Piano tuning

Piano tuner

For the novel see The 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.

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 performed first by checking the pitch of two notes to the right of middle C on a full size keyboard consisting of 88 keys of which 52 keys are naturals (white plastic or off-white plastic or simulated ivory or ivory) and 36 keys are sharps (black painted wood or black molded plastic). The two notes are A440 which is the "A" to the right of middle C and the other note is "C" to the right of the "A". The standard musical pitch for the United States of America is A-440 which means the string will vibrate 440 times per second. The C to the right of A-440 is vibrating at a rate of 523.3 times per second. The two check notes must be exactly on pitch for the next step in the process of tuning a piano can be achieved. The next step is the tuning of the temperament which is one octave of notes starting at the "F" to the left of middle "C" and ending with the "F" to the right of middle "C" so all intervals of notes between lower "F" and upper "F" are properly set using fourths and fifths chord combinations. Once all combinations have no errors then a smooth transition using thirds can be relied upon to show an increase in beats per second starting at 7 beats per second, 7.5 beats, 8.0 beats, 8.5 beats, etc. as the notes are played in the piano. This test using thirds should not have any slowing down of intervals or speeding up of intervals rather a consistent transition from note to note using third combinations for the piano to be considered a properly tuned piano. Once the temperament is properly tuned, then octaves are tuned starting in the middle register and working towards the bass section of the piano and then to the treble section of the piano. The next step is tuning unisons. There are three piano strings for each note in the middle section of the piano called the tenor section. The bass section is the left side of the piano and commonly viewed as copper wound wire wrapping around an inner core steel wire. Sometimes the copper winding is double with two layers of copper winding around the steel core wire. There are two bass wires for each note except the last 10 notes at the far left side of the piano. The last 10 notes have one wire each. The unison means each string of a note must be tuned exactly to sound like the octave wire of the same note. So when you hear one note there should be one note in the low bass with one wire that is the octave, the upper bass having two wires sounding together exactly in unison as if one piano wire, and the tenor or treble section having three piano wires each per note of which one wire is the octave and the other two piano wires are tuned in unison to the octave wire of the same note exactly. The "A" to the right of A-440 is a frequency nearly twice the speed or A-880. The "A" to the left of A-440 is a frequency nearly half the speed or A-220. There are seven and a half octaves in the full size keyboard. An image of the grand piano showing the hammer striking the piano wires in the tenor section and the tuning lever on the tuning pin with wires attached follows: [Cincinnati Piano Tuning, http://en.wikipedia.org/wiki/Image:DSCF0297_0013.jpg]

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.

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 some other mathematical property; in equal temperament, to temper a fifth one slightly narrows it 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, a term derived from early treatises on temperament which asserted that 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 the frequencies of any harmonics that coincide (for a fifth this would be the third harmonic of the lower note and the second harmonic of the upper note) and is heard clearly when these 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

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.

References

1. Helmholtz, Hermann. On the Sensations of Tone. Trans: Alexander Ellis. Dover Publications. New York, 1954 (1885). ISBN 0-486-60753-4
2. Jorgensen, Owen. Tuning. Michigan State University Press, 1991. ISBN 0-87013-290-3
3. Cincinnati Piano Tuning. Bob is an expert concert piano tuner who is professionally trained by world renown piano instructor and director Robert Perkins, Perkins School of Piano Tuning and Technology, Elyria, Ohio, USA, 1992, hands-on applied 1992 (Written by Bob, Saturday, February 24, 2007, http://cincinnatipianotuning.com)

External links

• Cincinnati Piano Tuning - Information on piano tuning, piano parts, frequently asked questions
• Piano Technicians Guild - Information on piano tuning and becoming a Registered Piano Technician.
• Piano tuning, music, mathematics and philosophy
• Let's Tune Up, by John W. Travis
• Fundamentals of Piano Practice, Chapter Two: Tuning Your Piano
• How to tune a piano yourself
• Frequency of every note; link to spreadsheet of beat rates for equally tempered intervals