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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.

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Neat question, but I would suggest retitling it since you already understand that an "actual" subharmonic is impossible. Sounds like you're instead trying to trick the ear into hearing notes below the range of the instrument. –  NReilingh Mar 26 '12 at 5:12
    
The idea of using a "tuned" tremolo is interesting. It would be fun to experiment with this. –  Ulf Åkerstedt Apr 21 '12 at 20:51
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@Ulf The tuned-termolo idea came from Paganini's Capriccio #6. A tempo the tremolo should be over 24 notes-per-second. Since there's a low A at 22.5hz, a little slower should coincide with a low G (the Key of the piece!) right at the threshold of pitch perception. But I'm still only about half that speed. :( –  luser droog May 13 '12 at 15:58
    
@luserdroog Aha. Bummer... :-) I was actually thinking of something more in terms of using a volume tremolo effect pedal that had tremolo frequency adaption based on the pitch played. What further options/ideas could there be? –  Ulf Åkerstedt May 23 '12 at 23:37
    
This would be an interesting twist on the question... en.wikipedia.org/wiki/Binaural_beats –  user6513 Jun 22 '13 at 4:48
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5 Answers 5

up vote 7 down vote accepted
+100

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.

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The true difference tone in the described case, with a fourth inbetween the played tones, is actually yet an other octave down. The perceived B, closest below the fourth interval, that I refer to, is called the cubic difference tone. See for instance 'Musical Acoustics' by Donald E. Hall, Brooks/Cole Publishing Company, Pacific Grove, CA. –  Ulf Åkerstedt Apr 21 '12 at 20:44
    
I'm immensely pleased to read this. .... I'll be in the shed. ... No calls. ... :) –  luser droog Apr 21 '12 at 22:55
    
You'll notice it wont be perfect - it wont sound just as the real thing - but hopefully you can produce a good enough illusion. –  Ulf Åkerstedt Apr 21 '12 at 22:59
    
Do you strike them as a slap or a sweep? You want a little more intensity on the B, less on the F#, right? I can get a softer overall attack with a slap, but I can get more difference of intensity with a sweep pluck. And all with the thumb. ... –  luser droog Apr 26 '12 at 16:26
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I just discovered this can be made to work with a strum if you apply (palm-)muting (a little more muffling on the lower note, but mute both to keep them in tone). –  luser droog Dec 30 '12 at 7:35
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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.

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That's awesome! Found a link with more: marikimura.com/subharmonics.html –  luser droog Jul 1 '13 at 4:50
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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
νi   =   ν0i      ∀ i ∊ {1,2,3...}
then there are frequency-differences
Δνi j   =   νiνj   ∊   { ν0ν0 , 2 ν0ν0 , ... , ν0 – 2 ν0 , ... }   =   { ... , -2 ν0 , -ν0 , 0 , ν0 , 2 ν0 , 3 ν0 , ... }
These get multiplied and put back on the frequencies in the signal.
fi j k = νi + Δνi jk     ∀ 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.

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This is exactly the principle of interferences, isn't it? –  Gauthier Mar 28 '12 at 1:26
    
Wow. So subharmonic difference tones happen with any chord? Perhaps that's the key to the word "chord" itself. A Cord from a Chorus. Perhaps chords work precisely because they stand in relation to an imaginary unifying pitch. Like watching a scaly lizard from above. You can only see a pattern of flashing shapes as the scales change their angle to the sun. But the pattern of movement is truly governed by the feet. –  luser droog Mar 28 '12 at 5:06
    
@luserdroog: "subharmonic difference tones happen with any chord" well, in a sense yes; but for chords played on "clean" instruments such as piano, acoustic guitar or an ensemble of melody instruments, the difference tones are just amplitude envelopes, which can not directly be perceived. Only when you apply distortion, they become actual sound vibrations. — Interesting analogy with the lizards, that's the poet's way of looking at it. Of course you can as well explain chords quite boringly by the fact that various harmonics of different strings fall on the same frequencies. –  leftaroundabout Mar 28 '12 at 9:32
    
I wonder if buzzing the strings against the frets could produce the desired distortion on a "clean" guitar... –  luser droog May 13 '12 at 16:04
    
@luserdroog: string buzzing is indeed comparable to distortion in a way, but as it happens separately for each string it does not cause any intermodulation. –  leftaroundabout May 13 '12 at 17:25
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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.

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IIRC, the clarinet has a relatively weak fundamental. –  luser droog Mar 26 '12 at 18:55
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The filtering caused by telephone microphones, speakers, and transmission processing eliminates most low partials. However the human hearing makes up for that and extrapolates the fundamental from the higher partials, and thus a deep voice to still be perceived as deep (but with a phone filter sound). This means you will still "hear" a sung low F as the very low F although the lowest frequency you actually hear might be an octaveor more above. –  Ulf Åkerstedt Apr 21 '12 at 16:11
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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.

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Anyone interested in "unrelated" frequencies shoulld first be brave enough to listen to Stria. It's the sound of The Mountains of Madness. –  luser droog Mar 26 '12 at 19:38
    
Short of signal processing, I guess the best you could do is some sort of external resonance chamber with "tuned" wolf-tones. Perhaps an arrangement of panels? –  luser droog Mar 26 '12 at 22:51
    
@DrMayhem: You can't do it acoustically using a single tone, but you can do it using two tones. See leftaroundabout and my answers. :-) –  Ulf Åkerstedt Apr 21 '12 at 21:09
    
@Ulf - very cool. Must try and play with that idea... –  Dr Mayhem Apr 21 '12 at 21:53
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I love the way this one goes "You can't do this, but...". –  luser droog Dec 30 '12 at 7:36
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