Piano Tuning - Temperament and Beating

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.

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.