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Piano Tone Production Theory


This article is copyright (c) AVGPA 2012.  Distribution in whole or part through any means is a violation of federal copyright law.


When a piano string is struck by a hammer, it vibrates along its entire length. This vibration represents the string's fundamental pitch, which we identify as a particular "note" such as "middle C ".

At the same time, the string also vibrates in numerous fractional parts, such as two separate parts of equal length, three parts, four parts, etc.  Normally, we can't hear these harmonics or "sub" pitches produced by partial string vibrations. 

In simpler terms, harmonics are extremely soft "notes" within each note you can hear, which combine to create the impression of beautiful tone.  Without harmonics, all musical instruments would sound very bland. 

Below is an illustration of this phenomenon.  Because of the illogical numbering system of harmonics, where "harmonic 1" is the fundamental, the word overtone will be used instead.

Whole string vibrating (the fundamental note or pitch):

String vibrating in halves, twice as fast as the fundamental, which produces the first overtone an octave higher than the fundamental:

String vibrating in thirds, which produces the second overtone an octave plus a fifth higher than the fundamental:

Fundamental note C1 plus the first 15 overtones above it span a range of three octaves:


The tone color of a piano note is determined by the volume levels of each of the overtones of a particular string.  For example, we hear a note as bright because it's higher numbered overtones are louder than those on a mellow note.

When you strike a piano key very firmly, you cause the string to produce louder higher overtones than when you strike the key more gently. 

When the higher overtones of a note are too loud, it sounds off-pitch because higher overtones are grossly out of tune with the fundamental.  For example, as seen in the previous chart, the 14th and 15th overtones of the note C are Bb and B.  If you play these notes together anywhere on the keyboard, you can hear a dissonant sound or "jangle" that doesn't sound like a pure note.




The note C2 (second C from the left of the keyboard) has the same fundamental pitch as the second harmonic of the lower C1 key to left of it (both C2s).  With dampers up, the second harmonic of C1 will vibrate in sympathy with the fundamental pitch of C2.  To hear this, hold down C1 without sounding it.  Then firmly play C2 and let go.  When you do, you will hear the strings of C1 vibrating in a chime-like way because they were triggered by the sounding of C1's second harmonic.



The sympathetically vibrating strings described above are associated with notes with the same name, i.e. Cs. 


In addition, sympathetic vibrations can occur between strings whose note names do not match.  For example, if you hold down C1, and play and release G2 (11 white keys above C1), the second overtone (G2) of C1 will sound.  


As a result, raising all the piano's dampers with the damper pedal and playing notes makes it sound louder, more airy, and more colorful in tone compared to not using the damper pedal.




Since all piano strings have flaws that can't be eliminated, they don't produce perfectly accurate overtone pitches listed in the first chart above.  In particular, they have a high level of tension that limits their flexibility.  Therefore, they do not always divide at the correct points along their length associated with each overtone.  In particular, the higher overtones are the most inaccurate, making them even more dissonant in comparison to the fundamental tone of a string.


As a simple, HYPOTHETICAL example, instead vibrating in two exactly equal parts for the first overtone, the dividing point of string might be slightly off-center (not in the middle).  Thus, the pitch of the first overtone would be wrong, i.e. not exactly an octave above the fundamental.


This is called inharmonicity, which causes problems with tuning the piano.  in the past, they were resolved solely by ear.  Now, we have software that identifies the inaccurate overtones of each string, then calculates a fundamental tuning pitch that compensates for or "levels out" the inharmonicity as much as possible. 


Good piano tunings are possible without this technology. Moreover, in the final analysis, the piano tuner must use their ear to judge and possibly correct the results of using the electronic tuner or software.


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