Piano tuning

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Piano tuner

Aural piano tuning is the art of making adjustments to the tensions in the strings of a piano so that the instrument is in tune.

1 Introduction
2 Temperament and beating
- 2.1 Stretched octaves
3 References
4 External links

[edit] Introduction

The meaning of the term in tune in the context of piano tuning is not as straightforward as it might seem, as it does not refer to the assignment of particular fixed set of pitches as it may with other instruments. Fine piano tuning requires an assessment of the interaction between 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. In the case of piano tuning, A440 is the usual standard.

Piano tuning is distinct from repairs or other maintenance that may be carried out (e.g. regulation of the action). There are typically about 220 strings in a full-sized piano, which may have a combined tension of about 20 tonnes. Tuning involves making minute adjustments to the tensions of these strings, in order to properly align the intervals between their tones.

[edit] 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 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 some other mathematical property; to temper a fifth, in this case, is to slightly narrow the interval by flattening its upper pitch slightly. A system of temperament can also be known as a set of bearings, from early treatises on temperament which suggested a fifth could be flattened "as much as it can bear".

Tempering an interval produces a beating, which is a fluctuation of the intensity of sound heard when an interval is played. The rate of beating is determined by the difference of their absolute frequencies, and is heard clearly when two pitches are close enough together that this difference 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 produce a beating at a higher pitch due to an interaction between their harmonic series. 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 (known to musicians as 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 a set of strings in the middle range of the piano to a temperament octave. 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 most easy interval to tune (having the simplest ratio of 2:1) after the unison (1:1). (These octaves are tuned to have no beating.)

The following 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 aurally. 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 bearings (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			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

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
Unison	1:1	Unison	Exact
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

[edit] Stretched octaves

The tuning described by the above bearing 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 on all piano strings. The harmonic series of strings does not fall exactly into whole-number multiples of a fundamental frequency, but rather each harmonic is slightly sharper than a whole-number ratio, and this sharpness increases as higher tones in the harmonic series are reached. This means that an aurally tuned octave will be slightly wider than the just 2:1 ratio assumed above, known as a stretched octave. The amount of stretching depends on the style of piano and is determined mainly by the length of the strings: shorter pianos such as baby grands and spinets will have octaves that are stretched farther than concert grands.

This has the effect that, on a piano, the notes in the higher register will end up slightly sharper than those in the lower octave. This is less apparent on longer pianos which have proportionally thinner strings (string inharmonicity is directly related to the ratio of string length to thickness). Despite this deviation from the simpler ideal equal temperament, this is considered the correct way to tune a piano because it maintains interval identity across the piano, which generally improves the sound of music played on it.

There are other factors, physical and psychoacoustic that affect the tuner's ability to achieve a temperament. There are additional inharmonic effects due to soundboard resonance in the bass strings, and there are other effects such as poorly manufactured strings or peculiarities of resonance or bridge shape which can cause beatings in some notes that are unrelated to the tuning and that the piano tuner cannot correct.

See also: Piano acoustics, Stretched octave and Stretched tuning.

[edit] 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