Piano tuning

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.
In all systems of tuning, every pitch may be derived from its relationship to a chosen fixed pitch, which is usually A440, the note A above middle C. For a classical piano and musical theory, the middle C is usually labelled as C₄ ; However, in the MIDI standard definition this middle C 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.

Background

Many factors cause pianos to go out of tune, particularly atmospheric changes. Changes in humidity will significantly affect the pitch of a piano. High humidity causes the sound board, crowned upward, to swell, crowning it further and pushing upward on the strings, causing the pitch to rise. Low humidity has the opposite effect. To a lesser degree, changes in temperature will 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 can become loose and not hold the piano in tune as well.
Frequent and hard playing can also cause a piano to go out of tune. Many piano manufacturers recommend that new pianos be tuned four times during the first year, mostly owing to string stretch, and twice a year thereafter.
An out-of-tune piano can often be identified by the characteristic "honky tonk", warbling, or beating sound it produces. This fluctuation in the sound intensity is a result of two tones of similar frequencies being played together. For example, if a piano string tuned to 440 Hz is played together with a 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 is played together with a string tuned at 442 Hz, the same 2 Hz beat is heard. 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. Piano strings are wrapped around tuning pins, which are turned to adjust the tension of the strings.

History

Piano tuning became a profession around the beginning of the 1800s, as the "pianoforte" became mainstream. 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. 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.
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.
Pianos are generally tuned to an A440 pitch standard that was adopted during the early 20th century in response to widely varying standards. 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.

Theory

Overtones and harmonics

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. In theory, the higher harmonics, also called overtones or partials, vibrate at integer multiples of the fundamental frequency. For example, 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.
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 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.

Temperament

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, 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, 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 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.
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 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