Piano tuning
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Piano tuning is the act of making minute adjustments to the tensions of the strings of a piano to properly align the intervals between their tones so that the instrument is 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 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 (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.
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 a minimum of twice a year. [1]
Temperament and beating
The relationship between two pitches, called an interval, is the ratio of their absolute frequencies. 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 — which have a simple whole-number ratio. The term temperament refers to a tuning system which tempers the just intervals (usually the perfect fifth 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, 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 (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, two strings which are close to a simple harmonic ratio such as a perfect fifth will beat at higher pitches (at their coincident harmonics), because of 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 of the lower string (an octave plus a 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, 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 (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.
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.
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, 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, 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 is aware of which octave needs "more" or "less" stretching. Subtle compromise between tonal brilliance, intonation and an awareness of gradation of tone through the compass of the instrument, gives the perceptive craftsman the necessary understanding to enable the appropriate single, double,triple & quadruple 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, 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.
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See also
- Piano acoustics
- Electronic tuner
- Institute of Musical Instrument Technology
- Pseudo-octave
- Stretched tuning
References
- Helmholtz, Hermann. On the Sensations of Tone. Trans: Alexander Ellis. Dover Publications. New York, 1954 (1885). ISBN 0-486-60753-4
- Jorgensen, Owen. Tuning. Michigan State University Press, 1991. ISBN 0-87013-290-3