— 2020 — Apr 16

Some Fun with Piano Harmonics

by Anton Schwartz

The other day I played the following chord on the piano:

D-B-E-A Chord

It sounded like this:

You might notate the chord as a D69(omit 3) or a D6sus2. It’s notable in that, unlike most of the chords we see, it doesn’t include a third.

Except that it sounded major to me. Clearly major. Do you hear it too? And that would mean that it has a major third. Which it doesn’t.

I double checked my fingers: yep, no third in the chord. What was going on?

What’s going on?

It seemed to me the only explanation was that I was hearing the major third as a harmonic of one of the other notes. For instance, the F# that’s 2+ octaves above the D is the fifth harmonic of D, but that’s a relatively weak overtone, and is present whenever there’s a D. It couldn’t explain what I was hearing. 

That same F# is also a harmonic of the B in the chord. And it’s a strong one: the third harmonic, which is the first non-octave overtone. Was that what I was hearing?

Some background about harmonics

First a little background about harmonics: when the B string vibrates, it produces a sound with a pitch corresponding to B — no surprises there. That’s caused by the string’s main way of vibrating. Here’s an image of how a string vibrates, exaggerated to make it easier to see. The white curves below represent how a B string might vibrate to sound like a B:

Modes of Vibration
First Three Modes of Vibration of a String

But the B string can also vibrate in two separate halves, as in the red curve, which produces a sound as though the string were half its length. That sound is an octave higher. It’s not as loud as the lower sound, but it’s crucial to making the note sound like a piano note and not, for instance, a clarinet note. Likewise, the string can vibrate in 3 parts, in 4 parts, in 5 parts, etc. We call these different modes of vibration. They give rise to the 3rd, 4th, 5th (etc.) harmonics of B.

What gives the piano much of its characteristic sound is the fact that in practice every string vibrates simultaneously in all of its different modes. So what we hear is a combination of all the string’s harmonics. The 1st harmonic, representing the simplest mode of vibration, is simply the most prominent of all those modes. We call that one the fundamental mode, vibrating at the fundamental frequency. (For example, the A above middle C vibrates at many frequencies, but its fundamental frequency is 440Hz.)

So… is it the 3rd (yellow) mode of vibration we hear, making the chord sound major?

We can test it out on the piano using a trick: Lift the damper of the B in the chord (by pressing that key very slowly so that the hammer does not strike the note) and strike staccato the F# in question. You’ll hear a brief, loud F# followed immediately by the same note only quieter:

The softer F# you’re hearing, once the F# string itself is quickly muted, is the B string vibrating in its 3rd mode, producing the F# as its third harmonic. Because we didn’t strike the B, we don’t hear its fundamental mode of vibration—just the F#, which is caused sympathetically by the fundamental vibration of the staccato F#.

Having heard the sound of the B string vibrating in its 3rd mode, scroll back up and listen to the original chord. Listen specifically for the F# you’ve just heard. Do you hear it in the chord now? It becomes particularly prominent about five seconds into the audio.

More piano tricks

There’s lots we can do with the trick of un-muting a string and making it vibrate by striking another string fleetingly. For instance, if you un-mute the low D instead of the B when you strike the F#, you’ll hear the F# as the 5th harmonic of that low string. Notice that this time, the overtone you hear is flat compared to the F# you strike! (If you hold on to the original F# for a bit and listen to it before you release it and hear just the overtone, you can hear the pitch change as the note goes flat.) That’s because in our equal-temperament system, major thirds are a larger interval than they are in nature—the ratio of pitches is larger than the 5:4 interval generated from the 5th overtone. The difference is about 14 cents—about a seventh of a half step.

By contrast, the perfect fifth of equal temperament is quite close to the 3:2 interval generated from the 3rd overtone. So the F# generated from the B string is much closer to actual pitch of the piano’s F#.

What did you do to my piano?!

This lets us take a perfectly in tune piano and make it sound out of tune! If you un-mute both the D and the B below middle C and strike the F# (two octaves higher), guess what? You hear the 5th harmonic of the D and the 3rd harmonic of the B, whose pitches are different enough that you can hear the pulsations of an out of tune piano. Next time you’re over at your pianist friend’s house, try that out on their newly tuned grand and see how they react!

But sorry, the trick only works on a real piano, not on a keyboard (unless it uses “physical modeling” technology). Why? Because there are no strings to sympathetically resonate.

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3 Responses

  1. I actually own a monochord, though it needs a needs a new movable bridge. It’s got a 100 cm measuring tape on it.

    • How cool, David! Have you ever used it to perform, or is it just for sciencey stuff?

      • David Valdez says:

        No, just as a research tool to check out different intervals. It’s also useful for tuning other instruments.

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