Piano tuning - AbsoluteAstronomy.com

Piano tuning is the act of making minute adjustments to the tensions of the strings of a piano

Piano

The piano is a musical instrument played by means of a keyboard. It is one of the most popular instruments in the world. Widely used in classical and jazz music for solo performances, ensemble use, chamber music and accompaniment, the piano is also very popular as an aid to composing and rehearsal...

to properly align the intervals between their tones so that the instrument is in tune

Musical tuning

In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...

. The meaning of the term in tune in the context of piano tuning is not simply a particular fixed set of pitches

Pitch (music)

Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

. Fine piano tuning requires an assessment of the 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

Equal temperament

An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

(see Piano key frequencies
Piano key frequencies
This is a virtual keyboard showing the absolute frequencies in hertz of the notes on a modern piano in twelve-tone equal temperament, with the 49th key, the fifth A , tuned to 440 Hz...

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.

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

Piano Technicians Guild

The Piano Technicians Guild , based in Kansas City, Kansas, is the official organization of the Registered Piano Technician . The PTG was formed in 1957 when the American Society of Piano Technicians and the National Association of Piano Tuners merged to form a single organization.- Membership...

Temperament and beating

The relationship between two pitches, called an interval

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

, is the ratio of their absolute frequencies

Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

. Two different intervals are perceived to be the same when the pairs of pitches involved share the same frequency ratio. The easiest intervals to identify, and the easiest intervals to tune, are those that are just

Just intonation

In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

— which have a simple whole-number ratio. The term temperament
Musical temperament
In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...

refers to a tuning system which tempers the just intervals

Just intonation

(usually the perfect fifth

Perfect fifth

In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

which has the ratio 3:2) in order to satisfy another mathematical property; in equal temperament, a fifth would be tempered by narrowing it slightly, achieved by flattening its upper pitch slightly, or raising its lower pitch slightly. A system of temperament can also be known as a set of bearings.

Tempering an interval causes it to beat

Beat (acoustics)

In acoustics, a beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies....

, which is a fluctuation in perceived sound intensity due to interference between close (but unequal) pitches. The rate of beating is equal to the frequency differences of any harmonics that are present for both pitches and that coincide or nearly coincide.

It is heard clearly when the difference in pitches of these coincident harmonics is small (less than 20 hertz

Hertz

The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

(Hz)). Because the actual tone of a vibrating piano string is not just one pitch, but a complex of tones arranged in a harmonic series

Harmonic series (music)

Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...

, two strings which are close to a simple harmonic ratio such as a perfect fifth will beat at higher pitches (at their coincident harmonics), due to the difference in pitch between their coincident harmonics. In the case of an interval that is close to a perfect fifth, the strongest beating will be heard at 3 times the fundamental frequency

Fundamental frequency

The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...

of the lower string (an octave

Octave

In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

plus a perfect fifth

Perfect fifth

up), and 2 times the frequency of the higher string (an octave up). Where these frequencies can be calculated, a temperament may be tuned aurally by timing the beatings of tempered intervals.

One practical method of tuning the piano begins with tuning each note of the chromatic scale in the middle range of the piano. This is referred to as the temperament octave. A beginning pitch is tuned from an external reference, usually an A440 tuning fork

Tuning fork

A tuning fork is an acoustic resonator in the form of a two-pronged fork with the prongs formed from a U-shaped bar of elastic metal . It resonates at a specific constant pitch when set vibrating by striking it against a surface or with an object, and emits a pure musical tone after waiting a...

, and the tuner successively adjusts each note's tempered intervallic relationships to other notes in the scale. During tuning it is common to assess fifths, fourths, thirds (both major and minor) and sixths (also major and minor), often in an ascending or descending pattern to easily hear whether an even progression of beat rates has been achieved.

Once these strings are tuned, the tuner may proceed to tune all other pitches by comparing octave intervals against this temperament octave. This is convenient, because the octave is the easiest interval to tune (having the simplest ratio of 2:1) after the unison

Unison

In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...

(1:1). It is unusual to tune 2:1 octaves on a piano. Often 6:3; 4:2 or other ratios are used. The octaves are tuned beatless at one partial only. On some pianos the 6:3 and 4:2 may happen to both be beatless--but it is rare for this to happen.

The followings table lists the 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. Begin by tuning one note to the other so that the beating disappears, temper that interval in the appropriate direction (either making the interval wider or narrower, see further below) until the desired beat rate is achieved. 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 over the range of the piano.

Equal temperament beatings (all figures in Hz)
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			B♭
11.8722	17.4610	.994304		1.48977	13.8588	19.9665			A
16.4810	.938498		1.40616	13.0810	18.8459			A♭
.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			E♭								Perfect fifth
		D									Perfect fourth
	C♯										Major third
C											Minor third

This 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.

The pitch of beatings
Interval	Approximate 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

Stretched octaves

The tuning described by the above beating plan will give a good approximation of equal temperament across the range of the temperament octave. If it were extended further, however, the actual tuning of the instrument would become increasingly inaccurate. This is due to a factor known as inharmonicity

Inharmonicity

In music, inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency....

, which is present in different amounts in all piano strings. Strings' harmonic series do not fall exactly into whole-number multiples of their fundamental frequency; instead each harmonic runs slightly sharp

Sharp (music)

In music, sharp, dièse , or diesis means higher in pitch and the sharp symbol raises a note by a half tone. Intonation may be flat, sharp, or both, successively or simultaneously...

, the sharpness increasing as higher tones in 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 tuned, not to the lowest coincidental overtone (second partial) of the note below, but to a higher one (often the 4th partial). This widens all intervals equally, thereby maintaining intervallic and tonal consistency.

All music, but classical literature in particular, requires this deviation from the theoretical equal temperament. It is because music is rarely played within a single octave. A pianist constantly plays notes spread over three and four octaves, so it is critical that the mid and upper range of the treble be stretched to conform to the inharmonic overtones of the lower registers. Since the stretch of octaves is perceived and not measured, the tuner has little sense for whether he is stretching "more" or "less". He merely tunes so that his single, double, and triple octaves are reasonably beatless, always ensuring that his M10s and especially M17s (octave doublings and triplings of the M3) are not "wild". The amount of stretching necessary to achieve this is a function of string scaling, a complex determination based on the string’s tension, length, and diameter. Some pianos—and some tuners—achieve this better than others.

It is commonly said that the octaves of a small piano need to be stretched more than those of a large piano. But from the concert tuner's perspective it is the opposite. Because smaller pianos' inharmonicity is so extreme, establishing octaves based on a triple octave makes the single octaves beat noticeably, and the wide, fast beating intervals in the upper treble—especially M17s—beat wildly. Of a necessity the tuner limits his stretch to what he deems acceptable. Concert grands' lesser inharmonicity allows a 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 ability contributes mightily to the response, brilliance and "singing" quality that concert grands offer, and concert artists require.

Stretched Fifths.

A serendipitous 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 (about one beat every two seconds) would double each ascending octave. At the top of the keyboard, then, the theoretically (and ideally) pure fifth would be beating as many as eight times per second. Modern 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

Psychoacoustics

Psychoacoustics is the scientific study of sound perception. More specifically, it is the branch of science studying the psychological and physiological responses associated with sound...

, 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.

External links

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Temperament and beating

Stretched octaves

See also

External links