Tuning a piano the classical way is a very complicated process. One has to possess the skills to handle the tuning hammer in the correct manner and to leave the tuning pin in a stable position. It is however even more difficult to determine the correct pitch of the strings.

Many instruments can be tuned using a simple tuning device. Each tone is then tuned exactly to a predefined frequency. This method can’t be used tuning a piano. The result would make the piano sound harsh and out of tune. The reason for this lies in the way stringed instruments produce their sound. sophisticated software  Piano Tuner analyzes your instrument and uses the results to calculate the optimal tuning for your piano in particular. Using this calculated setting, you can then tune your piano, in no time, to a professional level, without having to go through the classical, time consuming, process of ‘checking the intervals

Stretch

A string can vibrate in several ways. It can vibrate in one whole section; the middle of the string then oscillates and the ends are fixed. It can vibrate in two sections; the middle and the ends are then fixed and the string oscillates on 1/4 and on 3/4 of the length. This way a string can also vibrate in three, four, five or more sections.

All these vibrations occur at the same time in the string. Each vibration has its own frequency. A struck string therefore provides several frequencies at the same time. These are called the strings’ harmonics. When these harmonics are exactly 2, 3, 4 and 5 times the base frequency, then a simple tuning device can be used. For a piano this is unfortunately not the case. For a piano these factors are not exactly 2, 3, 4, and 5. The exact values of these factors determine the base frequencies of the piano strings, that is when the piano sounds most harmonic or ‘in tune’. In practice it shows that the low note need to be tuned a bit lower. The high tones need to be tuned a bit higher. The rate in which the low notes need to be tuned lower and the high notes higher is called the stretch of the piano. This means that each piano has to be tuned differently to achieve its optimal sound and performance.

Determine the right stretch

There are already tuning devices that take this stretch into account. You can choose a stretch at setup, where a small stretch is chosen for a large piano and a large stretch for a small piano. Such tunings devices can take the piano to an almost optimal tuning. But the specific design of the concerning instrument is not taken into account and therefore some of the factors, that decide the stretch, are left out. Because of this, the piano will not sound optimal and it will still be necessary to determine the best stretch ‘by ear’.

The piano technician does this by striking a lot of intervals (two keys at the same time) and aligning them. To do this properly, a lot of experience is an absolute necessity. The difficulty lies in the fact that you cannot tune the intervals entirely pure and the rate of purity is different for each and every interval. When an interval is not entirely correct, one of both notes will be adjusted accordingly. This note however is also part of several other intervals. Therefore all these intervals will have to change as well. All this produces a large complex puzzle that has to be solved.

Modern software solves this puzzle for me. In The Piano Tuner software all strings will be recorded by playing them all one by one, only one string per note the remaining strings muted. The tuning (and so is the stretch) is determined by the computer using these single string recordings. The tuner not only records the fundamentals, but also the harmonics. After this ‘single string’ recording, the tuner has all the data needed to calculate the optimum base setting of all strings. The tuner calculates then the purity of all possible intervals and aligns them. After this the strings can now be tuned one by one to the resulting notes from the tuner. The tuner recognizes the struck string automatically and shows how much the string deviates from its ideal pitch. The muted strings are lateron tuned equal to the earlier tuned string in the same unison. After tuning all strings the piano has been tuned with the best stretch. The piano now sounds as pure as possible for that particular piano. Each note needs only to be tuned once and it is no longer necessary to finish the tuning by ear.

Why does every piano need to be tuned differently?

This is quite a complex matter. I will explain it in a couple of steps.

1. Intervals beat
2. When two keys are pressed at the same time (this is called an interval), a piano will produce a combined sound. If you listen closely you will hear that this combined sound beats. The volume increases and decreases a few times per second (some intervals beat slower).
3. Each interval has to beat at the right speed
4. Each interval on a piano’s keyboard beats at a different rate. The rates of the beats for each interval are important. A piano will only sound good if these beats are exactly right. There are quite some rules that these beats have to respect. A major third beats faster than a fifth. An octave only beats very slowly. Also, if you play a major third somewhere on the keyboard, it will beat faster when you play it on a higher pitch (more to the right).
5. The beats are produced by higher harmonics
6. Okay, the interval’s beats are important but where do they come from? Well, if two tones with almost the same pitch (but different) sound together they will produce a beat. Let’s say tone A has pitch FA and tone B has pitch FB. When FB is only a little higher than FA together they will produce a beat with frequency F = FB – FA.

 7.    This is getting quite technical and not so important.

 So, two tones with pitches that are close to each other produce  a beat. That’s good to know but the two tones of an interval are not close at all! The pitch ratio of a major third (in just intonation), for example, is 1.25. These two pitches are far apart. They will never beat! What is it that’s producing the beat then?

The answer lies in the higher harmonics that vibrating strings produce. When a string is struck, it will produce many tones at the same time. It will vibrate at its fundamental (the lowest) pitch, but it will also vibrate at a frequency of 2, 3, 4, etc. times its fundamental frequency. The cause for this is that a string can vibrate in several ways. It can vibrate in one whole section; the middle of the string then oscillates and the ends are fixed. It can vibrate in two sections; the middle and the ends are then fixed and the string oscillates on 1/4 and on 3/4 of the length. This way a string can also vibrate in three, four, five, etc. sections. All these vibrations occur at the same time in the string. The animation below clarifies this. The sections are also called parts which is the reason that higher harmonics produced by strings are also called partials.

Now we can understand what is causing the beats in, for example, a major third. The interval from A4 to C#5 is a major third. If the A4 has a pitch of 440 Hz then the C#5 (in just intonation) has a pitch of 550 Hz. The fourth harmonic of the A4 (5 * 440 = 2200 Hz) is the same as the third harmonic of the C#5 (4 * 550 = 2200 Hz). So one of the beats that you hear in a major third is caused by these two higher harmonics. You actually can hear multiple beats caused by multiple pairs of different harmonics. You can imagine how hard it is to hear (and count) these beats by ear. Professional tuners practice years to get it right.

Inharmonicity In the previous section I explained that a string vibrates in different parts. This causes the higher harmonics (partials) to be produced. I talked about frequencies of 2, 3, 4, etc. times the fundamental frequency. That is actually not exactly the truth. Because of the physical properties of a string (length, diameter, stiffness, weight, imperfections, etc.) these factors are a little higher. A string needs to stretch a little in order to vibrate. A string needs to stretch even more to vibrate in more parts. That is one of the reasons why a higher harmonic frequency, in real life, has a higher pitch than you might expect.

Conclusion, All of the above means that the beat rate of an interval is dependent on the (huge amount of) physical properties of the piano. These properties are different for each piano. In order to get the beats of all intervals exactly right each piano needs to be tuned differently.

Complexed Piano Tuning Software solves the puzzle

It is an extremely complex puzzle to determine the right pitch for each string so each interval will beat exactly right and the piano will sound optimal. Piano Tuner software solves this puzzle for me. In the Piano Tuner all strings will be recorded by playing them all one by one, only one string per note the remaining strings muted. The necessary tuning is determined by the computer using these single string recordings. The tuner not only records the fundamentals, but also the harmonics. After this ‘single string’ recording, I the tuner has all the data needed to calculate the optimum base setting of all strings. The tuner calculates then the purity of all possible intervals and aligns them. After this the strings can now be tuned one by one to the resulting notes from the tuner.

Hertz and Cent

A tone which exists of exactly one frequency looks like a sine wave:

Hertz

The pitch of such a tone is expressed in Hertz: the number of waves per second. In the figure above two waves are shown.

Cent

The notes in the scale (equal temperament) increase in frequency. Every octave exists of 12 notes (semitones) and corresponds to a doubling in frequency. The A4 is 440 Hz and the A5 880 Hz. The frequency range (width) of a note is therefore larger if the pitch is higher. The A4 runs from 428 up to 453 HZ and the A5 runs from 855 up to 906 Hz. The width of a semitone is by definition (always) 100 Cent.

The ratio between Cent and Hertz

The width of a tone in Hertz increases as the pitch increases. The width of a semitone in Cent is always 100. The difference in Cent ∆ between two tones with frequencies f1 and f2 (in Hertz) can be calculated as follows:

∆ = 1200 log2 (f1 / f2)

If the frequency difference (f2 – f1 in Hertz) increases linearly than the difference in Cent increases logarithmic.

The deviation of a measured tone in Cent

If the tuner shows an error of a measured tone in Cent, then an error of 0 Cent means that the tone is exactly right. If the deviation is -50 Cent then the measured tone is exactly in the middle of the desired tone and the previous semitone. If the deviation is +50 Cent then the measured tone is exactly in the middle of the desired tone and the next semitone.

Beats in sound

Beating occurs when two tones with a small pitch difference are played at the same time. The waves of both tones then add up and influence each other. At some moments they amplify each other and at other moments they weaken each other. In the figure below, two tones (f1 and f2) with a small frequency difference are shown. In the bottom wave both tones are added up (f1 + f2). The occurring beating by the alternating amplifying and weakening of the combined signal is clearly visible. The frequency of the beating is exactly equal to the difference of the two tones (f2 – f1).

Beating of octaves

When two tones that lie approximately an octave apart from each other are played at the same time beating can also occur. In that case beating occurs from the addition of the higher octave and the first overtone of the lower octave. These are again, just like with ‘ordinary beating’, two frequencies which lie near at each other.

The equal temperament

The tuning or temperament of a scale is the way the frequencies of the notes are chosen. In Western music the equal temperament is most popular. Other temperaments are for example: the just intonation, the Pythagorean tuning, the mean tone temperament, the well temperament and the 31 equal temperament.

An octave is divided into 12 ‘proportionally increasing’ distances. The ratio of the frequencies of two successive semitones is always the same (approximately 1.0594631). Because of this, all intervals (second, third, fourth, fifth, sixth, seventh), except the octave, 

deviate from the just tuning. They cause beating. All equally named intervals sound equally false (they beat). The advantage of this tuning is that it remains the same when switched to another tone type (a number of semitones higher or lower), and it is therefore not needed to tune the instrument differently.

The intervals and the differences of the equal and the just temperament. The just temperament is the way to construct a scale where the frequency ratios are simple integers. This produces music which is experienced as pure (not false).

Can I tune a piano to another frequency than 440 Hz?

Yes, I can tune a piano to any frequency you like. I can even set the A4-frequency to the current frequency of your piano prior to recording the strings. Then after recording and before calculating the optimal stretch, I set the A4-frequency to the frequency you like.

Firstly all the piano strings need to be recorded, To be able to calculate the right stretch. This is needed only once per piano. Cost £25

Then every tune there after, tuning every string one by one  Cost £65

Use the contact form to get in touch