Is there actually an "undertone series"?

Question

We know there is an overtone series, or harmonic series, i.e. if a note A of 440 Hz is being played on a musical instrument, there will also be different amplitudes of sound of 880 Hz (440 Hz * 2), 1320 Hz (440 Hz * 3) etc. emitted at the same time.

But will there be a series of undertones (i.e. 220 Hz, 110 Hz...) emitted too?

I'm asking this question because:

• I can sometimes hear a note which is an octave lower than what's actually being played on a piano. This can be very significant if this note is in a chord, which can even make me think it's an inversion;
• In a discussion of the mechanism of the new Taylor V-Class bracing, an explanation which was claimed to be made by Andy Powers, the designer of the bracing, seems to believe the existence of undertones;
• However, the concept "undertone" is rarely mentioned elsewhere. The wikipedia page believes undertone series can only be produced with unconventional methods.

Answer 1

The answer is 'yes,' with a big "if". If there is something in the instrument which is capable of vibrating at some simple multiple of the tone's wavelength, then it'll happen. As Tim's test showed, playing a note on a piano will produce a little bit of sound at sub8va and other subharmonics.
In general, if you play, say a 440-A, the 110, 220, 880, 1760 A-strings will vibrate in sympathy. Now, the higher pitched strings don't support 440, so they will "dump" energy into their own fundamental. However, the lower-pitched strings do support 440 (as the 2nd, 4th, etc harmonics), so their principle resonance will be at 440. Due to mathematical magic :-), there will be some energy at each string's fundamental, but that's a weak secondary effect.

BTW, lower-brass players can, with a little embouchure hacking, produce subharmonics. It's a matter of forcing a quarter-wave stability on a tube which prefers to be stable at half-wave.

Answer 2

You have some good answers already but here's a test that I just performed on my upright acoustic piano.

I pressed A3 (A below middle C) gently so that the damper was lifted but no sound was made. If you cannot manage this then play as softly as you can and wait for the note to end. Assuming that my piano is tuned well, the fundamental of this string is 220Hz.

I stuck A2 (octave below) hard and then released it. So, this is 110Hz. The A3 string resonated afterwards but at its normal frequency 220Hz. To check that the note is coming from this string, let go of the key and you will find that the note stops.

Now, I stuck A4 (octave above) similarly. So, 440Hz. Again, the A3 string resonated but not at its normal frequency; it resonated at the frequency of the higher A4 key.

It also works two octaves away e.g. A1 and A5 but in the case of A1 the resonance from A3 is very brief. Again, A1 caused A3 to resonate at its normal frequency but A5 caused it to resonate at A5's frequency.

I also tried a twelveth away which is approximately the third harmonic. So, D2 and E5 with similar results to the two octave case. So, the match does not need to mathematically exact since assuming that my piano is tuned normally, these twelveths won't be exactly on the third harmonic.

However, even though I can induce the resonance with notes above or below, this can all be explained by overtones without undertones. My experiment does not disprove undertones but suggests that, if they exist, they are much weaker than overtones.

Finally, note that if you really wanted to do this experiment scientifically then you should not trust your ear as the measuring device. It is quite conceivable that the ear may falsely detect 110Hz from a 220Hz stimulus. Tim's piece of paper on the guitar string test is more objective but just proves that resonance can occur above or below and does not tell us the frequency of that resonance. You would need some good lab equipment to be sure.

Even more finally, harmonics are not always so simple. Look up stretch tuning for pianos and why most drums don't have a distinct pitch.

Answer 3

The "test" described is being used incorrectly to support a potentially false claim. In linear system you will NEVER excite an undertone. This is simply not possible. It could be done by some non-linear coupling that cause sub harmonics to be generated. In Tim's experiment there is a false conclusion being drawn from the fact that plucking the high e string caused the low e string to vibrate. Of course it will vibrate as Newton's law says that a force applied to the object will cause acceleration and hence motion.

What is really happening is that the high e string's fundamental is a natural harmonic of the low e string. Hence when the high e string is plucked the low e string vibrates in response to that frequency and the n = 4 harmonic (with n = 1 being the fundamental) is excited. This does NOT mean that there is a sub harmonic or under tone being generated. For that to be the case you would need to somehow prove that the lower octaves were excited by this.

EDIT: I had a mistake in my example. The high open e string is the n = 4 harmonic of the low E string and the n = 3 harmonic of the A string. This is not the E on the D string.

This is the case with all sympathetic resonance phenomenon and I think it is grossly misunderstood. E is also the 5th of the A string and hence equal to the n = 3 harmonic, and n = 6 as well. If you were to play the high e string you would "SEE" literally the low E and A strings vibrate. However this does not mean that the note A was produced, in fact it is the E that is the 7th fret harmonic that is excited in the A string. A is NOT a natural harmonic of E.

When you pluck a string (or excite vibration by some other means) the spectrum of that string is comprised of its harmonics as the motion has to obey the boundary conditions imposed (approximately) by the nut and bridge. There will NOT be any lower frequencies in that spectrum other than the fundamental of that string, or fretted note. Once those frequencies are excited and begin propagating through the instrument they will act as a driving force for the other strings but they will not drive them at the natural frequency of the string, only at the frequency present in the original vibration spectrum. If any of the pieces of the system have a natural harmonic that matches one of the harmonics in the original spectrum then that will respond on resonance. It should be noted that everything in the guitar vibrates at whatever driving force is present. So in a sense the E string will, if forced to, move at the frequency of an A, but it will not "resonate", it will not grow in amplitude on its own (a feature of the resonance phenomenon).

The only way that I can think of to prove that a sub harmonic is present is to capture the sound with a high quality mic (broad band flat response) and DAC and FFT it. If the sub harmonic is there then its there but that has never been my experience. As I mentioned in the beginning a non-linear coupling that caused frequency splitting or some other phenomenon could be at work but the linear models of instruments do a pretty good job describing what we see and hear.

Answer 4

Just done a mini-experiment. On a guitar, when a tiny strip of paper is rested on an open string - let's say the top e, and the bottom E is plucked, the paper gets vibrated off the string. Well-known, as the bottom E produces overtones, one of which is the top e pitch. Worth mentioning that if a bottom F, G etc., is played the paper stays put.

Now, turning the idea on its head, and the paper resting on the bottom, thick E, and plucking the top thin e, that paper falls off!

So yes, undertones must be there - maybe not as strong, as the basic top e is making the bottom E vibrate in sympathy, probably due to the opposite effect.

You can most likely only hear 'undertones' on a piano when the dampers aren't stopping the strings.

Answer 5

I mostly agree with Carl Witthoft's answer, though I'd say the procedure suggested there likely falls under what the question described as "unconventional methods." Basically, if a sound is created by one sounding body, and another body is present with a resonant frequency among its harmonics that matches the first sounding body, then you can get a special effect. For example, a string may sympathetically vibrate when one of its harmonics is sounding.

Note the first string doesn't respond sympathetically at its own fundamental (which might indicate the presence of true undertones). Instead, some rather small part of the vibrating energy will "leak" (due to the imperfect nature of real vibrating bodies) into the fundamental mode (as well as other modes/frequencies). If anything, this effect is further proof of the pervasive nature of the overtone series, which causes this all to happen. If the second body didn't have overtones at the right frequency, there would be no sympathetic vibration and no leakage to the fundamental.

This could easily be confirmed experimentally with some audio software capable of doing an FFT or even by damping lower vibrations on the second body to see what's really going on. For example, suppose you created a sound of 300 Hz and put it in the presence of a string with a fundamental frequency of 100 Hz. The string will resonate at 300 Hz sympathetically. But assuming the stimulus is strong enough, some energy will eventually leak into the other modes of the string, including 100 Hz. If this were truly evidence of undertones, you'd only see vibration at 100 Hz. However, a real-world string will begin to have some limited vibrations (due to similar leakage of energy) at other modes too, e.g., 200 Hz or 400 Hz, which have no relationship to the first string's frequency in terms of overtone or undertone series. (And no, you can't blame 200 Hz as sympathetically vibrating with the 600 Hz harmonic of the first sound, particularly if a 300 Hz sine wave is your stimulus.)

In sum, a true "undertone series" like the overtone series is not produced naturally by normal harmonic sounding bodies. Under certain circumstances, some sounding bodies can be manipulated to vibrate at unusual modes (including undertones), and other sympathetic bodies in the area could leak energy to other modes of vibration.

From the question:

I can sometimes hear a note which is an octave lower than what's actually being played on a piano. This can be very significant if this note is in a chord, which can even make me think it's an inversion.

There are a variety of effects that can create that sensation. Without knowing the exact circumstances, I can't say for certain, but my guess is that what you're hearing in a situation like that is the psychoacoustical phenomenon known as a missing fundamental.

This is a well-known cognitive effect where the brain creates the sensation of a fundamental that matches the harmonics in a stimulus even if that fundamental frequency isn't present in the real sound. The reason for this is presumably because the brain spends so much time processing harmonic stimuli and condensing all of those various overtone frequencies into a single fundamental "pitch" that we hear. So, when we hear a bunch of frequencies and no fundamental, the brain inserts it anyway. (Note that this effect is exploited all the time, e.g., in telephone speakers which are often incapable of producing the fundamental pitches of low male voices. Instead, the speaker transmits the higher partials, and your brain adds in the fundamental so you perceive the voice at the correct pitch.)

Anyhow, by listening carefully and playing notes of, say, a major triad, you can sometimes have a sensation of hearing the fundamental that would theoretically lie under those pitches, generally an octave or two lower (depending on chord voicing). This effect becomes much more pronounced when playing the triad in just intonation. And if played with pure tones (sine waves) in a 4:5:6 frequency ratio, many people would simply hear a fundamental pitch two octaves lower and no individual frequencies at all. (Piano tuners and other musicians who train themselves to listen to harmonics may not hear this "ghost" effect sometimes, however.)

I should note, in closing, that psychoacoustical phenomena such as the "missing fundamental" effect are likely one of the primary reasons why an "undertone series" was first postulated. There are many situations where the brain can seem to "hear" lower pitches than are actually present in a stimulus. Nowadays, with access to audio analysis software, it's easy to verify which frequencies are and aren't present in a given stimulus.

Answer 6

There are certain techniques on a Violin where if you bow the string with extreme pressure it will vibrate at one-half the fundamental frequency, although the tone produced is so scratchy you would only want use it in very select situations. The same technique can be used to vibrate the string at one-third and one-quarter of the fundamental frequency, so in this case Sub Harmonics are possible.