What happens when you pluck a taut string, exactly? The obvious answer is that it produces a note, but in fact the result of plucking a string is far more complex. What we think we hear as a note actually consists of many notes, most of which are significantly less audible than the fundamental one. Our brain mostly ignores the others and picks up on the fundamental because it is louder. But those “unheard” tones often do have a subconscious effect on the listener.
The sounds produced when you pluck a string include overtones, undertones, and sympathetic vibrations, which we will explore in this article. We will also explore a related phenomenon called “resonant frequency”.
In acoustic stringed instruments, a fundamental note rarely occurs alone, even though we are most often not aware that we are hearing more than one note. Acoustic instruments produce complex sounds from one plucked string, while single (pure) tones can be produced by certain electronic instruments – which is why they don’t sound as musical as acoustic instruments unless the sounds are “sampled” (recorded from an acoustic instrument rather than originating electronically).
The sound wave produced by a single pure tone is called a sine wave, and we can think of it as looking like this, as generated on an oscilloscope:
But we can think of the many tones of an acoustic plucked string as looking more like this:
Overtones and Undertones
The spectrum of sound produced by one plucked string include tones above, below, and within the range of human hearing. Overtones are higher than the fundamental note, and undertones are lower than the fundamental note.
You can hear overtones by playing harmonics: if you very lightly touch a string precisely at its midpoint and then pluck it at the same time, you will hear a note one octave higher than when you pluck the open string. (This can require some practice to get the note to sound clearly; it should sound like a chiming bell.) The octave note you hear is called the second harmonic, because the first harmonic is the fundamental that you hear when you pluck the open string.
If you create a harmonic at 1/3 of the string length, you will hear the third harmonic – a note that is one octave plus a third higher than the fundamental.
At ¼ of the string length, you will hear the next octave higher (two octaves higher than the fundamental), the fourth harmonic.
See the chart below for all of the most easily produced harmonics. The waves represent the vibrational width of a plucked string, and also the corresponding appearance of sound waves.
Few musicians realize that you can easily get more harmonics than just an octave. Experimenting with this is fun. Because string materials are not perfect, it may or may not be possible on your instrument to get all these harmonics, but you will probably get some of them.
When playing harmonics, what you are hearing are the overtones that occur whenever a string is plucked. They are there even when you do not “stop” (touch) the string at any point along its length. The reason you can hear them when you do stop the string is because you have stopped the fundamental note from sounding, so they are no longer disguised by its relative volume.
When playing harmonics, what you will not hear are the undertones. Those notes are lower than the fundamental and are very subtle. Making them audible requires some rather esoteric techniques as described in the Wikipedia article here: https://en.wikipedia.org/wiki/Undertone_series . Suffice it to say, however, that they do exist and can subconsciously affect a listener’s experience of the music they hear.
You can hear the difference between an acoustic instrument that has a “rich” sound (lots of overtones and undertones) and one that does not. The richness or lack of richness of an instrument may or may not be due to how well it’s made, but some are purposely made to emphasize the fundamental notes. Some people prefer them that way; others prefer richness.
Often, sustain (how long a string rings) is mistaken for richness (preponderance of audible overtones). But they are only related in that a longer sustain can make overtones more obvious.
A response often occurs when the sound waves emanating from a plucked string meet another easily-vibrating object, causing that object to also make a sound. Sympathetic vibrations are transmitted through the air or through matter, so this can occur whether or not they are touching each other. Hence, in the case of a plucked string, other strings usually sound because they pick up the vibration. When tuning a stringed instrument, one often must damp the strings not being plucked so you can accurately hear the one that is.
In the case of unrelated (nonmusical) objects vibrating sympathetically, usually it will be only one note that causes the phenomenon in any specific object. A lampshade or a window screen, for instance, might vibrate at 440Hz (A above middle C) but not at 441 or any other note. (Hz is a measure of frequency.)
Sympathetic vibration can be annoying if something in the room is vibrating, or when something on the instrument itself is buzzing when you pluck a string. Identifying the offending part or object can be challenging.
On the other hand, some pleasant sympathetic vibrations are those heard on an instrument such as a nyckelharpa, sitar, or Hardanger fiddle; these have extra courses of strings whose only function is to vibrate sympathetically. Overtones are produced in both the plucked strings and the sympathetic ones. The resulting resonance makes these instruments sound like they are being played in a cathedral. Of course, the sympathetic strings must be perfectly tuned or they will not sound at all because they won’t be prone to picking up the vibrations.
By the way, don’t get the sound of wind blowing through the strings of a harp or lyre confused with sympathetic vibration. Wind is an energy source that causes the strings to ring by exerting force upon them directly. That is different from vibration caused by sound waves meeting an object and causing it to vibrate.
All matter is frequency. This is known in physics. The illusion of solidity is produced by vibration (frequency), rather than the other way around. You may ask, if there is no such thing as solid matter, what vibrates? For one version of a detailed explanation, see http://nzetc.victoria.ac.nz/tm/scholarly/tei-Stout03a-t17-body-d1-d3.html . For purposes of this article, let’s simply work with the concept.
Because everything is composed of frequency, and objects and living things are composed of numerous elements, most objects and living things have immeasurable numbers of different frequencies. Living things also have many organs or organelles, each with a set of frequencies. We can think of these coexisting frequencies as being somewhat like orchestras. When the many instruments of an orchestra are in tune with each other, the sound produced is pleasing. When they are not in tune with each other, the sound is terrible. Likewise, the many frequencies of a body are healthy when they are “in tune” with each other.
In a state of illness or injury, one or more frequencies go “out of tune”. This article is not the place to discuss what causes a state of dissonance to occur, but it is known that when dissonance does occur, exposure to former healthy frequencies can “re-set” the dissonant ones. This is not an exact science, obviously, since every individual is different and will therefore respond to different input. But we do know that in general, finding and using general resonant frequencies can be therapeutic. In case this all sounds a bit too far-fetched, please read this: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4325896/
Musicians certified to play in therapeutic/medical settings often use the concept of resonant frequency, a practice that involves observing what notes seem to produce a positive response in a patient (patients may report a feeling of well-being, or three may be improvements in vital signs among patients who are seriously ill) and then using those notes predominantly in the music they offer to that person. (That is not, by the way, the only skill therapeutic musicians use – it’s just one of many.)
There is also a discipline called Vibroacoustic Harp Therapy which addresses resonant frequencies in more specific ways. See https://www.musiatry.com/vibroacoustic-harp-therapy.html and https://www.harpforhealing.com/ . Vibroacoustic Harp Therapy is usually done in private practice rather than in hospitals and hospices, because it requires solicitation of input from the patient.
In therapeutic music in general, richness of tone is important, because a preponderance of overtones will cover many of the possible resonant frequencies that can be helpful to a patient. An instrument with a broad range of notes is desirable, but remember that each note contains many overtone notes, so even a small stringed instrument can be therapeutically effective.
Obviously, not all frequencies are beneficial to living things. It’s not difficult to identify which sounds are harmful: these may be noise from traffic, machines, jets, sonar, angry vocal outbursts, overly loud rock music, explosions, gun shots, chaotic rhythms, and so on.
Usually it is man-made frequencies that are problematic, since most sounds in nature are pleasant and beneficial. That brings up another concept: entrainment, which deserves its own article at a future date. Meanwhile, suffice it to say that our natural body rhythms and frequencies tend to entrain to the sounds around us; you may notice if you live near an unpleasant sound source, you may eventually stop noticing it because your body has entrained to it. When we are surrounded continually by sound pollution, it affects us on a physical level.
Pythagoras said that all notes are contained within a single plucked string. He experimented with a one-stringed instrument (monochord) to find and document that all chords come from harmonic mathematics, and the diatonic and chromatic scales do as well, because when conditions such as string material are perfect, one plucked string produces all notes. Even in this imperfect world, we can utilize this concept for beneficial effects.
For more detailed info see my book “Singing the Universe Awake” on the books page of my website at www.laurieriley.com