Essays/PianoTuning
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 is very close to 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 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.
Script file for definitions of tables
Contributed by: Don Guinn