Piano Tuning and Harmony
by Carol Beigel, RPT

Harmony is a system of compromises, and different musical schemes have been used throughout musical history to tune stringed instruments.  Some of these harmonic systems used in piano music have had names like Meantone, Well-Temperament, or Equal Temperament, just to name a few.  Although there is a standard “exact” pitch for every note on electronic instruments, the tuning of strings cannot be so quantified.

The great mystery discovered by Pythagoras over 1500 years ago was that you cannot tune  strings by the numbers.  If you take a string and pluck it, you get a Fundamental tone.  If you pinch it in half you get a tone that is one octave higher; pluck that segment in half and you get another octave higher, etc.  Do that 8 times and the value of the frequency for  that highest tone equals “x”.  Now take the same string and this time pinch it in thirds (instead of halves).  Do that 12 times and you get the Circle of Fifths, which are the 12 tones that make up the octave.  However, the great mystery is that when you get to “x” it is now significantly higher in pitch! This significant deviation was known throughout musical history as the “Comma of Pythagoras” and threw any tuning of stringed instruments into chaos until the early twentieth century.

Where to start the tuning was an interesting problem.  No matter what key you chose to start the tuning, only half the tones were musically useable.  Keyboard instruments were finally laid out in the key of C and are still that way today.  All the tuning compromises were done on the black keys, or the “half tones”.  There was a huge difference between the color and mood  of each key signature, and you could only play music written for the way your piano was tuned.  So distinctive was the sound of each key signature you could listen to a piece of music and know in what key it was written.

To see a graphical illustration of a Yamaha 6ft. grand piano tuned in an Equal Temperament of the 20th century, or the same piano tuned using Handel's Temperament in 1700, or the way Pythagoras's calculations would have it tuned in 1300, see Equal and Historic Temperaments Illustrated.

As time went by, musicians wanted to make more interesting sounding music.  As each new system (temperament) reduced the  “Comma”, more notes and harmonic sounding intervals became musically available for composition.  Bach wrote his 26 Inventions because there were only 26 ways he could play with the harmony.  During his lifetime, a new tuning system appeared called Well Temperament, so he wrote music for it called the Well-tempered Clavichord. Mozart had more tones to work with, and Beethoven had even more harmony at his disposal because tuning system calculations had found a way to further reduce this “Comma”.  However, no tuning system was “all purpose” until the early 20th century with the use of Equal Temperament.  By this time a system for making each tone on the keyboard equidistant was in place.  Today, this tonal distance between each note on the piano is the twelfth root of 2.  However, now that we use a one-size-fits-all temperament, our music has lost the color that individual key signatures once gave it.  No matter what key signature the music is written, it all sounds the same in Equal Temperament.

Because piano wire has stiffness, and theory does not, the numbers will never match the theoretical values.  For instance, the A above middle C is set to a standard pitch of 440 hertz.  In theory, the A an octave below it would be tuned to 220 hertz.  A very clean stringing scale might have this lower A tuned to 222 hertz before the octave is “beatless”.  A tuner would then need to spread out this 2 hertz difference over the 12 tones in the octave.  This inharmonicity varies greatly among different models and brands of pianos.  A piano tuner uses a measurement called a "cent" to discern how flat or sharp a tone is from what it ought to be.  There are 100 cents between each note on the piano, i.e. C and C#.  A graph of a well-tuned piano that is pleasing to the ears does not look like a perfect horizontal line.  The line would be at an angle, with the bass tones starting about -30 cents flat and the high treble tones gradually increasing to +30 cents sharp.  Each piano will have its own anomalies.

A piano tuner sets the pitch of the piano – the standard international pitch being A=440 hertz.  If your piano sits in a humid room, and measures out at A=442 in the middle of the summer, it might be prudent to leave the pitch alone knowing that it will probably drop to A440 as soon as the heating season starts.  Often in the middle of the winter, pianos can be -25 cents flat.  A good number is 0, plus or minus 5 cents.  This is why piano technicians recommend climate control for the piano.  Tuning forks are not as reliable as an electronic pitch source because they change their pitch when the temperature changes, or if they are dropped or get bumped by the other tools in the tool kit. The dial tone on the telephone line is very close to A440.

Using a computer running a software program like Reyburn Cybertuner to tune a piano lets you see what you are hearing.  You can instantly see what is going on with the piano pitch; calculate the perfect tuning for that particular instrument and can show how the pitch is affected by the humidity or lack of it.  But the most wonderful thing a good tuning program will do is calculate the historic temperaments we almost lost so we can hear the music the way the composers heard it!

Piano tuning is still an art – even when using a computer.  There is a large window of what is considered “correct”.  Not only does theory come into play when tuning, but also the physical adjusting of the tuning pins and the rendering of the strings. It is the art that pleases the listener!

Carol Beigel, 2003

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