Stupid piano tricks

Your grandparents may have told you tales of a musical instrument called a "piano". This crude instrument had keys on it, much like today's pianos, but it lacked the ability to produce its own drum track, the ability to sound like a horn section, and could not (in most cases) play itself. How quaint! And if you had one of these old relics, you needed to hire someone every year or so to come to your house and tune the darn thing. I can't see why anyone would want one, really.

Some of the best piano players of all time -
Ray Charles, Elton John, Jerry Lee Lewis, Vladimir Horowitz, Bugs Bunny,
Victor Borge, Doctor John, Fats Waller, Linus Van Pelt, Liberace, and Dawg

But just in case you do have one, you could try a little experiment that I just tried on one of the two pianos in my house. Very slowly press down on middle C. If you are careful, you can get the key all the way down without the hammer touching the string, that is, you can do it silently. The cool thing is that while you hold the key down, the damper will be released from the string so that it is free to vibrate.

Now comes the truly amazing part. While you are holding down middle C, sharply strike the C below middle C. Let it up right away so that the damper clamps down again on that string. The piano will still be echoing the sound, but in an odd way. You will hear the sound of middle C, and not the sound of the note that was hit. On most pianos you will hear a lot of tinny vibrations from all over the piano, but the loudest will probably be middle C. If you let up on middle C, you will see that the middle C string was indeed vibrating.

First experiment in sympathetic vibrations

What we have discovered is called sympathetic vibration. (Sympathetic vibrations have nothing to do with the Beach Boy's Good Vibrations, in case you were wondering.) The middle C string has "sympathy" for the vibration of the lower C. Air and that big piece of cast iron that the strings are attached to both carry the vibration from one piano wire to the other and set that second wire to vibrating. 

So far we have learned that sympathetic vibrations cause middle C to play middle C. But that's a bit odd. It makes sense that if it's going to vibrate, it will vibrate at middle C. After all, that's how it was was designed to vibrate. But why does it vibrate at all? Can a piano wire vibrate sympathetically to any other note, or are there some rules to it?

Vibration modes of a piano wire

We can answer that by looking at how a piano wire (or guitar string, or vocal chord, or a column of air in an organ pipe) can vibrate. In the drawing below, we see one mode of vibration. The whole string flexes up and down as a whole. This is the fundamental tone; the vibration is at about 262 Hz (vibrations per second) for the middle C piano wire. 

Fundamental vibration of piano wire

But that's not the only mode of vibration for the middle C wire on the piano. It can also vibrate in the funny shape shown below where one half of the wire is flexing up and the other half down. This is called the first overtone, and it is at twice the frequency, or about 524 Hz. If we heard this all by itself, we would hear this as the C above middle C.

First harmonic vibration of piano wire

Is that it? Hardly! Piano wires (or practically anything else) are capable of all kinds of vibrations. Below we see the second harmonic vibration of the piano wire. If a middle C piano wire is set to vibrate in this mode, the frequency would be around 3 X 262 Hz, or about 786 Hz. If we heard this vibration all by itself, we would hear the G which is one and one-half octaves above middle C.

Second harmonic vibration of piano wire

Why does middle C sound when the lower C is played?

Back to the original question. When a piano wire vibrates, it normally vibrates in a collection of all the vibration modes. When middle C is played, there will generally be vibrations of 262 Hz, 524 Hz, 786 Hz, 1048 Hz, 1310 Hz, and so on. Our ear and brain conspire to hear a single note and not the complex chord.

So... for the purpose of this discussion, it is important that when the C below middle C is played, it vibrates not only at its fundamental of 131 Hz, but also at the first harmonic of 262 Hz. That first harmonic is what sets middle C to vibrating.

The rule of sympathy is thus pretty simple. A string will vibrate sympathetically to the frequency that it was tuned for.

But it gets a tiny bit more complicated

Let's go back to the piano and repeat the experiment, only this time we will plunk a different note. We will hold down middle C so that this wire is free to vibrate, but will plunk the F below. When the F is released we will hear a C, but this time the C is the C above middle C. One octave higher. 

Second experiment in sympathetic vibrations

Confused? It doesn't follow our first guess at the rule for sympathetic vibrations. 

But it shouldn't be all that confusing, provided you consider that the F below middle C has it's overtone series, and middle C has it's overtone series. From a previous blog post on tuning a piano, the avid JMG blog reader will recall that F, being a fifth below middle C, will have a frequency that is 2/3rds that of middle C, or about 175 Hz. That is the fundamental frequency for this wire, but it will also vibrate at 2 X 175 Hz (the F above middle C) and at 3 X 175 Hz (the C above middle C).

In other words, in the second experiment, the middle C did not vibrate at middle C because the F didn't provide that frequency in its overtone series. But the C above middle C (which the middle C wire likes to vibrate at) is a frequency that the F is more than happy to provide.

And another example

Here is another example that should help solidify the concept. While the note held down is middle C as before, the note plunked down in the third experiment is G below middle C.  From experimenting on my own piano, I found the following results.

Third experiment in sympathetic vibrations

This can be explained by a similar analysis, as organized in the following table. The first column shows all the frequencies that are made available by the G that is plunked. The second column shows all the frequencies at which the middle C wire would like to vibrate. As can be seen, the lowest frequency provided by the G that the C will respond to is 786 Hz, the second G above middle C. Hence, that's what we hear.

G	C
196.5 Hz (fund)
	262 Hz (fund)
393 Hz (1st)
	524 Hz (1st)
589.5 Hz (2nd)
786 Hz (3rd)	786 Hz (2nd)

Summary

Things like piano wires naturally vibrate at a number of different frequencies. Sympathetic vibration occurs when one vibrating object has some loose connection to another object. The sympathetic vibration that is induced in this way depends on finding a match between those frequencies that are available and those frequencies that the second object has an affinity for.