Is it possible to create the illusion of a sub-harmonic?

Question

The way I understand it, a normal plucked tone consists of a great many overtones in addition to the base tone. A pinched harmonic, however, shifts the base tone higher into the overtone series by establishing a node in the string's range of vibration, suppressing lower frequencies.

Can you go the other way? I'm imagining some sort of vibrato or tremolo in-phase with one of the missing lower overtones that the desired low tone would have, then you could potentially trick the ear into perceiving the resulting tone as a harmonic of a tone that isn't actually present.

So, is it possible to "play" a subharmonic?

Ps. One other scrap of an idea is to strike the string with a sort of slap-mute technique where you impact both the string and the bridge so the vibrating length of string is roughly half the length of the string+body system. Like a deep stroke on a bass drum. It sounds cool, regardless.

Answer 1

As leftaroundabout indicated in another answer, you could achieve subharmonics (specifically a difference tone) using two sound sources - such as two strings. See "Combination tone" on wikipedia for further specifics. Or see "Resultant tone" regarding the usage of this for organs, where sometimes two shorter organ pipes are used to sound simultaneously emulating the sound of a longer organ pipe.

I sometimes use this phenomenon on my four-string electric bass: Striking the lowest B (on the A string) and the even lower F# (on the E string) together, creates the illusion of a B one octave lower than the strung B. This makes it sound like if I had a five string electric bass with the low B. When doing this I strike the strings softly about midways up the string to try to feed the fundamentals of the strung notes as much as possible, and to eliminate as many higher partials as possible in order not to give away the mismatching partials of the F#. (In effect it is a powerchord, but I don't use any effect pedal, and I try not to use "power" when playing it. ;-) It also helps to disguise the trick if there are other instruments playing at the same time.

I suppose you could try this on the guitar also.

Answer 2

You can't actually do this - obviously you are aware that the whole pich/natural/articial harmonic bit doesn't create frequencies, it just cuts out some so the tonal quality changes.

You do have technological alternatives though, if you do want to build lower frequencies related to the original root.

Octave dividers are the simplest example - you can buy pedals which will give you an output at half or quarter the frequency you play. When mixed with the original signal this can give a nice rich sound, and when attenuated sufficiently, can support the root rather than replacing it.

With modern DSP technology, you can be even cleverer and add in frequencies that are not octaves, or even frequencies unrelated to the original.

Answer 3

In my answer to How do harmonics work?, I said:

[A] vibrating guitar string has components at many multiples of the base frequency (call it F). To your ear it still sounds like the fundamental, but mathematically it's more like this:

a*F + b*2F + c*3F + ...

The higher-frequency elements give the note it's timbre; this is how you can tell two instruments apart, or even tell different kinds of guitar strings apart. For example, the sound where a=1 b=0.6 c=0.3 will sound different than a=1 b=0.5 c=0.4. Note that a is always the largest coefficient, since F is the fundamental frequency. If it wasn't it would sound like you were playing a different note, or multiple notes.

That last bit is actually false; I glossed over it for the sake of simplicity. In fact, it's perfectly possible for the fundamental to be weaker than other components of the sound while the perceived note is still that of the fundamental.

By the same logic, it's also possible for a to be so weak that the lowest component is not perceived as the fundamental. In other words, F would not be the fundamental, and I think it would be perfectly accurate to call it a "sub-harmonic" in that case.

It might be difficult if not impossible to create such a sound with a single vibrating string (for example), but you could probably do something like play A440 quietly on one string and A880 loudly on another, and end up with the perceived fundamental at A880 with A440 as your sub-harmonic. Of course, the human ear is quite good and it would also be difficult to prevent the two notes from being perceived distinctly. This might be easier with electronic approaches, as Dr Mayhem talks about.

Answer 4

One possibility is to exploit intermodulation.

One way overtones come up has already been discussed: if you pluck a string, it vibrates in multiple modes from the beginning (explained here). But there is an important other mechanism: nonlinearities. If you feed a pure overtone-free sine wave through a distortion pedal*, what comes out has a whole bunch of overtones on it, in principle the same ones that also turn up in acoustic instruments: harmonics. (They sure sound quite different, but that has other reasons.)

One might think the same process is going on, standing waves etc., but that's not the case. In fact, not just multiples of the fundamental are created, rather every difference between any two frequencies in the input signal is "considered", and you get multiples of it created (I'm not going to prove this here, but it's possible).
Huh, then why don't we hear a complete mess of thousands of different frequencies when playing a tone that has already harmonics with distortion?
Let's look at it. Suppose the original signal has frequencies
ν_i   =   ν₀ ⋅ i      ∀ i ∊ {1,2,3...}
then there are frequency-differences
Δν_{i j}   =   ν_i – ν_j   ∊   { ν₀ – ν₀ , 2 ν₀ – ν₀ , ... , ν₀ – 2 ν₀ , ... }   =   { ... , -2 ν₀ , -ν₀ , 0 , ν₀ , 2 ν₀ , 3 ν₀ , ... }
These get multiplied and put back on the frequencies in the signal.
f_{i j k} = ν_i + Δν_{i j} ⋅ k     ∀ k ∊ {1,2,3...}
Here you can easily see that the resulting signal will also contain the frequencies { ... , -2 ν₀ , -ν₀ , 0 , ν₀ , 2 ν₀ , 3 ν₀ , ... }. Negative frequncies? Zero? Well, zero just means no sound, so you can't hear that one. And positive and negative phase velocity is indistinguishable, so effectively we have the frequencies { ν₀ , 2 ν₀ , 3 ν₀ , ... }. Exactly the same ones as were in the original tone! Just more powerful, because they are all doubled from the many difference-possibilities. That's why you can put distortion on virtually any melody-playing guitar: you get the same frequencies again, so it never sounds wrong (except too loud / unsubtle).

If you play multiple different notes simultaneously, things get a bit more complicated: with a clean sound, the ear can separate them again, and we hear a clear chord. But with distortion, there will be frequency differences that do not map again to frequencies originally present in the signal. If it was just a little tube overdrive, these frequencies make up the famous bluesey "dirt" and you can still well distinguish the original chord notes; if you play Jazz chords through a metal distortion pedal the extra frequencies won't be properly seperable and you can't distinguish anything at all anymore. Seldom desirable.

However, you know there is one kind of chord that works at any distortion level: exactly, the powerchord. And here's why:
In a perfect fifth, the frequencies have a ratio of 2:3. So if you play a two-stringed^† powerchord, you have those frequencies in the original signal:
{ ν₀ ,  ³⁄₂ ν₀ ,  2 ν₀ ,  ⁶⁄₂ ν₀ ,  3 ν₀ ,  ⁹⁄₂ ν₀ ,  4 ν₀ ,  ... }
where you can find the differences
{ 0 ν₀ ,  ½ ν₀ ,  1 ν₀ ,  2 ν₀ ,  2 ν₀ ,  ⁷⁄₂ ν₀ ,  3 ν₀ , ...
⁻½ ν₀ ,  0 ν₀ ,  ½ ν₀ , ³⁄₂ ν₀ , ³⁄₂ ν₀ ,  2 ν₀ ,  ⁵⁄₂ ν₀ , ...
}
You see what's going on here: we're getting components with half the original fundamental frequency! Subharmonics. And those are in fact the reason why powerchords sound so fat.

*It doesn't actually have to be electronic, mechanical distortion work as well. Such are found in the nonlinear air flow in all wind instruments, this effect is used by rock flautist who sing notes while playing, to create intermodulation.
^†Three-stringed is exactly the same, as the octave only backs up even harmonics of the low string.

Answer 5

The first subharmonic, an octave below, can be produced on the violin by applying the right amount of bowing pressure (more than usual) and also bowing slowly, if I recall this correctly. As far as I know, this technique requires lots of practice to master. Similarly, subharmonics can be produced by the voice if tensing the vocal cords, as you can try for yourself.

In some synthesis techniques, in particular when using nonlinearities and feedback, you get a whole cascade of subharmonics that usually end up with chaos. This is an inherent property of the system, and probably there are some underlying similarities with the way subharmonics are produced in acoustic instruments.

Answer 6

In a sense, you actually can. Go to 0:06 in this video and you'll see the technique used.

You'll need distortion, but basically what you do, is you play one natural harmonic on one string, and then another natural harmonic on a different string, but on the second string, you fret the note instead of lift your finger off right after you get the harmonic right. from there, bend your fretted finger and with distortion, you should hear a low sound that sounds a lot like a sub drop. Using these I can "play" certain notes lower than my 7-string in Drop A.

I don't know if that answered your question, but I think that's as close as you're going to get with just a guitar, a pick, distortion, and a speaker that can sound bass frequencies.

Answer 7

I've only ever seen Michael League (bassist from Snarky Puppy) talk about this, but you can get an OC2 effect by picking directly on the octave harmonic.

You can see him do this in his Reverb interview on bass-pedals.

3:38 is where he explains what he does. I have no clue what this is called.

Answer 8

i can produce subharmonics with a classical guitar tuned down a bit. i have the 6th string E tuned to C#. the 5th A down to G# and the 4th string D down to C#. Fret the 12th fret on the 6th string and the 12th fret on the 4th string leaving the 5th string open. When all three strings are plucked lightly you can clearly hear a low D# which normally can only be produced at the 2nd fret of the 6th string. There are similar "ghost tones" found all over the guitar but this is one of the most prominent to the ear on my guitar.

Answer 9

This is actually a multi faceted question with quite a bit of depth to it. I will add something to this but my answer will also be multi-faceted. I am not going to consider electronic enhancements. And I will pose some additional questions to you.

First: If you are asking about exciting a tone lower than the lowest natural frequency of the instrument but resonance AND the instrument is linear (obeys the physics of linear response) then I'd say no. Linear systems typically do not behave that way.

Second: It has been observed that sub-harmonics can be excited in horns with just the right kind of "blow" (for lack of a better term) in terms of force and direction. Similar statements have been made of stringed instruments.

This does not contradict the previous statement as we probably think of the lowest pitch string as defining the lowest vibrational frequency of the guitar as a system and this is not quite accurate. The basic model of string harmonics taught in elementary physics texts treat the string a being fixed at both ends. This is not how strings really work. On an acoustic guitar the bridge and top movement will change this model. So in theory it s possible to excite a longer wavelength in the instrument. The excitation is related to the driving force, whether bowed, blown, plucked or struck.

Third: In non-linear systems sub-harmioncs are more commonplace. Some musical instruments may contain such non-linearities (it is our tendency to simplify the system so we can understand it based on what we can solve and thus it may be that no instrument is truly linear, but can be expected to be approximately linear). We live with this approximation and forget the more complex reality, convincing ourselves that it cannot be done.

Fourth: There is a phenomenon called fundamental tracking that I believe is purely neurological, a brain function. If the brain is presented with a sequence of tones that obey the standard harmonic sequence it will fill in the gap, calculating and inserting the mathematical fundamental. This is a very interesting part of the human system. The ear is non-linear and when excited creates aural harmonics. As a result of this it is not likely that we can hear a "pure tone" generates by a function generator hooked up to a speaker. These two phenomenon have far reaching consequences w/r to our perception of tone.

As for actual guitar technique, I don't know of any for the acoustic. Some folks have offered possibilities but I haven't actually tried any. On purely theoretical grounds if you were to create a chord sequence that contained an implicit fundamental below the notes I suspect a person would hear that lower note, but this is not the same as generating it acoustically. Comparison of a spectral analysis of the sound and the claim made by the listener would differ.

Answer 10

Yes, you can. At least, you can make people hear a note lower, even if you don't actually make it. To start, tune a guitar to Drop-D. (You needn't always use this tuning, but it's the easiest to demonstrate with.) This makes the lowest two strings D2 and A2.

On only the lowest two strings, fret the 5th fret on the D string (G2) and the third on the A string (C3), palm-mute, and strum--you'll hear a C2. Because the G2 is the 5th of, and C3 is the full octave (8th) of, C2, your mind fills in the missing fundamental (C2). (4th and 2nd fret for B1, 3rd and open for A1). Note, you're not actually producing this note, it's just an auditory hallucination that most (not all) people will hear. Some folks will simply not hear anythin more than a G2 and a C3.

This works best when palm-muting to reduce the higher harmonics of the notes you're actually playing, but can be done easily on an acoustic.