, but a complex of tones arranged in a harmonic series, two strings that are close to a simple harmonic ratio such as a perfect fifth beat at higher pitches (at their coincident harmonics), because of the difference in pitch between their coincident harmonics. Where these frequencies can be calculated, a temperament may be tuned aurally by timing the beatings of tempered intervals.
A common method of tuning the piano begins with tuning all the notes in the "temperament" octave in the lower middle range of the piano, usually F3 to F4. A tuner starts by using an external reference, usually an A440 tuning fork, (or commonly a C523.23 tuning fork) to tune a beginning pitch, and then tunes the other notes in the "temperament" using tempered interval relationships. During tuning it is common to assess perfect fifths and fourths, major and minor thirds, and major and minor sixths, often playing the intervals in an ascending or descending pattern to hear whether an even progression of beat rates has been achieved.
Having established the 12 notes of the chromatic scale, the technician then replicates the temperament throughout the piano by tuning octaves and cross-checking with other intervals, to align each note with others that have already been tuned.
Electronic piano tuning devices are also commonly used. They are designed to adjust the same tonal complexities that the aural tuner encounters. The devices use sophisticated algorithms to continuously test the harmonic makeup of each string as it is sounded, and apply the derived information to determine its optimal pitch within the context of the entire instrument.
The following table lists theoretical beat frequencies between notes in an equal temperament octave. The top row indicates absolute frequencies of the pitches; usually only A440 is determined from an external reference. Every other number indicates the beat rate between any two tones (which share the row and column with that number) in the temperament octave. Slower beat rates can be carefully timed with a metronome, or other such device. For the thirds in the temperament octave, it is difficult to tune so many beats per second, but after setting the temperament and duplicating it one octave below, all of these beat frequencies are present at half the indicated rate in this lower octave, which are excellent for verification that the temperament is correct. One of the easiest tests of equal temperament is to play a succession of major thirds, each one a semitone higher than the last. If equal temperament has been achieved, the beat rate of these thirds should increase evenly in the temperament region. [14]
261.626 | 277.183 | 293.665 | 311.127 | 329.628 | 349.228 | 369.994 | 391.995 | 415.305 | 440.000 | 466.164 | 493.883 | 523.251 |
0.00000 | 14.1185 | 20.7648 | 1.18243 | 1.77165 | 16.4810 | 23.7444 | C | |||||
13.3261 | 19.5994 | 1.11607 | 1.67221 | 15.5560 | 22.4117 | B | ||||||
12.5781 | 18.4993 | 1.05343 | 1.57836 | 14.6829 | 21.1538 | A♯ | ||||||
11.8722 | 17.4610 | .994304 | 1.48977 | 13.8588 | 19.9665 | A | ||||||
16.4810 | .938498 | 1.40616 | 13.0810 | 18.8459 | G♯ | |||||||
.885824 | 1.32724 | 12.3468 | 17.7882 | G | Fundamental | |||||||
1.25274 | 11.6539 | 16.7898 | F♯ | Octave | ||||||||
1.18243 | 10.9998 | 15.8475 | F | Major sixth | ||||||||
10.3824 | 14.9580 | E | Minor sixth | |||||||||
14.1185 | D♯ | Perfect fifth | ||||||||||
D | Perfect fourth | |||||||||||
C♯ | Major third | |||||||||||
C | Minor third |
The next table indicates the pitch at which the strongest beating should occur for useful intervals. As described above, when tuning a perfect fifth, for instance, the beating can be heard not at either of the fundamental pitches of the keys played, but rather an octave and fifth (perfect twelfth) above the lower of the two keys, which is the lowest pitch at which their harmonic series overlap. Once the beating can be heard, the tuner must temper the interval either wide or narrow from a tuning that has no beatings.
Interval | Approximate frequency ratio | Beating above the lower pitch | Tempering |
---|---|---|---|
Octave | 2:1 | Octave | Exact |
Major sixth | 5:3 | Two octaves and major third | Wide |
Minor sixth | 8:5 | Three octaves | Narrow |
Perfect fifth | 3:2 | Octave and fifth | Slightly narrow |
Perfect fourth | 4:3 | Two octaves | Slightly wide |
Major third | 5:4 | Two octaves and major third | Wide |
Minor third | 6:5 | Two octaves and fifth | Narrow |
Unison | 1:1 | Unison | Exact |
The tuning described by the above beating plan provides a good approximation of equal temperament across the range of the temperament octave. If extended further, however, the actual tuning of the instrument becomes increasingly inaccurate because of deviation of the real partials from the theoretical harmonics, pianos' partials run slightly sharp, as increasingly higher orders of the harmonic series are reached. This problem is mitigated by "stretching" the octaves as one tunes above (and to an extent below) the temperament region. When octaves are stretched, they are not tuned to the lowest coincidental overtone (second partial) of the note below, but instead to a higher overtone (often the 4th partial). This widens all intervals equally, thereby maintaining intervallic and tonal consistency.
All western music, but western classical literature in particular, requires this deviation from the theoretical equal temperament because the music is rarely played within a single octave. A pianist constantly plays notes spread over three and four octaves, at least, so it is critical that the mid and upper range of the treble be stretched, or widened, to better match with the inharmonic overtones of lower registers. Since the stretch of octaves is perceived and not measured, the tuner determines which octave needs more or less octave stretching "by ear". Good tuning requires compromise between tonal brilliance, accurate intonation, and an awareness of gradation of timbre through the compass of the instrument. The name of this modification of the width of the scale is called the piano tuners' octave, as opposed to the simple 2:1 octave expected from a (theoretical) harmonic oscillator.
The amount of stretching necessary to achieve the desired compromise is a complicated determination described theoretically as a function of string scaling. String scaling considers the string's tension, length, diameter, weight per unit length, and in elasticity in either the strings' core wires or any overwinding used to modify the wire's weight. The overwindings are normally made from a denser, heavier, but less "springy" metal than the steel used for the core. Imperfect "springiness" anywhere in the string wire makes its partials deviate slightly from mathematically pure harmonics, and no real material used to generating musical tones is perfectly elastic. The Railsback curve is the result of measuring the fundamental frequencies of stretched tunings and plotting their deviations from unstretched equal temperament.
In small pianos the inharmonicity is so extreme that establishing a stretch based on a triple octave makes the single octaves beat noticeably, and the wide, fast beating intervals in the upper treble beat wildly – especially major 17ths (two octaves + a major 3rd). By necessity the tuner will attempt to limit the stretch. In large pianos like concert grands, less inharmonicity allows for a more complete string stretch without negatively affecting close octaves and other intervals. So while it may be true that the smaller piano receives a greater stretch relative to the fundamental pitch, only the concert grand's octaves can be fully widened so that triple octaves are beatless. This contributes to the response, brilliance, and "singing" quality that concert grands offer.
A benefit of stretching octaves is the correction of dissonance that equal temperament imparts to the perfect fifth. Without octave stretching, the slow, nearly imperceptible beating of fifths in the temperament region (ranging from little more than one beat every two seconds to about one per second) would double each ascending octave. At the top of the keyboard, then, the theoretically (and ideally) pure fifth would be beating more than eight times per second. Modern western ears easily tolerate fast beating in non-just intervals (seconds and sevenths, thirds and sixths), but not in perfect octaves or fifths. Happily for pianists, the string stretch that accommodates inharmonicity on a concert grand also nearly exactly mitigates the accumulation of dissonance in the perfect fifth.
Other factors, physical and psychoacoustic, affect the tuner's ability to achieve a temperament. Among physical factors are inharmonic effects due to soundboard resonance in the bass strings, poorly manufactured strings, or peculiarities that can cause "false beats" (false because they are unrelated to the manipulation of beats during tuning). The principal psychoacoustic factor is that the human ear tends to perceive the higher notes as being flat when compared to those in the midrange. Stretching the tuning to account for string inharmonicity is often not sufficient to overcome this phenomenon, so piano tuners may stretch the top octave or so of the piano even more.
Common tools for tuning pianos include the tuning lever or "hammer", a variety of mutes, and a tuning fork or electronic tuning device. [15] The tuning lever is used to turn and 'set' the tuning pins, increasing or decreasing the tension of the string. Mutes are used to silence strings that are not being tuned. While tuning the temperament octave, a felt strip is typically placed within the temperament (middle) section of the piano; it is inserted between each note's trichord, muting its outer two strings so that only the middle string is free to vibrate. A Papps mute performs the same function in an upright piano and is placed through the piano action to mute either the 2 left strings (of a trichord), or the 2 right strings similarly. After the center strings are all tuned (or right if a Papps mute is used) the felt strip can be removed note by note, tuning the outer strings to the center strings. Wedge-shaped mutes are inserted between two strings to mute them, and the Papps mute is commonly used for tuning the high notes in upright pianos because it slides more easily between hammer shanks.
In an aural tuning a tuning fork is used to tune the first note (generally A4) of the piano, and then a temperament octave is tuned between F3 and F4 using a variety of intervals and checks, until the tuner is satisfied that all the notes in the octave are correctly tuned. [16] The rest of the piano is then tuned to the temperament octave, using octaves and other intervals as checks.
If an electronic tuning device is used, the temperament step might be skipped, as it is possible for the tuner to adjust notes directly with the tuning device in any reasonable order.
An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system yields pitch steps perceived as equal in size, due to the logarithmic changes in pitch frequency.
A harmonic series is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a fundamental frequency.
In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth.
In music, there are two common meanings for tuning:
The piano is a keyboard instrument that produces sound when its keys are depressed, through engagement of an action whose hammers strike strings. Modern pianos have a row of 88 black and white keys, tuned to a chromatic scale in equal temperament.
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.
Meantone temperaments are musical temperaments, that is a variety of tuning systems, obtained by narrowing the fifths so that their ratio is slightly less than 3:2, in order to push the thirds closer to pure. Meantone temperaments are constructed similarly to Pythagorean tuning, as a stack of equal fifths, but they are temperaments in that the fifths are not pure.
Pitch is a perceptual property that allows sounds to be ordered on a frequency-related scale, or more commonly, pitch is the quality that makes it possible to judge sounds as "higher" and "lower" in the sense associated with musical melodies. Pitch is a major auditory attribute of musical tones, along with duration, loudness, and timbre.
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 2:1, that of the octave, but is perceived or treated as equivalent to this ratio, and whose pitches are considered equivalent to each other as with octave equivalency.
Piano acoustics is the set of physical properties of the piano that affect its sound. It is an area of study within musical acoustics.
This is a list of the fundamental frequencies in hertz (cycles per second) of the keys of a modern 88-key standard or 108-key extended piano in twelve-tone equal temperament, with the 49th key, the fifth A (called A4), tuned to 440 Hz (referred to as A440). Every octave is made of twelve steps called semitones. A jump from the lowest semitone to the highest semitone in one octave doubles the frequency (for example, the fifth A is 440 Hz and the sixth A is 880 Hz). The frequency of a pitch is derived by multiplying (ascending) or dividing (descending) the frequency of the previous pitch by the twelfth root of two (approximately 1.059463). For example, to get the frequency one semitone up from A4 (A♯4), multiply 440 Hz by the twelfth root of two. To go from A4 up two semitones (one whole tone) to B4, multiply 440 twice by the twelfth root of two (or once by the sixth root of two, approximately 1.122462). To go from A4 up three semitones to C5 (a minor third), multiply 440 Hz three times by the twelfth root of two (or once by the fourth root of two, approximately 1.189207). For other tuning schemes, refer to musical tuning.
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.
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."
12 equal temperament (12-ET) is the musical system that divides the octave into 12 parts, all of which are equally tempered on a logarithmic scale, with a ratio equal to the 12th root of 2. That resulting smallest interval, 1⁄12 the width of an octave, is called a semitone or half step.
In music, an electronic tuner is a device that detects and displays the pitch of musical notes played on a musical instrument. "Pitch" is the perceived fundamental frequency of a musical note, which is typically measured in Hertz. Simple tuners indicate—typically with an analog needle or dial, LEDs, or an LCD screen—whether a pitch is lower, higher, or equal to the desired pitch. Since the early 2010s, software applications can turn a smartphone, tablet, or personal computer into a tuner. More complex and expensive tuners indicate pitch more precisely. Tuners vary in size from units that fit in a pocket to 19" rack-mount units. Instrument technicians and piano tuners typically use more expensive, accurate tuners.
In music, the septimal minor third, also called the subminor third or septimal subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5. In 24-tone equal temperament five quarter tones approximate the septimal minor third at 250 cents. A septimal minor third is almost exactly two-ninths of an octave, and thus all divisions of the octave into multiples of nine have an almost perfect match to this interval. The septimal major sixth, 12/7, is the inverse of this interval.
Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory.
In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system. Tempering is the process of altering the size of an interval by making it narrower or wider than pure. "Any plan that describes the adjustments to the sizes of some or all of the twelve fifth intervals in the circle of fifths so that they accommodate pure octaves and produce certain sizes of major thirds is called a temperament." Temperament is especially important for keyboard instruments, which typically allow a player to play only the pitches assigned to the various keys, and lack any way to alter pitch of a note in performance. Historically, the use of just intonation, Pythagorean tuning and meantone temperament meant that such instruments could sound "in tune" in one key, or some keys, but would then have more dissonance in other keys.
In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2, 1.5, and may be approximated by an equal tempered perfect fifth which is 27/12. If the A above middle C is 440 Hz, the perfect fifth above it would be E, at (440*1.5=) 660 Hz, while the equal tempered E5 is 659.255 Hz.