# Tuning a Piano

## Abstract

This article describes the actual process of tuning a piano and the math behind it. It tells how piano tuners can tune precisely using only their hearing. This won't make you a piano tuner but it explains how tuners tune.

## Background

There are all sorts of tuning meters to aid piano tuners. There is even a shareware version for Macs. But these tools didn't exist one hundred years ago. Tuners had to do it by hearing alone. To me this is still the best way to tune a piano. Tuning is more than getting the strings to vibrate at exactly the right frequency. It is more important is that the notes blend together. In other words, it needs to sound good.

Even tone deaf people can tune pianos by ear. It's just a matter of knowing what to listen for.

Western music is based on the equal tempered scale. The theory behind this scale is covered in the article in Vector Jottings 46: Musical J-ers by Norman Thomson. You can familiarize yourself with the theory by reading his article and information on many other web sites.

As stated in his article, the equal tempered scale is based on the remarkable coincidence that ${\displaystyle 2^{(7/12)}}$ is very close to ${\displaystyle 3/2}$ and all notes within an octave can be generated through the circle of fifths. Tuning a piano relies on this circle of fifths, along with other intervals.

When the frequencies of note combinations are close to simple ratios, the sound is pleasing. We hear note combinations in two ways: One is in a sequence of notes as in melodies. The other is with several notes played simultaneously as in chords or harmonically. Our ears are much more sensitive to notes being out of tune when played in chords. Additionally, there is the heterodyne effect.

## The Heterodyne Effect

When two frequencies combine they produce additional frequencies which are the sums and the differences of the original frequencies. This property is used in electronics to step down the high frequencies of radio and satellite signals to frequencies which are easier to handle. The same thing happens with piano strings; however, piano strings do not produce just one frequency. They produce overtones, called harmonics since they are multiples of the lowest frequency called the fundamental. When two notes are played together all the different frequencies add and subtract to produce additional frequencies. If any of these frequencies are nearly equal then it is heard as a wavering of the sound called beats. This is what piano tuners listen for.

Find a piano and you can try it yourself. Play A and E around middle C together. If the piano is well tuned you should hear a wavering in the sound a little slower than once a second. This A has a fundamental frequency of 220 hertz. The fundamental frequency of E is about 329.628 Hz. The second harmonic of A is three times the fundamental or 660 Hz. The first harmonic of E is about 659.255 Hz. The difference between these two frequencies is about 0.745 Hz, or three beats in four seconds.

If the beat is faster or slower by much then one of the notes is out of tune. Say that E is tuned to 329.667 hertz, then the beat is one per second. This faster beat is easily heard, but is an error of less than ${\displaystyle 1/4}$ percent. This, according to Norman's article, is far below what even trained ears can detect musically.

## The Design of a Piano

There are 88 notes on a piano. Strings are strung over a steel frame in an X configuration. Bass strings strung from the upper left to lower right and treble strings from upper right to lower left. The strings pass over several pens for support and a bridge which passes the string vibrations to the sound board. The sound board is thin wood designed to transmit sound into the air. The sound board doesn't have much strength. The frame takes the force from the stretched strings, which can be as much as thirty tons.

The treble notes usually have three strings for each note. The bass strings are wrapped with coiled wire to add weight. The higher bass notes have two strings for each note. Multiple strings and wrapping increase the volume.

The upper end of each string goes to a tuning pin and passes over several pens and a bridge to the other end. To tune a string the tuning pin is turned with a tuning hammer, which is really a wrench. Turning the tuning pin clockwise tightens the string raising its pitch.

## Don't Break a String

It's really bad news if a string breaks when tuning. Replacing one is very time consuming and a new string requires several tunings and months before it will stay in tune. To minimize the chance of breaking a string, always lower the pitch first. This lets the tension of the string start it slipping over various pins along its length. Next raise its pitch above the desired pitch, then lower it to the desired pitch. The goal is to have the tension over the length of the string passing over several pins as uniform as possible. Any segment at a different tension will eventually slip to even the tension and then the string will be out of tune.

Strange as it may seem, it's easy to get lost and think that a note is too low when it's really too high. Lowering the pitch first avoids raising it even higher risking breaking the string before the error is obvious.

## Beats

The first step in tuning the piano is to tune A using a tuning fork. Eliminate the beats between the piano A and the tuning fork. It's important to avoid hitting the tuning fork on something hard. Hit it on your knee. That way you don't damage the fork by hitting it too hard.

The A below middle C is set to 220 hertz which is the current standard.

It's interesting to note that tuning forks used in high school physics classes have middle C set to 256 Hz instead of 261.626 Hz. Works better for the arithmetic, I guess.

Since strings produce harmonics as well as the fundamental, all intervals within the octave give off beats. When the beat is over five or six per second it gives more of a roughness to the sound rather than countable beats. With practice this roughness is easily heard as too fast or too slow.

For the temperament range I use, F to E around middle C, the frequencies for the notes are given by:

```  Fundamental      First     Second      Third     Fourth
Harmonic   Harmonic   Harmonic   Harmonic
F     174.614    349.228    523.842    698.456    873.071
F#    184.997    369.994    554.992    739.989    924.986
G     195.998    391.995    587.993    783.991    979.989
G#    207.652    415.305    622.957    830.609   1038.262
A     220.000    440.000    660.000    880.000   1100.000
A#    233.082    466.164    699.246    932.328   1165.409
B     246.942    493.883    740.825    987.767   1234.708
C     261.626    523.251    784.877   1046.502   1308.128
C#    277.183    554.365    831.548   1108.731   1385.913
D     293.665    587.330    880.994   1174.659   1468.324
D#    311.127    622.254    933.381   1244.508   1555.635
E     329.628    659.255    988.883   1318.510   1648.138
```

Look for beats in the table above. Compare the second harmonic of F to the first harmonic of C. This is the interval of a fifth. Next, the third harmonic of F to the second harmonic of A#. This is a fourth. And the fourth harmonic of F to the third of A. This is a major third.

The table below shows intervals used in tuning. Multiply by the frequency of a note, to get the beats which one would hear when played with the note above in the interval.

This table only looks at the first eight harmonics to find the beat. Even though there are harmonics beyond them which produce other beats, the volume decreases for the higher harmonics so those beats are not noticeable.

```Interval      Beat Factor      Ratio
-----------------------------------
minor third     _0.0540         6r5
major third      0.0397         5r4
fourth           0.0045         4r3
fifth           _0.0034         3r2
minor sixth     _0.0630         8r5
major sixth      0.0454         5r3
```

Positive values mean the interval is wider than perfect. Perfect meaning zero beats. Negative values mean the interval is narrower.

For the interval of a fifth, A to E:

```   220*_0.0034 NB. Beats per second
_0.748
```

## Laying the Temperament

Tuning is done one string at a time. Mutes are used to allow only one string to vibrate for notes with multiple strings. Mutes are made usually of rubber and are wedged between strings to be muted. They are moved as each note is tuned. A rubber or felt strip is used to mute all the notes within the temperament octave so once inserted, the temperament can be laid without having to mess with moving mutes.

Left: tuning hammer, next: tuning fork, various mutes which are wedges and a mute strip

Notice the red mute strip in the temperament octave.

Once A is tuned I lay the temperament, tune the twelve notes within the temperament octave. Then the rest of the piano can be tuned in octaves using the temperament as a base.

After lowering the pitch of E, raise it until the beat is eliminated. Now E is vibrating at 330 Hz. Lower it slightly until there are about three beats in four seconds. If the beats get too fast it's too low. Raise the pitch back to eliminate the beats then lower it again.

You should never raise the pitch to get the proper beat speed. Always lower the pitch to the proper speed of beats since this is more likely to leave the tension even over the length of the string.

The temperament is set using fourths and fifths within the octave. Other intervals are essential as checks to detect errors. I use F below middle C to E above for the temperament because these strings are closest to the ideal string on which Norman's article is based. They are very thin when compared with their lengths. Also, this range is easiest to hear and tune.

Except for tuning D, all tuning intervals go down by fourths and up by fifths. So after tuning D, all notes are tuned flat of perfect. First lower the pitch, next raise the pitch to eliminate the beat, then lower the pitch until the beat is at the desired rate.

The steps I use to lay the temperament are shown below. They consist of tuning using fifths and fourths and checking using other intervals. The final fifth from G to D is a final check and completes the circle of fifths. If any check doesn't sound right then start over.

```  Action           Note       Using        Beats
------------------------------------------------
Tune  octave       A  220.000 tuning fork  0.000
Tune  fourth       D  293.665 A  220.000   0.994
Tune  fifth        E  329.628 A  220.000  _0.745
Tune  fourth       B  246.942 E  329.628  _1.116
Check minor third  B  246.942 D  293.665  13.326
Tune  fourth       F# 184.997 B  246.942  _0.836
Check minor third  F# 184.997 A  220.000   9.983
Check minor sixth  F# 184.997 D  293.665  11.654
Tune  fifth        C# 277.183 F# 184.997  _0.626
Check major third  C# 277.183 A  220.000   8.731
Check minor third  C# 277.183 E  329.628  14.958
Tune  fourth       G# 207.652 C# 277.183  _0.938
Check minor third  G# 207.652 B  246.942  11.206
Check tritone      G# 207.652 D  293.665 _14.757
Check fourth       G# 207.652 C# 277.183  _0.938
Tune  fifth        D# 311.127 G# 207.652  _0.703
Check tritone      D# 311.127 A  220.000  15.635
Check major sixth  D# 311.127 F# 184.997   8.395
Tune  fourth       A# 233.082 D# 311.127  _1.053
Check major third  A# 233.082 F# 184.997   7.341
Check major third  A# 233.082 D  293.665  _9.250
Check minor third  A# 233.082 C# 277.183  12.578
Tune  fourth       F  174.614 A# 233.082  _0.789
Check minor third  F  174.614 G# 207.652   9.423
Check major third  F  174.614 A  220.000  _6.929
Check major sixth  F  174.614 D  293.665  _7.924
Check minor sixth  F  174.614 C# 277.183  11.000
Tune  fifth        C  261.626 F  174.614  _0.591
Check major third  C  261.626 G# 207.652   8.241
Check major third  C  261.626 E  329.628 _10.382
Check minor third  C  261.626 A  220.000 _11.872
Check minor third  C  261.626 D# 311.127  14.118
Tune  fourth       G  195.998 C  261.626  _0.886
Check major third  B  246.942 G  195.998   7.778
Check minor third  G  195.998 A# 233.082  10.577
Check major sixth  G  195.998 E  329.628  _8.894
Check tritone      G  195.998 C# 277.183 _13.929
Check fifth        G  195.998 D  293.665   0.664
```

About half way through the temperament there are three checks which provide a good measure as to how well the tuning is proceeding. It's the checks of A# to F#, D and C#. The verb "beats" shows how the beats in these checks changes as the pitch of A# is varied by 0.5 Hz.

```   NB. Calculate  A# beats F# D C#
{."1] 233.082 beats 184.997 293.665 277.183
_7.343 9.25 _12.577

NB. Raising A# by 0.5 Hz
{."1] 233.582 beats 184.997 293.665 277.183
_9.343 6.75 _15.577

NB. Lowering A# by 0.5 Hz
{."1] 232.582 beats 184.997 293.665 277.183
_5.343 11.75 _9.577

NB. Determine the error in cents
233.082 cent 233.582 232.582
3.70981 _3.71778
```

If A# is sharp by 0.5 Hz the beat speeds swap. That is, the lower major third's beat is faster than the upper one's. If A# is 0.5 Hz flat the beat speeds are too different and the beats for the minor third are slower than those of the major third. This amount of error is easily heard. Yet, this is an error of less than four cents. With practice, an error of much less than 0.5 Hz can be heard.

This assumes that F#, D and D# are correctly tuned. But if they are not it is very unlikely that this test would pass. Even if it did, the final check would not pass.

After completing the temperament tuning, make a final check by playing fifths chromatically (in ascending half-steps). Then repeat this with fourths, major thirds and minor thirds. The beat speeds for each interval must gradually increase as you move up the scale. Then play various cords within the temperament octave. These tests must be pleasing musically as well. After all, whether or not the math works out. If it doesn't sound good it's not good.

## Tuning the Rest of the Piano

Now the rest of the piano can now be tuned in octaves, eliminating the beat in the octave. Again, check with fifths, fourths and thirds as each note is tuned.

Starting in the last octave or so, the octaves need to be noticeably widened. If these notes are tuned to the proper frequency, they sound flat. This is particularly noticeable playing arpegeos. In addition, the extreme treble strings are so short that their thickness and stiffness become significant thereby distorting their harmonic structure. Sometimes a single string can beat with itself.

The bass strings are wrapped to add weight to give more volume. This wrapping distorts the harmonics also as the thickness becomes significant. On spinet pianos the bass strings are so short that sometimes it's hard to tell what the note really is. Grand pianos sound so much better because they have much longer bass strings.

## A Final Check

And then comes the final test. Play the piano. After all, that's what it's all about. And the customer likes to hear their freshly tuned piano played.

Contributed by: Don Guinn